text
stringlengths 1
2.34M
| meta
dict |
|---|---|
\section{Introduction}
Electron spin confined in a semiconductor quantum dot is a promising qubit candidate because of both the long dephasing time and the relative convenience for scalability~\cite{Loss1998,Hanson2007}. The spin dephasing time can be as long as a millisecond in isotopically purified Si quantum dot~\cite{veldhorst2014,veldhorst2015}. While in III-V semiconductor quantum dots, such as GaAs, the spin dephasing time is in the microsecond region~\cite{petta2005}, limited mainly by the hyperfine interaction between the electron and lattice nuclear spins~\cite{Yao2006,Cywinski2009}. Single qubit manipulation in the quantum dot can be achieved via either the electron spin resonance~\cite{koppens2006} or the electric-dipole spin resonance (EDSR)~\cite{Rashba2003,Golovach2006,LiRui2013,nowack2007,Nadj2012}. Two qubit manipulation can be naturally achieved by using the exchange interaction in a double quantum dot~\cite{Burkard1999,Hu2000}.
The manipulation time $T_{\rm Rabi}$ and the dephasing time $T^{*}_{2}$ are two important time scales for the qubit~\cite{Buluta2011}. The values of these two quantities determine whether a qubit candidate is suitable for quantum computing. An ideal quantum computer requires that enough number (about 1000) of single qubit manipulations should be completed in the qubit dephasing time~\cite{ladd2010}. Dephasing is a leading obstacle limiting all potential applications of the qubit. In order to alleviate the qubit suffering from dephasing caused by environmental noises, we should first understand various possible dephasing mechanisms~\cite{chan2018}.
There are both no internal spin-orbit coupling and negligible lattice nuclear spins in isotopically purified 28Si, such that Si quantum dot is expected to be one of the most feasible platforms for quantum computing~\cite{veldhorst2014,veldhorst2015}. The spin qubit in Si quantum dot is so separate from the external environment that single qubit manipulation becomes relatively inconvenient. Electron spin resonance in a quantum dot is proved to be technically challenging~\cite{koppens2006}. A feasible way is to integrate the quantum dot with a slanting magnetic field~\cite{pioro2008,Brunner2011,kawakami2014,Chesi2014,Forster2015,Scarlino2015}, such that single spin manipulation can be achieved via EDSR. However, as observed in experiments, the slanting field also brought the 1/f charge noise to the spin qubit~\cite{Kawakami2016,yoneda2018}. 1/f charge noise commonly exists in many nano-structures~\cite{Dutta1981,Weissman1988,Paladino2014}, and it has also been regarded as the main noise source that causes the dephasing of the qubit, such as Josephson qubit~\cite{Astafiev2004,You2007,bylander2011}, quantum dot charge qubit~\cite{Petersson2010,Shi2013}, spin qubit~\cite{chan2018,Kha2015}, singlet-triplet qubit~\cite{Culcer2009,Hu2006,Gamble2012}, etc.
In this paper, we study the slanting field mediated spin manipulation and spin dephasing in a Si quantum dot. In the spin manipulation via EDSR, the transverse slanting field mediates a transverse driving term which contributes to the periodic oscillation of the spin population inversion, while the longitudinal slanting field mediates a longitudinal driving term which gives a modulation to the spin population inversion. Fortunately, the effect of the modulation can be reduced by applying a large Zeeman field to the quantum dot. The pure dephasing is caused by the longitudinal spin-1/f-charge noise interaction, which is also mediated by the longitudinal slanting field. We propose prolonging the spin dephasing time by reducing the quantum dot size, lowering the experimental temperature, reducing the longitudinal slanting field, or using a dynamical decoupling scheme~\cite{Uhrig2007}. Under eight pulse sequences, the spin dephasing time $T_{2}$ can be prolonged to the sub-millisecond region. Finally, because the upper bound of the 1/f charge noise spectrum is usually less than the qubit level spacing in the quantum dot, the 1/f charge noise cannot contribute to the spin relaxation.
\section{The model}
\begin{table}
\centering
\caption{\label{tab}The parameters of the Si quantum dot used in our calculations. The values are taken from Ref.~\onlinecite{yoneda2018}}
\begin{ruledtabular}
\begin{tabular}{ccccccc}
$m/m_{0}$\footnote{$m_{0}$ is the free electron mass}&$g$&$B_{0}$~(T)&$\omega_{0}$~(THz)\footnote{$r_{0}=\sqrt{\hbar/(m\omega_{0})}=20$ nm}&$b_{t}$\footnote{in unit of (mT/nm), $z_{0}=g\mu_{B}b_{t}/(2m\omega^{2}_{0})=2.431\times10^{-2}$nm}&$b_{l}$\footnote{in unit of (mT/nm), $y_{0}=g\mu_{B}b_{l}/(2m\omega^{2}_{0})=0.4862\times10^{-2}$nm}&{\it T} (mK)\\
$0.2$&$2$&$0.5$&1.447&$1.0$&$0.2$&100
\end{tabular}
\end{ruledtabular}
\end{table}
We consider a realistic quantum dot model which is intimately related to the experimental situations demonstrated recently~\cite{yoneda2018,Yoneda2014}. The quantum dot has a two-dimensional harmonic confining potential on the $yz$ plane and is exposed to both static and slanting magnetic fields. The slanting field, which is used to assist the spin manipulation via an external electric-field, is created by covering a Co micromagnet on the quantum dot~\cite{yoneda2018,Yoneda2014,Neumann2015,Wu2014}. The model under consideration reads
\begin{equation}
H=\frac{p^{2}_{y}+p^{2}_{z}}{2m}+\frac{1}{2}m\omega^{2}_{0}(y^{2}+z^{2})+\frac{g\mu_{B}({\bf B}_{0}+{\bf B}_{\rm m})\cdot\boldsymbol{\sigma}}{2},
\end{equation}
where $m$ is the effective electron mass, $\omega_{0}$ is the frequency of the harmonic confining potential [the quantum dot characteristic length $r_{0}=\sqrt{\hbar/(m\omega_{0})}$], ${\bf B}_{0}=(0,0,B_{0})$ is an in-plane static field applied long the $z$ direction, and ${\bf B}_{\rm m}=(B^{x}_{\rm m},B^{y}_{\rm m},B^{z}_{\rm m})$ is the stray field induced by the Co micromagnet. One can expand the stray field up to the linear terms using Taylor's formula
\begin{eqnarray}
B^{x}_{\rm m}(z)&=&B^{x}_{\rm m}(0)+b_{t}z,\nonumber\\
B^{y}_{\rm m}(z)&=&B^{y}_{\rm m}(0)+b_{l}z,\nonumber\\
B^{z}_{\rm m}(x,y)&=&B^{z}_{\rm m}(0)+b_{l}y+b_{t}x,\label{eq_slantingfield}
\end{eqnarray}
where $b_{t}$ and $b_{l}$ are the slopes of the transverse and longitudinal fields~\cite{yoneda2018,Yoneda2014}, respectively. One can check that the above stray field does not violate Maxwell's equations $\nabla\cdot{\bf B}_{\rm m}=0$ and $\nabla\times{\bf B}_{\rm m}=0$ for a static system. The small $x,y$-components $B^{x,y}_{m}(0)$ of the stray field are neglected from consideration and the $z$-component $B^{z}_{m}(0)$ can be absorbed to the static Zeeman field $B_{0}$. After the above linear approximation, the quantum dot Hamiltonian can be written as
\begin{eqnarray}
H&=&\frac{p^{2}_{y}+p^{2}_{z}}{2m}+\frac{m\omega^{2}_{0}}{2}(y^{2}+z^{2}+2y_{0}y\sigma^{z})+\Delta\sigma^{z}\nonumber\\
&&+m\omega^{2}_{0}z\sqrt{z^{2}_{0}+y^{2}_{0}}(\sigma^{x}\cos\theta_{0}+\sigma^{y}\sin\theta_{0}),\label{eq_model2}
\end{eqnarray}
where $y_{0}=g\mu_{B}b_{l}/(2m\omega^{2}_{0})$ and $z_{0}=g\mu_{B}b_{t}/(2m\omega^{2}_{0})$ characterize the length scale of the longitudinal and transverse gradient fields, respectively, $\theta_{0}=\arctan(y_{0}/z_{0})$ and $\Delta=g\mu_{B}B_{0}/2$ is half of the Zeeman splitting. It should be noted that the vector potential $\textbf{A}=\textbf{B}_{0}\times\textbf{r}/2$ is perpendicular to the $yz$ plane, such that there are no vector potential components in the Hamiltonian (\ref{eq_model2}), i.e., $p_{y}=-i\hbar\partial_{y}$ and $p_{z}=-i\hbar\partial_{z}$. Also, we have assumed the quantum dot lies on the $x=0$ plane.
In line with the experimental investigation~\cite{yoneda2018}, here we choose Si as our quantum dot material. In our following calculations, unless otherwise stated, the parameters chosen are listed in table~\ref{tab}.
\section{Slanting field mediated electric-dipole spin resonance}
\begin{figure}
\centering
\includegraphics[width=8.5cm]{EDSR.eps}
\caption{\label{fig_Rabi_Si}EDSR described by the driving Hamiltonian (\ref{eq_edsr}) under the resonant condition $\hbar\omega=2\Delta$. The qubit population inversion is defined as $|c_{\Uparrow}(t)|^{2}-|c_{\Downarrow}(t)|^{2}$ for state: $|\varphi(t)\rangle=c_{\Uparrow}(t)|\!\!\Uparrow\rangle+c_{\Downarrow}(t)|\!\!\Downarrow\rangle$. The qubit is initially in state $|\varphi(0)\rangle=|\!\!\Uparrow\rangle$ and the driving strength is chosen as $E_{y}=E_{z}=4000$ V/m. The results at the external fields of $B_{0}=0.005$ T (a), $B_{0}=0.02$ T (b), and $B_{0}=0.5$ T (c).}
\end{figure}
The manipulation of the quantum-dot spin qubit is usually achieved via EDSR. Quantum-dot EDSR can be mediated by internal spin-orbit coupling~\cite{Rashba2003,Golovach2006,LiRui2013,nowack2007,Nadj2012,Khomitsky2012,Nowak2013,Romhanyi2015}, electron-nuclear hyperfine interaction~\cite{Laird2007,Rashba2008,LiRui2016}, and external slanting magnetic field~\cite{Tokura2006,Rancic2016}. In the earlier seminal work of Tokura and co-workers~\cite{Tokura2006}, only a transverse slanting field is proposed to mediate the EDSR. However, under realistic experimental circumstance, the micromagnet brings no only the transverse but also the longitudinal slanting fields to the quantum dot~\cite{pioro2008,Brunner2011,yoneda2018,Yoneda2014} [see Eq.~(\ref{eq_model2})]. Here we examine the impacts of the longitudinal slanting field on the spin manipulation.
Under the external electric-field driving, an additional electric-dipole interaction term $e\textbf{E}\cdot\textbf{r}\cos\omega\,t$ should be added to Hamiltonian (\ref{eq_model2}). When we focus only on the qubit Hilbert space spanned by $|\!\!\Uparrow\rangle\equiv|\Psi_{0,0,\uparrow}\rangle$ and $|\!\!\Downarrow\rangle\equiv|\Psi_{0,0,\downarrow}\rangle$, the electric-driving Hamiltonian can be reduced to the form of a two-level atom interacting with a classical field~\cite{scully1999quantum} (for details see Appendix \ref{appendix_A})
\begin{eqnarray}
H_{\rm dr}&=&\Delta\tau^{z}-eE_{y}y_{0}\tau^{z}\cos\omega\,t\nonumber\\
&&-eE_{z}\sqrt{z^{2}_{0}+y^{2}_{0}}(\tau^{x}\cos\theta_{0}+\tau^{y}\sin\theta_{0})\cos\omega\,t,\label{eq_edsr}
\end{eqnarray}
where $\tau^{z}=|\!\!\Uparrow\rangle\langle\Uparrow\!\!|-|\!\!\Downarrow\rangle\langle\Downarrow\!\!|$, $\tau^{x}=|\!\!\Uparrow\rangle\langle\Downarrow\!\!|+|\!\!\Downarrow\rangle\langle\Uparrow\!\!|$, $\tau^{y}=-i|\!\!\Uparrow\rangle\langle\Downarrow\!\!|+i|\!\!\Downarrow\rangle\langle\Uparrow\!\!|$, $E_{y}$ and $E_{z}$ are the $y$ and $z$ components of the driving-field, respectively, and $\omega$ is the frequency of the driving-field. This Hamiltonian is slightly different from the standard Rabi oscillation Hamiltonian in quantum optics~\cite{scully1999quantum} because of the presence of the second term, which is induced by the longitudinal slanting field given in Eq.~(\ref{eq_model2}).
Let us examine the influence of the longitudinal driving term [the second term in Eq.~(\ref{eq_edsr})] on the spin manipulation. Similar to the standard Rabi oscillation, the qubit is initially prepared in state $|\varphi(0)\rangle=|\!\!\Uparrow\rangle$. When the frequency of the driving field matches the qubit level spacing $\hbar\omega=2\Delta$, the spin population inversion is obtained by numerically solving the time-dependent Schr\"odinger equation governed by Hamiltonian (\ref{eq_edsr}). We find that, at small external magnetic field such as $B_{0}=0.005$ T, there is an apparent modulation on the spin population inversion [see Fig.~\ref{fig_Rabi_Si}(a)]. When the magnetic field is increased to $B_{0}=0.02$ T, the modulation becomes relative small[see Fig.~\ref{fig_Rabi_Si}(b)]. When the external magnetic field is large enough, such as $B_{0}=0.5$ T, the modulation becomes negligible (almost invisible) [see Fig.~\ref{fig_Rabi_Si}(c)]. Anyway, one can reduce the modulation via increasing the external magnetic field $B_{0}$. This is very reasonable, the longitudinal driving term can be regarded as a time-dependent Zeeman field applied to the spin qubit $(\Delta-eE_{y}y_{0}\cos\omega\,t)\tau^{z}$. The larger the static magnetic field, the smaller the relative ratio $eE_{y}y_{0}/\Delta$, hence the smaller the effect of the longitudinal driving term.
Next, let us analyze the strength of the Rabi frequency, which characterizes the qubit manipulation time. Note that the qubit is encoded to the lowest two energy levels of the quantum dot. Although the qubit Hilbert space is well separated from the other higher orbital levels in the quantum dot, i.e., the Zeeman splitting $2\Delta_{B_{0}=0.5~{\rm T}}$ (0.058 meV) is much smaller than the orbital splitting $\hbar\omega_{0}$ (0.95 meV), there still exist leakages from the qubit Hilbert space to the higher orbital states under the strong field driving. The spin dynamics in this case are totally nontrivial, and one has to consider the multi-level effects in the EDSR~\cite{Khomitsky2012}. In order to avoid the electron being excited to higher orbital states, here the electric field strength is constrained to $|\textbf{E}|\ll(\hbar\omega_{0})/(er_{0})=4.769\times10^{4}$ V/m. This result gives an upper bound on the Rabi frequency in our model $\Omega_{R}\ll\,eE_{\rm max}\sqrt{z^{2}_{0}+y^{2}_{0}}/h=286$ MHz, and agrees qualitatively well with the experimental observations~\cite{yoneda2018,Takedae2016}.
\section{\label{sec_IV}Charge noise induced pure-dephasing}
1/f charge noise has been observed in many quantum nano-structures~\cite{Dutta1981,Weissman1988,Paladino2014}, and it has also been regarded as the main noise limiting the dephasing time of many qubit candidates~\cite{Astafiev2004,You2007,bylander2011,Petersson2010,Shi2013,Kha2015,Culcer2009,Hu2006,Gamble2012,lirui2018a}. The physical origin of the charge fluctuation spectrum with 1/f distribution is still unclear, and many theoretical models have been proposed~\cite{Paladino2014}. Here we just assume that the charge field has a spectrum function $A^{2}/\omega$, and the value of $A$ is chosen to fit well with the experimental observation.
We assume the fluctuating charge field has a similar form as that of the vacuum electromagnetic field~\cite{scully1999quantum}
\begin{equation}
\textbf{E}(\textbf{r})=\sum_{k}\Xi_{k}\vec{e}_{k}(a_{k}e^{i\vec{k}\cdot\vec{r}}+a^{\dagger}_{k}e^{-i\vec{k}\cdot\vec{r}}),\label{eq_chargefield}
\end{equation}
where $\Xi_{k}$ is the charge field in the wavevector space, $\vec{e}_{k}$ is a unit vector, and $\vec{k}$ is the wavevector. The transverse character of the electromagnetic field gives rise to $\vec{e}_{k}\cdot\vec{k}=0$~\cite{scully1999quantum}. In order to simplify the complexity of the problem, we further assume the wave is propagating along the $x$ direction: $\vec{k}=k\vec{e}_{x}\perp\,yz$ plane, such that $\vec{e}_{k}$ is an in-plane unit vector, hence $\textbf{E}(\textbf{r})=\sum_{k}\Xi_{k}\vec{e}_{k}(a_{k}+a^{\dagger}_{k})$ (the quantum dot is confined on the $x=0$ plane). Replacing the classical field in Eq.~(\ref{eq_edsr}) with the above quantized electric-field, we obtain the total Hamiltonian describing the interaction between the spin qubit and the charge noise
\begin{eqnarray}
H_{\rm tot}&=&\Delta\tau^{z}-\sum_{k}e\Xi_{k}y_{0}\tau^{z}(a_{k}+a^{\dagger}_{k})\cos\Theta+\sum_{k}\hbar\omega_{k}a^{\dagger}_{k}a_{k}\nonumber\\
&&-\sum_{k}e\Xi_{k}(z_{0}\tau^{x}+y_{0}\tau^{y})(a_{k}+a^{\dagger}_{k})\sin\Theta,\label{eq_decoherence}
\end{eqnarray}
where $\Theta$ is the azimuth of the charge field on the $yz$ plane. The exact value of $\Theta$ is unknown, such that it is reasonable to average over all possible angle $\Theta$ for the obtained physical quantities, e.g., $\Gamma(t)\equiv\langle\Gamma(t)\rangle_{\Theta}=\int^{2\pi}_{0}\Gamma(t)d\Theta/2\pi$.
\begin{figure}
\centering
\includegraphics{dephasing.eps}
\caption{\label{fig_dephasing}The pure dephasing of the spin qubit due to the $1/f$ charge noise. We have chosen the noise spectrum strength $A_{r_{0}=20~{\rm nm},T=100~{\rm mK}}=35$ MHz in order to fit well with the experimental observation~\cite{yoneda2018}.}
\end{figure}
The pure-depasing of the qubit is caused by the longitudinal coupling between the qubit and the charge noise as illustrated by the second term in Eq.~(\ref{eq_decoherence}). This term can been traced back to the longitudinal slanting term in Eq.~(\ref{eq_model2}). If we model the qubit dephasing as ${\rm exp}\left[-\Gamma_{\rm ph}(t)\right]$, the decaying factor be written as~\cite{Palma1996}
\begin{equation}
\Gamma_{\rm ph}(t)=2\frac{y^{2}_{0}}{r^{2}_{0}}\int^{\omega_{\rm max}}_{\omega_{\rm min}}d\omega\,S(\omega)\frac{\sin^{2}(\omega\,t/2)}{(\omega/2)^{2}},\label{eq_dephasingrate}
\end{equation}
where $\omega_{\rm min (max)}$ is the lower (upper) bound of the noise frequency, and the spectrum function is defined as
\begin{eqnarray}
S(\omega)&=&\sum_{k}\frac{e^{2}r^{2}_{0}\Xi^{2}(\omega)[2n(\omega)+1]}{2\hbar^{2}}\delta(\omega-\omega_{k})\nonumber\\
&\approx&\sum_{k}\frac{e^{2}r^{2}_{0}\Xi^{2}(\omega)k_{B}T}{\hbar^{3}\omega}\delta(\omega-\omega_{k})\equiv\frac{A^{2}_{r_{0},T}}{\omega},\label{eq_noisespectrum}
\end{eqnarray}
with $A_{r_{0},T}$ being a parameter characterizing the strength of the charge noise. The lower bound of the noise spectrum is about $\omega_{\rm min}\approx\,10^{-2}$ Hz~\cite{yoneda2018}, and the upper bound of the noise spectrum is about $\omega_{\rm max}\approx\,5\times10^{5}$ Hz~\cite{yoneda2018}. We have also included the temperature effect in deriving Eq.~(\ref{eq_noisespectrum}) by writing the Bose occupation number as $n(\omega)=1/\left[{\rm exp}(\hbar\omega/k_{B}T)-1\right]\approx\,k_{B}T/(\hbar\omega)\gg1$, under the realistic temperature~\cite{yoneda2018} ($T=100$ mK) for all the low frequency noise modes ($\omega_{\rm max}\sim0.004$ mK). Note that $A_{r_{0},T}$ has the dimension of the frequency, in order to fit well with the experimental observed dephasing time $T^{*}_{2}\approx20\,\mu$s~\cite{yoneda2018}, we have chosen $A_{r_{0}=20~{\rm nm},T=100~{\rm mK}}=35$ MHz (see Fig.~\ref{fig_dephasing}). It is instructive to see for the time scale $t<1/\omega_{\rm max}=2 \mu$s, we can write the dephasing factor as the following simple form (a similar version of Ref.~\onlinecite{Schriefl2006})
\begin{equation}
\Gamma_{\rm ph}(t)=2A^{2}_{r_{0},T}t^{2}\frac{y^{2}_{0}}{r^{2}_{0}}\ln\frac{\omega_{\rm max}}{\omega_{\rm min}}.\label{eq_gaussdecay}
\end{equation}
Thus, the qubit dephasing at short time must be a Gauss decay. Actually, for time scale larger than $t>1/\omega_{\rm max}$ in our model, we find that the difference between the Gauss decay (\ref{eq_gaussdecay}) and the exact decay (\ref{eq_dephasingrate}) is very small (see Fig.~\ref{fig_dephasing}).
Let us discuss on the spectrum function defined in Eq.~(\ref{eq_noisespectrum}). Although our derivation of the spectrum function with 1/$\omega$ distribution has been made plausible, the difficulty lies in choosing reasonable $\Xi_{k}$ such that the second expression can be written as the third expression in the last line of Eq.~(\ref{eq_noisespectrum}). Actually, the physical mechanism of the charge spectrum with 1/$\omega$ distribution is still unclear~\cite{Paladino2014}. Here, we give a simple argument to realize the 1/f spectrum function. Note that the wavevector $\vec{k}$ is perpendicular to the $yz$ plane, and for the electromagnetic wave we have the dispersion relation $\omega_{k}=ck$, where $c$ is the speed of light. We make the following replacement in Eq.~(\ref{eq_noisespectrum}) $\sum_{k}\rightarrow\int\,d\omega_{k}L/(\pi\,c)$, where $L$ is the length of the space in the $x$ dimension ($V=L^{3}$). It is suggested that the charge field of wavevector $\Xi_{k}$ should be a constant $\Xi_{k}\equiv\Xi$, which is in stark contrast with that of the vacuum electromagnetic field~\cite{scully1999quantum}. Hence the spectrum function can be written as
\begin{equation}
S(\omega)=\frac{e^{2}r^{2}_{0}\Xi^{2}Lk_{B}T}{\pi\,c\hbar^{3}\omega},
\end{equation}
which is indeed of the 1/$\omega$ form. Note that the linear temperature dependence of the spectrum function is consistent with both theoretical~\cite{Dutta1981,Culcer2009} and experimental~\cite{Jung2004} investigations. Although we only study the low-frequency 1/f charge noise, it is still of interest to discuss the spectrum function in the high-frequency region under this argument. Note that the first line of Eq.~(\ref{eq_noisespectrum}) is valid in all frequency range. For the high-frequency noise modes $\hbar\omega\gg\,k_{B}T$, $n(\omega)=1/\left[{\rm exp}(\hbar\omega/k_{B}T)-1\right]\approx0$. Hence, the spectrum function in the high-frequency region should be
\begin{equation}
S(\omega)=\frac{e^{2}r^{2}_{0}\Xi^{2}L}{2\pi\,c\hbar^{2}}.
