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Tidal turbine array optimisation using the adjoint
approach
S.W. Funkea,b,, P.E. Farrella,c , M.D. Piggotta,b
arXiv:1304.1768v2 [math.OC] 12 Oct 2013
a Applied
Modelling and Computation Group, Department of Earth Science and
Engineering, Imperial College London, London, UK
b Grantham Institute for Climate Change, Imperial College London, London, UK
c Center for Biomedical Computing, Simula Research Laboratory, Oslo, Norway
Abstract
Oceanic tides have the potential to yield a vast amount of renewable energy.
Tidal stream generators are one of the key technologies for extracting and harnessing this potential. In order to extract an economically useful amount of
power, hundreds of tidal turbines must typically be deployed in an array. This
naturally leads to the question of how these turbines should be configured to
extract the maximum possible power: the positioning and the individual tuning
of the turbines could significantly influence the extracted power, and hence is
of major economic interest. However, manual optimisation is difficult due to
legal site constraints, nonlinear interactions of the turbine wakes, and the cubic
dependence of the power on the flow speed. The novel contribution of this paper
is the formulation of this problem as an optimisation problem constrained by a
physical model, which is then solved using an efficient gradient-based optimisation algorithm. In each optimisation iteration, a two-dimensional finite element
shallow water model predicts the flow and the performance of the current array
configuration. The gradient of the power extracted with respect to the turbine positions and their tuning parameters is then computed in a fraction of
the time taken for a flow solution by solving the associated adjoint equations.
These equations propagate causality backwards through the computation, from
the power extracted back to the turbine positions and the tuning parameters.
This yields the gradient at a cost almost independent of the number of turbines, which is crucial for any practical application. The utility of the approach
is demonstrated by optimising turbine arrays in four idealised scenarios and a
more realistic case with up to 256 turbines in the Inner Sound of the Pentland
Firth, Scotland.
Keywords: marine renewable energy, tidal turbines, gradient-based
optimisation, adjoint method, shallow water equations, array layout
Email address: s.funke09@imperial.ac.uk (S.W. Funke)
Preprint submitted to Renewable Energy
October 15, 2013
1. Introduction
With the increasing cost of energy, tidal turbines are becoming a competitive
and promising option for renewable electricity generation. A key advantage of
tidal energy is that the power extracted is predictable in advance, which is
highly attractive for grid management. In order to amortise the fixed costs of
installation and grid connection, arrays consisting of hundreds of tidal turbines
must typically be deployed at a particular site. This raises the question of
where to place the turbines within the site and how to tune them individually
in order to maximise the power output; finding the optimal configuration is
of huge importance as it could substantially change the energy captured and
possibly determine whether the project is economically viable. However, the
determination of the optimal configuration is difficult because of the complex
flow interactions between turbines and the fact that the power output depends
sensitively on the flow velocity at the turbine positions.
This problem has heretofore been addressed in two different ways. One approach is to simplify the tidal flow model such that the solutions are either
available as explicit analytical expressions, or are extremely fast to compute.
This means that the optimum can be analytically derived, or that the whole parameter space of possible configurations can be rapidly explored. For example,
Bryden and Couch (2007) and Garrett and Cummins (2008) optimised simplified models to derive an estimate for the maximum energy that can be extracted
from a tidal basin. Vennell (2010, 2011) used simple one-dimensional models to
demonstrate the importance of tuning each turbine individually to account for
the channel geometry, turbine positions, and the tidal forcing. Thus, optimisation of farms is a crucial step needed to achieve their full potential. However,
Vennell (2012b) observes that this optimisation requires many model runs (if
performed naively), thus making it computationally infeasible to use expensive,
physically-accurate flow models for this task. While this approach can provide a
coarse estimate for the power potential of a site, these simplified models cannot
accurately capture the complex nonlinear flow interactions between turbines.
The second approach is to use more complex flow models to accurately predict the tidal flow, the turbine wakes, and the resulting power output. These
models are usually formulated as numerical solutions to partial differential equations (PDEs). The computational expense of these models prohibits the exploration of the whole parameter space (Thomson et al., 2011). Consequently,
typically only a handful of manually identified turbine configurations are investigated in a given scenario (Adams et al., 2011). Divett et al. (2013) compared
the power output of four different layouts in a rectangular channel by solving
the two-dimensional nonlinear shallow water equations and was able to improve
the power outcome by over 50% compared to a regular layout. Lee et al. (2010)
used a three-dimensional model to investigate how the distance between adjacent rows in a regular array layout impacts the turbine efficiency and showed
an efficiency decay for distances of less than three times the turbine diameter. While these studies show the potential of improving the performance by
changing the turbine positions, such manual optimisation guided by intuition
2
and experience becomes difficult in a realistic domain with complex bottom
bathymetry, flow dynamics and hundreds of turbines.
In this paper, we present a novel technique for maximising the power extraction of array configurations that combines the physical fidelity of PDE-based
flow models with advanced automated optimisation techniques. This approach
allows the identification of optimal solutions in a computationally feasible number of iterations, circumventing the computational limitations noted in Vennell
(2012b). The turbine configuration problem is formulated as a PDE-constrained
optimisation problem, which is a major topic of research in applied mathematics
(Gunzburger, 2003; Hinze et al., 2009). The resulting maximisation problem is
solved using a gradient-based optimisation algorithm that takes orders of magnitude fewer iterations than genetic algorithms or simulated annealing approaches
(see e.g. Bilbao and Alba (2009)). In this paper, the power extracted by an array configuration is predicted using a two-dimensional nonlinear shallow water
model, which captures the interactions between the geometry, the turbines, and
the flow. The gradient of the power is efficiently computed using the adjoint
technique of variational calculus, which solves an auxiliary system that propagates causality backwards through the physical system. This yields the gradient
at a cost almost independent of the number of turbines to be optimised, which is
crucial for the method to be applied to large arrays. This gradient is used by the
optimisation algorithm to automatically reposition the turbines and to adjust
their tuning parameters. The flow solution is re-evaluated, and the algorithm
iterated until an optimum is found.
This approach has several key advantages. Firstly, it closes the optimisation
loop, by accounting for the effects of the turbines on the flow field itself. This
is necessary to find the actual optimum of the nonlinear optimisation problem.
Secondly, unlike gradient-free methods, the approach requires a relatively small
number of model evaluations and scales to large numbers of turbines, which is
necessary for the optimisation of industrial arrays. For example, in section 6,
an array of 256 turbines is optimised in a realistic domain at an approximate
cost of 200 flow solutions. Thirdly, the optimisation algorithm can incorporate complex constraints such as minimum separation distances, bathymetry
gradient constraints, and legal site restrictions. Finally, the same mathematical
framework extends naturally to more realistic flow models such as the Reynoldsaveraged Navier-Stokes equations, and to other functionals such as profit or
environmental impact.
The approach is implemented in an open-source software framework called
OpenTidalFarm; all code and examples from this paper are available at
http://opentidalfarm.org.
1.1. Optimisation algorithms
Optimisation algorithms can be divided into two categories: gradient-free
and gradient-based algorithms. Gradient-free optimisation algorithms use the
functional of interest (in this case, power extracted by the array) as a black box.
They proceed by evaluating the functional at many points in parameter space
and use these values to decide which areas merit further exploration. While
3
these methods tend to be robust and can, under certain smoothness conditions,
provably find globally optimal solutions (Rudolph, 1996), they typically require
a very large number of functional evaluations that scales linearly or superlinearly
with the number of parameters to be optimised. For example, Bilbao and Alba
(2009) used a genetic algorithm that mimics the process of natural evolution to
optimise the location of 8 wind turbines. The algorithm was able to improve
the power output by about 70% compared to the initial layout after 17, 300
functional evaluations. This large number of evaluations clearly introduces a
practical upper limit for the number of turbines that can be optimised. This
difficulty is compounded if a more realistic (and hence more expensive) model
is used.
By contrast, gradient-based optimisation algorithms use additional information to update the position in parameter space at each iteration: the first or
higher derivatives of the functional of interest with respect to the parameters.
Depending on the problem, this can lead to a significant reduction in the number
of iterations required compared to gradient-free algorithms, making these the
only feasible choice for large scale optimisation problems (Gunzburger, 2003).
One caveat of applying gradient-based optimisation algorithms is that they find
only local optima. This issue can be circumvented by using hybrid approaches
(Huang, 2009). The main difficulty of applying gradient-based methods is that
the implementation of the gradient computation can be difficult for complex
models, as it involves differentiating through the solution of a partial differential equation.
One way to obtain the derivative information is to approximate the gradient
using finite differences. However, a major disadvantage of this approach is that
a single gradient evaluation requires a large number of functional evaluations
that scales linearly with the number of optimisation parameters. This sets a
practical upper bound on the number of turbines to be optimised, and discards
the main advantage of gradient-based optimisation algorithms. Alternatively,
the tangent linearisation of the model (i.e. the derivative of the model evaluated
at a particular solution) can efficiently compute the derivative of all outputs
with respect to a single input, while the adjoint linearisation can efficiently
compute the derivative of a single output with respect to all inputs (Griewank
and Walther, 2008). For the turbine optimisation problem, we wish to maximise
a single output (the power extracted) with respect to many input parameters
(the positions and tuning parameters of the turbines); this means that the
adjoint approach is the natural choice, as the required gradient information can
be computed in a number of equation solves that is independent of the number
of turbines.
The development of adjoint models is generally considered as very complicated (Giles and Pierce, 2000; Naumann, 2012). However, this problem has
been solved in recent work for the case where the forward model is discretised
using finite elements, in the high-level FEniCS framework (Farrell et al., 2013).
This allows for the extremely rapid development of optimally efficient adjoint
models, which significantly reduces the development effort required to imple-
4
ment gradient-based optimisation algorithms for PDE-constrained optimisation
problems (Funke and Farrell, 2013).
To the best of our knowledge, this paper presents the first application of the
adjoint method to the optimisation of turbine arrays. While the examples are
shown in the marine context, it is expected that the presented techniques can
also be applied to the optimisation of wind farms. As the wind turbine layout
problem is both closely related and better studied, we next review techniques
proposed for its solution.
1.2. Wind farm optimisation
Layout optimisation for wind farms has been addressed in numerous studies,
most of which are based on gradient-free optimisation algorithms. In particular,
evolutionary methods (Bäck, 1996) are known to yield good results (SalcedoSanz et al. (2011) and the references therein). These algorithms mimic the
process of natural evolution by considering a population of candidate solutions
on which it executes an evolutionary process to find the “fittest” solution.
A related method is particle swarm optimisation (Kennedy and Eberhart,
1995), which considers a population of candidate solutions called particles, that
move through the parameter space and influence each other to drive the swarm
to the best solution. Wan et al. (2010) applied particle swarm optimisation on an
analytical wake model to optimise the location of 39 wind turbines and showed
that this approach can yield better optimal solutions than genetic algorithms.
Simulated annealing algorithms (Laarhoven and Aarts, 1987) are probabilistic
optimisation methods that exploit an analogy between the way in which a metal
heats and cools into a minimum energy crystalline structure (the annealing
process) and the search for a minimum in a more abstract system. Bilbao and
Alba (2009) used an analytical wake model to compare a simulated annealing
algorithm with a genetic optimisation algorithm. In a scenario with 47 turbines,
the number of model evaluations was reduced from 1, 036, 200 with the genetic
algorithm to 61, 802 with simulated annealing. However, such a large number
of evaluations would be infeasible for a more complex PDE-based model.
Few publications solve the layout problem with gradient-based optimisation
algorithms. Lackner and Elkinton (2007) optimised the position of two wind
turbines by applying a gradient-based optimisation algorithm to a simplified
energy production model with an analytical expression. Huang (2009) combined
a genetic algorithm with steepest ascent to accelerate convergence to an optimal
solution. With this additional derivative information, the number of iterations
was reduced by approximately an order of magnitude to less than 300 iterations,
for a similar power extraction. Finally, Fagerfjäll (2010) showed how mixed
integer linear programming techniques can be used to optimise for both the
number and position of turbines.
All of these publications use very simplified flow models for which the gradient is either available analytically or can be easily approximated. Furthermore,
none of the reviewed papers use gradient-based methods with the adjoint technique to find an optimal configuration in a physically realistic model.
5
While this prior research on wind farm optimisation is relevant, there are
key differences between wind and tidal turbine arrays. Firstly, the flow in a tidal
channel is dominantly driven by the predictable tidal forcing, while wind flow
modelling is inherently stochastic and needs to include the temporal uncertainty
in the magnitude and direction of the wind forcing. Secondly, the ratio of turbine
height to free-surface elevation is significantly different: while the rotor diameter
of a wind turbine is small compared to the height of the atmosphere, tidal
turbines typically have diameters of around 20 m and are deployed in water
depths of approximately 50 m or less. This leads to little undisturbed flow
above the turbine which could contribute to wake recovery and thus potentially
increases the length of the wake compared to wind turbines (Bryden et al.,
2004; Divett et al., 2013). Finally, if a tidal farm occupies a large fraction of
the channel’s width and depth then the presence of turbines significantly affects
the flow velocity upstream of the farm (Garrett and Cummins, 2005; Vennell,
2010). In wind farms, this is not the case (Vennell, 2012a). This interaction
adds an additional complication to the optimisation of tidal farms.
The remaining sections are organised as follows. In section 2, we formulate
the turbine configuration problem as an optimisation problem constrained by
the shallow water equations. Section 3 discusses the discretisation and implementation, which is then verified in section 4. In section 5, we demonstrate
the capabilities of the proposed approach on four idealised scenarios with 32
turbines. Section 6 presents an application of this approach where the positions
of up to 256 turbines are optimised in a geometry motivated by the Inner Sound
of the Pentland Firth, Scotland. Finally, we make some concluding remarks in
section 7.
2. Problem formulation
In this section we formulate the optimal configuration of turbines in an array
as a PDE-constrained optimisation problem in the following abstract form:
max J(z, m)
z,m
subject to F (z, m) = 0,
(1)
bl ≤ m ≤ bu ,
g(m) ≤ 0,
where J(z, m) ∈ R is the functional of interest, m are the design parameters,
F (z, m) is a PDE operator parameterised by m with solution z, bl and bu are
lower and upper bound constraints for the design parameters, and g(m) enforces
additional restrictions on the design parameters.
In this work z = (u, η) is the solution (horizontal velocity, free-surface displacement) of the shallow water equations written in the form F (z, m) = 0,
and m contains the configuration (position and/or tuning parameters) of the
turbines. The bounds bl and bu are used to enforce that the turbines remain in
a prescribed area (here assumed a rectangle for simplicity), while g(m) is used
6
to enforce a minimum distance between any two turbines (e.g. as a multiple of
the turbine diameter).
The optimisation problem (1) can be reduced by using the fact that the
constraint equation F (z, m) = 0 implicitly maps any choice of m to a unique
solution z. Hence, the solution z can be considered as an implicit function of
the optimisation parameters: z ≡ z(m) . By substituting this operator into the
functional of interest, we obtain the reduced optimisation problem:
max J(z(m), m)
m
(2)
subject to bl ≤ m ≤ bu ,
g(m) ≤ 0.
While it is possible to solve the optimisation problem in unreduced form
(1), we choose to solve the reduced form, as it is usually preferable for timedependent governing equations (Long et al., 2012).
2.1. The design parameters
For the turbine optimisation problem considered here, the design parameter
m is a vector containing the positions, and optionally the tuning parameters, of
the turbines.
If the turbine tuning parameters are fixed, then the design parameters contain only the (x, y) positions of the N turbines encoded in the form:
m=
x1
y1
x2
y2
...
xN
yN .
If additionally the turbines are to be individually tuned, then the vector m is
extended to contain the friction coefficient Ki of each turbine:
m=
K1
K2
...
KN
x1
y1
x2
y2
...
xN
yN .
This could be further generalised to account for any number of turbine parameters.
2.2. The PDE constraint
The constraint equation F (z, m) = 0 enforces the laws of physics in the optimisation problem (1). For gradient-based optimisation, the constraint equation
must fulfill some properties. Firstly, for every m it must yield a unique solution z so that z can be written as an implicit function of m. Secondly, F
must be differentiable and its derivative with respect to z must be continuously
invertible.
In this work, the physical laws are modelled by the nonlinear shallow water
equations. Let Ω be the domain of interest and let (0, T ) be the simulation
period. Then the equations read:
κ
∂u
cb + ct (m)
+ u · ∇u − ν∇2 u + g∇η +
kuku = 0,
∂t
H
∂η
κ
+ ∇ · (Hu) = 0,
∂t
7
(3)
where the unknowns u : Ω × (0, T ] → R2 and η : Ω × (0, T ] → R are the depthaveraged velocity and the free-surface displacement, respectively, H : Ω → R
is the water depth at rest, g ∈ R is the acceleration due to gravity, ν ∈ R is
the viscosity coefficient, and cb : Ω → R and ct (m) : Ω → R represent the
quadratic bottom friction and the turbine parameterisation, respectively. The
parameter κ ∈ {0, 1} specifies if the stationary (κ = 0) or the non-stationary
problem (κ = 1) is considered; in the stationary case the time-dependency of
the variable definitions above can be neglected.
The boundary conditions are as follows: on the domain inflow boundary
∂Ωin a Dirichlet boundary condition is applied to the velocity. On the outflow
boundary ∂Ωout the free-surface displacement is set to zero. Elsewhere, a noslip or a no-normal flow boundary condition with a free-slip condition for the
tangential components is imposed. Note that these boundary conditions imply
that the effect of the array on the free-stream flow is negligible, which is not
the case for large farms (Vennell, 2010): for large arrays, it would be more
appropriate to force the model with tidal boundary conditions on the free surface
displacement or to ensure that the size of the domain includes the far-field region
around the farm.
2.3. The turbine parameterisation
A turbine is modelled here via an increased bottom friction over a small
area representative of an individual turbine (Divett et al., 2013). The sum of
the bottom friction associated with all turbines is denoted as ct (m) in equation (3). This individual turbine approach is in contrast to previous work (e.g.
Garrett and Cummins (2005); Draper et al. (2010)), where uniformly increased
bottom drag over the array area was used to parameterise groups of turbines.
In Divett et al. (2013), the authors set the friction to a constant value at the
turbine locations and zero everywhere else. This constant parameterisation is
problematic in the context of gradient-based optimisation because the friction
becomes a non-differentiable function of the turbine position. For this reason,
the turbine parameterisation used in this work smoothly increases the friction
value at each turbine position. The associated friction function is constructed
from bump functions, i.e. smooth functions with finite support. A bump function in one dimension is:
(
x−p 2
e1−1/(1−k r k ) for k x−p
r k < 1,
(4)
ψp,r (x) ≡
0
otherwise,
where the two parameters p and r are the centre and the support radius of
the bump function, respectively. A two-dimensional bump function is obtained
by multiplying equation (4) in both independent dimensions. With that, the
friction function of a single turbine parameterised by a friction coefficient Ki
centred at a point (xi , yi ) is given by:
Ci (m)(x, y) ≡ Ki ψxi ,r (x)ψyi ,r (y).
8
(5)
0.8
0.6
0.4
0.2
0.0
20
0
15
10
5
10
5
15
20 0
Figure 1: The turbine is parameterised by a smoothly increasing friction coefficient towards the turbine centre, given by the bump function (4) multiplied in
the x and y-direction.
A plot of the resulting friction for Ki = 1, xi = 10, yi = 10 and r = 10 is shown
in figure 1.
The turbine friction function ct in the governing PDE (3) is defined to be
the sum of the friction functions (5) for all N turbines:
ct (m) ≡
N
X
Ci (m),
(6)
i=1
where a single value for r is used based on the assumption that the deployed
turbines are of equal size.
Note that turbine properties can be calibrated by modifying the friction parameter K in equation (5). For example, the amount of energy that is extracted
from an individual turbine (e.g. due to different pitch settings of the turbine
blades) can be controlled. As a further extension this could be used to handle
cut in/out velocities in which the turbines are operational, but this is not considered in this work. Finally, other more sophisticated turbine parameterisations
such as extensions of actuator disc theory could be employed (Roc et al., 2013).
2.4. The functional of interest
The functional of interest J in equation (1) defines the value of interest that
is to be maximised. For gradient-based optimisation, J must be differentiable.
A natural choice for the functional of interest is the time-averaged power extracted due to the increased friction by the turbines (Vennell, 2012a; Sutherland
et al., 2007). In the non-stationary case (κ = 1) this is expressed as:
1
J(u, m) =
T
Z TZ
0
ρct (m)kuk3 dx dt,
Ω
9
(7)
where ρ is the fluid density. In the stationary case (κ = 0) the functional is
defined to be the power extracted from the increased friction:
Z
J(u, m) =
ρct (m)kuk3 dx.
(8)
Ω
Note that this value represents kinetic power extraction rather than electrical
power generation, since it does not incorporate losses due to the turbine support
structures and the conversion to electricity.
More advanced functional choices could instead maximise the profit of the
turbine farm, by including installation and service costs depending on turbine
size and the deployment location (Adams et al., 2011). Another alternative
would be to incorporate potential environmental impacts. These are not considered in this study.
2.5. Box and inequality constraints
The box and inequality constraints in the generic optimisation problem (1)
are used to define the feasible values for the optimisation variables. In the
context of the turbine layout problem, a typical condition is to restrict the area
in which the turbines may be placed to the development site. The numerical
examples in this work have rectangular shaped deployment sites and therefore
box constraints are sufficient to enforce this restriction. For more general site
shapes appropriate inequality constraints can be used instead.
Another common condition is to ensure that individual turbines do not overlap. This is implemented by enforcing a minimum distance dmin between any
two turbines:
kpi − pj k2 ≥ d2min
∀ i, j : 1 ≤ i < j ≤ N.
(9)
In order for the optimisation to be well-posed, the constraints must satisfy a
constraint qualification (Hinze et al., 2009). The box constraints and inequality
constraint (9) are concave functions, and hence satisfy the Concave Constraint
Qualification (CCQ) (Carter, 2001, theorem 5.4).
More advanced constraints could for example enforce a minimum or maximum deployment depth or limit the maximum local bathymetry steepness where
turbines may be installed, but these are not investigated in this work.
3. Numerical setup
3.1. Optimisation algorithm
A typical gradient-based optimisation algorithm implements the following
iteration:
• Choose an initial guess m0 for the design parameters.
• for i = 0, 1, ...
10
1. Solve the forward problem F (z i , mi ) = 0 for z i and evaluate the
functional of interest J(z i , mi ).
2. Compute the functional gradient dJ/dm(z i , mi ).
3. Stop if the termination criteria are fulfilled.
4. Find improved design parameters mi+1 using the results of 1 and 2.
Optimisation algorithms differ mainly in their implementation of step 4. The
main task is to identify an improved parameter choice so that the algorithm converges quickly to the optimal solution while satisfying the imposed constraints.
In this paper we solve the turbine configuration problem using sequential
quadratic programming (SQP), which is considered to be one of the most efficient optimisation algorithms (Boggs and Tolle, 1995). The implementation
used here is the SLSQP algorithm available through the SciPy optimisation
package (Jones et al., 2001) and is described in detail in Kraft (1988). The
optimisation problem was formulated and solved with the PDE-constrained optimisation framework described in Funke and Farrell (2013).
The SQP implementation used is not scale-invariant, i.e. scaling the functional of interest can impact the convergence of the algorithm (even though
it does not change the optimal configuration). Preliminary numerical investigation found that such rescaling was necessary to achieve fast convergence.
Therefore, for each numerical experiment presented here the problem was internally rescaled such that the maximum absolute value of the initial functional
gradient with respect to the turbine positions was ten times the turbine radius1 .
3.2. The functional gradient computation with the adjoint approach
The second step of the optimisation algorithm requires the computation of
the functional derivative with respect to the optimisation parameters dJ/dm.
Its efficient computation is achieved using the adjoint approach. For reasons
of brevity, only the key equations are mentioned; for fuller explanations, see
Gunzburger (2003); Giles and Pierce (2000); Farrell et al. (2013).
Let |J| be the number of functional outputs (in this case 1, the power), let
|m| be the number of parameters, and let |z| be the number of degrees of freedom
of the discretised state vector. The adjoint approach computes the gradient in
two steps. Firstly, the adjoint equation is solved to obtain the adjoint solution
λ (with the matrix dimensions indicated below the braces):
∂F ∗
∂J ∗
λ =
,
∂z
|∂z
{z } |{z}
|{z}
|z|×|z| |z|×|J|
|z|×|J|
where ∗ denotes the Hermitian transpose. The adjoint solution plays the role
of the Lagrange multiplier enforcing the PDE constraint in the Lagrangian as1 More precisely, we compare the Riesz representer of the functional gradient with respect
to turbine positions, which has the same units of length as the turbine radius.
11
sociated with the optimisation problem (1). Secondly, the functional gradient
is obtained by:
∂F
dJ
∂J
= −λ∗
+
.
(10)
dm
∂m
∂m
|{z} |{z}
|{z}
|{z}
|J|×|m|
|J|×|z| |z|×|m|
|J|×|m|
The functional gradient is the key piece of information that makes the optimisation of large numbers of turbines tractable.
For the optimisation problem considered here, the adjoint shallow water
equations are:
−κ
∂λu
+ (∇u)∗ λu − (∇ · u) λu − u · ∇λu − ν∇2 λu
∂t
u · λu
∂J ∗
cb + ct (m)
kukλu +
u =
,
−H∇λη +
H
kuk
∂u
∂λη
− g∇ · λu = 0.
−κ
∂t
(11)
where λ ≡ (λu , λη ) is the vector containing the unknown adjoint velocity and
adjoint free-surface displacement, respectively. The derivation of these equations can be found in Funke (2012, appendix C). The non-stationary adjoint
equations (κ = 1) have a final-time condition for the adjoint velocity and freesurface displacement; this condition is homogeneous, as the functional J has no
term evaluated at the end of time. The adjoint equations are solved from the
final time to the initial time, propagating information backwards in time. The
boundary conditions for the adjoint equations are the homogeneous versions of
the boundary conditions of the forward equations, again because the functional
has no boundary integral terms. The functional derivative (∂J/∂u)∗ appears
as the source term for the adjoint velocity equation and is easy to evaluate as
the functional is available as an analytical expression. Note that the adjoint
equations are linear while the forward equations are nonlinear, and therefore
solving the adjoint equations is typically much cheaper. If the time-dependent
equations are solved, the entire forward trajectory is required to assemble the
adjoint equations; if the forward trajectory is too large to store at once, then
a checkpointing algorithm must be used (Griewank and Walther, 2000; Farrell
et al., 2013).
In this work, rather than deriving, discretising and implementing the adjoint equation (11) by hand, we apply the high-level algorithmic differentiation
approach described in Farrell et al. (2013). This efficiently and automatically
derives and implements the discrete adjoint model from the implementation of
the forward model (12), without user intervention. This significantly reduces
the effort and expertise required to implement adjoints of complex nonlinear
forward models.
The second step of the adjoint approach (equation (10)) evaluates the functional gradient using the adjoint solution λ. This step only requires the computation of a matrix-vector product and consequently its computational cost
is negligible. This allows for the computation of the gradient of the functional
12
with only the solution of one adjoint system. While the cost of the matrixvector product technically depends on the number of turbines, in practice the
cost of the adjoint PDE solution dominates, and so the adjoint technique yields
the gradient at a cost independent of the number of turbines. This is a key
property if many turbines are to be optimised.
3.3. Discretisation
The governing PDEs (3) are discretised with the finite element method. The
weak form is derived by multiplying the equations with test functions (Ψ, Φ)
from suitable function spaces, integrating over the computational domain and
applying integration by parts to selected terms. The weak form of equations (3)
is: find (u, η) such that ∀ (Ψ, Φ):
∂u
+ hu · ∇u, ΨiΩ + ν h∇u, ∇ΨiΩ
,Ψ
κ
∂t
Ω
cb + ct (m)
+g h∇η, ΨiΩ +
kuku, Ψ
= 0,
(12)
H
Ω
∂η
− hHu, ∇ΦiΩ + hHu · n, Φi∂Ωin ∪∂Ωout = 0,
,Φ
κ
∂t
Ω
where h·, ·iΩ denotes the L2 (Ω) inner product. The no-normal flow/free-slip
conditions on ∂Ω \ (∂Ωin ∪ ∂Ωout ) are weakly imposed by excluding the associated surface integrals. The Dirichlet boundary conditions are strongly imposed
by restricting the function spaces to functions that yield the correct boundary
values.
The discretised problem is obtained by choosing discrete function spaces in
the weak formulation (12). In this work, these are constructed from a suitable
triangulation of the computational domain Ω using the Taylor-Hood finite element pair which uses piecewise quadratic functions on each triangle for the
velocity and piecewise linear functions on each triangle for the free-surface displacement (Taylor and Hood, 1973).
If the non-stationary problem is considered (κ = 1), then the spatially discretised equations (12) must also be discretised in time. In this paper, the
time discretisation was performed using the implicit Euler method, due to its
unconditional stability and simplicity.
Finally, the integral evaluation of the functionals of interest (7) and (8) used
the same quadrature rules as used in the underlying finite element formulation
of the problem.
4. Verification
4.1. Verification of the forward model
The shallow water model implementation was verified by order-of-convergence
analysis. The analytical solution is constructed using the method of manufactured solutions (Salari and Knupp, 2000; Roache, 2002): a desired analytical
13
103
101
E error
E error
102
102
100
10−1 1
10
Spatial error
Second order
102
Element size [m]
Temporal error
First order
101 −1
10
103
(a) The spatial order of convergence
100
Time step [s]
101
(b) The temporal order of convergence
Figure 2: The expected and achieved orders of convergence for the forward
model.
solution is chosen and then substituted into the governing PDE, which yields a
non-zero remainder. By adding this remainder as a source term to the governing
equations, the selected solution becomes an analytical solution to the modified
PDE.
For the following tests, the analytical solution consists of sinusoidal functions
for both the velocity and the free-surface displacement:
p
√
η0 gH −1 cos kx − gHkt
uexact (x, y, t) =
,
0
(13)
p
ηexact (x, y, t) = η0 cos kx − gHkt ,
with k = π/640 m−1 , η0 = 2 m, H = 50 m,√ν = 3 m2 s−1 , ct = 0, cb = 0.0025,
g = 9.81 ms−2 and a final time of T = π/( gHk) ≈ 28.9 s which corresponds
to half a wave cycle. The computational domain Ω is defined to be a rectangle
of size 640 m × 320 m. Following the method of manufactured solutions, the
functions (13) are substituted into the shallow water equations (3) and the nonzero remainders are added as source terms. This ensures that (13) is a solution
of this modified system.
To determine the spatial order of convergence of the model, the time step
was fixed to a small value of ∆t = T /16800 s, to ensure that the numerical error
of the spatial discretisation dominates the overall discretisation error. Then, the
forward model with the added source term was run on four uniform, increasingly
fine meshes and the error in the numerical solution (u, η) measured as:
21
2
2
E = ||u − uexact ||Ω×(0,T ) + ||η − ηexact ||Ω×(0,T ) .
The resulting errors plotted in figure 2a show the second-order convergence that
is expected from the Taylor-Hood finite element pair.
For determining the temporal order of convergence, a mesh with 2.5 m element size in the x-direction and 160 m element size in the y-direction was
generated (the analytical solution does not vary in the y-direction and hence
14
a relatively large mesh element size can be used). This fine mesh resolution
ensures that the numerical error of the temporal discretisation dominates the
overall discretisation error. The forward model was then run with a set of different time steps. The resulting errors plotted in figure 2b show the first-order
convergence expected for the implicit Euler time discretisation.
4.2. Verification of the gradient computation
The gradient computation was verified using the Taylor remainder converˆ
gence test. Let J(m)
≡ J(z(m), m). The first-order Taylor expansion states
that:
dJˆ
ˆ + δm) − J(m)
ˆ
J(m
−
(14)
δm = O(kδmk2 ).
dm
Examining the convergence order of the remainder term is a strong test that
the adjoint model and the gradient evaluation are implemented correctly: for
nonlinear functionals, the gradient computed using the discrete adjoint approach
is correct if and only if the Taylor remainder converges at second order.
Firstly, a simple configuration with a single turbine was set up with the turbine parameterisation and the functional of interest as described above. The
Taylor remainder convergence test in many random perturbation directions δm
yielded the expected second-order convergence (not shown). Secondly, the Taylor remainder convergence test was applied on several of the numerical examples
presented in the following section (not shown). All yielded second-order convergence, giving confidence that the adjoint model and the gradient computation
are implemented correctly.
5. Examples
The following numerical examples solve the optimal layout problem in four
idealised scenarios motivated by Draper et al. (2010) (figure 3). Additionally,
in scenario 3 we optimise for the positions and the tuning of the individual turbines. In all examples, 32 turbines are to be deployed in a rectangular turbine
site of size 320 m × 160 m. The idealised domains simplify the subsequent interpretation of the optimised configurations. Nevertheless, the domains are chosen
such that they resemble structures that can be found in practical deployment
sites. The main three objectives for the numerical examples are to investigate:
by how much can optimisation increase the energy extraction? Can the optimisation algorithm reliably improve the energy extraction for different scenarios?
How does the choice of the optimisation variables and the constraints impact
the resulting configuration?
The parameter choices for the experiments are listed in table 1. The stopping
accuracy of the SLSQP optimisation algorithm was set to 10−6 unless otherwise
stated. The stopping criteria demand that the solution is feasible (i.e. that
the L1 norm of the constraint residuals is less than the tolerance) and that the
solution is optimal (i.e. that the gradient in the search direction is also less
than the tolerance). This tolerance is extremely tight, as can be seen in the
15
(a) Scenario 1
(b) Scenario 2
(c) Scenario 3
(d) Scenario 4
Figure 3: The four turbine optimisation scenarios considered in the numerical
examples, motivated by Draper et al. (2010). The dashed lines mark the 320 m×
160 m sized sites where 32 turbines are to be deployed. In scenarios 1 and 3 a
constant inflow velocity is enforced, while the other scenarios are driven by a
sinusoidal inflow.
Parameter
Value
Water depth
Viscosity coefficient
Turbine friction coefficient
Acceleration due to gravity
Water density
Bottom friction coefficient
Turbine radii
H = 50 m
ν = 3 m2 s−1
K = 21
g = 9.81 ms−2
ρ = 1, 000 kgm−3
cb = 0.0025
r = 10 m
Table 1: The parameter values used in the experiments of section 5. The nondimensional bottom friction coefficient is a common value for coastal modelling
(Vennell, 2012a).
16
convergence plots, and could be weakened for efficiency reasons. Scenarios 1
and 3 were modelled using the stationary shallow water equations with the following boundary conditions. On the inflow boundary a constant inflow velocity
of 2 ms−1 was enforced and on the outflow boundary the free-surface displacement was set to zero. A no-normal flow boundary condition with a free-slip
condition for the tangential components was applied on the remaining boundaries. Scenarios 2 and 4 solved the non-stationary shallow water equations with
boundary conditions explained in the associated sections.
All examples used unstructured meshes with a uniform mesh element size of
h = 20 m outside the site area. Inside the site area, the mesh was structured
with an element size of h = 2 m. The higher resolution in the turbine site ensures
that each individual turbine is well resolved, independent of its location within
the site; this obviates the need for regridding when the turbines are moved.
Doubling the resolution for the problem considered in section 5.1 changed the
power extracted by less than 0.5%. It is therefore assumed that the problems
are sufficiently well resolved. The resulting meshes consisted of approximately
33, 000 triangles for scenario 1 and 2, 63, 000 triangles for scenario 3 and 45, 000
triangles for scenario 4. All meshes were generated using Gmsh (Geuzaine and
Remacle, 2009).
In all numerical experiments, the optimisation algorithm was initialised with
the 32 turbines deployed in a regular 8 × 4 grid and with box constraints for the
turbine positions to ensure that the turbines remain inside the site areas.
5.1. A single turbine
As a preliminary test, a single turbine was deployed in the setup of scenario
1. The turbine was placed 640/3 m×320/2 m away from the bottom left corner,
as shown in figure 4a.
This setup was used to study the dependency of the power extraction J
on the friction coefficient K that occurs in the turbine parameterisation (5).
Figure 4c shows the power extraction for a range of K values. The graph shows
a defined single peak where the power extraction is maximised, and is similar
to previous studies (Garrett and Cummins, 2005; Vennell, 2011, 2012a). The
reason for this peak is that as K → 0, the turbine friction function ct approaches
0, which in turn results in no power extraction since the power function (8) is
multiplied by ct ; similarly, as K → ∞, the flow is deflected around the turbine
and results in the observed power drop. The power extraction peaks for K = 21,
which was used for all following numerical tests if not otherwise stated. With
this choice the single turbine extracted 3.2 MW from the flow.
5.2. Scenario 1
Firstly, the layout problem for scenario 1 (figure 3a) was solved without enforcing a minimum distance between turbines, i.e. the turbines can be placed
arbitrarily inside the site area and may even overlap. With that setup the optimisation algorithm terminated after 135 iterations (134 gradient evaluations,
231 functional evaluations). The results are shown in figure 5. Compared to the
17
Power output [MW]
(a) Turbine position
(b) Velocity magnitude
3
2
1
0
0
50
100
150
Friction coefficient K
200
(c) Dependency of the power extraction J on the friction coefficient K in
equation (5)
Figure 4: The results of deploying a single turbine in the domain of scenario 1.
initial regular layout (figure 5a) the optimisation algorithm was able to increase
the farm power extraction by 76% from 54.5 MW to 95.7 MW (figure 5c). The
optimised layout (figure 5b) has the turbines aligned in the shape of two As
with the open end facing the inflow. An intuitive interpretation of this layout
is that the water is funneled by the two sides of the As and then forced through
the dense turbine ‘wall’ at their closed ends. This interpretation is confirmed
by an increasing free-surface displacement (not shown) and velocity difference
along the sides of the As and the large jump along their closed end (figure 5d).
An additional experiment was performed where inequality constraints were
included to enforce a minimum distance of 30 m (3 turbine radii) between each
turbine. With this setup, the optimisation algorithm terminated after 54 iterations (53 gradient evaluations, 112 functional evaluations) and the farm power
extraction increased by 38% from 54.5 MW to 75.0 MW. The reduced optimised
power extraction compared to the previous setup is expected since the inequality constraints add further restrictions to the feasible turbine positioning. In
particular, the previous optimised turbine layout is not a feasible solution for
this setup.
The optimised alignment differs significantly from the previous one (figure
6b). The two main characteristic structures are a > shaped alignment close to
the inflow boundary and a wall of turbines near the outflow boundary. Furthermore, the turbines are staggered to avoid placing one turbine in the direct wake
of another turbine. Finally, the free-surface displacement (not shown) and the
velocity magnitude decrease more gradually towards the outflow compared to
the previous setup (figure 6d).
18
(a) Initial turbine positions
(b) Optimised turbine positions
Power [MW]
100
90
80
70
60
50
0
20
40 60 80 100 120 140
Optimisation iteration
(c) Optimisation convergence
(d) Velocity magnitude
Figure 5: Results of scenario 1 without minimum distance constraints, i.e. turbines may overlap.
Power [MW]
(a) Initial turbine positions
80
75
70
65
60
55
50
0
10
20
30
40
50
Optimisation iteration
(b) Optimised turbine positions
60
(c) Optimisation convergence
(d) Velocity magnitude
Figure 6: Results of scenario 1 with minimum distance constraints.
19
(a) Initial turbine positions
(b) Optimised turbine positions
22
20
18
16
14
12
10
(d) Velocity magnitude during the
flood tide
40
Power [MW]
Average power [MW]
(c) Velocity magnitude during the
ebb tide
0
20
40
60
80
Optimisation iteration
(e) Optimisation convergence
30
20
10
0
100
0
2
4
6
Time [min]
8
10
(f) Power extraction over time of the
optimised configuration
Figure 7: Results of the non-stationary scenario 2 without minimum distance
constraints.
5.3. Scenario 2
The layout problem for scenario 2 (figure 3b) was solved using the nonstationary shallow water equations.
The boundary conditions were as follows. On the top, bottom and right
boundaries a no-slip boundary condition was applied. On the left boundary a
Dirichlet boundary condition enforced a sinusoidal in-/outflow velocity:
−2 sin (2πt/P )
u(x, y, t) =
,
0
where P is the tidal period time. Due to the small basin size, a realistically
long tidal period would lead to an excessively large tidal range. Therefore, the
tidal period was defined to be P ≡ 10 minutes, which resulted in a tidal range
of ±12 m. The simulation time was set to one full tidal period with a time step
of ∆t = 12 s. No spin up phase was applied, as its effect is assumed to be small
due to the relatively short extent of the domain.
The optimisation was performed without enforcing a minimum distance between the turbines. After 97 optimisation iterations (96 gradient evaluations,
20
217 functional evaluations) the relative functional improvement in each iteration
dropped below the tolerance and the optimisation was terminated. The results
are shown in figure 7. The average power extracted during one tidal cycle increased by 96% from 10.5 MW to 20.6 MW (figure 7e). The optimised layout
(figure 7b) resembles the result of scenario 1 (figure 5b), with the difference that
the opening of the A structure faces the closed basin side.
5.4. Scenario 3
The domain of the third scenario is shown in figure 3c. First, only the
layout problem is solved. For this test, inequality constraints were applied to
enforce a minimum distance of 30 m between each turbine. The optimisation
algorithm terminated after 56 iterations (55 gradient evaluations, 73 functional
evaluations). The optimised farm layout extracts 40.6 MW, which corresponds
to an increase of 31% compared to the initial layout (30.9 MW) (figure 8c). The
optimised layout features a distinct ♦−shaped alignment with an opening on
the inflow facing side (figure 8b). Figure 8d shows the velocity magnitude and
suggests that this hole acts to trap and push the flow through the downstream
turbines similar to the previous examples.
Second, the optimisation parameters were extended to include the friction
coefficients K of each turbine in the parameterisation (5). Vennell (2010, 2011)
showed the necessity of varying the friction coefficients in order to achieve optimal farm performance. Each K coefficient was constrained such that 0 ≤ K ≤
21. With this setup, the optimisation algorithm terminated after 92 iterations
(88 gradient evaluations, 148 functional evaluations). The results are presented
in figure 9. Compared to the initial configuration, the farm power extraction
increased by 39% from 30.9 MW to 42.9 MW – the additional freedom of varying the K coefficients resulted in a higher optimised power extraction than the
previous setup.
The optimised turbine configuration is similar in shape to the previous solution but with a less distinct hole on the inflow facing side. Most notably, the
friction coefficients of most turbines are significantly reduced. Only the turbines
on the downflow edges of the ♦ take the maximum value, but nevertheless this
configuration extracts 6% more energy than the optimal solution of the previous setup where only the positions were optimised. This example shows that
optimising the friction coefficient (which can be viewed as reducing the size of
the turbine or controlling the blade pitch) can lead to a significant increase in
the power extraction of the farm.
5.5. Scenario 4
The final scenario (figure 3d) was solved with the non-stationary shallow
water equations. The simulation time consisted of one P = 12 h sinusoidal
period with a time step of ∆t = 864 s. No spin up phase was applied, as its
effect is assumed to be small due to the relatively short extent of the domain.
21
Power [MW]
(a) Initial turbine positions
42
40
38
36
34
32
30
0
10
20
30
40
50
Optimisation iteration
(b) Optimised turbine positions
60
(c) Optimisation convergence
(d) Velocity magnitude
Figure 8: Results of scenario 3 with minimum distance constraints.
Power [MW]
(a) Initial turbine positions and fric-(b) Optimised turbine positions and
tion coefficients
friction coefficients
44
42
40
38
36
34
32
30
0
20
40
60
80
Optimisation iteration
100
(c) Optimisation convergence
(d) Velocity magnitude
Figure 9: Results of scenario 3 with minimum distance constraints and optimising for both the turbine friction coefficients and the turbine positions.
22
(a) Initial turbine positions
(b) Optimised turbine positions
60
58
56
54
52
50
48
(d) Velocity magnitude at the
time when the input velocity
from the right reaches its maximum
Power [MW]
Average power [MW]
(c) Velocity magnitude at the
time when the input velocity
from the left reaches its maximum
0
10
20
30
40
50
Optimisation iteration
(e) Optimisation convergence
60
140
120
100
80
60
40
20
0
0
2
4
6
8
Time [h]
10
12
(f) Power extraction over time of the
optimised configuration
Figure 10: Results of the non-stationary scenario 4 with minimum distance
constraints.
Dirichlet boundary conditions on the left and right boundaries enforced the
following sinusoidal in-/outflow velocity:
−2 sin (2πt/P )
u(x, y, t) =
.
0
On the remaining boundaries a no-slip boundary condition was applied.
The setup optimised the turbine positions and applied the inequality constraints to enforce a minimum turbine distance of 30 m. The optimisation
algorithm terminated after 55 iterations (54 gradient evaluations, 75 functional
evaluations). The results are shown in figure 10.
The averaged power extracted during one cycle increased by 22% from
48.4 MW to 59.0 MW (figure 10e). Since the computational domain is sym23
metric and the simulation time covered one full period, the optimised layout is
expected to be symmetric in the x-direction. The numerical solution, shown in
figure 10b, indeed shows an almost symmetric result. The turbine alignment
consists of two distorted ∨ shapes whose open ends face the in/-outflow boundaries. Similar to the previous example, an interpretation of this alignment is to
divert the stream towards the corner of the ∨ where turbines can extract large
amounts of power. An additional row of turbines can be seen parallel to the
bottom of the domain. These turbines are positioned to capture energy from
the flow passing along the boundary.
6. Farm optimisation in the Inner Sound of the Pentland Firth
A key design feature of the framework is that it should scale to problems in
realistic oceanographic domains with large numbers of turbines in parallel. To
demonstrate this capability, the layout optimisation of a farm in a semi-idealised
geometry modelled on the Inner Sound of the Pentland Firth (figure 11a) was
conducted on the Stampede supercomputer. This site is one of the most promising locations in the UK, and is currently under development by MeyGen Ltd.
The computational domain is shown in figure 11b. The pink area marks the
turbine site location which roughly approximates the area used by the MeyGen
project. The discretised domain consists of a regular mesh in the turbine site
area with 2 m element size, and an unstructured mesh elsewhere with element
sizes ranging from 1.5 − 200 m. The mesh was generated using Gmsh (Geuzaine
and Remacle, 2009) using the GSHHS shoreline database (Wessel and Smith,
1996). The mesh consists of 1.25 × 106 elements, which induces a discretisation
with a total of 5.6 × 106 degrees of freedom with the Taylor-Hood finite element
pair. In this idealised problem, the bathymetry is assumed constant at H =
50 m. The addition of bathymetry is straightforward, and will be presented
in future work, but would obscure the physical interpretation of the results
presented here.
The farm optimisation was modelled during the flood tide using the stationary shallow water equations. The simulations were performed with the same
parameter settings as in the previous section (table 1), but with an increased
viscosity coefficient of ν = 30 m2 s−1 and a decreased turbine friction coefficient of K = 10.5. This was to ensure the existence of a steady-state solution.
Again, the unsteady case is straightforward and will be presented in future
work. For efficiency reasons, the convergence tolerance of the SLSQP optimisation algorithm was changed to 1. The same boundary conditions were used as
in the steady-state scenarios of the previous section, with the inflow condition
imposed on the western boundary and the outflow condition enforced on the
eastern boundary.
Two optimisation runs were conducted, with 128 and 256 turbines respectively. Both were performed on the Stampede supercomputer at the Texas
Advanced Computing Center on 64 cores; both took between 24 and 48 hours
of real time to complete (one functional evaluation took approximately 270 seconds, one gradient evaluation took approximately 90 seconds). The 128 turbine
24
(a)
(b)
760
850
720
800
Power [MW]
Power [MW]
Figure 11: (a): Satellite image of Stroma Island and Caithness (Bing Maps,
Microsoft). Satellite imagery is only available for the land and nearshore areas.
(b): Computational domain with the turbine site marked in pink.
680
640
600
0
50
100
150
Optimisation iteration
(a) Optimisation convergence
(128 turbines)
750
700
650
600
200
0
20
40 60 80 100 120 140
Optimisation iteration
(b) Optimisation convergence
(256 turbines)
Figure 12: Convergence of the Pentland Firth optimisation.
case converged in 197 iterations (197 gradient evaluations, 349 functional evaluations), and increased the power extraction from 600 MW to 741 MW, an
increase of 24% (figure 12a). The 256 turbine case converged in 133 iterations
(133 gradient evaluations, 185 functional evaluations), and increased the power
extraction from 607 MW to 804 MW, an increase of 33% (figure 12b). Even
though eight times as many turbines are to be optimised compared to the previous section, the number of iterations has hardly increased at all, suggesting
the suitability of the method for larger arrays.
The initial and optimised configurations are shown in figure 13. In both
cases, the optimisation algorithm builds very similar structures. The key features of the optimised layouts are the following:
• Walls of turbines aligned with the north and south boundaries; the southern wall extends for the entire zonal length, while the northern wall only
extends for the eastern two-thirds of the site. The turbine density on the
western side of the northern wall appears to decay in a similar way in both
cases.
25
(a) Initial turbine positions (128 turbines)
(b) Optimised turbine positions (128 turbines)
(c) Initial turbine positions (256 turbines)
(d) Optimised turbine positions (256 turbines)
Figure 13: Initial and optimised turbine positions for the Pentland Firth example.
• Eastern and western ‘barrages’ of densely packed turbines; in the 128
turbine case, the barrages consist of a single meridional column, while
for the 256 turbine case the extra turbines are used to form additional
columns. The optimisation algorithm automatically staggers these extra
columns to minimize wake shadowing (figure 13d). These barrages are
aligned perpendicular to the flow field (figures 14a and 14b) and extract
the majority of the power (figures 14c and 14d).
• ‘Spurs’ arcing from the southern wall in a north-easterly direction; in the
128 turbine case two spurs can be seen, while there are three spurs in the
256 turbine case. Again, the optimisation aligns the spurs perpendicular
to the flow field. The spur length is chosen such that the majority of
streamlines only intersect with two rows of turbines (figures 14a and 14b).
We hypothesise that these structures serve the following functions:
• The main purpose of the southern and northern walls is to funnel the flow
into and retain the flow inside the site boundary. Similar features are
found in all scenarios of the previous section. The water predominantly
flows in from the northwest, which is why the northern wall stops short
on the western side. We hypothesise that the decay of the turbine density
on the western side of the northern wall acts to funnel water through the
26
(a) Streamline flow visualisation (128 turbines)(b) Streamline flow visualisation (256 turbines)
(c) Turbine power map (128 turbines)
(d) Turbine power map (256 turbines)
Figure 14: The streamline and power results for the Pentland Firth example.
The power map displays the integrand of the functional J (section 2.4, equation (8)).
barrages. A similar density decay is visible in the optimised layout of
scenario 1 (figure 5b).
• The flow trapped inside the site domain attempts to escape through the
northern and southern walls, which causes the arcing of the western and
eastern barrages close to the northern and southern boundaries (figures 14a
and 14b). Since the prevailing incoming flow is towards the southeast,
more turbines are placed on the southern wall to retain the flow. This
motivates the deployment of the spurs on the western side of the southern
wall.
It is not physically meaningful to compare the optimised power extractions
for the 128 and 256 turbine cases, as a realistic power curve was not used. The
maximum velocity in the site for the 128 turbine case was 3.7 ms−1 , while it was
3.0 ms−1 for the 256 turbine case. Due to the cubic dependence of the power
extraction on the speed, the power per turbine is approximately doubled in the
128 turbine case, and so the total power extraction of the two cases are almost
27
the same. However, if the turbine model enforced a rated speed beyond which
no extra power was extracted, this approximate equality would not hold.
7. Conclusions
In this work, the optimal configuration of tidal turbine farms was formulated as an optimisation problem constrained by partial differential equations
describing the flow. This formulation allows the direct application of sophisticated mathematical techniques to its solution, particularly the adjoint technique
for rapidly evaluating gradients. The optimisation of tidal turbine farms is necessary to realise their full potential, but without gradient-based optimisation
techniques the computational cost is prohibitive (Vennell, 2012b).
The approach presented here has several key advantages. It fully accounts
for the nonlinear interactions between the geometry, the turbines, and the flow
throughout the optimisation. The use of gradient-based optimisation algorithms
combined with the adjoint technique enables the use of physically-realistic flow
models, even for a large number of turbines. Once the flow model inputs are
specified (domain, boundary conditions, initial array configuration, etc.), the
optimisation is fully automatic. The approach extends naturally to more realistic flow models, and to different functionals of interest such as profit or
environmental impact.
The algorithm was first applied to the optimisation of four idealised scenarios, both to demonstrate the capability of the method and to build physical
intuition. In all cases, the optimisation algorithm was successful in significantly
increasing the power extracted by the farm, at a computationally feasible cost.
The algorithm was then successfully applied to a more realistic optimisation
problem, involving a site of major industrial interest, accurate shoreline geometry, and an industrially relevant number of turbines.
Four main extensions are required to apply this in an industrial setting.
Firstly, the simulations must be driven by realistic tidal forcing, and incorporate a wider model domain around the site to be investigated. This also
includes extending the forward model to be forced by a head loss instead of a
fixed inflow velocity to incorporate the impact of large arrays on the free-stream
velocity (Vennell, 2010). Secondly, bathymetric effects must be accounted for.
Thirdly, the flow model must then be validated against real-world measurements. Finally, the wake modelling should be improved via a turbulence closure
and a realistic power curve used. All of these advances are the subject of ongoing
work.
The source-code of the turbine farm optimisation software and all examples
are open-source and available at http://opentidalfarm.org.
8. Acknowledgements
This work is supported by the Grantham Institute for Climate Change,
a Fujitsu CASE studentship, EPSRC grants EP/I00405X/1, EP/J010065/1,
28
EP/K503733/1 and EP/K030930/1, an EPSRC Pathways to Impact award, and
a Center of Excellence grant from the Research Council of Norway to the Center
for Biomedical Computing at Simula Research Laboratory. High-performance
Computing support was provided by the Texas Advanced Computing Center.
The authors would like to acknowledge helpful discussions with S. C. Kramer,
A. S. Candy, A. Avdis of Imperial College London, and R. Caljouw and S.
Crammond of MeyGen Ltd. The authors would also like to thank R. Vennell of
the University of Otago for his extremely thorough and constructive review.
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arXiv:1802.04834v1 [cs.LG] 13 Feb 2018
Challenging Images For Minds and Machines
Amir Rosenfeld, John K. Tsotsos
Department of Electrical Engineering and Computer Science
York University, Toronto, ON, Canada
amir@eecs.yorku.ca,tsotsos@cse.yorku.ca
February 15, 2018
Abstract
There is no denying the tremendous leap in the performance of machine learning methods in the past half-decade. Some might even say that specific sub-fields
in pattern recognition, such as machine-vision, are as good as solved, reaching
human and super-human levels. Arguably, lack of training data and computation
power are all that stand between us and solving the remaining ones. In this position paper we underline cases in vision which are challenging to machines and
even to human observers. This is to show limitations of contemporary models that
are hard to ameliorate by following the current trend to increase training data, network capacity or computational power. Moreover, we claim that attempting to do
so is in principle a suboptimal approach. We provide a taster of such examples in
hope to encourage and challenge the machine learning community to develop new
directions to solve the said difficulties.
1
Introduction
Once only known to a few outside of academia, machine-learning has become ubiquitous in both popular media and in the industry. Superhuman capabilities are now
being gradually recorded in various fields: in the game of GO, ([1, 2]), in face verification ([3, 4]), image categorization ([5]) and even in logical reasoning in simple scenes
([6, 7, 8]).
Most current leading methods involve some variant of deep learning. Consequentially, they require large amounts of hand-labeled data (with the exception of [2] - which
used self-play to gain experience). This has elicited a data-hungry era, with increasingly large-scale datasets painstakingly labeled for object classification/detection/segmentation,
image annotation, visual question-answering, and pose estimation ([9, 10, 11, 12, 13])
to name a few. This is accompanied by a growing demand for computational power.
We bring forward challenges in vision which do not seem to be solved by current
methods - and more importantly - by current popular methodologies, meaning that neither additional data, nor added computational power will be the drivers of the solution.
1
Figure 1: A children’s puzzle where the goal is to find six hidden words: Book, words,
story, pages, read, novel. For a machine this is far from child’s play. Could this be
solved by providing a million similar examples to a deep-learning system? Does a
human need such training?
Related Work
Imbalanced or Small Data: datasets tend to be naturally imbalanced, and there is
a long history of suggested remedies ([14, 15, 16]). Handling lack of training data
has also been treated by attempting to use web-scale data of lesser quality than handannotated dataset [17], simulating data [cite data for cars, text recognition in the wild,
captcha]. Transfer Learning: reusing features of networks trained on large is a useful
starting point (cf [18]) One-Shot-Learning: attempting to reduce the number of required training example, in extreme cases to one or even zero examples ([19]); DeepLearning Failures: recently, some simple cases where deep learning fails to work as
one would possibly expect were introduced, along with theoretical justifications ([20]).
2
Challenging Cases
We present two examples and then discuss them. They have a few common characteristics: humans are able to solve them on the first “encounter” - despite not having
seen any such images before. Incidentally - but not critically - the two examples are
from the domain of visual text recognition. Moreover, though humans know how to
recognize text as seen in regular textbooks, street-signs, etc, the text in these images is
either hidden, rendered, or distorted in an uncharacteristic manner.
Children’s games: the first case is well exemplified by a child’s game, hidden
word puzzles. The goal is to find hidden words in an image. Fig. 1 shows an arbitrarily
selected example. For a human observer this is a solvable puzzle, though it may take a
few minutes to complete. We applied two state-of-the-art methods for text recognition
2
Sub Image
[21]
“sned”
“vvoz”
“novees”
“teg”
[22]
“score”
∅
∅
∅
Table 1: Text detected by two state-of-the-art scene-text recognition methods applied
to sub-images of a children’s puzzle. ∅ means no text was detected by the method
(images scaled to fit figure).
Figure 2: Variants of textual CAPTCHA. Captchas are becoming increasingly difficult
(reproduced from [24])
in the wild with available code ([21]) or an on line-demo ([22]1 ) on the image in Fig.
1. As this did not work immediately, we focused on the word “NOVEL” (the “N” is
below the forearm of the left person, ending with an “L” below his foot), by cropping
it an rotating so the text is level, cropping more tightly , and even cropping only the
letter “L”. See Table 1 for the corresponding sub-images (including the entire image at
the top row) and the results output by the two methods.
This is by no means a systematic test and some may even claim that it isn’t fair and they would be right: these systems were not trained on such images; [21] was only
trained on a photo-realistic dataset of 8 million synthetic training images, and [22] was
only trained on tens of thousands of images from coco-text ([23]), or used powerful
pre-trained networks where training data was less available.
CAPTCHA: a well-known mechanism to thwart automated misuse of websites by
distinguishing between humans and machines ([25]). Textual captchas involve presenting an image of text which has to be read and written by the user. We focus on this
type of captcha, though others exist ([26]). The introduction of captchas immediately
triggered the invention of new automatic ways to break them ([27]), which eventually
sparked an “arms race” between increasingly complex captchas and correspondingly
powerful automated methods ([28]). This caused a state where on one-hand the best
leading textual captcha-solution methods involve training DNN’s over data with similar distortion characteristics as the desired types of captcha - though still these systems
have limited success rates (at times less than 50%) - and on the other hand the level of
distortion has become such that humans have a hard-time solving some of them.
3
Machines vs Humans as Supervised Learners
One can rule out the suggested examples by saying that they are simply out-of-sample
datapoints on behalf of a statistical learner’s perspective. Yet it seems that with what1 http://east.zxytim.com
3
ever supervision human-beings receive - they are usually able to solve them despite
not being especially exposed to this kind of stimulus. Moreover, precisely these kinds
of images are used routinely in human IQ testing, so they are a universally accepted
indicator for human performance. If these examples may seem esoteric, we can revert
to more common cases: as a child, how often is one exposed to bounding boxes of
objects? How often to delineations of objects with precise segmentation masks? How
often to pose-configurations, facial and bodily key-points, and dense-meshes of 3D objects overlayed on their field of view ([13])? More critically, for how many different
object types does this happen (if any), for how many different instances, with what
level of precision of annotation, and in how many modalities?
The granularity of visual supervision given to machines seems to be much finer
than that given to humans. As for the amount of directly supervised data, it does not
seem to really be the main limiting factor; as already noted several times, performance
either saturates with training data ([29, 30]) or at best grows logarithmically ([17, 31],
increasing mAP from 53% to 58% when growing from 10M to 300M examples) making the solution of more data for better performance simply impractical - even for those
with the most resources. And this is for “common” problems, such as object detection.
Humans who only ever read street-signs and textbooks are able to solve captchas
of various kinds without any special training on their first encounter with them. The
same is true for the “picture puzzles” mentioned above, as it is for other cases not
mentioned here. We do not claim that humans are not subject to supervised learning
in their early life, and in later stages. On the contrary, supervisory signals arise from
multiple sources: caretakers who provide supervisory signals by teaching, “internal supervision” provided by innate biases ([32]) and finally rewards stemming from results
of behaviour, such as suffering pain from hitting an object. But any such supervision is
interspersed within a vast, continuous stream of unsupervised data, most of which does
not have an easily measurable supervisory affect on the observer.
There is something fundamentally different about the way humans construct or
use internal representations, enabling them to reason about and solve new patternrecognition tasks. We hypothesize that these are approached by generating procedures
of a compositional nature when presented with a novel - or known - task (as suggested
by the Visual Routines of [33] or the Cognitive Programs of [34]. We intend to maintain
a collection of examples beyond the ones suggested above, to encourage the community to attempt to solve them, not by learning from vast amounts of similar examples,
but by learning from related, simpler subtasks and learning to reason and solve them
by composing the appropriate solutions.
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1
Action-Attending Graphic Neural Network
arXiv:1711.06427v1 [cs.CV] 17 Nov 2017
Chaolong Li*, Zhen Cui*, Member, IEEE, Wenming Zheng, Member, IEEE, Chunyan Xu, Member, IEEE,
Rongrong Ji, Senior Member, IEEE, and Jian Yang, Member, IEEE
Abstract—The motion analysis of human skeletons is crucial
for human action recognition, which is one of the most active
topics in computer vision. In this paper, we propose a fully
end-to-end action-attending graphic neural network (A2 GNN)
for skeleton-based action recognition, in which each irregular
skeleton is structured as an undirected attribute graph. To extract
high-level semantic representation from skeletons, we perform the
local spectral graph filtering on the constructed attribute graphs
like the standard image convolution operation. Considering not
all joints are informative for action analysis, we design an actionattending layer to detect those salient action units (AUs) by adaptively weighting skeletal joints. Herein the filtering responses are
parameterized into a weighting function irrelevant to the order
of input nodes. To further encode continuous motion variations,
the deep features learnt from skeletal graphs are gathered along
consecutive temporal slices and then fed into a recurrent gated
network. Finally, the spectral graph filtering, action-attending
and recurrent temporal encoding are integrated together to
jointly train for the sake of robust action recognition as well
as the intelligibility of human actions. To evaluate our A2 GNN,
we conduct extensive experiments on four benchmark skeletonbased action datasets, including the large-scale challenging NTU
RGB+D dataset. The experimental results demonstrate that our
network achieves the state-of-the-art performances.
Index Terms—Human action recognition, Skeleton-based action recognition, Convolutional neural networks, Attention mechanism.
I. I NTRODUCTION
H
UMAN action recognition is an active area of research
with wide applications such as video surveillance, games
console, robot vision, etc. Over the past few decades, human
action recognition from 2D RGB video sequences has been
extensively studied [1]. However, 2D cameras cannot fully
capture human motions, which are actually located in 3D
space. With the advent of depth sensors such as Microsoft
Kinect and Asus Xtion PRO LIVE, 3D action recognition
arises more attention of researchers due to its several advantages in segmenting foreground/background, resisting illumination variations, etc.
Recently several approaches have been developed to deal
with 3D human action recognition. Generally they fall into
two categories: depth map based or skeleton based. The depth
map based methods directly extract volumetric and temporal
features of overall point set from depth map sequences. The
* C. Li and Z. Cui have equal contributions.
C. Li and W. Zheng are with the Key Laboratory of Child Development
and Learning Science (Southeast University), Ministry of Education, School
of Biological Science & Medical Engineering, Southeast University, Nanjing
210096, China (e-mail: {lichaolong, wenming zheng}@seu.edu.cn).
Z. Cui, C. Xu and J. Yang are with the School of Computer Science and
Engineering, Nanjing University of Science and Technology, Nanjing 210094,
China (e-mail: {zhen.cui, cyx, csjyang}@njust.edu.cn).
R. Ji is with the School of Information Science and Engineering, Xiamen
University, Xiamen 361005, China (e-mail: rrji@xmu.edu.cn).
skeleton based methods utilize 3D coordinates of skeletal
joints estimated from depth maps to model actions of human
body. As human body can be viewed as an articulated system
of rigid bones connected by hinged joints, the actions of
human body principally reflect in human skeletal motions in
the 3D space [2]. Thereby, to model human skeletal joints
should be more effective for action recognition, as suggested
in the early work [3].
Skeleton based methods [4], [5], [6], [7], [8], [9], [10], [11],
[12], [13] have become prevalent for human action recognition
since Shotton et al. [14] successively estimated 3D coordinates
of skeletal joints from depth maps. To represent trajectories of
human body motions, the position, speed or even acceleration
of skeletal joints are often gathered from several consecutive
frames, and then modeled with some statistic methods (e.g.,
histogram). Further, the skeletal joints may be partitioned into
several parts according to the concurrent function of adjacent
joints, so human actions are represented with the motion
parameters of body parts. To encode temporal motions of
skeletal joints, linear/non-linear dynamical systems are often
used to model human action, e.g., Hidden Markov Models
(HMMs) [12] and Long Short Term Memory (LSTM) [15].
However, there still exist some key issues need to be studied
deeply. First, how to extract robust high-level semantic features
from irregular skeletons? The current skeleton-based methods
usually employ some simple statistical features on skeletal
joints or body parts. It is insufficient to describe and abstract
spatial structure of skeleton at each time slice, whereas nonlinear dynamic systems are only used to abstract high-level
motion information. Second, which/what action units (AUs)
identify a special human action? Most motions are produced
from only a few joints or body parts, e.g., waving with arm
and hand, kicking ball with leg and foot. Detecting salient
action units should be helpful to eliminate some useless motion
noises as well as provide a cognitive explanation to understand
human action.
To address the above problems, in this paper, we propose
a fully end-to-end action-attending graphic neural network
(A2 GNN) for skeleton-based action recognition. To extract
high-level semantic information from spatial skeletons, we represent each human skeleton with an undirected attribute graph,
and then perform the local spectral filtering on the structured
graph to laywisely abstract spatial skeletal information like
the classic convolutional neural network (CNN) [16]. Hence,
different from the recent graph-based method [13], which
only took the traditional technique line of graph matching,
A2 GNN should be a true deep network directly learning
from irregular skeletal structure. To detect those salient action
units, we design an action-attending layer to adaptively weight
skeletal joints for different human motions. Specifically, we
2
parameterize the weighting process as one dynamic function,
which takes the responses of local spectral filtering on skeletal
graphs as the input. Incidentally, we draw a conclusion that
the weighting process is irrelevant to the input order of
skeletal joints, which thus can be used for those general graph
tasks. After extracting high-level discriminant features at each
time slice, we finally stack a recurrent neural network on a
consecutive sequence to model temporal variations of different
human actions. All processes including spectral graph filtering,
action unit detection, temporal motion modeling are integrated
into one network framework to train jointly. To evaluate our
proposed method, we conduct extensive experiments on four
benchmark skeleton-based action datasets: the Motion Capture
Database HDM05 [17], the Florence 3D Action dataset [18],
the Large Scale Combined (LSC) dataset [19], the NTU
RGB+D dataset [20]. The experimental results demonstrate
our A2 GNN is competitive over the state-of-the-art methods.
In summary, our contributions are three folds:
1) Propose a fully end-to-end graphical neural network
framework to deal with skeleton-based action recognition,
where we model human skeletons as attribute graphs and
then introduce spectral graph filtering to extract high-level
skeletal features, .
2) Design a weighting way to adaptively detect salient
action units for different human actions, which not only
promotes human action recognition accuracy but also
favors our cognitive understanding to human actions.
3) Achieve the state-of-the-art performances on the four
benchmark datasets including the two large-scale challenging datasets: LSC and NTU RGB+D.
The reminder of this paper is organized as follows. In
Section II, we introduce related work on skeleton based
action recognition. In Section III, we first give an overview
of our proposed network and introduce three main modules
respectively. Implementation details of our proposed network
are stated in Section IV. Experimental results and discussion
are presented in Section V. Finally, we conclude this work in
Section VI.
II. R ELATED W ORK
The most related works to ours are those methods of
skeleton-based human action recognition. Here we briefly
review them from the view of skeletal representation, including
low-level statistical features and high-level semantic features.
Various low-level statistical features have been used to
describe skeleton data in the past years. Generally they contain
three categories: joint based, body part based and pose based
features. Hussein et al. [21] used covariance matrix of skeletal
joint coordinates over time to describe motion trajectories. In
the literature [22], histogram of oriented displacement was
used to represent the 3D trajectories of body joints. Ofli et
al. [23] chose a few most informative joints, and employed
some physical interpretable measures, such as the mean or
variance of joint angles, maximum angular velocity of joints
and so on, to encode skeletal motions. Considering jointassociated movements, Chaudhry et al. [24] divided human
skeleton into several smaller parts hierarchically and depicted
each part with certain bio-inspired shapes. Vemulapalli et
al. [10] formulated each body part over times into a curved
manifold and performed action recognition in the Lie group.
In view of skeletal postures, Zanfir et al. [25] proposed the
moving pose descriptor by considering position, speed and
acceleration information of body joints. Xia et al. [12] represented postures as histograms of 3D skeletal joint locations
by casting selected joints into corresponding spatial histogram
bins. These low-level statistical features are usually limited in
representing those complex human actions.
High-level sematic features are popular for encoding human
actions due to the robust representation ability. Especially, the
temporal pyramid is used to hierarchically encode temporal
dynamical variations, and all level features are gathered together to model actions [9], [10], [22], [26], [27]. To model
the temporal dynamics, Vemulapalli et al. [10] used dynamic
time warping (DTW) and Fourier temporal pyramid (FTP);
Wang et al. [27] employed FTP to encoded local occupancy
patterns over times; Chaudhry et al. [24] employed linear
dynamical systems (LDSs) to learn the dynamical variation of
features; Xia et al. [12] used HMMs to capture the temporal
action dynamics. In addition, more recently, recurrent neural
network has been employed to encode the temporal dynamic
variations [15], [28]. However, these methods mainly focus
on the deep encoding of temporal motions rather than spatial
layout of joints.
Until recently, graph was used to represent skeletal motion in the literature [13], although a few graph-based algorithms [29], [30] had been proposed for action recognition
on RGB videos. These graph-based methods usually took the
traditional graph matching strategy after modeling skeletal
joints or body parts into the graph structure. Therefore, two
crucial issues, the construction of graph and the definition
of graph kernel, need be conducted in these graph-based
methods. Different from them, we perform spectral graph
filtering on skeleton-induced graphs to extract high-level skeletal features from the spatial graphs. With a combination of
recurrent motion encoding, the spatial-temporal features of
skeletal motions are abstracted from the constructed end-toend network. In contrast to the existing skeleton-based action
recognition methods, another one important difference is that
an adaptive action-attending mechanism is introduced to detect
those salient action units w.r.t different human actions, which
can benefit the final human action recognition.
III. T HE P ROPOSED A2 GNN
In this section, we first give an overview of our proposed
A2 GNN, then we respectively introduce three involved modules: the learning of deep graphical features, the detection of
salient action units and the dynamic modeling of temporal
motions.
A. Overview
An overview of our proposed A2 GNN is illustrated in Fig. 1.
The input is a motion sequence of skeletons, in which each
skeletal joint is described as a 3D coordinate (x, y, z). For
each skeleton at one time slice, we model it into an undirected
3
Skeletal
Graphs
Dynamic weighting
Dynamic weighting
Prediction
FC
Spectral graph
filtering
(Section III. B)
Action unit
detecting
(Section III. C)
Spectral graph
filtering
(Section III. B)
Action unit
detecting
(Section III. C)
Softmax
Temporal recurrent
learning (LSTM)
(Section III. D)
Fig. 1. The illustration of our proposed A2 GNN architecture. An overall introduction can be found in Section III-A.
attribute graph, where each skeletal joint is one node and the
bone between two joints is considered as one connected edge.
For the weight between two nodes, we assign a connected
edge to 1, otherwise 0. In addition, several alternative ways to
weight edges are permissible, e.g., Gaussian kernel. Another
important characteristic is that each node is associated with an
observed signal vector (a.k.a. attribute), which is 3D spatial
coordinates of the joint. To reduce individual differences
of different samples, we normalize the input signals with
coordinating, scaling and rotating transformations.
For the constructed spatial skeletal graphs, in order to
extract high-level skeletal features, we expect to perform
convolutional filtering on them like on regular grid-shape
images. To the end, we introduce local spectral filtering on
graphs, inspired by signal theory on graphs [31] and the recent
graph convolution [32]. To avoid the eigenvalue decomposition
of Laplacian matrix, the original solution is approximated by
a polynomial of Laplacian matrix, in which each k-order term
derives a k-hop neighborhood subgraph like a local receptive
field. The details are introduced in Section III-B.
Specially, we design an action-attending layer to detect
salient action units by adaptively weighting skeletal joints
for different human actions. Usually, specific action units are
activated for different human actions, e.g., hands and arms
segments for clapping, hand, arm and head segments for
drinking. Hence, weighting skeletal joints may reduce the
disturbance of those useless joints, and benefit the final action
recognition. The related details may be found in Section III-C.
By alternately stacking the spectral graph filtering layer and
the action-attending layer, we may obtain the graph features
of skeletal joints. After concatenating features of all joints at
one time slice, we fed it into a recurrent network (LSTM) to
encode feature variations at all temporal slices. Please refer to
more details in Section III-D. Finally, a fully connected layer
is used to gather the outputs of recurrent network and learn
the skeleton sequence representation followed by a softmax
layer for classification. All processes including spectral graph
filtering, action unit detection, temporal motion modeling are
integrated into a network framework and jointly trained.
B. Learning Deep Graphical Features
We model a body skeleton into an undirected attribute graph
G = (V, A, X) of N nodes, where V = {v1 , ..., vN } is the set
of skeletal joints, A is the (weighted) adjacency matrix, and X
is the matrix of node signals/attributes. The adjacency matrix
A ∈ RN ×N encodes the connection between two nodes (or
joints), where if vi , vj are connected, then Aij = 1, otherwise
Aij = 0. If each joint is endowed with a vectorized signal of
3D coordinates, i.e., x : V → R3 , we may stack the signals of
all nodes to form the signal matrix X ∈ RN ×3 , where each
row is associated with one node.
In order to extract skeletal graph features, we aim to perform
convolutional filtering on these irregular attribute graphs like
on regular grid-shape images. As studied in [33], [34], it is
difficult to express a meaningful translation operator in the
vertex domain. According to the spectral graph theory [31],
the convolutional filtering on graphs depends on the graph
N ×N
Laplacian operator L = D − A, where
is the
P D ∈ R
diagonal degree matrix with Dii =
j Aij . For the graph
Laplacian matrix, the normalized version is often used, i.e.,
1
1
1
1
Lnorm = D− 2 LD− 2 = I − D 2 WD 2 ,
(1)
where I is the identity matrix. Unless otherwise specified, we
use the normalized Laplacian matrix below.
As a real symmetric positive definite (SPD) matrix, the
graph Laplacian matrix L may be decomposed into
L = UΛU> ,
(2)
where Λ = diag([λ1 , λ2 , · · · , λN ]) is a diagonal matrix of
nonnegative real eigenvalues {λl } (a.k.a spectrum), and the
4
orthogonal matrix U = [u1 , · · · , uN ] means the corresponding eigenvectors. Analogous to the classic Fourier transform,
the graph Fourier transform of a signal x in spatial domain
b = U> x, where x
b is the produced
can be defined as x
frequency signal. The corresponding inverse Fourier transform
is x = Ub
x.
Give any one filtering function g(·) of the graph L, we
can define the frequency responses on the input signal x as
zb(λl ) = x
b(λl )b
g (λl ), or the inverse graph Fourier transform,
z(i) =
N
X
x
b(λl )b
g (λl )b
ul (i),
(3)
l=1
where zb(λl ), x
b(λl ), gb(λl ) are the Fourier coefficients corresponding to the spectrum λl . Hence, the matrix description of
graph filtering is
z = gb(L)x = U diag([b
g (λ1 ), · · · , gb(λN )])U> x.
(4)
We need to learn the filtering function g(·), but the computational cost of Eqn. (2) and Eqn. (4) is expensive because the
eigenvalue decomposition need to be done.
To address this problem, we may parameterize the filtering
function g(·) with a polynomial approximation. As used in
the literature [32], we employ the Chebyshev expansion of
K order [35] by defining the recurrent relation Tk (x) =
2xTk−1 (x) − Tk−2 (x) with T0 = 1 and T1 = x. Any one
function in the space
∈ [−1, 1] can be expressed with the
Px
∞
expansion: f (x) = k=0 ak Tk (x). Suppose we take the Korder approximation for g(·), i.e.,
gb(λl ) =
K−1
X
θk Tk (λel ),
(5)
k=0
>
K
where θ = [θ1 , · · · , θK ] ∈ R is the parameter vector of
2
λl − 1 with λmax =
polynomial coefficients, and λel = λmax
max{λ1 , · · · , λN } ≤ 2. By substituting Eqn. (5) into Eqn. (4),
we can derive the following equation,
z=
K−1
X
k=0
θk Tk (
2
λmax
L − I)x,
(6)
where we use this basic equation,
Lk = U diag([λk1 , · · · , λkN ])U>
= (U diag([λ1 , · · · , λN ])U> )k .
(7)
2
L−I, if we denote the K filter bases about the
Let Le = λmax
e T1 (L),
e · · · , TK−1 (L)},
e the final
Laplacian matrix as {T0 (L),
local spectral graph filtering can be written as
e
e
e
Z = [T0 (L)X,
T1 (L)X,
· · · , TK−1 (L)X]Θ,
(8)
where Θ ∈ R3K×dz is the parameters to be learnt with the dz
output channels, and Z ∈ RN ×dz is the responses of spectral
graph filtering. As Lek encodes a k-hop local neighborhood of
each node, so the K-order polynomial in Eqn. (6) is a exactly
K-localized filter function of graphs. Correspondingly, Θ is
the K-localized filtering parameter need to be solved.
C. Detecting Salient Action Units
As observed that an action often occurs at a special body
part, the detection of salient action units is necessary to reduce
the disturbances of irrelevant joints as well as verify human
cognition on actions. To detect action units, we propose a new
layer named as action-attending layer to adaptively weight
skeletal joints for different human actions, inspired by the
attention mechanism used for various tasks, e.g., machine
translation [36], speech recognition [37], and image captioning [38], etc. Our purpose is to decide what action units
identify (or play a key role in) a special human action.
We stack this layer after the spectral graph filtering layer to
take advantage of high-level features. As the K-order filtering
has a K-hop receptive field as introduced in Section III-B,
the filtering responses of each node actually assemble certain
information of K-path neighbors around the node. Suppose
the feature responses after spectral filtering as Z ∈ RN ×dz
(in Eqn. 8), we attempt to learn a projecting matrix to weight
nodes. But considering different action cases, we expect that
the projecting matrix can dynamically change with the dif0
ferent input Z. That is, the dynamic matrix W ∈ RN ×N is
actually a parameterized function of the variable Z, formally,
e = W(Z)Z,
Z
(9)
N
X
s.t. ,Wij ≥ 0,
Wik = 1,
(10)
k=1
j = 1, · · · , N 0 .
i = 1, · · · , N,
The larger the matrix item Wij is, the more important the
j-the node is for action recognition. To model the dynamic
property, we define the dynamic function as
W(Z) = (tanh(ZQ + b> )V)> ,
exp(Wij )
,
Wij = PN
k=1 exp(Wik )
0
0
0
0
(11)
(12)
where Q ∈ Rdz ×d , V ∈ Rd ×N , b ∈ Rd are the parameters
to be solved. Hence, the dynamic function W takes advantage
of the input feature Z.
In addition to the detection of salient action units, the
dynamic weighting function has two extra advantages:
(1) Order-independency of nodes.
Suppose all nodes are ordered in {v1 , · · · , vN } and the
signal matrix X = [x1 , · · · , xN ]> is built, then we can
extract the filtering feature Z = [z1 , · · · , zN ]> in the same
order of nodes, i.e., zi is still associated with vi . However, if all nodes are disordered, e.g., {vN , vN −1 · · · , v1 },
correspondingly, we have X0 = [xN , xN −1 · · · , x1 ]> and
Z0 = [zN , zN −1 · · · , z1 ]> . If we employ a constant projection
W rather than the dynamic function W, it will result into
WZ 6= WZ0 . That means, different traversing ways on a
graph will produce different responses, which is unfeasible
e will be flatten
to feature comparisons, e.g., the feature Z
as the input to fed into recurrent neural network (See Section III-D). However, the dynamic weighting function W
is order-independent for graphical nodes, according to the
following theory.
5
Theorem 1. Given the dynamic function W in Eqn. (11), the
e in Eqn. (9) is irrelevant to the order of traversing
output Z
order of graphical nodes.
Proof. Suppose the input signal matrix X, correspondingly,
e
we can obtain the convolutional filtering feature Z and Z
according to Eqn. (8) and Eqn. (9) respectively. Given any one
permutation matrix P ∈ {0, 1}N ×N , PX equals to reorder all
e 0 when taking
signals in X. Let denote the feature Z0 and Z
PX as the input. According to Eqn. (8), we can have
e > )PX, · · · , TK−1 (P LP
e > )PX]Θ
Z0 = [T0 (P LP
e > PX, · · · , PTK−1 (L)P
e > PX]Θ
= [PT0 (L)P
e
e
= P[T0 (L)X,
· · · , TK−1 (L)X]Θ
= PZ.
(13)
Note that the Laplacian matrix is also reordered by the pere=Z
e 0.
mutation matrix P. Now we only need to prove that Z
According to Eqn. (9) and Eqn. (11), we have
states ct ∈ Rdh are intermediate vectors, formally, the motion
variations can be modeled as
i = σ(Wzie
zt + Whi ht−1 + wci ct−1 + bi ),
(17)
f = σ(Wzf e
zt + Whf ht−1 + wcf ct−1 + bf ),
(18)
ct = ft ct−1 + it tanh(Wzce
zt + Whc ht−1 + bc ),
(19)
ot = σ(Wzoe
zt + Who ht−1 + wco ct + bo ),
(20)
ht = o
tanh(ct ),
(21)
where σ(·) is the elementwise sigmoid function, i.e., σ(x) =
1/(1 + e−x ), denotes the Hadamard product and i, f , o, c ∈
Rdh are respectively the input gate, forget gate, output gate,
cell and cell input activation vectors. The weight matrices
0
{Wz· ∈ Rdh ×N dz , Wh· ∈ Rdh ×dh , wc· ∈ Rdh } and the bias
vectors {bi , bf , bc , bo ∈ Rdh } are the model parameters to
be solved. Finally, we use the output gate ot as the response
at the t-th time slice.
IV. I MPLEMENTATION D ETAILS
e 0 = W(Z0 )Z0 = W(PZ)PZ
Z
In this section, we will introduce our implementation details
including network architecture and data augmentation.
= (tanh(PZQ + b> )V)> PZ
= (P tanh(ZQ + b> )V)> PZ
= (tanh(ZQ + b> )V)> P > PZ
A. Network Architecture
= (tanh(ZQ + b> )V)> Z
e
= Z.
For the undirected graph, we simply construct edge connections based on human bones. That means, if two joints
are bridged with a bone, the edge weight is assigned to 1,
otherwise 0. Our plain network contains two spectral graph
filtering layers along with two action-attending layers followed
by a temporal recurrent layer, as shown in Fig. (1). In the
spectral filtering layers, the receptive fields are set to K = 10
as default, and the output signals have the length of 32
dimensions and 64 dimensions respectively for two layers.
In the action-attending layer, we take a simple design rule:
the dimensions remain invariant with regard to the input, i.e.,
N 0 = N, d0 = dz . In the temporal recurrent layer, we employ
the classic LSTM unit to model the temporal dynamics, where
the dimension of hidden units is set to 256. In the full
connected layer, the output has the same dimension to the
input. The network ends with a cross-entropy loss used for
classification. The learning rate of network is set to 0.02 with
a momentum of 0.9. More analysis/discussion of parameters
can be found in Section V-E. The concrete implementation
takes TensorFlow as the infrastructure.
(14)
(2) Dynamic pooling of nodes.
The dynamic function W may be regarded as a pooling
operation on nodes. Given a row Wi· of W, the output
Wi· Z is a new signal by weighting and combining nodes.
Correspondingly, we can update the adjacency matrix of nodes
and the Laplacian matrix,
A0 = WAW > ,
Le0 = I − D−1/2 A0 D−1/2 ,
(15)
(16)
0
0
where A0 , Le0 ∈ RN ×N . Thus, the new Laplacain matrix Le0
e can be fed into the next layer and make the
and the feature Z
neural network go deeper.
D. Modeling Temporal Motions
B. Data Processing
After extracting the spatial graphical features, we need
to model temporal motion variations of a skeletal sequence. Many non-linear dynamic models can be used to
solve this problem. Here we employ a special class of
recurrent neural networks (RNN), long short-term memory
(LSTM) [39], which can mitigate gradient vanishment when
back-propagating gradients. LSTM has demonstrated the powerful ability to model long-range dependencies [40], [41], [42].
Suppose the graphical feature of the t-th skeletal frame is
e t ) ∈ RN 0 dz , the cell output ht ∈ Rdh and
e
zt = vectorize(Z
As skeletal data is usually captured from multi-view points
and human actions are independent on the user coordinate
system, we modify the origin of the coordinate system as the
orthocenter
of joints for each frame of skeleton, i.e., O =
PN
1
x
,
where xi ∈ R3 is a 3D coordinate of the i-th
i
i=1
N
joint, N is the number of joints.
To enhance the robustness of model training, we perform
data augmentation as widely used in previous deep learning
literature [43], [20]. Concretely, for each action sequence, we
split the sequence into several equal sized subsequences, here
6
TABLE I
S UMMARIZATION OF FOUR ACTION RECOGNITION DATASETS .
Dataset
HDM05 [17]
Florence 3D [18]
LSC [19]
NTU RGB+D [20]
# Joints
31
15
15/20
25
# Actions
130
9
88
60
# Subjects
5
10
79
40
# Sequences
2337
215
3898
56880
12 segments, and then pick one frame from each segment
randomly to generate a large amount of training sequences.
In addition, we randomly scale the skeletons by multiplying
a factor in [0.98, 1.02] for the sake of the adaptive capability
of scaling.
V. E XPERIMENTS
To evaluate our proposed A2 GNN, we conduct extensive
experiments on four benchmark skeleton-based action datasets,
including HDM05 [17], Florence 3D [18], Large Scale Combined dataset (LSC) [19] and NTU RGB+D [20]. A brief
summarization about them is given in Table I. Below we will
compare our A2 GNN with the recent state-of-the-art methods,
then analyze confusion matrices and action unit detection, and
finally discuss some network parameters.
A. Datasets
1) HDM05 [17]: This dataset was captured by using an
optical marker-based Vicon system, and gathered 2337 action
sequences for 130 motion classes, which are performed by 5
non-professional actors named “bd”, “bk”, “dg”, “mm” and
“tr”. Each skeleton data is represented with 31 joints. Until
now, this dataset should involve the most skeleton-based action
categories to the best of our knowledge. Due to the intra-class
variations and large number of motion classes, this dataset is
challenging in action recognition.
2) Florence 3D [18]: This dataset was collected via a
stationary Microsoft Kinect camera. It consists of 215 action
sequences from 10 subjects for 9 actions: wave, drink from a
bottle, answer phone, clap, tight lace, sit down, stand up, read
watch, bow. Only 15 joints are recorded for each skeletal data.
As a few skeletal joints, some types of actions are difficult to
distinguish, such as drink from a bottle, answer phone and
read watch.
3) LSC [19]: This dataset combines nine publicly available datasets, including MSR Action3D Ext [44], [45], [46],
UTKinect-Action3D [12], MSR DailyActivity 3D [27], MSR
Action Pairs 3D [47], CAD120 [48], CAD60 [49], G3D [50],
[51], RGBD-HuDa [52], UTD-MHAD [53], and form a complex action dataset with 94 actions. As some samples have not
skeleton information, we remove them and construct a skeleton
dataset of 88 actions by following previous standard protocols.
As each individual dataset has its own characteristics in
the action execution manners, backgrounds, acting positions,
view angles, resolutions, and sensor types, the combination
of a large number of action classes makes the dataset more
challenging in suffering large intra-class variation compared
to each individual dataset.
TABLE II
C OMPARISONS ON HDM05 DATASET.
Method
RSR-ML [56]
Cov-RP [57]
Ker-RP [54]
SPDNet [55]
Lie Group [10]
LieNet [58]
P-LSTM [20]
A2 GNN
Protocol [54]
Accuracy
40.0%
58.9%
66.2%
70.4%
76.5%
Protocol [55]
Accuracy
61.45%±1.12
70.26%±2.89
75.78%±2.26
73.42%±2.05
84.47%±1.52
*Note that all 130 classes are used here.
TABLE III
C OMPARISONS ON F LORENCE DATASET.
Method
Multi-part Bag-of-Poses [18]
Riemannian Manifold [6]
Lie Group [10]
Graph-Based [13]
MIMTL [59]
P-LSTM [20]
A2 GNN
Accuracy
82.00%
87.04%
90.88%
91.63%
95.29%
95.35%
98.60%
4) NTU RGB+D [20]: This dataset is collected by Microsoft Kinect v2 cameras from different views. It consists of
56880 sequences and over 4 million frames for 60 distinct
actions, including various of daily actions and pair actions.
These actions were performed by 40 subjects aged between 10
and 35. The skeleton data is represented by 25 joints. As far
as we know, this dataset is currently the largest skeleton-based
action recognition dataset. The large intra-class and view point
variations make this dataset great challenging. Meanwhile, a
large amount of samples will bring a new challenging to the
current skeleton-based action recognition methods.
B. Comparisons with State-of-the-art Methods
1) The Results:
a) HDM05: To compare with those previous literatures,
we conduct two types of experiments by following two widelyused protocols. First, we follow the protocol used in [54] to
perform action recognition on all of the 130 classes. Actions
of two subjects named “bd” and “mm” are used to train
the model, and the remaining three ones for testing. The
comparison results are shown in the left column of Table II.
Second, to fairly compare the current deep learning methods,
we follow the settings of the literature [55] to conduct 10
random evaluations, each of which randomly selects half of
the sequences for training and the rest for testing. The results
are reported in the right column of Table II.
b) Florence 3D: We follow the experimental settings
of the literature [13] to perform leave-one-subject-out crossvalidation. In each round, skeletal data of 9 subjects is taken
for training and the remain one for testing. The experimental
results are reported in Table III.
7
TABLE IV
C OMPARISONS ON L ARGE S CALE C OMBINED DATASET.
Method
HON4D [47]
Dynamic Skeletons [60]
P-LSTM [20]
A2 GNN
Cross Sample
Precision
Recall
84.6%
84.1%
85.9%
85.6%
84.2%
84.9%
87.6%
88.1%
Cross Subject
Precision
Recall
63.1%
59.3%
74.5%
73.7%
76.3%
74.6%
84.0%
82.0%
1.0
10
Part A
0.9
20
0.8
30
0.7
40
Part B
50
0.6
60
0.5
70
0.3
100
0.2
110
0.1
120
110
100
90
80
70
60
50
40
30
120
0.0
(a)
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
5
6
7
8
9
10
11
12
13
14
15
16
(b)
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
c) LSC: We follow the protocol designed in the recent
work [19] to conduct two types of experiments, random cross
subject and random cross sample. The experimental results are
reported in Table IV.
d) NTU RGB+D: Different from the above three
datasets, we preprocess the joint coordinates in a way similar
to [20]. Concretely, after translating the original coordinates of
body joints as mentioned above, we rotate the x axis parallel
to the 3D vector from “right shoulder” to “left shoulder” and y
axis towards the 3D vector from “spine base” to “spine”. The
z axis is fixed as the new x × y. This dataset has two types
of standard evaluation protocols [20]. One is cross-subject
evaluation, for which half of the subjects are used for training
and the remaining ones for testing. The second is cross-view
evaluation, for which two viewpoints are used for training and
one is left out for testing. The experimental results are shown
in Table V.
2) The Analysis:
As shown in Table II∼V, we compare the current state-ofthe-art methods on different datasets, including the shallow
learning methods and the deep learning methods. From these
results, we have the following observations:
• Matrix-based descriptors (e.g., covariance or its variants)
are conventionally used to model spatial or temporal relationships. Moreover, these descriptors are often regarded
to be embedded on specific geometric manifolds [10],
[58], [55]. The advanced variant [54] improved the performance by constructing some robust SPD matrices. More
recently, SPDNet [58] and LieNet [55] attempted to learn
deep features from those raw matrix descriptors under
the assumption of manifold. Although the deep manifold
learning strategy on matrix descriptors raises a promising
90
20
Cross View
Accuracy
7.26%
52.76%
41.36%
65.22%
63.97%
66.95%
64.09%
67.29%
70.27%
77.7%
82.80%
10
HON4D [47]
Lie Group [10]
Skeletal Quads [61]
Dynamic Skeletons [60]
HBRNN [62]
LieNet [58]
Deep RNN [20]
Deep LSTM [20]
P-LSTM [20]
ST-LSTM [43]
A2 GNN
Cross Subject
Accuracy
30.56%
50.08%
38.62%
60.23%
59.07%
61.37%
56.29%
60.69%
62.93%
69.2%
72.74%
0
Method
0.4
80
5
6
7
8
9
10
11
12
13
14
15
16
TABLE V
C OMPARISONS ON NTU RGB+D DATASET.
(c)
Fig. 2. Confusion matrix of our A2 GNN on HDM05 dataset according to
the testing protocol in [55]. (b) and (c) are the close up of Part A and Part
B in (a). X-axis and Y-axis are associated with the indices of action classes.
** Indices of part classes: 5-depositFloorR; 6-depositHighR; 7-depositLowR;
8-depositMiddleR; 13-grabFloorR; 14-grabHighR; 15-grabLowR; 16grabMiddleR; 41-kickLFront1Reps; 42-kickLFront2Reps; 43-kickLSide1Reps;
44-kickLSide2Reps;
45-kickRFront1Reps;
46-kickRFront2Reps;
47kickRSide1Reps;
48-kickRSide2Reps;
50-punchLFront1Reps;
51punchLFront2Reps;
52-punchLSide1Reps;
53-punchLSide2Reps;
54punchRFront1Reps; 55-punchRFront2Reps; 56-punchRSide1Reps.
•
direction to some extent, the matrix-based representations
principally limit their capability of modeling dynamic
variations, because the only second-order statistic relationship of skeletal joints is preserved in the descriptors,
whereas first-order statistics is also informative [63].
Deep features are more effective than those shallow
features for skeleton-based action representation. The
advanced nonlinear dynamic networks, specifically DeepRNN, Deep-LSTM, P-LSTM [20] and ST-LSTM [43],
largely improve the action recognition performance, due
to the good encoding capability of gated network units.
Most of them use recurrent networks to model temporal
dynamics. Besides, ST-LSTM also attempted to model
spatial skeletal joints by taking a tree-structure traversal
way on spatial joints. Similar to them, we also use
recurrent neural network to model temporal dynamics.
But different from them, we directly extract high-level
8
Part A
0
5
10
15
20
25
30
35
40
45
50
55
59
Part B
0
5
10
15
20
25
30
35
40
45
50
55
59
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0.0
0.0
0
5
10
15
20
25
30
35
40
45
50
55
59
(a) Cross View
•
(b) Cross Subject
10
12
14
16
18
20
22
24
26
28
49
50
51
52
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
53
54
(c)
57
56
55
54
53
10
12
14
16
18
20
22
24
26
28
49
57
52
56
50
55
51
0
5
10
15
20
25
30
35
40
45
50
55
59
(d)
Fig. 3. Confusion matrix of our A2 GNN on NTU RGB+D dataset. (c) and (d)
are the close up of Part A and Part B in (a). X-axis and Y-axis are associated
with the indices of action classes.
** Indices of part classes: 10-reading; 11-writing; 15-wear a shoe; 16-take off
a shoe; 17-wear on glasses; 18-take off glasses; 19-put on a hat/cap; 20-take
off a hat/cap; 28-playing with phone/tablet; 29-typing on a keyboardp; 49punching/slapping other person; 50-kicking other person; 51-pushing other
person; 52-pat on back of other person; 53-point finger at the other person;
54-hugging other person; 55-giving something to other person; 56-touch other
person’s pocket; 57-handshaking.
•
1.0
wave 1.00
0.9
drink from a bottle
0.95 0.05
answer phone
1.00
0.8
0.7
•
1.00
clap
1.00
tight lace
0.5
0.4
1.00
sit down
0.3
1.00
stand up
read watch 0.04
0.6
0.04
0.2
0.91
motion parts (called motionlets). Second, the purpose of
the use of graph is to extract skeletal features for ours,
not model the relationship of motion parts. Third, our
method is a fully end-to-end deep learning architecture,
not abide by the conventional two-step way: i) construct
graphs of motion parts, and ii) compute the similarity
between graphs via subgraph-patten graph kernel. Besides, the action-attending mechanism is introduced into
our network architecture.
Our proposed A2 GNN greatly improves the current stateof-the-art on most datasets. On the Florence dataset, our
method achieves a nearly perfect performance 98.60%.
On the current largest dataset NTU RGB+D dataset,
the state-of-the-art performance is pushed to the higher
72.74% and 82.80%, from 69.2% and 77.7% for STLSTM. In summary, we can benefit from the deep graph
network architecture. The reason can be two folds: i) the
deep skeletal graph features, rather than simple spatial or
temporal features learnt by LSTM; and ii) the preservation of more original feature information (as signals of
each node), rather than only the second-order statistics of
skeletal joints.
Different performances occur on different datasets.
Among the four datasets, Florence 3D is the simplest one
with 215 sequences and 9 action classes, so most methods
can obtain a good accuracy. The most difficult dataset
should be the largest dataset NTU RGB+D dataset, which
consists of 56880 sequences and covers various of daily
actions and pair actions. The cross subject accuracy on
it just surpasses 70% due to various entangled actions as
analyzed in Section V-C. Specifically, the phenomenon of
entangled actions deteriorates in HDM05, which contains
many confusable classes, such as walk step number, walk
start with left or right, etc.
Cross subject is more difficult than cross view or cross
sample. The phenomenon is observed from Table IV,
Table V, and Table II (the left/right column w.r.t cross
subject/cross sample). It is easy to understand, each subject has itself action characteristics. In the cross subject
task, more unforeseeable information exists the testing
set, compared to the other tasks.
0.1
1.00
bow
ne
ve
ttle
wa a bo r pho
m swe
o
r
f
an
nk
dri
h
p
n
p
e
cla t lac dow nd u watc
h
sit
sta ead
tig
r
w
0.0
bo
Fig. 4. Confusion matrix of our A2 GNN on Florence 3D Actions dataset.
•
semantic features from spatial skeletal graphs like the
standard convolutional neural network.
The proposed deep graph method is superior to the
recent graph-based method [13]. As shown in Table III,
our A2 GNN has a large improvement (about 7%) in
contrast to the work [13]. In principle, our A2 GNN is
very different from this work [13], although graph is used
for both. First, the graph is used to describe skeleton at
each temporal slice for our method, not those segmented
C. Analysis of Confusion Matrices
To further reveal what classes are easy to confuse with
others, we show confusion matrices on HDM05, NTU RGB+D
and Florence datasets, repectively in Fig. 2, Fig. 3, and Fig. 4.
For LSC dataset, we don’t depict its confusion matrix due to
the different evaluation criterion (precision and recall).
For the HDM05 dataset, as shown in Fig. 2(a), we give the
confusion matrix of 130 classes in the case of recognition
rate 84.47% (i.e., the testing protocol follows the literature [55]). The diagonal characterizes the correct classification
for each action, and those non-diagonals depict the confusion
results cross different classes. In order to view them more
clearly, we provide two close up areas (Part A and Part
B) in Fig. 2(b) and Fig. 2(c). As observed from the two
subfigures, our proposed method suffers some failures in
9
(a) Drink from a bottle
(b) Answer phone
(c) Clap
(d) Tight lace
(e) Sit down
(f) Stand up
(g) Read watch
(h) Bow
(i) Wave
(j) Wave
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
(k) Colorbar
Fig. 5. Visual examples of detected salient action units of all 9 different action classes on Florence 3D. Higher weights in the colorbar means more important
to characterize the action. Note that the motion sequences in (i) and (j) are annotated as one class (“wave”) in this dataset, although one is left arm while the
other is right arm.
10
(a) Drink water
(b) Eat meal/snack
(c) Brushing teeth
(d) Use a fan (with hand or paper)
(e) Pickup
(f) Throw
(g) Wear a shoe
(h) Cheer up
(i) Hopping (one foot jumping)
(j) Jump up
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
(k) Colorbar
Fig. 6. Visual examples of detected salient action units on NTU RGB+D. Higher weights in the colorbar means more important to characterize the action.
11
TABLE VI
R ESULTS OF TUNING NETWORK MODULES .
A2 GNN filt. AU det.
A2 GNN AU det.
A2 GNN
HDM05
Protocol [54]
Protocol [55]
Acc.
Acc.
71.67%
77.74%±1.61
75.28%
83.01%±1.43
76.46%
84.47%±1.52
Florence
Acc.
93.49%
96.74%
98.60%
distinguishing those very similar actions. For example, the
depositing and grabbing are seriously confused, as “depositFloorR” and “grabFloorR” are almost visually consistent in
knees bent and arms stretch. For a basic kicking behavior,
the actions of left foot forwards (“kickLFront1Reps”) or left
foot sideways (“kickLSide1Reps”) are subtle with small interclass distance. Likewise, there are some other confusable
actions, such as “kickRFront1Reps” vs “kickRSide1Reps”,
“punchLFront1Reps” vs “punchLFront2Reps” and so on. Note
that the postfixes “1Reps” and “2Reps” means the repetitive
times of a action.
Different from HDM05 with finer-partitioned action classes,
NTU RGB+D contains many pairs of reversed actions, such as
“wear A” vs “take off A”, “put on A” vs “take off A”, etc. As
shown in Fig. 3(c), these reversed pairs are easy to be confused
when only considering human skeleton data. Besides, similar
to the observation from HDM05, the confusion cases occur in
those similar motions, such as “pat on back of other person” vs
“point finger at the other person”. Likewise, this phenomenon
happens in Florence 3D, as shown in Fig. 4. Although the
accuracies of 100% are achieved on seven actions (i.e., wave,
answer phone, clap, tight lace, sit down, stand up, bow), there
are a few uncorrect classifications including “drink from a
bottle” to “answer phone”, “read watch” to “wave”/“answer
phone”. A future possible solution to these above confusable
phenomenons is that the contextual information in the RGB
color space may be considered to compensate for the plain
coordinate information of skeletal joints to some extent.
D. Visualization of Action Units
To verify whether the action-attending layer detects those
salient action units, we provide some visualization examples in
Fig. 5 and Fig. 6, where skeletal joints are colored according
to the learnt weights. Note that here we exhibit the first actionattending layer because the learnt weights are directly associated with skeletal joints. From the visualization of Florence 3D
in Fig. 5, we can find that our A2 GNN is able to learn those
salient joints for all 9 actions. For examples, for the “wave”
action, those joints on the moving right arm are important; for
the “tight lace” action, the joints of hands and knees (bent)
is endowed with higher weights. Specifically, for the same
“wave” action, our method can still correctly detect the moving
units of the left arm or the right arm, as shown in Fig. 5(i)
and Fig. 5(j).
For the more complex action dataset, NTU RGB+D, we
can still observe that those detected salient action units are
almost matched to our intuitive understanding. For the “pick
LSC
Cross Sample
Cross Subject
Prec.
Rec.
Prec.
Rec.
79.03%
76.96%
73.21%
70.77%
86.56%
85.71%
79.97%
79.92%
87.61%
88.10%
83.98%
82.03%
Accuracy(%)
Method
NTU RGB+D
Cross Subject
Cross View
Acc.
Acc.
63.18%
68.82%
68.26%
80.08%
72.74%
82.80%
78
76
74
72
70
68
66
0
2
4
6
8
10
12
14
16
Size K of receptive fields
Fig. 7. The performance trend of different receptive fields K on HDM05.
up” action, the units of head, hands and knees are crucial
to identify this action. For the “hopping” action (right foot
jumping) in Fig. 6(i), besides those salient joints on head,
hands and knees, the joint on left shoulder is still more salient
than that of right shoulder due to the more drastic motion
variations on left shoulder compared to right shoulder. In
contrast to Florence 3D, as NTU RGB+D is configured with
more sensors to record human motions, thus more detailed
motions can be captured, e.g., the foe joint on the “wear a
shoe” action, the “hopping” action.
Consequently, the above observations indicate that the
action-attending layer can adaptively weight skeletal joints for
different actions, and the detected salient action units almost
conform to our cognitive understanding on human actions.
Moreover, the detection of action units can bring some gains
of the accuracy as analyzed in Section V-E.
E. Tuning of Our Network
In our proposed A2 GNN, there are three main modules:
spectral graph filtering, action unit detection, and temporal
motion modeling. The last module is an indispensable unit to
our network. To dissect our network architecture, we conduct
experiments on the four datasets by removing the former two
modules (i.e., A2 GNN filt. AU det.) or only removing AU
detection (i.e., A2 GNN AU det.). The comparison results
are reported in Table VI. As observed from this table, the
detection of action units can be helpful for action recognition
more or less, as skeletal joints are successfully activated with
different weights as analyzed in Section V-D. Further, the
spectral graph filtering module plays a tremendous role in
promoting recognition accuracy, as exhibited at the former
two rows in the table. The main reason should be that highlevel semantic features are extracted from graphs like the
conventional convolution network on grid-shape images.
Among the parameters of our network, the most key is
the size K of receptive fields. With the increase of K, the
local filtering region, i.e., covering the hopping neighbors,
12
will become larger. To check the influence of different K
values, we conduct an experiment on HDM05 dataset by
testing K = 2, 4, 6, 8, 10, 12, 14. The results are shown in
Fig. 7, we can find that, the performance improves with the
increase of K due to larger receptive fields, and then degrades
after K = 10, for which a possible reason is the locality of
salient action units.
VI. C ONCLUSION
In this paper, an end-to-end action-attending graphic neural
network (A2 GNN) is proposed to deal with the task of
skeleton-based action recognition. In order to extract deep
features from body skeletons, we model human skeletons into
undirected attribute graphs, and then perform spectral graph
filtering on skeletal graphs like the standard CNN. To detect
those salient action units crucial to identify human motions, we
further design an action-attending layer to adaptively weight
skeletal joints for different actions. Those extracted deep graph
features at consecutive frames are finally fed into a recurrent
network of LSTM. Extensive experiments and analyses have
indicated that the modules of spectral graph filtering and action
unit detection play an important role in the improvement on
action classification. Especially, the action-attending layer also
produces some interesting salient action units, which may be
understandable from the view of our cognition. Further, our
proposed A2 GNN has achieved the state-of-the-art results on
the four public skeleton-based action datasets, including the
current largest and most challenging NTU RGB+D dataset. In
the future, we will explore the fusion RGB color information
into our network.
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| 1 |
A Circuit-Based Approach to Efficient
Enumeration
Antoine Amarilli1 , Pierre Bourhis2 , Louis Jachiet3 , and Stefan
Mengel4
1
2
3
arXiv:1702.05589v2 [cs.DS] 5 May 2017
4
LTCI, Télécom ParisTech, Université Paris-Saclay; France
antoine.amarilli@telecom-paristech.fr
CRIStAL, CNRS UMR 9189 & Inria Lille; France
pierre.bourhis@univ-lille1.fr
Université Grenoble Alpes; France
louis.jachiet@inria.fr
CNRS, CRIL UMR 8188; France
mengel@cril.fr
Abstract
We study the problem of enumerating the satisfying valuations of a circuit while bounding the
delay, i.e., the time needed to compute each successive valuation. We focus on the class of
structured d-DNNF circuits originally introduced in knowledge compilation, a sub-area of artificial
intelligence. We propose an algorithm for these circuits that enumerates valuations with linear
preprocessing and delay linear in the Hamming weight of each valuation. Moreover, valuations
of constant Hamming weight can be enumerated with linear preprocessing and constant delay.
Our results yield a framework for efficient enumeration that applies to all problems whose
solutions can be compiled to structured d-DNNFs. In particular, we use it to recapture classical
results in database theory, for factorized database representations and for MSO evaluation. This
gives an independent proof of constant-delay enumeration for MSO formulae with first-order free
variables on bounded-treewidth structures.
1998 ACM Subject Classification F.2.2 Nonnumerical Algorithms and Problems
Keywords and phrases circuits; constant-delay; enumeration; d-DNNFs; MSO
Digital Object Identifier 10.4230/LIPIcs...
1
Introduction
When a computational problem has a large number of solutions, computing all of them at
once can take an unreasonable amount of time. Enumeration algorithms are an answer to
this challenge, and have been studied in many contexts (see [35] for an overview). They
generally consist of two phases. First, in a preprocessing phase, the input is read and indexed.
Second, in an enumeration phase that uses the result of the preprocessing, the solutions are
computed one after the other. The goal is to limit the amount of time between each pair of
successive solutions, which is called delay.
We focus on a well-studied class of efficient enumeration algorithms with very strict
requirements: the preprocessing must be linear in the input size, and the delay between
successive solutions must be constant. Such algorithms have been studied in particular for
database applications, to enumerate query answers (see [18, 5, 19, 6, 7, 23, 24] and the recent
survey [32]), or to enumerate the tuples of factorized database representations [28].
One shortcoming of these existing enumeration algorithms is that they are typically
shown by building a custom index structure tailored to the problem, and designing ad hoc
© Antoine Amarilli; Pierre Bourhis; Louis Jachiet; Stefan Mengel;
licensed under Creative Commons License CC-BY
Leibniz International Proceedings in Informatics
Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany
XX:2
A Circuit-Based Approach to Efficient Enumeration
Linear-time preprocessing phase (Sec. 3–5)
Enumeration phase (Sec. 6)
d-DNNF Prp. augmented Prp. normal Thm. normal Prp. compressed Prp.
satisfying
traces
3.9 d-DNNF 4.3 d-DNNF 5.4 d-DNNF 6.3
6.5 valuations
(Def.
3.6)
(Def.
4.2)
(Def.
6.1)
+OR-index
v-tree
Figure 1 Overview of the proof of Theorem 2.1
preprocessing and enumeration algorithms. This makes it hard to generalize them to
other problems, or to implement them efficiently. In our opinion, it would be far better if
enumeration for multiple problems could be done via one generic representation of the results
to enumerate, reusing general algorithms for the preprocessing and enumeration phases.
This paper accordingly proposes a new framework for constant-delay enumeration algorithms, inspired by the field of knowledge compilation in artificial intelligence. Knowledge
compilation studies how the solutions to computational problems can be compiled to generic
representations, in particular classes of Boolean circuits, on which reasoning tasks can then
be solved using general-purpose algorithms. In this paper, we show how this knowledge
compilation approach can be implemented for constant-delay enumeration, by compiling to a
prominent class of circuits from knowledge compilation called deterministic decomposable
negation normal form (in short, d-DNNF) [16]. These circuits generalize several forms of
branching programs such as OBDDs [17] and were recently shown to be more expressive
than Boolean circuits of bounded treewidth [12]. Further, there are many efficient algorithms
to compute d-DNNF representations of small width CNF formulae for a wide range of
width notions [11], and even software implementations to compute such representations for
given Boolean functions [29, 13]. d-DNNFs are also intimately related to state-of-the-art
propositional model counters based on exhaustive DPLL [20], to syntactically multi-linear
arithmetic circuits [31], and to probabilistic query evaluation in database theory [21].
Our main technical contribution is an efficient algorithm to enumerate the satisfying
valuations of a d-DNNF under a standard structuredness assumption, namely, assuming that
a so-called v-tree is given [30]: this assumption holds in all works cited above. Our first main
result (Theorem 2.1) shows that we can enumerate the satisfying valuations of such a circuit
with linear preprocessing and delay linear in the Hamming weight of each valuation. Further,
our second main result (Theorem 2.2) shows that, if we impose a constant bound on the
Hamming weight, we can enumerate the valuations with constant delay. In these results we
express valuations succinctly as the set of the variables that they set to true.
To show our results, we consider d-DNNFs under a semantics where negation is implicit, i.e.,
variables that are not tested must be set to zero. In analogy to zero-suppressed OBDDs [36],
we call this semantics zero-suppressed. The preprocessing phase of our algorithm rewrites
such circuits to a normal form (Section 4) and pre-computes a multitree reachability index on
them (Section 5), which allows us to enumerate efficiently the traces of the circuit, and obtain
the desired valuations (Section 6). To enumerate for d-DNNFs in standard semantics, we
show how to rewrite the input circuit to zero-suppressed semantics, using the structuredness
assumption, and using a new notion of range gates to make the process efficient (Section 3).
The overall proof is very modular; for an outline see Figure 1.
Our second contribution is to illustrate how our circuit-based framework and enumeration
results can be useful in database theory. As a proof of concept, we present two known results
that we can extend, or recapture with an independent proof. First, we re-prove with our
framework that the answers to MSO queries on trees and bounded-treewidth structures can
be enumerated with linear preprocessing and delay linear in each assignment, i.e., constant-
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
delay if the free variables are first-order. This was previously shown by Bagan [5] with a
custom construction, by Kazana and Segoufin [23] using a powerful result of Colcombet [14],
and by Courcelle [15] in a more general setting (but with O(n log n) preprocessing) using
AND/OR-DAGs (that share some similarities with DNNFs). Our proof follows our proposed
approach: we compute a circuit representation of the output following the provenance
constructions in [3], and simply apply our enumeration result to this circuit. Second, we show
how d-DNNFs generalize the deterministic factorized representations of relational instances
studied in database theory [28]. This allows us to give enumeration algorithms with linear
preprocessing and constant delay for arbitrary deterministic d-representations, extending the
enumeration result of [28].
The paper is structured as follows. Section 2 gives the main definitions and results.
We then describe the preprocessing phase of our algorithm: we reduce the input circuit
to zero-suppressed semantics in Section 3, rewrite it to a normal form in Section 4, and
compute the multitree index in Section 5. We then describe the enumeration algorithm in
Section 6. We present our two applications in Section 7 and conclude in Section 8. Due to
space restrictions, many details and the proofs are found in the appendix.
2
Preliminaries and Problem Statement
Circuits. A circuit C = (G, W, g0 , µ) is a directed acyclic graph (G, W ) whose vertices G
are called gates, whose edges W are called wires, which has an output gate g0 ∈ G, and where
each gate g ∈ G has a type µ(g) among ∧ (AND-gate), ∨ (OR-gate), ¬ (NOT-gate), or var
(variable). We represent the circuit with adjacency lists that indicate, for each gate g ∈ G,
the gates having a wire to g (called the inputs of g), and the gates of which g is an input; the
number of such gates is called respectively the fan-in and fan-out of g. The size |C| of this
representation is then |G| + |W |. We require that variables have fan-in zero, that NOT-gates
have fan-in one, and we will always work on negation normal form (NNF) circuits where the
input of NOT-gates is always a variable. A circuit without NOT-gates is called monotone.
We write Cvar for the set of variables of C. A valuation of Cvar is a function ν : Cvar →
{0, 1}. A circuit defines a Boolean function on Cvar , that is, a function φ that maps each
valuation of Cvar to {0, 1}. For any valuation ν, the image of ν by φ is defined by substituting
each gate in Cvar by its value according to ν, evaluating the circuit using the standard
semantics of Boolean operations, and returning the value of the output gate g0 . Note that
AND-gates (resp., OR-gates) with no inputs always evaluate to 1 (resp., to 0) in this process.
We call a gate unsatisfiable if it evaluates to 0 under all valuations (and satisfiable otherwise);
we call it 0-valid if it evaluates to 1 under the valuation which sets all variable gates to 0.
We say that ν satisfies C if φ maps ν to 1 (i.e., g0 evaluates to 1 under ν), and call ν a
satisfying valuation.
For enumeration, we represent a valuation ν of C as the set Sν of variables of Cvar that it
sets to 1, i.e., {g ∈ Cvar | ν(g) = 1}. We call Sν an assignment, and a satisfying assignment
if ν is a satisfying valuation. The Hamming weight |ν| of ν is the cardinality of Sν . Unlike
valuations, assignments of constant Hamming weight are of constant size, no matter the size
of Cvar . We write {} for the empty assignment, and write ∅ for an empty set of assignments.
The main class of circuits that we will study are d-DNNFs [16], of which we now recall
the definition. We say that an AND-gate g of a circuit C is decomposable if there is no pair
g1 6= g2 of input gates to g such that some variable g 0 ∈ Cvar has a directed path both to g1
and to g2 : intuitively, a decomposable AND-gate is a conjunction of inputs on disjoint sets of
variables. We say that an OR-gate g of C is deterministic if there is no pair g1 6= g2 of input
XX:3
XX:4
A Circuit-Based Approach to Efficient Enumeration
gates of g and valuation ν of C such that g1 and g2 both evaluate to 1 under ν: intuitively, a
deterministic OR-gate is a disjunction of mutually exclusive inputs. A circuit C is a d-DNNF
if all its AND-gates are decomposable, and all its OR-gates are deterministic.
We will further study the subclass of d-DNNFs called structured d-DNNFs, consisting
of the d-DNNFs having a v-tree [30]. A v-tree on a set S of variables is a rooted unranked
ordered tree T whose set of leaves is exactly S. We write <T for the order on T in which
the nodes are visited in a pre-order traversal. For a circuit C, we say that a v-tree T on the
set Cvar is a v-tree of C if there is a mapping λ from the gates of C to the nodes of T such
that: (i) λ maps the variables of C to themselves; (ii) for each wire (g, g 0 ) of C, the node λ(g)
is a descendant of λ(g 0 ) in T ; and (iii) for each AND-gate g of C with inputs g1 , . . . , gn (in
this order), the nodes λ(g1 ), . . . , λ(gn ) are descendants of λ(g), none of them is a descendant
of another, and we have λ(g1 ) <T · · · <T λ(gn ). Note that having a v-tree implies (by
point iii) that all AND-gates are decomposable. A structured d-DNNF is a d-DNNF C given
with a v-tree T of C.
Enumeration. As usual for efficient enumeration algorithms [32], we work in the RAM
model with uniform cost measure (see, e.g., [2]), where pointers, numbers, labels for vertices
and edges, etc., have constant size; thus an assignment has size linear in its Hamming weight.
An enumeration algorithm with linear-time preprocessing computes a set of results S(I)
from an input instance I. It consists of two parts. First, the preprocessing phase takes as
input an instance I and produces in linear time an indexed instance I 0 and an initial state.
Second, the enumeration phase repeatedly calls an algorithm A. Each call to A takes as input
the indexed instance I 0 and the current state, and returns a result and a new state: a special
state value indicates that the enumeration is over so A should not be called again. The
results produced by the calls to A must be exactly the elements of S(I), with no duplicates.
We say that the enumeration algorithm has linear delay if the time to produce each new
output element E is linear in the size of E (and independent of the input instance I). In
particular, when the output elements have constant size, each element must be produced
with constant delay, which we call constant-delay enumeration. The memory usage of an
enumeration algorithm is the maximum number of memory cells used during the enumeration
phase (not counting the indexed instance I 0 , which resides in read-only memory), expressed
as a function of the input instance size |I| and of the size |O| of the largest output (as in [5]).
Note that constant delay does not imply a bound on memory usage, because the state can
become large even if we only add a constant quantity of information at each step.
Main results. Our main theorem on circuit enumeration is the following:
I Theorem 2.1. Given a structured d-DNNF C with a v-tree T , we can enumerate its
satisfying assignments with linear-time preprocessing, linear delay, and memory usage
O(|O| · log |C|), where |O| is the Hamming weight of the largest assignment.
If we fix a maximal Hamming weight k ∈ N, we can show constant-delay enumeration:
I Theorem 2.2. For any k ∈ N, given a structured d-DNNF C with a v-tree T , we can
enumerate its satisfying assignments of Hamming weight ≤ k with preprocessing in time
O(|T | + k 2 · |C|), delay in O(k), and memory in O(k · log |C|), i.e., linear-time preprocessing
and constant delay for fixed k.
In both results, remember that |C| is the number of gates and wires of C. We prove our
two results in Sections 3–6: the first three sections present the three steps of the linear-time
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
preprocessing algorithm, and the last one presents the enumeration algorithm. We then use
the results for database applications in Section 7, in particular re-proving constant-delay
enumeration for MSO queries with free first-order variables on bounded-treewidth structures.
The memory bound in our results is not constant and depends logarithmically on the
input. While we think that this is reasonable, we also show constant-memory enumeration
for some restricted circuit classes: the details are deferred to Appendix F for lack of space.
3
Reducing to Zero-Suppressed Semantics
We start our linear preprocessing by rewriting the input circuit to an alternative zerosuppressed semantics where negation is coded implicitly. For this rewriting, we will use the
structuredness assumption on the circuit, in a weaker form called having a compatible order:
this is the first thing that we present. We will also extend slightly our circuit formalism, to
concisely represent sets of inputs with range gates that use this order: we present this second.
Last, we present the alternative semantics, and give our translation result (Proposition 3.9).
Compatible orders.
Our structuredness requirement is to have a compatible order:
I Definition 3.1. An order for a circuit C is a total order < on Cvar . For two variables
g1 , g2 ∈ Cvar , the interval [g1 , g2 ] consists of the variables g which are between g1 and g2
for <, i.e., g1 ≤ g ≤ g2 or g2 ≤ g ≤ g1 . The interval of a gate g is then [min(g), max(g)],
where min(g) denotes the smallest gate according to < that has a directed path to g, and
max(g) is defined analogously. In particular, the interval of any g ∈ Cvar is [g, g] = {g}.
We say that the order < is compatible with C if, for every AND-gate g with inputs
g1 , . . . , gn (in this order), for all 1 ≤ i < j ≤ n, we have max(gi ) < min(gj ); in particular,
the intervals of g1 , . . . , gn are pairwise disjoint.
Note that, if a circuit C has compatible order <, every AND-gate g is decomposable: if
some g 0 ∈ Cvar had a directed path to two inputs of g then their intervals would intersect.
Observe further that, given a structured d-DNNF C with a v-tree T , we can easily
compute a compatible order < for C in linear time in T . Indeed, let < be the restriction
to Cvar of the order <T on T given by pre-order traversal. Considering any suitable mapping λ
from C to T , for any gate g, we know that min(g) is no less than the first leaf of T in <
reachable from λ(g), and that max(g) is no greater than the last leaf reachable from λ(g).
The intervals of the inputs g1 , . . . , gn to an AND-gate are then pairwise disjoint, because
they are included in the sets of reachable leaves from the nodes λ(g1 ), . . . , λ(gn ) in the v-tree,
and none of these nodes is a descendant of another, so they cannot share any descendant
leaf. Hence, if we know a v-tree T for C then we know an order < for C.
Augmented circuits.
We use compatible orders to define circuits with a new type of gates:
I Definition 3.2. For k ∈ N, we define a k-augmented circuit C as a circuit with a compatible
order < and with k additional types of gates, called range gates: there are the = i-range gates
for 0 ≤ i < k, and the ≥ k-range gates. These gates must have exactly two inputs, which
must be variables of C (they are not necessarily different, so we allow multi-edges in circuits
for this purpose). We talk of augmented circuits when the value of k does not matter.
When evaluating a k-augmented circuit under a valuation ν, each = i-range gate g (resp.,
≥ k-range gate g) with inputs g1 and g2 evaluates to 1 if there are exactly i gates (resp., at
least k gates) in [g1 , g2 ] set to 1 by ν; note that g may be unsatisfiable if |[g1 , g2 ]| is too small.
XX:5
XX:6
A Circuit-Based Approach to Efficient Enumeration
Range gates are related to the threshold gates studied in circuit complexity (see e.g. [10]),
but we only apply them directly to variables. We can of course emulate range gates with
standard gates, e.g., ≥ 0-gates always evaluate to 1, and a ≥ 1-range gate on g1 and g2 can
be expressed as an OR-gate g having the interval [g1 , g2 ] as its set of inputs. However, the
point of range gates is that we can now write this in constant space, thanks to <. This will
be important to rewrite circuits in linear time to our alternative semantics.
Zero-suppressed semantics. We are now ready to introduce our alternative semantics for
augmented circuits. We will do so only on monotone augmented circuits, i.e., without
NOT-gates, because negation will be coded implicitly. We use the notion of traces:
I Definition 3.3. An upward tree T of a monotone augmented circuit C = (G, W, µ, g0 ) is a
subgraph (G0 , W 0 ) of C, with G0 ⊆ G and W 0 ⊆ W , which is a rooted tree up to reversing
the direction of the wires. For all (g 0 , g) ∈ W 0 , we call g 0 ∈ G0 a child of g ∈ G0 in T , and
call g the parent of g 0 in T ; note that g 0 is an input of g in C. A gate g ∈ G0 in T is an
internal gate of T if it has a child in T , and a leaf otherwise. T is a partial trace if its internal
gates are AND-gates and OR-gates and if its gates satisfy the following:
for every AND-gate g in T , all its inputs in C are children of g in T ;
for every OR-gate g in T , exactly one of its inputs in C is a child of g in T .
Note that T cannot contain OR-gates with no inputs, and that its leaves consist of range
gates, variable gates, and AND-gates with no inputs. We call T a trace of C if its root is g0 .
We define traces as trees, not general DAGs, because we cannot reach the same gate
in a trace by two different paths (remember that AND-gates in augmented circuits are
decomposable). We can see each trace (G0 , W 0 ) of C = (G, W, µ, g0 ) as an augmented circuit
(G0 , W 0 , µ, g0 ), up to adding to range gates in the trace their inputs in C, and we then have:
I Observation 3.4. A valuation ν of a monotone augmented circuit C satisfies C if and
only if ν satisfies a trace of C.
Observe that we can check if a valuation ν of C satisfies a trace T simply by looking at
the value of ν on the leaves of T ; the definition of ν outside the intervals of the leaves does
not matter. We will change this point to define zero-suppressed semantics, where ν can only
satisfy T if it maps to 0 all the other variables. We then call ν a minimal valuation of T :
I Definition 3.5. Let C be a monotone augmented circuit, ν be a valuation of C, and T be
a trace or partial trace of C. We call ν a minimal valuation of T if:
For every variable g in T , we have ν(g) = 1;
For every ./ i-range gate g in T with inputs g1 and g2 in C (where ./ ∈ {=, ≥} and i ∈ N),
the number n of variables in [g1 , g2 ] that are set to 1 by ν satisfies the constraint n ./ i;
All other variables of Cvar are set to 0 by ν.
Note that this implies that ν satisfies T . We call ν a minimal valuation for a gate g of C
(resp., for C) if it is a minimal valuation of a partial trace rooted at g (resp., at the output g0 ).
Note that C may have two minimal valuations ν1 and ν2 whose assignments S1 and S2 are
such that S1 ( S2 (see, e.g., Example 3.7 below). Minimality only imposes that, relatively to
a trace T , the valuation sets to 0 all variables that are not tested in T . Minimal valuations
allow us to define the zero-suppressed semantics of a monotone augmented circuit C: the
satisfying valuations of C in this semantics are those that are minimal for some trace.
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
I Definition 3.6. A monotone augmented circuit C in zero-suppressed semantics captures the
(generally non-monotone) Boolean function Φ mapping a valuation ν to 1 iff ν is a minimal
valuation for C. We call S(C) the set of satisfying assignments of C in this semantics.
We call C a d-DNNF in zero-suppressed semantics if it satisfies the analogue of determinism: there is no OR-gate g with two inputs g1 6= g2 and valuation ν of C that is a minimal
valuation for both g1 and g2 . (Decomposability again follows from the compatible order.)
I Example 3.7. Consider the monotone circuit C whose output gate is an OR-gate with
three inputs: x, y, and an AND-gate of y and z. The circuit C captures x ∨ y in standard
semantics, and it is not a d-DNNF. C has three traces, having one minimal valuation each.
In the zero-suppressed semantics, we have S(C) = {{x}, {y}, {y, z}}, and C captures the
Boolean function (x ∧ ¬y ∧ ¬z) ∨ (¬x ∧ y). Further, C is a d-DNNF in that semantics.
Zero-suppressed semantics makes enumeration easier, because it expresses negation
implicitly in a very concise way. The name is inspired by zero-suppressed OBDDs [36,
Chapter 8]: variables that are not tested when following a trace are implicitly set to 0. We
can equivalently define the assignments S(C) of C inductively as follows:
I Lemma 3.8. Let C be a monotone augmented circuit. Let us define inductively a set of
assignments S(g) for each gate g in the following way:
for all g ∈ Cvar , we set S(g) := {g};
for all ./ i-range gates g with inputs g1 and g2 , we set S(g) := {t ⊆ [g1 , g2 ] | |t| ./ i};
S
for all OR-gates g with inputs g1 , . . . , gn , we set S(g) := 1≤i≤n S(gi ) (with S(g) = ∅ if
g has no inputs);
for all AND-gates g with inputs g1 , . . . , gn , we set S(g) := {S1 ∪ · · · ∪ Sn | (S1 , . . . , Sn ) ∈
Q
1≤i≤n S(gi )} (with S(g) = {{}} if g has no inputs); observe that the unions are always
disjoint because C has a compatible order.
Then, for any gate g, the set S(g) contains exactly the assignments that describe a minimal
valuation for g. In particular, for g0 the output gate of C, the set S(g0 ) is exactly S(C).
We can now state our main reduction result for this section: we can rewrite any d-DNNF
to an equivalent d-DNNF in zero-suppressed semantics, by introducing ≥ 0-range gates to
write explicitly that the variables not tested in a trace are unconstrained:
I Proposition 3.9. Given a d-DNNF circuit C and a compatible order <, we can compute
in linear time a monotone 0-augmented circuit C ∗ having < as a compatible order, such
that C ∗ is a d-DNNF in zero-suppressed semantics and such that S(C ∗ ) is exactly the set of
satisfying assignments of C.
4
Reducing to Normal Form Circuits
In this section, given Proposition 3.9, we work on a monotone 0-augmented d-DNNF circuit C
in zero-suppressed semantics, with a compatible order < to define the semantics of range gates.
We present our next two preprocessing steps for the enumeration of the assignments S(C)
of C: restricting our attention to valuations of the right Hamming weight (for Theorem 2.2
only), and bringing C to a normal form that makes enumeration easier.
Homogenization. Our input augmented circuit C in zero-suppressed semantics may have
satisfying assignments of arbitrary Hamming weight. When proving Theorem 2.1, this is
intended, and the construction that we are about to describe is not necessary. However, when
proving Theorem 2.2 about enumerating valuations of constant weight, we need to restrict
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A Circuit-Based Approach to Efficient Enumeration
our attention to such valuations, to ensure constant delay. We do so using the following
homogenization result, adapted from the technique of Strassen [33]:
I Proposition 4.1. Given k ∈ N and a monotone augmented d-DNNF circuit C in zerosuppressed semantics with compatible order <, we can construct in time O(k 2 ·|C|) a monotone
augmented d-DNNF circuit C 0 in zero-suppressed semantics with compatible order < such
that S(C 0 ) = {t ∈ S(C) | |t| ≤ k}.
Proof sketch. We create k +2 copies of each gate g, with each copy capturing the assignments
of a specific weight from 0 to k inclusive (or, for the k + 2-th copy, the assignments with
weight > k). In particular, for ≥ 0-gates g, for 0 ≤ i ≤ k, we use an = i-gate for the copy of g
capturing weight i. We then re-wire the circuit so that weights are correctly preserved. J
Note that this is the only place where our preprocessing depends on k: in particular, for
constant k, the construction is linear-time. This result allows us to assume in the sequel
that the set of assignments of the circuit in zero-suppressed semantics contains precisely the
valuations that we are interested in, i.e., those that have suitable Hamming weight.
Normal form. Now that we have focused on the interesting valuations of our circuit C, we
can bring it to our desired normal form:
I Definition 4.2. A normal circuit C is a monotone augmented circuit such that:
C is arity-two, i.e., each gate has fan-in at most two.
C is ∅-pruned, i.e., no gate g is unsatisfiable (i.e., each gate has some minimal valuation).
C is {}-pruned, i.e., no gate g is 0-valid (i.e., the valuation that sets all variables to 0 is
not a minimal valuation for any gate).
C is collapsed, i.e., it has no AND-gate with fan-in 1.
C is discriminative, i.e., for every OR-gate g with an input that is not an OR-gate (we
call g an exit), g has fan-in 1, fan-out 1, and the one gate with g as input is an OR-gate.
C is a normal d-DNNF if it is additionally a d-DNNF in the zero-suppressed semantics.
The pruned requirements slightly weaken the expressiveness of normal circuits C, because
they forbid that S(C) = ∅ or that {} ∈ S(C). These cases will be easy to handle separately.
The main result of this section is then the following:
I Proposition 4.3. Given a monotone augmented d-DNNF circuit C in zero-suppressed
semantics with compatible order < and with S(C) 6= ∅ and S(C) 6= {{}}, we can build in
O(|C|) a normal d-DNNF C 0 , with < as a compatible order, such that S(C 0 ) = S(C)\{{}}.
Proof sketch. We reuse the construction of Proposition 4.1 with k = 1 to split the gates
so that they are not 0-valid, we eliminate bottom-up the unsatisfiable gates, we make C
arity-two in a straightforward way, we collapse all AND-gates with fan-in 1, and we make C
discriminative by inserting new OR-gates (i.e., the exits) on all wires from non-OR-gates to
OR-gates.
J
This result allows us to assume in the sequel that we are working with normal d-DNNFs.
5
Indexing OR-Components
This section presents the last step of our preprocessing. Remember that we now work with
a normal d-DNNF, and we want to enumerate its set of assignments. Intuitively, this last
preprocessing will help us to enumerate the choices that can be made at OR-gates. Formally,
we will work on the OR-components of our circuit:
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
I Definition 5.1. The OR-component K of an OR-gate g in a normal circuit C is the set of
OR-gates that can be reached from g by going only through OR-gates, following wires in
either direction. We abuse notation and also see K as a DAG, whose vertices are the gates
of K, and whose edges are the wires between them.
Recall from Definition 4.2 that, as C is discriminative, all gates of an OR-component K
with no inputs in K must be exits; we call them the exits of K. For a gate g in K, the exits
of g are the gates of K that have a directed path to g in K; intuitively, they are the “possible
choices” for a partial trace rooted at g. Our goal is to preprocess each OR-component of C
to be able to enumerate efficiently the exits of all OR-gates of C. This enumeration task is
tricky, however: exploring K naively when enumerating would take time dependent of C,
but materializing a reachability index would take quadratic preprocessing time. Thus, we
design an efficient indexing scheme, using the fact that OR-components are multitrees:
I Definition 5.2. A DAG G is a multitree if it has no pair n =
6 n0 of vertices such that there
0
are two different directed paths from n to n . In particular, forests are multitrees, and so are
polytrees (DAGs with no undirected cycles).
I Lemma 5.3. For any normal d-DNNF C, each OR-component of C is a multitree.
We can then prepare the enumeration of exits of gates in OR-components, by designing
an efficient and generic indexing scheme on multitrees (see Appendix C). We deduce:
I Theorem 5.4. Given a normal d-DNNF C, we can compute in O(|C|) a structure called
OR-index allowing us to do the following: given an OR-gate g of C, enumerate the exits of g
in its OR-component K, with constant delay and memory usage O(log |K|).
6
Enumerating Assignments
We have described in the previous sections our linear-time preprocessing on the input circuit:
this produces a normal d-DNNF C together with an OR-index, and we wish to enumerate
its assignments S(C) in zero-suppressed semantics. In this section, we show that we can
enumerate the elements of S(C), producing each assignment t with delay O(|t|).
To prove this, we will go back to our definition of zero-suppressed semantics in Section 3,
namely, the minimal valuations of the traces of C (recall Definition 3.3). We will proceed in
two steps. First, we use our preprocessing and the OR-index to show an efficient enumeration
scheme for the traces of C, in a compact representation called compressed traces. Second, we
show how to enumerate efficiently the minimal valuations of a compressed trace.
Compressed traces. We cannot enumerate traces directly because they can be arbitrarily
large (e.g., contain long paths of OR-gates) even for assignments of small weight. We
accordingly define compressed traces as a variant of traces that collapse such paths:
I Definition 6.1. An OR-path of a monotone augmented circuit C = (G, W, µ, g0 ) is a path
from g ∈ G to g 0 ∈ G where all intermediate gates are OR-gates; in particular if (g, g 0 ) ∈ W
then there is an OR-path from g to g 0 . A compressed upward tree of C is a pair (G0 , W 0 )
where G0 ⊆ G and where W 0 ⊆ G0 × G0 is such that for each (g, g 0 ) ∈ W 0 there is an OR-path
from g to g 0 : we require that T is a rooted tree up to reversing the direction of the edges. T
is a compressed partial trace if its internal gates are AND-gates and OR-gates such that:
for every AND-gate g in T , all its inputs in C are children of g in T ;
for every exit g in T (it is an OR-gate), its one input in C is a child of g in T ;
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A Circuit-Based Approach to Efficient Enumeration
for every non-exit OR-gate g in T , exactly one of its exits g 0 in C is a child of g in T .
We write |T | := |G0 |. We call T a compressed trace of C if its root is g0 . The minimal
valuations of a compressed trace are defined like for non-compressed traces (Definition 3.5).
The use of compressed traces is that their size is linear in that of their minimal valuations:
I Lemma 6.2. For any compressed trace T of a normal circuit C and minimal valuation ν
for T and C, we have |T | ≤ 6 · |ν|.
From a trace T in a normal d-DNNF C, we can clearly define a compressed trace T 0 with
the same leaves, as follows. Whenever T contains an OR-gate g whose parent gate g 0 in T is
not an OR-gate (or when g is the root of T ), as g cannot be an exit, we know that there is a
OR-path in T from g to an exit g 00 of g in its OR-component. We “compress” this OR-path
in T 0 as an edge from g to g 00 . Conversely, given a compressed trace T 0 , we can fill it to a
trace T with the same leaves, by replacing each edge from g to g 0 by a witnessing OR-path;
and there is only one way to do so because OR-components in C are multitrees (Lemma 5.3).
Hence, there is a bijection between traces and compressed traces that preserves the set of
leaves. As the minimal valuations of traces and compressed traces are defined in the same
way from their set of leaves, we can simply enumerate compressed traces instead of traces.
The following shows that we can perform enumeration of the compressed traces efficiently:
I Proposition 6.3. Given a normal d-DNNF C with its OR-index, we can enumerate its
compressed traces, with the delay to produce each compressed trace T being in O(|T |).
In particular, if all compressed traces have constant size, then the delay is constant.
Proof sketch. At each AND-gate, we enumerate the lexicographic product of the partial
traces of its two children; at each OR-gate, we enumerate its exits using the OR-index. J
Enumerating valuations of a compressed trace. We now show how, given a compressed
trace T , we can enumerate its minimal valuations (recall Definition 3.5). Restricting our
attention to the leaves of T , we can rephrase our problem in the following way:
I Definition 6.4. The assignment enumeration problem for a total order < on gates Cvar is
as follows: given pairwise disjoint intervals [g1− , g1+ ], . . . , [gn− , gn+ ], and cardinality constraints
./1 ii , . . . , ./n in , where 0 < ij ≤ [gj− , gj+ ] and ./j ∈ {=, ≥}, enumerate the values of the
products t1 × · · · × tn for all the assignments of the tj ⊆ [gj− , gj+ ] such that |tj | ./j ij for all j.
Indeed, remember that, as C is {}-pruned, the leaves of T consist of variables and range
gates, and their intervals are pairwise disjoint thanks to decomposability. A ./ i-gate with
inputs g − , g + codes the interval [g − , g + ] with cardinality constraint ./ i, and a variable g
simply codes [g, g] with constraint = 1. Further, thanks to {}-pruning, we know that no
range gate is labeled with = 0 or ≥ 0, and thanks to ∅-pruning, we know that no range gate
is labeled with an infeasible cardinality constraint. We claim:
I Proposition 6.5. We can enumerate the solutions to the assignment enumeration problem
for < on Cvar , with each solution t being produced with delay linear in its size |t|.
Again, this is constant-delay when all solutions have size bounded by a constant.
Proof sketch. We enumerate the possible assignments of weights to intervals with constantdelay, to reduce to the case where all cardinality constraints are equalities. We then enumerate
the assignments in lexicographic order, using an existing scheme [25, Section 7.2.1.3] to
enumerate the assignments in each interval.
J
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
We have now concluded the proof of Theorem 2.1 and 2.2: refer back to Figure 1 for an
overview of the proof of Theorem 2.1. Given our input d-DNNF C and v-tree T rewritten
to a compatible order, we rewrite C to an equivalent normal d-DNNF and compute the
OR-index. We then enumerate compressed traces, and enumerate the valuations for each
trace. The proof of Theorem 2.2 is the same except that we additionally use Proposition 4.1
before Proposition 4.3 to restrict to valuations of Hamming weight ≤ k.
7
Applications
We now present two applications of our main results. Our first application recaptures the
well-known enumeration results for MSO queries on trees [5, 22]. The second application
describes links to factorized databases and strengthens the enumeration result of [28].
MSO enumeration. Recall that the class of monadic second-order formulae (MSO) consists
of first-order logical formulae extended with quantification over sets, see e.g. [26]. The
enumeration problem for a fixed MSO formula φ(X1 , . . . , Xk ) with free second-order variables,
given a structure I, is to enumerate the answers of φ on I, i.e., the k-tuples (B1 , . . . , Bk )
of subsets of the domain of I such that I satisfies φ(B1 , . . . , Bk ). We measure the data
complexity of this task, i.e., its complexity in the input structure, with the query being fixed.
It was shown by Bagan [5] that MSO query enumeration on trees and bounded treewidth
structures can be performed with linear-time preprocessing and delay linear in each MSO
assignment; in particular, if the free variables of the formula are first-order, then the delay is
constant. This latter result was later re-proven by Kazana and Segoufin [23]. We show how
to recapture this theorem from our main results. From the results of Courcelle and standard
techniques (see, e.g., [22], Theorem 6.3.1 and Section 6.3.2), we restrict to binary trees.
I Definition 7.1. Let Γ be a finite alphabet. A Γ-tree T is a rooted unordered binary tree
where each node n ∈ T carries a label in Γ. We abuse notation and identify T to its node set.
MSO formulae on Γ-trees are written on the signature consisting of one binary predicate for
the edge relation and unary predicates for each label of Γ.
Let φ(X1 , . . . , Xk ) be an MSO formula on Γ-trees, and let T be a Γ-tree. We will show
our enumeration result by building a structured circuit capturing the assignments of φ on T :
I Definition 7.2. A singleton on X1 , . . . , Xk and T is an expression of the form hXi : ni
with n ∈ T . An assignment on X1 , . . . , Xk and T is a set S of singletons: it defines a k-tuple
(B1S , . . . , BkS ) of subsets of T by setting BiS := {n ∈ T | hXi : ni ∈ S} for each i. The
assignments of φ on T are the assignments S such that T satisfies φ(B1S , . . . , BkS ).
We will enumerate assignments instead of answers: this makes no difference because we
can always rewrite each assignment in linear time to the corresponding answer. We now
state the key result: we can efficiently build circuits (with singletons as variable gates) that
capture the assignments to MSO queries. (While these circuits are not augmented circuits,
they are decomposable, so the definition of zero-suppressed semantics clearly extends.)
I Theorem 7.3. For any fixed MSO formula φ(X1 , . . . , Xk ) on Γ-trees, given a Γ-tree T , we
can build in time O(|T |) a monotone d-DNNF circuit C in zero-suppressed semantics whose
set S(C) of assignments (as in Definition 3.6) is exactly the set of assignments of φ on T .
Proof sketch. We simplify φ to have a single free variable and limit to assignments on leaves
as in [5], and rewrite φ to a deterministic tree automaton A using the result of Thatcher and
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A Circuit-Based Approach to Efficient Enumeration
Wright [34], in time independent of T (though the runtime is generally nonelementary in φ).
We then compute our circuit as a variant of the provenance circuits in our earlier work [3],
observing that it is a d-DNNF thanks to the determinism of the automaton as in [4]. This
second step is in O(|A| · |T |), so linear in T . Appendix E.1 gives a self-contained proof. J
Note that the resulting circuit is already in zero-suppressed semantics, and has no range
gates. By continuing as in the proof of Theorem 2.1 (for free second-order variables) or
of Theorem 2.2 (for free first-order variables), we deduce the MSO enumeration results
of [5, 23]. Note that, once we have computed the tree automaton for the query and the
circuit representation, our proof of the enumeration result is completely query-agnostic: we
simply apply our enumeration construction on the circuit. Our proof also does not depend
on the factorization forest decomposition theorem of [14] used by [23]; it consists only of the
simple circuit manipulation and indexing that we presented in Sections 4–6. Note that the
delay is in O(k · |T |), with no large hidden constants, and O(k) for first-order variables.
A limitation of our approach is that our memory usage bound includes a logarithmic
factor in T , whereas [5, 23] show constant-memory enumeration. However, we can show that
the circuit computed in Theorem 7.3 satisfies an upwards-determinism condition that allow
us to replace the indexing scheme of Theorem 5.4 (our memory bottleneck) by a more efficient
index. We can thus reprove the constant-memory enumeration of [5, 23] (see Appendix F).
Factorized representations. Our second application is the factorized representations of [28],
a concise way to represent database relations [1] by “factoring out” common parts. The atomic
factorized relations are the empty relation ∅, the relation hi containing only the empty tuple,
and singletons hA : ai where A is an attribute and a is an element. Larger relations are built
using the relational union and Cartesian product operators on sub-relations with compatible
schemas. For example, hA1 : a1 i × (hA2 : a2 i ∪ hA2 : a02 i) is a factorized representation of the
relation on attributes A1 , A2 containing the tuples (a1 , a2 ) and (a1 , a02 ). A d-representation
is a factorized representation given as a DAG, to reuse common sub-expressions. We show
that d-representations can be seen as circuits in zero-suppressed semantics:
I Lemma 7.4. For any d-representation D, let C be the monotone circuit obtained by
replacing × and ∪ by AND and OR, replacing ∅ and hi by AND-gates and OR-gates with no
inputs, and keeping singletons as variables. Then all AND-gates of C are decomposable, and
S(C) (defined as in Section 3) is exactly the database relation represented by D.
Hence, our results in Theorem 2.2 can be rephrased in terms of factorized representations:
I Theorem 7.5. The tuples of a deterministic d-representation D over a schema S can be
enumerated with linear-time preprocessing, delay O(|S|), and memory O(|S| log |D|).
Note that the existing enumeration result on factorized representations (Theorem 4.11
of [28]) achieves a constant memory bound, unlike ours. However, this existing result applies
only to deterministic d-representations that are normal (Definition 4.6 of [28]), whereas
ours does not assume this. Normal d-representations are intuitively pruned and collapsed
circuits where no OR-gate is an input to an OR-gate, which avoids, e.g., the need for the
constructions of Section 5. Observe that the circuits that we build for MSO queries are not
normal in this sense, so we cannot prove Theorem 7.3 directly from Theorem 4.11 of [28].
8
Conclusion
We have studied how to enumerate satisfying valuations of circuits, under the structuredness, decomposability, and determinism conditions introduced in AI: we have shown that
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
enumeration can be performed with linear preprocessing and delay linear in each valuation
(so constant delay for valuations of constant Hamming weight). We have given two example
applications of this result: factorized databases, and an independent proof of the MSO query
enumeration results of [5, 22]. Beyond these applications, however, our method implies
efficient enumeration results for all problems studied in knowledge compilation, when they
can be compiled to structured d-DNNFs (refer back to the Introduction for examples).
A natural question is whether our constructions can be extended for other tasks, e.g.,
computing the i-th valuation [5, 8]; managing updates on the structure [27]; or enumerating
valuations in order of weight, or in lexicographic order: this latter problem is open for MSO
[32, Section 6.1] though results are known for factorized representations following an f-tree [9].
Another direction is to strengthen our result to constant-memory enumeration on all d-DNNF
circuits, or lift some hypotheses on the input circuits. We also intend to study a practical
implementation, which we believe to be realistic since our construction only performs simple
and modular transformations on the input circuits, and has no hidden large constants.
Acknowledgements. This work was partly funded by the French ANR Aggreg project,
by the CPER Nord-Pas de Calais/FEDER DATA Advanced data science and technologies
2015-2020, by the PEPS JCJC INS2I 2017 CODA, and by the Télécom ParisTech Research
Chair on Big Data and Market Insights.
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A Circuit-Based Approach to Efficient Enumeration
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A
Proofs for Section 3 (Reducing to Zero-Suppressed Semantics)
We now introduce some additional notation to be used throughout the appendix. We will
call a decomposable circuit or DNNF a circuit where all AND-gates are decomposable, and a
deterministic circuit a circuit where all OR-gates are deterministic.
We introduce an additional notation related to zero-suppressed semantics:
I Definition A.1. For a gate g in a monotone augmented circuit C, we will call its captured
set S(g) the set of the minimal valuations of g (written as assignments) in the sense of
Definition 3.5 (and for which an alternative characterization is given as Lemma 3.8). The
captured set S(g0 ) of the output gate g0 of C is then equal to its set of assignments S(C);
so we may also call S(C) the captured set of C.
We also introduce an additional definition related to partial traces (in particular, traces):
I Definition A.2. The variables tested by a partial trace T of C are the variables that occur
in T or occur in the interval of a range gate of T . Note that these are a subset of the interval
of the root of T .
We then show the auxiliary characterization of the set of assignments, which we will use
heavily in the proofs:
I Lemma 3.8. Let C be a monotone augmented circuit. Let us define inductively a set of
assignments S(g) for each gate g in the following way:
for all g ∈ Cvar , we set S(g) := {g};
for all ./ i-range gates g with inputs g1 and g2 , we set S(g) := {t ⊆ [g1 , g2 ] | |t| ./ i};
S
for all OR-gates g with inputs g1 , . . . , gn , we set S(g) := 1≤i≤n S(gi ) (with S(g) = ∅ if
g has no inputs);
for all AND-gates g with inputs g1 , . . . , gn , we set S(g) := {S1 ∪ · · · ∪ Sn | (S1 , . . . , Sn ) ∈
Q
1≤i≤n S(gi )} (with S(g) = {{}} if g has no inputs); observe that the unions are always
disjoint because C has a compatible order.
Then, for any gate g, the set S(g) contains exactly the assignments that describe a minimal
valuation for g. In particular, for g0 the output gate of C, the set S(g0 ) is exactly S(C).
Proof. We show the claim by induction. For the base cases:
For a variable g, the only partial trace rooted at g is {g}, and indeed its only minimal
valuation is {g}.
For a range gate g with inputs g1 and g2 , the only partial trace rooted at g is {g}, and
its minimal valuations are as defined.
For the induction cases:
For an OR-gate g, if g has no inputs, then there is no partial trace rooted at g, so S(g) = ∅
is correct. If g has inputs, then we can partition the partial traces rooted at g depending
on which input is retained. In particular, the set of leaves of the partial traces rooted
at g are exactly the union of the set of leaves of the partial traces rooted at the inputs
of g. Hence, the assignments describing the minimal valuations of g are exactly the union
of the corresponding assignments for the inputs of g, so we conclude by induction.
For an AND-gate g, if g has no inputs, then the only partial trace rooted at g is the
partial trace {g}, whose one minimal assignment is {}, which sets all variables to 0, and
S(g) = {{}} is correct. Otherwise, the partial traces rooted at g are obtained by taking g
and taking one partial trace rooted at each input of g. In particular, if there is an input
g 0 such that there is no partial trace rooted at g 0 , then there is no partial trace rooted
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
at g: now as by induction we have S(g 0 ) = ∅, so we have indeed set S(g) = ∅ which is
correct. Otherwise, remembering that an augmented circuit is decomposable (because it
has a compatible order), as g is an AND-gate, we know that the intervals for its inputs
must be pairwise disjoint. In particular, the partial traces for each input of g must be on
leaves having pairwise disjoint intervals. Hence, the minimal valuations of a partial trace
of g must be minimal valuations of some partial trace rooted at g 0 , and conversely any
choice of minimal valuation for partial traces rooted at the inputs of g can be combined
to a minimal valuation of a partial trace rooted at g thanks to the fact that the intervals
are disjoint. This allows us to conclude using the induction hypothesis.
J
Observe that Lemma 3.8 allows us to give an equivalent rephrasing of the determinism condition in zero-suppressed semantics: a d-DNNF in zero-suppressed semantics is a decomposable
circuit where, for each OR-gate g, the captured set S(g) is the disjoint union of the captured
sets of the inputs of g.
We now show the main result of this section:
I Proposition 3.9. Given a d-DNNF circuit C and a compatible order <, we can compute
in linear time a monotone 0-augmented circuit C ∗ having < as a compatible order, such
that C ∗ is a d-DNNF in zero-suppressed semantics and such that S(C ∗ ) is exactly the set of
satisfying assignments of C.
To prove the result, we will define complete circuits, where, intuitively, all variables are tested
so the semantics makes no difference:
I Definition A.3. Given an augmented circuit or decomposable circuit C with a compatible
order <, we call reach(g) the subset of Cvar of the gates g 0 that have a directed path to g, or
that are in the interval of a range gate that has a directed path to g. An AND-gate g of C is
complete if we have reach(g) = [min(g), max(g)]. An OR-gate g of C is complete if, for any
input g 0 to g, we have min(g 0 ) = min(g) and max(g 0 ) = max(g). The circuit C is complete if
every gate of C is complete, and if the interval of the output g0 of C is the complete set of
variables Cvar .
To prove Proposition 3.9, the first step is to rewrite the input d-DNNF to an equivalent
complete augmented d-DNNF (in the standard semantics). This will be done using ≥ 0-gates,
which are important to make this possible in linear time, and always evaluate to 1 so can be
added without changing the completed function. The second step is to rewrite the complete
augmented circuit to a monotone augmented circuit in the zero-suppressed semantics, whose
captured set is the set of satisfying assignments of the original circuit.
Let us first take care of the first step:
I Lemma A.4. Given a d-DNNF circuit C and a compatible order <, we can compute in
linear time a complete 0-augmented circuit C 0 that has < as a compatible order, computes
the same function as C, and is a d-DNNF (i.e., all its OR-gates are deterministic, with
decomposability being implied by the compatibility of <).
The completion procedure of Lemma A.4 is a standard routine for many forms of read
once branching programs, see e.g. [Weg00, Lemma 6.2.2]. Note that this lemma is actually
the only place where we use the compatible order < directly; in all later results, it will only
be used implicitly, to define the semantics of range gates (if any).
Proof of Lemma A.4. In a first step, it is convenient to make sure that every gate in C has
fan-in at most two. To this end, replace each gate of higher fan-in with a binary tree in the
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A Circuit-Based Approach to Efficient Enumeration
obvious way. Note that this can easily be done in linear time and in a way that preserves
compatibility with <.
In a second step we compute for every gate g in C the values min(g) and max(g). Again,
this can be done in linear time in a straightforward bottom-up fashion.
Note that the completeness requirement is trivial on gates g with no inputs, and it is
immediately satisfied on gates g with exactly one input if we assume that their one input
satisfies the requirement. Hence, it suffices to consider gates of fan-in exactly 2 in the
following. Also note that one direction of the completeness requirement is immediate: for any
AND- or OR-gate g, we have reach(g) ⊆ [min(g), max(g)], so only the converse implication
needs to be proven. We will write ≺ for the covering relation of <, i.e., we have g ≺ g 0 if
g < g 0 and there is no g 00 such that g < g 00 < g 0 . We will process the gates bottom-up to
ensure the completeness requirement, assuming by induction that it is satisfied on all input
gates. As the requirement is immediate for variables, NOT-gates and range gates, we first
explain the construction for AND-gates and second for OR-gates.
First, let g be an AND-gate with inputs g1 and g2 . Remember that the interval of g
is [min(g), max(g)] which is [min(g1 ), max(g2 )], and we know by compatibility of < that
max(g1 ) < min(g2 )]. If max(g1 ) ≺ min(g2 ), i.e., max(g1 ) is the predecessor of min(g2 ), then
there is nothing to do for g, as every gate in the interval of g is in that of g1 or in that of g2 ,
in which case we conclude that it is in reach(g1 ) or in reach(g2 ) by induction hypothesis, and
conclude.
If max(g1 ) 6≺ min(g2 ), let g10 and g20 be such that max(g1 ) ≺ g10 and g20 ≺ min(g2 ). Add a
fresh child g 0 to g, between g1 and g2 , which is a ≥ 0-range gate with inputs g10 and g20 . It is
clear that this does not violate compatibility of < with C, because the interval of g 0 is [g10 , g20 ]
and we have max(g1 ) < g10 and g20 < min(g2 ). Further, it does not change the computed
function, because g 0 always evaluate to 1 which is neutral for AND. Last, it is now clear that
g satisfies the condition of Definition A.3, because any gate g 00 in the interval of g is now
either a gate of the interval of g1 , of the interval of g2 , or of the interval of g 0 : we conclude
by induction as above in the first two cases, and in the third case we conclude because g 00 is
in the interval of the range gate g 0 which has a directed path to g.
Second, let g be an OR-gate with inputs g1 and g2 . We replace g1 with an AND-gate g10
computing the AND of the following:
if min(g) < min(g1 ), letting g1− be such that min(g) ≤ g1− ≺ min(g1 ), a ≥ 0-range gate
(g10 )− whose inputs are min(g) and g1− ;
g1 itself;
if max(g1 ) < max(g), letting g1+ be such that max(g1 ) ≺ g1+ ≤ max(g), a ≥ 0-range gate
(g10 )+ whose inputs are g1+ and max(g).
We do the analogous construction for g2 . It is clear that this does not violate the compatibility
of < with C, because the interval of the new AND-gates g10 and g20 are exactly the interval of g
by construction, and the interval of g is unchanged. It is also clear that these transformations
do not change the computed function because the ≥ 0-range gates evaluate to 1; further, the
transformations do not affect determinism at the OR-gate g for the same reason. Last, it
is now clear that g10 , g20 and g satisfy the conditions of Definition A.3. Indeed, the interval
of these three gates is [min(g), max(g)]. Now, to show the condition on g10 any gate of this
interval is either in the interval of g1 , of (g10 )− , or of (g10 )+ , we conclude using the induction
hypothesis in the first case and immediately in the two other cases. To show the condition
of g, we use the same proof. To show the condition on g20 , we use the analogous proof with
g2 , (g20 )− , and (g20 )+ .
We perform the above constructions for all gates. Last, if the interval of the output
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
gate g0 does not contain all variables of Cvar , we replace it by an AND-gate of g0 and of
≥ 0-range gates that capture the missing variables. The resulting circuit C 0 then computes
the same function as C and is complete. Moreover, C 0 has a compatible order < and all
OR-gates are deterministic, so it is a d-DNNF. Finally, for every gate g that we manipulated,
we only performed a construction that can be done in constant time, so the overall time of
the construction is linear in the size of C.
J
We now take care of the second step: given an augmented d-DNNF circuit C 0 which
is complete, rewrite it to a monotone augmented circuit C ∗ which is a d-DNNF in zerosuppressed semantics and captures the satisfying assignments of C 0 . We will do so simply by
substituting all NOT-gates with gates that always evaluate to 1:
I Definition A.5. Given an (augmented) circuit C, its monotonization is the monotone
(augmented) circuit C ∗ obtained by removing the input wire to each NOT-gate of C and
changing the type of these gates to AND-gates (which have no children, so they always
evaluate to 1).
This transformation can clearly be performed in linear time. We now claim that it
preserves some properties. First, we must make the following trivial observation:
I Observation A.6. For any augmented circuit C with compatible order <, its monotonization
C ∗ still admits < as a compatible order.
Second, the key observation is the following:
I Lemma A.7. For any complete augmented circuit C 0 capturing Boolean function Φ, its
monotonization C ∗ captures Φ under zero-suppressed semantics.
To prove this lemma, it will be useful to extend the definitions of upward trees and
traces (Definition 3.3) to augmented circuits which are not necessarily monotone; the only
difference is that the leaves of a trace can now also include NOT-gates. We define minimal
valuations like in Definition 3.5, except that we enforce that negated variables must be set
to 0; the variables tested by a trace also include the negated variables. We can now show the
important property of complete circuits:
I Observation A.8. Every trace T of a complete circuit C tests all variables of Cvar .
Proof. We simply prove by bottom-up induction that any trace rooted at a gate g of C tests
all variables of the interval of g. This is true of variable gates, NOT-gates, and range gates;
it is true of OR-gates because it is true of all their inputs and they have the same interval as
their inputs; it is true of AND-gates because the sub-traces on all inputs satisfy the property.
We conclude thanks to the fact that the interval of the output gate consists of all variables
of Cvar .
J
We can now show:
Proof of Lemma A.7. We observe the existence of a bijection between the traces of C ∗ and
of C 0 . Specifically, consider the mapping f from the traces of C 0 to traces of C ∗ obtained by
replacing each leaf which is a NOT-gate of C 0 by the corresponding AND-gate with no inputs
in C ∗ . It is clear that any upward tree in the image of this transformation is a trace of C ∗ .
Conversely, we can map the traces of C ∗ to traces of C 0 by replacing the fresh AND-gates
with no inputs by the corresponding NOT-gate, and again the upward trees in the image of
this transformation are indeed traces of C 0 . As this defines an inverse function for f , it is
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A Circuit-Based Approach to Efficient Enumeration
clear that f is a bijection. Further, as C 0 is complete, it is clear that the variables which are
not tested by f (T 0 ) are precisely the variables whose negation is a leaf of 0 T
For the forward direction, consider a satisfying valuation ν 0 of C 0 . By Observation 3.4,
there is a trace T 0 of C 0 of which ν 0 is a satisfying valuation. As C 0 is complete, by
0
Observation A.8, T 0 tests all variables of Cvar
. Hence, the satisfying valuation ν 0 of T 0 is
actually a minimal satisfying valuation of T (there are no variables implicitly set to 0). Now,
clearly ν 0 is also a satisfying valuation of f (T 0 ). The converse is shown in the same way
by considering a satisfying valuation ν ∗ of C ∗ , a witnessing trace T ∗ of C ∗ of which it is
a minimal valuation, and observing that ν ∗ is a satisfying valuation of the trace f −1 (T ∗ )
of C 0 .
J
The last claim to show is the preservation of d-DNNF through monotonization:
I Lemma A.9. For any complete augmented circuit C 0 which is a d-DNNF in the standard
semantics, its monotonization C ∗ is a d-DNNF in zero-suppressed semantics (in the sense
of Definition 3.6).
Proof. As in the proof of Lemma A.7, we know that the captured set of each gate g of C ∗
describes the satisfying assignments of g in C 0 in the standard semantics. Hence, any violation
of determinism in zero-suppressed semantics on C ∗ witnesses a violation of determinism in
the standard semantics in C 0 .
J
We are now ready to prove our main result for this section:
Proof of Proposition 3.9. We first apply to C the construction of Lemma A.4 to get in
linear time a complete 0-augmented circuit C 0 which is a d-DNNF in the standard sense,
has compatible order <, and computes the same function Φ as C. Now, we construct in
linear time the monotonization C ∗ of C 0 , which is a monotone 0-augmented circuit. By
Observation A.6, C ∗ still admits < as compatible order. By Lemma A.9, C ∗ is a d-DNNF
in the zero-suppressed semantics. By Lemma A.7, C ∗ captures Φ under zero-suppressed
semantics, which concludes the proof.
J
We conclude the section by a remark on zero-suppressed semantics: this semantics, as
well as the relevant definitions, can be defined not only for augmented circuits, but more
generally for decomposable (non-augmented) circuits:
I Remark A.10. The definition of upwards trees and traces (Definition 3.3) extend to
monotone (non-augmented) circuits which are decomposable, even if they do not have a
compatible order. Observation 3.4 extends to them, and the definition of minimal valuations
(Definition 3.5) also does. Further, the definition of zero-suppressed semantics extends
(Definition 3.6), as does the definition of captured sets in Definition A.1. Last, the alternative
characterization of the captured sets in Lemma 3.8 also extends.
Thanks to this remark, we will be able to talk about zero-suppressed semantics and
captured sets in decomposable monotone circuits with no range gates, even if they do not
have a compatible order. This will be especially useful in Section 7, where we directly produce
decomposable circuits in zero-suppressed semantics.
B
Proofs for Section 4 (Reducing to Normal Form Circuits)
Throughout this appendix, two monotone circuits C and C 0 are called equivalent (in zerosuppressed semantics) if S(C) = S(C 0 ).
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
B.1
Reduction to Arity-Two
We show that following easy general-purpose result which will be useful in several proofs.
I Lemma B.1. For any (augmented) monotone circuit C, we can compute in linear time
an equivalent arity-two (augmented) monotone circuit C 0 . Further, if C is a d-DNNF in
zero-suppressed semantics (resp., if it is ∅-pruned, if it is {}-pruned, if it has some order <
as a compatible order, if it is monotone), then the same is true of C 0 .
Proof. We use the standard construction of adding intermediate AND- and OR-gates,
leveraging the associativity of the Boolean ∧ and ∨ operations. It is clear that none of the
existing or additional gates is unsatisfiable or 0-valid if none of the original gates are, so
the process preserves being ∅-pruned and {}-pruned. The process adds no NOT-gates so
it clearly preserves monotonicity. Any violation of the d-DNNF condition on the rewritten
circuit witnesses a violation on the original circuit. Last, it is clear that any compatible order
is still compatible with the result of the rewriting, as any violation of compatibility would
witness a violation in the original circuit.
J
B.2
Homogenization
The proof will use the following definition:
I Definition B.2. For k ∈ N, a monotone augmented circuit C is k-homogenized if its gate
set G is partitioned as G = G=0 ∪ G=1 ∪ · · · ∪ G=k ∪ G>k , all these unions being disjoint,
such that for i ∈ [0, k] the G=i and G>k satisfy the following properties:
For every g ∈ G=i , for every t ∈ S(g), we have |t| = i, and
For every g ∈ G>k , for every t ∈ S(g), we have |t| > k.
Note that S(g) = ∅ is allowed in both cases. Note that in particular variable gates are all in
G=1 if k ≥ 1, and they are all in G>0 if k = 0.
C is a called a k-homogenization of an augmented circuit C 0 , if for every gate g of C 0
For every i ∈ [0, k], C contains a gate g =i such that S(g =i ) = {t ∈ S(C 0 ) | |t| = i}, and
C contains a gate g >k such that S(g >k ) = {t ∈ S(C 0 ) | |t| > k}.
We will now prove the following strengthening of Proposition 4.1:
I Proposition B.3. For every l ∈ N, given a monotone l-augmented C with compatible
order < and k ∈ N, we can construct in time O((k + 1)2 · |C|) a monotone max(l, k + 1)augmented circuit C 0 with compatible order < such that C 0 is a k-homogenization of C.
Further, if C is a d-DNNF in zero-suppressed semantics then C 0 also is.
It is clear that this result implies Proposition 4.1. Indeed, to impose the desired semantics
on the resulting circuit C 0 , one can simply add a fresh OR-gate as the output gate of the
k-homogenization C 0 as the OR of the G=i for 0 ≤ i ≤ k. It is then clear that the circuit
satisfies the required properties, and we easily check determinism on the fresh output gate
in the sense of Definition 3.6 thanks to the fact that the valuations are disjoint (they have
different weights).
We now prove Proposition B.3:
Proof of Proposition B.3. The construction essentially follows the classical homogenization
technique introduced by Strassen [Str73], and will not use the input compatible order except
for the definition of range gates. We first rewrite the input circuit C in linear time using
Lemma B.1 to ensure that it is arity-two. We will write in(g) to denote the inputs of a gate g.
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A Circuit-Based Approach to Efficient Enumeration
Now, for every gate g of C, for every i ∈ N, we define the sets S =i (g) := {t ∈ S(G) | |t| = i}
and S >i (g) := {t ∈ S(G) | |t| > i}. We create the k-homogenized circuit C 0 by associating,
to each gate g of C, k + 2 gates g =0 , . . . , g =k and one gate g >k in C 0 such that for i ∈ [0, k]
we have S(g =i ) = S =i (g) and S(g >k ) = S >k (g). To ensure this, we proceed iteratively as
follows:
If g is a variable, for all i ∈ [0, k] \ {1}, the gate g =i is an OR-gate with no inputs (so
S(g =0 ) = ∅). If k > 0, g =1 is a variable identified to g and g >k is an OR-gate with no
inputs. Otherwise, g >0 is a variable identified to g.
0
If g is a = k 0 -range gate, then if k 0 ≤ k we set g =k to be an = k 0 -range gate with the
same inputs as g, otherwise we have k < k 0 and set g >k to be an = k 0 -range gate with
the same inputs as g. All other gates are OR-gates with no inputs.
If g is a ≥ k 0 -range gate, for all i ∈ [0, k 0 − 1], the gate g =i is an OR-gate with no inputs.
For all i ∈ [k 0 , k], the gate g =i is a = i-range gate with the same inputs as g. Finally, g >k
is a ≥ (k + 1)-range gate identified with the same inputs as g.
If g is an OR-gate, for every i ∈ [0, k], the gate g =i is an OR-gate whose inputs are
{(g 0 )=i | g 0 ∈ in(g)}, and g >k is an OR-gate whose inputs are {(g 0 )>k | g 0 ∈ in(g)};
If g is an AND-gate, for every i ∈ [0, k], the gates g =i and g >k are defined as follows:
(remember that C is arity-two so the following cases are exhaustive):
if |in(g)| = 0, then g =i and g >k are AND-gates with in(g =i ) := ∅ and in(g >k ) := ∅;
if |in(g)| = 1, writing in(g) = {g 0 }, then g =i and g >k are AND-gates with in(g =i ) :=
{(g 0 )=i } and in(g =i ) := {(g 0 )>i };
if |in(g)| = 2, writing in(g) = {g10 , g20 }, then for i ∈ [0, k], the gate g =i is an ORgate with inputs g0=i , . . . , gi=i where each gj=i is an AND-gate with inputs (g10 )=j and
>k
(g20 )=(i−j) . Moreover, g >k is an OR-gate with inputs gi,j
for i, j ∈ [0, k] and gi>k,1 and
gi>k,2 for i ∈ [0, k] which are defined as follows:
>k
∗ if i + j > k, then gi,j
is an AND-gate with inputs (g10 )=i and (g20 )=j ,
>k
∗ if i + j ≤ k, then gi,j
is an OR-gate with no inputs,
∗ gi>k,1 is an AND-gate with inputs (g10 )>k and (g20 )=i
∗ gi>k,2 is an AND-gate with inputs (g10 )=i and (g20 )>k
By straightforward but somewhat cumbersome induction, it is easy to see that the gates
g =0 , . . . , g =k and g >k indeed compute the sets S =0 (g), . . . , S =k (g) and S >k (g), respectively.
Thus, C 0 is a k-homogenization of C as claimed. Moreover, the construction for every gate g
in C can be performed in time O(k 2 ), so the overall runtime of the algorithm is O(k 2 · |C|).
It is clear that the resulting circuit C 0 is monotone. If C has a compatible order <,
we first show that < is a compatible order for C 0 , so that C 0 is l-augmented. Indeed, all
AND-gates g ∧ with more than one input in C 0 derive from the construction for an AND-gate
g with inputs g10 and g20 in C. But then in g ∧ always has inputs (g10 )./1 i1 and (g20 )./2 i2 for
./1 , ./2 ∈ {=, >} and i1 , i2 ∈ [0, k]. Note that, by construction of C 0 , the variables having a
directed path to (g10 )./1 i1 and (g20 )./2 i2 in C 0 are subsets of those having a directed path to
g10 and g20 respectively in C. Hence, as < is compatible with C, from the condition on g with
inputs g10 and g20 in C, we deduce that the condition is satisfied for g ∧ in C 0 . Hence, < is a
compatible order for C 0 . We last observe that the range gates that we create are not labeled
with integer values that are larger than those of the original circuit or than k + 1, so C 0 is
indeed a monotone max(l, k + 1)-augmented circuit.
We last show that if C is a d-DNNF in zero-suppressed semantics then so is C 0 . To do
so, we show that the OR-gates of C 0 are deterministic in the sense of Definition 3.6. To this
end, consider an OR-gate g ∨ in C 0 with at least two inputs. Only two cases can occur:
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
g ∨ was introduced in the construction for an OR-gate g in C. In this case, the inputs
of g ∨ are {(g 0 )./i | g 0 ∈ in(g)} for ./ ∈ {=, >} and i ∈ [0, k]. As before S((g 0 )./i ) ⊆ S(g 0 )
for every input g 0 of g in C. Since g is deterministic, we know that, for all gates
g 0 , g 00 ∈ in(g) with g 0 =
6 g 00 , the sets S(g 0 ) and S(g 00 ) are disjoint. It follows that for every
(g 0 )./i , (g 00 )./i ∈ in(g ∨ ) with g 0 6= g 00 , the sets S((g 0 )./i ) and S((g 00 )./i ) are disjoint. Thus
g ∨ is deterministic.
g ∨ was introduced in the construction for an AND-gate g in C. In this case, the inputs
of g ∨ are OR-gates with no inputs (which can never cause a violation of determinism
because they capture the empty set), and AND-gates with inputs of the form (g10 )./1 i1
and (g20 )./2 i2 where ./1 , ./2 ∈ {=, >} and i1 , i2 ∈ [0, k]. Let now g and g 0 be two inputs
0 0
of g ∨ where g has the inputs (g10 )./1 i1 and (g20 )./2 i2 and g 0 has the inputs (g10 )./1 i1 and
0 0
(g20 )./2 i2 . By inspection of the construction, we see that we cannot have ./1 = ./01 and
./2 = ./02 and i1 = i01 and i2 = i02 at the same time, that is, one of these equalities must
be false. But then, depending on whether the false equality is on ./1 or i1 , or on ./2 or i2 ,
0 0
0 0
we have S((g10 )./1 i1 ) ∩ S((g10 )./1 i1 ) = ∅ or S((g20 )./2 i2 ) ∩ S((g20 )./2 i2 ) = ∅. Thus, in both
cases S(g) ∩ S(g 0 ) = ∅ by Lemma 3.8 and thus g ∨ is deterministic.
Since in both cases g ∨ is deterministic, it follows, as claimed, that C 0 is a d-DNNF in
zero-suppressed semantics.
J
B.3
Reduction to Normal Form
I Proposition 4.3. Given a monotone augmented d-DNNF circuit C in zero-suppressed
semantics with compatible order < and with S(C) 6= ∅ and S(C) 6= {{}}, we can build in
O(|C|) a normal d-DNNF C 0 , with < as a compatible order, such that S(C 0 ) = S(C)\{{}}.
In the rest of this section, we prove Proposition 4.3 in multiple steps to ensure each condition.
The first step is to make the circuit ∅-pruned and {}-pruned. To do so, it will be
convenient to reuse our homogenization process (Proposition B.3 of Appendix B.2) and our
notion of homogenization of circuits (Definition B.2). We first show how to make circuits
∅-pruned while preserving being a d-DNNF and being homogenized:
I Lemma B.4. For any l ∈ N and monotone l-augmented circuit C with compatible order <
such that S(C) 6= ∅, we can compute in linear time an equivalent ∅-pruned monotone laugmented circuit C 0 with compatible order <. Further, if C is a d-DNNF in zero-suppressed
semantics, then so is C 0 , and if C is a k-homogenized for some k ∈ N then so is C 0 .
Proof. We first compute which gates g of C are unsatisfiable. It is easily seen that these
gates are the following, which can be computed in linear time by processing C bottom-up:
range gates labeled with ./ i whose interval contains less than i variables (which we call
unsatisfiable range gates);
AND-gates with no inputs gates;
AND-gates where one input gate is unsatisfiable;
OR-gates where all input gates is unsatisfiable.
We define the circuit C 0 as C where we remove all unsatisfiable gates and all wires leading
out of these gates. This can clearly be computed in linear-time. Further, C 0 has a output
gate (namely, the same as C), because as S(C) 6= ∅, we know that we have not removed the
output gate of C.
We will show that, for every gate g of C 0 , the set S(g) in C 0 is the same as S(g) in C.
This claim implies in particular that C 0 is equivalent to C (when applying it to the output
gate), and it shows that C 0 is ∅-pruned: if some gate g is unsatisfiable in C 0 , then g is also
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A Circuit-Based Approach to Efficient Enumeration
unsatisfiable in C, so g should have been removed in C 0 . To see why the claim is true, observe
that the only wires from a removed gate to a non-removed gate, i.e., from an unsatisfiable
gate g to a satisfiable gate g 0 , must be such that g 0 is an OR-gate (otherwise it would be
unsatisfiable too), and clearly removing the wire from g to g 0 does not change the set captured
by g 0 .
Assuming now that C is k-homogenized for some k ∈ N, we can see from the previous
claim that the same is true of C 0 . Indeed, we can suitably partition the gates of C 0 using the
same partition as the one used for C.
Last, to see that C 0 admits < as a compatible order, and that C 0 is a d-DNNF in
zero-suppressed semantics if C is, observe that we construct C 0 from C by removing gates
and removing input wires to the remaining gates, so this cannot introduce violations of the
compatibility of < or the determinism of OR-gates.
J
We now show how to make the circuit ∅-pruned and {}-pruned, using the previous process
and the homogenization process (Proposition B.3 of Appendix B.2).
I Lemma B.5. For any l ∈ N, for any monotone l-augmented circuit C with compatible
order < such that S(C) 6= ∅ and S(C) 6= {{}}, we can compute in linear time a monotone
max(l, 1)-augmented circuit C 0 with compatible order < such that S(C 0 ) = S(C)\{{}} and
C 0 is ∅-pruned and {}-pruned. Further, if C is a d-DNNF in zero-suppressed semantics, then
so is C.
Proof. We use Proposition B.3 for k = 1 to compute in linear time in C a monotone max(l, 1)augmented circuit Chomog which is a 0-homogenization of C, which admits < as a compatible
order, and which is a d-DNNF according to zero-suppressed semantics iff C is. Recalling now
Definition B.2, we know that Chomog has a gate g >0 such that S(g >0 ) = {t ∈ S(C) | |t| > 0,
so choosing g >0 as the output gate of Chomog we have indeed that S(Chomog ) = S(C)\{{}};
in particular S(Chomog ) 6= ∅.
We now apply Lemma B.4 to Chomog , to obtain an equivalent ∅-pruned monotone
max(l, 1)-augmented circuit C∅ which is ∅-pruned, which is is still 0-homogenized, and which
is a d-DNNF in zero-suppressed semantics if C is.
We last rewrite C∅ to an equivalent circuit C 0 . Recall that, as C 0 is 0-homogenized, its
gate set is partitioned in G=0 and G>0 , such that all gates of G=0 capture {{}} (they cannot
capture ∅ as C∅ is ∅-pruned), and no gates of the latter capture a set containing {} (and G>0
contains in particular the output gate). We define our final circuit C 0 from C∅ by removing
all gates of G=0 and all wires leading out of them. Note that, by definition of homogenized
circuits, we do not remove variable gates or the output gate. The construction of C 0 is clearly
in linear-time, and C 0 is compatible with < and is a d-DNNF in zero-suppressed semantics if
C 0 is, because C 0 is constructed from C∅ by removing gates and input to remaining gates,
which cannot introduce violations of these requirements.
We now show that for every gate g of C 0 , its captured set S(g) in C 0 is the same as S(g)
in C∅ . This claim implies in particular that C 0 is equivalent to C∅ (when applying it to the
output gate), that C 0 is ∅-pruned (any violation of this in C 0 implies a violation of the fact
that C∅ is ∅-pruned), and it shows that C 0 is {}-pruned: if {} ∈ S(g) in C 0 for some gate g,
then the same holds of g in C∅ , so g ∈ G=0 and g should have been removed in C 0 . To see
why the claim is true, observe that when there is a wire from a gate g in C 0 to a gate g 0 in C 0 ,
and g is removed and g 0 is not, i.e., g ∈ G=0 and g 0 ∈ G>0 , then we have S(g) ⊆ {{}}, and
then as C 0 is ∅-pruned we must have S(g) = {{}}. Hence, we know that g 0 is an AND-gate,
because if it were an OR-gate we would have {} ∈ S(g 0 ) contradicting g 0 ∈ G>0 . Now as {}
is neutral for ×, we do not change the semantics by removing the wire.
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
Hence, C 0 is a monotone max(l, 1)-augmented circuit with compatible order < that is
∅-pruned and {}-pruned, we have S(C 0 ) = S(C∅ ) = Chomog = S(C)\{{}}, and if C is a
d-DNNF in the zero-suppressed semantics then Chomog , C∅ , and thus C 0 , also are. This
concludes the proof.
J
The third step is to ensure that the circuit is collapsed and discriminative, but this is
completely straightforward:
I Lemma B.6. For any l ∈ N and l-augmented circuit C with compatible order <, we can
compute in linear time an l-augmented circuit C 0 with compatible order < which is collapsed
and discriminative. Further, if C is arity-two (resp., is ∅-pruned, is {}-pruned, is a d-DNNF
in the zero-suppressed semantics, is monotone), then so is C 0 .
Proof. Simply merge all AND-gates with one input with their one input to make the
circuit collapsed. This is clearly linear-time, and does not affect compatibility with <,
OR-determinism, being arity-two, being ∅-pruned, or being {}-pruned.
Then, for every wire (g, g 0 ) where the input gate g is not an OR-gate but the output
gate g 0 is an OR-gate, rewrite the wire by inserting an intermediate OR-gate, i.e., we create
a fresh OR-gate g 00 (the exit), and replace the wire (g, g 0 ) by (g, g 00 ) and (g 00 , g 0 ). This is
clearly linear-time, ensures that the output is discriminative, and it does not affect any of
the requirements.
J
Using these results, we can conclude the proof of Proposition 4.3:
Proof of Proposition 4.3. We first apply Lemma B.5 to make the circuit ∅-pruned and
{}-pruned. We then apply Lemma B.1 to make it arity-two. We last apply Lemma B.6 to
make it collapsed and discriminative.
J
As in Appendix A, we will need in Section 7 to apply the process described in this section
to non-augmented circuits that have no compatible order but are decomposable. The claim
is as follows, and it is straightforward to verify, because all transformations described in this
section do not introduce range gates if their input does not contain range gates, and do not
depend on the compatible order except to define the semantics of range gates and to ensure
decomposability.
I Remark B.7. If the input circuit C to Propositions 4.1 or B.3 has no range gates and no
compatible order, but is decomposable, then the construction still works, and the output still
does not have range gates and is still decomposable. The same is true of Proposition 4.3.
C
Proofs for Section 5 (Indexing OR-Components)
I Lemma 5.3. For any normal d-DNNF C, each OR-component of C is a multitree.
Proof. Assume by contradiction that an OR-component is not a multitree, so it has two gates
g and g 0 such that there are two different directed paths π1 and π2 from g to g 0 . As π1 and
π2 are two different paths to the same gate g 0 , there must be a gate g 00 with inputs g100 6= g200
such that π1 goes through g100 and g 00 , and π2 goes through g200 and g 00 . As C is ∅-pruned,
S(g 0 ) is non-empty, so let t ∈ S(g 0 ). As C is {}-pruned, t is non-empty. The directed paths
π1 and π2 witness that t ∈ S(g100 ) and t ∈ S(g200 ), and this violates the determinism condition
on the OR-gate g 00 .
J
XX:25
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A Circuit-Based Approach to Efficient Enumeration
I Theorem 5.4. Given a normal d-DNNF C, we can compute in O(|C|) a structure called
OR-index allowing us to do the following: given an OR-gate g of C, enumerate the exits of g
in its OR-component K, with constant delay and memory usage O(log |K|).
In order to prove Theorem 5.4, thanks to Lemma 5.3, it suffices to show the following general
result on multitrees, where a leaf of a multitree T is a vertex n with no edge to a vertex of T :
I Theorem C.1. Given a multitree T , we can compute in linear time a data structure
allowing us to perform the following: given n ∈ T , enumerate the leaves of T that are
reachable from n with constant-delay and memory usage in O(log |T |).
This theorem allows us to compute the required OR-index. Indeed, we can compute the
OR-components of C in linear time, go over each OR-component K, and apply the theorem
to the reverse of K (in which exits are leaves), which is still a multitree. The result over
all OR-components is an index that allows us, given any OR-gate g in C, to enumerate the
exits of g with constant delay and with the claimed memory usage. This computation is
linear-time overall, and concludes the preprocessing of our input circuit. All that remains is
to prove Theorem C.1, which we do in the rest of this appendix.
Given a multitree T , we will show how to compute in linear time a multitree Q(T ) labeled
with leaves of T and a mapping q from T to Q(T ) such that the leaves reachable from a
node n ∈ T correspond (in a one-to-one correspondence) to the labels of the nodes reachable
from q(n) ∈ Q(T ). This ensures that, by enumerating the labels of the nodes reachable
from q(n) in Q(T ), we enumerate the leaves reachable from n in T .
We then show that this second task is easy, as the nodes of a multitree T reachable from
a node n ∈ T can be enumerated in constant delay with a simple tree traversal.
Finally, we improve on the tree traversal so that the memory usage is logarithmic in the
size of the multitree.
Transformation of T into Q(T ). Let T be a multitree, and let us explain how to construct Q(T ). We may assume without loss of generality that T is binary. Indeed, if this is
not the case, we simply consider all nodes of T with more than two children and replace
them by binary trees in the obvious way. This transformation does not change the leaves
that are reachable from the original nodes, so it suffices to solve our enumeration problem on
the new binary tree.
We create Q(T ) by a bottom-up traversal of T , and consider every node n of T from the
leaves to the root:
If n is a leaf, we introduce a leaf node q(n) in Q(T ).
If n is an internal node with a single child n0 , we introduce a node q(n) in Q(T ) and
connect it as a parent of the children of q(n0 ) in Q(T ) if they exist (note that they must
already have been constructed).
If n is an internal node with two children n1 and n2 , we introduce two nodes q(n) and
c(n) in Q(T ). We connect q(n) as a parent of c(n), and connect c(n) as a parent of
the children of q(n1 ) and q(n2 ) in Q(T ) if they exist (again, they have already been
constructed).
This completes the construction of Q(T ). Note that multiple internal nodes of Q(T ) may
share the same children, so it is not generally a tree.
We show that Q(T ) is a binary multitree. Indeed, it is immediate to see that whenever
there is an edge in Q(T ) from a node c(n) or q(n) to a node c(n0 ) or q(n0 ), then either n = n0
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
or n0 is a descendant of n in T . Hence, Q(T ) is acyclic, and if there is a path from c(n) or
q(n) to c(n0 ) or q(n0 ) in Q(T ), then there is a path from n to n0 in T , so any violation of
the fact that Q(T ) is a multitree would imply a violation in T . This shows that Q(T ) is
a multitree (but note that it is generally not a tree). Further, it is immediate to show by
induction that all nodes of the form q(n) have at most one child, and then the nodes of the
form c(n) have at most two children, so indeed Q(T ) is binary.
We now describe how to label each node n0 of Q(T ) with a leaf of T , which we write λ(n0 ).
Our construction will ensure the following property: for any node n ∈ T , the leaves reachable
from n in T are in a one-to-one correspondence with the labels of nodes reachable from q(n)
in T 0 . More precisely, for each leaf ` of T reachable from n, there is exactly one node
reachable from q(n) in T 0 such that λ(q(n)) = `, and, conversely, for every node n0 reachable
from q(n) in T 0 , the leaf λ(n0 ) is reachable from n in T . We describe the construction in a
bottom-up fashion on nodes n of T , and show that the property is verified for n:
If n is a leaf, we set λ(q(n)) := n. This clearly satisfies the property.
If n is an internal node with a single child n0 , we set λ(q(n)) := λ(q(n0 )), which was
defined before. Since we reach the same leaves from n and n0 in T , the property is satisfied
by induction.
If n is an internal node with two children n1 and n2 , we set λ(q(n)) := λ(q(n1 )), and
λ(c(n)) := λ(q(n2 )), which were both defined before. We now explain why this is correct.
The set of leaves reachable from n in T is the union of the leaves reachable from n1 and n2 ,
and this union is disjoint because T is a multitree. Now, the set of nodes reachable from
q(n) in T 0 contains q(n), c(n), and the nodes reachable from q(n1 ) or q(n2 ) except q(n1 )
and q(n2 ) themselves. So our choice of labels clearly guarantees the desired property by
induction.
Enumeration phase. Given a node n ∈ T , thanks to the property of Q(T ) that we just
showed, we can enumerate the leaves reachable from n in T simply by traversing the tree
rooted in q(n). Our enumeration state is a stack S of nodes in the multitree Q(T ) that have
yet to be processed. At the beginning of the enumeration, S = {q(n)}. At each step of the
enumeration, we pop a node n0 of T 0 from S, push the children of n0 (if any) back into S,
and then output λ(n0 ).
The stack S can be implemented with a linked list, so that we can push and pop elements
in constant time. It is immediate that this algorithm can indeed enumerate in constant delay
the labels of the nodes reachable from an node q(n) of Q(T ). So we have solved our initial
enumeration problem on T : given a node n ∈ T , we enumerate the nodes reachable from q(n)
in Q(T ) as we explained, and by the property that we showed, the process enumerates exactly
the leaves of T reachable from n.
Memory usage. We now explain how to refine the preprocessing and enumeration process
to satisfy the logarithmic memory bound. We define the weight w(n) of a node n in a
multitree to be the number of nodes reachable from a node n, including n itself.
The memory usage of our enumeration algorithm is the maximum size of the state
maintained during the enumeration, i.e., the maximum size reached by S. Given a node
n ∈ T , if the tree rooted in q(n) is unbalanced then S may contain as many as w(q(n))/2
nodes. We now show how to get a tighter, logarithmic bound on memory usage by choosing
the order in which we traverse Q(T ).
We first pre-compute the weight of all nodes of Q(T ) in linear time in a bottom-up
fashion, as part of our preprocessing. Now, at each step of the enumeration, we pop a node
XX:27
XX:28
A Circuit-Based Approach to Efficient Enumeration
from S (which is the last inserted still in S) and then push its children onto S. When there
are two children, we make sure that the child with greater weight is pushed first.
We claim that at every step of the enumeration for a node q(n) of T 0 corresponding to
n ∈ T , the weight of a node in S is greater or equal to the sum of the weights of nodes
in S that were inserted afterwards, i.e., that precede the node in S. We show the claim by
induction along the enumeration process:
At the beginning of the enumeration S contains only q(n) and the claim is vacuous.
At each step of the enumeration, we pop the first node and we push back its children, of
which there are at most two. The property holds for all the nodes of the new stack that
already existed in the old stack, since the total weight of the nodes we push back (i.e.
the children) is the weight of the popped node minus one. The property also holds for
the newly added nodes, because we add the node with bigger weight first.
Hence, we have shown our claim by induction. Thus, let us consider any point of the
enumeration algorithm, write the stack S = (s1 , . . . , sp ), and show that p = O(log |T |). We
assume in particular that p ≥ 2, otherwise there is nothing to show. Let us define a sequence
P
(Ti ) by T1 := 1 and Ti+1 := j≤i Tj : it is clear by induction from our previous claim that,
for all 1 ≤ q ≤ p, we have w(sq ) ≥ Tq . Now, it is easy to see that Ti = 2i−2 for i ≥ 2, so we
have w(sp ) ≥ 2p−2 . Remember now that the weight of a node in S cannot exceed w(q(n)),
because all nodes in S are reachable from q(n). So we must have w(sp ) ≤ w(q(n)), and
w(q(n)) ≥ 2p−2 . This clearly implies that p = O(log(w(q(n))), in particular p = O(log |T |).
Hence, the stack is always of size logarithmic in |T |, which proves the memory usage claim.
D
D.1
Proofs for Section 6 (Enumerating Assignments)
Compressed Traces
I Lemma 6.2. For any compressed trace T of a normal circuit C and minimal valuation ν
for T and C, we have |T | ≤ 6 · |ν|.
Proof. First observe that, as C is {}-pruned, C (hence T ) cannot contain AND-gates with
no children, or range gates labeled = 0 or ≥ 0. Hence, each leaf of T is either a variable gate
or a range gate capturing a non-empty set. Remember further that, as C has a compatible
order, no two leaves can share a common variable. Hence, each leaf of T contributes at least
one to the Hamming weight of a minimal valuation ν, so that, letting n be the number of
leaves of T , we have |ν| ≥ n.
As C is arity-two and collapsed, each AND-gate of T has exactly two children, and by
definition of a compressed trace each OR-gate of T has exactly one child. Letting n0 be the
number of AND-gates in T , it is then clear that n0 = n − 1. Call AND-gates, variable gates,
and range gates useful: their number is n + n0 . It suffices to show that the number n00 of
OR-gates of T is at most 2(n0 + n). This follows if we can show that, for each useful gate,
one of its parent, grandparent, and great-grandparent in T is also useful (or is undefined,
in the case of the root). Indeed, this implies that n00 ≤ 2(n0 + n) because, if each useful
gate covers its parent and grandparent, this guarantees that all non-useful gates (namely, all
OR-gates) are covered. The reason why a parent, grandparent, or great-grandparent of a
useful gate must be useful is that, whenever T contains an OR-gate, it is either an exit and
then its one child is not an OR-gate so it is useful, or it is not an exit, in which case its one
child is an exit. So we have shown the desired inequality, which concludes the proof.
J
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
I Proposition 6.3. Given a normal d-DNNF C with its OR-index, we can enumerate its
compressed traces, with the delay to produce each compressed trace T being in O(|T |).
Proof. We define inductively an algorithm to enumerate the sequence of partial compressed
traces in C rooted at a gate g as follows:
If g is a variable, produce the one element of its singleton sequence of compressed traces
and halt immediately.
If g is an OR-gate, enumerate with constant delay its sequence of reachable exits, using
the precomputed OR-index: as the circuit is normal, this sequence is non-empty. For
each reachable exit g 0 , letting g 00 be its one input gate, enumerate the sequence T (g 00 )
of partial compressed traces rooted at g 00 . For each such compressed trace C, produce
C ∪ {g, g 0 } (the union is disjoint). Halt when the enumeration of reachable exits has
halted with the last such gate g 0 , and the enumeration of partial compressed traces rooted
at g 0 has halted.
If g is an AND-gate, as the circuit is normal it has exactly two inputs. Enumerate the
sequence of partial compressed traces T (g1 ) rooted at its first input g1 . For each trace
C1 ∈ T (g1 ), enumerate the sequence T (g2 ) of partial compressed traces rooted at its
second input g2 . For each such trace C2 , produce {g} ∪ C1 ∪ C2 (the unions are disjoint).
Halt when the enumeration of compressed traces of T (g1 ) has halted with the last such
trace C1 and the enumeration of T (g2 ) has also halted.
Running the algorithm on C is simply running it on the output gate g0 .
We claim that the delay of this algorithm when producing a compressed trace T is
in O(|T |). To see why, observe that the state when we start to enumerate the next valuation
consists of gates of C where the enumeration has not yet halted, or (in the case of left
children of AND-gates) where enumeration has halted but where the previous compressed
trace will be reused in full. At each node that we consider in the algorithm, we perform a
constant amount of computation (in particular, for OR-gates, we use the OR-index), and
then we output that gate as part of the compressed trace. Hence, the algorithm performs a
constant number of steps at a set of gates which is a subset of the gates of the compressed
trace which is output, so the claim holds. (In particular, when the enumeration at one gate
halts, the end of the computation at that gate is accounted as part of the last compressed
trace using a recursive call at that gate, but it is not considered when producing the next
compressed trace (which may not include that gate).)
J
D.2
Enumerating Valuations of a Compressed Trace
I Proposition 6.5. We can enumerate the solutions to the assignment enumeration problem
for < on Cvar , with each solution t being produced with delay linear in its size |t|.
To prove this result, it will be convenient to enumerate assignments following the lexicographic
product of the individual orders:
I Definition D.1. Given two sets S1 , S2 with orders ≤1 , ≤2 , the lexicographic product
≤1 × ≤2 on S1 × S2 is defined by (a1 , a2 )(≤1 × ≤2 )(b1 , b2 ) if and only if
(a1 <1 b1 ), or
a1 = b1 and (a2 ≤2 b2 ).
The lexicographic product of two totally ordered sets is clearly a total order, and the
lexicographic product operation is clearly associative, so this definition extends to an arbitrary
number of sets and yields a total order.
XX:29
XX:30
A Circuit-Based Approach to Efficient Enumeration
We show the following general lemma about enumeration in the lexicographic order:
I Lemma D.2. Let S1 , . . . , Sn be non-empty sets that do not contain the empty assignment,
such that the elements of each Si can be enumerated in some total order ≤i , each element
being produced with delay linear in its size. Then the elements of the product S1 × · · · × Sn
can be enumerated in the lexicographic order ≤1 × · · · × ≤n , each element being produced
with delay linear in its total size.
Proof. We run the enumeration algorithm for each Sj . To produce the first enumeration
result, we enumerate the first element of each Sj with its algorithm, and we find the largest
1 ≤ j ≤ n such that the element of Sj that we enumerated is not the last one: this obeys the
delay bound, because, as the Sj do not contain the empty assignment, the time required to
iterate over all the Sj is linear in the total size of the enumerated solution.
Now, at each stage of the global enumeration algorithm, we remember the last element
of the product that we enumerated, the corresponding enumeration state in each Sj , and
the largest 1 ≤ j ≤ n such that the element of Sj that we enumerated is not the last one.
To produce the next element, enumerate the next element e of this Sj , and then compute
first element ej → for Sj → for j < j → ≤ n as when producing the first enumeration result
(this may be empty if j = n). Finally, we go over all Sj to update our value for j. We then
produce our enumeration result: it is composed of the element of the product of the Sj ←
for 1 ≤ j ← < j that we had enumerated in the round before, of the element e ∈ Sj that we
just enumerated, and the ej → for j < j → ≤ n. The delay of this is the delay of writing the
solution, which is linear in its total size, plus the delay of going over the Sj , which is linear
in the total size as above, and the delay of enumerating e ∈ Sj and the ej → , which is less
than the total size again, so we obey the bound.
J
We will use Lemma D.2 to enumerate the solutions to the assignment enumeration
problem (recall Definition 6.4), and we will do so in two steps. We will first reduce to the
case where all constraints ./j are equalities. Then we will enumerate valuations in this case.
To reduce to equalities, we will define the range Rj of the interval [gj− , gj+ ] for 1 ≤ j ≤ n
as the singleton {ij } if ./j is =, and the range {ij , ij + 1, . . . , [gj− , gj+ ] } if ./j is ≥; note that
this set is non-empty. The range R of the product is simply R1 × · · · × Rn . We can talk of an
element t1 × · · · × tn of the product of the intervals as realizing the vector (|t1 | , . . . , |tn |) of R,
which we call a histogram: clearly all such elements realize a histogram of R (so the values
of R partition the assignments), and conversely every value of R is the histogram of some
element (i.e., the classes of the partition are non-empty). Thus, to prove Proposition 6.5,
we can enumerate the histograms of the range R, and then enumerate the assignments
corresponding to this histogram.
It is clear that, for each range Rj , we can enumerate the integers that it contains with
constant delay since we are working in a RAM model. Hence, by Lemma D.2, we can
enumerate the histograms of R with delay linear in n, i.e., the number of entries in the
histograms. Note now that n is always less than the size of any assignment that realizes it
because the ij and thus the number of inputs chosen from each interval [gj− , gj+ ] is strictly
positive, so the delay when enumerating a histogram is within the allowed delay to enumerate
an assignment. Note that, as each histogram is realized by at least one assignment, the delay
when enumerating a histogram is paid at most once when enumerating an assignment.
Hence, it suffices to study the enumeration of assignments that satisfy a fixed histogram,
i.e., where ./j is = for each 1 ≤ j ≤ n. Let Sq for a set S and a non-negative integer q
denote the set of all subsets of size q of S. We make the following observation.
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
I Observation D.3. Let [g1− , g1+ ], . . . , [gn− , gn+ ] and = ii , . . . , = in be an instance of the assignment enumeration problem where all cardinality constraints are equalities. Then the
− +
− +
1 ] × . . . × [gn ,gn ] .
assignments to be enumerated on the given instance are exactly [g1 i,g
in
1
Proof. By definition of the assignment enumeration problem, when choosing for each j ∈ [n]
an assignment tj of size ij from [g1− , g1+ ], we have that t1 × . . . × tn has to be enumerated.
Conversely, every assignment a that has to be enumerated decomposes as a1 × . . . × an with
− +
− +
1 ] × . . . × [gn ,gn ] .
|aj | = ij and thus t ∈ [g1 i,g
J
in
1
Hence, applying Lemma D.2 again, it suffices to argue that we can enumerate the elements
[g − ,g +
of the ji j in delay linear in the size of the produced elements, i.e., linear in ij . We
j
will see the elements of [gj− , gj+ ] as ordered according to <, which allows us to define a
[g − ,g +
lexicographic order on ji j . It is then known that we can enumerate such elements with
j
delay linear in ij ; we refer the reader to e.g. [Knu05, Section 7.2.1.3] where implicitly the
following is shown.
I Proposition D.4. Given a set S of p ordered elements and q ∈ N, the following tasks can
be performed in time O(q):
compute the lexicographically minimal combination of q elements from S, and
given a combination of q elements from S, compute the lexicographically next such
combination if it exists.
Hence, by Lemma D.2 we can enumerate the assignments satisfying a histogram, producing
each assignment with delay linear in its total size. This concludes the proof of Proposition 6.5.
D.3
Putting Things Together
We are now ready to put things together to prove our main results. We first show:
I Proposition D.5. Given a normal d-DNNF C with its OR-index, we can enumerate the
elements of S(C), producing each assignment t with delay O(|t|).
Proof. The enumeration algorithm consists of two nested loops: In the outer loop, we
enumerate the compressed traces T of C with the help of Proposition 6.3. In the inner
loop, we enumerate for each T the satisfying assignments with Proposition 6.5. Since C is
deterministic, each satisfying assignments of C is captured by exactly once compressed trace.