\end{equation}
This spectrum function is irrelevant to the frequency $\omega$. In the noise theory, noise with this kind of spectrum is called white noise~\cite{Paladino2014}.
\section{Prolong the dephasing time}
The dephasing time $T^{*}_{2}$ is an important time scale for the qubit~\cite{Buluta2011}. A long dephasing time is always appreciated for almost all qubit candidates. Based on the spin dephasing theory built in the above section, here we study how to prolong the spin dephasing time in a Si quantum dot.
The first intuitional approach is to reduce the quantum dot characteristic length $r_{0}$~\cite{Bermeister2014}. The characteristic length is related to the electric dipole moment of the quantum dot, such that reducing $r_{0}$ obviously reduces the effective coupling between the spin and the charge noise in Eq.~(\ref{eq_decoherence}). However, the coupling between the spin and the classical field, i.e., the Rabi frequency in Eq.~(\ref{eq_edsr}), is reduced simultaneously. Therefore, reducing $r_{0}$ not only increases the dephasing time $T^{*}_{2}$ [see Fig.~\ref{fig_dephasingvsr0}(b)] but also increases the Rabi manipulation time $T_{\rm Rabi}$ [see Fig.~\ref{fig_dephasingvsr0}(a)]. The $r_{0}$ dependence of the dephasing can be roughly written as ${\rm T}^{*}_{2}\propto\,r^{-4}_{0}$. From this viewpoint, reducing $r_{0}$ may not be an effective way to prolong the dephasing time. Note that the spin dephasing time $T^{*}_{2}$ is obtained by solving $\Gamma_{\rm ph}( T^{*}_{2})=1$ in Eq.~(\ref{eq_dephasingrate}).
\begin{figure}
\includegraphics{T2vsr0.eps}
\caption{\label{fig_dephasingvsr0}The spin manipulation time (a) and the spin dephasing time (b) as a function of the quantum dot characteristic length $r_{0}$. The manipulation time is defined as $T_{\rm Rabi}=h/(2eE_{z}\sqrt{z^{2}_{0}+y^{2}_{0}})$, where $E_{z}=4000$ V/m, and the dephasing time $T^{*}_{2}$ is solved from $\Gamma_{\rm ph}(T^{*}_{2})=1$.}
\end{figure}
The second approach is to lower the environmental temperature $T$~\cite{Culcer2009}. Lower the temperature can remarkably reduce the average occupation number $n(\omega)\approx\,k_{B}T/(\hbar\omega)$ in the low frequency noise mode. The typical temperature in experiment is about $100$ mK~\cite{yoneda2018}. The effects of lowering the temperature are shown in Fig.~\ref{fig_dephasingvstemperature}(a). The temperature dependence of the dephasing can be roughly written as $T^{*}_{2}\propto1/\sqrt{T}$. A substantial improvement in the dephasing time is achievable if the experimental temperature can be lowered to the micro-Kelvin region.
\begin{figure}
\includegraphics{T2vsT.eps}
\caption{\label{fig_dephasingvstemperature}(a) The spin dephasing time as a function of the environment temperature $T$. (b) The spin dephasing time as a function of the longitudinal field gradient $b_{l}$.}
\end{figure}
The third approach is to engineer the slanting fields~\cite{Goldman2000,Neumann2015,Yoneda2015}. As can be seen from Eqs.~(\ref{eq_edsr}) and (\ref{eq_decoherence}), the longitudinal field gradient $b_{l}$ is detrimental to both the spin manipulation and the spin dephasing. While the transverse field gradient $b_{t}$ contributes to the Rabi frequency in EDSR. Thus, it is desirable to design a proper micromagnet structure, that can give rise to both an increased transverse slanting field (shorter $T_{\rm Rabi}$) and decreased longitudinal slanting field (longer $T^{*}_{2}$). The dependence of the dephasing $T^{*}_{2}$ on the longitudinal field slope $b_{l}$ is shown in Fig.~\ref{fig_dephasingvstemperature}(b). This dependence can be roughly written as $T^{*}_{2}\propto\,1/b_{l}$.
Of great interest is designing a proper micromagnet-quantum-dot structure such that the longitudinal field gradient is reduced. Let us consider a cuboid micromagnet, the dimensions of which along $x$, $y$, and $z$ are $W$, $D$, and $L$, respectively (see Fig.~\ref{fig_micromagnet}). The external magnetic field is applied along the $z$ direction, and we assume the micromagnet is fully polarized. The origin of the coordinate system is located at the geometric center of the micromagnet. We give two possible structures with one micromagnet involved [see Fig.~\ref{fig_micromagnet}(a)] and two micromagnets involved [see Fig.~\ref{fig_micromagnet}(b)]. The $y$-dimension of the micromagnet $D$ should be large enough such that there is no $y$-component of the field ($B^{y}_{m}=0$) near the quantum dot, only $x$ and $z$-components of the field are retained ($B^{x}_{m}\neq0$ and $B^{z}_{m}\neq0$). From Eq.~(\ref{eq_slantingfield}), these ideal structures give $b_{l}=0$.
\begin{figure}
\includegraphics[width=8.5cm]{micomagnet.eps}
\caption{\label{fig_micromagnet}Possible micromagnet-quantum-dot structures giving rise to reduced longitudinal field gradient ($d>W/2$). (a) The quantum dot is placed below the micromagnetic~\cite{Goldman2000}. (b) The quantum dot is placed below two identical micromagnets~\cite{yoneda2018}. }
\end{figure}
\begin{figure}
\includegraphics{micromagnet_field.eps}
\caption{\label{fig_micromagnet_field}The stray field and its gradient near the quantum dot. The structure parameters are $L=3$ $\mu$m, $W=0.3$ $\mu$m, $D=3.76$ $\mu$m and $s=0.2$ $\mu$m. (a) The results for single micromagnet design given in Fig.~\ref{fig_micromagnet}(a). (b) The results for two micromagnets design given in Fig.~\ref{fig_micromagnet}(b).}
\end{figure}
The Co micromagnet has a Curie temperature $T_{C}\approx1400$ K and a saturation magnetization $M_{s}=1.467\times10^{6}$ A/m~\cite{Neumann2015}. Assuming full polarization and neglecting the edge fluctuations of the micromagnet, one can obtain the field distribution using the analytical method given in Ref.~\onlinecite{Goldman2000}. Because the quantum dot is placed on the symmetrical line of the proposed micromagnet structure, from symmetry analysis, the stray field at
$(-d,0,0)$ in Fig.~\ref{fig_micromagnet_field}(a) or $(-d,0,(L+s)/2)$ in Fig.~\ref{fig_micromagnet_field}(b) must parallel with $\hat{z}$, i.e., $\bf{B}_{m}//\hat{z}$, and its strength depends on $d$. Hence, there is a transverse field gradient $b_{t}=\partial\,B^{z}_{m}/\partial\,x=\partial\,B^{x}_{m}/\partial\,z$. While the longitudinal field gradient $b_{l}=\partial\,B^{y}_{m}/\partial\,z=\partial\,B^{z}_{m}/\partial\,y=0$ is guaranteed by the large dimension $D$ of the micromagnet. In the single micromagnet design, the maximal transverse field gradient is about $0.6$ mT/nm (see Fig.~\ref{fig_micromagnet_field}(a)). Of course, a larger field gradient is achievable by reducing $L$. In the two micromagnets design, the transverse field gradient can be as large as $10\sim20$ mT/nm (see Fig.~\ref{fig_micromagnet_field}(b)). The structure with two micromagnets more easily produces a larger transverse slanting field.
The forth promising way is to use the dynamical decoupling scheme~\cite{Uhrig2007,Lee2008,Yang2008,Cywinski2008} as has also been used in experiments. The spirit of dynamical decoupling is to frequently flip the spin using pulse sequences, such that the effective spin-noise interaction is eliminated as being of high-order small. Certainly, the performance of dynamical decoupling depends on how many pulses are applied~\cite{Medford2012}. Consider $n$ pulses applied to the qubit at a serious instant time $0<\delta_{1}t<\delta_{2}t<\ldots<\delta_{n}t<t$, i.e., at each instant time the spin qubit is flipped by the pulse, we want to determine the qubit phase coherence at the time $t$. Note that here we only consider ideal pulses, i.e., each pulse has a delta-function shape, so that the spin flip is accomplished at the instant time of the pulse applied~\cite{Uhrig2007}.
Under $n$-pulse sequences, the dephasing of the spin qubit due to 1/f charge noise reads as~\cite{Uhrig2007}
\begin{equation}
\Gamma^{d}_{\rm ph}(t)=\frac{y^{2}_{0}}{2r^{2}_{0}}\int^{\omega_{\rm max}}_{\omega_{\rm min}}d\omega\,S(\omega)\frac{|y_{n}(\omega\,t)|^{2}}{(\omega/2)^{2}},
\end{equation}
where
\begin{equation}
y_{n}(\omega\,t)=1+(-1)^{n+1}e^{i\omega\,t}+2\sum^{n}_{l=1}(-1)^{l}e^{i\delta_{l}\omega\,t}.
\end{equation}
For Carr-Purcell-Meiboom-Gill (CPMG) pulse sequence, the $n$ pulses are applied at the following serious instant time $\delta_{l}=(l-1/2)/n$~\cite{Carr1954,Meiboom1958}, while for Uhrig pulse sequence $\delta_{l}=\sin^{2}(\frac{\pi\,l}{2n+2})$~\cite{Uhrig2007} ($l=1,\cdots\,n$). In principle, dynamical decoupling can prolong the qubit dephasing time to any desired time scale as long as enough number of pulses are applied~\cite{Medford2012}. The practical performance of dynamical decoupling is often limited by the fact that realistic pulses are impossible in delta-function shape, i.e., the flip of the spin must cost a finite time. The phase coherence of the spin qubit under dynamical decoupling is shown in Fig.~\ref{fig_dyndcoup}. As can be seen from the figure, the phase coherence time under spin echo is about $T^{\rm echo}_{2}\approx100$ $\mu$s. Under eight-pulse sequences, the spin dephasing time can be prolonged to $T_{2}\approx260$ $\mu$s. We also find that the CPMG-pulse sequences [see Fig.~\ref{fig_dyndcoup}(a)] perform a little better than the Uhrig-pulse sequences [see Fig.~\ref{fig_dyndcoup}(b)] in our model.
\begin{figure}
\includegraphics{dd.eps}
\caption{\label{fig_dyndcoup}The phase coherence of the spin qubit under dynamical decoupling. The noise spectrum strength is chosen as $A_{r_{0}=20~{\rm nm},T=100~{\rm mK}}=35$ MHz. (a) CPMG-pulse sequences. (b) Uhrig-pulse sequences.}
\end{figure}
\section{Relaxation of the spin qubit}
The relaxation time $T_{1}$, i.e., the lifetime, is also an important characteristic time of the qubit~\cite{Khaetskii2001}. Even if there is no pure-dephasing for the qubit, the phase coherence time $T_{2}$ can still be limited by the qubit relaxation $T_{2}=2T_{1}$~\cite{Golovach2004}. Here we examine whether the 1/f charge noise will give rise to spin relaxation~\cite{Huang2014} in our model. The possible relaxation mechanism comes from the third term in Hamiltonian (\ref{eq_decoherence}). There is no exact method in calculating the relaxation rate, instead, the Fermi golden rule is usually used to calculate this quantity~\cite{Khaetskii2001}
\begin{equation}
\Gamma_{\rm relax}=\frac{\pi}{\hbar^{2}}\int^{\omega_{\rm max}}_{\rm \omega_{min}}\,d\omega\rho(\omega)e^{2}\Xi^{2}(z^{2}_{0}+y^{2}_{0})\delta\left(\omega-\frac{2\Delta}{\hbar}\right),
\end{equation}
where $\rho(\omega)$ is density of state of the charge noise mode. It should be noted that the qubit level spacing is about 80 GHz and the maximal charge noise frequency is about $0.5$ MHz~\cite{yoneda2018}, such that there is no charge noise frequency can match the level spacing of the spin qubit. Here arises the problem of whether the upper bound of the charge noise spectrum is indeed in the MHz range~\cite{chan2018,Kawakami2016,bylander2011,kuhlmann2013}? Our simple argument in Sec.~\ref{sec_IV} suggests $\omega_{\rm max}\ll\,k_{B}T/\hbar$ ($\sim13$ GHz for $T=100$ mK). An upper bound of 20 KHz in a SiMOS quantum dot is reported in Ref.~\onlinecite{chan2018}. Even if the qubit level spacing lies in the range of the charge noise spectrum, i.e., $\omega_{\rm min}<2\Delta/\hbar<\omega_{\rm max}$, our following calculation shows that the spin relaxation time is actually very long. By making the replacement $\sum_{k}\rightarrow\int\,d\omega_{k}\rho(\omega_{k})$ in Eq.~(\ref{eq_noisespectrum}), we have $\rho(\omega)e^{2}r^{2}_{0}\Xi^{2}\equiv\hbar^{3}A^{2}_{r_{0},T}/(k_{B}T)$. Hence, the relaxation rate can be written as
\begin{equation}
\Gamma_{\rm relax}=\frac{\pi\hbar\,A^{2}_{r_{0},T}}{k_{B}T}\times\frac{z^{2}_{0}+y^{2}_{0}}{r^{2}_{0}}.
\end{equation}
For a Si quantum dot with the parameters given in Table~\ref{tab}, we have $\Gamma_{\rm relax}=0.4519$ Hz, hence ${\rm T}_{1}=2.2$ s, indeed is a very long relaxation time. Thus, based on the above analysis, we suggest that 1/f charge noise does not limit the spin relaxation time in a Si quantum dot integrated with a slanting field.
\section{Summary}
In summary, we have studied in detail the spin manipulation and the spin dephasing in a Si quantum dot integrated with a slanting magnetic field. The longitudinal slanting field not only gives rise to a modulated Rabi oscillation in the spin manipulation, but also mediates a longitudinal spin-charge interaction which leads to spin dephasing. Several practical strategies are also proposed to alleviate the spin dephasing. Also, 1/f charge noise does not limit the spin relaxation time due to the mismatching between the qubit level spacing and the charge noise frequency. Our study can help clarify the spin dephasing mechanism in Si quantum dot.
\section*{Acknowledgements}
This work is supported by the National Natural Science Foundation of China Grant No.~11404020, the Postdoctoral Science Foundation of China Grant No.~2014M560039, and Doctoral Fund of Yanshan University Grant No. BL18043.
|
{
"timestamp": "2019-04-30T02:17:30",
"yymm": "1804",
"arxiv_id": "1804.05476",
"language": "en",
"url": "https://arxiv.org/abs/1804.05476"
}
|
\section{introduction}
\label{sec:intro}
We consider the problem of approximating a binary matrix $X \in \{0,1\}^{m{\times}n}$ as the product of two other binary matrices $U \in \{0,1\}^{m{\times}p}$ and $V \in \{0,1\}^{n{\times}p}$ plus a third \emph{residual} matrix $E$,
\begin{equation}
X = UV^\intercal + E.
\label{eq:mf}
\end{equation}
\color{blue}
\mnote{1.1}The problem of Binary Matrix Factorization (BMF) arises naturally in many problems, in particular in the so called ``association matrices'' where a $X_{ij}=1$ indicates that some object $i$ belongs to group $j$~\cite{bmf-app-biclustering}, or some entity (e.g., a gene) $i$ is present in some species $j$~\cite{bmf-app-diversity,bmf-app-microbial}, or are related to some type of disease~\cite{bmf-app-tumor}. The latter examples show the increasing relevance of this type of data in genomics. Other similar, growing applications include recommender systems (client-product preferences, etc.).
\color{black}
The BMF problem dates back to at least the 1960s~\cite{bmf-oldest} and has been treated extensively
in the last three decades by various research communities, under quite different names. It was first
studied as a combinatorial problem as a particular case of the classic \emph{set covering} problem
(see~\cite{monson95} and references therein). It then received great attention from the data mining
community.
\color{blue}\mnote{2.2-3} Long known to be an NP-hard problem~\cite{asso}\color{black}, the earlier works in this field developed heuristics such as the \emph{tiling} or \emph{tile matching/searching} methods, where binary matrices are decomposed as Boolean or modulo-2 superpositions of rectangular tiles~\cite{proximus,tiling}; the BMF problem was later formulated as a matrix factorization problem in~\cite{bmf07}, with several works following that line since then.
\color{blue}
Being an NP-hard problem, the quality of the decompositions $(U,V)$ is commonly \mnote{1.2}measured in terms of their \emph{interpretability}, that is whether the columns of $U$ and $V$ exhibit patterns that are intuitive in some sense, or reflect \emph{a priori} knowledge about the problem. For example, in data mining problems, where the pairs $(U_i,V_i)$ are interpreted as \emph{association rules} (for example, a group of clients -- indicated by non-zeros in $U_i$ -- prefers a certain group of products -- indicated by non-zeros in $V_i$);
the examples in this paper are designed to show this interpretability in a visual way, by studying the results on visual patterns obtained on sets of images.
\color{black}
A thorough survey of BMF methods is beyond the scope of this paper; we refer the reader
to~\cite{bmf-comp} for a more in-depth review. We will however mention some works which are
representative of the diversity of formulations and tools that surround the treatment of this
problem, as well as the shortcomings that are common to the current state of the art and that
motivate the development of the tools that we present in this work.
\subsection{Brief overview of Binary Matrix Factorization}
We begin with the ASSO algorithm proposed in~\cite{asso}. The method begins by constructing the so
called \emph{association matrix} $C$, which is a thresholded version of a particular normalization
of the correlation matrix $X{^\intercal}X$. It then produces a series of increasing rank approximations
by adding to $U$ a column taken from $C$ and searching for a corresponding binary row $V_k$ whose
outer product with $U_k$ minimizes the number of non-zeros in the current approximation error
$E$. This method can be efficiently implemented with bitwise and integer operations. It is also a
popular and simple method with good performance. However, the complexity of each ASSO step is
$O(kn^2m)$, so that it does not scale well in applications where $n$ (the number of samples) is very
large, something very common in current data science problems, which is the target of our work.
\color{blue}
\mnote{0.1}
\mnote{2.2-3}
Many works approximate the BMF problem by a relaxed (non-convex) Non-negative Matrix Factorization (NMF) problem where $U$ and $V$ are allowed to take on real values, and then map the resulting (approximate) solution to the binary domain using some predefined rule. Examples of this are~\cite{bmf07,bmf13,bmf-nmf2}. In particular, the work~\cite{bmf13} develops a set of BMF \emph{identifiability} conditions, that is, conditions under which the binary factorization of $X$ is unique (up to permutations). However, as in~\cite{bmf07,bmf13}, their solution is an approximation based on a local minima of the NMF problem, so there are no guarantees that the binarized pair $(U,V)$ obtained coincides with the unique solution even if the identifiability conditions are satisfied.
Being based on non-linear optimization, the NMF methods are significantly more computationally demanding than binary methods such as ASSO.
\color{black}
The work~\cite{bmf-mp} stands out as an interesting alternative to BMF which formulates the
decomposition of $X$ as a Bayesian denoising problem with a particular prior on the
unobserved \emph{clean} matrix $\hat{X}$ ($X=\hat{X}+E$) and uses a \emph{Message Passing} algorithm
to find the maximum a posteriori estimation of $\hat{X}$. Message Passing is a mature technique
which in the form presented in this work can be quite computationally demanding. Approximate Message
Passing \cite{amp,gamp} techniques have since been developed which may provide significant
efficiency gains to the technique proposed in~\cite{bmf-mp}, but we are currently unaware of any
development in this direction.
An example of a tile-searching heuristic is the Proximus method~\cite{proximus}, which approximates
the first principal left and right binary components of the binary matrix $X$. The method can be
extended to produce a hierarchical representation of the matrix with further rank-$1$ components,
although these do not coincide with additional factors in a rank-$k$ factorization. Contrary to the
above methods, the focus of Proximus is on speed an scalability, and indeed is much faster and
scales better than any of the above methods. Also, as we will see in the next subsection, finding
the first principal component of a binary matrix is closely related to a crucial step in one of the
main dictionary learning methods. e will describe this method in detail later in this document.
\color{blue}
\subsection{Dictionary learning}
\emph{Dictionary learning} methods were first introduced in \cite{olshausen97}
and later adopted as a powerful extension of the \emph{transform analysis} concept, ubiquitous in
signal processing since the introduction of Fourier analysis and their related discrete variants
(DFT, DCT). In this setting, the matrix $X \in \ensuremath{\mathbb{R}}^{m{\times}n}$ consists of $n$ columns of
dimension $m$, and the decomposition obtained $X=DA+E$ is comprised of a \emph{dictionary} $D$, a
matrix of \emph{linear coefficients} $A$, plus a residual matrix $E$. Dictionary learning methods
are adaptive: given sufficient data samples, they can be trained to efficiently represent such
samples as a linear combination of very few basis elements or ``atoms'' (see~\cite{dl-review} for a
review). Despite coming from a very different community, dictionary learning methods can be seen as
matrix factorization methods which are specially tailored to the where $X$ is either extremely
``fat'' ($n \gg m$) or ``tall'' ($n \ll m$). Furthermore, many of these methods can be implemented on-line, that is,
they can process new data samples as they arrive, and adapt the dictionary along the
way~\cite{online-dl}. Another important aspect of dictionary learning methods is that they are not
restricted to low-rank decompositions; if fact, the solutions can be \emph{overcomplete}, meaning that the
number of columns $p$ in $D$ may be (much) larger than $m$, the dimension of the samples to be
represented.
\mnote{2.2-3}As with all BMF methods, dictionary learning problems are non-convex and their solution cannot be obtained exactly. Nevertheless, similarly to what happens with BMF, their enormous practical success in a wide range of signal processing and machine learning problems, and their ability to produce human-interpretable patterns, has led to their widespread adoption.
\color{black}
\subsection{Model selection}
As with any statistical model, the problem of model selection, (in this case, choosing $p$) is of
paramount importance to BMF. Various works~\cite{bmf-mdl,bmf-tiling-mdl,bmf-sel,panda} have
addressed this particular problem. In particular,~\cite{bmf-mdl} and~\cite{bmf-tiling-mdl} are based
on the Minimum Description Length (MDL) principle~\cite{mdl1,mdl2,mdl3}, which forms the basis of
our model selection strategy as well. As a side note, the work~\cite{bmf-tiling-mdl} represents a
recent example of the tiling approach. This problem has also been addressed in dictionary learning
in~\cite{dl-mdl}.
\subsection{Main contribution}
\color{blue}
To the best of our knowledge, the works in the existing literature on BMF make no particular
assumptions on the \emph{shape} of the matrices to be decomposed. In particular, many methods which
are efficient for the $n \approx m$ case do not scale well for $n \gg m$ and vice versa. Also, most
methods deal with the \emph{offline} analysis of readily available matrices, which makes them
unsuitable to many recent data processing tasks involving \emph{online} adaptation of the models.