Consequently, we enumerate every satisfying assignments of C exactly once, so the algorithm
is correct.
To analyze the delay of the algorithm, note that, to enumerate a valuation ν, in the worst
case we have to first enumerate the next compressed trace T of C and then compute the
valuation ν as a valuation of T . The first part takes time O(|T |) by Proposition 6.3 which by
Lemma 6.2 is O(|ν|). The second part takes time O(|ν|) by Proposition 6.5. So the overall
delay to produce ν is O(|ν|) as claimed.
J
We are now ready to prove our first main result:
Proof of Theorem 2.1. Given C, we first deal with two special cases. We first check if C
has any satisfying assignments. If not, we are done at this point and stop. Note that this
consistency check can be done in linear time [Dar01].
The second special case is that we check if C is satisfied exactly by {{}}. If so, we print
out {} and are done. This test can also be done in linear time as follows: First check if
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A Circuit-Based Approach to Efficient Enumeration
{} satisfies C. This can be done in linear time by substituting all inputs by 0 and then
evaluating C. Afterwards, we check if C is satisfied by exactly one valuation. Since satisfying
assignments of a d-DNNF can be counted in linear time [DM02], this is also a linear time
test.
In the remainder of the proof, we may now assume that the set S of valuations satisfying C
is such that S 6= ∅ and S 6= {{}}.
We now infer a compatible order < for C. As discussed in Section 3, this is easy to do
in linear time, assuming we are given a v-tree. Next, we proceed with Proposition 3.9 to
compute a monotone 0-augmented d-DNNF C 0 in zero-suppressed semantics having < as a
compatible order such that S(C 0 ) = S. We then use Proposition 4.3 to compute a 1-normal
d-DNNF C 00 which has < as a compatible order and is such that S(C 00 ) = S(C 0 ) \ {{}}.
Finally, we compute the OR-index of C 00 with Theorem C.1.
Before we start the enumeration phase, we check if {} satisfies C. If so, we enumerate {}
as the first valuation. Afterwards, we use Proposition D.5 to enumerate the valuations in
S \ {{}}.
By inspection of the individual results used in this algorithm, it is obvious that the
satisfying assignments of C are correctly enumerated. Moreover, the linear runtime bound
on the preprocessing follows by the fact that all individual steps can be performed in
time linear in their input size. The bound on the enumeration delay follows directly from
Proposition D.5.
J
The proof of Theorem 2.2 is identical to that of Theorem 2.1 except for the fact that we
make an additional preprocessing step. After using Proposition 3.9, we compute a circuit C k
that is satisfied exactly by the satisfying assignments of C with Hamming weight at most k
with Proposition 4.1. We then proceed as in the proof of Theorem 2.1. Note that this slightly
increases the runtime of the preprocessing from O(|C|) to O(k 2 · |C|).
E
E.1
Proofs for Section 7 (Applications)
Computing Circuit Representations of MSO Answers
I Theorem 7.3. For any fixed MSO formula φ(X1 , . . . , Xk ) on Γ-trees, given a Γ-tree T , we
can build in time O(|T |) a monotone d-DNNF circuit C in zero-suppressed semantics whose
set S(C) of assignments (as in Definition 3.6) is exactly the set of assignments of φ on T .
This appendix section proves Theorem 7.3; we later explain in Appendix E.2 how we can use
this result to deduce MSO enumeration results using our main results. The key ingredient
of the proof of Theorem 7.3 is our existing construction for provenance of MSO queries
on treelike instances [ABS15], using automaton determinism to obtain a d-DNNF [ABS16].
However, for readability, we give a self-contained proof of this result, which focuses on the
case of trees. The rewritten proof presented here is also useful to show upwards-determinism
and deduce constant memory bounds for enumeration (see Appendix F).
We introduce some additional notation. Given a Γ-tree T , we will write λ(n) to denote
the label in Γ of a node n of T ; in other words, the labeling function λ is part of the Γ-tree,
but we do not write it explicitly for brevity. We will write Leaf(T ) for the set of leaves of a
Γ-tree T . Remember that we often identify T with its set of nodes when no confusion can
ensue.
Further, we will write Assign(φ, T ) to denote the set of assignments of an MSO formula
φ(X1 , . . . , Xk ) on Γ-trees with free second-order variables on a Γ-tree T , i.e., the set of
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
assignments A on schema X = X1 , . . . , Xk and domain T such that T satisfies φ(A1 , . . . , Ak )
with the Ai defined as in Definition 7.2.
To prove Theorem 7.3, somewhat similarly to Sections 3.3.2 and Sections 3.3.3 of [Bag13],
it will be useful to assume that assignments are only considered on leaves, and that the
MSO formula only has one free second-order variable. We will explain how to do this, up to
extending the size of the alphabet.
I Definition E.1. Let Γ be a finite alphabet of labels, let X = X1 , . . . , Xk be a tuple of
second-order variables which we see as labels disjoint from Γ, and let ⊥ be a fresh node label.
Let ΓX := Γ ∪ {⊥, X1 , . . . , Xk }.
A ΓX -assignment tree (T, µ) is a ΓX -tree T and a mapping µ from Leaf(T ) to a domain
D called the domain of the assignment tree. We impose the following requirements:
The labels X1 , . . . , Xk are used only on leaf nodes, and conversely every leaf node carries
a label of this set. Formally, we require Leaf(T ) = {n ∈ T | λ(n) ∈ {X1 , . . . , Xk }}.
The mapping µ is computable in constant time, i.e., we can read the image by µ of a leaf
node of T directly from that node.
If µ(n) 6= µ(n0 ) for two leaves n 6= n0 of T , we require that λ(n) 6= λ(n0 ).
For any ΓX -assignment tree T and subset U ⊆ Leaf(T ), the X-assignment α(U ) of U is
defined as {hλ(n) : µ(n)i | n ∈ U }. Note that this set is without duplicates thanks to our
requirement on µ above, and it is an assignment on schema X and domain T .
We now claim that, up to increase the size of the formula, we can rewrite an MSO
formula so that it has only one free variable and only answers that include leaves need to be
considered:
I Lemma E.2. For any MSO formula φ(X1 , . . . , Xk ) on Γ-trees, we can compute an MSO
formula ψ(Y ) on ΓX -trees with one free second-order variable that has the following property:
given any Γ-tree T , we can compute in linear time a ΓX -assignment tree (T 0 , µ), whose domain
is the nodes of T , such that the assignments of φ on T are exactly the ΓX -assignments of the
answers of ψ on T 0 ; formally: Assign(φ, T ) = {α(U ) | U ⊆ Leaf(T 0 ), T 0 |= ψ(U )}.
Proof. We rewrite φ(X1 , . . . , Xk ) to an MSO formula ψ(Y ) on ΓX -trees, by creating the free
second-order variable Y and replacing each atom of the form Xi (x) for a first-order variable x
and free second-order variable Xi by ∃y Y (y) ∧ Xi (y) ∧ Φ(x, y), where Φ is a constant-sized
MSO subformula asserting that y is a descendant of x and the path from x to y in the tree
passes only through nodes labeled ⊥.
We now describe the linear-time rewriting of input trees. We rewrite an input Γ-tree
T to a ΓX -assignment tree (T 0 , µ) consisting of a ΓX -tree T 0 and function µ from Leaf(T 0 )
to T (written directly on the leaves to ensure constant-time computability). We do so by
adding, for every node n of T , k fresh descendants n1 , . . . , nk that we connect to n by a
binary tree of fresh nodes labeled ⊥. Each ni is labeled with Xi and mapped by µ to n. It is
clear that this process runs in linear time, remembering that k is a constant. Further, it is
clear that (T 0 , µ) uses Xi only on leaf nodes, and exactly on such nodes; and that µ satisfies
the requirement that it does not map to the same element of T two leaves of T 0 carrying the
same label.
Last, it is immediate that the answers of ψ on T 0 map to the assignments of φ on T in
the prescribed way. Indeed, the rewriting of φ to ψ clearly ensures that U ⊆ Leaf(T 0 ) is
an answer to ψ iff (U1 , . . . , Un ) is an answer to φ, where Ui contains the nodes of T whose
fresh descendant labeled Xi and connected by a ⊥-path in T 0 is in U . This is the case iff the
X-assignment of U on X and T is an assignment to φ.
J
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A Circuit-Based Approach to Efficient Enumeration
Thanks to this result, we can restrict our study to MSO formulae ψ(Y ) with only one
free variable, and to answers of ψ that only contain leaves of the tree. We will now state a
simple lemma that asserts that the interpretation of a free second-order variable in an MSO
formula can always be read off directly from the labels of the tree. We first introduce some
definitions:
I Definition E.3. A leaf valuation ν of a Γ-tree T is a function mapping the nodes of Leaf(T )
to {0, 1}; we will abuse notation and see them as valuations of T by extending them to map
every internal node to 0. We write LVal(T ) for the set of leaf valuations of T .
We write Γ to mean Γ × {0, 1}. For ν ∈ LVal(T ), we denote by ν(T ) the Γ-tree obtained
from T by relabeling each node n from λ(n) to (λ(n), ν(n)).
I Lemma E.4. Given an MSO formula ψ(Y ) on Γ-trees with one free variable, we can
compute an MSO formula χ on Γ-trees with no free variables (i.e., a Boolean formula) that
has the following property: for any Γ-tree T , for any leaf valuation ν ∈ LVal(T ), the Γ-tree
ν(T ) satisfies χ iff {hY : ni | ν(n) = 1} is an assignment of ψ(Y ).
Proof. We simply rewrite each atom L(x) for a node predicate L of Γ by ((L, 0))(x) ∨
((L, 1))(x), and we replace atoms Y (x) that use the free second-order variable Y with
W
L ((L, 1))(x) for all node predicates L in Γ. It is then clear that the additional label of a
Γ-tree indicates how the free second variable should be interpreted.
J
Remembering that we are only considering answers to the input MSO formula ψ(Y ) that
consist of leaf nodes, this lemma allows us to assume a Boolean formula χ on Γ-trees and to
study the leaf valuations ν of an input Γ-tree T such that the χ accepts ν(T ). Our goal is
to obtain a circuit which captures these leaf valuations (represented as assignments) under
zero-suppressed semantics. In other words, the circuit will have variable gates that correspond
to the nodes of T , and its captured set should be exactly the assignments corresponding to
leaf valuations of T that make it satisfy χ.
To compute this circuit, we will be going through tree automata. To this end, it will be
simpler to think of automata that read ordered trees, i.e., there is an order on the children of
each internal node; we will define automata accordingly but will ensure that this order is
inessential. It will also be simpler to assume that input trees are full, i.e., every node has
either 0 or 2 children. To do this, we can always add a fresh symbol ⊥0 to the alphabet,
with its two labeled versions (⊥0 , 0) and (⊥0 , 1), and add fresh leaves to Γ-trees labeled ⊥0 to
make them full. One would then rewrite the MSO formula to relativize quantification to
nodes that are not labeled ⊥0 (i.e., do not quantify over them), and add a constant-sized
formula asserting that these nodes are all labeled 0 so that they never occur in assignments.
We thus define deterministic bottom-up tree automata in the standard way:
I Definition E.5. A bottom-up deterministic tree automaton on Γ-trees that are full and
ordered (and binary), called a Γ-bDTA for brevity, is a tuple A = (Q, F, ι, δ) where:
1. Q is a finite set of states;
2. F is a subset of Q called the accepting states;
3. ι : Γ → Q is an initialization function which determines the state of the automaton on a
leaf node from the label of that node;
4. ∆ : Γ × Q2 → Q is a transition function which determines the state of the automaton on
an internal node from its label and the state of the automaton on its two children.
As our trees are unordered, we require that the order in which the automaton reads the children
of a node never matters, i.e., for every l ∈ Γ and q1 , q2 ∈ Q, we have δ(l, q1 , q2 ) = δ(l, q2 , q1 ).
Given a Γ-tree T , we define the run of A on T as the function f : T → Q defined by:
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
1. For each leaf l of T , set f (l) := ι(λ(l));
2. For each internal node n of T with children n1 and n2 , set f (l) := δ(λ(n), f (n1 ), f (n2 )).
We say that the bDTA A accepts a Γ-tree T if, letting nr be the root of T , the run f of A
on T is such that f (nr ) ∈ F .
We now use the well-known fact that Boolean MSO formulae on Γ-trees can be rewritten
to equivalent Γ-bDTAs, using the standard translation of Thatcher and Wright [TW68] and
standard techniques to determinize the automaton [CDG+ 07]:
I Theorem E.6 [TW68]. For any tree alphabet Γ and Boolean MSO formula χ on Γ-trees,
we can compute a Γ-bDTA A such that, for any Γ-tree T , we have that T satisfies χ iff T is
accepted by A.
Having fixed our Boolean formula χ on Γ-trees, let us compute accordingly such a
Γ-bDTA A. Remember that, given a Γ-tree T , we want to compute a circuit whose captured
set under zero-suppressed semantics is the set of assignments representing leaf valuations
ν of T such that A accepts ν(T ). We call this the assignment set of the automaton A on
the tree T . The following definition is inspired by the provenance notions in [ABS15], but
changed to work only on leaves.
I Definition E.7. Let A be a Γ-bDTA, and T be a Γ-tree. The assignment set α(A, T ) of A
on T is the set {α(ν) | ν ∈ LVal(T ), A accepts ν(T )}.
We then give a construction inspired to Proposition 3.1 of [ABS15], but rephrased in the
terminology of factorized representations, and simplified by limiting the uncertain labels to
leaves. We also observe that the result is deterministic thanks to the determinism of the
automaton, as in Theorem 6.11 of [ABS16].
I Proposition E.8. For any tree alphabet Γ, given a Γ-bDTA A = (Q, F, ι, δ) and a full
(binary) Γ-tree T , we can compute in time O(|A| · |T |) a monotone circuit C which is a
d-DNNF in zero-suppressed semantics, such that S(C) = α(A, T ).
Note that C is not an augmented circuit, but as it is decomposable, the set S(C) of
assignments of C in zero-suppressed semantics (in the sense of Definition 3.6, or Lemma 3.8)
is well-defined (recall Remark A.10).
Proof of Proposition E.8. We compute the circuit C in a bottom-up fashion on T . We
consider each node n of T with label λ(n) ∈ Γ.
If n is a leaf node, for b ∈ {0, 1} we let qb := ι((λ(n), b)), and we create the following
gates in C:
One OR-gate gnq for each q ∈ Q with the following inputs:
If q = q0 , one AND-gate with no inputs.
If q = q1 , one variable gate corresponding to the node n
If n is an internal node with children n1 and n2 , we create the following gates in C:
One AND-gate gnq1 ,q2 for each q1 , q2 ∈ Q whose inputs are gnq11 and gnq22 ;
One OR-gate gnq for each q ∈ Q with inputs the gnq1 ,q2 for each q1 , q2 ∈ Q such that
δ((λ(n), 0), q1 , q2 ) = q.
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A Circuit-Based Approach to Efficient Enumeration
The output gate g0 is a ∨-gate of the gnq r for q ∈ F , where nr is the root of T .
It is clear that the construction of C runs in the prescribed time bound, because the
processing that we perform at each node of T is linear in |A|, specifically, in the table of the
transition function δ of A.
It is clear that C is decomposable, because AND-gates that have inputs are of the form
gnq1 ,q2 for internal nodes n of T , in which case the inputs are gnq11 and gnq22 . Now, it is immediate
S
that, for i ∈ {1, 2}, only descendant leaves of ni can appear in S(gnqii ). As these sets
of descendant leaves for the two sibling nodes n1 and n2 are disjoint, the decomposability
condition is indeed satisfied.
It is now easy to show the following inductive correctness claim on C: for each q ∈ Q
and n ∈ T the assignment set S(gnq ) captured by the gate gnq precisely describes the leaf
valuations ν of the subtree Tn of T rooted at n such that the run of A on ν(Tn ) reaches q
on the root node n of ν(Tn ). Indeed, for a leaf node n of T and for q ∈ Q, the assignments
corresponding to the possible leaf valuations are {} and {n}, and we have {} ∈ S(gnq ) iff
q = ι((λ(n), 0) and {n} ∈ S(gnq ) iff q = ι((λ(n), 1)). For an internal node n of T with children
n1 and n2 and q ∈ Q, an assignment a corresponding to a leaf valuation ν belongs to S(gnq )
iff there is a pair q1 , q2 ∈ Q of states such that δ((λ(n), 0), q1 , q2 ) = q and, for each i ∈ {1, 2},
the assignment ai of the restriction νi of ν to the subtree Tni rooted at ni belongs to S(gnqii ).
By induction hypothesis, for any q1 , q2 ∈ Q, for each i ∈ {1, 2}, this happens iff the run of A
on ν(Tni ) reaches qi on the root node ni of ν(Tni ). Hence, the condition is equivalent to
requiring that there is q1 , q2 ∈ Q such that δ((λ(n), 0), q1 , q2 ) = q and, for all i ∈ {1, 2}, the
run of A on ν(Tni ) reaches qi on the root node ni . By definition of δ, this is the case iff the
run of A on the subtree Tn of T rooted at n reaches q on the root node n. This concludes
the inductive proof of the correctness claim.
This clearly implies that the set captured by the decomposable circuit C is the union of
the assignment sets ν such that the run of A on ν(T ) reaches a final state at the root, i.e.,
the assignments ν(T ) for which A accepts ν(T ), so the construction is correct.
It remains to show that C is deterministic. The only OR-gates that we introduce are
the gnq , and the output gate g0 . For a leaf node n ∈ T , it is clear from their definition that
the gnq are deterministic. For an internal node n ∈ T with children n1 and n2 , the fact that
the gnq are deterministic is thanks to the determinism of the automaton: for every valuation
ν ∈ LVal(T ), by the inductive invariant, for each i ∈ {1, 2}, there is exactly one qi ∈ Q such
that ν ∈ gnqii . Hence, there is exactly one q1 , q2 ∈ Q such that ν ∈ S(gnq1 ,q2 ). This implies
that there could not be a gate gnq for some q ∈ Q such that ν is in the captured set of two of
its inputs. For the output gate g0 , determinism follows again from the determinism of the
automaton, as for every leaf valuation ν of T the automaton A reaches exactly one state on
the root of ν(T ). Thus, C is deterministic. This concludes the proof.
J
This allows us to recap the proof of Theorem 7.3:
Proof of Theorem 7.3. Fix the MSO formula φ(X1 , . . . , Xk ) on Γ-trees, compute the MSO
formula ψ(Y ) on ΓX -trees by Lemma E.2, and the Boolean MSO formula χ on ΓX -trees by
Lemma E.4. Rewrite χ to χ0 by adding one fresh symbol ⊥0 that can be used to make input
trees full, relativizing quantification to exclude ⊥0 -nodes from consideration but asserting that
they never carry the label (⊥0 , 1), and let Γ0 be the resulting alphabet, where Γ0 = ΓX ∪ {⊥0 },
the union being disjoint as ⊥0 is fresh. Now, use Theorem E.6 to compute a Γ0 -bDTA for χ0 .
Given the input Γ-tree T , rewrite it in linear time following the process of Lemma E.2 to
a ΓX -tree T 0 , and complete it with ⊥0 -nodes to a Γ0 -tree T 00 which is binary and full. Now,
use Proposition E.8 to compute a deterministic circuit C that captures the assignments of A
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
on T 00 and is a d-DNNF in the zero-suppressed semantics. Finally, rewrite C in linear time
to C 0 by considering each variable gate n and doing the following:
If n is a leaf of T 00 which is not in T 0 (i.e., it was added just to make the tree full), replace
n with an OR-gate with no inputs. Recalling that χ0 enforces that such nodes are never
annotated with 1 in a valuation, this does not change the captured set of the circuit.
Further, it clearly cannot alter decomposability, nor can it alter determinism because
the captured set S(g) of each gate g after this transformation are a subset of the set
previously captured by gate g.
If n is a leaf of T 00 which is in T 0 , recalling that its label λ(n) is necessarily in X1 , . . . , Xk ,
replace the singleton n by hλ(n) : µ(n)i. By the condition on µ, this cannot break
decomposability or determinism, because it is a bijective renaming of the variable gates.
Hence, the result C 0 is a monotone d-DNNF in zero-suppressed semantics. We now
show that it captures the assignments of φ on T . For the forward direction, consider an
assignment A of φ on T . By Lemma E.2, there is a subset U of leaves of T 0 such that
α({hY : ni | n ∈ U }) = A and T 0 satisfies φ(U ). By Lemma E.4, the leaf valuation
νU obtained from U is such that νU (T 0 ) satisfies χ, and clearly if we expand νU to a
valuation of T 00 that sets to 0 the additional leaves of T 0 we know that νU (T 00 ) satisfies
χ0 . Hence, by Theorem E.6, we know that A accepts νU (T 00 ), so by Proposition E.8 the
assignment U corresponding to νU is captured by C. Now, our rewriting ensures that, as
A = α({hY : ni | n ∈ U }), the circuit C 0 captures A.
For the backward direction, consider an assignment A captured by the monotone d-DNNF
C 0 . Considering its preimage in C, this means that C captures an assignment U , i.e., a
set of leaves of T 00 , that are all in T 0 and such that α({hY : ni | n ∈ U }) = A. Now, by
Proposition E.8 we know that, letting νU be the leaf valuation of T 00 defined by setting
the nodes of U to 1 and setting all other nodes to 0, the automaton A accepts νU (T 00 ).
By Theorem E.6, this implies that νU (T 00 ) satisfies χ0 , hence νU (T 0 ) satisfies χ, hence, by
Lemma E.4, T 0 satisfies ψ(U ), and by Lemma E.2 we know that T satisfies φ(A). This
concludes the correctness proof.
J
E.2
Proof of MSO Enumeration Results
We now explain formally how our results can be used to re-prove the existing result of [Bag06,
KS13], once we have restricted to Γ-trees. Note that, unlike what we defined in the main
text, this result does not only focus on data complexity: the goal is to justify the O(k · |T |)
claim for the delay given in the main text.
I Theorem E.9. For any fixed tree alphabet Γ, given an MSO formula φ with k free variables
and a Γ-tree T , we can enumerate the answers to φ on T with the following complexities:
the preprocessing has linear data complexity, i.e., it is in O(f (|φ|) · |T |) for some fixed
function f ;
the delay is linear in each produced valuation and independent from the query except for k,
in particular, it is in O(k · |T |);
the memory usage is linear in the size of the largest valuation and again independent
from the query except for k, so again in particular in O(k · |T |).
If all free variables of φ are first-order, the delay and memory usage are in O(k).
Proof. For the preprocessing phase, we use Theorem 7.3 to compute in linear-time a monotone
circuit C which is a d-DNNF in zero-suppressed semantics and captures the assignments of φ
on T . Note that we have not shown a compatible order for C, but it has no range gates, so
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A Circuit-Based Approach to Efficient Enumeration
we know by Remark B.7 that we can apply the results of Section 4 to the circuit, and the
same is immediately true for Sections 5 and 6.
We further know that this circuit is upwards-deterministic by Claim F.3 (see Appendix F),
so we can apply the linear-time preprocessing scheme of Theorem F.2 as well as its enumeration
scheme. This runs in delay linear in each assignment, which is always in O(k · |T |), i.e.,
constant delay (O(k)) if the size of assignments is constant, which is in particular the case if
the free variables of φ are second-order translations of free first-order variables. We then
rewrite each assignment (set of singletons) to the answer that it represents, in time linear in
each assignment. The memory usage is linear in each assignment thanks to Theorem F.2. J
E.3
Factorized Representations
I Lemma 7.4. For any d-representation D, let C be the monotone circuit obtained by
replacing × and ∪ by AND and OR, replacing ∅ and hi by AND-gates and OR-gates with no
inputs, and keeping singletons as variables. Then all AND-gates of C are decomposable, and
S(C) (defined as in Section 3) is exactly the database relation represented by D.
Proof. The fact that C is decomposable, i.e., a DNNF, is thanks to the requirement on
d-representations which imposes that gates have a schema, with union always having input
gates of the same schema, and product always having input gates of disjoint schemas. This
requirement clearly disallows in particular that some singleton has a path to two different
inputs to a product gates. Note that C is not an augmented circuit, but as it is decomposable,
its set of assignments S(C) in zero-suppressed semantics (in the sense of Definition 3.6, or
Lemma 3.8) is well-defined (recall Remark A.10). The claimed result on S(C) then follows
immediately from Lemma 3.8.
J
I Theorem 7.5. The tuples of a deterministic d-representation D over a schema S can be
enumerated with linear-time preprocessing, delay O(|S|), and memory O(|S| log |D|).
Proof. Let D be a deterministic d-representation, and let C be the corresponding monotone
circuit as in the statement of Lemma 7.4, such that the set S(C) captured by C is the
relation represented by D: we know that C is decomposable. The circuit C is not exactly
deterministic because the determinism requirement of [OZ15] only requires that they are no
duplicate tuples in the captured set of the output gate g0 . However, it is easy to see that
this requirement implies that, for every OR-gate g, there are no duplicates when computing
S(g), unless g has no directed path to g0 or it is “absorbed” later in the circuit (i.e., we only
use its value conjoined with gates capturing ∅). Hence, we rewrite C to C 0 by removing
gates with no directed path to g0 , and by computing bottom-up in linear time which gates
capture exactly ∅ (as in Lemma B.4), and replace them by OR-gates with no inputs: this
does not change the set captured by C (indeed, the sets captured by all remaining gates),
and C 0 is still decomposable. Now, it is clear that the determinism requirement of [OZ15]
on S(g0 ) in C, hence on S(g0 ) in C 0 , imposes that all OR-gates are deterministic, because
any violation of determinism on a gate g would imply a duplicate in S(g), hence in S(g0 ),
following a directed path from g to g0 , and observing that the duplicate can never be lost
at an OR-gate along the path, or at an AND-gate (this uses the fact that no gate captures
∅). Hence, C 0 is a d-DNNF in zero-suppressed semantics such that S(C 0 ) is the relation
represented by D.
We note that C 0 does not have a compatible order, but again it is decomposable and
does not have range gates, so the process in Sections 4–6 still applies to it (see in particular
Remark B.7), because the process does not introduce range gates, and does not use the order
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
except to define the semantics of range gates and to guarantee decomposability. So we can
simply use Proposition 4.3 to compute a normal monotone circuit C 00 capturing the same set
as C 0 (it is not necessary to apply homogenization because C 0 already captures tuples of the
correct weight), we apply Theorem C.1, and last we enumerate following Proposition D.5. We
handle the special cases of {{}} and of circuits capturing ∅ like in the proof of Theorem 2.1
in Appendix D.3. Thus, we can enumerate the tuples of C 0 , hence of D, with linear-time
preprocessing, delay in O(|S|), and memory O(|S| log |D|) as in Theorem 2.1.
J
F
Constant-Memory Enumeration for Upwards-Deterministic Circuits
Remember that our enumeration results of Theorem 2.1 and Theorem 2.2 use memory in
O(|O| log |I|), where |I| is the size of the input circuit and |O| is the size of each output.
The factor in |O|, which is constant for constant-sized outputs, is obviously difficult to avoid.
However, the same is not true of the logarithmic factor in the input, which comes from the
indexing construction on multitrees of Theorem 5.4 in Section 5.
In this appendix, we explain how the memory usage of the enumeration phase of Theorem 2.1 and Theorem 2.2 can be improved to O(|O|), under an additional hypothesis on
the input circuit which allows us to bypass Theorem 5.4. We first present this condition,
called upwards-determinism, and claim that enumeration for such circuits can be performed
using memory linear in the size of each valuation (Theorem F.2). Second, we show that the
circuits produced for MSO enumeration in Theorem 7.3 are upwards-deterministic. Third,
we prove Theorem F.2.
F.1
Upwards-Deterministic Circuits
We define upwards-deterministic circuits in the following way:
I Definition F.1. A wire (g, g 0 ) of C is pure if g 0 is an OR-gate, or if g 0 is an AND-gate and
all its other inputs are 0-valid. A gate g is upwards-deterministic if g is unsatisfiable or there
is at most one gate g 0 such that (g, g 0 ) is a pure wire of C. We call C upwards-deterministic
if every AND-gate and OR-gate in C is upwards-deterministic.
In particular, when a wire (g, g 0 ) of a monotone circuit is pure, it intuitively means
that g 0 evaluates to 1 whenever g does, and S(g) ⊆ S(g 0 ) in zero-suppressed semantics.
Upwards-determinism imposes that g is an input to at most one such g 0 .
If we assume upwards-determinism, we can show the analogue of our main results of
Theorem 2.1 and Theorem 2.2, but with memory usage linear in each output. Namely:
I Theorem F.2. Given a structured upwards-deterministic d-DNNF C with its v-tree T ,
we can enumerate its satisfying assignments with linear-time preprocessing and delay and
memory usage linear in each valuation. Further, for any k ∈ N, we can enumerate the
satisfying assignments of Hamming weight ≤ k with preprocessing O(k 2 |C|) and with delay
and memory usage in O(k), i.e., constant delay and constant memory.
Proof sketch. We show that upwards-determinism can be preserved in our preprocessing in
Sections 3–4. Once the circuit is normal, upwards-determinism ensures that each OR-gate is
the input to at most one OR-gate, so OR-components in Section 5 are actually reversed trees,
and we can replace Theorem C.1 with a much simpler constant-memory indexing scheme.
J
The complete proof of Theorem F.2 is technical, and presented in Appendix F.3.
XX:39
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A Circuit-Based Approach to Efficient Enumeration
F.2
Upwards-Deterministic Circuits for MSO Enumeration
We now show the claim that Theorem 7.3 produces circuits whose underlying circuit is
upwards-deterministic. This implies the constant memory bound for MSO enumeration in
Theorem E.9, using Theorem F.2.
I Claim F.3. Theorem 7.3 produces upwards-deterministic circuits.
Proof. The input rewriting that we perform in the proof of Theorem 7.3 clearly cannot
influence the fact that the circuit is upwards-deterministic. Indeed, first, the bijective
renaming of inputs clearly has no effect. Second replacing some inputs by gates capturing ∅
ensures that the captured set of each gate is a subset of what it was before the rewriting:
so the set of unsatisfiable gates is a superset of what it was initially, and the set of 0-valid
gates is a subset of what it was initially, thus any violation of upwards-determinism in the
initial circuit implies the existence of a violation in the original circuit. From this, to show
the claim for Theorem 7.3, it suffices to show that the circuits produced in Proposition E.8
are upwards-deterministic. To show this, consider its application to an automaton with state
set A and to a Γ-tree T , and let C be the resulting circuit.
In the construction, the only gates that are used as input to multiple gates are the gnq for
q ∈ Q and n ∈ T when n is not the root of T . Let n0 be the parent of n in T , and assume
that n is the first child of n0 in T : the proof if n is the second child is symmetric. Let n2 be
2
the second child of n0 . The gates of C that have gnq as an input are then the gnq,q
for q2 ∈ Q,
0
q2
and the other input to each of them is gn2 . Now, by determinism of the automaton, using
the inductive invariant in the proof of Proposition E.8, we know that there is exactly one q2
such that {} ∈ gnq22 , i.e., gnq22 . Hence, the only outgoing wire of gnq which is pure is the one
2
to gnq,q
, so gnq does not violate upwards-determinism. This concludes the proof.
J
2
F.3
Proof of Theorem F.2
To show Theorem F.2, we revisit the proofs of Sections 3–6. Specifically:
1. We must show that the preprocessing steps of Sections 3–4 preserve upwards-determinism.
We must specifically show this for the reduction to zero-suppressed semantics (Proposition 3.9), the homogenization (Proposition 4.1), and the normalization (Proposition 4.3).
2. We must show that we can replace the use of Theorem C.1 in Section 5 by a constantmemory indexing result. To do this, we can use the assumption that OR-components are
reversed trees (i.e., rooted trees, where edges are reversed and go from the leaves to the
root), because this is guaranteed by upwards-determinism on normal circuits. Indeed, a
gate g with two different children in an OR-component would necessarily be satisfiable
(because a normal circuit is ∅-pruned), and its two outgoing wires in the OR-component
would be pure.
3. We must show that enumeration in Section 6 with the indexes of Proposition F.4 uses
linear memory.
We first show the second point:
I Proposition F.4. Given a reversed tree T , we can compute in linear time a data structure
allowing us to perform the following: given n ∈ T , enumerate in constant delay and constant
memory the leaves of T that have a directed path to n.
Proof. We traverse the tree in prefix order in linear time and store at each leaf a pointer to
the next leaf. We then traverse the tree bottom-up and store, for each internal node n of
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
the tree, a pointer to its first leaf in the prefix order (i.e., the first leaf that has a directed
path to n), and a pointer to its last leaf in the prefix order. This can clearly be performed in
linear time.
To perform the enumeration, given a node n, we jump to its first leaf n0 , remember its
last leaf n00 , and we enumerate the leaves in prefix order from n0 to n00 . This process is clearly
correct, constant delay, and uses only a constant amount of memory.
J
We next argue for the third point: the process of Section 6 takes memory linear in each
produced valuation (and in particular constant when valuations have bounded size). Indeed,
the only place in this section where memory usage did not satisfy this property was when
using the OR-indexes, but the indexes of Proposition F.4 only require constant memory, so
the overall memory usage is linear in the produced valuations.
We last take care of the first point. We first show that rewriting circuits to arity-two can
be performed in linear-time without breaking upwards-determinism, extending Lemma B.1:
I Claim F.5. Every upwards-deterministic Boolean circuit C can be rewritten in linear time
to an arity-two circuit C 0 that is equivalent to C in standard semantics (i.e., captures the
same function) and that is upwards-deterministic. We can further do so while preserving a
compatible order <.
For every k ∈ N, every upwards-deterministic monotone k-augmented Boolean circuit C
can be rewritten it in linear time to an arity-two monotone k-augmented Boolean circuit C 0
that is equivalent to C in zero-suppressed semantics and is upwards-deterministic. Further,
all properties preserved in Lemma B.1 are still preserved.
Proof. We will use the same construction to show the two claims, and it will essentially be
the same general construction that we used to show Lemma B.1: we rewrite each gate with
fan-in greater than 2 to a tree of gates of the same type with fan-in two. For this reason, we
will not argue that the same properties as before are preserved, because this will still be true
for the same reasons as before. To preserve upwards-determinism, we will simply be more
specific about the way in which we construct each tree.
We first preprocess the circuit once to compute which gates are 0-valid. This can clearly
be performed in linear time, as in Lemma B.5.
Whenever we wish to rewrite a gate g with input gates g1 , . . . , gn , with n > 2, the tree
of gates of the same type that we introduce will be linear (i.e., as unbalanced as possible).
Specifically, we remove the wires from gi to g for 1 ≤ i ≤ n, we introduce gates gi0 of the
same type as g for 1 < i < n − 1, we set the inputs of g to be g1 and g10 , the inputs of each
0
0
gi0 for 1 < i < n − 2 to be gi+1 and gi+1
, and the inputs of gn−2
to be gn−1 and gn . This
ensures that all gates have arity-two, and that the circuit is equivalent.
We now impose a constraint on the order in which the input gates g1 , . . . , gn should
be considered: we require that all gates that are 0-valid are enumerated first, so they are
attached as high in the tree as possible.
The only thing to show is that upwards-determinism is preserved. The new gates, i.e., the
gi0 introduced for each gate g, cannot introduce a violation of upwards-determinism, because
0
they have only one outgoing wire (to gi−1
, or to g). Hence, it suffices to consider outgoing
0
wires for gates of the rewritten circuit C that stand for gates of the input circuit C, i.e.,
using our terminology above, it suffices to consider the wires from the gi to the gj0 , or to g.
It clearly suffices to show that, whenever such a wire is pure, then the corresponding wire
(gi , g) is pure in C. Indeed, this implies that any violation of upwards-determinism in C 0 on
a gate g 0 (which also exists in C) would imply a violation of upwards-determinism on g 0 in C.
XX:41
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A Circuit-Based Approach to Efficient Enumeration
Hence, let us consider a wire (g 0 , g 00 ) in C 0 where g 0 exists in C, let g be the gate for
which g 00 was introduced: observe that the wire (g 0 , g) exists in C, and that g and g 00 have
the same type, in fact possibly we have g 00 = g 0 . Let us assume that (g 0 , g 00 ) is pure in C 0 ,
and show that it is pure in C. There are four possibilities:
The gate g is an OR-gate. In this case, the wire is pure in C, and there is nothing to
show.
The gate g is an AND-gate and all its inputs are 0-valid in C. In this case, the wire is
pure in C, and there is nothing to show.
The gate g is an AND-gate and only one of its inputs g ∗ is not 0-valid in C. In this case,
the only incoming pure wire of g in C is (g ∗ , g), and under our assumption that (g 0 , g 00 )
is pure in C 0 we must show that g 0 = g ∗ . From the construction we know that g ∗ is still
0-valid in C 0 , so we know that g ∗ was enumerated last in the inputs of g, so it is attached
to the lowest node in the tree of C 0 introduced for g. As g ∗ is still not 0-valid in C 0 , we
then know that g is not 0-valid in C 0 and none of the gi0 is 0-valid (because there is a path
from the gate g ∗ , which is not 0-valid, to all these gates that goes only via AND-gates).
So if the wire (g 0 , g 00 ) is pure, it must be the case that g 00 is the lowest node in the tree
of C 0 , and the other input to g 00 must be 0-valid so we must have g 0 = g ∗ which is what
we wanted to show.
The gate g is an AND-gate and at least two of its inputs are not 0-valid in C. In this
case, similarly to the above reasoning, g is not 0-valid in C and none of the gi0 are 0-valid
in C, Further, as two inputs that are not 0-valid were enumerated last, the lowest node
in the tree has two inputs that are not 0-valid. Hence, in fact, this case cannot occur
under our assumption that the wire (g 0 , g 00 ) is pure in C 0 .
This concludes the proof.
J
We then show:
I Claim F.6. The construction of Proposition 3.9 preserves upwards-determinism.
Proof. The construction first completes the input circuit using Lemma A.4, and then
computes its monotonization. We argue that monotonization on a decomposable circuit
cannot break upwards-determinism. Indeed, it does not change which gates are 0-valid, it
does not change the type of AND-gates or OR-gates except to introduce AND-gates with
no inputs, so it does not change which wires are pure; and further it cannot make any gate
unsatisfiable which wasn’t unsatisfiable. We thus focus on completion.
The completion construction in the proof of Lemma A.4 first rewrites the circuit to
an arity-two circuit C, which does not break upwards-determinism by Claim F.5. Then
it adds range gates as children to some AND-gates and rewrites inputs to OR-gates by
AND-gates of the original gates and some fresh range gates. We explain why this does not
break upwards-determinism.
It is clear that, in the new circuit C 0 , the wires going out of a new range gate or out of a
new AND-gate cannot violate upwards-determinism, because these gates are used as input
to only one gate. So it suffices to consider the wires (g 0 , g) going out of gates g 0 in C 0 that
correspond to gates that already existed in C. There are two cases: either g is an AND-gate
of C 0 that already existed in C, or g is an AND-gate introduced when rewriting an OR-gate
g 00 of C.
In the first case, we show that if the wire (g 0 , g) is pure in C 0 , then it was already pure
in C. But this is immediate: if the wire is pure, then all other inputs to g in C 0 are 0-valid,
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
and then from our rewriting it is clear that all inputs of g in C (which are a subset of those
in C 0 ) were already 0-valid.
In the second case, as g 00 was an OR-gate of C, the wire (g, g 00 ) was necessarily pure in C.
This allows us to conclude the proof. Indeed, assume by way of contradiction that there
is a gate g of C 0 that violates upwards-determinism. By our initial reasoning, g is necessarily
a gate that already exists in C. Further, g captures a non-empty set in C 0 , and by our
construction we know that the same is true of g in C. Now, let g1 6= g2 be the gates of C 0
such that the wires (g, g1 ) and (g, g2 ) are pure in C 0 . Let g10 , g20 be the gates that correspond
to g1 and g2 in C, i.e., gi0 = gi if gi0 exists in C, and otherwise gi0 is the OR-gate of C for which
the AND-gate gi was introduced. Our construction clearly ensures that g10 6= g20 : indeed, our
construction ensures that the gate g cannot have a wire both to a fresh AND-gate of C 0 and
to the original OR-gate (indeed no gates at all have wires to the original OR-gates), and g
cannot have a wire to two new AND-gates introduced for the same OR-gate (as we create
one AND-gate for each input). Now, our previous claim ensures that the wires (g, g10 ) and
(g, g20 ) are pure in C, so g witnesses that C is not upwards-deterministic, contradicting our
assumption and concluding the proof.
J
We then show the claim for Proposition 4.3. The construction of Lemma B.1 extends
thanks to Claim F.5. It is straightforward that Lemma B.6 does not break upwardsdeterminism. Indeed, wires to AND-gates that are collapsed are necessarily pure because
they have only one input, so collapsing the gates cannot cause a gate to have more than one
outgoing pure wire. Further, adding exits is not problematic, because wires to exits were
already to OR-nodes, so already pure, and each exit has exactly one outgoing wire.
The ∅-pruning process of Lemma B.4 preserves upwards-determinism. Indeed, any gate
in the output existed with the same type in the input, it is 0-valid in the output iff it is
0-valid in the input, all gates in the output are satisfiable but were already satisfiable in the
input, every wire in the output existed in the input, and it is not hard to see that if a wire
in the output is pure then is also pure in the input: indeed, the inputs to AND-gates are
unchanged, and changing the inputs to OR-gates is unproblematic because all their incoming
wires are always pure.
What must be shown is that upwards-determinism is preserved by the pruning construction
of Lemma B.5. In this lemma, the actual process of {}-pruning is unproblematic for similar
reasons as for ∅-pruning: note that removing input gates to AND-gates that are 0-valid
cannot cause any of the other input wires to become pure. The crux of the matter is to show
that Proposition B.3 preserves upwards-determinism. This also takes care of proving the
extension of Proposition 4.1. Hence, we claim:
I Claim F.7. If the input C to Proposition B.3 is upwards-deterministic, then its output C 0
also is.
Proof. Remember that the construction in the proof of Proposition B.3 first rewrites the
circuit to arity-two with Lemma B.1, which does not break upwards-determinism thanks
to Claim F.5; so we let C be the arity-two version of the input circuit, which is upwardsdeterministic.
The construction then produces C 0 by introducing, for each gate g of the original circuit C,
>k,1
>k
gates of the form g =i for 0 ≤ i ≤ k and g >k , as well as gates of the form gj=i and gi,j
, gi,j
,
>k,2
gi,j , which we call fresh gates of C 0 . We will define the original gate ω(g) of a gate g of C
as follows:
if g is of the form g0=i or g0>k , then ω(g) := g0
XX:43
XX:44
A Circuit-Based Approach to Efficient Enumeration
if g is a fresh gate created for a AND-gate g0 with two inputs in C, then ω(g) := g0 .
It is clear that fresh gates g in C 0 cannot violate upwards-determinism, because in the
construction any such gate g is used as input to only one gate in C 0 , specifically, a gate
whose original gate is the same as that of g. So it suffices to check upwards-determinism for
gates of C 0 which are not fresh gates, i.e., wires (g 0 , g) of C 0 where g 0 is not fresh, so that in
particular ω(g 0 ) 6= ω(g), and by construction (ω(g 0 ), ω(g)) is a wire of C.
We will show the following claim (*): for every wire (g00 , g0 ) of C such that g00 is an
AND-gate or an OR-gate, when considering every wire (g 0 , g) of C 0 such that ω(g) = g0 and
ω(g 0 ) = g00 , then (i) for each choice of g 0 , at most one g is such that the wire is pure, and
(ii) if one such wire is pure then (g00 , g0 ) is also pure in C. This claim implies that C 0 is
upwards-deterministic. Indeed, assume to the contrary that C 0 is not upwards-deterministic,
then it has an AND- or OR-gate g 0 which is not fresh, is satisfiable, and has two pure wires
(g 0 , g1 ) and (g, g2 ) with g1 6= g2 . The construction then ensures that ω(g 0 ) is an AND-gate
or an OR-gate and that (ω(g 0 ), ω(g1 )) and (ω(g 0 ), ω(g2 )) are wires of C. Further, by the
properties of C 0 , the set S(g 0 ) captured by g 0 in C 0 is a subset of the set S(ω(g 0 )) of ω(g 0 )
in C, so ω(g 0 ) is satisfiable. By (i), we know that we must have ω(g1 ) 6= ω(g2 ), and by (ii)
these two wires are pure in C, so ω(g 0 ) is not upwards-deterministic in C, a contradiction.
Hence, it suffices to show claim (*).
Let us show claim (*) by considering all possible wires (g00 , g0 ) of C:
If g0 is an OR-gate, then the wire (g00 , g0 ) is always pure so (ii) is vacuous. Further, for
each gate g 0 of C 0 with ω(g 0 ) = g00 , there is exactly one gate g of C 0 with ω(g) = g0 such
that the wire (g 0 , g) is in C 0 , so (i) holds.
If g0 is an AND-gate, then:
If g0 has no inputs, then there are no wires to consider so (i) and (ii) are vacuous.
If g0 has one input then the wire (g00 , g0 ) is always pure so (ii) is vacuous, and (i) holds
for the same reasons as for OR-gates.
If g0 has two inputs, let g000 be the input of g0 in C which is different from g00 , i.e., the
inputs of g0 in C are g00 and g000 . Observe that in the construction, for any wire (g 0 , g)
of C with ω(g) = g0 and ω(g 0 ) = g00 , the gate g is always a fresh AND-gate with two
inputs, and its other input is a non-fresh gate g 00 such that ω(g 00 ) = g000 . Hence, the
wire (g 0 , g) of C 0 is pure only if g 00 is 0-valid in C 0 . Recalling the properties of C 0 ,
remember that this can only happen if g 00 is the gate (g000 )=0 and if g000 is 0-valid in C,
so we have shown point (ii). Further, observe from the construction that the only such
wires (g 0 , g) in C 0 are:
00 =0
∗ For i ∈ {0, . . . , k}, the wire from (g00 )=i to (g0 )=i
.
i , whose other input is (g0 )
∗ The wire from (g00 )>k to (g0 )>k,1 , whose other input is (g000 )=0 .
So indeed, for each choice of g, there is at most one pure wire, so (i) holds too.
We have thus established claim (*), which concludes the proof.
J
With the above, we have finished the proof of Theorem F.2.
References for the Appendix
ABS15
ABS16
Antoine Amarilli, Pierre Bourhis, and Pierre Senellart. Provenance circuits for trees and
treelike instances. In ICALP, 2015.
Antoine Amarilli, Pierre Bourhis, and Pierre Senellart. Tractable lineages on treelike instances: Limits and extensions. In PODS, 2016.
Antoine Amarilli, Pierre Bourhis, Louis Jachiet, and Stefan Mengel
Bag06
Guillaume Bagan. MSO queries on tree decomposable structures are computable with
linear delay. In CSL, 2006.
Bag13
Guillaume Bagan. Algorithmes et complexité des problèmes d’énumération pour l’évaluation
de requêtes logiques. PhD thesis, Université de Caen, 2013.
CDG+ 07 H. Comon, M. Dauchet, R. Gilleron, C. Löding, F. Jacquemard, D. Lugiez, S. Tison,
and M. Tommasi. Tree automata: Techniques and applications, 2007. Available from
http://tata.gforge.inria.fr/.
Dar01
Adnan Darwiche. Decomposable negation normal form. J. ACM, 48(4), 2001.
DM02
Adnan Darwiche and Pierre Marquis. A knowledge compilation map. JAIR, 17, 2002.
Knu05
Donald E. Knuth. Art of Computer Programming. Volume 4a: Combinatorial Algorithms,
Part 1, 2005.
KS13
Wojciech Kazana and Luc Segoufin. Enumeration of monadic second-order queries on trees.
TOCL, 14(4), 2013.
OZ15
Dan Olteanu and Jakub Závodnỳ. Size bounds for factorised representations of query
results. TODS, 40(1), 2015.
Str73
Volker Strassen. Vermeidung von Divisionen. Journal für die reine und angewandte Mathematik, 264, 1973.
TW68
James W. Thatcher and Jesse B. Wright. Generalized finite automata theory with an
application to a decision problem of second-order logic. Math. Systems Theory, 2(1), 1968.
Weg00
Ingo Wegener. Branching programs and binary decision diagrams. SIAM, 2000.
XX:45
| 8 |
A Faster Cutting Plane Method and its
Implications for Combinatorial and Convex Optimization
arXiv:1508.04874v2 [cs.DS] 5 Nov 2015
Yin Tat Lee
MIT
yintat@mit.edu
Aaron Sidford
MIT
sidford@mit.edu
Sam Chiu-wai Wong
UC Berkeley
samcwong@berkeley.edu
Abstract
In this paper we improve upon the running time for finding a point in a convex set given
a separation oracle. In particular, given a separation oracle for a convex set K ⊂ Rn that is
contained in a box of radius R we show how to either compute a point in K or prove that K
does not contain a ball of radius using an expected O(n log(nR/)) evaluations of the oracle
and additional time O(n3 logO(1) (nR/)). This matches the oracle complexity and improves
upon the O(nω+1 log(nR/)) additional time of the previous fastest algorithm achieved over 25
years ago by Vaidya [103] for the current value of the matrix multiplication constant ω < 2.373
[110, 41] when R/ = O(poly(n)).
Using a mix of standard reductions and new techniques we show how our algorithm can be
used to improve the running time for solving classic problems in continuous and combinatorial
optimization. In particular we provide the following running time improvements:
• Submodular Function Minimization: let n be the size of the ground set, M be the
maximum absolute value of function values, and EO be the time for function evaluation.
Our weakly and strongly polynomial time algorithms have a running time of O(n2 log nM ·
EO + n3 logO(1) nM ) and O(n3 log2 n · EO + n4 logO(1) n), improving upon the previous
best of O((n4 · EO + n5 ) log M ) and O(n5 · EO + n6 ) respectively.
• Matroid Intersection: let n be the size of the ground set, r be the maximum size of
independent sets, M be the maximum absolute value of element weight, and Trank and
Tind be the time for each rank and independence oracle query.
We obtain a running time of O(nrTrank log n log(nM )+n3 logO(1) nM ) and O(n2 Tind log(nM )+
n3 logO(1) nM ), achieving the first quadratic bound on the query complexity for the independence and rank oracles. In the unweighted case, this is the first improvement since
1986 for independence oracle.
• Submodular Flow: let n and m be the number of vertices and edges, C be the maximum
edge cost in absolute value, and U be the maximum edge capacity in absolute value.
We obtain a faster weakly polynomial running time of O(n2 log nCU ·EO+n3 logO(1) nCU ),
improving upon the previous best of O(mn5 log nU · EO) and O n4 h min {log C, log U }
from 15 years ago by a factor of Õ(n4 ). We also achieve faster strongly polynomial time
algorithms as a consequence of our result on submodular minimization.
• Semidefinite Programming: let n be the number of constraints, m be the number of
dimensions and S be the total number of non-zeros in the constraint matrix.
We obtain a running time of Õ(n(n2 + mω + S)), improving upon the previous best of
Õ(n(nω + mω + S)) for the regime S is small.
1
Contents
Overview
4
1 Introduction
1.1 Paper Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
5
2 Overview of Our Results
2.1 Cutting Plane Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Convex Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3 Submodular Function Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
8
3 Preliminaries
3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Separation Oracles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
8
9
I
A Faster Cutting Plane Method
4 Introduction
4.1 Previous Work . . . . . . . . . .
4.2 Challenges in Improving Previous
4.3 Our Approach . . . . . . . . . . .
4.4 Organization . . . . . . . . . . .
10
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10
10
12
13
14
5 Preliminaries
14
5.1 Leverage Scores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
5.2 Hybrid Barrier Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
6 Our
6.1
6.2
6.3
6.4
6.5
Cutting Plane Method
Centering . . . . . . . . . . .
Changing Constraints . . . .
Hybrid Center Properties . .
The Algorithm . . . . . . . .
Guarantees of the Algorithm
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15
15
22
26
28
33
7 Technical Tools
38
7.1 Estimating Changes in Leverage Scores . . . . . . . . . . . . . . . . . . . . . . . . . 38
7.2 The Stochastic Chasing ~0 Game . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
II
A User’s Guide to Cutting Plane Methods
45
8 Introduction
45
8.1 Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
8.2 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
8.3 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
9 Preliminaries
50
2
10 Convex Optimization
53
10.1 From Feasibility to Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
10.2 Duality and Semidefinite Programming . . . . . . . . . . . . . . . . . . . . . . . . . . 55
11 Intersection of Convex Sets
11.1 The Technique . . . . . .
11.2 Matroid Intersection . . .
11.3 Submodular Flow . . . . .
11.4 Affine Subspace of Convex
III
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Set
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Submodular Function Minimization
60
61
66
67
69
71
12 Introduction
71
12.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
12.2 Our Results and Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
12.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
13 Preliminaries
13.1 Submodular Function Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Lovász Extension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13.3 Polyhedral Aspects of SFM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
74
74
76
14 Improved Weakly Polynomial Algorithms for SFM
77
15 Improved Strongly Polynomial Algorithms for SFM
15.1 Improved Oracle Complexity . . . . . . . . . . . . . . . . .
15.2 Technical Tools . . . . . . . . . . . . . . . . . . . . . . . . .
15.2.1 SFM over Ring Family . . . . . . . . . . . . . . . . .
15.2.2 Identifying New Valid Arcs . . . . . . . . . . . . . .
e 4 · EO + n5 ) Time Algorithm . . . . . . . . . . . . . . .
15.3 O(n
15.3.1 Consolidating A and f . . . . . . . . . . . . . . . . .
15.3.2 Deducing New Constraints xi = 0, xj = 1, xi = xj or
15.3.3 Running Time . . . . . . . . . . . . . . . . . . . . .
e 3 · EO + n4 ) Time Algorithm . . . . . . . . . . . . . . .
15.4 O(n
15.4.1 Partitioning Ground Set into Buckets . . . . . . . .
15.4.2 Separating Hyperplane: Project and Lift . . . . . . .
15.4.3 Deducing New Constraints xi = 0, xj = 1, xi = xj or
15.4.4 Running Time . . . . . . . . . . . . . . . . . . . . .
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80
80
81
81
83
87
88
89
95
95
96
97
99
102
16 Discussion and Comparison with Previous Algorithms
103
16.1 Open Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
3
Part
Overview
1
Introduction
The ellipsoid method and more generally, cutting plane methods,1 that is optimization algorithms
which iteratively call a separation oracle, have long been central to theoretical computer science. In
combinatorial optimization, since Khachiyan’s seminal result in 1980 [65] proving that the ellipsoid
method solves linear programs in polynomial time, the ellipsoid method has been crucial to solving
discrete problems in polynomial time [49]. In continuous optimization, cutting plane methods have
long played a critical role in convex optimization, where they are fundamental to the theory of
non-smooth optimization [45].
Despite the key role that cutting plane methods have played historically in both combinatorial
and convex optimization, over the past two decades progress on improving both the theoretical
running time of cutting plane methods as well as the complexity of using cutting plane methods
for combinatorial optimization has stagnated.2 The theoretical running time of cutting plane
methods for convex optimization has not been improved since the breakthrough result by Vaidya
in 1989 [103, 105]. Moreover, for many of the key combinatorial applications of ellipsoid method,
such as submodular minimization, matroid intersection and submodular flow, the running time
improvements over the past two decades have been primarily combinatorial; that is they have been
achieved by discrete algorithms that do not use numerical machinery such as cutting plane methods.
In this paper we make progress on these classic optimization problems on two fronts. First we
show how to improve on the running time of cutting plane methods for a broad range of parameters
that arise frequently in both combinatorial applications and convex programming (Part I). Second,
we provide several frameworks for applying the cutting plane method and illustrate the efficacy of
these frameworks by obtaining faster running times for semidefinite programming, matroid intersection, and submodular flow (Part II). Finally, we show how to couple our approach with the problem
specific structure and obtain faster weakly and strongly polynomial running times for submodular
function minimization, a problem of tremendous importance in combinatorial optimization (Part
III). In both cases our algorithms are faster than previous best by a factor of roughly Ω(n2 ).
We remark that many of our running time improvements come both from our faster cutting
method and from new careful analysis of how to apply these cutting plane methods. In fact, simply
using our reductions to cutting plane methods and a seminal result of Vaidya [103, 105] on cutting
plane methods we provide running times for solving many of these problems that improves upon
the previous best stated. As such, we organized our presentation to hopefully make it easy to apply
cutting plane methods to optimization problems and obtain provable guarantees in the future.
Our results demonstrate the power of cutting plane methods in theory and possibly pave the
way for new cutting plane methods in practice. We show how cutting plane methods can continue to
improve running times for classic optimization problems and we hope that these methods may find
further use. As cutting plane methods such as analytic cutting plane method [43, 10, 44, 87, 111, 45]
are frequently used in practice [48, 42], these techniques may have further implications.
1
Throughout this paper our focus is on algorithms for polynomial time solvable convex optimization problems
given access to a linear separation oracle. Our usage of the term cutting plane methods, should not be confused with
work on integer programming, an NP-hard problem.
2
There are exceptions to this trend. For example, [70] showed how to apply cutting plane methods to yield running
time improvements for semidefinite programming, and recently [15] showed how to use cutting plane methods to obtain
an optimal result for smooth optimization problems.
4
1.1
Paper Organization
After providing an overview of our results (Section 2) and preliminary information and notation
used throughout the paper (Section 3), we split the remainder of the paper into three parts:
• In Part I we provide our new cutting plane method.
• In Part II we provide several general frameworks for using this cutting plane method and
illustrate these frameworks with applications in combinatorial and convex optimization.
• In Part III we then consider the more specific problem of submodular function minimization
and show how our methods can be used to improve the running time for both strongly and
weakly polynomial time algorithms.
We aim to make each part relatively self contained. While each part builds upon the previous and
the problems considered in each part are increasingly specific, we present the key results in each
section in a modular way so that they may be read in any order. The dependencies between the
different parts of our paper are characterized by the following:
• Part I presents our faster cutting plane method as Theorem 31.
• Part II depends only on Theorem 31 of Part I and presents a general running time guarantee
for convex optimization problems as Theorem 42.
• The faster weakly polynomial time algorithm in Part III depends only on Theorem 42, Part II.
• The faster strongly polynomial time algorithm in Part III depends only on Theorem 31, Part I.
2
Overview of Our Results
Here we briefly summarize the contributions of our paper. For each of Part I, Part II, and Part III
we describe the key technical contributions and present the running time improvements achieved.
2.1
Cutting Plane Methods
The central problem we consider in Part I is as follows. We are promised that a set K is contained
a box of radius R and a separation oracle that given a point ~x in time SO either outputs that ~x
is in K or outputs a separating hyperplane. We wish to either find a point in K or prove that K
does not contain an ball of radius . The running times for this problem are given in Table 1.
Year
1979
1988
1989
1995
2004
2013
Algorithm
Ellipsoid Method [97, 112, 65]
Inscribed Ellipsoid [66, 88]
Volumetric Center [103]
Analytic Center [10]
Random Walk [13]
This paper
Complexity
+ n4 log κ)
O(nSO log κ + (n log κ)4.5 )
O(nSO log κ + n1+ω log κ)
O(nSO log2 κ + nω+1 log2 κ + (n log κ)2+ω/2 )
→ O(nSO log κ + n7 log κ)
O(nSO log κ + n3 logO(1) κ)
O(n2 SO log κ
Table 1: Algorithms for the Feasibility Problem. κ indicates nR/. The arrow, →, indicates that
it solves a more general problem where only a membership oracle is given.
5
In Part I we show how to solve this problem in O(nSO log(nR/) + n3 logO(1) (nR/)) time.
e
This is an improvement over the previous best running time of O(nSO
log(nR/) + nω+1 log(nR/))
for the current best known bound of ω < 2.37 [41] assuming that R/ = O(poly(n)), a common
assumption for many problems in combinatorial optimization and numerical analysis as we find in
Part II and Part III. (See Table 1 for a summary of previous running times.)
Our key idea for achieving this running time improvement is a new straightforward technique
for providing low variance unbiased estimates for changes in leverage scores that we hope will be of
independent interest (See Section 7.1). We show how to use this technique along with ideas from
e ω+1 log(D/)) overhead in the previous fastest algorithm [103].
[10, 104, 76] to decrease the O(n
2.2
Convex Optimization
In Part II we provide two techniques for applying our cutting plane method (and cutting plane
methods in general) to optimization problems and provide several applications of these techniques.
The first technique concerns reducing the number of dimensions through duality. For many
problems, their dual is significantly simpler than itself (primal). We use semidefinite programming
as a concrete example to show how to improve upon the running time for finding both primal and
dual solution by using the cutting planes maintained by our cutting plane method. (See Table 2.)
The second technique concerns how to minimize a linear function over the intersection of convex
sets using optimization oracle. We analyze a simple potential function which allows us to bypass
the typical reduction between separation and optimization to achieve faster running times. This
reduction provides an improvement over the reductions used previously in [49]. Moroever, we
show how this technique allows us to achieve improved running times for matroid intersection and
minimum cost submodular flow. (See Tables 2, 3, 4, and 5 for running time summaries.)
Authors
Years
Nesterov, Nemirovsky[89]
Anstreicher [7]
Krishnan, Mitchell [70]
This paper
1992
2000
2003
2015
Running times
√
Õ( m(nmω + nω−1 m2 ))
Õ((mn)1/4 (nmω + nω−1 m2 ))
Õ(n(nω + mω + S)) (dual SDP)
Õ(n(n2 + mω + S))
Table 2: Algorithms for solving a m × m SDP with n constraints and S non-zero entries
Authors
Edmonds [26]
Aigner, Dowling [2]
Tomizawa, Iri [102]
Lawler [72]
Edmonds [28]
Cunningham [21]
Years
1968
1971
1974
1975
1979
1986
This paper
2015
Complexity
not stated
O(nr2 Tind )
not stated
O(nr2 Tind )
not stated
O(nr1.5 Tind )
2
O(n log nTind + n3 logO(1) n)
O(nr log2 nTrank + n3 logO(1) n)
Table 3: Algorithms for (unweighted) matroid intersection. n is the size of the ground set, r is the
maximum rank of the two matroids, Tind is the time to check if a set is independent (membership
oracle), and Trank is the time to compute the rank of a given set (rank oracle).
6
Authors
Edmonds [26]
Tomizawa, Iri [102]
Lawler [72]
Edmonds [28]
Frank [33]
Orlin, Ahuja [91]
Brezovec, Cornuéjols, Glover[14]
Fujishige, Zhang [39]
Shigeno, Iwata [96]
Years
1968
1974
1975
1979
1981
1983
1986
1995
1995
This paper
2015
Running times
not stated
not stated
O(nr2 Tind + nr3 )
not stated
O(n2 r(Tcircuit + n))
not stated
O(nr(Tcircuit + r + log n))
O(n2 r0.5 log rM · Tind )
O((n + Tcircuit )nr0.5 log rM )
O(n2 log nM Tind + n3 logO(1) nM )
O(nr log n log nM Trank + n3 logO(1) nM )
Table 4: Algorithms for weighted matroid intersection. In addition to the notation in Table 3
Tcircuit is the time needed to find a fundamental circuit and M is the bit complexity of the weights.
Authors
Fujishige [35]
Grotschel, Lovasz, Schrijver[49]
Zimmermann [113]
Barahona, Cunningham [12]
Cunningham, Frank [22]
Fujishige [36]
Frank, Tardos [34]
Cui, Fujishige [108]
Fujishige, Röck, Zimmermann[38]
Chung, Tcha [18]
Zimmermann [114]
McCormick, Ervolina [82]
Wallacher, Zimmermann [109]
Iwata [52]
Iwata, McCormick, Shigeno [57]
Iwata, McCormick, Shigeno [58]
Fleischer, Iwata, McCormick[32]
Iwata, McCormick, Shigeno [59]
Fleischer, Iwata [30]
This paper
Years
1978
1981
1982
1984
1985
1987
1987
1988
1989
1991
1992
1993
1994
1997
1998
1999
1999
1999
2000
2015
Running times
not stated
weakly polynomial
not stated
not stated
→ O(n4 h log C)
not stated
strongly polynomial
not stated
→ O(n6 h log n)
not stated
not stated
O(n7 h∗ log nCU )
O(n8 h log nCU )
7
O(n
h log U ) 2
4
O n h min log nC, n log n
O n6 h min log nU, n2 log n
O n4 h min log U, n2 log n
O n4 h min log C, n2 log n
O(mn5 log nU · EO)
O(n2 log nCU · EO + n3 logO(1) nCU )
Table 5: Algorithms for minimum cost submodular flow with n vertices, maximum cost C and
maximum capacity U . The factor h is the time for an exchange capacity oracle, h∗ is the time
for a “more complicated exchange capacity oracle,” and EO is the time for evaluation oracle of
the submodular function. The arrow, →, indicates that it uses the current best submodular flow
algorithm as subroutine which was non-existent at the time of the publication.
7
2.3
Submodular Function Minimization
In Part III we consider the problem of submodular minimization, a fundamental problem in combinatorial optimization with many diverse applications in theoretical computer science, operations
research, machine learning and economics. We show that by considering the interplay between the
guarantees of our cutting plane algorithm and the primal-dual structure of submodular minimization we can achieve improved running times in various settings.
First, we show that a direct application of our method yields an improved weakly polynomial
time algorithm for submodular minimization. Then, we present a simple geometric argument
that submodular function can be solved with O(n3 log n · EO) oracle calls but with exponential
running time. Finally, we show that by further studying the combinatorial structure of submodular
minimization and a modification to our cutting plane algorithm we can obtained a fully improved
strongly polynomial time algorithm for submodular minimization. We summarize the improvements
in Table 6.
Authors
Grötschel, Lovász,
Schrijver [49, 50]
Cunningham [20]
Schrijver [93]
Iwata, Fleischer,
Fujishige[56]
Iwata, Fleischer [31]
Years
Running times
1981,1988
e 5 · EO + n7 ) [81]
O(n
1985
2000
O(M n6 log nM · EO)
O(n8 · EO + n9 )
O(n5 · EO log M )
O(n7 log n · EO)
O(n7 · EO + n8 )
O((n4 · EO + n5 ) log M )
O((n6 · EO + n7 ) log n)
O(n7 · EO + n8 )
O(n5 · EO + n6 )
O((n4 · EO + n5 ) log nM )
O((n5 · EO + n6 ) log n)
2
O(n log nM · EO + n3 logO(1) nM )
O(n3 log2 n · EO + n4 logO(1) n)
2000
2000
Iwata [54]
2003
Vygen [107]
Orlin [90]
2003
2007
Iwata, Orlin [60]
2009
Our algorithms
2015
Remarks
first weakly
and strongly
first combin. pseudopoly
first combin. strongly
first combin. strongly
current best weakly
current best strongly
Table 6: Algorithms for submodular function minimization.
3
Preliminaries
Here we introduce notation and concepts we use throughout the paper.
3.1
Notation
Basics: Throughout this paper, we use vector notation, e.g ~x = (x1 , . . . , xn ), to denote a vector
and bold, e.g. A, to denote a matrix. We use nnz(~x) or nnz(A) to denote the number of nonzero
entries in a vector or a matrix respectively. Frequently, for ~x ∈ Rd we let X ∈ Rd×d denote diag(~x),
the diagonal matrix such that Xii = xi . For a symmetric matrix, M, we let diag(M) √
denote the
def
vector corresponding to the diagonal entries of M, and for a vector, ~x, we let k~xkM = ~xT M~x.
8
Running Times: We typically use XO to denote the running time for invoking the oracle, where X
depends on the type of oracle, e.g., SO typically denotes the running time of a separation oracle, EO
e ) def
denotes the running time of an evaluation oracle, etc. Furthermore, we use O(f
= O(f logO(1) f ).
Spectral Approximations: For symmetric matrices N, M ∈ Rn×n , we write N M to denote that ~xT N~x ≤ ~xT M~x for all ~x ∈ Rn and we define N M, N ≺ M and N M analogously.
def
Standard Convex Sets: We let Bp (r) = {~x : ~x p ≤ r} denote a ball of radius r in the `p
norm. For brevity we refer to B2 (r) as a a ball of radius r and B∞ (r) as a box of radius r.
Misc: We let ω < 2.373 [110] denote the matrix multiplication constant.
3.2
Separation Oracles
Throughout this paper we frequently make assumptions about the existence of separation oracles
for sets and functions. Here we formally define these objects as we use them throughout the paper.
Our definitions are possibly non-standard and chosen to handle the different settings that occur in
this paper.
Definition 1 (Separation Oracle for a Set). Given a set K ⊂ Rn and δ ≥ 0, a δ-separation oracle
for K is a function on Rn such that for any input ~x ∈ Rn , it either outputs “successful” or a half
space of the form H = {~z : ~cT ~z ≤ ~cT ~x + b} ⊇ K with b ≤ δ ~c 2 and ~c 6= ~0. We let SOδ (K) be the
time complexity of this oracle.
For brevity we refer to a 0-separation oracle for a set as just a separation oracle. We refer to
the hyperplanes defining the halfspaces returned by a δ-separation oracle as oracle hyperplanes.
Note that in Definition 1 we do not assume that K is convex. However, we remark that it is
well known that there is a separation oracle for a set if and only if it is convex and that there is a
δ separation oracle if and only if the set is close to convex in some sense.
Definition 2 (Separation Oracle for a Function). For any convex function f , η ≥ 0 and δ ≥ 0, a
(η, δ)-separation oracle on a convex set Γ for f is a function on Rn such that for any input ~x ∈ Γ,
it either asserts f (~x) ≤ min~y∈Γ f (~y ) + η or outputs a half space H such that
def
{~z ∈ Γ : f (~z) ≤ f (~x)} ⊂ H = {~z : ~cT ~z ≤ ~cT ~x + b}
with b ≤ δ ~c and ~c 6= ~0. We let SOη,δ (f ) be the time complexity of this oracle.
9
(3.1)
Part I
A Faster Cutting Plane Method
4
Introduction
Throughout Part I we study the following feasibility problem:
Definition 3 (Feasibility Problem). Given a separation oracle for a set K ⊆ Rn contained in a box
of radius R either find a point ~x ∈ K or prove that K does not contain a ball of radius .
This feasibility problem is one of the most fundamental and classic problems in optimization.
Since the celebrated result of Yudin and Nemirovski [112] in 1976 and Khachiyan [65] in 1979
essentially proving that it can be solved in time O(poly(n) · SO · log(R/)), this problem has served
as one of the key primitives for solving numerous problems in both combinatorial and convex
optimization.
Despite the prevalence of this feasibility problem, the best known running time for solving this
problem has not been improved in over 25 years. In a seminal paper of Vaidya in 1989 [103], he
ω+1 log(nR/)) time. Despite interesting
e
showed how to solve the problem in O(n·SO·log(nR/)+n
generalizations and practical improvements [5, 92, 43, 10, 44, 87, 111, 45, 15], the best theoretical
guarantees for solving this problem have not been improved since.
In Part I we show how to improve upon Vaidya’s running time in certain regimes. We provide
a cutting plane algorithm which achieves an expected running time of O(n · SO · log(nR/) +
n3 logO(1) (nR/)), improving upon the previous best known running time for the current known
value of ω < 2.373 [110, 41] when R/ = O(poly(n)).
We achieve our results by the combination of multiple techniques. First we show how to use
techniques from the work of Vaidya and Atkinson to modify Vaidya’s scheme so that it is able to
tolerate random noise in the computation in each iteration. We then show how to use known numerical machinery [104, 99, 76] in combination with some new techniques (Section 7.1 and Section 7.2)
to implement each of these relaxed iterations efficiently. We hope that both these numerical techniques as well as our scheme for approximating complicated methods, such as Vaidya’s, may find
further applications.
While our paper focuses on theoretical aspects of cutting plane methods, we achieve our results
via the careful application of practical techniques such as dimension reduction and sampling. As
such we hope that ideas in this paper may lead to improved practical3 algorithms for non-smooth
optimization.
4.1
Previous Work
Throughout this paper, we restrict our attention to algorithms for the feasibility problem that have
a polynomial dependence on SO, n, and log(R/). Such “efficient” algorithms typically follow the
following iterative framework. First, they compute some trivial region Ω that contains K. Then,
they call the separation oracle at some point ~x ∈ Ω. If ~x ∈ K the algorithm terminates having
successfully solved the problem. If ~x ∈
/ K then the separation oracle must return a half-space
3
Although cutting plane methods are often criticized for their empirical performance, recently, Bubeck, Lee and
Singh [15] provided a variant of the ellipsoid method that achieves the same convergence rate as Nesterov’s accelerated
gradient descent. Moreover, they provided numerical evidence that this method can be superior to Nesterov’s accelerated gradient descent, thereby suggesting that cutting plane methods can be as aggressive as first order methods if
designed properly.
10
Year
1979
1988
1989
Algorithm
Ellipsoid Method [97, 112, 65]
Inscribed Ellipsoid [66, 88]
Volumetric Center [103]
1995
Analytic Center [10]
2004
2013
Random Walk [13]
This paper
Complexity
+ n4 log κ)
O(nSO log κ + (n log κ)4.5 )
O(nSO log κ + n1+ω log κ)
nSO log2 κ + nω+1 log2 κ
O
+ (n log κ)2+ω/2
→ O(nSO log κ + n7 log κ)
O(nSO log κ + n3 logO(1) κ)
O(n2 SO log κ
Table 7: Algorithms for the Feasibility Problem. κ indicates nR/. The arrow, →, indicates that
it solves a more general problem where only a membership oracle is given.
containing K. The algorithm then uses this half-space to shrink the region Ω while maintaining
the invariant that K ⊆ Ω. The algorithm then repeats this process until it finds a point ~x ∈ K or
the region Ω becomes too small to contain a ball with radius .
Previous works on efficient algorithms for the feasibility problem all follow this iterative framework. They vary in terms of what set Ω they maintain, how they compute the center to query the
separation oracle, and how they update the set. In Table 7, we list the previous running times for
solving the feasibility problem. As usual SO indicates the cost of the separation oracle. To simplify
def
the running times we let κ = nR/. The running times of some algorithms in the table depend on
R/ instead of nR/. However, for many situations, we have log(R/) = Θ(log(nR/)) and hence
we believe this is still a fair comparison.
The first efficient algorithm for the feasibility problem is the ellipsoid method, due to Shor [97],
Nemirovksii and Yudin [112], and Khachiyan [65]. The ellipsoid method maintains an ellipsoid as
Ω and uses the center of the ellipsoid as the next query point. It takes Θ(n2 log κ) calls of oracle
which is far from the lower bound Ω(n log κ) calls [86].
To alleviate the problem, the algorithm could maintain all the information from the oracle, i.e.,
the polytope created from the intersection of all half-spaces obtained. The center of gravity method
[77] achieves the optimal oracle complexity using this polytope and the center of gravity of this
polytope as the next point. However, computing center of gravity is computationally expensive and
hence we do not list its running time in Table 7. The Inscribed Ellipsoid Method [66] also achieved
an optimal oracle complexity using this polytope as Ω but instead using the center of the maximal
inscribed ellipsoid in the polytope to query the separation oracle. We listed it as occurring in year
1988 in Table 7 because it was [88] that yielded the first polynomial time algorithm to actually
compute this maximal inscribed ellipsoid for polytope.
Vaidya [103] obtained a faster algorithm by maintaining an approximation of this polytope and
using a different center, namely the volumetric center. Although the oracle complexity of this
volumetric center method is very good, the algorithm is not extremely efficient as each iteration
involves matrix inversion. Atkinson and Vaidya [10] showed how to avoid this computation in
certain settings. However, they were unable to achieve the desired convergence rate from their
method.
Bertsimas and Vempala [13] also gives an algorithm that avoids these expensive linear algebra
operations while maintaining the optimal convergence rate by using techniques in sampling convex
sets. Even better, this result works for a much weaker oracle, the membership oracle. However,
the additional cost of this algorithm is relatively high in theory. We remark that while there are
considerable improvemenst on the sampling techniques [79, 63, 76], the additional cost is still quite
high compared to standard linear algebra.
11
4.2
Challenges in Improving Previous Work
Our algorithm builds upon the previous fastest algorithm of Vaidya [105]. Ignoring implementation
details and analysis, Vaidya’s algorithm is quite simple. This algorithm simply maintains a polytope
P (k) = {x ∈ Rn : A~x − ~b ≥ ~0} as the current Ω and uses the volumetric center, the minimizer of
the following volumetric barrier function
arg min
~
x
1
log det AT S~−2
A
x
2
where
def
S~x = diag(A~x − ~b)
(4.1)
as the point at which to query the separation oracle. The polytope is then updated by adding shifts
of the half-spaces returned by the separation oracle and dropping unimportant constraints. By
choosing the appropriate shift, picking the right rule for dropping constraints, and using Newton’s
method to compute the volumetric center he achieved a running time of O(n·SO·log κ+n1+ω log κ).
While Vaidya’s algorithm’s dependence on SO is essentially optimal, the additional per-iteration
costs of his algorithm could possibly be improved. The computational bottleneck in each iteration
of Vaidya’s algorithm is computing the gradient
of log det which in turn involves computing the
def
T S−2 A −1 AT S−1 ), a commonly occurring quantity in nuleverage scores ~σ (~x) = diag(S−1
A
A
x
x
x
merical analysis and convex optimization [99, 19, 78, 76, 75]. As the best known algorithms for
computing leverage scores exactly in this setting take time O(nω ), directly improving the running
time of Vaidya’s algorithm seems challenging.
However, since an intriguing result of Spielman and Srivastava in 2008 [99], it has been well
known that using Johnson-Lindenstrauss transform these leverage scores can be computed up to
a multiplicative (1 ± ) error by solving O(−2 log n) linear systems involving AT S−2
x A. While
−2
ω
in general this still takes time O( n ), there are known techniques for efficiently maintaining
the inverse of a matrix so that solving linear systems take amortized O(n2 ) time [104, 75, 76].
Consequently if it could be shown that computing approximate leverage scores sufficed, this would
potentially decrease the amortized cost per iteration of Vaidya’s method.
Unfortunately, Vaidya’s method does not seem to tolerate this type of multiplicative error. If
leverage scores were computed this crudely then in using them to compute approximate gradients
for (4.1), it seems that any point computed would be far from the true center. Moreover, without
being fairly close to the true volumetric center, it is difficult to argue that such a cutting plane
method would make sufficient progress.
To overcome this issue, it is tempting to directly use recent work on improving the running time
of linear program [75]. In this work, the authors faced a similar issue where a volumetric, i.e. log det,
potential function had the right analytic and geometric properties, however was computational
expensive to minimize. To overcome this issue the authors instead computed a weighted analytic
center:
X
def
arg min −
wi log si (~x) where ~s(~x) = A~x − ~b .
~
x
i∈[m]
For carefully chosen weights this center provides the same convergence guarantees as the volumetric
potential function, while each step can be computed by solving few linear systems (rather than
forming the matrix inverse).
Unfortunately, it is unclear how to directly extend the work in [75] on solving an explicit
linear program to the feasibility problem specified by a separation oracle. While it is possible
to approximate the volumetric barrier by a weighted analytic center in many respects, proving
that this approximation suffices for fast convergence remains open. In fact, the volumetric barrier
12
function as used in Vaidya’s algorithm is well approximated simply by the standard analytic center
X
def
log si (~x) where ~s(~x) = A~x − ~b .
arg min −
~
x
i∈[m]
as all the unimportant constraints are dropped during the algorithm. However, despite decades of
research, the best running times known for solving the feasibility problem using the analytic center
are Vaidya and Atkinson algorithm from 1995 [10]. While the running time of this algorithm could
possibly be improved using approximate leverage score computations and amortized efficient linear
system solvers, unfortunately at best, without further insight this would yield an algorithm which
requires a suboptimal O(n logO(1) κ) queries to the separation oracle.
As pointed out in [10], the primary difficulty in using any sort of analytic center is quantifying
the amount of progress made in each step. We still believe providing direct near-optimal analysis of
weighted analytic center is a tantalizing open question warranting further investigation. However,
rather than directly address the question of the performance of weighted analytic centers for the
feasibility problem, we take a slightly different approach that side-steps this issue. We provide a
partial answer that still sheds some light on the performance of the weighted analytic center while
still providing our desired running time improvements.
4.3
Our Approach
To overcome the shortcoming of the volumetric and analytic centers we instead consider a hybrid
barrier function
X
def
arg min −
wi log si (~x) + log det(AT S−1
where ~s(~x) = A~x − ~b .
x A)
~
x
i∈[m]
for careful chosen weights. Our key observation is that for correct choice of weights, we can compute
the gradient of this potential function. In particular if P
we let w
~ = ~τ − ~σ (~x) then the gradient of
this potential function is the same as the gradients of i∈[m] τi log si (~x), which we can compute
efficiently. Moreover, since we are using log det, we can use analysis similar to Vaidya’s algorithm
[103] to analyze the convergence rate of this algorithm.
Unfortunately, this is a simple observation and does not immediately change the problem substantially. It simply pushes the problem of computing gradients of log det to computing w.
~ Therefore, for this scheme to work, we would need to ensure that the weights do not change too much
and that when they change, they do not significantly hurt the progress of our algorithm. In other
words, for this scheme to work, we would still need very precise estimates of leverage scores.
However, we note that the leverage scores ~σ (~x) do not change too much between iterations.
Moreover, we provide what we believe is an interesting technical result that an unbiased estimates
to the changes in leverage scores can be computed using linear system solvers such that the total
error of the estimate is bounded by the total change of the leverage scores (See Section 7.1). Using
this result our scheme simply follows Vaidya’s basic scheme in [103], however instead of minimizing
the hybrid barrier function directly we alternate between taking Newton steps we can compute,
changing the weights so that we can still compute Newton steps, and computing accurate unbiased
estimates of the changes in the leverage scores so that the weights do not change adversarially by
too much.
To make this scheme work, there are two additional details that need to be dealt with. First,
we cannot let the weights vary too much as this might ultimately hurt the rate of progress of our
algorithm. Therefore, in every iteration we compute a single leverage score to high precision to
13
control the value of wi and we show that by careful choice of the index we can ensure that no weight
gets too large (See Section 7.2).
Second, we need to show that changing weights does not affect our progress by much more than
the progress we make with respect to log det. To do this, we need to show the slacks are bounded
above and below. We enforce this by adding regularization terms and instead consider the potential
function
X
λ
1
2
wi log si (~x) + log det AT S−2
p~e (~x) = −
x 2
x A + λI +
2
2
i∈[m]
This allows us to ensure that the entries of ~s(~x) do not get too large or too small and therefore
changing the weighting of the analytic center cannot affect the function value too much.
Third, we need to make sure our potential function is convex. If we simply take w
~ = ~τ − ~σ (~x)
with ~τ as an estimator of ~σ (~x), w
~ can be negative and the potential function could be non-convex.
To circumvent this issue, we use w
~ = ce + ~τ − ~σ (~x) and make sure ~τ − ~σ (~x) ∞ < ce .
Combining these insights, using efficient algorithms for solving a sequence of slowly changing
linear systems [104, 75, 76], and providing careful analysis ultimately allows us to achieve a running
time of O(nSO log κ+n3 logO(1) κ) for the feasibility problem. Furthermore, in the case that K does
not contain a ball of radius , our algorithm provides a proof that the polytope does not contain a
ball of radius . This proof ultimately allows us to achieve running time improvements for strongly
polynomial submodular minimization in Part III.
4.4
Organization
The rest of Part I is organized as follows. In Section 5 we provide some preliminary information
and notation we use throughout Part I. In Section 6 we then provide and analyze our cutting plane
method. In Section 7 we provide key technical tools which may be of independent interest.
5
Preliminaries
Here we introduce some notation and concepts we use throughout Part I.
5.1
Leverage Scores
Our algorithms in this section make extensive use of leverage scores, a common measure of the
importance of rows of a matrix. We denote the leverage scores
Rn×d by ~σ ∈ Rn
−1 Tof a matrix A ∈n×d
def
T
and say the leverage score of row i ∈ [n] is σi = [A A A
A ]ii . For A ∈ R
, d~ ∈ Rn>0 , and
def
~ we use the shorthand ~σA (d)
~ to denote the leverage scores of the matrix D1/2 A. We
D = diag(d)
frequently use well known facts regarding leverage scores, such as σi ∈ [0, 1] and ~σ 1 ≤ d. (See
[99, 80, 78, 19] for a more in-depth discussion of leverage scores, their properties, and their many
applications.) In addition, we make use of the fact that given an efficient linear system solver of
AT A we can efficiently compute multiplicative approximations to leverage scores (See Definition 4
and Lemma 5 below).
Definition 4 (Linear System Solver). An algorithm S is a LO-time solver of a PD matrix M ∈ Rn×n
if for all ~b ∈ Rn and ∈ (0, 1/2], the algorithm outputs a vector S(~b, ) ∈ Rn in time O(LO·log(−1 ))
2
2
such that with high probability in n, S(~b, ) − M−1~b M ≤ M−1~b M .
Lemma 5 (Computing Leverage Scores [99]). Let A ∈ Rn×d , let ~σ denote the leverage scores of A,
e
and let > 0. If we have a LO-time solver for AT A then in time O((nnz(A)
+ LO)−2 log(−1 ))
we can compute ~τ ∈ Rn such that with high probability in d, (1 − )σi ≤ τi ≤ (1 + )σi for all i ∈ [n].
14
5.2
Hybrid Barrier Function
As explained in Section 4.3 our cutting plane method maintains a polytope P = {~x ∈ Rn : A~x ≥ ~b}
for A ∈ Rm×n and ~b ∈ Rn that contains some target set K. We then maintain a minimizer of the
following hybrid barrier function:
X
λ
1
def
2
(ce + ei ) log si (~x) + log det AT S−2
p~e (~x) = −
x 2
x A + λI +
2
2
i∈[m]
def
where ~e ∈ Rm is a variable we maintain, ce ≥ 0 and λ ≥ 0 are constants we fix later, ~s(~x) = A~x −~b,
def
and Sx = diag(~s(~x)). When the meaning is clear from context we often use the shorthand Ax =
S−1
x A.
Rather than maintaining ~e explicitly, we instead maintain a vector ~τ ∈ Rm that approximates
the leverage score
−1 T
~ x) def
ψ(~
= diag Ax ATx Ax + λI
Ax
.
~ x) is simply the leverage scores of certain rows of the matrix
Note that ψ(~
Ax
√
.
λI
and therefore the usual properties of leverage scores hold, i.e. ψi (~x) ∈ (0, 1) and ψi (~x) 1 ≤ n.
~ x) equivalently as ψ
~A or ψ
~P when we want the matrix to be clear. Furthermore,
We write ψ(~
x
def
def
~ x)) and µ(~x) = mini ψi (~x). Finally, we typically pick ~e using the function
we let Ψx = diag(ψ(~
def
~
~eP (~τ , ~x) = ~τ − ψ(~x). Again, we use the subscripts of Ax and P interchangeably and often drop
them when the meaning is clear from context.
2
We remark that the last term λ2 x 2 ensures that our point is always within a certain region
(Lemma 23) and hence the term (ce + ei ) log si (~x)i never gets too large. However, this `2 term
changes the Hessian of the potential function and hence we need to put a λI term inside both
~ instead of the
the log det and the leverage score to reflect this. This is the reason why we use ψ
standard leverage score.
6
Our Cutting Plane Method
In this section we develop and prove the correctness of our cutting plane method. We use the
notation introduced in Section 3 and Section 5 as well as the technical tools we introduce in
Section 7.
We break the presentation and proof of correctness of our cutting plane methods into multiple
parts. First in Section 6.1 we describe how we maintain a center of the hybrid barrier function
p~e and analyze this procedure. Then, in Section 6.2 we carefully analyze the effect of changing
constraints on the hybrid barrier function and in Section 6.3 we prove properties of an approximate
center of hybrid barrier function, which we call a hybrid center. In Section 6.4 we then provide
our cutting plane method and in Section 6.5 we prove that the cutting plane method solves the
feasibility problem as desired.
6.1
Centering
In this section we show how to compute approximate centers or minimizers of the hybrid barrier
function for the current polytope P = {~x : A~x ≥ ~b}. We split this proof up into multiple parts.
First we simply bound the gradient and Hessian of the hybrid barrier function, p~e , as follows.
15
def
Lemma 6. For f (~x) =
1
2
log det AT S−2
x A + λI , we have that
~ x)
∇f (~x) = −ATx ψ(~
and
ATx Ψ(~x)Ax ∇2 f (~x) 3ATx Ψ(~x)Ax
.
Proof. Our proof is similar to [4, Appendix] which proved the statement when λ = 0. This case
does not change the derivation significantly, however for completeness we include the proof below.
We take derivatives on ~s first and then apply chain rule. Let f (~s) = 12 log det AT S−2 A + λI .
We use the notation Df (~x)[~h] to denote the directional derivative of f along the direction ~h
d
at the point ~x. Using the standard formula for the derivative of log det, i.e. dt
log det Bt =
dB
Tr((Bt )−1 ( dtt )), we have
1
Tr((AT S−2 A + λI)−1 (AT (−2)S−3 HA))
2
X ψi hi
X hi
~1Ti S−1 A AT S−2 A + λI −1 AS−1~1i = −
= −
si
si
Df (~s)[~h] =
(6.1)
.
i
i
~ Now let P = S−1 A AT S−2 A + λI −1 AT S−1 .
Applying chain rules, we have ∇f (~x) = −ATx ψ.
Taking the derivative of (6.1) again and using the cyclic property of trace, we have
−1
−1
D2 f (~s)[~h1 , ~h2 ] = Tr AT S−2 A + λI
AT (−2)S−3 H2 A AT S−2 A + λI
AT S−3 H1 A
−1
−Tr AT S−2 A + λI
AT (−3)S−4 H2 H1 A
= 3Tr PS−2 H2 H1 − 2Tr PS−1 H2 PS−1 H1
X
X ~h1 (i)~h2 (i)
~h2 (j) ~h2 (i)
= 3
Pii
−2
Pji
Pij
2
sj
si
si
def
ij
i
= 3
X
i
X
~h1 (i)~h2 (i)
~
~
2 h2 (j) h2 (i)
ψi
−
2
P
ij
sj
si
s2i
ij
.
Consequently, D2 f (~x)[~1i , ~1j ] = [S−1 3Ψ − 2P(2) S−1 ]ij where P(2) is the Schur product of P with
itself.
Now note that
X
−1 T −2
−1 T −1
Pij2 = ~1j S−1 A AT S−2 A + λI
A S A AT S−2 A + λI
A S ~1j
i
−1 T −1
≤ ~1j S−1 A AT S−2 A + λI
A S ~1j = Pjj = Ψjj
.
Hence, the Gershgorin circle theorem shows that the eigenvalues of Ψ − P(2) are lies in union of
the interval [0, 2ψj ] over all j. Hence, Ψ − P(2) 0. On the other hand, Schur product theorem
shows that P(2) 0 as P 0. Hence, the result follows by chain rule.
Lemma 6 immediately shows that under our choice of ~e = ~eP (~x, ~τ ) we can compute the gradient of the hybrid barrier function, p~e (~x) efficiently. Formally, Lemma 6 immediately implies the
following:
Lemma 7 (Gradient). For ~x ∈ P = {~y ∈ Rn : A~y ≥ ~b} and ~e ∈ Rm we have
~P (~x)) + λ~x
∇p~e (~x) = −ATx (ce~1 + ~e + ψ
and therefore for all ~τ ∈ Rm , we have
∇p~e(~τ ,~x) (~x) = −ATx ce~1 + ~τ + λ~x.
16
Remark 8. To be clear, the vector ∇p~e(~τ ,~x) (~x) is defined as the vector such that
1
p~e(~τ ,~x) (~x + t~1i ) − p~e(~τ ,~x) (~x)
t→0 t
[∇p~e(~τ ,~x) (~x)]i = lim
.
In other words, we treat the parameter ~e(~τ , ~x) as fixed. This is the reason we denote it by subscript
to emphasize that p~e (~x) is a family of functions, p~e(~τ ,~x) is one particular function, and ∇p~e(~τ ,~x)
means taking gradient on that particular function.
Consequently, we can always compute ∇p~e(~τ ,~x) (~x) efficiently. Now, we measure centrality or
how close we are to the hybrid center as follows.
n
o
Definition 9 (Centrality). For ~x ∈ P = ~y ∈ Rn : A~y ≥ ~b and ~e ∈ Rm , we define the centrality
of ~x by
def
δ~e (~x) = ∇p~e (~x)
−1
H(~x)
def
where H(~x) = ATx (ce I + Ψ(~x)) Ax + λI. Often, we use weights w
~ ∈ Rm
>0 to approximate this
def
T
Hessian and consider Q(~x, w)
~ = Ax (ce I + W) Ax + λI.
Next, we bound how much slacks can change in a region close to a nearly central point.
n
o
p
Lemma 10. Let ~x ∈ P = ~y ∈ Rn : A~y ≥ ~b and ~y ∈ Rn such that ~x − ~y H(~x) ≤ ce + µ(~x)
for < 1. Then ~y ∈ P and (1 − )Sx Sy (1 + )Sx .
Proof. Direct calculation reveals the following:
S−1
s~y − ~sx )
x (~
∞
≤ Ax (~y − ~x)
2
1
≤p
Ax (~y − ~x)
ce + µ(~x)
1
≤p
~y − ~x
ce + µ(~x)
H(~
x)
≤
ce I+Ψ(~
x)
.
Consequently, (1 − )Sx Sy (1 + )Sx . Since y ∈ P if and only if Sy 0 the result follows.
Combining the previous lemmas we obtain the following.
Lemma 11. Let ~x ∈ P = {~y ∈ Rn : A~y ≥ ~b} and ~e, w
~ ∈ Rm such that ~e
p
1
Ψ(~x) W 34 Ψ(~x). If ~y ∈ Rn satisfies ~x − ~y Q(~x,w)
≤
ce + µ(~x), then
10
~
1
Q(~x, w)
~ ∇2 p~e (~y ) 8Q(~x, w)
~
4
and
∞
1
H(~x) H(~y ) 2H(~x)
2
≤
1
2 ce
≤ 1 and
.
Proof. Lemma 6 shows that
ATy (ce I + E + Ψ(~y )) Ay + λI ∇2 p~e (~y ) ATy (ce I + E + 3Ψ(~y )) Ay + λI .
(6.2)
p
Since W Ψ, we have that Q(~x, w)
~ H(~x) and therefore ~x −~y H(~x) ≤ ce + µ(~x) with = 0.1.
Consequently, by Lemma 10 we have (1 − )Sx Sy (1 + )Sx and therefore
(1 − )2
(1 + )2
Ψ(~
x
)
Ψ(~
y
)
Ψ(~x)
(1 + )2
(1 − )2
and
1
(1 − )2
(1 + )2
H(~x)
H(~
x
)
H(~
y
)
H(~x) 2H(~x)
2
(1 + )4
(1 − )4
17
Furthermore, (6.2) shows that
∇2 p~e (~y ) ATy (ce I + E + 3Ψ(~y )) Ay + λI
(1 + )2 T
A (ce I + E + 3Ψ(~x)) Ax + λI
(1 − )4 x
(1 + )2 T 3
A
ce I + 3W Ax + λI
(1 − )4 x 2
(1 + )2
3
Q(~x, w)
~ 8Q(~x, w)
~
(1 − )4
and
∇2 p~e (~y ) ATy (ce I + E + Ψ(~y )) Ay + λI
(1 − )4 T
A (ce I + E + Ψ(~x)) Ax + λI
(1 + )2 x
(1 − )4 T 1
3
A
ce I + W Ax + λI
(1 + )2 x 2
4
4
1 (1 − )
1
Q(~x, w)
~ Q(~x, w).
~
2
2 (1 + )
4
To analyze our centering scheme we use standard facts about gradient descent we prove in
Lemma 12.
Lemma 12 (Gradient Descent). Let f : Rn → R be twice differentiable and Q ∈ Rn×n be positive
def
definite. Let ~x0 ∈ Rn and ~x1 = ~x0 − L1 Q−1 ∇f (~x0 ). Furthermore, let ~xα = ~x0 + α(~x1 − ~x) and
suppose that µQ ∇2 f (~xα ) LQ for all α ∈ [0, 1]. Then,
1. ∇f (~x1 ) Q−1 ≤ 1 − Lµ ∇f (~x0 ) Q−1
2. f (~x1 ) ≥ f (~x0 ) −
1
L
∇f (~x0 )
2
Q−1
Proof. Integrating we have that
ˆ 1
ˆ
2
∇f (~x1 ) = ∇f (~x0 ) +
∇ f (~xα )(~x1 − ~x0 )dα =
0
0
1
1 2
Q − ∇ f (~xα ) Q−1 ∇f (~x0 )dα
L
Consequently, by applying Jensen’s inequality we have
ˆ 1
1
Q − ∇2 f (~xα ) Q−1 ∇f (~x0 )dα
∇f (~x1 ) Q−1 =
L
0
Q−1
ˆ 1
1
≤
Q − ∇2 f (~xα ) Q−1 ∇f (~x0 )
dα
L
0
Q−1
≤ Q−1/2 ∇f (~x0 )
[Q−1/2 (Q− L1 ∇2 f (~xα ))Q−1/2 ]
Now we know that by assumption that
1 2
µ
−1/2
0Q
Q − ∇ f (~xα ) Q−1/2 1 −
I
L
L
18
2
and therefore combining these (1) holds.
Using the convexity of f , we have
f (~x1 ) ≥ f (~x0 ) + h∇f (~x0 ), ~x1 − ~x0 i
≥ f (~x0 ) − ∇f (~x0 )
and since ~x1 − ~x0
Q
=
1
L
∇f (~x0 )
Q−1
Q−1
~x1 − ~x0
Q
, (2) holds as well.
Next we bound the effect of changing ~e on the hybrid barrier function p~e (~x).
Lemma 13. For ~x ∈ P = {~y ∈ Rn : A~y ≥ ~b}, ~e, f~ ∈ Rm , and w
~ ∈ Rm
>0 such that W Ψx
∇pf~(~x)
Q(~
x,w)
~ −1
≤ ∇p~e (~x)
Q(~
x,w)
~ −1
1
+p
f~ − ~e
ce + µ(~x)
2
Proof. Direct calculation shows the following
∇pf~(~x)
Q(~
x,w)
~ −1
=
~P (~x)) + λ~x
− ATx (ce~1 + f~ + ψ
≤ ∇p~e (~x)
Q(~
x,w)
~ −1
≤ ∇p~e (~x)
Q(~
x,w)
~ −1
≤ ∇p~e (~x)
Q(~
x,w)
~ −1
+ ATx (f~ − ~e)
(Formula for ∇pf~(~x))
Q(~
x,w)
~ −1
(Triangle inequality)
Q(~
x,w)
~ −1
1
+p
~
ATx (f~ − ~e) (AT Ax )−1 (Bound on Q(~x, w))
x
ce + µ(~x)
1
+p
f~ − ~e 2 (Property of projection matrix)
ce + µ(~x)
where in the second to third line we used Q(~x, w)
~ H(~x) (ce + µ(~x))ATx Ax .
We now have everything we need to analyze our centering algorithm.
Algorithm 1: (~x(r) , ~τ (r) ) = Centering(~x(0) , ~τ (0) , r, c∆ )
Input: Initial point ~x(0) ∈ P = {~y ∈ Rn : A~y ≥ ~b}, Estimator of leverage scores ~τ (0) ∈ Rn
Input: Number of iterations r > 0, Accuracy of the estimator 0 ≤ c∆ ≤ 0.01ce .
Given: ~e(0) ∞ ≤ 13 ce ≤ 13 where ~e(0) = ~e(~τ (0) , ~x(0) ).
p
1
Given: δ~e(0) (~x(0) ) = ∇p~e(0) (~x(0) ) H(~x(0) )−1 ≤ 100
ce + µ(~x(0) ).
Compute w
~ such that Ψ(~x(0) ) W 43 Ψ(~x(0) ) (See Lemma 5)
def
Let Q = Q(~x(0) , w).
~
for k = 1 to r do
~x(k) := ~x(k−1) − 18 Q−1 ∇p~e(k−1) (~x(k−1) ).
~ (k) ∈ Rn s.t.
Sample ∆
(k)
~ x(k) ) − ψ(~
~ x(k−1) ) and
~ ] = ψ(~
E[∆
with high probability in n,
~ x(k) ) − ψ(~
~ x(k−1) )) ≤ c∆ S−1
~ (k) − (ψ(~
∆
(~sx(k) − ~s~x(k−1) )
2
~
x(k−1) ~
(k)
(k−1)
(k)
~
~τ := ~τ
+∆ .
(k)
(k)
~e := ~e(~τ , ~x(k) ).
end
Output: (~x(r) , ~τ (r) )
19
2
(See Section 7.1)
Lemma 14. Let ~x(0) ∈ P = {~y ∈ Rn : A~y ≥ ~b} and let ~τ (0) ∈ Rm such that ~e(~τ (0) , ~x(0) ) ∞ ≤
p
1
1
1
x(0) ) ≤ 100
ce + µ(~x(0) ).
3 ce ≤ 3 . Assume that r is a positive integer, 0 ≤ c∆ ≤ 0.01ce and δ~e(0) (~
With high probability in n, the algorithm Centering(~x(0) , ~τ (0) , r, c∆ ) outputs (~x(r) , ~τ (r) ) such that
1 r
δ~e(0) (~x(0) ).
1. δ~e(r) (~x(r) ) ≤ 2 1 − 64
2
2. E[p~e(k) (~x(r) )] ≥ p~e(0) (~x(0) ) − 8 δ~e(0) (~x(0) ) .
3. E~e(r) = ~e(0) and ~e(r) − ~e(0)
4.
S~−1
(~s(~x(r) ) − ~s(~x(0) ))
x(0)
2
≤
2
≤
1
10 c∆ .
1
10 .
where ~e(r) = ~e(~τ (r) , ~x(r) ).
∇p~e(0) (~x(0) ) Q−1 . First, we use induction to prove that ~x(r) − ~x(0) Q ≤ 8η,
1 r
1
∇p~e(r) (~x(r) ) Q−1 ≤ 1 − 64
η and ~e(r) − ~e(0) 2 ≤ 10
c∆ for all r.
Clearly the claims hold for r = 0. We now suppose they hold for all r ≤ t and show that
they hold for r = t + 1. Now, since ~x(t) − ~x(0) Q ≤ 8η, ~x(t+1) = ~x(t) − 18 Q−1 ∇p~e(t) (~x(t) ), and
1 t
∇p~e(t) (~x(t) ) Q−1 ≤ 1 − 64
η ≤ η, we have
Proof. Let η =
~x(t+1) − ~x(0)
Q
≤ ~x(t) − ~x(0)
Q
+
1
∇p~e(t) (~x(t) )
8
Q−1
≤ 9η.
We will improve this estimate later in the proof to finish the induction on ~x(t+1) − ~x(0) Q , but
p
using this, η ≤ 0.01 ce + µ(~x(0) ), and ~e(t) ∞ ≤ ~e(t) − ~e(0) ∞ + ~e(0) ∞ ≤ c2e , we can invoke
Lemma 11 and Lemma 12 and therefore
1
∇p~e(t) (~x(t) ) Q−1 .
∇p~e(t) (~x(t+1) ) Q−1 ≤ 1 −
32
By Lemma 13 we have
∇p~e(t+1) (~x
(t+1)
)
To bound ~e(t+1) −~e(t)
~x(t)
− ~x(0)
Q
Q−1
2
1
≤ 1−
∇p~e(t) (~x(t) )
32
Q−1
1
+p
~e(t+1) − ~e(t)
(0)
ce + µ(~x )
2
, we note that Lemma 10 and the induction hypothesis ~x(t) −~x(0)
.
(6.3)
H(~
x(0) )
≤
≤ 8η shows that (1 − 0.1)Sx(0) Sx(t) (1 + 0.1)Sx(0) and therefore
S−1
(~s (t) − ~sx(t+1) )
x(t) x
2
1
(t)
(t+1)
S−1
A
~
x
−
~
x
(0)
2
1 − 0.1 x
1
1 −1
=
Q ∇p~e(t) (~x(t) )
1 − 0.1 8
AT S−2
≤
x(0)
≤
A
1
p
∇p~e(t) (~x(t) )
(0)
8 (1 − 0.1) ce + µ(~x )
Now since
~ x(t) )
~ x(t+1) ) − ~τ (t) − ψ(~
~e(t+1) − ~e(t) = ~τ (t+1) − ψ(~
~ x(t+1) ) − ψ(~
~ x(t) )
~ (t+1) − ψ(~
=∆
20
Q−1
(6.4)
Consequently, with high probability in n,
~e(t+1) − ~e(t)
2
~ x(t+1) ) − ψ(~
~ x(t) )
~ (t+1) − ψ(~
= ∆
≤ c∆
S−1
(~s (t+1)
x(t) x
− ~sx(t) )
2
2
c
p∆
≤
∇p~e(t) (~x(t) )
(0)
8 (1 − 0.1) ce + µ(~x )
Q−1
.
where in the last line we used mini∈[m] wi ≥ µ(~x(0) ). Since c∆ < 0.01ce , by (6.3), we have
1
0.01ce
(t+1)
∇p~e(t) (~x(t) )
∇p~e(t+1) (~x
) Q−1 ≤ 1 −
∇p~e(t) (~x(t) ) Q−1 +
32
8 (1 − 0.1) (ce + µ(~x(0) ))
1
≤ 1−
∇p~e(t) (~x(t) ) Q−1 .
64
Q−1
Furthermore, this implies that
~x(t+1) − ~x(0)
≤
Q
t
X
1
k=0
8
Q−1 ∇p~e(k) (~x(k) )
≤
Q−1
∞
1X
64
1 k
η ≤ η = 8η
1−
8
64
8
.
i=0
Similarly, we have that
(t+1)
~e
t
X
(0)
− ~e
2
1 k
c∆
p
1−
≤
∇p~e(0) (~x(0) ) Q−1
(0) )
64
8
(1
−
0.1)
c
+
µ(~
x
e
k=0
8c∆ η
8c
1
p
p∆
≤
≤
δ~e(0) (~x(0) ) ≤ c∆
10
(1 − 0.1) ce + µ(~x(0) )
(1 − 0.1) ce + µ(~x(0) )
where we used η = ∇p~e(0) (~x(0) )
Q−1
~x(t) − ~x(0)
≤ ∇p~e(0) (~x(0) )
and
tion on ∇p~e(t) (~x(t) ) Q−1 ,
Q
Hence, for all r, Lemma 11 shows that
H−1
~e(t) − ~e(0)
= δ~e(0) (~x(0) ) and this finishes the induc2
.
√
∇p~e(r) (~x(r) ) H(~x(r) )−1 ≤ 2 ∇p~e(r) (~x(r) ) H(~x(0) )−1
r
r
8
8
1 r
(r)
≤
∇p~e(r) (~x ) Q−1 ≤
1−
∇p~e(0) (~x(0) )
3
3
64
1 r
δ~e(0) (~x(0) ).
≤ 2 1−
64
δ~e(r) (~x(r) ) =
Q−1
Using that E~e(t+1) = ~e(t) , we see that the expected change in function value is only due to the
change while taking centering steps and therefore Lemma 12 shows that
∞
2
1X
1 2k
2
(r)
(0)
E[p~e(r) (~x )] ≥ p~e(0) (~x ) −
1−
∇p~e(0) (~x(0) ) Q−1 ≥ p~e(0) (~x(0) ) − 8 δ~e(0) (~x(0) ) .
8
64
k=0
Finally, for (4), we note that
s(~x(r) ) − s(~x(0) )
s(~x(0) )
= ~x(r) − ~x(0)
2
AT S−2
A
x(0)
21
1
≤p
~x(r) − ~x(0)
(0)
µ(~x ) + ce
Q−1
≤
1
.
10
6.2
Changing Constraints
Here we bound the effect that adding or a removing a constraint has on the hybrid barrier function.
Much of the analysis in this section follows from the following lemma which follows easily from the
Sherman Morrison Formula.
Lemma 15 (Sherman Morrison Formula Implications). Let B ∈ Rn×n be an invertible symmetric
matrix and let ~a ∈ Rn be arbitrary vector satisfying ~aT B−1~a < 1. The following hold:
−1
−1
T B−1
1. B ± ~a~aT
= B−1 ∓ B1±~a~aT~aB−1
.
~a
B−1~a~aT B−1
1±~aT B−1~a
T
−1
~a B ~a
1±~
B−1 .
aT B−1~a
3. log det B ± ~a~aT = ln det B + ln 1 ± ~aT B−1~a .
2. 0
Proof. (1) follows immediately from Sherman Morrison [95]. (2) follows since ~a~aT is PSD,
"
#
−1/2~
a~aT B−1/2
B−1~a~aT B−1
−1/2 B
B−1/2 ,
=B
1 ± ~aT B−1~a
1 ± ~aT B−1~a
and ~y~y T ~y
2
I
2
for any vector ~y . (3) follows immediately from the Matrix Determinant Lemma.
We also make use of the following technical helper lemma.
Lemma 16. For A ∈ Rn×m and all ~a ∈ Rn we have
X 1
−1 4 T
−1 2
A AT A + λI
~a ≤ ~a AT A + λI
~a
ψA [i]
i
i∈[m]
Proof. We have by Cauchy Schwarz that
2
~1Ti A AT A + λI −1 ~a ≤ ψA [i] · ~aT AT A + λI −1 ~a
and consequently
X
4
~1Ti A AT A + λI −1 ~a
ψA [i]
i∈[m]
2
−1 X
~1i A AT A + λI −1 ~a .
≤ ~aT AT A + λI
~a
i∈[m]
Since
2
X
~1Ti A AT A + λI −1 ~a = ~aT AT A + λI −1 AT A AT A + λI −1 ~a
i∈[m]
−1
≤ ~aT AT A + λI
~a,
we have the desired result.
We now bound the effect of adding a constraint.
22
n
o
def
Lemma 17. Let A ∈ Rm×n , ~b ∈ Rm , ~τ ∈ Rm , and ~x ∈ P = ~y ∈ Rn : A~y ≥ ~b . Let A ∈ R(m+1)×n
be A with a row ~am+1 added, let b ∈ Rm+1 be the vector ~b with an entry bm+1 added, and let
~aT (AT Ax +λI)−1~am+1
def
P = ~y ∈ Rn : A~y ≥ b . Let sm+1 = ~aTm+1 ~x − bm+1 > 0, ψa = m+1 x s2
.
m+1
Now, let ~υ ∈ Rm+1 be defined so that υm+1 =
ψa
1+ψa
and for all i ∈ [m]
−1 ~am+1 2
1
T
Ax Ax Ax + λI
.
υi = τi −
1 + ψa
sm+1 i
Then, the following hold
• [Leverage Score Estimation] eP (~υ , ~x)m+1 = 0 and eP (~υ , ~x)i = eP (~τ , ~x)i for all i ∈ [m].
• [Function Value Increase] p~eP (~υ,~x) (~x) = p~eP (~τ ,~x) (~x) − ce ln s(~x)m+1 + ln(1 + ψa ).
q
ψa
• [Centrality Increase] δ~eP (~υ,~x) (~x) ≤ δ~eP (~υ,~x) (~x) + (ce + ψa ) µ(~
x) + ψa .
Proof. By (1) in Lemma 15, we have that for all i ∈ [m]
−1 ~am+1 2
1
T
Ax Ax Ax + λI
ψP (~x)i = ψP (~x)i −
1 + ψa
sm+1 i
and that
ψP (~x)m+1 = ψa −
ψa2
ψa
=
.
1 + ψa
1 + ψa
Consequently [Leverage Score Estimation] holds. Furthermore, by (3) in Lemma 15 this then
implies that [Function Value Change] holds.
−1
To bound the change in centrality note that by (2) in Lemma 15 we have that H H−1 .
Therefore if let ~υ 0 ∈ Rm be defined so that ~υi0 = ~υi for all i ∈ [m] then by triangle inequality we
have
T
δ~ep (~υ,~x) (~x) = Ax (ce~1 + ~υ )
H
−1
T
≤ Ax (ce~1 + ~υ )
H−1
~am+1
≤ ATx (ce~1 + ~τ ) H−1 +
(ce + υm+1 )
+ ATx (~υ 0 − ~τ )
sm+1
−1
H
ψa
~am+1
= δ~eP (~τ ,~x) (~x) + ce +
+ ATx (~υ 0 − ~τ ) H−1
1 + ψa
sm+1 H−1
Now, since H−1
1
µ(~
x)
ATx Ax + λI
~am+1
sm+1
Since Ψ1/2 Ax ATx ΨAx
−1
−1
, we have that
~am+1
≤p
µ(~x) sm+1
s
1
H−1
H−1
=
(AT
x Ax +λI)
−1
ψa
.
µ(~x)
ATx Ψ1/2 is a projection matrix, we have Ψ−1 Ax ATx ΨAx
23
−1
ATx
Ax H−1 ATx . By Lemma 16, we have
2
ATx ~τ 0 − ~υ H−1 ≤ ~τ 0 − ~υ
=
X
i∈[m]
≤
2
Ψ−1
1
1 + ψa
1
ψ(~x)i
1
1 + ψa
2
~aTm+1
2 !2
~
a
−1
m+1
~1i Ax ATx Ax + λI
sm+1
!2
−1
2
ATx Ax + λI
~am+1
ψa
=
1 + ψa
s2m+1
Combining, we have that
s
ψa
ψa
ψa
δ~eP (~υ,~x) (~x) ≤ δ~eP (~τ ,~x) (~x) + ce +
+
1 + ψa
µ(~x) 1 + ψa
s
ψa
≤ δ~eP (~τ ,~x) (~x) + (ce + ψa )
+ ψa .
µ(~x)
We now bound the effect of removing a constraint.
def
Lemma 18 (Removing a Constraint). Let A ∈ Rm×n , ~b ∈ Rm , ~τ ∈ Rm , and ~x ∈ P = {~y ∈
Rn : A~y ≥ ~b}. Let A ∈ R(m−1)×n be A with row m removed, let b ∈ Rm−1 denote the first m − 1
def
coordinates of ~b, and let P = {~y ∈ Rn : A~y ≥ b}. Let ψd = ψP (~x)m .
Now, let ~υ ∈ Rm−1 be defined so that for all i ∈ [m − 1]
−1 T 2
1
υi = τi +
Ax ATx Ax + λI
Ax ~1m .
1 − ψd
i
Assume ψd ≤ 1.1µ(~x) ≤
1
10
and ~eP (~τ , ~x)
∞
≤ ce ≤ 12 , we have the following:
• [Leverage Score Estimation] eP (~υ , ~x)i = eP (~τ , ~x)i for all i ∈ [m − 1].
• [Function Value Decrease] p~ep (~υ,~x) (~x) = p~eP (~τ ,~x) (~x) + [ce + eP (~τ , ~x)m ] ln s(~x)m + ln(1 − ψd )
• [Centrality Increase] δ~ep (~υ,~x) (~x) ≤ √
1
δ~e (~τ ,~x) (~x)
1−2µ(~
x) P
+ 3(ce + µ(~x)).
Proof. By (1) in Lemma 15, we have that for all i ∈ [m − 1]
−1 T 2
1 ~ T
Ax ~1m .
ψP (~x)i = ψP (~x)i +
1i Ax ATx Ax + λI
1 − ψd
Consequently, [Leverage Score Estimation] holds. Furthermore, by (3) in Lemma 15, this then
implies that [Function Value Change] holds.
To bound the change in centrality we first note that by (1) and (2) in Lemma 15, we have that
the approximate Hessian for P , denoted H(~x), is bounded by
−1
H(~x)−1 = H(~x) − ATx (ce I + Ψx )1/2 ~1m~1Tm (ce I + Ψx )1/2 Ax
α
1
−1
1+
H(~x) =
H(~x)−1
1−α
1−α
24
def
where α = ~1Tm (ce I + Ψx )1/2 Ax H(~x)−1 ATx (ce I + Ψx )1/2 ~1m . Using ce + µ(~x) ≤ 12 +
have
−1
−1
H(~x)−1 ATx (ce + µ(~x))Ax + λI
(ce + µ(~x))−1 ATx Ax + λI
.
1
10
≤ 1, we
(6.5)
Using this, we have
α≤
−1 T
ce + ψd
ce + ψd ~ T
T
~
1m Ax Ax Ax + λI
Ax 1m =
ψd .
ce + µ(~x)
ce + µ(~x)
(6.6)
Now let ~τ 0 ∈ Rm−1 be defined so that τi0 = τi for all i ∈ [m − 1]. We have by above that
T
δ~ep (~υ,~x) (~x) = Ax (ce~1 + ~υ )
H
−1
≤√
1
T
Ax (ce~1 + ~υ )
1−α
H−1
and therefore, by triangle inequality
T
Ax (ce~1 + ~υ )
H−1
+ ATx ~1m (ce + τm ) H−1 + ATx (~τ 0 − ~υ )
= δ~eP (~τ ,~x) (~x) + (ce + τm ) ATx ~1m H−1 + ATx (~τ 0 − ~υ ) H−1 .
≤ ATx (ce~1 + ~τ )
H−1
H−1
Now, (6.5) shows that
ATx ~1m
H−1
ATx ~1m
≤p
ce + µ(~x)
Furthermore, since Ψ−1 Ax ATx ΨAx
ATx ~τ 0 − ~υ
s
1
2
H−1
−1
−1
(AT
x Ax +λI)
≤
ψd
ce + µ(~x)
ATx Ax H−1 ATx , by Lemma 16 we have
2
≤ ~τ 0 − ~υ Ψ−1
X 1 1
=
ψ(~x)i 1 − ψd
i∈[m]
2
1
~1Tm Ax
≤
1 − ψd
2 2
~1Ti Ax ATx Ax + λI −1 ATx ~1m
ATx Ax
+ λI
−1
ATx ~1m
2
=
ψd
1 − ψd
2
Combining, we have that
s
#
"
1
ψd
ψd
δ~ep (~υ,~x) (~x) ≤ √
.
δ~eP (~τ ,~x) (~x) + (ce + τm )
+
ce + µ(~x) 1 − ψd
1−α
Using the assumption ψd ≤ 1.1µ(~x) ≤
1.21µ(~x) and τm ≤ ψd + ce , and
δ~ep (~υ,~x) (~x) ≤
≤
1
10 ,
~eP (~τ , ~x)
∞
≤ ce and (6.6), we have α ≤ 1.1ψd ≤
h
i
√
1
p
δ~eP (~τ ,~x) (~x) + (ce + τm ) 1.1 + 1.2ψd
1 − 1.3µ(~x)
√
√
1
1
p
δ~eP (~τ ,~x) (~x) + q
1.1 · 2ce + ( 1.1 + 1.2 · 1.1)µ(~x)
1 − 2µ(~x)
1 − 1.3
10
25
6.3
Hybrid Center Properties
Here we prove properties of points near the hybrid center. First we bound the distance between
points in the H(~x) norm in terms of the `2 norm of the points.
m
m
~
~
Lemma 19. For A ∈ Rm×n and
p b ∈ R suppose that ~x ∈ P = {~y : A~y ≥ b} and ~e ∈ R such that
1
1
~e ∞ ≤ 2 ce < 20 and δ~e ≤ 0.1 ce + µ(~x). Then for all ~y ∈ P we have
~x − ~y
and
2
2
~x
2
2
12mce + 6n + 2λ ~y
p
≤
ce + µ(~x)
H(~
x)
≤ 4λ−1 (mce + n) + 2 ~y
2
2
(6.7)
.
def
def
~x , T def
= diag(~t), and Q = ATx (ce I + Ψx )Ax .
Proof. For notational simplicity let ~t = ce~1 + ~e + ψ
We have
X [~sx − ~sy ]2
X
[~sy ]2i
[~sy ]i
2
i
~x − ~y AT TAx =
ti
+
=
t
1
−
2
(6.8)
i
x
[~sx ]i [~sx ]2i
[~sx ]2i
i∈[m]
i∈[m]
and
[~sy ]2i
ti
[~sx ]2i
i∈[m]
X
≤
X
ti
i∈[m]
[~sy ]i
[~sy ]i
max
≤
[~sx ]i i∈[m] [~sx ]i
X
ti
i∈[m]
[~sy ]i
sy − ~sx )
1 + S−1
x (~
[~sx ]i
(6.9)
∞
and
S−1
sx − ~sy )
x (~
∞
≤ max ~1i S−1
x − ~y ) ≤ ~x − ~y
x A (~
i∈[m]
≤ (ce + µ(~x))−1/2 ~x − ~y
r
H(~
x)
x)−1 AT S−1
max S−1
x ii
x AH(~
i∈[m]
.
H(~
x)
(6.10)
P
Now, clearly i∈[m] ti [~sy ]i /[~sx ]i is positive and since ~e ∞ ≤ 21 ce we know that 12 Q ATx TAx .
Therefore, by combining, (6.8), (6.9), and (6.10) we have
~x − ~y H(~x)
X
X
[~
s
]
[~
s
]
1
2
y i
y i
p
~x − ~y Q ≤ ~t 1 −
ti
+
ti
2
[~sx ]i
[~sx ]i
ce + µ(~x)
i∈[m]
i∈[m]
~x − ~y H(~x)
X [~sy ]i
p
≤ ~t 1 +
ti
(6.11)
[~sx ]i
ce + µ(~x)
i∈[m]
~
Now since ∇p~e (~x) = −AT S−1
x we have
x T1 + λ~
X [~sx − ~sy ]i
h~x − ~y , ∇p~e (~x)i = −
ti
+ λ~xT (~x − ~y )
[~sx ]i
i∈[m]
and therefore by Cauchy Schwarz and ~xT ~y ≤ ~x
X
i∈[m]
ti
[~sy ]i
= ~t
[~sx ]i
≤ t
2
2
+
1
4
~y
2
,
2
1
− λ ~x
2
2
+ λ~xT ~y + h~x − ~y , ∇p~e (~x)i
(6.12)
1
+
λ
~y
4
2
2
+ ~x − ~y
(6.13)
26
δ (~x)
H(~
x) ~e
.
Now, using (6.11), (6.13) and the definition of H(~x), we have
1
~x − ~y
2
2
H(~
x)
1
λ
2
2
~x − ~y Q +
~x − ~y 2
2
2
~x − ~y H(~x) λ
X [~sy ]i
p
~t +
t
~x − ~y
+
i
1
[~sx ]i
2
ce + µ(~x)
=
≤
2
2
i∈[m]
2
t + λ ~y 2
≤ ~t 1 + p 1 4
~x − ~y
ce + µ(~x)
p
Using the fact that δe (~x) ≤ 0.1 ce + µ(~x), we have
2
~x − ~y H(~x) λ
+ δe (~x) p
+
~x − ~y
H(~
x)
2
ce + µ(~x)
2
.
2
2
t + λ ~y 2
1
λ
2
2
~x − ~y
~x − ~y H(~x) ≤ ~t 1 +
~x − ~y 2 + p 1 4
4
2
ce + µ(~x)
P
Furthermore, since i∈[m] ti [~sy ]i /[~sx ]i is positive, (6.12) shows that
λ~xT (~x − ~y ) = λ ~x
2
2
− λ~xT ~y ≤ ~t
1
+ h~x − ~y , ∇p~e (~x)i ≤ ~t
1
H(~
x)
+ ~x − ~y
.
(6.14)
δ (~x)
H(~
x) e
and hence
λ
~x − ~y
2
λ
λ
λ
2
2
2
~x − ~y 2 +
~x 2 = λ~xT (~x − ~y ) +
~y 2
2
2
2
λ
2
~y 2 + ~x − ~y H(~x) δe (~x) .
≤ ~t 1 +
2
p
Putting (6.15) into (6.14) and using the fact that δe (~x) ≤ 0.1 ce + µ(~x), we have
2!
λ
~
+
t
~
y
λ
1
2
2
2
~x − ~y H(~x) ≤ 2 ~t 1 +
~y 2 + 0.1 + p 1 4
~x − ~y H(~x) .
4
2
ce + µ(~x)
Now, using ~t
1
≤
(6.15)
≤ 2mce + n, we have
1
~x − ~y
4
Since
2
2
2
H(~
x)
≤ 2α + (0.1 + α) ~x − ~y
H(~
x)
for
2mce + n + λ4 ~y
p
α=
ce + µ(~x)
p
ce + µ(~x) ≤ 1.05, we have α ≥ 0.9 and hence
p
0.1 + α + (α + 0.1)2 + 2α
~x − ~y H(~x) ≤
≤ 6α
2 · 14
yielding (6.7).
p
We also have by (6.12) and the fact that δe (~x) ≤ 0.1 ce + µ(~x),
λ ~x
2
2
= t
+ λ~xT ~y + h~x − ~y , ∇p~e (~x)i −
1
X
i∈[m]
λ
~x
≤ t 1+
2
λ
≤ t 1+
~x
2
λ
+
~y
2
λ
2
+
~y
2
2
2
2
ti
[~sy ]i
[~sx ]i
2
2
+ ~x − ~y
2
2
p
+ 0.1 ce + µ(~x) ~x − ~y
27
δ (~x)
H(~
x) e
H(~
x)
2
2
.
Hence, using ~t
1
≤ 2mce + n and ~x − ~y
λ
~x
2
2
2
≤ t
+
1
≤ λ ~y
2
2
≤ 6α, we have
H(~
x)
λ
~y
2
2
2
λ
+ 0.6 2mce + n +
~y
4
2
2
+ 2(mce + n).
In the following lemma we show how we can write one hyperplane in terms of the others provided
that we are nearly centered and show there is a constraint that the central point is close to.
Lemma 20. Let A ∈ Rm×n and ~b ∈ Rm such that ai 2 = 1 for all i. Suppose that ~x ∈ P = {~y :
A~y ≥ ~b} and ~e ∈ Rm such that ~e ∞ ≤ 12 ce ≤ 12 . Furthermore, let = minj∈[m] sj (~x) and suppose
that i = arg minj∈[m] sj (~x) then
"
2
λ ~x
≤
(ce + µ(~x))
X s(x)i ce + ej + ψj (~x)
~aj
~ai +
s(x)j
ce + ei + ψi (~x)
j6=i
r
2
+ δe (~x)
#
mce + n
+λ .
2
2
Proof. We know that
~x ) + λ~x
~ e+ψ
∇pe (~x) = −AT S−1
x (ce 1 + ~
X (ce + ei + ψi )
= λ~x −
~ai
s(~x)i
i∈[m]
Consequently, by ~e
∞
≤ 21 ce , and ψi (~x) ≥ µ(~x)
X s(x)i ce + ej + ψj (~x)
~aj
~ai +
s(x)j
ce + ei + ψi (~x)
j6=i
Using ~ai = 1,
=
si (~x)
~x )
~ e+ψ
AT S−1
x (ce 1 + ~
(ce + ei + ψi (~x))
≤
2
λ ~x
(ce + µ(~x))
P
i ψi
2
+ ∇pe (~x)
2
.
≤ n, and si (~x) ≥ , we have
Tr(ATx (ce I + Ψx )Ax ) = Tr(Ax ATx (ce I + Ψx ))
2
X
ai 2
mce + n
=
(ce + ψi ) 2
≤
.
2
si (~x)
i
Hence, we have H(~x)
6.4
2
2
mce +n
2
+ λ I and ∇pe (~x)
2
(6.16)
q
≤ δe (~x) mce2+n + λ yielding the result.
The Algorithm
Here, we put all the results in the previous sections to get our ellipsoid algorithm. Below is a sketch
of the pseudocode; we use ca , cd , ce , c∆ to denote parameters we decide later.
In the algorithm, there are two main invariants we maintain. First, we maintain that the
centrality δP,~e (~x), which indicates how close ~x is to the minimum point of p~e , is small. Second, we
maintain that ~e(~τ , ~x) ∞ , which indicates how accurate the leverage score estimate ~τ is, is small.
In the following lemma we show that we maintain both invariants throughout the algorithm.
28
Algorithm 2: Our Cutting Plane Method
Input: A(0) ∈ Rm×n , ~b(0) ∈ Rm , > 0, and radius R > 0.
Input: A separation oracle for a non-empty set K ⊂ B∞ (R).
Check: Throughout the algorithm, if si (~x(k) ) < output P (k) .
Check: Throughout the algorithm, if ~x(k) ∈ K, output ~x(k) .
Set P (0) = B∞ (R).
(0)
Set ~x(0) := ~0 and compute τi = ψP (0) (~x(0) )i for all i ∈ [m] exactly.
for k = 0 to ∞ do
Let m(k) be the number of constraints in P (k) .
Compute w
~ (k) such that ΨP (k) (~x(k) ) W(k) (1 + c∆ )ΨP (k) (~x(k) ).
(k)
Let i(k) ∈ arg maxi∈[m(k) ] wi
(k+ 31 )
i(k)
Set τ
(k)
− τi
.
(k+ 31 )
= ψP (k) (~x(k) )i(k) and τj
(k)
if mini∈[m(k) ] wi
(k)
= τj
for all j 6= i(k) .
≤ cd then
(k)
Remove constraint with minimum wi
yielding polytope P (k+1) .
(k+ 2 )
Update ~τ according to Lemma 18 to get τj 3 .
else
Use separation oracle at ~x(k) to get a constraint {~x : ~aT ~x ≥ ~aT ~x(k) } with ~a 2 = 1.
q
−1/2
Add constraint {~x : ~aT ~x ≥ ~aT ~x(k) − ca
~aT (AT S~−2
A + λI)−1~a} yielding
x(k)
polytope P (k+1) .
(k+ 32 )
Update ~τ according to Lemma 17 to get τj
.
2
(~x(k+1) , ~τ (k+1) ) = Centering(~x(k) , ~τ (k+ 3 ) , 200, c∆ ).
end
√
cd
Lemma 21. Assume that ce ≤ cd ≤ 1016 , ca ca ≤ 10
3 , cd ≤ ca , and c∆ ≤ Cce / log(n) for some
small enough universal constant C. During our cutting plane method, for all k, with high probability
in n, we have
1
~e(~τ (k+ 3 ) , ~x(k) )
1.
2. δ
2
P (k) ,~e(~
τ (k+ 3 ) ,~
x(k) )
2
1
~e(~τ (k+ 3 ) , ~x(k) ) ∞ ≤ 1000
ce , ~e(~τ (k+1) , ~x(k+1) )
q
1
(~x(k) ) ≤ 100
ce + min µ(~x(k) ), cd .
∞
≤
1
1000 ce ,
3. δP (k+1) ,~e(~τ (k+1) ,~x(k+1) ) (~x(k+1) ) ≤
1
400
q
∞
≤
1
400 ce .
ce + min µ(~x(k+1) ), cd .
Proof. Some statements of the proof hold only with high probability in n; we omit mentioning this
for simplicity.
We prove by induction on k. Note that the claims are written in order consistent with the
algorithm and proving the statement for k involves bounding centrality at the point ~x(k+1) . Trivially
2
1
we define, ~τ (−1) = ~τ (− 3 ) = ~τ (− 3 ) = ~τ (0) and note that the claims then hold for k = −1 as we
compute the initial leverage scores, ~τ (0) , exactly and since the polytope is symmetric we have
δ~e(~τ (0) ,~x(0) ) (~0) = 0. We now suppose they hold for all r < t and show that they hold for r = t.
p
def
We first bound δ. For notational simplicity, let ηt = ce + min{µ(~x(t) ), cd }. By the induction
1
1
hypothesis we know that δP (t) ,~e(~τ (t) ,~x(t) ) (~x(t) ) ≤ 400
ηt . Now, when we update τ (t) to τ (t+ 3 ) , we set
29
~ei(t) to 0. Consequently, Lemma 13 and the induction hypothesis ~e(~τ (t) , ~x(t) )
δ
1
P (t) ,~e(~
τ (t+ 3 ) ,~
x(t) )
∞
≤
1
400 ce
show that
1
(~x(t) ) ≤ δP (t) ,~e(~τ (t) ,~x(t) ) (~x(t) ) + p
ei(t) (~τ (t) , ~x(t) )
ce + µ(~x(t) )
√
ce
1
ηt
≤
ηt +
≤
400
400
200
(6.17)
Next, we estimate the δ changes when we remove or add a constraint.
For the case of removal, we note that it happens only if µ(~x(t) ) ≤ mini wi ≤ cd ≤ 1016 . Also,
the row we remove has leverage score at most 1.1µ(~x(t) ) because we pick the row with minimum
w. Hence, Lemma 18 and ce ≤ 1016 show that
δ
2
P (t+1) ,~e(~
τ (t+ 3 ) ,~
x(t) )
1
δ (t) (t+ 1 ) (t) (~x(t) ) + 2.7(ce + µ(~x(t) ))
(~x(t) ) ≤ p
1 − 2µ(~x(t) ) P ,~e(~τ 3 ,~x )
η
1
ηt
t
≤√
+ 3(ce + µ(~x(t) )) ≤
−6
200
100
1 − 2 · 10
√
√
2
where we used the fact µ(~x(t) ) ≤ cd and hence ce + µ(~x(t) ) ≤ ce + cd ηt ≤ 1000
ηt .
(t)
For the case of addition, we note that it happens only if 2µ(~x ) ≥ mini wi ≥ cd . Furthermore, in
this case the hyperplane we add is chosen precisely so that ψa = ca . Furthermore, since ce ≤ cd ≤ ca
by Lemma 17 we have that
s
r
ηt
ψa
ca
(t)
+ ψa ≤
+ 4ca
.
δ (t+1) (t+ 23 ) (t) (~x ) ≤ δ (t) (t+ 13 ) (t) + (ce + ψa )
(t)
,~
x )
P ,~e(~
τ
,~
x )
P
,~e(~
τ
200
cd
µ(~x )
p
√
cd
Furthermore, since ca ca ≤ 1000
, µ(~x(t) ) ≥ cd /2, and cd ≤ 10−6 we know that 4ca ca /cd ≤
1
and consequently in both cases we have δ (t+1) (t+ 32 ) (t) (~x(t) ) ≤ 100
ηt .
P
,~e(~
τ
,~
x
1
200 ηt
)
Now, note that Lemmas 17 and 18 show that ~e does not change during the addition or removal
2
1
of an constraint. Hence, we have ~e(~τ (t+ 3 ) , ~x(t) ) ∞ ≤ ~e(~τ (t+ 3 ) , ~x(t) ) ∞ . Furthermore, we know
(k+ 1 )
2
the step “~τi(k) 3 = ψP (k) (~x(k) )i(k) ” only decreases ~e ∞ and hence we have ~e(~τ (t+ 3 ) , ~x(t) ) ∞ ≤
ce
~e(~τ (t) , ~x(t) ) ∞ ≤ 400
. Thus, we have all the conditions needed for Lemma 14 and consequently
δP (t+1) ,~e(~τ (t+1) ,~x(t+1) ) (~x
(t+1)
Lemma 14 also shows that that
Therefore, ηt ≤ 2ηt+1 and thus
1
)≤2 1−
64
200
s(~
x(t+1) )−s(~
x(t) )
s(~
x(t) )
2
δP (t+1) ,~e(~τ (t+1) ,~x(t+1) ) (~x(t+1) ) ≤
δ
2
x(t) )
P (t+1) ,~e(~
τ (t+ 3 ) ,~
≤
1
10
(~x(t) ) ≤
1
ηt
1000
.
and hence ψi (~x(t) ) ≤ 2ψi (~x(t+1) ) for all i.
q
ce + min cd , µ(~x(t+1) )
400
.
completing the induction case for δ.
Now, we bound ~e ∞ . Lemma 17 and 18 show that ~e does not change during the addition or
(k+ 1 )
removal of an constraint. Hence, ~e is affected by only the update step “τi(k) 2 = ψP (k) (~x(k) )i(k) ” and
1
the centering step. Using the induction hypothesis δ (r) (r+ 23 ) (r) (~x(r) ) ≤ 100
ηr and Lemma 14
P
shows that
E~e(~τ (r+1) , ~x(r+1) )
= ~e(~τ
(r+ 23 )
, ~x(r) )
and
30
,~e(~
τ
,~
x )
2
(r+1)
(r+1)
~e(~τ
, ~x
) − ~e(~τ (r+ 3 ) , ~x(r) )
2
≤
1
10 c∆
for
all r ≤ t. The goal for the update step is to decrease ~e by updating ~τ . In Section 7.2, we give a
self-contained analysis of the effect of this step as a game. In each round, the vector ~e is corrupted
by some mean 0 and bounded variance noise and the problem is how to update ~e such that ~e ∞
is bounded. Theorem 34 shows that we can do this by setting the ~ei = 0 for the almost maximum
coordinate in each iteration. This is exactly what the update step is doing. Hence, Theorem 34
shows that this strategy guarantees that after the update step, we have
1
~e(τ (r+ 3 ) , ~x(r) )
∞
= O (c∆ log (n))
1
1
for all r ≤ t. Now, by our choice of c∆ , we have ~e(~τ (t+ 3 ) , ~x(t) ) ∞ ≤ 1000
ce . Lemma 17 and
18 show that ~e does not change during the addition or removal of an constraint. Hence, we
2
1
have ~e(~τ (t+ 3 ) , ~x(t) ) ∞ ≤ 1000
ce . Now, we note that again Lemma 14 shows ~e(~τ (t+1) , ~x(t+1) ) −
2
1
1
~e(~τ (t+ 3 ) , ~x(t) ) 2 ≤ 10
c∆ ≤ 1000
ce , and we have
induction case for ~e ∞ and proves this lemma.
~e(~τ (t+1) , ~x(t+1) )
∞
ce
400 .
≤
This finishes the
Next, we show the number of constraints is always linear to n.
Lemma 22. Throughout our cutting plane method, there are at most 1 +
2n
cd
constraints.
cd
Proof. We only add a constraint if mini wi ≥ cd . Since 2ψi ≥ wi , we have
Pψi ≥ 2 for all i. Letting
m denote the number of constraints after we add that row, we have n ≥ i ψi ≥ (m − 1)(cd /2).
Using K 6= ∅ and K ⊂ B∞ (R), here we show that the points are bounded.
Lemma 23. During our Cutting Plane Method, for all k, we have ~x(k)
2
≤6
p
√
n/λ + 2 nR.
2
2
Proof. By Lemma 21 and Lemma 19 we know that ~x(k) 2 ≤ 4λ−1 (mce + n) + 2 ~y 2 for any
~y ∈ P (k) . Since our method never cuts out any point in K and since K is nonempty, there is some
2
~y ∈ K ⊂ P (k) . Since K ⊂ B∞ (R), we have ~y 2 ≤ nR. Furthermore, by Lemma 22 we have that
mce ≤ ce + 2n ≤ 3n yielding the result.
q
p
√
Lemma 24. si ~x(k) ≤ 12 n/λ + 4 nR + ca1λ for all i and k in the our cutting plane method.
Proof. Let ~x(j) be the current point at the time that the constraint corresponding to si , denoted
{~x : ~aTi ~x ≥ aTi ~x(j) − si (~x(j) )}, was added. Clearly
si (~x(k) ) = ~aTi ~x(k) − aTi ~x(j) + si (~x(j) ) ≤ ~ai · ~x(k) + ~aTi ~x(j) − si (~x(j) )
.
On the one hand, if the constraint for si comes from the initial symmetric polytope P (0) = B∞ (R),
we know ~aT ~x(j) − ~si (~x(j) ) ≤ R . On the other hand, if the constraint was added later then we
know that
q
(j)
−1/2
si (~x ) = ca
~aT (AT S~−2
A + λI)−1~a ≤ (ca λ)−1/2
x(j)
and ~aT ~x(j) − si (~x(j) ) ≤ ~ai · ~x(j) + si (~x(j) ) . Since ~ai 2 = 1 by design and ~x(j) 2 and
p
√
~x(k) 2 are upper bounded by 6 n/λ + 2 nR by Lemma 23, in either case the result follows.
Now, we have everything we need to prove that the potential function is increasing in expectation.
31
Lemma 25. Under the assumptions of Lemma 21 if λ =
then for all k we have
1
,
ca R2
ce =
cd
6 ln(17nR/) ,
and 24cd ≤ ca ≤
1
3
Ep~e(~τ (k+1) ,~x(k+1) ) (~x(k+1) ) ≥ p~e(~τ (k) ,~x(k) ) (~x(k) ) − cd + ln(1 + β)
where β = ca for the case of adding a constraint β = −cd for the case of removal.
Proof. Note that there are three places which affect the function value, namely the update step for
1
τ (k+ 3 ) , the addition/removal of constraints, and the centering step. We bound the effect of each
separately.
First, for the update step, we have
p
1
~e(~
τ (k+ 3 ) ,~
x(k) )
(~x(k) ) = −ei(k) log(si(k) (~x(k) )) + p~e(~τ (k) ,~x(k) ) (~x(k) ).
Lemma 24, the termination condition and λ =
1
ca R 2
ensure that
r
p
√
√
1
(k)
≤ si(k) (~x ) ≤ 12 n/λ + 4 nR +
≤ 17 nR
ca λ
and Lemma 21 shows that |ei(k) | ≤ ce . Hence, we have
p
1
~e(~
τ (k+ 3 ) ,~
x(k) )
(~x(k) ) ≥ p~e(~τ (k) ,~x(k) ) (~x(k) ) − ce log(17nR/).
For the addition step, Lemma 17 shows that
p
2
~e(~
τ (k+ 3 ) ,~
x(k) )
(~x(k) ) = p
1
(~x(k) ) − ce ln s(~x)m+1 + ln(1 + ca )
≥ p
1
(~x(k) ) − ce log(17nR/) + ln(1 + ca )
~e(~
τ (k+ 3 ) ,~
x(k) )
~e(~
τ (k+ 3 ) ,~
x(k) )
and for the removal step, Lemma 18 and |ei | ≤ ce shows that
p
2
~e(~
τ (k+ 3 ) ,~
x(k) )
(~x(k) ) ≥ p
1
(~x(k) ) − [ce + eP (~τ , ~x)m ] ln s(~x)m + ln(1 − cd )
≥ p
1
(~x(k) ) − 2ce log(17nR/) + ln(1 − cd )
~e(~
τ (k+ 3 ) ,~
x(k) )
~e(~
τ (k+ 3 ) ,~
x(k) )
After the addition or removal of a constraint, Lemma 21 shows that
q
1
(k)
δ (k) (k+ 23 ) (k) (~x ) ≤
ce + min µ(~x(k) ), cd
P ,~e(~
τ
,~
x )
100
and therefore Lemma 14 and ce ≤ cd show that
q
2
ce + min µ(~x(k) ), cd
Ep~e(~τ (k+1) ,~x(k+1) ) (~x(k+1) ) ≥ p (k+ 23 ) (k) (~x(k) ) − 8
~e(~
τ
,~
x )
100
≥ p
2
~e(~
τ (k+ 3 ) ,~
x(k) )
Combining them with ce =
cd
6 ln(17nR/) ,
(~x(k) ) −
cd
.
625
we have
Ep~e(~τ (k+1) ,~x(k+1) ) (~x(k+1) ) ≥ p~e(~τ (k) ,~x(k) ) (~x(k) ) − 3ce log(17nR/) −
≥ p~e(~τ (k) ,~x(k) ) (~x(k) ) − cd + ln(1 + β)
where β = ca for the case of addition and β = −cd for the case of removal.
32
cd
+ ln(1 + β)
625
(6.18)
Theorem 26. For ca = 10110 , cd = 10112 , ce =
enough universal constant C, then we have
cd
6 ln(17nR/) ,
c∆ =
Cce
log(n)
Ep~e(~τ (k+1) ,~x(k+1) ) (~x(k+1) ) ≥ p~e(~τ (k) ,~x(k) ) (~x(k) ) −
and λ =
1
ca R 2
for some small
1
9β
+
1011 1011
where β = 1 for the case of addition and β = 0 for the case of removal.
Proof. It is easy to see that these parameters satisfy the requirements of Lemma 25.
6.5
Guarantees of the Algorithm
In this section we put everything together to prove Theorem 31, the main result of this section,
providing the guarantees of our cutting plane method.
cd
For the remainder of this section we assume that ca = 10110 , cd = 10112 , ce = 6 ln(17nR/)
,
c∆ =
Cce
log(n)
and λ =
1
.
ca R2
~x
2
Consequently, throughout the algorithm we have
p
p
√
√
√
≤ 6 n/λ + 2 nR = 6 ca nR2 + 2 nR ≤ 3 nR.
(6.19)
Lemma 27. If si (~x(k) ) < for some i and k during our Cutting Plane Method then
max
~
y ∈P (k) ∩B∞ (R)
h~ai , ~y i −
min
~
y ∈P (k) ∩B∞ (R)
h~ai , ~y i ≤
8n
.
ca ce
2
Proof. Let ~y ∈ P (k) ∩ B∞ (R) be arbitrary. Since ~y ∈ B∞ (R) clearly ~y 2 ≤ nR2 . Furthermore, by
Lemma 22 and the choice of parameters mce + n ≤ 3n. Consequently, by Lemma 19 and the fact
that λ = ca1R2 and ca < 1 we have
~x − ~y
≤
H(~
x)
12mce + 6n + 2λ ~y
p
ce + µ(~x)
2
2
30n + 2 cna
4n
≤p
≤ √
c
ce + µ(~x)
a ce
and therefore
S−1
(s(~x(k) ) − s(~y ))
x(k)
Consequently, we have (1 −
1
≤√
S−1
(s(~x(k) ) − s(~y ))
x(k)
ce
∞
4n
x(k) )
ca ce )si (~
≤ si (~y ) ≤ (1 +
4n
x(k) )
ca ce )si (~
ce I+Ψ
≤
4n
ca ce
.
for all ~y ∈ P (k) ∩ B∞ (R).
Now let us show how to compute a proof (or certificate) that the feasible region has small width
on the direction ~ai .
Lemma 28. Suppose that during some iteration k for i = arg minj sj (~x(k) ) we have si (~x(k) ) ≤ . Let
(~x∗ , ~τ∗ ) = Centering(~x(k) , ~τ (k) , 64 log(2R/), c∆ ) where ~τ (k) is the τ at that point in the algorithm
and let
X
ce + ej (~x∗ , ~τ∗ ) + ψj (~x∗ )
s(~x∗ )i
∗
~a =
tj~aj where tj =
.
s(~x∗ )j
ce + ei (~x∗ , ~τ∗ ) + ψi (~x∗ )
j6=i
Then, we have that ~ai + ~a∗
2
≤
√
8 n
ca ce R
O(n)
X
j6=i
and tj ≥ 0 for all j. Furthermore, we have
T
tj aj ~x∗ −
O(n)
X
j6=i
33
tj bj ≤
3n
s(~x∗ )i
ce
.
p
Proof. By Lemma 14 and Lemma 21 we know that ~e(~x∗ , ~τ∗ ) ≤ 21 ce and δ~e(~x∗ ,~τ∗ ) ≤ R ce + µ(~x∗ ).
Since ~e(~x∗ , ~τ∗ ) ≤ 12 ce , we have tj ≥ 0 for all j. Furthermore, by Lemma 20 and (6.19), we then
have that with high probability in n
"
#
r
2
mc
+
n
e
λ ~x∗ 2 + δe (~x∗ )
+λ
~ai + ~a∗ 2 ≤
(ce + µ(~x∗ ))
2
r
√
2
1
n
3n
≤
(3 nR) +
+
ce ca R2
R 2
ca R2
" √
#
√
√
2 3 n
3n
2 n
.
≤
+
+√
ce ca R
4
ca R
Hence, we have
∗
~ai + ~a
2
√
8 n
≤
.
ca ce R
By Lemma 21 we know that ~e(~x∗ , ~τ∗ ) ≤ 21 ce and hence
T
O(n)
O(n)
O(n)
O(n)
X
X
X
X ce + ej (~x∗ , ~τ∗ ) + ψj (~x∗ )
tj aj
~x∗ −
tj bj =
tj s(~x∗ )j = si (~x∗ )
ce + ei (~x∗ , ~τ∗ ) + ψi (~x∗ )
j6=i
j6=i
j6=i
j6=i
O(n)
≤ si (~x∗ )
X
j6=i
3
2 ce
1
2 ce
+ ψj (~x∗ )
+ ψi (~x∗ )
O(n)
!
≤ si (~x∗ )
X
j6=i
3mce + 2n
ce
≤
3n
si (~x∗ )
ce
Lemma 29. During our Cutting Plane Method, if p~e (~x(k) ) ≥ n log( cna )+ 6n
x(k) ) ≤
ca , then we have si (~
for some i.
Proof. Recall that
p~e (~x(k) ) = −
X
(ce + ei ) log si (~x(k) )i +
i∈[m]
√
Using ~x(k) ≤ 3 nR (6.19) and λ =
p~e (~x(k) ) ≤ −
X
1
,
ca R2
λ
1
~x(k)
log det AT S−2
A
+
λI
+
x(k)
2
2
we have
(ce + ei ) log(s(~x(k) )i ) +
i∈[m]
Next, we note that ei ∞ ≤ ce ≤
(Lemma 24). Hence, we have
2
.
2
1
12 ln(17nR/)
5n
1
log det AT S−2
A
+
λI
+
.
(k)
x
2
ca
q
p
√
√
and si ~x(k) ≤ 12 n/λ + 4 nR + ca1λ ≤ 6 nR
6n
1
log det AT S−2
A
+
λI
+
.
(k)
x
2
ca
1
T S−2 A + λI ≥ n log( n ). Using < R, we
Since p~e (~x(k) ) ≥ n log( cna ) + 6n
,
we
have
log
det
A
ca
2
ca
x(k)
p~e (~x(k) ) ≤
have that
n2
c2a 2
≥
n2
2
+ λ and hence
X
i
n
log λi AT S−2
A
+
λI
≥
n
log
+
λ
.
x
2
34
Therefore, we have log λmax AT S−2
x A + λI ≥ log
such that ~v AT S−2
v + λ~v T ~v ≥ n2 + λ. Thus,
x A~
n
2
+ λ . Hence, we have some unit vector ~v
X (A~v )2
n
i
≥ 2.
2
(k)
s(~x )i
i
(A~v )2i
s(~
x(k) )2i
(k)
s(~x )i ≤ .
Therefore there is some i such that
h~ai , ~v i2 ≥ s(~x(k) )2i /2 and hence
≥
1
.
2
Since ~ai and ~v are unit vectors, we have 1 ≥
Lemma 30. With constant probability, the algorithm ends in 1024 n log( nR
) iterations.
4
Proof. Theorem 26 shows that for all k
Ep~e(~τ (k+1) ,~x(k+1) ) (~x(k+1) ) ≥ p~e(~τ (k) ,~x(k) ) (~x(k) ) −
9β
1
+ 11
11
10
10
(6.20)
where β = 1 for the case of adding a constraint and β = 0 for the case of removing a constraint.
Now, for all t consider the random variable
Xt = p~e(~τ (t) ,~x(t) ) (~x(t) ) −
4.5m(t)
3.5t
− 11
11
10
10
where m(t) is the number of constraints in iteration t of the algorithm. Then, since m(t+1) =
m(t) − 1 + 2β, (6.20) shows that
1
9β
4.5m(t+1) 3.5(t + 1)
+
−
−
1011 1011
1011
1011
1
9β
4.5(−1 + 2β)
3.5
= Xt − 11 + 11 −
− 11 = Xt .
11
10
10
10
10
EXt+1 ≥ p~e(~τ (t) ,~x(t) ) (~x(t) ) −
Hence, it is a sub-martingale. Let τ be the iteration the algorithm throws out error or outputs
P (k) . Optional stopping theorem shows that
EXmin(τ,t) ≥ EX0 .
√
Using the diameter of P (0) is
nR, we have
λ
1
~0
ce log si (~0) + log det AT S−2
0 A + λI +
2
2
i∈[m(0) ]
√
n
1
≥ −ce m(0) log( nR) + log
2
ca R2
√
≥ − n + ce m(0) log( nR).
p~0 (~0) = −
Using ce =
cd
6 ln(17nR/) ,
cd =
X
1
1012
(6.21)
2
2
and m(0) = 2n, we have
√
4.5m(0)
X0 ≥ − n + ce m(0) log( nR) −
1011
√
≥ −n log( nR) − 100n.
4
We have made no effort on improving this constant and we believe it can be improved to less than 300 using
techniques in [5, 6].
35
Therefore, (6.21) shows that for all t we have
− n(log(nR) + 100) ≤ EXmin(τ,t)
= pE Xmin(τ,t) |τ < t + (1 − p)E Xmin(τ,t) |τ ≥ t
(6.22)
def
where p = P(τ < t).
Note that
h
i 4.5m(t)
3.5t
E Xmin(τ,t) |τ ≥ t ≤ E p~e(~τ (t) ,~x(t) ) (~x(t) )|τ ≥ t −
− 11 .
1011
10
h
i 3.5t
≤ E p~e(~τ (t) ,~x(t) ) (~x(t) )|τ ≥ t − 11 .
10
Furthermore, by Lemma 29 we know that when p~e(~τ (t) ,~x(t) ) (~x(t) ) ≥ n log( cna ) +
that is too small and the algorithm terminates. Hence, we have
6n
ca ,
there is a slack
n
6n
3.5t
E Xmin(τ,t) |τ ≥ t ≤ n log(
)+
− 11 .
ca
ca
10
The proof of Lemma 21 shows that the function value does not change by more than 1 in one
iteration by changing ~x and can change by at most mce log( 3nR
) by changing τ . Since by Lemma 22
cd
2n
we know that m ≤ 1 + ca and ce = 6 ln(17nR/) , we have that p~e (~x) ≤ n log( cna ) + 7n
ca throughout
the execution of the algorithm. Therefore, we have
n
7n
E Xmin(τ,t) |τ ≤ t ≤ Eτ <t p~e(~τ (t) ,~x(t) ) (~x(t) ) ≤ n log(
)+
.
ca
ca
Therefore, (6.22) shows that
−n(log(nR) + 100) ≤ n log
n
ca
+
7n
3.5t
− (1 − p) 11 .
ca
10
Hence, we have
3.5t
(1 − p) 11
10
n
7n
≤ n log
+
+ n(log(nR) + 100)
ca
ca
2
Rn
7n
+ 100n
≤ n log
+
ca
ca
2
Rn
= n log
+ 8 · 1010 n.
ca
Thus, we have
1
P(τ < t) = p ≥ 1 −
t
2
n R
11
22
10 n log
+ 10 n .
Now, we gather all the result as follows:
Theorem 31 (Our Cutting Plane Method). Let K ⊆ Rn be a non-empty set contained in a box
of radius R, i.e. K ⊆ B∞ (R). For any ∈ (0, R) in expected time O(nSOΩ(/√n) (K) log(nR/) +
n3 logO(1) (nR/)) our cutting plane method either outputs ~x ∈ K or finds a polytope P = {~x :
A~x ≥ ~b} ⊇ K such that
36
1. P has O(n) many constraints (i.e. A ∈ RO(n)×n and ~b ∈ RO(n) ).
2. Each constraint of P is either an initial constraint from B∞ (R) or of the form h~a, ~xi ≥ b − δ
where h~a, ~xi ≥ b is a normalized hyperplane (i.e. ~a 2 = 1) returned by the separation oracle
and δ = Ω √n .
3. The polytope P has small width with respect to some direction ~a1 given by one of the constraints, i.e.
max
h~a1 , ~y i −
min
h~a1 , ~y i ≤ O (n ln(R/))
~
y ∈P ∩B∞ (R)
~
y ∈P ∩B∞ (R)
4. Furthermore, the algorithm produces a proof of the fact above involving convex combination
of the constraints, namely, non-negatives t2 , ..., tO(n) and ~x ∈ P such that
√
≤ 3 nR,
PO(n)
(b) ~a1 + i=2 ti~ai
(a)
~x
2
2
=O
√
R n log(R/)
,
(c) ~aT1 ~x − ~b1 ≤ ,
P
T
PO(n)
O(n)
(d)
t
a
~x − i=2 ti bi ≤ O(n log(R/))
i=2 i i
.
Proof. Our algorithm either finds ~x ∈ K or we have si (~x(k) ) < . When si (~x(k) ) < , we apply
PO(n)
Lemma 28 to construct the polytope P and the linear combination i=2 ti~ai .
Notice that each iteration of our algorithm needs to solve constant number of linear systems
~ x(k) ) − ψ(~
~ x(k−1) ). Theorem 33
~ (k) ∈ Rn s.t. E[∆
~ (k) ] = ψ(~
and implements the sampling step to find ∆
shows how to do the sampling in Õ(1) many linear systems. Hence, in total, each iterations needs
to solve Õ(1) many linear systems plus nearly linear work. To output the proof for (4), we use
Lemma 28.
Note that the linear systems the whole algorithm need to solve is of the form
−1
(AT S−2
x = ~y .
x A + λI) ~
T
where the matrix AT S−2
x A + λI can be written as A DA for the matrix A = [A I] and diagonal
matrix
−2
S
0
D=
.
0 λI
−1 (k+1)
1
S(k)
Note that Lemma 14 shows that
(~s
− ~s(k) ) 2 ≤ 10
for the k th and (k + 1)th linear
−1
1
systems we solved in the algorithm. Hence, we have
D(k)
(d~(k+1) − d~(k) ) 2 ≤ 10
. In [76],
2
they showed how to solve such sequence of systems in Õ(n ) amortized cost. Moreover, since our
algorithm always changes the constraints by δ amount where δ = Ω( √n ) an inexact separation
oracle SOΩ(/√n) suffices. (see Def 1). Consequently, the total work O(nSOΩ(/√n) (K) log(nR/) +
n3 logO(1) (nR/)). Note that as the running time holds with only constant probability, we can
restart the algorithm whenever the running time is too large.
To prove (2), we note that from the algorithm description, we know the constraints are either
from B∞ (R) or of the form ~aT ~x ≥ ~aT ~x(k) − δ where
s
~aT (AT S~−2
A + λI)−1~a
x(k)
δ=
.
ca
37
n
From the proof of Lemma 29, we know that if λmax (AT S−2
x A + λI) ≥ 2 , then there is si < .
2
−1
a is a unit vector, we have
Hence, we have λmin ((AT S−2
x A + λI) ) ≤ n . Since ~
s
s
−1~
~aT (AT S~−2
A
+
λI)
a
2
(k)
x
≥
.
ca
nca
7
Technical Tools
In this section we provide stand-alone technical tools we use in our cutting plane method in Section 6. In Section 7.1 we show how to efficiently compute accurate estimates of changes in leverage
scores using access to a linear system solver. In Section 7.2 we study what we call the “Stochastic
Chasing ~0 Game” and show how to maintain that a vector is small in `∞ norm by making small
coordinate updates while the vector changes randomly in `2 .
7.1
Estimating Changes in Leverage Scores
In previous sections, we needed to compute leverage scores accurately and efficiently for use in our
cutting plane method. Note that the leverage score definition we used was
√
−1 T √
ψ(w)
~ i = ~1Ti WA AT WA + λI
A W~1i
for some λ > 0 which is different from the standard definition
√
−1 T √
σ(w)
~ i = ~1Ti WA AT WA
A W~1i .
T
However, note that the matrix AT WA + λI can be written as A DA for the matrix A = [A I]
and diagonal matrix
W 0
D=
.
0 λI
and therefore computing ψ is essentially strictly easier than computing typical leverage scores.
Consequently, we use the standard definition σ to simplify notation.
In [99], Spielman and Srivastava observed that leverage scores can be written as the norm of
certain vectors
√
2
−1 T √
σ(w)
~ i=
WA AT WA
A W~1i
2
and therefore leverage scores can be approximated efficiently using dimension reduction. Unfortunately, the error incurred by this approximation is too large to use inside the cutting point method.
In this section, we show how to efficiently approximate the change of leverage score more accurately.
In particular, we show how to approximate σ(w)
~ − σ(~v ) for any given w,
~ ~v with log(w)
~ −
log(~v ) 2 1. Our algorithm breaks σ(w)
~ i − σ(~v )i into the sum of the norm of small vectors and
then uses the Johnson-Lindenstrauss dimension reduction to approximate the norm of each vector
separately. Our algorithm makes use of the following version of Johnson-Lindenstrauss.
Lemma 32 ([1]). Let 0 ≤ ≤ 12 and let ~x1 , ..., ~xm ∈ Rn be arbitrary m points. For k =
O(−2 log(m)) let Q be a k × n random matrix with each entry sampled from {− √1k , √1k } uniformly and independently. Then, E kQ~xi k2 = ~xi
we have that for all i ∈ [m]
(1 − ) ~xi
2
2
for all i ∈ [m] and with high probability in m
≤ kQ~xi k2 ≤ (1 + ) ~xi
38
2
.
Algorithm 3: b
h = LeverageChange(A, ~v , w,
~ , α)
m×n
m
Input: A ∈ R
, ~v , w
~ ∈ R>0 , ∈ (0, 0.5).
1
~ 2 ≤ 10
Given: V−1 (~v − w)
and AT VA and AT WA are invertible.
−2
Sample Qd ∈ √
RO( log(m))×n as in Lemma 32.
−1 T ~ 2
Let dˆi = Qd WA AT WA
A 1i 2 for all i ∈ [n].
−1
Let t = O log( ) .
−2
Sample Qf ∈ RO( log(mt))×n as in Lemma 32.
Pick positive integer u randomly such that Pr[u = i] = ( 21 )i .
for j ∈ {1, 2, · · · , t} ∪ {t + u} do
if j is even then
√
−1 T
−1 2j T
(j)
A ~1i
Let fˆi = Qf VA AT VA
A (V − W) A AT VA
2
.
2
else
def
Let ∆+ = (V − W)+ , i.e. the matrix V − W with negative entries set to 0.
def
Let ∆− = (W − V)+ , i.e. the matrix W − V with negative entries set to 0.
√
−1 j−1
2
(j)
Let α̂i = Qf ∆+ A(AT VA)−1 (AT (V − W) A AT VA ) 2 AT ~1i 2 .
√
j−1
2
(j)
Let β̂i = Qf ∆− A(AT VA)−1 (AT (V − W)A(AT VA)−1 ) 2 AT ~1i 2 .
(j)
(j)
(j)
Let fˆ = α̂ − β̂ .
i
i
i
end
end
P
(t+u)
Let fˆi = 2u fˆ
+ t
i
ˆ(j)
j=1 fi .
Output: ĥi = (wi − vi )dˆi + vi fˆi . for all i ∈ [m]
def
1
−1 v − w)
~ 2 ≤ 10
and both
Theorem 33. Let A ∈ Rm×n and ~v , w
~ ∈ Rm
>0 be such that α = V (~
T
T
A VA and A WA are invertible. For any ∈ (0, 0.5), Algorithm 3 generates a random variable b
h
b
such that Eĥ = σ(w)
~ − σ(~v ) and with high probability in m, we have kh − (σ(w)
~ − σ(~v )) k2 ≤ O (α).
e
Furthermore, the expected running time is O((nnz(A)
+ LO)/2 ) where LO is the amount of time
−1
−1
needed to apply AT VA
and AT WA
to a vector.
(j)
(j)
(j)
Proof. First we bound the running time. To compute dˆi , fˆi , α̂i , β̂i , we simply perform matrix
multiplications from the left and then consider the dot products with each of the rows of A. Naively
e + u)2 log(mt)(nnz(A) + LO)). However, we can reuse the computation
this would take time O((t
e + u) log(mt)(nnz(A) + LO)). Now since E[u]
in computing high powers of j to only take time O((t
is constant we see that the total running time is as desired. It only remains to prove the desired
properties of ĥ.
First we note that we can re-write leverage score differences using
h
h
−1 T i
−1
−1 T i
σ(w)
~ i − σ(~v )i = (wi − vi ) A AT WA
A
− AT VA
+ vi A AT WA
A
.
ii
ii
Consequently, for all i ∈ [m], if we let
di
fi
−1 T
= ~1Ti A AT WA
A ~1i ,
h
−1
−1 i T
def
= ~1Ti A AT WA
− AT VA
A ~1i .
def
then
σ(w)
~ i − σ(~v )i = (wi − vi )di + (vi )fi
39
.
(7.1)
We show that dˆi approximates d and fˆi approximate fˆ well enough to satisfy the statements in the
Theorem.
√
−1 T
2
First we bound the quality of dˆi . Note that di =
WA AT WA
A ~1i 2 . Consequently,
Lemma 32 shows that E[dˆi ] = di and that with high probability in m we have (1−)di ≤ dˆi ≤ (1+)di
for all i ∈ [m]. Therefore, with high probability in m, we have
(w
~ − ~v ) db − (w
~ − ~v ) d~
2
2
=
2
X
(wi − vi )2 d2i
(wi − vi )2 dbi − di ≤ 2
X
i∈[m]
i∈[m]
2
=
X
2
(wi − vi )
i∈[m]
σ(w)
~ i
w
~i
2
X wi − vi 2
≤ 2
vi
2
.
i∈[m]
−1/2 T
−1/2
def
Next we show how to estimate f . Let X = AT VA
A (V − W) A AT VA
. By the
assumption on α we know − 12 V ≺ V − W ≺ 21 V and therefore − 12 I ≺ X ≺ 12 I. Consequently we
have that
−1
−1/2
−1/2
AT WA
= AT VA
(I − X)−1 AT VA
∞
X
−1/2 j
−1/2
=
AT VA
X AT VA
.
j=0
and therefore
fi = ~1Ti A
∞
X
AT VA
−1/2
Xj AT VA
−1/2
− AT VA
−1
AT ~1i
j=0
=
∞
X
(j)
fi
where
(j)
fi
def
= ~1Ti A AT VA
−1/2
Xj AT VA
−1/2
AT ~1i
j=1
Furthermore, using the definition of X we have that for even j
(j)
fi
j
X 2 AT VA
=
T
=
A VA
−1/2
−1/2
AT ~1i
2
2
T
T
A (V − W) A A VA
−1 2j
2
T~
A 1i
2
√
=
VA AT VA
−1
AT (V − W) A AT VA
−1 2j
2
AT ~1i
2
For odd j, using our definition of ∆+ and ∆− we have that
(j)
fi
−1/2 j
−1/2 T
= ~1Ti A AT VA
X AT VA
A ~1i
j−1
−1 T
−1 T
2
= ~1Ti A AT VA
A (V − W) A
AT VA
A (V − W)
−1 T
−1 j−1
2
A (V − W) A AT VA
×A AT VA
AT ~1i
(j)
(j)
= αi − β i
40
.
where
(j)
√
def
=
αi
(j)
√
def
=
βi
∆+ A AT VA
∆− A AT VA
−1
−1
AT (W − V) A AT VA
AT (W − V) A AT VA
−1 j−1
2
−1 j−1
2
2
AT ~1i
,
2
2
AT ~1i
.
2
Consequently, by Lemma 32 and the construction, we see that
t
∞
∞
u
X
X
X
2
(j)
(j)
ˆ(t+u) =
f
Efˆi = E
fˆi +
fi = fi
2u i
u=1
j=1
j=1
and therefore Eĥ = σ(w)
~ − σ(~v ) as desired. All that remains is to bound the variance of fˆi .
−1/2 T
−1/2
To bound the variance of fˆ, let |X| = AT VA
A |W − V| A AT VA
. Note that
− 41 I − |X| X |X| 14 I and consequently for all j
−1/2
−1/2 T
= ~1Ti A AT VA
|X|j AT VA
A ~1i
−1/2
−1/2 T
1
|X| AT VA
A ~1i
≤ j−1 ~1Ti A AT VA
4
1 ~T
def
1 Pv ∆Pv~1i
=
vi 4j−1 i
(j)
def
gi
√
−1/2 T √
where Pv = VA AT VA
A V and ∆ is a diagonal matrix with ∆ii =
that 0 Pv I, we have that for all j
(4j−1 )2
m
X
(j) 2
vi gi
wi −vi
vi
. Using
m
2
X
~1Ti Pv ∆Pv~1i = Tr (Pv ∆Pv Pv ∆Pv )
=
i=1
i=1
≤ Tr (Pv ∆∆Pv ) = Tr (∆Pv Pv ∆)
m
X
wi − v i 2
2
≤ α2
≤ Tr ∆ =
vi
i=1
and thus V~g (j)
(j)
αi
(j)
≤ gi
2
Furthermore, since ∆+ |W − V| and ∆− |W − V| we have that
≤
4α
.
4j
(j)
≤ gi . Consequently, by Lemma 32 again, we have
and βi
(j)
Vfˆ(j) − Vf~(j)
2
2
=
(j)
(j) 2
vi2 fˆi − fi
X
i
≤ 2
X
X (j)
(j) 2
(j)
(j) 2
vi2 α̂i − αi
+2
vi2 β̂i − βi
i
i
≤ 22
X
≤ 22
X
vi2
(j)
αi
2
(j)
+ βi
2
i
(j) 2
vi gi
i
41
≤
2α2 2
(4j−1 )2
.
Putting this all together we have that
Vfˆ − Vf~
≤ 2 Vfˆ(t+u) +
u
2
t
X
Vfˆ(j) −
j=1
≤2
u
Vfˆ(t+u)
2
+
∞
X
Vf~(j)
2
j=1
t
X
Vfˆ(j) − Vf~(j)
j=1
2
+
∞
X
Vf~(j)
2
j=t+1
t √
∞
X
X
4α
2α
4α
≤ 2 t+u +
+
j−1
4
4
4j
j=1
j=t+1
α
= O α + t .
4
u
Consequently, since t = O(log(−1 )) we have the desired result.
7.2
The Stochastic Chasing ~0 Game
To avoid computing leverage scores exactly, in Section 7.1 we showed how to estimate the difference
of leverage scores and use these to update the leverage scores. However, if we only applied this
technique the error of leverage scores would accumulate in the algorithm and we need to fix it.
Naturally, one may wish to use dimension reduction to compute a multiplicative approximation to
the leverage scores and update our computed value if the error is too large. However, this strategy
would fail if there are too many rows with inaccurate leverage scores in the same iteration. In
this case, we would change the central point too much that we are not able to recover. In this
section, we present this update problem in a general form that we call Stochastic Chasing 0 game
and provide an effective strategy for playing this game.
The Stochastic chasing 0 game is as follows. There is a player, a stochastic adversary, and a
point ~x ∈ Rm . The goal of the player is to keep the point close to ~0 ∈ Rm in `∞ norm and the goal
of the stochastic adversary is to move ~x away from ~0. The game proceeds for an infinite number
of iterations where in each iteration the stochastic adversary moves the current point ~x(k) ∈ Rm to
~ (k) ∈ Rm and the player needs to respond. The stochastic adversary cannot
some new point ~x(k) + ∆
(k)
~
move the ∆
arbitrarily, instead he is only allowed to choose a probability distribution D(k) and
~ 2 ≤ c for some fixed c
~ (k) from it. Furthermore, it is required that E (k) ∆
~ = ~0 and ∆
sample ∆
D
2
~ ∈ D(k) . The player does not know ~x(k) or the distribution D(k) or the move ∆
~ (k) of the
and all ∆
(k)
n
(k)
stochastic adversary. All the player knows is some ~y ∈ R that is close to ~x in `∞ norm. With
(k+1)
this information, the player is allowed to choose one coordinate i and set xi
to be zero and for
(k+1)
(k)
(k)
other j, we have xj
= xj + ∆ j .
The question we would like to address is, what strategy the player should choose to keep ~x(k)
close to ~0 in `∞ norm? We show that there is a trivial strategy that performs well: simply pick the
42
largest coordinate and set it to 0.
Algorithm 4: Stochastic chasing ~0 game
Constant: c > 0, R > 0.
Let ~x(1) = ~0 ∈ Rm .
for k = 1 to ∞ do
~ = ~0 and ∆
~
~ ∈ D(k) .
Stochastic Adversary: Pick D(k) such that ED(k) ∆
≤ c all ∆
2
Stochastic Adversary: Pick ~y (k) ∈ Rm such that ~y (k) − ~x(k) ∞ ≤ R.
Player: Pick a coordinate i(k) using only ~y (k) .
~ (k) from D(k) .
Sample ∆
(k+1)
(k+1)
(k)
(k)
Set xi(k) = 0 and xj
= xj + ∆j for all j 6= i.
end
(k)
Theorem 34. Using the strategy i(k) = arg maxi yi
~x(k)
∞
, with probability at least 1 − p, we have
≤ 2(c + R) log 4mk 2 /p
for all k in the Stochastic Chasing ~0 Game.
P
P
Proof. Consider the potential function Φ(~x) = i eαxi + i e−αxi where α is to be determined.
2
Now for all x we know that ex ≤ 1 + x + x2 e|x| and therefore for all |δ| ≤ c, x and α, we have
1
eαx+αδ ≤ eαx + αδeαx + α2 δ 2 eαx+|α|c
2
.
Consequently,
~ ≤ Φ(~x(k) ) + αE ~ (k)
E∆∈D
x(k) + ∆)
(k) Φ(~
~
∆∈D
X
e
(k)
αxi
∆i −
i∈[m]
α2 α
e
+
2
~
~ = ~0 and ∆
Since ED(k) ∆
E∆∈D
(k)
~
X
(k)
αxi
e
∆2i
i
2
+
(k)
−αxi
e
e
(k)
−αxi
∆i
i∈[m]
~
∆
∞
E∆∈D
(k)
~
X
e
(k)
αxi
P
(k)
!
∆2i
≤ E∆∈D
(k)
~
i
.
∆2i
(k)
−αxi ∆
e
= 0 and
i
i
!
(k)
αxi
max e
i
(k)
(k)
+ max e
−αxi
i
(k)
≤ c2 max eαxi + max e−αxi
i
∆2i
P
i
e
i∈[m]
αxi ∆ −
i
ie
X
(k)
−αxi
X
∆2i +
i∈[m]
≤ c, we have E∆∈D
(k)
~
X
X
i
.
(k)
(k)
Letting η (k) = max maxi eαxi , maxi e−αxi , we then have
~ ≤ Φ(~x(k) ) + α2 eαc c2 η (k) .
x(k) + ∆)
E∆∈D
(k) Φ(~
~
(k)
Since i(k) = arg maxi yi
and ~y (k) − ~x(k)
(k+1)
∞
≤ R, the player setting xi(k)
at least e−α(R+c) η (k) . Hence, we have
E∆∈D
x(k+1) ) ≤ Φ(~x(k) ) + α2 eαc c2 η (k) − e−αR η (k) .
(k) Φ(~
~
43
= 0 decreases Φ by
Picking α =
have that
1
2(c+R) ,
we have e2α(c+R) (α(c + R))2 ≤ 1 and hence α2 eαc c2 ≤ e−α(R+c) . Therefore, we
E∆∈D
x(k+1) ) ≤ EΦ(~x(k) ) ≤ ... ≤ Φ(~x(1) ) = 2m .
(k) Φ(~
~
Consequently, by Markov’s inequality we have that Pr[Φ(~x(k) ) ≥ λk ] ≤ 2m
λk for any λk . Furthermore,
since clearly Φ(~x) ≥ eαk~xk∞ we have that Pr[k~x(k) k∞ ≥ log(λk )/α] ≤ 2m
λk for all k. Choosing
λk =
4mk2
p
and taking a union bound over all k, we have that
~x(k)
∞
≤ 2(c + R) log 4mk 2 /p
for all k with probability at least
1−
∞
X
2m
i=1
λk
=1−
∞
X
p
≥1−p
2k 2
k=1
44
.
Part II
A User’s Guide to Cutting Plane Methods
8
Introduction
Cutting plane methods have long been employed to obtain polynomial time algorithms for solving
optimization problems. However, for many problems cutting plane methods are often regarded
as inefficient both in theory and in practice. Here, in Part II we provide several techniques for
applying cutting plane methods efficiently. Moreover, we illustrate the efficacy and versatility of
these techniques by applying them to achieve improved running times for solving multiple problems
including semidefinite programming, matroid intersection, and submodular flow.
We hope these results revive interest in ellipsoid and cutting plane methods. We believe these
results demonstrate how cutting plan methods are often useful not just for showing that a problem
is solvable in polynomial time, but in many yield substantial running time improvements. We stress
that while some results in Part II are problem-specific, the techniques introduced here are quite
general and are applicable to a wide range of problems.
In the remainder of this introduction we survey the key techniques we use to apply our cutting
plane method (Section 8.1) and the key results we obtain on improving the running time for solving
various optimization problems (Section 8.2). We conclude in Section 8.3 by providing an overview
of where to find additional technical result in Part II.
8.1
Techniques
Although cutting plane methods are typically introduced as algorithms for finding a point in a
convex set (as we did with the feasibility problem in Part I), this is often not the easiest way
to apply the methods. Moreover, improperly applying results on the feasibility problem to solve
convex optimization problems can lead to vastly sub-optimal running times. Our central goal, here,
in Part II is to provide tools that allow cutting plane methods to be efficiently applied to solve
complex optimization problems. Some of these tools are new and some are extensions of previously
known techniques. Here we briefly survey the techniques we cover in Section 10 and Section 11.
Technique 0: From Feasibility to Optimization
In Section 10.1, we explain how to use our cutting plane method to solve convex optimization
problems using an approximate subgradient oracle. Our result is based on a result of Nemirovski [85]
in which he showed how to use a cutting plane method to solve convex optimization problems
without smoothness assumptions on the function and with minimal assumptions on the size of the
function’s domain. We generalize his proof to accommodate for an approximate separation oracle,
an extension which is essential for our applications. We use this result as the starting point for two
new techniques we discuss below.
Technique 1: Dimension Reduction through Duality
In Section 10.2, we discuss how cutting plane methods can be applied to obtain both primal and
dual solutions to convex optimization problems. Moreover, we show how this can be achieved while
only applying the cutting plane method in the space, primal or dual, which has a fewer number of
variables. Thus we show how to use duality to improve the convergence of cutting plane methods
while still solving the original problem.
45
To illustrate this idea consider the following very simple linear program (LP)
n
X
min
P
xi ≥0,
xi =1
wi x i
i=1
where ~x ∈ Rn and w
~ ∈ Rn . Although this LP has n variables, it should to be easy to solve purely
on the grounds that it only has one equality constraint and thus dual linear program is simply
max y ,
y≤wi ∀i
i.e. a LP with only one variable. Consequently, we can apply our cutting plane method to solve it
efficiently.
However, while this simple example demonstrates how we can use duality to decrease dimensions, it is not always obvious how to recover the optimal primal solution x variable given the
optimal dual solution y. Indeed, for many problems their dual is significantly simpler than itself
(primal), so some work is required to show that working in the space suffices to require a primal
solution.
One such recent example of this approach proving successful is a recent linear programming
result [75]. In this result, the authors show how to take advantage of this observation and get a
faster LP solver and maximum flow algorithm. It is interesting to study how far this technique can
extend, that is, in what settings can one recover the solution to a more difficult dual problem from
the solution to its easier primal problem?
There is in fact another precedent for such an approach. Grötschel, Lovász and Schrijver[50]
showed how to obtain the primal solution for linear program by using a cutting plane method to
solve the linear program exactly. This is based on the observation that cutting plane methods are
able to find the active constraints of the optimal solution and hence one can take dual of the linear
program to get the dual solution. This idea was further extended in [69] which also observed that
cutting plane methods are incrementally building up a LP relaxation of the optimization problem.
Hence, one can find a dual solution by taking the dual of that relaxation.
In Section 10.2, we provide a fairly general technique to recover a dual optimal solution from
an approximately optimal primal solution. Unfortunately, the performance of this technique seems
quite problem-dependent. We therefore only analyze this technique for semidefinite programming
(SDP), a classic and popular convex optimization problem. As a result, we obtain a faster SDP
solver in both the primal and dual formulations of the problem.
Technique 2: Using Optimization Oracles Directly
In the seminal works of Grötschel, Lovász, Schrijver and independently Karp and Papadimitriou
[49, 64], they showed the equivalence between optimization oracles and separation oracles, and gave
a general method to construct a separation oracle for a convex set given an optimization oracle for
that set, that is an oracle for minimizing linear functionals over the set. This seminal result led
to the first weakly polynomial time algorithm for many algorithms such as submodular function
minimization. Since then, this idea has been used extensively in various settings [62, 16, 17, 23].
Unfortunately, while this equivalence of separation and optimization is a beautiful and powerful
tool for polynomial time solvability of problems, in many case it may lead to inefficient algorithms.
In order to use this reduction to get a separation oracle, the optimization oracle may need to
be called multiple times – essentially the number of times needed to run a cutting plane method
and hence may be detrimental to obtaining small asymptotic running times. Therefore, it is an
interesting question of whether there is a way of using an optimization oracle more directly.
46
In Section 11 we provide a partial answer to this question for the case of a broad class of
problems, that we call the intersection problem. For these problems we demonstrate how to achieve
running time improvements by using optimization oracles directly. The problem we consider is
as follows. We wish to solve the problem for some cost vector ~c ∈ Rn and convex set K. We
assume that the convex set K can be decomposed as K = K1 ∩ K2 such that max~x∈K1 h~c, ~xi and
max~x∈K2 h~c, ~xi can each be solved efficiently. Our goal is to obtain a running time for this problem
comparable to that of minimizing K given only a separation oracle for it.
We show that by considering a carefully regularized variant, we obtain a problem such that
optimization oracles for K1 and K2 immediately yield a separation oracle for this regularized
problem. By analyzing the regularizer and bounding the domains of the problem we are able to
show that this allows us to efficiently compute highly accurate solutions to the intersection problem
by applying our cutting plane method once. In other words, we do not need to use a complicated
iterative scheme or directly invoke the equivalence between separation and optimization and thereby
save O(poly(n)) factors in our running times.
We note that this intersection problem can be viewed as a generalization of the matroid intersection problem and in Section 11.2, we show our reduction gives a faster algorithm in certain
parameter regimes. As another example, in Section 11.3 we show our reduction gives a substantial
polynomial improvement for the submodular flow problem. Furthermore, in Section 11.4 we show
how our techniques allow us to minimize a linear function over the intersection of a convex set and
an affine subspace in a number of iterations that depends only on the co-dimension of the affine
space.
8.2
Applications
Our main goal in Part II is to provide general techniques for efficiently using cutting plane methods
for various problems. Hence, in Part II we use minimally problem-specific techniques to achieve
the best possible running time. However, we also demonstrate the efficacy of our approach by
showing how techniques improve upon the previous best known running times for solve several
classic problems in combinatorial and continuous optimization. Here we provide a brief overview
of these applications, previous work on these problems, and our results.
In order to avoid deviating from our main discussion, our coverage of previous methods and
techniques is brief. Given the large body of prior works on SDP, matroid intersection and submodular flow, it would be impossible to have an in-depth discussion on all of them. Therefore, this
section focuses on running time comparisons and explanations of relevant preivous techniques.
Semidefinite Programming
In Section 10.2 we consider the classic semidefinite programming (SDP) problem:
max C • X s.t. Ai • X = bi (primal)
X0
min ~bT ~y s.t.
~
y
n
X
yi Ai C (dual)
i=1
def
where X, C, Ai are m × m symmetric matrices, ~b, ~y ∈ Rn , and A • B = Tr(AT B). For many
problems, n m2 and hence the dual problem has fewer variables than the primal. There are
many results and applications of SDP; see [106, 101, 83] for a survey on this topic. Since our focus
is on polynomial time algorithms, we do not discuss pseudo-polynomial algorithms such as the
spectral bundle method [51], multiplicative weight update methods [8, 9, 61, 3], etc.
47
Authors
Years
Nesterov, Nemirovsky[89]
Anstreicher [7]
Krishnan, Mitchell [70]
This paper
1992
2000
2003
2015
√
Running times
Õ( n(nmω + nω−1 m2 ))
Õ((mn)1/4 (nmω + nω−1 m2 ))
Õ(m(nω + mω + S)) (dual SDP)
Õ(m(n2 + mω + S))
Table 8: Previous algorithms for solving a n × n SDP with m constraints and S non-zeros entries
Currently, there are two competing approaches for solving SDP problems, namely interior point
methods (IPM) and cutting plane methods. Typically, IPMs require fewer iterations than the cutting plane methods, however each iteration of these methods is more complicated and possibly more
computationally expensive. For SDP problems,
Pn interior point methods require the computations of
the Hessian of the function − log det (C − i=1 yi Ai ) whereas cutting
plane methods usually only
P
need to compute minimum eigenvectors of the slack matrix C − ni=1 yi Ai .
In [7], Anstreicher provided the current fastest IPM for solving the dual SDP problem using
a method based on the volumetric barrier function. This method takes O((mn)1/4 ) iterations
and each iteration is as cheap as usual IPMs. For general matrices C, X, Ai , each iteration takes
O(nmω + nω−1 m2 ) time where ω is the fast matrix multiplication exponent. If the constraint
matrices Ai are rank one matrices, the iteration cost can be improved to O(mω + nm2 + n2 m)
[71]. If the matrices are sparse, then [40, 84] show how to use matrix completion inside the IPM.
However, the running time depends on the extended sparsity patterns which can be much larger
than the total number of non-zeros.
In [70], Krishnan and Mitchell
Pn observed that the separation oracle for dual SDP takes only
ω
O(m + S) time, where S =
i=1 nnz(Ai ) be the total number of non-zeros in the constant
matrix. Hence, the cutting plane method by [105] gives a faster algorithm for SDP for many
regimes. For ω = 2.38,
P the cutting plane method is faster when Ai is not rank 1 and the problem is
not too dense, i.e. ni=1 nnz(Ai ) < n0.63 m2.25 . While there are previous methods for using cutting
plane methods to obtain primal solutions[69] , to the best of our knowledge, there are no worst
case running time analysis for these techniques.
In Section 10.2, show how to alleviate this issue. We provide an improved algorithm for finding
the dual solution and prove carefully how to obtain a comparable primal solution as well. See
Figure 9.1 for a summary of the algorithms for SDP and their running times.
Matroid Intersection
In Section 11.2 we show how our optimization oracle technique can be used to improve upon the
previous best known running times for matroid intersection. Matroid intersection is one of the most
fundamental problems in combinatorial optimization. The first algorithm for matroid intersection
is due to the seminal paper by Edmonds [26]. In Figures 9.2 and 9.3 we provide a summary
of the previous algorithms for unweighted and weighted matroid intersection as well as the new
running times we obtain in this paper. While there is no total ordering on the running times of
these algorithms due to the different dependence on various parameters, we would like to point
out that our algorithms outperform the previous ones in regimes where r is close to n and/or the
oracle query costs are relatively expensive. In particular, in terms of oracle query complexity our
algorithms are the first to achieve the quadratic bounds of Õ(n2 ) and Õ(nr) for independence and
rank oracles. We hope our work will revive the interest in the problem of which progress has been
mostly stagnated for the past 20-30 years.
48
Authors
Edmonds [26]
Aigner, Dowling [2]
Tomizawa, Iri [102]
Lawler [72]
Edmonds [28]
Cunningham [21]
Years
1968
1971
1974
1975
1979
1986
This paper
2015
Running times
not stated
O(nr2 Tind )
not stated
O(nr2 Tind )
not stated
O(nr1.5 Tind )
2
O(n log nTind + n3 logO(1) n)
O(nr log2 nTrank + n3 logO(1) n)
Table 9: Previous algorithms for (unweighted) matroid intersection. Here n is the size of the ground
set, r = max{r1 , r2 } is the maximum rank of the two matroids, Tind is the time needed to check if
a set is independent (independence oracle), and Trank is the time needed to compute the rank of a
given set (rank oracle).
Authors
Edmonds [26]
Tomizawa, Iri [102]
Lawler [72]
Edmonds [28]
Frank [33]
Orlin, Ahuja [91]
Brezovec, Cornuéjols, Glover[14]
Fujishige, Zhang [39]
Shigeno, Iwata [96]
Years
1968
1974
1975
1979
1981
1983
1986
1995
1995
This paper
2015
Running times
not stated
not stated
O(nr2 Tind + nr3 )
not stated
2
O(n r(Tcircuit + n))
not stated
O(nr(Tcircuit + r + log n))
O(n2 r0.5 log rM · Tind )
O((n + Tcircuit )nr0.5 log rM )
O((n2 log nTind + n3 logO(1) n) log nM )
O((nr log2 nTrank + n3 logO(1) n) log nM )
Table 10: Previous algorithms for weighted matroid intersection. In additions to the notations used
in the unweighted table, Tcircuit is the time needed to find a fundamental circuit and M is the bit
complexity of the weights.
49
Minimum-Cost Submodular Flow
In Section 11.3 we show how our optimization oracle technique can be used to improve upon the
previous best known running times for (Minimum-cost) Submodular Flow. Submodular flow is
a very general problem in combinatorial optimization which generalizes many problems such as
minimum cost flow, the graph orientation, polymatroid intersection, directed cut covering [37]. In
Figure 9.4 we provide an overview of the previous algorithms for submodular flow as well as the
new running times we obtain in this paper.
Many of the running times are in terms of a parameter h, which is the time required for computing an “exchange capacity”. To the best of our knowledge, the most efficient way of computing
an exchange capacity is to solve an instance of submodular minimization which previously took
time Õ(n4 EO + n5 ) (and now takes Õ(n2 EO + n3 ) time using our result in Part III). Readers may
wish to substitute h = Õ(n2 EO + n3 ) when reading the table.
The previous fastest weakly polynomial algorithms for submodular flow are by [59, 30, 32], which
take time Õ(n6 EO + n7 ) and O(mn5 log nU · EO), assuming h = Õ(n2 EO + n3 ). Our algorithm
for submodular flow has a running time of Õ(n2 EO + n3 ), which is significantly faster by roughly
a factor of Õ(n4 ).
For strongly polynomial algorithms, our results do not yield a speedup but we remark that
our faster strongly polynomial algorithm for submodular minimization in Part III improves the
previous algorithms by a factor of Õ(n2 ) as a corollary (because h requires solving an instance of
submodular minimization).
8.3
Overview
After providing covering some preliminaries on convex analysis in Section 9 we split the remainder of Part II into Section 10 and Section 11. In Section 10 we cover our algorithm for convex
optimization using an approximate subgradient oracle (Section 10.1) as well as our technique on
using duality to decrease dimensions and improve the running time of semidefinite programming
(Section 10.2). In Section 11 we provide our technique for using minimization oracles to minimize
functions over the intersection of convex sets and provide several applications including matroid
intersection (Section 11.2), submodular flow (Section 11.3), and minimizing a linear function over
the intersection of an affine subspace and a convex set (Section 11.4).
9
Preliminaries
In this section we review basic facts about convex functions that we use throughout Part II. We also
introduce two oracles that we use throughout Part II, i.e. subgradient and optimization oracles, and
provide some basic reductions between them. Note that we have slightly extended some definitions
and facts to accommodate for the noisy separation oracles used in this paper.
First we recall the definition of strong convexity
Definition 35 (Strong Convexity ). A real valued function f on a convex set Ω is α-strongly
convex if for any ~x, ~y ∈ Ω and t ∈ [0, 1], we have
1
f (t~x + (1 − t)~y ) + αt(1 − t) ~x − ~y
2
Next we define an approximate subgradient.
50
2
≤ tf (~x) + (1 − t)f (~y ).
Authors
Fujishige [35]
Grötschel, Lovász, Schrijver[49]
Zimmermann [113]
Barahona, Cunningham [12]
Cunningham, Frank [22]
Fujishige [36]
Frank, Tardos [34]
Cui, Fujishige [108]
Fujishige, Röck, Zimmermann[38]
Chung, Tcha [18]
Zimmermann [114]
McCormick, Ervolina [82]
Wallacher, Zimmermann [109]
Iwata [52]
Iwata, McCormick, Shigeno [57]
Iwata, McCormick, Shigeno [58]
Fleischer, Iwata, McCormick[32]
Iwata, McCormick, Shigeno [59]
Fleischer, Iwata [30]
This paper
Years
1978
1981
1982
1984
1985
1987
1987
1988
1989
1991
1992
1993
1994
1997
1998
1999
1999
1999
2000
2015
Running times
not stated
weakly polynomial
not stated
not stated
→ O(n4 h log C)
not stated
strongly polynomial
not stated
→ O(n6 h log n)
not stated
not stated
O(n7 h∗ log nCU )
O(n8 h log nCU )
7
O(n
h log U ) 2
4
O n h min log nC, n log n
O n6 h min log nU, n2 log n
O n4 h min log U, n2 log n
O n4 h min log C, n2 log n
O(mn5 log nU · EO)
O(n2 log nCU · EO + n3 logO(1) nCU )
Figure 8.1: Previous algorithms for Submodular Flow with n vertices, maximum cost C and maximum capacity U . The factor h is the time for an exchange capacity oracle, h∗ is the time for
a “more complicated exchange capacity oracle” and EO is the time for evaluation oracle of the
submodular function. The arrow,→, indicates that it used currently best maximum submodular
flow algorithm as subroutine which was non-existent at the time of the publication.
51
Definition 36 (Subgradient). For any convex function f on a convex set Ω, the δ-subgradients of
f at x are defined to be
def
∂δ f (~x) = {~g ∈ Ω : f (~y ) + δ ≥ f (~x) + h~g , ~y − ~xi for all ~y ∈ Ω}.
Here we provide some basic facts regarding convexity and subgradients. These statements are
natural extensions of well known facts regarding convex functions and their proof can be found in
any standard textbook on convex optimization.
Fact 37. For any convex set Ω and ~x be a point in the interior of Ω, we have the following:
1. If f is convex q
on Ω, then ∂0 f (~x) 6= ∅ and ∂s f (~x) ⊆ ∂t f (~x) for all 0 ≤ s ≤ t.Otherwise, we
1
have ~g 2 > 2 Dδ . For any f (~y ) ≤ f (~x), we have δ ≥ h~g , ~y − ~xi and hence
2. If f is a differential convex function on Ω, then ∇f (~x) ∈ ∂0 f (~x).
3. If f1 and f2 are convex function on Ω, ~g1 ∈ ∂δ1 f1 (~x) and ~g2 ∈ ∂δ2 f1 (~x), then α~g1 + β~g2 ∈
∂αδ1 +βδ2 (~g1 + ~g2 )(~x).
4. If f is α-strongly convex on Ω with minimizer x∗ , then for any ~y with f (~y ) ≤ f (~x∗ ) + , we
2
have 12 α ~x∗ − ~y ≤ .
Next we provide a reduction from subgradients to separation oracles. We will use this reduction
several times in Part II to simplify our construction of separation oracles.
Lemma 38. Let f be a convex function. Suppose we have ~x and ~g ∈ ∂δ f (~x) with ~x 2 ≤ 1 ≤ D
q
q
√
1
δ
1
and δ ≤ 1. If ~g 2 ≤ 2 D , then f (~x) ≤ mink~y2 k2 ≤D f (~y ) + 2 δD and if ~g 2 ≤ 2 Dδ then
{ ~y
with d~ = ~g / ~g
2
2
√
≤ D : f (~y ) ≤ f (~x)} ⊂ {~y : d~T ~y ≤ d~T ~x + 2 δD}
√
√
. Hence, this gives a (2 δD, 2 δD)-separation oracle on the set { ~x
Proof. Let ~y such that ~y
2
2
≤ D}.
≤ D. By the definition of δ-subgradient, we have
f (~y ) + δ ≥ f (~x) + h~g , ~y − ~xi .
If ~g ≤
1
2
q
δ
D,
then, we have |h~g , ~y − ~xi| ≤
√
δD because ~x ≤ D and ~y
2
≤ D. Therefore,
√
min f (~y ) + 2 δD ≥ f (~x).
~
y
Otherwise, we have ~g
2
>
1
2
q
δ
D.
2
≤D
For any f (~y ) ≤ f (~x), we have δ ≥ h~g , ~y − ~xi and hence
√
2 δD ≥
*
+
~g
, ~y − ~x .
~g
At several times in Part II we will wish to construct subgradient oracles or separation oracles
given only the ability to approximately maximize a linear function over a convex set. In the
remainder of this section we formally define such a optimization oracle and prove this equivalence.
52
Definition 39 (Optimization Oracle). Given a convex set K and δ > 0 a δ-optimization oracle for
K is a function on Rn such that for any input ~c ∈ Rn , it outputs ~y such that
max h~c, ~xi ≤ h~c, ~y i + δ.
~
x∈K
We denote by OOδ (K) the time complexity of this oracle.
Lemma 40. Given a convex set K, any -optimization oracle for K is a -subgradient oracle for
def
f (~c) = max~x∈K h~c, ~xi .
Proof. Let ~xc be the output of -optimization oracle on the cost vector ~c. We have
max h~c, ~xi ≤ h~c, ~xc i + .
~
x∈K
~ we have and therefore
Hence, for all d,
D
E
~ + .
~xc , d~ − ~c + f (~c) ≤ f (d)
Hence, ~xc ∈ ∂δ f (~c).
Combining these lemmas shows that √
having
√ an -optimization oracle for a convex set K contained in a ball of radius D yields a O( D, D) separation oracle for maxx∈K h~c, ~xi. We use
these ideas to construction separation oracles throughout Part II.
10
Convex Optimization
In this section we show how to apply our cutting plane method to efficiently solve problems in
convex optimization. First, in Section 10.1 we show how to use our result to minimize a convex
function given an approximate subgradient oracle. Then, in Section 10.2 we illustrate how this
result can be used to obtain both primal and dual solutions for a standard convex optimization
problems. In particular, we show how our result can be used to obtain improved running times for
semidefinite programming across a range of parameters.
10.1
From Feasibility to Optimization
In this section we consider the following standard optimization problem. We are given a convex
function f : Rn → R ∪ {+∞} and we want to find a point ~x that approximately solves the
minimization problem
minn f (~x)
~
x∈R
given only a subgradient oracle for f .
Here we show how to apply the cutting plane method from Part I turning the small width
guarantee of the output of that algorithm into a tool to find an approximate minimizer of f . Our
result is applicable to any convex optimization problem armed with a separation or subgradient
oracle. This result will serve as the foundation for many of our applications in Part II.
Our approach is an adaptation of Nemiroski’s method [85] which applies the cutting plane
method to solve convex optimiziation problems, with only minimal assumption on the cutting
plane method. The proof here is a generalization that accommodates for the noisy separation
oracle used in this paper. In the remainder of this subsection we provide a key definition we will
use in our algorithm (Defintion 41), provide our main result (Theorem 42), and conclude with a
brief discussion of this result.
53
def
Definition 41. For any compact set K, we define the minimum width by MinWidth(K) =
mink~ak2 =1 max~x,~y∈K h~a, ~x − ~y i .
Theorem 42. Let f be a convex function on Rn and Ω be a convex set that contains a minimizer
of f . Suppose we have a (η, δ)-separation oracle for f and Ω is contained inside B∞ (R). Using
B∞ (R) as the initial polytope for our Cutting Plane Method, for any 0 < α < 1, we can compute
~x ∈ Rn such that
f (~x) − min f (~y ) ≤ η + α max f (~y ) − min f (~y )
~
y ∈Ω
~
y ∈Ω
~
y ∈Ω
.
(10.1)
with an expected running time of
nκ
nκ
O nSOη,δ (f ) log
+ n3 logO(1)
,
α
α
R
where δ = Θ αMinWidth(Ω)
. Furthermore, we only need the oracle defined on
and κ = MinWidth(Ω)
n3/2 ln(κ)
the set B∞ (R).
Proof. Let ~x∗ ∈ arg min~x∈Ω f (~x). Since B∞ (R) ⊃ Ω contains a minimizer of f , by the definition of
(η, δ)-separation oracles, our Cutting Plane Method (Theorem 31) either returns a point ~x that is
almost optimal or returns a polytope P of small width. In the former case we have a point ~x such
that f (~x) ≤ min~y f (~y ) + η. Hence, the error is clearly at most η + α (max~z∈Ω f (~z) − min~x∈Ω f (~x))
as desired. Consequently, we assume the latter case.
Theorem 31 shows MinWidth(P ) < Cn ln(R/) for some universal constant C. Picking
= C0
αMinWidth(Ω)
n ln nκ
α
(10.2)
for small enough constant C 0 , we have MinWidth(P (i) ) < αMinWidth(Ω). Let Ωα = ~x∗ +α(Ω−~x∗ ),
namely, Ωα = {~x∗ + α(~z − ~x∗ ) : ~z ∈ Ω}. Then, we have
MinWidth(Ωα ) = αMinWidth(Ω) > MinWidth(P ).
Therefore, Ωα is not a subset of P (i) and hence there is some point ~y ∈ Ωα \P . Since Ωα ⊆ Ω ⊆
B∞ (R), we know that ~y does not violate any of the constraints of P (0) and therefore must violate
one of the constraints added by querying the separation oracle. Therefore, for some j ≤ i, we have
E
D
E D
√
(j−1)
(j−1) (j−1)
+ cs / n .
~c
, ~y > ~c
, ~x
√
By the definition of (η, cs / n)-separation oracle (Definition 2), we have f (~y ) > f (~x(j−1) ). Since
~y ∈ Ωα , we have ~y = (1 − α)~x∗ + α~z for some ~z ∈ Ω. Thus, the convexity of f implies that
f (~y ) ≤ (1 − α)f (~x∗ ) + αf (~z).
Therefore, we have
min f (~x(k) ) − min f (~x) < f (~y ) − f (~x∗ ) ≤ α max f (~z) − min f (~x) .
1≤k≤i
~
x∈Ω
~
z ∈Ω
~
x∈Ω
Hence, we can simply output the best ~x among all ~x(j) and in either case ~x satisfies (10.1).
√
Note that we need to call (η, δ)-separation oracle with δ = Ω(/ n) to ensure we do not cut
out ~x∗ . Theorem 31 shows that the algorithm takes O(nSOη,δ (f ) log(nR/) + n3 logO(1) (nR/))
expected time, as promised. Furthermore, the oracle needs only be defined on B∞ (R) as our
cutting plane method guarantees ~x(k) ∈ B∞ (R) for all k (although if needed, an obvious separating
hyperplane can be returned for a query point outside B∞ (R) ).
54
Observe that this algorithm requires no information about Ω (other than that Ω ⊆ B∞ (R))
and does not guarantee that the output is in Ω. Hence, even though Ω can be complicated to
describe, the algorithm still gives a guarantee related to the gap max~x∈Ω f (~x) − min~x∈Ω f (~x). For
specific applications, it is therefore advantageous to pick a Ω as large as possible while the bound
on function value is as small as possible.
Before indulging into specific applications, we remark on the dependence on κ. Using John’s
ellipsoid, it can be shown that any convex set Ω can be transformed linearly such that (1) B∞ (1)
contains Ω and, (2) MinWidth(Ω) = Ω(n−3/2 ). In other words, κ can be effectively chosen as
3/2
O(n
). Therefore if we are able to find
such a linear transformation, the running time is simply
O(1)
3
O nSO(f ) log (n/α) + n log
(n/α) . Often this can be done easily using the structure of the
particular problem and the running time does not depend on the size of domain at all.
10.2
Duality and Semidefinite Programming
In this section we illustrate how our result in Section 10.1 can be used to obtain both primal and
dual solutions for standard problems in convex optimization. In particular we show how to obtain
improved running times for semidefinite programming.