The main motivation behind this work is the ability of dictionary learning methods to cope with the challenges mentioned above. We propose two dictionary learning-based BMF methods which are particularly suited to the treatment of extremely fat (or tall) matrices, and which are also suitable to online processing of samples. The first one, named Method of Binary Directions (MOB), is an adaptation of the method proposed in~\cite{online-dl}, itself an adaptation of the Method of Optimum Directions (MOD)~\cite{mod} (hence the name). The second method is based on the idea of sequential rank-one updates of the K-SVD method~\cite{aharon06}; we adopt the Proximus~\cite{proximus} algorithm as a fast approximation to the mentioned operation; we thus bring together tools from two inherently related but otherwise disconnected fields to obtain a novel formulation to the binary matrix factorization. Finally, both methods rely on a third novel algorithm which we call Binary Matching Pursuit (BMP). This is a binary adaptation of the Matching Pursuit method~\cite{mp} for approximating the sparsest solution to a least squares regression problem. \mnote{2.1}This adaptation, although conceptually analogous to MP, is non-trivial, as its efficiency relies on a careful combination of different algebraic operations.
\color{black}
Our methods construct binary dictionaries and binary coefficients matrices using efficient bitwise and a few integer operations. This is particularly relevant to the efficiency of our methods as recent processor architectures incorporate the ability to handle large number of bits through SIMD (Single Instruction Multiple Data) instructions. For example, a current off-the-shelf processor can perform a \emph{popcount} instruction (which counts the number of $1$s in a binary array) on a 256-bit register. This allows, for example, to compute the dot product between two binary vectors of dimension $256$ with just two processor instructions. \color{blue} \mnote{2.4}As we show in Section~\ref{sec:bdl}, our methods are also linear both in $n$ and $m$ and thus scale well for large matrices.\color{black}
\color{blue} The main objectives of this paper are to present our proposed methods, assess their interpretability on different datasets, to analyze their computational properties such as computational complexity and convergence rate, and to see how these properties are affected by the initial conditions. We thus focus our experiments on a small set of easily-interpretable datasets for which the patterns obtained can be easily recognized as salient features in the data. \color{black}
The rest of this document is organized as follows: Section~\ref{sec:background:dictionary-learning} provides the notation and background on the methods on which our methods are based. The proposed methods themselves are described in Section~\ref{sec:bdl}.
Section~\ref{sec:model-selection} describes the proposed MDL-based model selection algorithm for searching the best model order $p$. %
We present and discuss our results in Section~\ref{sec:results}, and provide concluding remarks in Section~\ref{sec:conclusion}.
\section{Background}
\label{sec:background}
\subsection{Notation}
\label{sec:background:notation}
\def\mathbf{1}{\mathbf{1}}
\def\mathrm{bool}{\mathrm{bool}}
\def\circ{\circ}
\def\lor{\lor}
\def\bigwedge{\bigwedge}
\def\bigvee{\bigvee}
\def\oplus{\oplus}
\def\otimes{\otimes}
\def\mathrm{mod}{\mathrm{mod}}
\newcommand{\iter}[1]{^{(#1)}}
\newcommand{\ensuremath{\quad\mathrm{s.t.}\quad}}{\ensuremath{\quad\mathrm{s.t.}\quad}}
\newcommand{\norm}[1]{\ensuremath{\left\|#1\right\|}}
\newcommand{\support}[1]{\mathrm{supp}(#1)}
\newcommand{\rankf}[1]{\mathrm{rank}(#1)}
\def\mathrm{rank}{\mathrm{rank}}
\newcommand{\fun}[1]{\mathrm{#1}}
\newcommand{\abs}[1]{\ensuremath{\left|#1\right|}}
\newcommand{\setdef}[1]{\ensuremath{\left\{#1\right\}}}
\newcommand{\ensuremath{\mathrm{span}}}{\ensuremath{\mathrm{span}}}
\newcommand{\svec}[1]{_{[#1]}}
\newcommand{\row}[1]{_{#1:}}
\newcommand{\col}[1]{_{:#1}}
\newcommand{\havg}[1]{\langle\!\langle{#1}\rangle\!\rangle}
We begin this section by establishing the notation to be used throughout the paper.
Standard operations such as addition or subtraction are denoted as usual, $1+1=2$, $1-1=0$,
etc. Given two binary values $a$ and $b$, we use $a \land b$ to denote their logical AND (Boolean product),
$a \lor b$ is their OR (Boolean sum), $a \oplus b$ is modulo-2 addition (Boolean eXclusive OR), and
$\neg a$ is the 1's complement (Boolean negation) of $a$. The same notation is used for element-wise operations between vectors and matrices of the same dimension.
Let $x$ and $y$ be two binary vectors and $A$ and $B$ two binary matrices.
We denote the standard inner and outer vector-vector, matrix-vector and matrix-matrix products as $x{^\intercal}y$, $xy^\intercal$, $Ax$ and $AB$.
The Boolean inner product between two vectors, $z = x^\intercal \circ y$ is defined as $\bigvee x_i \land y_i$. Similarly, the $C_{ij}$ element of the matrix product $C=A \circ B$ is defined as the Boolean product between the $i$-th row of $A$, $A\row{i}$, and the $j$-th column of $B$, $B\col{j}$. We define the modulo-2 inner product of $x$ and $y$ as $x \otimes y = (x+y)\;\mathrm{mod} \;2$. Finally, similarly to the Boolean case the $C_{ij}$ element of the modulo-2 product between two matrices, $C=A \otimes B$ is given by $A\row{i} \otimes B\col{j}$.
The cardinality of a set $J$ is denoted by $|J|$. Given a set of indexes $J$, $B\col{J}$ denotes the
sub-matrix of columns of $B$ indexed by $J$, and $B\row{J}$ denotes a subset of its rows. These can
be combined with single indexes or other sets, e.g., $B_{iJ}$ contains the elements of row $i$ and
column indexes in $J$. For a
vector $x$, its Hamming weight $h(x)$ is defined as the number of non-zero elements in $x$, that is
$h(x)=|\{i: x_i \neq 0\}|$. The same notation $h(A)$ is applied to count the non-zero elements of
matrices. The Hamming weight is usually referred to as the $\ell_0$ pseudo-norm,
$\|x\|_0=h(x)$. Note that, for binary vectors, $h(x)=\|x\|_0=\|x\|_1=\norm{x}_2^2$. Likewise, for
binary matrices, $\|A\|_0=\|A\|_{1,1}=\|A\|_F^2$. The function $\mathbf{1}(\cdot)$ is defined so
that $\mathbf{1}(cond)=1$ if $cond$ is true, and $0$ otherwise.
\color{blue}
\subsection{A note on computational complexity}
\mnote{2.4}For each algorithm we provide a brief computational complexity analysis using a
simplified form of the familiar ``big O'' notation. Here a procedure which has order $O(m)$ is said
to have \emph{linear complexity} in $m$, that is, it requires at most $am + b$ operations where $a$
and $b$ do not depend on $m$. Similarly $O(m^2)$ indicates \emph{quadratic complexity}, and
$O(m \log m)$ requires about $a(m \log m) + b$ operations in the worst case. This use is slightly
different than the formal Bachmann-Landau definition of $O(\cdot)$, which is defined in asymptotic
terms. For example, we might write $O(m \log m + p)$ as a shorthand to $O(m \log m) + O(p)$ to
indicate that a function requires about $am \log m + bp + c$ operations.
On the other hand, in all cases we will make a distinction as to the nature -- floating point,
integer, bitwise -- of the operations, as the constants involved in each case can vary greatly. For
example, as pointed out in the introduction, a bitwise inner product of two binary vectors of length
$256$ can computed in just \emph{two} CPU instructions, whereas it might require over $512$ instructions to compute the same operation on floating point or integer vectors.
\color{black}
\subsection{Dictionary Learning and sparsity}
\label{sec:background:dictionary-learning}
\color{blue}
As described before, dictionary learning methods seek a to decompose $X$ as $X=DA+E$, where $D$ is such that we can achieve $\|E\col{j}\| \ll \|X\col{j}\|$ by using just a few atoms of $D$, that is, with a number of non-zero elements in $A\col{j}$ much smaller than $p$. The latter requirement is called \emph{sparsity}. Thus, dictionary learning is often also referred to as \emph{sparse modeling}, and finding $A\col{j}$ given $D$ as \emph{sparse coding}.
A typical approach to the dictionary Learning problem is to obtain a local solution by \emph{alternate minimization} in $D$ and $A$ of a cost function $f(D,A) + g(A)$,
\begin{eqnarray}
A\iter{t+1} =& \arg\min_{A} \{ f(D\iter{t},A) + g(A) \} \\
D\iter{t+1} =& \arg\min_{D} \{ f(D,A\iter{t+1}) + g(A\iter{t+1}) \},
\label{eq:dl}
\end{eqnarray}
where $f(\cdot)$ is a \emph{data fitting} term, usually $\|DA-X\|_F^2$ (here $\|X\|_F$ denotes the Frobenius norm of matrix $X$), and $g(\cdot)$ is a \emph{regularization} term which promotes sparsity in the columns of $A$. We now describe the two methods on which our methods are inspired.
\color{black}
\subsubsection{Method of Optimal Directions (MOD)}
\label{sec:mod}
For the case $f(D,A)=\|DA-X\|_2^2$ and $g(A)=\sum_{j}\|A\col{j}\|_1$ the \emph{Method of Directions} (MOD)~\cite{mod}, is given by
\begin{eqnarray}
A_j\iter{t+1}\!\! &=&\!\! \arg\min_{a \in \ensuremath{\mathbb{R}}^p} \{ \|x_j - D\iter{t}a \|_2^2 + \|a\|_1 \}, \\
D_r\iter{t+1}\!\! &=&\!\! U_r/\min\{1,\|U_r\|_2\},\nonumber\\
\!\!&&U=\!\!X(A\iter{t+1})^\intercal\left(A\iter{t+1}({A}\iter{t+1})^\intercal\right)^{-1},\,
\label{eq:mod}
\end{eqnarray}
The first step corresponds to an $\ell_1$-regularized least squares regression problem on each column of $A$, also known as LASSO~\cite{lasso}, a non-differentiable convex problem whose solution has been extensively studied in recent years, with several efficient algorithms designed specifically for the task (see e.g.~\cite{fista}).
In the second step, each atom $D\col{r}$ of the dictionary corresponds to a normalized down version of the least squares solution $u_r$.
The MOD algorithm is well suited for online dictionary adaptation, as both $A{A}^\intercal$ and
$XA^\intercal$ can be efficiently updated when new columns are added to $X$. Furthermore, if new
samples arrive one at a time, the inverse of the Hessian matrix $({A}A^\intercal)^{-1}$ can be
efficiently updated via the Matrix Inversion Lemma~\cite{matrix-inv-lemma}. Moreover, as shown
in~\cite{online-dl}, excellent results can be still obtained if the Hessian is approximated by its
diagonal (in which case computing its inverse requires just $O(p)$ operations).
\color{blue}
\paragraph*{Computational complexity} The offline version of the MOD dictionary \mnote{2.4}update step presented in Algorithm~\ref{eq:mod} requires $O(mnp) + O(p^2n) + O(p^3)$ floating point operations, which will generally be dominated by the $(p^2n)$ term.
\color{black}
\subsubsection{Matching Pursuit and the K-SVD algorithm}
\label{sec:ksvd}
In this algorithm, proposed in~\cite{aharon06}, $f(E)=\norm{E}_F^2$ and $g(A)=h(A)$. The columns of $A$ are computed using a greedy method known as OMP (Orthogonal Matching Pursuit)~\cite{omp}, which under certain conditions can be shown to provide the actual solution to the corresponding $\ell_0$-penalized least squares problem (see~\cite{tropp07}). A simpler variant of this step uses the (non-orthogonal) Matching Pursuit (MP)~\cite{mp}, which is described next in Algorithm~\ref{alg:mp},
\begin{algorithm}[ht]
\caption{\label{alg:mp}Matching Pursuit}
\KwData{vector to encode $x$, dictionary $D$, maximum residual norm $\epsilon$, maximum coefficients weight $h_{\max}$}
\KwResult{Coefficients vector $a$}
Set iteration $t \leftarrow 0$, residual $r\iter{0} \leftarrow x$, initial coefficients $a\iter{0} \leftarrow 0$\;
Set $g\iter{0} \leftarrow D^\intercal{r\iter{0}}$, $G \leftarrow D^\intercal{D}$ \;
\While{$\norm{r\iter{t}} \geq \epsilon$ and $h(a) < h_{\max}$}{
$i = \arg\max \left\{g\iter{t} \right\} $ \;
${\Delta} \leftarrow D\col{i}^\intercal r\iter{t}$ \;
$a_i\iter{t+1} \leftarrow a_i\iter{t} + \Delta$ \;
$r\iter{t+1} \leftarrow r\iter{t} - {\Delta}D\col{i} $\;
$g\iter{t+1} \leftarrow g\iter{t} - {\Delta}G\col{i} $\tcc*{(a)}
$t \leftarrow t+1 $ \;
}
\Return $a \leftarrow a\iter{t}$ \;
\end{algorithm}
What MP does at each iteration is to project the residual onto the atom that is most correlated to it, and then remove the projection from the residual. For this to work well, the atoms must be normalized to have $\ell_2$ norm 1.
The vector $g$ keeps track of the correlation between the residual $r$ and the dictionary $D$. Its update $(a)$ is derived as follows:
\begin{eqnarray}
g\iter{t+1}
&=& D^\intercal{r\iter{t+1}}=D^\intercal(r\iter{t}-{\Delta}D_i)\nonumber\\
&=& g\iter{t}-{\Delta}D^\intercal{D}\col{i} =g\iter{t}-{\Delta}G\col{i}.
\label{eq:corr-up}
\end{eqnarray}
Note that, by means of \refeq{eq:corr-up}, updating $g$ requires only $O(p)$ floating point operations, whereas the na\"{\i}ve update requires $O(mp)$ operations. This same trick, with a few modifications, will also be useful in the binary case.
\color{blue} \mnote{2.4}As the Gramm matrix is computed only once, the overall complexity of MP for processing all $n$ samples is $O(p^2m + p + kmn)$ floating point operations.\color{black}
The dictionary update of K-SVD performs a simultaneous update of each atom $D\col{r}$ and the row of $A$ associated to it. This update is described in Algorithm~\ref{alg:ksvd}.
\begin{algorithm}
\KwData{Current iterate $(D\iter{t}$, $A\iter{t})$}
\KwResult{Next iterate $(D\iter{t+1}$,$A\iter{t+1}$}
\For{$r=1,\ldots,p$}{
$J \leftarrow \{j: A_{rj}\iter{t} \neq 0\}$ \;
$R \leftarrow X\col{J} - D\iter{t}A\col{J}\iter{t} + D\col{r}\iter{t}(A_{rJ}\iter{t})$ \;
$U{\Sigma}V^\intercal \leftarrow \mathrm{SVD}(R)$ \;
$D\col{r}\iter{t+1}\leftarrow U\col{1}$ \;
$A_{rJ}\iter{t+1} \leftarrow V\col{1}$ \;
}
\caption{\label{alg:ksvd}K-SVD Dictionary update.}
\end{algorithm}
\color{blue}
\paragraph*{Computational complexity} Each of the $p$ updates of $D$ requires $O(mn)$
\mnote{2.4} operations for updating the residual plus another $O(n^2)$ operations for computing the first
pair of singular eigenvectors. This gives a total $O(pn^2) + O(pmn)$ operations, which in general
will in our case ($n \gg m \approx p$) will be significantly more expensive than the $O(p^2n) + O(pmn) + O(p^3)$ complexity of MOD.
\color{black}
\section{Binary Dictionary Learning}
\label{sec:bdl}
Below we describe our dictionary learning methods, which assume a given fixed dictionary size $p$
and employ the traditional alternate descent approach to obtain the best model $(D,A,E)$ for that
$p$. We leave the description of the top-level model selection algorithm for choosing the best model
order $p$ to Section~\ref{sec:model-selection}.
Both BMF algorithms share a common coefficients update step, the Binary Matching Pursuit (BMP)
algorithm, and two choices for the dictionary update step: MOB (a binarized version of MOD) and
K-PROX (combining ideas of K-SVD and Proximus); these are detailed next.
\subsection{Coefficients update via Binary Matching Pursuit (BMP)}
\label{sec:bdl:bmp}
\begin{algorithm}
\KwData{sample to encode $x$, dictionary $D$, initial coefficients $a\iter{0}$, maximum coeffs. weight $h_{\max}$, maximum residual weight $w_{\max}$.}
\KwResult{Optimum coefficients for $x$, $a$}
Set iteration $t=0$, coefficients $a\iter{0}=a_0$ ,residual $r\iter{0}=x \oplus D{\otimes}a\iter{0}$\;
Set modulo-2 Gramm matrix $G \leftarrow D^\intercal \otimes D$\tcc*{(a)}
Set residual correlation $g\iter{0} \leftarrow D^\intercal{r\iter{0}}$\tcc*{(b)}
\While{$h(r\iter{t}) \geq w_{\max}$ \textbf{and} $t < h_{\max}$ }{
$k \leftarrow \arg \max_l \{\;|g\iter{t}_l|\;/\;\|D\col{l}\|_0\;\} $ \tcc*{(c)}
\If{$g\iter{t}_{k} = 0$} {
\Return $a \leftarrow a\iter{t}$ \;
}
$r\iter{t+1} \leftarrow r\iter{t} \oplus D\col{k} $\;
\If { $h(r\iter{t+1}) \geq h(r\iter{t})$ }
{
\Return $a \leftarrow a\iter{t}$ \;
}
\eIf {$a\iter{t+1}_{k} = 1$} {
$a\iter{t+1}_{k} \leftarrow 0$; $g\iter{t+1} \leftarrow g\iter{t} - G\col{k}$\tcc*{(d)}
} {
$a\iter{t+1}_{k} \leftarrow 1$; $g\iter{t+1} \leftarrow g\iter{t} + G\col{k}$ \tcc*{(d')}
}
}
\Return $a \leftarrow a\iter{t}$ \;
\caption{\label{alg:bmp} Binary Matching Pursuit.}
\end{algorithm}
In essence, BMP is a binarized version of the Matching Pursuit Algorithm~\ref{alg:mp}. For a given
sample $x$, we begin ($t=0$) with an initial coefficients vector $a\iter{0}=a_0$, a residual
$r\iter{0}_j=x \oplus D \otimes a\iter{0}$ and an initial vector $g\iter{0} = D^\intercal r\iter{0}$
which keeps track of the correlation between the columns of $D$ and $r\iter{t}$. Then, at each
iteration $t$ we determine the atom $D\col{k}$ which is most correlated to $r\iter{t}$. We
then \emph{toggle} the coefficient corresponding to that atom, $a_k\iter{t}$, and update
$r\iter{t}$ and $g\iter{t}$ accordingly. The pseudocode is given in Algorithm~\ref{alg:bmp}.
Some steps of this algorithm, marked as $(a)$, $(b)$, $(c)$ and $(d)$ (appearing twice) in
Algorithm~\ref{alg:bmp}, are not obvious from the overall description of the algorithm and need to
be clarified. In $(a)$, the \emph{modulo-2 Gramm matrix} of $D$ is computed; this matrix is used in
an analogous way in Algorithm~\ref{alg:mp} for the fast update of the correlations vector $g$ as
described in \refeq{eq:corr-up}. In $(b)$, we compute the standard correlation between the columns of
$D$ and $r$. In $(c)$, since the atoms are not normalized, the best candidate is chosen using a
form of normalized correlation called \emph{association accuracy}~\cite{association-accuracy},
$g\iter{t}_l / \|D\col{l}\|_2^2 = g\iter{t}_l / \|D\col{l}\|_0$.
Finally, in $(d)$ and $(d')$, despite the correlation vector $g$ being initially computed using
the \emph{standard} matrix-vector product, its updated has exactly the same form
as \refeq{eq:corr-up}, but the Gramm matrix $G$ is actually computed using the modulo-2;
$(d)$ corresponds to the case when $\Delta=1$, that is, when $a_k$ is switched from $0$ to $1 $, and
$(d')$ corresponds to $\Delta=-1$, when $a_k$ is switched off. (This curious result is easily verified by writing down the corresponding arithmetic.)
\color{blue}
\paragraph*{Computational complexity} The initialization of BMP requires $O(p^2m)$ binary
operations for computing the Gramm matrix and integer ones $O(p^2m)$ for the \mnote{2.4}correlation vector
$g$. Then, for each sample $X\col{j}$ a maximum of $h_{\max}$ iterations can be required, each one
requiring $O(p)$ floating point operations for finding the most correlated atom, another $O(p)$
integer operations for updating the residual $r$, and another $O(p)$ integer ones for updating $g$, for a total of $O(h_{\max}p)$ operations per sample. Overall, a whole pass over the $n$ data samples requires $O(p^2m)+ O(p) + O(h_{\max}pn)$ operations.
\color{black}
\subsection{MOB: Method Of Binary Directions}
\label{sec:bdl:mod}
Here we want to update $D$ so as to minimize the Hamming weight $h(E)$ of the residual matrix $E =
X \oplus D \otimes A$. Suppose we want to update the $r$-th atom at iteration $k$. The
affected columns will only be those for which the coefficients in $A$ corresponding to that atom
are non-zero. We define $J=\{j : A_{rj} \neq 0 \}$ to be the set of indexes of those columns. What
we want is to update $D\col{r}$ so that the weight of the columns of the residual
affected by it is minimized,
\begin{eqnarray}
D\col{r}\iter{t+1} &=& \arg\min_{d \in \{0,1\}^m} \sum_{j \in J} h(E\iter{t}\col{j} \oplus d) \nonumber\\
&=& \arg\min_d \sum_{j \in J} (\sum_i E\iter{t}_{ij} + |J| d_i).
\label{eq:mob1}
\end{eqnarray}
According to~\refeq{eq:mob1}, the optimization of $D\col{r}\iter{t+1}$ is separable in each of its elements,
\begin{eqnarray}
D_{ir}\iter{t+1} &=& \arg\min_{u \in \{0,1\}} \sum_j E_{ij}\iter{t} \oplus u \nonumber\\
&=& \arg\min_{u \in \{0,1\}} \left|\;\sum_{j\in J} \;E_{ij}\iter{t} - |J|u\;\right| \nonumber\\
&=& \arg\min_{u \in \{0,1\}} |\,h( E_{iJ}\iter{t} ) - |J|u\,|\nonumber\\
&=& \mathbf{1}\left( \frac{ h( E_{iJ}\iter{t} ) }{|J|} > \frac{1}{2} \right)
\label{eq:mob2}
\end{eqnarray}
Note that when $h(E_{iJ})/|J| = 1/2$ both $0$ and $1$ are optimum, in which case we use $0$. Also, note that \refeq{eq:mob2} can be rewritten as
$$D_{ir}\iter{t+1} = \mathbf{1}\left( \frac{h(EA\row{r}^\intercal)} {h(A\row{r})} > \frac{1}{2} \right).$$
As these statistics can be easily updated as new samples arrive, it follows that MOB is suited for fast online dictionary adaptation. (Our implementation of this feature is still work in progress.)
\color{blue}
\paragraph*{Computational complexity of MOB} Our current (offline) \mnote{2.4} implementation requires $O(mnp)$ bitwise operations for the modulo-2 product $E A^\intercal$, $O(np)$ integer operations for computing the weights of the rows of $A$, and $O(mp)$ integer comparisons for updating $D$, for a total of $O(mnp)$ binary plus $O(mp)$ integer operations.
\color{black}
\subsection{K-PROX: Dictionary Update via Proximus}
\label{sec:bdl:k-prox}
In this case, following the K-SVD concept, we want to obtain the best rank-one approximation to the
residual $E$ obtained after removing the contribution of $D\col{r}$. Let $J = \{j: A_{rj} \neq 0 \}$
and $R_{J}=D\col{r}{\otimes}A_{rJ} \oplus E_{J}$. We then have,
\begin{equation}
(D\col{r}\iter{t+1},A_{rJ}\iter{t+1}) = \arg\min_{u,v} h\left( R\col{J}\iter{t} \oplus uv^\intercal \right).
\label{eq:bsvd1}
\end{equation}
As the name K-PROX implies, our approximation to the NP-hard problem \refeq{eq:bsvd1} is based on the Proximus algorithm~\cite{proximus}, summarized in Algorithm~\ref{alg:proximus},
\begin{algorithm}
\caption{\label{alg:proximus} Proximus}
\KwData{matrix $X \in \{0,1\}^{m{\times}n}$, $u\iter{0} \in \{0,1\}^m$, $v\iter{0} \in \{0,1\}^n$ }
\KwResult{Vectors $u$, $v$ so that $X \approx uv^\intercal$}
Set iteration $k=0$\;
\Repeat {$u\iter{t+1}(v\iter{t+1})^\intercal = u\iter{t}(v\iter{t})^\intercal$} {
$u\iter{t+1}_i\!\! \leftarrow \mathbf{1} \left( X\row{i} v\iter{t} > h(v\iter{t})/2 \right),\;i=1,\ldots,m$ \;
$v\iter{t+1}_j\!\! \leftarrow \mathbf{1}\left( X\col{j}^\intercal{u\iter{t+1}}\!>\!h(u\iter{t+1})/2 \right),j=1,\ldots,n$ \;
$k \leftarrow k+1 $ \;
}
\end{algorithm}
Interestingly, Algorithm~\ref{alg:proximus} provides a local optimum to the rank-one approximation that we seek. This is stated in Proposition~1 below.
\begin{proposition}
The output $(u,v)$ of the Proximus Algorithm~\ref{alg:proximus} is a local optimum of the problem $\min \|X - uv^\intercal\|_0$.
\end{proposition}
\begin{proof}
Given $v\iter{t}$, it is easy to check that the update $u\iter{t+1}$ in Algorithm~\ref{alg:proximus} is the value of $u$ that \emph{globally} minimizes $\|X \oplus uv\iter{t+1} \|$ (if $ s\iter{t}_i = w\iter{t}/2$, both $0$ and $1$ are equally optima; in such case, we default to $0$). The same happens with the update $v\iter{t+1}$ given $u\iter{t+1}$. Therefore, $h(E\iter{t})=h(X \oplus u\iter{t}(v\iter{t})^\intercal)$ cannot increase with $k$. As $h(E\iter{t}) \geq 0$ is bounded, non-increasing, and the iterates can take on a finite number of values, the sequence $h(E\iter{t})$ must converge after a finite number of steps. Let $(u,v)$ be the arguments at which the stopping condition is satisfied. By definition of the algorithm, no change in $u$ or $v$ decreases the objective. This guarantees that $(u,v)$ is a local minimum in a Hamming ball of radius at least $1$.\footnote{We cannot guarantee that a simultaneous change in a single coordinate of $u$ and a single coordinate of $v$ will not decrease the cost function!.}
\end{proof}
\color{blue}
\paragraph*{Computational complexity of K-PROX} As with the K-SVD algorithm, we update $D$ one atom at a time. This requires $O(mn)$ operations for computing \mnote{2.4}the residual in \refeq{eq:bsvd1} and running Algorithm~\ref{alg:proximus}, which takes a finite number of iterations requiring $O(mn) + O(np)$ bitwise operations and the same number of integer comparisons.
\color{black}
\subsection{Initialization}
\label{sec:bdl:init}
Initialization is of paramount importance to the success of any non-convex matrix factorization method. At the same time, there is no provably optimum way of doing so, otherwise we would be contravening the NP-hard nature of the factorization problem itself. We are left with heuristics based on intuition and prior information, if any. Ultimately, \emph{initialization is an art}, and also an engineering decision which may depend on several aspects. As an example, we could use the resulting factorization obtained with \emph{any} of the existing methods mentioned in Section~\ref{sec:intro} as an initial point. In this case, to maintain scalability and simplicity, we experiment with two simple but generally effective methods drawn also from dictionary learning literature: given dictionary size $p$, the first draws $p$ atoms using a pseudo-random Bernoulli$(\theta)$ distribution; we use $\theta=1/2$, but other values could be used to reflect prior information on the problem.
The second method is to draw $p$ columns from $X$ at random and use them as the initial atoms. WE report on both strategies on Section~\ref{sec:results}.
\section{Model selection}
\label{sec:model-selection}
Beyond theoretical results and simulations, when confronted with real data,
the true underlying model governing the generation of data is rarely
available. In such situations, models can only be assumed to be tools for
understanding the data at hand, and the best choice is dictated by how much
regularity or structure each candidate model is capable to grasp from that
data. In particular, the complexity of a model is limited by the amount of
data available to estimate the different parameters of the model. An overly
complex model will tend to \emph{overfit} the data, whereas an overly simple
one will miss important details. The problem of model selection is that of
finding the best model for a given data set. Typically, this is done by
seeking a balance between the \emph{goodness of fit} of a model, usually
expressed in terms of the likelihood of the data given the model, and a
measure of the \emph{model complexity}; some popular examples in this
category are the Bayesian Information Criterion (BIC)~\cite{bic}, the
Akaike's Information Criterion~\cite{aic}, and the Minimum Description
Length (MDL) principle~\cite{mdl1,mdl2,mdl3}.
MDL translates the model
selection problem as one of data compression, where both the data and the
model have to be (hypothetically) transmitted and perfectly recovered using
some encoding mechanism. The tension between complexity is then resolved by
the number of bits (\emph{codelength}) required to describe the data in
terms of the model (\emph{stochastic complexity}) and the number of bits
required to describe the model itself (\emph{model complexity}). The
original version of MDL~\cite{mdl1} made this division explicit, and is
asymptotically equivalent to BIC. However, the modern version of
MDL~\cite{mdl2,mdl3} uses the more recent
information-theoretic \emph{universal coding theory} to make the
aforementioned balance implicit by producing an optimum, joint description
of both the model and the data which is furthermore independent of arbitrary
choices of, for example, the way the model is parameterized.
These advantages make MDL an appealing method for model selection.
In our case, the data $X$ is described by the triplet $(D,A,E)$. We describe
each of these components separately using universal codes, so that
$L(X)=L(D)+L(A)+L(E)$ is the total codelength of describing $X$. Here the
tension between a good fit and a simple model is represented respectively by
$L(E)$ and $L(D)+L(A)$.
\subsection{\revTwo{Forward selection algorithm}}
\color{blue}
Model selection is usually formulated as two nested problems. The inner
problem is how to search for the best model $(D,A,E)$ among all models of
\mnote{2.5} order $p$. The second problem is to choose the best model for $X$ among all
possible models of order $p \geq 0$. Given a codelength function $L(X)$,
our method uses a \emph{forward selection} strategy to sweep over all
models. Starting with a initial order $p=p_0$, we approximate the best model
given $p$ using one of our dictionary learning algorithms, and then
gradually increase $p$ until the codelength $L(X)$ is no longer diminished.
\mnote{2.4}
In going from $p$ to $p+1$ we employ a \emph{warm restart} strategy, that is, atoms and coefficients learned for model order $p$ are used as the starting point for learning the $p+1$-th order model. The overall procedure is summarized in Algorithm~\ref{alg:fwd},
\begin{algorithm}
\caption{\label{alg:fwd}MDL-based Forward Selection Algorithm}
\KwData{matrix $X \in \{0,1\}^{m{\times}n}$, initial order $p_0$ and model $(D\iter{0},A\iter{0},E\iter{0})$ }
\KwResult{Selected model $(D^*,A^*,E^*)$}
$p \leftarrow p_0$ \; $(D\iter{p},A\iter{p},E\iter{p}) \leftarrow
(D\iter{0},A\iter{0},E\iter{0})$ \; $L\iter{p} \leftarrow
L(D\iter{p})+L(A\iter{p})+L(E\iter{p})$ \;
\Repeat {$L\iter{p} \geq L\iter{p-1}$} {
Initialize the rank-$1$ model $(d,a)$ using $E\iter{p}$ as input \;
$(\tilde{D},\tilde{A},\tilde{E}) \leftarrow
([D\iter{p}|d],\,[(A\iter{p})^\intercal|a^\intercal]^\intercal\,,\,E\iter{p}-da^\intercal) $ \;
Adapt $(D\iter{p+1},A\iter{p+1},E\iter{p+1})$ using $(\tilde{D},\tilde{A},\tilde{E})$ as the starting point\;
$L\iter{p+1} \leftarrow L(D\iter{p+1})+L(A\iter{p+1})+L(E\iter{p+1})$ \;
$p \leftarrow p+1$ \;
}
$(D^*,A^*,E^*) \leftarrow (D\iter{p-1},A\iter{p-1},E\iter{p-1})$ \;
\end{algorithm}
\color{black}
\subsection{Codelength computation}
It remains to describe how we compute $L(X)$. The universal compression of
binary sources has been extensively studied in the literature. Moreover, we
do not need to perform a real encoding; we need only to compute the
codelength. One particularly simple method, for which the codelength is easy
to compute, is \emph{enumerative coding}~\cite{enum}. Given a binary string
$x$ of length $n$ and $r=h(x)$, its enumerative code is
composed of two parts. The first one describes $r$ with
$\lceil\log_2(n)\rceil$ bits, and the second one describes the index of $x$
in the lexicographically ordered list of all binary strings of length $n$ and weight $r$, which requires $\lceil\log_2{r \choose n}\rceil$ bits. The
total codelength is then
\begin{equation}
L(x) = \lceil\,\log_2(n)\,\rceil + \left\lceil\log_2{r \choose n}\right\rceil.
\label{eq:codelength}
\end{equation}
(Note that $\log{r \choose n}$ can be accurately approximated by using
Stirling's formula, i.e., requiring $O(1)$ operations.) As prior
information, we expect the different columns of $D$ to represent particular
patterns in the data, the corresponding rows of $A$ their appearance in $X$,
and the different rows (corresponding to different variables of the data
samples) of $E$ to exhibit different patterns also. Accordingly, we encode
each column of $D$ and each row of $E$ and $A$ with its own code,
\begin{equation}
L(X) = \sum_{i=1}^m L(E\row{i}) + \sum_{k=1}^p L(D\col{k}) + \sum_{k=1}^p L(A\row k)
\label{eq:model-codelength}
\end{equation}
\color{blue}
\paragraph*{Computational complexity} The cost of each forward selection step is \mnote{2.4-5}clearly dominated by the dictionary learning algorithms. The codelength evaluation is negligible, requiring $O(mn) + O(mp) + O(np)$ operations to count the number of zeros in each component $(D,E,A)$.
\color{black}
\section{Results and discussion}
\label{sec:results}
\color{blue}
The main objectives of the following experiments are three. The first is to
demonstrate the interpretability of the resulting models; we expect to
\mnote{1.2}\mnote{2.6} obtain atoms which exhibit recognizable patterns of the input data. The
second is to see the effect of different initialization strategies on the
final result. The third is to study the numerical properties of the methods, in particular their convergence rate and computational cost.\footnote{The version used in this
paper can be downloaded from \url{http://iie.fing.edu.uy/~nacho/bmf/bmf.zip}; this
includes scripts and data to reproduce all the results shown in this section
and many more not discussed in this paper for lack of space. The latest
version is available as a GIT
repository \url{https://gitlab.fing.edu.uy/nacho/bmf}.}
\color{black}
The first dataset is a binarized version of MNIST~\cite{mnist}, a set of $n=10000$ $17{\times}17$ images of handwritten digits;
here each columns of $X$ contains the vectorized version of a digit
($m=289$). The second consists of all the $m=4088$ non-overlapping
blocks of size $16{\times}16$ of the \emph{halftone Einstein}, a
binary $1160{\times}896$ image; the columns of $X$ are the $4088$
vectorized blocks ($m=256$) of the image. As can be seen in
Figure~\ref{fig:datasets}, both datasets have easily recognizable patterns.
\begin{figure*}[t]
\centering%
\includegraphics[height=1.8in]{fig/mnist_bin.png} %
\includegraphics[height=1.8in]{fig/einstein-and-stripe.png} %
\includegraphics[height=1.8in]{fig/einstein-patches-and-stripe-crop.png}
\caption{\label{fig:datasets} \color{blue} Left to right: a few samples of the MNIST dataset; halftone image of Einstein with the stripe used in Figure~\ref{fig:einstein-kprox} highlighted in magenta; detail of Einstein's left eye and the $16{\times}16$ blocks partition; including part of the stripe. the first case, we expect the atoms in the final dictionary to resemble numbers of different shapes (see e.g.~\cite{cvpr10}). In the case of Einstein, we expect the dictionary atoms to resemble the halftoning patterns observable in the $16{\times}16$ blocks shown on the right picture.\color{black}}
\end{figure*}
\begin{figure*}[t]
\centering%
\includegraphics[width=\textwidth]{fig/mnist-mob.pdf}%
\caption{\label{fig:mnist-mob}\color{blue} MNIST model obtained using Forward selection with MOB at each step. The model was initialized with $p_0=16$ random samples. The final dictionary, with $p=79$ atoms, is shown to the left as a mosaic from top to bottom and left to right. The three rows to the right of the dictionary show $40$ samples from $X$ (top), their final residual from $E$ (middle), and the corresponding coefficients from $A$ (bottom). Each column of $A$ is represented as a vertical pattern of $79$ bands: the top band corresponds to the first atom, and the bottom one to the 79th; a black band on the $i$-th row indicates that atom $i$ is being used to represent the sample on top of the same column, giving rise to the residual shown in the middle. %
In general, coefficients are not easy to interpret. However, in some cases the correspondence is clear. In this case the four digits ``0'' use the third atom (painted in red). Something similar happens with the ``6''s: three of them use the 5th atom (green) and one uses the 9th (blue). The corresponding coefficients are marked as red, green and blue bands in the bottom-right picture.
\color{black}}
\end{figure*}
\begin{figure*}[t]
\centering%
\includegraphics[width=1.0\textwidth]{fig/einstein-kprox.pdf}
\caption{\label{fig:einstein-kprox}\color{blue} Einstein model obtained using forward selection and K-PROX. Here too we used random samples for the initialization. As in Figure~\ref{fig:mnist-mob}, we show the resulting dictionary on the left, and the samples (top), residuals (middle) and coefficients (bottom) for the patches contained in the stripe highlighted in Figure~\ref{fig:datasets}. \color{black} }
\end{figure*}
\begin{figure}
\centering%
\includegraphics[height=1.0in]{fig/einstein-init-rand-init.png}
\includegraphics[height=1.0in]{fig/einstein-init-rand-mob.png}
\includegraphics[height=1.0in]{fig/einstein-init-rand-kprox.png}\\[1ex
\includegraphics[height=1.0in]{fig/einstein-init-samp-init.png}
\includegraphics[height=1.0in]{fig/einstein-init-samp-mob.png}
\includegraphics[height=1.0in]{fig/einstein-init-samp-kprox.png}\\[1ex
\caption{\label{fig:init-einstein} \color{blue} Effect of initialization on Einstein. \color{blue} Top row: initial dictionary drawn from a Bernoulli($1/2$) process (left), resulting MOB (center) and K-PROX (right) models for $p=36$. Bottom row: initial dictionary using random columns from $X$ (left), resulting MOB (center) and K-PROX (right) dictionaries. Both initialization methods worked well in this case using both MOB and K-PROX. In all cases, the dictionaries evolved so that some atoms resemble halftoning patterns. Other atoms were not changed, indicating that they were never used.
\color{black}}
\end{figure}
\begin{figure}[t]
\centering%
\includegraphics[height=1.0in]{fig/mnist-init-rand-init.png}
\includegraphics[height=1.0in]{fig/mnist-init-rand-mob.png}
\includegraphics[height=1.0in]{fig/mnist-init-rand-kprox.png}\\[1ex
\includegraphics[height=1.0in]{fig/mnist-init-samp-init.png}
\includegraphics[height=1.0in]{fig/mnist-init-samp-mob.png}
\includegraphics[height=1.0in]{fig/mnist-init-samp-kprox.png}\\[1ex
\caption{\label{fig:init-mnist} \color{blue} Effect of initialization on MNIST. Top row: initial pseudo-random dictionary using a Bernoulli($1/2$) model (left), result of MOB (center) and of K-PROX (right) for $p=36$. Bottom row: initial dictionary using random samples from the dataset (left), MOB (center) and K-PROX (right) dictionaries. Here the pseudo-random initialization does not work; no atom is ever used, and so both algorithms stop at iteration $1$. In the case of the samples-based initialization, both final models show some adaptation. \color{black}}
\end{figure}
\color{blue}
\subsection{Interpretability of the resulting models}
Figure~\ref{fig:mnist-mob} shows the MNIST model obtained using MOB and forward selection, The dictionary and a few columns from the coefficients and residual \mnote{1.3}\mnote{2.6}matrices $A$ and $E$; atoms from $D$ and samples from $E$ are represented as mosaics with each sample/atom displayed as a tile. As can be seen, many atoms look like numbers; other atoms exhibit number-like silhouettes. The results obtained with K-PROX (shown in the supplementary material) are very similar in all aspects. Figure~\ref{fig:einstein-kprox} shows the model obtained for the Einstein image using K-PROX; we show the dictionary, a few samples; again, the results are clearly interpretable in terms of the patterns that can be observed in the data. In this case too the results obtained with MOB (in the supplementary material) are very similar.
\subsection{Sensitivity to initialization}
Figures~\ref{fig:init-einstein} and~\ref{fig:init-mnist} show initial and final dictionaries of fixed size $p=36$ for MNIST and Einstein, using both dictionary learning methods, but using each of the two initialization methods described in Section~\ref{sec:bdl:init}. In this case, the random samples initialization does a good job in both cases, whereas the pseudo-random initialization fails miserably on the MNIST case. These results should be taken with a grain of salt, only to show how different initialization methods can work (or fail) under different circumstances.
\subsection{Numerical results}
\begin{figure}
\centering%
\includegraphics[width=0.235\textwidth]{fig/einstein_bps_vs_atoms_comp.pdf}\hspace{2ex}%
\includegraphics[width=0.23\textwidth]{fig/einstein_time_vs_atoms.pdf}\\[1.5ex]
\includegraphics[width=0.23\textwidth]{fig/einstein_bps_vs_atoms_detail.pdf}\hspace{2ex}%
\includegraphics[width=0.23\textwidth]{fig/einstein_bps_vs_atoms_kprox_detail.pdf}%
\caption{\label{fig:numerical-results-atoms}
\color{blue}%
Convergence of Forward Selection on Einstein. Top to bottom, left to right: MDL cost function (in avg. bits per sample) for MOB and K-PROX; computational cost (in seconds) for both variants; break-down of the cost function in its three parts ($E$, $A$ and $D$) for MOB; same for K-PROX. %
Both MOD and K-PROX produced similar models in the end. K-PROX, however, required more time to run ($14$s) than MOB ($4$s). Thanks to the warm-restart strategy, the cost per forward selection step is small ($0.11$s for MOB and $0.03$s for K-PROX) and approximately constant despite the growing model size $p$. Finally, both break-downs show typical MDL curves: as $p$ increases, the stochastic complexity ($L(E)$) decreases while the model cost $L(A)+L(D)$ increases.\color{black}}
\end{figure}
\begin{figure}
\centering%
\includegraphics[width=0.23\textwidth]{fig/einstein_bps_vs_iter.pdf}\hspace{2ex}%
\includegraphics[width=0.23\textwidth]{fig/einstein_time_vs_iter.pdf}\\[1.5ex]
\includegraphics[width=0.23\textwidth]{fig/einstein_change_vs_iter_mob.pdf}\hspace{2ex}%
\includegraphics[width=0.23\textwidth]{fig/einstein_change_vs_iter_kprox.pdf}%
\caption{\label{fig:numerical-results-iters}
\color{blue}%
Convergence of MOB and K-PROX on Einstein. Top to bottom, left to right: MDL cost function; execution time; change in the arguments $A$,$D$ and $E$ for MOB; same for K-PROX. Both methods converge quickly, with K-PROX reaching its near-optimum in just one iteration. In terms of execution time, however, just one iteration of K-PROX required $0.6$s (this is much larger than the $0.11$s reported in Figure~\ref{fig:numerical-results-atoms} since the model has to be learned from scratch), whereas all $10$ MOB iterations take an accumulated time smaller than $0.1$s. Note that the convergence is exact, not asymptotic: both algorithms stop when the change of all arguments is $0$. \color{black}}
\end{figure}
\color{blue}
In this set of experiments we are interested mainly in three aspects of the proposed methods. First, the convergence of the forward selection method in \mnote{1.3}\mnote{2.4-5}terms of overall cost function, the convergence rate of the dictionary learning algorithms, and the empirical computational complexity of both the selection and learning methods as measured in running time (seconds), both per iteration and accumulative.\footnote{The timing results were obtained on a Lenovo V310-14IKB notebook with an Intel i5-7200U (4 cores) processor, 8GB of RAM, running Lubuntu 18.04 64bits, with executables compiled from C++ code using GCC 7.3.0 with maximum optimization (``-O3''), multicore support (``-fopenmp'') and all SIMD functions enabled.} %
Figure~\ref{fig:numerical-results-atoms} shows that forward selection converges exponentially to a local minimum which is very close in cost function for both MOB and K-PROX. The forward selection mechanics shown in the break-down of the arguments are typical of methods such as MDL, which shows the correctness of the procedure.
Figure~\ref{fig:numerical-results-iters} shows that both MOB and K-PROX converge quickly both in cost function and the respective optimization variables (we recall that convergence is exact here -- there is no further change in the arguments). This same behavior was observed for all model sizes in Figure~\ref{fig:numerical-results-atoms}.
In terms of execution time, both Figure~\ref{fig:numerical-results-atoms} and ~\ref{fig:numerical-results-iters} show MOB as the fastest of the two methods in terms of computing speed. It should be noted however that our implementation for MOB is reasonably optimized, whereas that of K-PROX is not. By comparing the average time for learning a fixed order model in figures~\ref{fig:numerical-results-atoms} and \ref{fig:numerical-results-iters} it should be clear that the warm-restarts strategy used is crucial for efficiently searching through the different model sizes.
\color{black}
\section{Concluding remarks}
\label{sec:conclusion}
\color{blue}
In this paper we have presented two novel and efficient Binary Factorization Methods based in dictionary learning techniques through the combination of three novel algorithms, BMP for learning coefficients, and MOB and K-PROX for updating the dictionaries. We have provided theoretical guarantees on their convergence to local minima, and a simplified but rigorous analysis of their computational complexity.
\mnote{1.3}\mnote{2.4-5}Through experimentation on two very different datasets, we have demonstrated that our methods produce interpretable results in both cases, requiring very few iterations to converge, and with an overall computational complexity which is very competitive (even though we did not employ any speed-up strategies such as online or mini-batch learning), requiring as little as $0.1$s to learn a complete dictionary on a dataset of $10000$ $289$-dimensional samples from scratch. We have also shown the scalability of our model in terms of growing model size, allowing us to search over a large family (hundreds) of candidate models (dictionaries) in as little as $4$ seconds on a modest notebook.
In a follow up of this work, which is currently underway, we will present on-line implementations of the MOB and K-PROX algorithms developed here, with the objective of employing them on huge genomic datasets. Other possible lines of work include finding conditions under which our methods (or some of them) can be guaranteed to recover a given underlying model.\mnote{2.2-3}
\color{black}
\bibliographystyle{IEEEtran}
|
{
"timestamp": "2018-07-27T02:03:42",
"yymm": "1804",
"arxiv_id": "1804.05482",
"language": "en",
"url": "https://arxiv.org/abs/1804.05482"
}
|
\section{Introduction}
\label{sec:intro}
Peaks in the underlying mass density field are the most likely sites for the formation of halos where gas is expected to accrete and form galaxies
\citep{White1978}.