To explain our approach, consider the following minimax problem
D E
~ ~y
min max hA~x, ~y i + h~c, ~xi + d,
(10.3)
~
y ∈Y ~
x∈X
where ~x ∈ Rm and ~y ∈ Rn . When m n, solving this problem by directly using Part I could lead
to an inefficient algorithm with running time at least m3 . In many situations, for any fixed ~y , the
problem max~x∈X hA~x, ~y i is very easy and hence one can use it as a separation oracle and apply
Part I and this would gives a running time almost independent of m. However, this would only
give us the ~y variable and it is not clear how to recover ~x variable from it.
In this section we show how to alleviate this issue and give semidefinite programming (SDP)
as a concrete example of how to apply this general technique. We do not write down the general
version as the running time of the technique seems to be problem specific and faster SDP is already
an interesting application.
For the remainder of this section we focus on the semidefinite programming (SDP) problem:
max C • X s.t. Ai • X = bi
X0
and its dual
min ~bT ~y s.t.
~
y
n
X
yi Ai C
(10.4)
(10.5)
i=1
where X, C, Ai are m × m symmetric matrices and ~b, ~y ∈ Rn . Our approach is partially inspired
by one of the key ideas of [51, 70]. These results write down the dual SDP in the form
n
X
T
~
min b ~y − K min(λmin (
yi Ai − C), 0)
y
(10.6)
i=1
for some large number K and use non-smooth optimization techniques to solve the dual SDP
problem. Here, we follow the same approach but instead write it as a max-min problem min~y fK (~y )
where
*
+!
n
X
T
~b ~y + X, C −
fK (~y ) =
max
yi A i
.
(10.7)
TrX≤K,X0
i=1
55
Thus the SDP problem in fact assumes the form (10.3) and many ideas in this section can be
generalized to the minimax problem (10.3).
To get a dual solution, we notice that the cutting plane method maintains a subset of the primal
feasible solution conv(Xi ) such that
*
+
*
+
n
n
X
X
min ~bT ~y +
max
X, C −
yi Ai ∼ min ~bT ~y + max
X, C −
yi A i .
TrX≤K,X0
~
y
~
y
i=1
X∈conv(Xi )
i=1
Applying minimax theorem, this shows that there exists an approximation solution X in conv(Xi )
for the primal problem. Hence, we can restrict the primal SDP on the polytope conv(Xi ), this
reduces the primal SDP into a linear program which can be solved very efficiently. This idea of
getting primal/dual solution from the cutting plane method is quite general and is the main purpose
of this example. As a by-product, we have a faster SDP solver in both primal and dual! We remark
that this idea has been used as a heuristic to obtain [69] for getting the primal SDP solution and
our contribution here is mainly the asymptotic time analysis.
We first show how to construct the separation oracle for SDP. For that we need to compute
smallest eigenvector of a matrix. Below, for completeness we provide a folklore result showing we
can do this using fast matrix multiplication.
Lemma 43. Given a n × n symmetric matrix Y such that −RI Y RI, for any > 0,
with high probability in n in time O(nω+o(1) logO(1) (R/)) we can find a unit vector ~u such that
~uT Y~u ≥ λmax (Y) − .
def
Proof. Let B =
Bk+1 =
B2k
TrB2k
1
RY
+ I. Note that B 0. Now, we consider the repeated squaring B0 = B and
. Let 0 ≤ λ1 ≤ λ2 ≤ · · · ≤ λn be the eigenvalues of B and ~vi be the corresponding
λ2
k
eigenvectors. Then, it is easy to see the the eigenvalues of Bk are Pn i 2k .
i=1 λi
P
P 2
def
Let ~q be a random unit vector and ~r = Bk ~q. Now ~q = αi~vi for some αi such that
αi = 1.
Letting
P
2k v
i
λi >(1−δ)λn αi λi ~
p~ =
Pn
k
2
i=1 λi
we have
2k v
i
λi ≤(1−δ)λn αi λi ~
Pn
k
2
i=1 λi
2
P
k~r − p~k2 =
Letting k = log2
log(n3/2 /δ)
δ
2k
λi ≤(1−δ)λn λi
Pn
2k
i=1 λi
P
≤
k
≤ (1 − δ)2 n.
√
, we have k~r − p~k2 ≤ δ/ n. Since 0 B 2I, we have
q
p
√
~rT B~r ≥
p~T B~
p − (~r − p~)T B(~r − p~)
p
√
≥
p~T B~
p − 2δ/ n.
Note that p~ involves only eigenvectors between (1 − δ)λn to λn . Hence, we have
√
p
√
~rT B~r ≥ (1 − δ)λn p~ 2 − 2δ/ n.
56
√
√
With constant probability, we have αn = Ω(1/ n). Hence, we have p~ 2 = Ω(1/ n). Using
√
B 2I and p~ 2 ≥ ~r 2 − δ/ n we have that so long as δ is a small enough universal constant
√
~rT B~r
~r 2
Therefore, we have
~
rT Y~
r
2
~
r
√
− 2δ/ n
≥
√
p~ 2 + δ/ n
p
= (1 − O(δ)) λn − O(δ)
p
√
=
λn − O(δ R).
p
(1 − δ)λn p~
2
≥ λmax (Y) − O(Rδ). Hence, we can find vector ~r by computing k matrix
multiplications. [24] showed that fast matrix multiplication is stable under Frobenius norm, i.e.,
for any η > 0, using O(log(n/b)) bits, we can find C such that C − AB F ≤ 1b A B in
time O(nω+η ) where ω is the matrix multiplicative constant. Hence, this algorithm takes only
O(nω+o(1) logO(1) (δ −1 )) time. The result follows from renormalizing the vector ~r, repeating the
algorithm O(log n) times to boost the probability and taking δ = Ω(/R).
The following lemma shows how to compute a separation for fK defined in (10.7).
Lemma 44. Suppose that Ai F ≤ M and C F ≤ M . For any 0 < < 1 and ~y with ~y 2 =
O(L), with high probability in m, we can compute a (, )-separation of fK on { ~x 2 ≤ L} at ~y
ω+o(1) logO(1) (nKM L/)) where where S is the sparsity of the problem defined as
in time O(S
Pn+ m
nnz(C) + i=1 nnz(Ai ).
P
Proof. Note that −O(nM L)I C− ni=1 yi Ai O(nM L)I. Using Lemma 43, we can find a vector
~v with ~v 2 = K in time O(mω+o(1) logO(1) (nKM L/δ)) such that
~v T
C−
n
X
!
*
yi Ai ~v ≥
i=1
X, C −
max
TrX≤K,X0
n
X
+
yi Ai
− δ.
(10.8)
i=1
In other words, we have a δ-optimization oracle for the function
K . Lemma 40 shows this yields a
f√
√
δ-subgradient oracle and Lemma 38 then shows this yields a O( δL), O( δL) -separation oracle
on the set { ~x
2
≤ L}. By picking δ = 2 /L, we have the promised oracle.
With the separation oracle in hand, we are ready to give the algorithm for SDP:
Theorem 45. Given a primal-dual semidefinite programming problem in the form (10.4) and
(10.5), suppose that for some M ≥ 1 we have
1.
b
2
≤ M, C
F
≤ M and Ai
F
≤ M for all i.
2. The primal feasible set lies inside the region TrX ≤ M .
3. The dual feasible set lies inside the region ~y
∞
≤ M.
Let OPT be the optimum solution of (10.4) and (10.5). Then, with high probability, we can find X
and ~y such that
P
1. X 0, TrX = O(M ), i |bi − hX, Ai i| ≤ for all i and C • X ≥ OPT − .
P
2. ~y ∞ = O(M ), ni=1 yi Ai C − I and ~bT ~y ≤ OPT + .
57
where S is the sparsity of the problem
nS + n3 + nmω+o(1) logO(1) nM
Pn
defined as nnz(C) + i=1 nnz(Ai ) and ω is the fast matrix multiplication constant.
in expected time O
Proof. Let K ≥ M be some parameter to be determined. Since the primal feasible set is lies inside
the region TrX ≤ M ≤ K, we have
Pn
min
i=1 yi Ai C
~bT ~y =
C•X
max
X0,TrX≤K,Ai •X=bi
=
max
X0,TrX≤K
min C • X −
X
~
y
yi (Ai • X − bi )
i
!
= min
~
y
~bT ~y + (C −
max
X0,TrX≤K
X
yi Ai ) • X
i
= min fK (~y ).
~
y
Lemma 44 shows that it takes SOδ,δ (fK ) = O(S + mω+o(1) log(nKM L/δ)) time to compute a
(δ, δ)-separation oracle of fK for any point ~y with k~y k∞ = O(L)
where L is some parameter with L ≥
M . Taking the radius R = L, Theorem 42 shows that it takes O nSOδ,δ (fK ) log αn + n3 logO(1) αn
expected time with δ = Θ αn−3/2 L to find ~y such that
fK (~y ) −
min
~
y
Picking α =
∞
fK (~y ) ≤ δ + α max fK (~y ) −
≤L
7nM KL ,
~
y
∞
≤L
fK (~y ) ≤ δ + 2α (nM L + 2nKM L) .
min
~
y
∞
≤L
we have fK (~y ) ≤ min~y fK (~y ) + . Therefore,
~bT ~y + K max(λmax (C −
n
X
yi Ai ), 0) ≤ OPT + .
i=1
Let β = max(λmax (C −
Pn
i=1 yi Ai ), 0).
~bT ~y ≥
Pn
min
i=1
=
Pn
i=1 yi Ai
Then, we have that
yi Ai C−βI
max
X0Ai •X=bi
C − βI and
~bT ~y
(C − βI) • X
≥ OPT − βM
because TrX ≤ M . Hence, we have
OPT − βM + βK ≤ ~bT ~y + K max(λmax (C −
n
X
yi Ai ), 0) ≤ OPT +
i=1
Putting K = M + 1, we have β ≤ . Thus,
n
X
yi Ai C − I.
i=1
This gives the result for the dual with the running time O
58
nS + n3 + nmω+o(1) logO(1)
nM L
.
Our Cutting Plane Method accesses the sub-problem
X
max (C −
yi A i ) • X
X0,TrX≤K
i
O(n)
only through the separation oracle. Let ~z be the output of our Cutting Plane Method and {~vi~viT }i=1
be the matrices used to construct the separation for the P
O(n) hyperplanes the algorithm maintains
at the end. Let ~u be the maximum eigenvector of C − ni=1 zi Ai . Now, we consider a realization
of fK
*
+
n
X
f˜K (~y ) = ~bT ~y +
max
X, C −
yi A i .
X∈conv(K~
u~
uT ,~vi~viT )
i=1
Since applying our Cutting Plane Method to either fK or f˜K gives the same result, the correctness
of the our Cutting Plane Method shows that
f˜K (~z) ≤
f˜K (~y ) + .
min
~
y
∞
≤L
Note that the function f˜K is defined such that f˜K (~z) = fK (~z). Hence, we have
fK (~y ) ≤ fK (~z) ≤ f˜K (~z) ≤
min
~
y
∞
f˜K (~y ) + .
min
≤L
~
y
∞
≤L
Also, note that f˜K (~x) ≤ fK (~x) for all ~x. Hence, we have
min
~
y
∞
fK (~y ) − ≤
f˜(~y ) ≤
min
≤L
~
y
∞
fK (~y ).
min
≤L
~
y
∞
≤L
Now, we consider the primal version of f˜, namely
*
min ~bT ~y +
def
g(X) =
~
y
∞
X, C −
≤L
n
X
+
yi A i
.
i=1
Sion’s minimax theorem [98] shows that
OPT ≥
max
X∈conv(K~
u~
uT ,~vi~viT )
g(X) =
f˜(~y ) ≥ OPT − .
min
~
y
∞
≤L
Therefore, to get the primal solution, we only need to find ~u by Lemma 43 and solve the maximization problem on g. Note that
g(X) =
n
X
min
~
y
= −L
yi (bi − hX, Ai i) + hX, Ci
≤L i=1
∞
X
|bi − hX, Ai i| + hX, Ci .
i
For notation simplicity, we write K~u~uT = ~v0~v0T . Then, X =
αj ≥ 0. Substituting this into the function g, we have
g(~
α) = −L
X
j
bi −
X
αj ~vjT Ai~vj +
j
PO(n)
j=0
X
j
59
αj ~vj ~vjT for some
αj ~vjT C~vj .
P
αj = 1 and
Hence, this can be easily written as a linear program with O(n) variables and O(n) constraints in
time O(nS). Now, we can apply interior point method to find α
~ such that
g(~
α) ≥
max
X∈conv(K~
u~
uT ,~vi~viT )
g(X) − ≥ OPT − 2.
e = P αj ~vj ~v T . Then, we have
Let the corresponding approximate solution be X
j
D
E
X
e C −L
X,
|bi − hX, Ai i| ≥ OPT − 2.
i
D
E
e Ai . Then, we note that
Now, we let b̃i = X,
D
E
e C
X,
≤
=
C•X
max
X0Ai •X=b̃i
Pn
min
i=1
yi Ai C
≤ OPT + M
b̃Ti ~y
X
D
E
e Ai
bi − X,
i
≤ M . Hence, we have
D
E
D
E
D
E
X
X
e Ai ≥ X,
e C −L
e Ai ≥ OPT − 2.
OPT + (M − L)
bi − X,
bi − X,
because ~y
∞
i
i
Now, we put L = M + 2, we have
X
D
E
e Ai ≤ .
bi − X,
i
This gives the result for the primal. Note that it only takes O(n5/2 logO(1) (nM/)) to solve a
linear program with O(n) variables and O(n) constraints because we have an explicit interior point
1
for some parameter m [76].5 Hence, the running time is
deep inside the feasible set, i.e. αi = m
dominated by the cost of cutting plane method which is O nS + n3 + nmω+o(1) logO(1) nM
by putting L = M + 2.
We leave it as an open problem if it is possible to improve
the computation
this result by reusing
O(1) nM
3
2
in the separation oracle and achieve a running time of O nS + n + nm log
.
11
Intersection of Convex Sets
In this section we introduce a general technique to optimize a linear function over the intersection of two convex sets, whenever the linear optimization problem on each of them can be done
efficiently. At the very high level, this is accomplished by applying cutting plane to a suitably
regularized version of the problem. In Section 11.1 we present the technique and in the remaining
sections we provide several applications including, matroid intersection (Section 11.2), submodular
flow (Section 11.3), and minimizing over the intersection of an affine subspace and a convex set
(Section 11.4).
5
Without this, the running time of interior point method depends on the bit complexity of the linear programs.
60
11.1
The Technique
Throughout this section we consider variants of the following general optimization problem
max
~
x∈K1 ∩K2
h~c, ~xi
(11.1)
where ~x, ~c ∈ Rn , K1 and K2 are convex subsets of Rn . We assume that
max k~xk2 < M, max k~xk2 < M, k~ck2 ≤ M
~
x∈K1
(11.2)
~
x∈K2
for some constant M ≥ 1 and we assume that
K1 ∩ K2 6= ∅.
(11.3)
Instead of a separation oracle, we assume that K1 and K2 each have optimization oracles (see
Section 9).
To solve this problem we first introduce a relaxation for the problem (11.1) that we can optimize
efficiently. Because we have only the optimization oracles for K1 and K2 , we simply have variables
~x and ~y for each of them in the objective. Since the output should (approximately) be in the
2
intersection of K1 and K2 , a regularization term − λ2 ~x − ~y 2 is added to force ~x ≈ ~y where λ is
a large number to be determined later. Furthermore, we add terms to make the problem strong
concave.
Lemma 46. Assume (11.2) and (11.3). For λ ≥ 1, let
1
λ
1
1
1
2
2
2
h~c, ~xi + h~c, ~y i −
~x − ~y 2 −
~x 2 −
~y 2 .
(11.4)
2
2
2
2λ
2λ
There is an unique maximizer (~xλ , ~yλ ) for the problem max~x∈K1 ,~y∈K2 fλ (~x, ~y ). The maximizer
2
2
(~xλ , ~yλ ) is a good approximation of the solution of (11.1), i.e. ~xλ − ~yλ 2 ≤ 6M
λ and
def
fλ (~x, ~y ) =
max
~
x∈K1 ∩K2
h~c, ~xi ≤ fλ (~xλ , ~yλ ) +
M2
.
λ
(11.5)
Proof. Let ~x∗ be a maximizer of max~x∈K1 ∩K2 h~c, ~xi. By assumption (11.2), ~x∗ 2 ≤ M , and
therefore
2
~x∗ 2
M2
∗ ∗
∗
fλ (~x , ~x ) = h~c, ~x i −
≥ max h~c, ~xi −
.
(11.6)
λ
λ
~
x∈K1 ∩K2
This shows (11.5). Since fλ is strongly concave in ~x and ~y , there is a unique maximizer (~xλ , ~yλ ).
Let OPTλ = fλ (~xλ , ~yλ ). Then, we have
1
λ
1
~c 2 ~xλ 2 + ~c 2 ~yλ 2 −
~xλ − ~yλ
2
2
2
M2 M2 λ
2
≤
+
−
~xλ − ~yλ 2 .
2
2
2
On the other hand, using λ ≥ 1, (11.6) shows that
OPTλ ≤
OPTλ ≥ fλ (~x∗ , ~x∗ ) ≥
max
~
x∈K1 ∩K2
h~c, ~xi −
2
2
M2
≥ −2M 2 .
λ
Hence, we have
2
~xλ − ~yλ 2
≤
2 M 2 − OPTλ
6M 2
≤
.
λ
λ
61
(11.7)
Now we write max fλ (~x, ~y ) as a max-min problem. The reason for doing this is that the dual
approximate solution is much easier to obtain and there is a way to read off a primal approximate
solution from a dual approximate solution. This is analogous to the idea in [73] which showed how
to convert a cut solution to a flow solution by adding regularization terms into the problem.
Lemma 47. Assume (11.2) and (11.3). Let λ ≥ 2. For any ~x ∈ K1 and ~y ∈ K2 , the function fλ
can be represented as
fλ (~x, ~y ) =
min
gλ (~x, ~y , θ~1 , θ~2 , θ~3 )
(11.8)
(θ~1 ,θ~2 ,θ~3 )∈Ω
where Ω = {(θ~1 , θ~2 , θ~3 ) : θ~1
*
θ~2
≤ 2M,
2
2
θ~3
≤ M,
2
≤ M } and
+ *
+
~2
~3
θ
~
c
θ
λ ~
~c
+ λθ~1 + , ~x +
− λθ~1 + , ~y +
θ1
2
λ
2
λ
2
1 ~ 2
θ3 2 .
2λ
(11.9)
Let hλ (θ~1 , θ~2 , θ~3 ) = max~x∈K1 ,~y∈K2 gλ (~x, ~y , θ~1 , θ~2 , θ~3 ). For any (θ~10 , θ~20 , θ~30 ) such that hλ (θ~10 , θ~20 , θ~30 ) ≤
min(θ~1 ,θ~2 ,θ~3 )∈Ω hλ (θ~1 , θ~2 , θ~3 ) + , we know ~z = 21 (θ~20 + θ~30 ) satisfies
gλ (~x, ~y , θ~1 , θ~2 , θ~3 ) =
max
~
x∈K1 ∩K2
and ~z − ~xλ 2 + ~z − ~yλ
max~x∈K1 ,~y∈K2 fλ (~x, ~y ).
2
h~c, ~xi ≤ h~c, ~zi +
2
2
+
1 ~
θ2
2λ
2
2
+
20M 2
+ 20λ3 .
λ
q
√
2
≤ 4 2λ + 6M
xλ , ~yλ ) is the unique maximizer for the problem
λ where (~
Proof. Note that for any ξ~
2
≤ α, we have
−
1 ~
ξ
2
2
2
= min
θ~
2
≤α
D
E
~ ξ~ + 1 θ~
θ,
2
2
2
Using this and (11.2), we have (11.8) for all ~x ∈ K1 and ~y ∈ K2 as desired. Since Ω is closed
and bounded set and the function gλ is concave in (~x, ~y ) and convex in (θ~1 , θ~2 , θ~3 ), Sion’s minimax
theorem [98] shows that
max
~
x∈K1 ,~
y ∈K2
fλ (~x, ~y ) =
min
(θ~1 ,θ~2 ,θ~3 )∈Ω
hλ (θ~1 , θ~2 , θ~3 )
(11.10)
Since fλ is strongly concave, there is an unique maximizer (~xλ , ~yλ ) of fλ . Since hλ is strongly
convex, there is a unique minimizer (θ~1∗ , θ~2∗ , θ~3∗ ). By the definition of fλ and hλ , we have
hλ (θ~1∗ , θ~2∗ , θ~3∗ ) ≥ gλ (~xλ , ~yλ , θ~1∗ , θ~2∗ , θ~3∗ ) ≥ fλ (~xλ , ~yλ ) .
Using (11.10), the equality above holds and hence (θ~1∗ , θ~2∗ , θ~3∗ ) is the minimizer of gλ (~xλ , ~yλ , θ~1 , θ~2 , θ~3 )
over (θ~1 , θ~2 , θ~3 ). Since the domain Ω is large enough that (θ~1∗ , θ~2∗ , θ~3∗ ) is an interior point in Ω, the
optimality condition of gλ shows that we have θ~2∗ = ~xλ and θ~3∗ = ~yλ .
2
2
2
Since hλ is λ1 strongly convex, we have θ~10 − θ~1∗ 2 + θ~20 − θ~2∗ 2 + θ~30 − θ~3∗ 2 ≤ 2λ (Fact 37).
Since θ~2∗ = ~xλ and θ~3∗ = ~yλ , we have
θ~20 − ~xλ
2
2
+ θ~30 − ~yλ
62
2
2
≤ 2λ.
(11.11)
√
Therefore, we have ~xλ − ~yλ 2 ≥ θ~20 − θ~30 2 − 2 2λ, ~xλ 2 ≥
√
θ~30 2 − 2λ. Using these, ~xλ 2 ≤ M and ~yλ 2 ≤ M , we have
θ~20
2
−
√
2λ and
1 ~0 2
1 ~0
1 D ~0 E 1 D ~0 E λ ~0 ~0 2
~c, θ2 +
~c, θ3 −
θ2 − θ3 2 −
θ2 2 −
θ
2
2
2
2λ
2λ 3
√
1
1
h~c, ~xλ i + h~c, ~yλ i − M 2λ
≥
2
2
√ 2
λ
−
~xλ − ~yλ 2 + 2 2λ
2
√ 2
√ 2
1
1
−
~xλ 2 + 2λ −
~yλ 2 + 2λ
2λ
2λ
1
1
λ
1
1
2
2
=
h~c, ~xλ i + h~c, ~yλ i −
~xλ − ~yλ 2 −
~xλ 2 −
~yλ
2 √
2 √
2
2λ
2λ
−M 2λ − 2λ 2λ ~xλ − ~yλ 2 − 4λ2
√
√
1
1
− ~xλ 2 2λ − −
~yλ 2 2λ − .
λ
λ
fλ (θ~20 , θ~30 ) =
Using ~xλ − ~yλ
2
≤
q
6M 2
λ
(Lemma 46), ~xλ
2
< M and ~yλ
2
< M , we have
fλ (θ~20 , θ~30 ) ≥ fλ (~xλ , ~yλ )
√
√
−M 2λ − 2λ 2λ ~xλ − ~yλ 2 − 4λ2
√
√
1
1
− ~xλ 2 2λ − −
~yλ 2 2λ − .
λ
λ
≥ fλ (~xλ , ~yλ )
√
√
−M 2λ − 2λ 12M − 4λ2
r
−2M 2 − 2.
λ
Since λ ≥ 2, we have
√
fλ (θ~20 , θ~30 ) ≥ fλ (~xλ , ~yλ ) − 20M λ − 10λ2 .
Let ~z =
θ~20 +θ~30
2 .
Lemma 46 shows that
max
~
x∈K1 ∩K2
h~c, ~xi ≤
max
~
x∈K1 ,~
y ∈K2
fλ (~x, ~y ) +
M2
λ
√
M2
≤ fλ (θ~20 , θ~30 ) +
+ 20M λ + 10λ2
λ
20M 2
≤ h~c, ~zi +
+ 20λ3
λ
√
2
because 20M λ ≤ 10 Mλ + 10λ3 . Furthermore, we have
~z − ~xλ
2
+ ~z − ~yλ
2
+ θ~30 − ~yλ
2
r
√
6M 2
≤ 4 2λ +
.
λ
≤
θ~20 − ~xλ
63
2
+ θ~20 − θ~30
2
2
2
2
2
~yλ
2
≥
We now apply our cutting plane method to solve the optimization problem (11.1). First we
show how to transform the optimization oracles for K1 and K2 to get a separation oracle for hλ ,
with the appropriate parameters.
Lemma 48. Suppose we have a -optimization
√
√ oracle for K1 and K2 for some 0 < < 1. Then on
the set { θ~ 2 ≤ D}, we have a (O( λD), O( λD))-separation oracle for hλ with time complexity
OO (K1 ) + OO (K2 ).
Proof. Recall that the function hλ is defined by
hλ (θ~1 , θ~2 , θ~3 )
*
!
+ *
+
~2
~3
~c
1
1
θ
~
c
θ
λ
2
2
2
=
max
+ λθ~1 + , ~x +
− λθ~1 + , ~y +
θ~1 2 +
θ~2 2 +
θ~3 2
2
λ
2
λ
2
2λ
2λ
~
x∈K1 ,~
y ∈K2
*
+
*
+
~2
~3
~c
θ
~
c
θ
λ ~ 2
1 ~ 2
1 ~ 2
= max
+ λθ~1 + , ~x + max
− λθ~1 + , ~y +
θ1 2 +
θ2 2 +
θ3 2 .
2
λ
2
λ
2
2λ
2λ
~
x∈K1
~
y ∈K2
Lemma 40 shows how to compute the subgradient of functions of the form f (~c) = max~x∈K h~c, ~xi
using the optimization oracle for K. The rest of the term are differentiable so its subgradient is just
the gradient. Hence, by addition rule for subgradients (Fact 37), we have a O(λ)-subgradient oracle
for fλ using a O()-optimization oracle for K1 and K2 . The result then follows from Lemma 38.
Theorem 49. Assume (11.2) and (11.3). Suppose that we have -optimization oracle for every
> 0. For 0 < δ < 1, we can find ~z ∈ Rn such that
max
~
x∈K1 ∩K2
and ~z − ~x
2
+ ~z − ~y
2
h~c, ~xi ≤ δ + h~c, ~zi
≤ δ for some ~x ∈ K1 and ~y ∈ K2 in time
nM
3
O(1) nM
O n (OOη (K1 ) + OOη (K2 )) log
+ n log
δ
δ
δ O(1)
where η = Ω nM
.
2
7
Proof. Setting λ = 40M
and = 107δM 6 in Lemma 47 we see that so long as we obtain any
δ2
approximate solution (θ~10 , θ~20 , θ~30 ) such that
hλ (θ~10 , θ~20 , θ~30 ) ≤
min
(θ~1 ,θ~2 ,θ~3 )∈Ω
hλ (θ~1 , θ~2 , θ~3 ) + ,
then we obtain the point we want. To apply Theorem 42, we use
(
hλ (θ~1 , θ~2 , θ~3 ) if (θ~1 , θ~2 , θ~3 ) ∈ Ω
h̃(θ~1 , θ~2 , θ~3 ) =
.
+∞
else
~ by using
Lemma 48 shows that for any γ > 0 we can obtain a (γ, γ)-separation oracle of hλ (θ)
2
sufficiently accurate optimization oracles. Since Ω is just a product of ` balls, we can produce a
separating hyperplane easily when (θ~1 , θ~2 , θ~3 ) ∈
/ Ω. Hence, we can obtain a (γ, γ)-separation oracle
64
~ For simplicity, we use θ~ to represent (θ~1 , θ~2 , θ~3 ). Note that B∞ (2M ) ⊇ Ω and therefore we
of h̃(θ).
can apply Theorem 42 with R = 2M to compute θ~0 such
0
~ ≤ γ + α max h̃(θ)
~ − min h̃(θ)
~
h̃(θ~ ) − min h̃(θ)
~
θ∈Ω
in time O nSOγ,γ log
and κ =
2M
MinWidth(Ω)
nκ
α
~
θ∈Ω
+ n3 logO(1)
nκ
α
~
θ∈Ω
where γ = Ω αMinWidth(Ω)/nO(1) = Ω αM/nO(1)
= O(1). Using λ ≥ 1 and M ≥ 1, we have
~ − min h̃(θ)
~ ≤ O λM 2 ≤ O
max h̃(θ)
~
θ∈Ω
Setting α = Θ
δ9
M 10
~
θ∈Ω
M4
δ2
.
with some small enough constant, we have that we can find θ~0 such that
4
M
0
~
~
hλ (θ ) ≤ min hλ (θ) + γ + αO
δ2
~
θ∈P
7
~ +O δ
= min hλ (θ)
M6
~
θ∈P
~ +
= min hλ (θ)
~
θ∈P
nM
δ
+ n3 logO(1)
nM
δ
where γ = Ω
δ O(1)
nM
. Lemma 48 shows that
δ O(1)
the cost of (γ, γ)-separation oracle is just O(OOη (K1 ) + OOη (K2 )) where η = Ω nM
.
in time O nSOγ,γ log
Remark 50. Note that the algorithm does not promise that we obtain a point close to K1 ∩ K2 .
It only promises to give a point that is close to both some point in K1 and some point in K2 . It
appears to the authors that a further assumption is needed to get a point close to K1 ∩ K2 . For
example, if K1 and K2 are two almost parallel lines, it would be difficult to get an algorithm that
does not depend on the angle. However, as far as we know, most algorithms tackling this problem
are pseudo-polynomial and have polynomial dependence on the angle. Our algorithm depends on
the logarithmic of the angle which is useful for combinatorial problems.
This reduction is very useful for problems in many areas including linear programming, semidefinite programming and algorithmic game theory. In the remainder of this section we demonstrate
its power by applying it to classical combinatorial problems.
There is however one issue with applying our cutting plane algorithm to these problems. As
with other convex optimization methods, only an approximately optimal solution is found. On the
other hand, typically an exact solution is insisted in combinatorial optimization. To overcome this
gap, we introduce the following lemma which (1) transforms the objective function so that there
is only one optimal solution and (2) shows that an approximate solution is close to the optimal
solution whenever it is unique. As we shall see in the next two subsections, this allows us to round
an approximate solution to an optimal one.
Lemma 51. Given a linear program minA~x≥~b ~cT ~x where ~x, ~c ∈ Zn , ~b ∈ Zm and A ∈ Zm×n .
Suppose {A~x ≥ ~b} is an integral polytope (i.e. all extreme points are integral) contained in the
set { ~x ∞ ≤ M }. Then we can find a random cost vector ~z ∈ Zn with ~z ∞ ≤ O(n2 M 2 ~c ∞ )
such that with constant probability, minA~x≥~b ~zT ~x has an unique minimizer ~x∗ and this minimizer
is one of the minimizer(s) of minA~x≥~b ~cT ~x. Furthermore, if there is an interior point ~y such that
~zT ~y < minA~x≥~b ~zT ~x + δ, then ~y − ~x∗ ∞ ≤ 2nM δ.
65
Proof. The first part of the lemma follows by randomly perturbing the cost vector ~c. We consider a new cost vector ~z = 100n2 M 2~c + ~r where each coordinate of ~r is sampled randomly from
{0, 1, · · · , 10nM }. [67, Lem 4] shows that the linear program minA~x≥~b ~zT ~x has a unique minimizer
with constant probability. Furthermore, it is clear that the minimizer of minA~x≥~b ~zT ~x is a minimizer
of minA~x≥~b ~cT ~x (as ~ri 100n2 M 2 |~ci |).
Now we show the second part of the lemma. Given an interior point ~y of the polytope
{A~x ≥ ~b},
P
we canP
write ~y as a convex combination of the vertices of {A~x ≥ ~b}, i.e. ~y =
ti~vi . Note that
T
~z ~y =
ti~zT ~vi . If all ~vi are not the minimizer, then ~zT ~vi ≥ OPT + 1 and hence ~zT ~y ≥ OPT + 1
which is impossible. Hence, we can assume that ~v1 is the minimizer. Hence, ~zT ~vi = OPT if i = 1
T
and ~zT ~vi ≥ OPT + 1 otherwise. We then
P have ~z ~y ≥ OPT + (1 − t1 ) which gives 1 − t1 < δ. Finally,
the claim follows from ~y − ~v1 ∞ ≤ i6=1 ti ~vi − ~v1 ∞ ≤ 2nM δ.
11.2
Matroid Intersection
Let M1 = (E, I1 ) and M2 = (E, I2 ) be two matroids sharing the same ground set. In this section
we consider the weighted matroid intersection problem
min w(S).
~
S∈I1 ∩I2
def P
where w
~ ∈ RE and w(S) = e∈S we .
For any matroid M = (E, I), it is well known that the polytope of all independent sets has the
following description [28]:
conv(I1 ) = {~x ∈ RE s.t. 0 ≤ x(S) ≤ r(S) for all S ⊆ E}
(11.12)
where r is the rank function for M , i.e. r(S) is the size of the largest independent set that is
a subset of S. Furthermore, the polytope of the matroid intersection satisfies conv(I1 ∩ I2 ) =
conv(I1 ) ∩ conv(I2 ).
It is well known that the optimization problem
min w(S) and min w(S)
S∈I1
S∈I2
can be solved efficiently by the greedy method. Given a matroid (polytope), the greedy method
finds a maximum weight independent subset by maintaining a candidate independent subset S and
iteratively attempts to add new element to S in descending weight. A element i is added to S
if S ∪ {i} is still independent. A proof of this algorithm is well-known and can be found in any
standard textbook on combinatorial optimization.
Clearly, the greedy method can be implemented by O(n) calls to the independence oracle (also
called membership oracle). For rank oracle, it requires O(r log n) calls by finding the next element
to add via binary search. Therefore, we can apply Theorem 49 to get the following result (note
that this algorithm is the fastest if r is close to n for the independence oracle).
Theorem 52. Suppose that the weights w
~ are integer with w
∞
≤ M . Then, we can find
S ∈ arg min w(S)
S∈I1 ∩I2
in time O nGO log (nM ) + n3 logO(1) (nM ) where GO is the cost of greedy method for I1 and I2 .
66
Proof. Applying Lemma 51, we can find a new cost ~z such that
min
~
x∈conv(I1 )∩conv(I2 )
~zT ~x
has an unique solution. Note that for any ~x ∈ conv(I1 ), we have ~x ∞ ≤ 1. Hence, applying
theorem 49, we can find ~q such that ~qT ~z ≤ OPT + and ~q − ~x 2 + ~q − ~y 2 ≤ for some
~x ∈ conv(I1 ) and ~y ∈ conv(I2 ). Using (11.12), we have the coordinate wise minimum of ~x, ~y , i.e.
min{~x, ~y }, is in conv(I1 ) ∩ conv(I2 ). Since ~q − min{~x, ~y } 2 ≤ ~q − ~x 2 + ~q − ~y 2 ≤ , we have
(min{~x, ~y })T ~z ≤ OPT + nM .
Hence, we have a feasible point min{~x, ~y } which has value close to optimal and Lemma 51 shows that
min(~x, ~y )−~s ∞ ≤ 2n2 M 2 where ~s is the optimal solution. Hence, we have ~q−~s ∞ ≤ 2n2 M 2 +.
Picking = 6n21M 2 , we have ~q − ~s ∞ < 21 and hence, we can get the optimal solution by rounding
to the nearest integer.
Since optimization over I1 and I2 involves applying
greedy method on certain vectors,
it takes
O(1)
3
only O(GO) time. Theorem 49 shows it only takes O nGO log (nM ) + n log
(nM ) in finding
such ~q.
This gives the following corollary.
Corollary 53. We have O(n2 Tind log(nM )+n3 logO(1) nM ) and O(nrTrank log n log(nM )+n3 logO(1) nM )
time algorithms for weighted matroid intersection. Here Tind is the time needed to check if a subset
is independent, and Trank is the time needed to compute the rank of a given subset.
Proof. By Theorem 52, it suffices to show that the optimization oracle for the matroid polytope can
be implemented in O(nTind ) and O(rTrank log n) time. This is simply attained by the well-known
greedy algorithm which iterates through all the positively-weighted elements in decreasing order,
and adds an element to our candidate independent set whenever possible.
For the independence oracle, this involves one oracle call for each element. On the other
hand, for the rank oracle, we can find the next element to add by binary search which takes time
O(Trank log n). Since there are at most r elements to add, we have the desired running time.
11.3
Submodular Flow
Let G = (V, E) be a directed graphwith |E| = m, let f be a submodular function on RV with
|V | = n, f (∅) = 0 and f (V ) = 0, and let A be the incidence matrix of G. In this section we
consider the submodular flow problem
Minimize
subject to
hc, ϕi
(11.13)
l(e) ≤ ϕ(e) ≤ u(e) ∀e ∈ E
x(v) = (Aϕ)(v) ∀v ∈ V
X
x(v) ≤ f (S) ∀S ⊆ V
v∈S
where c ∈ ZE , l ∈ ZE , u ∈ ZE where C = ~c ∞ and U = max u ∞ , l ∞ , maxS⊂V |f (S)| . Here
c is the cost on edges, ϕ is the flow on edges, l and u are lower and upper bounds on the amount
of flow on the edges, and x(v) is the net flow out of vertex v. The submodular function f upper
bounds the total net flow out of any subset S of vertices by f (S).
67
Theorem 54. Suppose that the cost vector ~c is integer weight with ~c ∞ ≤ C and the capacity
vector and the submodular function satisfy U = max u ∞ , l ∞ , maxS⊂V |f (S)| . Then, we can
solve the submodular flow problem (11.13) in time O n2 EO log(mCU ) + n3 logO(1) (mCU ) where
EO is the cost of function evaluation.
Proof. First, we can assume l(e) ≤ u(e) for every edge e, otherwise, the problem is infeasible. Now,
we apply a similar transformation in [49] to modify the graph. We create a new vertex v0 . For every
vertex v in V , we create a edge from v0 to v with capacity lower bounded by 0, upper bounded
by 4nU , and with cost 2mCU . Edmonds and Giles showed that the submodular flow polytope
is integral [29]. Hence, there is an integral optimal flow on this new graph. If the optimal flow
passes through the newly created edge, then it has cost at least 2mCU − mCU because the cost
of all other edges in total has at least −mCU . That means the optimal flow has the cost larger
than mCU which is impossible. So the optimal flow does not use the newly created edges and
vertex and hence the optimal flow in the new problem gives the optimal solution of the original
problem. Next, we note that for any ϕ on the original graph such that l(e) ≤ ϕ(e) ≤ u(e), we can
send suitable amount of flow from v0 to v to make ϕ feasible. Hence, this modification makes the
feasibility problem trivial.
Lemma 51 shows that we can assume the new problem has an unique solution and it only blows
up C by a (mU )O(1) factors.
Note that the optimal value is an integer and its absolute value at most mCU . By binary
search, we can assume we know the optimal value OPT. Now, we reduce the problem to finding
a feasible ϕ with {hd, ϕi ≤ OPT + } with determined later. Let P be the set of such ϕ. Note
that P = K1, ∩ K2, where
l(e) ≤ ϕ(e) ≤ u(e) ∀e ∈ E
V
K1, =
x ∈ R such that x(v) = (Aϕ)(v) ∀v ∈ V
for some ϕ ,
hd, ϕi ≤ OPT +
(
)
X
X
y ∈ RV such that
y(v) ≤ f (S) ∀S ⊆ V,
y(v) = f (V ) .
K2, =
v∈S
v∈V
P
Note that the extra condition v y(v) = f (V ) is valid because v y(v) = v (Aϕ)(v) = 0 and
f (V ) = 0, and K1, has radius bounded by O((mCU )O(1) ) and K2, has radius bounded by O(nU ).
Furthermore, for any vector ~c ∈ RV , we note that
P
P
max hc, xi =
x∈K1,
hc, xi
max
l≤ϕ≤u,hd,ϕi≤OPT+,x=Aϕ
=
max
hc, Aϕi
l≤ϕ≤u,hd,ϕi≤OPT+
=
max
AT c, ϕ .
l≤ϕ≤u,hd,ϕi≤OPT+
To solve this problem, again we can do a binary search on hd, ϕi and reduce the problem to
AT c, ϕ
max
l≤ϕ≤u,hd,ϕi=K
for some value of K. Since AT c is fixed, this is a linear program with only the box constraints
and an extra equality constraint. Hence, it can be solved in nearly linear time [76, Thm 17, ArXiv
v1]. As the optimization oracle for K1, involves only computing AT c and solving this simple linear
program, it takes only O(n2 logO(1) (mCU/)) time. On the other hand, since K2, is just a base
68
polyhedron, the optimization oracle for K2, can be done by greedy method and only takes O(nEO)
time.
Applying Theorem 49, we can find q such that q − x 2 + q − y 2 ≤ δ for some x ∈ K1, ,
y ∈ K2, and δ to be chosen later. According to the definition of K1, , there is ϕ such that
l(e) ≤ ϕ(e) ≤ u(e) and x(v) = (Aϕ)(v) for all v and hd, ϕi ≤ OPT + . Since y − x 2 ≤ 2δ, that
means |y(v) − (Aϕ)(v)| ≤ 2δ for all v.
• Case 1) If y(v) ≥ (Aϕ)(v), then we can replace y(v) by (Aϕ)(v), note that y is still in K2,
because of the submodular constraints.
• Case 2) If y(v) ≤ (Aϕ)(v), then we can send a suitable amount of flow from v0 to v to make
ϕ feasible y(v) ≤ (Aϕ)(v).
Note that under this modification, we increased the objective value by (δn)(2mCU ) because the
new edge cost 2mCU per unit of flow. Hence, we find a flow ϕ which is feasible in new graph
1
with objective value + (δn)(2mCU ) far from optimum value. By picking δ = 2mnCU
, we have the
1
value 2 far from OPT. Now, we use Lemma 51 to shows that when is small enough, i.e, (mCU
)c
1
∗
∗
for some constant c, then we can guarantee that y − x ∞ ≤ 4 where x is the optimal demand.
Now, we note that q − y 2 ≤ δ and we note that we only modify y by a small amount, we in fact
have q − x∗ ∞ < 21 . Hence, we can read off the solution x∗ by rounding q to the nearest integer.
Note that we only need to solve the problem K1, ∩ K2, to (mCU1 )Θ(1) accuracy and the optimization
O(1)
2
oracle for K1, and K2, takes
(mCU )) and O(nEO)
respectively. Hence, Theorem
time O(n log
O(1)
2
3
49 shows that it takes O n EO log(mCU ) + n log
(mCU ) time to find x∗ exactly.
After getting x∗ , one can find ϕ∗ by solving a min cost flow problem using interior point method
√
[74], which takes O(m n logO(1) (mCU )) time.
11.4
Affine Subspace of Convex Set
In this section, we give another example about using optimization oracle directly via regularization.
We consider the following optimization problem
max
~
x∈K and A~
x=~b
h~c, ~xi
(11.14)
where ~x, ~c ∈ Rn , K is a convex subset of Rn , A ∈ Rr×n and ~b ∈ Rm . We suppose that r n
and thus, the goal of this subsection is to show how to obtain an algorithm takes only Õ(r) many
iterations. To do this, we assume a slightly stronger optimization oracle for K:
Definition 55. Given a convex set K and δ > 0. A δ-2nd-order-optimization oracle for K is a
function on Rn such that for any input ~c ∈ Rn and λ > 0, it outputs ~y such that
2
2
max h~c, ~xi − λ ~x
≤ δ + h~c, ~y i − λ ~y .
~
x∈K
(2)
We denote by OOδ,λ (K) the time complexity of this oracle.
The strategy for solving this problem is very similar to the intersection problem and hence some
details are omitted.
69
Theorem 56. Assume that max~x∈K k~xk2 < M , ~b 2 < M , ~c 2 < M , A 2 < M and λmin (A) >
1/M . Assume that K ∩ {A~x = ~b} =
6 ∅ and we have -2nd-order-optimization oracle for every > 0.
For 0 < δ < 1, we can find ~z ∈ K such that
max
~
x∈K and A~
x=~b
and A~z − ~b
2
h~c, ~xi ≤ δ + h~c, ~zi
≤ δ. This algorithm takes time
nM
(2)
3
O(1) nM
O rOOη,λ (K) log
+ r log
δ
δ
where r is the number of rows in A, η =
δ Θ(1)
nM
and λ =
δ Θ(1)
.
nM
Proof. The proof is based on the minimax problem
def
OPTλ =
D
E 1
~x
min max h~c, ~xi + ~η , A~x − ~b −
λ
x∈K
η
~ ≤λ ~
2
2
2
where λ =
δ
nM
c
for some large constant c. We note that
D
E 1
~x
OPTλ = max min h~c, ~xi + ~η , A~x − ~b −
λ
~
x∈K η
~ ≤λ
2
2
2
= max h~c, ~xi − λ A~x − ~b
~
x∈K
2
−
1
2
~x 2 .
λ
Since λmin (A) > 1/M and the set K is bounded by M , one can show that the saddle point (~x∗ , ~η ∗ )
of the minimax problem gives a good enough solution ~x for the original problem for large enough
constant c.
For any ~η , we define
D
E 1
2
~xη~ = arg max h~c, ~xi + ~η , A~x − ~b −
~x 2 .
λ
~
x∈K
Since the problem is strongly concave in ~x, one can prove that
nM O(c)
∗
~xη~ − ~x 2 ≤
~η − ~η ∗
δ
2
.
D
E
Hence, we can first find an approximate minimizer of the function f (~η ) = max~x∈K h~c, ~xi+ ~η , A~x − ~b −
1
λ
2
~x 2 and use the oracle to find ~xη~ .
To find an approximate minimizer of f , we note that the subgradient of f can be found using
the optimization oracle similar to Theorem 49. Hence, the result follows from our cutting plane
method and the fact that ~η ∈ Rr .
Remark 57. In [74], they considered the special case K = {~x : 0 ≤ xi ≤ 1} and showed that it can
√
be solved in Õ( r) iterations using interior point methods. This gives the current fastest algorithm
for the maximum flow problem on directed weighted graphs. Our result generalizes their result to
any convex set K but with Õ(r) iterations. This suggests the following open problem: under what
condition on K can one optimize linear functions over affine subspaces of K with r constraints in
√
Õ( r) iterations?
70
Part III
Submodular Function Minimization
12
Introduction
Submodular functions and submodular function minimization (SFM) are fundamental to the field
of combinatorial optimization. Examples of submodular functions include graph cut functions, set
coverage function, and utility functions from economics. Since the seminal work by Edmonds in
1970 [27], submodular functions and the problem of minimizing such functions (i.e. submodular
function minimization) have served as a popular modeling and optimization tool in various fields
such as theoretical computer science, operations research, game theory, and most recently, machine
learning. Given its prevalence, fast algorithms for SFM are of immense interest both in theory and
in practice.
Throughout Part III, we consider the standard formulation of SFM: we are given a submodular
function f defined over the subsets of a n-element ground set. The values of f are integers, have
absolute value at most M , and are evaluated by querying an oracle that takes time EO. Our goal is
to produce an algorithm that solves this SFM problem, i.e. finds a minimizer of f , while minimizing
both the number of oracle calls made and the total running time.
We provide new O(n2 log nM · EO + n3 logO(1) nM ) and O(n3 log2 n · EO + n4 logO(1) n) time
algorithms for SFM. These algorithms improve upon the previous fastest weakly and strongly
polynomial time algorithms for SFM which had a a running time of O((n4 · EO + n5 ) log M ) [54]
and O(n5 · EO + n6 ) [90] respectively. Consequently, we improve the running times in both regimes
by roughly a factor of O(n2 ).
Both of our algorithms bear resemblance to the classic approach of Grötschel, Lovász and
Schrijver [49, 50] using the Lovász extension. In fact our weakly polynomial time algorithm directly
uses the Lovász extension as well as the results of Part II to achieve these results. Our strongly
polynomial time algorithm also uses the Lovász extension, along with more modern tools from the
past 15 years.
At a high level, our strongly polynomial algorithms apply our cutting plane method in conjunction with techniques originally developed by Iwata, Fleischer, and Fujishige (IFF) [56]. Our
cutting plane method is performed for enough iterations to sandwich the feasible region in a narrow
strip from which useful structural information about the minimizers can be deduced. Our ability
to derive the new information hinges on a significant extension of IFF techniques.
Over the past few decades, SFM has drawn considerable attention from various research communities, most recently in machine learning [11, 68]. Given this abundant interest in SFM, we hope
that our ideas will be of value in various practical applications. Indeed, one of the critiques against
existing theoretical algorithms is that their running time is too slow to be practical. Our contribution, on the contrary, shows that this school of algorithms can actually be made fast theoretically
and we hope it may potentially be competitive against heuristics which are more commonly used.
71
12.1
Previous Work
Here we provide a brief survey of the history of algorithms for SFM. For a more comprehensive
account of the rich history of SFM, we refer the readers to recent surveys [81, 55].
The first weakly and strongly polynomial time algorithms for SFM were based on the ellipsoid
method [65] and were established in the foundational work of Grötschel, Lovász and Schrijver in
1980’s [49, 50]. Their work was complemented by a landmark paper by Cunningham in 1985 which
provided a pseudopolynomial algorithm that followed a flow-style algorithmic framework [20]. His
tools foreshadowed much of the development in SFM that would take place 15 years later. Indeed,
modern algorithms synthesize his framework with inspirations from various max flow algorithms.
The first such “flow style” strongly polynomial algorithms for SFM were discovered independently in the breakthrough papers by Schrijver [93] and Iwata, Fleischer, and Fujishige (IFF) [56].
Schrijver’s algorithm has a running of O(n8 · EO + n9 ) and borrows ideas from the push-relabel
algorithms [46, 25] for the maximum flow problem. On the other hand, IFF’s algorithm runs in
time O(n7 log n·EO) and O(n5 ·EO log M ), and applies a flow-scaling scheme with the aid of certain
proximity-type lemmas as in the work of Tardos [100]. Their method has roots in flow algorithms
such as [52, 47].
Subsequent work on SFM provided algorithms with considerably faster running time by extending the ideas in these two “genesis” papers [93, 56] in various novel directions [107, 31, 54, 90, 60].
Currently, the fastest weakly and strongly polynomial time algorithms for SFM have a running
time of O((n4 · EO + n5 ) log M ) [54] and O(n5 · EO + n6 ) [90] respectively. Despite this impressive
track record, the running time has not been improved in the last eight years.
We remark that all of the previous algorithms for SFM proceed by maintaining a convex combination of O(n) BFS’s of the base polyhedron, and incrementally improving it in a relatively local
manner. As we shall discuss in Section 12.2, our algorithms do not explicitly maintain a convex
combination. This may be one of the fundamental reasons why our algorithms achieve a faster
running time.
Finally, beyond the distinction between weakly and strongly polynomial time algorithms for
SFM, there has been interest in another type of SFM algorithm, known as fully combinatorial
algorithms in which only additions and subtractions are permitted. Previous such algorithms
include [60, 54, 53]. We do not consider such algorithms in the remainder of the paper and leave it
as an open question if it is possible to turn our algorithms into fully combinatorial ones.
12.2
Our Results and Techniques
In Part III we show how to improve upon the previous best known running times for SFM by
a factor of O(n2 ) in both the strongly and weakly polynomial regimes. In Table 11 summarizes
the running time of the previous algorithms as well as the running times of the fastest algorithms
presented in this paper.
Both our weakly and strongly polynomial algorithms for SFM utilize a convex relaxation of the
submodular function, called the Lovász extension. Our algorithms apply our cutting plane method
from Part I using a separation oracle given by the subgradient of the Lovász extension. To the best
of the author’s knowledge, Grötschel, Lovász and Schrijver were the first to formulate this convex
optimization framework for SFM [49, 50].
For weakly polynomial algorithms, our contribution is two-fold. First, we show that cutting
plane methods such as Vaidya’s [105] can be applied to SFM to yield faster algorithms. Second,
as our cutting plane method, Theorem 42, improves upon previous cutting plane algorithms and
consequently the running time for SFM as well. This gives a running time of O(n2 log nM · EO +
72
Authors
Grötschel, Lovász,
Schrijver [49, 50]
Cunningham [20]
Schrijver [93]
Iwata, Fleischer,
Fujishige[56]
Iwata, Fleischer [31]
Years
Running times
1981,1988
e 5 · EO + n7 )[81]
O(n
1985
2000
O(M n6 log nM · EO)
O(n8 · EO + n9 )
O(n5 · EO log M )
O(n7 log n · EO)
O(n7 · EO + n8 )
O((n4 · EO + n5 ) log M )
O((n6 · EO + n7 ) log n)
O(n7 · EO + n8 )
O(n5 · EO + n6 )
O((n4 · EO + n5 ) log nM )
O((n5 · EO + n6 ) log n)
2
O(n log nM · EO + n3 logO(1) nM )
O(n3 log2 n · EO + n4 logO(1) n)
2000
2000
Iwata [54]
2003
Vygen [107]
Orlin [90]
2003
2007
Iwata, Orlin [60]
2009
Our algorithms
2015
Remarks
first weakly
and strongly
first pseudopoly
first combin. strongly
first combin. strongly
current best weakly
current best strongly
Table 11: Algorithms for submodular function minimization. Note that some of these algorithms
were published in both conferences and journals, in which case the year we provided is the earlier
one.
n3 logO(1) nM ), an improvement over the previous best algorithm by Iwata [54] by a factor of almost
O(n2 ).
Our strongly polynomial algorithms, on the other hand, require substantially more innovation.
We first begin with a very simple geometric argument that SFM can be solved in O(n3 log n · EO)
oracle calls (but in exponential time). This proof only uses Grunbaum’s Theorem from convex
geometry and is completely independent from the rest of the paper. It was the starting point of
e 3 · EO + nO(1) ) for submodular minimization
our method and suggests that a running time of O(n
is in principle achievable.
To make this existence result algorithmic, we first run cutting plane, Theorem 31, for enough
iterations such that we compute either a minimizer or a set P containing the minimizers that
fits within in a narrow strip. This narrow strip consists of the intersection of two approximately
parallel hyperplanes. If our narrow strip separates P from one of the faces xi = 0, xi = 1, we can
effectively eliminate the element i from our consideration and reduce the dimension of our problem
by 1. Otherwise a pair of elements p, q can be identified for which q is guaranteed to be in any
minimizer containing p (but p may not be contained in a minimizer). Our first algorithm deduces
e 4 · EO + n5 ) time
only one such pair at a time. This technique immediately suffices to achieve a O(n
e 3 · EO + n4 ) by
algorithm for SFM (See Section 15.3). We then improve the running time to O(n
showing how to deduce many such pairs simultaneously. Similar to past algorithms, this structural
information is deduced from a point in the so-called base polyhedron (See Section 13).
Readers well-versed in SFM literature may recognize that our strongly polynomial algorithms
are reminiscent of the scaling-based approach first used by IFF [56] and later in [54, 60]. While
both approaches share the same skeleton, there are differences as to how structural information
about minimizers is deduced. A comparison of our algorithms and previous ones are presented in
Section 16.
Finally, there is one more crucial difference between these algorithms which we believe is responsible for much of our speedup. One common feature shared by all the previous algorithms is
73
that they maintain a convex combination of O(n) BFS’s of the base polyhedron, and incrementally
improve on it by introducing new BFS’s by making local changes to existing ones. Our algorithms,
on the other hand, choose new BFS’s by the cutting plane method. Because of this, our algorithm
considers the geometry of the existing BFS’s where each of them has influences over the choice of
the next BFS. In some sense, our next BFS is chosen in a more “global” manner.
12.3
Organization
The rest of Part III is organized as follows. We first begin with a gentle introduction to submodular
functions in Section 13. In Section 14, we apply our cutting plane method to SFM to obtain a
faster weakly polynomial algorithms. In Section 15 we then present our results for achieving better
e 4 · EO + n5 ) algorithm is given before the
strongly polynomial algorithms, where a warm-up O(n
e 3 · EO + n4 ) algorithm. Finally, we end the part with a discussion and comparison
full-fledged O(n
between our algorithms and previous ones in Section 16.
We note that there are a few results in Part III that can be read fairly independently of the
rest of the paper. In Theorem 67 we show how Vaidya’s algorithm can be applied to SFM to
obtain a faster weakly polynomial running time. Also in Theorem 71 we present a simple geometric
argument that SFM can be solved with O(n3 log n · EO) oracle calls but with exponential time.
These results can be read with only a working knowledge of the Lovász extension of submodular
functions.
13
Preliminaries
Here we introduce background on submodular function minimization (SFM) and notation that we
use throughout Part III. Our exposition is kept to a minimal amount sufficient for our purposes.
We refer interested readers to the extensive survey by McCormick [81] for further intuition.
13.1
Submodular Function Minimization
Throughout the rest of the paper, let V = {1, ..., n} = [n] denote a ground set and let f : 2V −→ Z
denote a submodular function defined on subsets of this ground set. We use V and [n] interchangedef
def
def
ably and let [0] = ∅. We abuse notation by letting S + i = S ∪ {i} and S − i = S\{i} for an
element i ∈ V and a set S ⊆ 2V . Formally, we call a function submodular if it obeys the following
property of diminishing marginal differences:
Definition 58 (Submodularity). A function f : 2V −→ Z is submodular if f (T + i) − f (T ) ≤
f (S + i) − f (S) for any S ⊆ T and i ∈ V \T .
For convenience we assume without loss of generality that f (∅) = 0 by replacing f (S) by
def
f (S) − f (∅) for all S. We also let M = maxS∈2V |f (S)|.
The central goal of Part III is to design algorithms for SFM, i.e. computing the minimizer of
f . We call such an algorithm strongly polynomial if its running time depends only polynomially
on n and EO, the time needed to compute f (S) for a set S, and we call such an algorithm weakly
polynomial if it also depends polylogarithmically on M .
13.2
Lovász Extension
Our new algorithms for SFM all consider a convex relaxation of a submodular function, known as
the Lovász extension, and then carefully apply our cutting plane methods to it. Here we formally
introduce the Lovász extension and present basic facts that we use throughout Part III.
74
The Lovász extension of fˆ : [0, 1]n −→ R of our submodular function f is defined for all ~x by
def
fˆ(~x) = Et∼[0,1] [f ({i : xi ≥ t})],
where t ∼ [0, 1] is drawn uniformly at random from [0, 1]. The Lovász extension allows us to reduce
SFM to minimizing a convex function defined over the interior of the hypercube. Below we state
that the Lovász extension is a convex relaxation of f and that it can be evaluated efficiently.
Theorem 59. The Lovász extension fˆ satisfies the following properties:
1. fˆ is convex and min~x∈[0,1]n fˆ(~x) = minS⊂[n] f (S);
(
1
2. f (S) = fˆ(IS ), where IS is the characteristic vector for S, i.e. IS (i) =
0
if i ∈ S
;
if i ∈
/S
3. If S is a minimizer of f , then IS is a minimizer of fˆ;
def
4. Suppose x1 ≥ · · · ≥ xn ≥ xn+1 = 0, then
fˆ(~x) =
n
X
n
X
f ([i])(xi − xi+1 ) =
(f ([i]) − f ([i − 1]))xi .
i=1
i=1
Proof. See [50] or any standard textbook on combinatorial optimization, e.g. [94].
Next we show that we can efficiently compute a subgradient of the Lovász or alternatively, a
separating hyperplane for the set of minimizers of our submdoular function f . First we remind the
reader of the definition of a separation oracle, and then we prove the necessary properties of the
hyperplane, Theorem 61.
Definition 60 (separation oracle, Defintion 1 restated for Lovász extension). Given a point x̄ and
a convex function fˆ over a convex set P , ~aT ~x ≤ b is a separating hyperplane if ~aT x̄ ≥ b and any
minimizer x∗ of fˆ over P satisfies ~aT x∗ ≤ b.
Theorem 61. Given a point x̄ ∈ [0, 1]n assume without loss of generality (by re-indexing the
coordinates) that x̄1 ≥ · · · ≥ x̄n . Then the following inequality is a valid separating hyperplane for
~x and f :
n
X
(f ([i]) − f ([i − 1]))xi ≤ fˆ(x̄)
i=1
i.e., it satisfies the following:
P
1. (separating) x̄ lies on ni=1 (f ([i]) − f ([i − 1]))xi ≤ fˆ(x̄).
P
P
2. (valid) For any ~x, we have ni=1 (f ([i]) − f ([i − 1]))xi ≤ fˆ(~x). In particular, ni=1 (f ([i]) −
f ([i − 1]))x∗i ≤ fˆ(x̄) for any minimizer ~x∗ , i.e. the separating hyperplane does not cut out
any minimizer.
Moreover, such a hyperplane can be computed with n oracle calls to f and in time O(n · EO + n2 ).
75
P
Proof. Note that by Theorem 59 we have that i∈[n] (f ([i]) − f ([i − 1]))xi = fˆ(x̄) and thus the
hyperplane satisfies the separating condition. Moreover, clearly computing it only takes time O(n ·
EO + n2 ) as we simply need to sort the coordinates and evaluate f at n points, i.e. each of the [i].
All that remains is to show that the hyperplane satisfies the valid condition.
def
Let L(t) = {i : xi ≥ t}. Recall that fˆ(~x) = Et∼[0,1] [f (Lt )]. Thus fˆ(~x) can be written as a
P
P
convex combination fˆ(~x) = t αt f (L(t) ), where αt ≥ 0 and t αt = 1. However, by diminishing
marginal differences we see that for all t
X
X
(f ([i]) − f ([i − 1])) (IL(t) )i =
(f ([i]) − f ([i − 1]))
i∈L(t)
i∈[n]
≤
X
f ([i] ∩ L(t) ) − f ([i − 1] ∩ L(t) )
i∈L(t)
(t)
= f (L ) − f (∅) = f (L(t) )
and therefore since
X
i∈[n]
13.3
P
t αt IL(t)
= ~x we have
(f [i] − f ([i − 1])xi =
X
αt
t
n
X
X
αt f (L(t) ) = fˆ(~x).
(f ([i]) − f ([i − 1])) (IL(t) )i ≤
t
i=1
Polyhedral Aspects of SFM
Here we provide a natural primal dual view of SFM that we use throughout the analysis. We
provide a dual convex optimization program to minimizing the Lovász extension and provide several
properties of these programs. We believe the material in this section helps crystallize some of the
intuition behind our algorithm and we make heavy use of the notation presented in this section.
However, we will not need to appeal to the strong duality of these programs in our proofs.P
Consider the following primal and dual programs, where we use the shorthands y(S) = i∈S yi
def
and yi− = min{0, yi }. Here the primal constraints are often called the base polyhedron B(f ) = {~y ∈
Rn : y(S) ≤ f (S)∀S 6⊆ V, y(V ) = f (V )} and the dual program directly corresponds to minimizing
the Lovász extension and thus f .
Primal
maxy (V )
Dual
−
y(S) ≤ f (S)∀S 6⊆ V
y(V ) = f (V )
minfˆ(~x)
0 ≤ ~x ≤ 1
Theorem 62. ~h is a basic feasible solution (BFS) of the base polyhedron B(f ) if and only if
hi = f ({v1 , ..., vi }) − f ({v1 , ..., vi−1 })
for some permutation v1 , ..., vn of the ground set V . We call v1 , ..., vn the defining permutation of
~h. We call vi precedes vj for i < j.
This theorem gives a nice characterization of the BFS’s of B(f ). It also gives the key observation
underpinning our approach: the coefficients of each separating hyperplane in Theorem 61
precisely corresponds to a primal BFS (Theorem 62). Our analysis relies heavily on this
connection. We re-state Theorem 61 in the language of BFS.
76
Lemma 63. We have ~hT ~x ≤ fˆ(~x) for any ~x ∈ [0, 1]n and BFS ~h.
Proof. Any BFS is given by some permutation. Thus this is just Theorem 61 in disguise.
We also note that since the objective function of the primal program is non-linear, we cannot
say that the optimal solution to the primal program is a BFS. Instead we only know that it is a
convex combination of the BFS’s that satisfy the following property. A proof can be found in any
standard textbook on combinatorial optimization.
Theorem 64. The above primal andPdual programs have
duality gap. Moreover, there always
P no
(k)
(k)
(k)
~
exists a primal optimal solution ~y = k λ h with k λ = 1 (a convex combination of BFS
~h(k) ) s.t. any i with yi < 0 precedes any j with yj > 0 in the defining permutation for each BFS
~h(k) .
Our algorithms will maintain collections of BFS and use properties of ~h ∈ B(f ), i.e. convex
combination of BFS. To simplify our analysis at several points we will want to assume that such
a vector ~h ∈ B(f ) is non-degenerate, meaning it has both positive and negative entries. Below, we
prove that such degenerate points in the base polytope immediately allow us to trivially solve the
SFM problem.
Lemma 65 (Degenerate Hyperplanes). If ~h ∈ B(f ) is non-negative then ∅ is a minimizer of f and
if ~h is non-positive then V is a minimizer of f .
Proof. While this follows immediately from Theorem 64, for completeness we prove this directly.
Let S ∈ 2V be arbitrary. If ~h ∈ B(f ) is non-negative then by the we have
X
f (S) ≥ ~h(S) =
hi ≥ 0 = f (∅) .
i∈S
On the other hand if ~h is non-positive then by definition we have
X
X
f (S) ≥ ~h(S) =
hi ≥
hi = h(V ) = f (V ) .
i∈S
14
i∈V
Improved Weakly Polynomial Algorithms for SFM
In this section we show how our cutting plane method can be used to obtain a O(n2 log nM · EO +
n3 logO(1) nM ) time algorithm for SFM. Our main result in this section is the following theorem,
which shows how directly applying our results from earlier parts to minimize the Lovász extension
yields the desired running time.
Theorem 66. We have an O(n2 log nM · EO + n3 logO(1) nM ) time algorithm for submodular
function minimization.
Proof. We apply Theorem 42 to the Lovász extension fˆ : [0, 1]n −→ R with the separation oracle
given by Theorem 61. fˆ fulfills the requirement on the domain as its domain Ω = [0, 1]n is symmetric
about the point (1/2, . . . , 1/2) and has exactly 2n constraints.
In the language of Theorem 42, our separation oracle is a (0, 0)-separation oracle with η = 0
and δ = 0.
77
We first show that δ = 0. Firstly, our separating hyperplane can be written as
n
n
X
X
(f ([i]) − f ([i − 1]))xi ≤ fˆ(x̄) =
(f ([i]) − f ([i − 1]))x̄i ,
i=1
i=1
where the equality follows from Theorem 59. Secondly, for any ~x with fˆ(~x) ≤ fˆ(x̄) we have by
Theorem 61 that
n
X
(f ([i]) − f ([i − 1]))xi ≤ fˆ(~x) ≤ fˆ(x̄)
i=1
which implies that ~x is not cut away by the hyperplane.
Next we show that η = 0. Our separating hyperplane induces a valid halfspace whenever it is
not nonzero, i.e. f ([i]) 6= f ([i − 1]) for some
Pi. In the case that it is zero f ([i]) = f ([i − 1])∀i, by
the same argument above, we have fˆ(x̄) = ni=1 (f ([i]) − f ([i − 1]))x̄i = 0 and
fˆ(~x) ≥
n
X
(f ([i]) − f ([i − 1]))xi = 0 = fˆ(x̄).
i=1
In other words, x̄ is an exact minimizer, i.e. η = 0.
Note that fˆ(~x) = Et∼[0,1] [f ({i : xi ≥ t})] ≤ M as M = maxS |f (S)|. Now plugging in α =
in the guarantee of Theorem 31, we can find a point x∗ such that
1
∗
fˆ(x ) − min fˆ(~x) ≤
max fˆ(~x) − min fˆ(~x)
4M ~x∈[0,1]n
~
x∈[0,1]n
~
x∈[0,1]n
1
(2M )
≤
4M
< 1
1
4M
We claim that mint∈[0,1] f ({i : x∗i ≥ t}) is minimum. To see this, recall from 59 that fˆ has an
integer minimizer and hence min~x∈[0,1]n fˆ(~x) = minS f (S). Moreover, fˆ(x∗ ) is a convex combination
of f ({i : x∗i ≥ t}) which gives
1 > fˆ(x∗ ) − min fˆ(~x) = fˆ(x∗ ) − min f (S) ≥ min f ({i : x∗i ≥ t}) − min f (S).
~
x∈[0,1]n
S
t∈[0,1]
S
Since f is integer-valued, we must then have mint∈[0,1] f ({i : x∗i ≥ t}) = minS f (S) as desired.
Since our separation oracle can be computed by n oracle calls and runs in time O(n · EO + n2 ), by
Theorem 42 the overall running time is then O(n2 log nM · EO + n3 logO(1) nM ) as claimed.
Needless to say the proof above completely depends on Theorem 42. We remark that one
can use the Vaidya’s cutting plane instead of ours to get a time complexity O(n2 log nM · EO +
nω+1 logO(1) n · log M ). There is actually an alternate argument that gives a time complexity of
O(n2 log M · EO + nO(1) · log M ). Thus it requires slightly fewer oracle calls at the expense of
slower running time. A proof is offered in this section, which can be skipped without any risk of
discontinuation. This proof relies the following cutting plane method.
Theorem 67 ([13] ). Given any convex set K ⊂ [0, 1]n with a separation oracle of cost SO, in
time O(kSO + knO(1) ) one can find either find a point ~x ∈ K or find a polytope P such that K ⊂ P
k
and the volume of K is at most 32 .
78
k
The Theorem allows us to decrease the volume of the feasible region by a factor of 23 after
k iterations. Similar to above, we apply cutting plane to minimize fˆ over the hypercube [0, 1]n for
O(n log M ) iterations, and outputs any integral point in the remaining feasible region P .
Lemma 68. Let x∗ achieve the minimum function value fˆ(x∗ ) among the points used to query the
separation oracle. Then
1. x∗ ∈ P (k) , the current feasible region.
2. Any ~x with fˆ(~x) ≤ fˆ(x∗ ) belongs to P (k) .
3. suppose x∗i1 ≥ · · · ≥ x∗in and let Sj = {i1 , . . . , ij }. Then Sl ∈ arg minSj f (Sj ) also belongs to
P (k) .
Proof. For any separating hyperplane ~hT x ≤ fˆ(x̄) given by x̄, we have by Lemma 63 that ~hT x∗ ≤
fˆ(x∗ ). Since fˆ(x∗ ) is the minimum among all fˆ(x̄), ~hT x∗ ≤ fˆ(x̄) and hence x∗ is not removed by
any new separating hyperplane. In other words, x∗ ∈ P (k) . The argument for (2) is analogous.
For (3), recall that by the definition of Lovász extension fˆ(x∗ ) is a convex combination of f (Sj )
and thus the indicator variable ISl for Sl satisfies f (ISl ) ≤ fˆ(x∗ ). By Lemma 63 again, this implies
~hT IS ≤ f (IS ) ≤ fˆ(x∗ ) ≤ fˆ(x̄) for any separating hyperplane ~hT x ≤ fˆ(x̄).
l
l
Theorem 69. Suppose that we run Cutting Plane in Theorem 67 for O(n log M ) iterations. Then
Sl from the last lemma also minimizes f .
Proof. We use the notations from the last lemma. After k = Kn log2/3 M iterations, the volume of
the feasible region P (k) is at most 1/M Kn . By the last lemma, ISl ∈ P (k) .
Suppose for the sake of contradiction that S minimizes f but f (S) < f (Sl ). Since f is integerdef
def
valued, f (S) + 1 ≤ f (Sl ). Let r = 1/6M . Consider the set B = {~x : 0 ≤ xi ≤ r ∀i ∈
/ S, 1 − r ≤
xi ≤ 1 ∀i ∈ S}. We claim that for ~x ∈ B,
fˆ(~x) ≤ f (S) + 1.
To show this, note that f ({i : xi ≥ t}) = f (S) for r < t ≤ 1 − r as xi ≤ r for i ∈
/ S and
xi ≥ 1 − r for i ∈ S. Now using conditional probability and |f (T )| ≤ M for any T ,
fˆ(~x) = Et∼[0,1] [f ({i : xi ≥ t})]
= (1 − 2r) E[f ({i : xi ≥ t})|r < t ≤ 1 − r] +
r (E[f ({i : xi ≥ t})|0 ≤ t ≤ r] + E[f ({i : xi ≥ t})|1 − r ≤ t ≤ 1]])
= (1 − r) f (S) + r (E[f ({i : xi ≥ t})|0 ≤ t ≤ r + E[f ({i : xi ≥ t})|1 − r ≤ t ≤ 1]])
≤ (1 − 2r) f (S) + 2rM
≤ f (S) + 4rM
≤ f (S) + 1
But now B ⊆ P (k) as fˆ(~x) ≤ f (S) + 1 ≤ f (Sl ) and by (2) of the last lemma. This would lead to a
contradiction since
1
1
vol(B) =
> Kn ≥ vol(P (k) )
(6M )n
M
for sufficiently large K.
79
Corollary 70. There is an O(n2 log M · EO + nO(1) log M ) time algorithm for submodular function
minimization.
Proof. This simply follows from the last lemma, Theorem 67, and the fact that our separation
oracle runs in time O(n · EO + n2 ).
Curiously, we obtained O(log M ) rather than O(log nM ) as in our algorithm. We leave it as an
open problem whether one can slightly improve our running time to O(n2 log M · EO + n3 logO(1) n ·
log M ). The rest of this paper is devoted to obtaining better strongly polynomial running time.
15
Improved Strongly Polynomial Algorithms for SFM
e 3 · EO + n4 )
In this section we show how our cutting plane method can be used to obtain a O(n
5
6
time algorithm for SFM, which improves over the currently fastest O(n · EO + n ) time algorithm
by Orlin.
15.1
Improved Oracle Complexity
We first present a simple geometric argument that f can be minimized with just O(n3 log n ·
EO) oracle calls. While this is our desired query complexity (and it improves upon the previous
best known bounds by a factor of O(n2 ) unfortunately the algorithm runs in exponential time.
Nevertheless, it does provide some insight into how our more efficient algorithms should proceed
and it alone, does suggests that information theoretically, O(n3 log n·EO) calls suffice to solve SFM.
In the rest of the paper, we combine this insight with some of the existing SFM tools developed
over the last decade to get improved polynomial time algorithms.
Theorem 71. Submodular functions can be minimized with O(n3 log n · EO) oracle calls.
Proof. We use the cutting plane method in Theorem 67 with the separation oracle given by Theorem 61. This method reduce the volume of the feasible region by a factor of ( 32 )k after k iterations
if the optimal has not found yet.
Now, we argue that after O(n log n) iterations of this procedure we have either found a minimizer
of f or we have enough information to reduce the dimension of the problem by 1. To see this, first
note that if the separation oracle ever returns a degenerate hyperplane, then by Lemma 65 then
either ∅ or V is the minimizer, which we can determine in time O(EO + n). Otherwise, after
100n log n iterations, our feasible region P must have a volume of at most 1/n10n . In this case, we
claim that the remaining integer points in P all lie on a hyperplane. This holds, as if this was not
the case, then there is a simplex 4, with integral vertices v0 , v1 , . . . , vn , contained in P . But then
vol(P ) ≥ vol(4) =
1
1
|det (v1 − v0 v2 − v0 . . . vn − v0 )| ≥
n!
n!
where the last inequality holds since the determinant of an integral matrix is integral, yielding a
contradiction.
In other words after O(n log n) iterations, we have reduced the dimension of all viable solutions
by at least 1. Thus, we can recurse by applying the cutting plane method to the lower dimensional
feasible region, i.e. P is (replaced by) the convex combination of all the remaining integer points.
There is a minor technical issue we need to address as our space is now lower dimensional and the
starting region is not necessarily the hypercube anymore and the starting volume is not necessarily
equal to 1.
80
We argue that the starting volume is bounded by nO(n) . If this is indeed the case, then our
previous argument still works as the volume goes down by a factor of 1/nO(n) in O(n log n) iterations.
√
Let v ∈ P be an integer point. Now the dim(P )-dimensional ball of radius n centered at v
√
must contain all the other integer points in P as any two points of {0, 1}n are at most n apart.
Thus the volume of P is bounded by the volume of the ball which is nO(n) . Now to get the volume
down to 1/n10n , the number of iterations is still O(n log n).
In summary, we have reduced our dimension by 1 using O(n log n) iterations which requires
O(n2 log n · EO) oracle calls (as each separating hyperplane is computed with n · EO oracle calls).
This can happen at most n times. The overall query complexity is then O(n3 log n · EO).
Note that the minimizer ~x obtained may not be integral. This is not a problem as the definition
of Lovász extension implies that if fˆ(~x) is minimal, then f ({i : xi ≥ t}) is minimal for any t ∈ [0, 1].
We remark that this algorithm does not have a polynomial runtime. Even though all the integral
vertices of P lie on a hyperplane, the best way we know of that identifies it takes exponential time
by checking for all the integer points {0, 1}n .
Remark 72. Note that this algorithm works for minimizing any convex function over the hypercube
that obtains its optimal value at a vertex of the hypercube. Formally, our proof of Theorem 71
holds whenever a function f : 2V −→ Rn admits a convex relaxation fˆ with the following properties:
1. For every S ⊆ V , fˆ(IS ) = f (S).
P
P
αS = 1, |S| =
2. Every fˆ(~x) can be written as a convex combination S∈S αS f (S), where
O(n), and S can be computed without any oracle call.
3. A subgradient ∂ fˆ(~x) of fˆ at any point ~x ∈ [0, 1]n can be computed with O(n · EO) oracle
calls.
In this case, the proof of Theorem 71, implies that fˆ and f can be minimized with O(n3 log n · EO)
oracle calls by using the separating hyperplane ∂ fˆ(x̄)T (~x − x̄) ≤ 0.
15.2
Technical Tools
To improve upon the running time of the algorithm in the previous section, we use more structure
of our submodular function f . Rather than merely showing that we can decrease the dimension of
our SFM problem by 1 we show how we can reduce the degrees of freedom of our problem in a more
principled way. In Section 15.2.1 we formally define the abstraction we use for this and discuss how
to change our separation oracle to accommodate this abstraction, and in Section 15.2.2 we show
how we can deduce these constraints. These tools serve as the foundation for the faster strongly
polynomial time SFM algorithms we present in Section 15.3 and Section 15.4.
15.2.1
SFM over Ring Family
For the remainder of the paper we consider a more general problem than SFM in which we wish to
compute a minimizer of our submodular function f over a ring family of the ground set V = [n]. A
ring family F is a collection of subsets of V such that for any S1 , S2 ∈ F, we have S1 ∪S2 , S1 ∩S2 ∈ F.
Thus SFM corresponds to the special case where F consists of every subset of V . This generalization
has been considered before in the literature and was essential to the IFF algorithm.
It is well known that any ring family F over V can be represented by a directed graph D = (V, A)
where S ∈ F iff S contains all of the descendants of any i ∈ S. An equivalent definition is that for
81
any arc (i, j) ∈ A, i ∈ S implies j ∈ S. It is customary to assume that A is acyclic as any (directed)
cycle of A can be contracted (see section 15.3.1).
We denote by R(i) the set of descendants of i (including i itself) and Q(i) the set of ancestors
of i (including i itself). Polyhedrally, an arc (i, j) ∈ A can be encoded as the constraint xi ≤ xj as
shown by the next lemma.
Lemma 73. Let F be a ring family over V and D = (V, A) be its directed acyclic graph representation. Suppose f : V −→ R is submodular with Lovász extension fˆ. Then the characteristic vector
IS of any minimizer S = arg minS∈F f (S) over F is also the solution to
minfˆ(~x)
xi ≤ xj ∀(i, j) ∈ A
(15.1)
0 ≤ ~x ≤ 1
Proof. Let x∗ be a minimizer, and L(t) = {i : x∗i ≥ t}. It is easy to check that the indicator
variable IL(t) satisfies (15.1) since x∗ does. Moreover, recall that fˆ(x∗ ) = Et∼[0,1] [f (Lt )]. Thus
P
P
fˆ(x∗ ) can be written as a convex combination fˆ(x∗ ) = t αt f (L(t) ) = t αt fˆ(IL(t) ), where αt > 0
P
and t αt = 1. Thus all such fˆ(IL(t) ) are minimal, i.e. (15.1) has no “integrality gap”.
We also modify our separation oracle to accommodate for this generalization as follows. Before
doing so we need a definition which relates our BFS to the ring family formalism.
Definition 74. A permutation (v1 , . . . , vn ) of V is said to be consistent with an arc (i, j) if j
precedes i in (v1 , . . . , vn ). Similarly, a BFS of the base polyhedron is consistent with (i, j) if j
precedes i in its defining permutation. (v1 , . . . , vn ) (or a BFS) is consistent with A if it is consistent
with every (i, j) ∈ A.
Readers may find it helpful to keep in mind the following picture which depicts the relative
positions between R(i), i, Q(i) in the defining permutation of ~h that is consistent with A:
· · · · · · R(i)\{i} · · · · · · i · · · · · · Q(i)\{i} · · · · · ·
In Theorem 61, given x̄ ∈ [0, 1]n our separating hyperplane is constructed by sorting the entries
of x̄. This hyperplane is associated with some BFS ~h of the base polyhedron. As we shall see
towards the end of the section, we would like ~h to be consistent with every arc (i, j) ∈ A.
This task is easy initially as x̄ satisfies xi ≤ xj for (i, j) ∈ A for the starting polytope of (15.1).
If xi < xj , nothing special has to be done as j must precede i in the ordering. On the other hand,
whenever xi = xj , we can always break ties by ranking j ahead of i.
However, a technical issue arises due to the fact that our cutting plane algorithm may drop
constraints from the current feasible region P . In other words, x̄ may violate xi ≥ 0, xj ≤ 1 or
xi ≤ xj if it is ever dropped. Fortunately this can be fixed by reintroducing the constraint. We
summarize the modification needed in the pseudocode below and formally show that it fulfills our
requirement.
Lemma 75. Our modified separation oracle returns either some BFS ~h = 0 or a valid separating
hyperplane, i.e.
1. x̄ either lies on the separating hyperplane or is cut away by it.
2. Any minimizer of (15.1) is not cut away by the separating hyperplane.
82
Algorithm 5: Modified Separation Oracle
Input: x̄ ∈ Rn and the set of arcs A
if x̄i < 0 for some i then
Output: xi ≥ 0
else if x̄j > 1 for some j then
Output: xj ≤ 1
else if x̄i > x̄j for some (i, j) ∈ A then
Output: xi ≤ xj
else
Let i1 , . . . , in be a permutation of V such that x̄i1 ≥ . . . ≥ x̄in and for all (i, j) ∈ A, j
precedes i in i1 , . . . , in .
Output: ~hT ~x ≤ fˆ(x̄), where ~h is the BFS defined by the permutation i1 , . . . , in .
Such a hyperplane can be computed with n oracle calls to f and in time O(n · EO + n2 ).
Proof. If we get xi ≥ 0, xj ≤ 1 or xi ≤ xj (if loop or the first two else loops), then clearly x̄ is cut
away by it and any minimizer must of course satisfy xi ≥ 0, xj ≤ 1 and xi ≤ xj as they are the
constraints in (15.1). This proves (1) and (2) for the case of getting xi ≥ 0, xj ≤ 1 or xi ≤ xj .
Thus it remains to consider the case ~hT ~x ≤ fˆ(x̄) (last else loop). First of all, x̄ lies on it as
fˆ(x̄) = ~hT x̄. This proves (1). For (2), we have from Lemma 63 that ~hT ~x ≤ fˆ(~x). If x∗ is a
minimizer of (15.1), we must then have ~hT x∗ ≤ fˆ(x∗ ) ≤ fˆ(x̄) as x̄ is also feasible for (15.1).
Finally we note that the running time is self-evident.
We stress again that the main purpose of modifying our separation oracle is to ensure that any
BFS ~h used to define a new separating hyperplane must be consistent with every (i, j) ∈ A.
15.2.2
Identifying New Valid Arcs
The reason for considering the ring family generalization of SFM is that our algorithms (and some
previous algorithms too) work by adding new arcs to our digraph D. This operation yields a
strongly polynomial algorithm since there are only 2 · n2 possible arcs to add. Of course, a new arc
(i, j) is valid only if i ∈ Smin =⇒ j ∈ Smin for some minimizer Smin . Here we show how to identify
such valid arcs by extracting information from certain nice elements of the base polyhedron.
This is guaranteed by the next four lemmas, which are stated in a way different from previous
works e.g. our version is extended to the ring family setting. This is necessary as our algorithms
require a more general formulation. We also give a new polyhedral proof, which is mildly simpler
than the previous combinatorial proof. On the other hand, Lemma 80 is new and unique to our
e 3 · EO + n4 ) time algorithm.
work. It is an important ingredient of our O(n
Recall that each BFS of the base polyhedron is defined by some permutation of the ground set
elements.
First, we prove the following two lemmas which show that should we ever encounter a nondegenerate point in the base polytope with a coordinate of very large value, then we can immediately
conclude that that coordinate must be or must not be in solution to SFM over the ring family.
Lemma 76. If ~y ∈ B(f ) is non-degenerate and satisfies y i > −(n − 1) minj y j , then i is not in any
minimizer of f (over the ring family A).
83
Proof. We proceed by contradiction and suppose that S is a minimizer of f that contains i. Now
since ~y is non-degenerate we know that minj yj ≤ 0 and by the definition of ~y we have the following
contradiction
X
0 < yi + (n − 1) min yj ≤
yj = ~y (S) ≤ f (S) ≤ f (∅) = 0 .
j
j∈S
Lemma 77. If ~y ∈ B(f ) is non-degenerate and satisfies y i < −(n − 1) maxj y j , then i is in every
minimizer of f (over the ring family A).
Proof. We proceed by contradiction and suppose that S is a minimizer of f that does not contain
i. Now since ~y is non-degenerate we know that maxj yj ≥ 0 and therefore
X
X
X
X
X
yj < −(n − 1) max yj +
yj + (|V | − |S| − 1) max yj ≤
yj .
yj = yi +
yj +
j∈[n]
j∈S
j∈V −(S+i)
j
j
j∈S
j∈S
However by the definition of ~y we have
X
X
yj = ~y (S) ≤ f (S) ≤ f (V ) =
yj .
j∈S
j∈[n]
Thus we have a contradiction and the result follows.
Now we are ready to present conditions under which a new valid arc can be added. We begin
def
def
with a simple observation. Let upper(i) = f (R(i)) − f (R(i) − i) and lower(i) = f (V \Q(i) + i) −
f (V \Q(i)). As the names suggest, they bound the value of hi for any BFS used.
Lemma 78. For any BFS ~h used to construct a separating hyperplane given by our modified
separation oracle, we have lower(i) ≤ hi ≤ upper(i).
Proof. Note that by Lemma 75, ~h is consistent with every (j1 , j2 ) ∈ A and hence i must precede
Q(i) and be preceded by R(i). Let S be the set of elements preceding i in the defining permutation
of ~h. Then hi = f (S + i) − f (S) ≤ f (R(i)) − f (R(i) − i) because of diminishing return and
R(i) − i ⊆ S. The lower bound follows from the same argument as Q(i) − i comes after i, and so
Q(i) ⊆ V \S.
In the following two lemmas, we show that if upper(i) is ever sufficiently positive or lower(i)
is sufficiently negative, then we find a new arc.
While these lemmas may appear somewhat technical but actually has an intuitive interpretation.
Suppose an element p is in a minimizer Smin of f over the ring family D. Then R(p) must also be
part of Smin . Now if f (R(p)) is very large relative to f (R(p) − p), there should be some element
q ∈ Smin \R(p) compensating for the discrepancy. The lemma says that such an element q can in
fact be found efficiently.
P
Lemma 79 (new arc). Let ~y = k λ(k) ~y (k) be a non-degenerate convex combination of O(n) base
polyhedron BFS’s ~y (k) which are consistent with every arc (i, j) ∈ A. If some element p satisfies
upper(p) > n4 max yj , then we can find, using O(n·EO) oracle calls and O(n2 ) time, some q ∈
/ R(p)
such that the arc (p, q) is valid, i.e. if p is in a minimizer, then so is q.
84
Proof. If max yj < 0 then we are immediately done by Lemma 65. We assume max yj ≥ 0 in
the proof. For all k let ~y 0(k) be the BFS obtained by taking the defining permutation of ~y (k) and
moving R(p) to the front while preserving the relative ordering of R(p) within each permutation).
def P
(k)
Furthermore, let ~y 0 = k λ(k) ~y 0(k) . Then since y 0 p = f (R(p)) − f (R(p) − p) = upper(p) we have
upper(p) = yp0 = f (R(p)) − f (R(p) − p). Moreover,
yj0 ≥ yj
∀j ∈ R(p) and yj0 ≤ yj
∀j ∈
/ R(p)
(15.2)
by diminishing marginal return.
Now, suppose p is in a minimizer Smin . Then R(p) ⊆ Smin by definition. We then define
f 0 (S) = f (S ∪ R(p)) for S ⊆ V \R(p). It can be checked readily that f 0 is submodular and
Smin \R(p) is a minimizer of f 0 (over the corresponding ring family). Note that now ~yV0 \R(p) (the
restriction of ~y 0 to V \R(p)) is a convex combination of the BFS’s of the base polyhedron B(f 0 ) of
f 0 . We shall show that ~yV0 \R(p) has the desired property in Lemma 77.
Note that y 0 (V \R(p) + p) ≤ y(V \R(p) + p) since
y 0 (V \R(p) + p) = y 0 (V ) − y 0 (R(p) − p) = y(V ) − y 0 (R(p) − p) ≤ y(V ) − y(R(p) − p) = y(V \R(p) + p).
But now since ~y is non-degenerate maxj yj ≥ 0 and therefore
y 0 (V \R(p)) ≤ y(V \R(p) + p) − yp0
= y(V \R(p) + p) − (f (R(p)) − f (R(p) − p))
(15.3)
≤ n max yj − (f (R(p)) − f (R(p) − p))
< (n − n4 ) max yj
Therefore by the Pigeonhole Principle some q ∈
/ R(p) must satisfy
yq0 < (n − n4 ) max yj /(n − 1)
= −(n3 + n2 + n) max yj
≤ −(n3 + n2 + n) max yj
j ∈R(p)
/
≤ −(n + n + n) max yj0
3
2
j ∈R(p)
/
by (15.2)
By Lemma 77, this q must be in any minimizer of f 0 . In other words, whenever p is in a minimizer
of f , then so is q.
Note however that computing all ~y 0 would take O(n2 ) oracle calls in the worst case as there are
O(n) ~y 0(k) ’s. We use the following trick to identify some q with yq0 < −(n − 1) max yj using just
O(n) calls. The idea is that we actually only want to have sufficient decreases in y 0 (V \R(p)) which
can be accomplished by having a large corresponding decrease in some ~y 0(k) .
For each k, by the same argument above (see (15.3))
y 0(k) (V \R(p)) − y (k) (V \R(p)) ≤ y (k)
p − (f (R(p)) − f (R(p) − p))
(k)
The “weighted decrease” λ(k) y p − (f (R(p)) − f (R(p) − p)) for ~y 0(k) sum up to
X
(15.4)
= yp − (f (R(p)) − f (R(p) − p)) < (1 − n4 ) max yj
λ(k) y (k)
−
(f
(R(p))
−
f
(R(p)
−
p))
p
Thus by the Pigeonhole Principle, some l will have
λ(l) y (l)
−
(f
(R(p))
−
f
(R(p)
−
p))
< (1 − n4 ) max yj /O(n) < −n2 max yj .
p
85
P
For this ~y (l) we compute ~y 0(l) . We show that ~y 00 = λ(l) ~y 0(l) + k6=l λ(k) ~y (k) has the same property
as ~y 0 above.
X
y 00 (V \R(p)) = λ(l) y 0(l) (V \R(p)) +
λ(k) y (k) (V \R(p))
k6=l
= y(V \R(p)) + λ(l) y 0(l) (V \R(p)) − y (l) (V \R(p))
≤ y(V \R(p)) + λ(l) y (l)
p − (f (R(p)) − f (R(p) − p))
by (15.4)
< (n − 1) max yj − n2 max yj
< (n − n2 ) max yj
Then some q ∈ V \R(p) must satisfy
yq00 <
n − n2
max yj = −n max yj
n−1
P
That is, the arc (p, q) is valid. This takes O(n) oracle calls as given ~y = k λ(k) ~y (k) , computing
~y 00 requires knowing only f (R(p)), f (R(p) − p), and ~y 0(l) which can be computed from ~y (l) with n
oracle calls. The runtime is O(n2 ) which is needed for computing ~y 00 .
P
Lemma 80. Let ~y = k λ(k) ~y (k) be a non-degenerate convex combination of base polyhedron BFS
~y (k) which is consistent with every arc (i, j) ∈ A. If lower(p) < n4 min yj , then we can find, using
O(n · EO) oracle calls and O(n2 ) time, some q ∈
/ Q(p) such that the arc (q, p) is valid, i.e. if p is
not in a minimizer, then q is not either.
Proof. It is possible to follow the same recipe in the proof of Lemma 79 but using Lemma 76 instead
of Lemma 77. Here we offer a proof which directly invokes Lemma 77 on a different submodular
function.
def
Let g be defined by g(S) = f (V \S) for any S, and Ag be the set of arcs obtained by reversing
the directions of the arcs of A. Consider the problem of minimizing g over the ring family Ag .
Using subscripts to avoid confusion with f and g, e.g. Rg (i) is the set of descendants of i w.r.t.
Ag , it is not hard to verify the following:
• g is submodular
• Rg (i) = Qf (i)
• g(Rg (p)) − g(Rg (p) − p) = − (f (V \Qf (p) + p) − f (V \Qf (p)))
• −~y (k) is a BFS of B(g) if and only if ~y (k) is a BFS of B(f )
• max(−yj ) = − min yj
By using the above correspondence and applying Lemma 79 to g and Ag , we can find, using O(n)
oracle calls and O(n2 ) time, some q ∈
/ Rg (p) = Q(p) such that the arc (p, q) is valid for g and Ag .
In other words, the reverse (q, p) will be valid for f and A.
These lemmas lay the foundation of our algorithm. They suggests that if the positive entries
of a point in the base polyhedron are small relative to some upper(p) = f (R(p)) − f (R(p) − p), a
new arc (p, q) can be added to A. This can be seen as a robust version of Lemma 65.
Finally, we end the section with a technical lemma that will be used crucially for both of our
algorithms. The importance of it would become obvious when it is invoked in our analyses.
86
Lemma 81. Let ~h00 denote a convex combination of two vectors ~h and ~h0 in the base polyhedron,
i.e. ~h00 = λ~h + (1 − λ)~h0 for some λ ∈ [0, 1]. Further suppose that
n
o
~h00 ≤ α min λ ~h , (1 − λ) ~h0
2
2
2
for some α ≤
1
√
.
2 n
Then for p = arg maxj (max{λ|hj |, (1 − λ)|h0j |}) we have
lower(p) ≤ −
1
√ · ~h00
2α n
∞
and
upper(p) ≥
1
√ · ~h00
2α n
.
∞
Proof. Suppose without loss of generality that λ|hp | ≥ (1 − λ)|h0p |. Then by assumptions we have
~h00
However, since α ≤
∞
1
√
2 n
≤ ~h00
2
n
≤ α · min λ ~h 2 , (1 − λ) ~h0
o
2
√
≤ α n |λhp |
.
we see that
λhp + (1 − λ)h0p ≤ h00
∞
√
1
≤ α n |λhp | ≤ |λhp |
2
Consequently, λhp and (1 − λ)h0p have opposite signs and (1 − λ)h0p ≥
1
2
.
λh0p . We then have,
1
1
lower(p) ≤ min hp , h0p ≤ min λhp , (1 − λ)h0p ≤ − |λhp | ≤ − √ h00
2
2α n
and
15.3
1
1
√ h00
upper(p) ≥ max hp , h0p ≥ max λhp , (1 − λ)h0p ≥ |λhp | ≥
2
2α n
∞
∞
.
e 4 · EO + n5 ) Time Algorithm
O(n
e 4 · EO + n5 ) time, i.e. strongly polynomial time algorithm, for SFM. We
Here we present a O(n
build upon the algorithm achieved in the section to achieve a faster running time in Section 15.4.
Our new algorithm combines the existing tools for SFM developed over the last decade with
our cutting plane method. While there are certain similarities with previous algorithms (especially
[54, 60, 56]), our approach significantly departs from all the old approaches in one important aspect.
All of the previous algorithms actively maintain a point in the base polyhedron and represent
it as a convex combination of BFS’s. At each step, a new BFS may enter the convex combination
and an old BFS may exit. Our algorithm, on the other hand, maintains only a collection of BFS’s
(corresponding to our separating hyperplanes), rather than an explicit convex combination. A
“good” convex combination is computed from the collection of BFS’s only after running Cutting
Plane for enough iterations. We believe that this crucial difference is the fundamental reason which
offers the speedup. This is achieved by the Cutting Plane method which considers the geometry
of the collection of BFS’s. On the other hand, considering only a convex combination of BFS’s
effectively narrows our sight to only one point in the base polyhedron.
Overview
87
Now we are ready to describe our strongly polynomial time algorithm. Similar to the weakly polynomial algorithm, we first run our cutting plane for enough iterations on the initial feasible region
{~x ∈ [0, 1]n : xi ≤ xj ∀(i, j) ∈ A}, after which a pair of approximately parallel supporting hyperplanes F1 , F2 of width 1/nΘ(1) can be found. Our strategy is to write F1 and F2 as a nonnegative
combination of the facets of remaining feasible region P . This combination is made up of newly
added separating hyperplanes as well as the inequalities xi ≥ 0, xj ≤ 1 and xi ≤ xj . We then argue
that one of the following updates can be done:
• Collapsing: xi = 0, xj = 1 or xi = xj
• Adding a new arc (i, j): xi ≤ xj for some (i, j) ∈
/A
The former case is easy to handle by elimination or contraction. If xi = 0, we simply eliminate i
from the ground set V ; and if xi = 1, we redefine f so that f (S) = f (S + i) for any S ⊆ V − i.
xi = xj can be handled in a similar fashion. In the latter case, we simply add the arc (i, j) to A.
We then repeat the same procedure on the new problem.
Roughly speaking, our strongly polynomial time guarantee
follows as eliminations and contrac
tions can happen at most n times and at most 2 · n2 new arcs can be added. While the whole
picture is simple, numerous technical details come into play in the execution. We advise readers to
keep this overview in mind when reading the subsequent sections.
Algorithm
Our algorithm is summarized below. Again, we remark that our algorithm simply uses Theorem 82
regarding our cutting plane and is agnostic as to how the cutting plane works, thus it could be
replaced with other methods, albeit at the expense of slower runtime.
1. Run cutting plane on (15.1) (Theorem 82 with τ = Θ(1)) using our modified separation oracle
(Section 15.2.1).
2. Identify a pair of “narrow” approximately parallel supporting hyperplanes or get some BFS
~h = 0 (in which case both ∅ and V are minimizers).
3. Deduce from the hyperplanes some new constraint of the forms xi = 0, xj = 1, xi = xj or
xi ≤ xj (Section 15.3.2).
4. Consolidate A and f (Section 15.3.1).
5. Repeat by running our cutting plane method on (15.1) with updated A and f . (Note that
Any previously found separating hyperplanes are discarded.)
We call step (1) a phase of cutting plane. The minimizer can be constructed by unraveling the
recursion.
15.3.1
Consolidating A and f
Here we detail how the set of valid arcs A and submodular function f should be updated once we
deduce new information xi = 0, xi = 1, xi = xj or xi ≤ xj . Recall that R(i) and Q(i) are the sets
of descendants and ancestors of i respectively (including i itself). The changes below are somewhat
self-evident, and are actually used in some of the previous algorithms so we only sketch how they
are done without a detailed justification.
Changes to the digraph representation D of our ring family include:
88
• xi = 0: remove Q(i) from the ground set and all the arcs incident to Q(i)
• xi = 1: remove R(i) from the ground set and all the arcs incident to R(i)
• xi = xj : contract i and j in D and remove any duplicate arcs
• xi ≤ xj : insert the arc (i, j) to A
• For the last two cases, we also contract the vertices on a directed cycle of A until there is no
more. Remove any duplicate arcs.
Here we can contract any cycle (i1 , . . . , ik ) because the inequalities xi1 ≤ xi2 , . . . , xik−1 ≤ xik , xik ≤
xi1 imply xi1 = . . . = xik .
Changes to f :
• xi = 0: replace f by f 0 : 2V \Q(i) −→ R, f 0 (S) = f (S) for S ⊆ V \Q(i)
• xi = 1: replace f by f 0 : 2V \R(i) −→ R, f 0 (S) = f (S ∪ R(i)) for S ⊆ V \R(i)
• xi = xj : see below
• xi ≤ xj : no changes to f needed if it does not create a cycle in A; otherwise see below
• Contraction of C = {i1 , . . . , ik }: replace f by f 0 : 2V \C+l −→ R, f 0 (S) = f (S) for S ⊆ V \C
and f 0 (S) = f ((S − l) ∪ C) for S 3 l
Strictly speaking, these changes are in fact not needed as they will automatically be taken care of
by our cutting plane method. Nevertheless, performing them lends a more natural formulation of
the algorithm and simplifies its description.
15.3.2
Deducing New Constraints xi = 0, xj = 1, xi = xj or xi ≤ xj
Here we show how to deduce new constraints through the result of our cutting plane method. This
is the most important ingredient of our algorithm. As mentioned before, similar arguments were
used first by IFF [56] and later in [54, 60]. There are however two important differences for our
method:
• We maintain a collection of BFS’s rather a convex combination; a convex combination is
computed and needed only after each phase of cutting plane.
• As a result, our results are proved mostly geometrically whereas the previous ones were proved
mostly combinatorially.
Our ability to deduce such information hinges on the power of the cutting plane method in Part I.
We re-state our main result Theorem 31 in the language of SFM. Note that Theorem 82 is formulated
in a fairly general manner in order to accommodate for the next section. Readers may wish to think
τ = Θ(1) for now.
Theorem 82 (Theorem 31 restated for SFM). For any τ ≥ 100, applying our cutting plane method,
Theorem 82, to (15.1) with our modified separation oracle (or its variant in Section 15.4) with high
probability in n either
1. Finds a degenerate BFS ~h ≥ ~0 or ~h ≤ ~0.
89
2. Finds a polytope P consisting of O(n) constraints which are our separating hyperplanes or
the constraints in (15.1). Moreover, P satisfies the following inequalities
~cT ~x ≤ M
~c0T ~x ≤ M 0 ,
and
both of which are nonnegative combinations of the constraints of P , where ||~c + ~c0 ||2 ≤
min{||~c||2 , ||~c0 ||2 }/nΘ(τ ) and |M + M 0 | ≤ min{||~c||2 , ||~c0 ||2 }/nΘ(τ ) .
Furthermore, the algorithm runs in expected time O(n2 τ log n · EO + n3 τ O(1) logO(1) n).
Proof. In applying Theorem 82 we let K be the set of minimizers of f over the ring family and
the box is the hypercube with R = 1. We run cutting plane with our modified separation oracle
(Lemma 75). The initial polytope P (0) can be chosen to be, say, the hypercube. If some separating
hyperplane is degenerate, then we have the desired result (and know that either ∅ or V is optimal).
Otherwise let P be the current feasible region. Note that P 6= ∅, because our minimizers of fˆ are
all in P (0) and P (k) as they are never cut away by the separating hyperplanes.
Let S be the collection of inequalities (15.1) as well as the separating hyperplanes ~hT ~x ≤
fˆ(x̄h ) = ~hT x̄h used. By Theorem 31, all of our minimizers will be contained in P , consisting of
O(n) constraints A~x ≥ ~b. Each such constraint ~aTi ~x ≥ bi is a scaling and shifting of some inequality
p~Ti ~x ≥ qi in S, i.e. ~ai = p~i /||~
pi ||2 and bi ≤ qi /||~
pi ||2 .
By taking = 1/nΘ(τ ) with sufficiently large constant in Θ, our theorem certifies that P
PO(n)
has a narrow width by ~a1 , some nonnegative combination
ai and point ~xo ∈ P with
i=2 ti~
√
√
||~xo ||∞ ≤ 3 nR = 3 n satisying the following:
O(n)
~a1 +
X
≤ 1/nΘ(τ )
ti~ai
i=2
2
0 ≤ ~aT1 ~xo − ~b1 ≤ 1/nΘ(τ )
T
O(n)
O(n)
X
X
0≤
ti ai
~xo −
ti bi ≤ 1/nΘ(τ )
i=2
i=2
def
We convert these inequalities to p~ and q. Let t0i = ti · ||~
p1 ||2 /||~
pi ||2 ≥ 0.
O(n)
p~1 +
X
t0i p~i
i=2
≤ ||~
p1 ||2 /nΘ(τ )
2
0 ≤ p~T1 ~xo − q1 ≤ ||~
p1 ||2 /nΘ(τ )
T
O(n)
O(n)
X
X
0
0≤
ti p~i
~xo −
t0i qi ≤ ||~
p1 ||2 /nΘ(τ )
i=2
We claim that6 ~c = −~
p1 , M = −q1 , ~c0 = −
ment.
i=2
PO(n)
i=2
t0i p~i , M 0 = −
PO(n)
i=2
t0i qi satisfy our require-
6
Minus signs is needed because we express our inequalities as e.g. ~hT ~
x ≤ ~hT x̄h whereas in Theorem 31, ~aTi ~
x ≥ bi
is used. We apologize for the inconvenience.
90
We first show that ||~c + ~c0 ||2 ≤ min{||~c||2 , ||~c0 ||2 }/nΘ(τ ) . We have ||~c + ~c0 ||2 ≤ ||~c||2 /nΘ(τ ) from
the first inequality. If ||~c||2 ≤ ||~c0 ||2 we are done. Otherwise, by triangle inequality
||~c0 ||2 − ||~c||2 ≤ ||~c + ~c0 ||2 ≤ ||~c||2 /nΘ(τ ) =⇒ 2||~c||2 ≥ ||~c0 ||2
and hence ||~c + ~c0 ||2 ≤ ||~c||2 /nΘ(τ ) ≤ ||~c0 ||2 /2nΘ(τ ) = ||~c0 ||2 /nΘ(τ ) .
We also need to prove |M + M 0 | ≤ min{||~c||2 , ||~c0 ||2 }/nΘ(τ ) . Summing the second and third
inequalities,
−||~c||2 /nΘ(τ ) ≤ (~c + ~c0 )T ~xo − (M + M 0 ) ≤ 0
√
Recall that we have ||~xo ||∞ ≤ 3 n. Then
|M + M 0 | ≤ |(~c + ~c0 )T ~xo − (M + M 0 )| + |(~c + ~c0 )T ~xo |
√
≤ ||~c||2 /nΘ(τ ) + 3 n||~c + ~c0 ||2
√
≤ ||~c||2 /nΘ(τ ) + 3 n||~c||2 /nΘ(τ )
= ||~c||2 /nΘ(τ )
as desired. Our result then follows as we proved 2||~c0 ||2 ≥ ||~c||2 .
Finally, we have the desired runtime as our modified separation oracle runs in time O(n · EO +
n2 logO(1) n).
Informally, the theorem above simply states that after O(nτ log n) iterations of cutting plane,
the remaining feasible region P can be sandwiched between two approximately parallel supporting
hyperplanes of width 1/nO(τ ) . A good intuition to keep in mind is that every O(n) iterations of
cutting plane reduces the minimum width by a constant factor.
Remark 83. As shown in the proof of Theorem 82, one of the two approximately parallel hyperplanes
can actually be chosen to be a constraint of our feasible region P . However we do not exploit this
property as it does not seem to help us and would break the notational symmetry in ~c and ~c0 .
Setup
In each phase, we run cutting plane using Theorem 82 with τ = Θ(1). If some separating hyperplane
used is degenerate, we have found the minimizer by Lemma 65.
Now assume none of the separating hyperplanes is degenerate. By Theorem 82, P is sandwiched
by a pair of approximately parallel supporting hyperplanes F, F 0 which are of width 1/10n10 apart.
The width here can actually be 1/nc for any constant c by taking a sufficiently large constant in
Theta.
Here, we show how to deduce from F and F 0 some xi = 0,, xj = 1,xi = xj , or xi ≤ xj constraint
on the minimizers of f over the ring family. Let
X
X
~cT ~x =
ci xi ≤ M and ~c0T ~x =
c0i xi ≤ M 0
be the inequality for F and F 0 such that
|M + M 0 |, ||~c + ~c0 ||2 ≤ gap,
def
where gap =
1
min{||~c||2 , ||~c0 ||2 }.
10n10
By the same theorem we can write ~cT ~x ≤ M as a nonnegative combination of the constraints
for P . Recall that the constraints for P take on four different forms: (1) −xi ≤ 0; (2) xj ≤ 1; (3)
P
−(xj − xi ) ≤ 0; (4) ~hT ~x =
hi xi ≤ fˆ(x̄h ). Here the first three types are present initially whereas
91
the last type is the separating hyperplane added. As alleged previously, the coefficient vector ~h
corresponds to a BFS of P
the base polyhedron for f . Our analysis crucially exploits this property.
Thus suppose ~cT ~x = i ci xi ≤ M is a nonnegative combination of our constraints with weights
αi , βj , γij , λh ≥ 0. The number of (positive) αi , βj , γij , λh is at most O(n). Here we denote separating hyperplanes by ~hT ~x ≤ fˆ(x̄h ). Let H be the set of BFS’s used to construct separating
hyperplanes.
X
X
X
X
X
X
γij (xi − xj ) +
λh~hT ~x and M =
~cT ~x = −
αi xi +
βj xj +
βj +
λh fˆ(x̄h ).
i
j
j
h∈H
(i,j)∈A
h∈H
(15.5)
Similarly, we write the inequality for
as a nonnegative combination of the constraints for P
0 , λ0 is O(n):
and the number of (positive) αi0 , βj0 , γij
h
F0
~c0T ~x = −
X
αi0 xi +
X
βj0 xj +
X
0
γij
(xi − xj ) +
X
λ0h~hT ~x
and
M0 =
X
βj0 +
h∈H
(i,j)∈A
X
λ0h fˆ(x̄h ).
h∈H
(15.6)
We also scale ~c, ~c0 , α, α0 , β, β 0 , γ, γ 0 , λ, λ0 so that
X
(λh + λ0h ) = 1
h∈H
as this does not change any of our preceding inequalities regarding F and F 0 .
Now that F, F 0 have been written as combinations of our constraints, we have gathered the
necessary ingredients to derive our new arc. We first give a geometric intuition why we would
expect to be able to derive a new constraint. Consider the nonnegative combination making up
F . We think of the coefficient βj as the contribution of xj ≤ 1 to F . Now if βj is very large, F is
“very parallel” to xj ≤ 1 and consequently F 0 would miss xj = 0 as the gap between F and F 0 is
small. P would then miss xj = 0 too as it is sandwiched between F and F 0 . Similarly, a large αi
and a large γij would respectively imply that xi = 1 and (xi = 0, xj = 1) would be missed. The
same argument works for F 0 as well.
But on the other hand, if the contributions from xi ≥ 0, xj ≤ 1, xi ≤ xj to both F and F 0
are small, then the supporting hyperplanes ~cT ~x ≤ ... and ~c0T ~x ≤ ... would be mostly made up
of separating hyperplanes ~hT ~x ≤ fˆ(x̄h ). By summing up these separating hyperplanes (whose
coefficients form BFS’s), we would then get a point in the base polyhedron which is very close
to the origin 0. Moreover, by Lemma 81 and Lemma 79 we should then be able to deduce some
interesting information about the minimizer of f over D.
The rest of this section is devoted to realizing the vision sketched above. We stress that while
the algebraic manipulations may be long, they are simply the execution of this elegant geometric
picture.
Now, consider the following weighted sum of ~hT ~x ≤ fˆ(x̄h ):
!
X
X
X
X
X
X
λ0 ~hT ~x =
λh~hT ~x +
λ0 ~hT ~x ≤
λh fˆ(x̄h ) +
λ0 fˆ(x̄h ).
λh~hT +
h
h
h∈H
h∈H
h∈H
h∈H
92
h
h∈H
h∈H
P
P
Observe that h∈H λh~hT + h∈H λ0h~hT is in the base polyhedron since it is a convex combination
of BFS ~h. Furthermore, using (15.5) and (15.6) this can also be written as
!
X
X
X
X
X
λh~hT +
γij (xj − xi )
λ0 ~hT ~x = ~cT ~x +
αi xi −
β j xj +
h
h∈H
h∈H
(i,j)∈A
+ ~c0T ~x +
X
αi0 xi −
X
βj0 xj +
X
(15.7)
0
γij
(xj − xi )
(i,j)∈A
and
X
h∈H
λh fˆ(x̄h ) +
X
X
X
λ0h fˆ(x̄h ) = M −
βj + M 0 −
βj0
h∈H
= (M + M 0 ) −
X
X
βj0
√
√
Furthermore, we can bound ~cT ~x + ~c0T ~x by ~cT ~x + ~c0T ~x ≥ −||~c + ~c0 ||1 ≥ − n||~c + ~c0 ||2 ≥ − ngap
as ~x ≤ 1. Since M + M 0 ≤ gap, we obtain
X
X
X
X
X
X
def
0
LHS =
αi xi +
αi0 xi −
β j xj −
βj0 xj +
γij (xj − xi ) +
γij
(xj − xi )
(i,j)∈A
√
≤ 2 ngap −
X
βj −
X
βj −
(i,j)∈A
βj0
Geometrically, the next lemma states that if the contribution from, say xi ≥ 0, to F is too
large, then F 0 would be forced to miss xi = 1 because they are close to one another.
P
P
√
0 ≥
Lemma 84. Suppose ~x satisfies (15.1) and LHS ≤ 2 ngap− βj − βj0 with αi , βj , γij , αi0 , βj0 , γij
0.
√
√
1. If αi > 2 ngap or αi0 > 2 ngap, then xi < 1.
√
√
2. If βj > 2 ngap or βj0 > 2 ngap, then xj > 0.
√
0 > 2√ngap, then 0 ≤ x − x < 1.
3. If γij > 2 ngap or γij
j
i
Proof. We only prove it for αi , βj , γij as the other case follows by symmetry.P
P 0
Using 0 ≤ x ≤ 1 and xi ≤ xj for (i, j) ∈ A, we have LHS ≥ αi xi −
βj −
βj . Hence
√
√
αi xi ≤ 2 ngap and we get xi <P
1 if αi > 2Pngap.
√
Similarly, LHS ≥ −βk xk − j6=k βj − βj0 which gives −βk xk ≤ 2 ngap − βk . Then xk > 0
√
if βk > 2 ngap.
P
P
√
Finally, LHS ≥ γij (xj − xi ) − βj − βj0 which gives γij (xj − xi ) ≤ 2 ngap. Then xj − xi < 1
√
if γij > 2 ngap. We have xi ≤ xj since (i, j) ∈ A.
So if either condition of Lemma 84 holds, we can set xi = 0 or xj = 1 or xi = xj since our
problem (15.1) has an integral minimizer and any minimizer of fˆ is never cut away by Lemma 75.
Consequently, in this case we can reduce the dimension by at least 1. From now on we may assume
that
√
0
max{αi , αi0 , βj , βj0 , γij , γij
} ≤ 2 ngap.
(15.8)
Geometrically, (15.8) says that if the supporting hyperplanes are both mostly made up of the
separating hyperplanes, then their aggregate contributions to F and F 0 should be small in absolute
value.
The next lemma identifies some p ∈ V for which f (R(p)) − f (R(p) − p) is “big”. This prepares
for the final step of our approach which invokes Lemma 79.
93
def
Lemma 85. Let ~y =
P
h∈H
def P
λh~h and ~y 0 = h∈H λ0h~h and let p ∈ arg maxl {max{|yl |, |yl0 |}} then
upper(p) ≥ n7 ~y + ~y 0
∞
assuming (15.8).
Proof. Recall that ~c + ~c0 2 ≤ gap where gap = 10n1 10 min{||~c||2 , ||~c0 ||2 },
X
X
X
X
X
X
0 ~
γij
(1i − ~1j ) .
γij (~1i − ~1j ) and ~c0 = ~y 0 −
αi0 ~1i +
βj0 ~1j +
βj ~1j +
~c = ~y −
αi~1i +
j
i
i
(i,j)
4
By (15.8) we know that ~c − ~y 2 ≤ 4n2 gap ≤ 10n
c
8 ~
Consequently, by the triangle inequality we have that
~y + ~y 0
and
~c
2
2
2
2
+ ~c − ~y
2
and ~c0 − ~y 0
+ ~c0 − ~y 0
2
4
~c 2 + ~y 2 ⇒
10n8
≤ 2 ~y 0 2 . Consequently since gap ≤
≤ ~c − ~y
Similarly, we have that ~c0
that
≤ ~c + ~c0
2
j
2
+ ~y
2
≤
2
min
~y
2
n8
and thus, invoking Lemma 81 yields the result.
~y + ~y 0
≤
2
, ~y 0
2
(i,j)
≤ 4n2 gap ≤
4
10n8
~c0
2
.
≤ 9n2 gap
~c
1
10n10
2
≤ 2 ~y
2
min{||~c||2 , ||~c0 ||2 }, we have
2
We summarize the results in the lemma below.
Corollary 86. Let P be the feasible region after running cutting plane on (15.1). Then one of the
following holds:
1. We found a degenerate BFS and hence either ∅ or V is a minimizer.
2. The integral points of P all lie on some hyperplane xi = 0, xj = 1 or xi = xj which we can
find.
3. Let H be the collection of BFS’s ~h used to construct our separating hyperplanes for P . Then
there is a convex combination ~y of H such that n4 |yi | < maxp upper(p) for all i.
Proof. As mentioned before, (1) happens if some separating P
hyperplane isP
degenerate. We have (2)
if one of the conditions in Lemma 84 holds. Otherwise, y = h∈H λh~h + h∈H λ0h~h is a candidate
for Case 3 by Lemma 85.
Let us revisit the conditions of Lemma 79 and explain that they are satisfied by Case 3 of the
last lemma.
• ~y is a convex combination of at most O(n) BFS’s. This holds in Case 3 since our current
feasible region consists of only O(n) constraints thanks to the Cutting Plane method.
• Those BFS’s must be consistent with every arc of A. This holds because Case 3 uses the
BFS’s for constructing our separating hyperplane. Our modified separation oracle guarantees
that they are consistent with A.
Thus in Case 3 of the last corollary, Lemma 79 allows us to deduce a new constraint xp ≤ xq for
some q ∈
/ R(p).
94
15.3.3
Running Time
Here we bound the total running time of our algorithm and prove the following.
Theorem 87. Our algorithm runs in time O(n4 log n · EO + n5 logO(1) n).
Proof. To avoid being repetitive, we appeal to Corollary 86. Each phase of cutting plane takes
time O(n2 log n · EO + n3 logO(1) n) (Theorem 82 with τ being a big constant. Given F and F 0
represented as a nonnegative combination of facets, we can check for the conditions in Lemma 84
in O(n) time as there are only this many facets of P . This settles Case 2 of Corollary 86. Finally,
Lemma 79 tells us that we can find a new arc in O(n · EO + n2 ) time for Case 3 of Corollary 86.
Our conclusion follows from the fact that we can get xi = 0, xi = 1, xi = xj at most n times and
xi ≤ xj at most O(n2 ) times.
15.4
e 3 · EO + n4 ) Time Algorithm
O(n
e 3 ·EO+n4 ). Our
Here we show how to improve our running time for strongly polynomial SFM to O(n
algorithm can be viewed as an extension of the algorithm we presented in the previous Section 15.3.
The main bottleneck of our previous algorithm was the time needed to identify a new arc, which
e 2 · EO + n3 ). Here we show how to reduce our amortized cost for identifying a valid arc
cost us O(n
e · EO + n2 ) and thereby achieve our result.
down to O(n
The key observation we make to improve this running time is that our choice of p for adding
an arc in the previous lemma can be relaxed. p actually need not be arg maxi upper(i); instead
0 }. For each such p a new constraint
it is enough to have upper(p) > n4 max{αi , αi0 , βj , βj0 , γij , γij
xp ≤ xq can be identified via Lemma 79. So if there are many p’s satisfying this we will be able to
obtain many new constraints and hence new valid arcs (p, q).
On the other hand, the bound in Lemma 85 says that our point in the base polyhedron is small
in absolute value. This is actually stronger than what we need in Lemma 79 which requires only
its positive entries to be “small”. However as we saw in Lemma 80 we can generate a constraint of
the form xq ≤ xp whenever lower(p) is sufficiently negative.
Using this idea, we divide V into different buckets according to upper(p) and lower(p). This
will allow us to get a speedup for two reasons.
First, bucketing allows us to disregard unimportant elements of V during certain executions
of our cutting plane method. If both upper(i) and lower(i) are small in absolute value, then i is
essentially negligible because for a separating hyperplane ~hT ~x ≤ fˆ(x̄), any hi ∈ [lower(i), upper(i)]
small in absolute value would not really make a difference. We can then run our cutting plane
algorithm only on those non-negligible i’s, thereby reducing our time complexity. Of course, whether
hi is small is something relative. This suggests that partitioning the ground set by the relative size
of upper(i) and lower(i) is a good idea.
Second, bucketing allows us to ensure that we can always add an arc for many edges simultaneously. Recall that we remarked that all we want is nO(1) |yi | ≤ upper(p) for some ~y in the base
polyhedron. This would be sufficient to identify a new valid arc (p, q). Now if the marginal differences upper(p) and upper(p0 ) are close in value, knowing nO(1) |yi | ≤ upper(p) would effectively
give us the same for p0 for free. This suggests that elements with similar marginal differences should
be grouped together.
The remainder of this section simply formalizes these ideas. In Section 15.4.1 we discuss how
we partition the ground set V . In Section 15.4.2, we present our cutting plane method on a subset
of the coordinates. Then in Section 15.4.3 we show how we find new arcs. Finally, in Section 15.4.4
we put all of this together to achieve our desired running time.
95
15.4.1
Partitioning Ground Set into Buckets
We partition the ground set V into different buckets according to the values of upper(i) and
lower(i). This is reminiscent to Iwata-Orlin’s algorithm [60] which considers elements with big
upper(i). However they did not need to do bucketing by size or to consider lower(i), whereas these
seem necessary for our algorithm.
Let N = maxi {max{upper(i), −lower(i)}} be the largest marginal difference in absolute value.
By Lemma (78), N ≥ 0. We partition our ground set V as follows:
B1 = {i : upper(i) ≥ N/n10 or lower(i) ≤ −N/n10 }
Bk = {i ∈
/ B1 ∪ . . . ∪ Bk−1 : N/n10k ≤ upper(i) < N/n10(k−1)
or − N/n10(k−1) < lower(i) ≤ −N/n10k },
k≥2
We call Bk buckets. Our buckets group elements by the values of upper(i) and lower(i) at 1/n10
“precision”. There are two cases.
• Case 1: the number of buckets is at most log n7 , in which case upper(i) > N/nO(log n) or
lower(i) < −N/nO(log n) for all i.
• Case 2: there is some k for which |B1 ∪ . . . ∪ Bk | ≥ |Bk+1 |.
This is because if there is no such k in Case 2, then by induction each bucket Bk+1 has at least
2k |B1 | ≥ 2k elements and hence k ≤ log n.
Case 1 is easier to handle, and is in fact a special case of Case 2. We first informally sketch the
treatment for Case 1 which should shed some light into how we deal with Case 2.
We run Cutting Plane for O(n log2 n) iterations (i.e. τ = Θ(log n)). By Theorem 82, our feasible
region P would be sandwiched by a pair of approximately parallel supporting hyperplanes of width
at most 1/nΘ(log n) . Now proceeding as in the last section, we would be able to find some ~y in the
base polyhedron and some element p such that nΘ(log n) |yi | ≤ upper(p). This gives
N
upper(p)
≤ Θ(log n) .
Θ(log
n)
n
n
Θ(log
n)
Θ(log
n)
Since upper(i) > N/n
or lower(i) < −N/n
for all i in Case 1, we can then conclude
that some valid arc (i, q) or (q, i) can be added for every i. Thus we add n/2 arcs simultaneously
in one phase of the algorithm at the expense of blowing up the runtime by O(log n). This saves
a factor of n/ log n from our runtime in the last section, and the amortized cost for an arc would
e · EO + n2 ).
then be O(n
On the other hand, in Case 2 we have a “trough” at Bk+1 .SRoughly speaking, this trough is
useful for acting
S as a soft boundary between B1 ∪ . . . ∪ Bk and l≥k+2 Bl . Recall that we are able
to “ignore” l≥k+2 Bl because their hi is relatively small in absolute value. In particular, we know
that for any p ∈ B1 ∪ . . . ∪ Bk and i ∈ Bl , where l ≥ k + 2,
nΘ(log n) |yi | ≤
max{upper(p), −lower(p)} ≥ n10 max{upper(i), −lower(i)}.
This is possible because Bk+1 , which is sandwiched in between, acts like a shield preventing Bl
to “mess with” B1 ∪ . . . ∪ Bk . This property comes at the expense of sacrificing Bk+1 which must
confront Bl .
Furthermore, we require that |B1 ∪ . . . ∪ Bk | ≥ |Bk+1 |, and run Cutting Plane on B = (B1 ∪
. . . ∪ Bk ) ∪ Bk+1 . If |Bk+1 | |B1 ∪ . . . ∪ Bk |, our effort would mostly be wasted on Bk+1 which is
sacrificed, and the amortized time complexity for B1 ∪ . . . ∪ Bk would then be large.
Before discussing the algorithm for Case 2, we need some preparatory work.
7
More precisely, Bk = ∅ for k > dlog ne.
96
15.4.2
Separating Hyperplane: Project and Lift
Our speedup is achieved by running our cutting plane method on the projection of our feasible
region onto B := (B1 ∪ · · · ∪ Bk ) ∪ Bk+1 . More precisely, we start by running our cutting plane
on P B = {~x ∈ RB : ∃~x0 ∈ RB̄ s.t. (~x, ~x0 ) satisfies (15.1)}, which has a lower dimension. However,
to do this, we need to specify a separation oracle for P B . Here we make one of the most natural
choices.
We begin by making an apparently immaterial change to our set of arcs A. Let us take the
transitive closure of A by adding the arc (i, j) whenever there is a path from i to j. Clearly this
would not change our ring family as a path from i to j implies j ∈ R(i). Roughly speaking, we do
this to handle pathological cases such as (i, k), (k, j) ∈ A, (i, j) ∈
/ A and i, j ∈ B, k ∈
/ B. Without
introducing the arc (i, j), we risk confusing a solution containing i but not j as feasible since we
are restricting our attention to B and ignoring k ∈
/ B.
Definition 88. Given a digraph D = (V, A), the transitive closure of A is the set of arcs (i, j) for
which there is a directed path from i to j. We say that A is complete if it is equal to its transitive
closure.
Given x̄ ∈ [0, 1]B , we define the completion of x̄ with respect to A as follows.
Definition 89. Given x̄ ∈ [0, 1]B and a set of arcs A, xC ∈ [0, 1]n is a completion of x̄ if xCB = x̄
and xCi ≤ xCj for every (i, j) ∈ A. Here xCB denotes the restriction of xC to B.
Lemma 90. Given x̄ ∈ [0, 1]B and a complete set of arcs A, there is a completion of x̄ if x̄i ≤ x̄j
for every (i, j) ∈ A ∩ (B × B). Moreover, it can be computed in O(n2 ) time.
Proof. We set xCB = x̄. For i ∈
/ B, we set
(
1
C
xi =
min(i,j)∈A,j∈B xCj
if @j ∈ B s.t. (i, j) ∈ A
otherwise
One may verify that xC satisfies our requirement as A is complete. Computing each xCi takes
O(n) time. Since |V \B| = |B̄| ≤ n, computing the whole xC takes O(n2 ) time.
This notion of completion is needed since our original separation oracle requires a full dimensional input x̄. Now that x̄ ∈ RB , we need a way of extending it to Rn while retaining the crucial
property that ~h is consistent with every arc in A.
Note that the runtime is still O(n · EO + n2 logO(1) n) as xC can be computed in O(n2 ) time by
the last lemma.
P
We reckon that the hyperplane ~hTB ~xB ≤ i∈B hi x̄i returned by the oracle is not a valid separating hyperplane (i.e. it may cut out the minimizers). Nevertheless,
we will show that it is a decent
P
T
C
~
ˆ
“proxy” to the true separating hyperplane h ~x ≤ f (x ) = i∈V hi xCi and is good enough to serve
our purpose of sandwiching the P
remaining feasible region in a small strip. To get a glimpse, note
that the terms missing ~hTB ~xB ≤ i∈B hi x̄i all involve hi for i ∈
/ B, which is “negligible” compared
to B1 ∪ · · · ∪ Bk .
P
P
P
One may try to make ~hTB ~xB ≤ i∈B hi x̄i valid, say, by ~hTB ~xB ≤ i∈B hi x̄i + i∈B
/ |hi |. The
P
T
~
problem
is that
P
P such hyperplanes would not be separating for x̄ anymore as hB x̄ = i∈B hi x̄i <
h
x̄
+
i∈B i i
i∈B
/ |hi |. Consequently, we lose the width (or volume) guarantee of our cutting
plane algorithm. Although this
P seems problematic, it is actually still possible to show a guarantee
sufficient for our purpose as i∈B
/ |hi | is relatively small. We leave it as a nontrivial exercise to
interested readers.
97
Algorithm 6: Projected Separation Oracle
Input: x̄ ∈ RB and a complete set of arcs A
if x̄i < 0 for some i ∈ B then
Output: xi ≥ 0
else if x̄j > 1 for some j ∈ B then
Output: xj ≤ 1
else if x̄i > x̄j for some (i, j) ∈ A ∩ B 2 then
Output: xi ≤ xj
else
Let xC ∈ Rn be a completion of x̄
Let i1 , . . . , in be a permutation of V such that xCi1 ≥ . . . ≥ xCin and for all (i, j) ∈ A, j
precedes i in i1 , . . .P
, in .
P
T
~
Output: hB ~xB = i∈B hi xi ≤ i∈B hi x̄i , where ~h is the BFS defined by the
permutation i1 , . . . , in .
In conclusion, it seems that one cannot have the best of both worlds: the hyperplane returned
by the oracle cannot be simultaneously valid and separating.
Algorithm
We take k to be the first for which |B1 ∪ . . . ∪ Bk | ≥ |Bk+1 |, i.e. |B1 ∪ . . . ∪ Bl | < |Bl+1 | for l ≤ k − 1.
Thus k ≤ log n. Let b = |B|, and so |B1 ∪ · · · ∪ Bk | ≥ b/2. Case 1 is a special case by taking B = V .
Our algorithm is summarized below. Here A is always complete as A is replaced its transitive
closure whenever a new valid arc is added.
1. Run Cutting Plane on P B = {x ∈ RB : ∃x0 ∈ RB̄ s.t. (x, x0 ) satisfies (15.1)} with the new
projected separation oracle.
2. Identify a pair of “narrow” approximately parallel supporting hyperplanes.
3. Deduce from the hyperplanes certain new constraints of the forms xi = 0, xj = 1, xi = xj or
xi ≤ xj by lifting separating hyperplanes back to Rn
4. Consolidate A and f . If some xi ≤ xj added, replace A by its transitive closure.
5. Repeat Step 1 with updated A and f . (Any previously found separating hyperplanes are
discarded.)
The minimizer can be constructed by unraveling the recursion.
First of all, to be able to run Cutting Plane on P B we must come up with a polyhedral
description of P B which consists of just the constraints involving B. This is shown in the next
lemma.
Lemma 91. Let P B = {~x ∈ RB : ∃~x0 ∈ RB̄ s.t. (~x, ~x0 ) satisfies (15.1)}. Then
P B = {~x ∈ RB : 0 ≤ ~x ≤ 1, xi ≤ xj ∀(i, j) ∈ A ∩ (B × B)}
Proof. It is clear that P B ⊆ {~x ∈ RB : 0 ≤ ~x ≤ 1, xi ≤ xj ∀(i, j) ∈ A ∩ (B × B)} as the constraints
0 ≤ x ≤ 1, xi ≤ xj ∀(i, j) ∈ A ∩ (B × B) all appear in (15.1).
Conversely, for any ~x ∈ RB satisfying 0 ≤ ~x ≤ 1, xi ≤ xj ∀(i, j) ∈ A ∩ (B × B), we know there is
some completion xC of ~x by Lemma 90 as A is complete. Now xC satisfies (15.1) by definition, and
hence ~x ∈ P B .
98
The only place where we have really changed the algorithm is Step (3).
15.4.3
Deducing New Constraints xi = 0, xj = 1, xi = xj or xi ≤ xj
Our method will deduce one of the following:
• xi = 0, xj = 1 or xi = xj
• for each p ∈ B1 ∪ · · · ∪ Bk , xp ≤ xq for some q ∈
/ R(p) or xp ≥ xq for some q ∈
/ Q(p)
Our argument is very similar to the last section’s. Roughly speaking, it is the same argument but
with “noise” introduced by i ∈
/ B. We use extensively the notations from the last section.
Our main tool is again Theorem 82. Note that n should be replaced by b in the Theorem
statement. We invoke it with τ = k logb n = O(log2 n) (using k ≤ log n) to get a width of
1/bΘ(τ ) = 1/nΘ(k) . This takes time at most O(bn log2 n·EO+bn2 logO(1) n). Again, this is intuitively
clear as we run it for O(kb log n) iterations, each of which takes time O(n · EO + n2 logO(1) n).
After each phase of (roughly O(kb log n) iterations) of Cutting Plane, P B is sandwiched between
a pair of approximately parallel supporting hyperplanes F and F 0 which have width 1/n20k . Let F
and F 0 be
~cT ~xB =
X
ci xi ≤ M,
~c0T ~xB =
i∈B
X
c0i xi ≤ M 0 ,
i∈B
such that
|M + M 0 |, ||~c + ~c0 ||2 ≤ gap,
where gap =
1
n20k
min{||~c||2 , ||~c0 ||2 }.
The rest of this section presents an execution of the ideas discussed above. All of our work is
e ·
basically geared towards bringing the amortized cost for identifying a valid arc down to O(n
2
EO + n ). Again, we can write these two constraints as a nonnegative combination. Here x̄Ch is
the completion of the point x̄h used to construct ~hTB ~xB ≤ ~hTB x̄Ch B . (Recall that x̄Ch B is the
restriction of x̄Ch to B.)
~cT ~xB = −
X
αi xi +
i∈B
~c0T ~xB = −
X
i∈B
X
j∈B
αi0 xi +
X
βj xj +
X
j∈B
γij (xi −xj )+
(i,j)∈A∩B 2
βj0 xj +
X
X
λh~hTB ~xB
and M =
(i,j)∈A∩B 2
X
βj +
j∈B
h∈H
0
γij
(xi −xj )+
X
λ0h~hTB ~xB
and M 0 =
h∈H
X
j∈B
X
λh~hTB x̄Ch
By adding all the constituent separating hyperplane inequalities, we get
99
B
.
h∈H
βj0 +
X
λ0h~hTB x̄Ch
h∈H
As we have discussed, the problem is that the separating hyperplanes ~hTB ~xB ≤ ~hTB x̄Ch B are not
actually valid. We can, however, recover their valid counterpart by lifting them back to ~hT ~x ≤ ~hT x̄Ch .
The hope is that ~hTB ~xB ≤ ~hTB x̄Ch B and ~hT ~x ≤ ~hT x̄Ch are not too different so that the arguments
will still go through. We show that this is indeed the case.
Again, we scale c, c0 , α, α0 , β, β 0 , γ, γ 0 , λ, λ0 so that
X
(λh + λ0h ) = 1.
h∈H
B
.
X
λh~hT ~x +
h∈H
X
λ0h~hT ~x ≤
h∈H
X
λh~hT x̄Ch +
h∈H
X
λ0h~hT x̄Ch
h∈H
Let
def
LHS =
X
αi xi +
X
αi0 xi −
X
β j xj −
X
βj0 xj +
X
γij (xj − xi ) +
X
0
γij
(xj − xi ).
Here we know that
X
X
X
X
λh~hT ~x +
λ0h~hT ~x = LHS + (~c + ~c0 )T ~xB +
λh~hTB̄ ~xB̄ +
λ0h~hTB̄ ~xB̄
h∈H
X
h∈H
λh~hT x̄Ch +
h∈H
X
h∈H
λ0h~hT x̄Ch = (M + M 0 ) +
h∈H
X
λh~hTB̄ x̄Ch
B̄
h∈H
h∈H
X
+
λ0h~hTB̄ x̄Ch
B̄
−
X
βj −
X
βj0
h∈H
Combining all yields
X
X
X
X
X
X
λh~hTB̄ ~xB̄ +
λ0h~hTB̄ ~xB̄ ≤ (M +M 0 )+
λh~hTB̄ x̄Ch B̄ +
λ0h~hTB̄ x̄Ch B̄ −
βj −
βj0
LHS+(~c+~c0 )T ~xB +
h∈H
h∈H
h∈H
h∈H
√
√
Here (~c + ~c0 )T ~xB can be bounded as before: (~c + ~c0 )T ~xB ≥ − n||~c + ~c0 ||2 ≥ − ngap. Since
M + M 0 ≤ gap, We then obtain
X
X
X
X
X
X
√
LHS +
λh~hTB̄ ~xB̄ +
λ0h~hTB̄ ~xB̄ ≤ 2 ngap +
λh~hTB̄ x̄Ch B̄ +
λ0h~hTB̄ x̄Ch B̄ −
βj −
βj0
h∈H
h∈H
h∈H
h∈H
We should expect the contribution from ~hB̄ to be small as hi for i ∈
/ B is small compared to
B1 ∪ . . . ∪ Bk . We formalize our argument in the next two lemmas.
P
P
Lemma 92. We have h∈H λh~hTB̄ x̄Ch B̄ + h∈H λ0h~hTB̄ x̄Ch B̄ ≤ N/n10(k+1)−1 .
P
P
Proof. We bound each component of h∈H λh~hTB̄ x̄Ch B̄ + h∈H λ0h~hTB̄ x̄Ch B̄ . For i ∈ B̄, we have
upper(i) ≤ N/n10(k+1) . By Lemma 78 hi ≤ upper(i). Therefore,
!
X
X
X
X
0
0
T
C
T
C
≤
λh +
λ N/n10(k+1) = N/n10(k+1) .
λ ~h x̄
λh~h x̄ +
i
h i
h i
h
h i
h∈H
h∈H
h∈H
h∈H
Our result then follows since
!
X
C
λh~hTB̄ x̄h
+
B̄
h∈H
Lemma 93. We have
X
λ0h~hTB̄ x̄h
h∈H
P
h∈H
C
B̄
=
X
i∈B̄
X
h∈H
C
λh~hTi x̄h
i
+
X
λ0h~hTi
x̄Ch i
.
h∈H
P
λh~hTB̄ ~xB̄ + h∈H λ0h~hTB̄ ~xB̄ ≥ −N/n10(k+1)−1 .
Proof. The proof is almost identical to the last lemma except that we use hi ≥ lower(i) instead of
hi ≤ upper(i), and lower(i) ≥ −N/n10(k+1) .
The two lemmas above imply that
X
X
X
X
√
LHS ≤ 2 ngap −
βj −
βj + 2N/n10(k+1)−1 = gap0 −
βj −
βj
√
where gap0 = 2 ngap + 2N/n10(k+1)−1 .
100
Lemma 94. Suppose x satisfies (15.1) and LHS ≤ gap0 −
0.
P
βj −
P
0 ≥
βj0 with αi , βj , γij , αi0 , βj0 , γij
1. If αi > gap0 or αi0 > gap0 , then xi < 1.
2. If βj > gap0 or βj0 > gap0 , then xj > 0.
0 > gap0 , then 0 ≤ x − x < 1.
3. If γij > gap0 or γij
j
i
√
Proof. The proof is exactly the same as Lemma 84 with 2 ngap replaced by gap0 .
From now on we may assume that
0
max{αi , αi0 , βj , βj0 , γij , γij
} ≤ gap0 .
def
Lemma 95. Let ~y =
P
h∈H
(15.9)
def P
λh~h and ~y 0 = h∈H λ0h~h and let p ∈ arg maxl∈B {max{|yl |, |yl0 |} then
0
N ≥ n10k+6 ~yB + ~yB
∞
assuming (15.9).
√
1
min{||~c||2 , ||~c0 ||2 } and gap0 = 2 ngap +
Proof. Recall that ~c +~c0 2 ≤ gap < gap0 where gap = n20k
2N/n10(k+1)−1 . Now there are two cases.
√
√
Case 1: 2 ngap ≥ 2N/n10(k+1)−1 . Then gap0 ≤ 4 ngap and we follow the same proof of
Lemma 85. We have
~c = ~yB −
X
αi~1i +
i
X
βj ~1j +
j
X
γij (~1i − ~1j )
0
and ~c0 = ~yB
−
0
~yB + ~yB
2
≤ ~c + ~c0
2
+ ~c − ~yB
and
2
≤ ~c − ~yB
Similarly, we have that ~c0
that
2
2
αi0 ~1i +
X
i
(i,j)
1
~c
By (15.9) we know that ~c − ~yB 2 ≤ 4n2 gap0 ≤ n17k
Consequently, by the triangle inequality we have that
~c
X
+ ~yB
0
≤ 2 ~yB
0
~yB + ~yB
j
0
and ~c0 − ~yB
0
+ ~c0 − ~yB
2
~c
. Consequently since gap0 ≤
1
n19k
2
≤
2
+ ~yB
18
min ~yB
17k
n
2
2
0
, ~yB
2
X
0 ~
γij
(1i − ~1j ) .
(i,j)
≤ 4n2 gap0 ≤
1
n17k
~c 2 .
≤ 9n2 gap0
⇒
2
2
1
~c
n17k
≤
2
2
βj0 ~1j +
2
≤ 2 ~yB
2
min{||~c||2 , ||~c0 ||2 }, we have
2
0
and thus, invoking Lemma 81 yields N ≥ upper(p) ≥ n16k ~yB + ~yB
, as desired.
∞
√
10(k+1)−1
Case 2: 2 ngap < 2N/n
. Then for any i ∈ B, |ci + c0i | ≤ ||~c + ~c0 ||2 ≤ gap <
10(k+1)−1
2N/n
. Since
X
X
X
X
X
X
0
0 ~
~yB + ~yB
= (~c + ~c0 ) +
αi~1i −
βj ~1j −
γij (~1i − ~1j ) +
αi0 ~1i −
βj0 ~1j −
γij
(1i − ~1j )
i
j
i
(i,j)
j
we have
0
~yB + ~yB
∞
≤ 2N/n10(k+1)−1 + 2n1.5 gap0 ≤ N/n10k+7 .
101
(i,j)
Corollary 96. Let P be the feasible region after running Cutting Plane on (15.1) with the projected
separation oracle. Then one of the following holds:
1. We found a BFS ~h with ~hB = 0.
2. The integral points of P all lie on some hyperplane xi = 0, xj = 1 or xi = xj .
3. Let H be the collection of BFS’s ~h used to construct our separating hyperplanes for P . Then
there is a convex combination ~y of H such that for p ∈ B1 ∪· · ·∪Bk , we have n4 |yi | < upper(p)
or lower(p) < −n4 |yi | for all i.
Proof. As mentioned before, (1) happens if some separating hyperplane satisfies ~hB = 0 when
running cutting plane on the non-negligible
coordinates.
We have (2) if some condition in Lemma
P
P
94 holds. Otherwise, we claim y = h λh h + h λ0h h is a candidate for Case 3. y is a convex
combination of BFS and by Lemma 95, for the big elements i ∈ B we have
|yi | ≤ N/n10k+6 ≤
1
max{upper(p), −lower(p)}.
n4
where the last inequality holds since for p ∈ B1 ∪ · · · ∪ Bk , max{upper(p), −lower(p)} ≥ N/n10k .
On the other hand, for the small elements i ∈
/ B, |yi | ≤ N/n10(k+1) ≤ n14 max{upper(p), −lower(p)}
as desired.
The gap is then smaller enough to add an arc for each p ∈ B1 ∪ · · · ∪ Bk by Lemmas 79 and
e
80. Therefore we can add a total of |B1 ∪ · · · ∪ Bk |/2 ≥ b/4 arcs with roughly O(kb log n) = O(b)
2
e
iterations of Cutting Plane, each of which takes O(n · EO + n ). That is, the amortized cost for
e · EO + n2 ). We give a more formal time analysis in below but it should be somewhat
each arc is O(n
clear why we have the desired time complexity.
Lemma 97. Suppose there is a convex combination ~y of H such that for p ∈ B1 ∪ · · · ∪ Bk , we
have n4 |yi | < upper(p) or lower(p) < −n4 |yi | for all i. Then we can identify at least b/4 new valid
arcs.
Proof. We have |H| = O(n) since H is the set of BFS’s used for the constraints of P which has
O(n) constraints. By Lemmas 79 and 80, for p ∈ B1 ∪ · · · ∪ Bk we can add a new valid arc (p, q)
or (q, p). However note that a new arc (p1 , p2 ) may added twice by both p1 and p2 . Therefore the
total number of new arcs is only at least |B1 ∪ · · · ∪ Bk |/2 ≥ b/4.
15.4.4
Running Time
Not much changes to the previous runtime analysis are needed. To avoid repetition, various details
already present in the corresponding part of the last section are omitted. Recall k ≤ log n, and of
course, b ≤ n.
For each (roughly) O(kb log n) iterations of Cutting Plane we either get xi = 0,xi = 1,xi = xj
or b/4 xi ≤ xj ’s. The former can happen at most n times while in the latter case, the amortized
cost of each arc is O(k log n) iterations of Cutting Plane. In the worst case the overall number
e 2 ). Thus our algorithm has a runtime of O(n
e 3 · EO + n4 ) since each
of iterations required is O(n
2
e
iteration is O(n · EO + n ) as shown below.
Theorem 98. Our algorithm runs in time O(n3 log2 n · EO + n4 logO(1) n).
102
Proof. We use Corollary 96. First we note that Case 1 can actually be integrated into Case 3 since
max{upper(p), −lower(p)} ≥ N/n10k = n10 N/n10(k+1) ≥ hi for i ∈
/ B.
As we have argued in the beginning of the last section, Theorem 82 with τ = k logb n implies
that the runtime for each phase is O(bn log2 n · EO + bn2 logO(1) n). In each phase we either get
xi = 0, xi = 1, xi = xj (Case 2) or b/4 xi ≤ xj ’s (Case 3), the latter of which follows from Corollary
96 and Lemma 97.
Case 2 can only happen n times. Thus the total cost is at most O(n3 log2 n · EO + n4 logO(1) n).
The overhead cost is also small. Similar to before, given F and F 0 represented as a nonnegative
combination of facets, we can check for the conditions in Lemma 94 in O(n) time as there are only
this many facets of P . This settles Case 2.
For case 3 the amortized cost for each arc is O(n log2 n · EO + n2 logO(1) n). Our desired runtime
follows since there are only O(n2 ) arcs to add. Unlike Case 2 some extra care is needed to handle
the overhead cost. The time needed to deduce a new arc (applying Lemmas 79 and 80 to ~y and
p ∈ B1 ∪ · · · ∪ Bk ) is still O(n · EO + n2 ). But as soon as we get a new arc, we must update A to
be its transitive closure so that it is still complete. Given A complete and a new arc (p, q) ∈
/ A, we
can simply add the arcs from the ancestors of p to q and from p to the descendants of q. There are
at most O(n) arcs to add so this takes time O(n2 ) per arc, which is okay.
16
Discussion and Comparison with Previous Algorithms
We compare and contrast our algorithms with the previous ones. We focus primarily on strongly
polynomial time algorithms.
Convex combination of BFS’s
All of the previous algorithms maintain a convex combination of BFS’s and iteratively improve
over it to get a better primal solution. In particular, the new BFS’s used are typically obtained by
making local changes to existing ones. Our algorithms, on the other hand, considers the geometry
of the existing BFS’s. The weighted “influences”8 then aggregately govern the choice of the next
BFS. We believe that this is the main driving force for the speedup of our algorithms.
Scaling schemes
Many algorithms for combinatorial problems are explicitly or implicitly scaling a potential
function or a parameter. In this paper, our algorithms in some sense aim to minimize the volume
of the feasible region. Scaling schemes for different potential functions and parameters were also
designed in previous works [56, 54, 60, 53]. All of these functions and parameters have an explict
form. On the contrary, our potential function is somewhat unusual in the sense that it has no
closed form.
Deducing new constraints
As mentioned in the main text, our algorithms share the same skeleton and tools for deducing
new constraints with [56, 54, 60, 53]. Nevertheless, there are differences in the way these tools
are employed. Our algorithms proceed by invoking them in a geometric manner, whereas previous
algorithms were mostly combinatorial.
Big elements and bucketing
Our bucketing idea has roots in Iwata-Orlin’s algorithm [60] but is much more sophisticated. For
instance, it is sufficient for their algorithm to consider only big elements, i.e. upper(i) ≥ N/nO(1) .
Our algorithm, on the other hand, must carefully group elements by the size of both upper(i) and
8
In the terminology of Part I, these weighted influences are the leverage scores.
103
lower(i). The speedup appears impossible without these new ideas. We do however note that it
is unfair to expect such a sophisticated scheme in Iwata-Orlin’s algorithm as it would not lead to
a speedup. In other words, their method is fully sufficient for their purposes, and the simplicity in
their case is a virtue rather than a shortcoming.
16.1
Open Problems
One natural open problem is improving our weakly polynomial algorithm to O(n2 log M · EO +
n3 logO(1) n · log M ) time. Our application of center of mass to SFM demonstrates that it should
be possible.
For strongly polynomial algorithms, the existential result of Theorem 71 shows that SFM can
be solved with O(n3 log n · EO) oracle calls. Unfortunately, our algorithm incurs an overhead of
log n as there can be as many as log n buckets each time. One may try to remove this log n overhead
by designing a better bucketing scheme or arguing that more arcs can be added.
The other log n overhead seem much trickier to remove. Our method currently makes crucial
use of the tools developed by [56], where the log n factors in the runtime seem inevitable. We
suspect that our algorithm may have an analogue similar to [93, 90], which do not carry any log n
overhead in the running time.
Perhaps an even more interesting open problem is whether our algorithm is optimal (up to
polylogarithmic factors). There are grounds for optimism. So far the best way of certifying the
optimality of a given solution S ⊆ V is to employ duality and express some optimal solution to the
base polyhedron as a convex combination of n + 1 BFS’s. This already takes n2 oracle calls as each
BFS requires n. Thus one would expect the optimal number of oracle calls needed for SFM to be
at least n2 . Our bound is not too far off from it, and anything strictly between n2 and n3 seems
instinctively unnatural.
Acknowledgments
We thank Matt Weinberg for insightful comments about submodular minimization and minimizing
the intersection of convex sets that were deeply influential to our work. We thank Yan Kit Chim,
Stefanie Jegelka, Jonathan A. Kelner, Robert Kleinberg, Pak-Hin Lee, Christos Papadimitriou,
and Chit Yu Ng for many helpful conversations. We thank Chien-Chung Huang for pointing out a
typo in an earlier draft of this paper. This work was partially supported by NSF awards 0843915
and 1111109, NSF grants CCF0964033 and CCF1408635, Templeton Foundation grant 3966, NSF
Graduate Research Fellowship (grant no. 1122374). Part of this work was done while the first two
authors were visiting the Simons Institute for the Theory of Computing, UC Berkeley. Lastly, we
would like to thank Vaidya for his beautiful work on his cutting plane method.
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111
| 8 |
Denoising Linear Models with Permuted Data
Ashwin Pananjady†
Martin J. Wainwright†,?
Thomas A. Courtade†
Department of Electrical Engineering and Computer Sciences†
Department of Statistics?
UC Berkeley
arXiv:1704.07461v1 [stat.ML] 24 Apr 2017
April 26, 2017
Abstract
The multivariate linear regression model with shuffled data and additive Gaussian
noise arises in various correspondence estimation and matching problems. Focusing on
the denoising aspect of this problem, we provide a characterization the minimax error rate
that is sharp up to logarithmic factors. We also analyze the performance of two versions
of a computationally efficient estimator, and establish their consistency for a large range
of input parameters. Finally, we provide an exact algorithm for the noiseless problem and
demonstrate its performance on an image point-cloud matching task. Our analysis also
extends to datasets with outliers.
1
Introduction
The linear model is a ubiquitous and well-studied tool for predicting responses y based on
a vector a of covariates or predictors. In this paper, we consider the multivariate version of
the model, with vector-valued responses yi ∈ Rm , and covariates ai ∈ Rd . In the standard
formulation of this problem, estimation is performed on the basis of a data set of n pairs
{ai , yi }ni=1 , in which each response yi is correctly associated with the covariate vector ai that
generated it. Our focus is instead on the following variant of the standard set-up: the input
consists of the permuted data set {ai , yπi }ni=1 , where π represents an unknown permutation.
The presence of this unknown permutation—which can be viewed as a nuisance parameter—
introduces substantial challenges to this problem.
It is convenient to introduce matrix-vector notation so as to state the problem more
precisely. If we form the matrices A ∈ Rn×d and Y ∈ Rn×m with aTi and yiT , respectively, as
their ith row, we arrive at the model
Y = Π∗ AX ∗ + W,
(1)
where Π∗ is an unknown n × n permutation matrix, X ∗ ∈ Rd×m is an unknown matrix of
parameters, and W is the additive observation noise1 . When m = 1, this reduces to the vector
linear regression model with an unknown permutation, given by
y = Π∗ Ax∗ + w,
(2)
which we refer to as the shuffled vector model.
The observation model (1) arises in multiple applications, which are discussed in detail
for the shuffled vector model (2) in our earlier work [PWC16]. Here let us describe two
applications that arise in the multivariate setting (m > 1), which we use as running examples
throughout the paper.
1
We refer to the setting W = 0 a.s. as the noiseless case.
1
Figure 1. Example of pose and correspondence estimation for 2D images. The image coordinates are related by an unknown resizing and rotation X. The unknown permutation represents
the correspondence between keypoints (white circles) obtained via corner-detection. The matrices Y and A represent coordinates of all keypoints, and approximately obey the relation (1)
because all the keypoints detected in the two images are not the same.
Example 1 (Pose and correspondence estimation). Our first motivating application is the
problem of pose and correspondence estimation in images [MSC09]; it is closely related to
point-cloud matching in graphics [Man93]. Suppose that we are given two images of a similar
object, with the coordinates of one image arising from an unknown linear transformation of
the coordinates of the second. In order to determine the linear transformation, keypoints are
detected in each of the images individually and then matched; see Figure 1 for an illustration.
We emphasize that in practice, the keypoint detection algorithm also returns features that
help in finding the matching permutation Π∗ , but our goal here is to analyze whether there
are procedures that are robust to such features being missing or corrupted. It is also worth
noting that while in this example we have d = m = 2, the model is also valid for higher (but
equal) parameters d and m, if we assume that in addition to the coordinates of the keypoints,
other attributes like pixel brightness, colour, etc. in the two images are also related by a
linear transformation.
Example 2 (Header-free communication). A second application is that of header-free communication in large communication networks [PWC16]. Suppose that we use multiple sensors
to take noisy measurements of a unknown matrix X ∗ of parameters; each measurement cor∗
>
responds to a noisy linear observation of the form a>
i X + wi . In very large networks,
such as those that arise in Internet of Things applications, it is often found that the bandwidth between a sensor and fusion center is mainly dominated by a header containing identity
information—that is, by a bitstring that identifies sensor i to the fusion center [KSF+ 09]. One
possible solution to this problem is header-free communication, meaning that the identities
of the sensors that sent the signal are no longer known to the fusion center. This absence can
be modeled by introducing the unknown permutation matrix as in our model. If we are still
able to achieve similar statistical performance without these headers, then such an approach
is clearly preferable from a bandwidth standpoint.
With this motivation in hand, let us now provide a high-level overview of the main results
of this paper. We focus on the multivariate model (1) with a fixed design matrix A, and
2
i.i.d.
b X)
b based on its “denoising”
Gaussian2 noise Wij ∼ N (0, σ 2 ). We evaluate an estimator (Π,
1 b
b − Π∗ AX ∗ |||2F .
capability, which we capture using the normalized prediction error nm
|||ΠAX
Our primary objective in this paper is to characterize the fundamental limits of denoising in
a minimax sense. In particular, an estimator is any measurable mapping of the input (y, A)
b X)
b of the permutation and regression matrix, and we measure the quality of
to estimates (Π,
these estimates via their uniform mean-squared error
b X)
b := 1
R(Π,
nm
sup
Π∗ ∈Pn
X ∗ ∈Rd×m
b X
b − Π∗ AX ∗ |||2 ,
E|||ΠA
F
(3a)
b X).
b
where the expectation is taken over the noise W , and any randomness in the estimator (Π,
b
b
Note that a control on this quantity ensures that the estimator (Π, X) performs uniformly
well over the full class of permutation and regression matrices. By taking an infimum over all
estimators, we arrive at the minimax risk associated with the problem, viz.
inf
b
Π∈P
n
d×m
b
X∈R
b X)
b =
R(Π,
inf
b
Π∈P
n
d×m
b
X∈R
1
nm
sup
Π∗ ∈Pn
X ∗ ∈Rd×m
b X
b − Π∗ AX ∗ |||2F .
E|||ΠA
(3b)
Our interest will be in upper and lower bounding this quantity as a function of the design
matrix A, dimensions (n, m, d) and the noise variance σ 2 . We also demonstrate an explicit
(but computationally expensive) algorithm that achieves the minimax risk up to a log(n)
factor, and analyze polynomial-time estimators with slightly larger prediction error.
In both of the examples discussed above, estimators with small minimax prediction error
are of interest. In the pose and correspondence estimation problem, obtaining low prediction
error is equivalent to obtaining near-identical keypoint locations on both images; in the sensor
network example, we are interested in obtaining a set of noise-free linear functions of the
input signal. It is important to note that depending on the application, multiple regimes of
the parameter triplet (n, m, d) are of interest. Therefore, in this paper, we focus on capturing
the dependence of denoising error rates on all of these parameters, and also on the structure
of the matrix A.
Our work contributes to the growing body of literature on regression problems with unknown permutations, as well as related row-space perturbation problems including blind deconvolution [LS15], phase retrieval [CLS15], and dictionary learning [TF11]. Regression problems with unknown permutations have been considered in the context of statistical seriation
and univariate isotonic matrix recovery [FMR16], and non-parametric ranking from pairwise
comparisons [SBGW17], which involves bivariate isotonic matrix recovery. Moreover, the
prediction error is used to evaluate estimators in both these applications.
Specializing to our setting, the shuffled vector model (2) was first considered in the context
of compressive sensing with a sensor permutation [EBDG14]. The first theoretical results were
provided by Unnikrishnan et al. [UHV15], who provided necessary and sufficient conditions
needed to recover an adversarially chosen x∗ in the noiseless model with a random design
matrix A. Also in the random design setting, our own previous work [PWC16] focused on
the complementary problem of recovering Π∗ in the noisy model, and showed necessary and
sufficient conditions on the SNR under which exact and approximate recovery were possible.
An efficient algorithm to compute the maximum likelihood estimate was also provided for the
special case d = 1.
2
Our results also extend to the case of i.i.d. sub-Gaussian noise.
3
1.1
Our contributions
First, we characterize the minimax prediction error of multivariate linear model with an
unknown permutation up to a logarithmic factor, by analyzing the maximum likelihood estimator. Since the maximum likelihood estimate is NP-hard to compute in general [PWC16],
we then propose a computationally efficient estimator based on singular value thresholding
and sharply characterize its performance, showing that it achieves vanishing prediction error
over a restricted range of parameters. We also propose a variant of this estimator that achieves
the same error rates, but with the advantage that it does not require the noise variance to
be known. Third, we propose an efficient spectral algorithm for the noiseless problem that
is exact provided certain natural conditions are met. We demonstrate this algorithm on an
image point cloud matching task. Finally, we extend our results to a richer class of models
that allows for outliers in the dataset. In the next section, we collect our main theorems and
discuss their consequences. Proofs are postponed to Section 3.
Notation: We use Pn to denote the set of permutation matrices. Let Id denote the identity
matrix of dimension d. We use the notation |||M |||F , |||M |||op , and |||M |||nuc to denote the Frobenius, operator, and nuclear norms of a matrix M , and c, c1 , c2 to denote universal constants
that may change from line to line.
2
Main results
In this section, we state our main results and discuss some of their consequences. We divide
our results into four subsections, having to do with minimax rates, polynomial time estimators,
efficient procedures for the noiseless problem, and an extension of the model (1) that allows
for outliers.
2.1
Minimax rates of prediction
Assuming that the noise W is i.i.d. Gaussian, so the maximum likelihood estimate (MLE) of
the parameters (Π∗ , X ∗ ) is given by
b ML , X
bML ) = arg min |||Y − ΠAX|||2F .
(Π
Π∈Pn
X∈Rd×m
(4)
This estimator is also sensible for non-Gaussian noise, as long as its tail behavior is similar
to the Gaussian case (as can be formalized by the notion of sub-Gaussianity).
In this section, we begin by providing an upper bound the prediction error achieved by the
maximum likelihood estimator for any design matrix A. In general, however, it is impossible
to prove a matching lower bound for an arbitrary matrix A. As an extreme example, suppose
that the matrix A with identical rows: in this case, the permutation matrix Π∗ plays no role
whatsoever, and the problem is obviously much easier than with a generic matrix A.
With this fact in mind, we derive lower bounds that apply provided the matrix A lies in a
restricted class, in order to define which we require some additional notation. For a vector v,
let v s denote the vector sorted in decreasing order, and let B2,n (1) denote the n-dimensional
`2 -ball of unit radius centered at 0. Define the matrix class
n
o
A(γ, ξ) = A ∈ Rn×d | ∃a ∈ range(A) ∩ B2,n (1) with asbγnc ≥ asbγnc+1 + ξ .
4
In rough terms, this condition defines matrices that are not “flat”, meaning that there is some
vector in their range obeying the (γ, ξ)-separation condition defined above. It can be verified
√
that a matrix A with i.i.d. sub-Gaussian entries lies in the class A(C1 , C2 / n) with high
probability for fixed constants C1 , C2 . We are now ready to state our first main result:
Theorem 1. For any triple (A, X ∗ , Π∗ ) ∈ Rn×d × Rd×m × Pn , we have
b ML AX
bML − Π∗ AX ∗ |||2
|||Π
1
F
2 rank(A)
≤ c1 σ
+ min {log n, m} ,
nm
n
m
(5a)
with probability greater than 1 − e−c(n log n+m rank(A)) .
√
b X),
b we have
Conversely, for any matrix A ∈ A(C1 , C2 / n), and any estimator (Π,
"
#
b X
b − Π∗ AX ∗ |||2F
|||ΠA
1
2 rank(A)
sup E
≥ c2 σ
,
(5b)
+
nm
n
m
Π∗ ∈Pn
X ∗ ∈Rd×m
where the constant c2 depends on the value of the pair (C1 , C2 ), but is independent of other
problem parameters.
Theorem 1 characterizes the minimax rate up to a factor that is at most logarithmic in
n. It shows that the MLE is minimax optimal for prediction error up to logarithmic factors
for all matrices that are not too flat. The bounds have the following interpretation, similar to
the results of Flammarion et al. [FMR16] on prediction error for unimodal columns. The first
term corresponds to a rate achieved even if the estimator knows the true permutation Π∗ ; the
second term quantifies the price paid for the combinatorial choice among n! permutations. As
a result, we see that if m log n, then the permutation does not play much of a role in the
problem, and the rates resemble those of standard linear regression. Such a general behaviour
is expected, since a large m means that we get multiple observations with the same unknown
b better.
permutation, and this should allow us to estimate Π
Clearly, a flat matrix is not influenced by the unknown permutation, and so the second
term of the lower bound need not apply. As we demonstrate in the proof, it is likely that the
flatness of A can also be incorporated in order to prove a tighter upper bound in this case,
but we choose to state the upper bound as holding uniformly for all matrices A, with the loss
of a logarithmic factor.
It is also worth mentioning that the logarithmic factor in the second term is shown to be
nearly tight for the problem of unimodal matrix estimation with an unknown permutation
[FMR16], suggesting that a similar factor may also appear in a tight version of our lower
bound (5b). For the specific case where m = 1 however, which corresponds to the shuffled
vector model (2), our bounds are tight up to constant factors, and summarized by the following
corollary.
√
Corollary 1. In the case m = 1, for any matrix A ∈ A(C1 , C2 / n), we have
1 b
2
∗
∗ 2
c2 σ ≤ inf
sup E
kΠAb
x − Π Ax k2 ≤ c1 σ 2 .
∗
n
b
Π
∈P
Π∈Pn
n
x
b∈Rd
x∗ ∈Rd
In other words, the normalized minimax prediction error for the shuffled vector model does
not decay with the parameters n or d, and so no estimator achieves consistent prediction for
every parameter choice (Π∗ , x∗ ). Again, this is a consequence of the fact that—unlike when
5
m is large—we do not get independent observations with the permutation staying fixed, and
herein lies the difficulty of the problem.
Both Theorem 1 and Corollary 1 provide non-adaptive minimax bounds. An interesting
question is whether the least squares estimator is also minimax optimal up to logarithmic
factors over finer classes of Π∗ and X ∗ , i.e., whether it is adaptive in some interesting way.
One would expect that the estimator adapts to the parameter κ(AX ∗ ), the number of distinct
entries in the matrix AX ∗ , similarly to the problem of monotone parameter recovery [FMR16].
2.2
Polynomial time estimators
As shown in our past work [PWC16], computing the MLE estimate (4) is NP-hard in general.
Accordingly, it is natural to turn our attention to alternative estimators, and in particular
ones that are guaranteed to run in polynomial time.
Here we analyze two simple methods for estimating the matrix Π∗ AX ∗ , based either on
singular value thresholding, and a closely related variant that uses an explicit regularization
based on the nuclear norm. It is well-known that such methods are appropriate when the
matrix is low-rank, or approximately low-rank. While the matrix Y ∗ is not low-rank, its rank
is bounded by that of the matrix A, a fact that we leverage in our bounds.
Pr
>
Given a matrix M with the singular value decomposition
Pr M = i=1 σi ui v>i , its singular
value thresholded version at level λ is given by Tλ (M ) = i=1 σi I(σi ≥ λ)ui vi , where I(·) is
the indicator function of its argument.
The singular value thresholding (SVT) operation serves the purpose of denoising the observation matrix, and has been analyzed in the context of more general matrix estimation
problems by various authors (e.g., [CCS10, Cha15]).
√
√
Theorem 2. For any matrices (Π∗ , X ∗ ), the SVT estimate with λ = 1.1σ( n + m) satisfies
1
|||Tλ (Y ) − Π∗ AX ∗ |||2F ≤ c1 σ 2 rank(A)
nm
1
1
+
n m
(6a)
with probability greater than 1 − e−cnm .
Conversely, for any matrix A with rank at most m, there exist matrices Π0 and X0 (that
may depend A) such that for any threshold λ > 0, we have
1
1
1
2
2
|||Tλ (Y ) − Π0 AX0 |||F ≥ c2 σ rank(A)
+
,
(6b)
nm
n m
with probability greater than 1 − e−cnm .
Comparing inequalities (5b) (which holds for any denoised matrix, not just those having
b X)
b and (6b), we see that the SVT estimator, while computationally efficient, may
the form ΠA
be statistically sub-optimal. However, it is consistent in the case where rank(A) is sufficiently
small compared to m and n, and minimax optimal when rank(A) is a constant. Intuitively,
the rate it attains is a result of treating the full matrix Π∗ A as unknown, and so it is likely
that better, efficient estimators exist that take the knowledge of A into account.
A potential concern is that the SVT estimator is required to know the noise variance
2
σ . This issue can be taken care of via the square-root LASSO “trick” [BCW11], which
ensures a self-normalization that obviates the necessity for a noise-dependent threshold level.
In particular, consider the estimate
Ybsr (λ) = arg min
|||Y − Y 0 |||F + λ|||Y 0 |||nuc .
0
Y
6
(7)
Using a choice of λ that no longer depends on σ, we have the following guarantee:
1
≤ 1/20, then for any choice
of parameters Π∗ and X ∗ , the
Theorem 3. If rank(A) n1 + m
√1
n
square-root LASSO estimate (7) with λ = 2.1
+
√1
m
satisfies
1 b
|||Ysr (λ) − Π∗ AX ∗ |||2F ≤ c1 σ 2 rank(A)
nm
1
1
+
n m
with probability greater than 1 − 2e−cnm .
We prove Theorem 3 in Section 3.3 for completeness. However, it should be noted that the
square-root LASSO has been analyzed for matrix completion problems [Klo14], and our proof
1
follows similar lines for our different observation model. The condition rank(A) n1 + m
≤
1/20 does not significantly affect the claim, since our bounds no longer guarantee consistency
of the estimate Ybsr (λ) when this condition is violated.
While the optimization problem (7) can be solved efficiently, there may be cases when the
noise is (sub)-Gaussian of known variance for which the SVT estimate can be computed more
quickly. Hence, the SVT estimator is usually preferred in cases where the noise statistics are
known.
2.3
Exact algorithm for the noiseless case
For the noiseless model, the only efficient algorithm known up to now is for the special case
d = m = 1, as presented in our past work [PWC16]. It turns out that this algorithm has a
natural generalization to higher dimensional problems, at least when certain conditions on the
input matrices (A, Y ) are satisfied. The higher dimensional generalization requires analyzing
certain spectral properties of the input matrices.
In order to state the theorem, we require require a few definitions. Given a matrix
> , where U
M ∈ Rn×d , consider its reduced singular value decomposition M = UM ΣM VM
M
is a matrix of its left singular vectors. The (left) leverage scores of the matrix M are given
the `2 -norms of the rows of the matrix UM ; in analytical terms, we can express them as the
> ), where the operator diag extracts the diagonal of
n-dimensional vector `(M ) = diag(UM UM
a square matrix. With this notation, the LevSort algorithm performs the following three steps
on the input pair (Y, A):
(i) Compute the leverage scores `(Y ) and `(A).
b lev ∈ arg minΠ k`(Y ) − Π
b lev `(A)k2 .
(ii) Find a permutation Π
2
blev = Π
b lev A † Y , where M † denotes the Moore-Penrose pseudoin(iii) Return the matrix X
verse of a matrix M .
Note that this algorithm runs in polynomial time, since it involves only spectral computations
and a matching step that can be computed in time O(n log n). As we demonstrate in the
b lev such that
proof, step (ii) for the noiseless model actually returns a permutation matrix Π
b
`(Y ) = Πlev `(A).
Theorem 4. Consider an instantiation of the noiseless model with rank(A) ≤ rank(X ∗ ), and
such `(A) and `(Y ) both have all distinct entries. Then the LevSort algorithm recovers the
parameters (Π∗ , X ∗ ) exactly.
7
(a)
(b)
Figure 2. Synthetic experiment illustrating exact pose and correspondence estimation by the
LevSort algorithm for a transformed “5” in panel (a), and a transformed fruit picture in panel
(b). In each panel, the right images are obtained via a linear tranformation of the coordinates
of the respective left images, and keypoints are generated according to the noiseless model (1);
keypoints are the same in the right and left image.
The LevSort algorithm is a generalization of our own algorithm [PWC16] to the matrix
setting. However, instead of a simple sorting algorithm, we now require an additional spectral
component. While showing the necessity of the condition rank(A) ≤ rank(X ∗ ) is still open,
an efficient algorithm that does not impose any conditions is unlikely to exist due to the
general problem being NP-hard [PWC16]. Note that the condition includes as a special case
all problems in which the matrices A and X ∗ are full rank, with d ≤ m.
In particular, the pose and correspondence estimation problem for 2D point clouds satisfies
the conditions of Theorem 4 under some natural assumptions. We have d = m = 2 for all
such problems, and rank(X ∗ ) = 2 unless the linear transformation is degenerate. Furthermore,
unless the keypoints are generated adversarially, the leverage scores of the matrix A and the
rows of Y are distinct. Thus, assuming that the noiseless version of model (1) exactly describes
the keypoints detected in the two images (which is an idealization that may not be true in
real data), we are guaranteed to find both the pose and the correspondence exactly.
In Figure 2, we demonstrate the guarantee of Theorem 4 on two image correspondence
tasks when the keypoints detected in the two images are identical and the transformation
between coordinates is linear.
2.4
Extensions to outliers
The results of Sections 2.1 and 2.2 also hold in a somewhat general setting, where the set
of perturbations to the rows of the matrix A is allowed to be larger than just the set of
permutation matrices Pn . In particular, defining the set of “clustering matrices” Cn as
Cn = {D ∈ {0, 1}n×n | D1 = 1},
we consider an observation model of the form
Y = D∗ AX ∗ + W,
(8)
where the matrices A, X ∗ , and W are as before, and D∗ ∈ Cn now represents a clustering matrix. Such a clustering condition ensures stochasticity of the matrix D∗ (not double
stochasticity, as in the permutation model), and corresponds to the case where multiple responses may come from the same covariate, and some of the data may be permuted. Such
a model is likely to better fit data from image correspondence problems when the keypoints
detected in the two images are quite different. Also, such a formulation is loosely related to
the k-means clustering problem with Gaussian data [ABC+ 15].
8
As it turns out, Theorems 1, 2 and 3 also hold for this model, with minor modifications
to the proofs. Defining the analogous MLE for this model as
bML = arg min |||Y − DAX|||2F ,
b ML , X
D
D∈Cn
X∈Rd×m
we have the following theorem.
Theorem 5.
have
(a) For any matrix A, and for all parameters X ∗ ∈ Rd×m and D∗ ∈ Cn , we
b ML AX
bML − D∗ AX ∗ |||2
|||D
F
≤ c1 σ 2
nm
rank(A)
1
+ min {log n, m} ,
n
m
with probability greater than 1 − e−c(n log n+m rank(A)) .
√
√
(b) For any choice of parameters D∗ and X ∗ , the SVT estimate with λ = 1.1σ( n + m)
satisfies
1
1
1
∗
∗ 2
2
|||Tλ (Y ) − D AX |||F ≤ c1 σ rank(A)
+
nm
n m
with probability greater than 1 − e−cnm .
(c) For any choice of parameters D∗ and X ∗ , the square-root LASSO estimate (7) with
λ = 2.1 √1n + √1m satisfies
1 b
|||Ysr (λ) − D∗ AX ∗ |||2F ≤ c1 σ 2 rank(A)
nm
1
1
+
n m
with probability greater than 1 − 2e−cnm .
Clearly, the lower bounds (5b) and (6b) hold immediately for the model (8) as a result of
the inclusion Pn ⊂ Cn .
3
Proofs
This section contains proofs of all our main results. We use C, c, c0 to denote absolute constants
that may change from line to line. We let σi (M ) denote the ith largest singular value of a
matrix M .
3.1
Proof of Theorem 1
We split the proof into two natural parts, corresponding to the upper and lower bounds,
respectively. The upper bound boils down to analyzing the Gaussian width [Pis99] of a
certain set, which we obtain via Dudley’s entropy integral [Dud67] and bounds on the metric
entropy of the observation space. The lower bound is obtained via a packing construction and
an application of Fano’s inequality.
9
3.1.1
Proof of upper bound
b ML AX
bML , we have by the optimality of Yb for problem (4)
Writing Y ∗ = Π∗ AX ∗ and Yb = Π
2
∗
2
b
b = Yb − Y ∗ satisfies
that |||Y − Y |||F ≤ |||Y − Y |||F , from which it follows that the error matrix ∆
the following basic inequality:
1 b 2
b W ii,
|||∆|||F ≤ hh∆,
2
(9)
where hhA, Bii denotes the trace inner product between two matrices A and B. We prove
inequality (5a) by proving the following claims.
(
Pr
(
Pr
b 2
|||∆|||
F
≥ 8σ 2
nm
)
b 2
|||∆|||
F
≥ c2 σ 2
nm
Proof of inequality (10a):
equality (9) yields
≤ e−
nm
8
, and
d log n
+
n
m
)
(10a)
≤ e−c(n log n+m rank(A)) .
(10b)
Applying the Cauchy Schwarz inequality to the RHS of in1 b
|||∆|||F ≤ |||W |||F .
2
(11)
Squaring both sides of inequality (11) and using standard sub-exponential tail bounds [Wai15]
yields inequality (10a).
Proof of inequality (10b): Without loss of generality, by rescaling as necessary, we may
assume that the noise W has standard normal entries (σ 2 = 1). We use Um (A) to denote the
set of matrices whose m columns lie in the range of ΠA for some permutation matrix Π, i.e.,
Um (A) = {Y ∈ Rn×m | Y = ΠAX for some Π ∈ Pn , X ∈ Rd×m }.
(12)
Also define the set
Udiff
m (A) = {Y | Y = Y1 − Y2 for Y1 , Y2 ∈ Um (A)},
as well as the function
Z(t) : =
sup
hhD, W ii.
D∈Udiff
m (A)
|||D|||F ≤t
Before proceeding with the proof, we state the definition of the covering number of a set.
Definition
1 (Covering number). A δ-cover of a set T with respect to a metric ρ is a set
1 2
θ , θ , . . . , θN ⊂ T such that for each θ ∈ T, there exists some i ∈ [N ] such that ρ(θ, θi ) ≤ δ.
The δ-covering number N (δ, T, ρ) is the cardinality of the smallest δ-cover.
The logarithm of the covering number is referred to as the metric entropy of a set. The
following lemma bounds the metric entropy of the set Udiff
m (A). Let BF (t) denote the Frobenius
norm ball of radius t centered at 0.
10
Lemma 1. The metric entropy of the set Udiff
m (A) ∩ BF (t) in the Frobenius norm metric is
bounded as
4t
diff
log N (δ, Um (A) ∩ BF (t), ||| · |||F ) ≤ 2 rank(A) · m log 1 +
+ 2n log n.
(13)
δ
We prove the lemma at the end of the section, taking it as given for the proof of inequality (10b).
Proof of inequality (10b). By definition of Z(t), is easy to see that we have
1 b 2
b F .
|||∆|||F ≤ Z |||∆|||
2
3
One can also verify that the set Udiff
m (A) is star-shaped , and so the following critical inequality
holds for some δn,m > 0:
E [Z(δn,m )] ≤
2
δn,m
.
2
(14)
We are interested in the smallest (strictly) positive solution to
pinequality (14). Moreover, we
b F ≤ c tδn,m with probability greater
would like to show that for every t ≥ δn,m , we have |||∆|||
0
than 1 − ce−c tδn,m .
Define the “bad” event
n
o
p
p
Et : = ∃∆ ∈ Udiff
(A)
|
|||∆|||
≥
tδ
and
hh∆,
W
ii
≥
2|||∆|||
tδ
.
(15)
F
n,m
F
n,m
m
Using the star-shaped property of Udiff
m (A), it follows by a rescaling argument that
p
Pr[Et ] ≤ Pr[Z(δn,m ) ≥ 2δn,m tδn,m ] for all t ≥ δn,m .
The entries of W are i.i.d. standard Gaussian, and the function W 7→ Z(t) is convex and
Lipschitz with parameter t. Consequently, by Borell’s theorem (see, for example, Milman and
Schechtman [MS86] for a simple proof), the following holds for all t ≥ δn,m :
p
Pr[Z(δn,m ) ≥ E[Z(δn,m )] + δn,m tδn,m ] ≤ 2e−ctδn,m .
p
2
By the definition of δn,m , we have E[Z(δn,m )] ≤ δn,m
≤ δn,m tδn,m for any t ≥ δn,m , and
consequently, for all t ≥ δn,m , we have
p
Pr[Et ] ≤ Pr[Z(δn,m ) ≥ 2δn,m tδn,m ] ≤ 2e−ctδn,m .
p
p
Now either |||∆|||F ≤ tδn,m , or we have k∆kF > tδn,m . In the latter case, conditioning
p
on the complementary event Etc , our basic inequality implies that 12 |||∆|||2F ≤ 2|||∆|||F tδn,m .
Consequently, we have
n
o
n
o
p
p
Pr |||∆|||F > 4 tδn,m ≤ Pr |||∆|||F > 4 tδn,m |Etc + Pr{Et }
≤ 2e−ctδn,m .
3
A set S is said to be star-shaped if t ∈ S implies that αt ∈ S for all α ∈ [0, 1]
11
Putting together the pieces yields
|||∆|||F ≤ c0
p
tδn,m
with probability at least 1 − 2e−ctδn,m for every t ≥ δn,m .
In order to determine a feasible δn,m satisfying the critical inequality (14), we need to
bound the expectation E[Z(δn,m )]. We now use Dudley’s entropy integral [Dud67] to bound
E[Z(t)]. In particular, for a universal constant C, we have
Z tq
1
log N (δ, Udiff
E [Z(t)] ≤
m (A) ∩ BF (t), ||| · |||F )dδ
C
0
Z ts
(i) p
p
2t
dδ
≤ t n log n + m rank(A)
log 1 +
δ
0
Z 1s
p
2
(ii) p
= t n log n + t m rank(A)
log 1 +
du
u
0
p
p
≤ t n log n + ct m rank(A),
where in step (i), we have made use of Lemma 1, and in step (ii), we have used the change of
variables u = δ/t. Now comparing with the critical inequality, we see that
p
p
δn ≤ c
n log n + m rank(A) .
Putting together the pieces then proves claim (10b).
It remains to prove Lemma 1.
Proof of Lemma 1. We begin by finding the δ-covering number of
n×m
UΠ
| Y = ΠAX for some X ∈ Rd×m }.
m (A) = {Y ∈ R
(16)
Note that UΠ
m is isomorphic to range(Im ⊗ ΠA), where ⊗ denotes the tensor product. Note
that range(Im ⊗ ΠA) is a linear subspace of dimension m · rank(A). Also, since the set
range(Im ⊗ ΠA) ∩ Bnm
2 (t) is an m · rank(A)-dimensional `2 -ball of radius t, we have by a
volume ratio argument that
2t m rank(A)
∩ BF (t), ||| · |||F ) ≤ 1 +
.
δ
S
By definition, we also have Um (A) = Π∈Pn UΠ
m (A), and so by the union bound, we have
N (δ, UΠ
m (A)
2t
N (δ, Um (A) ∩ BF (t), ||| · |||F ) ≤ n! 1 +
δ
m rank(A)
.
In order to complete the proof, we notice that
2
N (δ, Udiff
m (A) ∩ BF (t), ||| · |||F ) ≤ [N (δ/2, Um (A) ∩ BF (t), ||| · |||F )] ,
since it is sufficient to use two δ/2-covers of the set Um (A) ∩ BF (t) in conjunction in order to
obtain a δ-cover of the set Udiff
m (A) ∩ BF (t).
12
3.1.2
Proof of lower bound
As alluded to before, the bound follows from a packing set construction and Fano’s inequality,
which is a standard template used to prove minimax lower bounds. Suppose we wish to
estimate a parameter θ over an indexed class of distributions P = {Pθ | θ ∈ Θ} in the
square of a (pseudo-)metric ρ. We refer to a subset of parameters {θ1 , θ2 , . . . , θM } as a local
(δ, )-packing set if
ρ(θi , θj ) ≥ δ
min
and
i,j∈[M ],i6=j
1
M
2
X
D(Pθi kPθj ) ≤ .
i,j∈[M ]
Note that this set is a δ-packing in the ρ metric with the average KL-divergence bounded by
. The following result is a straightforward consequence of Fano’s inequality:
Lemma 2 (Local packing Fano lower bound). For any (δ, )-packing set of cardinality M , we
have
i δ2
h
+
log
2
∗
2
bθ ) ≥
inf sup E ρ(θ,
1−
.
(17)
2
log M
θb θ∗ ∈Θ
The remainder of argument is directed to establishing the following two claims:
b X
b − Π∗ AX ∗ |||2F
|||ΠA
rank(A)
≥ cσ 2
for all A, and
nm
n
X ∗ ∈Rd×m
b X
b − Π∗ AX ∗ |||2
√
|||ΠA
1
F
sup E
≥ c0 σ 2 if A ∈ A(C1 , C2 / n).
nm
m
X ∗ ∈Rd×m
sup
E
(18a)
(18b)
It is easy to see that both claims together prove the lemma.
Proof of claim (18a): This claim is consequence of classical minimax bounds on linear
regression. Since we are operating in the matrix setting, we include the proof for completeness.
The proof involves the construction of a packing set {ΠAXi }M
i=1 such that for all i 6= j ∈ [M ],
|||ΠAXi −ΠAXj |||F
|||ΠAXi |||F
√
√
we have
≤ 4δ and
≥ δ. Since we are effectively packing the space
nm
nm
√1
nm
range(Im ⊗ΠA), standard results show that there exists such a packing of this space with
log M ≥ rank(Im ⊗ ΠA) log 2.
Also note that with the underlying parameter Xi , our observations have the distribution
Pi = N (ΠAXi , σ 2 Inm ). Hence, the KL divergence between two observations i and j is simply
D(Pi kPj ) =
32δ 2 nm
1
2
kΠAX
−
ΠAX
k
≤
.
i
j
F
σ2
σ2
Substituting this into the bound of Lemma 2 with ρ(θ1 , θ2 ) = kθ1 − θ2 kF , we have
!
32δ 2 nm
+
log
2
δ2
2
σ
M≥
1−
,
2
m rank(A) log 2
where we have again used M to denote the minimax rate of prediction.
2
Setting δ 2 = c σ rank(A)
completes the proof of claim (18a). Note that the proof of this
n
√
claim did not require the assumption that A ∈ A(C1 , C2 / n).
13
Proof of claim (18b) For ease of exposition, we first prove claim (18b) for matrices in a
√
smaller class than A(C1 , C2 / n). We let 1pn denote the n-dimensional vector having 1 in its
first p coordinates and 0 in the remaining coordinates.
Now consider the class of matrices that have 1pn in their range. By multiplying with δ and
stacking m of these vectors up as columns, we have a matrix Ye 1 ∈ Rn×m whose first p rows
are identically δ and the rest are identically zero. Define the Hamming distance between two
binary vectors dH (u, v) = #{i : ui 6= vi }. We require the following lemma.
Lemma 3. There exists a set of binary n-vectors {vi }M
i=1 , each of Hamming weight p and
(np)
satisfying dH (vi , vj ) ≥ h, having cardinality M = Pb h−1 c
.
n−p p
2
(
)(
)
i=1
i
i
The lemma is proved at the end of this section.
Proof of claim (18b) Applying Lemma 3 and a rescaling argument, we see that there is a
packing set {Πi Ye 1 }M
i=1 such that
r
p
1
1
e
√
|||Πi Y |||F = δ
for i ∈ [M ], and
(19a)
n
nm
r
h
1
√
for i 6= j ∈ [M ].
(19b)
|||Πi Ye 1 − Πj Ye 1 |||F ≥ δ
n
nm
Fixing some constant γ ∈ (0, 1) and choosing p = γn and h = n2 min {γ/2, (1 − γ)/2}, it can
be verified that we obtain a packing set of size M ≥ eγ log(1/γ)n . We now have observation i
distributed as Pi = N (Πi Ye 1 , σ 2 Inm ), and so
D(Pi kPj ) =
2
1
e 1 − Πj Ye 1 |||2F ≤ c δ γnm .
|||Π
Y
i
σ2
σ2
Finally, substituting into the Fano bound of Lemma 2 yields
!
cδ 2 γnm
2
+
log
2
1 b b
δ
2
inf
sup E
|||ΠAX − Π∗ AX ∗ |||2F ≥
1− σ
.
nm
2
γ log(1/γ)n
b
Π∗ ∈Pn
Π∈P
n
d×m X ∗ ∈Rd×m
b
X∈R
2
Setting δ 2 = c(γ) σm for a constant c(γ) depending only on γ completes the proof provided
the vector 1pn ∈ range(A) for p = γn with γ ∈ (0, 1).
√
It remains to extend the proof to matrices in the class A(C1 , C2 / n), and to prove
Lemma 3.
√
By definition, if A ∈ A(C1 , C2 / n), then there exists a vector a ∈ range(A) ∩ B2 (1) such
√
that asC1 n ≥ asC1 n+1 + C2 / n. We may assume that kak2 = 1 by a rescaling argument, and
also that a = as . By definition, we have
(ai − aj )2 ≥ C22 /n for all i ≤ bC1 nc and j ≥ bC1 nc + 1.
(20)
It can also be verified that since kak2 = 1, we must have C2 ≤ 2. For the rest of
the proof, we assume for simplicity of exposition that C1 n is an integer. Fixing the value
= n2 min(C1 , 1 − C1 ), consider the -packing generated by permutations {Πi }M
i=1 of the vector
1 n , given by Lemma 3 by taking v = Π 1C1 n . Using these permutations, we observe that
1C
i
i n
n
kΠi a − Πj ak2 ≥
14
√ C2
√ ≥ c,
n
where c depends on the constants (C1 , C2 ), and we have used condition (20) along with the
fact that dH (vi , vj ) ≥ .
√
Following similar steps to before then proves lemma for all matrices A ∈ A(C1 , C2 / n).
It remains to prove Lemma 3.
Proof of Lemma 3 The proof follows by a volume ratio argument that underlies the proof
of the Gilbert-Varshamov bound. In particular, the number of permuted vectors of 1pn that are
Pb h−1 c n−p p
within a Hamming distance h − 1 of 1pn is given by ∆ = i=12
i
p−i . Now form a graph
p
n
with all p permuted vectors of 1n as vertices and connect two vertices if the corresponding
vectors have Hamming distance less than h. Then such a graph has uniform degree ∆ and
(n)
therefore contains an independent set of size ∆p .
3.2
Proof of Theorem 2
Again, we divide our proof into two parts, corresponding to the upper and lower bounds
respectively.
3.2.1
Proof of upper bound
For this proof, we use the shorthand Y ∗ = Π∗ AX ∗ . Also fix δ = 0.1, and let s be the number of
δ
singular values of Y ∗ greater than 1+δ
λ. Also, let Ys∗ denote the matrix formed by truncating
∗
Y to its top s singular values. By triangle inequality, we have
|||Tλ (Y ) − Y ∗ |||2F ≤ 2|||Tλ (Y ) − Ys∗ |||2F + 2|||Y ∗ − Ys∗ |||2F
≤ 2 rank(Tλ (Y ) −
Ys∗ )|||Tλ (Y
)−
Ys∗ |||2op
∗
+ 2 rank(Y )
2
δ
λ .
1+δ
Now note that by standard results in random matrix theory (see, for example, [Wai15, Theδ2
√
orem 6.1]), we have λ ≥ (1 + δ)|||W |||op with probability greater than 1 − e− 2 n(
condition on this event for the rest of the proof.
Consequently, for j ≥ s + 1, we have
√
n+ m)2
. We
σj (Y ) ≤ σj (Y ∗ ) + |||W |||op ≤ λ,
and so rank(Tλ (Y )) ≤ s. Additionally, we have
|||Tλ (Y ) − Ys∗ |||op ≤ |||Tλ (Y ) − Y |||op + |||Y − Y ∗ |||op + |||Y ∗ − Ys∗ |||op
δ
≤ λ + |||W |||op +
λ
1+δ
≤ 2λ.
Putting together the pieces yields
2
δ
|||Tλ (Y ) − Y |||F ≤ 16λ s + 2 rank(Y )
λ
1+δ
√
√
≤ Cσ 2 rank(Y ∗ )( n + m)2 ,
∗ 2
∗
2
a bound that holds with probability greater than 1 − e−cnm . In order to complete the proof,
we note that rank(Y ∗ ) ≤ rank(A).
15
3.2.2
Proof of lower bound
We split our analysis into two separate cases.
√
√
Case 1: First suppose that λ ≤ σ3 ( n + m). Consider any matrix Y ∗ = Π∗ AX ∗ , and
Y = Y ∗ + W . By definition of the thresholding operation, we have
|||Tλ (Y ) − Y |||2F ≤ min{n, m}|||Tλ (Y ) − Y |||2op ≤ min{n, m}λ2
√
√
1
≤ σ 2 min{n, m}( n + m)2 .
9
Triangle inequality yields
|||Tλ (Y ) − Y ∗ |||F ≥ |||Y − Y ∗ |||F − |||Tλ (Y ) − Y |||F
√
√
1 p
≥ |||W |||F − σ min{n, m}( n + m).
3
Now with probability greater than 1 − e−cnm , we have |||W |||2F ≥ σ 2 nm
2 , so that conditioned on
this event, we have
√
∗
|||Tλ (Y ) − Y |||F ≥ σ nm
2
1
√ −
2 3
,
which completes the proof.
√
√
Case 2: We now suppose that λ > σ3 ( n + m). Let the matrix A have the (reduced)
singular value decomposition A √
= U√A ΣA VA> , and introduce the shorthand r : = rank(A).
Form the diagonal matrix L = n+6 m Ir . Now let Π0 = In , and consider the parameter
>
matrix X0 = VA Σ−1
A LV , where V is an m × rank(A) dimensional matrix V with orthonormal
columns. Note that such a choice exists when rank(A) ≤ m.
We now have
|||Tλ (Y ) − Π0 AX0 |||2F = |||Tλ (UA LV > + W ) − UA LV > |||2F .
For two matrices A, B ∈ Rn×m with k = min{n, m}, it can be verified that
2
|||A − B|||F ≥
k
X
2
σi (A) − σi (B) .
i=1
By the definition of the thresholding operation, the top singular values of the matrix Tλ (UA LV > + W )
are all either greater than λ, or equal to 0. Hence, we have
r
n
o √n + √m 2
X
>
|||Tλ (Y ) − Π0 AX0 |||F ≥
λI σi (Tλ (UA LV + W )) ≥ λ −
6
2
i=1
≥ cr(n + m),
√
√
where the last step follows since λ > σ3 ( n + m), which completes the proof.
16
3.3
Proof of Theorem 3
It is again helpful to write the observation model in the form Y = Y ∗ +W , where Y ∗ = Π∗ AX ∗
represents the underlying matrix we are √
trying
√ to predict. Let us denote the choice of λ in
n+ m
. We use the shorthand R(M ) = |||Y − M |||F ,
the statement of Theorem 3 by λ0 = 2.1 √
nm
⊥ denote, respectively, the projection matrices onto the
and ∆ = Y ∗ − Ybsr (λ0 ). Let PM and PM
rowspace of the matrix M and its orthogonal complement.
We require the following auxiliary lemmas for our proof:
Lemma 4. We have
|||Y ∗ |||nuc − |||Ybsr (λ0 )|||nuc ≤ |||PY ∗ ∆|||nuc − |||PY⊥∗ ∆|||nuc .
|||W |||
Lemma 5. If λ0 ≥ 2 |||W |||op
, we have
F
|||PY⊥∗ ∆|||nuc ≤ 3|||PY ∗ ∆|||nuc .
We are now ready to prove Theorem 3.
Proof of Theorem 3. First, note that by standard results on concentration of χ2 -random variables and random matrices (see, for instance, Wainwright [Wai15]), we have
√
√
Pr{|||W |||op ≥ 1.01σ( n + m)} ≤ e−cnm , and
√
0
Pr{|||W |||F ≤ 0.99σ nm} ≤ e−c nm .
Hence, we have
|||W |||op
Pr λ0 ≥ 2
≥ 1 − 2ecnm .
|||W |||F
|||W |||
}.
For the rest of the proof, we condition on the event {λ0 ≥ 2 |||W |||op
F
Now, by definition of the quantity R(M ), we have
R(Ybsr (λ0 ))2 − R(Y ∗ )2 = hhY ∗ − Ybsr (λ0 ), Y ∗ − Ybsr (λ0 ) + 2W ii
= |||∆|||2F + 2hhW, ∆ii.
Some simple algebra yields
|||∆|||2F = −2hhW, ∆ii + (R(Ybsr (λ0 )) − R(Y ∗ ))(R(Ybsr (λ0 )) + R(Y ∗ )).
Now, from the definition of the estimate Ybsr (λ0 ), we have
R(Ybsr (λ0 )) + λ0 |||Ybsr (λ0 )|||nuc ≤ R(Y ∗ ) + λ|||Y ∗ |||nuc .
Rearranging terms yields
R(Ybsr (λ0 )) + R(Y ∗ ) ≤ 2R(Y ∗ ) + λ0 (|||Y ∗ |||nuc − |||Ybsr (λ0 )|||nuc )
(i)
≤ 2R(Y ∗ )+λ0 |||PY ∗ ∆|||nuc − |||PY⊥∗ ∆|||nuc ,
where step (i) follows from Lemma 4, and the fact that λ > 0.
17
(22)
Another rearrangement of inequality (22) yields
R(Ybsr (λ0 )) − R(Y ∗ ) ≤ λ0 |||Y ∗ |||nuc − |||Ybsr (λ0 )|||nuc
(ii)
≤ λ0 3|||PY ∗ ∆|||nuc − |||PY⊥∗ ∆|||nuc ,
where step (ii) follows from Lemma 4, and the fact that |||PY ∗ ∆|||nuc > 0. Thus, we have
established the upper bound (R(Ybsr (λ0 )))2 − (R(Y ∗ ))2 ≤ T1 T2 , where
T1 : = λ0 3|||PY ∗ ∆|||nuc − |||PY⊥∗ ∆|||nuc
and T2 : = 2R(Y ∗ ) + λ0 (|||PY ∗ ∆|||nuc − |||PY⊥∗ ∆|||nuc ) .
Expanding the product of the two terms yields
(R(Ybsr (λ0 )))2 − (R(Y ∗ ))2 ≤ 6λ0 R(Y ∗ )|||PY ∗ ∆|||nuc +3λ20 |||PY ∗ ∆|||2nuc −2λ0 R(Y ∗ )|||PY⊥∗ ∆|||nuc
+ λ20 |||PY⊥∗ ∆|||2nuc − 4λ20 |||PY ∗ ∆|||nuc |||PY⊥∗ ∆|||nuc
(iii)
≤ 6λ0 R(Y ∗ )|||PY ∗ ∆|||nuc +3λ20 |||PY ∗ ∆|||2nuc −2λ0 R(Y ∗ )|||PY⊥∗ ∆|||nuc ,
where step (iii) follows from Lemma 5, since λ20 |||PY⊥∗ ∆|||2nuc − 4λ20 |||PY⊥∗ ∆|||nuc |||PY ∗ ∆|||nuc ≤ 0.
We also note that
−2hhW, ∆ii ≤ 2|||W |||op |||∆|||nuc = 2|||W |||op (|||PY ∗ ∆|||nuc + |||PY⊥∗ ∆|||nuc ).
Combining with the fact that λ0 satisfies the inequality 2|||W |||op ≤ λ0 R(Y ∗ ), we find that
|||∆|||2F ≤ 7λ0 R(Y ∗ )|||PY ∗ ∆|||nuc + 2λ20 |||PY ∗ ∆|||2nuc
(iv)
p
≤ 7λR(Y ∗ ) rank(Y ∗ )|||∆|||F + 2λ20 rank(Y ∗ )|||∆|||F ,
where in step (iv), we have used the Cauchy Schwarz inequality and the fact that projections
are non-expansive to write
p
p
|||PY ∗ ∆|||nuc ≤ rank(PY ∗ ∆) |||PY ∗ ∆|||F ≤ rank(Y ∗ )|||∆|||F .
Rearranging yields
p
|||∆|||F 1 − 2λ20 rank(Y ∗ ) ≤ 7λ0 R(Y ∗ ) rank(Y ∗ ).
Squaring both sides, substituting the choice of λ0 , and using the condition rank(A)
completes the proof.
The only remaining detail is to prove Lemmas 4 and 5.
3.3.1
Proof of Lemma 4
We write
|||Ybsr (λ0 )|||nuc = |||Y ∗ + Ybsr (λ0 ) − Y ∗ |||nuc
= |||Y ∗ − PY⊥∗ ∆ − PY ∗ ∆|||nuc
≥ |||Y ∗ − PY⊥∗ ∆|||nuc − |||PY ∗ ∆|||nuc
= |||Y ∗ |||nuc + |||PY⊥∗ ∆|||nuc − |||PY ∗ ∆|||nuc .
Rearranging yields the claim.
18
1
n
+
1
m
≤ 1/20
3.3.2
Proof of Lemma 5
Rearranging the Cauchy Schwarz inequality for two matrices A and B yields
hhB, B − Aii
.
|||A|||F − |||B|||F ≥ −
|||B|||F
Now setting A = Y − Ybsr (λ0 ) and B = Y − Y ∗ , we have
R(Ybsr (λ0 )) − R(Y ∗ ) ≥ −
(i)
≥−
hhW, Ybsr (λ0 ) − Y ∗ ii
|||W |||F
λ0
|||W |||op |||∆|||nuc ,
2
|||W |||
where step (i) follows from Hölder’s inequality and choice of λ0 ≥ 2 |||W |||op
.
F
Combining this with the basic inequality (22) yields
λ0
λ0 (|||Ybsr (λ0 )|||nuc − |||Y ∗ |||nuc ) ≤
|||∆|||nuc .
2
Finally, using Lemma 4, we have
|||P ⊥∗ ∆|||nuc − |||PY ∗ ∆|||nuc ≤ |||Ybsr (λ0 )|||nuc − |||Y ∗ |||nuc
Y
1
≤ |||∆|||nuc
2
1
|||PY ∗ ∆|||nuc + |||PY⊥∗ ∆|||nuc ,
=
2
which completes the proof.
3.4
Proof of Theorem 4
> . We
We write the (reduced) singular value decomposition of a matrix M as M = UM ΣM VM
also adopt the shorthand rM = rank(M ) for the rest of this proof. The LevSort algorithm
clearly runs in polynomial time, since it involves a singular value decomposition and a sorting
operation, both of which can be accomplished efficiently. Let us now verify the exactness
guarantee.
Since the observation model (1) is noiseless and rA ≤ rX ∗ , we have rY = rA . Moreover,
by definition of the observation model, we have
Y > Y = (X ∗ )> A> AX ∗ .
Consequently, the unknown matrix X ∗ can be written as
>
X ∗ = VA Σ−1
A U ΣY VY ,
with U representing an unknown rA × rA unitary matrix (satisfying U > U = U U > = I).
Substituting this representation of X ∗ back into the noiseless observation model yields
UY ΣY VY> = Π∗ UA U ΣY VY> .
Now ΣY VY> has a full-dimensional row-space, and so we have UY = Π∗ UA U . We complete
the proof by observing that
UY UY> = Π∗ UA UA> (Π∗ )> ,
so that we have the equivalence `(Y ) = Π∗ `(A) as claimed. The uniqueness of the parameters
(Π∗ , X ∗ ) follows from the fact that the leverage score vectors `(A) and `(Y ) have distinct
entries.
19
3.5
Proof of Theorem 5
The proofs of Theorems 1, 2, and 3 apply to the model (8) with minor modifications. We
briefly mention these modifications here, leaving the details to the reader.
Part (a) follows by mimicking the proof of Section 3.1.1 as is, with a small modification
to the metric entropy of the observation space. In particular, the covering number of the
observation space is now upper bounded by nn · N (δ, UΠ
m (A) ∩ BF (t), ||| · |||F ), and the rest of
the proof follows as before.
Parts (b) and (c) follow by mimicking the proof of Sections 3.2.1 and 3.3, respectively, with
the definition Y ∗ = D∗ AX ∗ . Note that the clustering observation model can only decrease
the rank of Y ∗ from before.
4
Discussion
We conclude with a discussion of some possible future directions.
4.1
More general picture for regression problems
Multivariate linear regression is a specific case of the following problem with shuffled data
{(aπ(i) , yi )}ni=1 , with the covariates ai ∈ Rd and responses yi ∈ Rm related by the equation
yi = f aπ(i) + wi ,
(23)
where f represents a function from some parametric or non-parametric family F. The general
behaviour of prediction error for problems of this form should be similar to that seen in our
linear regression model, or the structured regression model of Flammarion et al. [FMR16]. In
particular, provided the data ai is sufficiently diverse and the function class F is sufficiently
expressive, the minimax rate of prediction for the permuted model should be given by the
sum of two terms: the minimax rate of the unpermuted model (or equivalently, with a known
permutation), and an additional constant/logarithmic term that accounts for the permutation.
4.2
Necessity of flatness condition and adaptivity
Our condition on the matrix A is a convenient one for the application of the Gilbert-Varshamov
type bound on distances between permuted binary vectors. However, this sufficient condition
may be far from necessary – we instead require some permutation codes of real numbers.
Conversely, the upper bound (5a) can be stated by explicitly taking the structure of the
matrix A into account; this will require bounds on the metric entropy of the union of subspaces
generated by permutations of the range space of A.
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22
| 10 |
ON THE GROUP OF AUTOMORPHISMS OF THE BRANDT λ0 -EXTENSION OF
A MONOID WITH ZERO
arXiv:1609.06085v1 [math.GR] 20 Sep 2016
OLEG GUTIK
Abstract. The group of automorphisms of the Brandt λ0 -extension Bλ0 (S) of an arbitrary monoid S
with zero is described. In particular we show that the group of automorphisms Aut(Bλ0 (S)) of Bλ0 (S) is
isomorphic to a homomorphic image of the group defines on the Cartesian product Sλ × Aut(S) × H1λ
with the following binary operation:
[ϕ, h, u] · [ϕ′ , h′ , u′ ] = [ϕϕ′ , hh′ , ϕu′ · uh′ ],
where Sλ is the group of all bijections of the cardinal λ, Aut(S) is the group of all automorphisms of
the semigroup S and H1λ is the direct λ-power of the group of units H1 of the monoid S.
1. Introduction and preliminaries
Further we shall follow the terminology of [2, 21].
Given a semigroup S, we shall denote the set of idempotents of S by E(S). A semigroup S with
the adjoined unit (identity) [zero] will be denoted by S 1 [S 0 ] (cf. [2]). Next, we shall denote the unit
(identity) and the zero of a semigroup S by 1S and 0S , respectively. Given a subset A of a semigroup
S, we shall denote by A∗ = A \ {0S }.
If S is a semigroup, then we shall denote the subset of idempotents in S by E(S). If E(S) is closed
under multiplication in S and we shall refer to E(S) a band (or the band of S). If the band E(S) is a
non-empty subset of S, then the semigroup operation on S determines the following partial order 6 on
E(S): e 6 f if and only if ef = f e = e. This order is called the natural partial order on E(S).
If h : S → T is a homomorphism (or a map) from a semigroup S into a semigroup T and if s ∈ S,
then we denote the image of s under h by (s)h.
Let S be a semigroup with zero and λ a cardinal > 1. We define the semigroup operation on the set
Bλ (S) = (λ × S × λ) ∪ {0} as follows:
(
(α, ab, δ), if β = γ;
(α, a, β) · (γ, b, δ) =
0,
if β 6= γ,
and (α, a, β) · 0 = 0 · (α, a, β) = 0 · 0 = 0, for all α, β, γ, δ ∈ λ and a, b ∈ S. If S = S 1 then the
semigroup Bλ (S) is called the Brandt λ-extension of the semigroup S [4]. Obviously, if S has zero then
J = {0} ∪ {(α, 0S , β) : 0S is the zero of S} is an ideal of Bλ (S). We put Bλ0 (S) = Bλ (S)/J and the
semigroup Bλ0 (S) is called the Brandt λ0 -extension of the semigroup S with zero [8].
If I is a trivial semigroup (i.e. I contains only one element), then we denote the semigroup I with
the adjoined zero by I 0 . Obviously, for any λ > 2, the Brandt λ0 -extension of the semigroup I 0 is
isomorphic to the semigroup of λ×λ-matrix units and any Brandt λ0 -extension of a semigroup with
zero which also contains a non-zero idempotent contains the semigroup of λ×λ-matrix units. We shall
denote the semigroup of λ×λ-matrix units by Bλ . The 2 × 2-matrix semigroup with adjoined identity
B21 plays an impotent role in Graph Theory and its called the Perkins semigroup. In the paper [20]
Perkins showed that the semigroup B21 is not finitely based. More details on the word problem of the
Perkins semigroup via different graphs may be found in the works of Kitaev and his coauthors (see
[17, 18]).
Date: January 22, 2018.
2010 Mathematics Subject Classification. 20M15, 20F29 .
Key words and phrases. Semigroup, group of automorphisms, monoid, extension.
1
2
OLEG GUTIK
We always consider the Brandt λ0 -extension only of a monoid with zero. Obviously, for any monoid
S with zero we have B10 (S) = S. Note that every Brandt λ-extension of a group G is isomorphic to the
Brandt λ0 -extension of the group G0 with adjoined zero. The Brandt λ0 -extension of the group with
adjoined zero is called a Brandt semigroup [2, 21]. A semigroup S is a Brandt semigroup if and only if
S is a completely 0-simple inverse semigroup [1, 19] (cf. also [21, Theorem II.3.5]). We shall say that
the Brandt λ0 -extension Bλ0 (S) of a semigroup S is finite if the cardinal λ is finite.
In the paper [14] Gutik and Repovš established homomorphisms of the Brandt λ0 -extensions of
monoids with zeros. They also described a category whose objects are ingredients in the constructions
of the Brandt λ0 -extensions of monoids with zeros. Here they introduced finite, compact topological
Brandt λ0 -extensions of topological semigroups and countably compact topological Brandt λ0 -extensions
of topological inverse semigroups in the class of topological inverse semigroups, and established the structure of such extensions and non-trivial continuous homomorphisms between such topological Brandt
λ0 -extensions of topological monoids with zero. There they also described a category whose objects
are ingredients in the constructions of finite (compact, countably compact) topological Brandt λ0 extensions of topological monoids with zeros. These investigations were continued in [10] and [9], where
established countably compact topological Brandt λ0 -extensions of topological monoids with zeros and
pseudocompact topological Brandt λ0 -extensions of semitopological monoids with zeros their corresponding categories. Some other topological aspects of topologizations, embeddings and completions
of the semigroup of λ×λ-matrix units and Brandt λ0 -extensions as semitopological and topological
semigroups were studied in [3, 5, 7, 11, 12, 13, 15, 16].
In this paper we describe the group of automorphisms of the Brandt λ0 -extension Bλ0 (S) of an arbitrary
monoid S with zero.
2. Automorphisms of the Brandt λ0 -extension of a monoid with zero
We observe that if f : S → S is an automorphism of the semigroup S without zero then it is obvious
that the map fb: S 0 → S 0 defined by the formula
(s)f, if s 6= 0S ;
(s)fb =
0S , if s = 0S ,
is an automorphism of the semigroup S 0 with adjoined zero 0S . Also the automorphism f : S → S of
the semigroup S can be extended to an automorphism fB : Bλ0 (S) → Bλ0 (S) of the Brandt λ0 -extension
Bλ0 (S) of the semigroup S by the formulae:
(α, s, β)fB = (α, (s)f, β),
for all α, β ∈ λ
and (0)fB = 0. We remark that so determined extended automorphism is not unique.
The following theorem describes all automorphisms of the Brandt λ0 -extension Bλ0 (S) of a monoid S.
Theorem 1. Let λ > 1 be cardinal and let Bλ0 (S) be the Brandt λ0 -extension of monoid S with zero.
Let h : S → S be an automorphism and suppose that ϕ : λ → λ is a bijective map. Let H1 be the group
of units of S and u : λ → H1 a map. Then the map σ : Bλ0 (S) → Bλ0 (S) defined by the formulae
(1)
((α, s, β))σ = ((α)ϕ, (α)u · (s)h · ((β)u)−1, (β)ϕ)
and
(0)σ = 0,
is an automorphism of the semigroup Bλ0 (S). Moreover, every automorphism of Bλ0 (S) can be constructed
in this manner.
Proof. A simple verification shows that σ is an automorphism of the semigroup Bλ0 (S).
Let σ : Bλ0 (S) → Bλ0 (S) be an isomorphism. We fix an arbitrary α ∈ λ.
Since σ : Bλ0 (S) → Bλ0 (S) is the automorphism and the idempotent (α, 1S , α) is maximal with the
respect to the natural partial order on E(Bλ0 (S)), Proposition 3.2 of [14] implies that ((α, 1S , α))σ =
(α ′ , 1S , α ′ ) for some α ′ ∈ λ.
Since (β, 1S , α)(α, 1S , α) = (β, 1S , α) for any β ∈ λ, we have that
((β, 1S , α))σ = ((β, 1S , α))σ · (α ′ , 1S , α ′ ),
ON THE GROUP OF AUTOMORPHISMS OF THE BRANDT λ0 -EXTENSION OF A MONOID WITH ZERO
3
and hence
((β, 1S , α))σ = ((β)ϕ, (β)u, α ′),
for some (β)ϕ ∈ λ and (β)u ∈ S. Similarly, we get that
((α, 1S , β))σ = (α ′ , (β)v, (β)ψ),
for some (β)ψ ∈ λ and (β)v ∈ S. Since (α, 1S , β)(β, 1S , α) = (α, 1S , α), we have that
(α ′ , 1S , α ′ ) = ((α, 1S , α))σ = (α ′ , (β)v, (β)ψ) · ((β)ϕ, (β)u, α ′) = (α ′ , (β)v · (β)u, α ′ ),
and hence (β)ϕ = (β)ψ = β ′ ∈ λ and (β)v · (β)u = 1S . Similarly, since (β, 1S , α) · (α, 1S , β) = (β, 1S , β),
we see that the element
((β, 1S , β))σ = ((β, 1S , α)(α, 1S , β))σ = (β ′ , (β)v · (β)u, β ′ )
is a maximal idempotent of the subsemigroup Sβ ′ ,β ′ of Bλ0 (S), and hence we have that (β)v · (β)u = 1S .
This implies that the elements (β)v and (β)u are mutually invertible in H1 , and hence (β)v = ((β)u)−1.
If (γ)ϕ = (δ)ϕ for γ, δ ∈ λ then
0 6= (α ′ , 1S , (γ)ϕ) · ((δ)ϕ, 1S , α ′ ) = ((α, 1S , γ))σ · ((δ, 1S , α))σ,
and since σ is an automorphism, we have that
(α, 1S , γ) · (δ, 1S , α) 6= 0
and hence γ = δ. Thus ϕ : λ → λ is a bijective map.
Therefore for s ∈ S \ {0S } we have
((γ, s, δ))σ = ((γ, 1S , α) · (α, s, α) · (α, 1S , δ))σ =
= ((γ, 1S , α))σ · ((α, s, α))σ · ((α, 1S , δ))σ =
= ((γ)ϕ, (γ)u, α ′) · (α ′ , (s)h, α ′ ) · (α ′ , ((δ)u)−1, (δ)ϕ)=
= ((γ)ϕ, (γ)u · (s)h · ((δ)u)−1 , (δ)ϕ).
Also, since 0 is zero of the semigroup Bλ0 (S) we conclude that (0)σ = 0.
Theorem 1 implies the following corollary:
Corollary 1. Let λ > 1 be cardinal and let Bλ (G) be the Brandt semigroup. Let h : G → G be an
automorphism and suppose that ϕ : λ → λ is a bijective map. Let u : λ → G be a map. Then the map
σ : Bλ (G) → Bλ (G) defined by the formulae
((α, s, β))σ = ((α)ϕ, (α)u · (s)h · ((β)u)−1, (β)ϕ)
and
(0)σ = 0,
is an automorphism of the Brandt semigroup Bλ (G). Moreover, every automorphism of Bλ (G) can be
constructed in this manner.
Also, we observe that Corollary 1 implies the following well known statement:
Corollary 2. Let λ > 1 be cardinal and ϕ : λ → λ a bijective map. Then the map σ : Bλ → Bλ defined
by the formulae
((α, β))σ = ((α)ϕ, (β)ϕ) and (0)σ = 0,
is an automorphism of the semigroup of λ×λ-matrix units Bλ . Moreover, every automorphism of Bλ
can be constructed in this manner.
The following example implies that the condition that semigroup S contains the identity is essential.
Example 1. Let λ be any cardinal > 2. Let S be the zero-semigroup of cardinality > 3 and 0S is
zero of S. It is easily to see that every bijective map σ : Bλ0 (S) → Bλ0 (S) such that (0)σ = 0 is an
automorphism of the Brandt λ0 -extension of S.
4
OLEG GUTIK
Remark. By Theorem 1 we have that every automorphism σ : Bλ0 (S) → Bλ0 (S) of the Brandt λ0 extension of an arbitrary monoid S with zero identifies with the ordered triple [ϕ, h, u], where h : S → S
is an automorphism of S, ϕ : λ → λ is a bijective map and u : λ → H1 is a map, where H1 is the group
of units of S.
Lemma 1. Let λ > 1 be cardinal, S be a monoid with zero and let Bλ0 (S) be the Brandt λ0 -extension
of S. Then the composition of arbitrary automorphisms σ = [ϕ, h, u] and σ ′ = [ϕ′ , h′ , u′ ] of the Brandt
λ0 -extension of S defines in the following way:
[ϕ, h, u] · [ϕ′ , h′ , u′] = [ϕϕ′ , hh′ , ϕu′ · uh′ ].
Proof. By Theorem 1 for every (α, s, β) ∈ Bλ0 (S) we have that
(α, s, β)(σσ ′) = (α)ϕ, (α)u·(s)h·((β)u)−1, (β)ϕ σ ′ =
−1
= ((α)ϕ)ϕ′, ((α)ϕ)u′ · (α)u · (s)h · ((β)u)−1 h′ · (((β)ϕ)u′ ) , ((β)ϕ)ϕ′ =
and since h′ is an automorphism of the monoid S we get that this is equal to
= ((α)ϕ)ϕ′ , ((α)ϕ)u′ · ((α)u) h′ · ((s)h) h′ · (((β)u)h′ )
−1
· (((β)ϕ)u′)
−1
= (α)(ϕϕ′ ), (α)(ϕu′ · uh′ ) · ((s)h) h′ · (β) (ϕu′ · uh′ ) , (β)(ϕϕ′) .
−1
, ((β)ϕ)ϕ′ =
This completes the proof of the requested equality.
Theorem 2. Let λ > 1 be cardinal, S be a monoid with zero and let Bλ0 (S) be the Brandt λ0 -extension
of S. Then the group of automorphisms Aut(Bλ0 (S)) of Bλ0 (S) is isomorphic to a homomorphic image
of the group defines on the Cartesian product Sλ × Aut(S) × H1λ with the following binary operation:
(2)
[ϕ, h, u] · [ϕ′ , h′ , u′] = [ϕϕ′ , hh′ , ϕu′ · uh′ ],
where Sλ is the group of all bijections of the cardinal λ, Aut(S) is the group of all automorphisms of
the semigroup S and H1λ is the direct λ-power of the group of units H1 of the monoid S. Moreover, the
inverse element of [ϕ, h, u] in the group Aut(Bλ0 (S)) is defined by the formula:
[ϕ, h, u]−1 = ϕ−1 , h−1 , ϕ−1 u−1 h−1 .
Proof. First, we show that the binary operation defined by formula (2) is associative. Let [ϕ, h, u],
[ϕ′ , h′ , u′] and [ϕ′′ , h′′ , u′′ ] be arbitrary elements of the Cartesian product Sλ × Aut(S) × H1λ . Then we
have that
[ϕ, h, u] · [ϕ′ , h′ , u′ ] · [ϕ′′ , h′′ , u′′ ] = [ϕϕ′ , hh′ , ϕu′ · uh′ ] · [ϕ′′ , h′′ , u′′] =
= [ϕϕ′ ϕ′′ , hh′ h′′ , ϕϕ′ u′′ · (ϕu′ · uh′ )h′′ ] =
= [ϕϕ′ ϕ′′ , hh′ h′′ , ϕϕ′ u′′ · ϕu′ h′′ · uh′ h′′ ]
and
[ϕ, h, u] · ([ϕ′ , h′ , u′] · [ϕ′′ , h′′ , u′′ ]) = [ϕ, h, u] · [ϕ′ ϕ′′ , h′ h′′ , ϕ′ u′′ · u′ h′′ ] =
= [ϕϕ′ ϕ′′ , hh′ h′′ , ϕ(ϕ′ u′′ · u′ h′′ ) · uh′ h′′ ] =
= [ϕϕ′ ϕ′′ , hh′ h′′ , ϕϕ′ u′′ · ϕu′ h′′ · uh′ h′′ ],
and hence so defined operation is associative.
Theorem 1 implies that formula (1) determines a map F from the Cartesian product Sλ ×Aut(S)×H1λ
onto the group of automorphisms Aut(Bλ0 (S)) of the Brandt λ0 -extension Bλ0 (S) of the monoid S,
and hence the associativity of binary operation (2) implies that the map F is a homomorphism from
Sλ × Aut(S) × H1λ onto the group Aut(Bλ0 (S)).
Next we show that [1Sλ , 1Aut(S) , 1H1λ ] is a unit element with the respect to the binary operation (2),
where 1Sλ , 1Aut(S) and 1H1λ are units of the groups Sλ , Aut(S) and H1λ , respectively. Then we have
ON THE GROUP OF AUTOMORPHISMS OF THE BRANDT λ0 -EXTENSION OF A MONOID WITH ZERO
that
5
[ϕ, h, u] · 1Sλ , 1Aut(S) , 1H1λ = ϕ1Sλ , h1Aut(S) , ϕ1H1λ · u1Aut(S) =
= ϕ, h, ϕ1H1λ · u1Aut(S) =
= ϕ, h, 1H1λ · u =
= [ϕ, h, u]
and
1Sλ , 1Aut(S) , 1H1λ · [ϕ, h, u] = 1Sλ ϕ, 1Aut(S) h, 1Sλ u · 1H1λ h = [ϕ, h, u],
because every automorphism h ∈ Aut(S) acts on the group H1λ by the natural way as a restriction of
global automorphism of the semigroup S on every factor, and hence we get that 1H1λ h = 1H1λ .
Also, similar arguments imply that
[ϕ, h, u] · [ϕ, h, u]−1 = [ϕ, h, u] · ϕ−1 , h−1 , ϕ−1 u−1 h−1 =
= ϕϕ−1 , hh−1 , (ϕϕ−1 )u−1h−1 · uh−1 =
= ϕϕ−1 , hh−1 , (1Sλ )u−1 h−1 · uh−1 =
= ϕϕ−1 , hh−1 , u−1h−1 · uh−1 =
= 1Sλ , 1Aut(S) , 1H1λ
and
[ϕ, h, u]−1 · [ϕ, h, u] = ϕ−1 , h−1 , ϕ−1 u−1 h−1 · [ϕ, h, u]=
= ϕ−1 ϕ, h−1 h, ϕ−1 u · ϕ−1 u−1 h−1 h =
= ϕ−1 ϕ, h−1 h, ϕ−1 u · ϕ−1 u−1 =
= 1Sλ , 1Aut(S) , 1H1λ .
This implies that the elements [ϕ−1 , h−1 , ϕ−1 u−1 h−1 ] and [ϕ, h, u] are invertible in Sλ × Aut(S) × H1λ,
and hence the set Sλ × Aut(S) × H1λ with the binary operation (2) is a group.
Let Id : Bλ0 (S) → Bλ0 (S) be the identity automorphism of the semigroup Bλ0 (S). Then by Theorem 1
there exist some automorphism h : S → S, a bijective map ϕ : λ → λ and a map u : λ → H1 into the
group H1 of units of S such that
(α, s, β) = (α, s, β)Id = ((α)ϕ, (α)u · (s)h · ((β)u)−1, (β)ϕ),
for all α, β ∈ λ and s ∈ S ∗ . Since Id : Bλ0 (S) → Bλ0 (S) is the identity automorphism we conclude that
(α)ϕ = α for every α ∈ λ. Also, for every s ∈ S ∗ we get that s = (α)u · (s)h · ((β)u)−1 for all α, β ∈ λ,
and hence we obtain that
1S = (α)u · (1S )h · ((β)u)−1 = (α)u · ((β)u)−1
for all α, β ∈ λ. This implies that (α)u = (β)u = u
e is a fixed element of the group H1 for all α, β ∈ λ.
We define
n
o
λ
−1
ker N = [ϕ, h, u
e] ∈ Sλ × Aut(S) × H1 : ϕ : λ → λ is an idemtity map, u
e(s)he
u
= s for any s ∈ S .
It is obvious that the equality u
e(s)he
u−1 = s implies that (s)h = u
e−1 se
u for all s ∈ S. The previous
arguments implies that [ϕ, h, u
e] ∈ ker N if and only if [ϕ, h, u
e]F is the unit of the group Aut(Bλ0 (S)),
and hence ker N is a normal subgroup of Sλ × Aut(S) × H1λ . This implies that the quotient group
(Sλ × Aut(S) × H1λ )/ ker N is isomorphic to the group Aut(Bλ0 (S)).
6
OLEG GUTIK
References
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[2] A. H. Clifford and G. B. Preston, The Algebraic Theory of Semigroups, Vols. I and II, Amer. Math. Soc. Surveys 7,
Providence, R.I., 1961 and 1967.
[3] S. Bardyla and O. Gutik, On a semitopological polycyclic monoid, Algebra Discr. Math. 21:2 (2016), 163–183.
[4] O. V. Gutik, On Howie semigroup, Mat. Metody Phis.-Mekh. Polya. 42:4 (1999), 127–132 (in Ukrainian).
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(2014), 59–85.
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Visnyk Lviv Univ., Ser. Mekh.-Math. 61 (2003), 98–105.
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49:3 (2006), 26–40.
[9] O. Gutik and K. Pavlyk, On pseudocompact topological Brandt λ0 -extensions of semitopological monoids, Topological
Algebra Appl. 1 (2013), 60–79.
[10] O. Gutik, K. Pavlyk, and A. Reiter, Topological semigroups of matrix units and countably compact Brandt λ0 extensions, Mat. Stud. 32:2 (2009), 115–131.
[11] O. V. Gutik, K. P. Pavlyk, and A. R. Reiter, On topological Brandt semigroups, Mat. Metody Fiz.-Mekh. Polya 54:2
(2011), 7–16 (in Ukrainian); English version in: J. Math. Sci. 184:1 (2012), 1–11.
[12] O. Gutik and O. Ravsky, On feebly compact inverse primitive (semi)topological semigroups, Mat. Stud. 44:1 (2015),
3–26.
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Mat. Metody Fiz.-Mekh. Polya 58:2 (2015), 20–37.
[14] O. Gutik and D. Repovš, On Brandt λ0 -extensions of monoids with zero, Semigroup Forum 80:1 (2010), 8–32.
[15] J. Jamalzadeh and Gh. Rezaei, Countably compact topological semigroups versus Brandt extensions and paragroups,
Algebras Groups Geom. 27:2 (2010), 219–228.
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Bull. Iran. Math. Soc. 39:1 (2013), 87–95.
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[19] W. D. Munn, Matrix representations of semigroups, Proc. Cambridge Phil. Soc. 53 (1957), 5–12.
[20] P. Perkins, Bases for equational theories of semigroups, J. Algebra 11:2 (1969), 298–314.
[21] M. Petrich, Inverse Semigroups, John Wiley & Sons, New York, 1984.
Faculty of Mathematics, National University of Lviv, Universytetska 1, Lviv, 79000, Ukraine
E-mail address: o gutik@franko.lviv.ua, ovgutik@yahoo.com
| 4 |
Crowdsourced correlation clustering with relative
distance comparisons
Antti Ukkonen
arXiv:1709.08459v1 [cs.DS] 25 Sep 2017
Department of Computer Science, University of Helsinki
Helsinki, Finland
Email: antti.ukkonen@helsinki.fi
Abstract—Crowdsourced, or human computation based clustering algorithms usually rely on relative distance comparisons,
as these are easier to elicit from human workers than absolute
distance information. A relative distance comparison is a statement of the form “item A is closer to item B than to item C”.
However, many existing clustering algorithms that use relative
distances are rather complex. They are often based on a twostep approach, where the relative distances are first used to learn
either a distance matrix, or an embedding of the items, and then
some standard clustering method is applied in a second step.
In this paper we argue that it should be possible to compute a
clustering directly from relative distance comparisons.
Our ideas are built upon existing work on correlation clustering, a well-known non-parametric approach to clustering.
The technical contribution of this work is twofold. We first
define a novel variant of correlation clustering that is based
on relative distance comparisons, and hence suitable for human
computation. We go on to show that our new problem is closely
related to basic correlation clustering, and use this property to
design an approximation algorithm for our problem. As a second
contribution, we propose a more practical algorithm, which we
empirically compare against existing methods from literature.
Experiments with synthetic data suggest that our approach can
outperform more complex methods. Also, our method efficiently
finds good and intuitive clusterings from real relative distance
comparison data.
I. I NTRODUCTION
Clustering is a classical unsupervised learning problem. The
task, in colloquial terms, is to divide a given set of items to
groups such that similar items are placed in the same group,
while dissimilar items end up in different groups. Clustering
has numerous practical applications, ranging from customer
segmentation to bioinformatics, and has attracted a lot of
attention from the research community for decades [1].
In this paper we study a novel approach to data clustering that is suitable for human computation [2], [3], i.e.,
an algorithmic process where humans carry out parts of the
computation, often using a crowdsourcing platform such as
Amazon Mechanical Turk. Human computation algorithms are
implemented via so called human intelligence tasks (HITs,
see also [4]). A HIT is defined as a piece of input data
together with instructions of what to do with the input. Human
computation algorithms operate by sending a large number of
HITs to a crowd that processes the tasks in parallel. Once
all tasks are completed, the algorithm collects the results and
possibly carries out some post-processing to obtain the final
result.
To motivate human computation approaches to clustering,
consider a scenario where we are given a collection of some
items, e.g. photographs or pieces of text, and ask human
labellers to assign each item to some category. Despite recent
advances in computer vision (e.g. [5]), the need for this type of
crowdsourced data analysis remains in scenarios where human
performance still exceeds that of machine learning. In simple
cases the categories (or labels) of interest are known e.g., they
can correspond to images different types of galaxies [6], or
to texts having positive / negative sentiment [7]. But in some
other situations, there may not be any predefined categories,
and the first task is to determine what kind of structure there
is in the data to begin with. This is a data exploration problem
to which clustering is a standard solution.
To design an efficient human computation algorithm we
should make sure that the required HITs a) are easy for
humans to solve, and b) can all be solved in parallel. Next we
discuss the technical motivation of our approach on the basis
of these two requirements.
At the core of most clustering algorithms is the notion
of distance. Indeed, similarity between two items is usually
defined in terms of a distance function that yields a small
numeric value when the items are similar, and increases as the
items become more dissimilar. A lot of the actual computation
carried out by a clustering algorithm involves calculating,
comparing or otherwise using these distances. Any human
computation algorithm for clustering must thus deal with
distances as well, and it must do this in a manner that satisfies
both requirements a) and b) above.
The main problem is that absolute (numeric) distances
between e.g. images can be difficult for humans to specify
in a consistent manner. Even a very simple distance function
that can take only two possible values, “similar” and “notsimilar”, can be problematic for human annotators [8]. Relative
distance comparisons, on the other hand, are often easier to
elicit. Rather than specifying distances on some absolute (and
arbitrary) scale, they represent the distance function in terms
of statements such as “item A is closer to item B than to
item C”, or “of items A, B, and C, item C is an outlier”.
Relative distance comparisons of this kind have been used
previously e.g. to compute the mean of a set of items [9],
density estimation [10], [11], distance/kernel learning [12],
[13], [14], [15], [16], and to compute embeddings [17], [18],
[19]. To satisfy requirement a) above, the clustering algorithm
must thus use relative distance comparisons only. Requirement
b) is satisfied as long as the distance comparisons can be
collected in one batch, i.e., there are no interdependencies
between the HITs.
Most existing human computation algorithms for clustering
that satisfy requirements a) and b) first use the relative distance
comparisons to either learn a distance/kernel matrix [13], [14],
or to compute an embedding of the items to Rn [17], and
as a second step apply some “regular” clustering algorithm
that uses distance matrices or embeddings. Such approaches
indeed can work well, and have the sometimes useful property
of learning features for the items (the embedding).
But we argue that clustering is a relatively simple combinatorial problem. If the ultimate task is to only compute a
clustering, and there is no other use for e.g. an embedding of
the items, it seems more desirable to compute the clustering
directly from the distance comparisons. The aim of this paper
is to devise an efficient method for doing this. We argue
that our approach is 1) conceptually simpler than competing
approaches, and 2) easier to implement and understand.
Our approach is based on C ORRELATION -C LUSTERING, a
problem originally proposed by Bansal et al [20]1 . It is a
parameter-free approach to clustering, where the distance function and item features have been replaced with “qualitative”
information about how pairs of items should relate to each
other in a good clustering. A lot of its theoretical properties are
known [22], [23], and unlike many other clustering methods,
it does not require setting the number of clusters in advance,
which is a nice practical advantage.
Our main result is a novel variant of correlation clustering
that uses relative distance comparisons only, and is thus
particularly well suited for human computation. The relative
distance comparisons we consider are expressed in terms of
triplets: out of a set of three items, one is designated as an
“outlier”, meaning that it’s distance from the two other items
is the largest.
We make the following contributions in this paper:
1) We define a variant of the C ORRELATION -C LUSTERING
problem that takes a set of relative distance comparisons
as input (Section II-B).
2) We analyse the problem, show it to be NP-hard (Section II-C), discuss its connections to the standard
C ORRELATION -C LUSTERING problem (Section II-D),
and propose an O(log |U |) approximation algorithm
(Section III), where U is the number of items being
clustered. This establishes that our problem is not harder
than C ORRELATION -C LUSTERING on general weighted
graphs from the point of view of approximation.
3) We also present a practical, simple, and very efficient local search algorithm for solving our problem
(Section IV), and carry out a number of experiments
where we compare our approach against two alternative
methods from literature (Section V).
1 Note that Böhm et al [21] use the term “correlation clustering” to describe
a different problem that is not to be confused with the one considered in this
paper.
II. P ROBLEM DEFINITIONS AND ANALYSIS
Let U denote a set of items, and let d denote a distance
function d : U × U → R+
0 between items in U . The triplet
(a, b, c), with a, b, c ∈ U , captures the following relative
distance comparison based on d:
d(a, b) ≤ min{d(a, c), d(b, c)}.
That is, of the items a, b, and c, item c is an outlier. (In our
notation, the outlier is always the third item of the triplet.) Let
T denote a set of such triplets over U . Our task is to cluster
the items in U given only T . In informal terms, we want to
partition U to disjoint subsets so that items in the same subset
are similar to each other, and items in different subsets are
different from one another in terms of the distance function
d. Let [1:n] denote the integers from 1 to n. A clustering
function f : U → [1:k] assigns to every item in U a cluster
label, i.e., an integer between 1 and k. Let I{·} denote the
indicator function.
A. Background about correlation clustering
The problem we define in this paper is closely related to the
C ORRELATION -C LUSTERING problem, as originally defined
in [20]. An instance of C ORRELATION -C LUSTERING is a
graph G where every edge (u, v) is associated some positive
weight w(u, v), as well as either the label + or the label −.
The objective is to find a clustering of the vertices so that
vertices connected by a + edge belong to the same cluster,
while vertices connected by a − edge are assigned to different
clusters. More formally:
Problem 1. (C ORRELATION -C LUSTERING) Given the graph
G = (U, {E + ∪ E − }, w), where U is a set of items, E +
and E − denote sets of edges that are labeled with a +
and −, respectively, and the edge weights are given by the
function w : E + ∪ E − → R+ . Find a clustering function f
that minimizes the cost
X
c(f, G) =
w(u, v) I{f (u) 6= f (v)} +
(u,v)∈E +
X
w(u, v) I{f (u) = f (v)}.
(u,v)∈E −
The variant of C ORRELATION -C LUSTERING given in Problem 1 is commonly known as the one that concerns “minimising disagreements in general weighted graphs”. Problem 1 is
NP-hard [20], as well as APX-hard [22].
B. Correlation clustering with relative distances
We proceed to define a novel variant of C ORRELATION C LUSTERING that is based on relative distance comparisons
in the form of a set T of triplets as defined above. Given T ,
the task is to compute a clustering function f . We first discuss
how relative distance comparisons and an underlying ground
truth clustering can interact.
Simply put (and somewhat exaggerated), a triplet (a, b, c)
can be understood as saying that “a and b are next to each
other, but c is further away”. From the point of view of
e
f
a
b
c
d
Fig. 1. A two-dimensional example with a ground truth clustering of size 3
(the gray circles), items a, b, c, d, e, and f at the locations shown, as well
as the triplets (a, b, c), (b, d, c), (a, b, e), and (a, f, c).
a clustering task, we argue that (a, b, c) thus provides three
(uncertain) pieces of information:
E + and E − will always yield a clustering function f such
that c(f, G) = 0, i.e. the hardness of Problem 1 is caused
by noisy constraints. (Indeed, without noise we can simply
remove all edges in E − and are left with cliques of E + edges
that correspond to the clusters.) With triplets no such easy
solutions exist even in a noise-free scenario, as the triplets do
not explicitly specify what items belong to the same cluster.
Moving on, we say that the triplet (a, b, c) is satisfied by the
clustering function f , if and only if I{f (a) = f (b) 6= f (c)} is
true. Otherwise (a, b, c) is unsatisfied. In Figure 1 all triplets
except (a, b, c) are unsatisfied by the clustering shown. We
consider the following problem:
Problem 2. Given a set of triplets T over items in the set U ,
find a clustering function f OPT that minimizes the cost
X
s(f, T ) =
I{ f (a) 6= f (b)∨
(a,b,c)∈T
f (a) = f (c) ∨ f (b) = f (c) },
1. items a and b might be in the same cluster,
i.e., f OPT = arg minf s(f, T ).
2. items a and c might be in different clusters, and
We have defined the objective function in Problem 2 to
minimize the number of unsatisfied triplets. Observe that the
number of clusters is not specified as part of the problem input.
As with traditional C ORRELATION -C LUSTERING, f OPT may
use any number of clusters that minimizes s(f OPT , T ).
3. items b and c might be in different clusters.
In practice this will not always be true, of course. Figure 1
illustrates this with a toy example with three clusters, six items
(letters a to f ), and four triplets. All four triplets reflect the
basic Euclidean distance between the items. Of the triplets,
(a, b, c) “correctly” reflects the cluster structure according to
the claim above: a and b indeed are in the same cluster, while
c is in a different cluster. The remaining three triplets show
how conflicts arise between the ground truth clustering and
relative distance comparisons.
First, distances between all items in the triplet may be
“long”, as is the case with b, c, and d, and intuition says that
the items should be put in different clusters. In Figure 1 this is
also the case. But if we interpret the triplet (b, d, c) as above,
we would put b and d in the same cluster. Second, all pairwise
distances may be “short”, as is the case with a, b and e, and
again intuitively it would make sense to put a, b and e in the
same cluster. In Figure 1 these items indeed do belong to the
same cluster, but our interpretation of (a, b, e) suggests that e
is in a different cluster. Third, it is possible that in an optimal
clustering, items that are closer to each other in fact belong to
different clusters, while the outlier belongs to the same cluster
as one of the other items. This is the case with items a, c and
f . Item c is an outlier, but nonetheless belongs to the same
cluster as f in the ground truth solution of Figure 1, while
the above interpretation of (a, f, c) would put a and f in the
same cluster, and c in a different cluster.
We leave a more fine-grained analysis of the effects of these
observations for future work, and in this paper simply assume
that the three pieces of information provided by the triplet
(a, b, c) (points 1–3 listed above) can be used to identify
useful clusterings. However, triplets inherently seem to make
the clustering task more challenging than the E + and E −
constraints of Problem 1. Because, in the absence of noise,
C. Problem complexity
Theorem 1. Problem 2 is NP-hard.
Proof. The proof is a simple reduction from the unweighted
variant of Problem 1. (This problem is also NP-hard.) Given an
instance of C ORRELATION -C LUSTERING determined by the
edge sets E + and E − , we create an instance of Problem 2, i.e.,
a set of triplets T as follows: for every (u, v) ∈ E + , insert the
triplet (u, v, xuv ), and for every (u, v) ∈ E − insert the triplet
(u, u0v , v). Here xuv and u0 are dummy items that each occur in
a single triplet only, and can hence be satisfied trivially. The
only real constraints to f OPT are thus determined by items
that appear in E+ and E−. Observe that an optimal solution
f OPT to Problem 2 immediately gives an optimal solution
to the C ORRELATION -C LUSTERING instance. Moreover, the
costs of the solutions are identical.
The reduction employed in the proof above has the implication that Problem 2 is at least as hard as Problem 1 (with
unit weights) also from the point of view of approximation.
Indeed, any approximation bound for Problem 2 also holds for
the unweighted variant of Problem 1.
D. Mapping Problem 2 to Problem 1
We continue by discussing further properties of Problem 2.
These will be useful when we design algorithms for the
problem. In short, we show how to “clean up” the set of triplets
T so that it is possible to construct a reguler C ORRELATION C LUSTERING instance from the triplets.
First, observe that if T contains both triplets t = (u, v, x1 )
and t0 = (u, x2 , v) for any u, v, x1 , x2 ∈ U , only one of
these can be satisfied by any clustering function f (including
f OPT ), because to satisfy t we must have f (u) = f (v), while
satisfying t0 requires f (u) 6= f (v). Some of the triplets in
T can (and in most cases will) thus be inconsistent with
each other. This is a natural consequence of how relative
distance information is encoded in the triplets: items u and
v may be “close” to each other in relation to item x1 , but “far
apart” when compared with item x2 . Indeed, inconsistencies
occur also in sets of triplets T that are fully noiseless,
as also discussed above. These inconsistencies are naturally
represented in terms of a constraint graph.
Definition 1. Two triplets are inconsistent when for some
items u and v and any clustering function f , one of the triplets
is satisfied when f (u) = f (v), while the other is satisfied when
f (u) 6= f (v). Let CT = (T , E) denote the constraint graph
associated with the set T . The vertices of CT are the triplets,
and the edge set E contains those pairs of triplets that are
inconsistent with each other.
A vertex cover of a graph is a subset of its vertices such
that for every edge, at least one endpoint belongs to the vertex
cover. Consider a vertex cover of CT , denoted vc(CT ). (Notice
that any vertex cover will do, it does not have to be one
of minimum size.) Let T 0 = T \ vc(CT ), i.e., T 0 contains
those triplets that are not part of the vertex cover vc(CT ). We
can show that in T 0 there are no triplets that would provide
conflicting information about any two items u and v:
Lemma 1. There are no inconsistent triplets in T
vc(CT ).
0
= T \
Algorithm 1 An approximation algorithm
1: Input: triplets T .
2: Construct the constraint graph CT according to Definition 1.
3: Find an approximate minimum vertex cover vcALG (CT )
by using the algorithm of Assumption 2.
4: Let T 0 ← T \ vcALG (CT ).
5: Construct GT 0 according to Definition 2.
6: Find an approximate solution f ALG to GT 0 by using the
algorithm of Assumption 1.
7: Return f ALG .
III. A N APPROXIMATION ALGORITHM
Above we showed that by finding a vertex cover of the
constraint graph CT , we can turn a given instance of Problem 2 into a regular C ORRELATION -C LUSTERING instance.
We make use of this to design an approximation algorithm for
Problem 2.
First, we assume that an approximation algorithm exists for
C ORRELATION -C LUSTERING in general weighted graphs.
OPT
denote the optimal solution to
Assumption 1. Let fCC
the C ORRELATION -C LUSTERING instance GT 0 . We assume
there exists a polynomial time algorithm for C ORRELATION C LUSTERING that finds the solution f ALG that satisfies
OPT
c(f ALG , GT 0 ) ≤ α c(fCC
, GT 0 ),
where α is some function of the input T .
Second, we assume that a constant factor approximation
algorithm exists for finding minimum vertex covers:
Proof. See Appendix A.
In practice this implies that all triplets in T 0 are guaranteed
to consistently suggest that items u and v either belong to
the same cluster, or that items u and v belong to different
clusters. Importantly, there are no two triplets in T 0 such that
one would prefer u and v in the same cluster, while the other
would prefer u and v in different clusters. We can thus map T 0
to an instance of C ORRELATION -C LUSTERING (Problem 1).
0
Definition 2. Given the set of triplets T = T \ vc(CT ),
we define the instance GT 0 = (U, {E + ∪ E − }, w) of
C ORRELATION -C LUSTERING as follows:
• The set of items U is the union of all items in the triplets
in T 0 .
0
• For every pair u, v ∈ U , let Tu,v
denote those triplets in
0
T that contain both items u and v.
0
• If for given u and v, all triplets in Tu,v
agree that neither
u or v is the outlier, we include {u, v} into E + .
0
• If for given u and v, all triplets in Tu,v
agree that either
u or v must be the outlier, we include {u, v} into E − .
0
• The weight w(u, v) is the size of Tu,v
.
Because there are no inconsistent triplets in T 0 , every pair
{u, v} will be assigned to either E + or E − .
Assumption 2. Let CT denote the constraint graph of Definition 1, and let vcmin (CT ) denote its vertex cover of minimum
size. We assume there exists a polynomial time algorithm that
finds the solution vcALG (CT ) that satisfies
| vcALG (CT )| ≤ β | vcmin (CT )|,
where β > 1 is some constant.
Our approximation algorithm is described in Algorithm 1.
In short, we first solve minimum vertex cover on CT , discard
triplets that belong to the found cover, and then solve the
remaining C ORRELATION -C LUSTERING instance. The algorithm runs in polynomial time as long as the algorithms of
assumptions 1 and 2 run in polynomial time. The intermediary
steps are trivially polynomial in the size of T .
We give the following theorem:
Theorem 2. Let f OPT denote the optimal solution to Problem 2, and denote by f ALG the solution found by Algorithm 1.
We have
s(f ALG , T ) ≤ (2α + β) s(f OPT , T ),
where α and β are the approximation factors in assumptions
1 and 2, respectively.
Algorithm 2 A local search heuristic
1: Input: a set of triplets T
2: f ← initialise (can be done in different ways)
3: knew ← maxu∈U f (u) + 1
4: f 0 ← ∅
5: while f 0 6= f do
6:
f0 ← f
7:
for u ∈ U do
8:
f (u) ← arg minf (u)∈[1:knew ] s(f, T )
9:
if f (u) = knew then
10:
knew ← knew + 1
11:
end if
12:
end for
13:
“clean up” f so that it maps U to the range [1:h], where
h = |{f (u)}u∈U |.
14:
knew ← maxu∈U f (u) + 1
15: end while
16: return f
Proof. See Appendix B.
The actual bound thus depends on both α and β. Currently best known approximation algorithms for solving
C ORRELATION -C LUSTERING in general weighted graphs and
finding minimum vertex covers have bounds of α = O(log n)
log n
[22] and β = 2 − log
2 log n [24], respectively. The graph associated with the C ORRELATION -C LUSTERING instance GT 0 has
|U | vertices, and hence we have α = O(log |U |) and obtain:
Corollary 1. Algorithm 1 is an O(log |U |) approximation
algorithm for Problem 2.
From the point of view of approximation, Problem 2 is
thus not asymptotically harder than Problem 1 (after omitting
constants). Indeed, Algorithm 1 is mainly of theoretical interest, as it shows that Problem 2, like regular C ORRELATION C LUSTERING, admits an approximation bound.
IV. A LOCAL SEARCH ALGORITHM
Next, we describe a more practical method, at the core of
which is a local search heuristic. It is, however, to a certain
extent inspired by Algorithm 1, as will become apparent below.
A. Outline of the algorithm
In short, we minimize the cost function s(f, T ) using a
simple greedy local search algorithm. The algorithm updates
the cluster assignment of a single item u ∈ U at a time, while
keeping the cluster assignment of all other items fixed. The
algorithm makes passes over all items until it reaches a fixed
point where the value of f (u) no longer changes for any u ∈
U . Details are shown in Algorithm 2.
We can initialise f in different ways on line 2 of Alg. 2. In
this paper we consider two approaches:
1) All equal: We set f (u) = 1 for every u ∈ U .
2) All different: We initialise f to be a bijection from U
to the integers [1:|U |], that is, every i ∈ [1:|U |] is the
initial cluster assignment one and only one u ∈ U .
When updating f (u) on line 8, the algorithm considers all
possible values in the range [1:knew ]. Here knew is the index
of a new cluster that does not yet exist in f . The update step
can thus introduce new clusters to f . On line 13 the algorithm
re-assigns (“cleans up”) f so that there are no gaps in the
cluster indices, meaning that the cluster indices must range
from 1 to the number of clusters.
B. Practical considerations
As discussed above in Section II-D, the set T of triplets
may (and in practice will) contain inconsistencies. These
inconsistencies are a fundamental property of relative distance
comparisons. Above we also showed that by finding a vertex
cover of the constraint graph CT , we can remove such
inconsistencies. While the local search heuristic described here
should in theory be unaffected by these inconsistencies, it
is conceivable that we will in practice obtain better results
after making T consistent. That is, in the experiments we will
consider an approach where the input to Algorithm 2 is in fact
T 0 = T \ vc(CT ).
This requires computing a vertex cover of CT . We will employ the simple and well-known 2-approximation algorithm2
that selects edges from the input graph one by one, while at
every step removing all other edges adjacent to the selected
edge (including the selected edge) and adding the endpoints of
the selected edge to the cover. The algorithm, called ApproxVC, terminates when the set of edges becomes empty. (For
details, see e.g. page 1025 of [26].)
Despite the constant factor approximation guarantee, this
algorithm can produce a fairly large cover. In theory this does
not really matter, since in our use-case any cover of CT will
result in a set of triplets T 0 that is free of inconsistencies.
However, in practice we would like there to be a sufficient
amount of information about the relative distances, and hence
T 0 should remain as large as possible. (Indeed, an empty set of
triplets is free of inconsistencies, but it is also rather useless.)
We will thus use a simple heuristic to further reduce the size
of the cover produced by Approx-VC. The idea is to remove
all such vertices from the cover that are redundant. A vertex is
redundant in a vertex cover if all of its neighbour vertices also
belong to the cover. We thus consider every vertex in vc(CT )
one-by-one, and remove it from the cover if (and only if) all
of its neighbours belong to the cover. This leaves us with a
proper vertex cover of CT , but the size of the cover will in
practice be substantially reduced.
V. E XPERIMENTS
We conduct experiments with the following variants of our
local search heuristic:
• Ls-EQ: Algorithm 2 with the All equal initialisation of
f.
• Ls-AD: Algorithm 2 with the All different initialisation
of f .
2 This algorithm was independently proposed by F. Gavril and M. Yannakakis, according to [25].
•
•
Ls-EQ-VC: Algorithm 2 with the All equal initialisation
of f . The input triplets are first cleaned up by running
the vertex cover heuristic on CT .
Ls-AD-VC: Algorithm 2 with the All different initialisation of f . The input triplets are first cleaned up by
running the vertex cover heuristic on CT .
Of these Ls-AD-VC is the one that we mainly promote and
study, others are shown for comparison. We implemented
both the local search as well as the vertex cover heuristic in
JavaScript. All experiments were run with Node.js.
We also consider two alternative approaches. Both first
compute an embedding of the items, and then run a “standard” clustering algorithm with this as the input. The first
method, called CrowdClust below, is the method described
in [13]. The second, called t-STE, is a well-known stochastic
neighbourhood embedding method adapted to relative distance
judgements in [17]. The clustering algorithm is the same
in both methods: a Dirichlet Process mixture model (MBVDP) [27]. We chose this, because like our algorithms, it
does not require setting the number of clusters in advance.
Of CrowdClust and MB-VDP we use the implementations
provided by Ryan Gomes3 , while of t-STE we use Michael
Wilber’s implementation4 .
Finally, we emphasise that the experiments have been carried out to illustrate the experience of a naive user, without
any extensive parameter tuning for the CrowdClust nor t-SNE
methods. The number of optimisation iterations in CrowdClust was capped at 20, and both methods compute a 4dimensional embedding. Otherwise we use default parameters
suggested by the authors. This is because we want to highlight
the simplicity of our algorithms that require no parameter
tuning of any kind, and thus competing approaches should
also work pretty much “out of the box”.
A. Experiment 1: Artificial data with known ground truth
Setup: We generate a set T of triplets from a known
ground truth clustering f ∗ over 160 items, and measure how
well the algorithms can recover f ∗ given the triplets. The
triplets are generated by first
constructing all possible triplets
(of which there are 160
= 669, 920 in this case), then
3
selecting a random subset of these to include in T , and finally
by introducing noise to some randomly chosen triplets by
swapping the outlying item with one of the two other items
(chosen at random). The process is thus parametrised by three
quantities:
k : the number of clusters in f ∗ ,
a : the fraction of triplets (out of all possible triplets) to
include in T , and
b : the fraction of “noisy” triplets in T .
We let k ∈ {2, 4, 8, 16}, a ∈ {0.005, 0.01, 0.02, 0.05, 0.1,
0.2, 0.5, 1.0}, and b ∈ {0, 0.1, 0.2}. When a = 1 and b = 0
the process outputs all possible triplets without any noise.
3 http://www.vision.caltech.edu/
gomes/software.html
4 https://github.com/gcr/tste-theano
With other choices of a and b the triplet generator is nondeterministic. For each of such combination of a and b we
generate 10 independent sets of triplets. This results in 924 test
cases (including repeated trials) in total. We run the algorithms
for all of these, and report averages over the 10 trials for given
values of a, b and k. We evaluate the algorithms by comparing
the found clusterings to a ground truth in terms of the adjusted
Rand index (RI, larger values better) [28].
Results: We find that in 894 out of the 924 test cases
(≈ 97%), Ls-AD-VC has at least the same RI value as CrowdClust. When compared against t-STE, Ls-AD-VC performs at
least as well in 751 out of 924 cases (≈ 81%) in terms of RI.
That is, in most cases our approach seems to do a better job
at reconstructing the true clustering f ∗ . When comparing LsAD-VC against Ls-AD (the pure local search heuristic without
vertex cover based pre-processing), we find that in 813 out of
the 924 test cases (≈ 88 percent) Ls-AD-VC has better (or
the same) performance in terms of RI. This suggests that preprocessing T so that there are no inconsistencies is useful.
Finally, when comparing the two initialisation strategies (all
equal, all different), we find that in 871 out of the 924 test
cases (≈ 94 percent) Ls-AD-VC outperforms Ls-EQ-VC. The
local search heuristic is thus more successful in reconstructing
f ∗ when it starts from a configuration where all points are in
different clusters.
A more fine-grained analysis of how different parameters of
the triplet generator affect performance is shown in Figure 2.
The left, middle, and right columns in Figure 2 show RI as a
function of a, k and b, respectively.
From the left column we observe that when only a small
fraction of triplets are available, and there is a fair amount
of erroneous triplets included in the input (b = 0.2), all
algorithms have difficulties reconstructing the original f ∗ . This
is especially true as the number of clusters in f ∗ increases, as
one might expect. With k = 4 (topmost panel) both Ls-ADVC and CrowdClust can recover f ∗ (almost) perfectly when
a ≥ 0.05, while t-STE does not do bad either. When k ≥ 8
(middle and bottom panel in left column) both Ls-AD-VC and
t-STE seem to produce reasonable results (for a ≥ 0.1), with
Ls-AD-VC finding a near-perfect clustering as long as a is
large enough.
However, in practical human computation applications small
values of a may be more relevant, as this corresponds to
fewer triplets, and hence less work by the human annotators.
This situation is highlighted in the middle column, where we
consider RI as a function of k with fixed a = 0.01, and
different values of b. In the absence of noise (b = 0, top panel),
t-STE is a clear winner. However, as noise is introduced,
Ls-AD-VC, as well as the other correlation clustering based
methods, outperform the two embedding approaches. For a
large number of clusters (say, k ≥ 8), the problem is very
hard for all methods, but when the ground truth clustering
only contains a few clusters, it seems that Ls-AD-VC can give
substantially better results than other methods.
Finally, the rightmost column of Figure 2 shows how the
fraction of erroneous triplets b affects the algorithms with fixed
0.100
0.500
0.8
0.0
2
4
6
8
10
12
14
16
0.00
0.10
0.15
k vs RI, a = 0.01, b=0.1
b vs RI, k = 8, a=0.10
0.100
0.500
0.0
0.4
RI
0.0
0.4
RI
0.4
0.020
2
4
6
8
10
12
14
16
0.00
0.05
0.10
0.15
k
b
a vs RI, k = 16, b=0.2
k vs RI, a = 0.01, b=0.2
b vs RI, k = 8, a=0.01
0.100
0.500
0.4
0.0
RI
0.4
0.0
0.4
RI
0.8
Ls-AD-VC
CrowdClust
t-STE
Ls-EQ-VC
Ls-AD
Ls-EQ
0.20
0.8
a
0.020
0.20
0.8
a vs RI, k = 8, b=0.2
0.8
b
0.8
k
0.0
0.005
0.05
a
0.8
0.005
0.4
RI
0.0
0.020
0.0
RI
0.4
RI
0.4
0.0
RI
0.005
RI
b vs RI, k = 8, a=1.00
0.8
k vs RI, a = 0.01, b=0.0
0.8
a vs RI, k = 4, b=0.2
2
4
6
a
8
10
k
12
14
16
0.00
0.05
0.10
0.15
0.20
b
Fig. 2. Results of Experiment 1. The panels show the adjusted Rand index as a function of different triplet generator parameters (a left, k middle and a
right). See Section V-A for details. Legend for all panels is shown in the center bottom panel.
k = 8 and different values of a. We observe that when there
are a lot of triplets (a ≥ 0.1), Ls-AD-VC is more or less
unaffected by the presence of noise, and t-STE is a close
second. When there are only few triplets (a = 0.01), t-STE
works very well when there is no noise, but its performance
rapidly decreases as b increases.
What kind of errors do the algorithms make, then? A
simple way to address this question is to consider the size
of the resulting clustering. Indeed, all algorithms should in
theory be able to find the correct number of clusters in ideal
settings. Table I shows the number of clusters (averages over
10 instances) in the solution returned by the algorithms for
different values of a, b and k. (Here we only consider Ls-ADVC, Ls-EQ-VC, CrowdClust, and t-STE.) We find that in
most cases, the algorithms tend to underestimate the number
of clusters. As suggested by the results in Fig. 2, Ls-AD-VC
performs very well when the number of triplets is large (a is
large), irrespectively of the amount of noise in the input. Also,
it tends to outperform t-STE for small values of a when there
is a lot of noise (b increases) and k is small.
TABLE I
E XPERIMENT 1: AVERAGE NUMBER OF CLUSTERS FOUND BY THE ALGORITHMS FOR DIFFERENT PARAMETERS OF THE TRIPLET GENERATOR .
b=0
b = 0.1
b = 0.2
Ls-AD-VC
Ls-EQ-VC
CrowdClust
t-STE
Ls-AD-VC
Ls-EQ-VC
CrowdClust
t-STE
Ls-AD-VC
Ls-EQ-VC
CrowdClust
t-STE
2
2
2
6
4
2
2
3.7
1.4
2
2
2.8
1
a=1
k
4
8
4
8
3
4
4
6
7
7
4
8
3
4
4
5.2
7.4
7
4
8
3
4.3
4
3.3
8.2
6.8
c1
16
16
6
6
14
16
5.3
5.6
11.6
16
4.6
5.3
10
2
2
2
3.5
4
2
2
2.8
1
2
2
2.5
1
a = 0.1
k
4
8
4
8
2.8
2.8
4
4.8
4.9
7
4
7.9
2.6
2.9
3.6
5.2
5.3
7
4
7.7
2.6
2.5
3.8
3.9
5.1
6.7
c7
c2
c3
16
14.3
2.4
3.8
13.4
12.7
2.2
1.8
12.4
12.4
2.4
1.3
5.4
2
2
2
3.1
4.2
2
2
2.4
1
2
2
1.8
1
a = 0.05
k
4
8
4
8
2.7
2.5
4
4
4
7.1
4
7.9
2.2
2.4
3.7
2.6
4
7.3
4
6.9
2.5
2.1
3.9
3.4
3.3
6.3
16
10.4
2.4
2
13.7
6.5
2.3
1.7
10.2
3.5
2.4
1
3.4
2
2
2
2.1
2.5
2
2
2.6
2
2
2
1.7
1
a = 0.01
k
4
8
3.7
4.1
2.5
2.2
3.4
3
4.3
7.5
3.8
3.5
2.4
2.3
2.5
1.2
4
5.9
3.4
2.8
2.4
2.4
2.1
1.2
1.9
1.2
16
3
2.2
1.4
11.9
2.8
2.3
1.2
3.9
2.4
2.3
1
1.1
c1
c2
c6
c3
c4
c4
c5
Fig. 3. Clustering of the Nature dataset found by Ls-AD-VC. The figure
shows a random sample of five items from clusters 1–5, and clusters 6 and 7
in full. Cluster 1 contains images of snowy mountains, cluster 2 of wooden
flat areas or forest, cluster 3 of drylands and deserts. Cluster 4 contains a
mixture of tropical trees and open water with a clearly visible horizon, cluster
5 contains images of rocky mountains. Clusters 6 and 7 seem like extra clusters
/ outliers that could also be merged with some of the larger clusters.
B. Experiment 2: Example clusterings with real triplet data
Setup: In our second experiment we present two case
studies that highlight how the Ls-AD-VC algorithm works
on real, crowdsourced data. We obtained two sets of relative
distance comparison triplets from the authors of [9] and
[29]. The first one (Nature), originally used in [9] to run
a crowd-powered k-means algorithm, contains a set of 3357
triplets over 120 images of natural scenes of four categories
(coast, open country, forest, mountain), from the Scene image
collection5 [30]. The second one (Food), collected by the
authors of [29] from Yummly.com, contains 190,376 triplets
over 100 images of various dishes of food.
Note that the two datasets are of different “densities” (recall
the parameter a from above): Nature contains only about 1
5 http://cvcl.mit.edu/database.htm
Fig. 4. Clustering of the Food dataset found by Ls-AD-VC. The figure shows
a random sample of five items from clusters 1–3, cluster 4 only contains a
single item. Cluster 1 contains images of savoury main courses, cluster 2
contains images of salads and other kinds of vegetables, cluster 3 corresponds
to desserts and sweet dishes, and cluster 4 is a single outlier with an image
of a cup of rice.
percent of all possible 120
triplets, while Food contains in
3
fact more than 100 percent of all possible triplets, i.e., Food
contains several instances of duplicated triplets.
We did not perform any kind of pre-processing or cleanup
of the data in either case. The triplets may thus be noisy,
conflicting (i.e., for the same set of three items, two triplets
may indicate different items as the outlier), etc. This decision
was deliberate, because we want to illustrate the experience
of a naive user, who wants results quickly, and as simply as
possible, e.g. without first running a complex consensus model
[31] to clean up erroneous inputs. Certainly there are situations
in which using those methods is really necessary, but here
we want to show how Ls-AD-VC fares with “raw” human
computation data. (One can always argue that results should
only improve with more complex pre-processing.) We thus run
the experiment simply by running Ls-AD-VC on all available
triplets.
Results: The found clusterings are shown in figures 3 and
4 for Nature and Food, respectively. Of clusters with more
than five items, we show only five randomly selected items.
From Nature the method finds 7 clusters, two of which are
very small, while from Food we find four clusters, one of
which contains only a single(!) item. As can be seen from the
figures, the found clusterings are very intuitive in both cases. In
particular, thanks to the large number of triplets, the result with
Food is extremely good, and really provides strong evidence
for the correlation clustering based method to be both simple
and very efficient. Also, it is rather remarkable that the method
can find a very good clustering from Nature, despite there
being very few triplets. However, this is in accordance with
the results from synthetic clusterings in Experiment 1, where
we show that Ls-AD-VC can find reasonable clusterings even
in this situation (a = 0.01) as long as the underlying clustering
is not too fine grained.
Finally, we want to point out that Ls-AD-VC is extremely
fast: the runtimes with Nature and Food are < 1 second and
≈ 11 seconds, respectively.
VI. C ONCLUSION AND F UTURE W ORK
We defined, analysed, and provided both an approximation
algorithm, as well as a practical local search algorithm for a
novel C ORRELATION -C LUSTERING variant based on relative
distance comparisons. We also showed empirically that the
approach has certain advantages over existing methods for
clustering with relative distance comparisons.
Our method is motivated by human computation approaches
to clustering. However, an interesting property of our approach
is that it does not require guessing the number of clusters in
advance. Given that an arbitrary distance matrix can always be
used to generate relative distance comparisons in a parameterfree manner, it seems interesting to investigate if and how the
approach could be used as a generic non-parametric clustering
algorithm. Moreover, it seems relevant to understand what
kind of clusterings our method can find when used in such
ways. For instance, does the distance between two clusters
and cluster diameter affect the outcome? Finally, implementing
agglomerative clustering approaches using relative distances
also seems of interest. First steps towards this have been taken
in [32].
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A PPENDIX
A. Proof of Lemma 1
Proof. For two triplets t1 and t2 to be inconsistent, there must
exist the items a and b such that t1 is satisfied only when
f (a) = f (b), and t2 is satisfied only when f (a) 6= f (b).
Consider only those edges in CT that exist due to inconsistencies induced by the items a and b. These edges must form
a bipartite clique. Since vc(CT ) is a vertex cover of CT , it
must contain a vertex cover of every bipartite clique in C.
A proper vertex cover of any bipartite clique must contain at
least all vertices from “one side” of the clique, otherwise it
cannot cover all edges in the clique. By the above reasoning, in
every bipartitie clique of C at least one of the “sides” must be
completely contained in vc(C). Since all triplets that belong
to the same “side” of a bipartite clique agree about the items
a and b, the triplets that remain in T \ vc(C) can not be
inconsistent.
Finally, because in every triplet t = (a, b, c) there is precisely
one pair, (a, b), that belongs to E + , while the other two pairs,
(a, c) and (b, c), belong to E − , we obtain:
X
c(f, GT 0 ) =
(I{f (a) 6= f (b)} +
(a,b,c)∈T 0
I{f (a) = f (c)} + I{f (b) = f (c)}).
B. Proof of Theorem 2
This has the same form as the cost s of Problem 2, with
the difference that in c(f, GT 0 ) the cost of every triplet
(a, b, c) ∈ T 0 is the sum of three indicator variables, instead
of a single indicator with a disjunction of three conditions, as
is the case in s(f, T 0 ). The conditions, however, are the same
in both cases. First, this implies that s(f, T 0 ) ≤ c(f, GT 0 )
for every f . Second, it is easy to see that for a given triplet
(a, b, c), at most two of these conditions can be satisfied
simultaneously. Whenever the triplet (a, b, c) incurs a cost of 1
in s(f, T 0 ), it therefore incurs at most a cost of 2 in c(f, GT 0 )
for every f . And if none of the conditions is satisfied, the
triplet (a, b, c) incurs a cost of zero in both cases. This implies
the 2nd inequality of the Lemma.
Before giving the proof, we present three technical lemmas
that are needed to establish the result.
Lemma 3. We have s(f, T ) ≤ | vc(CT )| + s(f, T 0 ) for any
vertex cover vc(CT ) and clustering function f .
Lemma 2. Let T 0 = T \ vc(CT ), and denote by GT 0 the
associated C ORRELATION -C LUSTERING instance. We have
s(f, T 0 ) ≤ c(f, GT 0 ) ≤ 2 s(f, T 0 )
for any clustering function f , where c and s are the cost
functions of Problems 1 and 2, respectively.
Proof. We rewrite the cost function c of Problem 1 as a sum
over T 0 . By definition of GT 0 , the weight w(u, v) is equal to
0
the size of Tu,v
, i.e., the number of triplets that contain both
items u and v. We can write
X
X
c(f, GT 0 ) =
I{f (u) 6= f (v)}+
0
(u,v)∈E + t∈Tu,v
X
X
I{f (u) = f (v)}.
0
(u,v)∈E − t∈Tu,v
Instead of summing over E + and E − separately, we simply
sum over all (u, v) and use an indicator function to select the
appropriate case. Note that any pair (u, v) can only belong to
either E + or E − but not both. This yields
X X
c(f, GT 0 ) =
(I{(u, v) ∈ E + } I{f (u) 6= f (v)}+
0
(u,v) t∈Tu,v
−
I{(u, v) ∈ E } I{f (u) = f (v)}).
We then change the order of summation to run first over T 0 ,
and then over the pairs in every triplet:
X X
c(f, GT 0 ) =
(I{(u, v) ∈ E + } I{f (u) 6= f (v)}+
t∈T 0 (u,v)∈t
I{(u, v) ∈ E − } I{f (u) = f (v)}).
Proof. By definition, vc(CT ) and T 0 constitute a disjoint
partition of T . Since the cost function s is simply a sum over
triplets in T , we have s(f, T ) = s(f, vc(CT )) + s(f, T 0 ) for
any f . Also, clearly s(f, vc(CT )) can be at most | vc(CT )| (all
triplets in vc(CT ) are violated), which leads to the inequality
of the Lemma.
Lemma 4. Let T be a set of triplets, CT denote the associated
constraint graph, vcmin (CT ) the minimum vertex cover of CT ,
and let f OPT denote the optimal solution to Problem 2. We
have | vcmin (CT )| ≤ s(f OPT , T ).
Proof. Every edge in CT consists of a pair of triplets, of which
at least one must be unsatisfied for any clustering f , including
f OPT . The minimum vertex cover vcmin (CT ) corresponds to
the smallest possible subset of triplets that must be unsatisfied,
resulting in a lower bound of s(f OPT , T ).
We conclude with the proof of Theorem 2:
Proof. The proof is a mechanical exercise with the lemmas
presented above. We start from Lemma 3 that holds for any
f and vc(CT ):
s(f ALG , T ) ≤ | vcALG (CT )| + s(f ALG , T 0 )
≤
| vcALG (CT )| + c(f ALG , GT 0 )
≤ β| vc
min
(CT )| +
≤ β| vc
min
(CT )| +
OPT
αc(fCC
, GT 0 )
OPT
αc(f
, GT 0 )
OPT
0
≤ β| vcmin (CT )| + 2αs(f
(Lemma 2)
(Assum. 1 and 2)
OPT
(fCC
is optimal)
, T ) (Lemma 2)
≤ β| vcmin (CT )| + 2αs(f OPT , T )
(T 0 ⊆ T )
≤ βs(f OPT , T ) + 2αs(f OPT , T )
(Lemma 4)
=
(2α + β) s(f
OPT
, T ).
| 8 |
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD
MANIFOLDS
arXiv:1801.08239v1 [math.GR] 24 Jan 2018
MICHAEL KAPOVICH AND BEIBEI LIU
Abstract. In this paper, we generalize Bonahon’s characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, C) to geometrically infinite discrete torsionfree subgroups Γ of isometries of negatively pinched Hadamard manifolds X. We then
generalize a theorem of Bishop to prove that every such geometrically infinite isometry
subgroup Γ has a set of nonconical limit points with cardinality of continuum.
1. Introduction
The notion of geometrically finite discrete groups was originally introduced by Ahlfors in
[1], for subgroups of isometries of the 3-dimensional hyperbolic space H3 as the finiteness
condition for the number of faces of a convex fundamental polyhedron. In the same paper,
Ahlfors proves that the limit set of a geometrically finite subgroup of isometries of H3 has
either zero or full Lebesgue measure in S 2 . The notion of geometric finiteness turned out
to be quite fruitful in the study of Kleinian groups. Alternative definitions of geometric
finiteness were later given by Marden [15], Beardon and Maskit [5], and Thurston [19].
These definitions were further extended by Bowditch [8] and Ratcliffe [18] for isometry
subgroups of higher dimensional hyperbolic spaces and, a bit later, by Bowditch [9] to
negatively pinched Hadamard manifolds. While the original Ahlfors’ definition turned out
to be too limited (when used beyond the hyperbolic 3-space), other definitions of geometric
finiteness were proven to be equivalent by Bowditch in [9].
Our work is motivated by the definition of geometric finiteness due to Beadon and Maskit
[5] who proved
Theorem 1.1. A discrete isometry subgroup Γ of H3 is geometrically finite if and only if
every limit point of Γ is either a conical limit point or is a bounded parabolic fixed point.
This theorem was improved by Bishop in [6]:
Theorem 1.2. A Kleinian group Γ < Isom(H3 ) is geometrically finite if and only if every
point of Λ(Γ) is either a conical limit point or a parabolic fixed point. Furthermore, if
Γ < Isom(H3 ) is geometrically infinite, Λ(Γ) contains a set of nonconical limit points with
cardinality of continuum.
The key ingredient in Bishop’s proof of Theorem 1.2 is Bonahon’s theorem1 [7]:
Theorem 1.3. A discrete torsion-free subgroup Γ < Isom(H3 ) is geometrically infinite if
and only if there exists a sequence of closed geodesics λi in the manifold M = H3 /Γ which
“escapes every compact subset of M ,” i.e., for every compact subset K ⊂ M ,
card ({i : λi ∩ K 6= ∅}) < ∞.
According to Bishop, Bonahon’s theorem also holds for groups with torsion, but it is
unclear to us if Bonahon’s proof extends to cover this case, as some of Bonahon’s arguments
Date: January 24, 2018.
1Bonahon uses this result to prove his famous theorem about tameness of hyperbolic 3-manifolds.
1
2
MICHAEL KAPOVICH AND BEIBEI LIU
require one to know that every nontrivial element of Γ is either loxodromic or parabolic.
However, for higher dimensional hyperbolic spaces Hn , we extend Bonahon’s proof and
prove that Bonahon’s theorem holds for discrete isometry subgroups with torsion, see [14].
Bowditch generalized the notion of geometric finiteness to discrete subgroups of isometries
of negatively pinched Hadamard manifolds [9]. A negatively pinched Hadamard manifold
is a complete, simply connected Riemannian manifold such that all sectional curvatures lie
between two negative constants. From now on, we use X to denote a negatively pinched
Hadamard manifold, ∂∞ X its visual (ideal) boundary, X̄ the visual compactification X ∪
∂∞ X, Γ a discrete subgroup of isometries of X, Λ = Λ(Γ) the limit set of Γ. The convex
core Core(M ) of M = X/Γ is defined as the Γ-quotient of the closed convex hull of Λ(Γ)
in X. Recall also that a point ξ ∈ ∂∞ X is a conical limit point 2 of Γ if for every x ∈ X
and every geodesic ray l in X asymptotic to ξ, there exists a positive constant A such that
the set Γ(x) ∩ NA (l) accumulates to ξ, where NA (l) denotes the A-neighborhood of l in X.
A parabolic fixed point ξ ∈ ∂∞ X (i.e. a fixed point of a parabolic element of Γ) is called
bounded if
(Λ(Γ) − {ξ})/Γξ
is compact. Here Γξ is the stabilizer of ξ in Γ.
Bowditch [9], gave four equivalent definitions of geometric finiteness for Γ:
Theorem 1.4. The followings are equivalent for discrete subgroups Γ < Isom(X):
(1) The quotient space M̄ (Γ) = (X̄ − Λ)/Γ has finitely many topological ends each of
which is a “cusp”.
(2) The limit set Λ(Γ) of Γ consists entirely of conical limit points and bounded parabolic
fixed points.
(3) The noncuspidal part of the convex core Core(M ) of M = X/Γ is compact.
(4) For some δ > 0, the uniform δ-neighbourhood of the convex core, Nδ (Core(M )), has
finite volume and there is a bound on the orders of finite subgroups of Γ.
If one of these equivalent conditions holds, the subgroup Γ < Isom(X) is said to be
geometrically finite; otherwise, Γ is said to be geometrically infinite.
The main results of our paper are:
Theorem 1.5. Suppose that Γ < Isom(X) is a torsion-free discrete subgroup. Then the
followings are equivalent:
(1) Γ is geometrically infinite.
(2) There exists a sequence of closed geodesics λi ⊂ M = X/Γ which escapes every
compact subset of M .
(3) The set of nonconical limit points of Γ has cardinality of continuum.
Corollary 1.6. If Γ < Isom(X) is a torsion-free discrete subgroup then Γ is geometrically
finite if and only if every limit point of Γ is either a conical limit point or a parabolic fixed
point.
We have the following conjecture regarding the Hausdorff dimension of the set of nonconical limit points of any geometrically infinite torsion-free discrete subgroup Γ < Isom(X).
Conjecture 1.7. Suppose that Γ < Isom(X) is a geometrically infinite torsion-free discrete
subgroup. Then the Hausdorff dimension of the set of nonconical limit points of Γ equals
the Hausdorff dimension of the limit set itself. Here, the Hausdorff dimension is defined
with respect to any of the visual metrics on ∂∞ X, see [17].
2Another way is to describe conical limit points of Γ as points ξ ∈ ∂ X such that one, equivalently,
∞
every, geodesic ray R+ → X asymptotic to ξ projects to a non-proper map R+ → M .
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
3
This conjecture is a theorem by Fernández and Melián [12] in the case of Fuchsian subgroups of the 1st kind, Γ < Isom(H2 ).
Below is an outline of the proof of Theorem 1.5. Our proof of the implication (1)⇒(2)
mostly follows Bonahon’s argument with the following exception: At some point of the
proof Bonahon has to show that certain elements of Γ are loxodromic. For this he uses
a calculation with 2 × 2 parabolic matrices: If g, h are parabolic elements of Isom(H3 )
generating a nonelementary subgroup then either gh or hg is non-parabolic. This argument
is no longer valid for isometries of higher dimensional hyperbolic spaces, let alone Hadamard
manifolds. We replace this computation with a more difficult argument showing that there
exists a number ` = `(n, κ) such that for every n-dimensional Hadamard manifold X with
sectional curvatures pinched between −κ2 and −1 and for any pair of parabolic isometries
g, h ∈ Isom(X) generating a nonelementary discrete subgroup, a certain word w = w(g, h)
of length ≤ ` is loxodromic (Theorem 8.5). Our proof of the implication (2)⇒(3) is similar
to Bishop’s but more coarse-geometric in nature. Given a sequence of closed geodesics λi
in M escaping compact subsets, we define a family of proper piecewise geodesic paths γτ
in M consisting of alternating geodesic arcs µi , νi , such that µi connects λi to λi+1 and is
orthogonal to both, while the image of νi is contained in the loop λi . If the lengths of νi are
sufficiently long, then the path γτ lifts to a uniform quasigeodesic γ̃τ in X, which, therefore,
is uniformly close to a geodesic γ̃τ∗ . Projecting the latter to M , we obtain a geodesic γτ∗
uniformly close to γτ , which implies that the ideal point γ̃τ∗ (∞) ∈ ∂∞ X is a nonconical
limit point of Γ. Different choices of the arcs νi yield distinct limit points, which, in turn
implies that Λ(Γ) contains a set of nonconical limit points with cardinality of continuum.
The direction (3)⇒(1) is a direct corollary of Theorem 1.4.
Organization of the paper. In Section 3, we review the angle comparison theorem
[9, Proposition 1.1.2] for negatively pinched Hadamard manifolds and derive some useful
geometric inequalities. In Section 5, we review the notions of elementary parabolic subgroups and elementary loxodromic subgroups of isometries of negatively pinched Hadamard
manifolds, [9]. In Section 6, we review the thick-thin decomposition in negatively pinched
Hadamard manifolds and some properties of parabolic subgroups, [9]. In Section 7, we use
the results in Section 3 to prove that certain piecewise geodesic paths in Hadamard manifolds with sectional curvatures ≤ −1 are uniform quasigeodesics. In Section 8, we explain
how to produce loxodromic isometries as words w(g, h) of uniformly bounded length, where
g, h are parabolic isometries of X with distinct fixed points. In Section 9, we generalize
Bonahon’s theorem, the implication (1)⇒(2) in Theorem 1.5. In Section 10, we construct
the set of nonconical limit points with cardinality of continuum and complete the proof of
Theorem 1.5.
Acknowledgements. The first author was partly supported by the NSF grant DMS16-04241 as well as by KIAS (the Korea Institute for Advanced Study) through the KIAS
scholar program. Some of this work was done during his stay at KIAS and he is thankful
to KIAS for its hospitality.
2. Notation
In a metric space (Y, d), we will use the notation B(a, r) to denote the open r-ball centered
at a in Y . For a subset A ⊂ Y and a point y ∈ Y , we will denote by d(y, A) the minimal
distance from y to A, i.e.
d(y, A) := inf{d(y, a) | a ∈ A}.
We use the notation Nr (A) for the closed r-neighborhood of A in Y :
Nr (A) = {y ∈ Y : d(y, A) ≤ r}.
4
MICHAEL KAPOVICH AND BEIBEI LIU
The Hausdorff distance hd(Q1 , Q2 ) between two closed subsets Q1 and Q2 of (Y, d) is the
infimum of r ∈ [0, ∞) such that Q1 ⊆ Nr (Q2 ) and Q2 ⊆ Nr (Q1 ).
Throughout the paper, X will denote a negatively pinched Hadamard manifold, unless
otherwise stated; we assume that all sectional curvatures of X lie between −κ2 and −1. We
let d denote the Riemannian distance function on X. We let Isom(X) denote the isometry
group of X.
For a Hadamard manifold X, the exponential map is a diffeomorphism, in particular, X
is diffeomorphic to Rn , where n is the dimension of X. Then X can be compactified by
adjoining the ideal boundary sphere ∂∞ X, and we will use the notation X̄ = X ∪ ∂∞ X for
this compactification. The space X̄ is homeomorphic to the closed n-dimensional ball.
In this paper, geodesics will be always parameterized by their arc-length; we will conflate
geodesics in X with their images.
Given a closed subset A ⊆ X and x ∈ X, we write
ProjA (x) = {y ∈ A | d(x, y) = d(x, A)}
as the projection of x to A. It consists of all points in A which are closest to x. If A is
convex, then ProjA (x) is a singleton.
Hadamard spaces are uniquely geodesic and we will let xy ⊂ X denote the geodesic
segment connecting x ∈ X to y ∈ X. Similarly, given x ∈ X and ξ ∈ ∂∞ X we will
use the notation xξ for the unique geodesic ray emanating from x and asymptotic to ξ;
for two distinct points ξ, η ∈ ∂∞ X, we use the notation ξη to denote the unique (up to
reparameterization) geodesic asymptotic to ξ and η.
Given ξ ∈ ∂∞ X, horospheres about ξ are level sets of a Busemann function h about
ξ. For details of Busemann functions, see [2, 9] (notice that Bowditch uses a nonstandard
notation for Busemann functions, which are negatives of the standard Busemann functions).
A set of the form h−1 ((−∞, r]) for r ∈ R is called a horoball about ξ. Horoballs are convex.
Given points P1 , P2 , · · · , Pn ∈ X we let [P1 P2 · · · Pn ] denote the geodesic polygon in X
which is the union of geodesic segments Pi Pi+1 , i taken modulo n.
Given two distinct points x, y ∈ X, and a point q ∈ xy, we define the normal hypersurface
Nq (x, y), i.e. the image of the normal exponential map to the segment xy at point q:
Nq (x, y) = expq (Tq⊥ (xy)),
where Tq⊥ (xy) ⊂ Tq X is the orthogonal complement in the tangent space at q to the segment
xy. In the special case when q is the midpoint of xy, Nq (x, y) is the perpendicular bisector of
the segment xy, and we will denote it Bis(x, y). Similarly, we define the normal hypersurface
Nq (ξ, η) for any point q in the biinfinite geodesic ξη.
Note that if X is a real-hyperbolic space, then Bis(x, y) is totally geodesic and equals the
set of points equidistant from x and y. For general Hadamard spaces, this is not the case.
However, if X is δ-hyperbolic, then each Np (x, y) is δ-quasiconvex, see Definition
3.12.
√
−1
We let δ denote the hyperbolicity constant of X; hence, δ ≤ cosh ( 2). We will use
the notation Hull(A) for the closed convex hull of a subset A ⊂ X, i.e. the intersection of
all closed convex subsets of X containing A. The notion of the closed convex hull extends
to the closed subsets of ∂∞ X as follows. Given a closed subset A ⊂ ∂∞ X, we denote by
Hull(A) the smallest closed convex subset of X whose accumulation set in X̄ equals A.
(Note that Hull(A) exists as long as A contains more than one point.)
For a subset A ⊂ X the quasiconvex hull QHull(A) of A in X is defined as the union of all
geodesics connecting points of A. Similarly, for a closed subset A ⊂ ∂∞ X, the quasiconvex
hull QHull(A) is the union of all biinfinite geodesics asymptotic to points of A. Then
QHull(A) ⊂ Hull(A), unless A is a singleton in ∂∞ X.
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
5
We will use the notation Γ for a discrete subgroup of isometries of X. We let Λ = Λ(Γ) ⊂
∂∞ X denote the limit set of Γ, i.e. the accumulation set in ∂∞ X of one (equivalently, any)
Γ-orbit in X. The group Γ acts properly discontinuously on X̄ \ Λ, [9, Proposition 3.2.6].
We obtain an orbifold with boundary
M̄ = Mc (Γ) = X̄ \ Λ /Γ.
If Γ is torsion-free, then M̄ is a partial compactification of the quotient manifold M = X/Γ.
We let π : X → M denote the covering projection.
3. Review of negatively pinched Hadamard manifolds
For any triangle [ABC] in (X, d), we define a comparison triangle [A0 B 0 C 0 ] for [ABC] in
(H2 , d0 ) as follows.
Definition 3.1. For a triangle [ABC] in (X, d), let A0 , B 0 , C 0 be 3 points in the hyperbolic
plane (H2 , d0 ) satisfying that d0 (A0 , B 0 ) = d(A, B), d0 (B 0 , C 0 ) = d(B, C) and d0 (C 0 , A0 ) =
d(C, A). Then we call [A0 B 0 C 0 ] a comparison triangle for [ABC].
In general, for any geodesic polygon [P1 P2 · · · Pn ] in (X, d), we define a comparison polygon [P10 P20 · · · Pn0 ] for [P1 · · · Pn ] in (H2 , d0 ).
Definition 3.2. For any geodesic polygon [P1 P2 · · · Pn ] in X, we pick points P10 , · · · , Pn0 in
0 ] is a comparison triangle for [P P P
0 0
0
H2 such that [P10 Pi0 Pi+1
1 i i+1 ] and the triangles [P1 Pi−1 Pi ]
0 ] lie on different sides of P 0 P 0 for each 2 ≤ i ≤ n − 1. The geodesic polygon
and [P10 Pi0 Pi+1
1 i
0
0
0
[P1 P2 · · · Pn ] is called a comparison polygon for [P1 P2 · · · Pn ].
Remark 3.3. Such a comparison polygon [P10 P20 · · · Pn0 ] is not necessarily convex and embedded. In the rest of the section, we have additional assumptions for the polygons [P1 P2 · · · Pn ].
Under these assumptions, their comparison polygons in H2 are embedded and convex, see
Corollary 3.7 and Corollary 3.9.
One important property of negatively pinched Hadamard manifolds X is the following
angle comparison theorem [10].
Proposition 3.4. [9, Proposition 1.1.2] For a triangle [ABC] in (X, d), let [A0 B 0 C 0 ] denote a comparison triangle for [ABC]. Then ∠ABC ≤ ∠A0 B 0 C 0 , ∠BCA ≤ ∠B 0 C 0 A0 and
∠CAB ≤ ∠C 0 A0 B 0 .
Proposition 3.4 implies some useful geometric inequalities in X:
Corollary 3.5. Consider a triangle in X with vertices ABC so that the angles at A, B, C
are α, β, γ and the sides opposite to A, B, C have lengths a, b, c, respectively. If γ ≥ π/2,
then
cosh a sin β ≤ 1.
Proof. Let [A0 B 0 C 0 ] be a comparison triangle for [ABC] in (H2 , d0 ). Let α0 , β 0 , γ 0 denote
the angles at A0 , B 0 , C 0 respectively as in Figure 1. By Proposition 3.4, d0 (A0 , B 0 ) =
0
c, d0 (A0 , C 0 ) = b, d0 (B 0 , C 0 ) = a and β 0 ≥ β, γ ≥ γ ≥ π/2. Take the point C 00 ∈ A0 B 0
such that ∠B 0 C 0 C 00 = π/2. In the right triangle [B 0 C 0 C 00 ] in H2 , we have cosh a sin β 0 =
cos(∠C 0 C 00 B 0 ), see [4, Theorem 7.11.3]. So we obtain the inequality:
cosh a sin β ≤ cosh a sin β 0 ≤ 1.
Remark 3.6. If A ∈ ∂∞ X, we use a sequence of triangles in X to approximate the triangle
[ABC] and prove that cosh a sin β ≤ 1 still holds by continuity.
6
MICHAEL KAPOVICH AND BEIBEI LIU
Figure 1.
Corollary 3.7. Let [ABCD] denote a quadrilateral in X such that ∠ABC ≥ π/2, ∠BCD ≥
π/2 and ∠CDA ≥ π/2 as in Figure 2(a). Then:
(1) sinh(d(B, C)) sinh(d(C, D)) ≤ 1.
(2) Suppose that ∠BAD ≥ α > 0. If cosh(d(A, B)) sin α > 1, then
cosh(d(C, D)) ≥ cosh(d(A, B)) sin α > 1.
Proof. Let [A0 B 0 C 0 D0 ] be a comparison quadrilateral for [ABCD] in (H2 , d0 ) so that [A0 B 0 C 0 ]
is a comparison triangle for [ABC] and [A0 C 0 D0 ] is a comparison triangle for [ACD]. By
Proposition 3.4, ∠A0 B 0 C 0 ≥ π/2, ∠A0 D0 C 0 ≥ π/2 and
∠B 0 C 0 D0 = ∠B 0 C 0 A0 + ∠A0 C 0 D0 ≥ ∠BCD ≥ π/2.
So 0 < ∠B 0 A0 D0 ≤ π/2 and [A0 B 0 C 0 D0 ] is an embedded convex quadrilateral.
We first prove that sinh d(B, C) sinh(d(C, D)) ≤ 1. In Figure 2(c), take the point H ∈
A0 B 0 such that ∠HC 0 D0 = π/2 and take the point G ∈ A0 H such that ∠GD0 C 0 = π/2. We
claim that ∠C 0 HA0 ≥ π/2. Observe that
∠C 0 HB 0 + ∠HB 0 C 0 + ∠B 0 C 0 H ≤ π
∠C 0 HA0 + ∠C 0 HB 0 = π.
Thus ∠C 0 HA0 ≥ ∠C 0 B 0 H ≥ π/2. We also have d0 (C 0 , H) ≥ d0 (C 0 , B 0 ) since
sinh(d0 (C 0 , B 0 ))
sinh(d0 (C 0 , H))
=
.
sin(∠C 0 B 0 H)
sin(∠C 0 HB 0 )
Take the point H 0 ∈ GD0 such that ∠C 0 HH 0 = π/2. In the quadrilateral [C 0 HH 0 D0 ],
cos(∠HH 0 D0 ) = sinh(d0 (H, C 0 )) sinh(d0 (C 0 , D0 )) [4, Theorem 7.17.1]. So we have
sinh(d(C, D)) sinh(d(B, C)) = sinh(d0 (C 0 , D0 ) sinh(d0 (B 0 , C 0 ))
≤ sinh(d0 (C 0 , D0 )) sinh(d0 (C 0 , H))
≤ 1.
Next, we prove that if cosh(d(A, B)) sin α > 1, then cosh(d(C, D)) ≥ cosh(d(A, B)) sin α.
In Figure 2(b), take the C 00 ∈ C 0 D0 such that ∠A0 B 0 C 00 = π/2. Observe that C 00 cannot be
on A0 D0 . Otherwise in the right triangle [A0 B 0 C 00 ], we have
cosh(d(A, B)) sin α ≤ cosh(d0 (A0 , B 0 )) sin(∠B 0 A0 D0 ) ≤ 1
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
7
which is a contradiction. Let EF denote the geodesic segment which is orthogonal to B 0 E
and A0 F . In the quadrilateral [A0 B 0 EF ], cosh(d0 (E, F )) = cosh(d0 (A0 , B 0 )) sin(∠B 0 A0 F ) by
hyperbolic trigonometry [4, Theorem 7.17.1]. So
cosh(d(C, D)) ≥ cosh(d0 (C 00 , D0 )) ≥ cosh(d0 (E, F )) ≥ cosh(d(A, B)) sin α.
Remark 3.8. If A ∈ ∂∞ X and ∠BAD = 0, we use quadrilaterals in X to approximate the
quadrilateral [ABCD] and prove that sinh(d(B, C)) sinh(d(C, D)) ≤ 1 by continuity.
Figure 2.
Corollary 3.9. Let [ABCDE] be a pentagon in X with each angle ≥ π/2 as in Figure
3(a). Then if d(A, B) → ∞, we have d(C, D) → ∞.
Proof. Let [A0 B 0 C 0 D0 E 0 ] be a comparison pentagon for [ABCDE] in (H2 , d0 ) as in Figure
3. By Proposition 3.4,
∠A0 B 0 C 0 ≥ π/2,
∠B 0 C 0 D0 ≥ π/2,
∠A0 E 0 D0 ≥ π/2
and
∠E 0 D0 C 0 ≥ π/2,
∠B 0 A0 E 0 ≥ π/2.
So the pentagon [A0 B 0 C 0 D0 E 0 ] is convex as in Figure 3.
Take the point C 00 ∈ C 0 D0 such that ∠A0 B 0 C 00 = π/2 as in Figure 3(b). Observe that C 00
cannot be in E 0 D0 if d(A, B) → ∞. Otherwise we will obtain a quadrilateral [A0 B 0 C 00 E 0 ].
By Corollary 3.7, sinh(d0 (A0 , B 0 )) sinh(d0 (A0 , E 0 )) ≤ 1. This is a contradiction when d(A, B)
is sufficiently large. Choose a point E 00 in H2 such that ∠E 00 A0 B 0 = π/2. Then either E 00 is
in E 0 D0 as in Figure 3(c) or E 00 is in C 0 D0 as in Figure 3(b).
If E 00 ∈ C 0 D0 , we have a quadrilateral [A0 B 0 C 00 E 00 ], and ∠C 00 B 0 A0 = ∠B 0 A0 E 00 = π/2.
So d0 (C 00 , E 00 ) ≥ d0 (A0 , B 0 ). If E 00 ∈ E 0 D0 , take the point F ∈ C 00 D0 such that ∠F E 00 A0 =
π/2 in Figure 3(c). Here F cannot be in B 0 C 00 . Otherwise we obtain a quadrilateral
[A0 B 0 F E 00 ] which is impossible if d(A, B) → ∞ by Corollary 3.7. Observe that d0 (A0 , E 00 ) ≥
d0 (A0 , E 0 ). Let GH denote the geodesic segment which is orthogonal to B 0 G and E 00 H.
Then d0 (C 00 , F ) ≥ d0 (G, H). In the pentagon [A0 E 00 HGB 0 ], we have cosh(d0 (G, H)) =
sinh(d0 (A0 , B 0 )) sinh(d0 (A0 , E 00 )), see [4, Theorem 7.18.1]. So
cosh(d(C, D)) ≥ cosh(d0 (G, H)) ≥ sinh(d(A, B)) sinh(d0 (A0 , E 0 ))
Thus in both cases, d(C, D) → ∞ if d(A, B) → ∞.
8
MICHAEL KAPOVICH AND BEIBEI LIU
Figure 3.
Another comparison theorem, the CAT(-1) inequality, can be used to derive the following
proposition (see [9]):
Proposition 3.10. [9, Lemma 2.2.1] Suppose x0 , x1 , · · · , xn ∈ X̄ are n + 1 points; then
x0 xn ⊆ Nλ (x0 x1 ∪ x1 x2 ∪ · · · ∪ xn−1 xn )
√
where λ = λ0 dlog2 ne, λ0 = cosh−1 ( 2).
Given a point ξ ∈ ∂∞ X, for any point y ∈ X, we use a map ρy : R+ → X to parametrize
the geodesic yξ by its arc-length. The following lemma is deduced from the CAT (−1)
inequality, see [9]:
Lemma 3.11. [9, Proposition 1.1.11]
(1) Given any y, z ∈ X, the function d(ρy (t), ρz (t)) is monotonically decreasing in t.
(2) For each r, there exists a constant R = R(r), such that if y, z ∈ X lie in the same
horosphere about ξ and d(y, z) ≤ r, then d(ρy (t), ρz (t)) ≤ Re−t for all t.
Next we discuss convex and quasiconvex sets in X.
Definition 3.12. A subset A ⊆ X is convex if xy ⊆ A for all x, y ∈ A. A closed subset
A ⊆ X is λ-quasiconvex if xy ⊆ Nλ (A) for all x, y ∈ A. Convex closed subsets are 0quasiconvex.
Remark 3.13. If A is a λ-quasiconvex set, then QHull(A) ⊆ Nλ (A).
Proposition 3.14. [9, Proposition 2.5.4] There is a function r : R+ → R+ such that for
every λ-quasiconvex subset A ⊆ X, we have
Hull(A) ⊆ Nr(λ) (A)
where the function r(λ) only depends on κ.
Remark 3.15. Note that, by the definition of the hyperbolicity constant δ of X, the quasiconvex hull QHull(A) is 2δ-quasiconvex for every closed subset A ⊆ X̄. Thus, Hull(A) ⊆
Nr (QHull(A)) for some uniform constant r ∈ [0, ∞).
Remark 3.16. For any closed subset A ⊆ ∂∞ X with more than one point, ∂∞ Hull(A) = A.
Lemma 3.17. Assume that ξ, η are distinct points in ∂∞ X and (xi ) ⊆ X is a sequence
of points which converges to ξ and (yi ) ⊆ X is a sequence of points which converges to η.
Then for every point p ∈ ξη ⊆ X, p ∈ N2δ (xi yi ) for all sufficiently large i.
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
9
Proof. Since (xi ) converges to ξ and (yi ) converges to η, then d(p, xi ξ) → ∞ and d(p, yi η) →
∞ as i → ∞. By δ-hyperbolicity of X,
p ∈ N2δ (xi yi ∪ xi ξ ∪ yi η).
Since d(p, xi ξ) → ∞ and d(p, yi η) → ∞, then
p ∈ N2δ (xi yi )
for sufficiently large i.
Remark 3.18. This lemma holds for any δ-hyperbolic geodesic metric space.
4. Escaping sequences of closed geodesics in negatively curved manifolds
In this section, X is a Hadamard manifold of negative curvature ≤ −1 with the hyperbolicity constant δ, Γ < Isom(X) is a torsion-free discrete isometry subgroup and M = X/Γ
is the quotient manifold. A sequence of subsets Ai ⊂ M is said to escape every compact
subset of M if for every compact K ⊂ M , the subset
{i ∈ N : Ai ∩ K 6= ∅}
is finite. Equivalently, for every x ∈ M , d(x, Ai ) → ∞ as i → ∞.
Lemma 4.1. Suppose that (ai ) is a sequence of closed geodesics in M = X/Γ which escapes
every compact subset of M and x ∈ M . Then, after passing to a subsequence in (ai ), there
exist geodesic arcs bi connecting ai , ai+1 and orthogonal to these geodesics, such that the
sequence (bi ) also escapes every compact subset of M .
Proof. Consider a sequence of compact subsets Kn := B̄(x, 7δn) exhausting M . Without
loss of generality, we may assume that ai ∩ Kn = ∅ for all i ≥ n.
We first prove the following claim:
Claim. For each compact subset K ⊂ M and for each infinite subsequence (ai )i∈I , I ⊂ N,
there exists a further infinite subsequence, (ai )i∈J , J ⊂ I, such that for each pair of distinct
elements i, j ∈ J, there exists a geodesic arc bij connecting ai to aj and orthogonal to both,
which is disjoint from K.
Proof. Given two closed geodesics a, a0 in M , we consider the set π1 (M, a, a0 ) of relative
homotopy classes of paths in M connecting a and a0 , where the relative homotopy is defined
through paths connecting a to a0 .
In each class [b0 ] ∈ π1 (M, a, a0 ), there exists a continuous path b which is the length
minimizer in the class. By minimality of its length, b is a geodesic arc orthogonal to a and
a0 at its end-points.
For each compact subset K ⊂ M , there exists m ∈ N such that for all i ∈ Im := I∩[m, ∞),
ai ∩ K 0 = ∅ where K 0 = N7δ (K). For i ∈ Im let ci denote a shortest arc between ai and
K 0 ; this geodesic arc terminates a point xi ∈ K 0 . By compactness of K 0 , the sequence
(xi )i∈Im contains a convergent subsequence, (xi )i∈J , J ⊂ Im and, without loss of generality,
we may assume that for all i, j ∈ J, d(xi , xj ) ≤ δ. Let xi xj denote a (not necessarily unique)
geodesic in M of length ≤ δ connecting xi to xj . For each pair of indices i, j ∈ J, consider
the concatenation
b0ij = ci ∗ xi xj ∗ c−1
j ,
which defines a class [b0ij ] ∈ π1 (M, ai , aj ). Let bij ∈ [b0ij ] be the length-minimizing geodesic
arc in this relative homotopy class. Then bij is orthogonal to ai and aj . By δ-hyperbolicity
of X,
bij ⊆ N7δ (ai ∪ ci ∪ cj ∪ aj ).
Hence, bij ∩ K = ∅ for any pair of distinct indices i, j ∈ J. This proves the claim.
10
MICHAEL KAPOVICH AND BEIBEI LIU
We now prove the lemma. Assume inductively (by induction on N ) that we have constructed an infinite subset SN ⊂ N such that:
For the N -th element iN ∈ SN , for each j > iN , j ∈ SN , there exists a geodesic arc bj in
M connecting aiN to aj and orthogonal to both, which is disjoint from KN −1 .
Using the claim, we find an infinite subset SN +1 ⊂ SN which contains the first N elements
of SN , such that for all s, t > iN , s, t ∈ SN +1 , there exists a geodesic bs,t in M connecting
as to at , orthogonal to both and disjoint from KN .
The intersection
\
S :=
SN
N ∈N
equals {iN : N ∈ N} and, hence, is infinite. We, therefore, obtain a subsequence (ai )i∈S
such that for all i, j ∈ S, i < j, there exists a geodesic bij in M connecting ai to aj and
orthogonal to both, which is disjoint from Ki−1 .
Remark 4.2. It is important to pass a subsequence of (ai ), otherwise, the lemma is false.
A counter-example is given by a geometrically infinite manifold with two distinct ends E1
and E2 where we have a sequence of closed geodesics ai (escaping every compact subset of
M ) contained in E1 for odd i and in E2 for even i. Then bi will always intersect a compact
subset separating the two ends no matter what bi we take.
5. Elementary groups of isometries
Every isometry g of X extends to a homeomorphism (still denoted by g) of X̄. We let
Fix(g) denote the fixed point set of g : X̄ → X̄. For a subgroup Γ < Isom(X), we use the
notation
\
Fix(Γ) :=
Fix(g),
g∈Γ
to denote the fixed point set of Γ in X̄. Typically, this set is empty.
Isometries of X are classified as follows:
(1) g is parabolic if Fix(g) is a singleton {p} ⊂ ∂∞ X. In this case, g preserves (setwise)
every horosphere centered at p.
(2) g is loxodromic if Fix(g) consists of two distinct points p, q ∈ ∂∞ X. The loxodromic
isometry g preserves the geodesic pq ⊂ X and acts on it as a nontrivial translation.
The geodesic pq is called the axis Ag of g.
(3) g is elliptic if it fixes a point in X. The fixed point set of an elliptic isometry is a
totally-geodesic subspace of X invariant under g. In particular, the identity map is
an elliptic isometry of X.
If g ∈ Isom(X) is such that Fix(g) contains three distinct points ξ, η, ζ ∈ ∂∞ X, then g
also fixes pointwise the convex hull Hull({ξ, η, ζ}) and, hence, g is an elliptic isometry of X.
For each isometry g ∈ Isom(X) we define its translation length l(g) as follows:
l(g) = inf d(x, g(x)).
x∈X
Proposition 5.1. Let hgi < Isom(X) be a cyclic group generated by a loxodromic isometry
g with translation length l(g) ≥ > 0. Let γ denote the simple closed geodesic Ag /hgi in
M where M = X/hgi. If w ⊆ M is a piecewise-geodesic loop freely homotopic to γ which
consists of r geodesic
√ segments, then γ is contained in some C-neighborhood of the loop w
where C = cosh−1 ( 2)dlog2 re + sinh−1 (2/) + 2δ.
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
11
Figure 4.
Proof. Let x ∈ w be one of the vertices. Connect this point to itself by a geodesic segment
α in M which is homotopic to w (rel {x}). The loop w ∗ α−1 lifts to a polygonal loop β ⊆ X
with consecutive vertices x0 , x1 , · · · , xr so that the geodesic segment α̃ := x0 xr covers α.
Let w̃ denote the union of edges of β distinct from α̃. By Proposition 3.10,√
α̃ is contained
in the λ-neighborhood of the piecewise geodesic path w̃ where λ = cosh−1 ( 2)dlog2 re. It
follows that α ⊆ Nλ (w).
Let h = d(α̃, Ag ). Choose a point A ∈ α̃ which is nearest to Ag . Let B ∈ Ag be the
nearest point to A. Let F = ProjAg (xr ). Then we obtain a quadrilateral [ABF xr ] with
∠ABF = ∠BF xr = ∠BAxr = π/2. By Corollary 3.7,
d(B, F ) ≤ sinh(d(B, F )) ≤ 1/ sinh(h).
Take the point D ∈ Ag which is closest to x0 . By a similar argument, we have d(B, D) ≤
1/ sinh(h). So d(F, D) ≤ 2/ sinh(h). The projection P rojAg is hgi-equivariant, thus F, D
are identified by the isometry g. Hence
≤ d(D, g(D)) = d(D, F ) ≤ 2/ sinh(h)
−1
and h ≤ sinh (2/).
Let E ∈ Ag be the nearest point to g(A). Then π(BE) in M = X/hgi is the geodesic loop
γ where π is the covering projection. By δ-hyperbolicity of X, BE is within the (h + 2δ)neighborhood of the lifts of α as in Figure 4. Thus γ is√within the (sinh−1 (2/) + 2δ)−1
neighborhood of α. Since α is contained
√ in the (cosh (−1 2)dlog2 re)-neighborhood of w,
−1
the loop γ is contained in the (cosh ( 2)dlog2 re + sinh (2/) + 2δ)-neighborhood of w.
A discrete subgroup Γ of isometries of X is called elementary if either Fix(Γ) 6= ∅ or if
Γ preserves set-wise some bi-infinite geodesic in X. (In the latter case, Γ contains an index
2 subgroup Γ0 such that Fix(Γ0 ) 6= ∅.) We are particularly interested in the following two
types of elementary subgroups.
Definition 5.2. A discrete elementary subgroup Γ < Isom(X) is parabolic if it contains a
parabolic isometry g and Fix(g) = Fix(Γ) = {p} ⊆ ∂∞ X.
Remark 5.3. Such Γ preserves setwise every horosphere centered at p. Thus, every parabolic subgroup consists of parabolic and elliptic elements.
Definition 5.4. A discrete elementary subgroup Γ < Isom(X) is loxodromic if it contains
a loxodromic element and preserves setwise its axis A.
Thus, every loxodromic subgroup Γ consists of loxodromic elements with the axis A and
elliptic elements.
12
MICHAEL KAPOVICH AND BEIBEI LIU
Consider a subgroup Γ of isometries of X. Given any subset Q ⊆ X̄, let
stabΓ (Q) = {γ ∈ Γ | γ(Q) = Q}
denote the setwise stabilizer of Q.
Definition 5.5. A point p ∈ ∂∞ X is called a parabolic fixed point of a subgroup Γ <
Isom(X) if stabΓ (p) is parabolic.
Remark 5.6. If p ∈ ∂∞ X is a parabolic fixed point of a discrete subgroup Γ < Isom(X),
then stabΓ (p) is a maximal parabolic subgroup of Γ, see [9, Proposition 3.2.1]. Thus, we
have a bijective correspondence between the Γ-orbits of parabolic fixed points of Γ and
Γ-conjugacy classes of maximal parabolic subgroups of Γ.
Consider an elementary loxodromic subgroup G < Γ with the axis β. Then stabΓ (β) is a
maximal loxodromic subgroup of Γ, see [9, Proposition 3.2.1].
Observe that the all isometries of finite order are elliptic and that a discrete subgroup
Γ < Isom(X) cannot contain elliptic elements of infinite order. Thus, a torsion-free discrete
subgroup Γ contains no elliptic elements besides the identity.
6. The thick-thin decomposition
Given p ∈ X, ε > 0, consider the set
Fε (p) = {γ ∈ Isom(X) | d(p, γp) ≤ ε}.
Given Γ < Isom(X), let Γε (p) = hΓ ∩ Fε (p)i denote the subgroup generated by all elements
γ ∈ Γ which move p a distance at most ε. Define the set Tε (Γ) = {p ∈ X | Γε (p) is infinite}.
It is a closed Γ-invariant subset of X.
Proposition 6.1 (The Margulis Lemma). There is a constant ε(n, κ) > 0 such that if
Γ < Isom(X) is discrete and p ∈ X, then Γε (p) is virtually nilpotent for all ε ≤ ε(n, κ).
Here, ε(n, κ) depends only on the dimension n of X and the lower curvature bound −κ2 .
See e.g. [3].
Remark 6.2. The constant ε(n, κ) is called the Margulis constant.
Lemma 6.3. Suppose that G < Isom(X) is a discrete parabolic subgroup and ε > 0. For
any z ∈ Tε/3 (G), we have B(z, ε/3) ⊆ Tε (G).
Proof. The set Fε/3 (z) = {γ ∈ G|d(z, γ(z)) ≤ ε/3} generates an infinite subgroup of G
since z ∈ Tε/3 (G). For any element γ ∈ Fε/3 (z) and z 0 ∈ B(z, ε/3), we have
d(z 0 , γ(z 0 )) ≤ d(z, z 0 ) + d(z, γ(z)) + d(γ(z), γ(z 0 )) ≤ ε/3 + ε/3 + ε/3 = ε.
So Fε (z 0 ) = {γ ∈ G|d(z 0 , γ(z 0 )) ≤ ε} also generates an infinite subgroup. Thus z 0 ∈ Tε (Gi )
and B(z, ε/3) ⊆ Tε (G).
Proposition 6.4. [9, Proposition 3.5.2] Suppose G < Isom(X) is a discrete parabolic subgroup with the fixed point p ∈ ∂∞ X, and ε > 0. Then Tε (G) ∪ {p} is starlike about p, i.e.
for each x ∈ X̄ \ {p}, the intersection xp ∩ Tε (G) is a ray asymptotpic to p.
Corollary 6.5. Suppose that G < Isom(X) is a discrete parabolic subgroup with the fixed
point p ∈ ∂∞ X. For every ε > 0, Tε (G) is a δ-quasiconvex subset of X.
Proof. By Proposition 6.4, Tε (G)∪{p} is starlike about p. Every starlike set is δ-quasiconvex,
[9, Corollary 1.1.6]. Thus Tε (G) is δ-quasiconvex for every discrete parabolic subgroup
G < Isom(X).
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
13
Remark 6.6. According to Proposition 3.14, there exists r ∈ [0, ∞) such that Hull(Tε (G)) ⊆
Nr (Tε (G)) for any ε > 0 and r depends only on κ.
Lemma 6.7. If G < Isom(X) is a discrete parabolic subgroup with the fixed point p ∈ ∂∞ X,
then ∂∞ Tε (G) = {p}.
Proof. By Lemma 3.11(2), for any p0 ∈ ∂∞ X \ {p}, both p0 p ∩ Tε (G) and X ∩ (p0 p \ Tε (G))
are nonempty [9, Proposition 3.5.2]. If p0 ∈ ∂∞ Tε (G), there exists a sequence of points
(xi ) ⊆ Tε (G) which converges to p0 . By Proposition 6.4, xi p ⊆ Tε (G). Since Tε (G) is closed
in X, then p0 p ⊆ Tε (G) which is a contradiction.
Proposition 6.8. Suppose that G < Isom(X) is a discrete parabolic subgroup with the fixed
point p ∈ ∂∞ X. Given r > 0 and x ∈ X with d(x, Hull(Tε (G))) = r, if (xi ) is a sequence
of points on the boundary of Nr (Hull(Tε (G))) and d(x, xi ) → ∞, then there exists zi ∈ xxi
such that the sequence (zi ) converges to p and for every ε > 0, zi ∈ Nδ (Tε (G)) for all
sufficiently large i.
Figure 5.
Proof. By δ-hyperbolicity of X, there exists a point zi ∈ xxi such that d(zi , px) ≤ δ and
d(zi , pxi ) ≤ δ. Let wi ∈ pxi and vi ∈ px be the points closest to zi , see Figure 5. Then
d(zi , wi ) ≤ δ, d(zi , vi ) ≤ δ and, hence, d(wi , vi ) ≤ 2δ.
According to Lemma 6.7, the sequence (xi ) converges to the point p. Hence, any sequence
of points on xi p converges to p as well; in particular, (wi ) converges to p. As d(wi , zi ) ≤ δ,
we also obtain
lim zi = p.
i→∞
Since d(zi , vi ) ≤ δ, it suffices to show that vi ∈ Tε (G) for all sufficiently large i. This
follows from the fact that d(x, vi ) → ∞ and that xp ∩ Tε (G) is a geodesic ray asymptotic
to p.
Given 0 < ε < ε(n, κ) and a discrete subgroup Γ, the set Tε (Γ) is a disjoint union of the
subsets of the form Tε (G), where G ranges over all maximal infinite elementary subgroups
of Γ, [9, Proposition 3.5.5]. For the quotient orbifold M = X/Γ, set
thinε (M ) = Tε (Γ)/Γ.
14
MICHAEL KAPOVICH AND BEIBEI LIU
This closed subset is the thin part3 of the quotient orbifold M . The thin part is a disjoint
union of its connected components, and that each such component has the form Tε (G)/G
where G ranges over all maximal infinite elementary subgroups of Γ. If G < Γ is a maximal
parabolic subgroup, Tε (G)/G is called a Margulis cusp. If G < Γ is a maximal loxodromic
subgroup, Tε (G)/G is called a Margulis tube.
The closure of the complement M/thinε (M ) is the thick part of M , denoted by thickε (M ).
Let cuspε (M ) denote the union of all Margulis cusps of M ; it is called the cuspidal part of
M . The closure of the complement M \cuspε (M ) is denoted by noncuspε (M ); it is called the
noncuspidal part of M . Observe that cuspε (M ) ⊆ thinε (M ) and thickε (M ) ⊆ noncuspε (M ).
If M is a manifold (i.e., Γ is torsion-free), the ε-thin part is also the collection of all points
x ∈ M where the injectivity radius of M at x is no greater than ε/2.
7. Quasigeodesics
In this section, X is a Hadamard manifold with sectional curvatures ≤ −1. We will
prove that certain concatenations of geodesics in X are uniform quasigeodesics, therefore,
according to the Morse Lemma, are uniformly close to geodesics.
Lemma 7.1. Let γ = γ1 ∗ · · · ∗ γn ⊆ X̄ be a piecewise geodesic path from x to y where each
γi is a geodesic. Assume that for each i, the length of γi is λi and for 1 ≤ i ≤ n − 1, the
angle between γi and γi+1 is αi . If for all i, λi ≥ L > 1 and cosh(L/2) sin(αi /2) > 1, then
γ is a (2L, 4L + 1)-quasigeodesic.
Proof. Let Bis(xi , xi+1 ) denote the perpendicular bisector of γi = xi xi+1 where x1 = x and
xn+1 = y. If the closures in X̄ of the bisectors Bis(xi , xi+1 ) and Bis(xi+1 , xi+2 ) intersect
each other, then we have the following quadrilateral [ABCD] with ∠DAB = ∠DCB = π/2
as in Figure 6(a), where B ∈ X̄. Connecting D, B by a geodesic segment (or a ray), we
get two right triangles [ADB] and [BCD], and one of the angles ∠ADB, ∠CDB is ≥ αi /2.
Without loss of generality, we can assume that ∠ADB ≥ αi /2. By Corollary 3.5 and
Remark 3.6, cosh(d(A, D)) sin ∠ADB ≤ 1. However, we know that
cosh(d(A, D)) sin ∠ADB ≥ cosh(L/2) sin(αi /2) > 1
which is a contradiction. Thus, the closures of Bis(xi , xi+1 ) and Bis(xi+1 , xi+2 ) are disjoint.
Figure 6.
Let C ∈ Bis(xi , xi+1 ), D ∈ Bis(xi+1 , xi+2 ) denote points (not necessarily unique) such
that d(C, D) minimizes the distance function between the points of these perpendicular
bisectors. Since CB ⊂ Bis(xi , xi+1 ), DE ⊂ Bis(xi+1 , xi+2 ), it follows that the segment
3more precisely, ε-thin part
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
15
CD is orthogonal to both CB and DE. The segment CD lies in a unique (up to reparameterization) bi-infinite geodesic ξη. Then A ∈ NP (ξ, η) for some point P ∈ ξη. We
claim that P ∈ CD. Otherwise, we obtain a triangle in X with two right angles which is a
contradiction. So the geodesic AP ⊆ NP (C, D) and AP is orthogonal to CD as in Figure
6(b). We get two quadrilaterals [ABCP ] and [AP DE]. Without loss of generality, assume
that ∠BAP ≥ αi /2. By Corollary 3.7,
d(C, D) ≥ d(C, P ) ≥ cosh(L/2) sin(αi /2) ≥ 1.
Now we prove that the piecewise geodesic path γ is a quasigeodesic. For each i, if
d(xi , xi+1 ) ≥ 2L, take the point yi1 ∈ γi such that d(xi , yi1 ) = L. If L ≤ d(yi1 , xi+1 ) < 2L,
we’ll stop. Otherwise, take the point yi2 ∈ γi such that d(yi1 , yi2 ) = L. If d(yi2 , xi+1 ) ≥ 2L,
we continue the process until we get yij such that L ≤ d(yij , xi+1 ) < 2L. So we get a
piecewise geodesic path γ = γ10 ∗ · · · ∗ γn0 0 satisfying the properties that the hyperbolic length
of each geodesic arc γi0 is no less than L and less than 2L, and adjacent geodesic arcs γi0 and
0
γi+1
meet at an angle either αi or π as in Figure 7(a).
Figure 7.
In order to prove that γ is a quasigeodesic, it suffices to show that there exist constants
λ and such that
1
length(γ|[ta ,tb ] ) − ≤ d(a, b) ≤ λ · length(γ|[ta ,tb ] ) +
λ
for any two points a, b ∈ γ where γ(ta ) = a and γ(tb ) = b. Suppose that a, b are both
endpoints of some geodesic arcs γi0 , γj0 as in Figure 7(b). The bisectors of the geodesic
segments in γ divide ab into several pieces, and each piece has hyperbolic length at least 1
except for the first piece and the last piece. So d(a, b) > j − i, while
(j − i + 1)L ≤ length(γ|[ta ,tb ] ) < 2(j − i + 1)L.
Let λ = 2L and ≥ 1. We have
1
length(γ|[ta ,tb ] ) − ≤ d(a, b).
λ
If at least one of a, b is not the endpoint of any geodesic arc γi0 , without loss of generality,
assume that a lies in the interior of some geodesic arc γi0 = x0i x0i+1 , and b ∈ γj0 = x0j x0j+1 as
16
MICHAEL KAPOVICH AND BEIBEI LIU
in Figure 7(c). Then we have
d(x0i , x0j+1 ) < d(a, b) + 4L
since d(x0i , a) < 2L and d(b, x0j+1 ) < 2L. By the previous argument, we have the following
inequalities
1
1
length(γ|[ta ,tb ] ) − ≤ length(γ|[ti ,tj+1 ] ) − ≤ d(x0i , x0j+1 ) ≤ d(a, b) + 4L.
λ
λ
where γ(ti ) = x0i and γ(tj+1 ) = x0j+1 . Thus let λ = 2L and = 4L + 1. For any two points
a, b ∈ γ, we have
1
length(γ|[ta ,tb ] ) − ≤ d(a, b) ≤ λ · length(γ|[ta ,tb ] ) + .
λ
Therefore γ is a (2L, 4L + 1)-quasigeodesic.
Proposition 7.2. Given θ > 0, there exist constants C, L < ∞ such that the following
holds. Suppose that γ = γ1 ∗ · · · ∗ γn ⊆ X̄ is a piecewise geodesic path from x to y. Assume
that each geodesic arc γi has length at least L, and adjacent geodesic arcs meet at an angle
≥ θ. Then the Hausdorff distance between the path γ and the geodesic xy is no greater than
C. Here C, L depend only on κ and θ.
Proof. We can choose L > 0 such that cosh(L/2) sin(θ/2) > 1. By Lemma 6.1, the piecewise
geodesic path γ is a (2L, 4L + 1)-quasigeodesic. So there is a constant C = C(2L, 4L + 1)
such that the Hausdorff distance between the piecewise geodesic path γ and the geodesic
xy is no greater than C [11, Lemma 9.38, Lemma 9.80].
Proposition 7.3 (Generalized version). Given θ, ε > 0, there exist constants C, L < ∞
such that the following holds. Suppose that γ = γ1 ∗ · · · ∗ γn ⊆ X̄ is a piecewise geodesic
path from x to y such that:
(1) Each geodesic arc γj has length either at least ε or at least L.
(2) If γj has length < L, then the adjacent geodesic arcs γj−1 and γj+1 have lengths at
least L and γj meets γj−1 and γj+1 at angles ≥ π/2.
(3) Other adjacent geodesic arcs meet at an angle ≥ θ.
Then the Hausdorff distance between γ and the geodesic xy is no greater than C. Here L
and C depend only on θ, ε and κ.
Figure 8.
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
17
Proof. Assume that γj has length < L for some j. Let Bis(xj−1 , xj ) and Bis(xj+1 , xj+2 )
be the perpendicular bisectors of γj−1 and γj+1 as in Figure 8. We claim that the closures of these bisectors in X̄ do not intersect each other. If they intersect, consider the
pentagon [ABCDE] where E ∈ X̄. The geodesic segment BC lies in a unique (up to
reparameterization) bi-infinite geodesic ξη. Then E ∈ NP (ξ, η) for some point P ∈ ξη.
Assume that P ∈ Bξ. Consider the quadrilateral [EP CD] in Figure 8(a). Then d(C, P ) ≥
ε. By Corollary 3.7 and Remark 3.8, sinh(d(C, P )) sinh(d(C, D)) ≤ 1. Therefore,
sinh(L/2) ≤ sinh(d(C, D)) ≤ 1/ sinh(ε)
which is a contradiction if we choose L > 2 arcsinh(1/ sinh(ε)). If P ∈ Cη, we consider the
quadrilateral [EP BA] and use a similar argument to get a contradiction. So P ∈ BC and
we have two quadrilaterals [AEP B] and [EP CD] as in Figure 8(b). Since d(B, C) ≥ ε,
one of d(B, C) and d(C, P ) has length at least ε/2. Without loss of generality, assume that
d(C, P ) ≥ ε/2. In the quadrilateral [EP CD], sinh(d(C, P )) sinh(d(C, D)) ≤ 1 by Corollary
3.7 and Remark 3.8. Therefore,
sinh(L/2) ≤ sinh(d(C, D)) ≤ 1/ sinh(ε/2).
This is a contradiction for L > 2 arcsinh(1/ sinh(ε/2)). Thus the closures of Bis(xj−1 , xj )
and Bis(xj+1 , xj+2 ) do not intersect each other by choosing L > 2 arcsinh(1/ sinh(ε/2)).
Let E ∈ Bis(xj−1 , xj ), F ∈ Bis(xj+1 , xj+2 ) denote points (not necessarily unique) such
that d(E, F ) minimizes the distance function between the points of these perpendicular
bisectors. The segment EF is orthogonal to both AF and DE, see Figure 8(c). Consider
the hexagon [ABCDEF ]. The segment EF lies in a unique (up to reparameterization)
bi-infinite geodesic ζθ. Then the midpoint H of the geodesic segment BC lies in NG (ζ, θ)
for some point G ∈ ζθ. We claim that G ∈ EF . Otherwise, we obtain a triangle in X with
two right angles which is contradiction. So HG ⊆ NG (E, F ) and HG is orthogonal to EF
at G as in Figure 8(c). Without loss of generality, assume that ∠BHG ≥ π/2. Consider the
pentagon [ABHGF ]. By Corollary 3.9, d(F, G) → ∞ as d(A, B) → ∞. For each positive
constant α, we can choose sufficiently large L < ∞ such that d(E, F ) ≥ α. By a similar
argument as in the proof of Proposition 7.2, we can show that γ is a uniform quasigeodesic
and there exists a constant C such that the Hausdorff distance hd(γ, xy) is no greater than
C by the Morse Lemma [11, Lemma 9.38, Lemma 9.80].
8. Loxodromic products
In order to prove our generalization of Bonahon’s theorem, we need to construct a loxodromic element with uniformly bounded word length in hf, gi where f, g are two parabolic
isometries generating a discrete nonelementary subgroup of Isom(X).
Lemma 8.1. [16, Theorem Σm ] Let F = {A1 , A2 , · · · , Am } be a family of open subsets of an
n-dimensional topological space X. If for every subfamily F 0 of size j where 1 ≤ j ≤ n + 2,
the intersection ∩F 0 is nonempty and contractible, then the intersection ∩F 6= ∅.
Proof. This lemma is a special case of the topological Helly theorem [16]. Here we give
another proof of the lemma. Suppose k is the smallest integer such that there exists a
subfamily F 0 = {Ai(1) , Ai(2) , · · · , Ai(k) } with size k with empty intersection ∩F 0 = ∅. By
the assumption, k ≥ n + 3. Then
[
U :=
Ai(j)
1≤j≤k
18
MICHAEL KAPOVICH AND BEIBEI LIU
is homotopy equivalent to the nerve N (F 0 ) [13, Corollary 4G.3], which, in turn, is homotopy
equivalent to S k−2 . Then Hk−2 (S k−2 ) ∼
= Hk−2 (U ) ∼
= Z which is a contradiction since
k − 2 ≥ n + 1 and X has dimension n.
Proposition 8.2. Let X be a δ-hyperbolic n-dimensional Hadamard space. Suppose that
B1 , · · · , Bk are convex subsets of X such that Bi ∩ Bj 6= ∅ for all i and j. Then there is a
point x ∈ X such that d(x, Bi ) ≤ nδ for all i = 1, ..., k.
Proof. For k = 1, 2, the lemma is clearly true.
We first claim that for each 3 ≤ k ≤ n + 2, there exists a point x ∈ X such that
d(x, Bi ) ≤ (k − 2)δ. We prove the claim by induction on k. When k = 3, pick points
xij ∈ Bi ∩ Bj , i 6= j. Then xij xil ⊂ Bi for all i, j, l. Since X is δ-hyperbolic, there exists
a point x ∈ X within distance ≤ δ from all three sides of the geodesic triangle [x12 x23 x31 ].
Hence,
d(x, Bi ) ≤ δ, i = 1, 2, 3
as well.
Assume that the claim holds for k − 1. Set Bi0 = Nδ (Bi ) and Ci = Bi0 ∩ B1 where
i ∈ {2, 3, · · · , k}. By convexity of the distance function on X, each Bi0 is still convex in X
and, hence, is a Hadamard space. Furthermore, each Bi0 is again δ-hyperbolic.
We claim that Ci ∩ Cj 6= ∅ for all i, j ∈ {2, 3, · · · , k}. By the nonemptyness assumption,
there exist points x1i ∈ B1 ∩ Bi 6= ∅, x1j ∈ B1 ∩ Bj 6= ∅ and xij ∈ Bi ∩ Bj 6= ∅. By δhyperbolicity of X, there exists a point y ∈ x1i x1j such that d(y, x1i xij ) ≤ δ, d(y, x2j xij ) ≤
δ.
Therefore, y ∈ B1 ∩ Nδ (Bi ) ∩ Nδ (Bj ) = Ci ∩ Cj . By the induction hypothesis, there exists
a point x0 ∈ X such that d(x0 , Ci ) ≤ (k − 3)δ for each i ∈ {2, 3, · · · , k}. Thus,
d(x0 , Bi ) ≤ (k − 2)δ, i ∈ {1, 2, · · · , k}
as required.
For k > n + 2, set Ui = Nnδ (Bi ). Then by the claim, we know that for any subfamily
of {Ui } of size j where 1 ≤ j ≤ n + 2, its intersection is nonempty and the intersection is
contractible since it is convex. By Lemma 8.1, the intersection of the family {Ui } is also
nonempty. Let x be a point in this intersection. Then d(x, Bi ) ≤ nδ for all i ∈ {1, 2, · · · , k}.
Proposition 8.3. There exists a function k : R+ × R+ → N with the following property.
Let g1 , g2 , · · · , gk be parabolic elements in a discrete subgroup Γ < Isom(X). For each gi let
Gi < Γ be the unique maximal parabolic subgroup containing gi , i.e. Gi = stabΓ (pi ), where
pi ∈ ∂∞ X is the fixed point of gi . Suppose that
Tε (Gi ) ∩ Tε (Gj ) = ∅
for all i 6= j. Then, whenever k ≥ k(D, ε), there exists a pair of indices i, j with
d(Tε (Gi ), Tε (Gj )) > D.
Proof. For each i, Hull(Tε (Gi )) is convex and by Remark 6.6, Hull(Tε (Gi )) ⊆ Nr (Tε (Gi )),
for some uniform constant r = r(κ). Suppose that g1 , g2 , · · · , gk and D are such that for all
i and j,
d(Tε (Gi ), Tε (Gj )) ≤ D.
Then d(Hull(Tε (Gi )), Hull(Tε (Gj ))) ≤ D.
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
19
Our goal is to get a uniform upper bound on k. Consider the D/2-neighborhoods
ND/2 (Hull(Tε (Gi ))). They are convex in X and have nonempty pairwise intersections.
Thus, by Proposition 8.2, there is a point x ∈ X such that
D
+ r, i = 1, ..., k.
d(x, Tε (Gi )) ≤ R1 := nδ +
2
Then
Tε (Gi ) ∩ B(x, R1 ) 6= ∅, i = 1, ..., k.
Next, we claim that there exists R2 ≥ 0, depending only on ε, such that
Tε (Gi ) ⊆ NR2 (Tε/3 (Gi )).
Choose any point y ∈ Tε (Gi ) and let ρi : [0, ∞) → X be the ray ypi . By Lemma 3.11,
there exists an absolute constant R = R() such that
d(ρi (t), g(ρi (t))) ≤ Re−t
whenever g ∈ Gi is a parabolic (or elliptic) isometry such that
d(y, g(y)) ≤ ε.
Let t = max{ln(3R/ε), 0}. Then d(ρi (t), g(ρi (t))) ≤ ε/3. So Tε (Gi ) ⊆ Nt (Tε/3 (Gi )) for
any i. Let R2 = t. By the argument above, B(x, R1 + R2 ) ∩ Tε/3 (Gi ) 6= ∅ for any i. Assume
that zi ∈ B(x, R1 + R2 ) ∩ Tε/3 (Gi ). Then B(zi , ε/3) ⊆ B(x, R3 ) where R3 = R1 + R2 + ε/3.
By Lemma 6.3, B(zi , ε/3) ⊆ B(x, R3 ) ∩ Tε (Gi ). Since Tε (Gi ) and Tε (Gj ) have empty
intersection for all i 6= j, B(zi , ε/3) and B(zj , ε/3) are disjoint. Let V (r, n) be the volume
of the uniform r-ball in Hn . Then for each i, B(zi , ε/3) has volume at least V (ε/3, n) [9,
Proposition 1.1.12]. The volume of B(x, R3 ) is at most V (κR3 , n)/κn , see [9, Propostion
V (κR3 , n)/κn
+ 1. If k ≥ k(D, ε), we obtain a contradiction. Thus
1.2.4]. Let k(D, ε) =
V (ε/3, n)
for k ≥ k(D, ε), there exist i and j such that d(Tε (Gi ), Tε (Gj )) > D.
Figure 9.
Proposition 8.4. Suppose that g1 , g2 are two parabolic elements. There exists a constant L
which only depends on ε, κ such that if d(Tε (g1 ), Tε (g2 )) > L, then h = g2 g1 is loxodromic.
Proof. Let Bi = Tε (gi ), so d(B1 , B2 ) > L. Consider the orbits of B1 and B2 under the
action of the cyclic group generated by g2 g1 as in Figure 10. Let x0 ∈ B1 , y0 ∈ B2 denote
points such that d(x0 , y0 ) minimizes the distance function between points of B1 and B2 .
For positive integers m > 0, we let
x2m−1 = (g2 g1 )m−1 g2 (x0 ),
x2m = (g2 g1 )m (x0 )
20
MICHAEL KAPOVICH AND BEIBEI LIU
and
y2m−1 = (g2 g1 )m−1 g2 (y0 ),
Similarly, for negative integers m < 0, we let
y2m = (g2 g1 )m (y0 ).
x2m+1 = (g2 g1 )m+1 g1−1 (x0 ),
x2m = (g2 g1 )m (x0 )
and
y2m+1 = (g2 g1 )m+1 g1−1 (y0 ), y2m = (g2 g1 )m (y0 ).
We construct a sequence of piecewise geodesic paths {γm } where γm = x−2m y−2m ∗
y−2m y−2m+1 · · · ∗ x0 y0 ∗ y0 y1 ∗ y1 x1 · · · ∗ x2m y2m for any positive integer m. Observe that
d(xi , yi ) = d(B1 , B2 ) > L and d(x2i−1 , x2i ) = ε, d(y2i , y2i+1 ) = ε for any integer i. By
convexity of B1 , B2 , the angle between any adjacent geodesic arcs in γm is at least π/2.
Let γ denote the limit of the sequence (γm ). By Proposition 7.3, there exists a constant
L > 0 such that the piecewise geodesic path γ : R → X is unbounded and is a uniform
quasigeodesic invariant under the action of h. By the Morse Lemma [11, Lemma 9.38,
Lemma 9.80], the Hausdorff distance between γ and the complete geodesic which connects
the endpoints of γ is bounded by a uniformly constant C. So g2 g1 fixes the endpoints of γ
and acts on the complete geodesic as a translation. Thus g2 g1 is loxodromic.
Figure 10.
Theorem 8.5. Suppose that g1 , g2 are two parabolic elements with different fixed points.
Then there exists a word w ∈ hg1 , g2 i such that l(w) ≤ 4k(L, ε) + 2 and w is loxodromic
where l(w) denotes the length of the word and k(L, ε) is the function in Proposition 8.3,
0 < ε < ε(n, κ) and L is the constant in Proposition 8.4.
Proof. Let pi denote the fixed point of the parabolic element gi where i = 1 or 2.
Assume that any element in hg1 , g2 i with word length no greater than 2k(L, ε)+1 is parabolic. Otherwise, there exists a loxodromic element w ∈ hg1 , g2 i with length ≤ 4k(L, ε) + 2.
Consider the parabolic elements g2i g1 g2−i ∈ hg1 , g2 i, 0 ≤ i ≤ k(L, ε). The fixed point of
each g2i g1 g2−i is g2i (p1 ). We claim that the points g2i (p1 ) and g2j (p1 ) are distinct for i 6= j. If
not, g2i (p1 ) = g2j (p1 ) for some i > j. Then g2i−j (p1 ) = p1 , and, thus, g2i−j has two distinct
fixed points p1 and p2 . This is a contradiction since any parabolic element has only one fixed
point. Thus, g2i g1 g2−i are parabolic elements with distinct fixed points for all 0 ≤ i ≤ k(L, ε).
Since 0 < ε < ε(n, κ), Tε (g2i g1 g2−i ), Tε (g2j g1 g2−j ) are disjoint for any pair of indices i, j [9]. By
Proposition 8.3, there exist 0 ≤ i, j ≤ k(L, ε) such that d(Tε (g2i g1 g2−i ), Tε (g2j g1 g2−j )) > L. By
Proposition 8.4, the element g2j g1 g2i−j g1 g2−i ∈ hg1 , g2 i is loxodromic, and its word length is no
greater than 4k(L, ε)+2. Thus we can find a word w ∈ hg1 , g2 i such that l(w) ≤ 4k(L, ε)+2
and w is loxodromic.
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
21
9. A generalization of Bonahon’s theorem
In this section, we use the construction in Section 8 to generalize Bonahon’s theorem for
any torsion-free discrete subgroup Γ < Isom(X) where X is a negatively pinched Hadamard
manifold.
Lemma 9.1. For every x̃ ∈ Hull(Λ(Γ)),
hd(QHull(Γx̃), QHull(Λ(Γ))) < ∞
Proof. By the assumption that x̃ ∈ Hull(Λ(Γ)) and Remark 3.15, there exists a universal
constant r1 = r(κ) ∈ [0, ∞) such that
QHull(Γx̃) ⊆ Hull(Λ(Γ)) ⊆ Nr1 (QHull(Λ(Γ)))
Next, we want to prove that there exists a constant r2 ∈ [0, ∞) such that QHull(Λ(Γ)) ⊆
Nr2 (QHull(Γx̃)).
Pick any point p ∈ QHull(Λ(Γ)). Then p lies on some geodesic ξη where ξ, η ∈ Λ(Γ) are
distinct points. Since ξ and η are in the limit set, there exist sequences of elements (fi ) and
(gi ) in Γ such that the sequence (fi (x̃)) converges to ξ and the sequence (gi (x̃)) converges
to η. By Lemma 3.17, p ∈ N2δ (fi (x̃)gi (x̃)) for all sufficiently large i. Let r = max{r1 , 2δ}.
Then hd(QHull(Γx̃), QHull(Λ(Γ))) = r < ∞.
Remark 9.2. Let γi = fi (x̃)gi (x̃). Then there exists a sequence of points (pi ) such that
pi ∈ γi and the sequence (pi ) converges to p.
If Γ < Isom(X) is geometrically infinite, then
Core(M ) ∩ noncuspε (M )
is noncompact, [9]. By Lemma 9.1, (QHull(Γx̃)/Γ) ∩ noncuspε (M ) is unbounded.
Now we generalize Bonahon’s theorem for any geometrically infinite torsion-free discrete
subgroup Γ < Isom(X) :
Proof of the implication (1) ⇒ (2) in Theorem 1.5: If there exists a sequence of closed
geodesics βi ⊆ M whose lengths go to 0 as i → ∞, then the sequence (βi ) escapes every
compact subset of M . From now on, we assume that there exists a constant > 0 such
that the length l(β) ≥ for any closed geodesic β in M .
Recall that a Margulis cusp is denoted by Tε (G)/G where G < Γ is a maximal parabolic
subgroup. There exists a universal constant r ∈ [0, ∞) such that Hull(Tε (G)) ⊆ Nr (Tε (G))
for any maximal parabolic subgroup G (see Section 5). Let B(G) = N2+4δ (Hull(Tε (G))).
Let M o be the union of all subsets B(G)/Γ where G ranges over all maximal parabolic
subgroups of Γ. We let M c denote the closure of Core(M ) \ M o . Since Γ is geometrically
infinite, the noncuspidal part of the convex core Core(M ) \ cuspε (M ) is unbounded by
Theorem 1.4. Then M c is also unbounded since M o ⊆ Nr+2+4δ (cuspε (M )).
Fix a point x ∈ M c . Let Cn = B(x, nR) = {y ∈ M c | d(x, y) ≤ nR} where R =
r + 2 + 4δ + ε. Let x̃ be a lift of x in X. By Lemma 9.1 (QHull(Γx̃)/Γ) ∩ M c is unbounded.
For every Cn , there exists a sequence of geodesic loops (γi ) connecting x to itself in Core(M )
such that the Hausdorff distance hd(γi ∩ M c , Cn ) → ∞ as i → ∞. Let yi ∈ γi ∩ M c be such
that d(yi , Cn ) is maximal on γi ∩ M c . We pick a component αi of γi ∩ M c such that yi ∈ αi .
c
c
Let δCn denote the relative boundary ∂Cn \ ∂Mcusp
of Cn where Mcusp
= M o ∩ Core(M ).
Consider the sequence of geodesic arcs (αi ).
After passing to a subsequence in (αi ), one of the following three cases occurs:
c
Case (a): Each αi has both endpoints x0i and x00i on ∂Mcusp
as in Figure 11(a). By
0
00
construction, there exist yi and yi in the cuspidal part such that d(x0i , yi0 ) ≤ r1 , d(yi0 , yi00 ) ≤ r1
where r1 = 2 + 4δ + r. Then we find short nontrivial geodesic loops αi0 , αi00 contained in the
22
MICHAEL KAPOVICH AND BEIBEI LIU
cuspidal part cuspε (M ) such that αi0 connects yi0 to itself and αi00 connects yi00 to itself and
the lengths l(αi0 ) ≤ ε, l(αi00 ) ≤ ε. Let
w0 = x0i yi0 ∗ αi0 ∗ yi0 x0i ∈ Ω(M, x0i )
and
w00 = αi ∗ x00i yi00 ∗ αi00 ∗ yi00 x00i ∗ αi−1 ∈ Ω(M, x0i )
where Ω(M, x0i ) denotes the loop space of M . Observe that w0 ∩Cn−1 = ∅ and w00 ∩Cn−1 = ∅.
Let g 0 , g 00 denote the elements of Γ = π1 (M, x0i ) represented by w0 and w00 respectively.
By the construction, g 0 and g 00 are both parabolic. We claim that g 0 and g 00 have different fixed points. Otherwise, g, g 00 ∈ G0 where G0 < Γ is some maximal parabolic
subgroup. Then yi0 , yi00 ∈ Tε (G0 )/Γ and x0i , x00i ∈ B(G0 )/Γ. Since Hull(Tε (G0 )) is convex,
B(G0 ) = N2+4δ (Hull(Tε (G0 ))) is also convex by convexity of the distance function. So
x0i x00i ⊆ B(G0 )/Γ. However, x0i x00i lies outside of B(G0 )/Γ by construction which is a contradiction.
By Theorem 8.5, there exists a loxordomic element ωn ∈ hg 0 , g 00 i < Γ = π1 (M, x0i ) with
the word length uniformly bounded by a constant C. Let wn be a concatenation of wi0 , wi00
and their reverses which represents ωn . Then the number of geodesic arcs in wn is uniformly
bounded by 5C. The piecewise geodesic loop wn is freely homotopic to a closed geodesic
wn∗ in M ; hence, by Proposition
5.1, wn∗ is contained in some D-neighborhood of the loop
√
−1
wn where D = cosh ( 2)dlog2 5Ce + sinh−1 (2/) + 2δ. Thus d(x, wn∗ ) ≥ (n − 1)R − D.
Figure 11.
c
Case (b): For each i, the geodesic arc αi connects x0i ∈ δCn to x00i ∈ ∂Mcusp
, as in Figure
00
00
00
11(b). For each xi , there exists a point yi ∈ cuspε (M ) such that d(xi , yi00 ) ≤ r1 and a
short nontrivial geodesic loop αi00 contained in the cuspidal part which connects yi00 to itself
and has length l(αi00 ) ≤ ε. Since δCn is compact, after passing to a further subsequence in
(αi ), there exists k ∈ N such that for all i ≥ k, d(x0i , x0k ) ≤ 1 and less than the injectivity
radius of M at x0k . Hence, there exists a unique shortest geodesic x0k x0i in the manifold
M . Let µi = x0k x00i denote the geodesic arc homotopic to the concatenation x0k x0i ∗ x0i x00i
rel. {x0i , x00i }. Then, by δ-hyperbolicity of X, the geodesic µi = x0k x00i is contained in the
(1 + δ)-neighborhood of αi .
Let
wk0 = αk ∗ x00k yk00 ∗ αk00 ∗ yk00 x00k ∗ αk−1 ∈ Ω(M, x0k )
and
wi0 = µi ∗ x00i yi00 ∗ αi00 ∗ yi00 x00i ∗ (µi )−1 ∈ Ω(M, x0k )
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
23
for all i > k. By the construction, wi0 ∩ Cn−1 = ∅ for each i ≥ k.
Let gi denote the element of Γ = π1 (M, x0k ) represented by wi0 , i ≥ k. Then each gi is
parabolic. We claim that there exists a pair of indices i, j ≥ k such that gi and gj have
distinct fixed points. Otherwise, assume that all parabolic elements gi have the same fixed
point p. Then x00i ∈ B(G0 )/Γ for any i ≥ k where G0 = StabΓ (p).
Since µi ∪ αk is in the (1 + δ)-neighborhood of M c , by δ-hyperbolicity of X we have that
00
xk x00i is in (1 + 2δ)-neighborhood of M c for every i > k. By the definition of M c , it follows
that
x00k x00i ∩ Nδ (Hull(Tε (G0 )))/Γ = ∅.
By the construction, the length l(αi ) → ∞ as i → ∞. Hence, the length l(µi ) → ∞ and
the length l(x00k x00i ) → ∞ as i → ∞. By Lemma 6.8, there exists points zi ∈ x00k x00i such that
zi ∈ Nδ (Tε (G0 ))/Γ for sufficiently large i. Therefore,
x00k x00i ∩ Nδ (Hull(Tε (G0 )))/Γ 6= ∅,
which is a contradiction.
We conclude that for some i, j ≥ k, the parabolic elements gi , gj of Γ have distinct fixed
points and, hence, generate a nonelementary subgroup of Isom(X). By Theorem 8.5, there
exists a loxodromic element ωn ∈ hgi , gj i with the word length uniformly bounded by a
constant C. By a similar argument as in Case (a), we obtain a closed geodesic wn∗ in M
such that d(x, wn∗ ) ≥ (n − 1)R − D.
Figure 12.
Case (c): We assume that for each i, the geodesic arc αi connects x0i ∈ δCn to x00i ∈ δCn .
The argument is similar to the one in Case (b). Since δCn is compact, after passing to
a further subsequence in (αi ), there exists k ∈ N such that for all i ≥ k, d(x0i , x0k ) ≤ 1,
d(x00i , x00k ) ≤ 1 and there are unique shortest geodesics x0k x0i and x00k x00i . For each i > k we
define a geodesic µi = x0k x00i as in Case (b), see Figure 12(a). Then, by δ-hyperbolicity of
X, each µi is in the (δ + 1)-neighborhood of αi . Let vi = αk ∗ x00k x00i ∗ (µi )−1 ∈ Ω(M, x0k ) for
i > k. By the construction vi ∩ Cn−1 = ∅.
Let hi denote the element in Γ = π1 (M, x0k ) represented by vi . If hi is loxodromic for
some i > k, there exists a closed geodesic wn∗ contained in the D-neighborhood of vi , cf.
Case (a). In this situation, d(x, wn∗ ) ≥ (n − 1)R − D.
Assume, therefore, that hi are not loxodromic for all i > k.
f0 be a lift of
We first claim that hi is not the identity for all sufficiently large i. Let x
k
f00 , x
f00 , xe0 and hi (x
f0 ) in X such that x
f0 x
f00 is a lift of αk , x
f00 x
f00
x0k in X. Pick points x
i
k i is a
k i
k
k k
24
MICHAEL KAPOVICH AND BEIBEI LIU
f00 is a lift of αi and xe0 hi (x
f0 ) is a lift of x0 x0 as in Figure 12(b) and Figure
lift of x00k x00i , xe0i x
i k
i
i
k
f0 ) = x
f0 and d(xe0 , x
f00 ) ≤ 2 + d(x
f0 , x
f00
12(c). If hi = 1, then hi (x
i i
k
k
k k ) as in Figure 12(b). By
f00 ) → ∞. Thus for sufficiently large
construction, the length l(αi ) → ∞ as i → ∞, so d(xe0i , x
i
f0 .
f0 ) 6= x
i, hi (x
k
k
Now we assume that hi are parabolic for all i > k 0 where k 0 > k is a sufficiently large
number. We claim that there exists a pair of indices i, j > k 0 such that hi and hj have
distinct fixed points. Otherwise, all the parabolic elements hi have the same fixed point p
f0 x
f00 f00 e0
f0 ) ⊆ N3δ+2 (x
f0 hi (x
for i > k 0 . By the δ-hyperbolicity of X, x
k k ∪ xi xi ). Since αk and αi
k
k
f0 ) lies outside of Nδ (Hull(Tε (G0 ))). Let
f0 hi (x
lie outside of B(G0 )/Γ where G0 = StabΓ (p), x
k
k
f0 , Hull(Tε (G0 ))). Then d(hi (x
f0 ), Hull(Tε (G0 ))) = r3 .
r3 = d(x
k
k
f0 hi (x
f0 )) → ∞
By the construction, the length l(αi ) → ∞ as i → ∞. Then the length l(x
k
k
0
0
f
f
as well. Observe that the points xk and hi (xk ) lie on the boundary of Nr3 (Hull(Tε (G)))
f0 hi (x
f0 ) such that zei ∈ Nδ (Tε (G0 ))
for all i > k 0 . By Lemma 6.8, there exist points zei ∈ x
k
k
for sufficiently large i, which is a contradiction. Hence, for some i > k 0 , j > k 0 , parabolic
isometries hi and hj have distinct fixed points.
By Theorem 8.5, there exists a loxodromic element ωn ∈ hhi , hj i of the word length
bounded by a uniform constant C. By a similar argument as in Case (a), there exists a
closed geodesic wn∗ such that d(x, wn∗ ) ≥ (n − 1)R − D.
Thus in all cases, for each n, the manifold M contains a closed geodesic wn∗ such that
d(x, wn∗ ) ≥ (n − 1)R − D. The sequence of closed geodesics {wn∗ }, therefore, escapes every
compact subset of M .
10. Continuum of nonconical limit points
In this section, using the generalized Bonahon theorem in Section 9, for each geometrically
infinite discrete torsion-free subgroup Γ < Isom(X) we find a set of nonconical limit points
with cardinality of continuum. This set of nonconical limit points is used to prove Theorem
1.5.
Theorem 10.1. If Γ < Isom(X) is a geometrically infinite discrete torsion-free isometry
subgroup, then the set of nonconical limit points of Γ has cardinality of continuum.
Proof. The proof is inspired by Bishop’s construction of nonconical limit points of geometrically infinite Kleinian groups in the 3-dimensional hyperbolic space H3 [6, Theorem
1.1]. Let π : X → M = X/Γ denote the covering projection. Pick a point x̃ ∈ X and let
x := π(x̃). If Γ is geometrically infinite, by the generalized Bonahon’s theorem in Section 9,
there exists a sequence of oriented closed geodesics (λi ) in M which escapes every compact
subset of M , i.e.
lim d(x, λi ) = ∞.
i→∞
Let L be the constant as in Proposition 7.2 when θ = π/2. Without loss of generality,
we assume that d(x, λ1 ) ≥ L and the minimal distance between any consecutive pair of
geodesics λi , λi+1 is at least L. For each i, let li denote the length of the closed geodesic λi
and let mi be a positive integer such that mi li > L.
We then pass to a subsequence in (λi ) as in Lemma 4.1 (retaining the notation (λi ) for
−
the subsequence) such that there exists a sequence of geodesic arcs µi := x+
i xi+1 meeting
λi , λi+1 orthogonally at its end-points, such that
lim d(x, µi ) = ∞.
i→∞
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
25
+
Let Di denote the length of the shortest positively oriented arc of λi connecting x−
i to xi .
We let µ0 denote the shortest geodesic in M connecting x to x−
1.
We next construct a family of piecewise geodesic paths γτ in M starting at x such that
the geodesic pieces of γτ are the arcs µi above and arcs νi whose images are contained in
λi and have the same orientation: Each νi wraps around λi certain number of times and
+
∞ → P (M ) where N∞ is the set
connects x−
i to xi . More formally, we define a map P : N
of sequences of positive integers and P (M ) is the space of paths in M as follows:
P : τ = (t1 , t2 , · · · , ti , · · · ) 7→ γτ = µ0 ∗ ν1 ∗ µ1 ∗ ν2 ∗ µ2 ∗ · · · ∗ νi ∗ µi ∗ · · ·
where the image of the geodesic arc νi is contained in λi and has length
l(νi ) = ti mi li + Di .
Observe that for i ≥ 1, the arc µi connects λi and λi+1 and is orthogonal to both, with
+
length l(µi ) ≥ L and νi starts at x−
i and ends at xi with length l(νi ) ≥ L.
For each γτ , we have a canonical lift γ̃τ in X, which is a path starting at x̃. We will use
the notation µ̃i , ν̃i for the lifts of the subarcs µi , νi respectively, see Figure 13(a, b). By the
construction, each γτ has the following properties:
(1) Each geodesic piece of γ̃τ has length at least L.
(2) Adjacent geodesic segments of γ̃τ make the angle equal to π/2 at their common
endpoint.
(3) The path γτ : [0, ∞) → M is a proper map.
By Proposition 7.2, γ̃τ is a (2L, 4L + 1)-quasigeodesic. Hence, there exists a limit
lim γ̃τ (t) = γ̃τ (∞) ∈ ∂∞ X,
t→∞
such that the Hausdorff distance between γ̃τ and xγ̃τ (∞) is bounded by a uniform constant
C, depending only on L and κ.
We claim that each γ̃τ (∞) is a nonconical limit point. Observe that γ̃τ (∞) is a limit of
loxodromic fixed points, so γ̃τ (∞) ∈ Λ(Γ). Let γτ∗ be the projection of xγ̃τ (∞) under π.
Then the image of γτ∗ is uniformly close to γτ . Since γτ is a proper path in M , so is γτ∗ .
Hence, γ̃τ (∞) is a nonconical limit point of Γ.
We claim that the set of nonconical limit points γ̃τ (∞), τ ∈ N∞ , has the cardinality of
continuum. It suffices to prove that the map
P∞ : τ 7→ γ̃τ (∞)
is injective.
Let τ = (t1 , t2 , · · · , ti ) and τ 0 = (t01 , t02 , · · · , t0i , · · · ) be two distinct sequences of positive
integers. Let n be the smallest positive integer such that tn 6= t0n . Then the paths γ̃τ , γ̃τ 0
can be written as concatenations
α̃τ ? ν̃n ∗ β̃τ ,
α̃τ ? ν̃n0 ∗ β̃τ 0 ,
where α̃τ is the common initial subpath
µ̃0 ∗ ν̃1 ∗ µ̃1 ∗ ν̃2 ∗ µ̃2 ∗ · · · ∗ ν̃n−1 ∗ µ̃n−1 .
The geodesic segments ν̃n , ν̃n0 have the form
+
ν̃n = x̃−
n x̃n ,
+
0
ν̃n0 = x̃−
n x̃ n .
Consider the bi-infinite piecewise geodesic path
+
0
σ := β̃τ−1 ? x̃+
n x̃ n ? β̃τ 0
26
MICHAEL KAPOVICH AND BEIBEI LIU
in X. Each geodesic piece of the path has length at least L and adjacent geodesic segments
of the path are orthogonal to each other. By Proposition 7.2, σ is a complete (2L, 4L + 1)quasigeodesic and, hence, it is backward/forward asymptotic to distinct points in ∂∞ X.
These points in ∂∞ X are respectively γ̃τ (∞) and γ̃τ 0 (∞). Hence, the map P∞ is injective.
We conclude that the endpoints of the piecewise geodesic paths γ̃τ yield a set of nonconical
limit points of Γ which has the cardinality of continuum.
Remark 10.2. This proof is a simplification of Bishop’s argument in [6], since, unlike [6],
we have orthogonality of the consecutive segments in each γτ .
Figure 13. Here Ai denotes a geodesic in X covering the loop λi , i ∈ N.
Here is an alternative way to see that the image of P∞ has the cardinality of continuum.
Let Gb be the set consisting of all infinite piecewise geodesic paths γ̃τ , τ ∈ N∞ .
As above, for each n ∈ N, we represent γ̃τ as the concatenation,
+
α̃τ ? ν̃n ? β̃τ , ν̃n = x̃−
n x̃n .
We define a new piecewise geodesic path γ̃τ,n by replacing ν̃n ? β̃τ with the unique geodesic
ray x̃−
n ξn containing ν̃n :
γ̃τ,n := α̃τ ? x̃−
n ξn .
Let Ga denote the set of such paths γ̃τ,n , τ ∈ N∞ , n ∈ N. As usual, we parameterize all
paths by their arclength. We obtain a subset G = Ga ∪ Gb in the space of paths P (X)
equipped with the topology of uniform convergence on compacts. It is clear that the subset
Gb is dense in G: For τ = (ti ),
(10.1)
γ̃τ,n = lim P(t1 , ..., tn−1 , k, tn+1 , ....).
k→∞
Similarly,
(10.2)
γ̃τ = lim γ̃τ,n .
n→∞
Lemma 10.3. G is closed in P (X).
Proof. By the denseness of Gb in G, it suffices to show that every sequence in Gb , after
extraction, converges to an element of G. We equip N∞ with the product topology; then
the map
P : N∞ → Gb
is continuous. The image of the product of finite subintervals in N under P is then compact.
Therefore, consider a sequence τj = (tij ) ∈ N∞ for which there exists the smallest integer
n such that
sup{tnj , j ∈ N} = ∞.
GEOMETRIC FINITENESS IN NEGATIVELY PINCHED HADAMARD MANIFOLDS
27
After extraction, we may assume that the first n − 1 coordinates of this sequence are
constant, equal (t1 , ..., tn−1 ) and that
lim tnj = ∞.
j→∞
Then
lim P(τj ) = γτ,n ∈ Ga .
j→∞
Each path α ∈ G is a (2L, 4L + 1)-quasigeodesic. Since the image of the geodesic ray
α∗ = x̃α(∞) is uniformly close to that of α, it follows that the map α 7→ α(∞) is continuous.
Hence, the set of limit points
G(∞) = {α(∞) : α ∈ G}
is closed, hence, compact.
Next, we show that G(∞) is perfect, i.e. has no isolated points. For each α = γ̃τ,n ∈ Ga ,
the ideal point α(∞) is a loxodromic fixed point (it is one of the ideal endpoints of a geodesic
in X projecting to the closed geodesic λn in M ). At the same time, according to (10.1),
α(∞) is the limit of nonconical limit points βk (∞) for some sequence βk ∈ Gb . Hence,
α(∞) is not an isolated point of G(∞). Similarly, for every τ ∈ N∞ , in view of (10.2),
the nonconical limit point γ̃τ (∞) is the limit of conical limit points γ̃τ,n (∞). Hence, γ̃τ (∞)
is not isolated in G(∞) either. Thus G(∞) also has no isolated points. Therefore, G(∞)
is a nonempty compact metrizable perfect space, hence, has the cardinality of continuum.
By the construction, Ga (∞) is countable, and, therefore, Gb (∞) has the cardinality of
continuum.
Proof of Theorem 1.5: The implication (1) ⇒ (2) (a generalization of Bonahon’s theorem) is the main result of Section 9. The implication (2) ⇒ (3) is the content of Theorem
10.1. It remains to prove that (3) ⇒ (1). If Γ is geometrically finite, by Theorem 1.4 Λ(Γ)
consists of conical limit points and bounded parabolic fixed points. Since Γ is discrete, it
is at most countable; therefore, the set of fixed points of parabolic elements of Γ is again
at most countable. If Λ(Γ) contains a subset of nonconical limit points of cardinality of
continuum, we can find a point in the limit set which is neither a conical limit point nor a
parabolic fixed point. It follows that Γ is geometrically infinite.
Proof of Corollary 1.6: If Γ is geometrically finite, by Theorem 1.4, Λ(Γ) consists of
conical limit points and bounded parabolic fixed points. Now we prove that if Λ(Γ) consists
of conical limit points and parabolic fixed points, then Γ is geometrically finite. Suppose
that Γ is geometrically infinite. By Theorem 1.5, there is a set of nonconical limit points
with cardinality of continuum. Since the set of parabolic fixed points is at most countable,
there exists a limit point in Λ(Γ) which is neither a conical limit point nor a parabolic fixed
point. This contradicts to our assumption. Hence, Γ is geometrically finite.
References
[1] L. V. Ahlfors, Fundamental polyhedrons and limit point sets of Kleinian groups. Proc. Nat. Acad. Sci.
U.S.A. 55 (1966), 251–254.
[2] W. Ballmann, “Lectures on spaces of nonpositive curvature.” With an appendix by Misha Brin. DMV
Seminar, vol. 25, Birkhäuser Verlag, Basel, 1995.
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Birkhäuser, Boston, 1985.
[4] A. Beardon, “The geometry of discrete groups.” Springer-Verlag, Berlin-New York 1983.
[5] A. Beardon and B. Maskit, Limit sets of Kleinian groups and finite sided fundamental polyhedra. Acta
Math. 132(1974), 1–12.
[6] C. J. Bishop, On a theorem of Beardon and Maskit. Annales Academiae Scientiarum Fennicae, Mathematica 21 (1996) 383–388.
28
MICHAEL KAPOVICH AND BEIBEI LIU
[7] F. Bonahon, Bouts des varietes hyperboliques de dimension 3. Ann. Math., 124 (1986) 71–158.
[8] B. H. Bowditch, Geometrical finiteness for hyperbolic groups. J. Funct. Anal. 113 (1993), 245–317.
[9] B. H. Bowditch, Geometrical finiteness with variable negative curvature. Duke Math. J. 77 (1995), no.
1, 229–274.
[10] J. Cheeger and D. G. Ebin, “ Comparison Theorem in Riemannian Geometry.” North-Holland Math.
Lib. 9, North-Holland, Amsterdam, 1975.
[11] C. Druţu and M. Kapovich, “Geometric group theory.”, To appear in AMS series Colloquium Publications, 2018.
[12] J. L. Fernández and M. V. Melián, Escaping geodesics of Riemannian surfaces, Acta Math. 187 (2001),
no. 2, 213–236.
[13] A.Hatcher, “Algebraic Topology.” Cambridge University Press, 2002.
[14] M. Kapovich, B. Liu, Geometrically infinite discrete isometry subgroups with torsion in Isom(Hn ), in
preparation.
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[16] L. Montejano, A new topological Helly Theorem and some transversal results. Discrete Comput. Geom.
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M.K.: Department of Mathematics, UC Davis, One Shields Avenue, Davis CA 95616, USA
KIAS, 85 Hoegiro, Dongdaemun-gu, Seoul 130-722, South Korea
E-mail address: kapovich@math.ucdavis.edu
B.L.: Department of Mathematics, UC Davis, One Shields Avenue, Davis CA 95616, USA
E-mail address: bxliu@math.ucdavis.edu
| 4 |
Compiling Quantum Circuits to Realistic Hardware
Architectures using Temporal Planners
Davide Venturelli1,2 , Minh Do3,4 , Eleanor Rieffel1 , Jeremy Frank4
arXiv:1705.08927v2 [quant-ph] 21 Dec 2017
1
NASA Ames Research Center, Quantum Artificial Intelligence Laboratory
USRA Research Institute for Advanced Computer Science (RIACS)
3
Stinger Ghaffarian Technologies (SGT Inc.)
4
NASA Ames Research Center, Planning and Scheduling Group
2
E-mail: davide.venturelli@nasa.gov
2
Abstract. To run quantum algorithms on emerging gate-model quantum hardware,
quantum circuits must be compiled to take into account constraints on the hardware. For
near-term hardware, with only limited means to mitigate decoherence, it is critical to
minimize the duration of the circuit. We investigate the application of temporal planners to
the problem of compiling quantum circuits to newly emerging quantum hardware. While
our approach is general, we focus on compiling to superconducting hardware architectures
with nearest neighbor constraints. Our initial experiments focus on compiling Quantum
Alternating Operator Ansatz
(QAOA) circuits whose high number of commuting gates allow great flexibility in the
order in which the gates can be applied. That freedom makes it more challenging to find
optimal compilations but also means there is a greater potential win from more optimized
compilation than for less flexible circuits. We map this quantum circuit compilation
problem to a temporal planning problem, and generated a test suite of compilation problems
for QAOA circuits of various sizes to a realistic hardware architecture. We report
compilation results from several state-of-the-art temporal planners on this test set. This
early empirical evaluation demonstrates that temporal planning is a viable approach to
quantum circuit compilation.
1. Introduction
We explore the use of temporal planners to optimize compilation of quantum circuits to
newly emerging quantum hardware. Currently only special purpose quantum hardware is
commercially available: quantum annealers that run only one type of quantum optimization
algorithm. The emerging gate-model processors, currently in prototype phase, are universal
in that, once scaled up, they can run any quantum algorithm. This facilitates expanding the
empirical exploration of quantum algorithms beyond optimization, as well as enabling the
exploration of a broader array of quantum approaches to optimization.
Quantum algorithms are usually specified as idealized quantum circuits that do not
take into account hardware constraints. This approach makes sense since the actual
physical constraints vary from architecture to architecture. With the advent of gate-model
processors, researchers have begun to explore approached to compiling idealized quantum
circuits to realistic hardware. For example, emerging superconducting quantum processors
have planar architectures with nearest-neighbor restrictions on the locations (qubits) to
which the gates can be applied. Such processors include the 5-qubit processor IBM
recently made publicly available the cloud [IBM, 2017a], recently updated to 20 qubits,
and processors being fabricated by other groups, such as Intel/TU Delft [Versluis et al.,
2016], UC Berkeley [Ramasesh et al., 2017], Rigetti Computing [Sete et al., 2016] [Reagor
et al., 2017], and Google [Boxio, 2016]. All cited groups have announced plans to build
3
gate-model quantum processors with 40 or more qubits in the near term. Idealized circuits
generally do not respect nearest neighbor constraints; idealized quantum circuits generaly
contain many gates between pairs of qubits that are not nearest neighbors and therefor
cannot be implemented directly on the processors. - see Figure 2 for more details. For this
reason, compiling idealized quantum circuits to superconducting hardware requires adding
supplementary gates that move qubit states to locations where the desired gate can act on
them.
Quantum computational hardware suffers from decoherence, which degrades the
performance of quantum algorithms over time. Especially for near-term hardware, with
only limited means to mitigate decoherence, it is critical to minimize the duration of
(execution of) the circuit that carries out the quantum computation, so as to minimize
the decoherence experienced by the computation. Other, more sophisticated, compilation
cost functions, such as figure-of-merits taking into account fidelity of operations [Bishop
et al., 2017] [Moll et al., 2017], could be used in the future within the temporal planning
approach to compilation we explore here. Optimizing the duration of compiled circuits is
a challenging problem due to the parallel execution of gates with different durations. For
quantum circuits with flexibility in when the gates can be applied, or when some gates
commute with each other (so can be applied in a different order while still achieving
the same computation) the search space for feasible compilations is larger than for less
flexible circuits. That freedom makes it more challenging to find optimal compilations but
also means there is a greater potential win from more optimized compilation than for less
flexible circuits.
While there has been active development of software libraries that server as a software
toolchain to compile an algorithm (specified in some standardized language) into idealized
quantum circuits (see Ref. [Chong et al., 2017] for a review, and [Wecker and Svore,
2014] [Smith et al., 2016a] [Steiger et al., 2016a] [Devitt, 2016] [Barends et al., 2016]
for the most relevant works), few approaches have been explored for compiling idealized
quantum circuits to realistic quantum hardware [Beals et al., 2013] [Brierley, 2015]
[Bremner et al., 2016], leaving the problem open for innovation. Recent studies explore
exact schemes [Wille et al., 2014], approximate tailored methods [Kole et al., 2017] or
formulations suited for off-the-shelf Mixed Integer Linear Programming (MILP) solvers
such as Gurobi [Bhattacharjee and Chattopadhyay, 2017]. The benchmarks in prior work
have mostly revolved around circuits composed by elementary gates relevant for faulttolerant quantum computing schemes, with attention shifting recently in the quantum
computing community towards algorithms to be run on near-term hardware. These
algorithms have circuits that often contain a large number of mutually commuting gates,
4
but not necessarily natively available in the hardware. Recently a tailored scheduling
heuristic approach has been published by [Guerreschi and Park, 2016] to schedule quantum
circuits with many commuting gates (but only for gates of unity duration and/or on a
linear architecture.) Prior work had not used a temporal planning approach, which can
be applied quite generally, enabling us to address, for the first time, gates with variable
durations, generic architectures with inhomogeneous spatial features, and efficiencies that
can be gained when large numbers of gates commute. An similar issue arising when
compiling classical programs is the register allocation problem, in which program variables
are assigned to machine registers to improve execution time; this problem reduces to graph
coloring [Fu et al., 2005].
In this paper, we apply temporal planning techniques to the problem of compiling
quantum circuits to realistic gate-model quantum hardware. Temporal planning is a
subdomain of Automated Planning and Scheduling, a branch of Artificial Intelligence
(AI) which is concerned with the identification of strategic decisions, among a large finite
set of possibilities, to achieve a specific goal that includes temporal constraints or time
optimization objectives.
As we will explain in Section 4, we use domain-independent AI planners to find
a parallel sequence of conflict-free instructions that, when executed, achieve the same
result as the machine-independet quantum circuit. Additional instructions specific to a
gate-model architecture are inserted. The temporal planners aim to provide a machinedependent plan that minimizes makespan, while respecting all machine-dependent
constraints (e.g. available gates to perform swap operations, physical layout of the gates,
duration of gates, exclusions on simultaneous operations.) While our approach is general,
we focus our initial experiments on circuits that have few ordering constraints and thus
allow highly parallel plans. We report on experiments using a diverse set of temporal
planners to compile circuits of various sizes to an architecture inpired by those currently
being built. This early empirical evaluation demonstrates that temporal planning is a viable
approach to quantum circuit compilation.
In Sec. 2, we describe the problem of compiling idealized quantum circuits to specific
hardware architectures in detail. Sec. 3 describes QAOA circuits, the class of circuits with
many commuting gates that we target for our initial invstigation. Section 4 explains our
mapping of the quantum circuit compilation problem to temporal planning problem. Sec. 5
presents our results applying state-of-the-art temporal planners to this circuit compilation
problem. In Sec. 6, we outline future research directions stemming from this study. The aim
is to write the paper to communicate clearly to both the quantum computing community
and the temporal planning and artificial intelligence communities, so it will necessarily
5
contain some material that is review for one or the other group.
2. Architecture-specific quantum circuit compilation problem
Quantum circuits for general quantum algorithms are often described on an idealized
architecture in which any 2-qubit gate can act on any pair of qubits. Physical constraints
impose restrictions on which pairs of qubits support gate interactions in an actual physical
architecture. In this work, we concentrate on superconducting qubit architectures, using an
abstract model that takes into account gate durations and the nearest neighbor topology
of the quantum processors which will be used as the basis for the temporal planning
formulation of the compilation problem.
In general, the model will specify different types of abstract quantum gates,
each taking different durations, with a duration depending on the specific physical
implementation in terms of primitive gates. Gate-model quantum architectures operate as
digital computers with a clock, and times could be expressed in terms of clock cycles.
Elementary or primitive gates are the gates that have been directly implemlented in
the hardware and carefully calibrated, so are generally the fastest possible operations.
However, from a sequence of primitive gates, composite gates can be synthesized, making
it possible to describe a chip in terms of the relevant gates for the algorithm, as long as we
take into account the number of clock cycles that are required to perform the wanted gate.
In the model, qubits in a quantum processor can be thought of as nodes in a graph, and
the relevant 2-qubit quantum gates are associated to edges. In many architectures more than
one 2-qubit gate may be implementable between a given pair of qubits, so this structure is
a multigraph (multiple edges allowed between two nodes). In general the gates are nonsymmetric so the multi-graph could be directed (i.e. distinguishing the roles of two qubits),
as for instance is the case if we include the primitive (calibrated) gates in the IBM Quantum
Experience chip [IBM, 2017b].
Gates that operate on distinct sets of qubits may be able operate concurrently, though
there can be additional restrictions on which operations can be done in parallel, such as requiring the sets in operation to be non-adjacent, due to cross-talk and frequency-crowding,
as in Google’s proposed architecture [Boxio, 2016]).
The swap gate and its synthesis: In this study, we make extensive use of a particular type
of a 2-qubit gate, the swap gate, which exchanges the state of two qubits, though other
gate choices are possible. In order for the computation specified by the idealized circuit
to be completed, quantum information must be moved to locations where the desired gates
=
X−π/2
X−π/2
iSWAP
=
iSWAP
=
iSWAP
6
X−π/2
Figure 1. A known decomposition of the swap gate using as primitive gates three CNOTs or replacing recursively the C-NOTs with their synthesis in terms of iSWAPs and
X-rotations.
can be carried out, and a sequence of swap gates can be used to move the contents of two
distant qubits to a location where a desired gate can be executed. For this reason, the swap
gate is a useful gate for compilation in sparsely-connected architectures.
This origin of the “duration” abstraction for gates is exemplified in Figure 1(left) for
the swap gate, which can be efficiently decomposed in three control-NOT (CNOT) gates
[Schuch and Siewert, 2003]. The swap gate should last at least three times as long as the
CNOT gate [Vatan and Williams, 2004]. Figure 1(right) shows a possible synthesis of the
same swap in terms of only iSWAP gates, which are the primitives available everywhere
across the chip described in [Sete et al., 2016] .
Beside the timings dictated by logical gate synthesis, different choices of synchronization and time-scales of executions are possible, leading to different possible durations.
For instance, in [Caldwell et al., 2017] the controlled-Z gate (CZ) could last 175 or 270
nanoseconds depending on which choices are made at calibration. Across the chip, calibration might result in different gate times depending on the location in the chip even if the
underlying circuitry is the same, as shown in [IBM, 2017b], where the CNOT gate duration
could vary up to a factor of 2.
For the purposes of this study, we consider a simplified model in which swap gates
are available between any two adjacent qubits on the chip and all swap gates have the same
duration, but our temporal planning approach can handle the more general cases.
2.1. Formal problem statement
An idealized quantum circuit consists of a set of nodes (qubits), which can be thought of
as memory locations, and a specification of start times for operations (gates), each acting on a single node or set of nodes. The idealized quantum circuit also includes implicit
or explicit specification of which operations commute (the order in which they are exe-
7
cuted can be switched without affecting the computation) either individually or as blocks.
A hardware architecture specification can be viewed as a weighted, labeled multigraph
on a set of nodes, corresponding to physical quantum memory locations (qubits) in the
hardware, where each edge represents an operation (quantum gate) that can be physically
implemented on the pair of qubits in the physical hardware, possibly as a composite gate,
with the labels indicating the type of quantum gate and the weight giving its duration. The
output of the compilation process is a circuit that can be used to perform a quantum computation. It does not perform the computation itself, and therefore the compilation step can
be carried out on a conventional (non-quantum) computer. In this work, we are concerned
with the efficiency and effectiveness of the compilation. A separate issue, which we do not
consider here, is the performance of the quantum algorithms we are compiling.
Ideal to hardware-specific quantum circuit compilation problem: The problem input
is an idealized quantum circuit and a hardware multigraph. The output is a time-resolved
hardware-specific circuit that implements the quantum computation described by idealized
quantum circuit. The objective is to minimize the makespan (the circuit duration) of the
resulting schedule.
2.2. Compilation examples
Fig. 2 shows a hypothetical chip design that we will use for our experiments on circuit
compilation. It is inspired by the architecture proposed by Rigetti Computing Inc. [Sete
et al., 2016]. Qubits are labeled with ni and the colored edges indicate the types of 2-qubit
gates available (considering just those relevant for the algorithm), in this case swap gates
and two other types of 2-qubit gate (further described in Section 4). Given an idealized
circuit consisting only of the non-swap gates, used to define general quantum algorithms,
the circuit compilation problem is to find a new architecture-specific circuit by adding swap
gates when required, and reordering commuting operations when desired. The objective is
to minimize the overall duration to execute all gates in the new circuit. To illustrate the
challenges of finding effective compilation, we present some concrete examples, with reference to the 8-qubit section in the top left of Fig. 2.
Example 1: Suppose that at the beginning of the compilation, each qubit location ni
is associated to the qubit state qi . Let us also assume that the idealized circuit requires
the application of a red gate to the states q2 and q4 , initially located on qubits n2 and n4 .
One way to achieve this task would be to swap the state in n4 with n1 , while at the same
8
n2
n3
0000
n1
n4
n5
n6
n7
n8
Figure 2. Left: A schematic for the hypothetical chip design, based on an architecture
proposed by Rigetti Computing used in our numerical experiments. Available relevant
2-qubit gates are represented by colored arcs in a weighted multigraph. Each color is
associated to a specified, distinct gate-type and duration: SWAP gates (black) and two other
types of 2-qubits gates (red and blue). The 1-qubit gates are present at each qubit (black
dot). Right: Dashed boxes indicate the three different chip sizes used in our empirical
evaluation (see Sec. 5). For visual clarity, only the label locations and the SWAP-gates for
the smaller chip size, corresponding to the top-left sector of the largest chip, are shown.
time swapping n2 with n3 . Another swap, between n1 and n2 , positions q4 in n2 where a
red-gate connects it to q2 (which is now in n3 ).
The sequence of gates to achieve the stated goal are:
{SWAPn4 ,n1 , SWAPn2 ,n3 } →
≡
RED (q2 , q4 )
SWAP n1 ,n2
→
RED n2 ,n3
(1)
The first line refers to the sequence of gate applications, while the second corresponds to
the algorithm objective specification (a task defined over the qubit states). The sequence in
Eq. (1) takes 2τswap + τred clock cycles where τ? represents the duration of the ?-gate.
Example 2:
Consider an idealized circuit that requires BLUE(q1 , q2 ) ∧ RED(q4 , q2 ), in no particular
order. If τblue > 3 × τswap , the compiler might want to execute BLUEn1 ,n2 while the qubit
state q4 is swapped all the way clockwise in five SWAPs from n4 to n3 where REDn2 ,n3 can
be executed. However, if τblue < 3×τswap , it is preferable to wait until the end of BLUEn1 ,n2
9
and then start to executthe instruction sequence in Eq. (1).
3. Compiling QAOA for the MaxCut problem
While our approach can be used to compile arbitrary quantum circuits to a wide range of
architectures, in this paper we concentrate on one particular case: the class of Quantum
Alternating Operator Ansatz (QAOA) circuits [Farhi et al., 2014a, Hadfield et al., 2017]
for MaxCut (defined in the later part of this section) to the above-mentioned architecture
inspired to Rigetti Computing Inc. [Sete et al., 2016]. We choose to work with QAOA
circuits because they have many gates that commute with each other (i.e., no ordering
enforced). Such flexibility in the ordering of the gates means that the compilation search
space is larger than for other less flexible circuits. This makes finding the optimal
compilation more challenging, but also means there is potential for greater compilation
optimization, compared to other less flexible classes of circuits.
QAOA circuits have been the focus of recent research [Farhi et al., 2014a] [Farhi et al.,
2014b] [Farhi and Harrow, 2016] [Wecker et al., 2016] [Yang et al., 2016] [Guerreschi
and Smelyanski, 2017] [Jiang et al., 2017] [Wang et al., 2017] [Hadfield et al., 2017]
in the quantum computing community since their introduction by Farhi et al. in [Farhi
et al., 2014a]. The acronym was reworked from “quantum approximate optimization
algorithm” to “quantum alternating operator ansatz” in [Hadfield et al., 2017] since QAOA
circuits have been applied to exact optimization and sampling as well as to approximate
optimization and there are other quantum approaches to approximate optimization.
Recently Google Inc. proposed an alternative quantum approximate optimization
approach
in fixed nearest-neighbor architectures explicitly to avoid the compilation step that is
the subject of our work [Farhi et al., 2017]. Their numerical results on MaxCut instances
show a small hit in performance. Furthermore, unlike the QAOA circuits we consider
here, in which the number of parameters is independent of the number of qubits, their
alternative approach has many more parameters, which increase with the number of qubits,
and these parameters must be optimized separately for each architecture. While their
approach makes good use of near-term hardware with numbers of qubits for which the
parameter optimization is tractable, ultimately one wants a scalable algorithm that can be
compiled to arbitrary architectures. Thus, while interesting, especially in the very nearterm, rather than obviating the need, their work serves to underscore the need for efficient
approaches to optimize compilation.
10
q5
q1
PS1
MX
PS2
q6
q3
q4
q7
Figure 3. Example of a 6-vertex MaxCut problem on a randomly generated graph (qstates
q2 and q8 are not appearing in this instance). The association of quantum states to every
node allows the definition of the compilation objectives in terms of gates, as exemplified
on the right panel for QAOA p = 2. Colored edges refer to Figure 6.
We chose MaxCut as the target problem of reference, as it is becoming one of the
de facto benchmark standards for quantum optimization of all types and it is considered a
primary target for experimentation in the architecture of [Sete et al., 2016].
MaxCut Problem: Given a graph G(V, E) with n = |V | vertices and m = |E| edges.
The objective is to partition the graph vertices into two sets such that the number of edges
connecting vertices in different sets is maximized.
A quadratic boolean objective function for MaxCut is:
1 X
(1 − si sj ),
(2)
U=
2 (i,j)∈E
where si are binary variables, one for each vertex vi , with values +1 or -1 indicating to
which partition the vertex vi is assigned.
Idealized QAOA circuits alternate between a phase separation step (PS), based on the
objective function, and a mixing step. The phase-separation step for QAOA for MaxCut is
simpler than for other optimization problems, consisting of a set of identical 2-qubit gates
that must be applied between certain pairs of qubits depending on the graph of the MaxCut
instance under consideration. Specifically, the idealized QAOA circuit for MaxCut requires
a 2-qubit gate for each quadratic term in the objective function of Eq. (2), as well as 1-qubit
gates for each vertex for the mixing step [Farhi et al., 2014a].
11
In Fig. 3 a 6-vertex graph is shown, providing an illustrative instance that will be used
to describe the compilation procedure. We will refer to these as p-s gates, and the main
goal of the compilation is to carry them out.
The p-s gates all commute with each other, implying that they can be carried out in
any order without changing the computation.
In the mixing phase, a set of 1-qubit operations are applied, one to each qubit‡
All p-s gates that involve a specific qubit q must be carried out before the mixing
operator on q can be applied. These two steps are repeated p times. We consider p = 1 and
p = 2 in our experiments (detailed in Section 5).
For every vertex i ∈ V , QAOA for MaxCut requires a quantum state qi to be assigned
on a qubit on the chip, and for every edge (i, j) ∈ E, the PS step of QAOA requires
executing a gate corresponding to P - S(qi , qj ). We ignore the final mixing step since it is
trivial to compile by just applying the 1-qubit mixing gate to each qubit as the last operation.
We chose the architecture proposed by Rigetti Compuing in [Sete et al., 2016] (see
Fig. 2) for our initial exploration because it offers a particularly interesting compilation,
and therefore planning, problem, due to the existence of two different kinds of nearest
neighbor relation in the proposed hardware. After the synthesis of the QAOA MaxCut
gates, these two different relations become two different durations of two-qubit gates,
which corresponds to the red and blue edges as described above.
In our problem specification, while there are two flavors of p-s gates (red, blue),
corresponding to two different durations of execution, the compilation goals (see figure 3)
do not care on which of these two types of gates carries out the required steps. For the
purpose of this proof-of-concept work, these durations we assign to the gates are not
derived from actual designs of ongoing experiments, but are realistic and serve to illustrate
possible future designs.
The constraints on the compilation problem can be understood, with reference to
Fig. 2, as:
•
SWAP
gates are located at every edge with τswap = 2.
• there are two kind of non-swap gates: P - S gates are 2-qubit gates and
1-qubit gates.
•
MIX
gates are
gates are located at every edge of the grid, but their duration τp−s can be 3 or 4
depending on their location (respectively blue or red edges in Fig.2).
P-S
‡ This is another simple feature of MaxCut, and it is due to the fact that all possible 2N si variable
assignments (see Eq. 2) are defining a valid cut. If this wasn’t the case, then the mixing phase would also
likely require the application of 2-qubit gates, further complicating the scheduling problem.
12
•
MIX
gates are located at every vertex with τmix =1.
• In an initialization stage, which is not considered as part of the compilation problem,
a quantum state is assigned to each qubit.
4. Compilation of a Quantum Circuit as Temporal Planning Problem
Planning is the problem of finding a conflict-free set of actions and their respective
execution times that connects the initial-state I and the desired goal state G. We now
introduce some key concepts that provide the background for approaching the problem of
compiling quantum circuits as a temporal planning problem.
Classical planning problems are expressed in terms of binary state variables and
actions. Examples of state variables for our problem are “The quantum state Ψ is assigned
to qubit number X” and “The quantum state Φ has been transformed by the application of
gate G present on qubits X and Y ,” which may be True or False. Actions consist of two
lists, a set of preconditions and a set of effects.
The effects of an action consists of a subset of state variables with the values they take
on if the action is carried out. For example, the action “State Ψ is now moved from qubit
X to qubit Y ” has one precondition, “State Ψ is assigned to X = True” and has two effects
“State Ψ is assigned to X = False” and “State Ψ is assigned to Y = True.”
A specific planning problem specifies an initial state, with values specified for all state
variables, and a goal, specified values for one or more state variables. As for preconditions,
goals are conventionally positive, so the goal value for the goal variables is True. Generally,
the goal specifies values for only a small subset of the state variables. A plan is a sequence
of actions.
A valid plan, or a solution to the planning problem, is a sequence of actions A1 , ..., AL
such that the state at time step ti−1 meets the preconditions for action Ai , the effects of
action Ai are reflected in the state at time step ti , and the state at the end has all of the
goal variables set to True. For an introduction on Automated Planning and Scheduling, see
[Ghallab et al., 2004].
Planners: A planner is software implementing a collection of algorithms; it takes as input
a specification of domain and a problem description and returns a valid plan if one exists.
Many different approaches have been implemented to find a viable plan, among them: (i)
heuristically search over the possible valid plan trajectories or over the library of partial
plans or (ii) compile the planning problem into another combinatorial substrate (e.g., SAT,
MILP, CSP) and feed the problem to off-the-shelf solvers.
13
Planning Domain Description Language (PDDL): PDDL is a modeling language that was
originally created to standardize the input for planners competing in the International Planning Competition (IPC). Over time, it has become the de facto standard for modeling languages used by many domain-independent planners. We use PDDL 2.1, which allows the
modeling of temporal planning formulation in which every action a has duration da , starting time sa , and end time ea = sa + da . Action conditions cond(a) are required to be
satisfied either (i) instantaneously at sa or ea or (ii) required to be true starting at sa and
remain true until ea . Action effects eff (a) may instantaneously occur at either sa or ea .
Actions can execute when their temporally-constrained conditions are satisfied, and when
executed, will cause state-change effects. The most common objective function in temporal planning is to minimize the plan makespan, i.e. the shortest total plan execution time.
This objective matches well with the objective of our targeted quantum circuit compilation
problem. To enable reuse of key problem features present in an ensemble of similar instances, the PDDL model of a planning problem is separated into two major parts: (i) the
domain description that captures the common objects and behaviors shared by all problem
instances of this planning domain and (ii) the problem instance description that captures
the problem-specific objects, initial state, and goal setting for each particular problem.
PDDL is a flexible language that offers multiple alternatives for modeling a planning problem. These modeling choices greatly affect the performance of existing PDDL
planners. For instance, many planners pre-process the original domain description before
building plans; this process is time-consuming, and may produce large ‘ground’ models depending on how action templates were written. Also, not all planners can handle all PDDL
language features effectively (or even at all). For this project, we have iterated through different modeling choices with the objective of constructing a PDDL model that: (i) contains
a small number of objects and predicates for compact model size; (ii) uses action templates
with few parameters to reduce preprocessing effort; while (iii) ensuring that the model can
be handled by a wide range of existing PDDL temporal planners.
Modeling Quantum Gate Compilation in PDDL 2.1: To apply a temporal planner to the
circuit compilation problem, we must represent the allowed gates as actions and the desired
circuit as a set of goal variables. We describe how to do so for the QAOA circuit compilation problem exemplified in Fig. 3, ensuring that for a plan to be valid, the required P - S or
MIX gates are scheduled for each step of the algorithm. At the high-level, in this domain,
we need to model: (i) how actions representing P - S, SWAP, and MIX gates affect qubits and
14
(:constants q1 q2 q3 q4 q5 q6 q7 q8 - qstate)
(:durative-action swap 1 2
:parameters (?q1 - qstate ?q2 - qstate)
:duration (= ?duration 2)
:condition (and (at start (located at 1 ?q1))
(at start (located at 2 ?q2)))
:effect (and (at start (not (located at 1 ?q1)))
(at start (not (located at 2 ?q2)))
(at end (located at 1 ?q2))
(at end (located at 2 ?q1))))
(:durative-action mix q5 at 1
:parameters ( )
:duration (= ?duration 1)
:condition (and (at start (located at 1 q5))
(at start (GOAL PS1 q1 q5))
(at start (GOAL PS1 q5 q6))
(over all (not (mixed q5))))
:effect (and (at start (not (located at 1 q5)))
(at end (located at 1 q5))
(at end (mixed q5))))
(:durative-action P-S 2ndPhaseSeparation at 6-7
:parameters (?q1 - qstate ?q2 - qstate)
:duration (= ?duration 3)
(:durative-action P-S 1stPhaseSeparation at 6-7
:condition (and (at start (located at 6 ?q1))
:parameters (?q1 - qstate ?q2 - qstate)
:duration (= ?duration 3)
(at start (located at 7 ?q2))
:condition (and (at start (located at 6 ?q1))
(at start (not (GOAL PS2 ?q1 ?q2)))
(at start (located at 7 ?q2))
(at start (GOAL PS1 ?q1 ?q2))
(at start (not (GOAL PS1 ?q1 ?q2)))
(at start (mixed ?q1))
:effect (and (at start (not (located at 6 ?q1)))
(at start (mixed ?q2)))
(at start (not (located at 7 ?q2)))
:effect (and (at start (not (located at 6 ?q1)))
(at end (located at 6 ?q1))
(at start (not (located at 7 ?q2)))
(at end (located at 7 ?q2))
(at end (located at 6 ?q1))
(at end (GOAL PS1 ?q1 ?q2))
(at end (located at 7 ?q2))
(at end (GOAL PS1 ?q2 ?q1)))))
(at end (GOAL PS2 ?q1 ?q2))
(at end (GOAL PS2 ?q2 ?q1)))))
Figure 4. PDDL model of actions representing some exemple of SWAP, MIX, and P - S
gates. The first line indicates that this compilation problem involves 8 qubit states. For
each action, the duration indicates how long the action takes. The first action, a swap at
qubits 1 and 2, has as parameters the two qstates to swap. The condition checks that these
states are indeed located at the qubits on which the swap will occur. The effect makes sure
the states have been swapped. The second action mixes qstate q5 at qubit 1, with conditions
that state q5 is indeed at qubit 1, and both the phase separation gates involving qstate q5
(see Fig. 3) have been carried out. The effects include setting mixed q5 to TRUE The
third and fourth actions are phase separation actions in a p = 2 circuit corresponding to the
first and second levels of the algorithm.
15
qubit states (qstate); (ii) the actual qubits and qstates involved in a particular compilation
problem, with their initial locations and final goal requirements, (iii) the underlying graph
structure (gates connecting different pairs of qubits). We follow the conventional practice
of modeling (i) in the domain description while (ii) is captured in the problem description.
One common practice is to model (iii) within the problem file. However, given that we
target a rather sparse underlying qubit-connecting graph structure (see Fig. 2), we decide to
capture it within the domain file to ease the burden of the “grounding” and pre-processing
step for existing planners, which can be very time-consuming. Specifically:
Objects: We need to model three types of object: qubits, qstates, and the location of the P - S
and SWAP gates (i.e., edges in the multigraph of Fig. 2 connecting different qubits). Since
qstates are associated (by means of the predicate located at, see Fig. 4 for concrete example) to specific qubits, they have been modeled explicitly as planning objects, while the
qubits and the gate locations (i.e., edges) are modeled implicitly. It is clear from the action
definitions in Fig. 4 that qubit locations are embedded explicitly within the action declaration. This approach avoids declaring qubits as part of the action parameters, significantly
reducing the number of ground actions to be generated. For 2-qubit actions, the potential
number of ground actions reduces from N 4 to N 2 × |E|, with N the number of qubits
in the chip (up to 40) and E the set of connections between qubits. While it’s true that
many modern planners will be able to filter out invalid ground actions during the grounding/preprocessing step, our empirical evaluation shows that capturing the graph structure
explicitly in the domain file speeds up the preprocessing time of all tested planners, sometime as significantly as 40x.
Actions: Temporal planning actions are created to model: (i) 2-qubit SWAP gates, (ii)
2-qubit P - S gates, and (iii) 1-qubit MIX gates. For reference, Fig. 4 shows the PDDL
description of a SWAP gate between qubits 1 and 2, the MIX gate of state q5 on qubit 1,
and the P - S gates between qubits 6 and 7 at the first and second phase separation. § In the
action’s condition list, we specify that gates are accomplished on the two qstates only if they
are located on the corresponding qubits. Note that some planners (e.g., CPT, POPF) can not
handle action preconditions that are specified negatively (e.g., (overall(not(mixedq5))) of
action mix q5 at 1 in Figure 4). For those planners, we’ve also created another PDDL
version of our domain where dummy predicates such as not mixed is introduced to
represent the opposite value of those that need to be negatively satisfied in some action’s
§ The full set of PDDL model for all our tested problems is available at: https://ti.arc.nasa.gov/
m/groups/asr/planning-and-scheduling/VentCirComp17_data.zip
16
precondition list. To prevent a qstate q currently belonging to qubit X from being addressed
by multiple gates at the same time (i.e. “mutex” relations in planning terminology), we
assign value FALSE to the predicate (located at X q) at the starting time of all actions
involving q.
The most complex constraint to model is the conditions to mix a qstate q given the
requirement that all P - S gates involving q in the previous phase separation step have been
executed. We explored several other choices to model this requirement such as: (i) use a
metric variable P Scount(q) to model how many P - S gates involving q have been achieved
at a given moment; or (ii) use ADL quantification and conditional effect constructs supported in PDDL. Ultimately, we decided to explicitly model all P - S gates that need to be
achieved as conditions of the MIX(q) action. This is due to the fact that alternative options
require using more expressive features of PDDL2.1 which are not supported by many effective temporal planners.k
Objective: For a level p QAOA circuit, the goal is to have all of the (GOAL P Si ?q1 ?q2)
predicates, for any q1 and q2 connected in the graph, for 1 ≤ i ≤ p set to TRUE and all
mixedi qj set to true for 1 ≤ j ≤ N and 1 ≤ i ≤ p − 1 (since the final mixing step can be
added by hand at the end). Since we only consider p = 1 and p = 2, so only have a mixing
step in the p = 2 case, we have simplified the notation to simply mixed qj. Further, we use
the standard temporal planning objective of minimizing the plan makespan. Minimizing
the makespane coincides with minimizing the circuit depth, which is the main objective of
the compilation problem.
Alternative models: Given that non-temporal planners can perform much better than
temporal planners on problems of the same size, we also created the non-temporal
version of the domain by discretizing action durations into consecutive “time-steps”
ti , introducing additional predicates next(ti , ti+1 ) enforcing a link between consecutive
time-steps. However, initial evaluation of this approach with the M/Mp SAT-based
planner [Rintanen, 2012] (which optimize parallel planning steps) indicated that the
performance of non-temporal planners on this discretized (larger) model is much worse
than the performance of existing temporal planners on the original model.
k Only one of six planners in the Temporal track of the latest IPC (2014) supports numeric variables and also
only one of six supports quantified conditions. Preliminary tests with our PDDL model using metric variables
to track satisfied goals involving qstate q using several planners shows that they perform much worse than on
non-metric version, comparatively. This is to be expected as currently, state-of-the-art PDDL planners still
do not handle metric quantities as well as logical variables.
17
Another option is to totally ignore the temporal aspect and encode it as a “classical”
planning problem where actions are instantaneous. A post-processing step is then
introduced to inject back the temporal constraints and schedule actions in the found
classical plans. While we do not believe this approach would produce good quality plans,
it’s another promising option to scale up to larger problems in this domain.
5. Empirical Evaluation
We modeled the QAOA circuit compilation problem as described in the previous sections
and tested them using various off-the-shelf PDDL 2.1 Level 4 temporal planners. The results were collected on a RedHat Linux 2.4Ghz machine with 8GB RAM.
Problem generation: We consider three problem sizes based on grids with N = 8, 21 and
40 qubits (dashed boxes in Fig. 2). The utilized chip layouts are representative of devices
to come in the next 2 years¶
For each grid size, we generated two problem classes: (i) p = 1 (only one PS-mixing
step) and (ii) p = 2 (two PS-mixing steps). To generate the graphs G for which a MaxCut
needs to be found, for each grid size, we randomly generate 100 Erdös-Rényi graphs G
[Erdös and Rényi, 1960]. Half (50 problems) are generated by choosing N of N (N − 1)/2
edges over respectively 7, 18, 36 qstates randomly located on the circuit of size 8, 21, and
40 qubits (referred to herafter as ‘Utilization’ u=90%). The other half are generated by
choosing N edges over 8, 21, and 40 qstates, respectively (referred to herafter as ‘Utilization’ u=100%). In total, we report tests on 600 random planning problems with size in the
range [1024-232000] for the number of grounded actions and [192-8080] for the number
of predicates.
Planner setup: Since larger N and p lead to more complex setting with more predicates,
ground actions, requiring planners to find longer plans, the allocated cutoff time for
different setting are as follow: (i) 10 minutes per instance for N = 8, (ii) 30 minutes per
instance for P = 1, N = 21; (iii) 60 minutes per instance for other cases. The timelimits
are comparable to what used in the previous International Planning Competitions. We
select planners that performed well in the temporal planning track of previous IPCs, while
at the same time representing a diverse set of planning technologies:
¶ A gate-model 8-qubit chip with the grid we used is currently available from Rigetti, however the gate set
is currently uncalibrated or calibrated with different durations depending on the edges, so our benchmarks do
not model actual hardware.
18
• LPG: which is based on local search with restarts over action graphs [Gerevini et al.,
2003]. Specifically, LPG incrementally builds a multi-level graph structure. Each
layer represented by a single action and each graph edge represents a supporting
connection between one action’s effect with a condition of another action appearing
in a later layer. The graph leaf nodes represent action conditions that have not been
supported (i.e., “connected”) by other action effects. At the beginning of the search
process, LPG starts with a two-layer graph consisting of two newly created actions:
(i) Ainit : which occupies the first layer of the graph, has an empty condition list,
and has an effect list represents state variables that are true in the initial states; (ii)
Agoal : which occupies the last layer, has an empty effect list, and has a condition list
represents all goals. At each search step, LPG generates the local search neighborhood
by considering all decisions of either: (i) establishing a new edge connecting an
existing action’s effect with an open condition of another action appears in a later
layer (without conflicting with negative effects of other actions), (ii) removing an
edge from the existing graph; (iii) adding another action to the graph; (iv) removing
an action from the graph. Each resulting candidate partial (i.e., incomplete) plan in the
local search neighborhood is evaluated by a heuristic function balancing between how
close that candidate is from being a complete plan (i.e., fewer unsatisfied conditions)
and how good the quality of the likely complete plan starting from that candidate
partial plan based on the user’s defined objective function (e.g., minimizing the
plan makespan). LPG then selects the best candidate partial plan from the search
neighborhood and starts a new search episode. This process is repeated until a
complete plan is found. When LPG is run in the “anytime” mode, it does not stop
when the first complete plan is found, but will restart its planning process with the
found plan(s) used as the baseline quality comparison on the subsequent trials.
• Temporal FastDownward (TFD): a heuristic forward state-space (FSS) search planner
with post-processing to reduce makespan [Eyerich et al., 2009]. In the FSS framework,
the planner starts from the initial state I with an empty plan P and tries to extends P
until the state resulted from applying P satisfies all goals+ . In each search step, FSS
planners will generate new search nodes by taking a state SP = Apply(P, I), reached
from applying P to the initial state I, and considers all actions A applicable in SP (i.e.,
all conditions of A are satisfied by SP ). All newly generated states S 0 = Apply(A, SP )
are put in the search queue, ordered by the heuristically evaluated “quality” of S 0 . The
heuristic value evaluating a given state S generally depends on two factors: (i) the
+
This is in contrast to the backward state-space (BSS) planners, which build the plan “backward” starting
from the goals until it reaches the initial state.
19
quality of the partial plan leading from I to S, and (ii) the estimation on the quality of
the remaining plan leading from S to the goals. In TFD, the second part is estimated
through analyzing a set of special structure called the domain-transition graphs (DTG)
and causal-graph (CG) that are statically built for each planning problem∗ . After
a valid plan P is found, TFD also tries to improve the final plan makespan by
rescheduling actions in P , pushing them to start as early as possible without violating
the various logical and temporal constraints between different actions in P such as
causal supports and potential conflicts caused by actions’ negative effects. This postprocessing step is done greedily and takes little time compared to the planning process.
• SGPlan: partition the planning problem into subproblems that can be solved
separately, while resolving the inconsistencies between partial plans using extended
saddle-point condition [Wah and Chen, 2004] [Chen and Wah, 2006]. Specifically,
SGPlan uses a sub-goal partitioning strategy in which a high-level planning problem
is divided into smaller planning problems, each one targets a smaller subset of goals.
Furthermore, if a “landmark” (i.e., a given state or condition that needs to be visited
by all plans when solving a given problem) is found for a subset of the goals, that
landmark can be used to further partition a sub-planning problem into a subset of
secondary sub-problems. Thus, the original planning problem can be partitioned into a
hierarchy of multi-level interconnected smaller sub-problems, each with its own initial
state and set of goals. Each sub-problem can be solved by any off-the-shelf planner. In
particular, SGPlan uses a slightly modified Metric-FF, a forward state-space planner,
and an earlier version of LPG to solve sub-planning problems.
• CPT: uses the Partial Order Causal Link (POCL) framework, which once dominated
planning research. POCL planners search through the space of partial plans; each one
consists of a list of actions and the causal-link between them. A causal-link indicates
that an action’s effect is used to support another action’s condition. CPT [Vidal and
Geffner, 2006] utilizes techniques from constraint-programming to create (1) effective
branching scheme to select what action to consider next during each search step; (2) a
makespan-bound automatically extracted from the planning problem to set the horizon
for the solution search.
At the moment, CPT is one of the most effective temporal planners that can minimize
plan makespan.
∗ A directed edge in the DTG connects two values v and v 0 of a given state variable s in which there exist
an action a that can make the “transition” from v to v 0 by deleting v and add v 0 when executed. There is an
edge in CG connecting two DTGs associated with two state variables s and s0 if there is an action a that has
a condition depends on s and an effect causing change of the value of s0 .
20
Utilization u
LPG
TFD
SGPlan
POPF
CPT
P1
P2
N8
N21
N40
N8
N21
0.9 1.0 0.9 1.0 0.9 1.0 0.9 1.0 0.9 1.0
50 50 50 50 10 14 50 50 50 50
50 50 50 50
50 50 50 50
50 50 50 50 50 50 50 50
50 50 48 50
8
19 50 50
4
6
50 50
-
Table 1. Summary of the solving capability of selected planners. Numbers indicate how
many random problems out of 50 have been solved.
• POPF: is a forward-chaining planner that combines forward-state-space search
framework with ideas from the POCL planning framework. In POCL planning, a
new action is a threat to a causal link if it removes the condition the action introduces
in support of another action. During the forward search, when applying an action to a
state, POPF [Coles et al., 2010] seeks to introduce only the ordering constraints needed
to resolve threats, rather than insisting the new action occurs after all of those already
in the plan. POPF mostly uses the relaxed-plan heuristic similar to other forward statespace planners, but has a dedicated Simple Temporal Network (STN) implementation
to handle the temporal constraints incurred from temporal actions interleaving in the
explored partial plans. Besides temporal constraints, POPF can also handle linear
continuous numeric effects on resources. It does so by offloading those constraints to
a dedicated MILP solver.
We ran SGPlan (Ver 5.22), TFD (Ver IPC2014), POPF, and CPT (latest versions at
time of writing) with their default parameters. Unlike other planners, which support a single mode, LPG (Ver TD 1.0) has three standard modes, which we all ran to collect empirical
results: (i) -speed that uses heuristic geared toward finding a valid plan quickly, (ii) -quality
that uses heuristic balancing plan quality and search steps, and (iii) -n 10 (k = 10) that will
try to find within the time limit up to 10 plans of gradually better quality by using the
makespan of previously found plan as upper-bound when searching for a new plan. Since
LPG (k = 10) option always dominates both LPG-quality and LPG-speed by solving more
problems with better overall quality for all setting, we will exclude results for LPG-quality
and LPG-speed from our evaluation discussion. For the rest of this section, LPG result is
represented by LPG (k = 10).
21
Utilization u
LPG
TFD
SGPlan
POPF
p=1, N8
0.9
1.0
0.88 0.89
0.87 0.87
0.67 0.68
0.81 0.81
p=1, N21
0.9
1.0
0.91 0.87
0.95 0.90
0.64 0.67
0.90 0.94
p=2, N8
0.9 1.0
0.50 0.52
0.99 0.99
0.75 0.79
0.87 0.91
Table 2. Plan quality comparison between different planners using IPC formula (higher
value indicates better plan quality). Highlighted results represent the best performance
in the problem class. CPT results for N=8 p=1 are proven optimal so they are implicitly
assigned 1.0.
Evaluation Result Summary: Table 1 shows the overall performance on the ability to
find a plan of different planners. SGPlan stops after finding one valid plan while TFD,
LPG, and POPF are “anytime” planners that exhaust the allocated time limit and try to find
gradually improving quality plans. While CPT can find multiple plans, it does not return
any until it can prove that the plan found is optimal. Since no planner was able to find a
single solution for N = 40 and p = 2 within the 60 minute cutoff, we omit the result for
this case from Table 1. Overall, LPG was able to solve the highest number of problems,
followed by TFD, SGPlan, and POPF. Being an optimal-guarantee planner, CPT can only
solve the smallest problem set (N = 8, p = 1) and can not find any solution for the other
sets. SGPlan can find a solution very quickly, compared to the time it takes other three
other anytime planners to find the first solution. It is the only planner that can scale up and
solve all 100 problem in the N = 40 for p = 1 (finding plans with 150-220 actions). Unfortunately, SGPlan stopped with an internal error for N = 21 and p = 2. TFD generally spent
a lot of time on preprocessing for p = 1, N = 21 (around 15 minutes) and p = 2, N = 21
(around 30 minutes) but when it is finished with the pre-processing phase] it can find a
solution quickly and also can improve the solution quality quickly. TFD spent all of the
60 minutes time limit on pre-processing for N = 40 problems. LPG can generally find
the first solution more quickly than POPF and much faster than TFD (but still much more
slowly than SGPlan) but does not improve the solution quality as quickly as TFD or POPF.