In the classical picture of \cite{PS}, matter in regions with linear density contrast above a threshold $\delta_c$ is assigned to
halos of mass larger than $M$, where $M$ defines the smoothing of the density field. This implies that halos of mass $M$ form at
peaks with $\delta=\delta_c$ in the smoothed density contrast $\delta$.
Naturally, most studies have focused on peaks associated with halos. Indeed,
statistical properties of local extrema \citep[e.g.][]{Adler1981} have gained a great deal of attention in cosmology \citep{BBKS} (hereafter BBKS).
Correlations of halos and their distribution in relation to the mass density field of the gravitationally dominant dark matter (DM), i.e. biasing
\citep[][]{Kaiser1984a}, have been studied
extensively with analytic methods and numerical simulations.
For Gaussian initial conditions and on sufficiently large scales, halos follow a linear biasing relation, $\delta_\mathrm{h}=b\delta$ between the
halo number density contrast, $\delta_\mathrm{h}$ and the mass density contrast $\delta$.
The bias factor $b$ depends on the height of the peaks associated with halos and on their mass.
An important result obtained in simulations \citep{Dalal:2007cu}, and confirmed by analytic techniques
\citep{grinstein/wise:1986,Dalal:2007cu,Matarrese:2008nc,Slosar:2008hx}, is that the presence
of initial local-type non-Gaussianity introduces a peculiar scale dependence in the bias factor dubbed ``non-Gaussian bias''.
The specific form of $b(k)$ ($k$ is the wavenumber of a given scale) opens the window for probing initial non-Gaussianity based on the clustering
properties of galaxies in planned large redshift surveys, e.g. Euclid \citep{EuclidRB} and DESI \citep{DESICollaboration2016a}.
It is well known that non-Gaussianity strongly affects the tails of density probability distributions \citep{Adler1981,catelan/etal:1988}.
Several authors have further specialized these results to local density maxima of non-Gaussian density fields, where the non-Gaussianity is either of a
generic form
\cite[e.g.][]{Catelan1988,Gay2012,codis/etal:2013,uhlemann/etal:2018} or developed via non-linear gravitational evolution of initial gaussian conditions
\citep[e.g.][]{Suginohara1991,matsubara:1994}.
In particular, \cite{Gay2012,codis/etal:2013} considered the effect of a generic non-Gaussianity on extrema counts and Minkowski functionals of the dark
matter density field.
In this work, we consider peaks and dips in cosmological density field smoothed on scales much larger than those of galactic and galaxy cluster halos
($\lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10$ Mpcs).
Using N-body simulations in large cosmological boxes, we focus on the total number of local extrema for density fields constructed from the {\it halo}
distribution, as a proxy for a galaxy catalogue. Earlier analyses (\cite{Croft:1997rv,desoma,De:2009uz}) have used this type of statistics for constraining parameters related to the linear matter power spectrum on smaller scales ($\lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10$ Mpcs).
Our goal is to assess the extent to which the abundance of extrema in three-dimensional (3D) fields inferred from current and forthcoming large galaxy redshift surveys can be used as
a cosmological tool and, more specifically, a probe of local primordial non-Gaussianity.
As we shall see, our main findings have a straightforward interpretation in terms of the non-Gaussian bias.
We adopt standard notation. The mean total and baryonic mass densities (in units of the critical density) are denoted by
$\Omega_\mathrm{m}$ and $\Omega_\mathrm{b}$, respectively. The Hubble constant is $H_0$ and $h=H_0/[100 \mathrm{\, {\rm km }\, {\rm s}^{-1}\; Mpc^{-1}} ]$.
The linear growth factor (normalized to unity at the present time) at redshift, $z$, is $D(z)$.
The outline of the paper is as follows. In \S\ref{sec:basics} we lay out known relations between the number of extrema and the underlying power spectrum for Gaussian fields.
A description of the N-body simulations is provided in \S\ref{sec:simulations} and the corresponding results for the abundance of local extrema identified in
smoothed density fields derived from the DM and halo distributions are in \S\ref{sec:results}.
In \S\ref{sec:prospects} we discuss the prospects for the application of the number of extrema as a test of cosmological parameters and conclude with a summary in
\S\ref{sec:discussion}.
\section{Definitions and Theoretical Expectations}
\label{sec:basics}
We define local maxima (peaks) in a smoothed random field, $f$, as points in space where the spatial gradient is $\partial_\alpha f=0$ and the Hessian
$\partial_\alpha\partial_\beta f$ is negative definite. Local minima (dips) are defined similarly but with a positive definite Hessian.
For a random Gaussian field, peaks and dips have an equal total number \textit{per unit volume}, which was computed by BBKS to be
\begin{equation}
n_{0}\approx 0.016 R_*^{-3}\; ,
\label{eq:npk}
\end{equation}
where
\begin{equation}
R_*=\sqrt{3} \frac{\sigma_1}{\sigma_2}\; ,
\label{eq:rst}
\end{equation}
and the spectral moments
\begin{equation}
\sigma^2_j=\int \frac{k^2\textrm{d} k}{2\pi^2}P(k) W_R^2(k) k^{2j}\; .
\label{eq:sigi}
\end{equation}
The expression for $n_0$ is independent of the clustering amplitude and it depends only on the shape of the power spectrum, $ P(k)$ of the field, and the smoothing
Kernel $W_R(k)$.
For $P(k)\sim k^n$, and a Gaussian smoothing window $W_R^2(k)=\exp(-k^2R^2)$, it is easy to see that,
\begin{equation}
\frac{R}{R_*}=\left(\frac{n+5}{6}\right)^{1/2}\; .
\label{eq:Rn}
\end{equation}
The total number of peaks is preserved under a local monotonous one-to-one mapping, $F(\delta)$, of the density field.
Thus we expect this quantity to be independent of time in the quasi-linear scales. On smaller scales, local extrema tend to merge and diffuse,
leading to deviations from expression Eq.~(\ref{eq:npk}) above.
In addition to the DM density field, we also examine peaks and dips in the smoothed distribution of halos.
The corresponding spectral moments $\sigma_j\equiv\sigma_{j,h}$ are given by
\begin{equation}
\label{eq:sigmahi}
\sigma_{j,h}^2 = \int\frac{k^2 dk}{2\pi^2}\, k^{2j} W_R^2(k) \Big[b^2(k) P(k) + \frac{1}{\bar n_h}\Big] \;.
\end{equation}
The expression in square brackets is a model for the power spectrum of the halo distribution
where $P(k)$ here refers to the underlying density field and $b(k)$ describes the scale-dependent halo bias.
The term $1/ {\bar n_h}$ is due to the finite number of halos and approximated as a Poisson discreteness noise.
Using the simulations described below we have found that the added discreteness variance is strongly suppressed for large smoothing and
is actually sub-Poissonian, in agreement with the findings of \cite{casas-miranda/etal:2002,Hamaus:2010im}.
On linear scales, the halo bias $b(k)$ is constant for Gaussian initial conditions but depends on the halo mass i.e. $b(k)= b^\mathrm{G}(M)$.
We also consider local-type non-Gaussianity \citep{Salopek1990,Gangui,KomatsuSpergel} for which
the Bardeen potential $\Phi$ deep in matter domination is expanded around a random Gaussian field $\phi$ as
\begin{equation}
\Phi(\mathbf{x}) = \phi(\mathbf{x})+f_{\rm NL} \left( \left[\phi(\mathbf{x})\right]^2-\langle \phi^2\rangle\right) \;.
\end{equation}
The bispectrum of $\Phi$ induces the following scale dependence in the bias factor,
\begin{equation}
\label{eq:bNG}
b(k) = b^\mathrm{G}(M) + \frac{\alpha(f_{_{\rm NL}})}{k^2T(k)}\; ,
\end{equation}
where
\begin{equation}
\alpha(f_{_{\rm NL}}) \equiv 3 f_{_{\rm NL}} \frac{\partial{\rm ln}\,\bar n_h}{\partial{\rm ln}\,\sigma_8} \frac{\Omega_m H_0^2}{D(z)c^2} \;,
\end{equation}
and $\bar n_h(M)$ is the abundance of halos (per unit $M$) computed for the Gaussian field without the $f_{_{\rm NL}}$ terms.
When implementing Eq. \eqref{eq:sigmahi} to compare it to data (see Section \S\ref{sec:results}), we use the following approximation
\begin{equation}\label{eq:universal}
\frac{\partial{\rm ln}\,\bar n_h}{\partial{\rm ln}\,\sigma_8} \approx \delta_c (b^{\rm G}(M)-1),
\end{equation}
with $\delta_c=1.687$, which is valid for universal mass functions and the spherical collapse model~\footnote{See \cite{Biagetti:2016ywx} for a
quantitative analysis on this approximation on the same set of simulations, sim 1, used here.}.
We do not include expressions \citep[e.g.][]{Gay2012} for the theoretical corrections to Eq.~(\ref{eq:npk}) due to $f_{_{\rm NL}}$ non-Gaussianity. Indeed, we will
see below that the expression remains accurate provided that the appropriate $\sigma_i$ is used.
\section{Simulations}
\label{sec:simulations}
Two sets of simulations, respectively in a $2h^{-1}\,{\rm Gpc}$ and a $3h^{-1}\,{\rm Gpc}$ box, are available for initial conditions generated from $\Lambda$CDM initial power spectra
with slightly different cosmological parameters, as described in the Table.
The simulations were run with the Gadget2 \citep{Gadget2} N-body code on the Baobab cluster at the University of Geneva.
The initial particle displacements were implemented at $z_i=99$ using the public code 2LPTic \citep{Crocce2006} for realizations with Gaussian initial conditions
and its modified version \citep{Scoccimarro12} for non-Gaussian initial conditions of the local type.
The transfer function for the smaller box (simulations 1, see Table) was obtained using the CLASS code \citep{Blas2011}.
This set contains runs for Gaussian initial conditions and two for local-type non-Gaussianity respectively, with $f_{_{\rm NL}}=250$ and $f_{_{\rm NL}}=-250$.
For each of these initial conditions, we obtain 8 random realizations corresponding to different random seeds.
The transfer function of the second set, simulations 2, was obtained using the CAMB code \citep{Lewis:1999bs}.
This set includes 3 types of models: Gaussian initial conditions ($f_{_{\rm NL}}=0$) and non-Gaussian initial conditions, respectively,
with $f_{_{\rm NL}}=100$ and $f_{_{\rm NL}}=-100$. For each type of models, we have 3 simulations corresponding to different random realizations of the initial conditions.
The Rockstar \citep{Behroozi2013} algorithm is employed to identify halos, with linking length $\lambda=0.28$.
\begin{table}
\centering
\begin{tabular}{c c c c c c c}
\hline \hline
& $L$ & $N_\mathrm{p}$& $M_\mathrm{halo}$& $\sigma_8 $ & $\Omega_m$& $\Omega_b $ \\
\hline
sim 1 & 2& $1536^3$& $3.67$ & 0.85 & 0.3 & 0.0455 \\
sim 2& 3 & $1024^3$& $37.9$ & 0.81 &0.272 & 0.0455 \\
\hline
\end{tabular}
\caption{ Simulation parameters, where $L$ is the box size (in unit of $h^{-1}\,{\rm Gpc}$), $N_\mathrm{p}$ number of simulation particles,
and $M_\mathrm{halo}$ is the minimum halo mass identified in the simulation (in unit of $10^{12}h^{-1}{\rm M}_{\odot}$).
Both, simulations 1\&2, include Gaussian and two choices for non-Gaussian initial conditions.
Outputs of simulations 1
are available at $z=0$ and $z=1$, while only the output at $z=0$ is available for simulations 2.
In all simulations the Hubble parameter is $h=0.7$ and the spectral index of the initial power spectrum at large scales is $n_s=0.967$.}
\end{table}
Density fields are interpolated from the DM and halo distributions in the simulation box
on a $512^3$ cubic grid using the Clouds-in-Cells (CIC) scheme.
The grid spacing is thus $3.9h^{-1}\,{\rm Mpc} $
and $5.85h^{-1}\,{\rm Mpc}$, for simulations 1 \& 2, respectively. The density fields were additionally smoothed with a Gaussian window
of 8 different widths in the range $20h^{-1}\,{\rm Mpc}$ to $500h^{-1}\,{\rm Mpc}$. For each smoothed field, local maxima (minima) were identified as grid points surrounded by
grid points with lower (higher) density values.
Fig.~\ref{fig:box} shows the total number of maxima
in the smoothed DM density field in the full boxes of simulations 1 \& 2 at $z=0$.
The theoretical predictions obtained from the BBKS expression (Eqs.~\ref{eq:npk}-\ref{eq:rst}) using the linear power spectrum $P(k)=P_\mathrm{L}(k)$
for the two models are also shown, as indicated in the figure\footnote{In performing the integration in Eq.~(\ref{eq:sigi}), it is important to impose a low $k$ cutoff
corresponding to the finite box size of the simulations. }.
The shaded area encompasses the expected range of (1$\sigma$) shot-noise for simulations 2.
The number drops like $R^{-3}$, consistently with Eq.~(\ref{eq:npk}) since $R_*\propto R$ upto a factor of $\mathcal{O}(1)$ which depends on the shape of the power
spectrum at scale $R$ (cf. Eq.~\ref{eq:Rn}).
The figure refers to the Gaussian simulations only. A similar figure can be found in \citep{2011MNRAS.413.1961L},
but for comparison of the theoretical expression with
peaks identified in the initial conditions of their simulations.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig1.pdf}
\vskip 0.0in
\caption{Total number of maxima versus the smoothing length, from the DM distribution in simulations 1 \& 2 for Gaussian initial conditions at redshift $z=0$
The lines represent the corresponding theoretical prediction using eq. \ref{eq:npk} and the shaded area
represents the $1\sigma$ shot-noise for
the larger simulation. }
\label{fig:box}
\end{figure}
\section{Results}
\label{sec:results}
\subsection{Total number of minima and maxima}
Gaussian initial conditions imply equal probability of producing peaks and dips, up-to fluctuations due to the finite box size.
However, on scales $\lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10$s of Mpcs, non-linear gravitational evolution breaks the initial symmetry through
the merging and smearing of dips and peaks.
For non-Gaussian initial conditions, the statistical symmetry between maxima and minima is already broken
initially.
We choose to first analyze the (total) number $n_\mathrm{1}$, per unit volume, of minima, $n_\mathrm{min}$, and maxima,
$n_\mathrm{max}$ in the simulations. The differences between the abundance of minima and maxima will be discussed at a later stage.
More precisely, we consider
\begin{equation}
\label{eq:n1}
n_\mathrm{1} =\frac{1}{2}\big(n_\mathrm{min}+n_\mathrm{max}\big) \; ,
\end{equation}
which is computed from the smoothed density fields for the various simulations. An advantage of $n_1$ is that it boosts the statistical significance of the measured abundance. For a Gaussian field, $n_1=n_0$ given in Eq.~(\ref{eq:npk}).
Inclusion of non-Gaussian terms modify the abundance of either the minima or maxima by a leading-order
correction proportional to the skewness of the density field and its derivatives \citep{Gay2012}.
The combined leading order correction for both minima and maxima cancel out in the expression of $n_\mathrm{1}$.
Consequently, the BBKS prediction Eq.~(\ref{eq:npk}) remains valid up to a small correction of order $f_{_{\rm NL}}^2$.
According to Fig.~\ref{fig:box}, differences in $n_1$ between the simulations are visually hard to examine directly.
Thus, we consider the statistic,
\begin{equation}
\label{eq:Rfromn}
\Upsilon\equiv \frac{n_1R^3}{0.016}
\end{equation}
where $R$ is the width of the smoothing window. According to Eq.~(\ref{eq:npk}), for a Gaussian field $\Upsilon=({R}/{R_*})^3$.
The three-panel Fig.~\ref{fig:panels} summarizes the main results.
The top panel plots $\Upsilon$, averaged over the 8 random realizations in simulations 1, against the smoothing length, $R$, for the DM density field.
The shaded area represents the 1$\sigma$ shot-noise in $n_1$ corresponding to the finite number of
peaks and dips in the simulation box.
It is estimated as $\sqrt{n_0 L^3/2}$ where $n_0$ is the theoretical
value according to Eq.~(\ref{eq:npk}) and the factor of $1/2$ arises from the definition of $n_1$ which involves both minima and maxima.
We have checked that the scatter from the 8 individual runs (not shown for clarity) is consistent with this estimate of the shot-noise.
For our Gaussian as well as non-Gaussian simulations, the results in the top panel
for $z=1$ and $z=0$ are almost identical.
The dotted line shows $(R/R_*)^3$ computed according to the theoretical expression Eq.~(\ref{eq:rst}) derived for Gaussian fields, where $\sigma_i$ are computed
using Eq.~(\ref{eq:sigi}) with the initial power spectrum $P_\mathrm{L}(k)$.
There is a reasonable match between the dotted curve and $(R/R_*)^3$ derived from $n_1$ for the Gaussian simulations (black and red solid curves).
Overall, the impact of $f_{_{\rm NL}}$ is very small, in agreement with the fact that, for dark matter, $n_1$ depends on $f_{_{\rm NL}}$ only at order $f_{_{\rm NL}}^2$.
The middle panel refers to results obtained from the halo distribution in simulations 1.
The solid curves corresponding to the Gaussian simulations at $z=1$ and $z=0$ are similar.
In great contrast to the upper panel, both $f_{_{\rm NL}}=250$ and $f_{_{\rm NL}}=-250$ models (dashed and dash-dotted lines) at the two redshifts are substantially
different.
It is interesting to check how well the BBKS expression in Eq.~(\ref{eq:rst}) fits the $\Upsilon$ computed from the halos in the non-Gaussian simulations.
To do that we compute $(R/R_*)^3$ using Eq.~(\ref{eq:rst}) for $\sigma_1$ and $\sigma_2$ computed directly from the halo
density fields.
The results are plotted as the plus signs and circles, respectively, for the $f_{_{\rm NL}}=250$ and $f_{_{\rm NL}}=-250$ simulations.
We present the $z=1$ case only but the excellent agreement of $\Upsilon$ with $(R/R_*)^3$ computed from $n_\mathrm{1}$ also holds at $z=0$ .
The bottom panel summarizes results for simulations 2 ($z=0$) of the larger box.
The halos in these simulations have a larger mass and therefore follow a different biasing relation than halos in simulations 1, yielding different
quantitative results.
For these simulations also, the BBKS expression (computed with $\sigma_i$ measured in the simulations), shown as the plus signs and circles, furnishes an
excellent match.
Therefore, despite the fact that relations Eqs.~(\ref{eq:npk}) and (\ref{eq:rst}) are formally obtained for Gaussian fields,
they remain accurate for the non-Gaussian fields considered here, provided the actual $\sigma_i$ are used.
In Fig.~\ref{fig:fnltheory} we compare the theoretical expectation of Eq. \eqref{eq:sigmahi} against $\Upsilon $ measured from the non-Gaussian simulations.
The theoretical curve fits good the data on scales $R \lesssim 100$Mpc/h and provides a qualitatively good description at all scales.
Deviations may be due in part to our approximation Eq. \eqref{eq:universal} and, especially at large scales, to the finite box size of the simulations.
\begin{figure}
\includegraphics[height=1.2\textwidth]{Fig3Panel.pdf}
\caption{The quantity $\Upsilon$ as estimated from Eq.~(\ref{eq:Rfromn}).
\textit{Top:} from the number of peaks and throughs in the dark matter distribution of simulations 1.
\textit{Middle: } The same the {\it Top}, but for the halo distribution.
\textit{Bottom:} For DM and halos for simulations 2, at $z=0$ only.
In all panels, the grey area represents the shot-noise estimated from the expression with using the theoretical linear power spectrum.}
\label{fig:panels}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig3.pdf}
\caption{A test of the analytic prediction for the non-Gaussian model. The dashed and dash-dotted curves are taken from the middle panel in the previous figure.
The curves with the circles plot $(R/R_*)^3$ computed with the approximate $\sigma_i$ given in Eqs.~(\ref{eq:sigmahi}-\ref{eq:bNG}) }
\label{fig:fnltheory}
\end{figure}
To conclude this Section, we note that the effect of $f_{_{\rm NL}}$ on $\Upsilon$ is only weakly degenerate with that of $\sigma_8$ because $n_1$ primarily depends on the
ratio of spectral moments $\sigma_{1,h}/\sigma_{2,h}$.
\subsection{Asymmetry and height distribution}
So far we have considered $n_1$, without distinguishing between minima and maxima.
In Fig.~\ref{fig:asymm}, we examine the asymmetry between the abundances of minima and maxima as a function of the smoothing width for simulations 2 at redshift $z=0$.
There is a clear excess of $N_\mathrm{max}$, which is significantly above the level of the shot-noise (grey area).
The trend is reversed at larger scales for both DM non-Gaussian models, but it becomes immersed in the shot-noise.
Results of the three individual runs for the Gaussian DM simulation are also shown.
It is clear that the shot-noise estimated theoretically as described above (grey area) is consistent with the scatter in the individual runs.
We explore the probability density distribution (PDF) of the value of the densities at the minima and maxima.
We define,
\begin{equation}
\nu=\frac{\delta}{\sigma_0}\quad {\rm and} \quad \nu_\mathrm{ln}=\frac{{\rm ln}\,(1+\delta)-\mu}{\sigma_\mathrm{ln}}
\end{equation}
where $\sigma_0$ is the rms of density field all over space while
$\mu $ and $\sigma_\mathrm{ln}$ are the mean and rms of the values of ${\rm ln}\,(1+\delta)$ at either the minima or maxima.
The quantity $\nu_\mathrm{ln}$ is motivated by the result that the PDF of the density field is well approximated by a log-normal distribution \citep[e.g.][]{Coles91,Kofman}.
In Fig.~\ref{fig:PDFnu} and \ref{fig:PDFnu80} we plot the PDF of $\nu $ (top) and
$\nu_\mathrm{ln}$ (bottom) for a smoothing of $R=20h^{-1}\,{\rm Mpc}$ and $80h^{-1}\,{\rm Mpc}$ for simulations 2.
The 3 curves of each line-style correspond to the Gaussian and 2 non-Gaussian simulations.
It is evident that the
PDF of densities at either maxima or minima is weakly sensitive to whether the initial conditions were Gaussian or not.
This is expected given that corrections arise at order $f_{\rm NL}^2$ as noted above.
Thus, for clarity, the plot does not indicate which of the simulations is shown.
For $R=20h^{-1}\,{\rm Mpc}$, the BBKS theoretical prediction for $P(\nu)$ (expression 4.3 in their paper) shown as the black in the top panel, is a poor fit to any of the PDFs measured in the simulations.
However, $P(\nu_\mathrm{ln}) $ for the DM density field (dotted), exhibit only minor differences at the tails, where the PDF for maxima is slightly skewed to positive values
relative to the Gaussian (black in the bottom panel), the distribution at minima is negatively skewed. The rather small differences between
the PDF from the halos and the corresponding DM are due to deviations from linear biasing.
Fig.~\ref{fig:PDFnu80} shows the same results, but for $R=80h^{-1}\,{\rm Mpc}$. This large smoothing greatly reduces the effect of non-linear evolution, bringing
the BBKS theoretical PDF (black curve, top panels) closer
to the measured PDF than it is for $R=20h^{-1}\,{\rm Mpc}$.
The log-normal curve (black, bottom) remains a good fit to $P(\nu_\mathrm{ln} $ for the DM although not as good as in the smaller smoothing. It is interesting that the log-normal describes the halo PDF fairly well for this smoothing.