Plan quality comparison: to compare the plan quality across planners, we use the formula
] The two most time-consuming parts in TFD’s pre-processing routine are “processing axioms” and
“invariant analysis”. While “processing axioms” are always consistently time-consuming, “invariant
analysis” is heuristically done and sometime can be quick while some other times can be very time
consuming.
22
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Figure 5. Instance-by-instance makespan comparison of the used planners on the problem
set for u=1.0 (results for u=0.9 are qualitatively similar). Scatterplots indicate on the xaxis a specific planner, compared on the y axis against another planner (color code in the
legend). The dashed line indicate equal makespan. The first row of plots shows N =8, p=1;
the second row N =8, p=2; the third row N =21, p=1.
employed by the IPCs to grade planners in the temporal planning track since IPC6 [Helmert
et al., 2008]: for each planning instance i, if the best-known makespan is produced by a
plan Pi , then for a given planner X that returns a plan PXi for i, the score of PXi is calculated
as: makespan(Pi ) divided by makespan(PXi ). A comparative value closer to 1.0 indicates
that planner X produces better quality plan for instance i. We use this formula and average
the score for our three tested planners over the instance ensembles that are completely
23
solved by the time cutoff. Table 2 shows the performance of different planners with regard
to plan quality. For N = 8 and p = 1, for which we know the optimal plans given that
CPT was able to solve all 100 problems, LPG found the best quality plans but TFD is only
slightly worse and POPF is not far behind.
The comparison results for N = 21 and p = 1 is similar in the sense that the three
anytime planners LPG, TFD, and POPF perform very similarly but in this case TFD and
POPF are slightly better than LPG. For N = 8 and p = 2, TFD nearly always produce the
best quality plan with POPF slightly behind while LPG perform significantly worse. The
mixing and the requirement to synchronize the two phase-separation seems to confuse the
local search heuristic in LPG. SGPlan, which unlike TFD, LPG, POPF only find a single
solution, produce lower quality plans, as expected. However, for the p = 2 case, SGPlan
produces overall better quality plans compared to LPG, even though LPG returns multiple
plans for each instance.
Fig. 5 shows in further detail the head-to-head makespan comparison between
different pairs of planners, specifically pairwise comparisons between TFD, SGPLan, LPG,
and POPF: TFD always dominates SGPlan, TFD dominates LPG majority of the times
(except for N = 8, p = 1 where the performances are comparable), and SGPlan dominates
LPG on bigger problems, but is slightly worse on smaller problems. POPF performance in
general is very similar with the one of TDF, especially for N = 8.
For a more detailed analysis with data reported for each specific instance, see [NASA,
2017].
Planning time comparison: Both TFD, LPG, and POPF use “anytime” search algorithms
and use all of their allocated time to try finding better gradually better quality plans. In contrast, SGPlan return a single solution and thus generally take a very short amount of time
with the median solving time for SGPlan in p=1|N8 , p=1|N21 , P =1|N40 and P =2|N8 are
0.02, 1, 25, and 0.05 seconds††. CPT, in general, set a very tight upper-bound on makespan
before trying to find a plan within the bound and prove that the solution is optimal. For the
smallest problem set when it can solve all 100 instances, it took a very short time between
1 to 2 seconds to find and prove optimality. However, for any other bigger set, its tight
upper bounds proven to be ineffective in finding a single solution.
Other planners: We have also conducted tests on: VHPOP, HSP*, and YASPH. While
LPG, SGPlan, TFD, and POPF were selected for their ability to solve large planning prob†† For comparison purpose, LPG-quality, which also try to returns a single solution of good quality, produces
the median solving time for P =1|N8 and P =2|N8 are 0.9 and 70 seconds respectively.
24
lems, we hoped that HSP* and VHPOP would return optimal plans to complement CPT in
providing a baseline for plan quality estimation. Unfortunately, HSP* and VHPOP failed
to find a single plan even for our smallest problems for various reason: VHPOP ran out
of memory quickly, while HSP* couldn’t find any plan for a cutoff time of 2 hours. We
have preliminary acceptable results with YAHSP up to N=40 but they will be discussed in
a future work.
Discussion: Our preliminary empirical evaluation shows that the test planners provide a
range of tradeoffs between scalability and plan quality. At one end, SGPlan can scale up to
large problem sizes and solve them in a short amount of time, providing reasonably good
quality plans (compared to the best known solutions). At the other end, TFD utilizes all
of the allocated time to find the best quality solutions but in general is the slowest by far
to obtain a valid solution. LPG and POPF balance between the two: they can either find
one solution quickly like SGPlan or can utilize the whole time prior to cutoff to find better
quality solutions.
Since planning is exponentially hard with regard to the problem size (i.e., number
of state variables and actions), being able to partition it into sub-problems of smaller
sizes definitely helps SGPlan to be find a valid solution quickly. However, there are
several reasons that TFD, LPG, and POPF can find overall better quality solutions: (i)
their anytime algorithms allow them to gradually find better quality plans, using the
previously found plans as baseline for pruning unpromising search directions; (ii) SGP’s
partitioning algorithm is based on logical relationship between state variables and actions
and ignores all temporal aspects. Thus, combining plans for sub-problems using logical
global constraints can lead to plans of lower quality for time-sensitive objective function
such as minimizing the plan makespan.
Fig. 6 shows a visualization of plans in a ‘Gantt chart’ format, putting the planners in
comparison for a single N =8 instance from Fig. 3. To illustrate the representation, we can
look at the shown p=2 case: consider the, qstate q1 initially located at n1 . The first gate it
is involved in is the phase separation gate shown in green between qstates q1 and q4. The
second gate is a phase separation gate between qubits 1 and 2 which contain qstates q1 and
q3 respectively, because states q2 and q3 were swapped in the previous step. State q1 is
then swapped with the state in qubit 2, prior to being involved in another phase separation
gate, between the contents of qubits 2 and 3, this time with state q5 that was swapped into
qubit 3 during time steps 3 − 4. It is then mixed while still located at qubit 2. Continuing to
read through the chart in this way, we see that qstate q1 undergoes the following sequence
25
n1
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n6
n7
n8
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1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35
Figure 6. The Gantt charts show compilations, labeled for each planner of the QAOA for
the MaxCut instance depicted in Fig. 3 on the N=8 processor in Fig. 2. Schedules have time
on the x-axis and qubit locations on the y-axis. Each row indicates what gate operates on
each qubit at a given time during the plan (Colored blocks represents p-s gates, of duration
3 or 4 depending on their location, with synchronized pair colors associated to edges in
Fig. 3 and white blocks are swap gates). The last schedule shows a compilation of p = 2
performed by TFD. Black blocks with numbers are mix gates acting on the corresponding
state. Gates marked with a + indicate superfluous gates that were inserted in the plan by
TFD, that could be detected and eliminated in post-processing.
26
of actions:
P - S 3 (q1 , q4 )
→ P - S3 (q1 , q3 ) → P - S4 (q1 , q5 ) →
→
→ P - S4 (q1 , q5 ) →
WAIT(4)
→ P - S3 (q1 , q3) →
WAIT(3)
MIX (q1 )
WAIT (2)
→ P - S3 (q1 , q4 )
where we denote the duration of the P - S gates in subscript and we introduced an WAIT gate
to indicate inaction times. The second mixing phase is trivially scheduled at the end of the
last tasks for each qstate.
In the shown case, TFD has found an optimal solution (same makespan as CPT).
Based on an “eye-test” and manual analysis, the best plans returned are usually of
good quality but not without defects. Note also that the plan found by TFD for p=2 is also
worse than the one that would be trivially obtained by replicating the optimal makespan
p=1 solution twice (the second time in reverse). The displayed output also contains some
unnecessary gates. Examples are the repeated swaps at time 11 and 30, and the mixing
of the un-utilized logical states q2 and q8 at times 1,5. These spurious gates/actions do
not affect the makespan, and they can be identified and eliminated by known plan postprocessing techniques [Do and Kambhampati, 2003]. We also believe a tighter PDDL
model will help eliminate extra gates.
6. Conclusion and Future Work
In this paper we presented a novel approach to the problem of compiling idealized quantum
circuits to specific quantum hardware, focusing our experiments on QAOA circuits. Our
presentation and tests have been focused on the pedagogical and practically relevant
example of MaxCut, but the approach is sufficiently general to be applied to QAOA circuits
for any discrete optimization problem, and to arbitrary quantum circuits more generally.
Because QAOA has so many commuting gates, we expect to obtain a bigger win for this
sort of algorithm than typical quantum algorithms. For most circuits, however, the problem
of determining which swaps to make when to ensure that the qstates are next each other
in order to carry out the desired gate sequence is highly non-trivial. Many swaps can be
done in parallel, and ideally would place qstates in such a way that multiple gates can be
carried out before having to swap again, or more generally in a way so as to minimize the
run time of the circuit. For these reasons, we expect the temporal planning approach to
enable significant gains over a brute force for circuits generally.
A handful of well-established temporal planners were able to compile the QAOA
circuits with reasonable efficiency, demonstrating the viability of this approach. We plan
27
to expand the portfolio of plannes we have tested, keeping up with latest AI planning
technology; the data used in our tests, as well as the PDDL models, will be made available
online at [NASA, 2017].
One thing that will be expanded in our analysis is the assessment of the quality
of the best plans found compared to optimal solutions. At the moment, there is no
published work on finding optimal solution for this problem and, as outlined in the
previous section, our current effort to get existing optimal-makespan planners to find
solutions has been successful only for N = 8, p = 1. Another important direction is to
support circuit optimization beyond the makespan objective: for instance there is currently
tremendous interest in the definition of figure-of-merits that could quantify the power of
near-term quantum devices to execute experiment of interests (i.e. quantum supremacy
experiments) [Bishop et al., 2017].
This work paves the way for future work on the use of AI planning methods for
quantum circuit compilation and design. In future work, we plan to further tune the
performance of the planners, including choosing an initial assignment of qstates to qubits
favorable for compilation, instead of using a random assignment. In order to scale reliably
to QAOA circuits with more levels and therefore larger plan sizes, we will develop
decomposition approaches in which p ¿ 1 could be divided into multiple p = 1 problems to
be solved independently and matched in a postprocessing phase. Initial results show that
the advantage of not decomposing the problem amounts to approximately 10% of reduction
of the makespan on average, and more than half of the test instances can be scheduled
optimally by focusing on the p = 1 problem.
We will also compare with other approaches to this compilation problem such as
sorting networks [Beals et al., 2013] [Brierley, 2015] [Bremner et al., 2016], constraint
programming (CP), or tailored scheduling heuristics [Guerreschi and Park, 2016] and
develop more advanced hybrid compilation methods building on the various strengths
of the temporal planning approaches and of other approaches. Once mature we will
integrate compilation with other software supporting experimentation with various nearterm processors. A virtue of the planning approach is that the temporal planning framework
is very flexible with respect to features of the hardware, including irregular graph structures
and diverse gate durations.
In the future, we can include in the PDDL modeling additional features that are
characteristics of quantum computer architectures, such as the crosstalk effects of 2-qubit
gates, realistic durations from optimal 2-qubit gate synthesis that could allow P - S and
SWAP gates to be performed as a single gate, or the ability to quantum teleport quantum
states across the chip [Copsey et al., 2003], and features of broad classes of quantum
28
algorithms including measurement and feedback, error correction, and fault tolerant gate
implementations. As hardware graphs, primitive gate sets, and gate durations for processors
build by experimental groups become available, we will apply this temporal planning
approach with these hardware parameters as input. Conversely, results from future work
comparing results of compilation to different potential architectures may suggest hardware
designs that take into account the ability to support efficient compiled circuits.
We will also consider other families of quantum circuits, including QAOA circuits
applied to problems beyond MaxCut [Hadfield et al., 2017], and to more typical quantum
circuits, with fewer pairwise commuting gates, but for which it remains difficult to fine
an optimal sequence of swap and goal gates. This temporal planning approach to quantum
circuit compilation should be of great interest to the community developing low-level quantum compilers for generic architectures [Steiger et al., 2016b] [Häner et al., 2016] and to
designers of machine-instructions languages for quantum computing [Smith et al., 2016b]
[Cross et al., 2017].
7. Acknowledgements
The authors acknowledge useful discussions with Will Zeng, Robert Smith, and Bryan
O’Gorman. The authors appreciate support from the NASA Advanced Exploration Systems program and NASA Ames Research Center (Sponsor Award No. NNX12AK33A and
contract No. NNA16BD14C). The views and conclusions contained herein are those of the
authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the U.S. Government. The U.S. Government is
authorized to reproduce and distribute reprints for Governmental purpose notwithstanding
any copyright annotation thereon.
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| 2 |
Causal Discovery in the Presence of Measurement Error:
Identifiability Conditions
Kun Zhang† , Mingming Gong?† , Joseph Ramsey† , Kayhan Batmanghelich? , Peter Spirtes† , Clark Glymour†
†
arXiv:1706.03768v1 [stat.ME] 10 Jun 2017
?
Department of philosophy, Carnegie Mellon University
Department of Biomedical Informatics, University of Pittsburgh
Abstract
Measurement error in the observed values of
the variables can greatly change the output of
various causal discovery methods. This problem has received much attention in multiple
fields, but it is not clear to what extent the
causal model for the measurement-error-free
variables can be identified in the presence of
measurement error with unknown variance. In
this paper, we study precise sufficient identifiability conditions for the measurement-errorfree causal model and show what information
of the causal model can be recovered from observed data. In particular, we present two different sets of identifiability conditions, based
on the second-order statistics and higher-order
statistics of the data, respectively. The former
was inspired by the relationship between the
generating model of the measurement-errorcontaminated data and the factor analysis
model, and the latter makes use of the identifiability result of the over-complete independent component analysis problem.
1
Introduction
Understanding and using causal relations among variables of interest has been a fundamental problem in various fields, including biology, neuroscience, and social
sciences. Since interventions or controlled randomized
experiments are usually expensive or even impossible
to conduct, discovering causal information from observational data, known as causal discovery (Spirtes et al.,
2001; Pearl, 2000), has been an important task and
received much attention in computer science, statistics,
and philosophy. Roughly speaking, methods for causal
discovery are categorized into constraint-based ones,
such as the PC algorithm (Spirtes et al., 2001), and
score-based ones, such as Greedy Equivalence Search
(GES) (Chickering, 2002).
Causal discovery algorithms aim to find the causal
relations among the observed variables. However, in
many cases the measured variables are not identical to
the variables we intend to measure. For instance, the
measured brain signals may contain error introduced by
the instruments, and in social sciences many variables
are not directly measurable and one usually resorts to
proxies (e.g., for “regional security" in a particular area).
In this paper, we assume that the observed variables
Xi , i = 1, ..., n, are generated from the underlying
measurement-noise-free variables X̃i with additional
random measurement errors Ei :
Xi = X̃i + Ei .
(1)
Here we assume that the measurement errors Ei are
independent from X̃i and have non-zero variances.
We call this model the CAusal Model with Measurement Error (CAMME). Generally speaking, because of
the presence of measurement errors, the d-separation
patterns among Xi are different from those among
the underlying variables X̃i . This generating process has been called the random measurement error
model in (Scheines & Ramsey, 2017). According
to the causal Markov condition (Spirtes et al., 2001;
Pearl, 2000), observed variables Xi and the underlying variables X̃i may have different conditional independence/dependence relations and, as a consequence,
the output of constraint-based approaches to causal
discovery is sensitive to such error, as demonstrated
in (Scheines & Ramsey, 2017). Furthermore, because of
the measurement error, the structural equation models
according to which the measurement-error-free variables X̃i are generated usually do not hold for the
observed variables Xi . (In fact, Xi follow error-invariables models, for which the identifiability of the underlying causal relation is not clear.) Hence, approaches
based on structural equation models, such as the linear,
non-Gaussian, acyclic model (LiNGAM (Shimizu et al.,
2006)), will generally fail to find the correct causal
direction and causal model.
In this paper, we aim to estimate the causal model
underlying the measurement-error-free variables X̃i
from their observed values Xi contaminated by random
measurement error. We assume linearity of the causal
model and causal sufficiency relative to {X̃i }ni=1 . We
particularly focus on the case where the causal structure
for X̃i is represented by a Directed Acyclic Graph
(DAG), although this condition can be weakened. In
order to develop principled causal discovery methods
to recover the causal model for {X̃i }ni=1 from observed
values of {Xi }ni=1 , we have to address theoretical issues
include
• whether the causal model of interest is completely
or partially identifiable from the contaminated
observations,
• what are the precise identifiability conditions, and
• what information in the measured data is essential
for estimating the identifiable causal knowledge.
We make an attempt to answer the above questions on
both theoretical and methodological sides.
One of the main difficulties in dealing with causal discovery in the presence of measurement error is because
the variances of the measurement errors are unknown.
Otherwise, if they are known, one can readily calculate
the covariance matrix of the measurement-error-free
variables X̃i and apply traditional causal discovery
methods such as the PC (Spirtes et al., 2001) or
GES (Chickering, 2002)) algorithm. It is worth noting
that there exist causal discovery methods to deal with
confounders, i.e., hidden direct common causes, such
as the Fast Causal Inference (FCI) algorithm (Spirtes
et al., 2001). However, they cannot estimate the causal
structure over the latent variables, which is what we aim
to recover in this paper. (Silva et al., 2006) and (Kummerfeld et al.) have provided algorithms for recovering
latent variables and their causal relations when each
latent variable has multiple measured effects. Their
problem is different from the measurement error setting we consider, where clustering for latent common
causes is not required and each measured variable is the
direct effect of a single "true" variable. Furthermore,
as shown in next section, their models can be seen as
special cases of our setting.
2
Effect of Measurement Error on
Conditional Independence /
Dependence
We use an example to demonstrate how measurement
error changes the (conditional) independence and dependence relationships in the data. More precisely,
we will see how the (conditional) independence and
independence relations between the observed variables
Xi are different from those between the measurementerror-free variables X̃i . Suppose we observe X1 , X2 ,
and X3 , which are generated from measurement-errorfree variables according to the structure given in Figure 1. Clearly X̃1 is dependent on X̃2 , while X̃1 and
X̃3 are conditionally independent given X̃2 . One may
consider general settings for the variances of the measurement errors. For simplicity, here let us assume that
there is only measurement error in X2 , i.e., X1 = X̃1 ,
X2 = X̃2 + E2 , and X3 = X̃3 .
X̃1
X̃2
X̃3
X1
X2
X3
Figure 1: A linear CAMME to demonstrate the effect
of measurement error on conditional independence and
dependence relationships. For simplicity, we consider
the special case where there is measurement error only
in X2 , i.e., X2 = X̃2 + E2 , but X1 = X̃1 and X3 = X̃3 .
Let ρ̃12 be the correlation coefficient between X̃1 and
X̃2 and ρ̃13,2 be the partial correlation coefficient between X̃1 and X̃3 given X̃2 , which is zero. Let ρ12
and ρ13,2 be the corresponding correlation coefficient
and partial correlation coefficient in the presence of
measurement error. We also let ρ̃12 = ρ̃23 = ρ̃ to make
the result simpler. So we have ρ13 = ρ̃13 = ρ̃12 ρ̃23 = ρ̃2 .
Std(E2 )
Let γ = Std(
. For the data with measurement error,
X̃ )
2
ρ12 =
Cov(X1 , X2 )
1/2
Var (X1 )Var1/2 (X2 )
Cov(X̃1 , X̃2 )
Var1/2 (X̃1 )(Var(X̃2 ) + Var(E2 ))1/2
ρ̃
=
;
(1 + γ 2 )1/2
ρ13 − ρ12 ρ23
=
(1 − ρ212 )1/2 (1 − ρ223 )1/2
=
ρ13,2
=
=
12 ρ̃23
ρ̃13 − ρ̃1+γ
2
1/2
1/2
ρ̃2
ρ̃2
1 − (1+γ 2 )
1 − (1+γ
2)
r2 ρ̃2
.
1 + γ 2 − ρ̃2
As the variance of the measurement error in X2 increases, γ become larger, and ρ12 decreases and finally
goes to zero; in contrast, ρ13,2 , which is zero for the
measurement-error-free variables, is increasing and finally converges to ρ̃2 . See Figure 2 for an illustration.
In other words, in this example as the variance of the
measurement error in X2 increases, X1 and X2 become more and more independent, while X1 and X3
are conditionally more and more dependent given X2 .
However, for the measurement-error-free variables, X̃1
One might apply other types of methods instead of the
constraint-based ones for causal discovery from data
with measurement error. In fact, as the measurementerror-free variables are not observable, X̃2 in Figure 1
is actually a confounder for observed variables. As a
consequence, generally speaking, due to the effect of
the confounders, the independence noise assumption
underlying functional causal model-based approaches,
such as the method based on the linear, non-Gaussian,
acyclic model (Shimizu et al., 2006), will not hold for
the observed variables any more. Figure 3 gives an
illustration on this. Figure 3(a) shows the scatter plot
of X1 vs. X2 and the regression line from X2 to X1 ,
where X̃2 , the noise in X̃1 , and the measurement error
E2 , are all uniformly distributed (ρ = 0.4, and γ = 1.4).
As seen from Figure 3(b), the residual of regressing
X1 on X2 is not independent from X2 , although the
residual of regressing X̃1 on X̃2 is independent from
X̃2 . As a result, the functional causal model-based
approaches to causal discovery may also fail to find the
causal structure of the measurement-error-free variables
from their contaminated observations.
ρ
ρ̃
12
ρ
ρ
ρ̃
ρ̃
2
2
X1
−2.5
−5
0
X2
5
(a)
2
0
−2
−4
−2
0
X2
2
4
(b)
Figure 3: Illustration on how measurement error leads
to dependence between regression residual and contaminated cause. (a) Scatter plot of X2 and X1 with
measurement error in X2 together with the regression
line. (b) Scatter plot of the regression residual and
X2 . Note that if we regress X̃1 on X̃2 , the residual is
independent from X̃2 .
the corresponding causal adjacency matrix for X̃i , in
which Bij is the coefficient of the direct causal influence
from X̃j to X̃i and Bii = 0. We have,
X̃ = BX̃ + Ẽ,
13, 2
4
6
8
γ = S td(E 2)/S td( X̃ 2)
10
Figure 2: The correlation coefficient ρ12 between X1
and X2 and partial correlation coefficient ρ13,2 between
X1 and X3 given X2 as functions of γ, the ratio of the
standard deviation of measurement error to the that of
X̃2 . We have assumed that the correlation coefficient
between X̃1 and X̃2 and that between X̃2 and X̃3 are
the same (denoted by ρ̃), and that there is measurement
error only in X2 .
3
0
13, 2
2
ρ
0
12
Data points
2.5Linear regression line
Residual of regressing X1 on X2
and X̃2 are dependent and X̃1 and X̃3 and conditionally independent given X̃2 . Hence, the structure given
by constraint-based approaches to causal discovery on
the observed variables can be very different from the
causal structure over measurement-error-free variables.
Canonical Representation of Causal
Models with Measurement Error
Let G̃ be the acyclic causal model over X̃i . Here we
call it measurement-error-free causal model. Let B be
(2)
where the components of Ẽ, Ẽi , have non-zero, finite
variances. Then X̃ is actually a linear transformation
of the error terms in Ẽ because (2) implies
X̃ = (I − B)−1 Ẽ.
| {z }
(3)
,A
Now let us consider two types of nodes of G̃, namely,
leaf nodes (i.e., those that do not influence any other
node) and non-leaf nodes. Accordingly, the noise term
in their structural equation models also has distinct
behaviors: If X̃i is a leaf node, then Ẽi influences only
X̃i , not any other; otherwise Ẽi influences X̃i and at
least one other variable, X̃j , j 6= i. Consequently, we
can decompose the noise vector into two groups: ẼL
consists of the l noise terms that influence only leaf
nodes, and ẼNL contains the remaining noise terms.
Therefore,
Equation (3) can be rewritten as
X̃ = ANL ẼNL + AL ẼL = X̃∗ + AL ẼL ,
(4)
where X̃∗ , ANL ẼNL , ANL and AL are n × (n − l)
and n × l matrices, respectively. Here both AL and
ANL have specific structures. All entries of AL are 0
or 1; for each column of AL , there is only one non-zero
entry. In contrast, each column of ANL has at least
two non-zero entries, representing the influences from
the corresponding non-leaf noise term.
Further consider the generating process of observed
variables Xi . Combining (1) and (4) gives
X = X̃∗ + AL ẼL + E = ANL ẼNL + (AL ẼL + E)
= ANL ẼNL + E∗
"
#
NL
ẼNL
I ·
= A
,
E∗
(5)
(6)
where E∗ = AL ẼL + E and I denotes the identity
matrix. To make it more explicit, we give how Xi∗ and
Ei∗ are related to the original CAMME process:
(
X̃i ,
if X̃i is not a leaf node in G̃;
∗
X̃i =
, and
X̃i − Ẽi , otherwise;
(7)
(
Ei ,
if X̃i is not a leaf node in G̃;
Ei∗ =
Ei + Ẽi , otherwise.
Clearly Ei∗ s are independent across i, and as we shall
see in Section 4, the information shared by difference
Xi is still captured by X̃∗ .
Proposition 1. For each CAMME specified by (2) and
(1), there always exists an observationally equivalent
representation in the form of (5) or (6),
The proof was actually given in the construction procedure of the representation (5) or (6) from the original
CAMME. We call the representation (5) or (6) the
canonical representation of the underlying CAMME
(CR-CAMME).
Example Set 1 Consider the following example with
three observed variables Xi , i = 1, 2, 3, for which X̃1 →
X̃2 ← X̃3 , with causal relations X̃2 = aX̃1 + bX̃3 + Ẽ2 .
That is,
0 0 0
B = a 0 b ,
0 0 0
and according to (3),
1 0 0
A = a 1 b .
0 0 1
X = X̃ + E = X̃∗ + E∗
E1
1 0
Ẽ
= a b · 1 + Ẽ2 + E2
Ẽ3
0 1
E3
Ẽ1
1 0
1 0 0
Ẽ3
0 1 0 · E1
= a b
.
0 0 1 Ẽ2 + E2
0 1
E3
In causal discovery from observations in the presence
of measurement error, we aim to recover information
of the measurement-error-free causal model G̃. Let us
define a new graphical model, G̃∗ . It is obtained by
replacing variables X̃i in G̃ with variables X̃i∗ . In other
words, it has the same causal structure and causal
parameters (given by the B matrix) as G̃, but its nodes
correspond to variables X̃i∗ . If we manage to estimate
the structure of and involved causal parameters in
G̃∗ , then G̃, the causal model of interest, is recovered.
Comparing with G̃, G̃∗ involves some deterministic
causal relations because each leaf node is a deterministic
function of its parents (the noise in leaf nodes has been
removed; see (7)). We defined the graphical model G̃∗
because we cannot fully estimate the distribution of
measurement-error-free variables X̃, but might be able
to estimate that of X̃∗ , under proper assumptions.
In what follows, most of the time we assume
A0. The causal Markov condition holds for G̃ and the
distribution of X̃i∗ is non-deterministically faithful
w.r.t. G̃∗ , in the sense that if there exists S, a
subset of {X̃k∗ : k 6= i, k 6= j}, such that neither
of X̃i∗ and X̃j∗ is a deterministic function of S and
X̃i∗ ⊥
⊥ X̃j∗ | S holds, then X̃i∗ and X̃j∗ (or X̃i and
X̃j ) are d-separated by S in G̃∗ .
This non-deterministically faithfulness assumption excludes a particular type of parameter coupling in the
causal model for X̃i . in Figure 4 we give a causal
model in which the causal coefficients are carefully
chosen so that this assumption is violated: because
X̃3∗ = aX̃1∗ + bX̃2∗ and X̃4∗ = 2aX̃1∗ + 2bX̃2∗ + E4∗ , we
have X̃4∗ = 2X̃3∗ + E4∗ , implying X̃4∗ ⊥
⊥ X̃1∗ | X̃3∗ and
∗
∗
∗
X̃4 ⊥
⊥ X̃2 | X̃3 , which are not given by the causal
Markov condition on G̃. We note that this nondeterministic faithfulness is defined for the distribution
of the constructed variables X̃i∗ , not the measurementerror-free variables X̃i . (Bear in mind their relationship
given in (7).) This assumption is generally stronger
than the faithfulness assumption for the distribution of
X̃i . In particular, in the causal model given in Figure 4,
the distribution of X̃i is still faithful w.r.t. G̃. Below
we call the conditional independence relationship between X̃i∗ and X̃j∗ given S where neither of X̃i∗ and
X̃j∗ is a deterministic function of S non-deterministic
conditional independence.
2a
X̃1
c
a
b
X̃2
2b
X̃4
d
X̃5
X̃3
Figure 4: A causal model in which X̃i∗ are not nondeterministically faithful w.r.t. G̃ because of parameter
coupling.
Now we have two concerns. One is whether essential
information of the CR-CAMME is identifiable from
observed values of X. We are interested in finding the
causal model for (or a particular type of dependence
structures in) X̃. The CR-CAMME of X, given by (5)
or (6), has two terms, X̃∗ and Ẽ∗ . The latter is independent across all variables, and the former preserves
major information of the dependence structure in X̃.
Such essential information of the CR-CAMME may
be the covariance matrix of X̃∗ or the matrix ANL , as
discussed in next sections. In the extreme case, suppose
such information is not identifiable at all, then it is
hopeless to find the underlying causal structure of G̃.
The other is what information of the original CAMME,
in particular, the causal model over the measurementerror-free variables, can be estimated from the above
identifiable information of the CR-CAMME. Although
the transformation from the original CAMME to a
CR-CAMME is straightforward, without further knowledge there does not necessarily exist a unique CAMME
corresponding to a given CR-CAMME: first, the CRCAMME does not tell us which nodes X̃i are leaf nodes
in G̃; second, even if X̃i is known to be a leaf node, it is
impossible to separate the measurement error Ei from
the noise Ẽi in Ei∗ . Fortunately, we are not interested
in everything of the original CAMME, but only the
causal graph G̃ and the corresponding causal influences
B.
Accordingly, in the next sections we will explore what
information of the CR-CAMME is identifiable from
the observations of X and how to further reconstruct
necessary information of the original CAMME. In the
measurement error model (1) we assumed that each
observed variable Xi is generated from its own latent
variable X̃i . We note that in case multiple observed
variables are generated from a single latent variable
or a single observed variable is generated by multiple
latent variables (see, e.g., (Silva et al., 2006)), we can
still use the CR-CAMME to represent the process. In
the former case, certain rows of ANL are identical.
For instance, if X1 and X2 are generated as noisy
observations of the same latent variable, then in (5) the
first two rows of ANL are identical. (More generally,
if one allows different coefficients to generate them
from the latent variable, the two rows are proportional
to each other.) Then let us consider an example in
the latter case. Suppose X3 is generated by latent
variables X̃1 and X̃2 , for each of which there is also
an observable counterpart. Write the causal model as
X3 = f (X̃1 , X̃2 ) + E3 and introduce the latent variable
X̃3 = f (X̃1 , X̃2 ), and then we have X3 = X̃3 + E3 .
The CR-CAMME formulation then follows.
4
Identifiability with Second Order
Statistics
The CR-CAMME (5) has a form of the factor analysis model (FA) (Everitt, 1984), which has been a
fundamental tool in data analysis. In its general
form, FA assumes the observable random vector X =
(X1 , X2 , ..., Xn )| was generated by
X = Lf + N,
(8)
where the factors f = (f1 , ..., fr )| satisfies Cov(f ) = I,
and noise terms, as components of N, are mutually
independent and also independent from f . Denote by
ΨN the covariance matrix of N, which is diagonal.
The unknowns in (8) are the loading matrix L and the
covariance matrix ΨN .
Factor analysis only exploits the second-order statistics,
i.e., it assumes that all variables are jointly Gaussian.
Clearly in FA L is not identifiable; it suffers from at
least the right orthogonal transformation indeterminacy.
However, under suitable conditions, some essential information of FA is generically identifiable, as given in
the following lemma.
Lemma 2. For the factor analysis model, when the
1/2
number of factors r < φ(n) = 2n+1−(8n+1)
, the
2
model is generically globally identifiable, in the sense
that for randomly generated (L, ΨN ) in (8), it is with
only measure 0 that there exists another representation
(L0 , Ψ0N ) such that (L0 , Ψ0N ) and (L, ΨN ) generate the
same covariance matrix for X and Ψ0N 6= ΨN .
This was formulated as a conjecture by (Shapiro, 1985),
and was later proven by (Bekker & ten Berge, 1997).
This lemma immediately gives rise to the following
generic identifiability of the variances of measurement
errors.1
1
We note that this “generic identifiability" is sightly
weaker than what we want: we want to show that for
certain (L, ΨN ) the model is necessarily identifiable. To
give this proof is non-trivial and is a line of our future
research.
Proposition 3. The variances of error terms Ei∗ and
the covariance matrix of X̃∗ in the CR-CAMME (5) are
generically identifiable when the sample size N → ∞
and the following assumption on the number of leaf
nodes l holds:
A1. The number of leaf variables l satisfies
l
(8n + 1)1/2 − 1
> c(n) ,
.
n
2n
(9)
Clearly c(n) is decreasing in n and c(n) → 0 as n → ∞.
To give a sense how restrictive the above condition
is, Fig. 5 shows how c(n) changes with n. In particular, when n = 4, c(n) = 59.3%, condition (9) implies
the number of leaf nodes is l > 2.4; when n = 100,
c(n) = 13.6%, condition (9) implies l > 13.6. Roughly
speaking, as n increases, it is more likely for condition (9) to hold. Note that the condition given in
Proposition 3 is sufficient but not necessary for the
identifiability of the noise variances and the covariance
matrix of the non-leaf hidden variables (Bekker & ten
Berge, 1997).
80%
c(n)
40%
20%
1
2
3
10
10
10
Number of variables n
4
10
Figure 5: c(n) as a function of n.
Now we know that under certain conditions, the covariance matrices of E∗ and X̃∗ in the CR-CAMME (5) are
(asymptotically) identifiable from observed data with
measurement error. Can we recover the measurementerror-free causal model G̃ from them?
4.1
Proposition 4. Suppose assumptions A0, A1, and A2
hold. Then as N → ∞, G̃ can be estimated up to the
equivalence class and, moreover, the leaf nodes of G̃
are identifiable.
Proofs are given in Appendix. The proof of this corollary inspires a procedure to estimate the information of
G̃ from contaminated observations in this case, which
is denoted by FA+EquVar. It consists of four steps. (1)
Apply FA on the data with a given number of leaf
nodes and estimate the variances of Ei∗ as well as the
covariance matrix of X̃∗ .2 (2) The (n − l) smallest values of the variances of Ei∗ correspond to non-leaf nodes,
and the remaining l nodes correspond to leaf nodes. (3)
Apply a causal discovery method, such as the PC algorithm, to the sub-matrix of the estimated covariance
matrix of X̃∗ corresponding to non-leaf nodes and find
the causal structure over non-leaf nodes. (4) For each
leaf node Xi∗ , find the subset of non-leaf nodes that determines Xi∗ , and draw directed edges from those nodes
to Xi∗ , and further perform orientation propagation.
4.2
60%
0 0
10
A2. The measurement errors in all observed variables
have the same variance.
Gaussian CAMME with the Same
Variance For Measurement Errors
In many problems the variances of the measurement
errors in different variables are roughly the same because the same instrument is used and the variables
are measured in similar ways. For instance, this might
approximately be the case for Functional magnetic resonance imaging (fMRI) recordings. In fact, if we made
the following assumption on the measurement error,
the underlying causal graph G̃ can be estimated at least
up to the equivalence class, as shown in the following
corollary.
Gaussian CAMME: General Case
Now let us consider the general case where we do not
have the constraint A2 on the measurement error. Generally speaking, after performing FA on the data, the
task is to discover causal relations among X̃i∗ by analyzing their estimated covariance matrix, which is,
unfortunately, singular, with the rank (n − l). Then
there must exist deterministic relations among X̃i∗ , and
we have to deal with such relations in causal discovery.
Here suppose we simply apply the Deterministic PC
(DPC) algorithm (Glymour, 2007; Luo, 2006) to tackle
this problem. DPC is almost identical to PC, and
the only difference is that when testing for conditional
independence relationship U ⊥
⊥ V | W, if U or V is a
deterministic function of W, one then ignores this test
(or equivalently we do not remove the edge between
U and V ). We denote by FA+DPC this procedure for
causal discovery from data with measurement error.
Under some conditions on the underlying causal model
G̃, it can be estimated up to its equivalence class, as
given in the following proposition. Here we use PA(X̃j )
to denote the set of parents (direct causes) of X̃j in G̃.
Proposition 5. Suppose Assumptions A0 and A1 hold.
As N → ∞, compared to G̃, the graph produced by the
above DPC procedure does not contain any missing edge.
In particular, the edges between all non-leaf nodes are
2
Here we suppose the number of leaf nodes is given. In
practice one may use model selection methods, such as BIC,
to find this number.
corrected identified. Furthermore, the whole graph of G̃
is identifiable up to its equivalence class if the following
assumption further holds:
A3. For each pair of leaf nodes X̃j and X̃k , there exists X̃p ∈ PA(X̃j ) and X̃q ∈ PA(X̃k ) that are
d-separated in G̃ by a variable set S1 , which may
be the empty set. Moreover, for each leaf node X̃j
and each non-leaf node X̃i which are not adjacent,
there exists X̃r ∈ PA(X̃j ) which is d-separated
from X̃i in G̃ by a variable set S2 , which may be
the empty set.
Example Set 2 and Discussion Suppose assumption A0 holds.
• G̃A , given in Figure 6(a), follows assumptions A1
and A3. According to Proposition 5, the equivalence class of this causal DAG can be asymptotically estimated from observations with measurement error.
• Assumptions A0, A1, and A3 are sufficient conditions for G̃ to be recovered up to its equivalence
class and, they, especially A3, may not be necessary. For instance, consider the causal graph
G̃B given in Figure 6(b), for which assumption A3
does not hold. If assumption A2 holds, G̃B can
be uniquely estimated from contaminated data.
Other constraints may also guarantee the identifiability of the underlying graph. For example,
suppose all coefficients in the causal model are
smaller than one in absolute value, then G̃B can
also be uniquely estimated from noisy data. Relaxation of assumption A3 which still guarantees
that G̃ is identifiable up to its equivalence class is
a future line of research.
• The causal graphs G̃C and G̃D , shown in Figure 6(c), do not follow A1, so generally speaking,
they are not identifiable from contaminated observations with second-order statistics. This is also
the case for G̃E , shown in Figure 6(d).
5
Identifiability with Higher Order
Statistics
The method based on second-order statistics exploits
FA and deterministic causal discovery, both of which
are computationally relatively efficient. However, if
the number of leaf-nodes is so small that the condition
in Proposition 3 is violated (roughly speaking, usually
this does not happen when n is big, say, bigger than
50, but is likely to be the case when n is very small,
G̃A :
G̃B :
X̃8
X̃1
X̃2
X̃5
X̃3
X̃6
X̃4
X̃2
X̃7
(b)
G̃E :
X̃4
G̃C (solid lines as its edges):
G̃D (all lines as its edges):
X̃2
X̃3
X̃4
X̃3
X̃4
(a)
X̃1
X̃1
X̃5
X̃6
(c)
X̃1
X̃2
X̃3
X̃5
X̃6
X̃8
X̃7
(d)
Figure 6: (a) G̃A : a causal DAG G̃ which follows
assumptions A1 and A3. (b) G̃B : a DAG which follows
assumption A1, but not A3; however, the structure is
still identifiable if either assumption A2 holds or we
know that all causal coefficients are smaller than one
in absolute value. (c) Two DAGs that do not follow
assumption A1; G̃C has only the solid lines as its edges,
and G̃D also includes the dashed line. (d) G̃E : another
DAG that does not follow assumption A1.
say, smaller than 10), the underlying causal model is
not guaranteed to be identifiable from contaminated
observations. Another issue is that with second-order
statistics, the causal model for X̃ is usually not uniquely
identifiable; in the best case it can be recovered up to
its equivalence class (and leaf nodes). To tackle these
issues, below we show that we can benefit from higherorder statistics of the noise terms.
In this section we further make the following assumption on the distribution of Ẽi :
A4. All Ẽi are non-Gaussian.
We note that under the above assumption, ANL in (6)
can be estimated up to the permutation and scaling
indeterminacies (including the sign indeterminacy) of
the columns, as given in the following lemma.
Lemma 6. Suppose assumption A4 holds. Given X
which is generated according to (6), ANL is identifiable
up to permutation and scaling of columns as the sample
size N → ∞.
Proof. This lemma is implied by Theorem 10.3.1 in
(Kagan et al., 1973) or Theorem 1 in (Eriksson &
Koivunen, 2004).
5.1
Non-Gaussian CAMME with the Same
Variance For Measurement Errors
We first note that under certain assumptions the underlying graph G̃ is fully identifiable, as shown in the
following proposition.
Proposition 7. Suppose the assumptions in Corollary 4 hold, and further suppose assumption A4 holds.
Then as N → ∞, the underlying causal graph G̃ is fully
identifiable from observed values of Xi .
5.2
Non-Gaussian CAMME: More General
Cases
In the general case, what information of the causal
structure G̃ can we recover? Can we apply existing methods for causal discovery based on LiNGAM,
such as ICA-LiNGAM (Shimizu et al., 2006) and
Direct-LiNGAM (Shimizu et al., 2011), to recover it?
LiNGAM assumes that the system is non-deterministic:
each variable is generated as a linear combination of its
direct causes plus a non-degenerate noise term. As a
consequence, the linear transformation from the vector
of observed variables to the vector of independent noise
terms is a square matrix; ICA-LiNGAM applies certain operations to this matrix to find the causal model,
and Direct-LiNGAM estimates the causal ordering by
enforcing the property that the residual of regressing
the effect on the root cause is always independent from
the root cause.
In our case, ANL , the essential part of the mixing
matrix in (6), is n × r, where r < n. In other words,
for some of the variables X̃i∗ , the causal relations are
deterministic. (In fact, if X̃k is a leaf node in G̃, X̃k∗ is
a deterministic function of X̃k ’s direct causes.) As a
consequence, unfortunately, the above causal analysis
methods based on LiNGAM, including ICA-LiNGAM
and Direct-LiNGAM, do not apply. We will see how to
recover information of G̃ by analyzing the estimated
ANL .
We will show that some group structure and the groupwise causal ordering in G̃ can always be recovered.
Before presenting the results, let us define the following recursive group decomposition according to causal
structure G̃.
Definition 8 (Recursive group decomposition).
Consider the causal model G̃∗ . Put all leaf nodes which
share the same direct-and-only-direct node in the same
group; further incorporate the corresponding direct-andonly-direct node in the same group. Here we say a
node X̃i∗ is the “direct-and-only-direct" node of X̃j∗ if
and only if X̃i∗ is a direct cause of X̃j∗ and there is no
other directed path from X̃i∗ to X̃j∗ . For those nodes
which are not a direct-and-only-direct node of any leaf
node, each of them forms a separate group. We call the
set of all such groups ordered according to the causal
ordering of the non-leaf nodes in DAG G̃∗ a recursive
group decomposition of G̃∗ , denoted by GG̃∗ .
Example Set 3 As seen from the process of recursive group decomposition, each non-leaf node is in one
and only one recursive group, and it is possible for
multiple leaf nodes to be in the same group. Therefore, in total there are (n − l) recursive groups. For
example, for G̃A given in Figure 6(a), a corresponding
group structure for the corresponding G̃∗ is GG̃∗ =
A
({X̃1∗ } → {X̃2∗ , X̃5∗ } → {X̃3∗ , X̃6 v} → {X̃4∗ , X̃7∗ , X̃8∗ }),
and for G̃B in Figure 6(b), there is only one group:
GG̃∗ = ({X̃1∗ , X̃2∗ , X̃3∗ , X̃4∗ }). For both G̃C and G̃D ,
B
given in Figure 6(c), a recursive group decomposition
is ({X̃1∗ } → {X̃2∗ , X̃3∗ } → {X̃4∗ } → {X̃5∗ , X̃6∗ }).
Note that the causal ordering and the recursive group
decomposition of given variables according to the graphical model G̃∗ may not be unique. For instance, if
G̃∗ has only two variables X̃1∗ and X̃2∗ which are
not adjacent, both decompositions (X̃1∗ → X̃2∗ ) and
(X̃2∗ → X̃1∗ ) are correct. Consider G̃∗ over three variables, X̃1∗ , X̃2∗ , X̃3∗ , where X̃1∗ and X̃2∗ are not adjacent
and are both causes of X̃3∗ ; then both (X̃1∗ → {X̃2∗ , X̃3∗ })
and (X̃2∗ → {X̃1∗ , X̃3∗ }) are valid recursive group decompositions.
We first present a procedure to construct the recursive
group decomposition and the causal ordering among
the groups from the estimated ANL . We will further
show that the recovered recursive group decomposition
is always asymptotically correct under assumption A4.
5.2.1
Construction and Identifiability of
Recursive Group Decomposition
First of all, Lemma 7 tells us that ÂNL in (6) is identifiable up to permutation and scaling columns. Let us
start with the asymptotic case, where the columns of
the estimated ANL from values of Xi are a permuted
and rescaled version of the columns of ANL . In what
follows the permutation and rescaling of the columns
of ANL does not change the result, so below we just
work with the true ANL , instead of its estimate.
X̃i∗ and X̃i follow the same causal DAG, G̃, and X̃i∗ are
causally sufficient, although some variables among them
(corresponding to leaf nodes in G̃∗ ) are determined by
their direct causes. Let us find the causal ordering
of X̃i∗ . If there are no deterministic relations and
the values of X̃i∗ are given, the causal ordering can
be estimated by recursively performing regression and
checking independence between the regression residual
and the predictor (Shimizu et al., 2011). Specifically,
if one regresses all the remaining variables on the root
cause, the residuals are always independent from the
predictor (the root cause). After detecting a root cause,
the residuals of regressing all the other variables on the
discovered root cause are still causally sufficient and
follow a DAG. One can repeat the above procedure to
find a new root cause over such regression residuals,
until no variable is left.
the above procedure on this new set of variance will
give the second root cause and its recursive group.
Applying this procedure repeatedly until no variable
is left finally discovers all recursive groups following
the causal ordering. The constructed recurse group
decomposition is asymptotically correct, as stated in
the following proposition.
However, in our case we have access to ANL but not
the values of X̃i∗ . Fortunately, the independence between regression residuals and the predictor can still be
checked by analyzing ANL . Recall that X̃∗ = ANL ẼNL ,
where the components of ẼNL are independent. Without loss of generality, here we assume that all components of ẼNL are standardized, i.e., they have a
zero mean and unit variance. Denote by ANL
the
i·
NL
∗ ∗
NL N L|
ith row of A . We have E[X̃j X̃i ] = Aj· Ai·
and
N L|
NL 2
E[X̃i∗2 ] = ANL
A
=
||A
||
.
The
regression
model
i·
i·
i·
for X̃j∗ on X̃i∗ is
Proposition 10. (Identifiable recursive group
decomposition) Let Xi be generated by the CAMME
with the corresponding measurement-error-free variables generated by the causal DAG G̃ and suppose
assumptions A0 and A4 hold. The recursive group
decomposition constructed by the above procedure is
asymptotically correct, in the sense that as the sample size N → ∞, if non-leaf node X̃i is a cause of
non-leaf node X̃j , then the recursive group which X̃i is
in precedes the group which X̃j belongs to. However,
the causal ordering among the nodes within the same
recursive group may not be identifiable.
X̃j∗ =
E[X̃j∗ X̃i∗ ]
E[X̃i∗2 ]
X̃i∗ + Rj←i =
N L|
ANL
j· Ai·
X̃i∗ + Rj←i .
2
||ANL
||
i·
Here the residual can be written as
Rj←i = X̃j∗ −
ANL
j·
=
|
N L|
ANL
j· Ai·
X̃i∗
2
||ANL
||
i·
N L| NL
ANL
Ai· NL
j· Ai·
Ẽ .
−
NL
||A ||2
{z i·
}
(10)
,αj←i
If for all j, Rj←i is either zero or independent from X̃i∗ ,
we consider X̃i∗ as the current root cause and put it
and all the other variables which are deterministically
related to it in the first group, which is a root cause
group. Now the problem is whether we can check for
independence between nonzero residuals Rj←i and the
predictor X̃i∗ . Interestingly, the answer is yes, as stated
in the following proposition.
Proposition 9. Suppose assumption A4 holds. For
variables X̃∗ generated by (5), regression residual Rj←i
given in (10) is independent from variable X̃i∗ if and
only if
|αj←i ◦ ANL
(11)
i· |1 = 0,
where ◦ denotes entrywise product.
So we can check for independence as if the values of X̃∗
were given. Consequently, we can find the root cause
group.
We then consider the residuals of regressing all the
remaining variables X̃k∗ on the discovered root cause as
a new set of variables. Note that like the variables X̃j∗ ,
these variables are also linear mixtures of Ẽi . Repeating
The result of Proposition 10 applies to any DAG structure in G̃. Clearly, the indentifiability can be naturally
improved if additional assumptions on the causal structure G̃ hold. In particular, to recover information of
G̃, it is essential to answer the following questions.
• Can we determine which nodes in a recursive group
are leaf nodes?
• Can we find the causal edges into a particular node
as well as their causal coefficients?
Below we will show that under rather mild assumptions,
the answers to both questions are yes.
5.2.2
Identifying Leaf Nodes and Individual
Causal Edges
If for each recursive group we can determine which variable is the non-leaf node, the causal ordering among
the variables X̃i∗ is then fully known. The causal structure in G̃∗ as well as the causal model can then be
readily estimated by regression: for a leaf node, its
direct causes are those non-leaf nodes that determine
it; for a non-leaf node, we can regress it on all non-leaf
nodes that precede it according to the causal ordering,
and those predictors with non-zero linear coefficients
are its parents. (Equivalently, its parents are the nodes
that causal precede it and in its Markov blanket.)
Now the problem is whether it is possible to find out
which variable in a given recursive group is a leaf node;
if all leaf nodes are found, then the remaining one is
the (only) non-leaf node. We may find leaf nodes by
“looking backward" and “looking forward"; the former
makes use of the parents of the variables in the consid-
ered group, and the latter exploits the fact that leaf
nodes do not have any child.
Proposition 11. (Leaf node determination by
“looking backward") Suppose the observed data were
generated by the CAMME where assumptions A0 and
A4 hold.3 Let the sample size N → ∞. Then if assumption A5 holds, leaf node O is correctly identified from
values of X (more specifically, from the estimated ANL
or the distribution of X̃∗ ); alternatively, if assumption
A6 holds, leaf nodes O and Q are correctly identified
from values of X.
A5. According to G̃∗ , leaf node O in the considered
recursive group, g (k) , has a parent which is not a
parent of the non-leaf node in g (k) .
∗
A6. According to G̃ , leaf nodes O and Q in
the considered recursive group, g (k) , are nondeterministically conditionally independent given
some subset of the nodes in g (1) , g (2) , ..., g (k) .
Example Set 4
hold.
Suppose assumptions A0 and A4
• For G̃A in Figure 6(a), assumption A6 holds for
X̃7∗ and X̃8∗ in the recursive group {X̃4∗ , X̃7∗ , X̃8∗ }:
they are non-deterministically conditionally independent given {X̃2∗ , X̃4∗ }; so both of them are identified to be leaf nodes from the estimated ANL or
the distribution of X̃∗ , and X̃4∗ can be determined
as a non-leaf node. (In addition, assumption A5
holds for X̃8∗ , allowing us to identify this leaf node
even if X̃7∗ is absent in the graph.)
• For both G̃C and G̃D in Figure 6(c), X̃6∗ , in the
recursive group {X̃5∗ , X̃6∗ }, follows assumption A5
and can be found to be a leaf node from the distribution of X̃i∗ ; accordingly, X̃5∗ has to be a non-leaf
node.
• For G̃E in Figure 6(d), assumption A5 holds for
X̃5∗ and X̃8∗ , which can then be found to be leaf
nodes.
Proposition 12. (Leaf node determination by
“looking forward") Suppose the observed data were
generated by the CAMME where assumptions A0 and
A4 hold. Then as the sample size N → ∞, we can
correctly identify the leaf node U in the considered recursive group g (k) from values of X if assumption A7
holds for it:
A7. For leaf node U in g (k) , there exists at least
one node causally following g (k) that 1) is dseparated from U by a subset of variables in
g (1) , ..., g (k−1) , g (k) which does not include all parents of U and 2) is a child of the non-leaf node in
g (k) .
Example Set 5
hold.
Suppose assumptions A0 and A4
• For data generated by G̃A in Figure 6(a), we already found X̃4∗ in recursive group {X̃4∗ , X̃7∗ , X̃8∗ }
to be a non-leaf node because of Proposition 11.
Proposition 12 further indicates that X̃2∗ (in group
{X̃2∗ , X̃5∗ }) and X̃3∗ (in group {X̃3∗ , X̃6∗ }) are nonleaf nodes, and all leaf nodes are identified.
• For G̃B in Figure 6(b), there is only one recursive
group, and it does not provide further information
by looking “backward" or “forward", and it is impossible to find the non-leaf node with Proposition
11 or 12.
• For both G̃C and G̃D in Figure 6(c), X̃6∗ was found
to be a leaf node due to Proposition 11; thanks
to Proposition 12, the other leaf node, X̃3∗ , was
also detected. In particular, in G̃C , for leaf node
X̃3∗ both X̃4∗ and X̃6∗ satisfy the two conditions
in assumption 12; however, in G̃D , for leaf node
X̃3∗ only X̃4∗ satisfies them. All leaf nodes were
successfully found.
• For G̃E in Figure 6(d), Proposition 11 already allows us to identify leaf nodes X̃5∗ and X̃8∗ . Because
assumption A7 holds for X̃4∗ (for it X̃7∗ satisfies
the two conditions), we can further identify this
leaf node.
We can also determine leaf nodes by looking at the
relationships between the considered variables and the
variables causally following them, as stated in the following proposition.
For contaminated data generated by any of G̃A , G̃C ,
G̃D , and G̃E , now we can find all leaf nodes in the
measurement-error-free causal model. One can then
immediately estimate the whole measurement-errorfree model, as seen next.
3
In this non-Gaussian case (implied by assumption A4),
the result reported in this proposition may still hold if
one avoids the non-deterministic faithfulness assumption
and assumes a weaker condition; however, for simplicity
of the proof we currently still assume non-deterministic
faithfulness.
The above two propositions are about the identifiably of
leaf nodes in the measurement-error-free causal model.
By applying them to all leaf nodes, we have the (sufficient) conditions under which the causal graph of G̃ is
fully identifiable.
Proposition 13. (Full identifiability) Suppose the
observed data were generated by the CAMME where
assumptions A0 and A4 hold. Assume that for each leaf
node in G̃∗ , at least one of the three assumptions, A5,
A6, and A7, holds. Then as the sample size N → ∞,
the causal structure in G̃ is fully identifiable from the
contaminated observations.
In the general case, the causal structure in G̃ might not
be fully identifiable, and the above propositions may
allow partial identifiability of the underlying causal
structure. Roughly speaking, the recursive group decomposition is identifiable in the non-Gaussian case;
with Propositions 11 and 12 one can further identify
some leaf nodes as well as their parents.
6
Conclusion and Discussions
For variables of interest in various fields, including social sciences, neuroscience, and biology, the measured
values are often contaminated by additional measurement error. Unfortunately, the outcome of existing
causal discovery methods is sensitive to the existence of
measurement error, and it is desirable to develop causal
discovery methods that can estimate the causal model
for the measurement-error-free variables without using
much prior knowledge about the measurement error. To
this end, this paper is concerned with the identifiability
conditions for the underlying measurement-error-free
causal model given contaminated observations. We
have shown that under various conditions, the causal
model is partially or even fully identifiable.
Table 6 summarizes the identifiability results presented
in this paper. Propositions 4 and 5 make use of secondorder statistics of the data, and Propositions 7 to 13
further exploit non-Gaussianity of the data. The identifiability conditions reported in this paper are sufficient
and might not be necessary. Below are some high-level
interpretations of the conditions.
• Roughly speaking, in the Gaussian case (Propositions 4 and 5) the identifiability conditions are
mainly about the number of leaf nodes in the
measurement-error-free causal model and the sparsity of the underlying causal structure. The
measurement-error-free causal model may be identifiable up to its equivalence class.
• In the non-Gaussian case, the conditions for full
identifiability are mainly about the sparsity of the
measurement-error-free causal structure.
• In the non-Gaussian case, the identifiability conditions of the recursive group decomposition (including causal ordering between groups) are rather
general (Proposition 7).
• The identifiability of the measurement-error-free
causal model may greatly benefit from additional
knowledge about the measurement error (e.g, that
all measurement errors have the same variance, as
discussed in Propositions 4 and 7).
• Suppose assumptions A0 and A4 hold (the nonGaussian case is considered); without additional
knowledge about the measurement error, we conjecture that the necessary and sufficient condition
for the non-leaf node to be identifiable is that at
least one of the three assumptions, A5, A6, and
A7, holds. To falsify or prove this conjecture is
part of our future work.
We note that in principle, all assumptions except
A0 (regarding the causal Markov condition and nondeterministic faithfulness assumption related to causal
model G̃∗ ) are testable from the observed data. This
suggests that it is possible to develop practical causal
discovery methods to deal with measurement error that
are able to produce reliable information at least in the
asymptotic case.
We have verified the validity of the algorithms
briefly given in the paper on large samples, including FA+EquVar and FA+DPC outlined in Sections 4.1
and 4.2, respectively, and the procedure to find recursive group decomposition given in Section 5.2. All of
them are two-step methods: in the first step, the first
two methods apply factor analysis on the data, and
the last procedure applies over-complete independent
component analysis; in the second step all the methods
do causal discovery with subsequent analysis on the estimated information of the canonical representation of
the CAMMA. These methods might not be statistically
efficient for the purpose of causal discovery because of
estimation errors in the first step. We are currently
studying their behavior on finite samples and aim at
developing statistically more efficient algorithms. Such
methods are also expected to be able to learn the optimal number of leaf nodes in the causal graph (in this
paper we assume this number is given).
It is worth noting that various kinds of background
knowledge of the causal model may further help improve the identifiability of the measurement-error-free
causal model. For instance, if one knows that all
causal coefficients are smaller than one in absolute
value, then the measurement-error-free causal model
in Figure 6(b) is immediately identifiable from contaminated data. Our future research also includes
establishing identifiability conditions that allow cycles
in the measurement-error-free causal model and developing efficient methods for particular cases where each
measurement-error-free variable has multiple measured
effects or multiplied measurement-error-free variables
Proposition #
Proposition 4
Proposition 5
Proposition 7
Proposition 10
Proposition 11
Proposition 12
Proposition 13
Table 1: Summary of the identifiability results.
Assumptions
What information of G̃ is identifiable?
up to the equivalence class; leaf nodes identifiA0, A1, and A2
able
A0, A1, and A3
up to the equivalence class
A0, A4, A1, and A2
Fully identifiable
Recursive group decomposition (including
A0 and A4
causal ordering between the groups)
A0, A4, and A5 or A6 for
Recursive group decomposition; the leaf nodes
some leaf nodes in G̃∗
A0, A4, and A7 for some
Recursive group decomposition; the leaf nodes
leaf nodes in G̃∗
A0, A4, and A5 or A6 or
Fully identifiable
A7 for each leaf node
generate a single measured effect.
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Appendix: Proofs
A.1: Proof of Proposition 4
Proof. According to Proposition 3, the variances of Ei∗
are identifiable. If they are not identical, then their
smallest value is the variance of measurement errors
Ei . If the estimated variance of Ej∗ is greater than the
smallest value, according to the definition of Ei∗ in (7),
X̃j must be a leaf node in G̃. There are in total three
types of edges in G̃.
1. First consider the edges between leaf nodes. According to the definition of leaf nodes, there is no
edge between them. Since we know which nodes
are leaf nodes, it is guaranteed that there is no
edge between them.
2. Then consider the edges between non-leaf nodes.
Remove all the leaf nodes, and we have the set of
non-leaf nodes, which is causally sufficient. The
subgraph of G̃ over this set of variables satisfies
the causal Markov condition and the faithfulness
assumption. Then by applying any consistent
constraint-based causal discovery method, such
as the PC algorithm, this subgraph is corrected
identified up to its equivalence class as N → ∞.
That is, the edges between non-leaf nodes are correctly identified.
3. Finally consider the edges between leaf nodes and
non-leaf nodes. Each leaf node X̃i∗ is a deterministic, linear function of its direct causes, which are
among non-leaf nodes. Denote by the set of the direct causes of X̃i∗ by Si . Recall that the covariance
matrix of non-leaf nodes is non-singular. As a consequence, X̃i∗ , as a linear combination of elements
of Si , cannot be represented as a deterministic, linear combination of other non-leaf nodes; otherwise
some leaf node can be written as a deterministic,
linear function of other non-leaf nodes, leading to
a contradiction. As a result, Si is identifiable and,
as a consequence, the edges between each leaf node
and non-leaf nodes are identifiable.
Therefore, G̃ is identifiable at least up to its equivalence
class. Furthermore, we know that for all edges between
leaf nodes and non-leaf nodes, the direction points to
the leaf nodes.
A.2: Proof of Proposition 5
Proof. The deterministic relations among X̃i∗ and their
conditional independence/dependence relations can be
seen from the covariance matrix of X̃i∗ . First, notice
that the conditional independence relations detected
by DPC are correct but may not be complete. Hence,
asymptotically speaking, the edges removed by DPC do
not exist in the true graph, i.e., DPC does not produce
any missing edge.
We then note that there is no deterministic relation
among non-leaf nodes and that two non-leaf nodes in
G̃ which are not adjacent are d-separated by a proper
subset of other non-leaf nodes; as a consequence, DPC
will successfully detect the conditional independence
relationships as well as the edges between non-leaf
nodes. Therefore, the edges between all non-leaf nodes
are identifiable.
Next, we shall consider whether DPC is able to correctly identify the edges between leaf nodes (which are
actually not adjacent in G̃) and those between leaf
nodes and non-leaf nodes. First consider the relationships between leaf nodes. According to the former part
of assumption A3, all paths between leaf nodes X̃j and
X̃k that go through at most one of X̃p and X̃q are
blocked by (PA(X̃j ) \ X̃p ) ∪ (PA(X̃k ) \ X̃q ); the paths
that go through both of X̃p and X̃q are blocked by S1 .
Therefore, all paths between X̃j and X̃k are blocked
by (PA(X̃j ) \ X̃p ) ∪ (PA(X̃k ) \ X̃q ) ∪ S1 , which does
not deterministically determine X̃j or X̃k , so such an
conditional independence relationship is detected by
DPC. Hence each pair of leaf nodes in G̃, X̃j and X̃k ,
are not adjacent in the output of DPC.
Finally, we consider the d-separation relationships between each leaf node X̃j and its non-adjacent non-leaf
node X̃i . According to the latter part of assumption A3,
all paths between X̃j and X̃i that do not go through
X̃r are blocked by PA(X̃j ) \ X̃r , and the paths between them that go through X̃r are blocked by S2 .
Therefore, all paths between X̃j and X̃i are blocked
by (PA(X̃j ) \ X̃r ) ∪ S2 . That is, the outcome of DPC
does not contain any extra edge between leaf nodes
and non-leaf nodes.
Hence, the skeleton given by DPC is the same as the
skeleton of G̃ under assumptions A0, A1, and A3. Furthermore, according to Lemma 4 in (Luo, 2006), DPC
correctly identifies all colliders in G̃. Therefore, under
such assumptions G̃ is recovered up to its equivalence
class.
A.3: Proof of Proposition 7
Proof. Denoted by ÂNL the estimate of ANL . First
note that according to Corollary 4, leaf nodes in G̃
are identifiable. Then for each leaf node X̃l , find the
combination of the non-leaf nodes which determines
X̃l in terms of ÂNL , which is achieved by finding the
combination of the rows of ÂNL corresponding to nonleaf nodes to determine the l-th row of ÂNL . All nodes
involved in this combination are direct causes of X̃l .
The solution in this step is unique because the rows of
ÂNL corresponding to non-leaf nodes are linearly independent, as implied by the non-deterministic relations
among X̃i (or non-zero variances of Ẽi ).
We have found all edges between leaf nodes and nonleaf nodes, and the remaining edges are those between
non-leaf nodes. If we remove all leaf nodes and edges
into them from G̃, we have the causal graph over nonleaf nodes and the graph is still acyclic, and the set of
non-leaf nodes is causally sufficient. Denote by ÂNL
s
the matrix consisting of the rows of ÂNL corresponding
to non-leaf nodes. According to the identifiability of
LiNGAM (Shimizu et al., 2006), the causal relations
among non-leaf nodes are uniquely determined by ÂNL
s
or its inverse.
A.4: Proof of Proposition 9
Proof. Both Rj←i and X̃i∗ are linear mixtures of independent, non-Gaussian variables Ẽi . According to
the Darmois-Skitovich theorem (Kagan et al., 1973),
Rj←i and X̃i∗ are statistically independent if and only
if the for any k, at most one of the kth entries of their
coefficient vectors, αj←i and ANL
i· , is non-zero, which
is equivalent to the condition (11).
A.5: Proof of Proposition 10
Proof. In the constructed recursive group decomposition, each group has one and only one non-leaf node.
Just consider the non-leaf nodes in the recursive decomposition. Combining Lemma 1 in (Shimizu et al., 2011)
and Proposition 9, one can see that the discovered
causal ordering among them must be correct.
A.6: Proof of Proposition 11
Proof. In each recursive group, there is a single nonleaf node, and all the others are leaf-nodes. Denote by
∗(k)
X̃q the qth node in the recursive group g (k) . Denoted
∗(k)
by X̃NL the only non-leaf node in the recursive group
∗(k)
g (k) . Denote by PA(X̃NL ) the set of direct causes of
∗(k)
X̃NL in G̃∗ .
First consider assumption A5. Let us regress each vari∗(k)
able X̃q
in this group on all variables X̃i∗ in the first
(k − 1) recursive groups; in this regression task, all pre∗(k)
dictors are causally earlier than X̃q
because of the
identifiable causal ordering among the recursive groups.
Although the realizations of variables X̃i∗ are unknown,
such regression models can be estimated from the estimated matrix ÂNL , as done in Section 5.2.1, or by
analyzing the estimated covariance matrix of X̃∗ , which
is ÂNL ÂN L| (we have assumed that Var(ẼiNL ) = 1
without loss of generality). There are two possible cases
to consider.
∗(k)
i) For the non-leaf node X̃NL in the kth group, all
predictors with non-zero coefficients are its direct
∗N L
causes, and their set is PA(X̃(k)
).
∗(k)
ii) Then consider a leaf node in this group, X̃q0 ,
∗(k)
∗(k)
∗(k)
X̃q0 6= X̃NL . Recall that when regressing X̃q0
on the variables in causally earlier recursive groups,
∗(k)
X̃NL is not among the predictors because it is
also in the kth group. First note that each node
∗(k)
∗(k)
in X̃NL is always d-connected to X̃q0 given any
∗(k)
variable set that does not include X̃NL . As a
∗(k)
consequence, in the regression model for X̃q0 , all
∗(k)
predictors in PA(X̃NL ) have non-zero coefficients,
so all predictors with non-zero coefficients form
∗(k)
a superset of PA(X̃NL ). Furthermore, under assumption A5, O has at least a direct cause that
∗(k)
is not in PA(X̃NL ). Therefore, in the regression
model for O, the set of predictors with non-zero
∗N L
coefficients is a proper superset of PA(X̃(k)
) (the
former has more elements).
∗(k)
That is, when regressing variables X̃q
in the considered group on variables in earlier groups, the non-leaf
node, as well as possibly some of the leaf nodes, always
has a smaller number of predictors with non-zero coefficients, compared to the model for leaf node O. Hence
we can determine O as a leaf node.
Then consider assumption A6. According to assumption A0, A6 implies that O and Q are d-separated by
a proper subset of the variables. Consequently, they
are not adjacent in G̃∗ . Bear in mind that the nonleaf node in g (k) is adjacent to all leaf nodes in the
same group in G̃∗ . Therefore both O and Q are leaf
nodes.
A.7: Proof of Proposition 12
Proof. We note that the recursive group decomposition
can be correctly identified from the values of X as N →
∞, as implied by Proposition 10. Denote by W the nonleaf node in g (k) . Let us first find a subset of the nodes
causally following g (k) in which each node, denoted by
S, is always non-deterministically dependent on at least
one of the nodes in g (k) relative to g (1) ∪g (2) ∪...∪g (k) ∪S.
Denoted by S this set of nodes. If assumption A7 holds,
S is non-empty.
We then see that the non-leaf node W is always nondeterministically dependent on every node in S. Suppose this is not the case, i.e., there is S ∈ S which
is non-deterministically independent from W given a
subset of g (1) ∪ g (2) ∪ ... ∪ g (k) . Denote by R1 this
subset. If R1 contains any leaf nodes in G̃∗ , let us
remove those leaf nodes from R1 and denote by R01 the
resulting variable set. Further note that S and W are
still de-separated by R01 . Then U 0 , a leaf node in g (k) ,
is always d-separated from S given R01 ∪ (PA(U 0 ) \ W ).
Since all nodes in R01 ∪(PA(U 0 )\W ) are non-leaf nodes,
W can not be represented as their linear combination;
thus U 0 is not their deterministic function. Furthermore, S is not a deterministic function of nodes in
R01 ∪ (PA(U 0 ) \ W ) either; otherwise, according to the
construction procedure of the recursive group decomposition, S will belong to g (1) ∪ g (2) ∪ ... ∪ g (k) because
all elements of R01 ∪ (PA(U 0 ) \ W ) belong to it. Hence
any leaf node U 0 in g (k) will be non-deterministically
independent from S relative to g (1) ∪ g (2) ∪ ... ∪ g (k) ∪ S,
so S is non-deterministically independent from every
node in g (k) given a subset of g (1) ∪ g (2) ∪ ... ∪ g (k) .
That is, S ∈
/ S, leading to a contradiction.
Next, we show that for leaf node U in g (k) , there exists
at least one element of S which is non-deterministically
conditionally independent from U given a subset of
g (1) ∪ g (2) ∪ ... ∪ g (k) . Denote by V one of the nodes
that causally follow g (k) and satisfy the two conditions
in assumption A7. Because of condition 2), V ∈ S.
Condition 1) states that V and leaf node U are dseparated by a subset of variables in g (1) , ..., g (k−1) , g (k)
that does not include all parents of U . Denote by R2
this variable set. If R2 contains any leaf nodes in G̃∗ ,
remove them from R2 and denote by R02 the resulting
variable set. V and U are still de-separated by R02 ,
but all elements of R02 are non-leaf nodes. Because
all non-leaf nodes in G̃∗ are linearly independent, the
parents of U that are not in R02 can not be written as
linear combinations of the elements of R02 . Therefore,
U is not a deterministic function of R02 . Moreover, V
is not a deterministic function of R02 either, because
otherwise V will not in groups causally following g (k) .
This means that leaf node U is non-deterministically
independent from V , as an element of S, given R02 .
That is, we can distinguish between leaf node U and the
non-leaf node in the same recursive group by checking
non-deterministic conditional independence relationships in X̃i∗ .
A.8: Proof of Proposition 13
Proof. First note that under the assumptions in the
proposition, the recursive group decomposition is identifiable, and all leaf nodes are asymptotically identifiable.
The causal ordering among the variables X̃i∗ is then
fully known. The causal graph G̃ as well as the causal
model can then be readily estimated by regression: for
a leaf node, its direct causes are those non-leaf nodes
that determine it; for a non-leaf node, we can regress
it on all non-leaf nodes that causally precede it according to the causal ordering, and those predictors with
non-zero linear coefficients are its parents.
| 2 |
arXiv:1703.01085v1 [math.RT] 3 Mar 2017
LOCAL REPRESENTATION THEORY OF
TRANSPORTER CATEGORIES
FEI XU
Abstract. We attempt to generalize the p-modular representation theory of finite groups to finite transporter categories, which
are regarded as generalized groups. We shall carry on our tasks
through modules of transporter category algebras, a type of Gorenstein skew group algebras. The Kan extensions, upgrading the
induction and co-induction, are our main tools to establish connections between representations of a transporter category and of
its transporter subcategories. Some important constructions and
theorems in local representation theory of finite groups are generalized.
1. Introduction
Let G be a finite group and P be a finite G-poset (we shall regard
a G-set as a G-poset with trivial relations). The transporter category
over P is a Grothendieck construction P ⋊ G, a specifically designed
finite category. It may be thought as a semi-direct product between G
and P, and is considered as a generalized group. This construction has
its roots in group theory, representation theory and algebraic topology.
Our initiative comes from the observation that there exists a category
equivalence (G/H) ⋊ G ≃ H for each subgroup H ⊂ G.
In our earlier work, we investigated homological properties of the
category algebra RP ⋊ G, where R is a commutative ring with identity. It is known that the category of finitely generated left modules,
RP ⋊ G-mod, is a symmetric monoidal category. Based on this, we
studied the representation theory of RP ⋊ G, and its connections with
representations of groups [10, 12, 13]. In this way, we generalized some
well-known results in group representations and cohomology, and provided new insights into certain existing results.
In the present paper, we examine transporter categories (as generalized groups) from a different point of view. Our treatment allows
Key words and phrases. G-poset, transporter category, category algebra, skew
group ring, Kan extension, vertex and source, defect category.
The author (徐斐) is supported by the NSFC grant No. 11671245.
1
2
FEI XU
some classical settings in local representation theory (of groups), and
the results that follow, to survive in this generality. To this end, let H
be a subgroup of G and Q be a H-subposet of P. The category Q ⋊ H
is called a transporter subcategory of P ⋊ G. We will discuss the structure theory of transporter categories, based on which we shall develop
a local representation theory. It means that we will establish connections between the representations of P ⋊ G and those of its transporter
subcategories. The idea of using Q ⋊ H to understand P ⋊ G may be
traced back to the Quillen stratification of the equivariant cohomology
ring H∗G (BP, k) ∼
= H∗ (EG ×G BP, k), in which EG ×G BP is indeed
homotopy equivalent to the classifying space B(P ⋊ G). Our ultimate
aim is to investigate representations of various local categories, arisen
in group representations and homotopy theory, and their applications,
see for instance [2].
We shall carry on the above mentioned tasks with the help of transporter category algebras kP ⋊ G, where k is an (algebraically closed)
field of characteristic p that divides the order of G. If H happens to
be a p-subgroup, we shall call Q ⋊ H a p-transporter subcategory. We
have the following comparison chart. The bulk of this paper contains a
theory of vertices and sources, as well as a theory of blocks, for transporter category algebras.
structure
substructure
algebra
canonical basis
modules
trivial module
restriction
left Kan extension
right Kan extension
enveloping category
diagonal category
block theory
defect
group representations
category representations
G
P ⋊G
H
Q⋊H
kG
kP ⋊ G
G = Mor G
Mor(P ⋊ G)
kG-mod, kH-mod
kP ⋊ G-mod, kQ ⋊ H-mod
k
k
P⋊G
G
↓H
↓Q⋊H
G
↑H
↑P⋊G
Q⋊H
P⋊G ∼ P⋊G
G
∼
)
⇑G
(
⇑
H =↑H
Q⋊H (6=↑Q⋊H )
e ∼
G = G×G
(P ⋊ G)e ∼
= P e ⋊ Ge
δ(G) ∼
F (P) ⋊ δ(G) ∼
=G
= F (P) ⋊ G
e
(P⋊G)e
G
∼
∼
kG = k ↑δ(G)
kP ⋊ G = k ↑F (P)⋊δ(G)
p-group D ⊂ G
p-category V ⋊ D ⊂ F (P) ⋊ G
To see a concrete example, we may choose P = Sp , the poset of nontrivial p-subgroups of G, with conjugation action. Then Sp ⋊ G is the
usual p-transporter category Trp (G), containing all p-local subgroups of
G, as automorphism groups of objects. By definition, a representation
TRANSPORTER CATEGORY ALGEBRAS
3
of Sp ⋊ G is a covariant functor from Sp ⋊ G to V ectk , the category
of finite-dimensional k-vector spaces. It can be thought as a diagram
of representations of local subgroups NG (P ), for a collection of P ∈
Ob Sp . As an application of local categories, one may find a new way
to reformulate the Alvis-Curtis duality when G is a Chevalley group in
[13].
The main results, whose proofs depend on the explicit calculation of
↑P⋊G
Q⋊H , include
(1) a generalized theory of vertices and sources, including a Mackey
formula (Theorem 4.18) and a Green correspondence (Theorem
4.26),
(2) a comparison between the theories for kG-modules and of constance kP ⋊ G-modules (Proposition 4.23 and Corollary 4.27),
and
(3) a generalized Brauer correspondence (Theorem 5.8).
To set our work into the historical context, we note that the transporter category algebras are skew group algebras, and thus are fully
group-graded algebras. This work is partially motivated by the papers on fully group-graded algebras by Boisen [4], Dade [5, 6, 7], and
Miyashita [8] (the latter in the context of G-Galois theory). Especially,
Dade conceived a theory of vertices and sources (for fully group-graded
algebras). However, his “vertices” seem to be too big, see Examples
4.15 and 4.24. Same problem occurs in Boisen’s definition of a “defect”
of a block, because a “defect” is a “vertex”, in the sense of Dade, of some
module. We shall propose a sharpened definition of a vertex, incorporating our earlier work on general EI category algebras [10], and prove
it is appropriate. For the reader’s convenience, some key constructions and results, from the previously mentioned papers, are quoted
here. The approach in this paper is mostly parallel to the standard one
for group representations. However the extra G-poset structure does
require more than mere technicality.
The paper is organized as follows. In Section 2, we recall relevant results for fully group-graded algebras. Then we examine local structures
of transporter categories in Section 3. Subsequently the Kan extensions
for investigating representations will be thoroughly discussed from the
beginning of Section 4. A generalized theory of vertices and sources
will be given. Finally in Section 5, we study the block theory of transporter category algebras.
Acknowledgement The author would like to thank Peter Webb for
stimulating discussions during his visits to Shantou in 2013 and 2014.
4
FEI XU
2. Results from fully group-graded algebras
In the present paper, we want to develop modular representation
theory of transporter category algebras. Some known results on fully
group-graded algebras of Boisen [4], Dade [5, 6, 7], and Miyashita [8],
will specialize to our situation and they will pave the way towards our
key constructions. We shall quote these results mainly for skew group
algebras. Some proof are given if they are needed in our presentation.
Let R be a commutative ring with identity. Suppose G is a group
and A is a G-graded R-ring. It means that, as R-modules, we have
M
A=
Ag ,
g∈G
satisfying Ag Ah ⊂ Agh . If A meets the extra condition that Ag Ah =
Agh , then we say A is fully G-graded.
L Suppose H is a subgroup of G.
We may define a subalgebra AH = h∈H Ah . Particularly A1 becomes
a subalgebra.
Suppose S is an R-ring that admits a G-action. We say S has a Gaction, if there exists a group homomorphism φ : G → Aut(S). Under
the circumstance, we also call S a G-ring. We usually denote the Gaction by g s = φ(g)(s) for all s ∈ S and g ∈ G. Then we may continue
to define the skew group ring S ⋊ G.PAs an R-module,
P it is simply
S ⊗R RG. For convenience, we write
sg, instead of
s ⊗ g, for an
element in the skew group ring. The multiplication is determined by
(sh)(tg) = sh thg, for s, t ∈ S and h, g ∈ G. This ring contains subrings
{s1 s ∈ S} ∼
= (Z/dZ)G for some
= S and {n1S g n ∈ Z, g ∈ G} ∼
d ∈ Z. We may wish to take a larger ring R ⊆ Z(S) fixed by G so that
RG ⊆ S ⋊ G. We assume S is free as an R-module. For the sake of
simplicity, for each s ∈ S and g ∈ G, we shall write g = 1S g and s = s1
as elements of S ⋊ G, when there is no confusion.
The skew group ring A = S ⋊ G is fully G-graded, if we put
Ag = Sg = {sg s ∈ S},
for each g ∈ G. Here we shall mainly recall constructions and results
by Dade [5, 6] and Boisen [4]. For future applications, we will only
state known results from [5, 6, 4] in the special forms for skew group
algebras.
We also note that Reiten and Riedtmann [9] studied the representation theory of skew group algebras over R = C, the complex numbers.
See [3] for another presentation.
If H ⊂ G, we have an inclusion S ⋊ H ⊂ S ⋊ G, and thus the
induction
S⋊G
M ↑S⋊H
:= S⋊G S ⋊ G ⊗S⋊H M
TRANSPORTER CATEGORY ALGEBRAS
5
and restriction
S⋊G
M ↓S⋊H
:= S⋊H S ⋊ G ⊗S⋊G M.
S⋊G
For instance S ⋊ G ∼
. In [5, 4], these two functors are denoted
= S ↑S⋊1
G
G
by symbols ↑H and ↓H since S is unchanged and it matches the special
case of groups. We refrain from using the latter in order to be consistent
throughout this paper.
In general, we have a decomposition
M
S⋊G
M ↑S⋊H
=
gi ⊗ M,
gi ∈[G/H]
and the S ⋊ G-modules structure is obtained by a “twisted permutation” of summands
−1
−1
−1
(sg)(gi ⊗ m) = ggi ⊗ (ggi ) sm = gj ⊗ (h(ggi ) sh)m = gj ⊗ (gj sh)m,
if ggi = gj h for some h ∈ H.
Parallel to this, if M ′ is a right S ⋊ H-module, then the induced
right S ⋊ G-module may be written as
M
S⋊G
M ′ ↑S⋊H
=
M ⊗ gi′ .
gi′ ∈[H\G]
It is a bit surprising, but the reasonable right S ⋊ G-action is
′
(m′ ⊗ gi′ )(sg) = mgi s ⊗ gi′ g,
for all m′ ∈ M ′ , s ∈ S and g ∈ G. This difference attributes to the fact
that usually sg 6= gs.
Given g ∈ [G/H] and a S ⋊ H-module N, we put
g
(S ⋊ H) := g(S ⋊ H)g −1.
The right hand side makes sense because we regard g as an element of
S ⋊ G and meanwhile S ⋊ H ⊂ S ⋊ G. It is also a skew group ring,
identified with g S ⋊ g H = S ⋊ g H via the following equation
g
(sh) = g(sh)g −1 = g sg h.
It follows that g ⊗ N becomes a g (S ⋊ H)-module, with
g(sh)g −1(g ⊗ n) = g ⊗ (sh)n.
Analogous to the situation of group representations, the underlying
space of N admits a g (S ⋊ H)-module structure via the linear isomorphism N ∼
= g ⊗ N. We shall denote it by g N.
6
FEI XU
Theorem 2.1 (Dade). Suppose H, K are two subgroups of G and M ∈
S ⋊ G-mod. There is a Mackey formula
M
S⋊G S⋊G
S⋊H
S⋊K
M ↑S⋊H
↓S⋊K =
[g (M ↓S⋊(K
g ∩H) )] ↑S⋊(K∩g H) .
g∈[K\G/H]
Proof. This comes from a L
decomposition of S⋊K S ⋊ GS⋊H . As a right
S ⋊ H-module, S ⋊ G =
gi ∈[G/H] gi S ⋊ H. Moreover gi S ⋊ H, for
gi ∈ KgH, is invariant under the action of S ⋊ K. The group K acts
on gS ⋊ H and its stabilizer is
StabK (g(S ⋊ H)) = {k ∈ K kg(S ⋊ H) = g(S ⋊ H)}
= {k ∈ K g −1 kg(S ⋊ H) = (S ⋊ H)}
= {k ∈ K g −1 kg ∈ H}
= K ∩ g H.
We readily verify that
M
S⋊K
gi (S ⋊ H) = [g(S ⋊ H)] ↑S⋊(K∩
g H) .
gi ∈[G:H]
gi ∈[K\g/H]
However since
S⋊H
g(S ⋊ H) = g (S ⋊ H ↓S⋊(K
g ∩H) )
as a S ⋊ (K ∩ g H)-module, our decomposition formula follows from
it.
In the proof, we actually showed that S ⋊ G is a free right S ⋊ HS⋊G
module. It means that ↑S⋊H
is exact.
Following Green’s approach to group representation theory, Dade introduced a concept of relative projectivity:
Let M be a S ⋊ G-module and H be a subgroup of G. If the S ⋊ Gmodule epimorphism
S⋊G S⋊G
M ↓S⋊H
↑S⋊H → M
is split, then M is said to be projective relative to S ⋊ H, or relatively
S ⋊ H-projective.
In [5, 4], M was called relatively H-projective, for the obvious reasons. We use the more cumbersome terminology because, again, we
want to be consistent with the rest of the paper. Dade proved several
equivalent conditions for M to be relatively S ⋊ H-projective, which
no doubt are parallel to the case of group representations. (Using a relative trace map of Miyashita, defined entirely analogous to the group
case, he also obtained a Higman’s criterion.)
TRANSPORTER CATEGORY ALGEBRAS
7
Theorem 2.2 (Dade). Let R be a field of characteristic p > 0. If M is
an indecomposable S ⋊ G-module, then there is a minimal p-subgroup
H, unique up to conjugacy, such that M is relatively S ⋊ H-projective.
If K is a subgroup such that M is relatively S ⋊ K-projective, then
S ⋊ K contains a conjugate of S ⋊ H.
If H is a Sylow p-subgroup of G, then every S ⋊ G-module is relatively S ⋊ H-projective.
Based on the previous theorem, in [5, 4], H was called a “vertex” of
the indecomposable S ⋊ G-module M. However, we only go down to
the subalgebra S ⋊ H, not RH. We shall see later on that this is a
reason why Dade’s construction may be improved for S = RP.
To study the block theory of a fully group-graded algebra, Boisen
introduced the concept of a diagonal subalgebra. We also recall it
for skew group algebras. Given A = S ⋊ G, we set S e = S ⊗R S op ,
Ge = G × Gop and δ(G) = {(g, (g −1)op ) g ∈ G}. Note that Ge
may be identified with the product group G × G, and there is a group
op
isomorphism δ(G) ∼
= G. The group Ge acts on S e via (g,h ) (s, top ) =
−1
(g s, (h t)op ), and the “diagonal subalgebra” of Ae = A⊗R Aop ∼
= S e ⋊Ge
is defined to be
M
M
Sg ⊗R (g −1 )op S op ∼
∆(A) =
Ag ⊗R (Ag−1 )op =
= S e ⋊ δ(G).
g∈G
g∈G
As an example, regarding kG as a fully G-graded algebra, we have
∆(kG) ∼
= kδ(G) ∼
= kG. Assume R is a field of charcteristic p > 0.
Boisen proved that an indecomposable summand B (a block) of the Ae module A is relatively S e ⋊ δ(H)-projective, for a minimal p-subgroup
H ⊂ G. In light of this, he called H a “defect group” of B. Analogous
to the group case, he subsequently defined a generalized Brauer correspondence and established a generalized Brauer’s First Main Theorem.
As in Dade’s treatment, his “defect” is also too big when S = RP.
3. Transporter categories and their algebras
We shall study a special class of skew group algebras, namely the
transporter category algebras kP ⋊ G, because P has “local structure”.
Like a group algebra or an incidence algebra, kP ⋊ G possesses a canonical base, which is the morphism set of the transporter category P ⋊ G.
This basis has an intrinsic structure and is crucial to our theory. Particularly it allows us to introduce transporter subcategories Q ⋊ H of
P ⋊ G, based on which we will be able to discuss the interactions between representations of P ⋊ G and those of Q ⋊ H.
8
FEI XU
3.1. Transporter categories and their subcategories. We shall
develop the structure theory of transporter categories, before going
into their representation theory. We follow standard terminologies in
category theory. For a category C, we show denote by Ob C and Mor C
its classes of objects and morphisms. If α and β are composable, then
α β
we write βα for the composite →→. If α is a morphism, we denote by
s(α) and t(α) the start (or domain) and the terminal (or codomain) of
α, respectively.
Let G be a group and P be a poset. We say P admits a G-action, or
is a G-poset, if there exists a group homomorphism φ : G → Aut(P).
We usually denote by g x = φ(g)(x) and g α = φ(g)(α), for all g ∈ G, x ∈
Ob P, α ∈ Mor P. The action is trivial if the image ℑφ = IdP .
A group is considered as a category with one object, each group
element giving an (auto)morphism, while a poset P is a category if
Ob P is the underlying set and we regard each x ≤ y as a morphism
x → y. The transporter category on P is by definition a Grothendieck
construction, some sort of “semi-direct product” between two small
categories.
Definition 3.1. Let G be a group and P be a G-poset. Then the transporter category P ⋊ G, of G on P, is a category, whose objects are the
same as those of P, and whose morphisms are given by HomP⋊G (x, y) =
{αg α ∈ HomP (g x, y)}, for any x, y ∈ Ob(P ⋊ G) = Ob P. It is required that αg = α′ g ′ if and only if α = α′ and g = g ′.
If two morphisms αg and βh are composable, in the sense that
(αg)(βh) ∈ Mor(P ⋊ G), then (αg)(βh) = (αg β)(gh).
If H ⊂ G and Q ⊂ P is a H-subposet, then we call Q ⋊ H a transporter subcategory of P ⋊ G.
Both groups and posets are examples of transporter categories. But
we are more interested in the others. Note that when the action of G
on P is trivial, we simply have P ⋊ G = P × G.
If x is an object of P ⋊ G, then we shall use hxi to denote the
set of objects that are isomorphic to x. It is easy to see that hxi =
G.x is exactly the G-orbit containing x. Subsequently, hxi ⋊ G is
a transporter subcategory of P ⋊ G, and furthermore is a groupoid.
The automorphism group AutP⋊G (x) is identified with the stabilizer
Gx = StabG (x). It follows that {x} × Gx is a skeleton of hxi ⋊ G (hence
hxi ⋊ G ≃ {x} × Gx as categories).
TRANSPORTER CATEGORY ALGEBRAS
9
Example 3.2. Let G = hg g 2 = 1i and H=1. Let P be the following
G-poset
1z
?⑧ z _❄❄
❄❄ β
α ⑧⑧
❄❄
⑧⑧
❄❄
⑧
⑧⑧
x
y f 1y
such that the group generator g ∈ G fixes z and exchanges x, y. On
morphisms g acts transitively on the two sets {α, β}, {1x , 1y }, and fixes
1z . The transporter category P ⋊ G is
1x
8
1z 1,1z g
1x 1
?⑧ z _❄❄
❄❄ β1
α1 ⑧⑧
⑧⑧ 1y g ❄❄❄
⑧
❄
⑧⑧
+y
k
x
f
8
1y 1
1x g
It is helpful to point out the existence of the following morphisms:
αg = (α1)(1x g) : y → z and βg = (β1)(1y g) : x → z. Choose Q to be
the subposet consisting of z. Then Q ⋊ H = Q × H is a transporter
subcategory consisting of exactly one object z and one morphism 1z 1.
In P ⋊ G, the objects x and y are isomorphic. Thus a skeleton of
P ⋊ G is
1z 1,1z g
α1
1x 1
8
x
6 Gz
βg
All transporter categories are EI categories, in the sense that every
endomorphism is an isomorphism [10]. We shall rely on the EI condition to introduce some crucial constructions. For instance, the EI
condition gurantees a partial order on the set of isomorphism classes
of objects.
Definition 3.3. Let C be an EI category and D be a full subcategory.
Given an object x ∈ Ob C, we define D≤x to be the full subcategory
of D consisting of objects {y ∈ Ob D HomC (y, x) 6= ∅}. Similarly we
can define D≥x . The subcategory D is said to be an ideal in C if, for
every x ∈ Ob D, C≤x ⊂ D. The subcategory D is said to be a coideal
in C if, for every x ∈ Ob D, C≥x ⊂ D. The subcategory D is said to be
convex if βα ∈ Mor D implies t(α) = s(β) ∈ Ob D.
10
FEI XU
We define Cx to be the convex subcategory consisting of all objects
isomorphic to x. Particularly, if C = P ⋊ G is a transporter category,
then (P ⋊ G)x = hxi ⋊ G.
Note that ideals and coideals in C are always convex. The intersection
of two convex (resp. ideal, coideal) subcategories is still convex (resp.
ideal, coideal) in C. These constructions were used to study general EI
categories. If C happens to be a poset, then every subposet D is full
as a subcategory. When we deal with transporter categories, we need
the following subcategories. By an ideal (or a coideal) H-subposet, we
mean an ideal (or a coideal) of P which is an H-subposet of P at the
same time. For brevity, we shall call an ideal (or a coideal) H-subposet
an H-ideal (or an H-coideal).
Definition 3.4. Let P ⋊ G be a transporter category. If H ⊂ G and
Q ⊂ P is an ideal (resp. a coideal) H-subposet, then we call Q ⋊ H a
weak ideal (resp. a weak coideal ) of P ⋊ G.
If H ⊂ G and Q ⊂ P is a convex H-subposet, then we call Q ⋊ H a
weakly convex transporter subcategory of P ⋊ G.
Weak ideals and coideals are weakly convex. Here Q ⋊ H is called
weakly convex because it is unnecessarily full in P ⋊ G. We shall
demonstrate that weak ideals and coideals, and weakly convex transporter subcategories, which reflect certain local structures of P ⋊ G,
are interesting subjects for investigation.
Lemma 3.5. Suppose D is a convex subcategory of P ⋊ G. Then there
is a unique convex G-subposet Q ⊂ P such that D = Q ⋊ G.
In fact, Q ⋊ G is convex (or an ideal, or a coideal) if and only if Q
is convex (or an ideal, or a coideal).
Proof. If x ∈ Ob D, then the isomorphism class of x is exactly G{x} ⊂
Ob D. Thus the subposet Q = P ∩ D is a G-subposet of P. It means
that D = Q ⋊ G. Since D is convex, Q has to be convex in P.
The preceding lemma implies, that Q ⋊ H is weakly convex (resp.
weak ideal/coideal) is equivalent to that Q ⋊ H is a (full) convex (resp.
ideal/coideal) subcategory of P ⋊ H, or that Q ⊂ P is convex (resp.
ideal/coideal).
Lemma 3.6. Let Q ⋊ H and R ⋊ K be two transporter subcategories
of P ⋊ G. Then (Q ⋊ H) ∩ (R ⋊ K) = (Q ∩ R) ⋊ (H ∩ K) is a transporter subcategory.
If both Q ⋊ H and R ⋊ K are weakly convex (resp. weak ideal/coideal),
so is (Q ∩ R) ⋊ (H ∩ K).
TRANSPORTER CATEGORY ALGEBRAS
11
Proof. Firstly, the objects of the intersection subcategory form the set
Ob(Q ∩ R). Secondly, any morphism of P ⋊ G has a unique way to be
written as αg, for some α ∈ Mor P and g ∈ G. Hence if αg belongs to
the intersection, we must have α ∈ Mor(Q ∩ R) and g ∈ H ∩ K. Our
first claim follows.
As to the second claim, we see Q ∩ R is convex (resp. an ideal/a
coideal) if both Q and R are.
We provide methods for constructing weak ideals and co-ideals.
Definition 3.7. Let P be a G-poset. For given subgroups K ⊂ H and
z}|{
a K-subposet Q, there is a smallest H-ideal H Q that contains Q. We
call it the H-ideal generated by Q. Similarly we also have H Q as the
|{z}
H-coideal generated by Q.
z}|{
Note that 1 Q and 1 Q are simply the smallest coideal and ideal,
|{z}
respectively, that contain Q. They are often simplified to |{z}
Q and
z}|{
z}|{
Q . Moreover, if Q is a K-subposet of P, then so are Q and |{z}
Q .
z}|{
From the proposed constructions, we see H Q ⋊H (resp. H Q ⋊H)
|{z}
is a weak ideal (resp. weak coideal) of P ⋊ G. We may characterize
z}|{
H Q ⊂ P as follows. Its objects form the set
{y ∈ Ob P ∃ y → h x, for some h ∈ H, x ∈ Ob Q}.
Similarly the objects of
H Q ⊂ P form the set
|{z}
{y ∈ Ob P ∃ h x → y, for some h ∈ H, x ∈ Ob Q}.
As a generalized group, we may define the conjugates of a transporter
subcategory of P ⋊ G.
Definition 3.8. Suppose Q ⋊ H ⊂ P ⋊ G is a transporter subcategory. Given g ∈ G, from Q ⋊ H we may define another transporter subcategory g (Q ⋊ H) as follows. Its objects are {g x x ∈ Ob(Q ⋊ H)},
and Homg (Q⋊H) (g x, g y) = {g αg h
αh ∈ HomQ⋊H (x, y)}. We call
g
(Q ⋊ H) a conjugate of Q ⋊ H in P ⋊ G.
The conjugate of a transporter subcategory is still a transporter subcategory.
Lemma 3.9. Suppose Q ⋊ H ⊂ P ⋊ G is a transporter subcategory.
For each g ∈ G, g Q is a g H-poset. There is an equality g (Q ⋊ H) =
g
Q ⋊ g H.
12
FEI XU
Proof. The g H-action on g Q is given by g x 7→ gh x on objects, and
g
α 7→ gh α on morphisms.
Since the two categories g (Q ⋊ H), g Q ⋊ g H share the same objects
and morphisms, we can identify them.
The conjugate of a weakly convex transporter subcategory (resp.
ideal/coideal) stays weakly convex (resp. ideal/coideal). For brevity,
we shall write g Q for g (Q ⋊ 1). It can be identified with a subposet of
P.
To study representations of transporter categories, it is necessary to
generalize some other constructions in group theory.
Definition 3.10. Let Q ⋊ H be a transporter subcategory of P ⋊ G.
We define NG (Q ⋊ H) = {g ∈ G g (Q ⋊ H) = Q ⋊ H}, called the
normalizer of Q ⋊ H in P ⋊ G. It follows that Q is a NG (Q ⋊ H)poset.
We also define CG (Q ⋊ H) = {g ∈ G g αg h = αh for all αh ∈
Q ⋊ H}, called the centralizer of Q ⋊ H in P ⋊ G, being a subgroup
of NG (Q ⋊ H). By definition, Q is a CG (Q ⋊ H)-poset with trivial
action, which implies Q ⋊ CG (Q ⋊ H) = Q × CG (Q ⋊ H).
For brevity, we write NG (P) for NG (P ⋊1) and CG (P) for CG (P ⋊1).
Then we find that H ⊂ NG (Q ⋊ H) = NG (Q) ∩ NG (H) and that
CG (Q ⋊ H) = CG (Q) ∩ CG (H).
Definition 3.11. Let G be a finite group and P be a G-poset. If H
is a p-subgroup, for a prime p that divides the order of G, then we
call Q ⋊ H a p-transporter subcategory. Particularly when S is a Sylow
p-subgroup of G, we call P ⋊ S a Sylow p-transporter subcategory of
P ⋊ G.
The following result justifies our terminology.
Proposition 3.12. Let G be a finite group and P ⋊ S be a Sylow
p-transporter subcategory of P ⋊ G. Then, for each x ∈ Ob(P ⋊ G),
AutP⋊S (x) is a Sylow p-subgroup of AutP⋊G (x).
Proof. Suppose S ′ is a Sylow p-subgroup of AutP⋊G (x) ∼
= StabG (x).
g
g ′
∼
Then S ⊂ S for some g ∈ G. Since x = x and hence AutP⋊G (g x) ∼
=
AutP⋊G (x), g S ′ is a Sylow p-subgroup of AutP⋊G (g x). Our first claim
follows from the fact that g S ′ ⊂ AutP⋊S (g x) ∼
= StabS (x).
It is easy to see that any two Sylow p-transporter subcategories of
P ⋊ G are conjugate.
TRANSPORTER CATEGORY ALGEBRAS
13
3.2. Enveloping categories and diagonal categories. These constructions were used in [11]. Let C e = C × C op be the product category,
between a small category C and its opposite, called the enveloping category. Given a small category C, its category of factorizations F (C)
is a small category, whose objects are the morphisms of C, that is
Ob F (C) = Mor C. To distinguish, when a morphism α is regarded as
an object of F (C), we shall use the symbol [α]. Let [α], [β] ∈ Ob F (C).
There is a morphism from [α] to [β] if and only if α is a factor of β (as
morphisms in C). More precisely, suppose β = µαγ for µ, γ ∈ Mor C.
Then we obtain a morphism (µ, γ op ) : [α] → [β] in F (C).
The category of factorization comes with functors t : F (C) → C,
s : F (C) → C op , and
δC : F (C) → C e = C × C op
such that t([α]) = t(α), the target of α; s([α]) = s(α), the source of α;
δC (α) = (t(α), s(α)) and δC (µ, γ op ) = (µ, γ op ).
The functor t : F (C) → C induces a homotopy equivalent between
classifying spaces BF (C) ≃ BC. See Remark 3.17 for further implications.
If C = G is a group, then F (G) is a groupoid, and has its skeleton
isomorphic to G.
α
β
Example 3.13. Let P = x→y →z. Then F (P) is the poset
[βα]
[1x ]
⑤>
⑤⑤
⑤
⑤⑤
⑤⑤
[α]
=
④④
④④
④
④
④④
a❈❈
❈❈
❈❈
❈❈
a❈❈
❈❈
❈❈
❈❈
④=
④④
④
④④
④④
[1y ]
[β]
`❆❆
❆❆
❆❆
❆❆
[1z ]
It is worth of mentioning that the subposet of F (P), with the object
[βα] removed, is not the factorization category of any subposet of P.
Lemma 3.14. Let P ⋊ G be a transporter category. If Q ⊂ P is a
subposet, then F (Q) ⊂ F (P) is a subposet. Indeed Q is convex if and
only if F (Q) is convex. Thus if Q ⋊ H is a weakly convex transporter
subcategory of P ⋊ G, then F (Q) ⋊ H ⊂ F (P) ⋊ G is a weakly convex
as well.
Proof. If Q is a subposet, then by definition Ob F (Q) = Mor Q is a
subset of Ob F (P) = Mor P. If [α], [β] ∈ Ob F (Q) and (u, v) : [α] →
14
FEI XU
[β] is a morphism in F (P), then (u, v) ∈ Mor F (Q) because the sources
and targets of both α and β are in Q.
If furthermore Q is convex, we easily verify that F (Q) is convex.
β
α
Conversely assume F (Q) to be convex. Let x→y →z be two morphisms
in P with x, z ∈ Ob Q. We want to show y belongs to Q too. But 1x
is a factor of α and α is a factor of βα. We obtain two morphisms
[1x ] → [α] → [βα] in Mor F (P). Since both [1x ] and [βα] belong to
F (Q), so does [α] which implies that its target belong to Ob Q.
Suppose Q ⋊ H is a weakly convex transporter subcategory. Then
Q is convex and thus F (Q) is convex. Our last claim follows.
Suppose P ⋊ G is transporter category. Then Ge is a group (which is
op
isomorphic to G×G) and P e admits a Ge -action, given by (g,h ) (x, y) =
−1
op
−1
(g x, h y) on objects, and (g,h ) (α, β op) = (g α, (h β)op ).
The enveloping category is also a transporter category.
Lemma 3.15. There is an isomorphism of categories P e ⋊ Ge ∼
=
e
(P ⋊ G) .
Proof. Since both categories have the same objects as P e , we define a
functor P e ⋊ Ge → (P ⋊ G)e to be identity on objects. On morphisms,
it is defined by the assignment
(α, β op )(g, hop) 7→ (αg, (h βh)op ).
One can readily verify that this functor is an isomorphism between
categories.
The category equivalence G → F (G) composes with F (G) → Ge
gives the well-known functor δG : G → Ge (abusing notations). Thus
G acts on P e via δG (G) = {(g, (g −1)op ) g ∈ G} ⊂ Ge .
Lemma 3.16. Let P be a G-poset. The functor δP : F (P) → P e is an
embedding of posets. Furthermore F (P) is a δ(G)-poset. Thus we may
identify F (P) with a coideal G-subposet of P e . Consequently, F (P) ⋊
δ(G) is identified with a coideal transporter subcategory of P e ⋊ δ(G).
Proof. That δP : F (P) → P e is an embedding of posets is easy to
verify. Indeed F (P) is identified with the subposet δP (F (P)) of P e
consisting of objects (y, x) such that HomP (x, y) 6= ∅. The subposet
is furthermore a coideal in P e . We readily check that F (P) can even
be regarded as a δ(G)-subposet of P e . Then F (P) ⋊ δ(G) is a full
subcategory of P e ⋊ δ(G).
TRANSPORTER CATEGORY ALGEBRAS
15
We emphasize that F (P) is not necessarily a Ge -poset. Nonetheless,
we obtain the following embeddings of transporter categories
δP
δG
e
e
F (P) ⋊ δ(G)−→P
⋊ δ(G)−→P
⋊ Ge ∼
= (P ⋊ G)e .
The two categories on the left are weakly convex inside (P ⋊ G)e . We
shall call F (P)⋊δ(G) the diagonal transporter subcategory of (P ⋊ G)e .
It plays a key role in our block theory.
Remark 3.17. Let P ⋊ G be a transporter category. The functor t :
F (P) → P induces a homotopy equivalence between the classifying
spaces BF (P) ≃ BP, which further gives rise to a homotopy equivalence B(F (P) ⋊ δ(G)) ≃ B(P ⋊ G), because they are homotopy equivalent to the Borel constructions EG ×G BF (P) and EG ×G BP, respectively. It has the consequence that F (P) ⋊ δ(G) and P ⋊ G have
the same number of connected components.
This fact will be useful in our investigation of block theory.
3.3. Category algebras. We want to study the representations of a
transporter category. The concept of a category algebra is key to us,
see [10] for an account. To this end, we shall recall some basics about
category algebras. Let C be a small category and R be a commutative
ring with identity. Then the category algebra RC is defined to be the
free R-module R Mor C, with multiplication determined by composites
of morphisms of C.
Theorem 3.18 (Mitchell). Let C be a small category such that Ob C is
finite. If R is a commutative ring with identity, then (R-mod)C ≃ RCmod.
The constant functor R : C → R-mod corresponds to the RC-module
afforded by the free R-module R Ob C. We shall call R the trivial kCmodule. It plays the role of R in group representations. The trivial
module is indecomposable if and only if C is connected.
If two small categories are equivalent, then their category algebras
are Morita equivalent. Note that, although BF (C) ≃ BC, RF (C) is
not Morita equivalent to RC, as RF (C) almost always has more simple
modules (up to isomorphism).
It is straightforward to verify that kC e ∼
= (kC)e = (kC) ⊗k (kC)op .
The “diagonal subalgebra” of (kP ⋊ G)e ∼
= kP e ⋊ Ge , in the sense of
Boisen, is a transporter category algebra, by the following result.
Lemma 3.19. There are isomorphisms of algebras kC e ∼
= (kC)e =
op
kC ⊗k (kC) . In the case of C = P ⋊ G, we have
kP e ⋊ δ(G) ∼
= ∆(kP ⋊ G).
16
FEI XU
Proof. The first isomorphism is known and is easy to produce. For the
second, we define a map on the base elements
−1
(α ⊗ β op )(g, g −1) 7→ (αg) ⊗ (g βg −1),
and then extend it linearly.
Suppose C is an EI category. When D ⊂ C is convex, we can regard
every kD-module as a kC-module. It partially explains why convex subcategories (weakly convex subcategories for a transporter subcategory)
play an important role in our theory.
Definition 3.20. Let C be an EI category. Given a kC-module M, an
object x ∈ C is said to be M-minimal if for any y ∈ Ob C admitting
a non-isomorphism y → x, we must have M(y) = 0. Similarly, an
object z ∈ C is said to be M-maximal if for any y ∈ Ob C admitting a
non-isomorphism z → y, we must have M(y) = 0.
We define CM to be the coideal whose minimal objects are exactly
|{z}
z}|{
all the M-minimal objects. We also define CM to be the ideal whose
maximal objects are exactly all the M-maximal objects. We define the
support of M to be the full subcategory supp M consisting of objects
{x ∈ Ob C M(x) 6= 0}. We define the convex support suppc M of
z}|{
M to be CM ∩ CM , which is the convex hull of supp M, the smallest
|{z}
convex subcategory of C that contains supp M.
When there is no confusion, sometimes we call the set Ob(supp M)
(or Ob(suppc M)) the support (or convex support) of M.
Remark 3.21. We are interested in the representation theory of C =
P ⋊ G. If N ∈ kQ ⋊ H-mod for a transporter subcategory Q ⋊ H ⊂
P ⋊ G, then, by Lemma 3.5, supp N = QN ⋊ H for a unique subposet
QN , and suppc N = QcN ⋊ H. Here QN = supp(N ↓Q⋊1 ) and QcN =
suppc (N ↓Q⋊1 ). Since supp N (resp. suppcN ) and supp(N ↓Q⋊1) (resp.
suppc (N ↓Q⋊1)) determine each other and share the same objects, we
may also refer to the latter as the support (resp. the convex support)
of N.
The two G-coideals G (QN ) and G (QcN ) of P (see Definition 3.7) are
| {z }
| {z }
identical, because QN shares the same minimal objects with QcN .
We shall prove that for an indecomposable kP ⋊ G-module M, there
is “no hole” in its convex support, in the sense that M(x) 6= 0 for every
x ∈ suppc M. In other words, for an indecomposable M, supp M =
TRANSPORTER CATEGORY ALGEBRAS
17
suppc M. It explains why we introduce the concept of the convex support, and it will be used when we develop a theory of vertices and
sources later on.
Lemma 3.22. Suppose P is a poset and M is an indecomposable kPmodule. Then for any x ∈ Ob suppc M, M(x) 6= 0.
Proof. If P has several components, then suppc M must be contained
in exactly one of the components. Without loss of generality, we may
assume P is connected and suppose suppc M = P. Assume there is
some x ∈ Ob P such that M(x) = 0. By the assumption, x can not be
either maximal or minimal. We may consider its projective cover PM .
Let π : PM → M be the epimorphism. Then (ker π)(x) = PM (x). But
PM ∼
= ⊕i Pynii for M-minimal objects yi (all satisfying yi ≤ x). Since,
in a poset, between any two objects there is at most one morphism, it
forces (ker π)(z) = PM (z) for all z ≥ x. In turn it implies M(z) = 0
for all z ≥ x, a contradiction to suppc M = P.
Proposition 3.23. Let P ⋊ G be a connected transporter category and
M an indecomposable kP ⋊ G-module. Then for any x ∈ suppc M,
M(x) 6= 0. Thus supp M = suppc M.
Proof. Without loss of generality, suppose suppc M = P ⋊ G. Then
the restriction decomposes as a direct sum of indecomposable kP ⋊ 1modules
n
M
P⋊G
Mi .
M ↓P⋊1 =
i=1
The group G acts on the set {M1 , · · · , Mn } of indecomposable kP ⋊ 1modules. The kP ⋊ G-module M is indecomposable if and only if
there is only one G-orbit on the set {M1 , · · · , Mn }. These Mi ’s have
conjugate supports.
Now assume M(x) = 0 (x not maximal or minimal, by assumption).
It implies that M(x′ ) = 0 and hence Mi (x′ ) = 0, ∀i, for all x′ ∼
= x in
Ob(P ⋊ G). Thus if there exists a morphism x → z in Mor(P ⋊ G),
there exists x′ → z in Mor P for some x′ ∼
= x. Applying Lemma 3.22,
we find that Mi (z) = 0 for all i. It means that M(z) = 0 for every
z with a morphism x → z, a contradiction to our assumption that
suppc M = P ⋊ G.
By the definitions of limits, it is straightforward to prove the following isomorphisms.
Lemma 3.24. Suppose C is finite EI and D ⊂ C is a full subcategory.
Let M ∈ kC-mod. Then
18
FEI XU
M ↓D ;
(1) if D is a coideal and contains supp M, limC M ∼
= lim
−→D
−→
and
M ↓D .
(2) if D is an ideal and contains supp M, limC M ∼
= lim
←−D
←−
4. Vertices and Sources
In Section 2, we briefly explained, given a skew group algebra S ⋊ G,
how Dade conceived a theory of vertices and sources, through comparing it with various subalgebras S ⋊ H, for H ⊂ G. When P = G/H
with left G-multiplication, kP ⋊ G is Morita equivalent to kH. If we
take the trivial module k ∈ kP ⋊ G-mod (corresponding to k ∈ kHmod), a “vertex” of k, in the sense of Dade, would be T , a Sylow
p-subgroup of H. However, it actually means that k is projective relative to kP ⋊ T , which is not Morita equivalent to kT , and thus does
not match the classical theory for group algebras. See Example 4.23 for
more details. We shall fix the problem and get a sharpened definition.
4.1. Inclusions and restrictions. Let P be a G-poset. Suppose H
is a subgroup of G. We may regard P as an H-poset. Assume Q
is a H-subposet of P. Then we have two faithful functors and their
composite, which are inclusions of transporter categories,
ιP⋊H
Q⋊H
ιP⋊G
P⋊H
P⋊G P⋊H
ιP⋊G
Q⋊H = ιP⋊H ιQ⋊H : Q ⋊ H −→P ⋊ H −→P ⋊ G.
We shall study their effects on representations of these categories.
If P ′ happens to be a G-subposet of P, there are two similar faithful
functors and their composite as follows
′
P ′ ⋊G
ιP⋊G
P ′ ⋊G ιP ′ ⋊H
P ⋊G
ιP
′ ⋊H
′
′
ιP⋊G
P ′ ⋊G
: P ⋊ H −→ P ⋊ G −→ P ⋊ G.
′
P ⋊G
P⋊G P⋊H
Obviously in this case, ιP⋊G
P ′ ⋊G ιP ′ ⋊H = ιP⋊H ιP ′ ⋊H .
In general, let D and C be two small categories and τ : D → C be
a functor. Then τ induces a restriction along τ , written as Resτ : kCmod → kD-mod. When C = G is a group and D = H is a subgroup,
the restriction along the inclusion is the usual restriction ↓G
H . In the
present paper, we will chiefly be interested in the situation where τ is
an inclusion. When this is the case, we shall denote Resτ M by M ↓CD ,
for all M ∈ kC-mod. Let P ⋊ G be a transporter category, and Q ⋊ H
be a transporter subcategory. We have
P⋊G P⋊H
M ↓P⋊G
Q⋊H = M ↓P⋊H ↓Q⋊H ,
for all M ∈ kP ⋊ G-mod. We comment that the restriction ↓P⋊H
Q⋊H =
1kQ⋊H · − is a brutal truncation, not coming from a unital algebra
homomorphism.
TRANSPORTER CATEGORY ALGEBRAS
19
4.2. Kan extensions. In group representations, the restriction has
isomorphic left and right adjoints, called the induction and the coinduction. They are actually special cases of the Kan extensions. Let
D and C be two small categories and τ : D → C be a functor. Then
the restriction Resτ : kC-mod → kD-mod possesses both left and right
adjoints (Kan extensions) LKτ , RKτ : kD-mod → kC-mod. These are
well-known constructions in homological algebra. However they are
seldom used in representation theory since they are usually extremely
hard to compute. We shall see, in the representation theory of transporter categories, that it is possible to understand the Kan extensions
and then to apply these constructions.
By definition, given N ∈ kD-mod, we can construct two kC-modules,
LKτ N and RKτ N. Suppose x ∈ Ob C. Then
[LKτ N](x) = limτ /x Ñ and [RKτ N](x) = limx\τ N̄ .
−→
←−
Here τ /x and x\τ are categories over and under x, respectively. We
often just refer to them as an overcategory or an undercategory. The
objects of τ /x are pairs (y, α), where y ∈ Ob D and α ∈ HomC (τ (y), x);
while a morphism f : (y, α) → (z, β) is a morphism f ∈ HomD (y, z),
such that α = βτ (f ). By comparison, the objects of x\τ are pairs
(γ, w), with w ∈ Ob D and γ ∈ HomC (x, τ (w)). The morphisms are
defined accordingly. Meanwhile Ñ is the restriction along the canonical
functor (a projection) τ /x → D and N̄ is the restriction along x\τ →
D. For convenience, we shall abbreviate the defining formulas of Kan
extensions to
[LKτ N](x) = limτ /x N and [RKτ N](x) = limx\τ N,
−→
←−
in the rest of the present paper, despite the fact that N is not defined
on the over- and undercategories.
When C = G is a group and D = H is a subgroup, the three functors
associated to the inclusion are the usual restriction Resι =↓G
H , induction
G
G
LKι =↑H and coinduction RKι =⇑H in group representations. We
emphasize that in general LKτ ∼
6= RKτ . The computations of these
functors usually amounts to analyzing the structures of all relevant
over- and undercategories.
To be consistent, we shall write M ↓CD = Resτ M for all M ∈ kC-mod,
and N ↑CD = LKτ N and N ⇑CD = RKτ N, for all N ∈ kD-mod.
Let P ⋊ G be a transporter category, and Q ⋊ H be a transporter
subcategory. By earlier discussions, we have
P⋊G P⋊H
↓P⋊G
Q⋊H =↓P⋊H ↓Q⋊H .
20
FEI XU
Consequently the left and right adjoints of ↓P⋊G
Q⋊H satisfy
P⋊H P⋊G
↑P⋊G
Q⋊H =↑Q⋊H ↑P⋊H
and
P⋊H P⋊G
⇑P⋊G
Q⋊H =⇑Q⋊H ⇑P⋊H .
At this point, we shall compute several Kan extensions in the context
of transporter categories. This will be a cornerstone for our upcoming
developments.
Theorem 4.1. Let P ⋊ G be a transporter category, and Q ⋊ H be
a transporter subcategory. Suppose ι = ιP⋊G
Q⋊H is the inclusion functor.
Choose a set of left coset representatives [G/H] = {g1 , · · · , gn }.
(1) For each x ∈ Ob(P ⋊ G), ι/x (resp. x\ι),
`if not empty, is the
disjoint union of full subcategories ι/x = gi ∈[G/H] (ι/x)i (resp.
`
x\ι = gi ∈[G/H] (x\ι)i ). Moreover each (ι/x)i (resp. (x\ι)i )
has its skeleton isomorphic to (gi Q)≤x (resp. (gi Q)≥x ). Consequently, given a kQ ⋊ H-module N,
M
M
∼
lim(gi Q) N.
[N ↑P⋊G
N
](x)
=
lim
=
Q⋊H
−→
−→(ι/x)i
≤x
gi ∈[G/H]
gi ∈[G/H]
(resp.
[N ⇑P⋊G
Q⋊H ](x) =
M
gi ∈[G/H]
lim(x\ι) N ∼
=
←−
i
M
gi ∈[G/H]
lim(gi Q) N.)
←−
≥x
If βs ∈ HomP⋊G (x, z), then it induces a functor ι/x → ι/z
(resp. z\ι → x\ι), given by (y, αg) 7→ (y, βsαg) = (y, β s αsg)
′
(resp. (α′ g ′ , y ′) 7→ (α′ g ′βs, y ′) = (α′g βg ′s, y ′)), which defines a
map
P⋊G
P⋊G
[N ↑P⋊G
Q⋊H ](βs) : [N ↑Q⋊H ](x) → [N ↑Q⋊H ](z)
(resp.
P⋊G
P⋊G
[N ⇑P⋊G
Q⋊H ](βs) : [N ⇑Q⋊H ](x) → [N ⇑Q⋊H ](z))
P⋊G
that gives rise to the βs-action on N ↑P⋊G
Q⋊H (resp. N ⇑Q⋊H ).
P⋊G
P⋊G ∼
(2) The left and right adjoints of ↓P⋊H are ↑P⋊H = kP ⋊ G⊗kP⋊H −
∼
and ⇑P⋊G
P⋊H = HomkP⋊H (kP ⋊ G, −), respectively.
P⋊H ∼
and
(3) The left and right adjoints of ↓P⋊H
Q⋊H are ↑Q⋊H = lim
−→Q≤
∼
, respectively.
⇑P⋊H
Q⋊H = lim
←−Q≥
Proof. We shall see that (2) and (3) are special cases of (1), although (2)
can be established by classical constructions. We will only compute the
TRANSPORTER CATEGORY ALGEBRAS
21
left Kan extensions, and leave the right Kan extensions to the interested
reader to check.
In order to prove (1), we first abbreviate the inclusion functor to ι.
Suppose ι/x is not empty. Let (y, αs) be an object of ι/x. Choose
g1 , · · · , gn ∈ [G/H] to be a set of left coset representatives. Then
s = gi h for some gi and h ∈ H. We see that (h y, αgi) also belongs
to ι/x and it is isomorphic to (y, αs) in ι/x. Up to isomorphism,
every object of ι/x is of the form (y ′, αgi ) for some y ′ ∈ Ob(Q ⋊ H),
α ∈ Mor P and gi ∈ [G/H]. Moreover if γh : (y1, α1 gi ) → (y2 , α2 gj ) is
a morphism, we obtain equalities α1 = α2 gj γ and gi = gj h. The latter
implies gi H = gj H. Particularly, it tells us that two objects (y, αs),
(z, βt) of ι/x are connected by a zigzag of morphisms, or lie in the same
connected component, if and only if sH = tH. Moreover, since both h
and γ are uniquely determined, it also tells us that, between any two
objects of ι/x, there is at most one morphism. Thus the skeleton of
ι/x must be a poset, with up to |G : H| connected components.
Denote by (ι/x)i the subcategory of ι/x consisting of objects of the
form (y, αgih), where h ∈ H. Our calculation means that ι/x is the
disjoint union of (ι/x)i , each indexed by a left coset representative
gi ∈ [G/H].
Now we define a functor, between posets, (ι/x)i → (gi Q)≤x by
(y, αgih) 7→ gi h y on objects. This functor has a quasi-inverse, given
−1
on objects by z 7→ (gi z, βgi ), if z ∈ gi Q and HomP (z, x) = {β}.
Hence (ι/x)i ≃ (gi Q)≤x .
As to (2), `
since this is a special case of Q = P in (1), we find
P⋊G
that ιP⋊H ≃ gi ∈[G/H] (gi P)≤x . In this case each (gi P)≤x has termi−1
nal objects {((gi h) x, 1x gi h) h ∈ H}. Given that gj ∈ [G/H] is the
unique coset representative satisfying ggi ∈ gj H, which sends the ter−1
−1
minal object (gi x, 1x gi ) to the terminal object ((ggi ) (g x), 1g x ggi ) ∼
=
gj−1 g
( ( x), 1g x gj ). Thus
[N ↑P⋊G
limιP⋊G /x N ∼
=
P⋊H ](x) = −
→ P⋊H
M
−1
N(gi x),
gi ∈[G/H]
−1
−1
−1
and the G-action is determined by N(gi x) → N(gj (g x)) ∼
= N(gj x).
Meanwhile, the restriction ↓P⋊G
P⋊H is induced by the injective unital algebra homomorphism kP ⋊ H → kP ⋊ G. So are its adjoints. The
functor that we just built is isomorphic to kP ⋊ G ⊗kP⋊H −, identified
with ↑G
H , in Section 2, used by Boisen and Dade.
22
FEI XU
We turn to (3). By (1), ιP⋊H
Q⋊H /x ≃ Q≤x , for each x ∈ Ob(P ⋊ G). It
follows that
N.
N ↑P⋊H
N](x) = limιP⋊H /x N ∼
= lim
Q⋊H (x) = [LKιP⋊H
−→Q≤x
−→ Q⋊H
Q⋊H
For the interested reader, when analysing x\ι 6= ∅, we should notice
that, for each (αs, y) ∈ Ob(x\ι), there is an isomorphism (αs, y) ∼
=
−1
−1
(s α, s y). The left coset representatives provide a set of right representatives [H\G] = {gi−1 }. Since s = hgi−1 for a unique i, we obtain
−1
−1
an isomorphism (αs, y) ∼
= (h αgi−1, h y) in x\ι. Hence two objects
(αs, y) and (βt, z), of x\ι, belong to the same connected component,
if and only if Hs = Ht. We can also verify that x\ι has a poset
as its skeleton. Denote by
Q (x\ι)i the connected component indexed
−1
by gi such that x\ι = g−1 ∈[H\G] (x\ι)i . It follows that the funci
−1
tor (x\ι)i → (gi Q)≥x , defined on objects by (αhgi−1, y) → gi h y is an
equivalence of categories.
Occasionally we will deal with the case where P ′ is a G-subposet
P ′ ⋊G
P⋊G P⋊H
of P. Under the circumstance, we will have ↓P⋊G
P ′ ⋊G ↓P ′ ⋊H =↓P⋊H ↓P ′ ⋊H ,
P⋊H P⋊G
P ′ ⋊G P⋊G
P⋊H P⋊G
P ′ ⋊G P⋊G
↑P ′ ⋊H ↑P ′ ⋊G =↑P ′ ⋊H ↑P⋊H , and ⇑P ′ ⋊H ⇑P ′ ⋊G =⇑P ′ ⋊H ⇑P⋊H .
From our proof of the above theorem, if the (convex) support of
an indecomposable kP ⋊ H-module L is Q ⋊ H, then the support of
g
L ↑P⋊G
P⋊H can be larger, and contains (Q ⋊ H), ∀g ∈ G.
Definition 4.2. Suppose Q ⋊ H is a transporter subcategory of P ⋊ G.
Let N ∈ kQ ⋊ H-mod. Fix an element g ∈ G. We define a k g (Q ⋊ H)module g N as follows. It equals N as a vector space, with g N(g x) =
N(x) for each x ∈ Ob(Q ⋊ H). While for n ∈ g N(g x), g x ∈ Ob g (Q ⋊ H),
and g (αh) : g x → g y, we define (g (αh))(n) = (αh)(n). We shall call g N
a conjugate of N.
If g ∈ NG (Q ⋊ H), then g N is a kQ ⋊ H-module.
Corollary 4.3. Let P ⋊ G be a transporter category and Q ⋊ H be a
transporter subcategory.
(1) Let N ∈ kQ ⋊ H-mod. Then there is a split surjection
M
gi
P⋊G ∼
(N ↑P⋊H
N ↑P⋊G
Q⋊H ).
Q⋊H ↓P⋊H =
gi ∈[G/H]
(2) Let N ∈ kQ ⋊ H-mod. Then there is a split surjection
P⋊G
p : N ↑P⋊G
Q⋊H ↓Q⋊H → N.
In particular if N ′ ∈ kP ⋊ H-mod, as kP ⋊ H-modules every
P⋊G
′
summand of N ′ ↑P⋊G
P⋊H ↓P⋊H is isomorphic to N .
TRANSPORTER CATEGORY ALGEBRAS
23
(3) If N is an indecomposable kQ ⋊ H-module with support Q ⋊ H,
then N ↑P⋊H
Q⋊H is an indecomposable kP ⋊H-module with support
Q
⋊H.
|{z}
(4) If L is an indecomposable kP ⋊H-module with supp L = P ⋊H,
then every indecomposable summand of L ↑P⋊G
P⋊H has P ⋊ G as
its convex support.
Proof. To prove the first statement, we write
P⋊H
P⋊G
N ↑P⋊G
Q⋊H = (N ↑Q⋊H ) ↑P⋊H ,
and then use Theorem 4.2 (2).
The second statement follows from the first, on restriction further
down to Q ⋊ H. The direct summand corresponding to gi = 1 is a
copy of N.
For 3), since Q ⋊ H is a full subcategory of P ⋊ H, N ↑P⋊H
Q⋊H is
P
c
indecomposable. By N ↑P⋊H
=
N
↑
and
supp
N
=
supp
N
=
Q
⋊ H,
Q
Q⋊H
P⋊H
we know N ↑Q⋊H has |{z}
Q ⋊H as its support.
If M is an indecomposable summand of L ↑P⋊G
P⋊H , then M ↓P⋊H is
P⋊G P⋊G ∼
|G:H|
. Thus every summand of
a direct summand of L ↑P⋊H ↓P⋊H = L
M ↓P⋊H is isomorphic to L. Then 4) follows from it.
Note that the last statement may not be true if supp L 6= P ⋊ H.
The first statement can be regarded as a generalization of a standard
result in group representations. However, usually the direct summands
P⋊G
of N ↑P⋊G
Q⋊H ↓Q⋊H need not be isomorphic to each other.
Example 4.4. Consider the transporter category P ⋊ G in Example
3.2
1z 1,1z g
1x 1
?⑧ z _❄❄
❄❄ β1
α1 ⑧⑧
❄❄
⑧
⑧
❄❄
⑧ 1y g
⑧⑧
+y
f
8xk
1y 1
1x g
Suppose K = 1 is the trivial subgroup of G and R = {x} is the
subposet consisting of a single object x. Then R ⋊ K = {x} × 1. Let
k x be the trivial k{x} × 1-module. It can also regarded as an (atomic)
kP × 1-module.
P⋊G
∼
(1) The induced module k x ↑P⋊G
{x}×1 is given by k x ↑{x}×1 (x) =
P⋊G
2
∼
k x ↑P⋊G
{x}×1 (y) = k as vector spaces and k x ↑{x}×1 (z) = k
because ι/z is the disjoint union of two trivial posets {(y, β1)}
24
FEI XU
and {(y, αg)}. One may check that k x ↑P⋊G
{x}×1 is indecomposable.
P×1
P⋊G
P⋊G
Moreover k x ↑{x}×1 = (k x ↑{x}×1 ) ↑P×1 , with k x ↑P×1
{x}×1 (y) = 0
P×1
and k x ↑{x}×1 (z) = k.
P⋊G
P⋊G
∼
(2) By comparison, k x ↑P⋊G
P×1 is given by k x ↑P×1 (x) = k x ↑P×1
(y) ∼
= k as vector spaces and k x ↑P⋊G
P×1 (z) = 0.
P⋊G P⋊G ∼
(3) On restriction to P × 1, k x ↑{x}×1 ↓P×1 = k Qx ⊕ k Qy , where
Qx = x → z and Qy = y → z, satisfying g Qx = Qy . Here k Qx
and k Qy are the trivial kQx ×1- and kQy ×1-modules, considered
as indecomposable kP × 1-module. Note that g k Qx ∼
= k Qy .
P⋊G P⋊G ∼
(4) On restriction to {x} × 1, k x ↑{x}×1 ↓{x}×1 = k x .
In the end, we record some technical statements that we need in
proving the Mackey formula.
Corollary 4.5. Let P ⋊ G be a transporter category, and Q be a Gsubposet of P. Also let H be a subgroup of G and R ⋊ K be a transporter subcategory of P ⋊ G. Then for every M ∈ kP ⋊ H-mod and
N ∈ kQ ⋊ H-mod, there exist isomorphisms
P⋊H Q⋊G
P⋊G ∼
(1) M ↑P⋊G
P⋊H ↓Q⋊G = M ↓Q⋊H ↑Q⋊H ;
P⋊(K∩H)
∼
↑
(2) N ↑P⋊H ↓P⋊H
; and
= N ↓Q⋊H
(3) N
Q⋊H P⋊(K∩H)
P⋊H P⋊H
↑Q⋊H ↓R⋊(K∩H) ∼
=
N
Q⋊(K∩H) Q⋊(K∩H)
P⋊(K∩H) P⋊(K∩H)
↓Q⋊H
Q⋊(K∩H) ↑Q⋊(K∩H) ↓R⋊(K∩H) .
Proof. For (1), we have
P⋊G
M ↑P⋊G
P⋊H ↓Q⋊G (x) =
L
gi ∈[G/H]
limP M
−→ ≤x
L
∼
= gi ∈[G/H] M(x)
=
L
gi ∈[G/H] [M
↓P⋊H
Q⋊H (x)]
L
∼
= gi ∈[G/H] lim
−→Q
≤x
M ↓P⋊H
Q⋊H
Q⋊G
= [M ↓P⋊H
Q⋊H ↑Q⋊H ](x),
for each x ∈ Ob(Q ⋊ G). Here we used the fact that x is the terminal
object of both P≤x and Q≤x . Then one may readily check that these
isomorphisms assemble to a module isomorphism.
As for (2), it comes from Theorem 4.1 (2) and the isomorphism
∼
[N ↓P⋊H
[limQ N] ↓P⋊H
P⋊(K∩H) ].
P⋊(K∩H) = lim
−→Q≤
−→ ≤
TRANSPORTER CATEGORY ALGEBRAS
25
Now (3) follows from (2), because
P⋊(K∩H)
P⋊H
P⋊H P⋊H
N ↑P⋊H
Q⋊H ↓R⋊(K∩H) = [N ↑Q⋊H ↓P⋊(K∩H) ] ↓R⋊(K∩H)
P⋊(K∩H)
P⋊(K∩H)
∼
= [N ↓Q⋊H
Q⋊(K∩H) ↑Q⋊(K∩H) ] ↓R⋊(K∩H) .
4.3. Relative projectivity. We want to develop a theory of vertices
and sources for transporter category algebras. It will generalize the
original theory for group algebras, when we regard groups as transporter categories. More precisely, let P ⋊ G be a transporter category
and M be an indecomposable kP ⋊ G-module. We shall define a vertex VM , of M, to be a weakly convex transporter subcategory Q ⋊ H,
unique up to conjugacy in P ⋊ G, and its source to be an indecomposable kQ ⋊ H-module, which is also unique up to conjugacy, such that
M N ↑P⋊G
Q⋊H .
Our generalization is motivated by two existing theories, one for fully
group-graded algebras [5, 4], and the other for EI category algebras [10].
It relies on the observation that transporter category algebras are both
fully group-graded algebras and EI category algebras.
Suppose C is a finite category and E is a subcategory. Then the
co-unit of the adjunction between ↓CE and ↑CE gives rise to a canonical
map ǫM : M ↓CE ↑CE → M for each M ∈ kC-mod. In general, ǫM is not
surjective, and one may easily construct examples in which ǫM = 0.
From now on, we shall assume C to be finite EI. We want to recall a
fraction of the theory of vertices and sources for EI category algebras
[10]. In fact, in order to get better results, we must modify and improve
it. First of all, we shall see ǫM can be surjective for some convex
subcategory E ⊂ C.
Lemma 4.6. The canonical map M ↓CE ↑CE → M is surjective if E contains suppc M.
Proof. We may assume without loss of generality that E = suppc M.
Then it follows from a basic property of the Kan extensions that the
counit gives an isomorphism M ↓CE ↑CE (x) ∼
= M(x), on every x ∈ Ob E.
In light of this lemma, we restrict M to its convex support in order to
sharpen the vertices (to non-full subcategories) of M given in [10]. This
restriction will not change the nature of our discussion, as the category
of kC-modules with support in D is canonically isomorphic to kD-mod,
as long as C is EI and D is (full) convex. Let us write kD-mod◦ for the
subcategory of kD-mod, consisting of modules whose convex supports
26
FEI XU
are exactly D. Thus kC-modules can be parametrized by their convex
supports. It means that kC-mod is patched up by kD-mod◦ , with D
running over the set of all convex subcategories of C. It motivates our
improved definition of the relative projectivity for category algebras.
Definition 4.7. Let C be a finite EI category and M be a kC-module.
Suppose D is a subcategory of C. Then we say M is projective relative
to D, or relatively D-projective, if the canonical map, still written as
ǫM ,
suppc M suppc M
(M ↓Csuppc M ) ↓D
↑D
→ M ↓Csuppc M
is a split surjection.
It is known from [10] that D contains all the M-minimal objects.
Lemma 4.6 is improved by the following statement in [10]. Here we
offer a different proof.
Proposition 4.8. Suppose M is a kC-module and M is relatively Dprojective for a full subcategory D ⊂ suppc M. Then
c
supp
(M ↓Csuppc M ) ↓D
M suppc M
↑D
→
M ↓Csuppc M
is an isomorphism. Under the circumstance suppc (M ↓D ) = D.
Proof. Without loss of generality, we assume suppc M = C. Then by
assumption M ↓D ↑C → M is split surjective. It implies that, for each
x ∈ Ob C, there is a map M(x) → M ↓D ↑C (x) = limι/x M ↓D . But it
−→
follows from the universal property of limits that this map has to be
an isomorphism, for every x. Since it is straightforward to check the
naturality, we obtain the claimed isomorphism of modules.
To show suppc (M ↓D ) = D, we only need to prove that M ↓D
takes non-zero values on maximal objects of D. In fact, assume x is
a maximal object of D and M ↓D (x) = M(x) = 0. Then, for each
y ∈ Ob suppc M that admits a morphism from x, we have M(y) = 0.
It implies that x 6∈ Ob suppc M, which is a contradiction.
When C is a transporter category, the following result is an immediate consequence of Theorem 4.1.
Proposition 4.9. Let P ⋊ G be a transporter category and P ′ be a
G-subposet of P. Assume M ∈ kP ⋊ G-mod. Then the following are
equivalent
(1) M is projective relative to P ′ ⋊ G;
(2) M ↓P⋊H is projective relative to P ′ ⋊ H for some H ⊂ G;
(3) M ↓P⋊1 is projective relative to P ′ ⋊ 1;
(4) M ↓P⋊K is projective relative to P ′ ⋊ K for every K ⊂ G.
TRANSPORTER CATEGORY ALGEBRAS
27
Proof. Without loss of generality, assume suppc M = P ⋊ G. As kvector spaces, M ↓P ′ ⋊H ↑P⋊H ∼
= M ↓P ′ ⋊G ↑P⋊G by Theorem 4.1 (3), for
every H ⊂ G.
Consider the natural kP ⋊ G-morphism ǫM
M ↓P ′ ⋊G ↑P⋊G → M ↓P⋊G .
Regarded as a kP ⋊ H-map, it is exactly the counit of adjunction
M ↓P ′ ⋊H ↑P⋊H = (M ↓P⋊H ) ↓P ′ ⋊H ↑P⋊H → M ↓P⋊H .
These two maps are identical as k-maps. Thus one of the morphism
being a k-isomorphism will imply the same for the other. However, such
a k-isomorphism, if exists, is automatically a module isomorphism.
For a module M, there usually exist proper subcategories of suppc M,
making ǫM split surjective. These subcategories do not have to be full.
We shall discuss the details in the context of transporter categories.
The following characterization (slightly modified from a result in [10])
will be used for C = P ⋊ G and D = Q ⋊ H ⊂ P ⋊ G.
Proposition 4.10. Let M be a kC-module. Suppose D ⊂ suppc M
such that the canonical map
c
supp
(M ↓Csuppc M ) ↓D
M suppc M
↑D
→
M ↓Csuppc M
is surjective. Then the following are equivalent:
suppc M suppc M
(1) (M ↓Csuppc M ) (M ↓Csuppc M ) ↓D
↑D
;
suppc M
C
(2) there is a kD-module N such that (M ↓suppc M ) N ↑D
;
(3) if 0 → A → B → C → 0 is an exact sequence of kC-modules,
with supports in suppc M, which splits upon restriction to kDsequences, then the sequence HomkC (M, B) → HomkC (M, C) →
0 is exact;
(4) if 0 → A → B → M → 0 is an exact sequence of kC-modules,
with supports in suppc M, which splits as an exact sequence of
kD-modules, then it splits as an exact sequence of kC-modules;
(5) M is relatively D-projective.
We state several examples of relative projectivity below.
Proposition 4.11. Every kP ⋊ G-module M is projective relative to
c
PM
⋊ S, where S is a Sylow p-subgroup of G.
Proof. The claim is due to Boisen (for fully group-graded algebra [4]).
The next result is actually [12, 2.3.1(2)]. We rewrite and include it
here as a generalization to a well-known statement in group representations. Note that kP ⋊ G is a Gorenstein algebra.
28
FEI XU
Proposition 4.12. Let M ∈ kP ⋊ G-mod. Then it is of finite projective dimension (equivalently, of finite injective dimension) if and only
if, for each x ∈ Ob(P ⋊ G), Mx is projective relative to {x} ⋊ 1.
When P is a point, the above statement says that M ∈ kG-mod is
of finite projective dimension (equivalently, projective) if and only if it
is projective relative to 1.
Example 4.13. We know from [10] that a simple kP ⋊ G-module is
written as Sx,V . It is determined by a simple kGx -module V = Sx,V (x),
and its support is hxi, consisting of the G-orbit of x. The projective
cover of Sx,V is Px,V . We know Px,V (x) is the projective cover of the
kGx -module V , and Px,V = Px,V (x) ↑P⋊G
{x}×Gx .
The module Px,V is relatively {x} × 1-projective. While Sx,V is relatively {x} × H-projective, for a p-subgroup H ⊂ Gx satisfying the
condition that V is projective relative to H. Comparing with [10], this
makes more sense.
We state the following result, also for the completion of the theory.
Proposition 4.14. An indecomposable kP ⋊ G-module P is projective
if and only if P is projective relative to {x}×1 for some x ∈ Ob P ⋊ G.
4.4. Vertices and sources. Using the relative projectivity, we introduce the concepts of vertices and sources. To this end, we shall establish
a Mackey formula.
Example 4.15. Consider Example 4.4 again. The module k x ↑P⋊G
{x}×1
is indecomposable. If we use the method of Dade and Boisen, then
its “vertex” can only be of the form P ⋊ H. Thus its “vertex” would
have to be P ⋊ G (by direct computation) and the “source” would be
itself. By contrast, it is more tempting to take {x} × 1 (or {y} × 1)
as a vertex while k x (or k y ) as a source. To show that this kind of
choices are feasible in general, we need a generalized Mackey formula
for transporter category algebras.
Suppose Q ⋊ H is a transporter subcategory of P ⋊ G. Let N ∈
kQ ⋊ H-mod. Fix an element g ∈ G. We defined the conjugate g N ∈
k g (Q ⋊ H)-module of N. If g ∈ NG (Q ⋊ H), then g N is a kQ ⋊ Hmodule. In general if M ∈ kP ⋊ G-mod is relatively Q ⋊ H-projective,
then M ∼
= g M is also relatively g (Q ⋊ H)-projective.
From Definition 3.7, if P ⋊ G is a transporter category and Q ⋊ H
z}|{
is a transporter subcategory, then H Q ⋊H becomes a weak ideal.
TRANSPORTER CATEGORY ALGEBRAS
29
Lemma 4.16. If Q ⋊ H is weakly convex in P ⋊ G, then Q ⋊ H bez}|{
comes a coideal of H Q ⋊H. Under the circumstance, we observe that
z}|{
NG (Q ⋊ H) ⊂ NG ( H Q ⋊H).
Proof. The first claim is by definition, so we turn to prove the second.
z}|{
In fact, suppose g ∈ NG (Q ⋊ H). Let y ∈ Ob( H Q ⋊H). There is
a (poset) morphism α1 : y → x for some x ∈ Ob(Q ⋊ H). Since
z}|{
g
(α1) = g α1 : g y → g x, by definition g y ∈ Ob( H Q ⋊H) because
z}|{
g
x ∈ Ob(Q ⋊ H). It follows that g (αh) ∈ Mor( H Q ⋊H) for every αh ∈
z}|{
Mor(Q ⋊ H) and g ∈ NG (Q ⋊ H). Thus NG (Q ⋊ H) ⊂ NG ( H Q ⋊H).
Now we are ready to establish a Mackey type formula. At first, we
provide an illuminating (more or less) example.
Example 4.17. Let P be a poset, Q and R be subposets. Suppose
N ∈ kQ-mod. We compute N ↑PQ ↓PR . To this end, we fix an object
x ∈ Ob R and analyse N ↑PQ ↓PR (x) = limQ N. For instance, in either
−→ ≤x
of the following two cases: N is the zero functor on Q≤x , or Q≤x = ∅,
the limit is zero. Meanwhile, if x happens to be an object of Q (in the
intersection Q ∩ R), the limit is just N(x) because x is the terminal
object of Q≤x .
In general Q≤x = Q ∩ P≤x . If Q≤x ⊂ R≤x , we will have Q≤x =
N ↓Q∩R ,
lim
(Q∩R)≤x . It has the consequence that limQ N ∼
−→ ≤x = −→(Q∩R)≤x
R
or N ↑PQ ↓PR (x) ∼
= N ↓Q
Q∩R ↑Q∩R (x). Thus if every x ∈ Ob R satisfies
the (stronger) condition that P≤x = R≤x , we obtain
N ↑P ↓P ∼
= N ↓Q ↑R .
Q R
Q∩R Q∩R
The property of R is equivalent to saying that R is an ideal in P (with
no reference to modules).
Set supp N = QN . In practice, we only need to ask ( QN )≤x ⊂ R≤x
|{z}
lim(Q∩R) N ↓Q∩R .
for every x ∈ Ob R, in order to get limQ N ∼
=
−→
−→ ≤x
≤x
The reason is that by Lemma 3.21
lim
limQ N ∼
N
−→ ≤x = −→Q≤x ∩Q
N
|{z}
because Q≤x ∩ QN is a coideal, containing supp N ↓Q≤x = (QN )≤x , in
|{z}
Q≤x . Moreover
limQ ∩Q N = limQ ∩(Q ) N = lim(Q∩R) ∩Q N
−→ ≤x |{z}
−→ ≤x |{z}
−→
N
N ≤x
N
≤x
|{z}
30
FEI XU
since the indexing posets are the same, and
N ↓Q∩R
lim(Q∩R) ∩Q N ∼
= lim
−→(Q∩R)≤x
−→
N
≤x
|{z}
as (Q∩R)≤x ∩ QN is a coideal in (Q∩R)≤x , containing supp N ↓(Q∩R)≤x .
|{z}
Note that QN = ( QN )c cannot be replaced by either QN or QcN .
|{z}
|{z}
With Theorem 4.1 (3), the above example can be readily extended.
Let P be a G-poset, and Q, R be two G-subposets. Given N ∈ kQ⋊Gmod such that supp N = QN ⋊ G and such that ( QN )≤x ⊂ R≤x for
|{z}
every x ∈ Ob R, then
Q⋊G
R⋊G
P⋊G ∼
N ↑P⋊G
Q⋊G ↓R⋊G = N ↓(Q∩R)⋊G ↑(Q∩R)⋊G .
Now we prove a generalized Mackey formula. The above special form
will also be used later on.
Theorem 4.18 (Mackey formula for transporter categories). Suppose
that Q ⋊ H is a transporter subcategory of P ⋊ G. Let N ∈ kQ ⋊ Hmod and supp N = QN ⋊ H. Assume R ⋊ K is a transporter subcategory of P ⋊ G such that g ( QN )≤x ⊂ R≤x , for all g ∈ G and x ∈ Ob R
|{z}
(which implies that R∩G (QcN ) is an ideal in G (QcN ) ⊃ suppc (N ↑P⋊G
Q⋊H ).)
| {z }
| {z }
Then
P⋊G
N ↑P⋊G
Q⋊H ↓R⋊K =
M
R⋊K
[g (N ↓Q⋊H
(Rg ∩Q)⋊(K g ∩H) )] ↑(R∩g Q)⋊(K∩g H) .
g∈[K\G/H]
Proof. Since g (QN ) = Qg N , the condition on R ⋊ K implies that, for
all g ∈ G and x ∈ Ob R = Ob(R ⋊ K),
(∗)
g
g
N ↓R∩g Q .
N∼
lim
= lim
−→(R∩g Q)≤x
−→g Q≤x
TRANSPORTER CATEGORY ALGEBRAS
31
We start with the Mackey formula for fully group-graded algebras,
and then analyse various functors that are involved.
P⋊G
N ↑P⋊G
Q⋊H ↓R⋊K
P⋊G P⋊G
P⋊K
= [(N ↑P⋊H
Q⋊H ) ↑P⋊H ↓P⋊K ] ↓R⋊K
L
g
P⋊H
P⋊K
P⋊K
(N ↑P⋊H
Q⋊H ↓P⋊(K g ∩H) )] ↑P⋊(K∩g H) ↓R⋊K
Thm 2.1
=
Cor 4.5(1)
L
g (P⋊H) g (P⋊H)
P⋊(K∩g H)
∼
= g∈[K\G/H] [g N ↑g (Q⋊H) ↓g (P⋊(K g ∩H)) ] ↓R⋊(K∩g H) ↑R⋊K
R⋊(K∩g H)
=
L
g∈[K\G/H] [
g∈[K\G/H] [
g
g
g
H P⋊ H
R⋊K
N ↑P⋊
g Q⋊g H ↓R⋊(K∩g H) ] ↑R⋊(K∩g H)
Cor 4.5(2)
L
g
gH
P⋊(K∩g H) P⋊(K∩g H)
R⋊K
∼
= g∈[K\G/H] [g N ↓g Q⋊
Q⋊(K∩g H) ↑g Q⋊(K∩g H) ↓R⋊(K∩g H) ] ↑R⋊(K∩g H)
(∗)
L
g Q⋊g H
R⋊(K∩g H)
R⋊K
∼
= g∈[K\G/H] [g N ↓(R∩
g Q)⋊(K∩g H) ↑(R∩g Q)⋊(K∩g H) ] ↑R⋊(K∩g H)
=
L
g∈[K\G/H]
=
L
g∈[K\G/H] [
g
g
g
Q⋊ H
R⋊K
N ↓(R∩
g Q)⋊(K∩g H) ↑(R∩g Q)⋊(K∩g H)
g
R⋊K
(N ↓Q⋊H
(Rg ∩Q)⋊(K g ∩H) )] ↑(R∩g Q)⋊(K∩g H) .
The conditions in the above theorem seem to be complicated and
asymmetric. However we will often find ourselves in a situation where
R ⊂ P is an ideal, and then the Mackey formula can be applied.
(Mackey formula) Suppose that Q ⋊ H is a transporter subcategory
of P ⋊ G. Let N ∈ kQ ⋊ H-mod. Assume R ⋊ K is a transporter
subcategory of P ⋊ G such that R ⊂ P is an ideal. Then
M
P⋊G
R⋊K
N ↑P⋊G
↓
=
[g (N ↓Q⋊H
Q⋊H R⋊K
(Rg ∩Q)⋊(K g ∩H) )] ↑(R∩g Q)⋊(K∩g H) .
g∈[K\G/H]
We shall demonstrate that the Mackey formula is as powerful as we
would have expected. The idea is to apply the Mackey formula to transc
porter subcategories of suppc M = PM
⋊ G. Fix an indecomposable
kP ⋊ G-module M. We shall show there exist minimal weakly convex
transporter subcategories (contained in suppc M), relative to which M
is projective, and furthermore they are conjugate in P ⋊ G. This will
be established in two steps.
Theorem 4.19. Let M be an indecomposable kP ⋊ G-module. Supc
pose suppc M = PM
⋊ G. Then
32
FEI XU
c
(1) there is a connected weak ideal of PM
⋊ G, denoted by Q ⋊ H,
unique up to conjugacy in P ⋊ G, such that M is relatively
Q ⋊ H-projective and such that M is relatively R ⋊ K-projective,
c
where R ⋊ K is a weak ideal of PM
⋊ G, if and only if R ⋊ K
contains a conjugate of Q ⋊ H.
(2) there is an indecomposable kQ ⋊ H-module L, unique up to
c ⋊G
PM
conjugacy in NG (Q ⋊ H), such that M ↓P⋊G
c ⋊G L ↑Q⋊H , and
PM
such that its convex support is exactly Q ⋊ H. Moreover L
M ↓Q⋊H .
c
Proof. Without loss of generality, we assume PM
= P. Suppose Q ⋊ H
is a minimal weak ideal, with respect to inclusion, such that M is
relatively Q ⋊ H-projective. Then M must be relatively g (Q ⋊ H)projective and moreover g (Q ⋊ H) has to be minimal as well, for every
g ∈ G. By choice, all these g (Q ⋊ H) have to be connected.
Let R ⋊ K be another weak ideal such that M is relatively R ⋊ KP⋊G P⋊G P⋊G
projective. We consider the module M ↓P⋊G
Q⋊H ↑Q⋊H ↓R⋊K ↑R⋊K . By assumption, M is a direct summand of it. It follows from the Mackey
formula that there exists some g ∈ [K\G/H] and an indecomposable
L′ ∈ k(R ∩ g Q) ⋊ (K ∩ g H)-mod, such that M L′ ↑P⋊G
(R∩g Q)⋊(K∩g H) .
g
g
By the minimality of (Q ⋊ H), we must have (R ∩ Q) ⋊ (K ∩ g H) =
g
(Q ⋊ H), that is, g (Q ⋊ H) ⊂ R ⋊ K.
−1
Set L = g L′ ∈ kQ ⋊ H-mod. Then it is indecomposable and M
P⋊G
P⋊G P⋊G ∼
n
′′
L ↑P⋊G
Q⋊H . From M ↓Q⋊H L ↑Q⋊H ↓Q⋊H = L ⊕ L (for some integer
n ≥ 1), such that L′′ is the direct sum of modules induced from proper
′′′
subcategories of Q ⋊ H, we deduce that L M ↓P⋊G
Q⋊H . If L is another
′′′
indecomposable kQ ⋊ H-module such that M L′′′ ↑P⋊G
Q⋊H , then L
P⋊G
∼
L ↑P⋊G
Q⋊H ↓Q⋊H . Applying the Mackey formula again, we find that L =
g ′′′
L for some g ∈ NG (Q ⋊ H).
The convex support of L has to be the whole Q ⋊ H, because if
L(x) = 0 at a maximal object of Q ⋊ H, then L ↑P⋊G
Q⋊H (y) = 0 for
all y ∈ Ob(P ⋊ G) that admits a morphism from x. It would imply
that M(y) = 0 for those objects y, and thus suppc M 6= P ⋊ G, a
contradiction.
c
Since every kP ⋊ G-module M is relatively PM
⋊ S-projective (Theorem 2.3), where S is a Sylow p-subgroup of G, the weak ideal Q ⋊ H
c
of PM
⋊ G in the preceding theorem must satisfy the condition that
H is a p-subgroup of G. We also emphasize that this Q ⋊ H is weakly
convex in P ⋊ G.
Given the generalized Mackey formula, we propose an explicit algorithm for finding a Q ⋊ H in the preceding theorem. Suppose M ∈
TRANSPORTER CATEGORY ALGEBRAS
33
kP ⋊ G-mod is indecomposable. Then according to Dade [5] and Boisen
[4], there exists a p-subgroup H ′ , minimal up to conjugations in G, such
c
that M is relatively PM
⋊ H ′-projective. By the Mackey formula, it is
′
easy to see that H must be conjugate to H in Theorem 4.19. For simc
plicity, we assume H ′ = H. Suppose N is an indecomposable kPM
⋊Hc
module (unique up to conjugations by NG (PM ⋊ H) = NG (H)), such
P c ⋊G
c ⋊G N ↑ M
. From [10], we know there exists the smallest
that M ↓PM
c
c
ideal D of PM ⋊ H, such that N ∼
= N ↓D ↑PM ⋊H . However, by Lemma
c
?, D must be of the form V ⋊ H, for some H-ideal V of PM
. The
c
subcategory V ⋊ H is a weak ideal in PM ⋊ G, and a weakly convex
transporter subcategory in P ⋊ G. (Different choices of N will result in
different V ′ ⋊ H which are conjugate to V ⋊ H, by elements of NG (H).)
We shall prove that V ⋊ H meets the requirements in Theorem 4.19.
The above V ⋊ H, and its conjugates, are actually minimal weakly
convex transporter subcategories, relative to which M is projective.
Proposition 4.20. Let M be an indecomposable kP ⋊ G-module. Supc
pose R ⋊ K is a weakly convex transporter subcategory of PM
⋊ G, relative to which M is projective. Then a conjugate of V ⋊ H, as in the
preceding paragraph, is contained in R ⋊ K.
Proof. Without loss of generality, we assume suppc M = P ⋊ G. The
module M is projective relative to both P ⋊ H and R ⋊ K. By the
Mackey formula, there exists g ∈ G such that H ⊂ g K and M is
projective relative to g R ⋊ H. Let L be an indecomposable k g R ⋊
H-module such that M
L ↑P⋊G
The module N = L ↑P⋊H is
g R⋊H .
indecomposable as a kP ⋊ H-module, satisfying that M N ↑P⋊G .
Since the indecomposable kP ⋊ H-module N is projective relative to
g
R ⋊ H, by the construction of V ⋊ H, we get V ⋊ H ⊂ g R ⋊ H. It
−1
implies that g (V ⋊ H) ⊂ R ⋊ K.
By the minimality, we know that V ⋊ H, constructed before Proposition 4.20, are conjugate to those Q ⋊ H in Theorem 4.19.
Now we are ready to introduce vertices and sources for indecomposable modules.
Definition 4.21. Let M be an indecomposable kP ⋊ G-module. Then
c
a minimal weakly convex transporter subcategory VM ⋊ H ⊂ PM
⋊
G, relative to which M is projective, is called a vertex of M. An
c ⋊G
PM
c ⋊G L ↑
indecomposable kVM ⋊ H-module L, such that M ↓PM
VM ⋊H , is
called a source for M.
The source L for M has (convex) support VM ⋊ H.
34
FEI XU
Proposition 4.22. If N is an indecomposable kR ⋊ K-module with
vertex Q ⋊ H and source U, then the (convex) support of N must be
K Q ⋊K.
|{z}
Proof. In fact, by direct calculations, U ↑R⋊K
Q⋊H , and hence N can only
possibly takes non-zero values on K Q ⋊K (state this fact in an earlier
|{z}
,
we
find N ↓R⋊K
subsection). From U N ↓R⋊K
Q⋊H
Q⋊H is non-zero on every
object of Q ⋊ H. It implies N is non-zero on every object of K Q ⋊ K,
and thus always non-zero on every object of K Q ⋊K.
|{z}
The upcoming result demonstrate connections between our constructions and the classical ones.
Proposition 4.23. Let P be a connected G-poset. Let M be an indecomposable kG-module with a vertex Q. Suppose κM is the restriction
of M along P ⋊ G → G. Then κM is an indecomposable kP ⋊ Gmodule with a vertex P ⋊ Q.
Proof. Since LKπ κM = M, κM is indecomposable if and only if M
is. Meanwhile the convex support of κM is the whole category P ⋊ G.
P⋊G ∼ G
∼ G
Thus ↑P⋊G
P⋊H =↑H and ↓P⋊H =↓H , for every subgroup H ⊂ G. It follows
that P ⋊ Q is a vertex of κM .
For any finite group G, Sp1 (the poset of all p-subgroups) is contractible (hence connected). If G has a non-trivial normal p-subgroup,
then Sp is contractible. If G is finite Chevalley group of characteristic
p and rank ≥ 2, then Sp is connected.
The vertices of an indecomposable module M are conjugate by elements of G. While given a vertex VM ⋊ H, the sources are unique up
to conjugation by elements of NG (VM ⋊ H). It is important to know
that the convex supports of sources are exactly vertices (not proper
subcategories). It means that our parametrization of indecomposable
modules via their convex supports makes sense.
Example 4.24. Based on Example 4.13, we see that the simple kP ⋊ Gmodule Sx,k has {x} × Tx as a vertex, where Tx is a Sylow p-subgroup
of Gx . Meanwhile its projective cover Px,k has {x} × 1 as a vertex.
More generally, we can deduce that Px,V has {x} × 1 as a vertex, and
Sx,V has {x} × Hx as a vertex, where Hx is a vertex (in the classical
sense) of the simple kGx -module V .
Let us provide one more concrete example. If H is a subgroup of G
and set P = G/H = {g1 H = H, · · · , gn H}, then the vertices of k are
identified with the classical vertices of the kH-module k, through the
category equivalence between P ⋊ G and H. In fact, fixing a Sylow
TRANSPORTER CATEGORY ALGEBRAS
35
p-subgroup S ⊂ G, there is some gi such that S gi ∩ H becomes a Sylow
p-subgroup of H. Then {H}×(S gi ∩H) is a vertex of k ∈ kP ⋊ G-mod.
Note that, under Dade’s construction, a “vertex” of k ∈ kP ⋊ Gmod would be P ⋊ (S gi ∩ H), which contains {H} × (S gi ∩ H) as a
proper subcategory. In fact, Dade asserted that every indecomposable
kP ⋊ G-module is projective relative to P ⋊ S gi . The connection between his approach and ours is established as follows. The (po)set
P is a disjoint union of S gi -orbits, and we denote by Γ = {Pt }t the
gi
set of all these
They are convex subposets of P, such that
` S -orbits.
gi
P ⋊ S = Γ Pt ⋊ S gi is a disjoint union of connected components,
which are groupoids. Suppose P1 = OS gi (H). Then the skeleton of
P1 ⋊ S gi is exactly {H} × (S gi ∩ H).
Next, we shall provide a generalized Green correspondence for modules. Based on our new definition of the vertex, it is necessary to note
that for posets, one should expect something different from the Green
correspondence for group modules.
Example 4.25. We may consider P : x → y → z and the subposets
Q = {x} and R : x → y. The kP-modules k → 0 → 0, k → k → 0 and
k → k → k all have Q as a vertex. In fact, they are all indecomposable
kP-modules with this property. If we consider kR-modules, then there
are two indecomposable modules with vertex Q. Thus there does not
exist a 1-1 correspondence between the set of isomorphism classes of
indecomposable kR-modules, with vertex Q, and that of isomorphism
classes of indecomposable kP-modules, with vertex Q.
However, we notice that either of the three modules k → 0 → 0,
k → k → 0 and k → k → k determines the other two via the restriction
or the left Kan extension. Thus if we consider the isomorphism classes
of indecomposable modules with vertex Q, of “maximal support”, then
there is a 1-1 correspondence (between k → k → 0 and k → k → k).
Moreover, the way they determine each other is clear.
We shall bear in mind that for a transporter category Q ⋊ H and
an indecomposable Q ⋊ H-module M, its support and convex support
are identical. Moreover, indecomposable kP ⋊ G-modules are stratified by their supports. It helps us to formulate a generalized Green
correspondence.
Theorem 4.26 (The Green correspondence for transporter categories).
Let P ⋊ G be a connected transporter category. Suppose Q ⋊ H is a
connected p-weakly ideal transporter subcategory of P ⋊ G. Let R ⋊ K
be a connected weakly convex transporter subcategory containing Q ⋊
NG (Q ⋊ H). Then there is a one-to-one correspondence between the set
36
FEI XU
of isomorphism classes of indecomposable kP ⋊ G-modules with vertex
Q ⋊ H and convex support G Q ⋊G, and the set of isomorphism classes
|{z}
of indecomposable kR ⋊ K-modules with vertex Q ⋊ H and convex support K Q ⋊K.
|{z}
Proof. At first, we assume R ⋊ K is a connected weak ideal of P ⋊ G.
On the one hand, let N be an indecomposable kR ⋊ K-module with
vertex Q ⋊ H and convex support R ⋊ K. We construct an indecomposable kP ⋊ G-module L with convex support P ⋊ G and vertex
Q ⋊ H. To this end, we show N ↑P⋊G
R⋊K has a unique summand with
vertex Q ⋊ H, while other summands are projective relative to transporter subcategories of the form (R ⋊ K) ∩ g (Q ⋊ H) for some g 6∈ K.
∼
Suppose U is a source for N. Then U ↑R⋊K
Q⋊H = N ⊕ V for some kR ⋊ KP⋊G ∼
P⋊G P⋊G ∼
′
module V . Put N ↑R⋊K ↓R⋊K = N ⊕ N ′ and V ↑P⋊G
R⋊K ↓R⋊K = V ⊕ V .
Then by the Mackey formula
P⋊G
U ↑P⋊G
Q⋊H ↓R⋊K =
M
R⋊K
{g [U ↓Q⋊H
(Rg ∩Q)⋊(K g ∩H) ]} ↑(R∩g Q)⋊(K∩g H) ,
g∈[K\G/H]
which is also isomorphic to N ⊕ N ′ ⊕ V ⊕ V ′ . There exists a g ∈ K
∼
and its corresponding summand is U ↑R⋊K
Q⋊H = N ⊕ V . The summands
′
′
of N and V are all projective relative to transporter subcategories of
the form (R ∩ g Q) ⋊ (K ∩ g H) for g 6∈ K.
Next we show N ↑P⋊G
R⋊K has a unique summand with vertex Q ⋊ H,
and the other summands have vertices contained in some (Q ⋊ H) ∩
g
(Q ⋊ H) for g 6∈ K. Let L be an indecomposable summand of N ↑P⋊G
R⋊K ,
which on restriction to kR ⋊ K has N as a summand. It must have
Q ⋊ H as its vertex. (It is projective relative to Q ⋊ H because N
is. However, its vertex cannot be conjugate to a proper weakly convex transporter subcategory of Q ⋊ H, since otherwise N would be
projective relative to this weakly convex transporter subcategory of
Q ⋊ H.) We want to prove that L is the unique summand having
Q ⋊ H as a vertex. Let L′ be another summand of N ↑P⋊G
R⋊K . Then
′
L′ ↓P⋊G
must
be
a
direct
summand
of
N
,
which
is
projective
relaR⋊K
g
g
tive to transporter subcategories of the form (R ∩ Q) ⋊ (K ∩ H) for
′
g 6∈ K. Since L′ is a summand of U ↑P⋊G
Q⋊H , L is projective relative
to Q ⋊ H. Thus L′ has a vertex Q′ ⋊ H ′ contained in Q ⋊ H. Since
Q′ ⋊ H ′ ⊂ R ⋊ K, L′ ↓P⋊G
R⋊K has a summand which on restriction to
Q′ ⋊ H ′ has a summand as a source for L′ . It follows that there is a
t
(Q′ ⋊ H ′), some t ∈ K, contained in one of the transporter subcategories, say (R ∩ s Q) ⋊ (K ∩ s H) for s 6∈ K. Thus Q′ ⋊ H ′ ⊂ g (Q ⋊ H)
TRANSPORTER CATEGORY ALGEBRAS
37
for g = t−1 s 6∈ K (which is not in NG (Q ⋊ H) ⊂ K). It means that
Q′ ⋊ H ′ ⊂ (Q ⋊ H) ∩ g (Q ⋊ H) Q ⋊ H.
By Proposition 4.22, L is supported on QG ⋊G.
|{z}
On the other hand, assume M is an indecomposable kP ⋊ G-module
with vertex Q ⋊ H, support G Q ⋊G and a source S ∈ kQ ⋊ H-mod.
|{z}
We construct an indecomposable kR ⋊ K-module W with vertex Q ⋊ H
P⋊G
and support K Q ⋊K. Since M is a summand of S ↑R⋊K
Q⋊H ↑R⋊K , there
|{z}
is a summand W S ↑R⋊K
W ↑P⋊G
Q⋊H such that M
R⋊K . The kR ⋊ Kmodule W must have vertex Q ⋊ H, because it it projective relative to
it, and moreover cannot be projective relative to a smaller transporter
P⋊G P⋊G
subcategory. The module M ↓P⋊G
R⋊K is a summand of W ↑R⋊K ↓R⋊K . By
our preceding arguments, it must have only one summand with vertex
Q ⋊ H, that is, W . This indecomposable module W , by Lemma 4.25,
has K Q ⋊K as its support.
|{z}
We readily verify that our previous constructions produce a 1-1 correspondence.
In the general situation, given a weakly convex R ⋊ K, we may proz}|{
duce a weak ideal K R ⋊K. Since it contains Q ⋊ NG (Q ⋊ H), by the
above discussions, there is a one-to-one correspondence between the
isomorphism classes of indecomposable kP ⋊ G-modules with vertex
Q ⋊ H and support G Q ⋊G, and the isomorphism classes of indecom|{z}
z}|{
posable k K R ⋊K-modules with vertex Q ⋊ H and support K Q ⋊K.
|{z}
z}|{
R⋊K
However, the Kan extension U ↑K
Q⋊H of any kQ ⋊ H-module U has its
support contained in R ⋊ K, because Q is inside a K-subposet R. It
implies that our correspondence is truly a correspondence between the
isomorphism classes of indecomposable kP ⋊ G-modules with vertex
Q ⋊ H and support G Q ⋊G, and the isomorphism classes of indecom|{z}
posable kR ⋊ K-modules with vertex Q ⋊ H and support K Q ⋊K.
|{z}
There are two special cases that we may apply the Green correspondence. One is P ⋊ H ⊂ P ⋊ G, for suitable H, and the other
is Q ⋊ H ⊂ P ⋊ H, for Q ⊂ P. These are already given by [5] and
[10], respectively. In light of Proposition 4.20, we may compose maps
in these special Green correspondences and obtain a result similar to
the above. However, if we were to use these procedures directly, we
would have to ask NG (H) ⊂ K, instead of the slightly weaker condition NG (Q ⋊ H) ⊂ K.
38
FEI XU
The following result establishes a clear connection between category
representations and group representations.
Corollary 4.27. Let P ⋊ G be a transporter category and x be an
object. Let H ⊂ Gx be a p-subgroup. Suppose K ⊃ NGx (H). Then
there is a one-to-one correspondence between the set of isomorphism
classes of indecomposable kP ⋊ G-modules with vertex {x} × H and
support G x ⋊ G, and the set of isomorphism classes of indecomposable
k{x} × K-modules with vertex {x} × H (and support {x} × K).
5. Block Theory
Boisen [4] studied the block theory of fully group-graded algebras. In
particular, based on Dade’s work, he defined “defect groups” of blocks,
and established a generalized Brauer’s First Main Theorem. These
constructions and results certainly are valid for transporter category
algebras. However, Boisen’s “defects” are not truly subgroups, and
they are too big in the case of transporter category algebras. We shall
improve a few of the existing results, and then propose some entirely
new constructions and theorems. Especially, for a transporter category
algebra, we can talk about defect transporter categories of its blocks.
Since there is a trivial representation k, we also have a notion of the
principal block of a transporter category algebra.
5.1. Defect transporter categories. As we mentioned in Section 2,
Boisen used a subalgebra, ∆(A) ⊂ Ae , to introduce the “defect groups”
of a block of a fully group-graded algebra A. His main observation is the
e
∼
∼
module isomorphism A1 ↑A
∆(A) = A. When A = kP ⋊ G, A1 = kP and
we have seen that ∆(kP ⋊ G) ∼
= kP e ⋊ δ(G). Boisen’s “defect groups”
would be minimal p-subgroups D ⊂ G such that kP ⋊ G is projective
relative to kP e ⋊ δ(D). We shall observe that Boisen’s isomorphism
e
e
P e ⋊Ge
∼
∼
kP ↑PP e ⋊G
⋊δ(G) = kP ⋊ G comes from another isomorphism k ↑F (P)⋊δ(G) =
kP. This prompts us to introduce the defect transporter subcategories
of a block of kP ⋊ G as subcategories of F (P)⋊δ(G). We shall explain
the ideas now.
In Section 3, we constructed the following transporter categories and
faithful functors
F (P) ⋊ δ(G) −→ P e ⋊ δ(G) −→ P e ⋊ Ge ∼
= (P ⋊ G)e .
Passing to module categories, the above functors give rise to restrictions
P e ⋊δ(G)
↓F (P)⋊δ(G)
e
e
e
⋊G
↓P
P e ⋊δ(G)
kF (P) ⋊ δ(G)-mod ←− kP ⋊ δ(G)-mod ←− kP e ⋊ Ge -mod,
TRANSPORTER CATEGORY ALGEBRAS
39
and their left adjoints
P e ⋊δ(G)
↑F (P)⋊δ(G)
e
e
e
⋊G
↑P
P e ⋊δ(G)
kF (P) ⋊ δ(G)-mod −→ kP ⋊ δ(G)-mod −→ kP e ⋊ Ge -mod,
Subsequently, we would like to demonstrate that F (P)⋊δ(G) ⊂ P e ⋊Ge
plays the role of the diagonal subgroup in the block theory of group
algebras. Since F (P) ⋊ δ(G) is a coideal, hence convex, in P e ⋊ δ(G),
every kF (P)⋊δ(G)-module is naturally a kP e ⋊δ(G)-module. It means
that
P e ⋊δ(G)
↑F (P)⋊δ(G) : kF (P) ⋊ δ(G)-mod → kP e ⋊ δ(G)-mod
is just an embedding.
In what follows, we shall regard F (P) ⋊ δ(G) as a weakly convex transporter subcategory of (P ⋊ G)e ∼
= P e ⋊ Ge . Consider the
e
(kP ⋊ G)e -module kP ⋊ G. Then suppc (kP ⋊ G) = PkP⋊G
⋊ Ge conop
sists of objects {(y, x ) HomP⋊G (x, y) 6= ∅}. Note that F (P), identie
fied with a subposet of P e , is convex but not ideal in PkP⋊G
.
P e ⋊δ(G)
Lemma 5.1. Let k ∈ kF (P) ⋊ δ(G)-mod. Then k ↑F (P)⋊δ(G) ∼
= kP as
e
(P⋊G)
kP e ⋊δ(G)-modules. Consequently k ↑F (P)⋊δ(G) ∼
= kP ⋊ G as (kP ⋊ G)e modules.
e
P e ⋊δ(G)
Proof. The first isomorphism follows from k ↑F (P)⋊δ(G) ∼
= k ↑PF (P) ∼
= kP,
see [11] for the latter isomorphism.
Along with Boisen’s result, our first statement gives rise to the second.
Remark 5.2. When talking about blocks of a category algebra kC, we
usually assume
C to be connected. If not, then kC becomes a direct
Q
product i kCi , where Ci rans over the set of connected components
of C. To study blocks of kC, it suffices to examine each kCi . There
is one more advantage to study connected categories. If C is (finite)
connected, then the blocks of kC, as kC e -modules, are non-isomorphic.
Now let P ⋊ G be connected. (We shall emphasize that the connectedness of P ⋊ G does not imply the connectedness of P.) Then the
kP e ⋊ δ(G)-module kP is indecomposable. The support of kP ⋊ G is
C whose objects are identified with those of the image of Ob F (P ⋊ G)
in Ob(P e ⋊ Ge ). Thus each block of kP ⋊ G has (convex) support
contained in C.
Suppose P ⋊ G is a connected transporter category. Let B be a
block of kP ⋊ G. Then B kP ⋊ G as (kP ⋊ G)e -modules. Since
(P⋊G)e
k ↑F (P)⋊δ(G) ∼
= kP ⋊ G, a vertex of B lies in F (P) ⋊ δ(G).
40
FEI XU
Definition 5.3. Suppose P ⋊ G is a connected transporter category.
Let B be a block of kP ⋊ G. Regarded as a (kP ⋊ G)e -module, a
vertex V ⋊ δ(D) of B that is contained in F (P) ⋊ δ(G) is called a defect
transporter category, or simply a defect, of B.
The block theory of transporter category algebras will be discussed
in a parallel paper. It is interesting, because there are enough blocks.
The simplest example will be that k(G/H) ⋊ G ≃ kH for a subgroup
H. Moreover, Peter Webb constructed examples where the blocks of
a group algebra biject with those of a certain transporter category
algebra (which is not a group algebra).
To finish off, we use a couple of examples to illustrate some features
of the theory. Unlike group representations, the defects of the block B
do not have to be conjugate by elements of δ(G).
αn−1
α
Example 5.4. Let Pn = x1 →1 x2 → · · · →xn−1 → xn . Then there is
only one block. Its defect is the vertex of k ∈ kF (Pn )-mod, which is
the following subposet V ⊂ F (Pn ).
[1x1 ]
③<
③③
③
③③
③③
[α1 ]
b❉❉
❉❉
❉❉
❉❉
[1x2 ]
③<
③③
③
③③
③③
[α2 ]
···
`❇❇
❇❇
❇❇
❇❇
❇
···
[αn−1 ]
②<
②②
②
②
②②
②②
c●●
●●
●●
●●
●
[1xn ]
Let M be an indecomposable kP ⋊ G-module that lies in a block B.
It is not necessarily true that a vertex VM of M satisfies the condition
that F (VM ) ⊂ DB , where DB is a defect of B.
Example 5.5. Let P be the following poset
Gz _❄❄
❄❄
✎✎✎
❄❄
✎✎
❄❄
✎✎✎
y _❄
❄❄
✎✎✎
❄❄
❄❄
✎✎
❄
✎
w
x
It is connected so there is only one block B0 (called the principal block).
Thus every module lies in the block B0 . The vertex of k ∈ kP-mod
is the whole poset P. However, when we examine the vertex of kP as
a kP e -module. Then we find that the defect of B0 = kP is a proper
subposet of F (P).
5.2. Brauer correspondent. Suppose A is a fully group-graded algebra. Boisen [4] introduced a Brauer correspondence between blocks
TRANSPORTER CATEGORY ALGEBRAS
41
of A and of AH for suitable subgroup H ⊂ G. Assume A = S ⋊ G is
a skew group algebra. Let b be a block of AH and B be a block of A.
Then B is said to correspond to b if B is the unique block such that
e
b B ↓A
AeH . We shall be interested in the case when A = kP ⋊ G, and
improve Boisen’s construction.
Definition 5.6. Suppose P ⋊ G is a connected transporter category
and Q ⋊ H is a connected transporter subcategory. Let B be a block
of kP ⋊ G and b be a block of kQ ⋊ H. We say B corresponds to b,
written as B = bP⋊G , if B is the unique block of kP ⋊ G such that
(P⋊G)e
b B ↓(Q⋊H)e .
If b is a block of kQ ⋊ H which has a block of kP ⋊ G corresponds
to it, then we say bP⋊G is defined.
Suppose P ⋊ G is a connected transporter subcategory and Q ⋊ H is
a connected weakly convex transporter subcategory. Let b be a block
of kQ ⋊ H. If Q ⋊ H is contained in some transporter subcategory
R ⋊ K while bR⋊K , (bR⋊K )P⋊G and bP⋊G are defined, then we have an
equality bP⋊G = (bR⋊K )P⋊G .
Based on his definition, Boisen [4] continued to establish a generalized Brauer’s First Main Theorem for fully group-graded algebras. We
state relevant consequences for a transporter category algebra kP ⋊ G.
(1) If CG (D) ⊂ H, then bP⋊G is defined for any block b of kP ⋊ H
which is projective relative to P e ⋊ D with D a minimal psubgroup with respect to this property.
(2) (Brauer correspondence for skew group algebras) Suppose H is
a subgroup and D is a p-subgroup of G such that NG (D) ⊂ H.
Then there is a one-to-one correspondence between the blocks
of kP ⋊ H that are relatively kP e ⋊D-projective for minimal D,
and the blocks of kP ⋊ G that are relatively kP e ⋊D-projective
for minimal D.
(3) With Lemma 5.1, (2) can be restated as follows. Under the same
assumptions, there is a one-to-one correspondence between the
blocks of kP ⋊ H that are relatively F (P) ⋊ D-projective for
minimal D, and the blocks of kP ⋊ G that are relatively F (P)⋊
D-projective for minimal D.
Boisen’s Brauer correspondence has a counterpart, when group is
fixed and subposets vary.
Lemma 5.7. Suppose P ⋊ G is a transporter category and Q ⋊ H is
a connected weakly convex transporter subcategory. Let b be a block of
kQ ⋊ H, with defect V ⋊ D. If NG (D) ⊂ H, then bP⋊H is defined. It
is the Green correspondent of b, and has V ⋊ D as its defect.
42
FEI XU
Proof. Let b1 be a block of kP ⋊ H with defect V ⋊ D. Then b1 ↓(Q⋊H)e
has a unique direct summand b, which is the Green correspondent of
(P⋊H)e
b1 . Since kP ⋊ H ↓(Q⋊H)e = kQ ⋊ H, it is clear that b is a block of
kQ ⋊ H with defect V ⋊ D.
Conversely if b is a block of kQ ⋊ H with defect V ⋊ D, then there
exists a block b2 of kP ⋊ H so that b b2 ↓(Q⋊H)e . Since the blocks of
kP ⋊ H (and of kQ ⋊ H) are non-isomorphic, b2 is unique.
Thus we have a one-to-one correspondence between blocks of kQ ⋊ H
with defect V ⋊ D, and blocks of kP ⋊ H with the same defect, via
b 7→ bP⋊H .
Along with the Brauer correspondence for group-graded algebras by
Boisen, we will obtain an improved correspondence between blocks of
transporter category algebras.
Theorem 5.8 (Brauer’s First Main Theorem for transporter categories). Suppose P ⋊ G is a (connected) transporter category and Q ⋊ H
is a (connected) weakly convex transporter subcategory. Let V ⋊ D be
a connected weakly convex p-transporter subcategory of F (P) ⋊ G, such
that V ⊂ F (Q) and NG (D) ⊂ H. Then there is a one-to-one correspondence between the blocks of kQ ⋊ H with defect V ⋊ D, and the
blocks of kP ⋊ G with defect V ⋊ D, given by letting b correspond to
bP⋊G .
Proof. Keep the following diagram of categories in mind, where each
arrow represents an inclusion.
(Q ⋊ H)e
V ⋊D
/
F (Q) ⋊ D
(P ⋊ H)e
/
O
/
O
/
(P ⋊ G)e
F (P) ⋊ D
Let B be a block of kP ⋊ G with defect V ⋊ D. Since it is projective
relative to P e ⋊ D, there is a unique block b1 of kP ⋊ H, which is
projective relative to P e ⋊ D, such that b1 B ↓(P⋊H)e . It means
bP⋊G
= B. We shall show b1 has V ⋊ D as a defect. Then Lemma
1
5.7 identifies a unique block b of kQ ⋊ H with defect V ⋊ D such that
bP⋊H = b1 . Since B = bP⋊G
= (bP⋊H )P⋊G = bP⋊G , we are done.
1
Let S ∈ kV ⋊ D-mod be a source for B. Then
b1 ↓P e ⋊D S ↑P
e ⋊Ge
↓P e ⋊D
Now by the Mackey formula, the right hand side is
M
e
e
P e ⋊Ge
[(g1 ,g2 ) (S ↓P[P e⋊G
)] ↑[P
e ⋊D]∩(g1 ,g2 ) (V⋊D) .
⋊D](g1 ,g2 ) ∩(V⋊D)
(g1 ,g2 )∈[D\Ge /D]
TRANSPORTER CATEGORY ALGEBRAS
43
By assumption D is minimal with respect to the property that b1 is
relatively P e ⋊D-projective. Since (P e ⋊D)∩ (g1 ,g2 ) (V ⋊ D) = (g1 ,g2 ) V ⋊
(D∩ (g1 ,g2) D), it implies that, for some (g1 , g2 ), D∩ (g1 ,g2 ) D = D. (While
the other intersection groups are strictly smaller than D.) It forces
D = (g1 ,g2 ) D and (g1 ,g2) V ⋊ (D ∩ (g1 ,g2 ) D) = (g1 ,g2 ) V ⋊ (g1 ,g2 ) D becomes a
conjugate of V ⋊ D. Thus b1 must have V ⋊ D as a defect.
Under the assumption of the above theorem, b is called the Brauer
correspondent of bP⋊G . Note that to guarantee the existence of bP⋊G ,
we only need to require CG (D) ⊂ H, by Boisen [4] and Lemma 5.7. In
a separate paper, we shall develop a ring-theoretic approach, and try
to generalize some other results, including the second and third main
theorems of Brauer.
References
[1] Alperin, J., Local Representation Theory, Cambridge Studies in Adv. Math.
11, Cambridge University Press 1986.
[2] Aschbacher, M., Kessar, R., Oliver, B., Fusion Systems in Algebra and Topology, LMS Lecture Note Series 391, Cambridge University Press 2011.
[3] Auslander, M., Reiten, I., Smalø, S., Representation Theory of Artin Algebras,
Cambridge Studies in Adv. Math. 36, Cambridge University Press 1997.
[4] Boisen, P., The representation theory of fully group graded algebras, J. Algebra
151 (1992), 160-179.
[5] Dade, E., Group graded-rings and modules, Math. Z. 174 (1980), 241-262.
[6] Dade, E., The equivalence of various generalizations of group rings and modules, Math. Z. 181 (1982), 335-344.
[7] Dade, E., Blocks of full group-grade rings, Pacific J. Math. 181 (1997), 85-122.
[8] Miyashita, Y., On Galois extensions and crossed products, J. Fac. Sci. Hokkaido
Univ. Ser. I 21 (1970), 97-121.
[9] Reiten, I., Riedtmann, C., Skew group algebras in the representation theory of
artin algebras, J. Algebra 92 (1985), 224-282.
[10] Xu, F., Representations of small categories and their applications, J. Algebra
317 (2007), 153-183.
[11] Xu, F., Hochschild and ordinary cohomology rings of small categories, Adv.
Math 219 (2008), 1872-1893.
[12] Xu, F., On local categories of finite groups, Math. Z. 272 (2012), 1023-1036.
[13] Xu, F., On local categories of finite groups II, preprint (2017).
E-mail address: fxu@stu.edu.cn
Department of Mathematics, Shantou University, Shantou, Guangdong 515063, China
| 4 |
arXiv:1701.07842v3 [cs.LO] 2 Mar 2018
DroidStar: Callback Typestates for Android Classes
Arjun Radhakrishna∗
Nicholas V. Lewchenko
Shawn Meier
Microsoft
University of Colorado Boulder
University of Colorado Boulder
Sergio Mover
Krishna Chaitanya Sripada
Damien Zufferey
University of Colorado Boulder
University of Colorado Boulder
Max Planck Institute for Software
Systems
Bor-Yuh Evan Chang
Pavol Černý
University of Colorado Boulder
University of Colorado Boulder
ABSTRACT
1
Event-driven programming frameworks, such as Android, are based
on components with asynchronous interfaces. The protocols for
interacting with these components can often be described by finitestate machines we dub callback typestates. Callback typestates are
akin to classical typestates, with the difference that their outputs
(callbacks) are produced asynchronously. While useful, these specifications are not commonly available, because writing them is
difficult and error-prone.
Our goal is to make the task of producing callback typestates
significantly easier. We present a callback typestate assistant tool,
DroidStar, that requires only limited user interaction to produce
a callback typestate. Our approach is based on an active learning
algorithm, L∗ . We improved the scalability of equivalence queries
(a key component of L∗ ), thus making active learning tractable on
the Android system.
We use DroidStar to learn callback typestates for Android
classes both for cases where one is already provided by the documentation, and for cases where the documentation is unclear. The
results show that DroidStar learns callback typestates accurately
and efficiently. Moreover, in several cases, the synthesized callback
typestates uncovered surprising and undocumented behaviors.
Event-driven programming frameworks interact with client code
using callins and callbacks. Callins are framework methods that the
client invokes and callbacks are client methods that the framework
invokes. The client-framework interaction is often governed by a
protocol that can be described by a finite-state machine we call
callback typestate. Callback typestates are akin to classical typestates [36], with the key difference that their outputs (callbacks) are
produced asynchronously. Our goal is to make the task of producing
callback typestates significantly easier for developers.
As an example of a callback typestate, consider a typical interaction between a client application and the framework when the
client wants to use a particular service. The client asks for the service to be started by invoking an startService() callin. After the
framework receives the callin, it asynchronously starts initializing
the service. When the service is started and ready to be used, the
framework notifies the client by invoking a onServiceStarted()
callback. The client can then use the service. After the client finishes using the service, it invokes a shutdownService() callin to
ask the framework to stop the service.
KEYWORDS
typestate, specification inference, Android, active learning
ACM Reference Format:
Arjun Radhakrishna, Nicholas V. Lewchenko, Shawn Meier, Sergio Mover,
Krishna Chaitanya Sripada, Damien Zufferey, Bor-Yuh Evan Chang,
and Pavol Černý. 2018. DroidStar: Callback Typestates for Android Classes.
In ICSE ’18: ICSE ’18: 40th International Conference on Software Engineering , May 27-June 3, 2018, Gothenburg, Sweden. ACM, New York, NY, USA,
11 pages. https://doi.org/10.1145/3180155.3180232
∗ This
work was done while Arjun Radhakrishna was employed at the University of
Pennsylvania.
Permission to make digital or hard copies of all or part of this work for personal or
classroom use is granted without fee provided that copies are not made or distributed
for profit or commercial advantage and that copies bear this notice and the full citation
on the first page. Copyrights for components of this work owned by others than the
author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or
republish, to post on servers or to redistribute to lists, requires prior specific permission
and/or a fee. Request permissions from permissions@acm.org.
ICSE ’18, May 27-June 3, 2018, Gothenburg, Sweden
© 2018 Copyright held by the owner/author(s). Publication rights licensed to the
Association for Computing Machinery.
ACM ISBN 978-1-4503-5638-1/18/05. . . $15.00
https://doi.org/10.1145/3180155.3180232
INTRODUCTION
Callback typestates. Callback typestates are useful in a number
of ways, but they are notoriously hard to produce. First, callback
typestates are a form of documentation. They tell client application
programmers in what order to invoke callins and which callback
to expect. Android framework documentation for some classes
already uses pictures very similar to callback typestates (Figure 1).
Second, callback typestates are useful in verification of client code.
They enable checking that a client uses the framework correctly.
Third, even though we infer the callback typestates from framework
code, they can be used for certain forms of framework verification.
For instance, one can infer typestates for different versions of the
framework, and check if the interface has changed.
Callback typestates are very hard to produce manually. On one
hand, inspecting code to see in what situation a callback arrives, and
what callins are enabled after that is error-prone. Even developers
familiar with the framework often miss corner-case behaviors. On
the other hand, obtaining the callback typestate with manual testing
is hard. One would need to run all sequences of callins, mixed in
sequence with the callbacks they produce. We systematize this
testing approach using an active learning algorithm.
Callback typestate assistant DroidStar. We present a tool that
makes producing callback typestates significantly easier. Our target user is a developer who wrote an Android class that interacts
ICSE ’18, May 27-June 3, 2018, Gothenburg, Sweden
Radhakrishna et al.
asynchronously using callbacks with client code. DroidStar is a
comprehensive framework for semi-automatically inferring callback typestates. The required user interaction happens in multiple
steps. In the first step, the user provides code snippets to perform local tasks, such as code for class initialization and code for invoking
each callin (similarly as in unit tests). This is sufficient as long as
certain widely applicable assumptions hold. First, we assume that
each sequence of callins produces a sequence of callbacks deterministically (this assumption fails when for instance a callback has a
parameter that is ignored at first by DroidStar but that influences
the typestate). Second, we assume that the resulting typestate is
finite. If these assumptions fail, in the following steps, DroidStar
asks the user for a solution to the problem. For instance, one way to
remove non-determinism is to refine one callback into two separate
logical callbacks, based on the parameter values. This design allows DroidStar to offer the user control over the final result while
requiring only limited, local, insight from the user. DroidStar is
available for download at https://github.com/cuplv/droidstar
Approach. We present a method for inferring typestates for Android classes. However, our method is equally applicable in other
contexts. The core algorithm is based on Angluin’s L∗ algorithm [5]
adapted to Mealy machines [32]. In this algorithm, a learner tries to
learn a finite-state machine — in our case a callback typestate — by
asking a teacher membership and equivalence queries. Intuitively,
a membership query asks for outputs corresponding to a sequence
of input callins, and the equivalence query asks if the learned typestate is correct. We note that the teacher does not need to know
the solution, but only needs to know how to answer the queries.
The key question we answer is how to implement oracles for the
membership and equivalence queries. We show how to implement
membership queries on Android classes using black-box testing.
Our main contribution here is an efficient algorithm for implementing the equivalence query using membership query. The insight
here is that the number of membership queries can be bounded
by a function of a new bound we call the distinguisher bound. We
empirically confirmed that for Android classes, the distinguisher
bound is significantly smaller than the state bound used in previous work [12, 16]. Given that the number of required membership
queries depends exponentially on the distinguisher bound, the novel
bound is what enables our tool to scale to Android classes.
Results. We use DroidStar to synthesize callback typestates for 16
Android framework classes and classes from Android libraries. The
results show that DroidStar learns callback typestates accurately
and efficiently. This is confirmed by documentation, code inspection, and manual comparison to simple Android applications. The
running time of DroidStar on these benchmarks ranged between
43 seconds and 72 minutes, with only 3 benchmarks taking more
than 10 minutes. The usefulness of the distinguisher bound was also
confirmed. Concretely, using previously known bounds, learning
the callback typestate for one of our examples (MediaPlayer) would
take more than a year, whereas with the distinguisher bound, this
example takes around 72 minutes. Furthermore, by inspecting our
typestates, we uncovered corner cases with surprising behavior
that are undocumented and might even be considered as bugs in
some cases. For instance, for the commonly used AsyncTask class, if
execute() is called after cancel() but before the onCancelled()
Figure 1: Part of MediaPlayer’s callback typestate from https://developer.
android.com/reference/android/media/MediaPlayer.html
callback is received, it will not throw an exception but will never
cause the asynchronous task to be run. Section 6 presents our results
in more detail.
Contributions. The contributions of this paper are: (a) We introduce the notion of callback typestates and develop an approach,
based on the L∗ algorithm, to infer them. (b) We show how to implement efficiently membership and equivalence oracles required by
the L∗ algorithm. (c) We evaluate our approach on examples from
the Android framework, and show its accuracy and effectiveness.
2
WORKFLOW AND ILLUSTRATIVE
EXAMPLE
We use the Android Framework’s MediaPlayer class to explain the
standard workflow for inferring callback typestate using DroidStar. This class is highly stateful—its interface includes many methods that are only meaningful or enabled in one or two particular
player states—and makes extensive use of callins and callbacks
to handle the delays of loading and manipulating large media
files. These properties make callback typestate a perfect fit; in fact,
MediaPlayer has one of the very few examples where we found a
complete callback typestate specification in the Android libraries
documentation. This callback typestate is shown in Figure 1.
In Figure 1, callins are represented by single arrows and callbacks
by double arrows. Let us look at one part of the protocol that
governs the client-framework interaction. The client first invokes
the callin setDataSource(), and the protocol transitions to the
Initialized state. In this state, the client can invoke the callin
prepareAsync(), and the protocol transitions to the Preparing
state. In the Preparing state, the client cannot invoke any callins,
but the framework can invoke the onPrepared() callback, and then
the protocol transitions to the Prepared state. At this point, the
client can invoke the start() callin, and the media starts playing.
DroidStar: Callback Typestates for Android Classes
Our goal is to semi-automatically infer the callback typestate
from the figure using the tool DroidStar. The developer interacts
with DroidStar in several steps, which we describe now.
2.1
Developer-Provided Snippets
To apply DroidStar to the MediaPlayer class, the developer provides a number of code snippets detailed below that act as an interface through which the tool can examine MediaPlayer instances.
Test object and environment instantiation. The main callback
typestate inference algorithm of DroidStar works roughly by repeatedly performing tests in the form of sequences of method calls
on an object of the given class, i.e., the MediaPlayer. Each test
must begin with an identical, isolated, class object, and if necessary,
a standard environment. In the first step, the developer provides a
snippet to initialize such an object and environment. In the case of
MediaPlayer, this snippet is as simple as discarding the previous
instance, creating a new one with new MediaPlayer(), and registering the necessary callback listeners (explained in the Callback
instrumentation paragraph below). In some cases this snippet is
more complex. As an example, we cannot create new instances of
the BluetoothAdapter class, so for that class this snippet would
need to bring the existing instance back to a uniform initial state.
Callin declaration. The next step is to declare the alphabet of “input symbols” that represent the callins in the interface of our class—
the final callback typestate will be written using these symbols—
and map each symbol to the concrete code snippet it represents. In
most cases, there is a one-to-one correspondence between input
symbols and callin methods. For example, the code snippets associated with the input symbols prepare, prepareAsync, and start are
prepare();, prepareAsync();, and start();, respectively.
In some cases, such as when a callin takes a parameter, the
developer may instead map a symbol to a set of code snippets representing alternative forms of the input which are suspected to have different behavior. In the MediaPlayer class, the
setDataSource() callin method takes a URL argument. The developer might (rightly) believe that depending on the validity and
reachability of the given URL, the behavior of the callin in the typestate may differ. In this case, the developer may provide the two snippets setDataSource(goodURL); and setDataSource(badURL);
for the same callin. DroidStar will consider both snippets for
generating tests, and further, it will indicate if they behave differently with respect to the typestate. In case a difference is detected,
the “non-determinism” is handled as explained later in this section.
The complete set of input symbols which would be declared
and mapped for the MediaPlayer class are setDataSource, prepare,
prepareAsync, start, stop, reset, release, and pause.
Callback instrumentation. As for the callin methods, which act
as the input symbols in the callback typestate, the callback methods
act as the output symbols in the callback typestate. The developer
specifies the set of output callback symbols and associated snippets to detect when callbacks occur. In most cases, this involved
adding the listeners for the callbacks in the initialization snippet
as mentioned above. In the MediaPlayer class, the output symbols
are onCompleted and onPrepared.
ICSE ’18, May 27-June 3, 2018, Gothenburg, Sweden
2.2
Automated Callback-Typestate Inference
Once the developer provides the input and output symbols and the
associated snippets, DroidStar attempts to automatically learn the
callback typestate following the framework of the L∗ algorithm.
L∗ inference. In L∗ , the learner tests sequences of inputs until she
can form a consistent hypothesis automaton. Each such test (or
sequence of inputs) is called a membership query. Once a hypothesis
automaton is produced, an equivalence query is performed; i.e.,
the hypothesis automaton is checked for equivalence with the true
callback typestate. If the two are equivalent, we are done; otherwise,
a counter-example test is returned from which the tool learns. This
process repeats until the produced hypothesis automaton is correct.
For MediaPlayer, the first set of membership queries each consist of a single different callin. Of these, only the query containing
setDataSource() succeeds. The learner continues with longer
membership queries while building the hypothesis automaton. For
instance, it learns that prepareAsync() and prepare() do not
lead to the same state: it is possible to invoke the start() after
prepare(), but not after prepareAsync(). Once the client receives
the callback onPrepared(), start() may be called. The learner
thus hypothesizes a transition from the Preparing to the Prepared
on onPrepared(). Once the hypothesis is complete, the learner asks
the equivalence query. Initially, a counter-example to equivalence
is returned using which the learner refines its hypothesis. The final
solution is found after 5 equivalence queries.
Answering Equivalence Queries. The equivalence query, i.e.,
checking if a learned callback typestate is in fact the true callback
typestate is undecidable in general. However, assuming a bound on
the size of the typestate, the equivalence query can be implemented
using further testing. However, equivalence queries are still expensive and to make them practical we present an new optimization
based on a distinguisher bound. We can observe in Figure 1 that for
any pair of states there is a transition in one state which leads to an
error in the other. This corresponds to a distinguisher bound of 1.
Small distinguisher bounds arise because typestates are not random
automata but part of an API designed for ease of use and robustness.
Such APIs are coded defensively and are fail-fast [35], i.e., errors are
not buffered but reported immediately. Each state in the typestate
has a specific function and an associated set of callins and callbacks.
In automata terms, the alphabet is roughly the same size as the
number of states and each state has only a few transitions, making
any two states easy to distinguish. In Section 4.3, we explain how
to use the distinguisher bound to implement equivalence queries
and discuss why distinguisher bounds are small in practice.
2.3
Obstacles to Inference and Solutions
The L∗ based callback typestate inference algorithm makes several
assumptions about the behavior of the class that do not always hold.
DroidStar is designed to detect these violations of assumptions and
notify the developer. Here, we discuss two such assumptions, the
exceptional situations that arise when the assumptions are violated,
and the additional developer intervention needed to handle such
cases.
Non-determinism. In input-output automata learning theory,
non-determinism makes learning impossible. Non-determinism is
ICSE ’18, May 27-June 3, 2018, Gothenburg, Sweden
the possibility of the same sequence of input callins producing different sequences of output callbacks across tests. Non-determinism
may be due to various controllable and non-controllable factors. Controllable factors include cases where behavior depends on if a file
exists, if a URL is reachable, etc. On the other hand, non-controllable
factors include random number generators, device sensors, etc. In
practice, most of the non-determinism was controllable.
The main technique for handling non-determinism is via refinement of input or output alphabets. Here, a single callin or callback is
split into multiple "logical" inputs or outputs.
(a) Controllable non-determinism can be eliminated by incorporating the controlling factor into the inputs. For example, in
the SQLiteOpenHelper class, the behavior of the constructor
callin changes depending on if a file exists. However, after splitting
the callin into two separate callins constructor/fileExists and
constructor/noFileExists, the behavior of each of each callin
becomes deterministic with respect to these callins.
(b) Another source of non-determinism is when the same callback
is used to notify logically different events. For example, a class
may use a generic onComplete callback which is passed a status
parameter that can have the values “Success” and “Failure”. Based
on this value, different further callins are enabled, leading to nondeterminism. Here, the developer may manually refine the callback
into two output symbols onEvent/Success and onEvent/Failure,
and the behavior is deterministic with respect to these.
In summary, for controllable non-determinism, the onus is on the
developer to identify the source of the detected non-determinism
and provide a refinement of the input or output alphabet and corresponding code snippets to control the source. No general technique
exists to handle non-controllable non-determinism, but specific
cases can be handled using techniques shown in Section 5.
Non-regularity. Another basic assumption that L∗ based inference algorithm makes is that the callback typestate under consideration is regular. This assumption is commonly violated in
request-response style behavior of classes where the number of
responses (output callbacks) invoked is exactly equal to the number
of requests (input callins). Our solution to this problem is to restrict
the learning to a subset of the class behavior, such as inputs with at
most one pending request callin using a learning purpose [1]. These
restrictions makes the behavior regular and amenable to learning.
3
THE CALLBACK TYPESTATE LEARNING
PROBLEM
We introduce formal models of interfaces, define the callback typestate learning problem, and present an impossibility result about
learning typestates. Callback typestates have both inputs (corresponding to callins) and outputs (corresponding to callbacks). In automata theory, callback typestates can be seen as interface automata.
Interface automata [14] are a well-studied model of automata that
can produce outputs asynchronously w.r.t. inputs. We use the name
callback typestates to emphasize that they are a generalization of
typestates as used in the programming languages literature.
3.1
Definitions and Problem Statement
Asynchronous interfaces. Let Σi and Σo be the set of callins and
callbacks of an asynchronous interface. We abstract away parameter
Radhakrishna et al.
and return values of callins and callbacks, and model a behavior of
the interface as a trace τi = σ0 . . . σn ∈ (Σi ∪ Σo )∗ . The interface I
is given by ⟨Σi , Σo , Πi ⟩ where Πi ⊆ {Σi ∪ Σo }∗ is the prefix-closed
set of all feasible traces of the interface.
Interface automata. We use interface automata [14] to represent
asynchronous interfaces. An interface automaton A is given by
⟨Q, q ι , Σi , Σo , ∆A ⟩ where: (a) Q is a finite set of states, (b) q ι ∈ Q
is the initial state, (c) Σi and Σo are finite sets of input and output
symbols, and (d) ∆A ⊆ Q × {Σi ∪ Σo } × Q are a set of transitions.
A trace τa of A is given by σ0 . . . σn if ∃q 0 . . . qn+1 : q 0 = q ι ∧
∀i.(qi , σi , qi+1 ) ∈ ∆A . Traces(A) is the set of all traces of A.
Problem statement. Given an interface I = ⟨Σi , Σo , Πi ⟩, the callback typestate learning problem is to learn an interface automaton A
such that Πi = Traces(A). We allow the learner to ask a membership
oracle MOracle[I] membership queries. For a membership query, the
learner picks mQuery = i 0i 1 . . . i n ∈ Σi∗ and the membership oracle
MOracle[I] returns either: (a) a trace τa ∈ Πi whose sequence of
callins is exactly mQuery, or (b) ⊥ if no such trace exists.
3.2
The Theory and Practice of Learning
Typestates
In general, it is impossible to learn callback typestates using only
membership queries; no finite set of membership queries fixes
a unique interface automaton. However, callback typestates can
be effectively learned given extra assumptions. We now analyze
the causes behind the impossibility and highlight the assumptions
necessary to overcome it.
Unbounded asynchrony. Membership queries alone do not tell
us if the interface will emit more outputs (callbacks) at any point
in time. Hence, we assume:
Assumption 1: Quiescence is observable.
This assumption is commonly used in ioco-testing frameworks [37].
In our setting, we add an input wait and an output quiet, where
quiet is returned after a wait only if there are no other pending
callbacks. In practice, quiet can be implemented using timeouts,
i.e., pending callbacks are assumed to arrive within a fixed amount
of time. If no callbacks are seen within the timeout, quiet is output.
Example 3.1. Using wait and quiet, in the MediaPlayer
example, we have that setDataSource() · prepareAsync()
· onPrepared() · wait · quiet is a valid trace, but
setDataSource() · prepareAsync() · wait · quiet is not.
Behavior unboundedness. For any set of membership queries,
let k be the length of the longest query. It is not possible to find
out if the interface exhibits different behavior for queries much
longer than k. This is a theoretical limitation, but is not a problem in
practice [7]; most callback typestates are rather small (≤ 10 states).
Assumption 2: An upper bound on the size of
the typestate being learned is known.
Non-determinism. We need to be able to observe the systems’
behaviors to learn them and non-determinism can prevent that.
Therefore, we assume:
Assumption 3: The interface is deterministic.
We assume that for every trace τa of the interface, there is at most
one output o ∈ Σo such that τa · o ∈ Πi . In practice, the nondeterminism problem is somewhat alleviated due to the nature of
DroidStar: Callback Typestates for Android Classes
callback typestates (see Section 5). See [1] for a detailed theoretical
discussion of how non-determinism affects learnability.
Example 3.2. Consider an interface with traces given by
(input · (out1 | out2))∗ . All membership queries are a sequence of
input’s; however, it is possible that the membership oracle never
returns any trace containing out2. In that case, no learner will be
able to learn the interface exactly.
4
LEARNING CALLBACK TYPESTATES
USING L∗
Given Assumption 1 and Assumption 3, we first build a “synchronous closure” of an asynchronous interface (Section 4.1). Then,
we show how to learn the synchronous closure effectively given
Assumption 2 (Section 4.2 and 4.3).
4.1
From Asynchronous to Synchronous
Interfaces
Using Assumption 1 and 3, we build a synchronous version of an
interface in which inputs and outputs strictly alternate following
[1]. For synchronous interfaces, we can draw learning techniques
from existing work [1, 5, 25, 32].
Define Σ̃i = Σi ∪ {wait} and Σ̃o = Σo ∪ {quiet, λ, err}. The
purpose of the extra inputs and outputs is discussed below. For any
τs ∈ (Σ̃i · Σ̃o )∗ , we define async(τs ) = τa ∈ (Σi ∪ Σo )∗ where τa is
had from τs by erasing all occurrences of wait, quiet, λ, and err.
Synchronous closures. The synchronous closure Is of an asynchronous interface I = ⟨Σi , Σo , Πi ⟩ is given by ⟨Σ̃i , Σ̃o , Πs ⟩ where Σ̃i
and Σ̃o are as above, and Πs ⊆ (Σ̃i · Σ̃o )∗ is defined as the smallest
set satisfying the following:
ϵ ∈ Πs
τs ∈ Πs ∧ async(τs ) · i ∈ Πi
=⇒ τs · i · λ ∈ Πs
τs ∈ Πs ∧ async(τs ) · o ∈ Πi
=⇒ τs · wait · o ∈ Πs
τs ∈ Πs ∧ async(τs ) · i < Πi
=⇒ τs · i · err ∈ Πs
τs ∈ Πs ∧ o ∈ Σo ∧ async(τs ) · o < Πi
=⇒ τs · wait · quiet ∈ Πs
τs ∈ Πs ∧ τs ends in err =⇒ τs · i · err ∈ Πs
Informally, in Is : (a) Each input is immediately followed by a
dummy output λ; (b) Each output is immediately preceded by a
wait input wait; (c) Any call to an input disabled in I is immediately
followed by an err. Further, all outputs after an err are err’s. (d) Any
call to wait in a quiescent state is followed by quiet.
Given MOracle[I] and Assumption 1, it is easy to construct the
membership MOracle[Is ]. Note that due to Assumption 3, there
is exactly one possible reply MOracle[Is ](mQuery) for each query
mQuery. Further, by the construction of the synchronous closure,
the inputs and outputs in MOracle[Is ](mQuery) alternate.
Mealy machines. We model synchronous interfaces using the
simpler formalism of Mealy machines rather than interface automata. A Mealy machine M is a tuple ⟨Q, q ι , Σ̃i , Σ̃o , δ, Out⟩ where:
(a) Q, q ι , Σ̃i , and Σ̃o are states, initial state, inputs and outputs,
respectively, (b) δ : Q × Σ̃i → Q is a transition function, and
(c) Out : Q × Σ̃i → Σ̃o is an output function. We abuse notation
and write Out(q, i 0 . . . i n ) = o 1 . . . on and δ (q, i 0 . . . i n ) = q ′ if
∃q 0 , . . . , qn+1 : q 0 = q ∧ qn+1 = q ′ ∧ ∀0 ≤ i ≤ n : δ (qi , i i ) =
qi+1 ∧ Out(qi , i i ) = oi . A sequence i 0o 0 . . . i n on ∈ (Σ̃i · Σ̃o )∗ is a
trace of M if Out(q ι , i 0 . . . i n ) = o 0 . . . on . We often abuse notation
ICSE ’18, May 27-June 3, 2018, Gothenburg, Sweden
and write M(i 0 . . . i n ) instead of Out(q ι , i 0 . . . i n ). We denote by
Traces(M) the set of all traces of M.
4.2
L∗ : Learning Mealy Machines
For the sake of completeness, we describe the classical L∗ learning
algorithm by Angluin [5] as adapted to Mealy machines in [32].
A reader familiar with the literature on inference of finite-state
machines may safely skip this subsection.
Fix an asynchronous interface I and its synchronous closure Is .
In L∗ , in addition to a membership oracle MOracle[Is ], the learner
has access to an equivalence oracle EOracle[Is ]. For an equivalence
query, the learner passes a Mealy machine M to EOracle[Is ], and
is in turn returned: (a) A counterexample input cex = i 0 . . . i n such
that M(cex) = o 0 . . . on and MOracle[Is ](cex) , i 0o 0 . . . i n on , or
(b) Correct if no such cex exists.
The full L∗ algorithm is in Algorithm 1. In Algorithm 1, the
learner maintains: (a) a set SQ ⊆ Σ̃i∗ of state-representatives (initially set to {ϵ }), (b) a set E ⊆ Σ̃i∗ of experiments (initially set to Σ̃i ),
and (c) an observation table T : (SQ ∪ SQ · Σ̃i ) → (E → Σ̃o∗ ). The observation table maps each prefix w i and suffix e to T (w i )(e), where
T (w i )(e) is the suffix of the output sequence of MOracle(w i · e) of
length |e |. The entries are computed by the sub-procedure FillTable.
Intuitively, SQ represent Myhill-Nerode equivalence classes of
the Mealy machine the learner is constructing, and E distinguish
between the different classes. For SQ to form valid set of MyhillNerode classes, each state representative extended with an input,
should be equivalent to some state representative. Hence, the algorithm checks if each w i · i ∈ SQ · Σ̃i is equivalent to some w i′ ∈ SQ
(line 3) under E, and if not, adds w i · i to SQ . If no such w i · i exists,
the learner constructs a Mealy machine M using the Myhill-Nerode
equivalence classes, and queries the equivalence oracle (line 5). If
the equivalence oracle returns a counterexample, the learner adds
a suffix of the counterexample to E; otherwise, it returns M. For the
full description of the choice of suffix, see [30, 32].
Theorem 4.1 ([32]). Let there exist a Mealy machine M with
n states such that Traces(M) is the set of traces of Is . Then, given
MOracle[Is ] and EOracle[Is ], Algorithm 1 returns M making at most
| Σ̃i | 2n + | Σ̃i |n2m membership and n equivalence queries, where m is
the maximum length of counterexamples returned by EOracle[Is ]. If
EOracle[Is ] returns minimal counterexamples, m ≤ O(n).
4.3
An Equivalence Oracle Using Membership
Queries
Given a black-box interface in practice, it is not feasible to directly
implement the equivalence oracle required for the L∗ algorithm.
Here, we demonstrate a method of implementing an equivalence
oracle using the membership oracle using the boundedness assumption (Assumption 2). As before fix an asynchronous interface I
and its synchronous closure Is . Further, fix a target minimal Mealy
machine M∗ such that Traces(M∗ ) is the set of traces of Is .
State bounds. A state bound of B State implies that the target Mealy
machine M∗ has at most B State states. Given a state bound, we can
replace an equivalence check with a number of membership queries
using the following theorem.
ICSE ’18, May 27-June 3, 2018, Gothenburg, Sweden
Algorithm 1 L∗ for Mealy machines
Input: Membership oracle MOracle, Equivalence oracle EOracle
Output: Mealy machine M
1: S Q ← {ϵ }; E ← Σ̃i ; T ← FillTable(S Q , Σ̃i , E,T )
2: while True do
3:
while ∃w i ∈ SQ , i ∈ Σ̃i : ∄w i′ ∈ SQ : T (w i · i) = T (w i′ ) do
4:
SQ ← SQ ∪ {w i · i}; FillTable(SQ , Σ̃i , E,T )
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
M ← BuildMM(SQ , Σ̃i ,T ); cex ← EOracle(M)
if cex = Correct then return M
E ← E ∪ AnalyzeCex(cex, M); FillTable(SQ , Σ̃i , E,T )
function BuildMM(SQ ,Σ̃i ,Σ̃o ,T )
Q ← {[w i ] | w i ∈ SQ }; q ι ← [ϵ]
∀w i , i : δ ([w i ], i) ← [w i′ ] if T (w i · i) = T (w i′ )
∀w i , i : Out([w i ], i) ← o if T (w i )(i) = o
return ⟨Q, q ι , Σ̃i , Σ̃o , δ, Out⟩
function AnalyzeCex(M,cex)
p
p
for all 0 ≤ i ≤ |cex| and w i , w is such that w i · w is =
p
cex ∧ |w i | = 1 do
p
p
p′
p
wo ← M(w i ); [w i ] ← δ ([ϵ], wo )
p′
wos ← last |w is | of Out(MOracle(w i · w is ))
p
if wo · wos , Out(MOracle(cex)) then return w is
procedure FillTable(SQ ,Σ̃i ,E,T )
for all w i ∈ SQ ∪ SQ · Σ̃i , e ∈ E do
T (w i )(e) ← Suffix of Out(MOracle(w i · e)) of length |e |
Theorem 4.2. Let M and M ′ be Mealy machines having k and
states, respectively, such that ∃w i ∈ Σ̃i∗ : M(w i ) , M ′ (w i′ ). Then,
there exists an input word w i′ of length at most k + k ′ − 1 such that
M(w i′ ) , M ′ (w i′ ).
k′
The proof is similar to the proof of the bound k + k ′ − 2 for
finite automata (see [33, Theorem 3.10.5]). We can check equivalence of M∗ and any given M by testing that they have equal
outputs on all inputs of length at most k M + B State − 1, i.e., using
O(| Σ̃i | BState +k−1 ) membership queries. While this simple algorithm
is easy to implement, it is inefficient and the number of required
membership queries make it infeasible to implement in practice.
Other algorithms based on state bounds have a similar problems
with efficiency (see Remark in Section 4.3). Further, the algorithm
does not take advantage of the structure of M. The following discussion and algorithm rectifies these short-comings.
Distinguisher bounds. A distinguisher bound of B Dist ∈ N implies
that for each pair of states q 1∗ , q 2∗ in the target Mealy machine M∗
can be distinguished by an input word w i of length at most B Dist ,
i.e., Out∗ (q 1∗ , w i ) , Out∗ (q 2∗ , w i ). Intuitively, a small distinguisher
bound implies that each state is “locally” different, i.e., can be distinguished from others using small length input sequences. The
following theorem shows that a state bound implies a comparable
distinguisher bound.
Theorem 4.3. State bound k implies distinguisher bound k − 1.
Small distinguisher bound. In practice, distinguishers are much
smaller than the bound implied by the state bound. For the mediaplayer, the number of states is 10, but only distinguishers of length
Radhakrishna et al.
1 are required. This pattern tends to hold in general due to the
following principles of good interface design:
• Clear separation of the interface functions. Each state in the
interface has a specific function and a specific set of callins
and callbacks. There is little reuse of names across state. The
typestate’s alphabet is roughly the same size as the number
of states.
• Fail-fast. Incorrect usage of the interface is not silently ignored but reported as soon as possible. This makes it easier
to distinguish states as disabled callins lead directly to errors.
• No buffering. More than just fail-fast, a good interface is interactive and the effect of callins must be immediately visible
rather than hidden. A good interface is not a combination
lock that requires many inputs that are silently stored and
only acknowledged at the very end.
This observation also is not specific to callbacks typestates and it
has been already observed for libraries [11].
Equivalence algorithm. Algorithm 2 is an equivalence oracle for
Mealy machines using the membership oracle, given a distinguisher
bound. First, it computes state representatives R : Q → Σ̃i∗ : for each
q ∈ Q, δ (q ι , R(q)) = q (line 1). Then, for each transition in M, the
algorithm first checks whether the output symbol is correct (line 5).
Then, the algorithm checks the “fidelity” of the transition up to the
distinguisher bound, i.e., whether the representative of the previous
state followed by the transition input, and the representative of
the next state can be distinguished using a suffix of length at most
B Dist . If so, the algorithm returns a counterexample. If no transition
shows a different result, the algorithm returns Correct.
Two optimizations further reduce the number of membership
queries: (a) Quiescence transitions. Transitions with input wait and
output quiet need not be checked at line 7; it is a no-op at the
interface level. (b) Error transitions. Similarly, transition with the
output err need not be checked as any extension of an error trace
can only have error outputs.
Remark. Note that if Algorithm 2 is being called from Algorithm 1,
the state representatives from L∗ can be used instead of recomputing R
in line 1. Similarly, the counterexample analysis stage can be skipped
in the L∗ algorithm, and the relevant suffix can be directly returned
(suffix in lines 10 and 11; and i in line 5).
Theorem 4.4. Assuming the distinguisher bound of B Dist for the
target Mealy machine M∗ , either (a) Algorithm 2 returns Correct
and ∀w i ∈ Σ̃i∗ : M(w i ) = M∗ (w i ), or (b) Algorithm 2 returns a
counterexample cex and M(cex) , M∗ (cex). Further, it performs at
most |Q | · | Σ̃i | BDist +1 membership queries.
Remark (Relation to conformance testing algorithms).
Note that the problem being addressed here, i.e., testing the equivalence
of a given finite-state machine and a system whose behavior can
be observed, is equivalent to the conformance testing problem from
the model-based testing literature. However, several points make the
existing conformance testing algorithms unsuitable in our setting.
Popular conformance testing algorithms, like the W-method [12]
and the Wp -method [16], are based on state bounds and have an
unavoidable O(| Σ̃i | BState ) factor in the complexity. In our experiments,
the largest typestate had 10 states and 7 inputs. The O(| Σ̃i | BState ) factor
leads to an infeasible (i.e., > 108 ) number of membership queries.
DroidStar: Callback Typestates for Android Classes
Algorithm 2 Equivalence oracle with distinguisher bound
Input: Mealy machine M = ⟨Q, q ι , Σ̃i , Σ̃o , δ, Out⟩, Distinguisher
bound B Dist , and Membership oracle MOracle
Output: Correct if M = M∗ , or cex ∈ Σ̃i∗ s. t. M(cex) , M∗ (cex)
1: for all q ∈ Q do R(q) ← w i | δ (q ι , w i ) = q s. t. |w i | is minimal
2: for all q ∈ Q, i ∈ Σ̃i do
3:
w i ← R(q) · i
4:
if Out(q, i) , last symbol of Out(MOracle(w i · i)) then
5:
return R(q) · i
6:
q ′ ← δ (q, i); w i′ ← R(q ′ )
7:
suffix ← check(w i , w i′ )
8:
if suffix , Correct then
9:
if M(R(q) · i · suffix) , Out(MOracle(R(q) · i · suffix))
then
10:
return R(q) · i · suffix
11:
else return R(q ′ ) · suffix
12: return Correct
′
13: function check(w i , w )
i
≤B Dist
14:
for all suffix ∈ Σ̃i
do
15:
wo ← Out(MOracle(w i · suffix))
16:
wo′ ← Out(MOracle(w i′ · suffix))
17:
if the last |suffix| symbols of wo and wo′ differ then
18:
return suffix
19:
return Correct
However, since distinguisher bounds are often much smaller than
state bounds, O(| Σ̃i | BDist ) membership queries are feasible (i.e., 103 ).
The W- and Wp -methods cannot directly use distinguisher bounds.
The other common algorithm, the D-method [20, 22], does not
apply in our setting either. The D-method is based on building a
distinguishing sequence, i.e., an input sequence which produces a
different sequence of outputs from every single state in the machine.
However, for callback typestates, such single distinguishing sequences
do not exist in practice. For similar reasons, conformance testing
algorithms such as the UIO-method [31] do not apply either.
In this light, we believe that Algorithm 2 is a novel conformance
testing algorithm useful in specific settings where resets are inexpensive and systems are designed to have small distinguisher bounds.
4.4
Putting It All Together
We now present the full callback typestate learning solution.
Theorem 4.5. Given a deterministic interface I with observable
quiescence and the membership oracle MOracle[I]. Assume there
exists an interface automaton A with n states with distinguisher bound
B Dist modeling the typestate of I. Interface automaton A can be learned
with O(|Σi | · n 3 + n · |Σi | BDist ) membership queries.
Proof sketch. Starting with an asynchronous interface I and
a membership oracle MOracle[I], using Assumption 1 and Assumption 3 we can construct the membership oracle MOracle[Is ]
for the synchronous closure Is of I. Given the distinguisher bound
(or a state bound using Assumption 2 and Theorem 4.3), we can
construct an equivalence oracle EOracle[Is ] using Algorithm 2. Oracles MOracle[Is ] and EOracle[Is ] can then be used to learn a Mealy
machine M with the same set of traces as Is . This Mealy machine
ICSE ’18, May 27-June 3, 2018, Gothenburg, Sweden
can be converted into the interface automata representing the callback typestate of I by: (a) Deleting all transitions with output err
and all self-loop transitions with output quiet, and (b) Replacing
all transitions with input wait with the output of the transition.
5
ACTIVE LEARNING FOR ANDROID
We implemented our method in a tool called DroidStar. In this
section we describe how it works, the practical challenges we faced
when working with Android, and our solutions to overcome them.
DroidStar is implemented as an Android application and learns
callback typestates from within a live Android system.
5.1
Designing an Experiment
To learn a typestate, a DroidStar user creates a test configuration
(an extension of the LearningPurpose class) providing necessary
information about a Java class under study. If known, the distinguisher bound can be provided here directly; otherwise, it can be
obtained from Assumption 2 by Theorem 4.3. The instrumented
alphabet, also defined here, specifies an abstract alphabet for the
learning algorithm and translation between the abstract alphabet
and concrete callins/callbacks of the class under study. Several other
options are available for adjusting the learning, the most important
being the quiescence timeout which determines Assumption 1.
5.2
Observing Asynchronous Callbacks
In our approach we assume bounded asynchrony (Assumption
1) and, therefore, we can observe when the interface does not
produce any new output (quiescence). We enforce this assumption
on a real system with timeouts: the membership query algorithm
waits for a new output for a fixed amount of time tmax , assuming
that quiescence is reached when this time is elapsed. However,
Android does not provide any worst case execution time for the
asynchronous operations and we rely on the user to choose a large
enough tmax . The membership query also assumes the existence of
a minimum time tmin before a callback occurs. This ensures that
we can issue a membership query with two consecutive callins (so,
without a wait input in between), i.e., we have the time to execute
the second callin before the output of the first callin.
Consider the MediaPlayer example from Section 2. The membership query setDataSource(URL) · wait · prepareAsync() · wait
may not return the onPrepared() if tmax is violated, i.e., if the
callback does not arrive before the timeout, and while testing it is
possible that the prepareAsync() · start() might not return an
error as expected if the lower bound tmin is violated. To avoid such
issues we try to control the execution environment and parameters
to ensure that callbacks occurred between tmin and tmax . In the
MediaPlayer case, we must pick the right media source file.
5.3
Checking and Enforcing Our Assumptions
The simplest experiment to learn a class’s callback typestate ties
a single input symbol to each of its callins and a single output
symbol to each of its callbacks. However, many Android classes
have behaviors which cause this simple experiment to fail and
require more detailed experiments to succeed.
The main challenges when designing an experiment are (a) Nondeterministic behaviors, i.e., the state of the device and external
ICSE ’18, May 27-June 3, 2018, Gothenburg, Sweden
getRDB
ctor
te
onCrea
onCrea
te
e
ctor/f
ctor/n
f
getRDB
onCreate
getRDB
onCreate
e
Figure 2: Eliminating non-determinism in SQLiteOpenHelper
events may influence an application. These elements are inherently
non-deterministic; however, non-determinism violates Assumption 3. (b) The parameter space required to drive concrete test cases
to witness a membership query is potentially infinite. Though we
have ignored callin parameters till now, they are a crucial issue
for testing. (c) The protocol we are learning may not be a regular
language. Note that this is a violation of Assumption 2.
Non-Deterministic Behavior. Non-deterministic behavior is disallowed by our Assumption 3. However, to make this assumption
reasonable we must make non-determinism straightforward to
eliminate when it arises. We explain two primary classes of nondeterministic behaviors and strategies to eliminate these behaviors.
The first class is related to controllable inputs and the second to
uncontrollable ones (such as inputs from the device sensors).
Because the learning algorithm cannot learn from nondeterministic systems, DroidStar will terminate if such behavior is detected. To assist in this process, DroidStar will report a
non-deterministic behavior is detected and display the disagreeing
sequences to the user. It detects this by caching all membership
queries as input/output sequence pairs. When a new trace is explored, DroidStar checks that the trace prefixes are compatible
with the previously seen traces.
In the first case, a hidden (not modeled) controllable input influences the typestate. We resolve this non-determinism by manually
adding the input value and create a finer alphabet that explicate
the previously hidden state of the environment. For example, in
the class SQLiteOpenHelper, the getReadableDatabase() may
either trigger a onCreate() callback or not, depending on the parameter value to a previous callin (constructor)was the name of
an existing database file. Hence, the behavior of the callin is nondeterministic, depending on the status of the database on disk. In
the SQLiteOpenHelper example, we split the constructor callin
into constructor/fileExists and constructor/noFileExists
and pass the right parameter values in each case. With this extra
modeling we can learn the interface automaton, since the execution getReadableDatabase() ends in two different states of the
automaton (see Figure 2).
The second class is the effect of the uncontrollable inputs on a
typestate. Such effects, by definition, cannot be controlled or made
explicit prior to the call. We can sometimes to remove this nondeterminism by merging different outputs, considering them to be
the same. This is the dual of the previous solution.
An example is the SpeechRecognizer, for which calling
startListening() produces different callbacks depending on the
environment. As the environment cannot be reasonably controlled,
we merge outputs to go to the same state. If outputs are erroneously
merged, the non-determinism will propagate and continue to manifest. Thus there is no risk of unsound results.
Radhakrishna et al.
Handling Callin Parameters. While parameter-less callins such
as start() and stop() are common in Android classes, many
parameterized callins exist. Because input symbols need to be listed
in the experiment definition, the full range of parameter values
cannot be explored. In practice, we found that parameters often
have little effect on the typestate automaton. In cases where they
do affect the automaton, multiple input symbols can be defined to
represent the same method called with several different parameters.
This solution is similar to splitting on environmental effects when
dealing with non-determinism.
Learning from Non-Regular Languages. An intrinsic limitation
of L∗ is that it learns only regular languages. However, some classes
expose non-regular protocols. Common cases include situations
where a request callin invoked n times trigger exactly n response callbacks. In the SpellCheckerSession class, callin getSuggestion()
and callback onGetSuggestions() follow this pattern.
However, even in such cases, it can be useful to build a regular approximation of the typestate. For example, restricting the typestate
to behaviors where there is at most one pending request (a regular
subset) provides all the information a programmer would need.
Hence, in such cases, we use the technique of learning purposes [1]
to learn a regular approximations of the infinite typestate.
6
EMPIRICAL EVALUATION
We evaluated our interface-learning technique, as implemented
in DroidStar, by using it to generate callback typestates for 16
classes, sampled from the Android Framework and popular thirdparty libraries. DroidStar is available at https://github.com/cuplv/
droidstar. For these experiments, DroidStar was run on an LG
Nexus 5 with Android framework version 23. Our evaluation was
designed to answer the following questions:
(1) Does our technique learn typestates efficiently?
(2) What size distinguisher bounds occur in practice? Do they
support the small distinguisher bound hypothesis?
(3) Do the callback typestates we learn reveal interesting or
unintended behavior in the interfaces?
Methodology. For each experimental class, we manually identified
a reduced alphabet of relevant callins and callbacks and provided
them (along with other necessary information as explained in Section 5) to DroidStar through instances of the LearningPurpose.
Relevant callins and callbacks for these experiments were those
which, according to the available documentation, appeared to trigger or depend on typestate changes (enabling or disabling of parts
of the interface). Each instance consisted of 50 − 200 lines of, mostly
boiler-plate, Java or Scala code.
To evaluate efficiency, we measured the overall time taken for
learning, as well as the number of membership (MQ) and equivalence queries (EQ). The number of queries is likely a better measure
of performance than running time: the running time depends on
external factors. For example, in the media player the running time
depends on play-length of the media file chosen during testing.
We validated the accuracy of learned callback typestates using
two approaches. First, for classes whose documentation contains a
picture or a description of what effectively is an callback typestate,
we compared our result to the documentation. Second, for all other
DroidStar: Callback Typestates for Android Classes
ICSE ’18, May 27-June 3, 2018, Gothenburg, Sweden
cancel()
classes we performed manual code inspection and ran test apps to
evaluate correctness of the produced typestates.
We used a distinguisher bound of 2 for our experiments; further,
we manually examined the learned typestate and recorded the
actual distinguisher bound. For our third question, i.e., does the
learned callback typestate reveal interesting behaviors, we manually
examined the learned typestate, compared it against the official
Android documentation, and recorded discrepancies.
Cancelling
Start
execute()
onCancelled()
execute()
Running
cancel()
Cancelling’
cancel()
Completed
onCancelled()
cancel()
onPostExecute()
6.1
Results
We discuss the results (in Table 1) and our three questions.
Question 1: Efficiency. The table shows that our technique is reasonably fast: most typestates learned within a few minutes. The
longest one takes 71 minutes, still applicable to nightly testing. The
numbers for membership queries are reported as X (Y )—X is the
number of membership queries asked by the algorithm, while Y
is the number actually executed by the membership oracle. This
number is lower as the same query may be asked multiple times,
but is executed only once and the result is cached. For each benchmark, the accuracy validation showed that the produced typestate
matched the actual behavior.
Question 2: Distinguisher Bounds. As mentioned before, we used a
distinguisher bound of 2 for all experiments. However, a manual
examination of the learned callback typestates showed that a bound
of 1 would be sufficient in all cases except the SQLiteOpenHelper
and the OkHttpCall where bounds of 2 are necessary. This supports
our conjecture that, in practice, interfaces are designed with each
state having a unique functionality (see Section 4.3).
Question 3: Interesting Learned Behavior. Of the three questions,
our experiments to examine the learned callback typestate for interesting behavior turned out to be the most fruitful, uncovering
several discrepancies, including corner cases, unintended behavior
and likely bugs, in the Android framework. These results reaffirm
the utility of our main goal of automatically learning callback typestate, and suggest that learning typestate can serve valuable roles
in documentation and validation of callback interfaces.
In 2 cases, the learned typestate and documented behavior differed in certain corner cases. We carefully examined the differences,
by framework source examination and manually writing test applications, and found that the learned typestate was correct and the
documentation was faulty. In 5 other cases, we believe the implemented behavior is not the intended behavior, i.e., these are likely
bugs in the Android implementation. These discrepancies mostly
fall into two separate categories:
Incorrect documentation. In such cases, it turned out that the discrepancy is minor and unlikely to produce bugs in client programs.
Race conditions. Several likely bugs were due to a specific category of
race conditions. These interfaces have (a) a callin to start an action
and a corresponding callback which is invoked when the action
is successfully completed; (b) a callin to cancel an already started
action and a corresponding callback which is invoked if the action
is successfully cancelled. When the start action and cancel action
callins are called in sequence, the expectation is that exactly one of
the two callbacks are called. However, when the time between the
two callins is small, we were able to observe unexpected behaviors,
including neither or both callbacks being invoked.
Figure 3: Learned typestate of the AsyncTask class
6.2
Selected Experiments
Of our 16 benchmarks, we briefly explain 5 here. The remaining
experiments are discussed in the technical report [29]1 .
MediaPlayer. This is the class from the example in Section 2. The
learned typestate differs from the existing documentation. The
learned typestate: (a) has the pause() callin enabled in the “playback completed” state, and (b) shows that onPrepared() is invoked
even after the synchronous callin prepare(). Though undocumented, these behaviors are unlikely to cause any issues.
AsyncTask. The AsyncTask class turns arbitrary computations
into callback operations with progress tracking and results are delivered via callbacks. For our experiment, the computation is a simple timer. A constructed AsyncTask object performs its task when
it receives the execute() callin, and then either returns the results
with the onPostExecute() callback, or returns an onCancelled()
if cancel() is called first. The object is single-use; after it has
returned a callback it will accept no further execute() commands.
Our experiment revealed an unexpected edge-case: if execute()
is after cancel() but before the onCancelled() callback is received, it will not throw an exception but will never cause the
callback task to be run. The learned interface is in Figure 3.
SpeechRecognizer. This class provides an example of uncontrollable environmental non-determinism. The particular callback that
signals the end of the speech session—either an onResults() or an
onError()—is determined by the environment (in particular, the
sound around the phone during the test). In this case, to reduce the
system to a deterministic one we can learn, we supposed that the
state after an onResults() or onError() is the same and merged
the two callbacks into a single onFinished() symbol.
Our results revealed two interesting corner cases for the ordering of inputs. First, if an app calls cancel() between calling startListening() and receiving the onReadyForSpeech()
callback (represented by our “starting” output symbol), calling
startListening() again will have no effect until after a certain
amount of time, as shown by the wait transition from state “Cancelling” to “Finished”. Delays in readiness like this can be generally
considered bugs; if a system will not be ready immediately for inputs it should provide a callback to announce when the preparations
are complete, so as not to invite race conditions.
Our second corner case is where the app calls stopListening()
as the very first input on a fresh SpeechRecognizer. This will not
throw an exception, but calling startListening() at any point
after will fail, making the object effectively dead.
1 http://arxiv.org/abs/1701.07842
ICSE ’18, May 27-June 3, 2018, Gothenburg, Sweden
Class name
AsyncTask
BluetoothAdapter
CountDownTimer
DownloadManager
FileObserver
ImageLoader (UIL)
MediaCodec
MediaPlayer
MediaRecorder
MediaScannerConnection
OkHttpCall (OkHttp)
RequestQueue (Volley)
SpeechRecognizer
SpellCheckerSession
SQLiteOpenHelper
VelocityTracker
LP LoC
79
161
94
84
134
80
152
171
131
72
79
79
168
109
140
63
Radhakrishna et al.
states
5
12
3
4
6
5
8
10
8
4
6
4
7
6
8
2
Time (s)
49
1273
134
136
104
88
371
4262
248
200
463
420
3460
133
43
98
MQ
372 (94)
839 (157)
232 (61)
192 (43)
743 (189)
663 (113)
1354 (871)
13553 (2372)
1512 (721)
403 (161)
839 (166)
475 (117)
1968 (293)
798 (213)
1364 (228)
1204 (403)
EQ
1
2
1
1
2
2
1
5
1
2
2
1
3
4
2
1
MQ per EQ
356 (0)
420 (16)
224 (0)
190 (0)
351 (8)
650 (33)
973 (482)
2545 (384)
1280 (545)
163 (57)
812 (13)
460 (0)
646 (35)
374 (8)
665 (6)
1156 (0)
B Dist (needed)
2 (1)
2 (1)
2 (1)
2 (1)
2 (1)
2 (1)
2 (1)
2 (1)
2 (1)
2 (1)
2 (2)
2 (1)
2 (1)
2 (1)
2 (2)
2 (1)
Table 1: DroidStar experimental results.
SQLiteOpenHelper. This class provides a more structured interface for apps to open and set up SQLite databases. It has callbacks
for different stages of database initialization, allowing apps to perform setup operations only as they are needed. When a database is
opened with getWritableDatabase(), a callback onConfigure()
is called, followed by an onCreate() if the database didn’t exist
yet or an onUpgrade() if the database had a lower version number
than was passed to the SQLiteOpenHelper constructor, all followed
finally by an onOpen() when the database is ready for reading. The
database can then be closed with a close().
Our experiment observed the callbacks when opening databases
in different states (normal, non-existent, and out of date) and performing the close() operation at different points in the sequence.
We found that once the getWritableDatabase() method is called,
calling close() will not prevent the callbacks from being run.
VelocityTracker. This class was a special case with no asynchronous behavior; it was a test of our tool’s ability to infer traditional,
synchronous typestates. The class has a recycle() method that
we expected to disable the rest of the interface, but our tool found
(and manual tests confirmed) that the other methods can still be
called after recycling. The documentation’s warning that “You must
not touch the object after calling [recycle]” is thus not enforced.
7
RELATED WORK
Works which automatically synthesize specifications of the valid
sequences of method calls (e.g. [3, 4, 18, 34]) typically ignore the
asynchronous callbacks.
Static analysis has been successfully used to infer typestates
specifications (importantly, without callbacks) [3, 23, 34]. The work
in [3] infers classical typestates for Java classes using L∗ . In contrast,
our approach is based on testing. Therefore, we avoid the practical
problem of abstracting the framework code. On the other hand, the
use of testing makes our L∗ oracles sound only under assumptions.
Similarly, [19] uses L∗ to infer classical typestates, including ranges
of input parameters that affect behavior. However, their tool is
based on symbolic execution, and thus would not scale to systems
as large and complex as the Android Framework.
Inferring interfaces using execution traces of client programs
using the framework is another common approach [2, 4, 13, 17, 28,
38, 40, 41]. In contrast to dynamic mining, we do not rely on the
availability of client applications or a set of execution traces. The
L∗ algorithm drives the testing.
The analysis of event-driven programming frameworks has recently gained a lot of attention (e.g. [6, 9, 10, 26]). However, none
of the existing works provide an automatic approach to synthesize
interface specifications. Analyses of Android applications mostly
focus on either statically proving program correctness or security
properties [6, 9, 15, 21, 39] or dynamically detecting race conditions [8, 24, 27]. These approaches manually hard-code the behavior of the framework to increase the precision of the analysis. The
callback typestate specifications that we synthesize can be used
here, avoiding the manual specification process.
Our work builds on the seminal paper of Angluin [5] and the
subsequent extensions and optimizations. In particular, we build
on L∗ for I/O automata [1, 32]. The optimizations we use include
the counterexample suffix analysis from [30] and the optimizations
for prefix-closed languages from [25]. The relation to conformance
testing methods [12, 16, 20, 22, 31] has been discussed in Section 4.3.
8
CONCLUSION
We have shown how to use active learning to infer callback typestates. We introduce the notion of distinguisher bound which take
advantage of good software engineering practices to make active
learning tractable on the Android system. Our method is implemented in the freely available tool called DroidStar. This paper
enables several new research directions. We plan to investigate mining parameters of callins from instrumented trace from real user
interactions, as well as the inference of structured typestates (for
instance, learning a typestate as a product of simpler typestates).
ACKNOWLEDGMENTS
This research was supported in part by DARPA under agreement
FA8750-14-2-0263. Damien Zufferey was supported in part by the
European Research Council Grant Agreement No. 610150 (ERC
Synergy Grant ImPACT (http://www.impact-erc.eu/)).
DroidStar: Callback Typestates for Android Classes
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| 6 |
Routing under Balance
arXiv:1603.09009v1 [cs.DS] 30 Mar 2016
Alina Ene∗
Gary Miller†
Jakub Pachocki‡
Aaron Sidford§
Abstract
We introduce the notion of balance for directed graphs: a weighted directed graph is αbalanced if for every cut S ⊆ V , the total weight of edges going from S to V \ S is within
factor α of the total weight of edges going from V \ S to S. Several important families of
graphs are nearly balanced, in particular, Eulerian graphs (with α = 1) and residual graphs of
(1 + ǫ)-approximate undirected maximum flows (with α = O(1/ǫ)).
We use the notion of balance to give a more fine-grained understanding of several well-studied
routing questions that are considerably harder in directed graphs. We first revisit oblivious
routings in directed graphs. Our main algorithmic result is an oblivious routing scheme for singlesource instances that achieve an O(α·log3 n/ log log n) competitive ratio. In the process, we make
several technical contributions which may be of independent interest. In particular, we give an
efficient algorithm for computing low-radius decompositions of directed graphs parameterized by
balance. We also define and construct low-stretch arborescences, a generalization of low-stretch
spanning trees to directed graphs.
On the negative side, we present new lower bounds for oblivious routing problems on directed
graphs. We show that the competitive ratio of oblivious routing algorithms for directed graphs
√ is
Ω(n) in general; this result improves upon the long-standing best known lower bound of Ω( n)
[HKRL07].
We also show that our restriction to single-source instances is necessary by showing
√
an Ω( n) lower bound for multiple-source oblivious routing in Eulerian graphs.
We also study the maximum flow problem in balanced directed graphs with arbitrary capacities. We develop an efficient algorithm that finds an (1 + ǫ)-approximate maximum flows in
2 2
e
α-balanced graphs in time O(mα
/ǫ ). We show that, using our approximate maximum flow algorithm, we can efficiently determine whether a given directed graph is α-balanced. Additionally,
we give an application to the directed sparsest cut problem.
∗
Department of Computer Science and DIMAP, University of Warwick, A.Ene@dcs.warwick.ac.uk.
Carnegie Mellon University, glmiller@cs.cmu.edu.
‡
Carnegie Mellon University, pachocki@cs.cmu.edu.
§
Microsoft Research New England, asid@microsoft.com.
†
1
Introduction
In this paper, we study several fundamental routing questions in directed graphs that are nearly
Eulerian. We introduce the notion of balance for directed graphs that quantifies how far away a
graph is from being Eulerian1 : a weighted directed graph is α-balanced if for every cut S ⊆ V , the
total weight of edges going from S to V \ S is within factor α of the total weight of edges going
from V \ S to S. Several important families of graphs are nearly balanced, in particular, Eulerian
graphs (with α = 1) and residual graphs of (1 + ǫ)-approximate undirected maximum flows (with
α = O(1/ǫ)).
We use the notion of balance to give a more fine-grained understanding of several well-studied
routing questions that are considerably harder in directed graphs. The first question that we address
is that of designing oblivious routing schemes for directed graphs. Oblivious routing schemes were
introduced in the seminal work of Räcke [Räc02]. They are motivated by practical applications in
routing traffic in massive networks such as the Internet, where it is necessary to route each request
independently of the other requests and the current traffic in the network. Oblivious routing
schemes were developed in a sequence of works [Räc02, ACF+ 03, BKR03, HKLR05, HKLR06,
HKRL07, Räc08, ER09]. In particular, if the graph is undirected, there exist oblivious routing
schemes that achieve competitive ratio O(log n) [Räc08], where n is the number of nodes, and this
result is optimal [BL99, MMVW97, MMVW97]. In contrast, Hajiaghayi et al. [HKRL07] show
√
a strong lower bound of Ω( n) on the competitive ratio of routing obliviously in directed graphs.
This lower bound holds even for single-source instances of bounded degree graphs, as well as for
instances with symmetric demands.
In this paper, we revisit oblivious routing in directed graphs, and we show that balanced graphs
bridge the gap between directed and undirected graphs (see Section 3). Our main algorithmic result
is an oblivious routing scheme for single-source instances that achieve an O(α·log 3 n/ log log n) competitive ratio. In the process, we make several technical contributions which may be of independent
interest. In particular, we give an efficient algorithm for computing low-radius decompositions of
directed graphs parameterized by balance. We also define and construct low-stretch arborescences,
a new concept generalizing low-stretch spanning trees to directed graphs. Given the far-reaching
implications of low-diameter decompositions and low-stretch spanning trees, we hope that our techniques may find other applications.
Our result is a generalization to directed graphs of Räcke’s influential work [Räc08] that established
a remarkable connection between oblivious routing in undirected graphs and metric embeddings
into trees.
On the negative side, we present new lower bounds for oblivious routing problems on directed
graphs. We show that the competitive ratio of oblivious routing algorithms for directed graphs
has to be Ω(n) in general; this result improves upon the long-standing best known lower bound
√
of Ω( n) [HKRL07]. We also show that the restriction to single-source instances is necessary by
√
showing an Ω( n) lower bound for multiple-source oblivious routing in Eulerian graphs.
The second question that we study is that of finding an approximate maximum flow in balanced
graphs. The maximum flow problem has received considerable attention in recent years, leading
to several breakthrough results. This line of work has led to the development of almost linear
time algorithms for approximate maximum flows in undirected graphs [KLOS14, She13] and the
1
A directed graph is Eulerian if, for each vertex, the total weight of its incoming edges is equal to the total weight
of its outgoing edges. An equivalent definition is that for each cut S ⊆ V , the total weight of edges from S to V \ S
is equal to the total weight of edges from V \ S to S.
1
subsequent improvement of [Pen14, RST14]. In contrast, progress on directed graphs has been
comparatively more modest, and the only improvements are the breakthrough results of Madry,
e 10/7 )-time algorithm for unit-capacity directed graphs with m edges [Mad13] and
yielding an O(m
e √n) for arbitrary directed graphs [LS13].
of Lee and Sidford, obtaining a running time of O(m
e min(√m, n2/3 )) given by Goldberg
These improve over the long-standing best running time of O(m
and Rao [GR98].
In this paper, we study the maximum flow problem in balanced directed graphs with arbitrary
capacities (see Section 5). We develop an efficient algorithm that finds an (1 + ǫ)-approximate
2 /ǫ2 ). Our algorithm builds on the work of
e
maximum flows in α-balanced graphs in time O(mα
Sherman [She13] and it can be viewed as an analogue of his result for directed graphs. The running
time of our algorithm degrades gracefully with the imbalance of the graph and thus it suggests that
balanced graphs provide a meaningful bridge between undirected and directed graphs.
We show that, using our approximate maximum flow algorithm, we can efficiently determine
whether a given directed graph is α-balanced (see Section 5.2). Additionally, we give an application to the directed sparsest cut problem (see Section 5.3).
1.1
Related Work
Oblivious Routing. Oblivious routing schemes are well-studied and several results are known;
we refer the reader to [Räc09] for a comprehensive survey of results for undirected graphs. As mentioned previously, in edge-weighted undirected graphs one can achieve a competitive ratio of O(log n)
[Räc08], and it is the best possible [BL99, MMVW97, MMVW97]. Hajiaghayi et al. [HKRL07] studied oblivious routing schemes in node-weighted undirected graphs and directed graphs. Their work
√
gives an Ω( n) lower bound on the competitive ratio for both node-capacitated undirected graphs
and directed graphs. They also show that these lower bounds still hold in more restricted settings,
such as single-source instances. On the positive side, they give oblivious routing scheme with com√
petitive
ratios of O( n log n) for single-source instances in bounded-degree directed graphs, and
√ 1/4
O( kn log n) for general instances in directed graphs, where k is the number of commodities
and in the worst case k = Θ(n2 ).
Maximum s-t Flows. The maximum flow problem is one of the most central problems in combinatorial optimization and has been studied extensively over the past several decades. Until recently,
most approaches have been based on combinatorial methods such as augmenting paths, blocking
flows, push-relabel, etc. This line of work culminated in the seminal algorithm of Goldberg and
Rao [GR98] that computes a maximum flow in time O(min(n2/3 , m1/2 ) log(n2 /m) log U ) in directed
graphs with integer weights that are at most U .
Over the past decade, a new approach emerged based on techniques drawn from several areas such
as continuous optimization, numerical linear algebra, and spectral graph theory. These approaches
led to a nearly-linear time algorithm for approximate maximum flows in undirected graphs [She13,
e 10/7 )-time algorithm for maximum flows in unit-capacity directed graphs
KLOS14, Pen14], an O(m
√
e
[Mad13] and an O(m n)-time algorithm for arbitrary directed graphs [LS13].
1.2
Organization
The rest of this paper is organized as follows. In Section 2, we give an overview of our main results
and introduce the definitions and notation we use throughout the paper. In Section 3, we give our
oblivious routing scheme for single-source instances. In Section 4, we state our lower bounds for
oblivious routing. In Section 5 we give our approximate maximum flow algorithm and applications.
Many proofs are deferred to the Appendix.
2
2
Overview
2.1
Basic Definitions
We study directed graphs G = (V, E, w, l) with edge set E ⊆ V × V , edge weights w : E → R+
and edge lengths l : E → R+ . Throughout this paper, we assume that G is strongly connected. In
several applications we deal with graphs without weights or lengths. For graphs with edge lengths,
we let d(u, v) denote the shortest path distance from u to v.
We associate the following matrices with the graph G. The matrix of edge weights is defined as
def
def
C = diag(w) and the vertex-edge incidence matrix B ∈ RV ×E is defined as Bs,(u,v) = −1 if
s = u, 1 if s = v and 0 otherwise. We are interested in finding flows that route demands with low
congestion. The congestion incurred by a flow f is C−1 f ∞ , and we say f routes demands b if
Bf = b. The problem of finding a minimum congestion flow for a given demand vector, and its
dual, the maximum congested cut, can be formulated as follows:
min.
f
kC−1 f k∞ s.t.
max. b⊤ v
v
Bf = d, f ≥ 0.
s.t. kC max(B⊤ v, 0)k1 ≤ 1.
We
P let OP Tb denote the optimum value of these problems. Throughout the paper, we let bS =
u∈S bu and w(S, T ) denote the total weight of edges from S to T . It is well-known that for the
second problem, one of the threshold cuts with respect to v achieves bS /w(S, V − S) ≥ b⊤ v.
2.2
Balance
We parameterize strongly connected directed graphs by their imbalance:
Definition 2.1 (Imbalance) Let G = (V, E, w) be a strongly connected directed graph. We define
its imbalance, bal(G), as the minimum α such that w(S, V \ S) ≤ α · w(V \ S, S) for every S ⊆ V .
Two canonical families of balanced graphs are Eulerian graphs. and residual graphs of approximate
undirected maximum flows.
Fact 2.2 A strongly connected directed graph G is Eulerian if and only if bal(G) = 1. If G is the
residual graph of a (1 + ǫ)-approximate undirected maximum flow, then bal(G) = O(ǫ−1 ).
Theorem 2.3 (Equivalent definitions of balance) Let G = (V, E, w) be a directed graph. The
following statements are equivalent:
1. bal(G) ≤ α.
2. There exists a circulation f on G with all edge congestions in [1, α].
3. Let d = B~1 be the residual degrees in G. Then −d can be routed with congestion α − 1.
2.3
Oblivious Routing Schemes
An oblivious routing scheme is a linear operator that, for each source-destination pair (s, t) ∈ V ×V ,
specifies how to route one unit of flow from s to t independently of the other pairs. Given a demand
vector d~ : D → R+ on a set D ⊆ V × V of source-sink pairs, one can produce a multi-commodity
flow that meets these demands by routing each demand pair using the (pre-specified) operator,
independently of the other demands. The competitive ratio of an oblivious routing scheme is the
worst ratio among all possible demand vectors between the congestion of the multi-commodity flow
given by the scheme and the congestion of the minimum congestion multi-commodity flow for the
3
given demand vector.
Our main positive result concerning oblivious routings, given in Section 3, is the existence of good
single-source oblivious routings for balanced graphs. A single-source oblivious routing with source
s ∈ V has D = {s} × V .
Theorem 2.4 (Single Source Oblivious Routings) Every strongly connected graph G admits
a single-source oblivious routing, from any source, with competitive ratio O(bal(G)·log3 n/ log log n).
We achieve this result by generalizing an algorithm for undirected graphs given by Racke [Räc08].
The core difficulty that we need overcome is to find a good way to cluster the vertices of a directed
balanced graph. We define the radius of a cluster C ⊆ V as minu∈C maxv∈C d(u, v).. The voldef P
ume vol(G) of G is defined as vol(G) = e∈E l(e)w(e). Our clustering algorithm is presented in
Section 3.1, and its guarantees can be formalized as follows:
Theorem 2.5 (Balanced Graph Clustering) Let G = (V, E, w, l) be a directed graph. Then
for every r > 0, V can be partitioned into clusters such that every cluster has radius at most r,
and the total weight of edges going between different clusters is O(bal(G)vol(G) log n/r). Moreover,
such a partition can be found in expected linear time.
The guarantees of Theorem 2.5 for undirected graphs match those given by prior work [Awe85,
AKPW95, Bar96, MPX13]. Extending the statement to directed graphs is nontrivial, as it requires
making the notion of cluster radii directed.
In Section 4 we give a new lower bound for all-pairs oblivious routings in directed graphs.
Theorem 2.6 No oblivious routing algorithm for directed graphs can guarantee competitive ratio
better than Ω(n).
We also show that restricting ourselves to single-source oblivious routings is necessary to achieve a
small competitive ratio even when bal(G) = 1.
Theorem 2.7 No oblivious routing algorithm for Eulerian graphs can guarantee competitive ratio
√
better than Ω( n).
2.4
Maximum Flows
Finally, we consider the maximum s-t flow problem in directed graphs parameterized by balance.
Given a source s and a destination t, the maximum s-t flow problem asks us to find a flow f that
routes as much flow as possible from s to t while sending at most we units of flow along each edge
e. In Section 5 we show the following result.
Theorem 2.8 (Approximate Maximum Flow) Given a strongly connected directed graph G,
a source s, and a sink t there is an algorithm that finds a (1 + ǫ)-approximate maximum s-t flow
e
and a (1 − ǫ)-approximate minimum s-t cut in G in time O(m
· bal(G)2 /ǫ2 ).
To achieve quadratic dependency on ǫ, in Section 5.4 we provide a general analysis of gradient
descent for composite function minimization under non-Euclidean norms.
We also show applications of this result to computing the sparsest cut (Section 5.3) and we prove
the following result on computing the imbalance of a graph (Section 5.2).
Lemma 2.9 There is an algorithm that either certifies that bal(G) ≤ α or shows that bal(G) >
2 /ǫ2 ).
e
(1 − ǫ)α in time O(mα
4
3
Oblivious Routing on Balanced Graphs
3.1
Low-radius Decompositions
Our algorithm for clustering directed graphs, presented in Figure 1, is based on the scheme given by
Miller, Peng and Xu [MPX13]. We first pick a start time xv for every vertex v from an exponential
distribution, and then explore the graph, starting the search from v at time xv and proceeding at
unit speed. Each vertex u is assigned to the vertex v that reached it first.
(V1 , V2 , . . .) = Cluster-Directed(G, r), where G = (V, E, l) is a directed graph and r > 0.
1. Set β := log n/(10r).
2. For every vertex v ∈ V pick xv ∼ Exp(β).2
3. For each vertex u ∈ V , assign u to the cluster rooted at the vertex v ∈ V which
minimizes −xv + d(v, u).
4. If any of the clusters has radius greater than r, return to step 2. Otherwise, return
the clusters.
Figure 1: The low-radius decomposition algorithm for directed graphs.
Our goal is to show that this procedure cuts few edges, i.e. assigns the endpoints of few edges to
different clusters. The original analysis of [MPX13] shows that for undirected graphs, this approach
guarantees cutting each edge e with low probability, namely O(l(e) log n/r). It turns out that even
in the case of unweighted Eulerian graphs such a guarantee no longer holds; there may exist edges
that are cut with very high probability. Consider for instance (Figure 2) a directed cycle of length
2
3k , with an undirected star of 2k leaves attached to one of its vertices, v. Set r := 2k . Let u be
the vertex preceding v on the cycle. It is now easy to verify by calculation that the edge (u, v) is
cut with probability arbitrarily close to 1 for a large enough k. With high probability, v will be
2
contained in a cluster rooted at one of the 2k leaves attached to it; also with high probability, no
such cluster will contain u.
u
v
..
.
...
Figure 2: An unweighted Eulerian graph where a particular edge is very likely to be cut by the
scheme of [MPX13].
This issue requires us to find a new way to guarantee that the total weight of cut edges is low. Our
key idea is to show that, for any fixed cycle, the expected number of edges in the cycle that are cut
is small. The desired guarantees then follow by noting that any graph G can be approximated up
to a factor bal(G) by a sum of cycles (Theorem 2.3).
2
Exp(β) is the exponential distribution with parameter β, with p.d.f. f (x) = βe−βx on x ≥ 0.
5
Lemma 3.1 Let P be the partition returned by Cluster-Directed(G, r). For any simple cycle
C in G, the expected number of edges in C that go between different clusters in P is an O(log n/r)
fraction of the length of C.
As the above example demonstrates, we cannot base the proof of Lemma 3.1 on the location of the
cuts, as it might depend strongly on the input graph. However, we can prove that, intuitively, cuts
occur infrequently as the graph is explored. This is the crucial idea of the proof: we analyze the
occurrence of cuts over time rather than bounding the probabilities of particular cuts. Then we
use the fact that a cycle of length L is fully explored within L time steps after the time it is visited
for the first time. The analysis is presented in Appendix B.
3.2
Low-stretch Arborescences
Let G be a directed graph and let s be a vertex in G. We say that a directed graph T is an
arborescence rooted at s for every vertex v, there is a unique directed path in T from s to v. In
this section, we define and construct low-stretch arborescences, which are a key intermediate step
between low-radius decompositions and oblivious routings.
Definition 3.2 Let G = (V, E, w, l) be a directed graph. We define the stretch of an edge (u, v) ∈ E
with respect to an arborescence T on the vertex set V as w(u, v) · dT (u, v), where dT (u, v) is the
distance between u and v in the undirected tree corresponding to T .
Following the notation of [Räc08], we define the load, loadT (e), of an edge e ∈ T as the sum of the
weights of edges (u, v) ∈ E(G) such that e is on the path between u and v in the undirected tree
corresponding to T . Note that the total load of the edges in T is equal to the total stretch of the
edges in G.
In order to construct low-stretch arborescences, we will recursively cluster V using the algorithm
from the previous section. The algorithm Find-Arborescence is defined and analyzed in Appendix C. It is similar to the scheme given by Bartal [Bar96]. One major difficulty is that the
clusters returned by Cluster-Directed may be very imbalanced; in particular, they need not be
strongly connected. In order to resolve this issue, we introduce the notion of additive imbalance
and prove that our clustering algorithms still give good guarantees for graphs with low additive
imbalance.
Theorem 3.3 Let G = (V, E, w, l) be a strongly connected directed graph. Let s ∈ V . Let T =
Find-Arborescence(G, s). Then:
• T has vertex set V and is rooted at s,
• every arc (u, v) in T can be mapped to a path from u to v in G of equal length, and
• the expected total stretch of G with respect to T is O(bal(G)vol(G) log3 n/ log log n).
Moreover, the algorithm works in expected O(m log n) time.
3.3
Constructing the Routing
Given an algorithm for constructing low-stretch arborescences, we can use it to compute a good
oblivious routings using the approach proposed by [Räc08]. The oblivious routing that we construct
for a given source s will be a convex combination of arborescences rooted at s, with the flow for
demand (s, u) being defined as the convex combination of the corresponding paths. The algorithm
is given in Figure 3.
The key idea we employ to extend the analysis of the algorithm to a directed graph G is to prove
that the routing scheme we construct is competitive even for the undirected graph underlying G.
6
((T1 , λ1 ), . . . , (Tk , λk )) = Find-Routing(G, s) where G = (V, E, w) is a strongly connected
directed graph and s ∈ V .
(0)
1. Set k :=
P0k and pe := 1 for all e ∈ E.
2. While i=1 λi < 1:
(a) k := k + 1.
(b) Let Gk = (V, E, lk ) be a copy G with edge lengths
!
X (k−1)
(k−1)
lk (e) := pe
/ w(e)
.
pe′
e′
(c) Tk := Find-Arborescence(G, s) (pick the minimum-stretch arborescence out
of O(log n) runs).
(d) ℓk := maxe{loadTk (e)/w(e)}.
Pk−1
(e) λk := min 1/ℓk , 1 − i=1
λi .
(f) For all edges e set:
(k−1)
· exp(λk · loadTk (e)/w(e)).
p(k)
e := pe
3. Return ((T1 , λ1 ), . . . , (Tk , λk )).
Figure 3: The algorithm for finding single-source oblivious routings on balanced graphs (adapted
from [Räc08]).
Lemma 3.4 ([Räc08], adapted) Let G be a strongly connected directed graph and s be a vertex in
G. Let ((T1 , λ1 ), . . . , (Tk , λk )) := Find-Routing(G, s). Then with high probability ((T1′ , λ1 ), . . . , (Tk′ , λk ))
is an O(bal(G) log 3 n/ log log n)-competitive oblivious routing for G′ , where T1′ , . . . , Tk′ , G′ are the
undirected counterparts of T1 , . . . , Tk and G, respectively, that we obtain by ignoring the directions.
In order to finish the analysis, we only need to note that ((T1 , λ1 ), . . . , (Tk , λk )) is an oblivious
routing for G.
We prove that for any s, the output of Find-Routing(G, s) satisfies
Proof of Theorem 2.4:
the criteria stated in the theorem statement. It follows from Lemma 3.4 that with high probability,
((T1′ , λ1 ), . . . , (Tk′ , λk )) is an O(bal(G) log 3 n/ log log n)-competitive oblivious routing for G′ , where
T1′ , . . . , Tk′ , G′ are undirected counterparts of T1 , . . . , Tk , G, respectively. In particular, it is also an
O(bal(G) log 3 n/ log log n)-competitive oblivious routing from s. Now it is enough to observe that
since T1 , . . . , Tk are directed away from s, ((T1 , λ1 ), . . . , (Tk , λk )) is an oblivious routing from s in G.
Since it is O(bal(G) log3 n/ log log n)-competitive in G′ , it must also be O(bal(G) log 3 n/ log log n)competitive in G.
4
Lower Bounds
We prove new lower bounds for oblivious routings in directed graphs. The constructions and proofs
are given in Appendix E.
Theorem 2.6 No oblivious routing algorithm for directed graphs can guarantee competitive ratio
better than Ω(n).
Theorem 2.7 No oblivious routing algorithm for Eulerian graphs can guarantee competitive ratio
7
s
t
..
.
..
.
Figure 4: The example from Theorem 2.6. The thick edges have weight n, the other edges have
weight 1. Any oblivious routing must put too much flow on the edge (s, t) when routing between
the vertices of the biclique.
...
√
Figure 5: The example from Theorem 2.7. The thick edges have weight n, the other edges have
weight 1. Any oblivious routing must put too much flow on the outer cycle when routing between
consecutive vertices of the inner cycle.
√
better than Ω( n).
5
5.1
Maximum Flow and Applications
Directed Maximum Flow
In this subsection we show how to efficiently compute an (1 + ǫ)-approximate maximum flow in
directed graphs given a good congestion-approximator.
Definition 5.1 An α-congestion-approximator for G is a matrix R such that for any demand
vector b, Rb ∞ ≤ OP Tb ≤ α Rb ∞ .
Since Rb
∞
=
− Rb
∞
, only well-balanced graphs admit good congestion approximators:
Fact 5.2 If G admits an α-congestion approximator, bal(G) ≤ α.
e
For undirected graphs, O(1)-congestion-approximators
can be computed in nearly linear time
e
[Mad10, She13, KLOS14, Pen14]. This implies that for directed G we can compute O(bal(G))congestion-approximators in nearly linear time by the following fact:
Fact 5.3 Let G be a directed graph and G′ be its undirected copy. Then for any demand vector b
OP Tb (G′ ) ≤ OP Tb (G) ≤ (1 + bal(G))OP Tb (G′ ).
8
Our main result is the following:
Theorem 5.4 Let G be a directed graph. Given an α-congestion-approximator R, we can compute
an (1 + ǫ)-approximate maximum flow and minimum congested cut for any demand vector in time
2 /ǫ2 ), assuming multiplication by R and R⊤ can be done in O(m)
e
e
O(mα
time.
Our algorithm is based very heavily on the approach for undirected graphs given by Sherman
[She13]. The main difference is the implementation of the key optimization procedure, presented
in Figure 6. Due to space constraints, in this section we only outline the main changes needed to
extend the algorithm of [She13] to balanced graphs.
Let G be a directed graph and b be
vector. Assume we are given an α-congestionP a xdemand
def
−x
i
i
approximator R. Let lmax(x) = ln i (e + e ) and define
def
µ(f ) = lmax(2αR(b − Bf ))
φ(f ) = C−1 f
def
∞
+ µ(f )
(f, v) = Almost-Route-Directed(b, ǫ, f0 )
ǫ
1. Initialize f := f0 , δ := 10α
2.
−1
2. Scale f and b so that C f ∞ + 2α R(b − Bf ) ∞ = 20ǫ−1 ln n.
3. Repeat while any of the following conditions is satisfied:
(a) if φ(f ) < 16ǫ−1 ln n, scale f and b up by 17/16 and restart step 3.
(b) let s be w(e) on the coordinates e where ∇µ(f ) is negative and 0 elsewhere. If
−∇µ(f )⊤ s > 1 + 4ǫ , set f := f + δs and restart step 3.
(c) if C−1 f ∞ + ∇µ(f )⊤ f > 4ǫ , set f := f − δf and restart step 3.
4. Set x := 2αR(b − Bf ).
5. Set p := ∇lmax(x).
6. Set v := R⊤ p.
Figure 6: The algorithm for computing the maximum flow and minimum congested cut.
Lemma 5.5 After Almost-Route-Directed(b, ǫ, f0 ) terminates, we have
φ(f ) ≤ (1 + ǫ)
b⊤ v
,
kC max(B⊤ v, 0)k1
assuming ǫ ≤ 1/2.
2 3
e
Lemma 5.6 Almost-Route-Directed(b, ǫ, f0 ) terminates within O(log(1+ǫ
0 )α /ǫ ) iterations,
where ǫ0 = max(φ(f0 )/OP Tb − 1, ǫ), assuming ǫ ≤ 1/2.
Note that Lemma 5.5 implies that v is a potential vector for a (1 + ǫ)-approximate minimum
congested cut. In order to recover the corresponding flow, we can employ the recursion described
in [She13]. The only additional component necessary for directed graphs is an O(poly(n, α))competitive oblivious routing. Since by 5.2 it must be that α ≥ bal(G), this can be obtained easily
by taking the maximum spanning in- and out-arborescences from any fixed vertex.
If we run Almost-Route-Directed with f0 = ~0, we can find (1+ǫ)-approximate solutions in time
2 /ǫ3 ). In order to improve the dependency on ǫ, we can employ a general form of composite
e
O(mα
9
function minimization, introduced in Section 5.4. Define
(
∞
if for some e, fe /w(e) ∈
/ [0, 50 ln n/ǫ]
def
ψ(f ) =
−1
C f ∞ otherwise.
The faster algorithm is presented in Figure 7.
(f, v) = Fast-Almost-Route(b, ǫ)
1. Set f0 using Almost-Route-Directed b, 12 , ~0 , keeping the rescaling.
2. Set K := ⌈α2 /ǫ2 ⌉.
3. For k = 0, . . . , K − 1 let
α2
fk+1 := argminf ∈RE ∇µ(fk )⊤ f +
C−1 (f − fk )
2
2
∞
4. Return Almost-Route-Directed(b, ǫ, fK ).
+ ψ(f ) .
Figure 7: Faster algorithm for computing the maximum flow and minimum congested cut.
If we apply the analysis from Section 5.4 (encapsulated in Theorem 5.9), we obtain the following.
2 /ǫ2 ) time, assuming ǫ ≤ 1/2.
e
Lemma 5.7 Fast-Almost-Route(b, ǫ) terminates in O(mα
5.2
Computing Imbalance
As verifying balance can be reduced to a maximum flow computation by Theorem 2.3, we obtain
the following result:
Lemma 2.9 There is an algorithm that either certifies that bal(G) ≤ α or shows that bal(G) >
2 /ǫ2 ).
e
(1 − ǫ)α in time O(mα
5.3
Application to Directed Sparsest Cut
In this subsection, assume G = (V, E) is a directed graph that is unweighted, strongly connected,
\S)
simple, and with an even number of vertices. We define the sparsity of a cut (S, V \ S) as w(S,V
|S|·|V \S| ,
where w(S, V \ S) is the number of edges going from S to V \ S. Note that under this definition,
no cut can have sparsity greater than one.
As a second application of our maximum flow algorithm, we get the following sparsest cut algorithm.
While blocking flows could also possibly be used for our purpose, our approach is clean and may
easily generalize to weighted graphs. We defer the details to the appendix.
Lemma 5.8 Given φ ≤ 1, we can find a cut of sparsity φ in G or determine that all cuts in G
2 ).
e
have sparsity Ω(φ/ log2 n) in time O(m/φ
5.4
Composite Function Minimization
In this section, we provide a non-Euclidean gradient descent method for minimizing a composite
def
function f (x) = g(x) + ψ(x), where g and ψ have specific properties. The algorithm and its
convergence guarantee are encapsulated in the following theorem, and they build on several works
in convex optimization, such as [Nes13, RT14].
def
Theorem 5.9 Let f : Rn → R be a convex function given by f (x) = g(x) + ψ(x) where g is
10
convex and L-smooth3 with respect to some norm · . Moreover, assume that f (x) is only finite
on some region of diameter D in · . Starting with some x0 ∈ Rn for all k let
xk+1 := argminx∈Rn
Then for all k ≥ 1 we have
ǫk ≤ max
(
▽ g(xk ), x +
2 · L · D2
,
⌊ k−1
2 ⌋+4
L
x − xk
2
2
+ ψ(x) .
⌊ k−1 ⌋ )
2
1
ǫ0
2
where ǫk = f (xk ) − minx f (x).
Note that the norm we use is arbitrary and we get a gradient descent analysis without appealing
to the dual norm. Also we do not require convex ψ we only require convex f .
Acknowledgments
We thank Yin-Tat Lee for several helpful discussions, and in particular for his help with the results
in Section 5. This work was partially supported by NSF awards 0843915, 1065106 and 1111109, NSF
Graduate Research Fellowship (grant no. 1122374) and Sansom Graduate Fellowship in Computer
Science. Part of this work was done while authors were visiting the Simons Institute for the Theory
of Computing, UC Berkeley.
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A
Missing proofs from Section 2
Lemma A.1 Let G be a strongly connected directed graph. If demand vector d can be routed in G
with congestion c, then −d can be routed in G with congestion at most bal(G) · c.
Proof:
Note that for any v ∈ Rn
kU max(Bv, 0)k1 ≤ bal(G)kU max(−Bv, 0)k1
follows from the definition of balance.
Hence it is easily seen that the optimum value for the dual problem is within a factor bal(G) for
demands d and −d. Our theorem now follows from strong duality to the original problem.
Lemma A.2 Let l, r ∈ R with l ≤ r. Let C ⊆ Rm be a convex set such that for any S ⊆
{1, 2, . . . , m} there exists a point x ∈ C such that xi is at least l for i ∈ S and xi is at most r for
i∈
/ S. Then there exists a point in C with all coordinates in [l, r].
Proof:
Let Pi (S), for i ∈ {0, . . . , m}, S ⊆ {1, . . . , m} be the subset of points x ∈ C that satisfy
• xj ∈ [l, r] for j ≤ i and
• xj ≥ l for j ∈ S and
• xj ≤ r for j ∈
/ S.
We prove that Pi (S) is nonempty for every i and S by induction on i. The base case i = 0 follows
from the assumption on C. Assume i ∈ {0, . . . , m − 1} and the thesis holds for i. Let S be any subset of {1, . . . , m}. Let SL := S ∪ {i + 1}, SR = S \ {i + 1}. Pick any xL ∈ Pi (SL ) and xR ∈ Pi (SR ).
Then a convex combination of xL and xR must belong to Pi+1 (S). Since S was arbitrary, this
concludes the proof.
13
Proof of Theorem 2.3:
The implication (2. → 1.) and the equivalence (2. ↔ 3.) are easy to
check (note that the circulation of 2. is the sum of ~1 and a routing of −d.). We now prove that if
bal(G) ≤ α there exists a circulation in G with each congestion in [1, α].
Note that for any subset S of edges of G we can route the residual degree dS induced by these edges
with congestion 1. Hence by Lemma A.1 we can route −dS with congestion at most α. Adding
these flows yields a circulation with congestion in [1, α + 1] on edges in S and in [0, α] on the other
edges. Since the choice of S was arbitrary, the thesis follows by Lemma A.2.
We now prove the following lemma, implying 2.2.
Lemma A.3 Let G = (V, E, w) be an undirected graph and s, t ∈ V . Let d be a demand vector
that can be routed in G with congestion at most 1. Let f be a flow from s to t in G with congestion
not exceeding
1 satisfying demands (1 − ǫ)d. Let H be the residual graph of f in G. Then bal(H) ≤
2ǫ−1 − 1 .
Proof: We use the third equivalent definition of balance from Theorem 2.3. The residual degrees
in H are 2(ǫ − 1)d. Since there exists a flow satisfying demands ǫd with congestion 1, 2(1 − ǫ)d can
be routed in H with congestion 2(1 − ǫ)ǫ−1 = 2ǫ−1 − 2.
B
Missing proofs from Section 3.1
Before we prove Lemma 3.1, we shall study the properties of a class of two-way infinite sequences.
For better understanding, we now attempt to provide the intuition on how the sequences defined
below are used in the proof. For simplicity, assume we are trying to analyze the clustering of a path
rather than a cycle. Imagine that every vertex v in the graph sends a runner of unit speed to every
vertex of the path, starting at time −xv . After reaching the path, the runner keeps running on it
until they reach the end. We will call a runner a local leader if they were the first one to reach the
end of the path out of all the runners that entered the path at a no later position. It is easy to see
that the sequence of origins of local leaders in the order they reach the end of the path is the same
as the sequence of roots of clusters into which the path is partitioned by the algorithm. Therefore,
it is enough to observe the local leaders as they reach the end of the path. It can be shown that in
any time interval [y, y + ǫ] the probability of the origin of the last local leader to reach the end of
the path changing is O(βǫ). Unfortunately, the entire process could take an arbitrary amount of
time in the case of a path.
To apply the above reasoning to a cycle, we will ’unroll’ it into a two-way infinite path. We will set
the ’finish line’ at an arbitrary vertex (at position 0) and observe the local leaders for any period
of time [y, y + L].
Assume the length of the cycle is L and it has l vertices. Let i ∈ {0, . . . , l − 1}, i ∈ Z. Then, the
i · n + j-th element of the sequence s will intuitively be equal to the time the runner sent from the
j-th vertex of the graph to the i-th vertex of the unrolled cycle reaches vertex 0 of the unrolled
cycle. The sequence a will simply label the origin of the runner relevant to the current index (and
so ai = (i mod n)). The sequence c will label the cluster to which the relevant vertex of the cycle
is assigned (the origin of the first runner to reach it). The function f (y) will give the origin of the
runner that reached vertex 0 before time y and entered the cycle at the earliest position. Since
only such runners will correspond to clusters, our goal will be to bound the frequency with which
f may change.
14
B.1
Periodically Decreasing Sequences
Let k, n ∈ N and L ∈ R+ .
Let si be a two-way infinite sequence of real numbers indexed by i ∈ Z with the property
∀i si+k = si − L.
Let ai be a two-way infinite sequence of integers in {0, . . . , n − 1} indexed by i ∈ Z, periodic with
period k, that is
∀i ai+k = ai .
We construct the sequence ci by defining
ci = aj ,
where j is the minimum q that minimizes the value of sq among q ≤ i.
We can similarly construct f : R → {0, . . . , n − 1} by setting for every real number y
f (y) = aj ,
where j is the minimum q that satisfies sq ≤ y.
Fact B.1 The sequence ci is periodic with period k.
Fact B.2 The function f is periodic with period L.
Fact B.3 For any i ∈ Z and y ∈ R, the number of times the sequence ci , ci+1 , . . . , ci+k changes
values is equal to the number of times f changes values on the interval [y, y + L].
B.2
Random Periodically Decreasing Sequences
Let k, n ∈ N, L ∈ R+ .
Let ti be a two-way infinite sequence of real numbers indexed by i ∈ Z with the property
∀i ti+k = ti − L.
Let ai be a two-way infinite sequence of integers in {0, . . . , n − 1} indexed by i ∈ Z, periodic with
period k, that is
∀i ai+k = ai .
Let x0 , x1 , . . . , xn−1 be independent random variables drawn from the exponential distribution with
parameter β.
We define for every i ∈ Z:
si = ti − xai
We define the function f : R → {0, . . . , n − 1} as in the previous section, that is
f (y) = aj ,
15
where j is the minimum q that satisfies sq ≤ y.
In the following lemmas, our goal will be to bound the expected number of times the value of f
changes on any interval.
Lemma B.4 For any y ∈ R, ǫ ∈ R+ , the probability that f is not constant on the interval [y, y + ǫ]
is bounded by O(βǫ).
Proof: Fix y and ǫ. We condition on the value of f (y + ǫ); assume it is k. We also condition on
xi for all i 6= k. Now the condition f (y + ǫ) = k is equivalent to assuming xk ≥ c for some constant
c. Because we have no more information about xk , the conditional probability that xk ≥ c + ǫ is
1 − O(βǫ). This implies the thesis.
In order to exploit Lemma B.4 to bound the expected number of changes in f we will attempt to
condition on the event Dǫ .
Definition B.5 Let ǫ ∈ R+ . The event Dǫ occurs iff for all pairs i, j ∈ Z such that ai 6= aj or
ti 6= tj it holds that
|si − sj | > ǫ.
Fact B.6
lim P (Dǫ ) = 1.
ǫ→0
Using B.6, we pick an ǫ > 0 that satisfies
P (Dǫ ) ≥ 1 − min
1 βL
,
2 k
and
L
∈N.
ǫ
Lemma B.7 Assume ǫ is chosen as above. Conditioning on Dǫ , for any y ∈ R, the probability
that f is not constant on the interval [y, y + ǫ] is bounded by O(βǫ).
Proof: Because P (Dǫ ) ≥ 12 , the conditional probability is at most two times larger than in the
case where we do not condition on Dǫ . The thesis follows from Lemma B.4.
Lemma B.8 Assume ǫ is chosen as above. Conditioning on Dǫ , for any y ∈ R, the expected
number of times f changes values in [y, y + L] is bounded by O(βL).
Proof: Because we assume Dǫ , we know that f can change at most once on any interval of length
ǫ. Hence it follows from Lemma B.7 that the expected number of time f changes on any interval
of length ǫ is bounded by O(βǫ). Because L/ǫ ∈ N, we can cover the interval [y, y + L] with L/ǫ
intervals of length ǫ. Because of linearity of expectation, the expected number of times f changes
values on [y, y + L] is therefore bounded by O(βL).
Lemma B.9 For any y ∈ R, the expected number of times f changes values in [y, y + L] is bounded
by O(βL).
Proof: It follows from B.3 that f cannot change values more than k times on an interval of
length L. For ǫ chosen as above, we can apply this observation together with Lemma B.8 to see
16
that the expected number of changes is bounded by
P (Dǫ )O(βL) + (1 − P (Dǫ ))k = O(βL) +
βL
· k = O(βL)
k
B.3
Low-radius Decompositions
Recall that we are considering the clustering algorithm Cluster-Directed applied to a directed
graph G = (V, E). Consider a cycle C in G. Assume the length of C is L and the number of
vertices on C is l. Let the vertices on the cycle be u0 , . . . , ul−1 , in order, with u0 chosen arbitrarily.
For i ∈ {0, . . . , l − 1}, define pi to be the distance from u0 to ui when going along the cycle.
Let k = l · n. We now define the two-way infinite sequence t as follows, for z ∈ Z, i ∈ {0, . . . , m −
1}, j ∈ {0, . . . , n − 1}:
tz·k+i·n+j = d(vj , ui ) − zL − pi .
We define the two-way infinite sequence a for z ∈ Z, j ∈ {0, . . . , n − 1}:
az·n+j = j.
Fact B.10 Let i ∈ {0, . . . , l−1}. Assume j is a (possibly negative) integer such that j ≤ i·n+n−1.
Then there exists a path from vaj to ui of length tj + pi .
Fact B.11 Let i ∈ {0, . . . , l − 1}, q ∈ {0, . . . , n − 1}. There exists an integer j ≤ i · n + n − 1 such
that aj = q and
tj + pi = d(vq , ui ).
Recall that in Cluster-Directed we associate with each vertex vi an independent random variable
xi drawn from the exponential distribution with parameter β. We now define the two-way infinite
sequence s as
si = ti − xai .
As in Section B.1 we construct the sequence ci by defining
ci = aj ,
where j is the minimum q that minimizes the value of sq among q ≤ i.
Lemma B.12 For i ∈ {0, . . . , l − 1}, ci·n+n−1 is the index of the vertex to whose cluster ui is
assigned by Cluster-Directed.
Proof:
This follows from Facts B.10 and B.11.
We are now ready to prove the main theorem.
17
Proof of Lemma 3.1:
By Lemma B.12, it is enough to bound the number of times c0 , . . . , ck
changes values. By B.3 this reduces to bounding the number of times the associated function
f : R → {0, . . . , n − 1} changes values on any interval of length L. This is shown to be O(βL) in
expectation in Lemma B.9.
First note that with high probability max(x1 , . . . , xn ) ≤ r, and so the
Proof of Theorem 2.5:
radius of the computed clusters is at most r. To bound the number of cut edges, it is enough to
note that by Theorem 2.3, we can find a set of simple cycles C1 , . . . , Ck such that their total volume
is at most bal(G)vol(G) and has weight at least w(e) on every edge e ∈ E. The thesis follows by
applying Lemma 3.1.
C
Missing proofs from Section 3.2
Definition C.1 We define the additive imbalance abal(G) of a directed graph G as the minimum
ι such that it is possible to add edges of total weight ι to G to make it Eulerian.
In order to make the running time of our algorithm independent of the diameter of the graph, we
will attempt to collapse very short edges in the upper levels of the recursion, that is, contract their
endpoints into a single vertex. This is similar to the scheme proposed in [CMP+ 14, CKM+ 14].
However, this operation is not always feasible in directed graphs; thus, we will only perform the
contraction if both endpoints of the edge can reach each other by following only very short edges.
Definition C.2 Let G = (V, E, w, l) be a directed graph and xL , xR ∈ R be such that 0 < xL < xR .
We construct G collapsed to [xL , xR ] by:
• merging any vertices that can reach each other while following only arcs of length at most xL ,
and
• reducing the length of all arcs longer than xR to xR .
T = Find-Arborescence(G, s), where G = (V, E, l) is a directed graph and s ∈ V is such
that all vertices in G are reachable from s.
1. If n = 1, return a single-vertex graph.
2. Let r := maxv∈V dG (s, v).
3. Let r ′ := r/(c · log n).
4. Let G′ be the graph G collapsed to [r ′ /n, 2r ′ ]. Let s′ be the vertex in G′ corresponding
to s.
5. Let V1′ , V2′ , . . . , Vk′ := Cluster-Directed-Rooted(G′ , s′ , r ′ ).
6. Expand the clusters V1′ , . . . , Vk′ back into G, obtaining V1 , . . . , Vk .
7. Let Gi be the graph induced by Vi , for i = 1, . . . k, and ui denote the center of cluster
Vi (with uS
1 = s1 ).
8. Let T ′ := ki=1 Find-Arborescence(Gi , ui ).
9. Let T be T ′ with the arcs (s, ui ) of length dG (s, ui ) added for each i = 2, . . . , k.
10. Return T .
Figure 8: The low-stretch arborescence finding algorithm.
Lemma C.3 Let G = (V, E, w, l), s ∈ V, r > 0. Let V1 , . . . , Vk = Cluster-Directed-Rooted(G, s, r).
Then:
• each cluster Vi has radius at most r,
18
(V1 , V2 , . . .) = Cluster-Directed-Rooted(G, s, r), where G = (V, E, l) is a directed
graph, s ∈ V and r > 0.
1. Choose r ′ uniformly at random from [0, r].
2. Let V1 be the set of vertices at distance at most r ′ from s.
3. Let G′ be the induced graph on V − V1 .
4. Let V2 , V3 , . . . Vk := Cluster-Directed(G′ , r).
5. Return V1 , V2 , . . . , Vk .
Figure 9: The decomposition algorithm with a specified root.
• the cluster V1 containing s has radius at most r from s,
• the expected total weight of edges going between different clusters is O(vol(G) log n/r+abal(G) log n),
and
• the expected total additive imbalance of the clusters is O(vol(G) log n/r + abal(G) log n).
Moreover, the algorithm works in expected linear time.
Proof: First, note that the expected total weight of edges between V1 and V − V1 is O(vol(G)/r).
Hence the expected additive imbalances of the cluster on V1 and that of G′ are both O(abal(G) +
vol(G)/r).
By the definition of additive imbalance, we can add edges of expected total weight O(abal(G) +
vol(G)/r) to G′ to make it Eulerian. We obtain the graph G′′ by adding such edges, each with length
2r. The expected volume of G′′ is O(vol(G)) + O(abal(G) + vol(G)/r) · 2r = O(vol(G) + abal(G) · r).
Now by Theorem 2.5 we can partition G′′ into clusters of radius at most r, with the expected total weight of edges going between clusters O(vol(G′′ ) log n/r) = O(vol(G) log n/r + abal(G) log n).
Note that if we remove the added edges, the radii of these clusters cannot change, as the edges
have length greater than r; at the same time, their total additive imbalance can increase by at most
O(abal(G) + vol(G)/r) in expectation. To complete the analysis, observe that in fact the edges
added in the above reasoning are ignored by the decomposition algorithm. Hence, they are only
necessary for the analysis.
Lemma C.4 Let G = (V, E, w, l) be a directed Eulerian graph and xL , xR ∈ R be such that 0 <
xL < xR . Let G′ = (V ′ , E ′ , w′ , l′ ) be G collapsed to [xL , xR ]. Then vol(G′ ) is at most
X
2·
w(e) min(l(e), xR ).
e∈E:l(e)>xL /n
Proof: Since G′ is Eulerian, it can be represented as a sum of simple cycles of uniform weight.
Consider any such decomposition and take any cycle C in it. Then C must contain an edge of
length at least xL , and it contains at most n edges of length not exceeding xL /n. Hence, the length
of C is at most two times greater than the sum of its edge lengths greater than xL /n. Summing
over all the cycles yields the desired bound.
Proof of Theorem 3.3: First, note that by Theorem 2.3 the edge weights in G can be increased
to obtain an Eulerian graph with volume at most bal(G)vol(G). Since the algorithm is oblivious to
weights, it is enough to consider Eulerian graphs in the proof; from now on we assume bal(G) = 1.
19
Properties 1 and 2 are easy to verify. Assume the constants hidden in the big-oh notation in
Lemma C.3 are bounded by c0 . We set c := 2c0 + 4.
Consider the i-th level (numbering from 0) of the tree of recursive calls of Find-Arborescence(G, s).
Let ri = r/(c log n)i . It can easily be shown by induction that the radii of the graphs in the i-th
level are at most ri , and the radii of the returned arborescences are at most 2ri , since c ≥ 4. Let
νi be the total volume of the collapsed graphs at level i.
By Lemma C.3 the additive imbalance of the graphs in the i-th level can be bounded by
(c0 log n)i · ν0 /r1
+(c0 log n)i−1 · ν1 /r2
+(c0 log n)i−2 · ν2 /r3
+...
+(c0 log n)1 · νi−1 /ri .
Since c > 2c0 , the above sum is bounded by
(c log n)i+1
X
j<i
νj /2i−j .
Hence, the total weight of edges cut at level i is at most
X
X
(c0 log n) νi /ri+1 + (c log n)i+1
νj /2i−j .
νj /2i−j ≤ (c log n)i+2 /2 ·
j<i
j≤i
Since the radius of the arborescence returned at level i is at most 2ri , we have that the total stretch
incurred at level i is at most
X
X
νj /2i−j .
νj /2i−j . ≤ (c log n)2 ·
2ri · (c log n)i+2 /2 ·
j≤i
j≤i
Hence the total stretch is at most
(c log n)2 ·
XX
i
j≤i
X
X
νj 2j
νj /2i−j = (c log n)2 ·
2−i
j
≤ 2(c log n)2 ·
X
i≥j
νj .
j
Observe that all the collapsed graphs at level j are subgraphs of G collapsed to [rj+1 /n, 2rj+1 ].
Hence, by Lemma C.4, we have
X
νj ≤ 2 ·
w(e) min(l(e), 2rj+1 ).
e∈E:l(e)>rj+1 /n2
20
Hence
X
j
νj ≤ 2 ·
X
X
e∈E j:rj+1
w(e) min(l(e), 2rj+1 )
<l(e)·n2
= O(vol(G) log n/ log log n).
Combining this with the previous bound yields the thesis.
D
Missing proofs from Section 3.3
Proof of Lemma 3.4:
It is enough to note that in step 2c) of Figure 3, with high probability,
Tk′ is a tree with total stretch O(bal(G) log 3 n/ log log n) in G′k , where Tk′ and G′k are undirected
counterparts of Tk and Gk , respectively. Hence, the analysis of [Räc08] can be applied to complete
the proof.
E
Missing proofs from Section 4
Proof of Theorem 2.6:
Let k ≥ 1. Let G be a directed graph on the vertex set
V = S ∪ T ∪ {s} ∪ {t}, where |S| = |T | = k
and edge set
E = S × T with weight 1
∪ S × {s} with weight k
∪ {(s, t)} with weight k
∪ {t} × T with weight k.
Assume some oblivious routing A achieves competitive ratio c on G. Let u ∈ S and v ∈ T . The
optimal congestion for the unit flow from u to v is at most 1/k, which can be achieved by routing
the flow through s and t. Therefore, A must achieve congestion at most c/k, hence putting at least
1 − c/k units of flow on the edge (s, t).
The optimal congestion for the multicommodity flow with unit demand between every pair in S × T
is clearly at most 1. Simultaneously, by the above argument, A must put at least k(k − c) flow on
the edge (s, t). Hence we have c ≥ k − c, implying c ≥ k/2. As n = 2k + 2 we have c = Ω(n).
Let n ≥ 2. Let G be a directed graph on the vertex set
Proof of Theorem 2.7:
V = {v1 , . . . , vn }
and edge set
E = C1 ∪ C2 , where
C1 = {(v1 , v2 ), (v2 , v3 ), . . . , (vn−1 , vn ), (vn , v1 )} with weight 1, and
√
C2 = {(vn , vn−1 ), (vn−1 , vn−2 ), . . . , (v2 , v1 ), (v1 , vn )} with weight n.
Note that G is Eulerian. Assume some oblivious routing A achieves competitive ratio c on G. Let
√
i < n. The optimal congestion for the unit flow from vi to vi+1 is at most 1/ n, which can be
21
√
achieved by routing the flow through C2 . Therefore, A must achieve congestion at most c/ n,
√
hence putting at least 1 − c/ n units of flow on the edge (vn , 1).
The optimal congestion for the multicommodity flow with unit demand between every such pair
(vi , vi+1 ) is clearly at most 1. Simultaneously, by the above argument, A must put at least
√ (n −
√
√
1)(1 − c/ n) flow on the edge (vn , 1). Hence we have c ≥ (n − 1)/ n − c, implying 2c ≥ n − 1.
√
Therefore c = Ω( n).
F
Missing proofs from Section 5
Proof of Lemma 5.5:
We have
▽µ(f ) = −2αB⊤ v.
Therefore
ǫ
2α · kC max(B⊤ v, 0)k1 ≤ 1 + .
4
It also holds that
C−1 f
∞
+ 2αv ⊤ (b − Bf ) = C−1 f
∞
+ pT x
≥ φ(f ) − 4 ln n
ǫ
φ(f ).
≥ 1−
4
Simultaneously, we have
ǫ
ǫ
φ(f ) ≥ ≥ C−1 f
4
4
= C−1 f
∞
+ ▽µ(f )T f
∞
− 2αf T B⊤ v.
Hence
and so
ǫ
2αv T b ≥ 1 −
φ(f ),
2
φ(f )
bT v
≥
.
⊤
kC max(B v, 0)k1
1+ǫ
Let us call the iterations between each scaling in step 2a) a phase.
Proof of Lemma 5.6:
Since the initial scaling gives us the correct scale to within factor 1 + ǫ0 , we will scale at most
O(log(1 + ǫ0 )) times. Moreover, if ǫ0 < 1/10, step 2a) will never be executed.
22
If step 2b) is about to be executed, then
φ(f + δs) ≤ φ(f ) + δ + δ ▽ µ(f )⊤ s + 2α2 δ2
ǫδ
+ 2α2 δ2 .
≤ φ(f ) −
4
If step 2c) is about to be executed, then
φ(f − δf ) ≤ φ(f ) − δ C−1 f ∞ − δ ▽ µ(f )⊤ f + 2α2 δ2
ǫδ
≤ φ(f ) −
+ 2α2 δ2 .
4
In both cases we have
ǫ2
ǫ2
ǫδ
− 2α2 δ2 ≥
−
4
40α2
50α2
2
ǫ
.
=
200α2
Hence each iteration of steps 2b) and 2c) decreases φ(f ) by at least Ω(ǫ2 α−2 ).
For ǫ0 ≥ 1/10, every scaling in step 2a) increases φ(f ) by at most ǫ−1 ln n. Hence, for such ǫ0 there
e
can be at most O(log(1
+ ǫ0 )α2 ǫ−3 ) iterations in total.
For ǫ0 < 1/10, step 2a) will never be executed. Moreover, the φ(f ) after the initial scaling must
e 0 ǫ−1 ). Hence steps 2b) and 2c) can be executed at most O(ǫ
e 0 α2 ǫ−3 ) =
be at most OP Tb + O(ǫ
e
O(log(1
+ ǫ0 )α2 ǫ−3 ) times.
Proof of Lemma 5.7:
2 ) time by Lemma 5.6.
e
e
As φ(~0) = O(OP
Tb α), step 1. works in O(mα
Now note that we can apply Theorem 5.9 to ψ ′ (x) + g(x) with ψ ′ (x) = ψ(Cx), g(x) = µ(Cx), L =
e
α2 , D = 50ǫ−1 ln n. This yields that φ(fK ) ≤ (1 + O(ǫ))OP
Tb . Hence by Lemma 5.6, step 4. runs
2
−2
e
in O(mα ǫ ) time.
e
The only remaining thing to show is that we can solve the optimization problem in step 3. in O(m)
time. It can be reformulated by introducing an auxiliary variable z:
minimize
f,z
subject to
1
▽ µ(fk )⊤ f + α2 z 2 + ψ(f )
2
−1
C (f − fk ) ∞ ≤ z.
For a fixed z, the problem can easily be solved in O(m log m) time by sorting. Hence we can employ
e
ternary search over z to achieve O(m)
runtime.
1
to G. Note
Proof of Lemma 2.9:
Construct G′ by adding the reverse of G multiplied by 4α
′
′
′
that bal(G ) ≤ 4α. Let b be the residual degrees in G . Now by Theorem 2.8 we can compute a
2 ). Note that we
e
2-overestimate c′ to the minimum congestion to route −b′ in G′ , in time O(mα
23
have
bal(G′ ) ≤ c′ − 1 ≤ 2bal(G′ ) ≤ 2bal(G).
Hence if c′ − 1 > 2α we can conclude that bal(G) > α and return the corresponding cut.
Otherwise, we must have bal(G) ≤ 2α. Hence we can compute a (1 + ǫ)-overestimate c to the
2 ǫ−2 ), where b are the residual degrees in G.
e
minimum congestion to route −b in G, in time O(mα
Now we have
bal(G) ≤ c − 1 ≤ (1 + ǫ)bal(G),
and so if c−1 ≤ α then bal(G) ≤ α, and otherwise we can return a cut proving that bal(G) > (1−ǫ)α.
First, we can use Lemma 2.9 to check whether bal(G) ≤ φ−1 . If it is
Proof of Lemma 5.8:
not, we can return the smaller direction of the imbalanced cut as the result. Otherwise, we use
4 and reduce the
can apply the cut-matching game algorithm given by Louis [Lou10] for φ′ = nφ
4
e
problem to a sequence of O(1)
maximum flow queries. Each of the queries fixes some S ⊆ V with
|S| = n/2 and asks for a flow in G with demands −φ′ on S and φ′ on V \ S. We can compute the
2-approximate minimum congestion flow for such a query. If the returned congestion is at most 1,
we return the flow. Otherwise, we have a set T ⊆ V which achieves
1
,
2
w(T, V − T ) ≤ 2bT
bT /w(T, V \ T ) ≥
≤ 2φ′ min(|T |, |V \ T |)
n
≤ φ min(|T |, |V \ T |)
2
≤ φ|T | · |V \ T |.
G
Missing proofs from Section 5.4
We break the proof into 2 parts, first we prove a lemma about the progress of each gradient step
and then we use this to prove the theorem. Let X∗ be the set of all optimal solutions to minx f (x).
Lemma G.1 (Gradient Descent Progress) For all k ≥ 0 we have that for all x∗ ∈ X∗
!2
1
f (xk ) − f (x∗ )
ǫk
,
f (xk+1 ) ≤ f (xk ) − min
2L
2
xk − x∗
Proof:
By the smoothness of g we know that for all x ∈ Rn we have
f (x) ≤ g(xk ) + ∇g(xk ), x − xk +
4
L
x − xk
2
2
+ ψ(x) .
The rescaling by n is used due to a slightly different definition of sparsity in [Lou10].
24
By definition of xk+1 we then have that
L
x − xk
f (xk+1 ) ≤ min g(xk ) + ∇g(xk ), x − xk +
x
2
Now it follows from the convexity of g that
2
+ ψ(x) .
g(x) ≥ g(xk ) + ∇g(xk ), x − xk ,
and combining these yields that
L
x − xk
f (xk+1 ) ≤ minn f (x) +
x∈R
2
2
(7.1)
Since f is convex, for all α ∈ [0, 1] and x∗ ∈ X∗ we have
f (αx∗ + (1 − α)xk ) ≤ αf (x∗ ) + (1 − α)f (xk ) = f (xk ) − α(f (xk ) − f (x∗ )).
Consequently
L
Lα2
minn f (x) +
x − xk ≤ min f (xk ) − α(f (xk ) − f (x∗ )) +
xk − x∗
x∈R
2
2
α∈[0,1]
2
.
By taking the derivative with respect to α of the expression on the right hand side above and
setting it to zero, we see that the optimal α satisfies
and thus using α = min
−(f (xk ) − f (x∗ )) + αL xk − x∗
)
(
f (xk ) − f (x∗ ) ≥ L xk − x∗
f (xk )−f (x∗ )
2
L y−x∗
2
,1
2
=0
yields the result, since f (xk ) − f (x∗ ) ≥ 0, and when
, we have
L
L
f (xk ) − f (x∗ )
2
minn f (x) +
≤ f (x∗ ) +
xk − x
xk − x∗ ≤ f (x∗ ) +
.
x∈R
2
2
2
Using the lemma, we can complete the proof of the theorem as follows.
Proof of Theorem 5.9:
By Lemma G.1 we have that ǫk+1 ≤ ǫk for all k and
1 ǫk 2 ǫk
ǫk+1 ≤ max ǫk −
,
2L D
2
Consequently for k ≥ 1 such that ǫk −
1
2L
ǫ k 2
D
≥
ǫk
2
we have
1
1
ǫk+1 − ǫk
1
1
ǫk
1
−
≤
≤−
· 2·
≤−
ǫk
ǫk+1
ǫk ǫk+1
2L D ǫk+1
2LD 2
Summing yields that
Nk
1
1
≤−
−
ǫ1 ǫk
2LD 2
25
1
where Nk is the number of steps k ≥ 1 for which ǫk − 2L
we have that
L
ǫ1 ≤ D 2
2
and thus
2LD 2
ǫk ≤
Nk + 4
On the other hand we have that
ǫk+1
ǫk 2
D
≥
ǫk
2
. Furthermore, clearly by G.1
k−1−Nk
1
≤
2
k−1
and noting that either Nk ≥ ⌊ k−1
2 ⌋ or k − 1 − Nk ≥ ⌊ 2 ⌋ then yields the result.
26
| 8 |
A Probabilistic Theory of Deep Learning
arXiv:1504.00641v1 [stat.ML] 2 Apr 2015
Ankit B. Patel, Tan Nguyen, Richard G. Baraniuk
Department of Electrical and Computer Engineering
Rice University
{abp4, mn15, richb}@rice.edu
April 2, 2015
A grand challenge in machine learning is the development of computational algorithms that match or outperform humans in perceptual inference tasks such
as visual object and speech recognition. The key factor complicating such tasks
is the presence of numerous nuisance variables, for instance, the unknown
object position, orientation, and scale in object recognition or the unknown
voice pronunciation, pitch, and speed in speech recognition. Recently, a new
breed of deep learning algorithms have emerged for high-nuisance inference
tasks; they are constructed from many layers of alternating linear and nonlinear processing units and are trained using large-scale algorithms and massive
amounts of training data. The recent success of deep learning systems is impressive — they now routinely yield pattern recognition systems with nearor super-human capabilities — but a fundamental question remains: Why do
they work? Intuitions abound, but a coherent framework for understanding,
analyzing, and synthesizing deep learning architectures has remained elusive.
We answer this question by developing a new probabilistic framework for deep
learning based on a Bayesian generative probabilistic model that explicitly captures variation due to nuisance variables. The graphical structure of the model
enables it to be learned from data using classical expectation-maximization
techniques. Furthermore, by relaxing the generative model to a discriminative
one, we can recover two of the current leading deep learning systems, deep
convolutional neural networks (DCNs) and random decision forests (RDFs),
providing insights into their successes and shortcomings as well as a principled route to their improvement.
1
Contents
1
Introduction
4
2
A Deep Probabilistic Model for Nuisance Variation
2.1 The Rendering Model: Capturing Nuisance Variation . . . . . . . . . . . .
2.2 Deriving the Key Elements of One Layer of a Deep Convolutional
Network from the Rendering Model . . . . . . . . . . . . . . . . . . . . .
2.3 The Deep Rendering Model: Capturing Levels of Abstraction . . . . . . . .
2.4 Inference in the Deep Rendering Model . . . . . . . . . . . . . . . . . . .
2.4.1 What About the SoftMax Regression Layer? . . . . . . . . . . . .
2.5 DCNs are Probabilistic Message Passing Networks . . . . . . . . . . . . .
2.5.1 Deep Rendering Model and Message Passing . . . . . . . . . . . .
2.5.2 A Unification of Neural Networks and Probabilistic Inference . . .
2.5.3 The Probabilistic Role of Max-Pooling . . . . . . . . . . . . . . .
2.6 Learning the Rendering Models . . . . . . . . . . . . . . . . . . . . . . .
2.6.1 EM Algorithm for the Shallow Rendering Model . . . . . . . . . .
2.6.2 EM Algorithm for the Deep Rendering Model . . . . . . . . . . . .
2.6.3 What About DropOut Training? . . . . . . . . . . . . . . . . . . .
2.7 From Generative to Discriminative Classifiers . . . . . . . . . . . . . . . .
2.7.1 Transforming a Generative Classifier into a Discriminative One . .
2.7.2 From the Deep Rendering Model to Deep Convolutional Networks .
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New Insights into Deep Convolutional Networks
3.1 DCNs Possess Full Probabilistic Semantics . . . . . . . . . . . . . . . . . . .
3.2 Class Appearance Models and Activity Maximization . . . . . . . . . . . . . .
3.3 (Dis)Entanglement: Supervised Learning of Task Targets Is
Intertwined with Unsupervised Learning of Latent Task Nuisances . . . . . . .
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From the Deep Rendering Model to Random Decision Forests
4.1 The Evolutionary Deep Rendering Model: A Hierarchy of Categories
4.2 Inference with the E-DRM Yields a Decision Tree . . . . . . . . . . .
4.2.1 What About the Leaf Histograms? . . . . . . . . . . . . . . .
4.3 Bootstrap Aggregation to Prevent Overfitting Yields A Decision
Forest . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 EM Learning for the E-DRM Yields the InfoMax Principle . . . . . .
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5
6
Relation to Prior Work
5.1 Relation to Mixture of Factor Analyzers . . . . . . . . . . . . .
5.2 i-Theory: Invariant Representations Inspired by Sensory Cortex
5.3 Scattering Transform: Achieving Invariance via Wavelets . . . .
5.4 Learning Deep Architectures via Sparsity . . . . . . . . . . . .
5.5 Google FaceNet: Learning Useful Representations with DCNs .
5.6 Renormalization Theory . . . . . . . . . . . . . . . . . . . . .
5.7 Summary of Key Distinguishing Features of the DRM . . . . .
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New Directions
6.1 More Realistic Rendering Models . . . . . . . . . . . . . . . . . . . . .
6.2 New Inference Algorithms . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Soft Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Top-Down Convolutional Nets: Top-Down Inference via the DRM
6.3 New Learning Algorithms . . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Derivative-Free Learning . . . . . . . . . . . . . . . . . . . . . .
6.3.2 Dynamics: Learning from Video . . . . . . . . . . . . . . . . . .
6.3.3 Training from Labeled and Unlabeled Data . . . . . . . . . . . .
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A Supplemental Information
A.1 From the Gaussian Rendering Model Classifier to Deep DCNs . . . . . . . . .
A.2 Generalizing to Arbitrary Mixtures of Exponential Family
Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.3 Regularization Schemes: Deriving the DropOut Algorithm . . . . . . . . . . .
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1
Introduction
Humans are expert at a wide array of complicated sensory inference tasks, from recognizing
objects in an image to understanding phonemes in a speech signal, despite significant variations
such as the position, orientation, and scale of objects and the pronunciation, pitch, and volume
of speech. Indeed, the main challenge in many sensory perception tasks in vision, speech, and
natural language processing is a high amount of such nuisance variation. Nuisance variations
complicate perception, because they turn otherwise simple statistical inference problems with
a small number of variables (e.g., class label) into much higher-dimensional problems. For example, images of a car taken from different camera viewpoints lie on a highly curved, nonlinear
manifold in high-dimensional space that is intertwined with the manifolds of myriad other objects. The key challenge in developing an inference algorithm is then how to factor out all of
the nuisance variation in the input. Over the past few decades, a vast literature that approaches
this problem from myriad different perspectives has developed, but the most difficult inference
problems have remained out of reach.
Recently, a new breed of machine learning algorithms have emerged for high-nuisance inference tasks, resulting in pattern recognition systems with sometimes super-human capabilities (1). These so-called deep learning systems share two common hallmarks. First, architecturally, they are constructed from many layers of alternating linear and nonlinear processing
units. Second, computationally, their parameters are learned using large-scale algorithms and
massive amounts of training data. Two examples of such architectures are the deep convolutional neural network (DCN), which has seen great success in tasks like visual object recognition and localization (2), speech recognition (3), and part-of-speech recognition (4), and random
decision forests (RDFs) (5) for image segmentation. The success of deep learning systems is
impressive, but a fundamental question remains: Why do they work? Intuitions abound to explain their success. Some explanations focus on properties of feature invariance and selectivity
developed over multiple layers, while others credit raw computational power and the amount
of available training data (1). However, beyond these intuitions, a coherent theoretical framework for understanding, analyzing, and synthesizing deep learning architectures has remained
elusive.
In this paper, we develop a new theoretical framework that provides insights into both the
successes and shortcomings of deep learning systems, as well as a principled route to their design and improvement. Our framework is based on a generative probabilistic model that explicitly captures variation due to latent nuisance variables. The Rendering Model (RM) explicitly
models nuisance variation through a rendering function that combines the task-specific variables of interest (e.g., object class in an object recognition task) and the collection of nuisance
variables. The Deep Rendering Model (DRM) extends the RM in a hierarchical fashion by ren4
dering via a product of affine nuisance transformations across multiple levels of abstraction. The
graphical structures of the RM and DRM enable inference via message passing, using, for example, the sum-product or max-sum algorithms, and training via the expectation-maximization
(EM) algorithm. A key element of the framework is the relaxation of the RM/DRM generative
model to a discriminative one in order to optimize the bias-variance tradeoff.
The DRM unites and subsumes two of the current leading deep learning based systems as
max-sum message passing networks. That is, configuring the DRM with two different nuisance
structures — Gaussian translational nuisance or evolutionary additive nuisance — leads directly
to DCNs and RDFs, respectively. The intimate connection between the DRM and these deep
learning systems provides a range of new insights into how and why they work, answering
several open questions. Moreover, the DRM provides insights into how and why deep learning
fails and suggests pathways to their improvement.
It is important to note that our theory and methods apply to a wide range of different inference tasks (including, for example, classification, estimation, regression, etc.) that feature a
number of task-irrelevant nuisance variables (including, for example, object and speech recognition). However, for concreteness of exposition, we will focus below on the classification
problem underlying visual object recognition.
This paper is organized as follows. Section 2 introduces the RM and DRM and demonstrates
step-by-step how they map onto DCNs. Section 3 then summarizes some of the key insights
that the DRM provides into the operation and performance of DCNs. Section 4 proceeds in a
similar fashion to derive RDFs from a variant of the DRM that models a hierarchy of categories.
Section 6 closes the paper by suggesting a number of promising avenues for research, including
several that should lead to improvement in deep learning system performance and generality.
The proofs of several results appear in the Appendix.
2
A Deep Probabilistic Model for Nuisance Variation
This section develops the RM, a generative probabilistic model that explicitly captures nuisance
transformations as latent variables. We show how inference in the RM corresponds to operations in a single layer of a DCN. We then extend the RM by defining the DRM, a rendering
model with layers representing different scales or levels of abstraction. Finally, we show that,
after the application of a discriminative relaxation, inference and learning in the DRM correspond to feedforward propagation and back propagation training in the DCN. This enables us to
conclude that DCNs are probabilistic message passing networks, thus unifying the probabilistic
and neural network perspectives.
5
2.1
The Rendering Model: Capturing Nuisance Variation
Visual object recognition is naturally formulated as a statistical classification problem.1 We
are given a D-pixel, multi-channel image I of an object, with intensity I(x, ω) at pixel x and
channel ω (e.g., ω ={red, green, blue}). We seek to infer the object’s identity (class) c ∈ C,
where C is a finite set of classes.2 We will use the terms “object” and “class” interchangeably.
Given a joint probabilistic model p(I, c) for images and objects, we can classify a particular
image I using the maximum a posteriori (MAP) classifier
ĉ(I) = argmax p(c|I) = argmax p(I|c)p(c),
c∈C
(1)
c∈C
where p(I|c) is the image likelihood, p(c) is the prior distribution over the classes, and p(c|I) ∝
p(I|c)p(c) by Bayes’ rule.
Object recognition, like many other inference tasks, is complicated by a high amount of
variation due to nuisance variables, which the above formation ignores. We advocate explicitly
modeling nuisance variables by encapsulating all of them into a (possibly high-dimensional)
parameter g ∈ G, where G is the set of all nuisances. In some cases, it is natural for g to be a
transformation and for G to be endowed with a (semi-)group structure.
We now propose a generative model for images that explicitly models the relationship between images I of the same object c subject to nuisance g. First, given c, g, and other auxiliary
parameters, we define the rendering function R(c, g) that renders (produces) an image. In image
inference problems, for example, R(c, g) might be a photorealistic computer graphics engine
(c.f., Pixar). A particular realization of an image is then generated by adding noise to the output
of the renderer:
I|c, g = R(c, g) + noise.
(2)
We assume that the noise distribution is from the exponential family, which includes a large
number of practically useful distributions (e.g., Gaussian, Poisson). Also we assume that the
noise is independent and identically distributed (iid) as a function of pixel location x and that
the class and nuisance variables are independently distributed according to categorical distributions.3 With these assumptions, Eq. 2 then becomes the probabilistic (shallow) Rendering
1
Recall that we focus on object recognition from images only for concreteness of exposition.
The restriction for C to be finite can be removed by using a nonparametric prior such as a Chinese Restaurant
Process (CRP) (6)
3
Independence is merely a convenient approximation; in practice, g can depend on c. For example, humans
have difficulty recognizing and discriminating upside down faces (7).
2
6
Model (RM)
c ∼ Cat({πc }c∈C ),
g ∼ Cat({πg }g∈G ),
I|c, g ∼ Q(θcg ).
(3)
Here Q(θcg ) denotes a distribution from the exponential family with parameters θcg , which
include the mixing probabilities πcg , natural parameters η(θcg ), sufficient statistics T (I) and
whose mean is the rendered template µcg = R(c, g).
An important special case is when Q(θcg ) is Gaussian, and this defines the Gaussian Rendering Model (GRM), in which images are generated according to
I|c, g ∼ N (I|µcg = R(c, g), Σcg = σ 2 1),
(4)
where 1 is the identity matrix. The GRM generalizes both the Gaussian Naı̈ve Bayes Classifier
(GNBC) and the Gaussian Mixture Model (GMM) by allowing variation in the image to depend
on an observed class label c, like a GNBC, and on an unobserved nuisance label g, like a GMM.
The GNBC, GMM and the (G)RM can all be conveniently described as a directed graphical
model (8). Figure 1A depicts the graphical models for the GNBC and GMM, while Fig. 1B
shows how they are combined to form the (G)RM.
Finally, since the world is spatially varying and an image can contain a number of different
objects, it is natural to break the image up into a number of (overlapping) subimages, called
patches, that are indexed by spatial location x. Thus, a patch is defined here as a collection of
pixels centered on a single pixel x. In general, patches can overlap, meaning that (i) they do not
tile the image, and (ii) an image pixel x can belong to more than one patch. Given this notion of
pixels and patches, we allow the class and nuisance variables to depend on pixel/patch location:
i.e., local image class c(x) and local nuisance g(x) (see Fig. 2A). We will omit the dependence
on x when it is clear from context.
The notion of a rendering operator is quite general and can refer to any function that maps
a target variable c and nuisance variables g into a pattern or template R(c, g). For example, in
speech recognition, c might be a phoneme, in which case g represents volume, pitch, speed, and
accent, and R(c, g) is the amplitude of the acoustic signal (or alternatively the time-frequency
representation). In natural language processing, c might be the grammatical part-of-speech, in
which case g represents syntax and grammar, and R(c, g) is a clause, phrase or sentence.
To perform object recognition with the RM via Eq. 1, we must marginalize out the nuisance
variables g. We consider two approaches for doing so, one conventional and one unconventional. The Sum-Product RM Classifier (SP-RMC) sums over all nuisance variables g ∈ G and
7
A
B
Naive Bayes Mixture
Classifier
Model
c
g
I
I
C
Rendering
Deep Rendering
Model
Model
c
g
I
μL
gL
μL–1
gL–1
...
μ1
g1
I
Figure 1. Graphical depiction of the Naive Bayes Classifier (A, left), Gaussian Mixture Model (A, right),
the shallow Rendering Model (B) and the Deep Rendering Model (C). All dependence on pixel location
x has been suppressed for clarity.
then chooses the most likely class:
1 X
p(I|c, g)p(c)p(g)
|G| g∈G
c∈C
1 X
= argmax
exphη(θcg )|T (I)i,
|G| g∈G
c∈C
ĉSP (I) = argmax
(5)
where h·|·i is the bra-ket notation for inner products and in the last line we have used the definition of an exponential family distribution. Thus the SP-RM computes the marginal of the
posterior distribution over the target variable, given the input image. This is the conventional
approach used in most probabilistic modeling with latent variables.
An alternative and less conventional approach is to use the Max-Sum RM Classifier (MSRMC), which maximizes over all g ∈ G and then chooses the most likely class:
ĉMS (I) = argmax max p(I|c, g)p(c)p(g)
c∈C
g∈G
= argmax maxhη(θcg )|T (I)i.
c∈C
g∈G
(6)
The MS-RMC computes the mode of the posterior distribution over the target and nuisance
variables, given the input image. Equivalently, it computes the most likely global configuration
of target and nuisance variables for the image. Intuitively, this is an effective strategy when
there is one explanation g ∗ ∈ G that dominates all other explanations g 6= g ∗ . This condition
8
is justified in settings where the rendering function is deterministic or nearly noise-free. This
approach to classification is unconventional in both the machine learning and computational
neuroscience literatures, where the sum-product approach is most commonly used, although it
has received some recent attention (9).
Both the sum-product and max-sum classifiers amount to applying an affine transformation
to the input image I (via an inner product that performs feature detection via template matching), followed by a sum or max nonlinearity that marginalizes over the nuisance variables.
Throughout the paper we will assume isotropic or diagonal Gaussian noise for simplicity,
but the treatment presented here can be generalized to any distribution from the exponential
family in a straightforward manner. Note that such an extension may require a non-linear transformation (i.e. quadratic or logarithmic T (I)), depending on the specific exponential family.
Please see Supplement Section A.2 for more details.
2.2
Deriving the Key Elements of One Layer of a Deep Convolutional
Network from the Rendering Model
Having formulated the Rendering Model (RM), we now show how to connect the RM with deep
convolutional networks (DCNs). We will see that the MS-RMC (after imposing a few additional
assumptions on the RM) gives rise to most commonly used DCN layer types.
Our first assumption is that the noise added to the rendered template is isotropically Gaussian (GRM) i.e. each pixel has the same noise variance σ 2 that is independent of the configuration (c, g). Then, assuming the image is normalized kIk2 = 1, Eq. 6 yields the max-sum
Gaussian RM classifier (see Appendix A.1 for a detailed proof)
1
1
µcg I − 2 kµcg k22 + ln πc πg
ĉMS (I) = argmax max
2
g∈G
σ
2σ
c∈C
≡ argmax max hwcg |Ii + bcg ,
(7)
c∈C
g∈G
where we have defined the natural parameters η ≡ {wcg , bcg } in terms of the traditional parameters θ ≡ {σ 2 , µcg , πc , πg } according to4
1
1
µcg = 2 R(c, g)
2
σ
σ
1
bcg ≡ 2 kµcg k22 + ln πc πg .
2σ
wcg ≡
(8)
Note that we have suppressed the parameters’ dependence on pixel location x.
4
Since the Gaussian distribution of the noise is in the exponential family, it can be reparametrized in terms of
the natural parameters. This is known as canonical form.
9
We will now demonstrate that the sequence of operations in the MS-RMC in Eq. 7 coincides
exactly with the operations involved in one layer of a DCN (or, more generally, a max-out neural
network (10)): image normalization, linear template matching, thresholding, and max pooling.
See Fig. 2C. We now explore each operation in Eq. 7 in detail to make the link precise.
First, the image is normalized. Until recently, there were several different types of normalization typically employed in DCNs: local response normalization, and local contrast normalization (11, 12). However, the most recent highly performing DCNs employ a different form of
normalization, known as batch-normalization (13). We will come back to this later when we
show how to derive batch normalization from a principled approach. One implication of this is
that it is unclear what probabilistic assumption the older forms of normalization arise from, if
any.
Second, the image is filtered with a set of noise-scaled rendered templates wcg . The size
of the templates depends on the size of the objects of class c and the values of the nuisance
variables g. Large objects will have large templates, corresponding to a fully connected layer in
a DCN (14), while small objects will have small templates, corresponding to a locally connected
layer in a DCN (15). If the distribution of objects depends on the pixel position x (e.g., cars are
more likely on the ground while planes are more likely in the air) then, in general, we will need
different rendered templates at each x. In this case, the locally connected layer is appropriate.
If, on the other hand, all objects are equally likely to be present at all locations throughout the
entire image, then we can assume translational invariance in the RM. This yields a global set of
templates that are used at all pixels x, corresponding to a convolutional layer in a DCN (14) (see
Appendix A.2 for a detailed proof). If the filter sizes are large relative to the scale at which the
image variation occurs and the filters are overcomplete, then adjacent filters overlap and waste
computation. In these case, it is appropriate to use a strided convolution, where the output of the
traditional convolution is down-sampled by some factor; this saves some computation without
losing information.
Third, the resulting activations (log-probabilities of the hypotheses) are passed through a
pooling layer; i.e., if g is a translational nuisance, then taking the maximum over g corresponds
to max pooling in a DCN.
Fourth, recall that a given image pixel x will reside in several overlapping image patches,
each rendered by its own parent class c(x) and the nuisance location g(x) (Fig. 2A). Thus
we must consider the possibility of collisions: i.e. when two different parents c(x1 ) 6= c(x2 )
might render the same pixel (or patch). To avoid such undesirable collisions, it is natural to
force the rendering to be locally sparse: i.e. we must enforce that only one renderer in a local
neighborhood can be “active”.
To formalize this, we endow each parent renderer with an ON/OFF state via a switching
variable a ∈ A ≡ {ON, OFF}. If a = ON = 1, then the rendered image patch is left untouched,
10
B
A
Probabilistic Model
(Deep Rendering Model) Inference
C
Factor Graph
(Inference via Max-Sum)
..
.
..
.
µ cl+1
I l+1
c
c
g l+1
1
3
4
2
ON
OF F
µ cl = ⇤l+1 g l+1 µ cl+1
1
µ c
|µ c , g
1
2
4
x
l
hW l+1 |·i
cl
x
l
Rendering
I0
noise
Inf erence
l+1
g l+1
1
3
4
2
F eature M ap
ON
OF F
al+1
DCN Convolution
1
M ax
I
l
Sum
M essage
cl
..
.
Output
xl+1
cl+1
2
4
3
l
Sum
M axP ool + RELU
g l+1
l+1
..
.
M essage
max {·}
F actor
3
M ax
l+1
al+1
Discriminative
Counterpart
I l+1
xl+1
xl+1
l+1
Neural Network
(Deep Convolutional
Network)
2
4
3
x
l
cl
Input
F eature M ap
Il
..
.
Inf erence
I0
..
.
I0
Figure 2. An example of mapping from the Deep Rendering Model (DRM) to its corresponding factor
graph to a Deep Convolutional Network (DCN) showing only the transformation from level ` of the
hierarchy of abstraction to level ` + 1. (A) DRM generative model: a single super pixel x`+1 at level
` + 1 (green, upper) renders down to a 3 × 3 image patch at level ` (green, lower), whose location
is specified by g `+1 (red). (B) Factor graph representation of the DRM model that supports efficient
inference algorithms such as the max-sum message passing shown here. (C) Computational network that
implements the max-sum message passing algorithm from (B) explicitly; its structure exactly matches
that of a DCN.
11
whereas if a = OFF = 0, the image patch is masked with zeros after rendering. Thus, the
switching variable a models (in)active parent renderers.
However, these switching variables have strong correlations due to the crowding out effect:
if one is ON, then its neighbors must be OFF in order to prevent rendering collisions. Although
natural for realistic rendering, this complicates inference. Thus, we employ an approximation by instead assuming that the state of each renderer ON or OFF completely at random and
thus independent of any other variables, including the measurements (i.e., the image itself).
Of course, an approximation to real rendering, but it simplifies inference, and leads directly
to rectified linear units, as we show below. Such approximations to true sparsity have been
extensively studied, and are known as spike-and-slab sparse coding models (16, 17).
Since the switching variables are latent (unobserved), we must max-marginalize over them
during classification, as we did with nuisance variables g in the last section (one can think of a
as just another nuisance). This leads to (see Appendix A.3 for a more detailed proof)
1
1
aµcg I − 2 (kaµcg k22 + kIk22 )) + ln πc πg πa
ĉ(I) = argmax max max
2
g∈G a∈A
σ
2σ
c∈C
≡ argmax max max a(hwcg |Ii + bcg ) + bcga
c∈C
g∈G
a∈A
= argmax max ReLU (hwcg |Ii + bcg ) ,
c∈C
g∈G
(9)
where bcga and bcg are bias terms and ReLu(u) ≡ (u)+ = max{u, 0} denotes the soft-thresholding
operation performed by the Rectified Linear Units (ReLU) in modern DCNs (18). In the last
line, we have assumed that the prior πcg is uniform so that bcga is independent of c and g and
can be dropped.
2.3
The Deep Rendering Model: Capturing Levels of Abstraction
The world is summarizable at varying levels of abstraction, and Hierarchical Bayesian Models
(HBMs) can exploit this fact to accelerate learning. In particular, the power of abstraction allows
the higher levels of an HBM to learn concepts and categories far more rapidly than lower levels,
due to stronger inductive biases and exposure to more data (19). This is informally known as
the Blessing of Abstraction (19). In light of these benefits, it is natural for us to extend the RM
into an HBM, thus giving it the power to summarize data at different levels of abstraction.
In order to illustrate this concept, consider the example of rendering an image of a face, at
different levels of detail ` ∈ {L, L−1, . . . , 0}. At level ` = L (the coarsest level of abstraction),
we specify only the identity of the face cL and its overall location and pose g L without specifying
any finer-scale details such as the locations of the eyes or type of facial expression. At level
` = L − 1, we specify finer-grained details, such as the existence of a left eye (cL−1 ) with a
12
Figure 3. This sculpture by Henri Matisse illustrates the Deep Rendering Model (DRM). The sculpture
in the leftmost panel is analogous to a fully rendered image at the lowest abstraction level ` = 0. Moving
from left to right, the sculptures become progressively more abstract, until the in the rightmost panel we
reach the highest abstraction level ` = 3. The finer-scale details in the first three panels that are lost
in the fourth are the nuisance parameters g, whereas the coarser-scale details in the last panel that are
preserved are the target c.
certain location, pose, and state (e.g., g L−1 = open or closed), again without specifying any
finer-scale parameters (such as eyelash length or pupil shape). We continue in this way, at
each level ` adding finer-scaled information that was unspecified at level ` − 1, until at level
` = 0 we have fully specified the image’s pixel intensities, leaving us with the fully rendered,
multi-channel image I 0 (x` , ω ` ). Here x` refers to a pixel location at level `.
For another illustrative example, consider The Back Series of sculptures by the artist Henri
Matisse (Fig. 3). As one moves from left to right, the sculptures become increasingly abstract,
losing low-level features and details, while preserving high-level features essential for the overall meaning: i.e. (cL , g L ) = “woman with her back facing us.” Conversely, as one moves from
right to left, the sculptures become increasingly concrete, progressively gaining finer-scale details (nuisance parameters g ` , ` = L−1, . . . , 0) and culminating in a rich and textured rendering.
We formalize this process of progressive rendering by defining the Deep Rendering Model
(DRM). Analogous to the Matisse sculptures, the image generation process in a DRM starts at
the highest level of abstraction (` = L), with the random choice of the object class cL and overall
pose g L . It is then followed by generation of the lower-level details g ` , and a progressive levelby-level (` → ` − 1) rendering of a set of intermediate rendered “images” µ` , each with more
detailed information. The process finally culminates in a fully rendered D0 ≡ D-dimensional
13
image µ0 = I 0 ≡ I (` = 0). Mathematically,
cL ∼ Cat(π(cL )),
g `+1 ∼ Cat(π(g `+1 )),
cL ∈ C L ,
g `+1 ∈ G `+1 ,
` = L − 1, L − 2, . . . , 0
µ(cL , g) = Λ(g)µ(cL ) ≡ Λ1 (g 1 ) · · · ΛL (g L ) · µ(cL ),
I(cL , g) = µ(cL , g) + N (0, σ 2 1D ) ∈ RD .
g = {g ` }L`=1
(10)
Here C ` , G ` are the sets of all target-relevant and target-irrelevant nuisance variables at level
`, respectively. The rendering path is defined as the sequence (cL , g L , . . . , g ` , . . . , g 1 ) from
the root (overall class) down to the individual pixels at ` = 0. µ(cL ) is an abstract template
Q
for the high-level class cL , and Λ(g) ≡ ` Λ` (g ` ) represents the sequence of local nuisance
transformations that renders finer-scale details as one moves from abstract to concrete. Note
that each Λ` (g ` ) is an affine transformation with a bias term α(g ` ) that we have suppressed
for clarity5 . Figure 2A illustrates the corresponding graphical model. As before, we have
suppressed the dependence of c` , g ` on the pixel location x` at level ` of the hierarchy.
We can cast the DRM into an incremental form by defining an intermediate class c` ≡
(cL , g L , . . . , g `+1 ) that intuitively represents a partial rendering path up to level `. Then, the
partial rendering from level ` + 1 to ` can be written as an affine transformation
µ(c` ) = Λ`+1 (g `+1 ) · µ(c`+1 ) + α(g `+1 ) + N (0, Ψ`+1 ),
(11)
where we have shown the bias term α explicitly and introduced noise6 with a diagonal covariance Ψ`+1 . It is important to note that c` , g ` can correspond to different kinds of target
relevant and irrelevant features at different levels. For example, when rendering faces, c1 (x1 )
might correspond to different edge orientations and g 1 (x1 ) to different edge locations in patch
x1 , whereas c2 (x2 ) might correspond to different eye types and g 2 (x2 ) to different eye gaze
directions in patch x2 .
The DRM generates images at intermediate abstraction levels via the incremental rendering
functions in Eq. 11 (see Fig. 2A). Hence the complete rendering function R(c, g) from Eq. 2
is a composition of incremental rendering functions, amounting to a product of affine transformations as in Eq. 10. Compared to the shallow RM, the factorized structure of the DRM
Q
results in an exponential reduction in the number of free parameters, from D0 |C L | ` |G` | to
P
|C L | ` D` |G ` | where D` is the number of pixels in the intermediate image µ` , thus enabling
more efficient inference and learning, and most importantly, better generalization.
5
This assumes that we are using an exponential family with linear sufficient statistics i.e. T (I) = (I, 1)T .
However, note that the family we use here is not Gaussian, it is instead a Factor Analyzer, a different probabilistic
model.
6
We introduce noise for two reasons: (1) it will make it easier to connect later to existing EM algorithms for
factor analyzers and (2) we can always take the noise-free limit to impose cluster well-separatedness if needed.
Indeed, if the rendering process is deterministic or nearly noise-free, then the latter is justified.
14
The DRM as formulated here is distinct from but related to several other hierarchical models,
such as the Deep Mixture of Factor Analyzers (DMFA) (20) and the Deep Gaussian Mixture
Model (21), both of which are essentially compositions of another model — the Mixture of
Factor Analyzers (MFA) (22). We will highlight the similarities and differences with these
models in more detail in Section 5.
2.4
Inference in the Deep Rendering Model
Inference in the DRM is similar to inference in the shallow RM. For example, to classify images
we can use either the sum-product (Eq. 5) or the max-sum (Eq. 6) classifier. The key difference
between the deep and shallow RMs is that the DRM yields iterated layer-by-layer updates, from
fine-to-coarse abstraction (bottom-up) and from coarse-to-fine abstraction (top-down). In the
case we are only interested in inferring the high-level class cL , we only need the fine-to-coarse
pass and so we will only consider it in this section.
Importantly, the bottom-up pass leads directly to DCNs, implying that DCNs ignore potentially useful top-down information. This maybe an explanation for their difficulties in vision
tasks with occlusion and clutter, where such top-down information is essential for disambiguating local bottom-up hypotheses. Later on in Section 6.2.2, we will describe the coarse-to-fine
pass and a new class of Top-Down DCNs that do make use of such information.
Given an input image I 0 , the max-sum classifier infers the most likely global configuration
{c` , g ` }, ` = 0, 1, . . . , L by executing the max-sum message passing algorithm in two stages: (i)
from fine-to-coarse levels of abstraction to infer the overall class label ĉLMS and (ii) from coarse`
at all intermediate levels `.
to-fine levels of abstraction to infer the latent variables ĉ`MS and ĝMS
As mentioned above, we will focus on the fine-to-coarse pass. Since the DRM is an RM with
a hierarchical prior on the rendered templates, we can use Eq. 7 to derive the fine-to-coarse
15
max-sum DRM classifier (MS-DRMC) as:
ĉM S (I) = argmax max hη(cL , g)|Σ−1 |I 0 i
cL ∈C
g∈G
= argmax max hΛ(g)µ(cL )|(Λ(g)Λ(g)T )† |I 0 i
cL ∈C
g∈G
L
= argmax max hµ(c )|
cL ∈C
g∈G
= argmax hµ(cL )|
cL ∈C
1
Y
`=L
1
Y
`=L
Λ` (g ` )† |I 0 i
max Λ` (g ` )† |I 0 i
g ` ∈G `
Λ1 (g 1 )† |I 0 i
= argmax hµ(c )| max ΛL (g L )† · · · max
1 ∈G 1
L ∈G L
g
g
L
c ∈C
|
{z
}
L
≡ I1
2 2 †
Λ (g ) |I 1 i
≡ argmax hµ(cL )| max ΛL (g L )† · · · max
2 ∈G 2
L ∈G L
g
g
L
c ∈C
|
{z
}
≡ I2
3 3 †
L †
Λ (g ) |I 2 i
≡ argmax hµ(c )| max Λ (g ) · · · max
3
3
L
L
g L ∈G L
cL ∈C
g ∈G
..
.
(12)
≡ argmax hµ(cL )|I L i,
cL ∈C
where Σ ≡ Λ(g)Λ(g)T is the covariance of the rendered image I and hx|M |yi ≡ xT M y. Note
the significant change with respect to the shallow RM: the covariance Σ is no longer diagonal
due to the iterative affine transformations during rendering (Eq. 11), and so we must decorrelate
the input image (via Σ−1 I 0 in the first line) in order to classify accurately.
Note also that we have omitted the bias terms for clarity and that M † is the pseudoinverse
of matrix M . In the fourth line, we used the distributivity of max over products7 and in the last
lines defined the intermediate quantities
I `+1 ≡
=
max h(Λ`+1 (g `+1 ))† |I ` i
|
{z
}
g `+1 ∈G `+1
max hW
g `+1 ∈G `+1
≡W `+1
`+1 `+1
(g
)|I ` i
≡ MaxPool(Conv(I ` )).
(13)
Here I ` = I ` (x` , c` ) is the feature map output of layer ` indexed by channels c` and η(c` , g ` ) ∝
µ(c` , g ` ) are the natural parameters (i.e., intermediate rendered templates) for level `.
7
For a > 0, max{ab, ac} = a max{b, c}.
16
If we care only about inferring the overall class of the image cL (I 0 ), then the fine-to-coarse
pass suffices, since all information relevant to determining the overall class has been integrated.
That is, for high-level classification, we need only iterate Eqs. 12 and 13. Note that Eq. 12
simplifies to Eq. 9 when we assume sparse patch rendering as in Section 2.2.
Coming back to DCNs, we have see that the `-th iteration of Eq. 12 or Eq. 9 corresponds to
feedforward propagation in the `-th layer of a DCN. Thus a DCN’s operation has a probabilistic
interpretation as fine-to-coarse inference of the most probable global configuration in the DRM.
2.4.1
What About the SoftMax Regression Layer?
It is important to note that we have not fully reconstituted the architecture of modern a DCN
as yet. In particular, the SoftMax regression layer, typically attached to the end of network, is
missing. This means that the high-level class cL in the DRM (Eq. 12) is not necessarily the
same as the training data class labels c̃ given in the dataset. In fact, the two labels c̃ and cL are
in general distinct.
But then how are we to interpret cL ? The answer is that the most probable global configuration (cL , g ∗ ) inferred by a DCN can be interpreted as a good representation of the input
image, i.e., one that disentangles the many nuisance factors into (nearly) independent components cL , g ∗ . 8 Under this interpretation, it becomes clear that the high-level class cL in the
disentangled representation need not be the same as the training data class label c̃.
The disentangled representation for cL lies in the penultimate layer activations: âL (In ) =
ln p(cL , g ∗ |In ). Given this representation, we can infer the class label c̃ by using a simple linear
classifier such as the SoftMax regression9 . Explicitly, the Softmax regression layer computes
p(c̃|âL ; θSoftmax ) = φ(W L+1 âL + bL+1 ), and then chooses the most likely class. Here φ(·) is the
softmax function and θSoftmax ≡ {W L+1 , bL+1 } are the parameters of the SoftMax regression
layer.
2.5
2.5.1
DCNs are Probabilistic Message Passing Networks
Deep Rendering Model and Message Passing
Encouraged by the correspondence identified in Section 2.4, we step back for a moment to
reinterpret all of the major elements of DCNs in a probabilistic light. Our derivation of the
DRM inference algorithm above is mathematically equivalent to performing max-sum message
passing on the factor graph representation of the DRM, which is shown in Fig. 2B. The factor
8
In this sense, the DRM can be seen as a deep (nonlinear) generalization of Independent Components Analysis
(23).
9
Note that this implicitly assumes that a good disentangled representation of an image will be useful for the
classification task at hand.
17
graph encodes the same information as the generative model but organizes it in a manner that
simplifies the definition and execution of inference algorithms (24). Such inference algorithms
are called message passing algorithms, because they work by passing real-valued functions
called messages along the edges between nodes. In the DRM/DCN, the messages sent from
finer to coarser levels are the feature maps I ` (x` , c` ). However, unlike the input image I 0 , the
channels c` in these feature maps do not refer to colors (e.g, red, green, blue) but instead to
more abstract features (e.g., edge orientations or the open/closed state of an eyelid).
2.5.2
A Unification of Neural Networks and Probabilistic Inference
The factor graph formulation provides a powerful interpretation that the convolution, MaxPooling and ReLu operations in a DCN correspond to max-sum inference in a DRM. Thus,
we see that architectures and layer types commonly used in today’s DCNs are not ad hoc; rather
they can be derived from precise probabilistic assumptions that entirely determine their structure. Thus the DRM unifies two perspectives — neural network and probabilistic inference. A
summary of the relationship between the two perspectives is given in Table 1.
2.5.3
The Probabilistic Role of Max-Pooling
Consider the role of max-pooling from the message passing perspective. We see that it can
be interpreted as the “max” in max-sum, thus executing a max-marginalization over nuisance
variables g. Typically, this operation would be intractable, since there are exponentially many
configurations g ∈ G. But here the DRM’s model of abstraction — a deep product of affine
transformations — comes to the rescue. It enables us to convert an otherwise intractable
max-marginalization over g into a tractable sequence of iterated max-marginalizations over
abstraction levels g ` (Eqs. 12, 13).10 Thus, the max-pooling operation implements probabilistic
marginalization, so is absolutely essential to the DCN’s ability to factor out nuisance variation.
Indeed, since the ReLu can also be cast as a max-pooling over ON/OFF switching variables,
we conclude that the most important operation in DCNs is max-pooling. This is in conflict with
some recent claims to the contrary (27).
2.6
Learning the Rendering Models
Since the RM and DRM are graphical models with latent variables, we can learn their parameters from training data using the expectation-maximization (EM) algorithm (28). We first
develop the EM algorithm for the shallow RM from Section 2.1 and then extend it to the DRM
from Section 2.3.
10
This can be seen, equivalently, as the execution of the max-product algorithm (26).
18
Aspect!
Neural'Nets'Perspective'!
Deep$Convnets$(DCNs)!
Probabilistic+Perspective+!
Deep$Rendering$Model$(DRM)!
Model!
Weights and biases of filters at a
given layer
Partial Rendering at a given
abstraction level/scale
!
Number of Layers
Number of Abstraction Levels
!
Number of Filters in a layer
Number of Clusters/Classes at a
given abstraction level
!
Implicit in network weights; can be
computed by product of weights over
all layers or by activity maximization
Inference!
Forward propagation thru DCN
!
Input and Output Feature Maps
!
Template matching at a given layer
(convolutional, locally or fully
connected)
Max-Pooling over local pooling region
Category prototypes are finely
detailed versions of coarser-scale
super-category prototypes. Fine
details are modeled with affine
nuisance transformations.
Exact bottom-up inference via MaxSum Message Passing (with MaxProduct for Nuisance Factorization).
Probabilistic Max-Sum Messages
(real-valued functions of variables
nodes)
Local computation at factor node (loglikelihood of measurements)
!
!
Rectified Linear Unit (ReLU).
Sparsifies output activations.
Learning!
Stochastic Gradient Descent
!
N/A
!
Batch-Normalized SGD (Google stateof-the-art [BN])
Max-Marginalization over Latent
Translational Nuisance
transformations
Max-Marginalization over Latent
Switching state of Renderer. Low prior
probability of being ON.
Batch Discriminative EM Algorithm
with Fine-to-Coarse E-step + Gradient
M-step. No coarse-to-fine pass in Estep.
Full EM Algorithm
Discriminative Approximation to Full
EM (assumes Diagonal Pixel
Covariance)
!
Table 1. Summary of probabilistic and neural network perspectives for DCNs. The DRM provides an
exact correspondence between the two, providing a probabilistic interpretation for all of the common
elements of DCNs relating to the underlying model, inference algorithm, and learning rules. [BN] =
reference (25).
19
2.6.1
EM Algorithm for the Shallow Rendering Model
Given a dataset of labeled training images {In , cn }N
n=1 , each iteration of the EM algorithm consists of an E-step that infers the latent variables given the observed variables and the “old” paold
rameters θ̂gen
from the last M-step, followed by an M-step that updates the parameter estimates
according to
old
E-step: γncg = p(c, g|In ; θ̂gen
),
M-step: θ̂ = argmax
θ
XX
n
γncg L(θ).
(14)
(15)
cg
Here γncg are the posterior probabilities over the latent mixture components (also called the
P
responsibilities), the sum cg is over all possible global configurations (c, g) ∈ C × G, and
L(θ) is the complete-data log-likelihood for the model.
For the RM, the parameters are defined as θ ≡ {πc , πg , µcg , σ 2 } and include the prior probabilities of the different classes πc and nuisance variables πg along with the rendered templates
µcg and the pixel noise variance σ 2 . If, instead of an isotropic Gaussian RM, we use a fullcovariance Gaussian RM or an RM with a different exponential family distribution, then the
sufficient statistics and the rendered template parameters would be different (e.g. quadratic for
a full covariance Gaussian).
When the clusters in the RM are well-separated (or equivalently, when the rendering introduces little noise), each input image can be assigned to its nearest cluster in a “hard” E-step,
wherein we care only about the most likely configuration of the c` and g ` given the input I 0 . In
`
this case, the responsibility γncg
= 1 if c` and g ` in image In are consistent with the most likely
configuration; otherwise it equals 0. Thus, we can compute the responsibilities using max-sum
message passing according to Eqs. 12 and 14. In this case, the hard EM algorithm reduces to
`
Hard E-step: γncg
= J(c, g) = (c∗n , gn∗ )K
`
M-step: N̂cg
=
X
(16)
`
γncg
n
`
N̂cg
=
N
1 X ` `
µ̂`cg =
γncg In
`
N̂cg
n
1 X `
2 `
(σ̂cg
) =
γncg kIn` − µ`cg k22 ,
`
N̂cg n
`
π̂cg
(17)
where we have used the Iversen bracket to denote a boolean expression, i.e., JbK ≡ 1 if b is true
and JbK ≡ 0 if b is false.
20
2.6.2
EM Algorithm for the Deep Rendering Model
For high-nuisance tasks, the EM algorithm for the shallow RM is computationally intractable,
since it requires recomputing the responsibilities and parameters for all possible configurations
τ ≡ (cL , g L , . . . g 1 ).
Q
There are exponentially many such configurations ( |C L | ` |G ` |), one for each possible
rendering tree rooted at cL . However, the crux of the DRM is the factorized form of the rendered
templates (Eq. 11), which results in a dramatic reduction in the number of parameters. This
enables us to efficiently infer the most probable configuration exactly11 via Eq. 12 and thus
avoid the need to resort to slower, approximate sampling techniques (e.g. Gibbs sampling),
which are commonly used for approximate inference in deep HBMs (20, 21). We will exploit
this realization below in the DRM E-step.
Guided by the EM algorithm for MFA (22), we can extend the EM algorithm for the shallow
RM from the previous section into one for the DRM. The DRM E-step performs inference,
finding the most likely rendering tree configuration τn∗ ≡ (cLn , gnL , . . . , gn1 )∗ given the current
training input In0 . The DRM M-step updates the parameters in each layer — the weights and
biases — via a responsibility-weighted regression of output activations off of input activations.
This can be interpreted as each layer learning how to summarize its input feature map into a
coarser-grained output feature map, the essence of abstraction.
In the following it will be convenient to define and use the augmented form12 for certain
parameters so that affine transformations can be recast as linear ones. Mathematically, a single
11
Note that this is exact for the spike-n-slab approximation to the truly sparse rendering model where only one
renderer per neighborhood is active, as described in Section 2.2. Technically, this approximation is not a tree, but
instead a so-called polytree. Nevertheless, max-sum is exact for trees and polytrees (29).
12
y = mx + b ≡ m̃T x̃, where m̃ ≡ [m|b] and x̃ ≡ [x|1] are the augmented forms for the parameters and input.
21
EM iteration for the DRM is then defined as
E-step:
(18)
γnτ = Jτ = τn∗ K where τn∗ ≡ argmax {ln p(τ |In )}
τ
E µ` (c` ) = Λ` (g ` )† (In`−1 − α` (g ` )) ≡ W ` (g ` )In`−1 + b` (g ` )
E µ` (c` )µ` (c` )T = 1 − Λ` (g ` )† Λ` (g ` )+
(19)
(20)
Λ` (g ` )† (In`−1 − α` (g ` ))(In`−1 − α` (g ` ))T (Λ` (g ` )† )T
M-step:
π(τ ) =
1 X
γnτ
N n
(21)
Λ̃` (g ` ) ≡ Λ` (g ` ) | α` (g ` )
=
X
n
Ψ` =
T
γnτ In`−1 E µ̃` (c` )
1
diag
N
(
X
n
γnτ
!
X
n
γnτ E µ̃` (c` )µ̃` (c` )T
` ` `−1 T
`−1
` `
In − Λ̃ (g )E µ̃ (c ) (In )
)
!−1
(22)
,
(23)
where Λ` (g ` )† ≡ Λ` (g ` )T (Ψ` + Λ` (g ` )(Λ` (g ` ))T )−1 and E µ̃` (c` ) = E µ` (c` ) | 1 . Note
that the nuisance variables g ` comprise both the translational and the switching variables that
were introduced earlier for DCNs.
Note that this new EM algorithm is a derivative-free alternative to the back propagation
algorithm for training DCNs that is fast, easy to implement, and intuitive.
A powerful learning rule discovered recently and independently by Google (25) can be
seen as an approximation to the above EM algorithm, whereby Eq. 18 is approximated by
normalizing the input activations with respect to each training batch and introducing scaling
and bias parameters according to
E µ` (c` ) = Λ` (g ` )† (In`−1 − α` (g ` ))
≈ Γ · I˜n`−1 + β
`−1 ¯
I
− IB
≡Γ· n
+ β.
(24)
σB
Here I˜n`−1 are the batch-normalized activations, and I¯B and σB are the batch mean and standard
deviation vector of the input activations, respectively. Note that the division is element-wise,
22
since each activation is normalized independently to avoid a costly full covariance calculation.
The diagonal matrix Γ and bias vector β are parameters that are introduced to compensate
for any distortions due to the batch-normalization. In light of our EM algorithm derivation
for the DRM, it is clear that this scheme is a crude approximation to the true normalization
step in Eq. 18, whose decorrelation scheme uses the nuisance-dependent mean α(g ` ) and full
covariance Λ` (g ` )† . Nevertheless, the excellent performance of the Google algorithm bodes
well for the performance of the exact EM algorithm for the DRM developed above.
2.6.3
What About DropOut Training?
We did not mention the most common regularization scheme used with DCNs — DropOut (30).
DropOut training consists of units in the DCN dropping their outputs at random. This can be
seen as a kind of noise corruption, and encourages the learning of features that are robust to
missing data and prevents feature co-adaptation as well (18, 30). DropOut is not specific to
DCNs; it can be used with other architectures as well. For brevity, we refer the reader to the
proof of the DropOut algorithm in Appendix A.7. There we show that DropOut can be derived
from the EM algorithm.
2.7
From Generative to Discriminative Classifiers
We have constructed a correspondence between the DRM and DCNs, but the mapping defined
so far is not exact. In particular, note the constraints on the weights and biases in Eq. 8. These
are reflections of the distributional assumptions underlying the Gaussian DRM. DCNs do not
have such constraints — their weights and biases are free parameters. As a result, when faced
with training data that violates the DRM’s underlying assumptions (model misspecification),
the DCN will have more freedom to compensate. In order to complete our mapping and create
an exact correspondence between the DRM and DCNs, we relax these parameter constraints,
allowing the weights and biases to be free and independent parameters. However, this seems an
ad hoc approach. Can we instead theoretically motivate such a relaxation?
It turns out that the distinction between the DRM and DCN classifiers is fundamental: the
former is known as a generative classifier while the latter is known as a discriminative classifier (31, 32). The distinction between generative and discriminative models has to do with
the bias-variance tradeoff. On the one hand, generative models have strong distributional assumptions, and thus introduce significant model bias in order to lower model variance (i.e., less
risk of overfitting). On the other hand, discriminative models relax some of the distributional
assumptions in order to lower the model bias and thus “let the data speak for itself”, but they do
so at the cost of higher variance (i.e., more risk of overfitting) (31, 32). Practically speaking, if
a generative model is misspecified and if enough labeled data is available, then a discriminative
23
model will achieve better performance on a specific task (32). However, if the generative model
really is the true data-generating distribution (or there is not much labeled data for the task),
then the generative model will be the better choice.
Having motivated the distinction between the two types of models, in this section we will
define a method for transforming one into the other that we call a discriminative relaxation. We
call the resulting discriminative classifier a discriminative counterpart of the generative classifier.13 We will then show that applying this procedure to the generative DRM classifier (with
constrained weights) yields the discriminative DCN classifier (with free weights). Although we
will focus again on the Gaussian DRM, the treatment can be generalized to other exponential
family distributions with a few modifications (see Appendix A.6 for more details).
2.7.1
Transforming a Generative Classifier into a Discriminative One
Before we formally define the procedure, some preliminary definitions and remarks will be
helpful. A generative classifier models the joint distribution p(c, I) of the input features and
the class labels. It can then classify inputs by using Bayes Rule to calculate p(c|I) ∝ p(c, I) =
p(I|c)p(c) and picking the most likely label c. Training such a classifier is known as generative
learning, since one can generate synthetic features I by sampling the joint distribution p(c, I).
Therefore, a generative classifier learns an indirect map from input features I to labels c by
modeling the joint distribution p(c, I) of the labels and the features.
In contrast, a discriminative classifier parametrically models p(c|I) = p(c|I; θd ) and then
trains on a dataset of input-output pairs {(In , cn )}N
n=1 in order to estimate the parameter θd . This
is known as discriminative learning, since we directly discriminate between different labels c
given an input feature I. Therefore, a discriminative classifier learns a direct map from input
features I to labels c by directly modeling the conditional distribution p(c|I) of the labels given
the features.
Given these definitions, we can now define the discriminative relaxation procedure for converting a generative classifier into a discriminative one. Starting with the standard learning
objective for a generative classifier, we will employ a series of transformations and relaxations
13
The discriminative relaxation procedure is a many-to-one mapping: several generative models might have the
same discriminative model as their counterpart.
24
to obtain the learning objective for a discriminative classifier. Mathematically, we have
X
max Lgen (θ) ≡ max
ln p(cn , In |θ)
θ
θ
(a)
= max
θ
(b)
n
X
n
= max
θ,θ̃:θ=θ̃
(c)
≤ max
θ
(d)
X
n
X
|n
= max
η:η=ρ(θ)
(e)
≤ max
η
n
ln p(cn |In , θ) + ln p(In |θ̃)
ln p(cn |In , θ)
{z
≡Lcond (θ)
X
X
|
ln p(cn |In , θ) + ln p(In |θ)
n
}
ln p(cn |In , η)
ln p(cn |In , η),
{z
≡Ldis (η)
}
(25)
where the L’s are the generative, conditional and discriminative log-likelihoods, respectively.
In line (a), we used the Chain Rule of Probability. In line (b), we introduced an extra set of
parameters θ̃ while also introducing a constraint that enforces equality with the old set of generative parameters θ. In line (c), we relax the equality constraint (first introduced by Bishop,
LaSerre and Minka in (31)), allowing the classifier parameters θ to differ from the image generation parameters θ̃. In line (d), we pass to the natural parametrization of the exponential family
distribution I|c, where the natural parameters η = ρ(θ) are a fixed function of the conventional
parameters θ. This constraint on the natural parameters ensures that optimization of Lcond (η)
yields the same answer as optimization of Lcond (θ). And finally, in line (e) we relax the natural parameter constraint to get the learning objective for a discriminative classifier, where the
parameters η are now free to be optimized.
In summary, starting with a generative classifier with learning objective Lgen (θ), we complete steps (a) through (e) to arrive at a discriminative classifier with learning objective Ldis (η).
We refer to this process as a discriminative relaxation of a generative classifier and the resulting
classifier is a discriminative counterpart to the generative classifier.
Figure 4 illustrates the discriminative relaxation procedure as applied to the RM (or DRM).
If we consider a Gaussian (D)RM, then θ simply comprises the mixing probabilities πcg and
the mixture parameters λcg , and so that we have θ = {πcg , µcg , σ 2 }. The corresponding relaxed
discriminative parameters are the weights and biases ηdis ≡ {wcg , bcg }.
Intuitively, we can interpret the discriminative relaxation as a brain-world transformation
applied to a generative model. According to this interpretation, instead of the world generating
25
A
B
⇡cg
c g
✓
I
cg
I
⌘brain
⇢(·)
X"
c g
✓ ⌘ ✓brain
discrimina+ve"
relaxa+on" X"
✓˜ ⌘ ✓world
Figure 4. Graphical depiction of discriminative relaxation procedure. (A) The Rendering Model (RM) is
depicted graphically, with mixing probability parameters πcg and rendered template parameters λcg . The
brain-world transformation converts the RM (A) to an equivalent graphical model (B), where an extra
set of parameters θ̃ and constraints (arrows from θ to θ̃ to η) have been introduced. Discriminatively
relaxing these constraints (B, red X’s) yields the single-layer DCN as the discriminative counterpart to
the original generative RM classifier in (A).
images and class labels (Fig. 4A), we instead imagine the world generating images In via the
rendering parameters θ̃ ≡ θworld while the brain generates labels cn , gn via the classifier parameters ηdis ≡ ηbrain (Fig. 4B). Note that the graphical model depicted in Fig. 4B is equivalent to
that in Fig. 4A, except for the relaxation of the parameter constraints (red ×’s) that represent
the discriminative relaxation.
2.7.2
From the Deep Rendering Model to Deep Convolutional Networks
We can now apply the above to show that the DCN is a discriminative relaxation of the DRM.
First, we apply the brain-world transformation (Eq. 25) to the DRM. The resulting classifier is
precisely a deep MaxOut neural network (10) as discussed earlier. Second, we impose translational invariance at the finer scales of abstraction ` and introduce switching variables a to
model inactive renderers. This yields convolutional layers with ReLu activation functions, as in
Section 2.1. Third, the learning algorithm for the generative DRM classifier — the EM algorithm in Eqs. 18–23 — must be modified according to Eq. 25 to account for the discriminative
relaxation. In particular, note that the new discriminative E-step is only fine-to-coarse and corresponds to forward propagation in DCNs. As for the discriminative M-step, there are a variety
of choices: any general purpose optimization algorithm can be used (e.g., Newton-Raphson,
conjugate gradient, etc.). Choosing gradient descent this leads to the classical back propagation
algorithm for neural network training (33). Typically, modern-day DCNs are trained using a
variant of back propagation called Stochastic Gradient Descent (SGD), in which gradients are
computed using one mini-batch of data at a time (instead of the entire dataset). In light of our
developments here, we can re-interpret SGD as a discriminative counterpart to the generative
batch EM algorithm (34, 35).
This completes the mapping from the DRM to DCNs. We have shown that DCN classi26
fiers are a discriminative relaxation of DRM classifiers, with forward propagation in a DCN
corresponding to inference of the most probable configuration in a DRM.14 We have also reinterpreted learning: SGD back propagation training in DCNs is a discriminative relaxation of
a batch EM learning algorithm for the DRM. We have provided a principled motivation for
passing from the generative DRM to its discriminative counterpart DCN by showing that the
discriminative relaxation helps alleviate model misspecification issues by increasing the DRM’s
flexibility, at the cost of slower learning and requiring more training data.
3
New Insights into Deep Convolutional Networks
In light of the intimate connection between DRMs and DCNs, the DRM provides new insights
into how and why DCNs work, answering many open questions. And importantly, DRMs also
show us how and why DCNs fail and what we can do to improve them (see Section 6). In this
section, we explore some of these insights.
3.1
DCNs Possess Full Probabilistic Semantics
The factor graph formulation of the DRM (Fig. 2B) provides a useful interpretation of DCNs: it
shows us that the convolutional and max-pooling layers correspond to standard message passing operations, as applied inside factor nodes in the factor graph of the DRM. In particular, the
max-sum algorithm corresponds to a max-pool-conv neural network, whereas the sum-product
algorithm corresponds to a mean-pool-conv neural network. More generally, we see that architectures and layer types used commonly in successful DCNs are neither arbitrary nor ad hoc;
rather they can be derived from precise probabilistic assumptions that almost entirely determine
their structure. A summary of the two perspectives — neural network and probabilistic — are
given in Table 1.
3.2
Class Appearance Models and Activity Maximization
Our derivation of inference in the DRM enables us to understand just how trained DCNs distill
and store knowledge from past experiences in their parameters. Specifically, the DRM generates
rendered templates µ(cL , g) ≡ µ(cL , g L , . . . , g 1 ) via a product of affine transformations, thus
implying that class appearance models in DCNs (and DRMs) are stored in a factorized form
across multiple levels of abstraction. Thus, we can explain why past attempts to understand
how DCNs store memories by examining filters at each layer were a fruitless exercise: it is the
14
As mentioned in Section 2.4.1, this is typically followed by a Softmax Regression layer at the end. This layer
classifies the hidden representation (the penultimate layer activations âL (In )) into the class labels c̃n used for
training. See Section 2.4.1 for more details.
27
product of all the filters/weights over all layers that yield meaningful images of objects. Indeed,
this fact is encapsulated mathematically in Eqs. 10, 11. Notably, recent studies in computational
neuroscience have also shown a strong similarity between representations in primate visual
cortex and a highly trained DCN (36), suggesting that the brain might also employ factorized
class appearance models.
We can also shed new light on another approach to understanding DCN memories that
proceeds by searching for input images that maximize the activity of a particular class unit
(say, cat) (37), a technique we call activity maximization. Results from activity maximization
on a high performance DCN trained on 15 million images from (37) is shown in Fig. 5. The
resulting images are striking and reveal much about how DCNs store memories. We now derive
a closed-form expression for the activity-maximizing images as a function of the underlying
DRM model’s learned parameters. Mathematically, we seek the image I that maximizes the
score S(c|I) of a specific object class. Using the DRM, we have
1
µ(c` , g ` )|Ii
I
g∈G
σ2
∝ max max hµ(c` , g)|Ii
max S(c` |I) = max max h
I
g∈G
I
= max max · · · max hµ(c` , g)|
g∈G
= max
g∈G
= max
g∈G
=
X
Pi ∈P
IP 1
IP p
X
Pi ∈P
X
Pi ∈P
X
Pi ∈P
IPi i
max hµ(c` , g)|IPi i
IP i
hµ(c` , g)|IP∗ i (c` , g)i
hµ(c` , g)|IP∗ i (c` , gP∗ i i,
(26)
=
g ∗ (c` , Pi )
≡
where IP∗ i (c` , g)
≡
argmaxIP hµ(c` , g)|IPi i and gP∗ i
i
`
∗
`
argmaxg∈G hµ(c , g)|IPi (c , g)i. In the third line, the image I is decomposed into P
patches IPi of the same size as I, with all pixels outside of the patch Pi set to zero. The
maxg∈G operator finds the most probable gP∗ i within each patch. The solution I ∗ of the activity
maximization is then the sum of the individual activity-maximizing patches
X
X
I∗ ≡
IP∗ i (c` , gP∗ i ) ∝
µ(c` , gP∗ i ).
(27)
Pi ∈P
Pi ∈P
Eq. 27 implies that I ∗ contains multiple appearances of the same object but in various poses.
Each activity-maximizing patch has its own pose (i.e. gP∗ i ), in agreement with Fig. 5. Such
images provide strong confirming evidence that the underlying model is a mixture over nuisance
(pose) parameters, as is expected in light of the DRM.
28
dumbbell
cup
dalmatian
bell pepper
lemon
husky
washing machine
computer keyboard
kit fox
goose
ostrich
limousine
Figure 1: Numerically computed images, illustrating the class appearance models, learnt by a
Figure 5. Results
of activity
ImageNet
dataset.
For
a given
class
activity-maximizing
ConvNet,
trained maximization
on ILSVRC-2013.on
Note
how different
aspects of
class
appearance
arec,captured
in
a
single
image.
Better
viewed
in
colour.
inputs are superpositions of various poses of the object, with distinct patches Pi containing distinct poses
∗ , as predicted by Eq. 27. Figure adapted from (37) with permission from the authors.
gP
i
3
29
3.3
(Dis)Entanglement: Supervised Learning of Task Targets Is
Intertwined with Unsupervised Learning of Latent Task Nuisances
A key goal of representation learning is to disentangle the factors of variation that contribute
to an image’s appearance. Given our formulation of the DRM, it is clear that DCNs are discriminative classifiers that capture these factors of variation with latent nuisance variables g. As
such, the theory presented here makes a clear prediction that for a DCN, supervised learning
of task targets will lead inevitably to unsupervised learning of latent task nuisance variables.
From the perspective of manifold learning, this means that the architecture of DCNs is designed
to learn and disentangle the intrinsic dimensions of the data manifold.
In order to test this prediction, we trained a DCN to classify synthetically rendered images
of naturalistic objects, such as cars and planes. Since we explicitly used a renderer, we have
the power to systematically control variation in factors such as pose, location, and lighting. After training, we probed the layers of the trained DCN to quantify how much linearly separable
information exists about the task target c and latent nuisance variables g. Figure 6 shows that
the trained DCN possesses significant information about latent factors of variation and, furthermore, the more nuisance variables, the more layers are required to disentangle the factors. This
is strong evidence that depth is necessary and that the amount of depth required increases with
the complexity of the class models and the nuisance variations.
In light of these results, when we talk about training DCNs, the traditional distinction between supervised and unsupervised learning is ill-defined at worst and misleading at best. This
is evident from the initial formulation of the RM, where c is the task target and g is a latent variable capturing all nuisance parameters (Fig. 1). Put another way, our derivation above shows
that DCNs are discriminative classifiers with latent variables that capture nuisance variation.
We believe the main reason this was not noticed earlier is probably that latent nuisance variables
in a DCN are hidden within the max-pooling units, which serve the dual purpose of learning
and marginalizing out the latent nuisance variables.
4
From the Deep Rendering Model to Random Decision
Forests
Random Decision Forests (RDF)s (5, 38) are one of the best performing but least understood
classifiers in machine learning. While intuitive, their structure does not seem to arise from a
proper probabilistic model. Their success in a vast array of ML tasks is perplexing, with no
clear explanation or theoretical understanding. In particular, they have been quite successful in
real-time image and video segmentation tasks, the most prominent example being their use for
pose estimation and body part tracking in the Microsoft Kinect gaming system (39). They also
30
of Classifying Classes and Latent Variables vs Layer
1.0 Accuracy
obj_accu_rate
Accuracy Rate
0.8
slant_accu_rate
tilt_accu_rate
locx_accu_rate
locy_accu_rate
locz_accu_rate
energy_accu_rate
0.6
0.4
0.2
0.0
0
1
2
Layer
3
4
1.0 Accuracy of Classifying Classes and Latent Variables vs Layer
Accuracy Rate
0.8
0.6
0.4
0.2
0.0
0
obj_accu_rate
slant_accu_rate
tilt_accu_rate
locx_accu_rate
locy_accu_rate
1
2
Layer
3
4
Figure 6. Manifold entanglement and disentanglement as illustrated in a 5-layer max-out DCN trained
to classify synthetically rendered images of planes (top) and naturalistic objects (bottom) in different
poses, locations, depths and lighting conditions. The amount of linearly separable information about
the target variable (object identity, red) increases with layer depth while information about nuisance
variables (slant, tilt, left-right location, depth location) follows an inverted U-shaped curve. Layers
with increasing information correspond to disentanglement of the manifold — factoring variation into
independent parameters — whereas layers with decreasing information correspond to marginalization
over the nuisance parameters. Note that disentanglement of the latent nuisance parameters is achieved
progressively over multiple layers, without requiring the network to explicitly train for them. Due to
the complexity of the variation induced, several layers are required for successful disentanglement, as
predicted by our theory.
31
have had great success in medical image segmentation problems (5, 38), wherein distinguishing
different organs or cell types is quite difficult and typically requires expert annotators.
In this section we show that, like DCNs, RDFs can also be derived from the DRM model,
but with a different set of assumptions regarding the nuisance structure. Instead of translational
and switching nuisances, we will show that an additive mutation nuisance process that generates
a hierarchy of categories (e.g., evolution of a taxonomy of living organisms) is at the heart of
the RDF. As in the DRM to DCN derivation, we will start with a generative classifier and then
derive its discriminative relaxation. As such, RDFs possess a similar interpretation as DCNs in
that they can be cast as max-sum message passing networks.
A decision tree classifier takes an input image I and asks a series of questions about it. The
answer to each question determines which branch in the tree to follow. At the next node, another
question is asked. This pattern is repeated until a leaf b of the tree is reached. At the leaf, there
is a class posterior probability distribution p(c|I, b) that can be used for classification. Different
leaves contain different class posteriors. An RDF is an ensemble of decision tree classifiers
t ∈ T . To classify an input I, it is sent as input to each decision tree t ∈ T individually,
and each decision tree outputs a class posterior p(c|I, b, t). These are then averaged to obtain
P
an overall posterior p(c|I) = t p(c|I, b, t)p(t), from which the most likely class c is chosen.
Typically we assume p(t) = 1/|T |.
4.1
The Evolutionary Deep Rendering Model: A Hierarchy of Categories
We define the evolutionary DRM (E-DRM) as a DRM with an evolutionary tree of categories.
Samples from the model are generated by starting from the root ancestor template and randomly
mutating the templates. Each child template is an additive mutation of its parent, where the specific mutation does not depend on the parent (see Eq. 29 below). Repeating this pattern at each
child node, an entire evolutionary tree of templates is generated. We assume for simplicity that
we are working with a Gaussian E-DRM so that at the leaves of the tree a sample is generated
by adding Gaussian pixel noise. Of course, as described earlier, this can be extended to handle
other noise distributions from the exponential family. Mathematically, we have
cL ∼ Cat(π(cL )),
g `+1 ∼ Cat(π(g `+1 )),
cL ∈ C L ,
g `+1 ∈ G `+1 ,
` = L − 1, L − 2, . . . , 0
µ(cL , g) = Λ(g)µ(cL ) ≡ Λ1 (g 1 ) · · · ΛL (g L ) · µ(cL )
= µ(cL ) + α(g L ) + · · · + α(g 1 ),
I(cL , g) = µ(cL , g) + N (0, σ 2 1D ) ∈ RD .
32
g = {g ` }L`=1
(28)
Here, Λ` (g ` ) has a special structure due to the additive mutation process: Λ` (g ` ) = [1 | α(g ` )],
where 1 is the identity matrix. As before, C ` , G ` are the sets of all target-relevant and targetirrelevant nuisance variables at level `, respectively. (The target here is the same as with the
DRM and DCNs — the overall class label cL .) The rendering path represents template evolution and is defined as the sequence (cL , g L , . . . , g ` , . . . , g 1 ) from the root ancestor template
down to the individual pixels at ` = 0. µ(cL ) is an abstract template for the root ancestor
P
cL , and ` α(g ` ) represents the sequence of local nuisance transformations, in this case, the
accumulation of many additive mutations.
As with the DRM, we can cast the E-DRM into an incremental form by defining an intermediate class c` ≡ (cL , g L , . . . , g `+1 ) that intuitively represents a partial evolutionary path up to
level `. Then, the mutation from level ` + 1 to ` can be written as
µ(c` ) = Λ`+1 (g `+1 ) · µ(c`+1 ) = µ(c`+1 ) + α(g `+1 ),
(29)
where α(g ` ) is the mutation added to the template at level ` in the evolutionary tree.
As a generative model, the E-DRM is a mixture of evolutionary paths, where each path starts
at the root and ends at a leaf species in the tree. Each leaf species is associated with a rendered
template µ(cL , g L , . . . , g 1 ).
4.2
Inference with the E-DRM Yields a Decision Tree
Since the E-DRM is an RM with a hierarchical prior on the rendered templates, we can use
Eq. 7 to derive the E-DRM inference algorithm as:
ĉM S (I) = argmax max hη(cL , g)|I 0 i
cL ∈C L
g∈G
= argmax max hΛ(g)µ(cL )|I 0 i
cL ∈C L
g∈G
= argmax max
· · · max hµ(cL ) + α(g L ) + · · · + α(g 1 )|I 0 i
1
1
cL ∈C L
g ∈G
= argmax max
···
1
1
cL ∈C L
g ∈G
g L ∈G L
max
g L−1 ∈G L−1
h µ(cL ) + α(g L∗ ) + · · · + α(g 1 )|I 0 i
{z
}
|
≡µ(cL ,g L∗ )=µ(cL−1 )
= argmax max
···
1
1
cL ∈C L
g ∈G
max
g L−1 ∈G L−1
hµ(cL−1 ) + α(g L−1 ) + · · · + α(g 1 )|I 0 i
..
.
(30)
≡ argmax hµ(cL , g ∗ )|I 0 i,
cL ∈C L
Note that we have explicitly shown the bias terms here, since they represent the additive mutations. In the last lines, we repeatedly use the distributivity of max over sums, resulting in the
33
iteration
g `+1 (c`+1 )∗ ≡ argmax hµ(c`+1 , g `+1 ) |I 0 i
{z
}
g `+1 ∈G `+1 |
= argmax hW
g `+1 ∈G `+1
≡W `+1
`+1 `+1
(c
, g `+1 )|I 0 i
≡ ChooseChild(Filter(I 0 )).
(31)
Note the key differences from the DRN/DCN inference derivation in Eq. 12: (i) the input
to each layer is always the input image I 0 , (ii) the iterations go from coarse-to-fine (from root
ancestor to leaf species) rather than fine-to-coarse, and (iii) the resulting network is not a neural
network but rather a deep decision tree of single-layer neural networks. These differences
are due to the special additive structure of the mutational nuisances and the evolutionary tree
process underlying the generation of category templates.
4.2.1
What About the Leaf Histograms?
The mapping to a single decision tree is not yet complete; the leaf label histograms (5, 38) are
missing. Analogous to the missing SoftMax regression layers with DCNs (Sec 2.4.1), the highlevel representation class label cL inferred by the E-DRM in Eq. 30 need not be the training
data class label c̃. For clarity, we treat the two as separate in general.
But then how do we understand cL ? We can interpret the inferred configuration τ ∗ =
(cL∗ , g ∗ ) as a disentangled representation of the input, wherein the different factors in τ ∗ , including cL , vary independently in the world. In contrast to DCNs, the class labels c̃ in a decision
tree are instead inferred from the discrete evolutionary path variable τ ∗ through the use of the
leaf histograms p(c̃|τ ∗ ). Note that decision trees also have label histograms at all internal (nonleaf) nodes, but that they are not needed for inference. However, they do play a critical role in
learning, as we will see below.
We are almost finished with our mapping from inference in Gaussian E-DRMs to decision
trees. To finish the mapping, we need only apply the discriminative relaxation (Eq. 25) in order
to allow the weights and biases that define the decision functions in the internal nodes to be
free. Note that this is exactly analogous to steps in Section 2.7 for mapping from the Gaussian
DRM to DCNs.
4.3
Bootstrap Aggregation to Prevent Overfitting Yields A Decision
Forest
Thus far we have derived the inference algorithm for the E-DRM and shown that its discriminative counterpart is indeed a single decision tree. But how to relate to this result to the entire
34
forest? This is important, since it is well known that individual decision trees are notoriously
good at overfitting data. Indeed, the historical motivation for introducing a forest of decision
trees has been in order to prevent such overfitting by averaging over many different models,
each trained on a randomly drawn subset of the data. This technique is known as bootstrap aggregation or bagging for short, and was first introduced by Breiman in the context of decision
trees (38). For completeness, in this section we review bagging, thus completing our mapping
from the E-DRM to the RDF.
In order to derive bagging, it will be necessary in the following to make explicit the dependence of learned inference parameters θ on the training data DCI ≡ {(cn , In )}N
n=1 , i.e.
θ = θ(DCI ). This dependence is typically suppressed in most work, but is necessary here as
bagging entails training different decision trees t on different subsets Dt ⊂ D of the full training
data. In other words, θt = θt (Dt ).
Mathematically, we perform inference as follows: Given all previously seen data DCI and
an unseen image I, we classify I by computing the posterior distribution
X
p(c, A|I, DCI )
p(c|I, DCI ) =
A
=
X
A
p(c|I, DCI , A)p(A)
≡ EA [p(c|I, DCI , A)]
(a) 1 X
p(c|I, DCI , At )
≈
T t∈T
Z
(b) 1 X
=
dθt p(c|I, θt ) p(θt |DCI , At )
T t∈T
(c)
1X
≈
p(c|I, θt∗ ) , θt∗ ≡ max p(θ|DCI (At )).
θ
T t∈T
|
{z
}
(32)
Decision Forest Classifier
Here At ≡ (atn )N
n=1 is a collection of switching variables that indicates which data points are
included, i.e., atn = 1 if data point n is included in dataset Dt ≡ DCI (At ). In this way, we have
randomly subsampled the full dataset DCI (with replacement) T times in line (a), approximating
the true marginalization over all possible subsets of the data. In line (b), we perform Bayesian
Model Averaging over all possible values of the E-DRM/decision tree parameters θt . Since this
is intractable, we approximate it with the MAP estimate θt∗ in line (c). The overall result is
that each E-DRM (or decision tree) t is trained separately on a randomly drawn subset Dt ≡
DCI (At ) ⊂ DCI of the entire dataset, and the final output of the classifier is an average over the
individual classifiers.
35
4.4
EM Learning for the E-DRM Yields the InfoMax Principle
One approach to train an E-DRM classifier is to maximize the mutual information between the
given training labels c̃ and the inferred (partial) rendering path τ ` ≡ (cL , g L , . . . , g l ) at each
level. Note that c̃ and τ ` are both discrete random variables.
This Mutual Information-based Classifier (MIC) plays the same role as the Softmax regression layer in DCNs, predicting the class labels c̃ given a good disentangled representation τ `∗
of the input I. In order to train the MIC classifier, we update the classifier parameters θMIC in
each M-step as the solution to the optimization:
L
L
1
· · · max
max M I(c̃, (c , g , . . . , g )) = max
1
θ
θL
θ
=
=
1
X
l=L
1
X
l=L
1
X
l=L
M I(c̃, gnl |gnl+1 ; θl )
max M I(c̃, gnl |gnl+1 ; θl )
θl
max H[c̃] − H[c̃|gnl ; θl ] .
{z
}
θl |
(33)
≡Information Gain
Here M I(·, ·) is the mutual information between two random variables, H[·] is the entropy of a
random variable, and θ` are the parameters at layer `. In the first line, we have used the layerby-layer structure of the E-DRM to split the mutual information calculation across levels, from
coarse to fine. In the second line, we have used the max-sum algorithm (dynamic programming)
to split up the optimization into a sequence of optimizations from ` = L → ` = 1. In the third
line, we have used the information-theoretic relationship M I(X, Y ) ≡ H[X] − H[Y |X]. This
algorithm is known as InfoMax in the literature (5).
5
5.1
Relation to Prior Work
Relation to Mixture of Factor Analyzers
As mentioned above, on a high level, the DRM is related to hierarchical models based on the
Mixture of Factor Analyzers (MFA) (22). Indeed, if we add noise to each partial rendering step
from level ` to ` − 1 in the DRM, then Eq. 11 becomes
I `−1 ∼ N Λ` (g ` )µ` (c` ) + α` (g ` ), Ψ` ,
(34)
where we have introduced the diagonal noise covariance Ψ` . This is equivalent to the MFA
model. The DRM and DMFA both employ parameter sharing, resulting in an exponential reduction in the number of parameters, as compared to the collapsed or shallow version of the
models. This serves as a strong regularizer to prevent overfitting.
36
Despite the high-level similarities, there are several essential differences between the DRM
and the MFA-based models, all of which are critical for reproducing DCNs. First, in the DRM
the only randomness is due to the choice of the g ` and the observation noise after rendering.
This naturally leads to inference of the most probable configuration via the max-sum algorithm,
which is equivalent to max-pooling in the DCN. Second, the DRM’s affine transformations Λ`
act on multi-channel images at level ` + 1 to produce multi-channel images at level `. This
structure is important, because it leads directly to the notion of (multi-channel) feature maps in
DCNs. Third, a DRM’s layers vary in connectivity from sparse to dense, as they give rise to
convolutional, locally connected, and fully connected layers in the resulting DCN. Fourth, the
DRM has switching variables that model (in)active renderers (Section 2.1). The manifestation
of these variables in the DCN are the ReLus (Eq. 9). Thus, the critical elements of the DCN
architecture arise directly from aspects of the DRM structure that are absent in MFA-based
models.
5.2
i-Theory: Invariant Representations Inspired by Sensory Cortex
Representational Invariance and selectivity (RI) are important ideas that have developed in
the computational neuroscience community. According to this perspective, the main purpose
of the feedforward aspects of visual cortical processing in the ventral stream are to compute
a representation for a sensed image that is invariant to irrelevant transformations (e.g., pose,
lighting etc.) (40, 41). In this sense, the RI perspective is quite similar to the DRM in its basic
motivations. However, the RI approach has remained qualitative in its explanatory power until
recently, when a theory of invariant representations in deep architectures — dubbed i-theory
— was proposed (42, 43). Inspired by neuroscience and models of the visual cortex, it is the
first serious attempt at explaining the success of deep architectures, formalizing intuitions about
invariance and selectivity in a rigorous and quantitatively precise manner.
The i-theory posits a representation that employs group averages and orbits to explicitly
insure invariance to specific types of nuisance transformations. These transformation must possess a mathematical semi-group structure; as a result, the invariance constraint is relaxed to a
notion of partial invariance, which is built up slowly over multiple layers of the architecture.
At a high level, the DRM shares similar goals with i-theory in that it attempts to capture
explicitly the notion of nuisance transformations. However, the DRM differs from i-theory in
two critical ways. First, it does not impose a semi-group structure on the set of nuisance transformations. This provides the DRM the flexibility to learn a representation that is invariant to a
wider class of nuisance transformations, including non-rigid ones. Second, the DRM does not
fix the representation for images in advance. Instead, the representation emerges naturally out
of the inference process. For instance, sum- and max-pooling emerge as probabilistic marginal37
ization over nuisance variables and thus are necessary for proper inference. The deep iterative
nature of the DCN also arises as a direct mathematical consequence of the DRM’s rendering
model, which comprises multiple levels of abstraction.
This is the most important difference between the two theories. Despite these differences,
i-theory is complementary to our approach in several ways, one of which is that it spends a good
deal of energy focusing on questions such as: How many templates are required for accurate
discrimination? How many samples are needed for learning? We plan to pursue these questions
for the DRM in future work.
5.3
Scattering Transform: Achieving Invariance via Wavelets
We have used the DRM, with its notion of target and nuisance variables, to explain the power
of DCN for learning selectivity and invariance to nuisance transformations. Another theoretical
approach to learning selectivity and invariance is the Scattering Transform (ST) (44, 45), which
consists of a series of linear wavelet transforms interleaved by nonlinear modulus-pooling of
the wavelet coefficients. The goal is to explicitly hand-design invariance to a specific set of
nuisance transformations (translations, rotations, scalings, and small deformations) by using
the properties of wavelet transforms.
If we ignore the modulus-pooling for a moment, then the ST implicitly assumes that images
can be modeled as linear combinations of pre-determined wavelet templates. Thus the ST approach has a maximally strong model bias, in that there is no learning at all. The ST performs
well on tasks that are consistent with its strong model bias, i.e., on small datasets for which
successful performance is therefore contingent on strong model bias. However, the ST will be
more challenged on difficult real-world tasks with complex nuisance structure for which large
datasets are available. This contrasts strongly with the approach presented here and that of the
machine learning community at large, where hand-designed features have been outperformed
by learned features in the vast majority of tasks.
5.4
Learning Deep Architectures via Sparsity
What is the optimal machine learning architecture to use for a given task? This question has
typically been answered by exhaustively searching over many different architectures. But is
there a way to learn the optimal architecture directly from the data? Arora et al. (46) provide
some of the first theoretical results in this direction. In order to retain theoretical tractability,
they assume a simple sparse neural network as the generative model for the data. Then, given the
data, they design a greedy learning algorithm that reconstructs the architecture of the generating
neural network, layer-by-layer.
38
They prove that their algorithm is optimal under a certain set of restrictive assumptions.
Indeed, as a consequence of these restrictions, their results do not directly apply to the DRM or
other plausible generative models of natural images. However, the core message of the paper
has nonetheless been influential in the development of the Inception architecture (13), which
has recently achieved the highest accuracy on the ImageNet classification benchmark (25).
How does the sparse reconstruction approach relate to the DRM? The DRM is indeed also a
sparse generative model: the act of rendering an image is approximated as a sequence of affine
transformations applied to an abstract high-level class template. Thus, the DRM can potentially
be represented as a sparse neural network. Another similarity between the two approaches is the
focus on clustering highly correlated activations in the next coarser layer of abstraction. Indeed
the DRM is a composition of sparse factor analyzers, and so each higher layer ` + 1 in a DCN
really does decorrelate and cluster the layer ` below, as quantified by Eq. 18.
But despite these high-level similarities, the two approaches differ significantly in their overall goals and results. First, our focus has not been on recovering the architectural parameters;
instead we have focused on the class of architectures that are well-suited to the task of factoring out large amounts of nuisance variation. In this sense the goals of the two approaches are
different and complementary. Second, we are able to derive the structure of DCNs and RDFs
exactly from the DRM. This enables us to bring to bear the full power of probabilistic analysis for solving high-nuisance problems; moreover, it will enable us to build better models and
representations for hard tasks by addressing limitations of current approaches in a principled
manner.
5.5
Google FaceNet: Learning Useful Representations with DCNs
Recently, Google developed a new face recognition architecture called FaceNet (47) that illustrates the power of learning good representations. It achieves state-of-the-art accuracy in face
recognition and clustering on several public benchmarks. FaceNet uses a DCN architecture, but
crucially, it was not trained for classification. Instead, it is trained to optimize a novel learning
objective called triplet finding that learns good representations in general.
The basic idea behind their new representation-based learning objective is to encourage
the DCN’s latent representation to embed images of the same class close to each other while
embedding images of different classes far away from each other, an idea that is similar to the
NuMax algorithm (48). In other words, the learning objective enforces a well-separatedness
criterion. In light of our work connecting DRMs to DCNs, we will next show how this new
learning objective can be understood from the perspective of the DRM.
The correspondence between the DRM and the triplet learning objective is simple. Since
rendering is a deterministic (or nearly noise-free) function of the global configuration (c, g), one
39
explanation should dominate for any given input image I = R(c, g), or equivalently, the clusters (c, g) should be well-separated. Thus, the noise-free, deterministic, and well-separated
DRM are all equivalent. Indeed, we implicitly used the well-separatedness criterion when
we employed the Hard EM algorithm to establish the correspondence between DRMs and
DCNs/RDFs.
5.6
Renormalization Theory
Given the DRM’s notion of irrelevant (nuisance) transformations and multiple levels of abstraction, we can interpret a DCN’s action as an iterative coarse-graining of an image, thus relating
our work to another recent approach to understanding deep learning that draws upon an analogy
from renormalization theory in physics (49). This approach constructs an exact correspondence
between the Restricted Boltzmann Machine (RBM) and block-spin renormalization — an iterative coarse-graining technique from physics that compresses a configuration of binary random
variables (spins) to a smaller configuration with less variables. The goal is to preserve as much
information about the longer-range correlations as possible, while integrating out shorter-range
fluctuations.
Our work here shows that this analogy goes even further as we have created an exact mapping between the DCN and the DRM, the latter of which can be interpreted as a new real-space
renormalization scheme. Indeed, the DRM’s main goal is to factor out irrelevant features over
multiple levels of detail, and it thus bears a strong resemblance to the core tenets of renormalization theory. As a result, we believe this will be an important avenue for further research.
5.7
Summary of Key Distinguishing Features of the DRM
The key features that distinguish the DRM approach from others in the literature can be summarized as: (i) The DRM explicitly models nuisance variation across multiple levels of abstraction
via a product of affine transformations. This factorized linear structure serves dual purposes:
it enables (ii) exact inference (via the max-sum/max-product algorithm) and (iii) it serves as a
regularizer, preventing overfitting by a novel exponential reduction in the number of parameters. Critically, (iv) the inference is not performed for a single variable of interest but instead
for the full global configuration. This is justified in low-noise settings, i.e., when the rendering
process is nearly deterministic, and suggests the intriguing possibility that vision is less about
probabilities and more about inverting a complicated (but deterministic) rendering transformation.
40
6
New Directions
We have shown that the DRM is a powerful generative model that underlies both DCNs and
RDFs, the two most powerful vision paradigms currently employed in machine learning. Despite the power of the DRM/DCN/RDF, it has limitations, and there is room for improvement.
(Since both DCNs and RDFs stem from DRMs, we will loosely refer to them both as DCNs in
the following, although technically an RDF corresponds to a kind of tree of DCNs.)
In broad terms, most of the limitations of the DCN framework can be traced back to the
fact that it is a discriminative classifier whose underlying generative model was not known.
Without a generative model, many important tasks are very difficult or impossible, including
sampling, model refinement, top-down inference, faster learning, model selection, and learning
from unlabeled data. With a generative model, these tasks become feasible. Moreover, the
DCN models rendering as a sequence of affine transformations, which severely limits its ability
to capture many important real-world visual phenomena, including figure-ground segmentation,
occlusion/clutter, and refraction. It also lacks several operations that appear to be fundamental
in the brain: feed-back, dynamics, and 3D geometry. Finally, it is unable to learn from unlabeled
data and to generalize from few examples. As a result, DCNs require enormous amounts of
labeled data for training.
These limitations can be overcome by designing new deep networks based on new model
structures (extended DRMs), new message-passing inference algorithms, and new learning
rules, as summarized in Table 2. We now explore these solutions in more detail.
6.1
More Realistic Rendering Models
We can improve DCNs by designing better generative models incorporating more realistic assumptions about the rendering process by which latent variables cause images. These assumptions should include symmetries of translation, rotation, scaling (44), perspective, and non-rigid
deformations, as rendered by computer graphics and multi-view geometry.
In order to encourage more intrinsic computer graphics-based representations, we can enforce these symmetries on the parameters during learning (50, 51). Initially, we could use local
affine approximations to these transformations (52). For example, we could impose weight tying based on 3D rotations in depth. Other nuisance transformations are also of interest, such
as scaling (i.e., motion towards or away from a camera). Indeed, scaling-based templates are
already in use by the state-of-the-art DCNs such as the Inception architectures developed by
Google (13), and so this approach has already shown substantial promise.
We can also perform intrinsic transformations directly on 3D scene representations. For example, we could train networks with depth maps, in which a subset of channels in input feature
maps encode pixel z-depth. These augmented input features will help define useful higher-level
41
Area
Problem (DCN)
Proposed Solution (DRM)
Model
Rendering model applies nuisance
transformations to extrinsic
representation (2D images or
feature maps).
Difficulty handling occlusion,
clutter and classifying objects that
are slender, transparent, metallic.
Modify DRM so that rendering applies
nuisance transformations to intrinsic
representations (e.g. 3D geometry).
Model is static and thus cannot
learn from videos.
Inference
Learning
Infers most probable global
configuration (max-product),
ignoring alternative hypotheses.
No top-down inference/feedback is
possible, so vision tasks involving
lower-level variables (e.g. clutter,
occlusion, segmentation) are
difficult.
Hard-EM Algorithm and its
discriminative relaxation tend to
confuse signal and noise
Nuisance variation makes learning
intrinsic latent factors difficult.
Discriminative models cannot
learn from unlabeled data.
Modify DRM to include intrinsic
computer-graphics-based
representations, transformations and
phototrealistic rendering.
Incorporate time into the DRM
(Dynamic DRM).
Use softer message-passing, i.e.
higher temperature or sum-product, to
encode uncertainty.
Compute contextual priors as topdown messages for low-level tasks.
Use Soft-EM or Variational Bayes-EM.
Use Dynamic DRM with movies to
learn that only a few nuisance
parameters change per frame.
Use DRM to do hybrid generativediscriminative learning that
simultaneously incorporates labeled,
unlabeled, and weakly labeled data.
Table 2. Limitations of current DCNs and potential solutions using extended DRMs.
42
features for 2D image features, and thereby transfer representational benefits even to test images that do not provide depth information (53). With these richer geometric representations,
learning and inference algorithms can be modified to account for 3D constraints according to
the equations of multi-view geometry (53).
Another important limitation of the DCN is its restriction to static images. There is no notion
of time or dynamics in the corresponding DRM model. As a result, DCN training on large-scale
datasets requires millions of images in order to learn the structure of high-dimensional nuisance
variables, resulting in a glacial learning process. In contrast, learning from natural videos should
result in an accelerated learning process, as typically only a few nuisance variables change from
frame to frame. This property should enable substantial acceleration in learning, as inference
about which nuisance variables have changed will be faster and more accurate (54). See Section 6.3.2 below for more details.
6.2
6.2.1
New Inference Algorithms
Soft Inference
We showed above in Section 2.4 that DCNs implicitly infer the most probable global interpretation of the scene, via the max-sum algorithm (55). However, there is potentially major component missing in this algorithm: max-sum message passing only propagates the most likely
hypothesis to higher levels of abstraction, which may not be the optimal strategy, in general,
especially if uncertainty in the measurements is high (e.g., vision in a fog or at nighttime). Consequently, we can consider a wider variety of softer inference algorithms by defining a temperature parameter that enables us to smoothly interpolate between the max-sum and sum-product
algorithms, as well as other message-passing variants such as the approximate Variational Bayes
EM (56). To the best of our knowledge, this notion of a soft DCN is novel.
6.2.2
Top-Down Convolutional Nets: Top-Down Inference via the DRM
The DCN inference algorithm lacks any form of top-down inference or feedback. Performance
on tasks using low-level features is then suboptimal, because higher-level information informs
low-level variables neither for inference nor for learning. We can solve this problem by using
the DRM, since it is a proper generative model and thus enables us to implement top-down
message passing properly.
Employing the same steps as outlined in Section 2, we can convert the DRM into a top-down
DCN, a neural network that implements both the bottom-up and top-down passes of inference
via the max-sum message passing algorithm. This kind of top-down inference should have a
dramatic impact on scene understanding tasks that require segmentation such as target detection
43
with occlusion and clutter, where local bottom-up hypotheses about features are ambiguous. To
the best of our knowledge, this is the first principled approach to defining top-down DCNs.
6.3
6.3.1
New Learning Algorithms
Derivative-Free Learning
Back propagation is often used in deep learning algorithms due to its simplicity. We have
shown above that back propagation in DCNs is actually an inefficient implementation of an
approximate EM algorithm, whose E-step consists of bottom-up inference and whose M-step
is a gradient descent step that fails to take advantage of the underlying probabilistic model
(the DRM). To the contrary, our above EM algorithm (Eqs. 18–23) is both much faster and
more accurate, because it directly exploits the DRM’s structure. Its E-step incorporates bottomup and top-down inference, and its M-step is a fast computation of sufficient statistics (e.g.,
sample counts, means, and covariances). The speed-up in efficiency should be substantial, since
generative learning is typically much faster than discriminative learning due to the bias-variance
tradeoff (32); moreover, the EM-algorithm is intrinsically more parallelizable (57).
6.3.2
Dynamics: Learning from Video
Although deep NNs have incorporated time and dynamics for auditory tasks (58–60), DCNs
for visual tasks have remained predominantly static (images as opposed to videos) and are
trained on static inputs. Latent causes in the natural world tend to change little from frameto-frame, such that previous frames serve as partial self-supervision during learning (61). A
dynamic version of the DRM would train without external supervision on large quantities of
video data (using the corresponding EM algorithm). We can supplement video recordings of
natural dynamic scenes with synthetically rendered videos of objects traveling along smooth
trajectories, which will enable the training to focus on learning key nuisance factors that cause
difficulty (e.g., occlusion).
6.3.3
Training from Labeled and Unlabeled Data
DCNs are purely discriminative techniques and thus cannot benefit from unlabeled data. However, armed with a generative model we can perform hybrid discriminative-generative training (31) that enables training to benefit from both labeled and unlabeled data in a principled
manner. This should dramatically increase the power of pre-training, by encouraging representations of the input that have disentangled factors of variation. This hybrid generativediscriminative learning is achieved by the optimization of a novel objective function for learning, that relies on both the generative model and its discriminative relaxation. In particular, the
44
learning objective will have terms for both, as described in (31). Recall from Section 2.7 that
the discriminative relaxation of a generative model is performed by relaxing certain parameter
constraints during learning, according to
max Lgen (θ; DCI ) = max Lnat (η; DCI )
θ
η:η=ρ(θ)
≤ max Lcond (η; DC|I )
η:η=ρ(θ)
≤ max Ldis (η; DC|I ),
η
(35)
where the L’s are the model’s generative, naturally parametrized generative, conditional, and
discriminative likelihoods. Here η are the natural parameters expressed as a function of the
traditional parameters θ, DCI is the training dataset of labels and images, and DC|I is the training
dataset of labels given images. Although the discriminative relaxation is optional, it is very
important for achieving high performance in real-world classifiers as discriminative models
have less model bias and, therefore, are less sensitive to model mis-specifications (32). Thus,
we will design new principled training algorithms that span the spectrum from discriminative
(e.g., Stochastic Gradient Descent with Back Propagation) to generative (e.g., EM Algorithm).
Acknowledgments
Thanks to CJ Barberan for help with the manuscript and to Mayank Kumar, Ali Mousavi,
Salman Asif and Andreas Tolias for comments and discussions. Thanks to Karen Simonyan
for providing the activity maximization figure. A special thanks to Xaq Pitkow whose keen
insight, criticisms and detailed feedback on this work have been instrumental in its development. Thanks to Ruchi Kukreja for her unwavering support and her humor and to Raina Patel
for providing inspiration.
45
A
A.1
Supplemental Information
From the Gaussian Rendering Model Classifier to Deep DCNs
Proposition A.1 (MaxOut NNs). The discriminative relaxation of a noise-free GRM classifier
is a single layer NN consisting of a local template matching operation followed by a piecewise
linear activation function (also known as a MaxOut NN (10)).
Proof. In order to teach the reader, we prove this claim exhaustively. Later claims will have
simple proofs that exploit the fact that the RM’s distribution is from the exponential family.
ĉ(I) ≡ argmax p(c|I)
c∈C
= argmax {p(I|c)p(c)}
c∈C
(
)
X
= argmax
p(I|c, h)p(c, h)
c∈C
h∈H
(a)
= argmax max p(I|c, h)p(c, h)
h∈H
c∈C
= argmax max exp (ln p(I|c, h) + ln p(c, h))
h∈H
c∈C
(
!)
X
(b)
= argmax max exp
ln p(I ω |c, h) + ln p(c, h)
c∈C
h∈H
ω
!)
1X ω
D
ω
ω
= argmax max exp −
I − µωch |Σ−1
ln |Σch |
ch |I − µch + ln p(c, h) −
h∈H
2
2
c∈C
ω
(
!)
X
ω
= argmax max exp
hwch
|I ω i + bωch
(c)
(
c∈C
h∈H
c∈C
h∈H
ω
(d)
≡ argmax exp max {wch ?LC I}
h∈H
c∈C
= argmax max {wch ?LC I}
= Choose {MaxOutPool(LocalTemplateMatch(I))}
= MaxOut-NN(I; θ).
In line (a), we take the noise-free limit of the GRM, which means that one hypothesis (c, h)
dominates all others in likelihood. In line (b), we assume that the image I consists of multiple channels ω ∈ Ω, that are conditionally independent given the global configuration (c, h).
46
Typically, for input images these are color channels and Ω ≡ {r, g, b} but in general Ω can
be more abstract (e.g. as in feature maps). In line (c), we assume that the pixel noise covariance is isotropic and conditionally independent given the global configuration (c, h), so that
Σch = σx2 1D is proportional to the D × D identity matrix 1D . In line (d), we defined the locally
connected template matching operator ?LC , which is a location-dependent template matching
operation.
Note that the nuisance variables h ∈ H are (max-)marginalized over, after the application
of a local template matching operation against a set of filters/templates W ≡ {wch }c∈C,h∈H
Lemma A.2 (Translational Nuisance →d DCN Convolution). The MaxOut template matching and pooling operation (from Proposition A.1) for a set of translational nuisance variables
H ≡ GT reduces to the traditional DCN convolution and max-pooling operation.
Proof. Let the activation for a single output unit be yc (I). Then we have
yc (I) ≡ max {wch ?LC I}
h∈H
= max {hwcg |Ii}
g∈GT
= max {hTg wc |Ii}
g∈GT
= max {hwc |T−g Ii}
g∈GT
= max {(wc ?DCN I)g }
g∈GT
= MaxPool(wc ?DCN I).
Finally, vectorizing in c gives us the desired result y(I) = MaxPool(W ?DCN I).
Proposition A.3 (Max Pooling DCNs with ReLu Activations). The discriminative relaxation
of a noise-free GRM with translational nuisances and random missing data is a single convolutional layer of a traditional DCN. The layer consists of a generalized convolution operation,
followed by a ReLu activation function and a Max-Pooling operation.
Proof. We will model completely random missing data as a nuisance transformation a ∈ A ≡
{keep, drop}, where a = keep = 1 leaves the rendered image data untouched, while a =
drop = 0 throws out the entire image after rendering. Thus, the switching variable a models
missing data. Critically, whether the data is missing is assumed to be completely random and
thus independent of any other task variables, including the measurements (i.e. the image itself).
Since the missingness of the evidence is just another nuisance, we can invoke Proposition A.1
to conclude that the discriminative relaxation of a noise-free GRM with random missing data is
also a MaxOut-DCN, but with a specialized structure which we now derive.
47
Mathematically, we decompose the nuisance variable h ∈ H into two parts h = (g, a) ∈
H = G × A, and then, following a similar line of reasoning as in Proposition A.1, we have
ĉ(I) = argmax max p(c, h|I)
h∈H
c∈C
= argmax max {wch ?LC I}
h∈H
c∈C
(a)
0
0
= argmax max max a(hwcg |Ii + bcg ) + bcg + ba + bI
g∈G a∈A
c∈C
(b)
0
0
0
= argmax max {max{(wc ?DCN I)g , 0} + bcg + bdrop + bI }
g∈G
c∈C
(c)
0
= argmax max {max{(wc ?DCN I)g , 0} + bcg }
g∈G
c∈C
(d)
= argmax max {max{(wc ?DCN I)g , 0}}
c∈C
g∈G
= Choose {MaxPool(ReLu(DCNConv(I)))}
= DCN(I; θ).
In line (a) we calculated the log-posterior
ln p(c, h|I) = ln p(c, g, a|I)
= ln p(I|c, g, a) + ln p(c, g, a)
1
1
= 2 haµcg |Ii − 2 (kaµcg k22 + kIk22 )) + ln p(c, g, a)
2σx
2σx
≡ a(hwcg |Ii + bcg ) + b0cg + ba + b0I ,
where a ∈ {0, 1}, ba ≡ ln p(a), b0cg ≡ ln p(c, g), b0I ≡ − 2σ1 2 kIk22 . In line (b), we use Lemma A.2
x
to write the expression in terms of the DCN convolution operator, after which we invoke the
identity max{u, v} = max{u − v, 0} + v ≡ ReLu(u − v) + v for real numbers u, v ∈ R. Here
we’ve defined b0drop ≡ ln p(a = keep) and we’ve used a slightly modified DCN convolution
p(a=keep)
operator ?DCN defined by wcg ?DCN I ≡ wcg ? I + ln p(a=drop) . Also, we observe that all
the primed constants are independent of a and so can be pulled outside of the maxa . In line(c),
the two primed constants that are also independent of c, g can be dropped due to the argmaxcg .
Finally, in line (d), we assume a uniform prior over c, g. The resulting sequence of operations
corresponds exactly to those applied in a single convolutional layer of a traditional DCN.
Remark A.4 (The Probabilistic Origin of the Rectified Linear Unit). Note the origin of
the ReLu in the proof above: it compares the relative (log-)likelihood of two hypotheses
48
a = keep and a = drop, i.e. whether the current measurements (image data I) are available/relevant/important or instead missing/irrelevant/unimportant for hypothesis (c, g). In this
way, the ReLu also promotes sparsity in the activations.
A.2
Generalizing to Arbitrary Mixtures of Exponential Family
Distributions
In the last section, we showed that the GRM – a mixture of Gaussian Nuisance Classifiers –
has as its discriminative relaxation a MaxOut NN. In this section, we generalize this result to an
arbitrary mixture of Exponential family Nuisance classifiers. For example, consider a Laplacian
RM (LRM) or a Poisson RM (PRM).
Definition A.5 (Exponential Family Distributions). A distribution p(x; θ) is in the exponential
family if it can be written in the form
p(x; θ) = h(x) exp(hη(θ)|T (x)i − A(η)),
where η(θ) is the vector of natural parameters, T (x)is the vector of sufficient statistics,
A(η(θ)) is the log-partition function.
By generalizing to the exponential family, we will see that derivations of the discriminative
relations will simplify greatly, with the key roles being played by familiar concepts such as natural parameters, sufficient statistics and log-partition functions. Furthermore, most importantly,
we will see that the resulting discriminative counter parts are still MaxOut NNs. Thus MaxOut
NNs are quite a robust class, as most E-family mixtures have MaxOut NNs as d-counterparts.
Theorem A.6 (Discriminative Counterparts to Exponential Family Mixtures are MaxOut Neural Nets). Let Mg be a Nuisance Mixture Classifier from the Exponential Family. Then the
discriminative counterpart Md of Mg is a MaxOut NN.
Proof. The proof is analogous to the proof of Proposition A.1, except we generalize by using
the definition of an exponential family distribution (above). We simply use the fact that all
exponential family distributions have a natural or canonical form as described above in the
Definition A.5. Thus the natural parameters will serve as generalized weights and biases, while
the sufficient statistic serves as the generalized input. Note that this may require a non-linear
transformation i.e. quadratic or logarithmic, depending on the specific exponential family.
A.3
Regularization Schemes: Deriving the DropOut Algorithm
Despite the large amount of labeled data available in many real-world vision applications of
deep DCNs, regularization schemes are still a critical part of training, essential for avoiding
49
overfitting the data. The most important such scheme is DropOut (30) and it consist of training
with unreliable neurons and synapses. Unreliability is modeled by a ‘dropout’ probability pd
that the neuron will not fire (i.e. output activation is zero) or that the synapse won’t send its
output to the receiving neuron. Intuitively, downstream neurons cannot rely on every piece of
data/evidence always being there, and thus are forced to develop a robust set of features. This
prevents the co-adaptation of feature detectors that undermines generalization ability.
In this section, we answer the question: Can we derive the DropOut algorithm from the
generative modeling perspective? Here we show that the answer is yes. Dropout can be derived from the GRM generative model via the use of the EM algorithm under the condition of
(completely random) missing data.
Proposition A.7. The discriminative relaxation of a noise-free GRM with completely random
missing data is a DropOut DCN (18) with Max-Pooling.
Proof. Since we have data that is missing completely at random, we can use the EM algorithm
to train the GRM (56). Our strategy is to show that a single iteration of the EM-algorithm
corresponds to a full epoch of DropOut DCN training (i.e. one pass thru the entire dataset).
Note that typically an EM-algorithm is used to train generative models; here we utilize the EMalgorithm in a novel way, performing a discriminative relaxation in the M-step. In this way, we
use the generative EM algorithm to define a discriminative EM algorithm (d-EM).
The d-E-step is equivalent to usual generative E-step. Given the observed data X and the
current parameter estimate θ̂t , we will compute the posterior of the latent variables Z = (H, A)
where A is the missing data indicator matrix i.e. Anp = 1 iff the p-th feature (e.g. pixel
intensity) of the input data In (e.g. natural image) is available. H contains all other latent
nuisance variables (e.g. pose) that are important for the classification task. Since we assume a
noise-free GRM, we will actually execute a hybrid E-step: hard in H and soft in A. The hard-E
step will yield the Max-Sum Message Passing algorithm, while the soft E-step will yield the
ensemble average that is the characteristic feature of Dropout (18).
In the d-M-step, we will start out by maximizing the complete-data log-likelihood
`(θ; H, A, X), just as in the usual generative M-step. However, near the end of the derivation we will employ a discriminative relaxation that will free us from the rigid distributional
assumptions of the generative model θg and instead leave us with a much more flexible set of
assumptions, as embodied in the discriminative modeling problem for θd .
Mathematically, we have a single E-step and M-step that leads to a parameter update as
50
follows:
n
o
`(θ̂new ) ≡ max EZ|X [`(θ; Z, X)]
θ
n
o
= max EA EH|X [`(θ; H, A, X)]
θ
n
h
io
= max EA EH|X `(θ; C, H|I, A) + `(θ; I) + `(θ; A)
θ
n
h
io
= max EA EH|X `(θd ; C, H|I, A) + `(θg ; I)
θd ∼ d θg
n
h
io
≤ max EA EH|X `(θd ; C, H|I, A)
θd
n
h
io
= max EA MH|X `(θd ; C, H|I, A)
θd
io
n h
∗
≡ max EA `(θd ; C, H |I, A)
θd
nX
o
∗
= max
p(A) · `(θd ; C, H |I, A)
θd
≈ max
θd
= max
θd
A
nX
A∈T
nX
A∈T
o
p(A) · `(θd ; C, H |I, A)
∗
p(A) ·
X
dropout
n∈DCI
o
ln p(cn , h∗n |Indropout ; θd ) .