At $R=20h^{-1}\,{\rm Mpc}$ and $80h^{-1}\,{\rm Mpc}$ the halo bias in the Gaussian and non-Gaussian simulations are small (cf. Eq.~(\ref{eq:bNG}). This explains the similarity between the halo PDFs in the simulations irrespective of the initial statistic.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig4.pdf}
\caption{ The relative difference between the total number of maxima and minima in simulations 2, versus the smoothing width, at redshift $z=0$.}
\label{fig:asymm}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{PDFnu.pdf}
\includegraphics[width=0.45\textwidth]{PDFnulog.pdf}
\caption{ \textit{Top:} The PDF of $\nu$ at minima and maxima in simulations 2, as indicated in the figure.
The black curve is the theoretical prediction for $P(\nu)$ given in BBKS.
\textit{Bottom:} The PDF of $\nu_\mathrm{ln}$ at minima and maxima for simulations 2. Here
the black line is a Gaussian with zero mean and unit variance. }
\label{fig:PDFnu}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{PDFnu80.pdf}
\includegraphics[width=0.45\textwidth]{PDFnulog80.pdf}
\caption{ The same as the previous figure, but for $R=80h^{-1}\,{\rm Mpc}$. }
\label{fig:PDFnu80}
\end{figure}
\section{Abundance of extrema as a cosmological test}
\label{sec:prospects}
We offer a preliminary assessment of using total number of peaks and dips as a test of cosmological models.
A proper analysis should take into account the covariance between the abundances corresponding to different smoothing scales.
However, this task is beyond the scope of the current paper.
Instead, we will focus on the expected discriminatory power of extrema abundance at distinct scales.
As an example, we consider the {\small Euclid} mission \citep{EuclidRB}, which will target emision line galaxies in
the redfshift range $0.9<z<1.8$ across $\sim 35\%$ of the sky.
For Planck's cosmological parameters, the corresponding survey volume is $48 (h^{-1}\,{\rm Gpc})^3$. Furthermore,
the typical host halo mass is $\sim 10^{11-12}h^{-1}{\rm M}_{\odot}$, in broad agreement with the minimum halo mass resolved
in simulations 1.
We wish to assess the ability that a measured total number $N$ of
extrema in a survey can reject a certain model given the hypothesis of an assumed fiducial underlying model.
For this purpose, we assume that $N$ follows a Poisson distribution
\begin{equation}
\label{eq:Poisson}
P_{\bar N}(N)= \frac{{\bar N}^N}{N!} \mathrm{e}^{-\bar N}\; ,
\end{equation}
where $\bar N$ is the mean number expected in a particular given model.
Given an observed $N$,
the preferred of two competing models with expected mean numbers ${\bar N}_1$ and ${\bar N}_2$, respectively, is determined by
\begin{eqnarray}
\label{eq:D}
D_{_{\bar N_1 \bar N_2}}&=&-2{\rm ln}\, \frac{P_{{\bar N}_1}}{P_{{\bar N}_2}}\nonumber \\
&=& 2 N{\rm ln}\,\frac{\bar N_2}{\bar N_1}+2({\bar N_1-\bar N_2})\; .
\end{eqnarray}
The mean value of $D$ over all measurements, which we loosely denote by $\Delta \chi^2$ is
\begin{eqnarray}
\label{eq:chis}
\Delta \chi^2&=&\sum_N P_{\bar N} D_{_{\bar N_1 \bar N_2}} \nonumber \\
&=&2\bar N {\rm ln}\,\frac{\bar N_2}{\bar N_1}+2({\bar N_1-\bar N_2})\; ,
\end{eqnarray}
where we have used $\sum_N P_{\bar N}(N)=1$ and $\sum_N N P_{\bar N}(N)=\bar N$.
For $\bar N_2=\bar N$, the quantity $\Delta \chi^2$ yields the confidence level with which
a model with $\bar N_1$ can be rejected if the underlying model is $\bar N$.
We use this statistic to assess whether the abundance of dips and peaks
can be used to reject certain models given a Gaussian cosmological model with fiducial cosmological parameter.
We focus on $\Omega_m$ and $f_{_{\rm NL}}$, separately.
Fig.~\ref{fig:chiomega} examines $\Delta \chi^2$ as a function of the matter density $\Omega_\mathrm{m}$.
Here, $\bar N $ is computed using Eq.~(\ref{eq:npk}-\ref{eq:sigi}) for fiducial DM linear
power spectrum with the cosmological parameters corresponding to simulations 1.
The same parameters with the exception of $\Omega_\mathrm{m}$ are used in the same expression to derive $\bar N_1 $.
This figure, therefore, refers to a Gaussian model ($f_{_{\rm NL}}=0$) and, in addition to DM density fields,
it is also relevant for halos with linear constant bias with respect to the DM.
Only two filtering scales are considered, as indicated in the figure.
It is remarkable that for $R=50h^{-1}\,{\rm Mpc}$ the $1\sigma $ level ($\Delta \chi^2=1$) is at $\Delta \Omega\approx\pm 0.01$ from the fiducial $\Omega_m=0.3$.
It should be pointed out that for the $\Lambda$CDM linear power spectrum the abundance on a filtering scale given in $h^{-1}\,{\rm Mpc}$ is degenerate with respect
to $\Omega_m h$ and $\Omega_b h$.
This sensitivity to $\Omega_m$ declines rapidly at $R=100h^{-1}\,{\rm Mpc}$ due to the $1/R^3$ dependence of the number of dips and peaks.
The sensitivity to $f_{_{\rm NL}}$ is demonstrated in Fig.~\ref{fig:chifnl} plotting $\Delta \chi^2$ with $\bar N$ from the fiducial model and
$\bar N_1$ for $f_{_{\rm NL}}\ne 0$ but with all other parameters fixed at the fiducial values. These curves refer to filtered halo distribution
where the theoretical expressions in
Eqs.~(\ref{eq:sigmahi}-\ref{eq:bNG}) are used in Eq.~(\ref{eq:npk}) to derive the mean number of dips and peaks $\bar N$ in a Euclid
volume survey at $z=1$.
In these calculations, we consider a halo mass distribution consistent with simulations 1, with a minimum mass of $3.67\times 10^{12}h^{-1} M_\odot$.
For this mass threshold, we have seen in the previous section that the theoretical
predictions are in reasonable agreement with the simulations.
The sensitivity to $f_{_{\rm NL}}$ is improved for the larger filtering widths, $R$ thanks to the stronger $f_{_{\rm NL}}$-dependence of halo bias
on larger scales. For $R=300h^{-1}\,{\rm Mpc}$, we find $\Delta \chi^2=1$ for deviations $\Delta f_{_{\rm NL}}\approx \pm 25 $.
This is encouraging especially if combined with measurements as a function of filtering scales and different halo masses.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig6.pdf}
\caption{ Abundance of dips and peaks as a cosmological test for estimating $\Omega_\mathrm{m}$ from a survey like {\small Euclid}.
Values of $\Delta \chi^2=1$ correspond to $1\sigma$ limits from the fiducial value of $\Omega_\mathrm{m}$. }
\label{fig:chiomega}
\end{figure}
\begin{figure}
\includegraphics[width=0.45\textwidth]{Fig7.pdf}
\caption{ The same as the previous figure but for $f_{_{\rm NL}}$ instead of $\Omega_\mathrm{m}$. }
\label{fig:chifnl}
\end{figure}
\section{Discussion and Conclusions}
\label{sec:discussion}
Locating points of maxima and minima is straightforward even for 3D density fields estimated from realistic galaxy redshift surveys.
Since the total abundance is computed irrespective of height,
it should be robust against the details of how the density field is estimated from the data.
The total abundance is also insensitive to redshift space distortions,
which in any case can be modeled with standard perturbation theory for smoothing widths $R\gtrsim 50h^{-1}\,{\rm Mpc}$ \citep{codis/etal:2013}
\citep[see also][]{lam/etal:2010}.
Further, it depends explicitly only on the shape of the power spectrum.
Any dependence on the amplitude (e.g. $\sigma_8$) is indirectly encoded
in non-linear corrections to the shape of the gravitationally evolved power spectrum.
The lack of sensitivity of this abundance statistics on the amplitude thereby implies
that it breaks most of the degeneracy
between $f_{_{\rm NL}}$ and the primordial amplitude of scalar perturbations, which arises in measurements of
galaxy clusters counts and shear peaks in weak lensing maps for instance
\citep[this degeneracy can also be broken by combining clusters and voids, see][]{kamionkowski:voids}.
We have demonstrated that a primordial non-Gaussianity of the local-$f_{_{\rm NL}}$ type imprints a strong signal in the abundance of peaks
and dips of the halo density field owing to the non-Gaussian bias.
An important result of the current paper is that the BBKS prediction derived for Gaussian density field can account for this effect reasonably well,
provided that the matter power spectrum is replaced by the halo power spectrum.
Therefore, the abundance of peaks and dips (a 1-point statistics) is sensitive to the scale-dependent bias in the halo power spectrum (a
2-point statistics), like the covariance of cluster counts \citep{cunha/etal:2010}.
This effect disappears when the density field perfectly traces the matter distribution as is the case for shear peaks for instance.
We have made a preliminary assessment of the applicability of the total abundance statistics as a test of $f_{_{\rm NL}}$ for a survey with specifications similar to those of the {\small Euclid}
mission \citep{EuclidRB}.
From a measurement at a single smoothing scale $R$, we obtain an uncertainty of $\Deltaf_{_{\rm NL}}=25$ (for $R=300h^{-1}\,{\rm Mpc}$) and 40
(for $R=50h^{-1}\,{\rm Mpc}$).
This suggests that a measurement combining different smoothing scales and halo masses should be able to achieve a sensitivity of
$\Deltaf_{_{\rm NL}}\lesssim 10$. While the sensitivity of this approach will likely be worse than the limits set by the latest CMB measurements
from Planck, $f_{_{\rm NL}}=0.8\pm 5$ \citep{PlanckPNG}, this approach should be competitive with galaxy clusters and shear peak counts in
weak-lensing maps,
for which the forecasted uncertainty is $\Deltaf_{_{\rm NL}}\sim 9$ \citep[e.g.][for a galaxy survey like {\small eROSITA}]{pillepich/etal:2012}
and $\Deltaf_{_{\rm NL}}\sim 13$ \citep[e.g.,][for a weak-lensing survey with Euclid specifications]{marian/etal:2011}, respectively.
However, our approach may also be affected by the Eddington bias that plagues galaxy cluster counts or shear peaks. Namely, additive
noise in the data will presumably increase the number of peaks while reducing the number of dips, which would mimic a small positive
$f_{_{\rm NL}}$. We will defer a more detailed study of this effect to future work.
The abundance of extrema depends on the cosmological parameters of the background cosmology. Here we explored the dependence on $\Omega_m$ alone with very encouraging results of an accuracy at the level of $\Delta \Omega_\mathrm{m}\sim 0.01$.
For a given filtering scale given in $h^{-1}\,{\rm Mpc}$, the abundance depends is nearly degenerate with the combination $\Omega_\mathrm{m}h$. Thus, this result regarding $\Omega_\mathrm{m}$ could alternatively by expressed as an accuracy of
$0.7\, {\rm km }\, {\rm s}^{-1}$ on $H_0$ if all other parameters are fixed.
\section*{Acknowledgements}
This research was supported by the I-CORE Program of the Planning and Budgeting Committee,
THE ISRAEL SCIENCE FOUNDATION (grants No. 1829/12 and No. 203/09 for AN; No. 1395/16 for VD)
and the Asher Space Research Institute.
M.B. acknowledges support from Delta ITP consortium, a program of the Netherlands Organisation for Scientific Research (NWO)
that is funded by the Dutch Ministry of Education, Culture and Science (OCW).
\bibliographystyle{mnras}
|
{
"timestamp": "2018-04-23T02:02:47",
"yymm": "1804",
"arxiv_id": "1804.05328",
"language": "en",
"url": "https://arxiv.org/abs/1804.05328"
}
|
\section{Introduction}\label{sec1}
In recent years, precision medicine has become an important topic in both industry and academia. It aims to find a mechanism of treatment assignment such that the patient can benefit the most. A lot of researchers have conducted vast investigations through different approaches. For example, the framework of outcome weighted learning is proposed to identify the individualized treatment rule (ITR) in \cite{Zhao2012}. Alternatively, Zhang et al. \cite{Zhang2012a} \cite{Zhang2012b} propose another general framework of estimating optimal treatment regimes with robustness from the perspective of classification. Fu, Zhou and Faries \cite{Fu2016} develop a comprehensive binary search approach, which is easy to interpret and apply in clinical study. Chen \cite{Chen2017} further extends the outcome weighted learning to other models and loss functions. However, these methods only focus on binary treatment recommendation. To handle multiple treatments, Zhang et al. \cite{Zhang2017} develop a multi-category outcome weighted learning approach using angle based classifiers.
Their model essentially assumes that each patient takes one out of multiple treatments. However, in reality, patients can benefit from taking multiple treatments simultaneously. Therefore, it is of great interest to develop methods for precision medicine in the context of combination therapies.
The adoption of combination therapies in medication is inevitable, especially in chronic diseases. For example, patients with type 2 diabetes often receive multiple medications, because single treatment may be insufficient to effectively control the blood glucose level. Hence, medications exerting their effects through different mechanisms are needed. For instance, DPP4 increases incretin level, which inhibits glucagon release. Sulfonylurea increases insulin release from $\beta$-cell in pancreas. DDP4 and Sulfonylurea function through different biological pathways, while serving the same purpose of reducing blood glucose. Thus, we can expect better control in blood glucose when taking both medications. In addition, diabetes may also cause other complications which require additional medications. Due to these two reasons, patients with type 2 diabetes are very likely to receive a combination of multiple treatments, and so is the case in other chronic diseases such as cancer, and chronic heart failure (CHF).
However, existing algorithms may not be adequate to precision medicine with combination therapies. Algorithms for multi-class classification are not scalable with the increasing number of treatments, that is, if we have $K$ treatments, there are $2^K$ possible combinations, which implies at least $2^K-1$ classifiers. This is the consequence of treating each combination individually \adapt{without considering the underlying structures among those treatments}. For example, DDP4 and Sulfonylurea can reduce A1c by $0.5\%$ and $1.0 \%$, respectively. \adapt{If the treatment effects of DDP4 and Sulfonylurea are additive, combining these two medications together can reduce A1c by $1.5\%$ . If small interaction exists, the reduction is expected to lie between $1.0\%$ and $2.0\%$.} {\adapt Our proposed method is able to leverage this information to improve the learning efficiency, which will be elaborated in Section \ref{sec5} and \ref{sec6}.
Different from the traditional supervised learning, the estimand is the optimal treatment assignment which is not directly observed from the patients. For example, in a classification problem, the correct label $Y$ is observed for each observed covariates $X$. Similarly, in a regression problem, an outcome $Y$ is also observed for each observed covariates $X$. In our treatment assignment problem, information about optimal treatment assignment rule is only available indirectly through the outcome due to the fact that only one potential outcome can be observed. In this paper, we extend deep learning algorithms, and propose an outcome weighted loss function to estimate optimal treatment assignment rule for combination therapies.
Our paper is organized in the following manner. Section \ref{sec2} provides a brief review to outcome weighted learning and multi-label classification problem. It explains the necessity of using multi-label classification rather than multi-class classification, and also {\adapt addresses some requirements that a good method needs to achieve}. Section \ref{sec3} incorporates multi-label classification with outcome weighted learning, and illustrates basic properties of our proposed method. Section \ref{sec4} further provides theoretical justifications on the fisher consistency under general cases and a special case with additive treatment effect and small interactions. Section \ref{sec5} and \ref{sec6} show the advantages of our method through simulations and real data analysis. In section \ref{sec7}, our method is further extended to a family of loss function which can be adaptive to the treatment interactions. We also discuss other possible future extensions in section \ref{sec7}.
\section{Review}\label{sec2}
\label{review:multi-label}
In this section, we will review some literature related to precision medicine and multi-label classification. In the meantime, the keys to solve this problem are identified.
Precision medicine has been a hot topic for years. On the one hand, outcome weighted learning proposed in \cite{Zhao2012} is one of the most popular method. Outcome weighted learning finds the optimal decision rule that maximizes the conditional expectation of the outcome given the decision rule. Let $X$ be the covariates, function $D(\cdot)$ be the decision rule, which is a mapping from space of covariates $\mathcal{X}$ to the space of treatment $\mathcal{A}$. The conditional expectation of the outcome $R\in \mathbb{R}$ given treatment assignment $D(X)$ can be written as:
\begin{equation}\label{eq:owl}
E^{D}(R)=E\left[\frac{R}{\pi_A}I\{A=D(X)\}\right],
\end{equation}
where $\pi_a=\Pr(A=a|X)$, where $A$ is the random variable of treatment assignment and $a\in \mathcal{A}$ is a treatment. In this framework, searching for the best treatment assignment $D$ to maximize $E^{D}(R)$ is converted into a classification problem. {\adapt Similar to the multi-class classification problem, the outcome weighted learning framework \eqref{eq:owl} can also be applied to multiple treatments problems using angel based learning \cite{Zhang2017}.} On the other hand, researchers also propose another general framework in \cite{Zhang2012a,Zhang2012b}, which estimates the contrast function of the treatment effect directly. The proposed methods in \cite{Zhang2012a} and \cite{Zhang2012b} also enjoy the advantage of double robustness which allows for misspecification on either potential outcome model or propensity score model. Chen \cite{Chen2017} further extends the ideas in \cite{Zhao2012} and \cite{Zhang2012b} to more general loss functions.
However, different from traditional multi-class classification problem where each patient is assigned to one of the treatments, multi-label classification problem allows patients to be assigned to a combination of multiple treatments. A naive implementation called Label Powerset (LP) transforms a multi-label classification problem with $K$ treatments (or classes) into a multi-class classification problem with $2^K$ classes, therefore, the dimension increases exponentially with the number of treatment $K$. To avoid the curse of dimensionality and to increase efficiency, two strategies are often adopted. The first strategy is Binary Relevance (BR) \cite{Luaces2012}. It transforms the original problem to a $K$ independent binary classification problem. For example, suppose $\mathcal{A}=\{a_1,a_2,\cdots,a_K\}$, BR type of methods build $K$ classifiers to individually and independently decide whether $a_k$, $k=1,\cdots,K$, should be adopted.
In this case, assumed linear classification rule with $p$-dimensional covariates, at least $Kp$ coefficients need to be estimated, which is much smaller than that of LP. However, it is broadly criticized by its independent assumption on treatments (or classes).
Therefore, ranking by pairwise comparison \cite{Hüllermeier2008} is proposed. {\adapt However, this method only provides a ranking of the treatments. It does not provide any `zero' point to separate treatments which are beneficial from those are harmful.}
Besides those attempts to convert the multi-label classification problem into other problems,
an alternative is adopting an appropriate loss. One of the commonly adopted loss function is Hamming loss.
Let $A=(A_1,\cdots,A_K)\in \mathcal{A}$ represents a vector of length $K$, where $\mathcal{A}=\{-1,1\}^K$. If the $k$th treatment is adopted, then $A_k=1$; If not, then $A_k=-1$. Accordingly, decision rule $D(X)=(D_1(X),\cdots,D_K(X))\in \{-1,1\}^K$. Hamming loss of a decision rule $D(\cdot)$ given the sample $(X,A)$ is defined as
\begin{equation}
\frac{1}{K}\sum_{k=1}^{K}I\{A_k\not=D_k(X)\}.
\end{equation}
And it can be interpreted as the proportion of the misclassified labels. Say there are only 3 treatments (labels), the relationship between Hamming loss and 0-1 loss is shown in Figure~\ref{fig:compare_loss}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.62]{compare_loss.eps}
\caption{The left is the 0-1 loss, the right is the Hamming loss. Hamming loss is a step function and bounded by 0-1 loss.\label{fig:compare_loss}}
\end{figure}
Another interesting topic is the shared subspace.
The idea proposed by \cite{Yan2007} assumes that all the decision rules are directly depend on the same subspace of covariates. Similar to central mean space estimation in sufficient dimension reduction \cite{Cook2005},
shared subspace essentially assumes that for certain $B$, there exist $\bar{D}_k$'s such that $D_k(X)=\bar{D}_k(B^\top X)$, for all $k\in 1,\cdots,K$.
Through shared subspace, we can always borrow some efficiency from other treatments, and thus facilitate the estimation of unknown non-linear relationship.
In addition, scalability of the algorithm is also important because heavy computation complexity can undermine its practical application.
Multi-label classification enables the scalable computation with respect to the number of treatments. The algorithm still needs to be scalable with the increasing sample size.
From this review of the literature, it's clear that
the method we are looking for should fully address the following issues:
\begin{enumerate}
\item \adapt{Applicability to precision medicine}.
\item Appropriate loss with the framework of multi-label classification.
\item Shared subspace.
\item Scalable computation with both number of treatments and sample size.
\end{enumerate}
\section{Method}\label{sec3}
In this section, a loss function is proposed within outcome weighted learning framework, which takes advantage of multi-label classification. Our classifier based on neural networks (NN) is introduced and combined with the proposed loss function. NN as a classifier naturally satisfies the requirement of shared subspace. And its algorithm is also scalable with the sample size.
\subsection{Outcome weighted learning with multi-label classification}
Let $X_i\in \mathcal{X}$ be the column vector of $p$-dimensional covariates for $i$th patient among total $n$ patients, $A_i=(A_{i1},\cdots,A_{iK})\in \{-1,1\}^K$ is the treatment assignment of the $i$th patient, and $R_i\in \mathbb{R}$ is the observed outcome of the $i$th patient. Again, decision rule is denoted as $$D(X)=(D_1(X),\cdots,D_K(X))\in \{-1,1\}^K.$$
Considering the following loss function for a given decision rule $D(X)$,
\begin{equation}\label{eq:proposed_loss}
L(D)=\frac{1}{n}\sum_{i=1}^{n}\frac{R_i}{\pi_{A_i}}\frac{1}{K}\sum_{k=1}^KI\{A_{ik}\not=D_k(X_i)\},
\end{equation}
where $\pi_{A_i}$ may be known in clinical trial or estimated in observational study.
When the problem is a multi-class classification problem, the loss above is exactly equal to the outcome weighted 0-1 loss proposed in \cite{Zhao2012}. When the problem is a multi-label classification problem, the proposed loss \eqref{eq:proposed_loss} is upper bounded by the outcome weighted 0-1 loss in outcome weighted learning. In addition, if $\frac{R_i}{\pi_{A_i}}$ is a constant, it reduces to Hamming loss. The loss function proposed in \eqref{eq:proposed_loss} combines the loss in outcome weighted learning and Hamming loss. Thus, we call it outcome weighted Hamming loss.
One of the difficulty in optimizing the proposed loss \eqref{eq:proposed_loss} is the non-smoothness and non-convexity of indicator functions. Hinge loss \cite{Cortes1995} or logistic loss can be adopted as surrogate losses for indicator functions. The fisher consistency of proposed loss in \eqref{eq:proposed_loss} and its surrogate loss are provided in the following sections. As proved in Section \ref{sec4}, our proposed loss is fisher consistent under small amount of interactions.
\adapt{
Another difficulty of the proposed method comes from the estimation of $\pi_{A_i}$. We provide two solutions to this issue when $K$ is large. First solution assumes that the treatment assignment $A$ is independent with the covariates $X$. In this case, $P(A=a|X)=P(A=a)$ can be estimated by the proportion of the patients with $A=a$ in the sample. Second solution assumes that each treatment assignment $A_k$ are independent with each other given covariates $X$. In this case, $P(A=a|X)=\prod_{k=1}^K P(A_k=a_k|X)$ and each $P(A_k=a_k|X)$'s can be estimated by logistic regression because $A_k$ are binary. When $K$ is small, multinomial regression can also be used to estimate $P(A=a|X)$ directly. For simplicity, the first solution is adopted in Section \ref{sec5} and Section \ref{sec6}.}
\subsection{Decision rule with deep learning}
In the previous section, outcome weighted Hamming loss is defined. Admittedly any suitable classifier can fit into our framework, the classifier adopted in this paper is the Neural Network (NN) for its advantages in shared subspace and scalable computation.