Here we have defined the conditional likelihood `(θ; D1 |D2 ) ≡ ln p(D1 |D2 ; θ), and D =
(D1 , D2 ) is some partition of the data. This definition allows us to write `(θ; D) =
`(θ; D1 |D2 ) + `(θ; D2 ) by invoking the conditional probability law p(D|θ) = p(D1 |D2 ; θ) ·
p(D2 |θ). The symbol MH|X [f (H)] ≡ maxH {p(H|X)f (H)} and the reduced dataset
dropout
DCI
(A) is simply the original dataset of labels and features less the missing data (as specified
by A).
The final objective function left for us to optimize is a mixture of exponentially-many discriminative models, each trained on a different random subset of the training data, but all sharing
parameters (weights and biases). Since the sum over A is intractable, we approximate the sums
by Monte Carlo sampling of A (the soft part of the E-step), yielding an ensemble E ≡ {A(i) }.
The resulting optimization corresponds exactly to the DropOut algorithm.
51
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56
| 9 |
Latent Factor Analysis of Gaussian Distributions
under Graphical Constraints
arXiv:1801.03481v3 [cs.IT] 20 Jan 2018
Md Mahmudul Hasan, Shuangqing Wei, Ali Moharrer
Abstract—In this paper, we explored the algebraic structures
of solution spaces for Gaussian latent factor analysis when the
population covariance matrix Σx is generated due to a latent
Gaussian star graph. In particular, we found sufficient and
necessary conditions under which the solutions to constrained
minimum trace factor analysis (CMTFA) and constrained minimum determinant factor analysis (CMDFA) is still star. The
later one (CMDFA) is also the problem of Wyner’s common
information, which has been under extensive study in recent
years. In addition, we further showed that the solution to CMTFA
under the star constraint can only have two cases, i.e. the number
of latent variable can be only one (star) or n − 1, where n is the
dimension of the observable vector.
Index Terms—Factor Analysis, MTFA, CMTFA, CMDFA
I. INTRODUCTION
Factor Analysis (FA) is a commonly used tool in multivariate statistics to represent the correlation structure of a set
of observables in terms of significantly smaller number of
variables called “latent factors”. With the growing use of data
mining, high dimensional data and analytics, factor analysis
has already become a prolific area of research [1] [2].
In traditional factor analysis of real n-dimensional random
vector X ∈ Rn with Gaussian distribution N (0, Σx ), where
Σx is the known population covariance matrix, the objective
is to decompose Σx as Σx = (Σx − D) + D, where Σx − D is
Gramian and D is diagonal. Classical approaches to solve this
problem are Minimum Rank Factor Analysis (MRFA) [3] and
Minimum Trace Factor Analysis (MTFA) [4]. As the name
suggests MRFA seeks to minimize the rank or Σx − D and
MTFA minimizes the trace of Σx − D. MTFA solution could
lead to negative values for the diagonal entries of the matrix
D. To solve this problem Constrainted Minimum Trace Factor
Analysis (CMTFA) was proposed [5], that imposes extra constraint of requiring D to be Gramian. Computational aspects
of CMTFA and uniqueness of its solution was discussed in
[6].
Since X is Gaussian random vecor, factor analysis of X
can be modelled by the following equation,
X = AY + Z
(1)
where An×k is a real matrix, Yk×1 , k < n is the vector of
independent latent variables and Zn×1 is a Gaussian vector of
zero mean and covariance matrix Σz = D. We have,
I(X; Y) = H(X) − H(X|Y) = H(X) − H(Z)
(2)
1 Md M Hasan, S. Wei and A. Moharrer are with the school of Electrical Engineering and Computer Science, Louisiana State University, Baton
Rouge, LA 70803, USA (Email: mhasa15@lsu.edu, swei@lsu.edu, alimoharrer@gmail.com).
where I(X; Y) is the mutual information between X and
Y, H(X), H(Z) are entrophies of X and Z and H(X|Y)
is the conditional entrophy of X given Y. Characterizing
the common information between X and Y [7] [8] [9]
minA,Σz I(X; Y) is an equivalent problem to maxZ H(Z) .
Moharrer and Wei in [10] established relationship between
the common information problem and MTFA, and named the
problem Constrained Minimum Determinant Factor Analysis
(CMDFA), because minA,Σz I(X; Y) for the above model is
equivalent to minz −log|Σz |.
In [11] the same condition was found on the subspace of Σx
for MTFA solution to be a star as the one we give through this
paper for CMTFA solution to be a star. For clarification, to
make a star Σx must be decomposable as a sum of a rank
1 matrix plus a diagonal matrix The condition they found
for MTFA was only a sufficient condition, we proved the
same condition both sufficient and necessary for CMTFA.
Another major difference between their work and ours is that
we also characterized the solution of CMTFA and analysed
the conditions on Σx when the CMTFA solution of Σx is not
a star, which they did not address in their MTFA solution.
In this paper we generate a certain family by requiring Σx
has a latent star structrure as shown in (59). We analysed the
solution space of both CMTFA and CMDFA for Σx . The novel
contribution of this paper is we prove that there are only two
solutions possible for CMTFA, one of which we prove is still
a star, which means the optimal solution has the dimension
k = 1. We also give the necessary and sufficient condition for
the structure of the solution when it is not a star. For CMDFA
we give the if and only if condtion for the solution to be a
star.
The rest of the paper is organized as follows: section II
gives the necessary and sufficient conditions for two possible
CMTFA solutions of Σx . Section III gives if and only if
condition for CMDFA solution of Σx to be a star. The last
section has the conclusion.
II. S OLUTION
TO
CMTFA P ROBLEMS
C ONSTRAINT
UNDER A
S TAR
In line with the affine model in (1), we impose the following
prior constraints on the generation of X ∈ Rn .
X1
α1
Z1
.. .. ..
(3)
. = . Y + .
Xn
where
αn
Zn
•
•
•
Y ∼ N (0, 1)
0 < |αj | < 1, j = 1, 2, . . . , n.
{Zj } are independent Gausian random varables with
Zj ∼ N (0, 1 − α2j )
We denote the real column vector α
~ = [α1 , . . . , αn ]′ ∈ Rn .
Using (59) we have,
′
Σx = α
~α
~ + Σz
(4)
If for any element of α
~ the following condition holds, we-call
it a non-dominant element, otherwise its a dominant element.
X
|αi | ≤
|αj |
i = 1, 2, . . . , n
(5)
j6=i
It is easy to see that there can be only one dominant element
in a vector and that has to be the element with the biggest
absolute value among all. We call α
~ dominant if its biggest
element in terms of absolute value is dominant, otherwize α
~
is non-dominant.
Let us, without the loss of generality, assume that α1 has
the largest absolute value and thus all dominance is defined
with respect to it.
The following necessary and sufficient condition for
CMTFA solution was set in [12],
The point d∗ is a solution of the CMTFA problem if and
only if λ(d∗ ) = 0, d∗ ≥ 0 and there exist ~ti ∈ N (Σx −
D∗ ), i = 1, ...., r such that the following holds,
1=
r
X
i=1
~t2i −
X
µj ξ~j
(6)
j∈I(d∗ )
where r ≤ n, 1 is n dimensional column vector with all the
components equal to 1, {~ti ∈ N (Σx − D∗ ), i = 1, ...., r}
are n dimensional column vectors forming the rank (n − k)
matrix T , ~t2i is the Hadamard product of vector ~ti with itself,
λ(d∗ ) is the minimum eigenvalue of (Σx − D∗ ), d∗ is the
vector having all the diagonal entries of the diagonal matrix
D∗ , I(d∗ ) = {i : d∗i = 0, i ≤ n}, {µj , j ∈ I(d∗ )} are
non-negative numbers and {ξ~j , j ∈ I(d∗ )} are column vectors
in Rn with all the components equal to 0 except for the jth
component which is equal to 1.
We have proved that if we apply CMTFA to Σx in (7) then
the solution is either the rank 1 matrix Σt,N D given in (8) or
the rank n − 1 matrix Σt,DM given in (9).
1
α1 α2 . . . α1 αn
α2 α1
1
. . . α2 αn
Σx = .
(7)
..
..
..
..
.
.
.
αn α1
Σt,N D
α21
α2 α1
= .
..
αn α1
αn α2
...
α1 α2
α22
..
.
...
...
..
.
αn α2
...
1
α1 αn
α2 αn
..
.
α2n
(8)
Σt,DM
(Σt,DM )11
α2 α1
=
..
.
α1 α2
(Σt,DM )22
..
.
...
...
..
.
α1 αn
α2 αn
..
.
αn α2
...
(Σt,DM )nn
αn α1
where
(Σt,DM )11 = |α1 |
X
i6=1
(9)
|αi |
(Σt,DM )ii = |αi | |α1 | −
X
j6=i,1
|αj | ,
i = 2, . . . , n
In next two subsections we present the analytical details of
those two solutions in terms of vector α
~ being domainant or
non-dominant.
A. Dominant Case
In this subsection we analyse the conditions under which the
CMTFA solution of Σx is not a star. Following two Lemmas
are essential to understand the Theorem to follow.
Lemma 1. Σt,DM is a rank n − 1 matrix.
Pn
We basically showed that,
i=1 si (Σt,DM )i = 0 where,
(Σt,DM )i is the ith row of Σt,DM and si ∈ {1, −1}. The
detailed proof is given in Appendix D.
Lemma 2. There exists a column
[Φ1 , Φ2 , ...., Φn ]′ such that Σt,DM Φ
Φi ∈ {−1, 1}, 1 ≤ i ≤ n.
vector Φ
=
=
0, where
In the two chambered proof of Lemma 2 we first find a
vector Φ, Φi ∈ {1, −1} such that (Σt,DM )1 Φ = 0. Then
we show that the same vector is orthogonal to the other rows
of Σt,DM as well. The detailed proof is given in Appendix E.
Theorem 1. Σt,DM is the CMTFA solution of Σx if and only
if α
~ is dominant.
Proof of Theorem 1. To prove the Theorem we refer to necessary and sufficient condition set in (6). Rank of Σt,DM is
n−1, so its minimum eigenvalue is 0. Since each 0 < |αi | < 1,
0 < (Σt,DM )ii ) < 1, i = 1, . . . , n. Hence all the diagonal
entries di of D are positive. As a result, the set I(d∗ ) is empty
and the second term in the right hand side of (6) vanishes.
The dimension of the null space of Σt,DM is 1. It will
suffice for us to prove the existence of a column vector Φn×1 ,
Φi ∈ {1, −1}, 1 ≤ i ≤ n such that Σt,DM Φ = 0. Lemma 2
gives that proof.
B. Non-Dominant Case
This subsection is dedicated to the analytical details of the
conditions under which the CMTFA solution of star structured
Σx is also star. The following Lemma is essential to prove the
Theorem to follow.
Lemma 3. There exists rank n − 1 matrix Tn×n such that the
column vectors of T are in the null space of Σt,N D and the
L2 -norm of each row of T is 1.
Proof of Lemma 3. Its trivial to find the following basis
vectors for the null space of Σt,N D ,
α2
− α1
1
v1 = 0 ,
.
..
0
α3
−α
1
0
v2 = 1 , . . . ,
.
..
0
We define matrix V as,
α3
. . . − ααn1
− α2 − α
1
α1
1
0
...
0
0
1
.
.
.
0
V =
..
..
..
.
.
.
.
.
.
0
0
...
1
vn−1
αn
− α1
0
= 0
.
..
1
(10)
α2
αn
− c2 α
+
·
·
·
+
c
n
α1
1
c2
c3
..
.
cn
(11)
The columns of V given in (11) span the null space of
Σt,N D . To prove the lemma, it will suffice for us to find a
diagonal matrix Bn×n such that the following holds.
Tn×n = Vn×n Bn×n
(12)
where, L2 -norm of each row of T is 1. Using (12),
T T ′ = V BB ′ V ′
(13)
We define the symmetric matrix β = BB ′ , and we require
the diagonal matrix β to have only non-negative entries.
Since we want each diagonal element of T T ′ to be 1, we
have the following n equations,
α2
α2
α22
β11 + 23 β22 + · · · + n2 βn−1,n−1 +
2
α1
α1
α1
2
α2
α3
αn
c2
+ c3
+ · · · + cn
βnn = 1
α1
α1
α1
βii + c2i+1 βnn = 1,
i = 1, . . . , n − 1
(14)
(15)
Solving, (14) we get,
βnn =
α21 − α22 − α23 − · · · − α2n
P
i6=j,i6=1,j6=1 ci cj αi αj
(16)
Since the diagonal entries of β can only be non-negative,
we have the following three cases.
α21 − α22 − α23 − · · · − α2n = 0
α21 − α22 − α23 − · · · − α2n > 0
α21 − α22 − α23 − · · · − α2n < 0
It will suffice for us to prove that for all of the above cases
there exist {cj , 2 ≤ j ≤ n} that make βii ≥ 0, i = 1, . . . , n.
Case 1: is straightforward. If α21 − α22 − α23 − · · · − α2n = 0
Then using (14) and (15) we get, βnn = 0 and β11 = β22 =
· · · = βn−1,n−1 = 1.
Before we move on to the remaining two cases, without the
loss of generality, we can re-arrange the elements of α
~ such
that,
|α1 | ≥ |α2 | ≥ · · · ≥ |αn |
(17)
Normalizing each element by |α1 | gives us the following,
1 ≥ |e
α2 | ≥ |e
α3 | ≥ · · · ≥ |e
αn |
where α
ej =
αj
α1 , 1
(18)
≤ j ≤ n. We define,
Smin = min
A
X
j∈A
|e
αj | −
X
j∈Ac
|e
αj |
(19)
where A ⊂ {2, 3, . . . , n} and Ac = {2, 3, . . . , n} − A
Let, of all the possibilities A = A∗ be the subset of
{2, 3, . . . , n} that gives us Smin . Assuming that A∗ has l
elements, let the set A∗ be A∗ = {a1 , a2 , ..., al }. We define
the set F as,
F = {Fa1 , . . . , Fal ,
αai |, ai ∈ A∗ }
Fai = |e
Now under this ordered and normalized settings, we have
Case 2: 1 − α
e22 − α
e23 − · · · − α
e2n > 0.
We can select c2 , c3 , . . . , cn in a way such that ci α
ei = |e
αi |
to make βnn > 0. Equation (15) dictates that to ensure the
other diagonal entries of β are non-negative, the following
must hold,
1−α
e2 − α
e23 − · · · − α
e2n
P 2
≤1
ei α
ej
i6=j,i6=1,j6=1 ci cj α
⇒1 ≤ (|e
α2 | + |e
α3 | + · · · + |e
αn |)2
⇒1 ≤ |e
α2 | + |e
α3 | + · · · + |e
αn |
(20)
which means such βnn exists if and only if α
e1 is non-dominat.
Because of the ordered representation, that essentially means
α
~ has to be non-dominant.
Case 3: 1 − α
e2 − · · · − α
e2n < 0
Using the lemma we proved in Appendix
C of this paper, if
Pn
we
select
c
∈
{1,
−1}
such
that
c
α
ej = Smin then,
i
j
j=2
P
c
c
α
e
α
e
<
0.
And
for
such
selection
of ci we
i
j
i
j
i6=j,i6=1,j6=1
have,
X
α
e22 + α
e23 + · · · + α
e2n +
ci cj α
ei α
ej
=
n
X
i=2
ci α
ei
!2
i6=j,i6=1,j6=1
2
= Smin
≤1
(21)
The last inequality is due to the lemma we proved in Appendix
A of this paper, that shows Smin ≤ Fai , ai ∈ A∗ . So, we have,
X
α
e22 + α
e23 + · · · + α
e2n +
ci cj α
ei α
ej ≤ 1
i6=j,i6=1,j6=1
⇒1 −
α
e22
−
α
e23
− ···−
α
e2n
≥
X
i6=j,i6=1,j6=1
ci cj α
ei α
ej
e22 − α
e23 − · · · − α
e2n
PBoth the terms 1 − α
ei α
ej are negative. Hence,
i6=j,i6=1,j6=1 ci cj α
1−α
e2 − α
e23 − · · · − α
e2n
βnn = P 2
≤1
ei α
ej
i6=j,i6=1,j6=1 ci cj α
and
i
(22)
Theorem 2. Σt,N D is the CMTFA solution of Σx if and only
if α
~ is non-dominant.
The theorem states that the CMTFA solution to a star
connected network is a star itself if and only if there is no
dominant element in the vector α
~.
Proof of Theorem 2:. We still use the same necessary and
sufficient condition set in (6). Σt,N D is rank 1, so its minimum
eigenvalue is 0. Since each 0 < |αi | < 1, 1 − α2i > 0, 1 ≤ i ≤
n. As a result the set I(d∗ ) is empty. So, the second term on
the right side of (6) vanishes.
The dimention of the null space of Σt,N D is n − 1. Lemma 3
proves that there exists rank n − 1 matrix Tn×n such that the
column vectors of T are in the null space of Σt,N D and the
L2 -norm of each row of T is 1. That essentially completes the
proof.
III. C ONDITIONS UNDER WHICH CMDFA S OLUTION
Σx P RODUCES A S TAR
Lemma 4. There exists rank n − 1 matrix Tn×n such that the
column vectors of T are in the null space of Σt,N D and the
1
L2 -norm of the ith row of T is 1−α
2 , 1 ≤ i ≤ n.
Proof of Lemma 4:. Since, it is the same Σt,N D as was the
solution for CMTFA non-dominant case, the basis vectors for
the null space remain the same v1 , v2 , . . . , vn . The matrix V
remain the same except for ci s. Here we define them as,
e
ci
,
ci = p
1 − α2i
i = 2, . . . , n
(24)
where e
ci ∈ {1, −1}.
Still the columns of V with the newly defined ci s, span the
null space of Σt,N D . To prove this Lemma, it will suffice for
us to find a diagonal matrix Bn×n such that the following
holds.
Tn×n = Vn×n Bn×n
where the L2 -norm of the ith row of T is
(25)
1
.
1−α2i
Using (25),
T T ′ = V BB ′ V ′ = V βV ′
(26)
Like before, we require the diagonal matrix β to have only
non-negative entries. Based on the conditions imposed on the
matrix T , we have the following n equations,
OF
This section analyses the conditions under which the
CMDFA solution of Σx is a star. The definition of dominance
and non-dominance slightly differs in CMDFA than it was
in CMTFA. We defne the real column vector, θ~ ∈ Rn as
~
θ = [θ1 , . . . , θn ]′ , θi = √ αi 2 , 1 ≤ i ≤ n. We define
1−αi
the element θi to be non-dominant, if equation (23) holds,
otherwise θi is dominant.
X
|θi | ≤
|θj |
i = 1, 2, . . . , n
(23)
α22
α2
α2
β11 + 23 β22 + · · · + n2 βn−1,n−1 +
2
α1
α1
α1
2
α3
αn
1
α2
+ c3
+ · · · + cn
βnn =
c2
α1
α1
α1
1 − α21
βii + c2i+1 βnn =
i = 1, . . . , n − 1
(28)
Solving, (27) with the help of (28) we get,
j6=i
In CMTFA the dominance was defined in terms of individual
vector component, in CMDFA it is defined
√ in terms of the
square root of their signal to noise ratio ( SNR). Note that
there can be only one dominant element in vector θ~ and that
has to be the element with the biggest absolute value among
all. We call ~θ dominant if its biggest element in terms of
absolute value is dominant, otherwize ~
θ is non-dominant.
Let us, without the loss of generality, assume θ1 has the
largest absolute value and thus all dominance is defined with
respect to it. The following necessary and sufficient condition
for CMDFA solution was set in [10].
The point d∗ is a solution of the CMDFA problem if and
only if λ(d∗ ) = 0, and their exists matrix T such that its
column vectors are in the null space of (Σx − D∗ ) and the L2
1
∗
norm of the ith row of T is 1−α
2 , where λ(d ) is the minimum
i
∗
∗
eigenvalue of (Σx − D ) and d is the vector having all the
diagonal entries of the diagonal matrix D∗ .
The following Lemma is needed to prove the Theorem to
follow.
1
,
1 − α2i+1
(27)
βnn =
α21
1−α21
P
−
α22
1−α22
− ···−
α2n
1−α2n
i6=j,i6=1,j6=1 ci cj αi αj
θ12 − θ22 − · · · − θn2
= P
ci e
c j θi θj
i6=j,i6=1,j6=1 e
(29)
To ensure that the diagonal entries of β are non-negative, we
need to consider the following three cases.
θ12 − θ22 − · · · − θn2 = 0
θ12 − θ22 − · · · − θn2 > 0
θ12 − θ22 − · · · − θn2 < 0
It will suffice for us to prove that for all of the above cases
there exist {e
ci , 2 ≤ i ≤ n} that make βii ≥ 0, i = 1, . . . , n.
Case 1: is straightforward. If θ12 − θ22 − · · · − θn2 = 0
Then using (27) and (28) we get, βnn = 0 and βii =
1
, i = 1, . . . , (n − 1).
1−α2i+1
Before we move on to other two cases, without the loss of
generality, we can arrange the elements of θ~ such that,
|θ1 | ≥ |θ2 | ≥ · · · ≥ |θn |
(30)
Normalizing each element by |θ1 | gives us the following,
where, θej =
We define,
1 ≥ |θe2 | ≥ |θe3 | ≥ · · · ≥ |θen |
θj
θ1 , 1
(31)
≤ j ≤ n.
Smin = min
A
X
j∈A
|θej | −
X
j∈Ac
|θej |
(32)
where A ⊂ {2, 3, . . . , n} and Ac = {2, 3, . . . , n} − A.
Let, of all the possibilities A = A∗ be the subset of
{2, 3, . . . , n} that gives us Smin from (32). Assuming that
A∗ has l elements, let the set A∗ be A∗ = {a1 , a2 , ..., al }. We
define the set F as,
F = {Fa1 , . . . , Fal ,
Fai = |θeai |, ai ∈ A∗ }
Now, under the above ordered and normalized settings,
Case 2: 1 − θe22 − θe32 − · · · − θen2 > 0
We can select e
c2 , e
c3 , . . . , e
cn in a way such that e
ci θei = |θei |
to make βnn > 0. Equation (28) dictates that to ensure the
other diagonal entries of β are non-negative, the following
must hold,
1 − θe22 − θe32 − · · · − θen2
≤1
P
ci e
cj θei θej
i6=j,i6=1,j6=1 e
⇒1 ≤ (|θe2 | + |θe3 | + · · · + |θen |)2
⇒1 ≤ |θe2 | + |θe3 | + · · · + |θen |
=
i=2
cei θei
!2
(33)
i6=j,i6=1,j6=1
2
= Smin
≤1
(34)
The last inequality is due to the lemma we proved in Appendix
A of this paper. So, we have,
X
θe22 + θe32 + · · · + θen2 +
e
ci e
cj θei θej ≤ 1
i6=j,i6=1,j6=1
⇒1 − θe22 − θe32 − · · · − θen2 ≥
X
i6=j,i6=1,j6=1
e
ci e
cj θei θej
terms 1 − θe22 − θe32 − · · · − θen2
ci e
cj θei θej are negative. Hence,
i6=j,i6=1,j6=1 e
PBoth
the
βnn
1 − θe2 − θe32 − · · · − θen2
= P 2
≤1
ci e
cj θei θej
i6=j,i6=1,j6=1 e
The theorem states that the CMDFA solution to a star
connected network is a star itself if and only if there is no
dominant element in the vector θ.
Proof of Theorem 3:. Now we refer back to the necessary
and sufficient condition for CMDFA solution at the begining of
this section. Since, Σt,N D in rank 1, it’s minimum eigenvalue
is 0. And Lemma 4 proves the existance of rank n − 1 matrix
Tn×n such that the column vectors of T are in the null space
1
of Σt,N D and the L2 -norm of the ith row of T is 1−α
2,1 ≤
i
i ≤ n. That completes the proof of Theorem 3.
IV. C ONCLUSION
In this paper we characterized the solution space of both
CMTFA and CMDFA. We showed that the CMTFA solution
of a star structured population matrix can have either a rank 1
or a rank n − 1 solution and nothing in between. We proved
both the solutions with sufficient and necessary conditions. We
established the necessary and sufficient conditons for CMDFA
solution to a star structured population matrix to be a star.
A PPENDIX A
Which means such βnn exists if and only if θe1 is non-dominat.
Because of the ordered representation, that essentially means
the vector ~θ has to be non-dominant.
Case 3, 1 − θe2 − · · · − θen2 < 0
Based on the proof in Appendix
PnC of ethis paper, if we
select e
ci ∈ {1, −1} such that
cj θj = Smin then,
j=2 e
P
e
e
ci e
cj θi θj < 0.
i6=j,i6=1,j6=1 e
And for such selection of e
ci we have,
X
θe22 + θe32 + · · · + θen2 +
ci e
e
cj θei θej
n
X
Theorem 3. CMDFA solution of Σx is Σt,N D if and only if
~θ is non-dominant.
Let e1 , e2 , . . . , en be a set of n positive numbers.
We define,
Smin = min
A
i∈A
ei −
X
ej
(36)
j∈Ac
where A ⊂ {1, 2, 3, . . . , n} and Ac = {1, 2, 3, . . . , n} − A
Let of all the possibilities A∗ be the subset of
{1, 2, 3, . . . , n} that gives us Smin . Assuming the set A∗ has
l elements, let the sets A∗ and (A∗ )c be A∗ = {a1 , a2 , ..., al }
and (A∗ )c = {ac1 , ac2 , ..., acn−l }. Let F and G be following
two sets,
F = {Fa1 , . . . , Fal , Fai = eai , ai ∈ A∗ }
G = {Gac1 , . . . , Gacn−l , Gaci = eaci , aci ∈ (A∗ )c }
We define,
M + Smin =
X
Fai ,
ai ∈A∗
1
(M + Smin ),
l
= min∗ Fai
Favg =
and
X
Fmin
ai ∈A
M=
X
Gaci
(37)
aci ∈(A∗ )c
Gavg =
1
M
n−l
(38)
(39)
Lemma 5. Smin ≤ Fmin
(35)
Proof of Lemma 5:. Let us assume Fmin < Smin and has
the value Fmin = Smin − ǫ where 0 < ǫ < Smin .
Now, If we deduct Fmin from set F and add it to the set
G, then we will have,
If
Pp−1
g=1
X
|(M + Smin − Fmin ) − (M + Fmin )| = |Smin − 2Fmin |
i6=j
= |Smin − 2ǫ|
< Smin
which is not possible. So, Fmin ≥ Smin .
Lemma 6. For any set of positive numbers e1 , e2 , . . . , en ,
X
ei ej ≤ n(n − 1)e2avg
(40)
i6=j
1
n
Pn
i=1 ei .
Proof of Lemma 6:. Without the loss of generality, we can
write the set of numbers in terms of their average in the
following way: eavg + k1 , eavg + k2 , . . . , eavg + kp , eavg −
j1 , eavg − j2 , . . . , eavg − jq , where, p + q = n, ki ≥ 0, ji ≥ 0.
P
P
It is straightforward to see, pi=1 ki = qi=1 ji . We define,
ψ1 =(eavg + k1 ) [(eavg + k2 ) + · · · + (eavg + kp )+
(eavg − j1 ) + (eavg − j2 ) + · · · + (eavg − jq )]
=(eavg + k1 ) [(p + q − 1)eavg + (k2 + · · · + kp )−
(j1 + · · · + jq )]
(41)
ψ2 =(eavg + k2 ) [(p + q − 2)eavg + (k3 + · · · + kp )
..
.
−(j1 + · · · + jq )]
(42)
ψp−1 =(eavg + kp−1 ) [(q + 1)eavg + kp − (j1 + · · · + jq )]
(43)
ψp =(eavg + kp ) [qeavg − (j1 + · · · + jq )]
ψp+1 =(eavg − j1 ) [(q − 1)eavg − (j2 + · · · + jq )]
..
.
ψp+q−1 =(eavg − jq−1 ) [eavg − jq ]
X
i6=j
Pp
h=g+1
Pp−1
g=1
kg
Pp
h=g+1
Smin = min
A
−eavg (p + q − 1)
+
q−1
X
g=1
jg
q
X
h=g+1
jh −
q
X
ji
i=1
!
p
X
i=1
+
ki
kg
g=1
!
q
X
i=1
p
X
i=1
ki
!
i∈A
F = {Fa1 , . . . , Fal ,
We define,
Fmin
h=g+1 jh
q
X
i=1
!2
ji
(49)
ei −
X
ej
(50)
j∈Ac
X
Fai ,
Fai = eai , ai ∈ A∗ }
Gaci = eaci , aci ∈ (A∗ )c }
M=
X
Gaci
(51)
aci ∈(A∗ )c
∈A∗
1
(M + Smin ),
l
= min∗ Fai
Gavg =
1
M
n−l
ai ∈A
(52)
(53)
Lemma 7. If we select {cj }nj=1 , cj ∈ {1, −1} such that,
n
X
kh
cj ej = Smin
(54)
ci cj e i e j < 0
(55)
j=1
h=g+1
!
ji
(48)
where A ⊂ {1, 2, 3, . . . , n} and Ac = {1, 2, 3, . . . , n} − A
Let of all the possibilities A∗ be the subset of
{1, 2, 3, . . . , n} that gives us Smin . Assuming the set A∗ has
l elements, let the sets A∗ and (A∗ )c be A∗ = {a1 , a2 , ..., al }
and (A∗ )c = {ac1 , ac2 , ..., acn−l }. Let F and G be following
two sets,
Favg =
p−1
X
X
G = {Gac1 , . . . , Gacn−l ,
p
X
g=1 jg
Pq
i=1
!2
ki
Let e1 , e2 , . . . , en be a set of n positive numbers.
We define,
M + Smin =
= 2 (1 + · · · + (p + q − 1))e2avg + eavg (p + q − 1)
kh <
Pq−1
p
X
A PPENDIX C
ei ej = 2[ψ1 + · · · + ψp+q−1 ]
"
h=g+1 jh
i6=j
(44)
(45)
(46)
Pq
Combining (48) and (50) we have,
X
ei ej ≤ n(n − 1)e2avg
ai
i6=j
g=1 jg
q−1
q
X
X
n(n − 1) 2
jg
eavg + 2
jh −
ei ej ≤ 2
2
g=1
h=g+1
"
#
q
X
n(n − 1) 2
2
=2
ji
eavg −
2
i=1
Using the above equations,
X
Pq−1
kh ≥
p−1
p
X
X
n(n − 1) 2
ei ej ≤ 2
kg
eavg + 2
km −
2
g=1
h=g+1
#
"
p
X
n(n − 1) 2
2
ki
eavg −
=2
2
i=1
Else if,
A PPENDIX B
where, eavg =
kg
then,
(47)
X
i6=j
We can write the left hand side of the equation (55) as,
X
X
Gaci Gacj
Fai Faj +
=
aci ,acj ∈(A∗ )c ,aci 6=acj
ai ,aj ∈A∗ ,ai 6=aj
(56)
− 2(Fa1 + ... + Fal )(Gac1 + ... + Gacn−l )
P
For l = 1 the term ai ,aj ∈A∗ ,ai 6=aj Fai Faj does not exist.
P
Similarly for n − l = 1 the term ac ,ac ∈(A∗ )c ,ac 6=ac Gaci Gacj
i
j
i
j
does not exist. For l ≥ 2 applying Lemma 6 in equation (56)
we get,
X
ci cj e i e j
i6=j
2
1)Favg
1)G2avg
≤ l(l −
+ (n − l)(n − l −
− 2M (M + Smin )
l−1
n−l−1 2
=
(M + Smin )2 +
M − 2M (M + Smin )
l
n−l
(57)
Fmin is the smallest element in the set F , so we can write,
M + Smin − Fmin
(58)
Fmin ≤
l−1
M
Now, applying Lemma 5 in (58) we get Smin ≤ l−1
.
Using (57) we have,
2
X
n−l−1
2 (l − 1) + 1
+
−2
ci cj ei ej ≤M
(l − 1)l
n−l
i6=j
l−1
+ 2M Smin
−1
l
<0
because,
(l−1)2 +1
(l−1)l
≤ 1 for l ≥ 2 and that completes the proof.
A PPENDIX D
Proof of Lemma 1:. Let γi ∈ {−1, 1} be the sign of αi , i.e.
αi = γi |αi |.
For the 1st column of Σt,DM ,
n
X
γ1 γg (Σt,DM )g1 =
g=2
=
n
X
g=2
n
X
g=2
γ1 γg γ1 γg |αg ||α1 |
n
X
g=2
|αg |
!
= (Σt,DM )11
X
γ1 γg (Σt,DM )g
g=2
⇒ (Σt,DM )1 −
⇒ (Σt,DM )1 −
where
n
X
g=2
n
X
(−1)Sg (Σt,DM )g = 0
g=2
(
Sg =
γ1 γg (Σt,DM )g = 0
1, γ1 γg = −1
2, γ1 γg = 1
A PPENDIX E
Proof of Lemma 2:. It is obvious to see that the following
selection of the elements of vector Φ makes it orthogonal to
(Σt,DM )1 , i.e. (Σt,DM )1 Φ = 0. Where (Σt,DM )1 is the 1st
row of Σt,DM .
(
−1, α1 αi > 0, i 6= 1
Φi =
1,
otherwise
Now it will be sufficient to prove that any vector Φ orthogonal
to (Σt,DM )1 is also orthogonal to all the other rows of Σt,DM ,
i.e. (Σt,DM )i Φ = 0, 2 ≤ i ≤ n. Let Φ = [Φ1 , Φ2 , ..., Φn ]′ be
a column vector such that Φi ∈ {−1, 1}, 1 ≤ i ≤ n and
(Σt,DM )1 Φ = 0.
Let γi ∈ {−1, 1} be the sign of αi , i.e. αi = γi |αi |
Now for any row g, g 6= 1,
X
(Σt,DM )g Φ = Φg (Σt,DM )gg +
Φh (Σt,DM )gh
= Φg |αg | |α1 | −
g6=h
X
i6=g,1
γ1 γh |αh ||αm | +
|αi | +
X
i6=g,1
X
Φh αg αh
g6=h
Φg |αg ||αi | +
X
X
X
(γg γh Φh − Φg )|αg ||αh |
h6=g,1
(59)
Else if Φg =
6 Φh ⇒ γ1 γg 6= γ1 γh ⇒ γg 6= γh ⇒
γg γh Φh − Φg = 0.
γ1 γm γm γh |αh ||αm |
Similarly, If Φg = Φ1 ⇒ α1 αg < 0 ⇒ γ1 6= γg ⇒
X
X
Φg + Φ1 γ g γ 1 = 0
= γ1 γh |α1 ||αh | −
γ1 γh |αh ||αm | +
γ1 γh |αh ||αm |
m6=h,1
m6=h,1
= γ1 γh |α1 ||αh |
= (Σt,DM )1h
m6=h,1
m6=h,1
Φh αg αh
h6=g,1
If Φg = Φh ⇒ γ1 γg = γ1 γh ⇒ γg = γh ⇒ γg γh Φh − Φg =
0.
γ1 γg (Σt,DM )gh
= γ1 γh |α1 ||αh | −
n
X
= (Φg + Φ1 γg γ1 )|αg ||α1 | +
For the hth (h 6= 1) column of Σt,DM ,
g=2
(Σt,DM )1 =
= Φg |αg ||α1 | + Φ1 αg α1 −
|αg ||α1 |
= |α1 |
n
X
Combining the above two results,
Else if Φg 6= Φ1 ⇒ α1 αg > 0 ⇒ γ1 = γg ⇒
Φg + Φ1 γ g γ 1 = 0
Plugging these results in equation (59), we get
(Σt,DM )g Φ = 0
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| 7 |
A Programming Model and Runtime System for
Significance-Aware Energy-Efficient Computing
Vassilis Vassiliadis1
Spyros Lalis5
1,2,4,5,6
Konstantinos Parasyris2
Charalambos Chalios3
Christos D. Antonopoulos4
Nikolaos Bellas6
Hans Vandierendonck7
Dimitrios S. Nikolopoulos8
Electrical and Computer Eng. Dept.
University of Thessaly, Greece
1,2,4,5,6
Centre for Research and Technology Hellas
(CE.R.T.H.), Greece
arXiv:1412.5150v1 [cs.PL] 15 Dec 2014
{vasiliad1 ,koparasy2 ,cda4 ,lalis5 ,nbellas6 }@uth.gr
3,7,8
Queen’s University Belfast
United Kingdom
{cchalios013 ,h.vandierendonck7 ,d.nikolopoulos8 }@qub.ac.uk
Abstract
1.
Reducing energy consumption is one of the key challenges in
computing technology. One factor that contributes to high energy
consumption is that all parts of the program are considered equally
significant for the accuracy of the end-result. However, in many
cases, parts of computations can be performed in an approximate
way, or even dropped, without affecting the quality of the final
output to a significant degree.
In this paper, we introduce a task-based programming model
and runtime system that exploit this observation to trade off the
quality of program outputs for increased energy-efficiency. This is
done in a structured and flexible way, allowing for easy exploitation of different execution points in the quality/energy space, without code modifications and without adversely affecting application
performance. The programmer specifies the significance of tasks,
and optionally provides approximations for them. Moreover, she
provides hints to the runtime on the percentage of tasks that should
be executed accurately in order to reach the target quality of results.
The runtime system can apply a number of different policies to decide whether it will execute each individual less-significant task in
its accurate form, or in its approximate version. Policies differ in
terms of their runtime overhead but also the degree to which they
manage to execute tasks according to the programmer’s specification.
The results from experiments performed on top of an Intelbased multicore/multiprocessor platform show that, depending on
the runtime policy used, our system can achieve an energy reduction of up to 83% compared with a fully accurate execution and up
to 35% compared with an approximate version employing loop perforation. At the same time, our approach always results in graceful
quality degradation.
Energy consumption has become a major barrier, not only for tetherless computing – the traditional energy-constrained environment
– but also for other computing domains, including big science.
Building an exascale machine with today’s technology is impractical due to the inordinate power draw it would require, hampering
large-scale scientific efforts. Likewise, current technologies are too
energy-inefficient to realize smaller and smarter embedded/wearable devices for a wide range of ubiquitous computing applications
that can greatly benefit society, such as personalized health systems.
One factor that contributes to the energy footprint of current
computer technology is that all parts of the program are considered
to be equally important, and thus are all executed with full accuracy.
However, as shown by previous work on approximate computing,
in several classes of computations, not all parts or execution phases
of a program affect the quality of its output equivalently. In fact, the
output may remain virtually unaffected even if some computations
produce incorrect results or fail completely. Data intensive applications and kernels from multimedia, data mining, and visualization
algorithms, can all tolerate a certain degree of imprecision in their
computations. For example, Discrete Cosine Transform (DCT), a
module of popular video compression kernels, which transforms a
block of image pixels to a block of frequency coefficients, can be
partitioned into layers of significance, owing to the fact that human
eye is more sensitive to lower spatial frequencies, rather than higher
ones. By explicitly tagging operations that contribute to the computation of higher frequencies as less-significant, one can leverage
smart underlying system software to trade-off video quality with
energy and performance improvements.
In this paper, we introduce a novel, significance-driven programming environment for approximate computing, comprising a
programming model, compilation toolchain and runtime system.
The environment allows programmers to trade-off the quality of
program outputs for increased energy-efficiency, in a structured
and flexible way. The programming model follows a task-based approach. For each task, the developer declares its significance depending on how strongly the task contributes to the quality of the
final program output, and provides an approximate version of lower
complexity that returns a less accurate result or just a meaningful
default value. Also, the developer controls the degradation of output quality, by specifying the percentage of tasks to be executed
accurately. In turn, the runtime system executes tasks on available
cores in a significance-aware fashion, by employing the approximate versions of less-significant tasks, or dropping such tasks altogether. This can lead to shorter makespans and thus to more energy-
Keywords Energy Saving, Approximate Computing, Controlled
Quality Degradation, Programming Model, Runtime System Evaluation.
Introduction
[Copyright notice will appear here once ’preprint’ option is removed.]
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2014/12/17
• to allow programmers to express the significance of computa-
efficient executions, without having a significant impact on the results of the computation.
The main contributions of this paper are the following: (i) We
propose a new programming model that allows the developer to
structure the computation in terms of distinct tasks with different
levels of significance, to supply approximate task versions, and to
control the degradation of program outputs; (ii) We introduce different runtime policies for deciding which tasks to execute accurately to meet the programmer’s specification; (iii) We implement
compiler and runtime support for the programming model and the
runtime policies. (iv) We experimentally evaluate the potential of
our approach, as well as the performance of the runtime policies.
Previous work has already explored the potential of approximate computing for specific algorithms and software blocks. Our
work is largely complementary to these efforts, as we introduce a
programming model that makes it possible to apply such techniques
in task-based programs that can exploit the parallelism of modern
many-core platforms. There are also major differences with other
approximate computing frameworks. For instance, the granularity
of approximation is at the level of tasks, rather than individual data
types, variables or arithmetic operations. Our programming model
operates not only at a different granularity but also at a different
level of abstraction for approximate computing –relative significance of code blocks–, which enables the compiler and runtime system to implement different policies that trade energy savings with
quality. Also, one can explore different points in the quality/energy
space in an easy and direct way, without code modifications, simply by specifying the percentage of tasks that should be executed
accurately - this can be an open parameter of a kernel or an entire
application, which can take different values in each invocation, or
be changed interactively by the user.
The rest of the paper is structured as follows. Section 2 introduces the programming model. Section 3 discusses the runtime
system, and the different policies used to drive task execution. Section 4 presents the experimental evaluation on top of an Intel-based
16-way multiprocessor/multicore platform, using a set of benchmark kernels that were ported to our programming model. Section 5
gives an overview of related work. Finally, Section 6 concludes the
paper and identifies directions for future work.
2.
tions in terms of their contribution to the quality of the endresult;
• to allow programmers to specify approximate alternatives for
selected computations;
• to allow programmers to express parallelism, beyond signifi-
cance;
• to allow programmers to control the balance between energy
consumption and the quality of the end-result, without sacrificing performance;
• to enable optimization and easy exploration of trade-offs at
execution time;
• to be user friendly and architecture agnostic.
Programmers express significance semantics using #pragma
compiler directives. Pragmas-based programming models facilitate
non-invasive and progressive code transformations, without requiring a complete code rewrite. We adopt a task-based paradigm,
similarly to OmpSS [3] and the latest version of OpenMP [9].
Task-based models offer a straightforward way to express communication across tasks, by explicitly defining inter-task data dependencies. Parallelism is expressed by the programmer in the form of
independent tasks, however the scheduling of the tasks is not explicitly controlled by the programmer, but is performed at runtime,
also taking into account the data dependencies among tasks.
Listing 1 illustrates the use of our programming model, using
the Sobel filter as a running example.
1
2
3
# pragma omp task [ significant ( expr (...) ) ]
[ approxfun ( function () ) ]
[ label (...) ] [ in (...) ] [ out (...) ]
Listing 2: #pragma omp task
Tasks are specified using the #pragma omp task directive (Listing 2), followed by a function which is equivalent to the task body.
The significance of the task is specified through the significant()
clause. Significance takes values in the range [0.0, 1.0] and characterizes the relative importance of tasks for the quality of the endresult of the application. Depending on their (relative) significance,
tasks may be approximated or dropped at runtime. The special values 1.0 and 0.0 are used for tasks that must unconditionally be executed accurately and approximately, respectively.
For tasks with significance less than 1.0, the programmer may
provide an alternative, approximate task body, through the approxfun() clause. This function is executed whenever the runtime opts
for a non-accurate computation of the task. It typically implements
a simpler, approximate version of the computation, which may even
degenerate to just setting default values to the output. If a task is
selected by the runtime system to be executed approximately, and
the programmer has not supplied an approxfun version, it is simply dropped by the runtime. It should be noted that the approxfun
function implicitly takes the same arguments as the function implementing the accurate version of the task body.
Programmers explicitly specify data flow to the task through the
in() and out() clauses. This information is exploited by the runtime
to automatically determine the dependencies among tasks.
Finally, label() can be used to group tasks, and to assign the
group a common identifier (name), which is in turn used as a
reference to implement synchronization at the granularity of task
groups (see next).
For example, in lines 51- 56 of Listing 1 a separate task is created to compute each row of the output image. The significance of
the tasks ranges between 0.1 and 0.9 in a round-robin way (line 53),
Programming Model
Improving energy consumption by controllably reducing the quality of application output has been already identified as an attractive
option in the domain of power-sensitive HPC programming. Part
of the problem of energy inefficiency is that all computations are
treated as equally important, despite the fact that only a subset of
these computations may be critical in order to achieve an acceptable quality of service (QoS). A key challenge though is how to
identify and tag computations of the program which must be executed accurately from those that are of less importance and thus
can be executed approximately.
In this section we introduce a programming model that allows the programmer to express her perspective on the significance of the contribution of each computation to the quality of
the final output.Highly significant computations are executed accurately, whereas non-significant computations can be executed approximately, at the expense of errors, or can be totally dropped.
Our vision is to elevate significance characterization as a first
class concern in software development, similar to parallelism and
other algorithmic properties traditionally being in the focus of
programmers. To this end, the main objectives of the proposed
programming model are the following:
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1
2
3
4
5
6
int sblX ( const unsigned char img [] , int y , int x ) {
return img [( y -1) * WIDTH +x -1]
+ 2* img [ y * WIDTH +x -1] + img [( y +1) * WIDTH +x -1]
- img [( y -1) * WIDTH + x +1]
- 2* img [ y * WIDTH + x +1] - img [( y +1) * WIDTH + x +1];
}
1
2
Listing 3: #pragma omp taskwait
7
The proposed programming model supports explicit barriertype synchronization through the #pragma omp taskwait directive
(Listing 3). A taskwait can serve as a global barrier, instructing
the runtime to wait for all tasks spawned up to that point in the
code. Alternatively, it can implement a barrier at the granularity of
a specific task group, if the label() clause is present; in this case the
runtime system waits for the termination of all tasks of that group.
Finally, the on() clause can be used to instruct the runtime to wait
for all tasks that affect a specific variable or data construct.
Furthermore, the omp taskwait barrier can be used to control the
minimum quality of application results. Through the ratio() clause,
the programmer can instruct the runtime to execute (at least) the
specified percentage of all tasks – either globally or in a specific
group, depending on the existence of the label() clause – in their
accurate version, while respecting task significance (i.e., a more
significant task should not be executed approximately, while a less
significant task is executed accurately). The ratio takes values in
the range [0.0, 1.0] and serves as a single, straightforward knob to
enforce a minimum quality in the performance / quality / energy
optimization space. Smaller ratios give the runtime more energy
reduction opportunities, however at a potential quality penalty.
For example, line 57 of Listing 1 specifies a barrier for the tasks
of the sobel task group. The runtime needs to ensure that at least the
35% most significant tasks of the group will be executed accurately.
The compiler for the programming model is implemented based
on a source-to-source compiler infrastructure [26]. It recognizes
the pragmas introduced by the programmer and lowers them to
corresponding calls of the runtime system discussed in Section 3.
8
9
10
11
12
13
14
15
int sblX_appr ( const unsigned char img [] ,
int y , int x ) {
return /* img [( y -1) * WIDTH +x -1]
Ommited taps */
+ 2* img [ y * WIDTH +x -1] + img [( y +1) * WIDTH +x -1]
/* - img [( y -1) * WIDTH + x +1]
Ommited taps */ /
- 2* img [ y * WIDTH + x +1] - img [( y +1) * WIDTH + x +1];
}
16
17
/* sblY and sblY_appr are similar */
18
19
20
21
void sbl_task ( unsigned char res [] ,
const unsigned char img [] , int i ) {
unsigned int p , j ;
22
for ( j =1; j < WIDTH -1; j ++) {
p = sqrt ( pow ( sblX ( img , i , j ) ,2) +
pow ( sblY ( img , i , j ) ,2) ) ;
res [ i * WIDTH + j ] = ( p > 255) ? 255 : p ;
}
23
24
25
26
27
28
}
29
30
31
32
33
void sbl_task_appr ( unsigned char res [] ,
const unsigned char img [] , int i ) {
unsigned int p , j ;
34
for ( j =1; j < WIDTH -1; j ++) {
/* abs instead of pow / sqrt ,
approximate versions of sblX , sblY */
p = abs ( sblX_appr ( img , i , j ) +
sblY_appr ( img , i , j ) ) ;
res [ i * WIDTH + j ] = ( p > 255) ? 255 : p ;
}
35
36
37
38
39
40
41
42
# pragma omp taskwait [ on (...) ] [ label (...) ]
[ ratio (...) ]
}
43
3.
44
45
46
47
double sobel ( void ) {
int i ;
unsigned char img [ WIDTH * HEIGHT ] , res [ WIDTH * HEIGHT ];
48
/* Initialize img array and reset res array */
...
for ( i =1; i < HEIGHT -1; i ++)
#pragma omp task label( sobel ) in( img ) out( res ) \
significant(( i %9 + 1) /10.0) approxfun( sbl_task_appr )
sbl_task ( res , img , i ) ; /* Compute a single
output image row */
}
#pragma omp taskwait label( sobel ) ratio(0.35)
49
50
51
52
53
54
55
56
57
58
Runtime
We demonstrate how to extend existing runtime systems to support
our programming model for approximate computing. To this end,
we extend a task-based parallel runtime system that implements
OpenMP 4.0-style task dependencies [23].
Our runtime system is organized as a master/slave work-sharing
scheduler. The master thread starts executing the main program
sequentially. For every task call encountered, the task is enqueued
in a per-worker task queue. Tasks are distributed across workers
in round-robin fashion. Workers select the oldest tasks from their
queues for execution. When a worker’s queue runs empty, the
worker may steal tasks from other worker’s queues.
The runtime system furthermore implements an efficient mechanism for identifying and enforcing dependencies between tasks
that arise from annotations of the side effects of tasks with in(...)
and out(...) clauses. Dependence tracking is however not affected
by our approximate computing programming model. As such, we
provide no further details on this feature.
}
Listing 1: Programming model use case: Sobel filter
3.1
Runtime API Extension
The runtime exposes an API that matches with the pragma-based
programming model. Every pragma in the program is translated in
one or more runtime calls. The runtime API is extended to convey
the new information in the programming model. Task creation is
extended to indicate the task group and significance of the task,
as well as an alternative (approximate) task function. On the first
use of a task group, the compiler inserts a call to tpc init group()
to create support data structures in the runtime for the task group.
This API call also conveys the per-group ratio of tasks that must be
executed accurately.
which ensures that there will not be extreme, apprehensible quality fluctuations across different areas of the output image. Care has
also been taken in this case to avoid using the special values 0.0 and
1.0. Moreover, an approximate version of the task body is implemented by the sbl task appr function (lines 31–42). This function
implements a light-weight version of the computation, substituting
complex arithmetic operations with simpler ones (line 38), while
at the same time skipping some filter taps (lines 11, 13). All tasks
created in the specific loop belong to the sobel task group, using
img as input and res as output (line 52).
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1
2
3
time can end up buffering all tasks until the corresponding synchronization barrier is encountered, and thus take a fully correct
decision as to which tasks to run accurately/approximately. In our
implementation, the buffer size is a configurable parameter passed
to the runtime system at compile time.
The global buffer policy has the potential disadvantage that it
slows done the program by postponing task execution until the
buffer is full and sorted. In the extreme case, the runtime system
needs to wait for all tasks to be issued and sorted in the buffer before starting their execution. This overhead can be mitigated by using a smaller window size and tasks of coarse enough granularity,
so that the runtime system can overlap task issue with task execution. Using smaller window sizes will incur the cost of not making fully correct decisions for approximate execution. Section 4
demonstrates that the GTB policy sustains low overhead in practice.
TaskDesc buffer [ BUFFER_SIZE ]; // to analyze tasks
size_t task_count = 0;
// buffer occupation
float group_ratio ;
// set by # pragma
4
5
6
7
8
9
10
void buffer_task ( TaskDesc t ) { // called by master
thread
buffer [ task_count ] = t ;
task_count ++;
if ( task_count == BUFFER_SIZE )
flush_buffer () ;
}
11
12
13
14
15
16
17
18
19
20
21
void flush_buffer () { // when tasks need to execute
sort ( buffer ) ; // sort by increasing significance
for ( i =0; i < task_count ; i ++) {
if ( i < group_ratio * task_count )
i s s u e _a c c u r a t e _ t a s k ( buffer [ i ]) ;
else
i s s u e _ a p p r o x i m a t e _ t a s k ( buffer [ i ]) ;
}
task_count = 0;
}
3.4
Listing 4: Global task buffering policy to choose the accuracy of a
task
An additional waiting API call is created. Next to the API call
tpc wait all(), which waits for all tasks to finish, we create the API
call tpc wait group() to synchronize on the completion of a task
group.
3.2
Runtime Support for Approximate Computing
The job of the runtime system is to selectively execute a subset
of the tasks approximately while respecting the constraints given
by the programmer. The relevant information consists of (i) the
significance of each task, (ii) the group a task belongs to, and
(iii) the fraction of tasks that may be executed approximately for
each task group. Moreover, preference should be given to approximating tasks with lower significance values as opposed to tasks
with high significance values.
The runtime system has no a priori information on how many
tasks will be issued in a task group, nor what the distribution is
of the significance levels in each task group. This information must
be collected at runtime. In the ideal case, the runtime system knows
this information in advance. Then, it is straightforward to execute
approximately those tasks with the lowest significance in each task
group. The policies we design must however work without this information, and estimate it at runtime. We define two policies, one
globally controlled policy based on buffering issued tasks and analyzing their properties, and a policy that estimates the distribution
of significance levels using per-worker local information.
3.3
Local Queue History (LQH)
The local queue history policy avoids the step of task buffering.
Tasks are issued to worker queues immediately as they are created.
The worker decides whether to approximate a task right before
it starts its execution, based on the distribution of significance
levels of the tasks executed so far, and the target ratio of accurate
tasks (supplied by the programmer). Hereto, the workers track the
number of tasks at each significance level as they are executed.
Formally, the local queue history policy operates as follows.
Let tg (s) indicate the number of tasks in task group g observed
by a worker with significance s or less. These statistics are updated for every executed task. Note that the significance levels s
are constrained to the range 0.0 to 1.0. In the runtime system,
we implement 101 discrete (integer) levels to simplify the implementation, ranging from 0.0 to 1.0 (inclusive) in steps of 0.01. By
construction, tg (1.0) equals the total number of tasks executed so
far. Let Rg be the target ratio of tasks that should be executed
accurately in task group g, as set by the programmer. Then, assuming a task has significance level s, it is executed accurately if
tg (s) > (1 − Rg )tg (1.0), otherwise it is executed approximately.
This policy attempts to achieve a ratio of accurately executed
tasks that converges to Rg and also approximates those tasks with
the lowest significance level, as stipulated by the programming
model.
The local queue history algorithm is performed independently
by each worker using only local information from the tasks that
appear in their work queue. Tasks of one group are distributed
among the workers via pushing of tasks to different local queues
by the master and work-stealing. As a result, each worker has only
partial information about each group.
The overhead of the local queue history algorithm is the bookkeeping of the statistics that form the execution history of a group.
This happens every time a task is executed. Updating statistics includes accessing an array of size equal to the number of distinct
significance levels (101 in the runtime), which is negligible compared to the granularity of the task.
The local queue history algorithm requires no global snapshot
of all tasks in the program and no synchronization between workers and the master. It is thus more realistic and scalable than global
task buffering. However, given that each worker has only a localized view of the tasks issued, the runtime system can only approximately enforce the quality requirements set by the programmer.
Global Task Buffering (GTB)
In the first policy the master thread buffers a number of tasks as
it creates them, postponing the issue of the tasks in the worker
queues. When the buffer is full, or when the a call to tpc wait all()
or tpc wait group() is made, the tasks in the buffer are analyzed and
sorted by significance. Given a per-group ratio of accurate tasks Rg ,
and a number of B tasks in the buffer, then the Rg · B tasks with
the highest significance level are executed accurately. The tasks are
subsequently issued to the worker queues. The policy is described
in Listing 4 for a single task group. The variables described (buffer,
task count and per-group accuracy ratio) are replicated over all task
groups introduced by the programmer.
The task buffering policy is parameterized by the task buffer
size. A larger buffer size allows the runtime to take more informed
decisions. Notably, if the buffer size is sufficiently large, the run-
4.
Experimental Evaluation
We performed a set of experiments to investigate the performance
of the proposed programming model and runtime policies, using
different benchmark codes that were re-written using the task-based
pragma directives. In particular, we evaluate our approach in terms
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Benchmark
Sobel
DCT
MC
Kmeans
Jacobi
Fluidanimate
Approximate
or Drop
A
D
D, A
A
D, A
A
Approx Degree
Mild
Med
Aggr
80%
30%
0%
80%
40%
10%
100% 80%
50%
80%
60%
40%
10−4 10−3
10−2
50%
25% 12.5%
methodology is used to decide how far from the current location the
next step of a random walk should be.
K-means clustering aims to partition n observations in a multidimensional space into k clusters by minimizing the distance of
cluster members to a cluster representative. In each iteration the
algorithm spawns a number of tasks, each being responsible for
a subset of the entire problem. All tasks are assigned the same
significance value. The degree of approximation is controlled by
the ratio used at taskwait pragmas. Approximated tasks compute a
simpler version of the euclidean distance, while at the same time
considering only a subset (1/8) of the dimensions. Only accurate
results are considered when evaluating the convergence criteria.
Jacobi is an iterative solver of diagonally dominant systems of
linear equations. We execute the first 5 iterations approximately, by
dropping the tasks (and computations) corresponding to the upper
right and lower left areas of the matrix. This is not catastrophic,
due to the fact that the matrix is diagonally dominant and thus
most of the information is within a band near the diagonal. All
the following steps, until convergence, are executed accurately,
however at a higher target error tolerance than the native execution
(see Table 1).
Fluidanimate, a code from the PARSEC benchmark suite [2],
applies the smoothed particle hydrodynamics (SPH) method to
compute the movement of a fluid in consecutive time steps. The
fluid is represented as a number of particles embedded in a grid.
Each time step is executed as either fully accurate or fully approximate, by setting the ratio clause of the omp taskwait pragma to
either 0.0 or 1.0. In the approximate execution, the new position
of each particle is estimated assuming it will move linearly, in the
same direction and with the same velocity as it did in the previous
time steps.
Three different degrees of approximation are studied for each
benchmark: Mild, Medium, and Aggressive (see Table 1). They
correspond to different choices in the quality vs. energy and performance space. No approximate execution led to abnormal program
termination. It should be noted that, with the partial exception of
Jacobi, quality control is possible solely by changing the ratio parameter of the taskwait pragma. This is indicative of the flexibility
of our programming model. As an example, Figure 1 visualizes the
results of different degrees of approximation for Sobel: the upper
left quadrant is computed with no approximation, the upper right
is computed with Mild approximation, the lower left with Medium
approximation, whereas the lower right corner is produced when
using Aggressive approximation.
The quality of the final result is evaluated by comparing it to
the output produced by a fully accurate execution of the respective
code. The appropriate metric for the quality of the final result differs according to the computation. For benchmarks involving image processing (DCT, Sobel), we use the peak signal to noise ratio
(PSNR) metric, whereas for MC, Kmeans, Jacobi and Fluidanimate
we use the relative error.
In the experiments, we measure the performance of our approach for the different benchmarks and approximation degrees,
for the two different runtime policies GTB and LQH. For GTB, we
investigate two cases: the buffer size is set so that tasks are buffered
until the synchronization barrier (referred to as Max Buffer GTB);
the buffer size is set to a smaller value, depending on the computation, so that task execution can start earlier (referred to as GTB).
As a reference, we compare our approach against:
Quality
PSNR
PSNR
Rel. Err.
Rel. Err.
Rel. Err.
Rel. Err.
Table 1: Benchmarks used for the evaluation. For all cases, except
Jacobi, the approximation degree is given by the percentage of accurately executed tasks. In Jacobi, it is given by the error tolerance
in convergence of the accurately executed iterations/tasks (10−5 in
the native version).
Figure 1: Different levels of approximation for the Sobel benchmark
of: (i) The potential for performance and energy reduction; (ii) The
potential to allow graceful quality degradation; (iii) The overhead
incurred by the runtime mechanisms. In the sequel, we introduce
the benchmarks and the overall evaluation approach, and discuss
the results achieved for various degrees of approximation under
different runtime policies.
4.1
Approach
We use a set of six benchmarks, outlined in Table 1, where we apply different approximation approaches, subject to the nature/characteristics of the respective computation.
Sobel is a 2D filter used for edge detection in images. The
approximate version of the tasks uses a lightweight Sobel stencil
with justq
2/3 of the filter taps. Additionally, it substitutes the costly
formula sblx 2 + sbly2 with its approximate counterpart |sblx | +
|sbly |. The way of assigning significance to tasks ensures that the
approximated pixels are uniformly spread throughout the output
image.
Discrete Cosine Transform (DCT) is a module of the JPEG
compression and decompression [20] algorithm. We assign higher
significance to tasks that compute lower frequency coefficients.
MC [24] applies a Monte Carlo approach to estimate the boundary of a subdomain within a larger partial differential equation
(PDE) domain, by performing random walks from points of the
subdomain boundary to the boundary of the initial domain. Approximate configurations drop a percentage of the random walks
and the corresponding computations. A modified, more lightweight
• A fully accurate execution of each application, using a signifi-
cance agnostic version of the runtime system.
• An execution using loop perforation [19], a simple yet usually
effective compiler technique for approximation. Loop perforation is also applied in three different degrees of aggressiveness.
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2014/12/17
formance and energy consumption, yet resulting in higher quality
results1 . This is due to the fact that our model offers more flexibility
than perforation in defining the relative significance of code regions
in DCT. The problematic performance of GTB(Max Buffer) is discussed later in this Section, when evaluating the overhead of the
runtime policies and mechanisms.
The approximate version of MC significantly outperforms the
original accurate version, without suffering much of a penalty on
its output quality. Randomized algorithms are inherently susceptible to approximations without requiring much sophistication. It is
characteristic that the performance of our approach is almost identical to that of blind loop perforation. We observe that the LQH
policy attains slightly better results. In this case, we found that the
LQH policy undershoots the requested ratio, evidently executing
fewer tasks 2 . This affects quality, which is lower than that achieved
by the rest of the policies.
Kmeans behaves gracefully as the level of approximation increases. Even in the aggressive case, all policies demonstrate relative errors less than 0.45%. The GTB policies are superior in terms
of execution time and energy consumption in comparison with the
perforated version of the benchmark. Noticeably, the LQH policy
exhibits slow convergence to the termination criteria. The application terminates when the number of objects which move to another cluster is less than 1/1000 of the total object population. As
mentioned in the Section 4.1, objects which are computed approximately do not participate in the termination criteria. GTB policies behave deterministically, therefore always selecting tasks corresponding to specific objects for accurate executions. On the other
hand, due to the effects dynamic load balancing in the runtime and
its localized perspective, LQH tends to evaluate accurately different objects in each iteration. Therefore, it is more challenging for
LQH to achieve the termination criterion. Nevertheless, LQH produces results with the same quality as a fully accurate execution
with significant performance and energy benefits.
Jacobi is a particular application, in the sense that approximations can affect its rate of convergence in deterministic, yet hard
to predict and analyze ways. The blind perforation version requires
fewer iterations to converge, thus resulting in lower energy consumption than our policies. Interestingly enough, it also results in a
solution closer to the real one, compared with the accurate execution.
The perforation mechanism could not be applied on top of the
Fluidanimate benchmark. This is because if the evaluation of the
movement of part of the particles during a time-step is totally
dropped, the physics of the fluid are violated leading to completely
wrong results. Our programming model offers the programmer the
expressiveness to approximate the movement of the liquid for a set
of time-steps. Moreover, in order to ensure stability, in is necessary
to alternate accurate and approximate time steps. In our programming model this is achieved in a trivial manner, by alternating the
parameter of the ratio clause at taskbarrier pragmas between 100%
and the desired value in consecutive time steps. It is worth noting
that Fluidanimate is so sensitive to errors that only the mild degree
of approximation leads to acceptable results. Even so, the LQH policy requires less than half the energy of the accurate execution, with
the 2 versions of the GTB policy being almost as efficient.
Following, we evaluate the overhead of the runtime policies and
mechanisms discussed in Sections 3.3 and 3.4. We measure the performance of each benchmark when executed with a significanceagnostic version of the runtime system, which does not include
the execution paths for classifying and executing tasks according
Figure 3: Different levels of perforation for the Sobel benchmark.
Accurate execution, Perforation of 20%, 70% and 100% of loop
iterations on the upper left, upper right, lower left and lower right
quadrants respectively.
The perforated version executes the same number of tasks as
those executed accurately by our approach.
The experimental evaluation is carried out on a system equipped
with 2 Intel(R) Xeon(R) CPU E5-2650 processors clocked at 2.00
GHz, with 64 GB shared memory. Each CPU consists of 8 cores.
Although cores support SMT execution (hyper-threading), we deactivated this feature during our experiments. We use Centos 6.5
Linux Operating system with the 2.6.32 Linux kernel. Each execution pinned 16 threads on all 16 cores.
Finally the energy and power are measured using likwid [22] to
access the Running Average Power Limit (RAPL) registers of the
processors.
4.2
Experimental Results
Figure 2 depicts the results of the experimental evaluation of our
system. For each benchmark we present execution time, energy
consumption and the corresponding error metric.
The approximated versions of the benchmarks execute significantly faster and with less energy consumption compared to their
accurate counterparts. Although the quality of the application output deteriorates as the approximation level increases, this is typically done in a graceful manner, as it can be observed in Figure 1
and the ’Quality’ column of Figure 2.
The GTB policies with different buffer sizes are comparable
with each other. Even though Max buffer GTB postpones task issue
until the creation of all tasks in the group, this does not seem to
penalize the policy. In most applications tasks are coarse-grained
and are organized in relatively small groups, thus minimizing the
task creation overhead and the latency for the creation of all tasks
within a group. LQH is typically faster and more energy-efficient
than both GTB flavors, except for Kmeans.
In the case of Sobel, the perforated version seems to significantly outperform our approach in terms of both energy consumption and execution time. However the cost of doing so is unacceptable output quality, even for the mild approximation level as shown
in Figure 3. Our programming model and runtime policies achieve
graceful quality degradation, resulting in acceptable output even
with aggressive approximation, as illustrated in Figure 1.
DCT is friendly to approximations: it produces visually acceptable results even if a large percentage of the computations is
dropped. Our policies, with the exception of the Max Buffer version
of GTB, perform comparably to loop perforation in terms of per-
1 Note
that PSNR is a logarithmic metric
2 4.6% and 5.1% more that requested tasks are approximated for the aggres-
sive and the medium case respectively.
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0.06
20
0.04
10
0.02
0
Medium
Mild
100
0.05
80
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60
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40
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20
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0
0
DCT
2
1
Medium
Mild
10
5
0
Aggr
Medium
Medium
Mild
Medium
Mild
50
2500
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40
2000
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30
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1000
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10
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0
Aggr
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Mild
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3
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0
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200
30
20
15
10
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0
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Medium
Mild
Aggr
Medium
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0
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Aggr
Medium
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Aggr
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1200
1000
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1
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Aggr
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0
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Aggr
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1200
1000
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600
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Aggr
P SN R−1
Aggr
Rel.Error
Mild
20
MC
30
Rel.Error
Medium
25
Kmeans
0.08
1.5
Aggr
Jacobi
0.1
40
0
Aggr
Fluidanimate
50
Rel.Error
1.4
1.2
1
0.8
0.6
0.4
0.2
0
Quality
lower is better
P SN R−1
Energy (Joules)
lower is better
80
60
Rel.Error
Sobel
Execution time (secs)
lower is better
40
20
0
Aggr
Medium
Mild
Figure 2: Execution time, energy and quality of results for the benchmarks used in the experimental evaluation under different runtime
policies and degrees of approximation. In all cases lower is better. Quality is depicted as PSNR−1 for Sobel and DCT, relative error (%)
is used in all others benchmarks. The accurate execution and the approximate execution using perforation are visualized as lines. Note that
perforation was not applicable for Fluidanimate.
7
2014/12/17
that implement approximate computation, and other approaches,
including domain-specific frameworks and hardware support for
approximate computation. Finally, we review prior work on runtime energy optimization of parallel programs, that does not employ approximation.
5.1
Several frameworks for approximate computing discard parts of
code at runtime, while asserting that the quality of the result complies with quality criteria provided by the programmer. Green [1]
is an API for loop-level and function approximation. Loops are approximated with a reduction of the loop trip count. Functions are
approximated with multi-versioning. The API includes calibration
functions that build application-specific QoS models for the outputs of the approximated blocks of code, as well as re-calibration
functions for correcting unacceptable errors that may incur due to
approximation. Sloan et al. [21] provide guidelines for manual control of approximate computation and error checking in software.
These frameworks delegate the control of approximate code execution to the programmer. We explore an alternative approach where
the programmer uses a higher level of abstraction for approximation, namely computational significance, while the system software
translates this abstraction into energy- and performance-efficient
approximate execution.
Loop perforation [19] is a compiler technique that classifies
loop iterations into critical and non-critical ones. The latter can be
dropped, as long as the results of the loop are acceptable from a
quality standpoint. Input sampling and code versioning [28] also
use the compiler to selectively discard inputs to functions and substitute accurate function implementations with approximate ones.
Similarly to loop perforation and code versioning, our framework
benefits from task dropping and the execution of approximate versions of tasks. However, we follow a different approach whereby
these optimizations are driven from user input on the relative significance of code blocks and are used selectively in the runtime
system to meet user-defined quality criteria.
EnerJ [16] implements approximate data types and supports
user-defined “approximable” methods, without tying these abstractions to a specific approximate execution model. To achieve energy
savings, the prototype implementation of EnerJ uses a simulated
environment where it stores approximate data types in DRAM with
low refresh rate and SRAM with low supply voltage. Approximable
methods are executed on aggressively voltage-scaled processors,
with ISA extensions for approximation [4, 17]. Similarly to our
framework, EnerJ provides abstractions that allow the programmer
to provide hints on where approximate execution can be safely used
in a program. Contrary to our framework, EnerJ does not use a runtime substrate for approximation on general-purpose hardware and
does not consider code dropping or task-parallel execution.
Figure 4: The normalized execution time of benchmarks under
different task categorization policies, with respect to that over the
significance-agnostic runtime system
Benchmark
Sobel
DCT
MC
KMeans
Jacobi
FluidAnimate
(%) Inversed Significance Tasks
LQH GTB(UD) GTB (MB)
2.7
0
0
2.7
0
0
4.8
0
0
0
0
0
0
0
0
0
0
0
LQH
0.07
0.18
0.17
0.9
0.12
0
Average Ratio Diff
GTB(UD) GTB (MB)
0
0
0
0
0
0
0
0
0
0
0
0
Table 2: Degree of accuracy of the proposed policies.
to significance. We then compare it with the performance attained
when executing the benchmarks with the significance-aware version of the runtime. All tasks are created with the same significance
and the ratio of tasks executed accurately is set to 100%, therefore eliminating any benefits of approximate execution. Figure 4
summarizes the results. It is evident that the significance-aware
runtime system typically incurs negligible overhead. The overhead
reaches in the order of 7% in the worst case (DCT under the GTB
Max Buffer policy). DCT creates many lightweight tasks, therefore
stressing the runtime. At the same time, given that for DCT task
creation is a non-negligible percentage of the total execution time,
the latency between task creation and task issue introduced by the
Max Buffer version of the GTB policy results in a measurable overhead.
The last step of our evaluation focuses on the accuracy of the
policies in terms of respecting the significance of tasks and the
user-supplied ratio of accurate tasks to be executed. Table 2 summarizes the results. The average offset in the ratio of accurate tasks
executed is calculated by the following formula:
ratio dif f =
PGroups
i=1
|requestedratioi −providedratioi |
T otalGroups
The two versions of GTB respect perfectly task significance
and the user-specified ratio. This is totally expected for the Max
Window version of GTB. The version of GTB using a limited
window benefits by the relatively small task groups created by
the applications and the smoothly distributed significance values
in tasks of each group. LQH, in turn, is inherently more inaccurate,
due to its localized perspective. It manages to avoid significance
inversion only in cases where all tasks within each task group
have the same significance (Kmeans, Jacobi, Fluidanimate). Even
in these cases, LQH may slightly deviate from the specified ratio,
due to the loose collaboration of policy modules active on different
workers.
5.
General-Purpose Approximation Frameworks
5.2
Parallel Approximation Frameworks
Quickstep [8] is a tool that approximately parallelizes sequential
programs. The parallelized programs are subjected to statistical accuracy tests for correctness. Quickstep tolerates races that occur
after removing synchronization operations that would otherwise
be necessary to preserve the semantics of the sequential program.
Quickstep thus exposes additional parallelization and optimization
opportunities via approximating the data and control dependencies
in a program. On the other hand, QuickStep does not enable algorithmic and application-specific approximation, which is the focus
of our work.
Variability-aware OpenMP [11] is a set of OpenMP extensions
that enable a programmer to specify blocks of code that can be
computed approximately, The programmer may also specify error
tolerance in terms of the number of most significant bits in a vari-
Related Work
We classify related work in approximate computation into generalpurpose frameworks, parallel programming and execution models
8
2014/12/17
cance to implicitly indicate code that can be approximated and the
runtime system implements selective approximation. In our framework, accurate and approximate code may run on any core for load
balancing purposes.
able which are guaranteed to be correct. Variability-aware OpenMP
applies approximation only to specific FPU operations, which execute on specialized FPUs with configurable accuracy. Our framework applies selective approximation at the granularity of tasks,
using the significance abstraction. Our programming and execution
model thus provides additional flexibility to drop or approximate
code, while preserving output quality. Furthermore, our framework
does not require specialized hardware support.
Variation-tolerant OpenMP [10] uses a runtime system that
characterizes OpenMP tasks in terms of their vulnerability to errors. The runtime system assesses error vulnerability of tasks online, similarly to our LQH policy for significance characterization. The variation-tolerant OpenMP runtime uses a hardware error
counter to apportion errors to tasks and estimate task vulnerability
to errors. The scheduler is a variant of an FCFS, centralized scheduler that uses task vulnerability to select the cores on which each
task runs, in order to minimize the number of instructions that are
likely to incur errors. Variation-tolerant OpenMP does not consider
explicitly identified approximate code and its selective execution
for quality-aware energy and performance optimization.
5.3
6.
Conclusions
We introduced a programming model that supports approximate
computing at the granularity of tasks. Tasks are widely used to
express parallelism in a high-level and platform-neutral way. We
believe that tasks can also be used to introduce approximate versions of specific parts of the computation in a structured way that is
amenable to flexible runtime scheduling to achieve energy-efficient
program execution at a controllable degradation of output quality.
We also introduced extensions to a task-based runtime system
to exploit significance information, along with a set of significancecentric scheduling policies for elastically deciding which tasks to
execute accurately and which approximately, while at the same
time respecting programmer’s specifications.
We have performed a first evaluation of our implementation on
an Intel-based multiprocessor consisting of twin multicore sockets.
The results across several different benchmark codes are encouraging, and show that the programmer can easily target different
energy-quality trade-offs, by adjusting in the majority of cases a
single parameter: the percentage of tasks to execute accurately.
In the future, we wish to explore more optimization scenarios,
such as DFVS in conjunction with suitable runtime policies for
executing approximate (and more light-weight) task versions on
the slower but also less power-hungry CPUs, as well as for using
more such cores to make up for this slower execution. We are
also interested in extending our programming model to support
approximate computing on top of ultra low-power but unreliable
hardware.
Other Approximation Frameworks
Several software and hardware schemes for approximate computing follow a domain-specific approach. ApproxIt [27] is a framework for approximate iterative methods, based on a lightweight
quality control mechanism. Unlike our task-based approach, ApproxIt uses coarse-grain approximation at a minimum granularity
of one solver iteration. Gschwandtner et al. use a similar iterative
approach to execute error-tolerant solvers on processors that operate with near-threshold voltage (NTC) and reduce energy consumption by replacing cores operating at nominal voltage with NTC
cores [6]. Schmoll et al. [18] present algorithmic and static analysis
techniques to detect variables that must be computed reliably and
variables that can be computed approximately in an H.264 video
decoder. Although we follow a domain-agnostic approach in our
approximate computing framework, we provide sufficient abstractions for implementing the aforementioned application-specific approximation methods.
SAGE [14] is a compiler and runtime environment for automatic
generation of approximate kernels in machine learning and image
processing applications. Paraprox [15] implements transparent approximation for data-parallel programs by recognizing common
algorithmic kernels and replacing them with approximate equivalents. ASAC [12] provides sensitivity analysis for automatically
generated code annotations that quantify significance. We do not
explore automatic generation of approximate code in this work.
However, our techniques for quality-aware, selective execution of
approximate code are directly applicable to scenarios where the approximate code is derived from a compiler, instead of source code
annotations.
Hardware support for approximate computation has taken the
form of programmable vector processors [25], neural networks
that approximate the results of code regions in hardware [5], and
low-voltage probabilistic storage [13]. These frameworks assume
non-trivial, architecture-specific support from the system software
stack, whereas we depend only on compiler and runtime support
for task-parallel execution, which is already widely available on
commodity multi-core systems. ESRA [7] is a multi-core architecture where cores are either fully reliable or have relaxed reliability. Programs running on ESRA divide their code into critical
(typically control code) and non-critical (typically data processing
code) parts and assign these to reliable or unreliable cores, respectively. Therefore, ESRA uses an explicit and application-specific
assignment of code to cores with different levels of reliability. We
follow a different approach whereby the programmer uses signifi-
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| 6 |
Topology Learning of Radial Dynamical Systems with Latent Nodes
arXiv:1803.02793v1 [cs.SY] 7 Mar 2018
Saurav Talukdar1 , Deepjyoti Deka2 , Michael Chertkov2 and Murti Salapaka3
Abstract— In this article, we present a method to reconstruct
the topology of a partially observed radial network of linear dynamical systems with bi-directional interactions. Our approach
exploits the structure of the inverse power spectral density
matrix and recovers edges involving nodes up to four hops
away in the underlying topology. We then present an algorithm
with provable guarantees, which eliminates the spurious links
obtained and also identifies the location of the unobserved
nodes in the inferred topology. The algorithm recovers the exact
topology of the network by using only time-series of the states at
the observed nodes. The effectiveness of the method developed
is demonstrated by applying it on a typical distribution system
of the electric grid.
I. I NTRODUCTION
Networks provide a convenient framework for analysis
of the behavior of large scale systems and have found
applications in neuroscience [1], biology [2], finance [3] and
many more. In this regard, an important question of interest is
to estimate the unknown topology or the interaction structure
amongst the entities of the network using measurements.
There are both active [4] and passive approaches [5] to infer
the unknown influence structure/ topology from measurements. Active approaches include performing interventions
on the network, which are not always possible. For example:
it is not viable to perform experiments by removing links in a
power distribution network. In this article we are interested
in passive approaches to topology learning from observed
time series measurements.
The problem of topology or structure learning under the
assumption that the entities are random variables is an
active area of research for the past few decades and a good
summary is available in [6], [7], [8], [9]. However, the
random variables framework does not capture the dynamics
amongst the entities and hence is not useful for application
where lagged dependencies are present. Such applications
include power distribution networks, seasonality in climate
systems [10], finance, thermal dynamics of buildings [11]
and many more.
Recently, there has been considerable interest in the topology learning for linear dynamical systems from time series
measurements. It this regard some of the notable works
are [12], [13], [14], [15], [16], [17]. However, these works
assume that all nodes in the network are observed or full
network observability. Topology reconstruction from passive
measurements for a network of linear dynamical systems
1 Saurav Talukdar is with Department of Mechanical Engineering, University of Minnesota, Minneapolis, USA, sauravtalukdar@umn.edu
2 Deepjyoti Deka and Michael Chertkov are with Los Alamos National
Lab, Los Alamos, USA, deepjyoti,chertkov@lanl.gov
3 Murti V. Salapaka is with Department of Electrical and Computer
Engineering, Minneapolis, USA, murtis@umn.edu
with unobserved nodes is discussed in [18], [19]. The problem formulation in [18] is focused on directed poly-tree
network of linear dynamical systems with unobserved nodes.
The framework presented in [19] is restricted to Gaussian
stationary time series and does not include consistency of
identification results. In this article, we present an algorithm
which leads to recovering the exact topology of the network
under partial observation of the nodes. In particular, we
are interested in radial linear dynamical systems, which are
characterized by a tree topology with undirected (that is bidirected) edges between neighbors rather than uni-directed
edges. Indeed the directed loops are a part of the framework
presented here unlike [18].
Radial linear dynamical systems (RLDS) [20] represent an
important class of networks in engineering systems. Among
others, RLDS can model dynamics in power distribution
systems. An algorithm for exact topology learning for RLDS
with all nodes being observed is presented in [20]. We
show that for RLDS, under the assumption that unobserved
nodes are ‘deep into the network’such that their effects are
felt through observed nodes, it is possible to recover the
underlying interaction topology exactly. In this regard, we
build upon topological separation ideas of [20] and phase
response properties of [21], to devise an algorithm which
provably recovers the exact topology of the RLDS. Our
algorithm uses only the time series measurements from the
nodes and does not use any knowledge of system parameters
as well as the exogenous injections. The efficacy of the
algorithm is demonstrated on a 39 bus radial power network
with linearized swing dynamics [22].
In the next section, we introduce definitions and notations
useful for the subsequent discussion, following which in
Section III we present an algorithm for inference of topology
with unobserved nodes using inverse power spectral density.
In Section IV, we present algorithms for exact topology
learning of RLDS with partial observability, followed by
results in Section V and conclusions in Section VI.
II. P RELIMINARIES
Consider the continuous time linear dynamical system,
l
X
m=1
am,i
dm xi
=
dtm
n
X
j=1,j6=i
bij (xj (t) − xi (t)) + pi (t),
(1)
,i ∈ {1, 2, .., n}, where, the exogenous forcing pi (t) is a zero
mean wide sense stationary process uncorrelated with pj (t)
for j 6= i. Here, xi (t) ∈ R is a state of the system, am,i ∈ R
and bij ∈ R≥0 . Assuming that discrete time samples of the
state xi are available as an output, we discretize the above
1
1
2
3
4
5
2
7
4
6
3
5
7
6
(a)
(b)
Fig. 1. (a) Graphical representation of a linear dynamical system where
Assumption 1 holds, and (b) its associated topology.
continuous time dynamics using z transform to obtain,
Si (z)Xi (z) =
n
X
bij Xj (z) + Pi (z),
j=1,j6=i
where, Si (z) is the frequency domain operator determined
by the time derivatives of xi . We rewrite the above equation
as,
m
X
Xi (z) =
Hij (z)Xj (z) + Ei (z)
(2)
j=1,j6=i
b
1
where, Hij (z) = Siij
(z) , Ei (z) = Si (z) Pi (z). Note that, for
j 6= i, ei (k) are uncorrelated with ej (k) (ei is the inverse z
transform of Ei (z) for all i = 1, ..., n) and is a zero mean
wide sense stationary sequence. Then, the dynamics of the
entire network can be written as,
X(z) = H(z)X(z) + E(z), where ,
X(z) = [X1 (z) X2 (z) ... Xn (z)]T
E(z) = [E1 (z) E2 (z) ... En (z)]T , H(z)(i, j) = Hij (z).
We assume that I − H(z) is invertible almost everywhere.
Since we are interested in bi-directed linear dynamical system, we make the following assumption in the rest of the
paper.
Assumption 1: Hij (z) 6= 0 almost surely implies
Hji (z) 6= 0 almost surely.
Assumption 1 is valid in linearized models of diverse
engineering systems around an operating/ equilibrium point.
For example - swing dynamics for power systems, lumped
parameter RC network models for heat transfer dynamics and
consensus networks. Note that the transfer functions Hij (z)
and Hji (z) need not be same. We now associate a graphical
model to the transfer function matrix H(z).
Graphical Representation: Consider a directed graph
G = (V, E) with V = {1, ..., n} and E = {(i, j)|Hij (z) 6=
0}. Each node i ∈ V is representative of the measured time
series xi (k). In the graph G, there is a directed edge from j
to i if Hij (z) 6= 0. It follows from Assumption 1 that, there
is a directed edge from i to j as well. Thus G is a bi-directed
graph. We call G to be the generative graph of the measured
time series. Given generative graph G, its topology is defined
as the undirected graph GT = (V, ET ) obtained by removing
the orientation on all its edges, and avoiding repetition. An
example of bi-directed generative graph and its topology are
shown in Figure 1(a) and Figure 1(b) respectively. Next,
we present terminology for undirected graphs which will be
useful in the subsequent discussion.
Definition 1: (Path) A path between two nodes
x0 , xk in an undirected graph GT = (V, ET ) is a
set of unique nodes {x0 , x1 , · · · , xk } ⊆ V where
{(x0 , x1 ), (x1 , x2 ), · · · , (xk−1 , xk )} ⊆ ET . We will
denote a path by x0 − x1 − x2 − · · · − xk−1 − xk . The length
of a path is one less than the number of nodes in the path.
For example: 1 − 2 − 4 − 6 is a path of length three between
node 1 and 6 in the undirected graph of Figure 1(b).
Definition 2: (n Hop Neighbor) In an undirected graph
GT = (V, ET ), j ∈ V is a n hop neighbor of i ∈ V, if there
is a path of length n between i and j in GT . For example:
1 and 6 are three hop neighbors in the undirected graph in
Figure 1(b). If n = 1, i and j are termed neighbors in GT .
Definition 3: (Tree) A connected undirected graph without cycles is called a tree. There is a unique path between
any two nodes in a tree.
Definition 4: (Leaf Node/ non-leaf Node of a Tree) In a
tree T , a node with degree 1 is called a leaf node. Nodes
with degree greater than 1 are called non-leaf nodes.
Next we present the formal definition of a radial linear
dynamical system (RLDS).
Definition 5: (Radial Linear Dynamical System) Consider
a generative graph G with the associated topology being a
tree, which is denoted by T . A linear dynamical system
with the above properties is referred to as a Radial Linear
Dynamical System(RLDS). Figure 1(a) shows a RLDS with
the corresponding topology T shown in Figure 1(b).
Definition 6: (Power Spectral Density(PSD) Matrix) For
a n dimensional collection of WSS time series x(k) =
{x1 (k), ..., xn (k)}T , P
the power spectral density matrix is
∞
defined as ΦX (ω) = k=−∞ E(x(k)x(0)T )e−ιωk .
In this article, we will focus on learning the topology
of radial linear dynamical systems following Assumption 1.
The only information available for topology estimation are
time series measurements obtained from a subset of nodes in
the system, while certain nodes are unobserved. Our analysis
uses properties of inverse power spectral density of linear
dynamical systems, which is presented next.
III. T OPOLOGY L EARNING USING I NVERSE PSD
Let X(z) ∈ Cn denote the vector of z transform of
n nodal states, with X(z) = [Xo (z)T , XhT (z)]T , where,
Xo (z) ∈ Cm and Xh (z) ∈ Cn−m are the z transform of
the nodal states corresponding to m observed and n − m
unobserved nodes respectively. The network dynamics is
represented in a compact form as,
Xo (z)
Hoo (z) Hoh (z)
Xo (z)
Eo (z)
+
=
Xh (z)
Hho (z) Hhh (z)
Xh (z)
Eh (z)
where, Eo (z) and Eh (z) denote the exogenous inputs at the
observed and hidden nodes respectively. We assume that the
unobserved nodes are not neighbors in GT , that is, Hhh (z) =
0. Let Vo denote the set of observed nodes and Vh denote
the set of unobserved nodes and V = Vo ∪ Vh .
For notational simplicity we drop the argument z in the
discussion below. Let ΦX denote the power spectral density
matrix of the nodal states, that is,
ΦX = (I − H)−1 ΦE (I − H)−∗ ,
(3)
where, ΦE is the diagonal matrix of power spectral densities
of exogenous inputs and ∗ denotes the Hermittian operator.
The objective of the following analysis is to show that inverse
of the power spectral density of the states at the observed
nodes (denoted by Φoo ) leads to a graph with spurious edges
connecting up to four hop neighbors in GT . Let J denote the
inverse power spectral density matrix, that is,
−1
Joo (z) Joh (z)
Φoo (z) Φoh (z)
−1
J=
= ΦX =
,
Jho (z) Jhh (z)
Φho (z) Φhh (z)
= (I − H(z))∗ Φ−1
E (I − H(z)).
Using the matrix inversion lemma [23] it follows that,
Φ−1
oo
−1
= Joo − Joh Jhh
Jho
=: Γ + ∆ + Σ
Ψ(ku , j)
(4)
where,
Γ =
∆ =
Σ =
Λ =
Ψ =
∗
(I − Hoo
)Φ−1
Eo (I − Hoo ),
−1
∗
Hho ΦEh Hho , and,
−Ψ∗ Λ−1 Ψ, where
−1
∗
Hoh
Φ−1
Eo Hoh + ΦEh ,
−1
∗
Φ−1
Hoh
Eo (I − Hoo ) + ΦEh Hho ,
p=1
(5)
1) Suppose i and j are observed nodes and suppose in GT
(i) there is no path of the form i − k − j with k also
observed and (ii) i − j is not present, then Γ(i, j) = 0.
2) If in GT there is no path between two observed nodes
i and j, connected via a single unobserved node, ku ,
of the form i − ku − j, then ∆(i, j) = 0.
3) Suppose in GT there is no path between two unobserved nodes with a single intermediate observed node;
of the form ku − i − ku0 where ku and ku0 are not
observed and i is observed, then Λ is real and diagonal.
4) If in GT for j in the observed set of nodes and ku in
the unobserved set of nodes; (i) j − ku is not present
and (ii) there is no path of the form j − p − ku with p
being a observed node, then Ψ(k, j) = 0.
5) Suppose Λ is diagonal, and if in GT , for observed
nodes i and j and unobserved node ku , there are no
paths of the form i − p − ku or i − ku and j − p0 − ku
or j − ku for any p and p0 being observed, then
Σ(i, j) = 0.
Proof: 1) From (5),
=
−1
−1
−1
∗
∗
Φ−1
Eo − ΦEo Hoo − Hoo ΦEo + Hoo ΦEo Hoo .
Note that ΦEo is diagonal for i 6= j; from which it follows
that,
Γ(i, j)
=
−1
−Φ−1
Eo (i, i)Hoo (i, j) − Hoo (j, i)ΦEo (j, j)
P
m
−1
+ k=1 Hoo (k, i)Hoo (k, j)ΦEo (k, k).
The first two terms are zero if i − j is not present in GT and
the third term is zero if a path of the form i − k − j with k
being a observed node is not present in GT .
∗
2) Note that ∆ = Hho
Φ−1
Eh Hho and thus for i and j in the
observed set
X
∆(i, j) =
Hho (ku , i)Φ−1
Eh (ku , ku )Hho (ku , j).
ku ∈Vh
−1
−1
∗
∗
= [Hoh
Φ−1
Eo ](ku , j) − [Hoh ΦEo Hoo ](ku , j) + [ΦEh Hho ](ku , j)
m
X
Hoh (p, ku )Φ−1
(j,
j)
−
= Hoh (j, ku )Φ−1
Eo
Eo (p, p)Hoo (p, j)
+ Φ−1
Eh (ku , ku )Hho (ku , j).
Lemma 3.1: The following assertions hold
Γ
Thus if there is no path of the form i − ku − j where ku is
unobserved, then ∆(i, j) = 0.
0
0
3) Suppose ku 6= ku with ku and ku being unobserved
0
0
−1
∗
nodes. Note that Λ(ku , ku ) = [Hoh
Φ−1
Eo Hoh + ΦEh ](ku , ku ).
P
0
0
∗
Thus Λ(ku , ku ) = i∈Vo Hoh
(ku , i)Φ−1
Eo (i, i)Hoh (i, ku ) =
P
0
−1
i∈Vo Hoh (i, ku )ΦEo (i, i)Hoh (i, ku ), which is zero if
0
there is no path of the from ku − i − ku with
iP being an observed node. Moreover, Λ(ku , ku ) =
−1
Hoh (i, ku )Φ−1
=
Eo (i, i)Hoh (i, ku ) + ΦEh (ku , ku )
Pi∈Vo −1
−1
2
Φ
(i,
i)|H
(i,
k
)|
+
Φ
(k
,
k
)
∈
R.
oh
u
u
u
Eh
i∈Vo Eo
4) Note that
(6)
The first and the last term are zero if j − ku is not present
in GT and the second term is zero if there exist no path of
the form j − p − ku in GT , with p being an observed node.
5) Note that if Λ is diagonal, then,
X
Ψ(ku , i)Λ−1 (ku , ku )Ψ(ku , j),
Σ(i, j) = −
ku ∈Vh
Thus if there is no unobserved node ku with paths of the
form i − p − ku or i − ku and j − p0 − ku or j − ku for any
p and p0 being observed in GT , then from 4) of Lemma 3.1,
Ψ(ku , i)Ψ(ku , j) = 0 for every unobserved node ku , which
will imply Σ(i, j) = 0. This completes the proof.
We use the above lemma to present a result on topology
inference using the inverse of the power spectral density
of the observed time series. In this regard we make the
following assumption in the rest of the article.
Assumption 2: The unobserved nodes in topology GT are at
least four or more hops away from each other.
Remark 1: All the results presented in this article assume
that the latent nodes are at least three or more hops away
from each other except in Theorem 4.3, which requires that
unobserved nodes are four or more hops away from each
other.
Theorem 3.1: Consider a linear dynamical system with
topology GT such that Assumption 2 holds. Then
Φ−1
oo (i, j)(ω) 6= 0 almost surely for ω ∈ [0, 2π), implies
that, i and j are within four hops of each other in the graph
GT .
The proof follows from Lemma 3.1 and is omitted here due
to space restriction.
Remark 2: Note that the non-zero values in Σ(i, j) (and
subsequently in Φ−1
oo (i, j)) for three and four hop observed
nodes i, j result from paths of the form i − q − ku − j, i −
ku − p − j and i − q − ku − p − j in GT , with q and p being
observed neighbors of i and j respectively and ku being an
unobserved node.
Remark 3: Note that the system transfer functions have to
take very specific forms in order for Γ(i, j)+∆(i, j)+Σ(i, j)
to be zero even though i, j are either neighbors or two hop
neighbors. Thus, except for these pathological cases, if i
and j are neighbors, two hop neighbors (with the common
neighbor being observed or unobserved) Φ−1
oo (i, j) 6= 0
almost surely. Furthermore, for i, j being three or four hop
neighbors, Γ(i, j) = 0 and ∆(i, j) = 0. The second term of
Ψ(i, j) is a contributor to three/ four hop contributions and
is non zero in a large number of applications. For example:
suppose that bij ≥ 0, am,i ≥ 0 for all i, j in (1) (which is true
for engineering networks like power distribution systems, RC
networks etc.); then it is not possible that Φ−1
oo (i, j) = 0 if
i and j are three or four hop neighbors in GT . We assume
that the systems of interest do not belong to the small set of
pathological cases.
If we form a graph Gm using the non-zero values in
Φ−1
oo (i, j)(ω) as the adjacency matrix, we obtain all links
up to four hop neighbors in GT . This is summarized as
Algorithm 1 and is the first step in our topology learning
scheme. The next objective is to identify the true links as well
as eliminate the spurious links and identify the location of the
unobserved nodes in Gm obtained from Algorithm 1. Note
that Theorem 3.1 does not depend on the linear dynamical
system being radial. However in the subsequent analysis, we
will explicitly use the fact that GT = T is a RLDS.
Algorithm 1 Topology Learning using Power Spectrum
Input: Time series xi (k) from observed nodes
Output: Gm = (Vo , EGm ).
1: Edge set EGm ← {}
2: Compute Φ−1
oo (ω)
3: for all i ∈ {1, 2, ..., m}, i 6= j do
4:
if Φ−1
oo (j, i)(ω) 6= 0 then
5:
EGm ← EGm ∪ {(i, j)}
6:
end if
7: end for
IV. E XACT T OPOLOGY R ECOVERY IN R ADIAL L INEAR
DYNAMICAL S YSTEMS
To recover the exact topology of the radial linear dynamical system (RLDS) there the two tasks: one is to determine
the set of true edges in the graph obtained from Algorithm 1
and the next is to determine the location of the unobserved
nodes. We accomplish these tasks in the following two
subsections.
A. True Edge Discovery between Observed Nodes
Consider a RLDS with a tree topology T . Let the unobserved nodes be at least four or more hops away from each
other as per Assumption 2. The graph Tm obtained using
Algorithm 1 has edges between observed nodes that are up
to four hops away in T . The objective of this section is
to design an algorithm to identify the true links as well as
eliminate the spurious links and detect the location of the
missing nodes in Tm to recover T from Tm . In this regard,
we introduce the following assumption and the notion of
separation in graphs.
Assumption 3: Each unobserved node is at least three hops
away from all leaf nodes in T .
Based on the above assumption, it is clear that all leaf
nodes in T are observed nodes. Put differently, each unobserved node is buried deep into the network so that their
effect is ‘felt’ at multiple observed nodes.
Definition 7: (Separation in Graph) In an undirected graph
U, the set of nodes Z is said to separate the path between
nodes i and j, if there exist no path between i and j in
U after removing the set of nodes Z. We will use the
notation sep(i, Z, j), which is to be read as Z separates the
path between i and j in U. For example, in Figure 1(b)
sep(1, {2, 4}, 6) holds.
Next, we present a result, which enables us to categorize
true and spurious edges between observed non-leaf nodes in
Tm . The proofs of the results presented below are omitted
due to space restrictions.
Theorem 4.1: Consider a RLDS with a tree topology T
such that Assumption 2 and 3 hold. Let Tm be such that there
are links between any two observed nodes that are within
four hops in the underlying topology T (that is no link up
to four hops is undetected as discussed in Remark 3). There
exist observed nodes c, d distinct from observed nodes a, b
such that sep(c, {a, b}, d) holds in Tm if and only if a − b
is an edge (and thus a true edge) in T and a, b are non-leaf
nodes.
Remark 4: The above theorem provides a topological test
on Tm (which can be performed in polynomial time) to
identify the observed non-leaf nodes, Vnl,o and the true
edges between them. All other observed nodes in the graph
Vl := Vo \Vnl,o then are leaf nodes.
Note that of all edges connected to a leaf node in Tm , only
one edge is connected to its true non-leaf neighbor in T (rest
are spurious edges). From Assumption 3, each leaf node is
at least three hops away from any unobserved node in T .
By Lemma 3.1 and Remark 2, it clear that spurious edges
connected with a leaf node in Tm include those to its twohop neighbors. The next result utilizes the phase response of
the entries of Φ−1
oo to determine the true and spurious edges
associated with leaf nodes.
Theorem 4.2: Consider a RLDS such that Assumption 2
and 3 hold. Let a be a leaf node in T and let v be a non-leaf
neighbor of a in Tm . Then, ∠Φ−1
a,v (ω) = 0 for all ω ∈ [0, 2π)
if and only if a, v are two hop neighbors in T .
The proof uses algebraic expansions of the expressions for
Φ−1
a,v (ω) for leaf v and non-leaf a. We use the theorems
mentioned above to devise Algorithm 2 that identifies all
true edges between observed nodes in the system.
The last task that remains is to locate the unobserved
nodes, which is discussed in the next subsection.
B. Location of Unobserved Nodes
After application of Algorithm 1 followed by Algorithm
2, we end up with a graph T of observed nodes and edges
between them. However the discovered network will have
multiple disconnected radial components, with the disconnections being at the locations of unobserved nodes. We will
refer to T j as a discovered disconnected component. For
example, consider a tree T with just one unobserved node
l as shown in Fig. 2. Let l be between observed nodes c, e.
Then there exist the path C − c − l − e − E in T , with
Algorithm 2 True Edge Set Discovery Algorithm
Input: Tm = (V, ETm ) generated by Algorithm 1
Output: T = (V, ET )
1: Edge set ET ← {}
2: for all edge a − b in ETm do
3:
if Z := {a, b} there exist I 6= {φ} and J 6= {φ} such that
sep(I, Z, J) holds in Tm then
4:
Vnl ← Vnl ∪ {a, b}, ET ← ET ∪ {(a, b)}
5:
end if
6: end for
7: Vl ← V − Vnl
8: for all a ∈ Vl , b ∈ Vnl with (a, b) ∈ ETm do
9:
if ∠Φ−1
oo (a, b) 6= 0 for all ω ∈ [0, 2π) then
10:
ET ← ET ∪ {(a, b)}
11:
end if
12: end for
C := {v ∈ V|path between v, l involves node c},
E := {v ∈ V|path between v, l involves node e}.
Using Algorithm 2 leads to discovery of the individual
components C −c and e−E in T . Based on our assumptions,
it can be shown that each such component has at least
three observed nodes. Since, T is a connected graph, the
discovered disconnected components need to be connected
by locating the unobserved node in T . Next, we present the
result which enables us to do so.
Theorem 4.3: Let Tm be such that there are links between
any two observed nodes that are within four hops in the
topology T . Consider two discovered disconnected components T 1 , T 2 in T with observed nodes c ∈ T 1 and e ∈ T 2 .
If ∀ b ∈ T 1 , ∀f ∈ T 2 such that b − c and e − f are edges
in T and b, c, e, f form a clique in Tm , then, there exists an
unobserved node l such that c − l − e is a path in T .
Based on Theorem 4.3, we present Algorithm 3 that inserts
hidden nodes by considering spurious edges between pairs
of disconnected components in the discovered network. As
each observed node can have a maximum of only one hidden
node as neighbor, we merge hidden nodes that may have been
duplicated in Algorithm 3 while checking for Theorem 4.3
between multiple disconnected components connected to the
same hidden node.
Algorithm 3 Unobserved Node Placement Algorithm
Input: T = (VT , ET ) = ∪hj=1 T j
Output: T̃ = (VT̃ , ET̃ ).
1: Node set VT̃ ← VT
2: Edge set ET̃ ← ET
3: for all j ∈ {1, 2, ..., h} do
4:
for all i ∈ {j + 1, ..., h} do
5:
if there exist a pair of nodes a, b such that a ∈ T j and
b ∈ T i such that all their neighbors in T are connected in Tm
then
6:
VT̃ ← VT̃ ∪ lj
7:
ET̃ ← ET̃ ∪ {(a, lj ), (lj , b)}
8:
end if
9:
end for
10: end for
11: Merge hidden nodes that are neighbors of the same observed
node.
V. R ESULTS
Topology inference for power distribution networks can
be applied towards fault isolation, control and flow opti-
Fig. 2. Illustration of the application of Algorithm 1, Algorithm 2 and
Algorithm 3 in succession. The red node is the latent node and green edges
denote the spurious edges up to four hop neighbors.
mization. Penetration of devices like Phasor Measurement
Units(PMUs) enable real time measurement of phases of
various nodes and facilitate inverse problems like topology
inference and state estimation [24]. However, these meters
cannot be deployed at all nodes and partial network observability is indeed the situation.
We demonstrate the efficacy of the algorithm presented by
testing it on data obtained from simulations of the linearized
swing equations (see (7) below) on a 39 bus radial topology.
This radial system is obtained by deleting a few edges from
the IEEE 39 bus system and is shown in Figure 3(a). The
state xi (t) denotes the fluctuations of the phase angles of
node i from equilibrium values while pi (t) denote the nodal
injections due to generation and losses. The nodal injections
pj are colored noise and are generated by filtered version of
a white noise sequence. Four nodes are unobserved in this
study. For i = 1, 2, ..., 39,
mi ẍi + di ẋi =
39
X
j=1,j6=i
bij (xj (t) − xi (t)) + pi (t),
(7)
Here, mi , di , bij ∈ R≥0 for all i, j ∈ {1, 2, ..., 39}.
The error proportion is defined as the ratio of the sum
of number of true links undetected and number of false
links detected to the total number of true links. The error
proportion for the RLDS in Figure 3(a) using the algorithms
presented previously is shown in Figure 3(b). As the samples
per observed node is increased it is seen that the error proportion decreases rapidly. We reemphasize that our algorithms
do not use any knowledge of the system parameters and
noise injections. Moreover, the noise injections used in the
simulation is colored noise unlike white noise models used
in previous studies.
1
Error Proportion
0.8
0.6
0.4
0.2
0
0
2
4
6
8
10
12
Samples of each observed node
×10 5
(a)
(b)
Fig. 3. (a) A RLDS obtained from the IEEE 39 bus system, (b) error
proportion against the number of samples in topology inference of the
system shown in Figure 2(a).
VI. C ONCLUSION
We presented algorithms, which when applied in succession, leads to the exact topology recovery of a RLDS in the
‘sufficient statistics’(large number of data samples) regime
under partial observation of the network. The proofs involve
a synergy of tools from signal processing and probabilistic
graphical models. Algorithm 2 and Algorithm 3 being graph
based checks, can be executed in polynomial time. Among
all the algorithms presented, Algorithm 1 is computationally most intensive due to computation of the inverse. We
demonstrated the performance of the algorithm on a 39 node
radial power distribution network. This work also provides
insights on placement of sensors for observing the network
for monitoring and fault detection applications.
In future, we plan to relax the assumptions of unobserved
nodes being at least four hops away from each other as well
as at least three hops away from any leaf node.
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| 3 |
Facial Emotion Recognition using Min-Max
Similarity Classifier
Olga Krestinskaya and Alex Pappachen James
arXiv:1801.00451v1 [cs.CV] 1 Jan 2018
School of Engineering, Nazarbayev University, Astana
www.biomicrosystems.info/alex
Email: apj@ieee.org
Abstract—Recognition of human emotions from the imaging
templates is useful in a wide variety of human-computer interaction and intelligent systems applications. However, the automatic
recognition of facial expressions using image template matching
techniques suffer from the natural variability with facial features
and recording conditions. In spite of the progress achieved
in facial emotion recognition in recent years, the effective
and computationally simple feature selection and classification
technique for emotion recognition is still an open problem. In
this paper, we propose an efficient and straightforward facial
emotion recognition algorithm to reduce the problem of interclass pixel mismatch during classification. The proposed method
includes the application of pixel normalization to remove intensity
offsets followed-up with a Min-Max metric in a nearest neighbor
classifier that is capable of suppressing feature outliers. The
results indicate an improvement of recognition performance
from 92.85% to 98.57% for the proposed Min-Max classification
method when tested on JAFFE database. The proposed emotion
recognition technique outperforms the existing template matching methods.
Index Terms—Face emotions, Classifier, Emotion recognition,
spatial filters, gradients
I. I NTRODUCTION
In recent years, the human-computer interaction challenge
has led to the demand to introduce efficient facial and speech
recognition systems [1]–[5]. Facial emotion recognition is
the identification of a human emotion based on the facial
expression and mimics [6]. The facial emotion recognition has
a wide range of appliction prospects in different areas, such as
medicine [7], robotics [8], [9], computer vision, surveillance
systems [1], machine learning [10], artificial intelligence,
communication [11], [12], psychological studies [4], smart
vehicles [9], security and embedded systems [13].
There are two main approaches for facial expression recognition: geometry-based and appearance-based methods [2].
The geometry-based methods extract main feature points and
their shapes from the face image and calculate the distances
between them. While, appearance-based methods focus on
the face texture using different classification and template
matching methods [14], [15]. In this paper, we focus on facial
emotion recognition based on template matching techniques
that remains a challenging task [16]–[18]. Since the orientation
of pixel features are sensitive to the changes in illumination, pose, scale and other natural imaging variabilities, the
matching errors tend to be high [4], [19], [20]. Pixel matching
methods are known to be useful when the images has missing
features because imaging matrices become sparse and feature
computation process is not trivial. As facial expressions cause
a mismatch of intra-class features due to their orientation
variability, it is difficult to map them between the imaging
templates.
Facial emotion recognition accuracy depends on the robustness of a feature extraction method to intra-class variations and
classifier performance under noisy conditions and with various
types of occlusions [10]. Even thought a variety of approaches
for the automated recognition of human expressions from
the face images using template matching methods have been
investigated and proposed over the last few years [14], the
emotion recognition method with robust feature extraction
and effective classification techniques accompanied by low
computational complexity is still an open research problem
[21]. Therefore, in this paper, we address the issues of
matching templates through pixel normalization followed by
the removal of inter-image feature outliers using a Min-Max
similarity metric. We apply Gaussian normalization method
with local mean and standard deviation to normalize the
pixels and extract relevant face features followed by MinMax classification method to determine an emotion class. The
simulation is performed in Matlab for the Japanese Female
Facial Expression (JAFFE) database [22] and the emotion
recognition accuracy is calculated using leave-one-out crossvalidation method.
The main contributions of this work are the following:
•
•
•
We develop a simplified approach for facial emotion
recognition with template matching method using Gaussian normalization, mean and standard deviation based
feature extraction and Min-Max classification approach.
We present simple and effective facial emotion recognition algorithm having low computational complexity
and ability to suppress the outliers and remove intensity
offsets.
We conduct the experiments and simulations on JAFFE
database to demonstrate the efficiency of the proposed
approach and highlight its advantages, comparing to the
other existing methods.
The paper is organized as follows. Section II presents the
overview of the existing methods for facial emotion recognition, their drawbacks and reasons to propose a new method.
In Section III, we show normalization, feature extraction
and classification parts of the proposed method, present the
algorithm and describe the experiments. Section IV contains
the simulation results and comparison of the obtained results
with the existing methods. In Section V, we discuss the
benefits and drawbacks of the proposed method, in comparison
to the traditional methods. Section VI concludes the paper.
II. BACKGROUND AND RELATED WORKS
To address the problem of facial emotion recognition, several template matching methods have been proposed in the last
decades [1], [8], [23]–[25]. In most of the cases, the process of
emotion recognition from human face images is divided into
two main stages: feature extraction and classification [1], [8].
The main aim of feature extraction methods is to minimize
intra-class variations and maximize inter-class variations. The
most important facial elements for human emotion recognition
are eyes, eyebrows, nose, mouth and skin texture. Therefore, a
vast majority of feature extraction methods focus on these features [2], [26]. The selection of irrelevant face image features
or insufficient number of them would lead to low emotion
recognition accuracy, even applying effective classification
methods [21]. The main purpose of the classification part is
to differentiate the elements of different emotion classes to
enhance emotion recognition accuracy.
The commonly used feature extraction methods include twodimensional Linear Discriminant Analysis (2D-LDA) [8], [25],
two-dimensional Principle Component Analysis (2D-PCA)
[27], Discrete Wavelet Transform (DWT) [6], [8], [28], Gabor
based methods [29], [30] and wavelets-based techniques [23],
[31]. In 2D-LDA method, the two-dimensional image matrix
is exploited to form scatter matrices between the classes and
within the class [8]. 2D-LDA method can be applied for facial
features extraction alone or accompanied with the other feature
extraction method, as DWT [8], [25]. In 2D-PCA feature
extraction method, the covariance matrix representation of the
image is derived directly from the original image [8], [27]. The
size of the derived principle component matrix is smaller than
the original image size that allows to decrease the amount
of processing data, and consequently, reduce the required
computational memory [32]. However, 2D-LDA and 2D-PCA
methods applied in template matching techniques require an
additional processing of the image, dimensionality reduction
techniques or application of another feature extraction method
to achieve higher recognition accuracy, which leads to the
increase in processing time.
The other feature extraction method is DWT. This method
is based on the low-pass and high-pass filtering, therefore,
it is appropriate for the images with different resolution
levels [8]. In the emotion recognition task, DWT is applied
for the extraction of useful features from the face images
and can be replaced with its Orthogonal Wavelet Transform
(OWT) and Biorthogonal Wavelet Transform (BWT) having
the advantages of orthogonality [6]. Another method for facial emotion recognition is Gauss-Laguerre wavelet geometry
based technique. This method represents the processed image
in polar coordinates with the center at a particular pivot point.
The degree of freedom is one of the advantages that GaussLaguerre approach provides, which in turn allows to extract of
features of the desirable frequency from the images [23], [31].
However, DWT and Gauss-Laguerre approaches are complex
and require time and memory consuming calculations.
The classification of the extracted features can be implemented using Support Vector Machine (SVM) algorithm [8],
[28], K-Nearest Neighbor (KNN) method [23], [33], Random
Forest classification method [7], [34] and Gaussian process
[24]. The SVM principle is based on non-linear mapping and
identification of a hyperplane for the separation of data classes.
SVM classifier is used with the application of different kernel
functions, such as linear, quadratic, polynomial and radial
basis functions, to optimize the SVM performance [8], [28].
KNN approach is based on the numerical comparison of a
testing data sample with training data samples of each class
followed by the determination of the similarity scores. The
data class is defined by K most similar data samples based
on the minimum difference between train and test data [23],
[33]. KNN and SVM classifiers are simple and widely used for
emotion recognition, however these classifiers do not suppress
the outliers that leads to lower recognition accuracy.
The Random Forest classification method is based on the decision making tree approach with randomized parameters [7],
[34]. To construct the decision tree, Random Forest algorithm
takes a random set of options and selects the most suitable
from them [35]. Random Forest classifier is robust and has a
high recognition rate for the images of large resolution [36].
The drawback of Random Forest classifier is its computational
complexity. The other classification method is the Gaussian
process approach. Gaussian process is based on the predicted
probabilities and can be used for facial emotion recognition without application of feature selection algorithms. The
Gaussian process allows a simplified computational approach,
however has a smaller emotion recognition rate, comparing to
the other methods [24].
Even thought the a number of methods for feature extraction
and classification have been proposed, there is a lack of template matching methods that allow to achieve high recognition
accuracy with minimum computational cost. Therefore, the
method that we propose has a potential to reduce the computational complexity of facial emotion recognition operation and
increase the recognition accuracy due to the simplicity of the
algorithm, effective feature extraction and ability to suppress
outliers. The proposed algorithm can be implemented using
small computational capacity devices keeping facial emotion
recognition operation fast and accurate.
III. M ETHODOLOGY
The main idea of the proposed method is to extract the
spatial change of standardized pixels in a face image and
detect the emotion class of the face using a Min-Max similarity Nearest Neighbor classier. The images from the JAFFE
database [22] are used for the experiments. This database
contains 213 images of 10 female faces comprising 6 basic
facial expressions and neutral faces. The original images from
the database have the size of 256 × 256 pixels and in our
experiments they are cropped to a size of 101 × 114 pixels
retaining only the relevant information of the face area. A
block diagram of the proposed method is shown in Fig. 1.
JAFFE database does not contain the face images with different illumination conditions, the illumination change was
created by adding and subtracting the value of 10 from the
original image. Fig. 2 (b) illustrates the respective images after
Gaussian local normalization. Irrespective of the illumination
conditions, the three locally normalized images appear similarly with the minimum pixel intensity variation.
(b)
(a)
(c)
Fig. 2. (a) Sample image from JAFFE database with different lighting
conditions obtained by adding and subtracting a value of 10 from the original
image.(b) Normalized images of above sample images obtained by performing
Gaussian normalization using local mean and local standard deviation taken
over a window of size N=11. (c) Feature detected images from the normalized
image by performing local standard deviation using a window of size M=11.
Fig. 1. Outline of the proposed emotion recognition system
B. Feature detection
A. Pre-processing
Illumination variability introduces the inter-class feature
mismatch resulting in the inaccuracies in the detection of
emotion discriminating features from the face images. Therefore, image normalization is essential to reduce the inter-class
feature mismatch that can be viewed as intensity offsets. Since
the intensity offsets are uniform within a local region, we
perform Gaussian normalization using local mean and standard
deviation. The input image is represented as x(i, j), and y(i, j)
is the normalized output image, where i and j are the row and
column number of the processed image. The normalized output
image is calculated by Eq. 1 [37], where µ is a local mean
and σ is a local standard deviation computed over a window
of N × N size.
The feature parts useful for the facial emotion recognition
are eyes, eyebrows, cheeks and mouth regions. In this experiment, we perform the feature detection by calculating local
standard deviation of normalized image using a window of
M×M size. Eq. 4 is applied for the feature detection with
b = (M − 1)/2.
v
u
b
b
X
u 1 X
[y(k + i, h + j) − µ0 (i, j)]2 (4)
w(i, j) = t 2
M
k=−b h=−b
In Eq. 4 the parameter µ0 refers to the mean of the
normalized image y(i, j) and can be calculated by Eq. 5.
b
b
1 X X
µ (i, j) = 2
y(k + i, h + j)
M
0
y(i, j) =
x(i, j) − µ(i, j)
6σ(i, j)
(1)
The parameters µ and σ are calculated using Eq. 2 and 3
with a = (N − 1)/2.
µ(i, j) =
a
a
1 X X
x(k + i, h + j)
N2
(2)
k=−a h=−a
v
u
a
a
X
u 1 X
σ(i, j) = t 2
[x(k + i, h + j) − µ(i, j)]2 (3)
N
k=−a h=−a
An example image from the JAFFE database with three
different lighting conditions is shown in Fig. 2 (a). As the
(5)
k=−b h=−b
Fig. 2 (c) shows the results of feature detection corresponding to the normalized images.
C. Emotion Classification
For the recognition stage, we propose a Min-Max similarity metric in a Nearest Neighbor classifier framework. This
method is based on the principle that the ratio of the minimum
difference to the maximum difference of two pixels will
produce a unity output for equal pixels and an output less than
unity for unequal pixels. The proposed method is described in
Algorithm 1. The algorithm parameter trainlen refers to the
number of train images, N corresponds to the normalization
window size, and M indicates the feature detection window
size. Each cropped image is of m × n pixel dimension. The
parameter train is a feature array of trainlen×(m × n) size,
where each row corresponds to the processed train images.
After normalization and feature detection, test images are
stored in to a vector test of 1×(m × n) size. A single test
image is compared pixel-wise with processed train images of
all the classes in the feature array using the proposed Min-Max
classifier:
s(i, j) = [
min[train(i, j), test(1, j)] α
] ,
max[train(i, j), test(1, j)]
(6)
where a parameter α controls the power of exponential to
suppress the outlier similarities. Outliers are the observations
that come inconsistent with the remaining observations and
are common in real-time image processing. The presence of
outliers may cause misclassification of an emotion, since sample maximum and sample minimum are maximally sensitive
to them. In order to remove the effects of the outliers, α = 3 is
selected to introduce the maximum difference to the inter-class
images and minimum difference to the intra-class images.
After Min-Max classification, a column vector z of
trainlen × 1 size containing the weights obtained after comparing the test image with each of the trainlen number of train
images is calculated using Eq. 7.
z(i) =
m×n
X
s(i, j)
IV. R ESULTS AND COMPARISON
To benchmark the performance of the proposed algorithm,
leave-one-out cross-validation method has been used. In this
method, one image of each expression of each person is
applied for testing and the remaining images are used for
training [23]. The cross-validation is repeated 30 times to
obtain a statistically stable performance of the recognition
system and to ensure that all the images in JAFFE database are
used for testing at least once. The overall emotion recognition
accuracy of the system is obtained by averaging the results of
the cross-validation trials. Fig. 3 shows the different accuracy
rates obtained for each trial by varying feature detection
window size M from 3 to 21 and keeping normalization
window size at N = 11. It is shown that the maximum emotion
recognition accuracy this normalization window size can be
achieved with the detection window size of 11.
(7)
j=1
The classification output out shown in Eq. 8 is the maximum
value of z corresponding to the train image that shows the
maximum match. The recognized emotion class is the class of
the matched train image.
out = max(z(i))
(8)
Fig. 3. The accuracy rates obtained for four trials of leave-one-out crossvalidation method for different feature detection window size M ranging from
3 to 21.
Fig. 4 shows the recognition accuracy rates obtained for
Algorithm 1 Emotion Recognition using Min-Max classifier different normalization and feature detection window sizes
Require: Test image Y,Train images Xt ,trainlen,window size ranging from 3 to 21 for a single trial.
N and M
1: Crop the images to a dimension of m × n
2: for t = 1 to trainlen do
3:
C(i, j) = Xt (i,j)−µ(i,j)
q 6σ(i,j)
Pb
Pb
4:
W (i, j) = M12 k=−b h=−b [C(k + i, h + j) − µ(i, j)]2
5:
Store the value of W to an array train of dimension
trainlen × m × n
6: end for
7: for t = 1 to trainlen do
8:
V (i, j) = Y (i,j)−µ(i,j)
6σ(i,j)
q
Pb
Pb
9:
test(i, j) = M12 k=−b h=−b [V (k + i, h + j) − µ(i, j)]2
min[train(t,j),test(1,j)] 3
s(t, j) = [ max[train(t,j),test(1,j)]
]
Pm×n
11:
z(t) = j=1 s(t, j)
12:
out = max(z(t))
13: end for
10:
Fig. 4. Graph shows the accuracy rates obtained for different normalization
and feature detection window sizes ranging from M=N=3 to 21 for one trial
of leave-one-out cross-validation method.
To evaluate the performance of the proposed Min-Max
classifier, it has been compared with the other classifiers,
such as Nearest Neighbor [38] and Random Forest [39], after
normalization and feature detection. The obtained accuracy
values are shown in Table I.
TABLE I
T ESTING OF FEATURE DETECTED IMAGES ON OTHER CLASSIFIERS
Classif ier
Nearest Neighbor
Random Forest
Proposed Min-Max classifier
Accuracy(%)
92.85
88.57
98.57
The proposed method achieves a maximum accuracy of
98.57% for a window size of M = N = 11 which outperforms
the other existing methods in the literature for leave-one-out
cross-validation method. Table II shows the performance comparison of the proposed method with the other existing systems
on the same database using leave-one-out cross-validation
method. Applying the proposed method, we achieved emotion
recognition accuracy of 98.57% proving the significance of
emotion detection method with Min-Max similarity classifier.
TABLE II
C OMPARISON OF PROPOSED EMOTION RECOGNITION SYSTEM WITH
OTHER EXISTING SYSTEM BASED ON LEAVE - ONE - OUT CROSS - VALIDATION
methods. The effect of the proposed Min-Max classification
method on recognition accuracy is also important. Table I
shows the application of the other classification method for the
same approach where proposed Min-Max classifier illustrates
the performance improvement. In comparison to the existing
method, the proposed Min-Max classifier has an ability to
suppress the outliers that significantly impacts overall performance of this approach.
The simplicity and straightforwardness of the proposed
approach are also important due to the resultant low computational complexity. Most of the existing methods use complicated feature extraction and classification approaches that
double the complexity of the facial recognition process and
require the device with large computational capacity to process
the images. We address this problem applying direct local
mean and standard deviation based feature detection methods
and simple Min-Max classification method. In comparison to
the existing feature detection methods, such as PCA [27] and
LDA [8], the proposed method is straightforward and does
not require dimensionality reduction. Moreover, simple MinMax classification method also reduce the computational time,
comparing to the traditional classification approaches, such
as SVM [28], KNN [33], Gaussian process [24] and Neural
Network [41]. Therefore, the algorithm can be run on the
device with low computational capacity.
METHOD
Existing systems
Cheng et. al [24]
Hamester et. al [40]
Frank et. al [8]
Poursaberi et. al [23]
Hegde et. al [29]
Proposed method
M ethod used
Gaussian Process
Convolutional Neural
Network
DWT + 2D-LDA
+SVM
Gauss
Laguerre
wavelet+ KNN
Gabor and geometry
based features
Min-Max classifier
Accuracy(%)
93.43
95.80
95.71
96.71
97.14
98.57
In addition, comparing to the proposed emotion recognition
system, the other existing methods require specialized feature
extraction and dimensionality reduction techniques before
classification stage. The main advantages of the proposed
emotion recognition system are its simplicity and straightforwardness.
V. D ISCUSSION
The main advantages of the proposed facial motion recognition approach are high recognition accuracy and low computational complexity. To achieve high recognition accuracy,
the effective feature selection is required. In the existing
methods, the complex algorithms for feature selection are
applied without normalizing the image. The normalization
stage is important and has a considerable effect on the accuracy
of the feature selection process. In the proposed algorithm,
we apply simple pre-processing methods to normalize the
images and eliminate intensity offsets that effects the accuracy
of the feature selection process and leads to the increase of
emotion recognition accuracy, in comparison to the existing
VI. C ONCLUSION
In this paper, we have represented the approach to improve the performance of emotion recognition task using
template matching method. We have demonstrated that the
pixel normalization and feature extraction based on local
mean and standard deviation followed up by the Mix-Max
similarity classification can result in the improvement of
overall classification rates. We achieved emotion recognition
accuracy of 98.57% that exceeds the performance of the
existing methods for the JAFFE database for leave-one-out
cross-validation method. The capability of the algorithm to
suppress feature outliers and remove intensity offsets results
in the increase of emotion recognition accuracy. Moreover,
the proposed method is simple and direct, in comparison to
the other existing methods requiring the application of dimensionality reduction techniques and complex classification
methods for computation and analysis. Low computational
complexity is a noticeable benefit of the proposed algorithm
that implies the reduction of computational time and required
memory space. This method can be extended to the other
template matching problems, such as face recognition and
biometric matching. The drawback of the proposed method, as
in any other template matching method, is the metric learning
requiring to create the templates for each class that, in turn,
consumes additional memory space to store the templates.
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| 1 |
How to be correct, lazy
and efficient ?
C. Recanati
Université Paris 13,
av. JB. Clément,
1. Lambdix and the semantic problems
93430, Villetaneuse,
France
of Lisp
This paper is an introduction to
Lambdix, a lazy Lisp interpreter
implemented at the Research Laboratory
of the University of Paris XI (Laboratoire
de Recherche en Informatique, Orsay).
Lambdix was devised in the course of an
investigation into the relationship between
the semantics of programming languages
and their implementation; it was used to
demonstrate that in the Lisp domain,
semantic correctness is consistent with
efficiency, contrary to what has often been
claimed.
The first part of the paper is an
overview of well-known semantic
difficulties encountered by Lisp as well as
an informal presentation of Lambdix; it is
shown that the difficulties which Lisp
encouters do not arise in Lambdix. The
second part is about efficiency in
implementation models. It explains why
Lambdix is better suited for lazy
evaluation than previous models. The
section ends by giving comparative
execution time tables.
In this section the semantical defects of
lisp1 are reviewed and shown not to exist
in Lambdix. These defects are
characteristic of early versions of lisp, but
they are still present in current versions
(although, of course, not all defects are
present in all versions); this is why we
think this little overview is not out of date
even if some of points we make are now
well-known.
1.1. The functional argument problem
Lisp was first thought of as being an
implementation of the lambda calculus. Its
syntax allows the definition of functions
by means of lambda expressions, as for
instance
(lambda (x y) (+ x y))
1
By 'lisp' here we mean a family of
languages rather than a particular language
belonging to this family.
page 1
for addition. This is a fundamental feature
when the function returns the lambda
of Lisp. Now serious problems arise when
expression; consequently, when the
these lambda expressions are used as
anonymous lambda function is applied, x
functional arguments - when a function
recovers its previous value - which is 456
takes another function as argument, and
or not defined.
also when a function returns such a
function as value.
Example 2
$ (define apply (f x) (f x) )
Example 1
$ ( define Identity ( x )
$ ( define BuildConstFunc (x)
( apply (lambda (y) x) 2))
( lambda(y) x ))
The Identity function defined here
The function BuildConstFunc is
is constructed by applying a function
supposed to return a function lambda of y,
returning x. In Lambdix, a call to (
which returns x. This lambda function
Identity 45) returns 45 but in lisp:
ought to be a constant function since its
argument y is not used. Then a call to (
$ ( Identity 45)
(BuildConstFunc 0) 1)
=> 2
ought to return 0, as well as
( (BuildConsFunc 0) 2). This is of course
the case in Lambdix, but not in Lisp
seems bound to 2. The source of the
trouble is the use of the name x in the
where:
definition of apply. Had apply been
$ ((BuildConsFunc 0) 1)
defined as
=> ** error - x not defined **
or
This is even more surprising since only y
$ (setq x 456)
$ ( define apply (f z)
$ ((BuildConsFunc 0) 1)
(f z) )
=> 456
the problem would have disappeared. Here
The reason for this surprising
it is not because the stack has been popped
answer is the following. In most lisps
too early that the binding fails, but
environments (bindings between names
because an intermediate binding has been
and values) are represented in a stack
inserted: x is first bound to 45; then the
which is popped when the execution of the evaluation of apply inserts the binding of
function terminates. The binding of x to 0 x to 2. The lambda is evaluated with y
in the application of BuildConsFunc is lost
page 2
bound to 2 and it returns x - which is now
part of the benefits of functional
bound to 2.
programming is lost.
Because of these functional
argument problems, Lisp is not a true
1.2. Lexical scope vs dynamic scope
second order functional language1, and
The foregoing examples show that the
1
Some lisps proposed a function to solve
the functional argument problem. This
function, sometimes called closure, takes
two arguments - a list of formal
parameters and an expression - and returns
the application of a lambda expression.
For instance, the value returned by the
importance in Lisp. This feature is due to
the type of variable binding used in Lisp,
called 'most recent binding' (MRB, for
short). MRB was perhaps, as Gowan has
wittily said, the 'most recent error'
([GOW72]). Originally motivated by the
technical advantages of the
closure function applied to the list '(x)
implementation, dynamic scoping of the
and an expression Expr - in an
type illustrated by MRB has disastrous
environment where x is bound to 2 - is
consequences for the safety of the
given by :
language. How can you trust a language
$ ( closure '(x) 'Expr ) when x =2
=> ((lambda (x) Expr)
names of formal parameters are of major
in which the names of formal parameters
'2 )
must be taken into account?
The use of dynamic scoping in
Although this can punctually solve the
problem, many critics can be made to this
solution. First, it is not very efficient
because it adds function calls and requires
the evaluation of all the parameters to be
saved. Moreover, it is an ad hoc solution.
It requires the programmer to know when
the closure function is necessary since
Lisp conflicts with the original model
introduced by Church, which was at the
source of Lisp. In lambda calculus, a rule
known as the ß-rule allows the reduction
of terms along the following pattern:
( ( λ x . expr ) val )
→
the call to the closure function must be
expr [ x ← val ]
explicit. Furthermore, lisp distinguishing
between an f-value and a c-value, some
particular attention must be given on the
way the interpreter pass the arguments.
value interpretation and this additional call
This requires the use of a special function
must also be explicit.
(called funcall ) to force the functional
page 3
The ß-rule takes as input the application to
allows the use of functional arguments and
a value val of a lambda expression with
is therefore a true second order language.
formal parameter x and yields as output
the same expression in which all tokens of
1.3. Evaluation order
x have been replaced by val. The
substitution is supposed to be determined
1.3.1. Call by need and call by value
by the inner lambda binding in the
(program) text, i.e. lexically.
In lambda calculus, a term t defined by
In most cases, dynamic scope and
(λ x y . x) A ((λ u . u u) (λ u . u u))
lexical scope yield the same result, but it
is very easy to construct cases with
reduces to A because the first reduction
(the substitution of A to x) yields a term, λ
diverging answers1 . This is why - like
many functional languages today
y . A, which always reduces to A because
(Common Lisp, Scheme, ML, etc.) -
y does not occur in the core of this lambda
Lambdix consistently uses lexical scope
expression. But in Lisp the interpreter
for formal parameters. Lambdix also
computes the values of the arguments
before the core of the function. This
strategy of parameter evaluation is known
For instance the term t defined by
t ≡ ( λ x . ( λ y . ( λ x . y) B ) x) A
as call by value. Since in the evaluation of
would reduced to A by ß-reduction
t
→
( λ y . ( λ x . y) B ) A)
calculation of the second argument - the
term (( λ u . u u) ( λ u . u u)) - generates
1
→
( λ x . A) B )
→
A
t all arguments are calculated first, the
an infinite loop:
y
≡ (( λ u . u u) ( λ u . u u))
while it would reduced to B in the
→ (( λ u . u u) ( λ u . u u))
dynamic model:
( λ x . ( λ y . ( λ x . y) B ) x) A:
→ (( λ u . u u) ( λ u . u u))
( λ y . ( λ x . y) B ) x)
( λ x . y) B )
y
→
Stack
[x–A]
...
In the framework of the lambda
[ x – A ][ y – x ]
calculus, call by value can be understood
[ x – A ][ y – x ][ x – B ]
as a strategy governing the order in which
B
the ß-reductions are performed. This
A formal demonstration of the non
strategy guarantees that if there is a unique
equivalence of the two models can be
solution, the calculus will converge on it.
found in A. Eick and E. Fehr [EIC85].
Unfortunately it also guarantees that the
page 4
calculus will never return if there is also
arguments and if the latter were calculated
an infinite derivation. Thus for terms
only when necessary. Such a strategy of
having both a finite and an infinite
evaluation is known as call by need. For
derivation, this strategy guarantees that the instance, the function f defined by
finite derivation will not be found. Hence
( de f (x y)
it is not a winning strategy. The winning
(if
strategy of the lambda-calculus requires
(< x 0)
that the leftmost inner term be reduced
1
first. If there is a finite solution, the
(f (- x 1) (f x y))))
calculus will converge on it (the
confluence property guarantees the unicity
of the finite solution). This strategy
will never terminate from a call to (f 1 2)
if the interpreter conforms to the call by
value strategy, but it returns 1 if the
corresponds to call by need.
interpreter conforms go the other strategy
Functions in Lisp are strict1,
(call by need ).
because of the strategy of the interpreter
The strategic choice made by Lisp
(call by value). But there is a Lisp
is not necessarily objectionable on
function which is not strict: the
semantic grounds, because strictness can
conditional test. By definition, the if
be seen as a positive feature; moreover, it
function does not calculate all its
arguments: the computation of the second
and the third argument is determined by
has the advantage of clarity. But it entails
a loss of expressive power, because the set
of defined functions is smaller than the set
the result of the computation of the first.
Consequently, nonstrict functions could be
constructed if the core of the functions
of convergent terms of lambda calculus.
The reason why Lisp has made this choice
is again efficiency: more often than not,
was evaluated before the values of the
implementations of call by need are
utterly inefficient.
1
A function is strict if, when applied to an
undetermined value, the result is
undetermined. This property is usually
written by the equation
F (⊥) = ⊥
where ⊥ denotes an undetermined value.
In case of several arguments, a function is
strict when:
F ( ... , ⊥ , ... ) = ⊥
page 5
functions we are talking about. Laziness
certainly decreases efficiency in
1.3.2. Lazy evaluation
connection with some functions, but it
yields fascinating results in connection
An interpreter conforming to the call by
need strategy is called a lazy interpreter.
with other functions. Well exploited, lazy
evaluation becomes very efficient in data
There are various degrees of laziness,
however. Pure laziness corresponds to the
situation in which the only arguments that
oriented algorithms, for instance in expert
systems or prolog interpreters.
A lazy cons allows the
are evaluated are the arguments of the
printing functions. Such a form of laziness
manipulation of potentially infinite lists
called streams. In Lambdix, we can
is very inefficient and "lazy evaluation"
define1 an infinite sequence of 1 by:
generally denotes the following pattern:
$ ( de x (cons 1 x))
and nevertheless have
◊ call by need for user defined
$ (cadr x)
functions
=1
◊ delayed evaluation for the function
cons
An infinite list of integers can be also
◊ strict primitive functions
defined as a function from:
(+, -, etc.) stand strict
It is the second point which makes the real
$ ( de (from x)
(cons x (from (+ x 1))))
difference between simple call by need
for user defined functions and lazy
The Lambdix interpreter will have no
evaluation. The introduction of a lazy
problem with
constructor on lists sometimes greatly
improves efficiency. Lists are the most
$ (print (cadr (from 2)))
important structures in Lisp and lazy
evaluation offers new ways of handling
=3
them.
Though laziness is a richer model,
1
This is a recursive definition - which is
distinct from the traditional setq of lisp:
it has always been considered less
efficient than the other evaluation patterns.
$ (setq x 2)
But is it really less efficient? This question
$ (setq x (cons 1 x))
cannot be answered in a straighforward
$ (cadr x)
manner, for it all depends on which
=2
page 6
Another difference between Lisp and
while Lisp gets trapped into an infinite
Lambdix is that Lambdix distinguishes
loop:
between program and text. Nevertheless,
Lambdix provides two operators for an
= 1 2 3 4 5 6 7 8 9 10 ....
explicit conversion between text and
Infinite structures can be very pleasant and program. This makes it possible for
programs to 'modify their own text' during
efficient in many programs. For instance
in numeric application, lazy evaluation is
execution.
Primitive objects of Lambdix are
well suited for computing with formal
series. The use of infinite structures can
typed (numbers, strings, characters, lists,
also simplify algorithms. It can suppress
booleans, primitive arithmetic operations,
tests and makes algorithms shorter.
primitive boolean operations, etc.). But
For instance, it suppresses the need for
iterators in unification programs.
there are two levels of language: the level
of the program text and that of its
semantic interpretation (i.e. its value). In
Stream processing is essential in
some simulation programs. It provides an
Lambdix quote does not indicate the
alternative to programming with
absence of evaluation, as it does in Lisp.
assignments. The FlipFlop RS gives an
Quote performs an operation which
interesting example of this feature:
converts any piece of an already computed
part of the program into a list. Lists in
Lambdix are constant values; they are not
$ (de (FlipFlop R S)
interpreted as forms. A list is a piece of
(let ( (de Q (NAnd S QBAR))
text which remains constant until the
(de QBAR (NAnd R Q)))
operation excla is applied. Excla is the
(cons Q QBAR)))
dual of quote. When applied to a list, the
Note that the definitions of Q and QBAR
list is interpreted in the program text as if
are mutually recursive and that the NAnd
introduced here will be an operator on
the corresponding piece of text had been
written there. For instance, the function
streams.
mapfun which constructs the list of the
applications of a function f to all the
elements of a list l can be written as1:
1.4. Reflexivity in Lisp
1
An obvious advantage of this definition
is that it is perfectly general; there is no
need for a distinction between f-subr, f-
page 7
be defined. Top level names in Lisp are
mutually defined2.
$ (de (mapfun f l)
(if (nullist l) ()
Nevertheless Lisp provides mutual
(cons ( ! (cons f (car l)))
definitions only at top level. In Lisp the
(mapfun f (cdr l))))))
expression
$ (mapfun + '((1 2) (2 3) (3 4)))
=(357)
(let ( (var1 expr1)
... (varN exprN) )
Here the characters ! and ' stand for the
body )
operators excla and quote respectively.
is equivalent to the application
1.5. Naming variables
( (lambda (var1 ... varN) body )
A fundamental difference between Lisp
expr1 ... exprN )
and the lambda calculus is that in
Lambdix function call is not the only way
ExprN are calculated in the calling
of binding names to values1. Most lisps
environment. Thus cross references cannot
have at least three other binding
be made without additional functional
mechanisms: affectation (setq ), function
arguments. Since Lisp has problems with
definition (introduced by
second order functions, proper recursive
de ) and local definitions (introduced by
definitions cannot be really introduced at
let ).
this particular level.
In Lambdix a recursive let has
been introduced in order to make local
1.5.1. Naming functions
Names introduced by de allow a term to
refer to itself in its own definition in such
2
a way that self applicative functions can
operator allows self reference and then,
This is not a special property of lisp. In
the lambda calculus, the fixed point
since function can be pass as arguments, is
is possible to write cross referenced terms.
expr, or macro functions as possible
In fact, it is lisp which offers restricted
argument values.
possibilities, since functions cannot be
1
passed as arguments (then cross references
Without this alternative, the choice of
dynamic scoping would really be
cannot be done outside the top level
meaningless
definitions).
page 8
function definitions mutual like top-level
the terms Q and QBAR must be mutually
definitions. The distinction between
defined.
function value and cell value has also been
removed from Lambdix. In Lambdix a
1.5.2. Naming stores: the assignment
term has a unique value. Thus thede
problem
function can be used for setting any value:
Names in Lisp are not used merely as
$ (de x 3)
tools for referencing argument values or
=3
functions. The basic concept in Lisp is not
$ (de (f x) (+ x 1))
that of function but that of location. A
=f
variable is viewed as a name for a location
at which a value can be stored. The set of
Terms introduced by de have mutual
all bindings at some point in a program is
references within their own lexical level of known as the environment at this point.
definition. The top-level is a special case
Definitions, lambda expressions and let
because top level definitions can be added
expressions are viewed as mechanisms
at any time whereas at other levels
which create new locations and bind
definitions come first. Levels can be
variables to those locations. Thus formal
imbricated. A reference to a name is first
parameters are not only viewed as
looked up at the same level. Parameter
potential values but also as stores. In this
names have priority over local definitions
section we shall consider the problems
when they conflict at the same lexical
related to the setq instruction.
level1.
First note that setq is a sequential
A recursive let is very useful in a
instruction which must be put inside an
functional language because even if it is
enclosing control instruction as progn,
possible to handle mutual recursion by
while, etc. This means that function call is
means of second order functions, this is
quite inefficient and not very clear. Some
not the only way of controlling data flow.
This new control level has nothing to do
problems cannot be dealt with without
with the lambda calculus - which was our
mutual recursion. For instance in the
guide up to now.
preceding definition of the FlipFlop RS,
The benefit of assignment is
efficiency. What is lost is, once again,
semantic clarity. The introduction of
1
If not found as parameter or local
stores in the semantic model decreases the
definition, the reference is searched in the
level of abstraction. For instance, if you
ancestors.
page 9
compare the recursive Fibonacci function
arguments of functions are evaluated does
definition and the iterative one - based on
not count. For instance, in the
assignments - it is obvious that the
interpretation of:
functional definition is very close to the
mathematical specification while the other
$ (f (setq x 5) (setq x 6))
requires a proof.
In Lisp, the top level is an implicit
the order of evaluation must be taken into
progn in which definitions occur.
account. Note that if a setq instruction is
Definitions are very similar to affectations
performed within the body of g, we would
since they change the values of variables.
have the same problem with
Is it possible to introduce a setq
instruction without radically changing the
$ (f (g 5) (g 7))
language? One way of doing so would be
to accept the use of setq in the body of a
In traditional lisps, the problem is solved
let while adding an implicit progn in the
because the order of evaluation is
core of the let. We could then perform
perfectly determined. In parallel lisps the
setq instructions on local variables
evaluation order is arbitrary (depending on
introduced by let . This move would not
processors allocation). Thus we must
significally change the language since the
ascertain that in such languages
let in question would only mimic the top-
assignments on globals are used only to
level.
guide the calculation (for instance to
To perform assignments on formal
synchronize processes) rather than to
contribute to it1.
parameters, we might add sequentiality
through an explicit (or implicit) control
In a lazy language, the situation is
instruction in the body of functions. This
intermediate. Shared variables are used by
would obscure the formal specification in
only one function at a time and the order
embedding a new control level.
Nevertheless, if parameters are passed by
of evaluation is determined at run time.
Then no special problem of concurrent
value, the meaning of the function being
defined could not be too much alterated.
This is not so, however, if assignments on
1
Global variables are effectively shared
and their content can be override by any
free variables are concerned.
If we accept assignment on global
processes at any time. Hence some more
variables, we will loose an interesting
global convention on the way processes
property: the fact that the order in which
may affect these shared variables must be
specified.
page 10
access arises as with parallelism but the
The original model of Lisp 1.5
used of global stores must also be
implementation used an access variable
carefully specified since the order of
mechanism called deep binding. This
evaluation may vary from one execution
model was very efficient with respect to
to another.
the switching of environments created by
Another annoying effect of global
function calls, but also fundamentally
assignments is that we could obscure the
inadequate to solve the funarg problem
meaning of the function being defined by
and totally inappropriate for lazy
setting a new value to the name introduced evaluation. In this model, the cost of a
within the body of its own definition1. For variable access is proportional to the
all these reasons, setq on global variables
distance (in the tree of dynamic
has not been allowed in standard
environments) between the current
Lambdix.
environment and the one where the
variable has been bound. Then the cost of
a variable access is not bound: the access
to a global variable from a recursive call
2. Implementation
requires a time proportional to the
recursion depth.
The main originality of Lambdix is its
The MacLisp interpreter has solved
implementation model. This section shows
that this model can bear comparison with
other models on call by value, and that it
is far better suited for lazy evaluation.
this problem by means of a new model
called shallow binding. This new model
was used in most Lisp implementations
thereafter. In this model, each variable has
a corresponding cell containing its value
2.1. Previous implementation models
in any environment. Therefore in any
environment the access to a value is direct
and constant. Thus variable accesses are
very efficient.
1
For instance
Nevertheless on function call, the
(de (f x)
old values of the argument list must be
....
saved into a stack and the new values
(f (setq f ...)) ... )
stored in the cells so as to reflect the
To forbid setq on the term being defined
bindings introduced by the function call.
would be illusory since the meaning of
Similarly when coming back to the
term relies anyway on the meaning of all
previous environment, the content of the
other variables defined at the same level.
page 11
cells corresponding to the argument list
What makes an implementation
must be exchanged with the corresponding efficient is the cost of variable access and
stacked values. This makes the
the cost of environment switching. In
binding/unbinding process much more
traditional lisp, the cost of environment
expensive than in the deep binding model. switching was not too important because it
So an unexpected consequence of this
only concerned neighbour environments
binding model is that context switching is
and could be handled by a stack
very expensive.
mechanism.
The original shallow binding
Conversely, in second order
model was not designed to solve the
languages (allowing functional arguments)
funarg problem. However Interlisp-10 has
as well as in lazy languages the cost of
implemented a solution maintaining a tree
environment switching is very important.
of environments instead of a stack.
In both cases, context switching can
Context switching is done by repeating the
frequently arise between two distant
binding/unbinding process along the path
environments.
from the current environment to the future
In a lazy language, context
one. A common ancestor must be first
switches are numerous. This is so because
determined on the two paths from these
the request for the calculation of an
environments to the root and then half of
argument occurs within the body of the
the path must be gone through while
unbinding the variables, and the second
part while rebinding them up to the target
way from the root to this environment and
environment. Here the cost of context
on each node, proceed to the reversal of
switching is proportional to the bindings
the pointer and to the exchange of binding
between the source environment and the
values of the environment with the content
target one. This means that both the
of their cells. At the end of this process,
number of function calls and the number
of arguments are to be taken into account1.
all identifiers have the correct values in
the cells. This model is known as
rerooting and an inductive proof of its
algorithm is given in [BAC78]. Although
1
Baker proposed an alternate solution also
this method does not require the search of
maintaining a tree of environment. In its
a common ancestor in the dynamic tree of
solution, the tree is an inverted tree, where
environments, context switching is still
the root is always the current environment. proportional to the bindings from the
To switch from the current environment to
current environment to the target one and
another one, one must follow the unique
consequently, still very expensive.
page 12
function, whose environment is by
substitution1 corresponds to a partial
definition different from that of the
compilation of lexical references as
function call. Therefore, each time a
illustrated by figure 1.
parameter must be evaluated, a context
f
switching must be performed.
If (revised) shallow binding is the
most efficient of the classical
x y
(+ x y)
implementation models, it is not very
good with respect to laziness. This is so
because the cost of a switch between two
fig. 1
environments is not limited by a constant
but is proportional to the bindings between Furthermore, any internal lambda
the environments in the dynamic tree.
structure contains a pointer to its lexical
parent structure, and this substitution is
2.2. Lambdix implementation model
performed at an arbitrary level of
definition. For instance, the function
2.2.1. Variable access
double-incr defined by the following topAs in the shallow binding schema, each
level definitions
parameter has a cell. But a cell is not
attributed to a name, but rather to a formal
$ ( de (double-incr x)
(twice (incr x))))
parameter. The binding of lexical
parameter is known in advance by lexical
$ ( de (twice f)
analysis. A reference to a parameter in the
core of a function is a direct reference to
the corresponding parameter cell. The
implementation uses internal structures
containing formal parameter cells and the
body of the lambda expression. A textual
substitution is performed by the function
read of the interpreter on the body of the
function. All names are replaced by the
corresponding parameter cells. This
1
This implementation of the ß-reduction
puts Lambdix near the implementation
prompted by the De Bruinj notation (like
Automath) - where the occurrences of
variables were replaced by the level of
their binding. But contrary to the De
Bruinj model, Lambdix takes care of the
names of the variables. This information
stands accessible for meta computation.
(this allows a certain degree of dynamicity
as shown by use of the QUOTE and
EXCLA operators).
page 13
(lambda (x) (f (f x)))))
by changing only dynamic pointers
$ ( de (incr x)
corresponding to function calls. In
(lambda (y) (+ y x)))
particular this makes the cost of
environment switching independent of the
will be represented by the internal
number of parameters introduced by
structures of figure 2:
double-incr
function calls.
x
(
(
x
!
x
))
( (
BLOCK
5
6
incr
twice
f
dynamic
pointer
f
x y
(+ x y)
!
))
y
(+
)
fig. 2
fig. 3
2.2.2. Representation of environments
In the same way, local definitions are
parsed and lexical references to them are
replaced by direct pointers to analogous
The dynamic pointers to the blocks are
used to access values and to represent
environments1. Environments are
internal structures. We shall not detail
these cases here.
In fact, the cell parameters do not
directly contain the values. A pointer to a
block is attributed to each function call
1
The concept of environment generally
varies from one implementation to the
and the called values are kept into it. Then
other. Traditionally environments have
the cells give access to their corresponding been represented as association list - i.e. as
values by means of pointers to the block
functions mapping Identificators to
(called the dynamic pointers of the
Values:
internal lambda structures) and offsets
(into the block). This scheme is not as fast
What we call environment in this paper
as pure shallow binding but in this frame
corresponds to the same notion, while in
variable access is also constant and very
[REC86], Lambdix environments were
efficient. This indirection is interesting
introduced by a recursive equation:
sigma: (Id -> Val)
because it induces environment switching
page 14
ENV = Id -> ( ENV -> Val)
transformed by function calls and function
occurs (G2). From this point G calls H
returns. This provides an order on the
(H2) which calls G again (G3).
environments - an order that is usually
P1
called 'dynamic'. Athough it is distinct
from the lexical ordering of the
definitions, they are related. Suppose the
H1
H2
functions P, H and G to be lexically
organized with P at the top, H local to P
G1
and G local to H. The lambda-structures
will reflect this lexical organization by
G2
G3
fig. 5
something like fig.4:
From this last call to G, one must access
P
H
G
the parameters of the call (G3) and those
of its lexical ancestors (H2 and P1). This
x y body
y z body t
body
block list (G3,H2,P1) combined with the
lambda structures G, H, P suffices for
fig. 4
representing the variable bindings required
Now a call to H always appears within a
during the execution of G3. To install or
call to P. The same thing holds for G and
reinstall this particular environment later,
H. The dynamic environment
one has only to make the dynamic pointers
corresponding to a call to G will be
of the lambda structures G, H and P point
defined by the arguments values of this
to the value blocks G3, H2 and P1
call and those corresponding to its
respectively. This is illustrated by figure 6.
ancestors. In figure 6, the dynamic
P1
ordering of the first three calls was: P
calling H calling G . Then this first call to
H1
G (G1) terminates and a new call to G
G1
These two views are not really opposed;
the second one fit better one of the aspects
P
of the implementation model: in a given
H2
G2
G3
H
G
environment, names give access to
function from environments to values and
x y body
y z body t
fig. 6
a certain calculation must be performed
before accessing a value.
page 15
body
closures given as functional argument)
2.2.3. Cost of environment switching
will also be limited because it will stop as
soon as an ancestor pointer is correctly set
In Lambdix implementation, a functional
- which means that in the worst case, it
value is represented by a lambda structure
requires the comparison of all dynamic
and a block corresponding to its
pointers to the top-level. But here, the
environment of definition. This
number of tests and assignments involved
environment corresponds to an
is limited by the lexical level of the
environment of call of its direct ancestor.
definition. It is not a dynamic depth that is
The tree of dynamic environment blocks is involved in this process. It is a lexical
maintained by associating each block with
depth, which corresponds to the number of
its dynamic (lexical) father block. One can
levels introduced by let - which usually
then find all dynamic ancestors pointers
reduces to 1.
from one block.
What is important is that the cost
To install an environment of
of the installation of an environment does
definition on the lambda structures, all
not depend on the current environment
ancestor dynamic pointers must be set in
since it is limited by a value that is
the lambda structures to their
independent. It does not matter from
corresponding blocks in the dynamic tree.
which environment the interpreter comes,
To do this, one must simply compare the
but how deep this environment is in the
present values of these dynamic pointers
lexical sense. Furthermore, if the
with those given by the tree of calling
environment of definition is the same as
environments. This calculation can be
the previous environment (as for instance
shortened by the convention that if a
in recursive call) environment switching is
pointer is correctly set, all the ancestors
reduced to a test.
will also be. This property can be easily
implemented: it only requires that the
previous environment be restored when a
In the general case, the cost of
function returns. In this way the cost of
environment switching does not depend
environment switching is limited to one
on the number of formal parameters, as in
assignment and one test when a function
theshallow binding schema. Thus,
call occurs within the same branch of the
functions of multiple arguments are not
lexical tree.
disadvantaged.
Now the switching from one
The combination of these two
environment to an arbitrary other (case of
characteristics — constant variable access
page 16
and environment switching independent of
implementation language, to the
the current environment — makes our
programmer's ability, or to the model
model well suited to lazy evaluation
itself.
because the computation can be postponed
without too drastic a supplementary cost.
To test our implementation model,
we have built two Lambdix interpreters:
the first supporting only call by value and
the second performing only call by need.
2.3. Execution timetables
Both interpreters are built on the same
implementation schema.
In this section we give some execution
timetables, although everyone knows that
2.3.1. Performances on call by value
such results cannot not be taken too
seriously.
We compare Lambdix with lisps that are
This is the case, first of all,
the most used at the L.R.I.: last Lisp
because languages have different
(called lal ), Frantz Lisp and Lelisp.
semantics. This means that some functions
defined in one language may be not
These lisps all suffer from the semantical
defects listed in the first section2.
defined in another. This situation arises
for Lambdix both with second order
functions and lazy functions manipulating
infinite lists1. It is nevertheless possible to
2
compare execution times of those
software in Frantz Lisp because it was
functions that can be defined in both
used in the industry and because it was the
languages.
only one to have a compiler.
Secondly, if we want to compare
Originally, a team has developed
Unfortunately, the interpreter was very
two models for the implementation of
slow because its implementation was
environments, we should compare
interpreters written in the same
based on the deep binding model. Thus
implementation language and as far as
efficient interpreter compatible with
possible by the same programmer;
Frantz Lisp. Lelisp has been chosen by
otherwise, we will not know if the results
another team for the efficiency of its
are due to the efficiency of the
interpreter and its good library (which by
Last lisp has been designed to be an
the way contains a closure function).
1
We only have compared the very same
programs. We did not take advantage of
implementation is based on the shallow
binding schema.
possible simplification of lazy program.
page 17
Last Lisp was written in C at the
Even if it is never the best (on call
L.R.I. in 1986 by a very good
by value ), our interpreter can bear
programmer, Patrick Amar, and the
comparison with its two rivals. It gives
interpreter is optimized by assembly code
better results than last Lisp on functions
insertions. The LeLisp interpreter was
using lists and is close to LeLisp on
directly written in Assembler. Given that
recursive functions manipulating numbers.
our implementation is presumably less
What is shown by these results is clearly
optimized than those of its two
that the Lambdix implementation model
competitors, the results1 shown in the
can compete with the shallow binding
following table look very good for the
model.
Lambdix environment model:
Comparison with the Frantz Lisp
interpreter and compiler (fig. 8) illustrates
lal
Tak
38.5
42.5
40
Fib
8.8
13.8
13.8
LComp 15.5
11.1
9.1
◊ good interpreters can compete with
8.1
6.5
compilers:
Sieve
9.1
Lambdix
two other points:
Prog
lelisp
◊ deep binding is not an efficient model
fig. 7
The Fibonacci test (Fib) is executed with a
Prog
best
call to (Fib 20). The well known Tak
Frantz Lisp
interp.
comp.
function is tested with a call to (Tak 18 12
Tak
38.5
overfl.
37
6). The last two tests are small programs
Fib
8.8
108
6
containing recursive calls to functions
LComp
9.1
overfl.
10
which manipulate lists. LComp is a
Sieve
6.5
overfl.
program which compares trees leaf by leaf
(same fringe problem). Sieve constructs
5.5
fig. 8
the list of the first 400 prime numbers with
the traditional algorithm of Eratosthenes.
2.3.2. Efficiency vs laziness
1
We have built two Lambdix interpreters:
The experimentation has been done on a
VAX 750, and time is expressed in
the first one supporting only call by value
second. To have the equivalent order on a
and the second one performing only call
Sun3.5 you must divided these numbers
by need. The comparison of these two
by a factor 2.
interpreters is surely very instructive
page 18
because the two implementations both
Lazy evaluation gives better results
share the same implementation language,
because the sum is computed only on the
the same model (adapted for lazy
first elements.
evaluation) and the same programmer.
As shown by this table, the bad
Figure 9 gives a comparison between
cases of lazy evaluation are between 10%
Lambdix with call by need and Lambdix
and 25% slower than traditional call by
with call by value. We have not shown the
need while the good cases are 95% better.
performances of other lisps because they
The worst case is that of the Tak function
can be found in the previous tables and are
which has three arguments and calculates
in any case of the same order as Lambdix
them separately - each time from another
with call by value.
environment. But such cases are not very
frequent and the price of laziness can be
Lambdix
Prog
% of
evaluated in our model as a 15% loss in
diff.
efficiency with strict functions. Such a
by val
by need
Fib
13.8
15.7
- 12
result is very important and could not have
Fib2
19.5
21.7
- 10
been obtained with other models because
Tak
42.5
57
- 25
of the cost of environment switching.
Comp
11.1
0.7
+ 94
Sieve
8.1
9.5
- 15
LSum
16.1
0.03
+ 99
fig. 9
The programs tested in this table are the
same as the previous ones except for Fib2
and LSum. Fib2 is a modified Fibonacci
function which takes three arguments1.
LSum calculates the sum of the terms of
BIBLIOGRAPHY
two lists generated by the Sieve program.
[1] J. ALLEN, Anatomy of Lisp,
McGraw-Hill Computer Science series,
1978.
1
Other lisps were very sensible to these
[2] H.G. BAKER, "Shallow binding in
additional arguments and gave all a result
Lisp 1.5", CACM, vol 21, Nb 7, July
around 20. Note that this is not the case
1978.
with the Lambdix model which is less
sensible to parameter addition.
page 19
[3] H.G. BAKER, "List processing in real
Symposia in applied mathematics, vol 19,
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21, Nb 4, April 1978.
[14] M.P. GEORGEFF & S.F. BODNAR,
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higher-order functions in LISP", Report
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[15] Mc GOWAN, "The 'most recent'
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définition interprétative et à
with nameless dummies, a tool for
l'implémentation des lambda-langages,
automatic formula manipulation", Indag
Thèse de Doctorat d'Etat, N° 78-2, Nov.
Math. 34, 1972.
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ABRAMSKY, "The theory of Strictness
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[19] G. PLOTKIN, "The Category of
Curie, Paris 1980.
complete partial orders: a tool for making
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meanings", Summer School on foundation
Manuel de Réference, Février 1984.
of Artificial Intelligence and Computer
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Science, Pisa 19-30 June 1978.
lambda-interprètes: application aux
langages LISP, PLASMA et SMALLTALK,
[20] T.W. PRATT, Programming
Languages: design and implementation.
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Printice-Hall, 1975.
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[21] C. RECANATI, Lambdix: un
Combinatory Logic, North-Holland, 1958.
interprète LISP à liaison lexicale et
[12] EICK & E. FEHR, "Inconsistencies
évaluation paresseuse, Thèse de 3ème
of pure LISP", Springer LNCS N° 145.
cycle de l'Université de Paris-Sud, N°
[13] R.W. FLOYD, "Assigning meanings
4072, Dec. 1986.
to programs", Proc. Amer. Math. Soc.
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[22] R. SETHI, "Semantics of computer
programs: Overview of language
definition methods", Sept. 1977.
[23] D. SCOTT & C. STRACHEY,
"Towards a formal semantics", Formal
language description languages for
computer programming, Steel. et al. eds,
North-Holland, Amsterdam 1966.
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[26] G.L. STEELE Jr, "Macaroni is better
than spaghetti", SIGPLAN Notices, vol
12, Nb 8, Aug. 1977.
[27] G.L. STEELE Jr & G.J. SUSMAN,
The Art of the Interpreter, AI Memo Nb
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aspects of functional programming",
Functional Programming and its
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page 21
| 6 |
Outage Performance Analysis of Multicarrier Relay
Selection for Cooperative Networks
Shuping Dang, Justin P. Coon, Gaojie Chen and David E. Simmons
arXiv:1708.05868v1 [cs.IT] 19 Aug 2017
Department of Engineering Science, University of Oxford, Oxford, UK, OX1 3PJ
Email: {shuping.dang, justin.coon, gaojie.chen, david.simmons}@eng.ox.ac.uk
Abstract—In this paper, we analyze the outage performance
of two multicarrier relay selection schemes, i.e. bulk and persubcarrier selections, for two-hop orthogonal frequency-division
multiplexing (OFDM) systems. To provide a comprehensive analysis, three forwarding protocols: decode-and-forward (DF), fixedgain (FG) amplify-and-forward (AF) and variable-gain (VG) AF
relay systems are considered. We obtain closed-form approximations for the outage probability and closed-form expressions for
the asymptotic outage probability in the high signal-to-noise ratio
(SNR) region for all cases. Our analysis is verified by Monte
Carlo simulations, and provides an analytical framework for
multicarrier systems with relay selection.
Keywords—Multicarrier relay selection, parallel fading channel,
cooperative systems, outage performance.
I.
I NTRODUCTION
Cooperative communications and relay-assisted systems
have attracted a large amount of attention since they were
proposed and comprehensively analyzed in [1]. In such a
network, relays are employed as intermediate communication
nodes to assist the transmission between source and destination, so that the communications can be maintained when
the direct transmission link between source and destination
undergoes deep fading [2], [3]. It has been proved that with
the help of relays and proper forwarding protocols, the network
coverage expansion, energy efficiency, system reliability and
quality of service (QoS) can be significantly enhanced [4]–
[7]. In particular, relay selection can further enhance the
system performance and obtain an extra diversity [8]. A variety
of selection schemes have been proposed and analyzed for
different types of relays and channel conditions [9]–[14].
Meanwhile, multicarrier communications, especially orthogonal frequency-division multiplexing (OFDM), is also
proposed and utilized in relay-assisted networks, which is capable of providing a better system performance and bandwidth
efficiency over frequency selective channels [15], [16]. But
relay selection in multicarrier systems is not straightforward,
because this is a two-tier selection/allocation problem involving relay selection and subcarrier allocation. In [17], two relay
selection schemes for OFDM systems, i.e. bulk selection (a
single relay is selected for transmission on all subcarriers) and
per-subcarrier selection (selection is treated independently for
each subcarrier, thus, potentially, leading to transmission via
multiple relays) are proposed and analyzed.
However, all previous works on multicarrier relay selection
are carried out based on an assumption that the outage con-
dition regarding each subcarrier is treated individually, which
corresponds to the block fading channel model when a user
is allocated a contiguous block of carriers that lie within a
coherence bandwidth. To be more realistic, we should consider
the parallel fading channel model that general OFDM systems
are well approximated by. The parallel fading channel model
is utilized to model a fading channel consisting of a finite
number of flat independent and identically distributed (i.i.d.)
fading subchannels [18]. The most unique property of the parallel fading channel relative to the conventional block fading
channel is the definition of outage probability. Considering the
coding over the parallel fading channel, the outage probability
is defined as the probability that the mutual information of
all subchannels is smaller than a target transmission rate
[19]. However, the mathematical tractability of the outage
probability over the parallel fading channel is rather poor,
because the distribution of the summation of a finite number of
random variables cannot be derived in a generic closed-form
expression [20]. In [21], an oversimplified case of a parallel
fading channel with only two subchannels is analyzed and an
integral formula for the outage probability is given. Also, upper
and lower bounds as well as an approximation for the outage
probability for any number of subchannels are determined in
[18] and [22], respectively.
On the other hand, to the best of the authors’ knowledge,
the work considering multicarrier relay selection for twohop OFDM systems approximated by parallel fading channel
model has not been fully treated. To fill this gap, we analyze
this scenario in a Rayleigh fading condition. Note, the method
applied in this paper can be easily tailored to analyze other
channel conditions.
The main contributions of this paper are summarized infra:
•
We obtain closed-form generic approximations for
outage probabilities of two-hop OFDM systems
when applying bulk and per-subcarrier relay selection
schemes.
•
We derive closed-form approximations for the outage probability of decode-and-forward (DF), fixedgain (FG) amplify-and-forward (AF) and variable-gain
(VG) AF relay systems with bulk and per-subcarrier
relay selection schemes.
•
We obtain closed-form asymptotic expressions for the
outage probability with different selection schemes at
high SNR and also derive diversity gains.
The rest of this paper is organized as follows. In Section
II, the system model and fundamentals are given. Then,
the outage performance is analyzed in Section III. Specific
applications including DF, FG AF and VG AF relay systems
are investigated in Section IV. The analysis is numerically
verified by Monte Carlo simulations and further discussed in
Section V. Finally, Section VI concludes the paper.
II.
S YSTEM M ODEL
A. Parallel fading channel and outage probability
In this paper, we consider a two-hop OFDM system with
K orthogonal subcarriers and M relays gathered in a relay
cluster, the size of which is relatively small compared to the
distance between source and destination. Therefore, K parallel
fading subchannels are constructed at each hop, and for each
subcarrier, there are M interfering channels. We denote the
sets of subcarriers and relays as K = {1, 2, . . . , K} and
M = {1, 2, . . . , M }. Also, we assume the entire two-hop
OFDM system operates in a half-duplex protocol and the direct
transmission link between source and destination does not
exist due to deep fading. As a result, two orthogonal time
slots are required for one complete transmission from source
to destination via relay(s). Here, we denote i.i.d. Rayleigh
channel coefficients in the first and second hops by h1 (m, k)
and h2 (m, k) ∀m ∈ M and k ∈ K. The channel gain
|hi (m, k)|2 , where i ∈ {1, 2}, is exponentially distributed with
the mean of µi . Therefore, its probability density function
(PDF) and cumulative distribution function (CDF) are
f|hi |2 (x) = e−x/µi /µi ⇔ F|hi |2 (x) = 1 − e−x/µi .
(1)
Subsequently, the end-to-end SNR transmitted on the kth
subcarrier and forwarded by the mth relay is denoted as
γ(m, k). Accordingly, the outage probability over the parallel
fading channel can be defined as
(K
)
X
(2)
Pout (s) = P
ln(1 + γ(mk , k)) < s
k=1
where P {·} denotes the probability of the enclosed; mk is
the index of the relay selected to forward the kth subcarrier;
we also let s = 2ξ for the convenience of the following
analysis, where ξ is the mutual information outage threshold1 .
In addition, we assume all noise statistics are assumed to be
zero-mean, complex Gaussian random variables with variance
N0 /2 per dimension, from which the noise power can be
expressed by N0 .
B. Forwarding protocols
We assume equal bit and power allocation schemes are
applied and the average transmit power per subcarrier at
source and relay is the same, denoted by Pt . Therefore, the
instantaneous end-to-end SNR of the kth subcarrier forwarded
by the mth relay using a DF protocol is written as
γDF (m, k) = γ̄ min |h1 (m, k)|2 , |h2 (m, k)|2 ,
(3)
where γ̄ = Pt /N0 . Similarly, for FG AF relaying that is
blind to all channel conditions and is only able to amplify the
1 The factor ‘2’ comes from the fact that two time slots are required for one
complete transmission in two-hop systems.
Fig. 1. Illustration of (a) bulk, (b) per-subcarrier relay selection schemes for
single source,
Psingle destination and M clustered relays with K subcarriers.
Note, K = M
m=1 km for km ∈ N, ∀ 1 ≤ m ≤ M .
received signal by a fixed gain, the instantaneous end-to-end
SNR is written as
γF G (m, k) =
γ̄ 2 |h1 (m, k)|2 |h2 (m, k)|2
.
γ̄µ1 + γ̄|h2 (m, k)|2 + 1
(4)
For VG AF relaying which is able to estimate channel conditions and amplify accordingly, the instantaneous end-to-end
SNR is thereby
γV G (m, k) =
γ̄ 2 |h1 (m, k)|2 |h2 (m, k)|2
.
γ̄|h1 (m, k)|2 + γ̄|h2 (m, k)|2 Pt + 1
(5)
C. Relay selection schemes
1) Bulk selection: By the bulk selection scheme, the source
only selects one out of M relays by which all subcarriers are
forwarded according to the selection criterion
(K
)
X
bulk
L
= arg max
ln(1 + γ(m, k)) ,
(6)
m∈M
k=1
where Lbulk is the set (of cardinality one) denoting the only
selected relay. This selection scheme is easy to implement
and will not involve an overcomplicated coordination protocol,
because there is only one selected relay. However, it is obvious
that the outage performance cannot be optimized in this case.
2) Per-subcarrier selection: The per-subcarrier selection
scheme selects L relays from M relays in a per-subcarrier
manner and 1 ≤ L ≤ min(M, K). Therefore, the relay selected to forward the kth subcarrier is individually determined
by
lps (k) = arg max ln(1 + γ(m, k)) = arg max γ(m, k), (7)
m∈M
m∈M
where lps (k) is the set of the selected relay corresponding
to the kth subcarrier. Then, this selection process will be
repeatedly applied for all subcarriers and finally L relays
are selected.
SK The set of all L selected relays is denoted by
Lps = k=1 {lps (k)}. Note, it is allowed that lps (k) = lps (n)
for k 6= n, i.e. one relay is allowed to assist in forwarding
two or more subcarriers at the same time. Obviously, this
selection scheme is optimal in terms of outage performance,
but this optimality will result in a high system complexity
[23]. For illustration purposes, these two selection schemes
are illustrated in Fig. 1.
III.
O UTAGE P ERFORMANCE A NALYSIS
A. Bulk selection
According to (2) and (6), the a posteriori outage probability
with bulk selection can be defined by
(
(K
)
)
X
Pout (s) = P max
ln(1 + γ(m, k)) < s
m∈M
=
M
Y
m=1
(
P
K
X
k=1
)
(8)
(a)
M
= (FI (s)) ,
ln(1 + γ(m, k)) < s
k=1
o
nP
K
where FI (s) := P
k=1 ln(1 + γ(m, k)) < s , ∀m ∈ M
and ∀k ∈ K; (a) is valid, because all parallel subchannels are
assumed to be i.i.d.
Assume the CDF and PDF of γ(m, k) are Fγ (s) and fγ (s),
respectively. Now following the Proposition 1 proved in [22],
we also make a hypothesis that fγ (s) can be expanded as a
power series about zero as
fγ (s) = sq (g0 + g1 s + O(s2 )),
(9)
where q is a non-negative integer representing the inherent
subchannel diversity; g0 , g1 are non-zero functions of γ̄ and
satisfy the condition g1 (γ̄) = o(g0 (γ̄)) when γ̄ → ∞. This
is a common assumption applicable to most two-hop channel
cases [24]. Then, FI (s) can be written as [22]
Kq+1
S0 (K, s, q)
FI (s) =
(S1 (K, s, q) − S0 (K, s, q))Kq
K
1
S1 (K, s, q) − S0 (K, s, q)
+O
× fγ
.
S0 (K, s, q)
γ̄ K(q+1)+2
(10)
The coefficient Sx (K, s, q) is defined as
q
X
X
q
K −1
j
Sx (K, s, q) :=
(−1)
j
a0 , a1 , . . . , aq
j=0
ap ∈N, ∀0≤p≤q
P
q
p=0 ap =K−1
#
ap # "
Z
q
sz
Y
q
1
e
dz
×
,
(−1)p
·
Q
2πi C K
p
p=0
k=0 (z − βk )
"
(11)
where βk is the kth element in
B = (0, q + 1, . . . , q + 1, q, . . . , q , . . . , 1, . . . , 1, q + 1 + x − j)
{z
} | {z }
| {z }
|
a0 terms
a1 terms
aq terms
(12)
and C is the contour which encloses all poles of the integrand.
R
sz
1
QKe dz
It should be noted that 2πi
is equivalent to
C
k=0
QK(z−βl )
the inverse Laplace transform of 1/ k=0 (z − βk ), which can
be expressed by the closed form as follows [25]:
Z
K
βk s
X
1
esz dz
e
Q
. (13)
=
Q
2πi C K
(β
−
β
)
1≤n≤K
k
n
(z
−
β
)
k
k=0
k=0
n6=k
However, this closed-form expression given above is only valid
if all poles of the integrand on the are simple (first-order poles),
i.e. βn 6= βk , for n 6= k. Closed-form expressions exist for
the cases with higher-order poles, but cannot be written as a
general form. This is the reason why we still keep the integral
form in (11) for generality. Another reason for keeping the
integral form is because this integral form can be efficiently
evaluated by computer-based simulations using the residue
theorem [26].
As a result, the a posteriori outage probability by bulk
selection as defined in (8) can be determined by
S0 (K, s, q)M (Kq+1)
bulk
Pout
(s) = (FI (s))M =
(S1 (K, s, q) − S0 (K, s, q))M Kq
M K
1
S1 (K, s, q) − S0 (K, s, q)
+O
× fγ
.
S0 (K, s, q)
γ̄ M K(q+1)+2
(14)
Furthermore, we can derive the asymptotic outage probability
at high SNR by power series and obtain
M
bulk
bulk
Pout
(s) ∼ P̃out
(s) = S0 (K, s, q)g0K
.
(15)
Note, because of the condition g1 (γ̄) = o(g0 (γ̄)) when γ̄ →
∞, the diversity order can be derived from (15) by Gbulk
=
d
bulk
log Pout
(s)
− limγ̄→∞
=
M
K(q
+
1).
log γ̄
B. Per-subcarrier selection
According to (2) and (7), the a posteriori outage probability
when per-subcarrier selection is employed can be defined by
(K
)
X
(16)
Pout (s) = P
ln(1 + max γ(m, k)) < s .
m∈M
k=1
Denote Ψ(m, k) = maxm∈M γ(m, k). We can obtain the CDF
and PDF of Ψ(m, k) as
FΨ (s) = (Fγ (s))
M
⇔ fΨ (s) = M (Fγ (s))
M −1
fγ (s). (17)
Following the assumption given in (9), fΨ (s) can be further
expressed by
M g0M
sM (q+1)−1
fΨ (s) =
(q + 1)M −1
#
"
(M − 1)g0M −1 g1
g0M −1 g1
+
sM (q+1) (18)
+M
(q + 1)M −1
(q + 1)M −2 (q + 2)
+ O sM (q+1)+1 .
Therefore, if we denote
q 0 = M (q + 1) − 1
M g0M
g00 = (q+1)M
h M−1−1
1
0
g = M g0 Mg−1
+
1
(q+1)
(19)
(M −1)g0M −1 g1
(q+1)M −2 (q+2)
i
fΨ (s) can also be written as
0
fΨ (s) = sq (g00 + g10 s + O(s2 )),
(20)
which also aligns with the form given in (9). This result
indicates that we can similarly employ Proposition 1 derived
in [22] to analyze the outage performance of per-subcarrier
relay selection over the parallel fading channel, and the only
modification is to replace fγ (·) with fΨ (·). Hence, the outage
0
10
performance of per-subcarrier relay selection over the parallel
fading channel as defined in (16) can be calculated by
0
Furthermore, we can derive the asymptotic outage probability
at high SNR by power series and obtain
ps
ps
Pout
(s) ∼ P̃out
(s) = S0 (K, s, q 0 )g00K .
−1
10
Outage Probability
S0 (K, s, q 0 )Kq +1
ps
Pout
(s) =
(S1 (K, s, q 0 ) − S0 (K, s, q 0 ))Kq0
K
1
S1 (K, s, q 0 ) − S0 (K, s, q 0 )
+
O
× fΨ
.
S0 (K, s, q 0 )
γ̄ K(q0 +1)+2
(21)
M=2 ;K=2
−2
10
−3
10
M=3 ;K=3
−4
10
(22)
Similarly to the case for bulk selection, we can also obtain
the diversity of per-subcarrier selection from (22) to be Gps
d =
M K(q + 1) as expected.
IV.
DF Numerical
DF Approx
FG AF Numerical
FG AF Approx
VG AF Numerical
VG AF Approx
A PPLICATIONS
−5
10
15
20
Fig. 2. Bulk selection case: numerical results and analytical approximations
with different system configurations of M and K.
0
10
DF Numerical
DF Approx
FG AF Numerical
FG AF Approx
VG AF Numerical
VG AF Approx
−1
10
Outage Probability
which can be expanded by power series as γ̄ → ∞ to
the standard expanded form given in (9), and thus can be
substituted into (14) and (21) to yield the approximated outage
probability with bulk and per-subcarrier selections over the
parallel fading channel.
10
γ (dB)
A. DF relay networks
According to (1) and (3), we can derive the PDF of the
end-to-end SNR γDF (m, k) for DF relay networks by
1 1
1
−s 1 + 1
fγDF (s) =
(23)
+
e γ̄ µ1 µ2 ,
γ̄ µ1
µ2
5
M=2; K=2
−2
10
−3
10
M=3; K=3
−4
10
B. FG AF and VG AF relay networks
By (1) and (4), we can similarly derive the PDF of the endto-end SNR γF G (m, k) for FG AF relay networks by [27]
s
"
!
s
s(1 + γ̄µ1 )
2(1
+
γ̄µ
)
1
− γ̄µ
FG
1
fγ (s) = e
K0 2
γ̄ 2 µ1 µ2
γ̄ 2 µ1 µ2
s
s
(24)
!#
2
s(1 + γ̄µ1 )
s(1 + γ̄µ1 )
+
K1 2
,
γ̄µ1
γ̄ 2 µ1 µ2
γ̄ 2 µ1 µ2
where Kx (·) is the xth order modified Bessel function of the
second kind.
Similarly, by (1) and (5), the PDF of the end-to-end SNR
γV G (m, k) for VG AF relay networks is given by [27]
s
"
!
2(1 + 2sµ2 )
s (1 + sµ2 )
− γ̄s µ1 + µ1
VG
1
2
K0 2
fγ (s) = e
γ̄ 2 µ1 µ2
γ̄ 2 µ1 µ2
s
s
.
!#
2 1
1
s (1 + sµ2 )
s (1 + sµ2 )
+
+
K1 2
γ̄ µ1
µ2
γ̄ 2 µ1 µ2
γ̄ 2 µ1 µ2
(25)
Although (24) and (25) cannot be expanded by power series
to the form as given in (14), these two PDFs can still be
substituted into (14) and (21) to yield the approximated outage
probability with a divergent error at low SNR (see Corollary
3 in [22]).
−5
10
5
10
15
20
γ (dB)
Fig. 3.
Per-subcarrier selection case: numerical results and analytical
approximations with different system configurations of M and K.
V.
N UMERICAL R ESULTS
To verify our analysis in Section III and Section IV, we
carry out Monte Carlo simulations and present the numerical
results in this section. To simplify simulations, without losing generality, we normalize the two-hop system by letting
µ1 = µ2 = 1 and s = 2 (i.e. ξ = 1 as the mutual information
outage threshold). Then, we can plot the relation among γ̄,
analytical approximations for outage probability (as given in
(14) and (21)) and numerical outage probabilities for different
combinations of M and K. We present the simulation results
for bulk selection and per-subcarrier selection in Fig. 2 and
Fig. 3, respectively. From these two figures, it is clear that the
proposed approximations for the outage probability with bulk
and per-subcarrier selections over the parallel fading channel
have been verified to be effective to approximate the outage
performance of DF, FG AF and VG AF relay systems at high
SNR. Note, although DF, FG AF and VG AF relay systems
have the same diversity gain determined by M K(q + 1), the
convergence rate of FG AF case to the asymptotic region is
smaller due to different correction terms. Another minor point
is that the similarity of outage performance between DF and
VG AF relay systems in the high SNR region as valid over
block fading channels [24], can still be found valid over the
parallel fading channel. In addition, by comparing Fig. 2 and
Fig. 3, we can find that the performance difference between
bulk and per-subcarrier selections is not so significant as that
in block fading channels [28]. This is because the outage
event over the parallel fading channel depends on the mutual
information over all subcarriers, instead of each individual
subcarrier.
VI.
[10]
[11]
[12]
[13]
C ONCLUSION
In this paper, we analyzed multicarrier relay selection for
two-hop OFDM systems and obtained closed-form approximations for the outage probabilities when applying bulk and
per-subcarrier relay selections. It has been numerically shown
that the derived approximations for DF, FG AF and VG AF
relay networks are valid and able to effectively approximate the
exact outage performance at high SNR. Meanwhile, we also
derived the generic asymptotic expressions for the outage probabilities at high SNR when applying bulk and per-subcarrier
relay selections. All these results provide an insight into the
outage performance of multicarrier relay systems over parallel
fading channels.
[14]
[15]
[16]
[17]
[18]
[19]
ACKNOWLEDGMENT
This work was supported by the SEN grant (EPSRC grant
number EP/N002350/1) and the grant from China Scholarship
Council (No. 201508060323).
[20]
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| 7 |
Improved Submatrix Maximum Queries in Monge Matrices
Pawel Gawrychowski1 , Shay Mozes2? , and Oren Weimann3∗
1
MPII, gawry@mpi-inf.mpg.de
IDC Herzliya, smozes@idc.ac.il
University of Haifa, oren@cs.haifa.ac.il
2
arXiv:1307.2313v4 [cs.DS] 12 Oct 2017
3
Abstract. We present efficient data structures for submatrix maximum queries in Monge matrices and
Monge partial matrices. For n × n Monge matrices, we give a data structure that requires O(n) space
and answers submatrix maximum queries in O(log n) time. The best previous data structure [Kaplan
et al., SODA‘12] required O(n log n) space and O(log2 n) query time. We also give an alternative data
structure with constant query-time and O(n1+ε ) construction time and space for any fixed ε < 1. For
n × n partial Monge matrices we obtain a data structure with O(n) space and O(log n · α(n)) query
time. The data structure of Kaplan et al. required O(n log n · α(n)) space and O(log2 n) query time.
Our improvements are enabled by a technique for exploiting the structure of the upper envelope of
Monge matrices to efficiently report column maxima in skewed rectangular Monge matrices. We hope
this technique will be useful in obtaining faster search algorithms in Monge partial matrices. In addition,
we give a linear upper bound on the number of breakpoints in the upper envelope of a Monge partial
matrix. This shows that the inverse Ackermann α(n) factor in the analysis of the data structure of
Kaplan et. al is superfluous.
1
Introduction
A matrix M is a Monge matrix if for any pair of rows i < j and columns k < ` we have that Mik +
Mj` ≥ Mi` + Mjk . Monge matrices have many applications in combinatorial optimization and computational
geometry. For example, they arise in problems involving distances in the plane [20,24,26,28], and in problems
on convex n-gons [2,3]. See [9] for a survey on Monge matrices and their uses in combinatorial optimization.
In this paper we consider the following problem: Given an n × n Monge matrix M , construct a data
structure that can report the maximum entry in any query submatrix (defined by a set of consecutive
rows and a set of consecutive columns). Recently, Kaplan, Mozes, Nussbaum and Sharir [21] presented an
Õ(n) space4 data structure with Õ(n) construction time and O(log2 n) query time. They also described
an extension of the data structure to handle partial Monge matrices (where some of the entries of M are
undefined, but the defined entries in each row and in each column are contiguous). The extended data
structure incurs larger polylogarithmic factors in the space and construction time. Both the original and the
extended data structures have various important applications. They are used in algorithms that efficiently
find the largest empty rectangle containing a query point, in dynamic distance oracles for planar graphs, and
in algorithms for maximum flow in planar graphs [6]. See [21] for more details on the history of this problem
and its applications.
Note that, even though explicitly representing the input matrix requires N = Θ(n2 ) space, the additional
space required by the submatrix maximum data structure of [21] is only Õ(n). In many applications (in
particular [6,21]), the matrix M is not stored explicitly but any entry of M can be computed when needed
in O(1) time. The space required by the application is therefore dominated by the size of the submatrix
maximum data structure. With the increasing size of problem instances, and with current memory and
cache architectures, space often becomes the most significant resource.
For general (i.e., not Monge) matrices, a long line of research over the last three decades including [5,13,14,17,29] achieved Õ(N ) space and Õ(1) query data structures, culminating with the O(N )-space
O(1)-query data structure of Yuan and Atallah [29]. Here N = n2 denotes the total number of entries in the
matrix. It is also known [8] that reducing the space to O(N/c) incurs an Ω(c) query-time. Tradeoffs requiring O(N/c) additional space and Õ(c) query-time were given in [7,8]. When the matrix has only N = o(n2 )
?
4
Mozes and Weimann supported in part by Israel Science Foundation grant 794/13.
The Õ(·) notation hides polylogarithmic factors in n.
nonzero entries, the problem is known in computational geometry as the orthogonal range searching problem
on the n × n grid. In this case as well, various tradeoffs with Õ(N )-space and Õ(1)-query appear in a long
history of results including [4,10,11,15,17]. In particular, a linear O(N )-space data structure was given by
Chazelle [11] at the cost of an O(logε n) query time. See [25] for a survey on orthogonal range search.
Contribution. Our first contribution is in designing O(n)-space O(log n)-query data structures for submatrix maximum queries in Monge matrices and in partial Monge matrices (see Section 3). Our data structures
improve upon the data structures of Kaplan et al. in both space and query time. Consequently, using our
data structures for finding the largest empty rectangle containing a query point improves the space and
query time by logarithmic factors.
We further provide alternative data structures with faster query-time; We achieve O(1) query-time at
the cost of O(n1+ε ) construction time and space for an arbitrarily small constant 0 < ε < 1 (see Section 5).
Our results are achieved by devising a data structure for reporting column maxima in m × n Monge
matrices with many more columns than rows (n >> m). We refer to this data structure as the micro
data structure. The space required by the micro data structure is linear in m, and independent of n. Its
construction-time depends only logarithmically on n. The query-time is O(log log n), the time required for
a predecessor query in a set of integers bounded by n. We use the micro data structure in the design of our
submatrix maximum query data structures, exploiting its sublinear dependency on n, and an ability to trade
off construction and query times.
For partial Monge matrices, we provide a tight O(m) upper bound on the complexity of the upper envelope
(see Section 4). The best previously known bound [27] was mα(m), where α(m) is the inverse Ackermann
function. This upper bound immediately implies that the α(m) factor stated in the space and construction
time of the data structures of Kaplan et al. is superfluous.
Notice that the upper envelope of a full m × n Monge matrix also has complexity O(m). The famous
SMAWK algorithm [2] can find all column maxima in O(n + m) time. However, this is not the case for
partial Monge matrices. Even for simple partial Monge matrices such as triangular, or staircase matrices,
where it has been known for a long time that the complexity of the upper envelope is linear, the fastest known
algorithm for finding all column maxima is the O(nα(m) + m) time algorithm of Klawe and Kleitman [22].
We hope that our micro data structure will prove useful for obtaining a linear-time algorithm. The known
algorithms, including the (nα(m) + m)-time algorithm of Klawe and Kleitman [22], partition the matrix into
skewed rectangular matrices, and use the SMAWK algorithm. It is plausible that our micro data structure
will yield a speed up since it is adapted to skewed matrices.
2
Preliminaries and Our Results
In this section we overview the data structures of [21] and highlight our results.
A matrix M is a Monge matrix if for any pair of rows i < j and columns k < ` we have that Mik + Mj` ≥
Mi` + Mjk . A matrix M is totally monotone in columns if for any pair of rows i < j and columns k < ` we
have that if Mik ≤ Mjk then Mi` ≤ Mj` . Similarly, M is totally monotone in rows if for any pair of rows i < j
and columns k < ` we have that if Mik ≤ Mi` then Mjk ≤ Mj` . Notice that the Monge property implies total
monotonicity (in columns and in rows) but the converse is not true. When we simply say totally monotone
(or TM) we mean totally monotone in columns (our results symmetrically apply to totally monotone in
rows).
A matrix M is a partial matrix if some entries of M are undefined, but the defined entries in each row
and in each column are contiguous. We assume w.l.o.g. that every row has at least one defined element and
that the defined elements form a single connected component (i.e., the defined column intervals in each pair
of consecutive rows overlap). If this is not the case then only minor changes are needed in our algorithms.
A partial TM (resp., Monge) matrix is a partial matrix whose defined entries satisfy the TM (resp., Monge)
condition.
The following propositions are easy to verify:
Proposition 1. An m×n matrix M is Monge iff Mi,j +Mi+1,j+1 ≥ Mi+1,j +Mi,j+1 for all i = 1, 2, . . . , m−1
and j = 1, 2, . . . , n − 1.
Proposition 2. If a matrix M is partial Monge, then it remains partial Monge after replacing any element
of M by a blank, so long as the defined (non-blank) entries in each row and in each column remain contiguous.
2
Proposition 3. If an m-by-n matrix M is (partial) Monge, then the (m + 1)-by-n matrix resulting by
replacing any row of M by two identical copies of that row is also (partial) Monge. An analogous statement
holds for duplicating any column of M .
We consider m × n matrices, but for simplicity we sometimes state the results for n × n matrices.
For a Monge matrix M , denote r(j) = i if the maximum element in column j lies in row i. (We assume
this maximum element is unique. It is simple to break ties by, say, taking the highest index.) The upper
envelope E of all the rows of M consists of the n values r(1), . . . , r(n). Since M is Monge we have that
r(1) ≤ r(2) ≤ . . . ≤ r(n) and so E can be implicitly represented in O(m) space by keeping only the r(j)s
of O(m) columns called breakpoints. Breakpoints are the columns j where r(j) 6= r(j + 1). The maximum
element r(π) of any column π can then be retrieved in O(log m) time by a binary search for the first
breakpoint-column j after π, and setting r(π) = r(j).
The first data structure of [21] is a balanced binary tree Th over the rows of M . A node u whose subtree
contains k leaves (i.e., k rows) stores the O(k) breakpoints of the k × n matrix M u defined by these k
rows and all columns of M . A leaf represents a single row and requires no computation. An internal node
u obtains its breakpoints by merging the breakpoints of its two children: its left child u1 and its right u2 .
By the Monge property, the list of breakpoints of u starts with a prefix of breakpoints of u1 and ends with
a suffix of breakpoints of u2 . Between these there is possibly one new breakpoint j. The prefix and suffix
parts can be found easily in O(k) time by linearly comparing the lists of breakpoints of u1 and u2 . The new
breakpoint j can then be found in additional O(log n) time via binary search. Summing O(k + log n) over
all nodes of Th gives O(m(log m + log n)) time. The total size of Th is O(m log m).
Note that the above holds even if M is not Monge but only TM. This gives rise to a data structure that
answers subcolumn (as opposed to submatrix) queries:
Subcolumn queries in TM matrices [21]. Given a n × n TM matrix, one can construct, in O(n log n)
time, a data structure of size O(n log n) that reports the maximum in a query column and a contiguous range
of rows in O(log n) time.
The maximum entry in a query column π and a contiguous range of rows R is found using Th by identifying
O(log m) canonical nodes of Th . A node u is canonical if u’s set of rows is contained in R but the set of
rows of u’s parent is not. For each such canonical node u, we find in O(log m) time the maximum element in
column π amongst all the rows of u. The output is the largest of these and the total query time is O(log2 m).
The query time can be reduced to O(log m) by using fractional cascading [12].
The first results of our paper improve the above subcolumn query data structure of [21], as indicated in
Table 1 under subcolumn query in TM matrices. The next data structure of [21] extends the queries from
subcolumn to submatrix (specified by ranges R of consecutive rows, and C of consecutive columns.)
Submatrix queries in Monge matrices [21]. Given a n × n Monge matrix, one can construct, in
O(n log n) time, a data structure of size O(n log n) that reports the maximum entry in a query submatrix in
O(log2 n)) time.
To obtain O(log2 n) = O(log m(log m + log n)) query time, note that R is the disjoint union of O(log m)
canonical nodes of Th . For each such canonical node u, we use u’s list of breakpoints {j1 , j2 , . . . , jk } to
find in O(log m + log n) time the maximum element in all rows of u and the range of columns C. This is
done as follows: we first identify in O(log m) time the set I = {ja , ja+1 , . . . , jb } of u’s breakpoints that are
fully contained in C. The columns of C that are to the left of ja all have their maximum element in row
r(ja ). To find the maximum of these we construct, in addition to Th , a symmetric binary tree B that can
report in O(log n) time the maximum entry in a query row and a contiguous range of columns. B is built in
O(n(log m + log n)) time and O(n log n) space using the subcolumn query data structure on the transpose
of M . This is possible since M is Monge.5 Similarly, we find in O(log n) time the maximum in all columns
of C that are to the right of jb .
To find the maximum in all columns between ja and jb , let m(ji ) denote the maximum element in the
columns interval (ji−1 , ji ] (note it must be in row r(ji )). We wish to find max{m(ja+1 ), . . . , m(jb )} which
corresponds to a Range Maximum Query in the array Au = {m(j1 ), . . . , m(jk )}. We compute the array Au
(along with a naive RMQ data structure with logarithmic query time) of every node u during the construction
5
In fact it suffices that M is a TM matrix whose transpose is also TM.
3
of Th . Most of the entries of Au are simply copied from u’s children arrays Au1 and Au2 . The only new m(·)
value that u needs to compute is for the single new breakpoint j (that is between the prefix from u1 and the
suffix from u2 ). Since m(j) must be in row r(j) it can be computed in O(log n) time by a single query to B.
Overall, we get a query time of O(log m+log n) per canonical node u for a total of O(log m(log m+log n)).
Building Th (along with all the RMQ arrays Au ) and B takes total O((m + n)(log m + log n)) time and
O(m log m+n log n) space. Our two improvements to this bound of [21] are stated in Table 1 under submatrix
queries in Monge matrices.
The next data structures of [21] extend the above subcolumn and submatrix data structures from full
to partial TM matrices. The construction is very similar. Merging the breakpoints of the two children u1 ,
u2 of a node u of Th is slightly more involved now, since the envelopes may cross each other multiple
times. The number of breakpoints of any subset of consecutive k rows is O(k · α(k)) [27], and so there are
O(m log m · α(m)) breakpoints in total over all nodes of Th (as opposed to O(m) in full matrices). This
implies the following
Subcolumn queries in partial TM matrices [21]. Given a partial TM n × n matrix, one can construct,
in O(n log2 n · α(n)) time, a data structure of size O(n log n · α(n)) that reports the maximum entry in a
query column and a contiguous range of rows in O(log n) time.
We improve this data structure to the same bounds we get for full matrices. i.e, we show that our bounds
for full matrices also apply to partial matrices. This is stated in Table 1 under subcolumn query in Partial
TM matrices. Finally, [21] extended their submatrix data structure from full to partial Monge matrices. It
uses a similar construction of Th and B as in the case of full matrices, but again requires the additional
O(log m · α(m) + log n · α(n)) multiplicative factor to store the breakpoints of all nodes of Th and B.
property
TM
TM
TM
Monge
Monge
Monge
Monge
Partial TM
Partial TM
Partial TM
Partial Monge
Partial Monge
Partial Monge
query type
subcolumn
subcolumn
subcolumn
submatrix
submatrix
submatrix
submatrix
subcolumn
subcolumn
subcolumn
submatrix
submatrix
submatrix
space
construction time query time
O(n log n)
O(n log n)
O(log n)
O(n)
O(n log n/ log log n) O(log n)
O(n1+ε )
O(n1+ε )
O(1)
O(n log n)
O(n log n)
O(log2 n)
O(n)
O(n log n)
O(log n)
O(n)
O(n log n/ log log n) O(log1+ε n)
O(n1+ε )
O(n1+ε )
O(1)
2
O(n log n · α(n)) O(n log n · α(n)) O(log n)
O(n)
O(n log n/ log log n) O(log n)
O(n1+ε )
O(n1+ε )
O(1)
O(n log n · α(n)) O(n log2 n · α(n)) O(log2 n)
O(n)
O(n log n)
O(log n · α(n))
O(n)
O(n log n/ log log n) O(log1+ε n · α(n))
Table 1. Our results compared to [21].
Lemma 3.1 in [21]
Lemma 2 here
Lemma 8 here
Theorem 3.2 in [21]
Theorem 1 here
Corollary 1 here
Theorem 4 here
Lemma 3.3 in [21]
Lemma 3 here
Lemma 3 here
Theorem 3.4 in [21]
Theorem 2 here
Corollary 2 here
Submatrix queries in partial Monge matrices [21]. Given a n × n partial Monge matrix, one can
construct, in O(nα(n) log2 n) time, a data structure of size O(nα(n) log n) that reports the maximum entry
in a query submatrix in O(log2 n) time.
We remove the O(log n · α(n)) multiplicative factor and obtain the bounds stated in the bottom of Table 1.
The α(n) factor is removed by showing that the number of breakpoints in the upper envelope of a partial
Monge matrix is linear.
3
Linear-Space Data Structures
In this section we present our data structures that improve the space to O(n) and the query time to O(log n).
We begin by introducing a new data structure for the case where a query is composed of an entire column (as
4
opposed to a range of rows). This new data structure (which we call the micro data structure) is designed to
work well when the number of rows in the matrix is much smaller than the number of columns. We denote
by pred(x, n) = O(min{log x, log log n}) the time to query a predecessor data structure with x elements from
{1, . . . , n}.
Lemma 1 (the micro data structure). Given a x × n TM matrix and r > 0, one can construct in
O(x log n/ log r) time, a data structure of size O(x) that given a query column can report the maximum entry
in the entire column in O(r + pred(x, n)) time.
Proof. Out of all n columns of the input matrix M , we will designate O(x) columns as special columns. For
each of these special columns we will eventually compute its maximum element. The first x special columns
of M are columns 1, n/x, 2n/x, 3n/x, . . . , n and are denoted j1 , . . . , jx .
Let X denote the x × x submatrix obtained by taking all x rows but only the x special columns j1 , . . . , jx .
It is easy to verify that X is TM. We can therefore run the SMAWK algorithm [2] on X in O(x) time and
obtain the column maxima of all special columns. Let r(j) denote the row containing the maximum element
in column j. Since M is TM, the r(j) values are monotonically non-decreasing. Consequently, r(j) of a nonspecial column j must be between r(ji ) and r(ji+1 ) where ji < j and ji+1 > j are the two special columns
bracketing j (see Figure 1).
n
ji
m x
xi
j
r(ji)
Mi
ji+1
r(ji+1)
n/x
Fig. 1. An x×n matrix inside an m×n matrix. The black columns are the first x special columns. The (monotonically
non-decreasing) gray cells inside these special columns are the column maxima (i.e., the r(ji ) values of breakpoints
ji ). The maximum element of column j in the x × n matrix must be between r(ji ) and r(ji+1 ) (i.e., in matrix Mi ).
Tuesday, July 2, 13
For every i, let xi = r(ji+1 ) − r(ji ). If xi ≤ r then no column between ji and ji+1 will ever be a special
column. When we will query such a column j we can simply check (at query-time) the r elements of j between
rows r(ji ) and r(ji+1 ) in O(r) time. If, however, xi > r, then we designate more special columns between
ji and ji+1 . This is done recursively on the xi × (n/x) matrix Mi composed of rows r(ji ), . . . , r(ji+1 ) and
columns ji , . . . , ji+1 . That is, we mark xi evenly-spread columns of Mi as special columns, and run SMAWK
in O(xi ) time on the xi × xi submatrix Xi obtained by taking all xi rows but only these xi special columns.
We continue recursively until either xi ≤ r or the number of columns in Mi is at most r. In the latter case,
before terminating, the recursive call runs SMAWK in O(xi + r) = O(xi ) time on the xi × r submatrix Xi
obtained by taking the xi rows and all columns of Mi (i.e., all columns of Mi will become special).
After the recursion terminates, every column j of M is either special (in which case we computed its
maximum), or its maximum is known to be in one of at most r rows (these rows are specified by the r(·) values
of the two special columns bracketing j). Let s denote the total number of columns that are marked as special.
We claim that s = O(x log n/ log r). To see this, notice that the number of columns in every recursive call
decreases by a factor of at least r and so the recursion
depth is O(logr n) = O(log n/ log r). In every recursive
P
level, the number of added special columns is
xi over all x0i s in this level that are at least r. In every
5
recursive level, this sum is bounded by 2x because each one of the x rows of M can appear in at most two Mi ’s
(as the last row of one and the first row of the other). Overall, we get 2x · O(log n/ log r) = O(x log n/ log r).
Notice that s = O(x log n/ log r) implies that the total time complexity of the above procedure is also
O(x log n/ log r). This is because whenever we run SMAWK on a y × y matrix it takes O(y) time and y new
columns are marked as special. To complete the construction, we go over the s special columns from left to
right in O(s) time and throw away (mark as non-special) any column whose r(·) value is the same as that
of the preceding special column. This way we are left with only O(x) special columns, and the difference in
r(·) between consecutive special columns is at least 1 and at most r. In fact, it is easy to maintain O(x) (and
not O(s)) space during the construction by only recursing on sub matrices Mi where xi > 1. We note that
when r = 1, the eventual special columns are exactly the set of breakpoints of the input matrix M .
The final data structure is a predecessor data structure that holds the O(x) special columns and their
associated r(·) values. Upon query of some column j, we search in pred(x, n) time for the predecessor and
successor of j and obtain the two r(·) values. We then search for the maximum of column j by explicitly
checking all the (at most r) relevant rows of column j. The query time is therefore O(r + pred(x, n)) and
the space O(x).
t
u
A linear-space subcolumn data structure.
Lemma 2. Given a m × n TM matrix, one can construct, in O(m(log n + log m)/ log log m) time, a data
structure of size O(m) that can report the maximum entry in a query column and a contiguous range of rows
in O(log m) time.
Proof. Given an m×n input matrix M we partition it into m/x matrices M 1 , M 2 , . . . , M m/x where x = log m.
Every M i is an x × n matrix composed of x consecutive rows of M . We construct the micro data structure of
Lemma 1 for each M i separately choosing r = xε for any constant 0 < ε < 1. This requires O(x log n/ log r) =
O(x log n/ log x) construction time per M i for a total of O(m log n/ log log m) time. We obtain a (micro)
data structure of total size O(m) that upon query (i, j) can report in O(xε + pred(x, n)) = O(logε m) time
the maximum entry in column j of M i .
0
Now, consider the (m/x) × n matrix M 0 , where Mij
is the maximum entry in column j of M i . We
0
0
cannot afford to store M explicitly, however, using the micro data structure we can retrieve any entry Mij
ε
0
in O(log m) time. We next show that M is also TM.