To simply illustrate our idea, we start from a $3$-layer NN. The first layer on the bottom is the input layer where $X_i=(X_{i1},\cdots,X_{ip})^\top$ is the input vector of covariates. The layer in the middle is the hidden layer with $d$ hidden variables. The top layer consists of $K$ output variables. An example of the graph structure of this NN is presented in Figure~\ref{fig:NN}. In this toy example, only 3 treatments are considered, say treatment $E$, treatment $F$, and treatment $G$. Thus, $\mathcal{A}=\{-1,1\}^3$. For example, $(1,1,-1)$ represents $EF$ which is the combination of treatment $E$ represented by $(1,-1,-1)$, and $F$ represented by $(-1,1,-1)$. The $k$th output in the top layer is $\tilde{D}_k(X_i)$ given $X_i$ in the input layer. The sign of $\tilde{D}_k(X_i)$ indicates the treatment assignment of $k$th treatment, which is $D_k(X_i)$. As shown in Figure~\ref{fig:NN}, the adjacent layers are fully connected and no variables in the same layer are connected.
Given the graphic structure described above, many NNs can fit into this framework. For example, Deep Belief Nets (DBN) proposed in \cite{Hinton2006}, which consists of stacks of Restricted Boltzmann Machines (RBMs) and a classifier based on the very top hidden layer. Another common choice is \adapt{Deep Neural Network (DNN)}, which is a large NN without probabilistic modeling. DNN can be obtained by firstly training a DBN and then fine-tuning by back-propagation. This approach of building DNN often times can help avoid local minimizers and obtain better generalization error. However, the topologies of DBN and DNN are slightly different. Although the skeleton for both are the same as shown in Fig~\ref{fig:DNN_DBN}, the directions of the connected lines can be different. In DBN, besides the top two layers, all connections between hidden layer and hidden layer or hidden layer and visible layer are bi-directional (or undirected), and only connections between top two layers are directed from the hidden layer to the output. In DNN, all connections are directed from lower layers to higher layers. Directed connections are defined by relationship similar to \eqref{eq:nn_build1} and \eqref{eq:nn_build2}, which have no probabilistic framework. Bi-directional (or undirected) connections are defined through undirected graphical models, such as RBM \cite{Goodfellow-et-al-2016}. Thus, the nodes connected by bi-directional (or undirected) connections are random variables. The value of the node is either a random sample from the defined conditional distribution or the conditional expectation given other nodes. For example, if the lower layer on the right in Fig~\ref{fig:DNN_DBN} is defined by RBM and denote the input layer as $v$, the first hidden layer as $h$, the joint density function of $(v,h)$ is defined as
\begin{equation*}
f_{(v,h)}(v,h)=\frac{1}{Z(\theta)}{\exp\{-E(v,h;\theta)\}},
\end{equation*}
where $\theta$ is unknown parameter, $Z(\theta)$ is a normalization constant, $E(v,h;\theta)$ is energy function which has certain forms depending on the type of the model \cite{Goodfellow-et-al-2016}. If $\theta$ is known, given an input $v$, the conditional distribution of $h$ given $v$ can be calculated. Either a random sample from this conditional distribution or conditional expectation of $h$ given $v$ can be used as the input for next layers. The density function of $v$ is
\begin{equation*}
f_{v}(v)=\int\frac{1}{Z(\theta)}{\exp\{-E(v,h;\theta)\}}dh.
\end{equation*}
Then $\theta$ can be estimated by the maximize likelihood estimation (MLE) given observed data $v$. However, In this paper, we focus on DNN in both simulations and real data analysis.
DNN is defined in the following fashion. For clarification, we explicitly define those variables and weights of connections given toy example in Fig \ref{fig:NN}. Let the input layer in the very bottom be $v=(v_1,\cdots,v_p)\in \mathbb{R}^p$, the hidden layer (consists of $d$ hidden variables) in the middle be $h=(h_1,\cdots,h_d)\in\mathbb{R}^d$, and the output layer on the very top be $\tilde{D}=(\tilde{D}_1,\cdots,\tilde{D}_K)\in\mathbb{R}^K$. Let $W_1$ be the $p\times d$ matrix representing the weights on connections between input layer and hidden layer. For example, the entry of $W_1$ on Row $3$ and Column $4$ is the weight on connection between $v_3$ and $h_4$. Similarly, let $W_2$ be the $d\times K$ matrix representing the weights on connections between hidden layer and output layer. For example, the entry of $W_2$ on Row $3$ and Column $2$ is the weight on connection between $h_3$ and $\tilde{D}_2$. Now, with these notations, the relationship between these variables can be defined as the following:
\begin{eqnarray}\label{eq:nn_build1}
h&=&ReLU(W_1^\top v+h_0),\\ \label{eq:nn_build2}
\tilde{D}&=&W_2^\top h+\tilde{D}_0,
\end{eqnarray}
where $ReLU(\cdot)$ is ReLU function defined as $ReLU(t)=t1\{t>0\}$, for any $t\in \mathbb{R}$. $ReLU(O)$ with a vector $O$ represents a vector with ReLU function applied to each entry of $O$. $h_0$ is a constant $d$-dimensional vector and $\tilde{D}_0$ is a constant $K$-dimensional vector. Apparently, from the above relationship, $\tilde{D}$ can be written as $\tilde{D}(v)=(\tilde{D}_1(v),\cdots,\tilde{D}_K(v))$. Having the indicator function replaced by Hinge loss \cite{Cortes1995}, we can formulate the optimization problem as follows,
\begin{equation}\label{eq:optimization}
\textrm{minimize}_{\theta}\frac{1}{n}\sum_{i=1}^{n}\frac{R_i}{\pi_{A_i}}\frac{1}{K}\sum_{k=1}^K\left[1-A_{ik}\tilde{D}_k(X_i)\right]_+,
\end{equation}
where $\theta=(W_1,h_0, W_2,\tilde{D}_0)$ and $[\cdot]_+$ represents taking non-negative part. This optimization problem can be solved directly by back-propagation and SGD. Note that the structure of DNN and the activation function can be modified based on real data. The final estimated decision rule, $\hat{D}$, is the sign of $\tilde{D}$ given the minimizer of \eqref{eq:optimization}, $\hat{\theta}$.
\begin{figure}
\centering
\includegraphics[scale=0.5]{DNNDBNFig.eps}
\caption{The left is the structure of DNN and the right is that of DBN. The solid connections represent a directed edge from the lower layer to the top layer. The dashed connections represent bi-directional (or undirected) connections.\label{fig:DNN_DBN}}
\end{figure}
NN is a natural choice in this setting for the following reasons. First, the optimization of NN given the proposed loss in \eqref{eq:proposed_loss} can be implemented by back-propagation \cite{Rumelhart1986} and stochastic gradient descent (SGD) \cite{Bottou2010}, which is scalable in terms of sample size due to the nature of SGD. Second, the decisions are made based on hidden layers which depend on some shared linear directions of $X_i$ and a pre-specified non-linear activation functions \cite{LeCun2015}. Thus, the subspace of these $K$ decision rules are shared. Third, the hidden variables can capture complicated correlations among treatments, which is quite clear in the point of graphical model that given $X_i$, $\tilde{D}_k$'s are not independent with each other.
Based on universal approximation theorem proved in \cite{Barron1993}, hidden variables also introduce more flexibility into the model. Without the hidden layers, NN is equivalent to a linear model.
In most applications, the structure of hidden layers such as the number of hidden layers and the number of hidden variables in each hidden layer is adjustable. More complicated the hidden structure is, more flexible the decision rule can be. Thus, our NN decision rule with proposed loss function in \eqref{eq:proposed_loss} can fully satisfy our requirements. Another ad hoc view of the hidden layers in Fig~\ref{fig:NN} is that those hidden variables may represent certain biological pathways through which certain treatment can affect the outcome. Thus, the decision rule is reasonable to depend on certain hidden variables.
\begin{figure}
\centering
\includegraphics[scale=0.5]{NNFig.eps}
\caption{The first layer on the bottom is the input layer where $X_i=(X_{i1},\cdots,X_{ip})^\top$ is the input. The layer on the top is hidden layer with $d$ hidden variables. The top layer consists of $K$ output random variables.\label{fig:NN}}
\end{figure}
\subsection{Overfitting}
Due to strong representative power of DNN, how to avoid over-fitting has become a big issue. To avoid over-fitting, a commonly used strategy is regularization through penalization. Mathematically, we can consider to solve the following problem:
\begin{equation}
\textrm{minimize}_{\theta}\frac{1}{n}\sum_{i=1}^{n}\frac{R_i}{\pi_{A_i}}\frac{1}{K}\sum_{k=1}^K[1-A_{ik}\tilde{D}_k(X_i)]_++\lambda p(W_1,W_2),
\end{equation}
where $\theta=(W_1,h_0, W_2,\tilde{D}_0)$. $p(W_1,W_2)$ is the penalty on $W_1$ and $W_2$. For example, $p(W_1,W_2)=\|W_1\|_F^2+\|W_2\|_F^2$ is the ridge penalty, where $\|\cdot\|_F$ is the Frobenius norm, and $p(W_1,W_2)=\|W_1\|_1+\|W_2\|_1$ is lasso penalty, where $\|\cdot\|_1$ represents the sum of absolute value of all entries. Both ridge penalty and lasso penalty can shrink the weights towards $0$, but the advantage of lasso penalty is that it can produce sparse solutions. Another popular method to prevent over-fitting is the so-called dropout proposed in \cite{Srivastava2014}. Wager et al. \cite{Wager2013} argue that dropout is closely related to adaptive penalization and they also clarify its relationship with ridge penalty.
In general, both regularization and dropout are possible choices to prevent over-fitting.
\adapt{In both regularization and dropout, how to choose the tuning parameter has been investigated for years\cite{Srivastava2014, tibshirani1996regression,zou2005regularization}. Cross-validation or training-validation-testing data split can be used to evaluate the prediction error and select tuning parameters with the best performance. }
\adapt{Other parameters such as number of layers and number of nodes each layer also play an important role in balancing under-fitting and over-fitting. Of course, cross-validation or training-validation-testing data split can be applied to tune the NN's structure, but it is extremely computational intensive and impractical some times. In general, how to design NN's structure is still an open problem, but extensive researches have been done in this field. On the one hand, some researchers are working on using genetic algorithm and reinforcement learning to decide these parameters \cite{leung2003tuning,zoph2016neural}. On the other hand, sparsity induced by lasso penalty can partially solve this issue. Suppose that all weights connected to a particular variable are $0$, it is equivalent to excluding this variable in the structure of NN. Additionally, other penalties, for example, in \cite{Scardapane:2017:GSR:3067301.3067328}, weights connected to a particular variable can be forced to be $0$ and thus the node is excluded in the structure. Thus, in practice, we may suggest a structure as more complex as possible considering the sample size and employ lasso penalty or other penalties to help select nodes included in the structure automatically.}
\section{Algorithm}
In this section, we introduce the algorithm to solve the proposed optimization problem. A straightforward solution is back-propagation with stochastic gradient descent (SGD). Back-propagation proposed in \cite{Goodfellow-et-al-2016} is an efficient algorithm to numerically compute the derivatives with respect to certain weight given a NN. SGD provides an computational effect alternative to gradient descent. Beyond directly implementing SGD, pre-training can also be used to prevent local minimizers and facilitate convergence of the algorithm. One of the pre-training procedure utilizes Restricted Boltzmann Machine (RBM) \cite{Goodfellow-et-al-2016}. In this procedure, we firstly train stacks of RBM in such fashion that the first RBM is trained on observed covariates and next RBM is trained on the top of the first one, taking the hidden layer in the first RBM as the visible layer in the next RBM, and so on. Then, for the two layers on the very top, a multi-label classifier based on outcome weighted Hamming loss is trained. After this pre-training procedure, the whole network is fine-tuned by back-propagation and SGD.
\subsection{Stochastic gradient descent}
\adapt{
In this section, we briefly introduce the stochastic gradient descent (SGD) under general empirical risk minimization. Say we want to minimize the following loss function
\begin{equation}
\min_{\theta} \frac{1}{n}\sum_{i=1}^{n} l(Y_i, X_i;\theta),
\end{equation}
where $\theta$ is the general notation for all parameters of interest, $(X_i, Y_i) ,~i=1,\cdots,n$ is the observed data. In gradient descent, we have
\begin{equation}
\theta^{(m+1)}\leftarrow \theta^{(m)}-t\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l(Y_i, X_i;\theta),
\end{equation}
where $\theta^{(m)}$ is the value of $\theta$ at the $m$ step, and $t>0$ is a scalar called learning rate (step size). In stochastic gradient descent, we have
\begin{equation}
\theta^{(m+1)}\leftarrow \theta^{(m)}-t\frac{1}{|\mathcal{S}^{(m)}|}\sum_{i\in \mathcal{S}^{(m)}}\nabla_{\theta}l(Y_i, X_i;\theta),
\end{equation}
where $\mathcal{S}^{(m)}$ is a random sample from $\{1,\cdots, n\}$ with or without replacement, and $|\mathcal{S}^{(m)}|$ is the cardinality of the set $\mathcal{S}^{(m)}$, which is a pre-specified batch size. It is easy to see that
\begin{equation}
E_{\mathcal{S}}\left [\frac{1}{|\mathcal{S}^{(m)}|}\sum_{i\in \mathcal{S}^{(m)}}\nabla_{\theta}l(Y_i, X_i;\theta)\right ]=\frac{1}{n}\sum_{i=1}^{n}\nabla_{\theta}l(Y_i, X_i;\theta),
\end{equation}
where $E_{\mathcal{S}}[\cdot]$ is the expectation of random sampling.} Because each update only depends on a small size of random sample, it is scalable with the increase of sample size.
\adapt{In each update of SGD, gradient is computed on a small batch based on a sub-sample of training dataset. On the one hand, the stochastic induced by sub-sampling prevents the algorithm from falling into local minimizers. On the other hand, it reduces the computation complexity dramatically compared with usual gradient descent. In general, SGD converges faster than usual gradient descent, especially for statistical problems \cite{bousquet2008tradeoffs}. However, due to the stochastic nature of SGD, the calculated gradient based on a batch is mostly not zero even at the global minimizer. Thus, the variance of sampling small batches has an impact on the updates of each iteration. To reduce this variance, SVRG and other techniques \cite{Johnson2013} are proposed. Another simple strategy is to gradually increase the size of the batch in gradient calculation and decrease learning rate (step size) exponentially in gradient update. The convergence of SGD has been proved in strictly convex problem \cite{bousquet2008tradeoffs}, while generally the convergence of SGD is still an open problem.
\subsection{Implementation}
\adapt{In this section, we introduce some packages to implement SGD. A well known package in Python is called Tensorflow developed by Google. Tensorflow provides an extremely powerful tool to customize NN (number of layers and number of nodes each layer) and SGD (batch size and step size). It also includes some well-developed advanced algorithm for SGD and keeps updating. Compared with Tensorflow, Keras is a high-level NN API which is more user-friendly. It allows researchers to build a very flexible NN with a few lines of code. The down side is that it may not be easy to customize everything using Keras. Recently, R initiated an access to Tensorflow, which is more friendly to statistical programmers. An experimental R package to implement the Keras is also available on GitHub. Readers can get more information on R versions of Tensorflow and Keras on https://tensorflow.rstudio.com.}
\section{Theoretical result}\label{sec4}
In this section, the theoretical justification of our proposed loss is provided. In the first part, we prove the fisher consistency of outcome weighted Hamming loss, which implies that minimizing the proposed loss is equivalent to minimizing the original outcome weighted 0-1 loss. In the second part, surrogate loss for outcome weighted Hamming loss is proposed and its consistency with outcome weighted Hamming loss is also provided, which indicates that minimizing the surrogate loss is also equivalent to minimizing the outcome weighted Hamming loss. Combining these two parts of theoretical results provides an overall theoretical support for our proposed method.
Without loss of generality, we assume that outcome $R$ is non-negative.
\subsection{Fisher consistency of Hamming loss}
We establish the fisher consistency of outcome weighted Hamming loss in this section, indicating the equivalence of minimizing outcome weighted Hamming loss and outcome weighted 0-1 loss. Let $D^*(X)=(D_1^*(X), \cdots,D_K^*(X))\in \{-1,1\}^K$ be the decision rule that minimizes outcome weighted 0-1 loss:
\begin{equation}
\mathcal{R}(D)=E\left[\frac{R}{\pi_A}I\{A\not = D(X)\}\right].
\end{equation}
It is easy to see that $D^*(X)=\{a:\max_a E[R|A=a, X]\}$. Again, outcome weighted Hamming loss is defined as
\begin{equation}
\mathcal{R}_H(D)=E\left[\frac{R}{\pi_A}\frac{1}{K}\sum_{k=1}^KI\{A_k\not = D_k(X)\}\right].
\end{equation}
\begin{theorem}[Fisher consistency of outcome weighted Hamming loss]\label{thm1}
Suppose $$D_k^*(X)=\textrm{sign}\left\{\sum_{\{a:a_k=1\}}E[R|A=a,X]-\sum_{\{a:a_k=-1\}}E[R|A=a,X]\right\},$$
then any function $f$ such that $$\mathcal{R}_H(f)=\inf_D\{\mathcal{R}_H(D)\}$$ satisfies
$$\mathcal{R}(f)=\inf_D\{\mathcal{R}(D)\}.$$
\end{theorem}
Note that in many cases, condition $D_k^*(X)=\textrm{sign}\left\{\sum_{\{a:a_k=1\}}E[R|A=a,X]-\sum_{\{a:a_k=-1\}}E[R|A=a,X]\right\}$ holds. For example, a sufficient condition is when treatment effects are additive and there is no interaction, i.e. $E[R|A=a, X]=\sum_{\{k:a_k=1\}}T_{e_k}(X) + m(X)$, where $T_{e_k}, k=1,\cdots,K$, is the treatment effect given only treatment $k$, $e_k$ is a $K$-dimensional vector having the $k$th entry equals $1$ and all other entries equal $-1$.
Moreover, small interactions are tolerable, as stated in Theorem \ref{thm2}.
\begin{theorem}[Fisher consistency in special case]\label{thm2}
Suppose that
\begin{equation*}
E[R|A=a, X]=\sum_{\{k:a_k=1\}}T_{e_k}(X) + r_a(X) + m(X),
\end{equation*}
where $T_{e_k}(X)$ is the treatment effect of only $k$th treatment being adopted, $r_a(X)$ is the additional interaction, and $m(X)$ is the main effect. If $2\sup_a|r_a(X)|< \inf_k|T_{e_k}(X)|$ for all $X$, then any function $f$ such that $$\mathcal{R}_H(f)=\inf_D\{\mathcal{R}_H(D)\}$$ satisfies
$$\mathcal{R}(f)=\inf_D\{\mathcal{R}(D)\}.$$
\end{theorem}
Theorem \ref{thm2} provides a sufficient condition for fisher consistency of outcome weighted Hamming loss. It essentially provides a theoretical guarantee of the robustness of our method against small interactions. When the condition is violated, our method is not necessarily consistent and its performance depends on the magnitude of interactions and amounts of patient affected. In simulation, we compare the performance our method with other methods under small violation of this condition
\subsection{Multi-label consistency of surrogate loss}
Minimizing the proposed loss \eqref{eq:proposed_loss} is still very hard due to the non-smoothness and non-convexity of indicator functions. Therefore, it is natural to replace the indicator functions in outcome weighted Hamming loss with some surrogate loss. For example, a common choice for outcome weighted Hamming loss is
\begin{equation}\label{eq:surrogate_hamming}
\Phi_H(\tilde{D})=E\left[\frac{R}{\pi_A}\frac{1}{K}\sum_{k=1}^K\phi(A_k\tilde{D}_k(X))\right],
\end{equation}
where $\phi$ is a pre-defined convex function. Still, it is critical that minimizing the surrogate loss is equivalent to minimizing the original outcome weighted Hamming loss. Unfortunately, not every $\phi$ satisfies this condition. As shown in the following theorem, \eqref{eq:surrogate_hamming} is consistent with outcome weighted Hamming loss if $\phi$ is one of the following:
\begin{enumerate}
\item Exponential: $\phi(x)=e^{-x}$;
\item Hinge: $\phi(x)=(1-x)_+$;
\item Least squares: $\phi(x)=(1-x)^2$;
\item Logistic Regression: $\phi(x)=\ln(1+e^{-x})$.
\end{enumerate}
Formally, we have the following theorem.
\begin{theorem}[Multi-label consistency of surrogate loss]\label{thm3}
Suppose $$\phi'(0)<0,$$
for any function $\tilde{f}$ such that $$\Phi_H(\tilde{f})=\inf_{\tilde{D}}\{\Phi_H(\tilde{D})\},$$ let $f=\textrm{sign}(\tilde{f})$, then $f$ satisfies
$$\mathcal{R}_H(f)=\inf_D\{\mathcal{R}_H(D)\}.$$
\end{theorem}
\section{Simulation}\label{sec5}
\subsection{Simulation with correctly specified model}\label{subsec5.1}
\label{Sec: correctmodel}
In this section, we will illustrate the performance of two DNN approaches under outcome weighted Hamming loss in \eqref{eq:optimization} by comparing with a naive method through simulations. In the following simulation settings, $K=5$, so in total there are $2^5=32$ combinations of treatments. The dimension of covariates is set to be $p=30$, which is common in clinical trials. As explained in Section \ref{review:multi-label}, the $K$ treatment multi-label problem can be decomposed to a multi-class problem with $2^K$ classes. The naive method further converts the multi-class problem to a series of two-class classification problem, where for each, it directly learns a binary classifier with linear decision rule through outcome weighted learning. To form a multi-class classifier with $32$ different classifiers with intercepts, $2^K(2^K-1)(p+1)/2=15376$ coefficients in $2^K(2^K-1)/2=496$ different linear decision rules have to be estimated. The first DNN approach, DNN-simple, is a DNN with only input and output layer. For any monotone activation function, DNN-simple is equivalent to $K=5$ binary linear classifiers with no shared subspace. In general, the only difference between DNN-simple and naive method is how to form the classifier for multi-label classification. In DNN-simple, only $K\times p+K=155$ parameters needs to be estimated. The comparison of DNN-simple and naive method essentially shows the efficiency boosting by adopting our proposed outcome weighted Hamming loss. The other DNN approach, DNN-1hdd, adds one hidden layer between input layer and output layer as shown on the left in Figure \ref{fig:DNN_DBN}. The hidden layer is fully connected with the input layer and the output layer.The NN structure allows for more flexibility and hence more parameters. In DNN-1hdd method, suppose the number of hidden variable is $n_h$, the total number of parameters to be estimated is $p(n_h+1)+n_h(K+1)=p+n_h(p+K+1)=30+36n_h$. By comparing DNN-simple and DNN-1hdd, it is easy to tell the loss and gain to accommodate a more flexible model. In addition, the Bayes rule is also evaluated, simply to quantifies the signal-to-noise ratio in our simulation settings. The Bayes rule is the treatment assignment which gives the largest conditional expectation of the potential outcomes, i.e.
\begin{equation}
D_{\rm B}(X)=\arg \max_aE\left[R|A=a,X\right].
\label{Eq:bayesrule}
\end{equation}
The Bayes rule is impossible to implement in reality, but in simulations, since we know the data generating procedure, we can directly evaluate $E\left[R|A=a,X\right]$.
In the simulation, every patient has the same probability to receive one of the treatment combination, and for each treatment, the treatment effect is generated from a NN with one hidden layer as shown in the left of Figure~\ref{fig:DNN_DBN}. The number of the hidden variables in the hidden layer is $n_h=45$. Firstly, we define the treatment effect of the $k$th treatment for $i$th patient $T_{k,i}$. Let
\begin{eqnarray}
h&=&ReLU(W_1^\top X_i),\\
T_{k,i}&=&\textrm{sign}\left (W_{2,\cdot,k}^\top h\right )\left\{\frac{0.05\exp\left\{W_{2,\cdot,k}^\top h\right\}}{1+\exp\left\{W_{2,\cdot,k}^\top h\right\}}+2.0\right\},
\end{eqnarray}
where $h$ is the hidden layer which is a $n_h$-dimensional vector, $W_{2,\cdot,k}$ is the $k$th column of $W_2$, and $\textrm{sign}(\cdot)$ is the function of taking sign coordinate-wise. All entries in $W_1$ and $W_2$ are generated from a standard normal distribution independently. Secondly, the main effect is defined by the following:
\begin{equation}
M_i=0.05\frac{\exp\left\{\gamma^\top X_i\right\}}{1+\exp\left\{\gamma^\top X_i\right\}}-2.05,
\end{equation}
where $\gamma$'s are the coefficients for main effect, whose entries are also generated from a standard normal distribution independently. Let $A_i\in\{-1,1\}^K$ represents the combination of assigned treatment to $i$th patient. Note that $R_{i,A_i}$, the potential outcome when given $A_i$ is defined by the following:
\begin{equation}
R_{i,A_i}=\sum_{\{k: A_{ik}=1\}}T_{k,i}+M_i+\sigma\epsilon_{i,A_i},
\end{equation}
where $\epsilon_{i,A_i}$ follows standard normal distribution and is independent with any other random variables. It can be simply verified that $\forall \beta_{p,k}, \beta_{n,k}, $ and $\gamma$, given any $A_i$, $\sum_{\{k: A_{ik}=1\}}T_{k,i}+M_i\in[-12,8]$. Let $A_i^{\rm opt}$ represents the selected combination of treatment, given which the potential outcome is maximized. In our setting, $\sigma$ is chosen to be $1.1$. In fact, it can be observed that the Bayes rule as defined in Equation \ref{Eq:bayesrule} is the sign of the $T_{k,i}$ for each $k$. In addition, based on Theorem \ref{thm2} and \ref{thm3}, our proposed loss is consistent with outcome weighted 0-1 loss in this setting.
In both DNN approaches, the activation function between hidden layer and output layer, or input layer and output layer is fixed to be a simple centered monotone transformation. The activation function between input layer and hidden layer in DNN-1hdd is chosen to be ReLU. $L_1$ penalty is applied to all the weights in both of DNN approaches.
To compare different methods, three scores are defined to quantify their performance. Two of the scores are based on misclassification rate. Let $\hat{A}_i$ be the predicted combination of treatments. Usually, the misclassification rate is defined as
\begin{equation}
MCR=\frac{1}{n}\sum_{i=1}^n 1\{\hat{A}_i\not = A_i^{\rm opt}\}.
\end{equation}
To account for the fact that $P(A_i^{\rm opt}=a)$ is not the same, for all $a\subset\{1,\cdots,K\}$, the average of proportion of misclassification rate is also proposed as following
\begin{equation}
AMCR=\frac{1}{2^K}\sum_{a\in\{-1,1\}^K}\frac{\sum_{i=1}^n 1\{\hat{A}_i\not=a, A_i^{\rm opt}=a\}}{\sum_{i=1}^n 1\{A_i^{\rm opt}=a\}}.
\end{equation}
To adjust MCR and AMCR based on the total number of combinations of treatments and make their performance comparable with binary classifier for binary classification problem, adjusted MCR and AMCR is calculated by $1-(1-MCR)^{1/K}$ and $1-(1-AMCR)^{1/K}$, respectively. Because in the framework of personalized medicine, the ultimate goal is not classification, but better clinical outcome. Thus, it is primary to consider the average benefit which defines as following
\begin{equation}
\label{Eq:AB}
AB=\frac{1}{n}\sum_{i=1}^n R_{i,\hat{A}_i}.
\end{equation}
For MCR (or adjusted MCR) and AMCR (or adjusted AMCR), lower is better. However, higher AB is preferred.
The simulation procedure works as follows. First, a training set and a validation set with the same sample size $n_{train}K$ are generated separately, and a sequence of candidate tuning parameters are pre-specified, in this case, 0.1, 0.01, 0.001, 0.0001. Then for each candidate tuning parameter, the Naive method, DNN-simple, and DNN-1hdd are trained on the training dataset and tested on the validation set, where we compute the misclassification rate on the validation dataset for each method. The tuning parameter gives the lowest misclassification rate on the validation set is selected, respectively for each method. At last, we evaluate the three scores (MCR, AMCR, AB) under the selected tuning parameter on testing dataset which is independently generated with sample size $n_{test}=10n_{train}K$. The above procedure is repeated 100 times. The mean and standard error (SE) of the scores are reported in Table \ref{tab2}. Figure \ref{Fig:comparison} shows a summary for adjusted scores and AB over 100 repeats.
Overall, DNN-1hdd performs the best in terms of MCR (or adjusted MCR) and AB. Note that Bayes method is infeasible in practice, its superiority is because it takes advantage of the true model. The Bayes method serves as a reference to quantify the best possible performance we could get in this problem. Although DNN-simple can only produce linear decision rules, which don't include the true decision rule, its performance is still competitive with DNN-1hdd because linear decision rule can explain the true non-linear model to some extent, especially in small samples when the information is limited. The fact that DNN-1hdd allows more flexibility also introduces additional variation. In general, DNN-1hdd performs slightly better than DNN-simple with this bias and variance trade-off. In terms of AMCR (or adjusted AMCR), DNN-simple performs the best among all three methods. From Table \ref{tab2} and Figure \ref{Fig:comparison}, as the sample size increases, the misclassification decreases and average benefit increases for all methods, which agrees with our consistency result proved in Theorem \ref{thm2}. However, the decreasing in MCR (or adjusted MCR) becomes slower as sample size increases, because (1) the scores are lower bounded by Bayes rule; (2) the algorithm is terminated after a certain times of iterations and takes the final update as the minimizer. In practice, one possible strategy is to use a validation dataset to monitor the algorithm and pick the update with the best performance in validation dataset.
\begin{center}
\begin{table}[t]%
\centering
\caption{Comparison of naive method, DNN-simple, and DNN-1hdd by adjusted scores.\label{tab2}}%
\begin{tabular*}{500pt}{@{\extracolsep\fill}ccccc@{\extracolsep\fill}}
\toprule
$n_{train}$ & Method & adjusted MCR & adjusted AMCR & AB\\
\midrule
200 & Naive &0.4708(0.0026) &0.4652(0.0012) &-1.5171(0.1004)\\
200 & DNN-simple & 0.2824(0.0022) &0.3555(0.0022)& 0.5765(0.1172)\\
200 & DNN-1hdd & 0.2782(0.0028) &0.3709(0.0025)& 0.6188(0.1200)\\
200 & Bayes & 0.0657(0.0001)&0.1108(0.0027) &3.1173(0.1143)\\ \hline
1000 & Naive &0.3517(0.0024) &0.3729(0.0012) &-0.0897(0.0948)\\
1000 & DNN-simple & 0.2125(0.0016) &0.3008(0.0028)& 1.3609(0.1153)\\
1000 & DNN-1hdd & 0.2142(0.0015) &0.3240(0.0041)& 1.3376(0.1145)\\
1000 & Bayes & 0.0657(0.0001)&0.1111(0.0027) &3.0871(0.1133)\\ \hline
4000 & Naive &0.3083(0.0029) &0.3396(0.0018) &0.3502(0.0883)\\
4000 & DNN-simple & 0.1907(0.0014) & 0.2902(0.0034) & 1.7257(0.1166)\\
4000 & DNN-1hdd & 0.1806(0.0012) & 0.2991(0.0050) & 1.8435(0.1134)\\
4000 & Bayes & 0.0656(0.0000)&0.1140(0.0029) &3.1965(0.1127)\\ \hline
10000 & Naive &0.2996(0.0035) &0.3305(0.0015) &0.4934(0.0766)\\
10000 & DNN-simple & 0.1846(0.0014) & 0.2865(0.0033) &1.7272(0.1008)\\
10000 & DNN-1hdd & 0.1617(0.0012) &0.2889(0.0063) &1.9858(0.0986)\\
10000 & Bayes & 0.0657(0.0001)&0.1111(0.0027) & 3.1272(0.0977)\\
\bottomrule
\end{tabular*}
\end{table}
\end{center}
\begin{figure}
\centering
\includegraphics[height=9cm, width=12cm]{Comparison_new}
\caption{Comparison of naive method, DNN-simple, and DNN-1hdd by adjusted MCR, AMCR, and AB.\label{Fig:comparison}}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[height=9cm, width=12cm]{Comparison_mis}
\caption{Comparison of naive method, DNN-simple, and DNN-1hdd by adjusted MCR, AMCR, and AB with model mis-specification.\label{Fig:comparison_mis}}
\end{figure}
\subsection{Simulation with model mis-specification}
In this section, we compare our two DNN approaches with a naive method when the condition in Theorem \ref{thm2} is not satisfied. The simulation procedure is the same as in Section \ref{Sec: correctmodel} except for the way to generate $R_{i, A_i}$. In this simulation, given $T_{k,i}$ and $M_i$, the potential outcome is generated by
\begin{equation}
R_{i,A_i}=\sum_{\{k: A_{ik}=1\}}T_{k,i}-\gamma (\sum_{\{k: A_{ik}=1\}}T_{k,i})^2+M_i+\sigma\epsilon_{i,A_i},
\end{equation}
where $\gamma=0.1$ and $\sigma=0.2$. In this setting, it is easy to see that $\sup_A|\gamma (\sum_{\{k: A_{ik}=1\}}T_{k,i})^2|\approx10$ and $\inf_k|T_{k,i}|\approx2$. Thus, the condition in Theorem \ref{thm2} is violated. The simulation procedure is the same as that described in Section \ref{subsec5.1}.
The averages of Adjust MCR, AMCR, and AB over 100 repeats with their standard deviations are reported in Table \ref{tab2_mis} and Figure \ref{Fig:comparison_mis}, which show that our proposed methods DNN-1hdd and DNN-simple outperform Naive method with respect to all three scores. Overall, DNN-1hdd has the best performance with respective to AB and adjusted MCR, while DNN-simple performs the best when considering the adjusted AMCR. Although the Naive method is consistent, it performs very bad under finite sample because of the high dimensionality of this problem. Our two DNN approaches still perform well when the small interaction assumption is violated.
\begin{center}
\begin{table}[t]%
\centering
\caption{Comparison of naive method, DNN-simple, and DNN-1hdd by adjusted scores with model mis-specification.\label{tab2_mis}}%
\begin{tabular*}{500pt}{@{\extracolsep\fill}ccccc@{\extracolsep\fill}}
\toprule
$n_{train}$ & Method & adjusted MCR & adjusted AMCR & AB\\
\midrule
200 & Naive &0.4863(0.0024) &0.4720(0.0017) &-2.6427(0.0921)\\
200 & DNN-simple & 0.3048(0.0053) &0.3703(0.0029)& -0.9197(0.0516)\\
200 & DNN-1hdd & 0.2994(0.0060) &0.3829(0.0028)& -0.8746(0.0491)\\
200 & Bayes & 0.1157(0.0064)&0.1726(0.0027) &0.2124(0.0199)\\ \hline
1000 & Naive &0.4074(0.0063) &0.4096(0.0022) &-1.2951(0.0470)\\
1000 & DNN-simple & 0.2432(0.0063) &0.3244(0.0036)& -0.4939(0.0308)\\
1000 & DNN-1hdd & 0.2433(0.0067) &0.3462(0.0043)& -0.4885(0.0307)\\
1000 & Bayes & 0.1127(0.0065)&0.1732(0.0029) &0.2136(0.0164)\\ \hline
4000 & Naive &0.3761(0.0070) &0.3898(0.0044) &-1.1061(0.0424)\\
4000 & DNN-simple & 0.2323(0.0064) & 0.3166(0.0040) & -0.3592(0.0267)\\
4000 & DNN-1hdd & 0.2254(0.0070) & 0.3275(0.0051) & -0.3151(0.0247)\\
4000 & Bayes & 0.1199(0.0066)&0.1761(0.0028) &0.2293(0.0161)\\ \hline
10000 & Naive &0.3890(0.0084) & 0.4059(0.0020)& -1.1827(0.0370)\\
10000 & DNN-simple &0.2210(0.0053) &0.3102(0.0037)&-0.3050(0.0279)\\
10000 & DNN-1hdd &0.2016(0.0060)&0.3128(0.0060)&-0.1966(0.0247)\\
10000 & Bayes &0.1146(0.0054) &0.1761(0.0027)& 0.2354(0.0172)\\
\bottomrule
\end{tabular*}
\end{table}
\end{center}
\section{Real data analysis}\label{sec6}
In this section, we apply our method to an electronic health record (EHR) data for type 2 diabetes patients from Clinical Practice Research Datalink (CPRD). 1139 patients are included in the dataset. For each patient, 21 covariates are collected before treatment assignment, which include demographical variables such as gender, BMI, HDL, and LDL, and also indicator of complications such as stroke and hypertension. The primary endpoint is change in A1c. Because A1c typically drops after applying the treatments, the primary endpoint is always negative and smaller is preferable. Thus, the proposed method can be easily implemented by considering negative change in A1c as patient outcome.
In this dataset, 4 treatments are considered for each patient, DDP4, sulfonylurea (SU), metformin (MET), and TZD. These 4 treatments function via four different biological processes. DDP4 increases incretin levels, which inhibits glucagon release. SU increases insulin release from $\beta$-cell in pancreas. MET decreases glucose production by the liver and increase the insulin sensitivity of body tissue. TZD makes cells more dependent on oxidation of carbohydrates. These 4 treatments target on different cells or functional organs, so it is reasonable to assume little treatment interaction among them and additive treatment effects.
In this real dataset, $K=4$, resulting in $16$ possible combinations of treatments. Because the original average benefit score can not be directly calculated in real settings, the naive method, DNN-1hdd, and DNN-simple are evaluated under a weighted version of average benefit defined as follows,
\begin{eqnarray}
T&=&\frac{\sum_{i=1}^{n} w_iR_i}{\sum_{i=1}^{n} w_i},\nonumber
\end{eqnarray}
where $w_i=\frac{I\{\hat{A}_i=a_i\}}{P(A_i=a_i|X_i)}$, $a_i$ is the observed treatment assignment, $R_i$ is the negative change in A1c, $\hat{A}_i$ is the predicted treatment assignment for the $i$th patient. Intuitively, $T$ is the weighted average of change in A1c over all the patients with the same treatment assignment as our predicted treatment assignment, which estimates the change in A1c if the fitted treatment assignment is adopted. If our proposed methods can recover the underlying optimal decision rule to some extent, their $T$ are expected to be lower than that of the naive method, indicating higher efficacy of our treatment recommendations. In addition, the number of patients $N$ satisfying $a_i=\hat{A}_i$ is also reported.
Multiple imputation is adopted to deal with missing data. Because of the randomness of the multiple imputation, $5$ different imputed datasets are analyzed following the same procedure, and then the scores from these $5$ imputed data are summarized by average.
For each imputed dataset, we do the following. Firstly, the whole dataset is randomly split into two datasets, training set and testing set. Training set contains $912$ patients, and testing dataset contains $227$ patients. All three methods are fitted on training set respectively, and the score for each of the method is calculated based on the testing set. This procedure is repeated $100$ times, each time the score is recorded. After summarizing these scores across $5$ imputed datasets, the mean of $T$, $N$ and their standard errors (SEs) are reported in Table~\ref{tab3}, and all results over $100$ repeats are shown by boxplots in Figure~\ref{Fig:realdata}.
It can be observed that both DNN-simple and DNN-1hdd have lower $T$ compared with naive method. Moreover, both DNN methods show significant effect on A1c while naive method does not. Comparing between two DNN methods, DNN-1hdd performs slightly better than DNN-simple in terms of $T$. In addition, $N$, which is the size of the subgroup with the same treatment assignment as predicted for our proposed methods, is much larger than that for naive method. Overall, our proposed methods have better performance than naive method, and DNN-1hdd is slightly better than DNN-simple.
\begin{center}
\begin{table}[t]
\caption{Comparison of Naive method, DNN-simple, and DNN-1hdd by scores in real data example. $T$ is the weighted average of change in A1c over all the patients with the same treatment assignment as the method suggests. $N$ is the number of the patients whose treatment assignment coincide with the predicted.\label{tab3}}%
\begin{tabular*}{500pt}{@{\extracolsep\fill}ccc@{\extracolsep\fill}}
\toprule
Method & $T$ & $N$ \\ \midrule
Naive & -1.534(0.078) & 4.410(0.107)\\
DNN-simple & -2.605(0.058) & 23.058(0.371) \\
DNN-1hdd & -2.695(0.057) & 25.790(0.394)\\
\bottomrule
\end{tabular*}
\end{table}
\end{center}
\begin{figure}
\centering
\includegraphics[height=9cm, width=12cm]{realdata}
\caption{Boxplots of scores for Naive method, DNN-simple, and DNN-1hdd over 100 repeats .\label{Fig:realdata}}
\end{figure}
\section{Conclusions and discussions}\label{sec7}
In this paper, an outcome weighted deep learning framework is proposed to estimate optimal combination therapies. Both simulation and real data analysis provide solid evidence on the power of our proposed method. Essentially, the proposed loss, outcome weighted Hamming loss, can be applied to any occasion, even when deep learning is not a desired classifier. For example, linear classifiers can be used in the case when efficiency and convexity is very important. Although other nonlinear classifiers can also be adopted, when adopting these methods, advantages of deep learning such as sharing subspace may disappear unless inducing certain techniques such as dimension reduction. Deep learning framework can also be applied under other losses. For example, partial ranking loss proposed in \cite{Gao2013} can also be combined with deep learning approach. However, for ranking based loss function, one of the significant drawback is the lack of `zero' point to distinguish between good treatments and bad treatments. In other word, ranking based loss function can only provide a rank of treatments instead of recommending treatment. In this paper, the advantages of adopting deep learning approach and the loss function has been articulated . Our method enjoys all of these advantages.
Furthermore, the proposed method is critical for future research. It can be extended in multiple directions. The first possible extension focuses on loss functions. In previous sections, it is easy to observe that Hamming loss provides an approximation to the original 0-1 loss. As we have shown, this approximation is done by overlooking strong interactions among treatments. While original 0-1 loss is very flexible so that it is generally consistent, the huge number of parameters to be estimated may undermine the efficiency and lead to huge computation costs. Thus, a natural extension of our method is to design a family of loss functions such that our method can be adaptive to certain amount of interactions among treatments. For example, Hamming loss is the proportion of mis-classified labels, and can be rewritten into the following
\begin{equation*}
1-\frac{1}{K}\sum_{k=1}^K 1\left\{A_k=D_k\right\}.
\end{equation*}
0-1 loss can also be rewritten into the following
\begin{equation*}
1-\prod_{k=1}^K 1\left\{A_k=D_k\right\}.
\end{equation*}
Naturally, a family of loss functions can be defined as
\begin{equation*}
1-\frac{1}{\binom{\tau}{K}}\sum_{\{k_1\cdots,k_{\tau}\}\subset\{1,\cdots,K\}} \prod_{q=1}^{\tau}1\left\{A_{k_q}=D_{k_q}\right\},
\end{equation*}
where $\tau$ is a parameter. We call it $\tau$th order Hamming loss. When $\tau=1$, it is the same as Hamming loss, which has the most strict conditions on treatment interactions in order to guarantee its fisher consistency. When $\tau=K$, it is actually 0-1 loss, which has the fewest conditions on treatment interactions in order to guarantee its fisher consistency. With the increasing of $\tau$, the loss function can accommodate more and more interactions, but may lead to more and more parameters to be estimated. Thus, the existence of $\tau$ allows us to choose loss function adaptively to the data.
The second possible extension focuses on the dynamic assignment of multiple treatments. Combination therapies considered in this paper do not involve treatment transition problem. For the real data analysis and simulation, all the treatment are assumed to be applied to the patient at the same time. However, this is not always true in real life. Patients typically change from one treatment to another. Thus, instead of deciding a static treatment assignment, a dynamic treatment assignment with transition scheduling is a more realistic and reasonable solution to precision medicine. During this process, multiple outcomes may also be involved in the analysis. The process of single outcome over time may also play an important role in this extension.
The third extension focuses on how to combine our proposed technique with methods in \cite{Zhang2012a} and \cite{Zhang2012b} such that we can directly estimate the contrasts of treatment effects. In this case, it is critical to model the potential outcomes and propensity scores in an efficient way, especially for combination therapies. In general, our work in this paper is the keystone to all these potentials.
One of the limitations in our proposed method is that our method may fail under large interactions between treatments. When the interactions between treatments share the same direction of treatment recommendations, our method still holds, even the condition in \ref{thm2} fails. For example, in the simulation with model mis-specification, if $\gamma\leq 0$, it can be shown that our method is still consistent but the condition in \ref{thm2} fails. When the interactions between treatments are large and have the opposite direction of treatment recommendations, for example, it is extremely harmful to take two good medications simultaneously. Each combination therapy should be considered as a totally new treatment whose effect is irrelevant to the treatment effects when taking medications separately. In this case, treatment recommendation should be considered as a multi-class classification problem rather than a multi-label classification problem, and no information can be borrowed to improve efficiency.
\section*{Acknowledgments}
This work was one of research topics in Eli Lilly and Company summer intern program. All supports were provided by Eli Lilly and Company. Thank Yebin Tao for suggestions on this work.
|
{
"timestamp": "2018-04-17T02:11:15",
"yymm": "1804",
"arxiv_id": "1804.05378",
"language": "en",
"url": "https://arxiv.org/abs/1804.05378"
}
|
"\\section{Introduction}\n\\label{sec:introduction}\n\nTemporal networks can be seen as an extension(...TRUNCATED)
| {"timestamp":"2018-04-17T02:06:25","yymm":"1804","arxiv_id":"1804.05219","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction} \\label{intro}\nAs the demand for localization services increases, indoor l(...TRUNCATED)
| {"timestamp":"2019-10-14T02:05:15","yymm":"1804","arxiv_id":"1804.05347","language":"en","url":"http(...TRUNCATED)
|
"\n\\section{Detailed Results for individual pipelines of BFSS}\n\\label{sec:appendix}\n\nAs mention(...TRUNCATED)
| {"timestamp":"2018-05-21T02:08:01","yymm":"1804","arxiv_id":"1804.05507","language":"en","url":"http(...TRUNCATED)
|
"\n\\section{Introduction}\n\\label{sec:intro} \n\nThe field of similarity computation is more than (...TRUNCATED)
| {"timestamp":"2018-04-17T02:11:52","yymm":"1804","arxiv_id":"1804.05420","language":"en","url":"http(...TRUNCATED)
|
"\\section*{ACKNOWLEDGMENT}\n{\\footnotesize\nThis work has been partly funded by the Intelligence (...TRUNCATED)
| {"timestamp":"2018-04-17T02:08:39","yymm":"1804","arxiv_id":"1804.05276","language":"en","url":"http(...TRUNCATED)
|
"\\section{Introduction}\nBicycle Sharing Schemes (BSS) have become increasingly vital elements of u(...TRUNCATED)
| {"timestamp":"2018-08-20T02:01:24","yymm":"1804","arxiv_id":"1804.05584","language":"en","url":"http(...TRUNCATED)
|
End of preview. Expand
in Data Studio
No dataset card yet
- Downloads last month
- 8