0
0
For any pair of rows i < j and any pair of columns k < ` we need to show that if Mik
≤ Mjk
then
0
0
0
0
0
0
Mi` ≤ Mj` . Suppose that Mik , Mjk , Mi` , and Mj` correspond to entries Mak , Mbk , Mc` , and Md` respectively.
We assume that Mak ≤ Mbk and we need to show that Mc` ≤ Md` . Notice that Mck ≤ Mak because Mak is
the maximal entry in column k of M i and Mck is also an entry in column k of M i . Since Mck ≤ Mak and
Mak ≤ Mbk we have that Mck ≤ Mbk . Since Mck ≤ Mbk , from the total monotonicity of M , we have that
Mc` ≤ Mb` . Finally, we have Mb` ≤ Md` because Md` is the maximal entry in column ` of M j and Mb` is
also an entry in column ` of M j . We conclude that Mc` ≤ Md` .
Now that we have established that the matrix M 0 is TM, we can use the subcolumn data structure of [21]
0
(see previous section) on M 0 . Whenever an entry Mij
is desired, we can retrieve it using the micro data
structure. This gives us the macro data structure: it is of size O(m/x · log(m/x)) = O(m) and can report in
O(log m) time the maximum entry of M 0 in a query column and a contiguous range of rows. It is built in
O(m/x · (log(m/x) + log n) · xε ) time which is O(m(log n + log m)/ log log m) for any choice of ε < 1.
To complete the proof of Lemma 2 we need to show how to answer a general query in O(log m) time.
Recall that a query is composed of a column of M and a contiguous range of rows. If the range is smaller than
log m we can simply check all elements explicitly in O(log m) time and return the maximum one. Otherwise,
the range is composed of three parts: a prefix part of length at most log m, an infix part that corresponds
to a range in M 0 , and a suffix part of length at most log m. The prefix and suffix are computed explicitly in
O(log m) time. The infix is computed by querying the macro data structure in O(log m) time.
t
u
A linear-space submatrix data structure.
Theorem 1. Given a m × n Monge matrix, one can construct, in O((m + n)(log n + log m)) time, a data
structure of size O(m + n) that can report the maximum entry in a query submatrix in O(log m + log n) time.
6
Proof. Recall from Section 2 that the submatrix data structure of [21] is composed of the tree Th over the
rows of M and the tree B over the columns of M . Every node u ∈ Th stores its breakpoints along with the
RMQ array Au (where Au [j] holds the value of the maximum element between the (j − 1)’th and the j’th
breakpoints of u). If u has k breakpoints then they are computed along with Au in O(k + log n) time: O(k)
to copy from the children of u and O(log n) to find the new breakpoint and to query B. As opposed to [21],
we don’t use a naive RMQ data structure but instead one of the existing linear-construction constant-query
RMQ data structures such as [19].
To prove Theorem 1 we begin with two changes to the above. First, we build Th on the rows of the
0
(m/x) × n matrix M 0 instead of the m × n matrix M (again, when an entry Mij
is desired, we retrieve it
ε
using the micro data structure in O(x ) time). Second, for B we use the data structure of Lemma 2 applied
to the transpose of M . B’s construction requires O(n(log m + log n)/ log log n) time and O(n) space. After
this, constructing Th (along with the Au arrays) on M 0 requires O(m/x · log(m/x)) = O(m) space and
O((m/x)(log(m/x) + log n) · xε ) = O(m(log m + log n)/ log log m) time by choosing x = log m and any ε < 1.
Finally, we construct a data structure Tv that is symmetric to Th but applied to the transpose of M . Notice
that Tv is built on the columns of an m×(n/ log n) matrix M 00 instead of the m×n matrix M . The construction
of Tv , from a symmetric argument to the previous paragraph, also takes O((m + n)(log n + log m)/ log log m)
time and O(m + n) space.
We now describe how to answer a submatrix query with row range R and column range C. Let R0 be
the set of consecutive rows of M 0 whose corresponding rows in M are entirely contained in R. Let Rp be the
prefix of O(log m) rows of R that do not correspond to rows of R0 . Let Rs be the suffix of O(log m) rows of R
that do not correspond to rows of R0 . We define the subranges C 0 , Cp , Cs similarly (with respect to columns
and to M 00 ). The submatrix query (R, C) can be covered by the following: (1) a submatrix query (R0 , C) in
M 0 , (2) a submatrix query (R, C 0 ) in M 00 , and (3) four small O(log m) × O(log n) submatrix queries in M for
the ranges (Ri , Cj ), i, j ∈ {p, s}. We find the maximum in each of these six ranges and return the maximum
of the six values.
We find the maximum of each of the small O(log m)×O(log n) ranges of M in O(log m+log n) time using
the SMAWK algorithm. The maximum in the submatrix of M 0 is found using Th as follows (the maximum in
the submatrix of M 00 is found similarly using Tv ). Notice that R0 is the disjoint union of O(log m) canonical
nodes of Th . For each such canonical node u, we use binary-search on u’s list of breakpoints {j1 , j2 , . . . , jk }
to find the set {ja , ja+1 , . . . , jb } of u’s breakpoints that are fully contained in C. Although this binary-search
can take O(log m) time for each canonical node, using fractional cascading, the searches on all canonical
nodes take only O(log m) time and not O(log2 m). The maximum in all rows of u and all columns between
ja and jb is found by one query to the RMQ array Au in O(1) time. Over all canonical nodes this takes
O(log m) time.
The columns of C that are to the left of ja all have their maximum element in row r(ja ) of M 0 (that
is, in one of O(log m) rows of M ) . Similarly, the columns of C that are to the right of jb all have their
maximum element in row r(jb+1 ) of M 0 . This means we have two rows of M 0 , r(ja ) and r(jb+1 ), where we
need to search for the maximum. We do this only after we have handled all canonical nodes. That is, after
we handle all canonical nodes we have a set A = a1 , a2 , . . . of 2 log m rows of M 0 in which we still need
to find the maximum. We apply the same procedure on Tv which gives us a set B = b1 , b2 , . . . of 2 log n
columns of M 00 in which we still have to find the maximum. Note that we only need to find the maximum
among the elements of M that lie in rows corresponding to a row in A and in columns corresponding to a
column in B. This amounts to finding the maximum of the O(log m) × O(log n) matrix M̄ , with M̄ij being
the maximum among the elements of M in the intersection of the x rows corresponding to row ai of M 0 ,
and of the x columns corresponding to column bj of M 00 .
An argument similar to the one in Lemma 2 shows that M̄ is Monge. Therefore we can find its maximum
element using the SMAWK algorithm. We claim that each element of M̄ can be computed in O(1) time,
which implies that SMAWK finds the maximum of M̄ in O(x) time.
It remains to show how to compute an element of M̄ in constant time. Recall from the proof of Lemma 2
that M is partitioned into x-by-n matrices M i . During the preprocessing stage, for each M i we compute and
store its upper envelope, and an RMQ array over the maximum elements in each interval of the envelope
(similar to the array Au ). Computing the upper envelope takes O(x log n) time by incrementally adding one
row at a time and using binary search to locate the new breakpoint contributed by the newly added row.
Finding the maximum within each interval of the upper envelope can be done in O(x log n) time using the
tree B. We store the upper envelope in an atomic heap [16], which supports predecessor searches in constant
7
time provided x is O(log n). Overall the preprocessing time is O(m log n), and the space is O(m). We repeat
the same preprocessing on the transpose of M .
Now, given a row ai of M 0 and column bj of M 00 , let [ca , cb ] be the range of x columns of M that
correspond to bj . We search in constant time for the successor ca0 of ca and for the predecessor cb0 of cb in
the upper envelope of M ai . We use the RMQ array to find in O(1) time the maximum element y among
elements in all rows of M corresponding to ai and columns in the range [ca0 , cb0 ). The maximum element
in columns [ca , ca0 ) and [cb0 , cb ] is contributed by two known rows r1 , r2 . We repeat the symmetric process
for the transpose of M , obtaining a maximum element y 0 , and two columns c1 , c2 . M̄ai ,bj is the maximum
among six values: y, y 0 and the four elements Mr1 c1 , Mr1 c2 , Mr2 c1 , Mr2 c2 .
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Notice that in the above proof, in order to obtain an element of M̄ in constant time, we loose the
O(log log m) speedup in the construction time. This is because we found the upper envelope of each M i . To
get the O(log log m) speedup we can obtain an element of M̄ in O(xε ) time using the micro data structure.
Corollary 1. Given a m×n Monge matrix, one can construct, in O((m+n)(log n+log m)/ log log m) time, a
data structure of size O(m+n) that reports the maximum entry in a query submatrix in O((log m+log n)1+ε )
time for any fixed 0 < ε < 1.
A linear-space subcolumn data structure for partial matrices. We next claim that the bounds of
Lemma 2 for TM matrices also apply to partial TM matrices. The reason is that we can efficiently turn any
partial TM matrix M into a full TM matrix by implicitly filling appropriate constants instead of the blank
entries.
Lemma 3. The blank entries in an m × n partial Monge matrix M can be implicitly replaced in O(m + n)
time so that M becomes Monge and each Mij can be returned in O(1) time.
Proof. Let si (resp. ti ) denote the index of the leftmost (resp. rightmost) column that is defined in row i.
Since the defined (non-blank) entries of each row and column are continuous we have that the sequence
s1 , s2 , . . . , sm starts with a non-increasing prefix s1 ≥ s2 ≥ . . . ≥ sa and ends with a non-decreasing suffix
sa ≤ sa+1 ≤ . . . ≤ sm . Similarly, the sequence t1 , t2 , . . . , tn starts with a non-decreasing prefix t1 ≤ t2 ≤
. . . ≤ tb and ends with a non-increasing suffix tb ≥ tb+1 ≥ . . . ≥ tm .
We partition the blank region of M into four regions: (I) entries that are above and to the left of M [i, si ]
for i = 1, . . . , a, (II) entries that are below and to the left of M [i, si ] for i = a + 1, . . . , m, (III) entries that
are above and to the right of M [i, ti ] for i = 1, . . . , b, (IV) entries that are below and to the right of M [i, ti ]
for i = b + 1, . . . , n. We first describe how to replace all entries in region I to make them non-blank and
obtain a valid partial Monge matrix (whose blank entries are only in regions II, III, and IV). The remaining
regions are handled in a similar manner, one after the other.
We describe our method for filling in the blank entries in region I in two steps. In the first step we
show how to implicitly fill in the blanks in a lower right triangular Monge matrix so that each filled blank
entry can be computed in O(1) time. By a lower right triangular Monge matrix we mean a partial Monge
square matrix with n rows and columns, such that, for all 1 ≤ i ≤ n, s(i) = n − i + 1. In the second step
we explain that any m-by-n partial Monge matrix whose blank entries are in region I can be turned into
a lower right triangular Monge matrix with at most m + n rows and columns. The only operations used
in the transformation are duplicating rows, duplicating columns, and turning elements into blanks. We will
show an O(m + n) procedure for computing two tables. One specifying, for each row index 1 ≤ i ≤ m, the
corresponding row index in the larger O(m + n) triangular matrix. The second is an analogous table for the
columns indices. The lemma then follows for the blank entries in region I. The other regions are treated by
reducing to the region I case, one after the other.
We now describe how to fill in the blank regions in a lower right triangular Monge matrix. Let W denote
the largest absolute value of any non-blank entry in M (We can find W by applying the algorithm of Klawe
and Kleitman [23]). Intuitively, we would like to make every M [i, j] in the upper left triangle very large.
However, we cannot simply assign the same large value to all of them, because then the Monge inequality
would not be guaranteed to hold if more than one of the four considered elements belongs to the replaced
part of the matrix. A closer look at all possible cases shows that setting all the entries of each diagonal to
the same value does work. More precisely, we replace the blank element M [i, j] with 3W [2(n − i − j) + 1].
Thus, each element in the first diagonal off the main diagonal (i + j = n) is set to 3W , the elements of the
8
second diagonal off the main diagonal are set to 9W , etc. Note that the maximum element in the resulting
matrix is O(nW ). To prove that the resulting new matrix M 0 is Monge, it suffices, by Proposition 1, to show
that, for all 1 ≤ i, k < n, M 0 [i, k] + M 0 [i + 1, k + 1] − M 0 [i, k + 1] − M 0 [i + 1, k] ≥ 0. To this end we consider
the following cases:
1. i + k > n, so all M [i, k], M [i + 1, k + 1], M [i, k + 1], M [i + 1, k] are non-blank, and the inequality holds
because M is partial Monge.
2. i + k = n, so M [i, k] is blank and M [i + 1, k + 1], M [i, k + 1], M [i + 1, k] are non-blank. Then
M 0 [i, k] + M 0 [i + 1, k + 1] − M 0 [i, k + 1] − M 0 [i + 1, k] = 3W + M 0 [i + 1, k + 1] − M 0 [i, k + 1] − M 0 [i + 1, k] ≥
3W − 3W = 0.
3. i + k = n − 1, so M [i, k], M [i, k + 1], M [i + 1, k] are blank, and M [i + 1, k + 1] is non blank. Then
M 0 [i, k] + M 0 [i + 1, k + 1] − M 0 [i, k + 1] − M 0 [i + 1, k] = 9W + M [i + 1, k + 1] − 3W − 3W ≥ 3W − W ≥ 0.
4. i + k < n − 1, so all M [i, k], M [i + 1, k + 1], M [i, k + 1], M [i + 1, k] are blank. Then,
M 0 [i, k] + M 0 [i + 1, k + 1] − M 0 [i, k + 1] − M 0 [i + 1, k] = 0.
Hence the new matrix M 0 is indeed Monge.
Next, we describe how to turn any m-by-n partial Monge matrix M whose blank entries are in region I
into a slightly larger lower right triangular matrix M 0 . This is done by duplicating rows or columns of M
and replacing by blanks a nonempty prefix in all but a single copy. Thus, each row r (column c) of M has
exactly one appearance in M 0 in which no elements are replaced by blanks. We say that r (c) is mapped to
this appearance in M 0 . Propositions 2 and 3 guarantee that M 0 is partial Monge. For ease of presentation
we describe the process as if we actually transform the M into M 0 .
The assumption that the blank entries are in region I implies that s1 ≥ s2 ≥ · · · ≥ sm and that
t1 = t2 = · · · = tm . We first guarantee that the si ’s are strictly decreasing. We do this by iterating through
the si ’s. If si = si−1 , we duplicate the column s[i] of M , make M [i − 1, s[i]] blank, and mark the column
currently at index s[i] as a duplicate (the index of this column might change later if columns with smaller
indices will be duplicated). This column duplication has the effect of increasing by 1 all s[j]’s for j < i.
At the end of this process we construct a table c[·] which keeps track of the mapping of columns of M to
M 0 by recording for each non-duplicate row its original index in M and its index after this process. Clearly,
computing c[·] and updating the si ’s can be done in O(m+n) time without actually duplicating the columns.
We may now assume that the si ’s are strictly decreasing, and we use n to denote the number of columns
of M after the transformation that ensured the strict monotonicity. We use a table r[·] to keep track of the
mapping from rows of M to rows of M 0 . For convenience, we define s0 = n+1, and r[0] = 0. We iterate through
the sequence s1 , s2 , . . . , sm . We add to M 0 si−1 − si copies of row i of M , and, for j = 1, 2, . . . , si−1 − si − 1,
replace the prefix of length j from the j’th copy by blanks, so only the last copy remains unchanged. We
therefore set r[i] to r[i − 1] + si−1 − si . Clearly, we can compute the table r[·] in O(m) time without actually
constructing M 0 .
Finally, to obtain the value with which the blank entry at M [i, j] should be replaced when converting M
into a full Monge matrix, we return 3W [2(n − r[i] − c[j]) + 1].
Regions II, III, and IV can be handled symmetrically to region I. To handle undefined entries in region
II, we implicitly reverse the order of the rows and negate all the elements of the matrix. It is easy to verify
that the resulting matrix is Monge with undefined entries in region I. We then implicitly fill in the undefined
values using the method described above, negate all the elements and revert the order of rows to its original
order. The transformation for region III is reversing the order of columns and negating all elements, and
the transformation for region IV is reversing the order of both rows and columns. Note that to make M full
Monge we first need to fill the blanks in region I, then calculate the new value of W and fill the blanks in
region II accordingly, and so on.
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The above lemma means we can (implicitly) fill the black entries in M so that M is a full TM matrix. We
can therefore apply the data structure of Lemma 2. Note that the maximum element in a query (a column
π and a range of rows R) might now appear in one of the previously-blank entries. This is easily overcome
by first restricting R to the defined entries in the column π and only then querying the data structure of
Lemma 2.
A linear-space submatrix data structure for partial matrices. Given a partial matrix M , the above
simple trick of replacing appropriate constants instead of the blank entries does not work for submatrix
9
queries because the defined (i.e., non-blank) entries in a submatrix do not necessarily form a submatrix.
Instead, we need a more complicated construction, which yields the following theorem.
Theorem 2. Given a m × n partial Monge matrix, one can construct, in O((m + n) log(m + n)) time, a data
structure of size O(m + n) that reports the maximum entry in a query submatrix in O((log m + log n)α(m +
n)) time.
Proof. As before, we partition M into m/x matrices M 1 , M 2 , . . . , M m/x , where x = log m and M i is an
x × n matrix composed of x consecutive rows of M . We wish to again define the (m/x) × n matrix M 0 such
0
that Mij
is equal to the maximum entry in column j of M i . However, it is now possible that some (or all) of
0
the entries in column j of M i are undefined. We therefore define M 0 so that Mij
is equal to the maximum
i
i
0
entry in column j of M only if the entire column j of M is defined. Otherwise, Mij
is undefined. We also
0
0
i
define the sparse matrix S so that Sij is undefined if column j of M is either entirely defined or entirely
0
undefined. Otherwise, Sij
is equal to the maximum entry among all the defined entries in column j of M i .
Using a similar argument as before, it is easy to show that M 0 is also a partial Monge matrix. The matrix
0
S , however, is not partial Monge, but it is a sparse matrix with at most two entries per column. It has
additional structure on which we elaborate in the sequel.
We begin with M 0 . As before, we cannot afford to store M 0 explicitly. Instead, we use the micro data
structure on M 1 , . . . , M m/x (after implicitly filling the blanks in M using Lemma 3). This time we use
r = 1 and so the entire construction takes O((m/x)x log n/ log r) = O(m log n) time and O(m) space,
after which we can retrieve any entry of M 0 in O(pred(x, n)) time. We then build a similar data structure
to the one we used in Theorem 1. That is, we build Th on M 0 , and for B we use the data structure of
Lemma 2 applied to the transpose of M (after implicitly filling the blanks). B’s construction therefore
requires O(n(log m + log n)/ log log n) time and O(n) space.
After constructing B, constructing Th (along with the RMQ arrays Au ) on M 0 is done bottom up. This
time, since M 0 is partial Monge, each node of Th can contribute more than one new breakpoint. However,
as we show in Section 4 (Theorem 3), a node whose subtree contains k leaves (rows) can contribute at most
O(k) new breakpoints. Each new breakpoint can be found in O(log n) time via binary search. Summing
O(k · log n · pred(x, n)) over all m/x nodes of Th gives O((m/x) log(m/x) · log n · pred(x, n))) = O(m log n)
time and O(m/x · log(m/x)) = O(m) space. Notice we use atomic heaps here to get pred(x, n) = O(1).
Similarly to what was done in Theorem 1, we repeat the entire preprocessing with the transpose of
M (that is, we construct Tv on the columns of the m × (n/ log n) matrix M 00 , along with the RMQ data
structures, and also construct the corresponding sparse matrix S 00 ). This takes O(n log m) time and O(n)
space.
We now describe how to answer a submatrix query with row range R and column range C. Let
R0 , Rs , Rp , C 0 , Cs , Cp be as in Theorem 1. The submatrix query (R, C) can be covered by the following:
(1) a submatrix query (R0 , C) in M 0 , (2) a submatrix query (R0 , C) in S 0 , (3) a submatrix query (R, C 0 ) in
M 00 , (4) a submatrix query (R, C 0 ) in S 00 , and (5) four small O(log m) × O(log n) submatrix queries in M
for the ranges (Ri , Cj ), i, j ∈ {p, s}. We return the overall maximum among the maxima in each of these
queries.
We already described how to handle the queries in items (1), (3), and (5) in the proof of Theorem 1. The
only subtle difference is that in Theorem 1 we used the SMAWK algorithm on O(log m) × O(log n) Monge
matrices while here we have partial Monge matrices. We therefore use the Klawe-Kleitman algorithm [22]
instead of SMAWK which means the query time is O((log m + log n)α(n)) and not O(log m + log n).
We next consider the query to S 0 . The query to S 00 is handled in a similar manner. Recall from the proof
of Lemma 3 the structure of a partial matrix M . Let si (resp. ti ) denote the index of the leftmost (resp.
rightmost) column that is defined in row i. Since the defined (non-blank) entries of each row and column are
continuous we have that the sequence s1 , s2 , . . . , sm starts with a non-increasing prefix s1 ≥ s2 ≥ . . . ≥ sa
and ends with a non-decreasing suffix sa ≤ sa+1 ≤ . . . ≤ sm . Similarly, the sequence t1 , t2 , . . . , tn starts with
a non-decreasing prefix t1 ≤ t2 ≤ . . . ≤ tb and ends with a non-increasing suffix tb ≥ tb+1 ≥ . . . ≥ tm . See
Fig. 2 for an illustration. It follows that the defined entries of S 0 can be partitioned into four sequences,
such that the row and column indices in each sequence are monotone. We focus on one of these monotone
sequences in which the set of defined entries is in coordinates (r1 , c1 ), (r2 , c2 ), . . . such that ri+1 ≥ ri and
ci+1 ≤ ci . The other monotone sequences are handled similarly. Notice that any query range that includes
(ri , cj ) and (rj , cj ) for some i < j must include entries (rk , ck ) for all i < k < j. Given a range query (R, C),
10
we find in pred(n, n) time the interval [i1 , i2 ] of indices that are inside R. Similarly, we find the interval [i01 , i02 ]
of indices that are inside C. We can then use a (1-dimensional) RMQ data structure on the O(n) entries in
this sequence to find the maximum element in the intersection of these two ranges in O(1) time. Overall,
handling the query in S 0 takes pred(n, n) = O(log log n) time.
To conclude the proof of Theorem 2, notice that our data structure requires O(m+n) space, is constructed
in O(m log n + n log n/ log log n + n log m + n log n) time which is O(n log n), and has O((log m + log n)α(n))
query time.
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Finally, for the same reasons leading to Corollary 1 we can get a log log m speedup in the construction-time
with a logε n slowdown in the query-time.
Corollary 2. Given a m × n partial Monge matrix, one can construct, in O((m + n) log(m + n)/ log log m)
time, a data structure of size O(m + n) that reports the maximum entry in a query submatrix in O((log m +
log n)1+ε α(m + n)) time for any fixed 0 < ε < 1.
4
The Complexity of the Upper Envelope of a Totally Monotone Partial
Matrix
In this section we prove the following theorem, stating that the number of breakpoints of an m × n TM
partial matrix is only O(m).
Theorem 3. Let M be a partial m × n matrix in which the defined entries in each row and in each column
are contiguous. If M is TM (i.e., for all i < j, k < ` where Mik , Mi` , Mjk , Mj` are all defined, Mik ≤
Mjk =⇒ Mi` ≤ Mj` ), then the upper envelope has complexity O(m).
The proof relies on a decomposition of M into staircase matrices. A partial matrix is staircase if the
defined entries in its rows either all begin in the first column or all end in the last column. It is well known
(cf. [1]) that by cutting M along columns and rows, it can be decomposed into staircase matrices {Mi } such
that each row is covered by at most three matrices, and each column is covered by at most three matrices.
For completeness, we describe such a decomposition below.
Lemma 4. A partial matrix M can be decomposed into staircase matrices {Mi } such that each row is covered
by at most three matrices, and each column is covered by at most three matrices.
Proof. Let si and ti denote the smallest and largest column index in which an element in row i is defined,
respectively. The fact that the defined entries of M are contiguous in both rows and columns implies that the
sequence s1 , s2 , . . . , sm consists of a non-increasing prefix and a non-decreasing suffix. Similarly, the sequence
t1 , t2 , . . . , tm consists of a non-decreasing prefix and a non-increasing suffix. It follows that the rows of M can
be divided into three ranges - a prefix where s is non-increasing and t is non-decreasing, an infix where both
s and t have the same monotonicity property, and a suffix where s is non-decreasing and t is non-increasing.
The defined entries in the prefix of the rows can be divided into two staircase matrices by splitting M at
t1 , the largest column where the first row has a defined entry. Similarly, the defined entries in the suffix of
the rows can be divided into two staircase matrices by splitting it at tm , the largest column where the last
row has a defined entry. The defined entries in the infix of the rows form a double staircase matrix. It can
be broken into staircase matrices by dividing along alternating rows and columns as shown in Figure 2.
It is easy to verify that, in the resulting decomposition, each row is covered by at most two staircase
matrices, and each column is covered by at most three staircase matrices. Also note that every set of consecutive columns whose defined elements are in exactly the same set of rows are covered in this decomposition
by the same three row-disjoint staircase matrices.
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We next prove the fact that, if M is a TM staircase matrix with m rows, then the complexity of its upper
envelope is O(m).
Lemma 5. The number of breakpoints in the upper envelope of an m × n TM staircase matrix is at most
2m.
11
Fig. 2. Decomposition of a partial matrix into staircase matrices (defined by solid thick black lines) and into blocks
of consecutive columns with the same defined entries (indicated by thin vertical red lines).
Proof. We focus on the case where the defined entries of all rows begin in the first column and end in
non-decreasing columns. In other words, for all i, si =1 and ti ≤ ti+1 . The other cases are symmetric.
A breakpoint is a situation where the maximum in column c is at row r1 and the maximum in column
c + 1 is at a different row r2 . We say that r1 is the departure row of the breakpoint, and r2 is the entry row
of the breakpoint. There are two types of breakpoints: decreasing (r1 < r2 ), and increasing (r1 > r2 ). We
show that (1) each row can be the entry row of at most one decreasing breakpoint, and (2) each row can be
the departure row of at most one increasing breakpoint.
(1) Assume that row r2 is an entry row of two decreasing breakpoints: One is the pair of entries
(r1 , c1 ), (r2 , c1 + 1) and the other is the pair (r3 , c2 ), (r2 , c2 + 1). We know that r1 < r2 , r3 < r2 ,
and wlog c2 > c1 + 1. Since the maximum in column c1 + 1 is in row r2 , we have Mr3 ,c1 +1 < Mr2 ,c1 +1 .
However, since the maximum in column c2 is in row r3 , we have Mr3 ,c2 > Mr2 ,c2 , contradicting the total
monotonicity of M . Note that Mr2 ,c2 is defined since Mr2 ,c2 +1 is defined.
(2) Assume that row r1 is a departure row of two increasing breakpoints: One is the pair of entries
(r1 , c1 ), (r2 , c1 + 1) and the other is the pair (r1 , c2 ), (r3 , c2 + 1). We know that r1 > r2 and r1 > r3 .
Since the maximum in column c1 is in row r1 , we have Mr2 ,c1 < Mr1 ,c1 . However, since the maximum
in column c1 + 1 is in row r2 , we have Mr2 ,c1 +1 > Mr1 ,c1 +1 , contradicting the total monotonicity of M .
Note that Mr1 ,c1 +1 is defined since Mr1 ,c2 is defined.
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Using Lemmas 3 and 5 we can now complete the proof of Theorem 3. Let bp(Mi ) denote the number of
breakpoints in the upper
P envelope of Mi . Let mi denote the number of rows in Mi . Since each row appears in
at most
three
M
s,
i
P
P i mi = O(m). The total number of breakpoints in the envelopes of all of Mi s is O(m)
since i bp(Mi ) = i O(mi ) = O(m).
Consider now a partition of M into rectangular blocks Bj defined by maximal sets of contiguous columns
whose defined entries are at the same set of rows. There are O(m) such blocks.PThe upper envelope of M is
just the concatenation of the upper envelopes of all the Bj ’s. Hence, bp(M ) = j bp(Bj ) + O(m) (the O(m)
term accounts forP
the possibility of a new breakpoint between every two consecutive blocks). Therefore, it
suffices to bound j bp(Bj ).
Consider some block Bj . As we mentioned above, the columns of Bj appear in the same three row-disjoint
staircase matrices M1 , M2 , M3 in the decomposition of M . The column maxima of Bj are a subset of the
12
column maxima of M1 , M2 , M3 . Assume wlog that the indices of rows covered by M1 are smaller than those
covered by M2 , which are smaller than those covered by M3 .
The breakpoints of the upper envelope of Bj are either breakpoints in the envelope of M1 , M2 , M3 , or
breakpoints that occur when the maxima in consecutive columns of Bj originate in different Mi . However,
since Bj is a (non-partial) TM matrix, its column maxima are monotone. So once a column maximum
originates in Mi , no maximum in greater columns will ever originate in Mj for j < i. It follows that the
numberP
of breakpoints
Pin Bj that are not breakpoints of M1 , M2 , M3 is at most two. Since there are O(m)
blocks, j bp(Bj ) ≤ i bp(Mi ) + O(m) = O(m). This completes the proof of Theorem 3.
5
Constant Query-Time Data Structures
In this section we present our data structures that improve the query time to O(1) at the cost of an nε factor
in the construction time and space for any constant 0 < ε < 1.
We use the following micro data structure that slightly modifies the one of Lemma 1.
Lemma 6 (another micro data structure). Given a TM matrix of size x × n, one can construct in
O(xnε ε−1 ) time and space a data structure that given a query column can report the maximum entry in the
entire column in O(log(ε−1 )) time for any 1 > ε ≥ log log n/ log n.
Proof. Recall that the data structure of Lemma 1, for r = 1, finds in O(x log n) time a set of O(x) values
(breakpoints) in the range {1, . . . , n}. A query is performed in O(pred(x, n)) time using a standard predecessor data structure on these O(x) values. Since now we can allow an nε factor we use a non-standard
predecessor data structure with faster O(ε−1 ) query-time. We now describe this data structure.
Consider the complete tree of degree nε over the leaves {1, . . . , n}. We do not store this entire tree. We
only store the leaf nodes corresponding to the O(x) existing values and all ancestors of these leaf nodes. Since
the height of the tree is O(ε−1 ) we store only O(xε−1 ) nodes. At each such node we keep two arrays, each
of size nε . The first array stores all children pointers (including null-children). The second array stores for
each child u (including null-children) the value pred(u) = the largest existing leaf node that appears before
u in a preorder traversal of the tree.
The y = O(xε−1 ) nodes are stored in a hash table. We use the static deterministic hash table of Hagerup
et al. [18] that is constructed in O(y log y) = O(xε−1 log(xε−1 )) worst case time and can be queried in O(1)
worst case time. Upon query, we binary-search (using the hash table) for the deepest node v on the root-toquery path whose child u on the root-to-query path is null. To find the predecessor we use v’s second array
and return pred(u).
The total construction time is O(x log n + xnε ε−1 + xε−1 log(xε−1 )) which is O(xnε ε−1 ) since we assume
ε ≥ log log n/ log n. The query time is O(log(ε−1 )) since we binary-search on a path of length ε−1 and each
lookup takes O(1) time using the hash table.
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We use the above micro data structure to obtain the following data structure.
Lemma 7. Given a TM x × n matrix, one can construct in O(x3 nε ε−1 ) time a data structure of size
O(x3 nε ε−1 ) that can report the maximum entry in a query column and a contiguous range of rows in
O(log(ε−1 )) time.
Proof. For each of the O(x2 ) row intervals, construct the data structure of Lemma 6.
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A constant-query subcolumn data structure.
Lemma 8. Given a TM matrix of size m × n, one can construct, in O(mnε ε−2 ) = O(n1+ε ε−2 ) time and
space a data structure that can report the maximum entry in a query column and a contiguous range of rows
in O(ε−1 log(ε−1 )) time.
Proof. The first idea is to use a degree-x tree, with x = mε/4 , instead of the binary tree Th . The height of the
tree is O(log m/ log x) = O(ε−1 ). The leaves of the tree correspond to individual rows of M . For an internal
node u of this tree, whose children are u1 , u2 , . . . , ux and whose subtree contains k leaves (i.e., k rows), recall
that M u is the k × n matrix defined by these k rows and all columns. Let M̂ u be the x × n matrix whose (i, j)
element is the maximum in column j among the rows of M ui . In other words, M̂ u (i, j) = max` M ui (`, j).
13
Working bottom up, for each internal node u, instead of explicitly storing the matrix M u (whose size is
O(kn)), we build the O(knε ε−1 )-sized micro data structure of Lemma 6 over the k rows of M u . This way,
any element M̂ u (i, j) can be obtained in O(log(ε−1 )) time by querying the data structure of ui . Once we
can obtain each M̂ u (i, j) in O(log(ε−1 )) time, we use this to construct the data structure of Lemma 7 over
the x = mε/4 rows of M̂ u .
Constructing the micro data structure of Lemma 6 for an internal node with k leaf descendants takes
O(knε ε−1 ) time and space. Summing this over all internal nodes in the tree, the total construction takes
O(mnε ε−2 ) time and space. After this, we construct the Lemma 7 data structure for each internal node
but we use ε/2 and not ε so the construction takes O(x3 nε/2 ε−1 · log(ε−1 )) = O(m3ε/4 nε/2 ε−1 log(ε−1 ))
time and space. The total construction time over all O(m/x) = O(m1−ε/4 ) internal nodes is thus
O(m1+ε/2 nε/2 ε−1 log(ε−1 )) = O(n1+ε ε−1 log(ε−1 )) time and space.
We now describe how to answer a query. Given a query column and a row interval I, there is an induced
set of O(log m/ log x) = O(ε−1 ) canonical nodes. Each canonical node u is responsible for a subinterval of I
(that includes all descendant rows of ui , u1+1 . . . , uj for some two children ui , uj of u). We find the maximum
in this subinterval with one query to u’s Lemma 7 data structure in O(log(ε−1 )) time. The total query time
is thus O(ε−1 log(ε−1 )).
t
u
A constant query submatrix data structure.
Theorem 4. Given a Monge matrix of size m × n, one can construct, in O(n1+ε ε−3 log(ε−1 )) time and
space, a data structure that can report the maximum entry in a query submatrix in O(ε−2 log(ε−1 )) time.
Proof. As in the proof of Lemma 8, we construct a degree-x tree Th over the rows of M , with x = mε/4 .
Recall that Th includes, for each internal node u, (i) the data structure of Lemma 6, which enables queries to
elements of M̂ u in O(log(ε−1 )) time, and (ii) the breakpoints of all possible row intervals of the x × n matrix
M̂ u . In addition to the breakpoints we store, for each of these O(x2 ) intervals, a RMQ data structure over
the maximum elements between breakpoints. The construction of those RMQ data structures is described
in the sequel.
For each level ` > 0 of the O(ε−1 ) levels of the tree Th (the leaves of Th are considered to be at level 0),
we construct the symmetric data structure of Lemma 8 over the (m/x`−1 ) × n matrix formed by the union
of M̂ u over all level-i nodes u in Th . We denote these data structures by B` . Their construction takes total
O(ε−1 · mnε ε−2 · log(ε−1 )) = O(n1+ε ε−3 log(ε−1 )) time and space. For notational convenience we define B0
to be equal to B1 .
We now describe how to construct the RMQ data structures for an internal node u at level ` of Th
with children u1 , . . . , ux . We describe how to construct the RMQ for the interval consisting of all rows of
M̂ u . Handling the other intervals is similar. We need to show how to list the maximum among the column
maxima of M̂ u between every two consecutive breakpoints of M̂ u . All the column maxima between any two
consecutive breakpoints are contributed by a single known child u0 of u. In other words, we are looking for
the maximum element in the range consisting of a single row of M̂ u and the range of columns between the
two breakpoints. This maximum can be found by querying the B` data structure in O(ε−1 log(ε−1 )) time.
There are O(x) such queries for each of the O(x2 ) intervals at each of the O(m/x) internal nodes. Therefore,
the total construction time of the RMQs is O(m1+ε · ε−1 log(ε−1 )). This completes the description of our
data structure.
We finally discuss how to answer a query (a range in M of rows R and columns C). A query induces a set
of O(ε−1 ) canonical nodes u. For a canonical node u ∈ Th and an induced row interval Ru , we use the list of
breakpoints of Ru in M̂ u to identify the breakpoints that are fully contained in C. This takes O(log(ε−1 ))
time. The maximum element in those columns is found by querying the RMQ data structure of Ru in u.
In addition to that, there are at most two column intervals C 0 and C 00 in M̂ u that intersect C but are not
fully contained in C. The maximum in C 0 ∩ C and C 00 ∩ C is contributed by two known children u0 , u00 of u,
respectively. In other words, each of them is the maximum element in the range consisting of a single row
of M̂ u and a range of columns. If u is a level-` node of the tree then we find them by two queries to B` : one
for the row of u0 and columns C 0 and one for the row of u00 and columns C 00 . The total query time is thus
O(ε−1 · ε−1 log(ε−1 )) = O(ε−2 log(ε−1 )).
t
u
14
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15
| 8 |
Computational Optimal Transport: Complexity by Accelerated Gradient
Descent Is Better Than by Sinkhorn’s Algorithm
Pavel Dvurechensky 1 Alexander Gasnikov 2 3 Alexey Kroshnin 2 3
arXiv:1802.04367v1 [cs.DS] 12 Feb 2018
Abstract
We analyze two algorithms for approximating the
general optimal transport (OT) distance between
two discrete distributions of size n, up to accuracy ε. For the first algorithm, which is based
on the celebrated Sinkhorn’s
algorithm, we prove
2
n
e
the complexity bound O ε2 arithmetic operations1 . For the second one, which is based on our
novel Adaptive Primal-Dual Accelerated Gradient
Descent (APDAGD) algorithm,
weo
prove the
n
e min n9/4 , n22
complexity bound O
arithmeε
ε
tic operations. Both bounds have better dependenceon
ε than the state-of-the-art result given
n2
e
by O ε3 . Our second algorithm not only has
better dependence on ε in the complexity bound,
but also is not specific to entropic regularization
and can solve the OT problem with different regularizers.
1. Introduction
Optimal transport (OT) distances between probability measures or histograms, including the earth mover’s distance
(Werman et al., 1985; Rubner et al., 2000) and MongeKantorovich or Wasserstein distance (Villani, 2008), play
an increasing role in different machine learning tasks, such
as unsupervised learning (Arjovsky et al., 2017; Bigot et al.,
2017), semi-supervised learning (Solomon et al., 2014),
clustering (Ho et al., 2017), text classification (Kusner et al.,
2015), as long as in image retrieval, clustering and classification (Rubner et al., 2000; Cuturi, 2013; Sandler &
Lindenbaum, 2011), statistics (Ebert et al., 2017; Panaretos & Zemel, 2016), and other applications (Kolouri et al.,
2017).
1
Weierstrass Institute for Applied Analysis and Stochastics,
Berlin, Germany 2 Moscow Institute of Physics and Technology,
Dolgoprudny, Moscow Region, Russia 3 Institute for Information
Transmission Problems RAS, Moscow, Russia. Correspondence
to: Pavel Dvurechensky <pavel.dvurechensky@wias-berlin.de>.
1
e hides polylogarithmic factors (ln n)c , c > 0.
O
Our focus in this paper is on the computational aspects of
OT distances for the case of two discrete probability measures with support of equal2 size n. The state-of-the-art
approach (Cuturi, 2013) for this setting is to apply Sinkhorn’s algorithm to the entropy-regularized OT optimization
problem. As it was recently shown in (Altschuler et al.,
2017), this approach allows
to find an ε-approximation for
n2
e
an OT distance in O ε3 arithmetic operations. In terms
of the dependence on n, this result improves on the come 3 ) achieved by the network simplex method or
plexity O(n
interior point methods (Pele & Werman, 2009), applied directly to the OT optimization problem, which is a linear
program (Kantorovich, 1942). Nevertheless, the cubic dependence on ε prevents approximating OT distances with
good accuracy.
On the other hand, in image color transfer (Pitié et al., 2007)
or domain adaptation (Courty et al., 2017) not only the OT
distance, but also the optimal transportation plan is of interest. Recent work (Blondel et al., 2017) advocates that
entropic regularization of the OT problem leads to a dense
transportation plan, which is in contrast to the sparse transportation plan obtained by solving the unregularized OT
problem. Motivated by this observation, they study general
regularization by a strongly convex function, e.g. squared
euclidean norm, and show that this leads to a sparse transportation plan. In this situation, Sinkhorn’s algorithm becomes
inapplicable since it is specific to entropic regularization.
Our goal in this paper is, first, to obtain better than stateof-the-art complexity bounds for approximating the OT distance and, second, propose a flexible algorithm for solving
the OT problem with different types of regularization.
Approximating the OT distance amounts to solving the OT
problem (Kantorovich, 1942):
min hC, Xi,
X∈U (r,c)
U(r, c) := {X ∈ Rn×n
: X1 = r, X T 1 = c},
+
(1)
where X is transportation plan, C ∈ Rn×n
is a given
+
ground cost matrix, r, c ∈ Rn are given vectors from the
2
This is done for simplicity and all the results easily generalize
to the case of measures with different support size.
Complexity of Optimal Transport Distances
probability simplex ∆n , 1 is the vector of all ones. The
regularized OT problem is
min hC, Xi + γR(X),
X∈U (r,c)
(2)
where γ > 0 is the regularization parameter and R(X)
is a strongly convex regularizer, e.g. negative entropy or
squared Euclidean norm.
b ∈ U(r, c) such that
Our goal is to find X
b ≤
hC, Xi
min hC, Xi + ε.
X∈U (r,c)
(3)
b is an ε-approximation for the OT disIn this case, hC, Xi
b
tance and X is an approximation for the transportation plan.
Related work. We focus on the general case with C being
a non-negative dense matrix with bounded entries. In this
case, (1) is a linear programming problem with best theoree 5/2 ) (Lee & Sidford, 2014) and best
tical complexity O(n
e 3 ) (Pele & Werman, 2009), which
practical complexity O(n
is problematic when n is larger than 103 .
A natural alternative is to approximate (1) by (2) with a
small γ. Starting with the work (Cuturi, 2013), the widely
used practical implementation of this idea is to use entropy
regularization, i.e. solve (2), where R(X) is negative entropy of a matrix X. The special structure of this problem
allows to use the balancing algorithm (Bregman, 1967) also
known as Sinkhorn’s algorithm (Sinkhorn, 1974) and RAS
(Kalantari & Khachiyan, 1993). The best known complex
e n32 to
ity bound in the literature for this approach is O
ε
obtain (3) (Altschuler et al., 2017), Theorem 1. They also
show that the regularization parameter should be chosen proportional to ε, which necessitates working with the matrix
exp(−C/ε) and leads to problems with numerical stability
of the algorithm. Several ways to overcome this instability
issue were proposed in (Schmitzer, 2016), but with limited
theoretical analysis. While the entropy-regularized OT problem allows to use other matrix-scaling algorithms such as
(Allen-Zhu et al., 2017; Cohen et al., 2017) with theoretical
guarantees, the authors do not provide any experimental results, so the practical implementability of these algorithms
is questionable. In (Genevay et al., 2016), stochastic gradient descent is applied to solve the entropy-regularized OT
problem, but the complexity for approximating OT distance
in the sense of (3) is not studied. In any case, Sinkhorn’s
and other mentioned algorithms are very specific to entropic
regularization in (2).
A flexible alternative can be to use some general purpose
optimization method to solve (2), which is a particular case
of a minimization problem with linear constraints. When
n is large, the natural choice is the class of first-order methods. Since our focus is on complexity analysis, Conjugate
Table 1. Comparison of algorithms for (2).
A LGORITHM
(B ECK & T EBOULLE , 2014)
(C HAMBOLLE & P OCK , 2011)
(M ALITSKY & P OCK , 2016)
(T RAN -D INH & C EVHER , 2014)
(Y URTSEVER ET AL ., 2015)
(PATRASCU ET AL ., 2015)
(G ASNIKOV ET AL ., 2016)
(L I ET AL ., 2016)
(L AN ET AL ., 2011)
(O UYANG ET AL ., 2015)
(X U , 2016)
T HIS PAPER (A LG . 3)
R ATES
×
×
×
√
√
√
√
√
×
×
√
√
LS
√
E NTR .
√
×
√
×
×
√
√
√
√
√
√
×
×3
×
×
×
×
√
×
√
×
×
√
Gradients and quasi-Newton methods, i.e. L-BFGS, are
not suitable. Due to the presence of linear constraints, the
most common approach involves the construction of the
Lagrange dual problem and primal-dual updates during the
algorithm’s progress. The tricky part of this approach is
to prove accelerated (Nesterov, 2004) convergence rates
separately for the primal objective residual and linear constraints feasibility. On the other hand, first-order methods
use the Lipschitz constant of the objective’s gradient to define the stepsize. The theoretical value for this constant is
usually an overestimation and leads to small stepsize and
slow convergence in practice. Thus, an algorithm should use
a line-search strategy to adapt to the local value of this constant and converge faster. Finally, entropy regularization in
(2) is an important particular case and an algorithm should
be able to deal with this non-Lipschitz-smooth regularizer.
We analyzed a bunch of algorithms in the literature (see Table 1) and none of them combine all three described features,
namely, a) accelerated convergence rates separately for the
primal objective and constraints feasibility, b) line-search,
c) entropy friendliness.
Our contributions can be summarized as follows.
• Improved analysis
2 of the Sinkhorh’s algorithm and
e
complexity O nε2 arithmetic operations for approximating the OT distance in the sense of (3).
• An Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD) algorithm, which incorporates a linesearch strategy and has accelerated convergence rates
separately for the primal objective and constraints feasibility in (2) with a general strongly convex regularizer.
o
n
e min n9/4 , n22
• Improved complexity O
arithmetic
ε
ε
operations for approximating the OT distance in the
3
Their algorithm uses Lipschitz constant in the stopping criterion and, hence, is not completely adaptive.
Complexity of Optimal Transport Distances
sense of (3), based on our APDAGD method.
• Numerical illustration of the practical performance of
these algorithms for approximating the OT distance.
Notation. For a general finite-dimensional real vector space
E, we denote by E ∗ its dual, given by linear pairing hg, xi,
x ∈ E, g ∈ E ∗ ; by k · kE the norm in E and by k ·
kE,∗ the norm in E ∗ , which is dual to k · kE . For a linear
operator A : E → H, we define its norm as kAkE→H =
maxx∈E,u∈H ∗ {hu, Axi : kxkE = 1, kukH,∗ = 1}. We
say that a function f : E → R is γ-strongly convex on
a set Q ⊆ E w.r.t. a norm in E iff, for any x, y ∈ Q,
f (y) ≥ f (x) + h∇f (x), y − xi + γ2 kx − yk2E , where ∇f (x)
is any subgradient of f (x) at x.
For a matrix A and a vector a, we denote eA , ea , ln A, ln a
their entrywise exponents and natural logarithms respectively. For a vector a ∈ Rn , we denote by kak1 the sum of
absolute values of its elements, and by kak2 its Euclidean
norm, and by kak∞ the maximum absolute value of its elements. Given a matrix A ∈ Rn×n , we denote by vec(A)
2
the vector in Rn , which is obtained from A by writing
its columns one below another. For a matrix A ∈ Rn×n ,
we denote kAk1 = kvec(A)k1 and kAk∞ = kvec(A)k∞ .
Further, we define the entropy of a matrix X ∈ Rn×n
by
+
H(X) := −
n
X
X ij ln X ij .
(4)
i,j=1
For two matrices A, B, we denote their Frobenius inner
product by hA, Bi. We denote by ∆n := {a ∈ Rn+ : aT 1 =
1} the standard simplex in Rn . For p, q ∈ ∆n , we define
the Kullback–Leibler divergence between p and q to be
KL(pkq) :=
n
X
i=1
pi ln
pi
.
qi
2. Sinkhorn’s Algorithm
In this section, our goal is to refine the complexity analysis
of the Sinkhorn’s algorithm, then, based on
analysis,
this
n2 kCk3∞ log n
improve the existing complexity bound O
3
ε
for approximating the OT distance in the sense of (3) and
obtain new complexity
2
n kCk2∞ log n
O
.
ε2
2.1. Improved Complexity of the Sinkhorn’s Algorithm
We consider the Sinkhorn–Knopp algorithm listed as Algorithm 1, which solves (Cuturi, 2013), Lemma 2, the minimization problem
n
o
minn ψ(u, v) := 1T B(u, v)1 − hu, ri − hv, ci , (5)
u,v∈R
Algorithm 1 Sinkhorn’s Algorithm
Input: Accuracy ε0 .
1: Set k = 0, u0 = v0 = 0
2: repeat
3:
if k mod 2 = 0 then
4:
uk+1 = uk + ln r − ln(B(uk , vk )1)
5:
vk+1 = vk
6:
else
7:
vk+1 = vk + ln c − ln(B(uk , vk )T 1)
8:
uk+1 = uk
9:
end if
10:
k =k+1
11: until kB(uk , vk )1 − rk1 + kB(uk , vk )T 1 − ck1 ≤ ε0
Output: B(uk , vk ).
where K := e−C/γ and
B(u, v) := diag eu K diag ev ,
diag(a) being the diagonal matrix with the vector a on the
diagonal. Problem (5) is the dual to (2) with a particular
choice R(X) = −H(X).
To improve the complexity of the Sinkhorn’s algorithm, first,
we obtain some bounds for the iterates uk , vk and an optimal
solution (u∗ , v ∗ ) for (5). Then, using these bounds, for each
iteration of the algorithm, we upper bound the objective
value ψ(uk , vk ) by kB(uk , vk )1 − rk1 + kB(uk , vk )T 1 −
ck1 . Finally, the latter bound is used, to prove the main
theorem of this subsection with new complexity result for
the Sinkhorn’s algorithm.
The following lemma provides the bounds for uk , vk , u∗
and v ∗ .
Lemma 1. Let k ≥ 0 and uk , vk be generated by Algorithm 1 and (u∗ , v ∗ ) be a solution of (5). Then
maxi uik − mini uik ≤ R,
maxj vkj − minj vkj ≤ R (6)
and
maxi (u∗ )i −mini (u∗ )i ≤ R,
where
and
maxj (v ∗ )j −minj (v ∗ )j ≤ R,
R := − ln ν mini,j {ri , cj }
ν := mini,j K ij = e−kCk∞ /γ .
(7)
(8)
Proof. First, we prove the bound for uk . Obviously, the
stated inequality holds for k = 0. Let k − 1 be even.
Then the variable u is updated on the iteration k − 1 and
B(uk , vk )1 = r by the algorithm construction. Hence, for
each i ∈ [1, n], we have
X i
j
i
euk νh1, evk i ≤
euk K ij evk = [B(uk , vk )1]i = ri ≤ 1
j
Complexity of Optimal Transport Distances
and
maxi uik ≤ − ln (νh1, evk i) .
(9)
On the other hand, since K ij ≤ 1, for each i ∈ [1, n],
uik
vk
e h1, e i ≥
X
uik
j
ij vk
e K e
max ui +mini uik
= [B(uk , vk )1]i = ri
j
, by Hölder’s inequality and
ri
h1, evk i
= ln
mini ri
h1, evk i
≤ kuk − a1k∞ kBk 1 − rk1
=
.
maxi uik − mini uik
R
kBk 1 − rk1 ≤ kBk 1 − rk1 .
2
2
Using the same arguments, we bound h−u∗ , Bk 1 − ri,
hvk , BkT 1 − ci and h−v ∗ , BkT 1 − ci in (10) and finish the
proof of the lemma.
The latter inequality and (9) give
maxi uik − mini uik ≤ − ln ν mini ri ≤ R.
Since the next iteration k, which is odd, updates the variable
v and leaves the variable u unchanged, the obtained bound
for uk holds for any k ≥ 0. The bound in (6) for vk is
proved in the same way. Finally, since (u∗ , v ∗ ) is an optimal
solution of (5), the gradient of the objective in (5) vanishes
at this point. Hence, B(u∗ , v ∗ )1 = r and B(u∗ , v ∗ )T 1 = c.
Using these equalities and repeating the same arguments as
in the proof of bounds for uk and vk , we prove the bounds
for u∗ and v ∗ .
The following lemma, for each iteration of Algorithm 1,
relates the objective ψ(uk , vk ) in (5) and kB(uk , vk )1 −
rk1 + kB(uk , vk )T 1 − ck1 . To simplify derivations, we
define
Now we are ready to improve the iteration complexity
bound for the Sinkhorn’s algorithm.
Theorem 1. Algorithm 1 outputs a matrix B(uk , vk ) satisfying kB(uk , vk )1 − rk1 + kB(uk , vk )T 1 − ck1 ≤ ε0
in
4R
k ≤2+ 0 .
ε
iterations.
Proof. Assume that k ≥ 1 is even. As before, we donote
Bk = B(uk , vk ). Since h1, Bk 1i = h1, Bk+1 1i = 1 and
vk+1 = vk , we have
ψ(uk , vk ) − ψ(uk+1 , vk+1 )
= h1, Bk 1i−h1, Bk+1 1i+huk+1 −uk , ri+hvk+1 −vk , ci
e v) := ψ(u, v) − ψ(u∗ , v ∗ )
ψ(u,
= h1, B(u, v)1i−h1, B(u∗ , v ∗ )1i+hu∗ −u, ri+hv ∗ −v, ci.
Lemma 2. Let k ≥ 1 and uk , vk be generated by Algorithm 1. Then
e k , vk ) ≤ R kBk 1 − rk1 + kB T 1 − ck1 ,
ψ(u
k
where Bk := B(uk , vk ).
Proof. Let us fix k ≥ 1 and consider the convex function of
(û, v̂)
h1, B(û, v̂)1i − hû, B(uk , vk )1i − hv̂, B(uk , vk )T 1i.
Since its gradient vanishes at (û, v̂) = (uk , vk ), the point
(uk , vk ) is its minimizer. Hence,
e k , vk ) = h1, Bk 1i − huk , Bk 1i − hvk , B T 1i
ψ(u
k
− h1, B(u∗ , v ∗ )1i − hu∗ , Bk 1i − hv ∗ , BkT 1i
∗
i k
Taking a =
2
Lemma 1, we obtain
huk , Bk 1 − ri = huk − a1, Bk 1 − ri
and
mini uik ≥ mini ln
Next, we bound the r.h.s. of this inequality. Since, on each
iteration of the Sinkhorn’s algorithm, either Bk 1 = r, or
BkT 1 = c, we have that h1, Bk 1i = 1 and h1, Bk 1−ri = 0.
By Pinsker’s inequality and Lemma 2, since BkT 1 = c, we
obtain
e k , vk ) − ψ(u
e k+1 , vk+1 ) = KL (rkBk 1)
ψ(u
(
)
e k , vk ) (ε0 )2
ψ(u
1
2
,
,
≥ kBk 1 − rk1 ≥ max
2
2R2
2
where we also used that, as soon as the stopping criterion
2
is not yet fulfilled and BkT 1 = c, kBk 1 − rk1 ≥ (ε0 )2 .
The same inequality can be proved for the case of odd k.
Therefore (Nesterov, 2004), §2.1.5, for any k ≥ 1,
e k+1 , vk+1 )
e k , vk )
ψ(u
ψ(u
≤
−
2
2R
2R2
e k , vk )
ψ(u
2R2
2
2
where ` = e 2R . Thus k ≤ 1 + e 2R
ψ(u1 ,v1 )
ψ(uk ,vk )
the other hand,
∗
+ huk − u , Bk 1 − ri + hvk − v , BkT 1 − ci
≤ huk − u∗ , Bk 1 − ri + hvk − v ∗ , BkT 1 − ci.
= hr, uk+1 − uk i = hr, ln r − ln(Bk 1)i = KL(rkBk 1)
(10)
!2
1
,
k+`
(11)
2
− e 2R . On
≤
ψ(u1 ,v1 )
0 2
e k+m , vk+m ) ≤ ψ(u
e k , vk ) − (ε ) m
ψ(u
2
(12)
Complexity of Optimal Transport Distances
Algorithm 2 Approximate OT by Sinkhorn
Input: Accuracy ε.
ε
ε
0
1: Set γ = 4 ln
n , ε = 8kCk∞ .
n
2: Find r̃, c̃ ∈ ∆ s.t. kr̃ − rk1 ≤ ε0 /4, kc̃ − ck1 ≤ ε0 /4
and mini r̃i ≥ ε0 /(8n), minj c̃j ≥ ε0 /(8n).
3: Calculate B by Algorithm 1 with marginals r̃, c̃ and
accuracy ε0 /2.
b as the projection of B on U(r, c) by Algorithm 2
4: Find X
in (Altschuler et al., 2017).
b
Output: X.
for all k, m ≥ 0.
To combine the two estimates (11) and (12), we consider a switching strategy, parametrized by number s ∈
e 1 , v1 )]. First, using (11), we estimate the number
(0, ψ(u
e v) from ψ(u
e 1 , v1 ) to s. Then,
of iterations to reduce ψ(u,
using (12), we estimate the number of iterations to reduce
e v) from s to zero, keeping in mind that ψ(u,
e v) ≥ 0
ψ(u,
by its definition. Minimizing the sum of these two estimates
e 1 , v1 )], we conclude that the total number of
in s ∈ (0, ψ(u
iterations k satisfies
k≤
min
e 1 ,v1 )
0<s≤ψ(u
=
2 +
2 +
2R2
2R2
2s
2+
−
+ 0 2
e
s
(ε
)
ψ(u1 , v1 )
4R
2R2
e 1 ,v1 ) ,
ε0 − ψ(u
e 1 ,v1 )
2ψ(u
,
(ε0 )2
In both cases, we have k ≤ 2 +
4R
ε0 .
!
e 1 , v1 ) ≥ Rε0 ,
ψ(u
e 1 , v1 ) < Rε0 .
ψ(u
2.2. Complexity of OT Distance by Sinkhorn
Now we apply the result of the previous subsection to derive
b ∈ U(r, c) satisfying (3).
a complexity estimate for finding X
The pseudocode of our procedure for approximating the OT
distance by the Sinkhorh’s algorithm is listed as Algorithm
2.
b ∈ U(r, c) satisfying
Theorem 2. Algorithm 2 outputs X
(3) in
2
n kCk2∞ ln n
O
ε2
arithmetic operations.
Before we prove the theorem, we compare our result with
the best known in the literature, which is given by (Altschuler et al., 2017), Theorem 1:
2
n kCk3∞ ln n
O
.
ε3
As we see, our result has better dependence on ε and kCk∞ .
Proof of Theorem 2. Following the same steps as in the
proof of Theorem 1 in (Altschuler et al., 2017), we obtain
b ≤ hC, X ∗ i + 2γ ln n
hC, Xi
+ 4(kB1 − rk1 + kB T 1 − ck1 )kCk∞ , (13)
b is the output of Algorithm 2, X ∗ is a solution to
where X
the OT problem (3), and B is the matrix obtained in step 3
of this Algorithm 2. At the same time, we have
kB1 − rk1 + kB T 1 − ck1
≤ kB1 − r̃k1 + kr̃ − rk1 + kB T 1 − c̃k1 + kc̃ − ck1 ≤ ε0
Setting γ =
we obtain from the
b
above inequality and (13) that X satisfies inequality (3).
ε
4 ln n
and ε0 =
ε
8kCk∞ ,
It remains to estimate the complexity of Algorithm 2. By
Theorem 1, when ε0 is sufficiently small, the number of
iterations of
the Sinkhorn’s algorithm in step 3 of Algorithm
2 is O εR0 , where, according to (7) and (8),
i j
R = − ln ν min{r̃ , c̃ }
i,j
0
kCk∞
ε
−kCk∞ /γ
i j
= − ln e
min{r̃ , c̃ } ≤
−ln
.
i,j
γ
8n
ε
Since γ = 4 lnε n and ε0 = 8kCk
, we obtain that R =
∞
kCk∞ ln n
O
. Inserting this into the estimate k = O εR0 ,
ε
we obtain that the total number of Sinkhorn’s algorithm
iterations is bounded by
kCk2∞ ln n
O
.
ε2
Obviously, r̃ and c̃ in step 2 of Algorithm 2 can be found in
O(n) time. Since each iteration of the Sinkhorn’s algorithm
requires O(n2 ) arithmetic
operations,
the total complexity
of Algorithm 2 is O
n2 kCk2∞ ln n
ε2
.
Note that, as a byproduct, we obtained a theoretical justification of a commonly used in practice heuristic trick of
changing zero values of measures r, c to some small positive
values.
3. Accelerated Gradient Descent
In this section, our goal is to propose a flexible algorithm
for solving the regularized OT problem (2) with a general
strongly convex regularizer and,
this o
algorithm,
based
n on
n9/4 n2
e
obtain a complexity bound O min
for apε , ε2
proximating the OT distance in the sense of (3). To achieve
this goal, we consider a general optimization problem, of
Complexity of Optimal Transport Distances
which (2) is a particular case, and provide an Adaptive
Primal-Dual Accelerated Gradient Descent (APDAGD) method for this problem together with its convergence rate. Finally, we apply this algorithm to the entropy-regularized OT
problem and obtain the desired complexity.
3.1. General Problem and Algorithm
In this subsection, we consider the optimization problem
min {f (x) : Ax = b} ,
x∈Q⊆E
(14)
where E is a finite-dimensional real vector space, Q is a
simple closed convex set, A is a given linear operator from
E to some finite-dimensional real vector space H, b ∈ H
is given, f (x) is a γ-strongly convex function on Q with
respect to some chosen norm k · kE on E.
The Lagrange dual problem for (14), written as a minimization problem, is
T
min∗ ϕ(λ) := hλ, bi + max −f (x) − hA λ, xi .
x∈Q
λ∈H
(15)
Note that ∇ϕ(λ) is Lipschitz-continuous (Nesterov, 2005)
k∇ϕ(λ1 ) − ∇ϕ(λ1 )kH ≤ Lkλ1 − λ2 kH,∗ ,
kAk2
E→H
where L ≤
. This estimate can be pessimistic and
γ
our algorithm does not use it and adapts automatically to
the local value of the Lipschitz constant.
We assume that the dual problem (15) has a solution and
there exist some R > 0 such that kλ∗ k2 ≤ R < +∞, where
λ∗ is the solution to (15) with minimum value of kλ∗ k2 .
Note that the algorithm does not need any estimate of R and
the value R is used only in the convergence analysis.
This algorithm can be considered as a primal-dual extension
of accelerated mirror descent (Tseng, 2008; Lan et al., 2011).
The difference to the literature consists in incorporating linesearch and an online stopping criterion based only on the
duality gap and constraints infeasibility. We provide a more
detailed discussion in the supplementary material.
Theorem 3. Assume that the objective in the primal problem (14) is γ-strongly convex and that the dual solution λ∗
satisfies kλ∗ k2 ≤ R. Then, for k ≥ 1, the points x̂k , ηk in
Algorithm 3 satisfy
f (x̂k ) − f ∗ ≤ f (x̂k ) + ϕ(ηk ) ≤
16kAk2E→H R
,
γk 2
8 kAkE→H R
kx̂k − x∗ kE ≤
,
k
γ
kAx̂k − bk2 ≤
16kAk2E→H R2
, (16)
γk 2
(17)
(18)
Algorithm 3 Adaptive Primal-Dual Accelerated Gradient
Descent (APDAGD)
Input: Accuracy εf , εeq > 0, initial estimate L0 s.t. 0 <
L0 < 2L.
1: Set i0 = k = 0, M−1 = L0 , β0 = α0 = 0, η0 = ζ0 =
λ0 = 0.
2: repeat {Main iterate}
3:
repeat {Line search}
Set Mk = 2ik −1 Mk , find αk+1 s.t. βk+1 := βk +
4:
2
αk+1 = Mk αk+1
. Set τk = αk+1 /βk+1 .
5:
λk+1 = τk ζk + (1 − τk )ηk .
6:
ζk+1 = ζk − αk+1 ∇ϕ(λk+1 ).
7:
ηk+1 = τk ζk+1 + (1 − τk )ηk .
8:
until
ϕ(ηk+1 ) ≤ϕ(λk+1 ) + h∇ϕ(λk+1 ), ηk+1 − λk+1 i
Mk
+
kηk+1 − λk+1 k22 .
2
9:
x̂k+1 = τk x(λk+1 ) + (1 − τk )x̂k .
10:
Set ik+1 = 0, k = k + 1.
11: until f (x̂k+1 ) + ϕ(ηk+1 ) ≤ εf , kAx̂k+1 − bk2 ≤ εeq .
Output: x̂k+1 , ηk+1 .
where x∗ and f ∗ are respectively an optimal solution and
the optimal value in (14). Moreover, the stopping criterion
in step 11 is correctly defined.
A stronger statement of the theorem and its proof can be
found in the supplementary material.
Note that APDAGD is indeed flexible. For the case of
entropy regularization, we set f (X) = hC, Xi − γH(X)
and immediately get an algorithm to solve (2) since −H(X)
is strongly convex w.r.t. k · k1 . For the case of Euclidean
norm regularization, we set f (X) = hC, Xi + γkXk22 and
obtain strong convexity w.r.t. the Euclidean norm. Other
strongly convex regularizes, such as group-lasso, are also
suitable.
3.2. Complexity of OT Distance by APDAGD
Now we apply the result of the previous subsection to derive
b ∈ U(r, c) satisfying
a complexity estimate for finding X
(3). We use entropic regularization of problem (1) and
consider the regularized problem (2) with the regularizer
R(X) = −H(X), where H(X) is given in (4). We define
2
2
E = Rn , k · kE = k · k1 , and variable x = vec(X) ∈ Rn
to be the vector obtained from a matrix X by writing each
column of X below the previous column. Also we set
2
f (x) = hC, Xi − γH(X), Q = Rn+ , bT = (rT , cT ) and
2
A : Rn → R2n defined by the identity (A vec(X))T =
((X1)T , (X T 1)T ). With this setting, we solve problem (14)
bk be defined by identity vec(X
bk ) =
by our APDAGD. Let X
Complexity of Optimal Transport Distances
Algorithm 4 Approximate OT by APDAGD
Input: Accuracy ε.
ε
1: Set γ = 5 ln
n.
2: for k = 1, 2, ... do
bk and ηk .
3:
Make step of APDAGD and calculate X
b
b
Find X as the projection of Xk on U(r, c) by Algo4:
rithm 2 in (Altschuler et al., 2017).
b −X
bk i ≤ ε and f (x̂k ) + ϕ(ηk ) ≤ ε then
5:
if hC, X
10
10
b
6:
Return X.
7:
else
8:
k = k + 1 and continue.
9:
end if
10: end for
b∈
x̂k , where x̂k is generated by APDAGD. We also define X
b
U(r, c) to be the projection of Xk onto U(r, c) constructed
by Algorithm 2 in (Altschuler et al., 2017). The pseudocode
of our procedure for approximating the OT distance is listed
as Algorithm 4.
b ∈ U(r, c) satisfying
Theorem 4. Algorithm 4 outputs X
(3) in
)!
(
p
n9/4 RkCk∞ ln n n2 RkCk∞ ln n
,
O min
ε
ε2
(19)
arithmetic operations.
Before we prove the theorem, we compare our result with
the best known in the literature, which is given by (Altschuler et al., 2017), Theorem 1:
2
n kCk3∞ ln n
O
.
ε3
As we see, our result in (19) has much better dependence
on ε and kCk∞ , which comes for a reasonable price of
n1/4 . We also underline that the complexity (19) obtained
with the accelerated gradient descent Algorithm 4 has better
dependence on ε and kCk∞ than our improved bound for
the Sinkhorn’s algorithm given in Theorem 2:
2
n kCk2∞ ln n
O
.
ε2
Proof of Theorem 4. Let X ∗ be the solution of the OT problem (1) and Xγ∗ be the solution of the regularized problem
(2). Then, we have
b =hC, X ∗ i + hC, X ∗ − X ∗ i
hC, Xi
γ
bk − Xγ∗ i + hC, X
b −X
bk i.
+ hC, X
(20)
Now we estimate the second and third term in the r.h.s.
Since, for any X ∈ U(r, c), −H(X) ∈ [−2 ln n, 0], we
have
hC, Xγ∗ i =
X∈U (r,c)
≤
X∈U (r,c)
min {hC, Xi − γH(X)}
min hC, Xi + 2γ ln n
(21)
and hC, Xγ∗ − X ∗ i ≤ 2γ ln n.
Further, since APDAGD solves problem (14) with f (x) =
hC, Xi − γH(X) and Xγ∗ is the solution, we have
bk − Xγ∗ i = (hC, X
bk i − γH(X
bk ))
hC, X
bk ) − H(X ∗ ))
− (hC, Xγ∗ i − γH(Xγ∗ )) + γ(H(X
γ
(16)
≤ f (x̂k ) + ϕ(ηk ) + 2γ ln n,
(22)
where we used again that −H(X) ∈ [−2 ln n, 0] for X ∈
U(r, c). Combining (20), (21) and (22), we obtain
b ≤hC, X ∗ i + hC, X
b −X
bk i
hC, Xi
+ f (x̂k ) + ϕ(ηk ) + 4γ ln n.
(23)
We immediately see that, when the stopping criterion in
b ∈ U(r, c)
step 5 of Algorithm 4 is fulfilled, the output X
satisfies (3).
It remains to obtain the complexity bound. First, we estimate the number of iterations in Algorithm 4 to guarantee
b −X
bk i ≤ ε and, after that, estimate the number of
hC, X
10
ε
iterations to guarantee f (x̂k ) + ϕ(ηk ) ≤ 10
. By Hölder’s
b −X
bk i ≤ kCk∞ kX
b −X
bk k1 . By
inequality, we have hC, X
Lemma 7 in (Altschuler et al., 2017),
b −X
bk k1 ≤ 2 kX
bk 1 − rk1 + kX
bkT 1 − ck1 . (24)
kX
Next, we obtain two estimates for the r.h.s of this inequality.
First, by the definition of the operator A and vector b,
√
bk 1 − rk1 + kX
b T 1 − ck1 ≤ 2nkAvec(X
bk ) − bk2
kX
k
√
√
(17) 16RkAk2
32R 2n
E→H 2n
≤
≤
. (25)
γk 2
γk 2
2
Here we used the choice of the norm k · k1 in E = Rn
and the norm k · k2 in H = R2n . Indeed, in this setting
kAkE→H = kAk1→2 and this norm is equal to the maximum Euclidean norm of a column of A. By definition, each
column of A contains only two non-zero
√ elements, which
are equal to one. Hence, kAk1→2 = 2.
Second, since Xγ∗ ∈ U(r, c), we have
bk 1 − rk1 = k(X
bk − X ∗ )1k1 ≤ kX
b k − X ∗ k1
kX
γ
γ
b T 1 − ck1 . Combining these
and a similar estimate for kX
k
estimates with (18) and an estimate for kAkE→H , we obtain
√
(18) 16R 2
T
∗
bk 1 − rk1 + kX
bk 1 − ck1 ≤ 2kX
bk − Xγ k1 ≤
kX
.
γk
(26)
Complexity of Optimal Transport Distances
Combining (24), (25) and (26), we obtain
(
√ )
√
32R 2n 16R 2
b
b
,
.
hC, X − Xk i ≤ 2kCk∞ min
γk 2
γk
ε
Setting γ
=
we have that, to obtain
5 ln n ,
ε
b
b
hC, X − Xk i ≤ 10 , it is sufficient to choose
(
)!
p
n1/4 RkCk∞ ln n RkCk∞ ln n
k = O min
,
.
ε
ε2
(27)
√
At the same time, since kAkE→H = 2,
(16)
f (x̂k ) + ϕ(ηk ) ≤
ε
4, the inequality f (x̂k ) + ϕ(ηk ) ≤ 10
is fulfilled faster than
ε
b −X
bk i ≤ . At the same time, the latter inequality
hC, X
10
ε
bk 1 − rk1 + kX
b T 1 − ck1 ≤
follows from kX
k
10kCk∞ . Thus,
we run Algorithm 4 until the latter inequality is fulfilled.
Figure 1 illustrates the working time of two algorithms. It
is worth noting that, in practice, the working time of the
Algorithm 2 is approximately proportional to 1ε . The reason
could be in a pessimistic theoretical bound R in Lemma 1.
32R2
.
γk 2
Since we set γ = 5 lnε n , we conclude that in order to obtain
ε
f (x̂k ) + ϕ(ηk ) ≤ 10
, it is sufficient to choose
!
√
R ln n
.
(28)
k=O
ε
To estimate the total number of iterations, we should take
maximum of (27) and (27). Normalizing the cost matrix
C, we can set kCk∞ = 1. At the same time, as one can
see from (5), the change of the dual variables u → u + t1,
v → v − t1, for any t ∈ R does not change the value of the
dual objective. Thus, without loss of generality, we can set
R ≤ 1. Hence, the maximum of (27) and (28) is attained by
(27).
Since each iteration of APDAGD uses only operations with
matrices of the size n × n and vectors of the size 2n, each
iteration requires O(n2 ) arithmetic operations. At the same
time, according to Lemma 7 in (Altschuler et al., 2017), the
bk on U(r, c) by their Algorithm 2
complexity of projecting X
is O(n2 ). Thus, to obtain the total complexity of Algorithm
4 as in the Theorem statement, we just multiply (27) by
n2 .
4. Experiments
In this section, we provide an empirical illustration of the
work of Algorithm 2 and Algorithm 4. We run experiments
on randomly chosen real images from the MNIST dataset.
This dataset contains images of handwritten digits of the
size 28 by 28 pixels. We change all the zero elements in
the measures, representing these images, to 10−4 and, then,
normalize them, so that they sum up to one. We choose
several values of accuracy ε ∈ [0.025, 0.12]. For each value,
we randomly choose 10 pairs of images and run Algorithm 2
and Algorithm 4, and average the results. We run Algorithm
2 until the stopping criterion kB1 − rk1 + kB T 1 − ck1 ≤
ε
8kCk∞ is fulfilled. As we can see from the proof of Theorem
Figure 1. Comparison of working time of Algorithm 2 (Sinkhorn’s
algorithm) and Algorithm 4 (APDAGD).
5. Conclusion
We analyze two algorithms for approximating the general
OT distances between two discrete distributions. Our first
algorithm is based on the entropic regularization of the OT
problem and
algorithm. We prove the complexity
2Sinkhorn’s
n
e
bound O ε2 arithmetic operations. The second algorithm
is based on the entropic regularization of the OT problem
and our novel Adaptive Primal-Dual Accelerated
Gradient
o
n 9/4
n
n2
e
method. We obtain the complexity O min
,
2
ε
ε
arithmetic operations for this algorithm. Both complexity
bounds
are better than the state-of-the-art result given by
e n32 . Our APDAGD can be of a separate interest for
O
ε
solving strongly convex problems with linear constraints,
since it is not specific to the entropic regularization, incorporates a line-search strategy and has an accelerated rate of
convergence.
As a future work, we consider changing, in our second approach, the APDAGD method for some method with cheaper
iterations. For, example, this could be a greedy or randomized coordinate descent. Motivated by GREENKHORN
algorithm in (Altschuler et al., 2017), we expect this modification to have the same theoretical complexity, but cheaper
iteration cost.
Complexity of Optimal Transport Distances
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Supplementary Material
1
Adaptive Primal-Dual Accelerated Gradient Descent
(APDAGD) Convergence Analysis
We provide the missing convergence rate proofs for the Adaptive Primal-Dual Accelerated
Gradient Descent method for constrained convex optimization problem, which was considered in Section 3 of the main part of the paper. This is a separate document and, if not
explicitly stated, all the references refer to formulas, algorithms, lemmas and theorems in
this document.
We consider the problem
(P1 )
min {f (x) : Ax = b} ,
x∈Q⊆E
where f (x) is a γ -strongly convex function on Q. The Lagrange dual problem to Problem
(P1 ) in a form of a minimization problem is
T
min ϕ(λ) := hλ, bi + max −f (x) − hA λ, xi ,
λ∈Λ
x∈Q
where Λ is the space of Lagrange multipliers and, hence is unbounded.
Our Adaptive Primal-Dual Accelerated Gradient Descent method can be considered as
an Adaptive Accelerated Gradient Descent applied to the dual problem and supplied with
a procedure to reconstruct the primal iterate. Since it can be of independent interest, we
rst, in subsection 1.1 consider Adaptive Accelerated Gradient Descent method for a general
convex optimization problem and prove in Theorem 1 its convergence rate in a primal-dual
friendly fashion. Then, in subsection 1.2 we use this result to analyze our Adaptive PrimalDual Accelerated Gradient Descent method.
1.1
Adaptive Accelerated Gradient Descent for Convex Optimization
In this section, we consider a general optimization problem
min ϕ(λ),
λ∈Λ
1
(1)
where Λ ∈ H ∗ is a closed convex, generally speaking, unbounded, set, ϕ(λ) is a convex
function with L-Lipschitz-continuous gradient, i.e.
ϕ(η) ≤ ϕ(λ) + h∇ϕ(λ), η − λi +
1.1.1
L
kη − λk2H,∗ ,
2
∀η, λ ∈ H ∗ .
(2)
Proximal Setup
In this subsection, we introduce proximal setup, which is usually used in proximal gradient
methods, see e.g. Ben-Tal and Nemirovski [2015]. We choose some norm k · k on the space
of vectors λ and a prox-function d(λ) which is continuous, convex on Λ and
1. admits a continuous in λ ∈ Λ0 selection of subgradients ∇d(λ), where λ ∈ Λ0 ⊆ Λ is
the set of all λ, where ∇d(λ) exists;
2. is 1-strongly convex on Λ with respect to k · k, i.e., for any λ ∈ Λ0 , η ∈ Λ, d(η) − d(λ) −
h∇d(λ), η − λi ≥ 21 kη − λk2 .
We de ne also the corresponding Bregman divergence V [ζ](λ) := d(λ)−d(ζ)−h∇d(ζ), λ−ζi,
λ ∈ Λ, ζ ∈ Λ0 . It is easy to see that
1
V [ζ](λ) ≥ kλ − ζk2 ,
2
λ ∈ Λ, ζ ∈ Λ0 .
(3)
Standard proximal setups, i.e. Euclidean, entropy, `1 /`2 , simplex, nuclear norm, spectahedron can be found in Ben-Tal and Nemirovski [2015].
1.1.2
Algorithm and Complexity Analysis
In this subsection, we present Adaptive Accelerated Gradient Descent (see Algorithm 1 below) and prove its convergence rate theorem. Our algorithm in its form is very close to
[Tseng, 2008, Alg.1] and [Lan et al., 2011, "Variant of Nesterov's algorithm"]. Nevertheless,
the algorithms in those two papers assume the Lipschitz constant L to be known and explicitly use it in the algorithm. Our algorithm is free of this drawback. Another distinction
of our algorithm is that we prove convergence rate in a primal-dual-friendly manner. As we
show in subsection 1.2, this allows us to apply our AAGD to the Lagrange dual problem for
(P1 ), and reconstruct also primal iterates. In his paper, Tseng obtains primal-dual rates, but
only for the case of bounded set Λ. In our case this analysis is inapplicable since the feasible
set of the Lagrange dual problem is unbounded. Lan, Lu and Monteiro, consider a special
problem of minimizing a linear function and do not prove primal-dual rates for their variant
of Nesterov's algorithm.
We denote by ηk , ζk , λk three sequences of iterates of the algorithm and by αk , βk two
sequences of numbers. The convergence rate is proved for the points ηk .
Algorithm 1 is de ned correctly in the sense that the inner cycle of checking the
inequality (8) is nite.
Lemma 1.
2
Algorithm 1
Adaptive Accelerated Gradient Descent (AAGD)
starting point λ0 ∈ Λ0 , initial guess 0 < L0 < 2L, prox-setup: d(λ) 1-strongly convex
w.r.t. k · k, V [ζ](λ) := d(λ) − d(ζ) − h∇d(ζ), λ − ζi, λ ∈ Λ, ζ ∈ Λ0 .
1: Set k = 0, β0 = α0 = 0, η0 = ζ0 = λ0 .
Input:
2:
3:
4:
5:
repeat
Set Mk = Lk /2.
repeat
Set Mk = 2Mk , nd αk+1 as the largest root of the equation
2
βk+1 := βk + αk+1 = Mk αk+1
.
6:
λk+1 =
αk+1 ζk + βk ηk
.
βk+1
(5)
7:
ζk+1 = arg min{V [ζk ](λ) + αk+1 (ϕ(λk+1 ) + h∇ϕ(λk+1 ), λ − λk+1 i)}.
λ∈Λ
8:
ηk+1 =
9:
αk+1 ζk+1 + βk ηk
.
βk+1
until
ϕ(ηk+1 ) ≤ ϕ(λk+1 ) + h∇ϕ(λk+1 ), ηk+1 − λk+1 i +
10:
11:
(4)
(6)
(7)
Mk
kηk+1 − λk+1 k2 .
2
(8)
Set Lk+1 = Mk /2, k = k + 1.
Option 1: k = kmax .
Option 2: R2 /βk ≤ ε.
Option 3:
until
( k
)
X αi
ϕ(ηk ) −
min
(ϕ(λi ) + h∇ϕ(λi ), λ − λi i) ≤ ε.
βk
λ∈Λ:V [ζ0 ](λ)≤R2
i=0
Here R is such that V [ζ0 ](λ∗ ) ≤
Output: The point ηk+1 .
R2
and ε is the desired accuracy.
Proof. Since, before each check of the inequality (8) on the step k , we multiply Mk by
2, after nite number of these multiplications, we will have Mk ≥ L. Since ϕ has LLipschitz-continuous gradient, due to (2), we obtain that (8) holds after nite number of
these repetitions.
Let the sequences {λk , ηk , ζk , αk , βk }, k ≥ 0 be generated by Algorithm 1. Then,
for all λ ∈ Λ, it holds that
Lemma 2.
αk+1 h∇ϕ(λk+1 ), ζk − λi ≤ βk+1 (ϕ(λk+1 ) − ϕ(ηk+1 )) + V [ζk ](λ) − V [ζk+1 ](λ).
Proof. Note that, from the optimality condition in (6), for any λ ∈ Λ, we have
h∇V [ζk ](ζk+1 ) + αk+1 ∇ϕ(λk+1 ), λ − ζk+1 i ≥ 0.
3
(9)
(10)
By the de nition of V [ζ](λ), we obtain, for any λ ∈ Λ,
V [ζk ](λ) − V [ζk+1 ](λ) − V [ζk ](ζk+1 ) =d(λ) − d(ζk ) − h∇d(ζk ), λ − ζk i
− (d(λ) − d(ζk+1 ) − h∇d(ζk+1 ), λ − ζk+1 i)
− (d(ζk+1 ) − d(ζk ) − h∇d(ζk ), ζk+1 − ζk i)
= h∇d(ζk ) − ∇d(ζk+1 ), ζk+1 − λi
= h−∇V [ζk ](ζk+1 ), ζk+1 − λi.
(11)
Further, for any λ ∈ Λ,
αk+1 h∇ϕ(λk+1 ), ζk − λi = αk+1 h∇ϕ(λk+1 ), ζk − ζk+1 i + αk+1 h∇ϕ(λk+1 ), ζk+1 − λi
(10)
≤ αk+1 h∇ϕ(λk+1 ), ζk − ζk+1 i + h−∇V [ζk ](ζk+1 ), ζk+1 − λi
(11)
= αk+1 h∇ϕ(λk+1 ), ζk − ζk+1 i + V [ζk ](λ) − V [ζk+1 ](λ) − V [ζk ](ζk+1 )
(3)
1
≤ αk+1 h∇ϕ(λk+1 ), ζk − ζk+1 i + V [ζk ](λ) − V [ζk+1 ](λ) − kζk − ζk+1 k2
2
β2
(5),(7)
kλk+1 − ηk+1 k2
= βk+1 h∇ϕ(λk+1 ), λk+1 − ηk+1 i + V [ζk ](λ) − V [ζk+1 ](λ) − k+1
2
2αk+1
Mk
(4)
2
kλk+1 − ηk+1 k + V [ζk ](λ) − V [ζk+1 ](λ)
= βk+1 h∇ϕ(λk+1 ), λk+1 − ηk+1 i −
2
(8)
≤ βk+1 (ϕ(λk+1 ) − ϕ(ηk+1 )) + V [ζk ](λ) − V [ζk+1 ](λ).
Let the sequences {λk , ηk , ζk , αk , βk }, k ≥ 0 be generated by Algorithm 1. Then,
for all λ ∈ Λ, it holds that
Lemma 3.
βk+1 ϕ(ηk+1 ) − βk ϕ(ηk ) ≤ αk+1 (ϕ(λk+1 ) + h∇ϕ(λk+1 ), λ − λk+1 i) + V [ζk ](λ) − V [ζk+1 ](λ).
(12)
Proof. For any λ ∈ Λ,
αk+1 h∇ϕ(λk+1 ), λk+1 − λi = αk+1 h∇ϕ(λk+1 ), λk+1 − ζk i + αk+1 h∇ϕ(λk+1 ), ζk − λi
(4),(5)
= βk h∇ϕ(λk+1 ), ηk − λk+1 i + αk+1 h∇ϕ(λk+1 ), ζk − λi
conv-ty
≤
(9)
βk (ϕ(ηk ) − ϕ(λk+1 )) + αk+1 h∇ϕ(λk+1 ), ζk − λi
≤ βk (ϕ(ηk ) − ϕ(λk+1 )) + βk+1 (ϕ(λk+1 ) − ϕ(ηk+1 )) + V [ζk ](λ) − V [ζk+1 ](λ)
= αk+1 ϕ(λk+1 ) + βk ϕ(ηk ) − βk+1 ϕ(ηk+1 ) + V [ζk ](λ) − V [ζk+1 ](λ).
(13)
Rearranging terms, we obtain the statement of the Lemma.
4
Let the sequences {λk , ηk , ζk , αk , βk }, k ≥ 0 be generated by Algorithm 1. Then,
for all k ≥ 0, it holds that
Theorem 1.
βk ϕ(ηk ) ≤ min
λ∈Λ
( k
X
i=0
)
αi (ϕ(λi ) + h∇ϕ(λi ), λ − λi i) + V [ζ0 ](λ) .
(14)
The number of inner cycle iterations after an iteration k ≥ 0 does not exceed
4k + 4 + 2 log2
L
L0
(15)
,
where L is the Lipschitz constant for the gradient of ϕ.
Proof. Let us change the counter in Lemma 2 from k to i and sum all the inequalities for
i = 0, ..., k − 1. Then, for any λ ∈ Λ,
βk ϕ(ηk ) − β0 ϕ(η0 ) ≤
k−1
X
i=0
αi+1 (ϕ(λi+1 ) + h∇ϕ(λi+1 ), λ − λi+1 i) + V [ζ0 ](λ) − V [ζk ](λ). (16)
Whence, since β0 = α0 = 0 and V [ζk ](λ) ≥ 0,
βk ϕ(ηk ) ≤
k
X
i=0
αi (ϕ(λi ) + h∇ϕ(λi ), λ − λi i) + V [ζ0 ](λ),
λ ∈ Λ.
(17)
Taking in the right hand side the minimum in λ ∈ Λ, we obtain the rst statement of the
Theorem.
The second statement of the Theorem is proved in the same way as in Nesterov and
Polyak [2006], but we provide the proof for the reader's convenience. Let us again change
the iteration counter in Algorithm 1 from k to i. Let ji ≥ 1 be the total number of checks of
0
and, for i ≥ 1, Mi = 2ji −1 Li =
the inequality (8) on the step i ≥ 0. Then, j0 = 1 + log2 M
L0
i
, i ≥ 1. Further, by the same reasoning as in Lemma 2,
2ji −1 Mi−1
. Thus, ji = 2 + log2 MMi−1
2
we obtain that Mi ≤ 2L, i ≥ 0. Then, the total number of checks of the inequality (8) is
k
X
i=0
k
M0 X
Mi
Mk
L
ji = 1 + log2
+
2 + log2
= 2k + 1 + log2
≤ 2k + 2 + log2 .
L0
Mi−1
L0
L0
i=1
At the same time, each check of the inequality (8) requires two oracle calls. This proves the
second statement of the Theorem.
Let the sequences {λk , ηk , ζk , αk , βk }, k ≥ 0 be generated by Algorithm 1. Then,
for all k ≥ 0, it holds that
Corollary 1.
ϕ(ηk ) − min ϕ(λ) ≤
λ∈Λ
V [ζ0 ](λ∗ )
,
βk
(18)
where λ∗ is the solution of minλ∈Λ ϕ(λ) s.t. V [ζ0 ](λ∗ ) is minimal among all the solutions.
5
Proof. Let λ∗ be the solution of minλ∈Λ ϕ(λ) s.t. V [ζ0 ](λ∗ ) is minimal among all the solutions.
Using convexity of ϕ, from Theorem 1, we obtain
βk ϕ(ηk ) ≤
Since βk =
Pk
i=0
k
X
i=0
αi ϕ(λ∗ ) + V [ζ0 ](λ∗ ).
αi , we obtain the statement of the Corollary.
The following Corollary justi es the stopping criteria in Algorithm 1.
Let λ∗ be a solution of minλ∈Λ ϕ(λ) such that V [ζ0 ](λ∗ ) is minimal among
all the solutions. Let R be such that V [ζ0 ](λ∗ ) ≤ R2 and ε be the desired accuracy. Let
the sequences {λk , ηk , ζk , αk , βk }, k ≥ 0 be generated by Algorithm 1. Then, if one of the
following inequalities holds
Corollary 2.
R2 /βk ≤ ε,
then
(19)
)
( k
X αi
(ϕ(λi ) + h∇ϕ(λi ), λ − λi i) ≤ ε,
ϕ(ηk ) −
min
λ∈Λ:V [ζ0 ](λ)≤R2
β
i=0 k
(20)
ϕ(ηk ) − min ϕ(λ) ≤ ε.
(21)
λ∈Λ
Proof. If the inequality (19) holds, the statement of the Corollary follows from inequality
V [ζ0 ](λ∗ ) ≤ R2 Corollary 1.
Since V [ζ0 ](λ∗ ) ≤ R2 , the point λ∗ is a feasible point in the problem
)
( k
X αi
(ϕ(λi ) + h∇ϕ(λi ), λ − λi i) .
min
λ∈Λ:V [ζ0 ](λ)≤R2
β
i=0 k
Then, by convexity of ϕ, we obtain
( k
)
k
X αi
X
αi
min
(ϕ(λi ) + h∇ϕ(λi ), λ − λi i) ≤
(ϕ(λi ) + h∇ϕ(λi ), λ∗ − λi i)
2
λ∈Λ:V [ζ0 ](λ)≤R
β
β
i=0 k
i=0 k
≤ ϕ(λ∗ ).
This and (20) nishes the proof.
Let us now obtain the lower bound for the sequence βk , k ≥ 0, which will give the rate
of convergence for Algorithm 1.
Lemma 4.
it holds that
Let the sequence {βk }, k ≥ 0 be generated by Algorithm 1. Then, for all k ≥ 1
(k + 1)2
,
8L
where L is the Lipschitz constant for the gradient of ϕ.
βk ≥
6
(22)
Proof. As we mentioned in the proof of Theorem 1, Mk ≤ 2L, k ≥ 0. For k = 1, since α0 = 0
and A1 = α0 + α1 = α1 , we have from (4)
β1 = α 1 =
1
1
≥
.
M1
2L
Hence, (22) holds for k = 1.
Let us now assume that (22) holds for some k ≥ 1 and prove that it holds for k + 1. From
(4) we have a quadratic equation for αk+1
2
Mk αk+1
− αk+1 − βk = 0.
Since we need to take the largest root, we obtain,
s
p
r
1
1
βk
1
βk
1 + 1 + 4Mk βk
=
+
+
≥
+
αk+1 =
2
2Mk
2Mk
4Mk
Mk
2Mk
Mk
≥
1 k+1
1
k+2
√
+√
,
=
4L
4L
2L 2 2L
where we used the induction assumption that (22) holds for k . Using the obtained inequality,
from (4) and (22) for k , we get
βk+1 = βk + αk+1 ≥
(k + 1)2 k + 2
(k + 2)2
+
≥
.
8L
4L
8L
Let the sequences {λk , ηk , ζk }, k ≥ 0 be generated by Algorithm 1. Then, for
all k ≥ 1, it holds that
Corollary 3.
ϕ(ηk ) − min ϕ(λ) ≤
λ∈Λ
8LV [ζ0 ](λ∗ )
,
(k + 1)2
(23)
where λ∗ is the solution of minλ∈Λ ϕ(λ) s.t. V [ζ0 ](λ∗ ) is minimal among all the solutions.
1.2
Adaptive Primal-Dual Accelerated Gradient Descent for Constrained Convex Optimization
In this section, we return to the constrained convex optimization problem, which was considered in Section 3 of the main part of the paper. For the reader's convenience, we repeat
the problem statement and some details.
1.2.1
Preliminaries
We consider convex optimization problem of the following form
(P1 )
min {f (x) : Ax = b} ,
x∈Q⊆E
7
where f (x) is a γ -strongly convex function on Q with respect to some chosen norm k · kE on
E and A : E → H is a linear operator, b ∈ H .
The Lagrange dual problem to Problem (P1 ) is
T
(D1 )
max −hλ, bi + min f (x) + hA λ, xi .
x∈Q
λ∈Λ
Here we denote Λ = H ∗ the space of Lagrange multipliers. It is convenient to rewrite
Problem (D1 ) in the equivalent form of a minimization problem
T
(P2 ) min hλ, bi + max −f (x) − hA λ, xi .
x∈Q
λ∈Λ
It is obvious that
Opt[D1 ] = −Opt[P2 ],
(24)
Opt[P1 ] ≥ Opt[D1 ].
(25)
where Opt[D1 ], Opt[P2 ] are the optimal function value in Problem (D1 ) and Problem (P2 )
respectively. The following inequality follows from the weak duality
We denote
ϕ(λ) = hλ, bi + max −f (x) − hAT λ, xi .
x∈Q
(26)
Since f is strongly convex, ϕ(λ) is a smooth function and its gradient is equal to (see e.g.
Nesterov [2005])
∇ϕ(λ) = b − Ax(λ),
(27)
where x(λ) is the unique solution of the strongly-convex problem
max −f (x) − hAT λ, xi .
x∈Q
(28)
Note that ∇ϕ(λ) is Lipschitz-continuous (see e.g. Nesterov [2005]) with constant
L≤
kAk2E→H
.
γ
We also assume that the dual problem (D1 ) has a solution λ∗ and there exists some R > 0
such that
kλ∗ k2 ≤ R < +∞.
(29)
1.2.2
Adaptive Primal-Dual Accelerated Gradient Descent
Now we are ready to apply Algorithm 1 to the problem (P2 ) and incorporate in the algorithm
a procedure, which allows to reconstruct also an approximate solution of the problem (P1 ).
We choose Euclidean proximal setup, which means that we introduce euclidean norm k · k2
in the space of vectors λ and choose the prox-function d(λ) = 21 kλk22 . Then, we have
8
V [ζ](λ) = 12 kλ − ζk22 . We state here as Algorithm 2 a more detailed version of Algorithm
3 in the main part of the paper. The rst dierence is that here we do not introduce an
auxiliary sequence τk = αk+1 /βk+1 . The second dierence is that here we use an equivalent
form
1
2
kλ − ζk k2 + αk+1 (ϕ(λk+1 ) + h∇ϕ(λk+1 ), λ − λk+1 i)
ζk+1 = arg min
λ∈Λ
2
of the step
ζk+1 = ζk − αk+1 ∇ϕ(λk+1 ).
The third dierence consists in the observation that, since, by de nition, βk =
x̂k+1
k+1
αk+1 x(λk+1 ) + βk x̂k
1 X
=
=
αi x(λi ).
βk+1
βk+1 i=0
Pk
i=0
αk ,
Finally, here we use a stronger stopping rule
|f (x̂k+1 ) + ϕ(ηk+1 )| ≤ εf
than in the main part of the paper. The reason is that, to obtain complexity result for
approximating OT distance, it is enough to satisfy
f (x̂k+1 ) + ϕ(ηk+1 ) ≤ ε
with a special choice of ε.
Assume that the objective in the problem (P1 ) is γ -strongly convex and that
the dual solution λ∗ satis es kλ∗ k2 ≤ R. Then, for k ≥ 1, the points x̂k , ηk in Algorithm 2
satisfy
Theorem 2.
16LR2
16LR2
≤
f
(x̂
)
−
Opt[P
]
≤
f
(x̂
)
+
ϕ(η
)
≤
,
k
1
k
k
k2
k2
16LR
kAx̂k − bk2 ≤
,
k2
s
−
kx̂k − x∗ kE ≤
8
k
LR2
,
γ
(35)
(36)
(37)
where x∗ and Opt[P1 ] are respectively an optimal solution and the optimal value in the problem
kAk2E→H
(P1 ), and L ≤
. Moreover, the stopping criterion in step 11 is correctly de ned and
γ
the number of inner cycle iterations after an iteration k ≥ 0 does not exceed
4k + 4 + 2 log2
where L ≤
kAk2E→H
γ
L
L0
,
is the Lipschitz constant for the gradient of ϕ.
9
(38)
Algorithm 2
Input:
Adaptive Primal-Dual Accelerated Gradient Descent (APDAGD)
starting point λ0 = 0, initial guess L0 > 0, accuracy εf , εeq > 0.
1: Set k = 0, β0 = α0 = 0, η0 = ζ0 = λ0 = 0.
2: repeat
3:
Set Mk = Lk /2.
4:
repeat
5:
Set Mk = 2Mk , nd αk+1 as the largest root of the equation
2
βk+1 := βk + αk+1 = Mk αk+1
.
6:
Calculate
λk+1 =
7:
λ∈Λ
1
2
kλ − ζk k2 + αk+1 (ϕ(λk+1 ) + h∇ϕ(λk+1 ), λ − λk+1 i) .
2
Calculate
ηk+1 =
9:
αk+1 ζk+1 + βk ηk
.
βk+1
until
ϕ(ηk+1 ) ≤ ϕ(λk+1 ) + h∇ϕ(λk+1 ), ηk+1 − λk+1 i +
10:
(31)
Calculate
ζk+1 = arg min
8:
αk+1 ζk + βk ηk
.
βk+1
(30)
Set
x̂k+1 =
1
βk+1
k+1
X
αi x(λi ) =
i=0
(32)
(33)
Mk
kηk+1 − λk+1 k22 .
2
(34)
αk+1 x(λk+1 ) + βk x̂k
.
βk+1
Set Lk+1 = Mk /2, k = k + 1.
until |f (x̂k+1 ) + ϕ(ηk+1 )| ≤ εf , kA1 x̂k+1 − b1 k2 ≤ εeq .
Output: The points x̂k+1 , ηk+1 .
11:
12:
Proof. From Theorem 1 with speci c choice of the Bregman divergence, since ζ0 = 0, we
have, for all k ≥ 0,
( k
)
X
1
βk ϕ(ηk ) ≤ min
αi (ϕ(λi ) + h∇ϕ(λi ), λ − λi i) + kλk22
(39)
λ∈Λ
2
i=0
Let us introduce a set ΛR = {λ : kλk2 ≤ 2R} where R is given in (29). Then, from (39), we
10
obtain
( k
X
βk ϕ(ηk ) ≤ min
λ∈Λ
i=0
≤ min
λ∈ΛR
≤ min
λ∈ΛR
1
αi (ϕ(λi ) + h∇ϕ(λi ), λ − λi i) + kλk22
2
( k
X
i=0
( k
X
i=0
)
1
αi (ϕ(λi ) + h∇ϕ(λi ), λ − λi i) + kλk22
2
)
αi (ϕ(λi ) + h∇ϕ(λi ), λ − λi i)
)
+ 2R2 .
(40)
On the other hand, from the de nition (26) of ϕ(λ), we have
ϕ(λi ) = hλi , bi + max −f (x) − hAT λi , xi
x∈Q
= hλi , bi − f (x(λi )) − hAT λi , x(λi )i.
Combining this equality with (27), we obtain
ϕ(λi ) − h∇ϕ(λi ), λi i = ϕ(λ) − h∇ϕ(λ), λi i
= hλi , bi − f (x(λi )) − hAT λi , x(λi )i
− hb − Ax(λi ), λi i = −f (x(λi )).
Summing these inequalities from i = 0 to i = k with the weights {αi }i=1,...k , we get, using
the convexity of f ,
k
X
i=0
αi (ϕ(λi ) + h∇ϕ(λi ), λ − λi i)
=−
k
X
αi f (x(λi )) +
i=0
k
X
i=0
αi hb − Ax(λi ), λi
≤ −βk f (x̂k ) + βk hb − Ax̂k , λi.
Substituting this inequality to (40), we obtain
βk ϕ(ηk ) ≤ − βk f (x̂k ) + βk min {hb − Ax̂k , λi} + 2R2 .
λ∈ΛR
Finally, since
max {h−b + A1 x̂k , λi} = 2RkAx̂k − bk2 ,
λ∈ΛR
we obtain
ϕ(ηk ) + f (x̂k ) + 2RkAx̂k − bk2 ≤
11
2R2
.
βk
(41)
Since λ∗ is an optimal solution of Problem (D1 ), we have, for any x ∈ Q
Opt[P1 ] ≤ f (x) + hλ∗ , Ax − bi.
Using the assumption (29), we get
(42)
f (x̂k ) ≥ Opt[P1 ] − RkAx̂k − bk2 .
Hence,
ϕ(ηk ) + f (x̂k ) = ϕ(ηk ) − Opt[P2 ] + Opt[P2 ] + Opt[P1 ] − Opt[P1 ] + f (x̂k )
(24)
= ϕ(ηk ) − Opt[P2 ] − Opt[D1 ] + Opt[P1 ] − Opt[P1 ] + f (x̂k )
(25)
(42)
(43)
≥ −Opt[P1 ] + f (x̂k ) ≥ −RkAx̂k − bk2 .
This and (41) give
RkAx̂k − bk2 ≤
2R2
.
βk
Hence, we obtain
ϕ(ηk ) + f (x̂k )
(43),(44)
≥
On the other hand, we have
(41)
ϕ(ηk ) + f (x̂k ) ≤
(44)
2R2
−
.
βk
(45)
2R2
.
βk
(46)
2R2
.
βk
(47)
Combining (44), (45), (46), we conclude
kAx̂k − bk2 ≤
2R
,
βk
|ϕ(ηk ) + f (x̂k )| ≤
At the same time,
ϕ(ηk ) + Opt[P1 ] = ϕ(ηk ) − Opt[P2 ] + Opt[P2 ] + Opt[P1 ]
(24)
(25)
= ϕ(ηk ) − Opt[P2 ] − Opt[D1 ] + Opt[P1 ] ≥ 0.
Hence,
f (x̂k ) − Opt[P1 ] ≤ f (x̂k ) + ϕ(ηk ).
(48)
(k+1)2
From (47), (48), by Lemma 4, stating that, for any k ≥ 0, βk ≥ 8L , we obtain
inequalities (35) and (36) in the Theorem statements.
It remains to prove inequality (37). By the optimality condition for Problem (P1 ), we
have
h∇f (x∗ ) + AT λ∗ , x̂k − x∗ i ≥ 0, Ax∗ = b,
12
where ∇f (x∗ ) ∈ ∂f (x∗ ). Then
h∇f (x∗ ), x̂k − x∗ i ≥ −hAT λ∗ , x̂k − x∗ i
≥ −hλ∗(1) , Ax̂k − bi
(44)
≥ −RkAx̂k − bk2 ≥ −
2R2
,
βk
(49)
where we used the same reasoning as while deriving (42). Using this inequality and γ strong
convexity of f , we obtain
(47),(48) 4R2
γ
∗
∗
∗ 2
kx̂k − x kE ≤ f (x̂k ) − Opt[P1 ] − h∇f (x ), x̂k − x i ≤
.
2
βk
Since, by Lemma 4, for any k ≥ 0, βk ≥
(k+1)2
,
8L
we obtain inequality (37).
References
Aaron Ben-Tal and Arkadi Nemirovski. Lectures on Modern Convex Optimization (Lecture
Notes). Personal web-page of A. Nemirovski, 2015. URL http://www2.isye.gatech.
edu/\~nemirovs/Lect\_ModConvOpt.pdf.
Guanghui Lan, Zhaosong Lu, and Renato D. C. Monteiro. Primal-dual rst-order methods
with O(1/ε) iteration-complexity for cone programming. Mathematical Programming, 126
(1):129, Jan 2011. ISSN 1436-4646. doi: 10.1007/s10107-008-0261-6. URL https:
//doi.org/10.1007/s10107-008-0261-6.
Yurii Nesterov. Smooth minimization of non-smooth functions. Mathematical Programming,
103(1):127152, 2005. ISSN 1436-4646. doi: 10.1007/s10107-004-0552-5. URL http:
//dx.doi.org/10.1007/s10107-004-0552-5.
Yurii Nesterov and Boris Polyak. Cubic regularization of newton method and its global
performance. Mathematical Programming, 108(1):177205, 2006. ISSN 1436-4646. doi:
10.1007/s10107-006-0706-8. URL http://dx.doi.org/10.1007/s10107-006-0706-8.
Paul Tseng. On accelerated proximal gradient methods for convex-concave optimization.
Technical report, MIT, 2008. URL http://www.mit.edu/~dimitrib/PTseng/papers/
apgm.pdf.
13
| 8 |
1
Analysis of Network Robustness for Finite Sized
Wireless Sensor Networks
arXiv:1609.06888v1 [cs.SY] 22 Sep 2016
Sateeshkrishna Dhuli, Chakravarthy Gopi, Yatindra Nath Singh, Senior Member, IEEE
Abstract—Studying network robustness for wireless sensor
networks(WSNs) is an exciting topic of research as sensor nodes
often fail due to hardware degradation, resource constraints, and
environmental changes. The application of spectral graph theory
to networked systems has generated several important results.
However, previous research has often failed to consider the
network parameters, which is crucial to study the real network
applications. Network criticality is one of the effective metrics to
quantify the network robustness against such failures and attacks.
In this work, we derive the exact formulas of network criticality
for WSNs using r-nearest neighbor networks and we show the
effect of nearest neighbors and network dimension on robustness
using analytical and numerical evaluations. Furthermore, we also
show how symmetric and static approximations can wrongly
designate the network robustness when implemented to WSNs.
Index Terms—Wireless sensor networks, r-nearest neighbor
networks, Network Robustness, Network Criticality, Spectral
graph theory
I. I NTRODUCTION
IRELESS sensor networks play a significant role
in applications such as healthcare monitoring,
environmental sensing, fire detection, disaster prevention, etc.
However, nodes in WSNs prone to failures due to hardware
degradation, resource constraints, and environmental changes
[1]. The reliability of WSNs is rely on continuing its
operations when a fraction of nodes are damaged [2], [3].
Structural robustness of a network is defined as the ability
of a network to maintain its connectivity against failures
or attacks [4], [5]. Studies of network robustness under
intentional attacks and failures have received a growing
interest in the recent years (see, e.g., [6]- [7]). In [6], authors
argued that node connectivity is the most suitable graph
theoretic metric to study the robustness in the face of node
failures. Based on different heuristics, various measures
have been proposed to quantify the network robustness,
such as natural connectivity [8], network criticality [9],
algebraic connectivity [10], etc. Although, these measures
are useful to quantify the network robustness, they cannot
be applied to large-scale networks due to high computational
complexity. This also implies that these measures are of no
great practical use within the context of WSNs. Many studies
have been discussed to optimize the network robustness. In
[11], a tabu search algorithm has been proposed to optimize
the network robustness by rewiring the links. Altering the
edge connections between low degree nodes can improve
the network robustness against intentional attacks [12].
Nonetheless, these approaches are only suitable for small
to medium-scale networks. To capture the small topological
changes in network caused by network component’s failures, a
W
new measure has been proposed based on information theory
[7]. A measure for evaluating network robustness for time
varying networks has been proposed in [13] and it has been
shown that temporal robustness gives more realistic results
over static approaches. Although there have been several
studies on network robustness, there appear to be inadequate
analytical tools to study the robustness for WSN scenarios. In
our work, we derive the exact formulas of network criticality
to study the network robustness for WSNs. We adopt the
network criticality metric to quantify network robustness and
derive a new measure of robustness in terms of network
parameters. To study the network robustness of WSNs, we
use r-nearest neighbor networks [14], [15], a well known
class of distance-regular networks with a varying number
of direct neighbors. Lattice networks (see, e.g., [16], [17],
[18], and [19]) and r-nearest neighbor networks are simple
structures which allow theoretical analysis that incorporates
important parameters like connectivity, scalability, network
size, and node failures. These structures are quite useful for
measuring and monitoring purposes in sensor networks [20].
Our analytic approach drastically decreases the computational
complexity over the existing graph-theoretic metrics. These
kind of analyses play a vital role in the initial stage of wireless
networks design and also more reliable than simulation based
evaluations. Furthermore, we use the probability switching
[21] and weight design approaches to study the effects of node
dynamics and asymmetric weights on network robustness
respectively. The structure of a WSN is highly dynamic, as
the WSNs are subject to node or link failures due to the
low-power batteries of sensors or environmental changes.
Hence, assuming WSN as a static network cannot model
the applications which involve mobility [1], [13]. Wireless
channels in low power wireless networks (such as WSNs)
are known to be time-varying, unreliable, and asymmetric
(see, e.g., [22], [23], [24], and [25]). Hence, modeling WSN
as a undirected graph may wrongly designate the network’s
performance. To show the effects of asymmetric link weights,
we consider ring topology for the ease of evaluation.
In summary, our contributions can be summarized as follows:
1) In Section III, we derive the exact formulas of network
criticality using r-nearest neighbor networks to compute
network criticality in terms of network parameters.
2) In Section IV, we study the effect of link failures on
network robustness by means of probability switching.
3) In Section V, we introduce the parameter and derive the
network criticality expression for asymmetric ring network.
4) In Section VI, we verify our analytical expressions
with the extensive simulations in MATLAB and propose a
2
optimization framework to minimize the network robustness
while limiting the power consumption.
II. S PECTRAL G RAPH T HEORY
Let G = (V, E) be an undirected graph with the set of nodes
V = {1, 2, ......n}, edge set E ⊆ V × V, and an adjacency
matrix A consists of non-negative elements aij . The degree
n
P
aij .
matrix D is expressed as D = diag[di ], where di =
j=1
The Laplacian matrix L is a n × n symmetric matrix defined
as L = D − A, where each entry in L is expressed as
deg(vi ) if j = i,
lij = lji =
(1)
−aij
if j 6= i.
Inspiring from Darwin’s survival value, the theory of network
criticality has been developed in [9]. A survival value quantifies the adaptability of network to unexpected variations.
Network criticality is defined as the average random walk
betweenness of a link or node normalized by its weight.
Random Walk Betweenness measures how many times a node
appears on random walks between all node-pairs in the graph.
A random-walk starts from a source node s, chooses one of
its neighbors with equal probabilities and continues traveling
until it reaches the destination d. So the betweenness bst (d)
of a node t for source-destination pair s-d is the expected
number of times that a packet passes via node t in it’s journey
from source s to destination d. Node criticality ηk is defined
as the random-walk betweenness of node over the weight of
the node.
bk
ηk =
,
wk
where bk , wk are the betweenness and weight of node k.
Similarly, link betweenness ηij is defined as the betweenness
of the link over its weight.
ηij =
bij
,
wij
Fig. 1.
2-nearest neighbor cycle
III. r-N EAREST N EIGHBOR N ETWORKS
In r-nearest neighbor networks, a communication link exists
between every pair of nodes that are r hops away. In this
section, we derive the network criticality expressions for
r-nearest neighbor cycle, r-nearest neighbor torus, and m
dimensional r-nearest neighbor torus networks.
Def inition 2: The (j + 1)th eigenvalue [26] of a circulant
matrix circ{a1 , a2 ......an } is defined as
λj = a1 + a2 ω j + .............. + an ω (n−1)j ,
where ω = e
matrix.
2πi
n
and {ai }ni=1 are row entries of circulant
A. r-Nearest Neighbor Cycle
The one dimensional wireless sensor network topology can
be modeled by r-nearest neighbor cycle Cnr .
Lemma 1: The (j + 1)th eigenvalue of a Laplacian matrix L
of Cnr is
r
X
2πj
cos
λj (L) = 2r − 2
,
(4)
n
j=1
where j = 0, 1, 2......(n − 1).
P roof : The Laplacian matrix of Cnr can be written as
L = circ{2r −1 − 1 − 1 ......0, 0..... −1 − 1 − 1}.
| {z }
| {z }
r times
where bij , wij are the betweenness and weight of link (i, j).
The parameter weight captures link quality and models the
QOS parameters like Bandwidth, Packet loss etc.
The probability of passing node or link k in the next step is
expressed as
(
pst (d) =
0
Pwst
wsq
τ (Cnr ) =
n−1
X
j=1
r times
2n
2r + 1 −
sin
(2r+1)πj
n
sin πj
n
.
(6)
P roof : Substituting the equation (4) in (2), we get
q∈A(s)
where A(s) is the direct neighbor nodes of s and wst is the
weight of link (s, t).
Def inition 1: Network criticality (τ ) [9] quantifies the
network robustness that captures the effect of environmental
changes in communication networks. It is expressed as
+
(5)
Using equation (3) and (5), we obtain equation (4).
T heorem 1: Network criticality τ of a r-nearest neighbor
cycle Cnr is given by
if s = d
Otherwise,
τ = 2nT r(L+ ),
(3)
(2)
where T r(L ) represents trace of the Moore-Penrose inverse
of Laplacian matrix.
τ (Cnr ) =
n−1
X
j=1
2n
r
P
2r − 2
cos
i=1
2πji
n
.
Def inition 3: Dirichlet kernel is defined as
r
X
sin r + 21 x
.
1+2
cos(jx) =
sin x2
j=1
We obtain (6), by substituting equation (8) in (7).
(7)
(8)
3
P roof : Using (2) and (12), the network criticality of Tkr1 ,k2
can be written as
τ (Tkr1 ,k2 ) =
Fig. 2.
kP
1 −1 kP
2 −1
j1 =0 j2 =1 4r−2
r
P
cos
i=1
2n
.
r
P
2πj2 i
−2
cos
k
2πj1 i
k1
1
i=1
(15)
To get (14), we further simplify equation (15) using (8).
Without loss of generality, we write the network criticality of
Tkr1 ,k2 ,....km as
Torus Network
τ (Tkr1 ,k2 ,....km ) =
B. r-Nearest Neighbor Torus
The cartesian product of two r-nearest cycles results in rnearest neighbor torus. A two dimensional 1-nearest neighbor
torus is as shown in Fig. 2.
Lemma 2: The (j1 + j2 + 1)th eigenvalue of Laplacian matrix
of r-nearest neighbor torus Tkr1 ,k2 is
X
r
r
X
2πj2 i
2πj1 i
−2
,
cos
λj1 ,j2 (Lrk1 ,k2 ) = 4r−2
cos
k1
k1
i=1
i=1
(9)
where j1 = 0, 1, 2, ...(k1 − 1), j2 = 0, 1, 2, ...(k2 − 1).
P roof : The Laplacian matrix of Tkr1 ,k2 can be written as
Ik Ik ..... Ik1 ....... Ik1 Ik1 ..... Ik1 }
}
{z
}
| 1 1{z
|
L = circ{Lk1 + 2rIk1
r times
r times
(10)
From (3) and (10), we obtain
λj1 ,j2 (Lrk1 ,k2 )
=
λj1 (Lrk1 )
+ 2r − 2
r
X
cos
i=1
2πj2 i
k2
(11)
By substituting equation (4) into (11), we obtain equation (9).
C. m-dimensional r-nearest neighbor torus
Lemma 3: The generalized expression for eigenvalue of
Laplacian matrix of m-dimensional r-nearest neighbor torus
is
r X
m
X
2πji
r
.
λj1 ,j2 ,...jm (Lk1 ,k2 ,..km ) = 2mr − 2
cos
ki
j=1 i=1
(12)
where ji = 0, 1, 2......(ki − 1).
P roof : The (j1 +j2 +j3 +1)th eigenvalue of Laplacian matrix
L for 3-dimensional r-nearest neighbor torus Tkr1 ,k2 ,k3 is
3 X
r
X
2πji
λ(Lrk1 ,k2 ,k3 ) = 6r − 2
cos
.
(13)
ki
i=1 j=1
Without loss of generality, by observing (13) and (9), we can
write (12) for variable m.
T heorem 2: The network criticality of r-nearest neighbor
torus Tkr1 ,k2 between every arbitrary pair of nodes is
τ (Tkr1 ,k2 ) =
kP
1 −1 kP
2 −1
j1 =1 j2 =0
2n
sin
(4r+2)−
(2r+1)πj1
k1
πj1
sin
k1
sin
−
(2r+1)πj2
k2
πj2
sin
k2
!
.
(14)
kP
2 −1
1 −1 kP
......
j1 =1 j2 =0
km
P−1
jm
=0
(2r+1)m−
2
m
P
sin
i=1
(2r+1)πji
ki
πji
sin
ki
.
(16)
D. Computational Complexity
O(n) denotes the asymptotic upper bound and it says that
the limit of a function when the argument tends towards
infinity. Time complexity for calculating τ using equation
(2) is O(n3 ), which is prohibited for large scale WSNs.
Specifically, our approach overcomes this disadvantage and
also effective to study the network robustness for large sized
networks.
E. Asymptotic Results
In this section, we derive the network criticality expressions
for n → ∞.
T heorem 3: The network criticality of Cnr when n → ∞ is
τ (Cnr ) ≈
n3
.
2r(r + 1)(2r + 1)
(17)
P roof : Proof is technical and deferred to Appendix A.
T heorem 4: The network criticality of Tkr1 ,k2 when r n is
τ (Tkr1 ,k2 ) ≈
3n3 Θ(log n)
8r(r + 1)(2r + 1)π 2
(18)
P roof : Proof is technical and deferred to Appendix B.
IV. DYNAMIC N ETWORK A NALYSIS
To study the effect of link failures and switching neighbors,
we use the probability switching method proposed in [21].
Here, we consider a n node ring network, where every time
instant, graph Gi is selected with probability pi . Since at each
time instant, graph topology is selected identically distributed
and independent of the previous topologies, the average of the
stochastic matrices will be evaluated.
A. Random Communication Links
In a n-node ring network, if each link exists between
any two nodes with probability q, then the link is selected
independently. Then adjacency matrix A of a random ring
network is written as
A = circ{0 q q .............q },
|
{z
}
n−1
(19)
4
D = circ{q(n − 1) 0 0 .............0},
(20)
Using equations (19) and (20), L is expressed as
L = circ{q(n − 1) −q − q ............. − q }.
|
{z
}
(21)
n−1
From equations (3) and (21), (j + 1)th eigenvalue can be
written as
(n−1)
X 2πijk
(22)
λj = q (n − 1) −
e n .
k=1
Fig. 3.
Asymmetric Ring Network
Thus, substituting equation (22) in equation (2) gives
τ=
n−1
X
1
j=1
(n−1)
P
q (n − 1) −
!
e
2πijk
n
(23)
.
k=1
For n = odd, equation (23) can be further simplified as
τ=
n−1
X
2n
j=1
q n−
sin πj
sin πj
n
.
bidirectionally with equal probability. In this case, adjacency
matrix A is
2
2
2
}.
(30)
A = circ{0
.............
n−1 n−1
n−1
{z
}
|
n−1
(24)
Hence, the degree matrix D is given by
D = circ{2 0 0 .............0}
B. Identically Independent Link Losses due to Communication
Failures
To study the link failures in WSNs, we assume that each
link in a ring network is fails with a probability p. These link
failures occur independently with respect to other link failures.
Then adjacency matrix A can be written as
Ā = circ{0 (1 − p) |0 0........
{z 0 0}(1 − p)},
(25)
n−3
Degree matrix D is given by
.
(31)
Using equations (30) and (31), we get Laplacian matrix L as
2
2
L = circ{2 −
............. −
}.
n−1
n−1
|
{z
}
(32)
n−1
From equations (3) and (32), we obtain (j + 1)th eigenvalue
as
n−1
2 X 2πjki
e n .
λj = 2 −
(33)
n − 1 i=1
Substituting equation (33) in equation (2) results in
D = circ{2(1 − p) 0 0 .............0},
(26)
τ=
Using equation (25) and equation (26), we get L as
n−1
X
j=1
L̄ = circ{2(1 − p) − (1 − p) |0 0........
{z 0 0} −(1 − p)},
n−3
(27)
Using equation (3) and equation (27), we obtain (j + 1)th
eigenvalue as
λj = 2(1 − p)(1 − cos
2πj
).
n
τ=
j=1
n
.
(1 − p) 1 − cos 2πj
n
1
(n−1)
.
e
(34)
2πkij
n
i=1
For n = odd, τ can be further simplified as
τ=
n−1
X
j=1
1−
1
(n−1)
n
sin πk
sin πk
n
.
−1
(35)
(28)
V. A SYMMETRIC W EIGHT A NALYSIS
Substituting the equation (28) in equation (2), results in
n−1
X
1−
n
n−1
P
(29)
C. Topology switch due to changing neighbors
The frequent topology changes due to node failures is quite
often in wireless sensor network operations. To study the effect
of change in neighbors, we consider a n-node ring network
with n-time slots. Every time each node chooses two neighbors
In this section, we compute the network criticality of a
asymmetric ring network. Asymmetric ring network can be
modeled as a directed graph, where we assume that forward
link weight is ‘10 and the backward link weight is ‘0 as shown
in Fig. 3.
T heorem 5: The network criticality τ of asymmetric ring
network is given by
τ=
n−1
X
1
j=2
2πj
n
1 + ε − (1 + ε) cos
− i(1 − ε) sin 2πj
n
(36)
5
5
P roof : Laplacian matrix L of a asymmetric ring network is
x 10
(37)
Network Criticality (τ)
L = circ{(1 + ε) − 1 0| 0.....0
{z 0} −ε}
2
n−3
Using equation (3) and equation (37), (j +1)th eigenvalue can
be expressed as
2πj
2πj
− i(1 + ε) sin
(38)
λj (L) = 1 + ε − (1 + ε) cos
n
n
Substituting equation (38) in (2), proves the theorem.
A. Network Robustness-Overhead Optimization
As shown in the Figs.5 and 7, network criticality τ increases
with r. But the node’s power consumption P [17] is
α
r
,
(39)
P = √
n
1
0.5
0
0
20
40
60
Number of Nodes (n)
80
100
Fig. 4. Comparison of Theoretical and Simulation results of τ versus n for
r-nearest neighbor cycle (r = 1).
5
14
x 10
Theortical
Simulation
Network Criticality (τ)
12
10
8
6
4
2
0
0
5
10
Nearest Neighbors (r)
15
20
Fig. 5. Comparison of Theoretical and Simulation results of τ versus r for
r-nearest neighbor cycle (n = 300).
Network Criticality (τ)
VI. N UMERICAL RESULTS AND D ISCUSSION
In this section, we present the analytical results to study
the effect of network parameters, link losses, topological
changes, random links, and asymmetric links on the network
robustness. The effectiveness of of analytical expressions have
been verified with the extensive simulations using MATLAB.
We have plotted τ versus n, for r=1 in Fig. 4. We observe
that τ increases exponentially from n=25. So it has to be
noted that large scale sensor networks exhibit less robustness
to topology changes. In Fig. 5, τ versus r, has been plotted
and we can observe a rapid decrease of τ values at r=5.
Because the node or link betweenness decreases with the
nearest neighbors when n is constant. From r=5, τ values are
too low and almost constant despite the increase in r values.
This result is justified as the r values increase the network
connectivity and a fully connected network always ensures
high robustness to topology changes. Similarly, to understand
the network criticality of a torus network, we have plotted
τ versus k1 and k2 as shown in Fig. 6. We observe that,
τ increases exponentially with the number of nodes in two
dimensions. For k1 =k2 =300, τ versus r plot can be seen in
Fig. 7, and we find that for r-nearest neighbor torus network,
a significant decay of τ values for r=5. We have also noticed
that from r = 5, τ values are almost constant. To investigate
the effect of network dimension m on τ , we have plotted the
Fig. 8, for k1 =16, k2 =18, k3 =20, and k4 =22. We have noticed
that WSN applications work in multiple dimensions with high
r values exhibit high robustness to topology changes. In Fig. 9,
we have compared the static ring network with the topologies
discussed in Section IV for p = q = 0.2. We have observed that
network robustness is drastically increases with the topology
switching between nodes and random WSNs exhibits more
robustness than static networks. In Fig. 10, we have plotted the
τ against n for p = q = 0.7 to see the effect of communication
link probabilities p and q on τ . We have observed that increase
in probability q of link existence results in increase of network
robustness, whereas probability of link failures p reduces the
network robustness exponentially. As shown in Fig. 11, we
have plotted τ versus n, varying values from 0 to 1. We
observe that τ values increases with the , which reveals that
network robustness increases with asymmetric link weights.
Theortical
Simulation
1.5
3000
2000
1000
0
20
15
10
5
0 0
k
5
15
20
k
2
Fig. 6.
10
1
τ versus k1 and k2 for r-nearest neighbor cycle (r = 2).
where α is a path-loss exponent. Since WSNs consist of
limited-battery powered nodes, it is necessary to minimize the
τ without effecting the power consumption P . So to handle
this trade off, an optimization framework has been proposed
to minimize the τ subject to total power consumption P
constraint or minimizing the P subject to τ .
minimize
τ
subject to r ≤ rmax , P ≤ Pmax ,
or
minimize
P
subject to τ ≤ τmax , r ≤ rmax .
6
6
3
4
x 10
Theortical
Simulation
x 10
Static ring network, analytical
Random links, analytical
IID links, analytical
Topology switch, analytical
Static ring network, simulation
Random links, simulation
IID links, simulation
Toplogy switch, simulation
6
Network Criticality (τ)
2.5
Network Criticality (τ)
7
2
1.5
1
0.5
5
4
3
2
1
0
0
5
10
Nearest Neighbors (r)
15
0
0
20
Fig. 7. Comparison of Theoretical and Simulation results of τ versus r for
r-nearest neighbor torus (k1 = k2 =300).
Fig. 10.
10
20
30
Number of Nodes (n)
40
50
τ versus n for 1-nearest neighbor cycle (p = q = 0.7).
5
2
160
x 10
r=1
r=2
r=3
r=4
120
100
Network Criticality (τ)
Network Criticality (τ)
140
80
60
40
Symmetric Network, theoretical
Asymmetric Network, theoretical
Symmetric Network, simulation
Asymmetric Network, simulation
1.5
1
0.5
20
0
1
1.5
2
2.5
3
Network Dimension (m)
3.5
0
0
4
Fig. 8. τ versus m for r-nearest neighbor torus (k1 = 16, k2 = 18,
k3 = 20, k4 = 22).
Fig. 11.
1.5
2
Static ring network, analytical
Random links, analytical
IID links, analytical
Topology switch, analytical
Static ring network, simulation
Random links, simulation
IID links, simulation
Toplogy switch, simulation
Network Criticality (τ)
Network Criticality (τ)
2
80
100
5
x 10
2.5
40
60
Number of Nodes (n)
τ versus n for Asymmetric Ring Network.
4
3
20
1
x 10
1.5
1
Symmetric Network
Asymmetric Network, ε=0
Asymmetric Network , ε=0.2
Asymmetric Network, ε=0.4
Asymmetric Network, ε=0.6
Asymmetric Network, ε=0.8
Asymmetric Network, ε=1
0.5
0.5
0
0
Fig. 9.
10
20
30
Number of Nodes (n)
40
0
0
50
τ versus n for 1-nearest neighbor cycle (p = q = 0.2).
here rmax , τmax , and Pmax are the maximum allowed values
defined based on WSN resource requirements. The above
problems can be solved by suitable optimization tools.
VII. C ONCLUSIONS
In this paper, we have derived the analytical expressions
of network criticality for m-dimensional WSNs to study the
network robustness. The derived analytical expressions reduce
the computational complexity compared to the existing metrics
based on graph-theoretic concepts. We have also studied the
effect of number of nodes, nearest neighbors, and network dimension on network robustness. We have shown that network
Fig. 12.
20
40
60
Number of Nodes (n)
80
100
τ versus n for variation.
robustness decreases with the number of nodes in large scale
WSNs and exponentially increases with the nearest neighbors.
This result is well justified as the number of nearest neighbors
or network dimension is inversely proportional to the node or
link betweenness. Since the sensor node’s power consumption
is increases with the nearest neighbors, WSNs face strict tradeoff between the network robustness and power consumption.
We have proved that probability of link failures drastically
reduces the network robustness. We have also proved that
asymmetric link weights increase the network robustness.
Furthermore, the proposed optimization framework can be
used in network control and optimization problems.
7
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A PPENDIX A
P ROOF OF T HEOREM 3
at x = 0, can be
Taylor series expansion for sin(2r+1)x
sin x
written as (2r + 1) − 23 r(r + 1)(2r + 1)x2 .
So, we can rewrite equation (6) as
τ (Cnr ) =
n
X
j=2
≈
2n
(2r + 1) −
sin
(2r+1)πj
n
sin πj
n
n
X
3n3
1
2
r(r + 1)(2r + 1)π j=2 j 2
(40)
Substituting the below identity in equation (40) proves the
theorem.
∞
X
1
π2
.
(41)
=
2
n
6
n=1
A PPENDIX B
P ROOF OF T HEOREM 4
After substituting the k1 = k2 =
obtain
τ (Tnr ) =
n
2
n
n
2 −1 2 −1
2n
P P
j1 =0 j2 =1
in equation (14), we
(4r+2)−
sin
(2r+1)2πj1
n
2πj1
sin
n
−
sin
(2r+1)2πj2
n
2πj2
sin
n
!
(42)
From Taylor series expansion, we can rewrite equation (42) as
n
n
2 −1 2 −1
τ (Tnr ) =
3n3
8r(r + 1)(2r + 1)π 2 (j12 + j22 )
=1
X X
j1 =0 j2
n
n
1
2
2 −1 X
2 −1
X
1
3n3
≈
8r(r + 1)(2r + 1)π 2 j =0 j =1 (j12 + j22 )
n
2 −1
X Z
3n3
1
≈
8r(r + 1)(2r + 1)π 2 j =0
(j12 + j22 )
n
2 −1
1
for 1 n and 0 < tan−1 x ≤
Zn Zn
0
1
1
dj1 dj2 ≈
j12 + j12
π
2,
Zn
Θ
(43)
j2 =1
we get
1
dj1 = Θ(log n),
j1
(44)
1
So equation(44) can be approximated as
τ (Tnr ) ≈
3n3 Θ(log n)
.
8r(r + 1)(2r + 1)π 2
(45)
| 3 |
BPS/CFT CORRESPONDENCE:
arXiv:1512.05388v2 [hep-th] 2 Jan 2016
NON-PERTURBATIVE
DYSON-SCHWINGER EQUATIONS
AND qq-CHARACTERS
NIKITA NEKRASOV
Abstract. We study symmetries of quantum field theories involving topologically distinct sectors of the field space. To exhibit these symmetries we define special gauge
invariant observables, which we call the qq-characters. In the context of the BPS/CFT
correspondence, using these observables, we derive an infinite set of Dyson-Schwingertype relations. These relations imply that the supersymmetric partition functions in
the presence of Ω-deformation and defects obey the Ward identities of two dimensional conformal field theory and its q-deformations. The details will be discussed in
the companion papers.
Contents
1. Introduction
1.1. Dyson-Schwinger equations
1.2. Non-perturbative Dyson-Schwinger identities
1.3. Organization of the presentation
1.4. Acknowledgements
2. The BPS/CFT correspondence
2.1. N = 2 partition functions
2.2. Defect operators and lower-dimensional theories
2.3. The Y- and X-observables
2.4. The physics of X-observables
2.5. Hidden symmetries
2.6. Some notations.
2.7. Equivariant virtual Chern polynomials
3. Supersymmetric gauge theories
3.1. Quivers
3.2. Quivers with colors
3.3. The symmetry groups
3.4. The parameters of Lagrangian
3.5. The group H
3.6. Perturbative theory
3.7. Realizations of quiver theories
4. Integration over instanton moduli spaces
1
4
4
6
8
10
12
12
13
14
14
17
17
22
22
22
22
22
24
25
26
28
30
2
NIKITA NEKRASOV
4.1. Instanton partition function
4.2. Characters, tangent spaces
4.3. Integral representation.
4.4. Full partition functions
5. The Y − observables
5.1. The bulk Y-observables
5.2. Q-observables
6. Enter the qq − characters
6.1. The main theorem
7. Examples of qq − characters
7.1. A-type theories: one factor gauge group
7.2. A-type theories: linear quiver theories
7.3. The D-type theories
8. The qq − character formula
8.1. Nakajima quiver variety
8.2. The bi-observables
8.3. The formula
8.4. Five dimensional theory
8.5. The symmetry
8.6. Convergence of the integrals
8.7. Reduction to the fixed loci
9. More on the Physics of qq − characters
9.1. Characters from supersymmetric quantum mechanics
9.2. Gauge theory realization of the qq-characters.
9.3. Other realizations of X-observables
10. The first applications
10.1. Effective prepotentials and superpotentials
10.2. Instanton fusion
10.3. Undressing the U (1) legs
10.4. Fractional instantons and quantum differential equations
11. Discussion and open questions
References
30
32
33
34
34
35
39
40
40
40
41
42
44
45
45
48
48
50
51
51
52
55
55
56
62
63
63
63
63
66
67
68
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
In memory of Lev Borisovich Okun (1929 − 2015)
3
4
NIKITA NEKRASOV
1. Introduction
1.1. Dyson-Schwinger equations. The correlation functions of Euclidean quantum field
theory are defined by the path integral:
Z
1
1
DΦ e − h̄ S[Φ] O1 (x1 ) . . . On (xn ) ,
(1)
hO1 (x1 ) . . . On (xn )i =
Z Γ
suitably regularized and renormalized. The classical theory is governed by the EulerLagrange equations, which are derived from the variational principle:
δS[Φcl ] = 0
(2)
These equations are modified in the quantum theory: consider an infinitesimal transformation
(3)
Φ −→ Φ + δΦ
Assuming (3) preserves the measure DΦ in (1) (no anomaly), then
(4) hO1 (x1 ) . . . On (xn )δS[Φ]i =
n
X
h̄
hO1 (x1 ) . . . Oi−1 (xi−1 )δO(xi )Oi+1 (xi+1 ) . . . On (xn )i
i=1
In (3) the change of variables can be also interpreted as a small modification of the
integration contour Γ in (1), Γ → Γ′ = Γ + δΓ, as in the picture below
Fig.1
The small change of contour does not change the integral of a closed form.
The usefulness of the Dyson-Schwinger equations depends on whether one can find
a convenient set of observables Oi in (4) and perhaps also take a limit in order to get
a closed system of equations. Formally, the loop equations [69], [72] are an example of
such a system. Another, related example, is the matrix model, i.e. the zero dimensional
gauge theory. The simplest model is the single matrix integral:
#
"
Z
DΦ
− 1 Tr V (Φ)
e gs N p+1
(5)
Z=
LieU (N ) VolU (N )
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
5
with the polynomial potential
(6)
Vp+1 (x) =
p
X
k=0
tk
xk+1
(k + 1)!
The convenient observable is
1
− V ′ (x)
x−Φ
In the limit N → ∞, gs → 0, with h̄ = gs N fixed, the expectation value Y (x) = h Y(x) i
obeys:
(7)
(8)
Y(x) = gs Tr N
′
Y (x)2 = Vp+1
(x)2 + f p−1 (x)
where f p−1 (x) is a degree p − 1 polynomial of x:
!+
*
V ′ (Φ) − V ′ (x)
(9)
f p−1 (x) = Tr N
Φ−x
whose coefficients encode the expectation values of degree ≤ p − 1 Casimirs of Φ. One
can reformulate (8) somewhat more invariantly by stating that the singularities of Y(x)
disappear in h Y(x) i2 , in the planar limit N → ∞. For finite N ,h̄ the Dyson-Schwinger
equation has the form
D
E
′
(10)
Y(x)2 − gs ∂x Y(x) = Vp+1
(x)2 + f p−1 (x)
Although the equation (10) is not a closed system of equations per se, it illustrates a
principle, which we shall generalize below: given the basic operator Y(x) which, as a
function of the auxiliary parameter x has singularities, one constructs an expression,
e.g.
(11)
T(x) = Y(x)2 − h̄∂x Y(x)
whose expectation value has no singularities in x for finite x, cf. (10). We shall be able
to generalize this procedure for the supersymmetric gauge theories in various spacetime
dimensions.
6
NIKITA NEKRASOV
1.2. Non-perturbative Dyson-Schwinger identities. Let us now study the identities,
which can be interpreted as the analogs
of (4), (10)P
corresponding to non-trivial perP
′
mutations of homology classes Γ = a na Γa −→ Γ = a n′a Γa , where (Γa ) is some basis
in the relative homology, cf. [7]
H 1 dim (F C , F+C ).
2
where F C is the space of complexified fields, and F+C ⊂ F C is the domain, where
ReS[Φ] ≫ 0.
H. Nakajima [76] discovered that the cohomology of the moduli spaces of instantons
carries representations of the infinite-dimensional algebras (this fact was used in the
first strong coupling tests of S-duality of maximally supersymmetric gauge theories
[118]). These algebras naturally occur in physics as symmetries of two dimensional
conformal theories. This relation suggests the existence of a novel kind of symmetry in
quantum field theory which acts via some sort of permutation of the integration regions
in the path integral. The transformations of cohomology classes do not, typically,
come from the point symmetries of the underlying space. Indeed, the infinitesimal
symmetries, e.g. generated by some vector field v ∈ Vect(X) act trivially on the de
Rham cohomology H ∗ (X) of X, as closed differential forms change by the exact forms:
(12)
δω = Liev ω = d(ιv ω) =⇒ [δω] = 0 ∈ H ∗ (X)
The symmetries of the cohomology spaces come, therefore, from the large transformations f : X → X or, more generally, the correspondences L ⊂ X × X:
(13)
φ L = t∗ (δL ∧ s∗ ) : H ∗ (X) → H ∗ (X)
where δL is the Poincare dual to L, the maps s, t are the projections
(14)
s
ւ
X
X×X
ցt
X
and we assume the compactness and smoothness. There exist generalizations which
relax these assumptions.
The physical realization of the symmetries generated by (14) is yet to be understood. It was proposed in [89, 97, 98] that there are symmetries acting between different quantum field theories, for example changing the gauge groups. Conjecturally [81]
supersymmetric domain walls in quantum field theory separating different phases of
one theory or even connecting, e.g. in a supersymmetric fashion, two different quantum
field theories can be used to generate the generalized symmetries of the sort we discussed earlier. More precisely, one exchanges the spatial and the temporal directions,
producing the S-brane [48] version of the domain wall.
This paper deals with another type of large symmetries. They are generated by
the transformations (3) changing the topological sector, i.e. mapping one connected
component of the space of fields to another. We shall be concerned with gauge theories,
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
7
i.e. the Yang-Mills theory on the space-time N,
(15)
Z=
Z
1
DA exp − 2
4g
Z
iϑ
Tr FA ∧ ⋆FA + 2
8π
N
Z
N
Tr FA ∧ FA
!
with the gauge group Gg , and its supersymmetric generalizations. The connected components of the space of gauge fields are labeled by the topology types of the principal
Gg -bundles, and measured, in particular, by the instanton charge
(16)
1
n=− 2
8π
Z
N
Tr FA ∧ FA .
Gauge theory path integral is the sum over n of the path integrals over the space of
fields of fixed topology:
Fig.2
Path integral in U(3) gauge theory in the sector with k = 14 instanons.
(1)
(1)
(2)
(2)
(3)
(3)
The labels (λ1 , λ2 , . . .)(λ1 , λ2 , . . .)(λ1 , λ2 , . . .)
denote various instanton configurations
The analog of the contour deformation (3) is the discrete deformation, as in the
picture:
8
NIKITA NEKRASOV
Fig.3
Path integral in U(3) gauge theory in the sector with k = 14 instanons,
and discrete deformation to account for k = 15 instantons
There is no a priori way to deform a connection A0 on a principal bundle P0 to
a connection A1 on a principal bundle P1 , which is not isomorphic to P0 . However,
imagine that we modify P0 in a small neighborhood of a point x ∈ N so that it becomes
isomorphic to P1 . It means that outside a small disk Dx ⊂ N there is a gauge transformation, which makes A0 deformable to A1 . One can loosely call such a modification
adding a point-like instanton at x
(17)
(1)
A −→ A + δx A
(1) (1)
One can imagine a successive application of the modifications δx1 δx2 , which add pointlike instantons at two distinct points x1 6= x2 ∈ N, or adding two instantons at the same
(2)
point, A −→ A + δx A, and so on.
The specific realization of such modifications is possible in the string theory context, where the gauge theory instantons are the codimension four D-branes dissolved
inside another brane [28]. The modification changing the instanton number is then a
transition where, say, a point-like instanton becomes a D0-brane departed from the
D4-brane.
1.3. Organization of the presentation. We want to study such modifications in the
gauge theory language. Specifically, we shall work in the context of N = 2 supersymmetric gauge theories subject to Ω-deformation. We explore these theories using
the special observables X and Y, which will help us to organize the non-perturbative
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
9
Dyson-Schwinger identities reflecting the invariance of the path integral with respect
to the transformations (17). We shall see that these identities are organized in a structure, the
qq-characters, which suggest a deformation of the q-deformed Kac-Moody
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
symmetry. The latter is familiar from the study of lattice and massive integrable field
theories in two dimensions [111, 112, 115, 109, 110, 10, 53, 30, 32, 33, 31, 114, 21, 113,
39, 22, 60, 23, 38, 36, 20, 35].
The qq-characters are local observables, the corresponding operators can be inserted
at a point in space-time. One can also define and study non-local observables, which are
associated to two-dimensional surfaces in space-time. These will be studied elsewhere.
It is worth revealing at this point that the qq-characters (and the analogous surface
operators) can be defined most naturally in the context of string theory, where the
gauge theory in question arises as a low energy limit of the theory on a stack of D3branes (the ‘physical’ branes) in some supersymmetric background. The qq-characters
in this realization are the low-energy limits of the partition function of the auxiliary
theory, which lives on a stack of D3-branes intersecting the physical branes transversely
at a point. We define the qq-character operators in the presence of the surface operators
in [103].
In the companion paper [100] these constructions of gauge theories with and without
surface defects, as well as the qq-character operators are given a unified treatment using
what we call the gauge origami, a generalized gauge theory, which is best thought of
a low-energy limit of a theory on a stack of Dp-branes in type II string theory, which
span the coordinate C2 -planes inside C4 times a common flat 2p −4-dimensional space.
Orbifolding this construction by discrete symmetries, preserving supersymmetry and
the Ω-deformation, leads to more examples of qq-observables and defect operators in
quiver gauge theories on asymptotically locally Euclidean spaces.
These constructions can be realized mathematically with the help of novel moduli
spaces, which we call the crossed instantons and the spiked instantons. The space of
crossed instantons describes the low-energy modes of open strings connecting k D(−1)
instantons and two stacks of D3-branes, spanning two transversely intersecting copies
of R4 inside R8 . When one of the two stacks is empty the moduli space coincides
with the ADHM moduli space of (noncommutative) instantons on R4 , together with
the obstruction bundle, isomorphic, in this case, to the cotangent bundle. The space
of spiked instantons is the further generalization, describing the low-energy modes of
the open strings stretched between the D(−1)-instantons and six stacks of D3-branes
spanning the coordinate complex 2-planes in C4 , a local model of the maximal number
of complex surfaces intersecting at a point in a Calabi-Yau fourfold.
In the next section we recall the relevant details about the ✿✿✿✿✿
BPS side of the BPS/CFT
correspondence, the supersymmetric partition functions of N = 2 theories. In this
paper we discuss the bulk partition functions, in the companion papers [103], [100]
we study the theories with defects. We also give a rough definition of the X and Y
observables, and some physics behind them. In the section 3 we review quiver gauge
theories with unitary gauge groups, which are superconformal in the ultraviolet. The
section 4 gives the mathematical expression for the integrals over instanton moduli
spaces. The path integrals in the quiver gauge theories under consideration, with
10
NIKITA NEKRASOV
and without defects, reduce to those finite dimensional integrals by localization. The
section 5 defines the Y-observables in gauge theory, both in the physical theory and
in the mathematical problem of integration over the instanton moduli. The section 6
introduces informally the X-observables, the qq-characters, and formulates the main
theorem. The section 7 presents the examples of qq-characters. The section 8 defines
the qq-characters rigorously, by explicit formulas.
1.4. Acknowledgements. I have greatly benefited from discussions with H. Nakajima
and A. Okounkov. The suggestion of V. Pestun that the results of [85], [86] must
generalize to the case of the general Ω-deformaton was instrumental in pursuing the
constructions presented below. Special thanks are to O. Tsymbalyuk who read the
preliminary versions of this manuscript and suggested lots of improvements. Part
of the work was done while the author visited the Euler International Mathematical
Institute in Saint-Petersburg and the Imperial College London.
Research was supported in part by the NSF grant PHY 1404446.
The results of the paper, notably the formulae for the qq-characters and their consequences, the equations on the gauge theory correlation functions, were reported at 1.
The preliminary version of this paper was published under the title “Non-Perturbative
Schwinger-Dyson Equations: From BPS/CFT Correspondence to the Novel Symmetries of Quantum Field Theory” in the proceedings [47] of the ITEP conference (June
2013) in honor of the 100-th anniversary of Isaac Pomeranchuk. It took us a long time
to write up all the details of the story. While the paper was being prepared, several
publications have appeared with some degree of overlap. The paper [56] supports the
validity of our main theorem in the case of the A-type quivers. The papers [117], [116]
contain some discussion of the (0, 4)-sigma model on our moduli space M(n, w, k) (for
ζ = 0). The paper [16] contains the first few instanton checks of some of the results
b0 theory). The paper [41] studies the codimension defects usof our paper (for the A
ing the superconformal index and sphere partition functions, and the RG flows from
1various conferences in 2013-2015:
• “Facets of Integrability: Random Patterns, Stochastic Processes, Hydrodynamics, Gauge Theories
and Condensed Matter Systems”, workshop at the SCGP, Jan 21-27, 2013
• Gelfand Centennial Conference: A View of 21st Century Mathematics, MIT, Sept 2013
• ITEP conference in honor of the 100-th anniversary of Isaac Pomeranchuk, June 2013
• MaximFest, IHES, June 2013
• Strings’2014, Princeton, June 2014
• ‘Frontiers in Field and String Theory’, Yerevan Physics Institute, Sept 2014
• “Gauged sigma-models in two dimensions”, workshop at the SCGP, Nov 3-7, 2014
• “Wall Crossing, Quantum Integrable Systems, and TQFT”, Nov 17-21, 2014
• “Recent Progress in String Theory and Mirror Symmetry”, FRG workshop at Brandeis, Mar 6-7,
2015
• “Resurgence and localization in string theory and quantum field theory”, workshop at the SCGP,
Mar 16-20, 2015
• “Algebraic geometry and physics”, workshop at the Euler Mathematics Institute, Saint-Petersburg,
May 2015
seminars in 2013-2015:
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
11
vortex constructions, also at the level of the first few instanton checks. The paper [14]
discusses the algebra of our Y -observables in the A-type quiver theories.
12
NIKITA NEKRASOV
2. The BPS/CFT correspondence
We start by briefly reviewing the BPS/CFT correspondence [83] between supersymmetric field theories with eight supercharges in four, five, and six dimensions, and
conformal and integrable theories in two dimensions. It is based on the observation
that the supersymmetric partition functions [94] are the remarkable special functions,
which generalize all the known special functions given by the periods, matrix integrals, matrix elements of group, Kac-Moody, and quantum group representations etc.
[94, 67, 84]. The particular implementations of this correspondence are well-known
under the names of the AGT conjecture [2, 120], and the Bethe/gauge correspondence
[89, 96] (see [74, 44, 43] for the prior work). For details the interested reader may
consult the references in, e.g. [86].
2.1. N = 2 partition functions. For the definition and some details see [94, 84]. The
supersymmetric partition function of N = 2 theory
(18)
Z(a; m; τ ; ε) = Z tree (a; m; τ ; ε) Z 1−loop (a; m; ε) Z inst (a; m; q; ε)
depends on the vacuum expectation value a of the adjoint Higgs field in the vector
multiplet, it belongs to the complexified Cartan subalgebra of the gauge group of the
theory, the set m of complex masses of the matter multiplets, and the set τ of the
complexified gauge couplings,
ϑ 4πi
,
τ=
+
2π g 2
one per simple gauge group factor (we shall not discuss the issue of SU (n) versus U (n)
gauge factors in this paper). We denote by q the set of the exponentiated couplings,
the instanton factors,
q = exp 2πiτ
The non-perturbative factor Z inst (a; m; q; ε) in (18) has the q-expansion, for small |q|:
X
(19)
Z inst (a; m; q; ε) =
qk Zk (a; m; ε)
k
Finally, ε = (ε1 , ε2 ) ∈ C2 are the complex parameters of the Ω-deformation of the theory
[94].
2.1.1. ✿✿✿✿✿✿✿✿✿✿✿✿✿
Asymptotics✿✿✿
of✿✿✿✿✿✿✿✿✿✿
partition✿✿✿✿✿✿✿✿✿✿
functions. The function (18) contains non-trivial information about the theory. For example, the asymptotics at ε → (0, 0), for generic a,
produces the prepotential [106, 107] of the low-energy effective action of the theory:
(20)
Z(a; m; τ ; ε) ∼ exp
1
F (a; m; τ ) + less singular in ε1 , ε2 .
ε1 ε2
the low-energy effective action being given by the superspace integral
Z
eff
(21)
S =
d 4 xd 4 ϑ F (a + ϑψ + ϑϑF − + . . .; m; τ )
R4|4
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
13
The prepotential F (a; m; τ ) as a function of a determines the special geometry [34] of
the moduli space Mvector of Coulomb vacua:
a
(22)
d = periods of ̟ C
∂F
∂a
along the 1-cycles on the abelian variety Ab , the fiber p −1 (b) of the Lagrangian projection
p : P −→ Mvector
(23)
of a complex symplectic manifold (P, ̟ C ), the moduli space of vacua of the same
gauge theory, compactified on a circle [108]. The manifold P is actually the phase
space of an algebraic integrable system [27]. The first example of this relation, the
periodic Toda chain for the SU (2) pure super-Yang-Mills theory, was found in [46].
The asymptotics of (18) at ε2 → 0 with ε1 = h̄ fixed, for generic a, gives the effective
twisted superpotential
(24)
Z(a; m; τ ; (h̄, ε2 )) ∼ exp
1
W (a; m; τ ;h̄) + less singular in ε2 ,
ε2
ε2 → 0
of a two dimensional effective theory. This function plays an important role in quantization of the symplectic manifold P and the Bethe/gauge correspondence [89, 96, 90,
87, 86].
The asymptotics (24), (20) are modified in an intricate way when the genericity
assumption on a is dropped. The interesting non-generic points are where a and ε1
(with ε2 → 0) are in some integral relation. The behavior near such special points and
its rôle in the Bethe/gauge correspondence will be discussed elsewhere.
2.2. Defect operators and lower-dimensional theories. In addition to the Z-functions,
which are the partition functions of the theory on R4 , in [103] we also consider the
partition functions Ψ of the same gauge theory in the presence of defects preserving
some fraction of supersymmetry. These defects could be point-like, or localized along
surfaces. We derive the differential equations, which can be used to relate the theory
with a surface operator to the theory without one. As a by-product we get the explicit
realization of the Bethe/gauge correspondence [89, 96] with an additional bonus: the
gauge theory produces not only the equations, characterizing the spectrum of the
quantum integrable system, but also gives an expression for the common eigenfunction
of the full set of quantum integrals of motion. In particular, we shall show [102] in that
a class of surface defect operators in the N = 2 theory with U (n) gauge group and 2n
fundamental hypermultiplets solves the Knizhnik-Zamolodchikov (KZ) equation, which
is obeyed by the 4-point conformal block of the SU (n) Wess-Zumino-Witten theory on
the sphere; that a class of surface defect operators in the N = 2∗ theory with the U (n)
gauge group solves the Knizhnik-Zamolodchikov-Bernard (KZB) equation, which is
obeyed by the 1-point conformal block of the SU (n) Wess-Zumino-Witten theory on the
torus. In the ε2 → 0 limit these surface operators become the eigenfunctions of Gaudin
14
NIKITA NEKRASOV
Hamiltonian and the elliptic Calogero-Moser Schrödinger equation, respectively, in
agreement with the conjectures in [96], [3] and earlier ideas.
In addition to the codimension two defects in four or five dimensional theories we
can also consider lower dimensional theories. For example, gauge theory on the AdS3
space with appropriate boundary conditions can be viewed as the U (1)-orbifold of a
four dimensional superconformal gauge theory. We study these cases in [100].
2.3. The Y- and X-observables. The main tools in our analysis are the gauge invariant
observables Yi (x) and Xi (x), defined for each simple factor U (ni ) of the gauge group.
Here i belongs to the set Vertγ , which in our story is the set of vertices of a quiver. The
Yi (x) are the suitable generalizations of the characteristic polynomials of the adjoint
Higgs field,
(25)
Yi (x) ∼ Det(x − Φi ) .
They are the gauge theory analogues of the matrix model resolvents (7). As a function
of x, each operator Yi (x) has singularities, i.e. the relation (25) is modified. This
modification is due to the mixing between the adjoint scalar and gluinos, e.g.
(26)
αj ′
βj ′′
∗
−1
Φi ∼ Φcl
i + εαβ εj ′ j ′′ (dAi dAi ) [ψi , ψi ]
which have zero modes in the presence of gauge instantons, leading to the poles in x, in
a way we make much more precise below. The Xi (x) are composite operators, built out
of Y’s. They are Laurent polynomials or series in Yi (x)’s with shifted arguments and
their derivatives. They are the analogues of the matrix model operators (11). Their
main property is the absence of singularities in h Xi (x) i for finite x, similarly to the
matrix model case, cf. (10). In the weak coupling limit Xi (x) → Yi (x) → (25). We
Vertγ
define also the observables Xw,ν (x), labelled by the string w = (wi )i∈Vertγ ∈ Z≥0 of
νi )i∈Vertγ , ν~i ∈ Cwi of complex numbers. Here
non-negative integers and the string ν = (~
D
E
Vertγ is the set of simple gauge group factors. The expectation values Xw (x) also
have no singularities, while in the limit of zero gauge couplings Xw,ν (x) approach
wi
YY
i
Yi (x + νi,f ).
f =1
We call Xw,ν (x) the qq-characters. For w = (δi,j )j∈Vertγ , and ν = 0, we call Xw,ν (x) =:
Xi (x) the fundamental qq-characters. The reason for and the meaning of the terms will
hopefully become clear in the coming chapters.
2.4. The physics of X-observables. The X-observables can be interpreted as the partition functions of the auxiliary gauge theory living on a space, transverse to the spacetime of our gauge theory, the “physical space-time”. The auxiliary theory has massive
degrees of freedom coupled to the degrees of freedom of our gauge theory at some point
p ∈ R4 in the physical space-time, so that integrating them out induces an operator
Ow,ν,x (p) inserted at p. The data w is the choice of the auxiliary gauge theory while
ν and x fix its vacuum.
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
15
The dimensionality of the auxiliary gauge theory is a somewhat subtle issue. Most of
the theories we study in the paper, such as the N = 2 quiver theories with affine quivers
have the X-observables which come from a four dimensional auxiliary theory. The
theories with finite quivers can be viewed as a sector in the auxiliary four dimensional
theory corresponding to an affine quiver. In fact, the finite quivers of A-type can
b∞ theory, which corresponds to the orbifold of C2
be realized as a subsector of the A
by U (1). Gauge theory living on such an orbifold can be viewed either as a three
dimensional theory on a manifold with corners (or, conformally, on the AdS3 ), or, for
the purposes of supersymmetric partition functions, as a two dimensional sigma model
[100].
Here is a sketch of the string theory construction. Consider IIB string theory on
the ten-dimensional manifold of the form R2φ × N × W/Γ, where N = R4 , W = R4 , and
Γ a finite subgroup of SU (2) (see [54] for the discussion of IIB string theory on ALE
spaces).
Recall [29] that N = 2 quiver gauge theories with affine A, D, E quivers can be
realized as the low energy limit of the theory on a stack of n D3-branes located at
ϕ × N × 0, with ϕ ∈ R2φ a point, and 0 the tip of the W/Γ singularity, with Γ being the
discrete subgroup of SU (2), McKay dual [71] to the corresponding A, D, E simple Lie
group.
The worldvolume of these D3 branes
is a copy of N. Let us now add a stack
of w D3-branes located at x × 0 × W/Γ,
with the worldvolume being a copy of W/Γ.
Here x ∈ R2φ ≈ C is a complex number,
and 0 ∈ N is a fixed point.
Here is the picture:
Fig.4
The low energy configurations in this system of two orthogonal stacks of D3 branes
are labelled by the separation of branes along R2φ encoded in ν, and by the choice of
flat U (w) connection at infinity S3 /Γ of W/Γ, which is equivalent to the choice of the
string w.
16
NIKITA NEKRASOV
Fig.5
Gauge theory with X-observables in the brane picture
The qq-character Xw,ν (x) is simply the observable in the original theory on the stack
of N D3-branes living along N, which is obtained by integrating out the degrees of
freedom on the transversal D3-branes, in the vacuum corresponding to the particular
asymptotic flat connection w and the vacuum expectation values ν of the scalars in
the vector multiplets living on W/Γ.
Fig.6
Gauge theory with the observables X(x1 ), X(x2 ), X(x3 )
The next piece of our construction is the Ω-deformation using a subgroup of the spin
cover of the group Spin(8) of rotations of W × N which commutes with Γ, preserves the
configuration of branes, and some supersymmetry. This subgroup generically has rank
two, which enhances to three for Γ of A type. The parameters of the Ω-deformation are
generically two complex numbers ε = (ε1 , ε2 ), and for Γ of A-type there is an additional
b0 -case,
parameter m. This parameter is the mass of the adjoint hypermultiplet in the A
bk case for
and the sum of masses of all k + 1 bi-fundamental hypermultiplets in the A
k > 0. It is convenient to introduce four ε-parameters, εa , a = 1, 2, 3, 4, which sum to
zero:
(27)
ε1 + ε2 + ε3 + ε4 = 0
so that ε3 = m, ε4 = −m − ε, ε = ε1 + ε2 . Together they parametrize the generic SU (4)
Ω-deformation.
The K-theoretic and elliptic versions of qq-characters correspond to the five- and sixdimensional theories, which are engineered in the analogous fashion, with R2φ replaced
by S1 × R1 and S1 × S1 , respectively. In the five dimensional case we use IIA string
and the D4 branes wrapped on S1 instead of D3’s, in the six dimensional case we are
back in the IIB realm with D5 branes wrapped on S1 × S1 .
The configuration of D3 branes which we described above can be generalized, by
considering other orbifolds of the ten dimensional Euclidean space R2φ × N × W. For
example, the orbifolds R2φ × N/ΓN × W/ΓW with D3 branes wrapping N ⊔ W define
]N .
the qq-characters relevant for the γΓW -quiver gauge theory on the ALE space N/Γ
The most general orbifold we could employ is by the subgroup Γ = ΓN × Γ∆ × ΓW ⊂
SU (2)N,L × SU (2)∆ × SU (2)W,L ⊂ Spin(4)N × Spin(4)W ⊂ Spin(8). We explain the rôle
of this group and its subgroups in the following chapters. Using this construction we
also realize various defect operators in various quiver gauge theories on conical spaces.
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
17
The final piece of the construction is turning on the appropriate B-field which makes
the configurations where the D(−1)-instantons are separate from the D3-branes nonsupersymmetric. The D(−1)-instantons bound to the two orthogonal stacks of D3
branes give rise to what we call the crossed instantons. We can also study the generalization involving six stacks of D3 branes spanning complex 2-planes inside R8 ≈ C4 .
The complex coordinate x parametrizes the remaining R2 ≈ C1 , which is orthogonal
to C4 in the ten dimensional Euclidean space-time of the type
E
D IIB string.
The ✿✿✿✿✿✿
main ✿✿✿✿✿✿
claim, i.e. the absence of singularities in x of Xw,ν (x) , is the statement that the combined system of the intersecting stacks of D3 branes has no phase
transitions and no runaway flat directions at special values of x, in the presence of
Ω-deformation. Mathematically, the argument is the compactness of the moduli space
of crossed (for two orthogonal stacks of branes) and spiked instantons (for six stacks
of branes), the supersymmetric configurations of the combined system of branes, with
the Ω-deformation and appropriate B-field turned on. We describe the moduli spaces
in [101].
One can also apply the orientifold projection (which, unfortunately, would not be
consistent with the B-field we are using) to arrive at the theory of crossed instantons
for the orthogonal and symplectic groups.
2.5. Hidden symmetries. The IIB string theory on R4 /Γ × R1,5 contains the nonabelian tensionless strings [119] with the A, D, E tensor symmetry (it becomes the gauge
symmetry of the A, D, E type upon compactification on a circle, i.e. when Σ = S1 ×R1 ).
In the limit ε1 , ε2 → 0 our qq-characters approach the ordinary characters for the
b D,
bE
b for affine quivers,
Kac-Moody group built on the quiver (i.e. the affine Lie group A,
and the simple A, D, E groups for the finite quivers), [85]. The non-abelian tensor
symmetry seems to be realized on the worldvolume of D3 branes by the large field
deformations which lead to the non-perturbative Dyson-Schwinger equations we discuss
in this paper. The qq-character observables may teach us something important about
the nature of the tensor symmetry representation. See [58] for the discussion of the
qq-deformed W -algebras and their gauge theory realizations.
2.6. Some notations..
2.6.1. Finite
sets. We use the following notations for certain finite sets:
✿✿✿✿✿✿✿✿✿✿
(28)
and
(29)
[p] ≡ {1, 2, . . . , p},
[0, q] ≡ {0, 1, 2, . . . , q},
p ∈ Z+ ,
q ∈ Z≥0
(xi )i∈I ≡ { xi | i ∈ I }
Also for the set (zi )i∈I of complex numbers indexed by the set I we use the notation
Y
(30)
zI =
zi
i∈I
for their product. This is consistent with the notation (40).
18
NIKITA NEKRASOV
2.6.2. Roots
of unity.
✿✿✿✿✿✿✿✿✿✿✿✿✿✿
√
i = −1 ,
(31)
and
̟p = exp
(32)
2πi
p
so that i = ̟4 , −i = ̟43 .
2.6.3. ✿✿✿✿✿✿✿✿✿✿✿✿
Parameters✿✿✿
of✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
Ω-deformations. In four dimensions, we have two parameters
2
ε = (ε1 , ε2 ) ∈ C . We also use their sum
ε = ε1 + ε2 ,
(33)
their exponents
q1 = e βε1 , q2 = e βε2 , q = e βε = q1 q2 ,
(34)
and the virtual characters
(35)
P = (1 − q1 )(1 − q2 ) ,
P ∗ = (1 − q1−1 )(1 − q2−1 )
The parameter β is the circumference of the circle of compactification of a 4+1 dimensional supersymmetric theory.
In the context of the BPS/CFT correspondence, the parameter
b 2 = ε1 /ε2
(36)
is useful.
In eight dimensions, or for the theories in four dimensions with adjoint matter, it
will be useful to have four parameters
ε̄ = (ε, ε̃) ≡ (ε1 , ε2 , ε3 , ε4 ) ∈ C4 ,
(37)
which sum to zero:
ε3 + ε4 = −ε
(38)
We denote 4 = {1, 2, 3, 4} and
qa = e βεa ,
qa∗ = qa−1 ,
(39)
Pa = (1 − qa ),
Pa∗ = (1 − qa−1 ),
For any subset S ⊂ 4 we define S̄ = 4\S, and:
Y
Y
∗
(40)
qS =
qa , qS = qS̄ ,
PS =
Pa ,
a∈S
so that q∅ = q4 = 1.
a∈S
a∈4
PS∗ = (−1)|S| qS̄ PS
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
19
2.6.4. Chern
Characters and Euler Classes. Let E → X be the rank m = rkE complex
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
vector bundle, and ci (E) ∈ H 2i (X, Z) the corresponding Chern classes. Then e(E) =
ctop (E) = cm (E) is the Euler class of E, and
ǫz (E) = e(E) + zcm−1 (E) + z 2 cm−2 (E) + . . . z m
(41)
is the Chern polynomial.
2.6.5. Weights
from characters. For a virtual representation R of a Lie group H, the
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
weights w are computed using its character as follows:
(42)
R = R+ ⊖ R− ,
Tr R± (e θ ) =
X
e w(θ) ,
w∈W (R± )
Tr R (e θ ) = Tr R+ (e θ ) − Tr R− (e θ ) ,
where R± are the vector spaces, the actual representations of H, and W (R± ) are the
sets of the corresponding weights (the linear functions on Lie(H) which take integer
values on the root lattice).
2.6.6. Chern
polynomials from characters. We denote by ǫθ (R) the following Weyl✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
invariant rational function on the Cartan subalgebra hC of Lie(HC ):
Q
w∈W (R− ) w(θ)
(43)
ǫθ (R) = Q
, θ ∈ hC ,
w∈W (R+ ) w(θ)
where the weights w(θ) are given by (42). Note that in (43) the weights of R+ are in
the denominator. It follows from the definition (42) that
(44)
ǫθ (R)ǫθ (−R) = 1 ,
if R does not contain ±1 as a summand, and, more generally:
(45)
ǫθ (R1 ⊕ R2 ) = ǫθ (R1 )ǫθ (R2 ) .
Also,
(46)
ǫθ (R∗ ) = ǫ−θ (R) = (−1)dR ǫθ (R)
where
(47)
dR = dimC R+ − dimC R−
functions from characters. In our story we occasionally encounter the
2.6.7. Chern
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
generalizations of the formulas like (43) where the representations R± are infinite dimensional. In order for (43) to make sense in this case we use the
20
NIKITA NEKRASOV
2.6.8. ζ-function
regularization. The map from the character (42) to ǫθ (R) can be
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
given in the integral form:
Z∞
dβ s
Λs
d
β Tr R e βθ
(48)
ǫθ (R) = exp
ds s=0 Γ(s) 0 β
where one chooses s and θ with the real part in the appropriate domain to ensure
the convergence of the integral in the right hand side of (48), and then analytically
continues. For finite dimensional R the result does not depend on Λ. For infinite
dimensional R the left hand side of (48) really is defined by the right hand side. More
precisely, we assume R is graded,
∞
M
(49)
R=
Rn ,
n=0
with the finite dimensional virtual subspaces Rn = R+n −R−n , dimR±n < ∞, whose superdimensions grow at most polynomially with n. We define
Z∞
∞
dβ s X −βt(n+1)
d
Λs
(50)
ǫθ (R) = Limt→+0 exp
e
Tr Rn e βθ
β
ds s=0 Γ(s) 0 β
n=0
where the integral in the right hand side converges for sufficiently large ℜ(t), ℜ(s),
defining an analytic function, whose asymptotics near s, t = 0 defines the left hand side.
For example, take R = C[z1 , z2 , z3 , . . . , zδ ] with H = (C× )δ+1 acting via:
(51)
t : f 7→ f t ,
f t (z) = t0 f (t1−1 z1 , t2−1 z2 , . . . , tδ−1 zδ )
The character Tr R e βθ
Tr R e βθ = e βθ0
(52)
δ
Y
i=1
1
1 − e βθi
and the refined character (where we define the grading by the polynomial degree)
(53)
∞
X
e −βt(n+1) Tr Rn e βθ = e β(θ0 −t)
n=0
δ
Y
i=1
1
1 − e β(θi −t)
are easy to compute.
The integral on the right hand side of (50) absolutely converges for ℜt > maxδi=0 ℜθi
and ℜs > δ.
2.6.9. Asymptotics
of the ζ-regularized ǫθ ’s. Let R be a virtual representation, θ ∈ hC ,
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
βθ
and χR,θ (β) = Tr R e . For β → 0 the function χR,θ (β) has an expansion:
(54)
χR,θ (β) =
+∞
X
β n χR,θ,n
n=−δR
where χR,θ,n is a homogeneous rational function of θ of degree n, obeying:
(55)
χR∗ ,θ,n = χR,−θ,n = (−1)n χR,θ,n
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
21
We are interested in the large x asymptotics of
Z∞
dβ s −βx
Λs
β e Tr R e βθ ∼
β
s=0 Γ(s) 0
−n
0
X
X
n
x
1
(−x)
log
−
∼ exp −
χR,θ,n
(−n)!
Λ
k
d
(56) ǫ−x+θ (R) ≡ exp
ds
n=−δR
k=1
∞
X
(n
−
1)!
× exp
χR,θ,n
xn
n=1
(since the variable x shifts the auxiliary variable t used to regularize an infinite trace,
we can safely set t = 0 in (56)). Thus,
0
X
n
(−x)
(57)
ǫx+θ (R∗ ) = ǫ−x+θ (R) × exp −πi
χR,θ,n
(−n)!
n=−δR
2.6.10. Flips
{. We shall also use a notation, for a virtual representation R = R1 ⊕ R2 ,
✿✿✿✿✿✿✿✿
R { R1 ⊕ R∗2
(58)
and similarly for their characters:
TrR { TrR1 + Tr∗R2
(59)
where we also use the convention
X
(60)
χ∗ =
e −w(θ) ,
for
w∈W (R)
χ=
X
e w(θ)
w∈W (R)
Sometimes, when the choice of the element θ ∈ hC is understood, we denote the trace
TrR (e θ ) in the representation R by the same letter R.
For example
(61)
(1 + q−1 )(1 − q1 ) { 1 − q1 + q1 q2 − q2 = P
The multiplicative measures of the finite dimensional virtual representations R, given
by the products (43) of weights w(θ) and their K-theoretic analogues, given by the
w(θ)
products of 2sin 2 do not change, up to a sign, under the { modifications:
(62)
ǫθ (R′ ) = (−1)dR2 ǫθ (R),
R { R′
In the infinite dimensional case the multiplicative anomaly of the measure (48) follows
from (57).
22
NIKITA NEKRASOV
2.7. Equivariant virtual Chern polynomials. Let R be a virtual representation as above,
and
(63)
R = ⊕w∈W (R+ ) R+w ⊖ ⊕w∈W (R− ) R−w
be the corresponding weight decomposition. Let Ew , with the weights w ∈ W (R+ ) ∪
W (R− ) be some vector bundles over X, and
(64)
E = ⊕w∈W (R+ ) R+w ⊗ Ew+ ⊖ ⊕w∈W (R− ) R−w ⊗ Ew−
be the associated virtual bundle over X. We denote by (cf. (41), (43))
Q
w∈W (R− ) ǫw(θ) (Ew− )
(65)
ǫθ (E) = Q
w∈W (R+ ) ǫw(θ) (Ew+ )
the rational function on the Cartan subalgebra hC with values in H ∗ (X, C).
3. Supersymmetric gauge theories
In this section we go back to the gauge theory narrative. Our gauge theories are
characterized by a quiver diagram. Let us start by reviewing what we mean by them.
3.1. Quivers. A quiver is an oriented graph γ, with the set Vertγ of vertices and the
set Edgesγ of oriented edges. We have two maps s, t : Edgesγ −→ Vertγ , sending each
edge e to its source s(e) and the target t(e), respectively.
We shall also use an unconventional term arrow
which is a pair (e, σ), where e ∈ Edgesγ , σ = ±1.
The set Arrowsγ = Edgesγ × 2Edgesγ of arrows
is equipped with two maps
s̄, t¯ : Arrowsγ → Vertγ , defined by:
s(e), if σ = +1
s̄(e, σ) =
t(e), if σ = −1
Note that the source and the target
of an edge may coincide,
t(e),
¯t (e, σ) =
s(e),
b0 example above.
as in the A
if σ = +1
if σ = −1
3.2. Quivers with colors. In addition to the quiver diagram, the gauge theory is characterized by the vectors n, m, sometimes called the colorings of the quiver:
(66)
Vertγ
n = (ni )i∈Vertγ ∈ Z>0 ,
to which we associate the vector spaces Ni = Cni , Mi = Cmi .
3.3. The symmetry groups.
Vertγ
m = (mi )i∈Vertγ ∈ Z≥0
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
23
3.3.1. The
gauge group. The gauge group Gg of the theory is the product
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
(67)
Gg =
U (ni )
i∈Vertγ
3.3.2. The
flavor symmetry. The theory has the global symmetry which is usually
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
called the flavor symmetry. The flavor symmetry group Gf is a quotient:
(68)
Gf =
U (mi ) × U (1)Edgesγ / U (1)Vertγ
i∈Vertγ
where U (1)Vertγ acts on
i∈Vertγ
U (mi ) × U (1)Edgesγ
as follows:
(69)
−1
)e∈Edgesγ
(ui )i∈Vertγ : (gi )i∈Vertγ , (ue )e∈Edgesγ 7→ (ui gi )i∈Vertγ , (us(e) ue ut(e)
This action is equivalently both left and right, therefore U (1)Vertγ is a normal subgroup
of ×i∈Vertγ U (mi ) × U (1)Edgesγ . In fact, the flavor group Gf occasionally enhances. For
example, the N = 4 theory, viewed as an N = 2 supersymmetric theory, is a particular
b0 with one vertex v, and
example of the quiver theory, corresponding to the quiver A
one edge e, connecting this vertex to itself s(e) = t(e) = v. The flavor symmetry is
enhanced from U (1) to SU (2) in this case. This is a subgroup of the R-symmetry
group SU (4) which commutes with the SU (2) × U (1)A R-symmetry of the particular
N = 2 subalgebra of the N = 4 theory.
3.3.3. Rotational
symmetries. Our four dimensional gauge theories, in the absence
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
of defects to be discussed below, are Poincare invariant. In what follows we shall be
breaking the translational invariance by deforming the theory in a rotationally covariant
way. The spin cover Spin(4)N of the group of rotations is the product
(70)
Spin(4)N = SU (2)N,L × SU (2)N,R
The regularization of the instanton integrals which we employ in [94] and here breaks
the Spin(4)N invariance down to its subgroup Grot = SU (2)N,L × U (1)N,R ≈ U (2) ⊂
Spin(4)N which is the group of rotations of the Euclidean space-time N = R4 , preserving the identification of the latter with the complex vector space C2 .
±
Let SN
be the defining two dimensional representations (chiral spinors) of SU (2)N,L
+
−
−
and SU (2)N,R , respectively, so that NC = SN
⊗ SN
. Under Grot , SN
splits as LN ⊕ L−1
N .
2
Let us denote the two dimensional representation of Grot by QN ≈ C . Then
(71)
+
Q N = SN
⊗ LN .
24
NIKITA NEKRASOV
3.4. The parameters of Lagrangian. The field content of the theory is the set of N = 2
vector multiplets Φi = (Φi , . . . , Ai ), i ∈ Vertγ , transforming in the adjoint representation of Gg , the set Qi = (Qi , . . . , Q̃i ), i ∈ Vertγ of hypermultiplets transforming in the
fundamental representation Cni of Gg , and the antifundamental representation Cmi of
Gf , and the set Qe , e ∈ Edgesγ of hypermultiplets transforming in the bi-fundamental
representation Cns(e) , Cnt(e) of Gg .
The Lagrangian L of the theory is parametrized by the complexified gauge couplings
τ = (τi )i∈Vertγ ,
via
Z
1 X
iReτi
Tr Ni FAi ∧ FAi +
L= − 2
8π
N
+Imτi
Z
i∈Vertγ
N
Tr Ni FAi ∧ ⋆FAi + Tr Ni DAi Φi ∧ ⋆DAi Φ̄i + Tr Ni [Φi , Φ̄i ]2 + . . .
and the masses
m = (me )e∈Edgesγ ⊕ (mi )i∈Vertγ ,
where
me ∈ C, mi = diag(mi,1 , . . . , mi,mi ) ∈ End(Cmi ) .
(72)
which enter the superpotential (in the N = 1 language)
X
X
W=
Tr Mi mi Qi Q̃i +
me Tr Ns(e) Q̃e Qe +
e∈Edgesγ
i∈Vertγ
X
i∈Vertγ
Tr Mi Qi Φi Q̃i +
X
e∈Edgesγ
Tr Ns(e) Q̃e Φt(e) Qe − Q̃e Qe Φs(e) ,
i.e. we view the scalars in the hypermultiplet Qi as the linear maps, the matrices:
and those in Qe as
Qi : Ni → Mi ,
Qe : Ns(e) → Nt(e) ,
Q̃i : Mi → Ni ,
Q̃e : Nt(e) → Ns(e) .
The vacua of the theory are parametrized by the Coulomb moduli
(73)
a = (ai )i∈Vertγ ,
ai = diag(ai,1 , . . . , ai,ni ) ∈ End(Cni ) ,
so that
h Φi ia = ai .
It is convenient to package the masses mi and the Coulomb moduli ai into the polynomials:
mi
ni
Y
Y
(74)
Pi (x) =
(x − mi,f ) ,
Ai (x) =
(x − ai,α )
f =1
α=1
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
25
We also use the characters
(75)
Ni =
ni
X
α=1
e
βai,α
,
Mi =
mi
X
e βmi,f
f =1
which contain the same information about the masses and Coulomb moduli as the
polynomials (74).
3.5. The group H. Define
(76)
H = Gg × Gf × Grot ,
The complexification of the Lie algebra of the maximal torus TH of this group is parameterized by (a; m; ε). It is the domain of definition of the supersymmetric partition
functions Zk in the Eq. (19).
26
NIKITA NEKRASOV
3.6. Perturbative theory.
3.6.1. Perturbative
consistency and asymptotic freedom. The theory defined by the
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
quiver data is perturbatively asymptoticaly free if the one-loop beta function of all
gauge couplings is not positive. For this to be possible we must restrict the gauge
group to be the product of special unitary groups
(77)
Gg −→
SU (ni )
i∈Vertγ
since the abelian factors are not asymptotically free, if there are fields charged under
them. For the SU (ni ) gauge coupling the beta function is easy to compute:
X
X
d
(78)
βi = µ τi = −2ni + mi +
ns(e) +
nt(e)
dµ
−1
−1
e∈t
(i)
e∈s
(i)
The requirement βi ≤ 0 for all i ∈ Vertγ implies (see [51, 57, 65, 85] for details) that γ
is a Dynkin graph of finite or affine type of a simply-laced finite dimensional or affine
Lie algebra gγ . In the latter case m = 0 (not to be confused with m 6= 0).
Fig.7
Affine A,D,E quivers with their n-coloring
Finite A,D,E quivers are obtained by removing the green node
Examples.
3.6.2. ✿✿✿✿✿✿✿✿✿✿✿
(1) The Ar -type quiver γ, with r ≥ 1, has:
(79)
Vertγ = [r], Edgesγ = [r − 1],
s(e) = e, t(e) = e + 1,
e = 1, . . . , r − 1 .
br , with r ≥ 0, has (see the Fig.6)
(2) The quiver A
(80)
Vertγ = Edgesγ = [r + 1],
s(e) = e, t(e) = 1 + (e mod(r + 1)) .
(3) The D and E-type quivers have a single tri-valent vertex, see the Fig.6.
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
27
3.6.3. Perturbative
partition function. The description of the tree level and the per✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
turbative contributions to the partition function (the latter is given exactly by one loop
computation) can be found in [86].
Here we just quote the results.
Y − 1 P n i a2
α=1 i,α
2ε ε
tree
(81)
Zγ (a; m; τ ; ε) =
qi 1 2
,
i∈Vertγ
and
1−loop
(82)
where (cf. (75)):
(83)
Zγ
pert
(a; m; ε) = ǫa,m,ε (−Tγ
)
X
1
pert
(Mi − Ni ) Ni∗ +
Tγ =
−βε
−βε
1
2
(1 − e
)(1 − e
)
i∈Vertγ
X
e∈Edgesγ
∗
e βme Nt(e) Ns(e)
The character (83) is not a finite sum of exponents as in (42), so the map ǫ from the
sums of exponents to the products of weights is defined by analytic continuation, cf.
(50):
Z∞
dβ s pert
d
Λs
pert
(84)
ǫa,m,ε (−Tγ ) = −
β Tγ
ds s=0 Γ(s) 0 β
There are subtle points of the regularization of (84) related to boundary conditions in
gauge theory. These will be discussed elsewhere. The ultraviolet, Λ → ∞ asymptotics
of (84), has, a priori, the terms proportional to Λ2 , Λ, Λ2 logΛ, ΛlogΛ, and logΛ.
The physically relevant terms are in the last one, they correspond to the one-loop
beta-function of τi if the coefficient of logΛ contain the terms proportional to
ni
X
(85)
ch2 (Ni ) ≡
a2i,α
α=1
Thus, these terms are absent precisely when (78) holds.
3.6.4. Beyond
asymptotic freedom. If the asymptotic freedom/conformality conditions
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
are not obeyed, our partition functions are defined as formal power series in the qi
couplings, and some additional couplings, which we call the higher times.
3.6.5. ✿✿✿✿
The✿✿✿✿✿✿✿✿✿✿
extended✿✿✿✿✿✿✿✿✿
coupling✿✿✿✿✿✿
space. Gauge theory can be deformed, in the ultraviolet,
by the irrelevant (higher degree) operators, which preserve N = 2 supersymmetry. One
adds to the tree level prepotential the terms of the form:
∞
XX
1
tree
(86)
F
=
τ Tr Φl+2
i
(l + 2)! i,l
i
l=0
The parameters τi,l with l > 0 are bosonic, in general nilpotent, variables. Actually,
for some observables one can make sense of the parameters τi,1 in a finite domain near
′
′′
zero [70]. One can also add the multi-trace operators ∼ Tr Φli Tr Φlj etc. which can be
analyzed with the help of Hubbard-Stratonovich transformation.
28
NIKITA NEKRASOV
3.7. Realizations of quiver theories.
3.7.1. Affine
quivers and McKay correspondence. For affine quivers γ, the choice of
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
gauge group Gg is characterized by a single integer N , for the equation βi = 0 for all
i ∈ Vertγ implies:
ni = N ai
(87)
where ai ≥ 1 solves
2ai =
X
as(e) +
e∈t −1 (i)
X
at(e)
e∈s−1 (i)
with the normalization, that for some 0 ∈ Vertγ , a0 = 1. It is well-known, that the
numbers ai = dimRi are the dimensions of the irreducible representations of some
finite subgroup Γ ∈ SU (2).
The Ar -type subgroup of SU (2) is Zr+1 , whose generator ΩAr acts on C2 via (cf.
(32)):
−1
(88)
ΩAr : (z1 , z2 ) 7→ ̟r+1 z1 , ̟r+1
z2 ,
so that Ωr+1
Ar = 1. The Dr -type subgroup of SU (2) (r ≥ 4) is the product Z2(r−2) ×Z2 Z4 ,
whose generators ΩDr and ΞDr act on C2 via:
−1
ΞDr : (z1 , z2 ) 7→ (z2 , −z1 )
z2 ,
(89)
ΩDr : (z1 , z2 ) 7→ ̟2(r−2) z1 , ̟2(r−2)
2
4
so that Ωr−2
Dr = ΞDr , ΞDr = 1.
The E6,7,8 -type subgroups SU (2) are the binary covers of the symmetry groups of the
three platonic solids (and their duals, see [85] for more details):
Fig.8
The quiver γ is associated to Γ as follows: the set Vertγ is identified with Γ∨ , the set
of irreducible representations of Γ, 0 ∈ Vertγ corresponds to the trivial representation
R0 = C1 . The set Edgesγ of edges is recovered from the tensor products as follows:
define the matrix A : Vertγ × Vertγ → Z≥0 by
M
(90)
Ri ⊗ S =
CAij ⊗ Rj
j∈Vertγ
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
29
where S ≈ C2 is the defining two dimensional representation of SU (2). The matrix A is
symmetric. There exists another matrix E : Vertγ × Vertγ → Z≥0 such that E + E t = A.
Then
G
(91)
Edgesγ =
[Ei,j ] × (i, j) s (k × (i, j)) = i, t (k × (i, j)) = j
(i,j)∈Vertγ ×Vertγ
The choice of E given A is the choice of the orientation of edges of γ. Note that this
b0 .
definition associates to Γ = 1 the quiver A
The N = 2 quiver four dimensional gauge theory corresponding to such quiver γ can
be described most simply by starting with the N = 4 super-Yang-Mills theory with the
gauge group U (N |Γ|), with the fields Aµ ∈ E ⊗ E ∗ , Ψα ∈ T ⊗ E ⊗ E ∗ , Φ ∈ Λ2 T ⊗ E ⊗ E ∗ ,
α = 1, 2, µ = 0, 1, 2, 3, with E = CN |Γ| the defining representation of U (N |Γ|) and T ≈ C4
the defining representation representation of the R-symmetry group SU (4). Now the
space of fields is endowed with the action of Γ:
M
CN ai ⊗ Ri
(92)
T = C2 ⊗ R0 ⊕ S,
E = CN ⊗ CΓ =
i∈Γ∨
One then defines the new theory by imposing the Γ-invariance constraint on the fields of
the original theory. The N = 4 supersymmetry reduces to N = 2, with U (N ai )-valued
vector multiplets Φi labelled by i ∈ Vertγ , and bi-fundamental hypermultiplets labelled
by e ∈ Edgesγ . The Lagrangian of the original N = 4 theory can be then deformed,
preserving the N = 2 supersymmetry. Since the gauge group U (N |Γ|) becomes the
product
(93)
U (N |Γ|) −→
U (N ai) ,
i
the gauge couplings τi can be chosen independently:
X
τ Tr N |Γ| F 2 −→
τi Tr N ai Fi2
i∈Vertγ
We have reviewed this well-known construction here because in what follows we shall
use its variants on several occasions.
3.7.2. Finite
quivers. Some of the finite quiver theories can be obtained as limits of
✿✿✿✿✿✿✿✿✿✿✿✿✿✿
the affine quiver theories. The rest is related to the ones we shall describe below by
analytic continuation, sometimes through a strong coupling region.
(1) The Ar type theory with n1 = . . . = nr = N , and m1 = mr = N , mi = 0 for 2 ≤ i ≤
br+1 theory, where one sends q0 → 0, qr+1 → 0. Then
r − 1, is the limit of the A
a0,α − m0 − ε, ar+1,α + mr become the masses of the fundamental hypermultiplets,
charged under U (n1 ) and U (nr ), respectively.
30
NIKITA NEKRASOV
Fig.9
b2
A1 theory as a limit of A
(2) A particular Dr type theory can be obtained by taking the limit q0 → 0 limit
br theory. The next-to-last node 2 with n2 = 2N has m2 = N .
of D
There are other ways of arriving at the quiver N = 2 theories corresponding to finite
quivers.
4. Integration over instanton moduli spaces
In this chapter we recall the mathematical definition of the instanton partition function Z inst of the bulk theory. In [103] we define the defect partition functions Ψinst .
We give the practical definition first, without actually describing the relevant instanton
moduli spaces. In [101] we describe the moduli spaces Mγ (n, k) whose contributions
dominate the gauge theory path integral, explicitly, via modified ADHM construction.
More precisely, the gauge theory path integral localizes to the integral of 1 over the
virtual fundamental cycle of degree (dimension) zero Mγ (n, k) which is represented, in
the perfect obstruction theory language of [11] by a smooth (super)-variety Mγ (n, k)c
(c stands for coarse) and H-equivariant vector bundle Obsγ → Mγ (n, k)c . The kinstanton contribution to the gauge theory partition function is the Euler class
Z
Z
inst
(94)
Zk =
1=
ǫ(Obsγ ) ,
Mγ (n,k)
Mγ (n,k)c
where we omitted the equivariant parameters.
We shall see that for the affine quiver theories (a representative example is the
N = 2∗ U (n) theory) the underlying variety Mγ (n, k)c is bosonic, while for the finite
quiver theories (a representative example is the U (n) theory with 2n fundamental
hypermultiplets) Mγ (n, k)c is a split super-manifold, a vector bundle over an ordinary
smooth variety with odd fibers.
The same pattern holds for the theories with defects.
4.1. Instanton partition function.
Vertγ
4.1.1. ✿✿✿✿✿
The ✿✿✿✿✿
bulk✿✿✿✿✿✿✿✿✿✿
partition ✿✿✿✿✿✿✿✿✿
function✿✿✿✿✿✿
Z inst . Let k = (ki )i∈Vertγ ∈ Z+
be the vector of
instanton charges for the gauge group Gg . We denote by Mγ (n, k) the moduli space
of framed quiver-graded torsion free sheaves E γ = (Ei )i∈Vertγ on CP2 . More precisely,
for each i ∈ Vertγ , Ei is a torsion free sheaf on CP2 = C2 ∪ CP1∞ , with the charge
ch2 (Ei ) = ki , and the framing at infinity:
(95)
∼
Ei |CP1∞ −→
Ni
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
Set theoretically,
Mγ (n, k)c =
(96)
31
M(ni , ki )
i∈Vertγ
is the product of ADHM moduli spaces of U (ni ) instantons of charge ki .
Let Ei be the universal i’th sheaf over Mγ (n, k)c × CP2 , and π : Mγ (n, k)c × CP2 −→
Mγ (n, k)c the projection onto the first factor. Define the obstruction sheaf Obsγ over
G
c
(97)
Mγ (n) =
Mγ (n, k)c
k
by:
(98)
Obsγ = Rπ∗
M
e∈Edgesγ
Hom(Es(e) , Et(e) ) ⊕
M
Hom(Ei , Mi )
i∈Vertγ
The sheaves above are all HC -equivariant, where H was defined in (76).
∼
The complexification of Gg acts on the isomorphisms Ei |CP1∞ −→
Ni , the complexification of Gf acts on the fibers of (98) in the natural way, the complexification GL(2, C)
of Grot acts by the symmetries of CP2 , with the fixed point 0 ∈ C2 = CP2 \CP1∞ . Let
TH ⊂ H, THC denote the maximal torus of H and its complexification, respectively.
The Coulomb moduli a belong to LieTGCg , the masses m belong to LieTGC . The Ωf
deformed theory has two additional complex parameters ε = (ε1 , ε2 ) which belong to
the Cartan subalgebra of Grot C , ε ∈ LieTGCrot ≈ C2 .
In [101] we shall discuss the modification of the ADHM construction [8] producing the
moduli spaces Mγ (n, k) (cf. [77], [88]) and the obstruction sheaf. More precisely, there
is a moduli space of solutions to a system of matrix equations, determined by the quiver
data, which depends on the choice of the Fayet-Illiopoulos (stability) parameters ζ~ ∈
RVertγ . It is the choice of these Fayet-Illiopoulos parameters which breaks the rotation
symmetry from Spin(4) down to Grot . When ζ~ is in certain chamber C ⊂ RVertγ the
space of solutions to this system of equations coincides with Mγ (n, k). The linearization
of the equations at the particular solution defines the obstruction sheaf, as the space
of solutions to the dual linear system.
The instanton factor in the partition function can be shown to reduce to the generating function of the equivariant integrals
Z
X
inst
k
(99)
Zγ (a; m; q; ε) =
q
ǫa;m;ε (Obsγ ) ,
k
with
qk =
Mγ (n,k)c
Y
k
qi i
i∈Vertγ
Mathematically (99) is just a definition of the left hand side. Each term of the qexpansion is a rational function on Lie(HC ), of negative degree of homogeneity for the
32
NIKITA NEKRASOV
asymptotically free theories, and degree zero (i.e. they are homogeneous functions) for
the asymptotically conformal theories.
4.1.2. ✿✿✿✿✿✿✿✿✿✿✿✿✿
Localization✿✿✿✿✿
and ✿✿✿✿✿✿
fixed ✿✿✿✿✿✿✿
points. The fixed points M(n, k)H of the TH -action on
M(n, k) are the sheaves which split as direct sums of monomial ideals:
(100)
TH
E ∈ Mγ (n, k)
⇔ Ei =
ni
M
α=1
Ii,α ,
where Ii,α = Iλ(i,α) , Thus, the set of fixed points Mγ (n, k)TH is in one-to-one correspondence with the set of quiver n-colored partitions:
ni
X
(i,α)
(i,α)
(i,α)
(101)
Eλ ↔ λ =
i
∈
Vert
,
α
∈
[n
],
λ
is
a
partition,
λ
|λ
|=
k
γ
i
i
α=1
These points are also the fixed points of the action of TH on the moduli space of Γinvariant instantons. The fixed point formula expresses the gauge theory path integral
as the sum over the set of quiver n-colored partitions. We shall present the explicit
formula in the next section.
Now that the path integration is reduced to a finite sum, the non-perturbative field
redefinitions involving adding a point-like instanton can be discussed rigorously.
4.2. Characters, tangent spaces. The contribution of a given fixed point to the partition function can be conveniently expressed using the characters of various vector
spaces involved in the local analysis of the path integral measure. The instanton partition function can be then written as:
X
(102)
Zγinst (a; m; ε; q) =
qλ µλ (a; m; ε)
λ
where
(103)
µλ (a; m; ε) = ǫa;m;ε −Tλ ,
and
α∈[ni ]
λ = λ(i,α)
,
(104)
i∈Vertγ
and
(105)
q
λ
=
ni
Y Y
i∈Vertγ α=1
|λ(i,α) |
qi
,
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
33
and
X
Ni Ki∗ + Ni∗ Ki q − PKi Ki∗
(106) Tλ =
i∈Vertγ
X
X
∗
∗
∗
.
−
Mi∗ Ki +
e βme Nt(e) Ks(e)
+ Ns(e)
Kt(e) q − PKt(e) Ks(e)
e∈Edgesγ
i∈Vertγ
In writing (106) we adopted a convention where the characters of the vector spaces are
denoted by the same letters as the vector spaces themselves. We are thus using the
notations (75) and
ni
X
X
βa
βc
e i,α
e
(107)
Ki =
α=1
∈λ(i,α)
In (107) we use the convention (60).
Note that the Eqs. (107) identify Ni , Ki , Mi with representations of TH . While Ni , Mi
are Weyl-invariant, and correspond to representations of H, the spaces Ki do not, in
general, carry a representation of H.
4.3. Integral representation.. The measure (102) can be also given an integral representation:
Y
Y
X qk I
inst
Υi
Υe ,
(108)
Zγ (a; m; q; ε) =
k! Γγ
i∈Vertγ
k
e∈Edgesγ
where
(109)
Υi =
^ni
α=1
ns(e)
Υe =
Y
α ′ =1
Y
dφ i,α Pi (φ i,α )
ε
1
,
ε1 ε2 Ai (φ i,α )Ai (φ i,α + ε) ′ ′′ S(φ i,α ′ − φ i,α ′′ )
α 6=α
ns(e) nt(e)
nt(e)
At(e) (φ
s(e),α ′
− me )
Y
α ′′ =1
t(e),α ′′
+ ε + me )
S(x) = 1 +
ε1 ε2
x(x + ε)
As(e) (φ
where
(110)
and the choice of the contour
(111)
Γγ ≈
Y Y
α ′ =1 α ′′ =1
S(φ t(e),α ′′ − φ s(e),α ′ + me )
R ni
i∈Vertγ
will be discussed elsewhere.
For generic values of the parameters ai , ε etc. the fixed point formula (102) can be
used. This is equivalent to the statement that (108) can be evaluated by computing
the residues at simple poles. The contributions of the particular toric instanton configuration λ (104) are rational functions with lots of poles. These poles lead to potential
34
NIKITA NEKRASOV
divergencies of the instanton partition function. For example, whenever the ratio (36)
is a positive rational number, b 2 ∈ Q+ some of the individual terms µλ (a; m; ε) blow
up. However the divergencies cancel between several terms. The contour integral representation is more convenient in this case, as it remains finite, as long as the contour
Γγ does not get pinched between two approaching poles.
Let us briefly explain the reason why these apparent poles occur, and why they
potentially cancel between themselves. A rational relation between the Coulomb parameters and the Ω-deformation parameters means that the symmetry group used in
the equivariant localization is strictly smaller then the maximal torus of H. Reduction
of the symmetry group means a potential enhancement of the fixed point locus. For
example, instead of a set of isolated points one may find a copy of CP1 or a more
complicated positive dimension subvariety. Each component of the fixed point locus
contributes an integral to the instanton partition function. This contribution is finite
if the component is compact. In the extreme case a = ε = 0, the symmetry group is
trivial. The fixed point locus in this case is the whole original moduli space Mγ (n, k),
and the integral diverges.
4.4. Full partition functions. The full partition functions are the products of the instanton partition functions and the tree and one-loop partition functions. They are
given by the product of (81), (82) and (94) leading to the following simple formulas
X
(112)
Zγ (a, m, ε; q) =
Q(Tλ )ǫa,m,ε (−T [λ])
λ
where
(113)
X
1
(Mi − Si [λ]) Si∗ [λ] +
T [λ] =
−βε
−βε
1
2
(1 − e
)(1 − e
)
i∈Vertγ
and
(114)
Q(T [λ]) =
Y
− ε 1ε ch2 (Si [λ])
qi
1 2
i∈Vertγ
X
e∈Edgesγ
∗
βme
e St(e) [λ]Ss(e) [λ]
Y − 1 Pni a2
α=1 i,α
2ε ε
× qλ
=
qi 1 2
i∈Vertγ
where
n
(115)
i
1X
ch2 (Si [λ]) =
a2i,α − ε1 ε2 ki [λ]
2
α=1
5. The Y − observables
The measures (102) µλ (a, m; ε) define the complexified statistical models which can
be studied without a reference to the original gauge theory. To any function F = F[λ]
on the space of quiver n-colored partitions one associates its normalized expectation
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
35
value:
(116)
h F iγ =
1 X
Zγinst
qλ µλ (a, m; ε) F[λ]
λ
Sometimes we shall also use the un-normalized expectation value
X
1−loop
qλ µλ (a, m; ε) F[λ] = Zγ h F iγ
(117)
⟪F⟫γ = Zγtree Zγ
λ
Sometimes, in what follows we shall view such a function F as an operator, acting in the
infinite-dimensional vector space H with the basis eλ labelled by the quiver n-colored
partitions,
(118)
Feλ = F[λ]eλ
The rôle of H in gauge theory will be discussed elsewhere.
The functions F which do come from gauge theory will be called observables. An
example of observable is the i’th instanton charge:
(119)
ki [λ] =
ni
X
λ(i,α)
α=1
5.1. The bulk Y-observables. The important observables are the characteristic polynomials of the adjoint Higgs fields:
(120)
Yi (x) = xni exp
∞
X
l=1
−
1
Tr (Φi |0 )l
l
lx
Here we denote by Φi |0 the lowest component of the vector multiplet Φi corresponding
to the node i ∈ Vertγ , evaluated at the specific point 0 ∈ C2 in the Euclidean spacetime. This is the fixed point of the rotational symmetry Spin(4)N of which the maximal
torus U (1) × U (1) is generated by the rotations in the two orthogonal two-planes.
In the N = 2 theory the gauge-invariant polynomials of the scalar components of
the vector multiplets, i.e.
(121)
Ol (x) = Tr Φli (x) ,
for x ∈ R4 are invariant under some supersymmetry transformations, which are nilpotent on the physical states. Moreover, the x-variation of such operators is in itself a
supersymmetry variation. Therefore, in the cohomology of such a supercharge, the observable Ol (x) is x-independent. The supersymmetry of the Ω-deformed N = 2 gauge
theory is such that the operator Ol (x) is invariant only at x = 0, i.e. at the fixed point
of the rotations.
Classically, i.e. for the ordinary matrix-valued function Φi (x) the exponential (120)
evaluates to the characteristic polynomial of this matrix (cf. (74)):
(122)
Yi (x)tree = Detni (x − Φi |0 ) = Ai (x)
36
NIKITA NEKRASOV
5.1.1. Y-observables
from sheaves. Mathematically Yi (x) is defined using the virtual
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
Chern polynomials of the universal sheaves, localized at the point 0 ∈ C2 :
h
i
(123)
Yi (x) = cx (Rπ∗ Ei → Ei ⊗ TP2 → Ei ⊗ ∧2 TP2 )
Here we used the Koszul resolution of the skyscrape sheaf S0 supported at 0 ∈ C2 :
(124)
0 → ∧2 TP∗2 → TP∗2 → OP2 → S0
where the second and the third maps are the contraction with the Euler vector field
z 1 ∂z 1 + z 2 ∂z 2 .
5.1.2. Y-observables
from noncommutative gauge fields. The proper physical definition
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
of the observable (120) is also subtler then the naive expression (122). In computing the
instanton partition function one uses the non-commutative deformation of the gauge
theory, in order to make the instanton moduli space smooth with isolated fixed points
[88]. In the noncommutative world, the notion of a particular point x = 0 in the spacetime R4θ makes no sense. The gauge fields and the adjoint scalar Φi are the operators
in the Hilbert space H,
M
H=
Ni ⊗ H
i∈Vertγ
where H is the 2-oscillator Fock space representation of the algebra of functions on
xµ , µ = 1, . . . , 4, obeying the Heisenberg algebra [b
xµ ,b
xν ] = iϑ µν , with
R4θ , generated by b
constant antisymmetric (non-degenerate) matrix θ:
µ
Ai,µ (x) 7→ Xi = b
xµ + ϑ µν Ai,ν (b
x)
where hµν
1
µ
Φi 7→ Φi = hµν Xi Xνi + φ i (b
x)
2
is the symmetric matrix, obeying
hµν ϑ να = Ωαµ
with Ω being the matrix of infinitesimal rotation of R4 , preserving both the metric
and ϑ. In the vacuum
n2 )
n 1 + ε2 b
Φi = diag(ai,1 , . . . , ai,ni ) ⊗ 1H + 1Ni ⊗ (ε1b
where b
nξ = a+ξ aξ are the oscillator number operators in H, ξ = 1, 2. The observables
like (121) are defined, cf. [84], as the ratio of infinite dimensional determinants,
(125)
Yi (x) =
DetH (x − Φi ) DetH (x − Φi − ε1 − ε2 )
DetH (x − Φi − ε1 ) DetH (x − Φi − ε2 )
or, equivalently, via a limiting procedure involving the regularized traces
TrH e −tΦi
partly explaining the non-triviality of what follows. Without going into detail, let us
quote the results which we shall need in this paper.
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
37
5.1.3. Y-observables
from Chern classes. Another, equivalent, definition of Yi (x) is the
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
following. We have the vector bundles Ni , Ki , i ∈ Vertγ , over Mγ (n, k). Topologically
Ni are trivial bundles, while Ki are, in general, not. These bundles are H-equivariant.
Then:
ǫx−ε1 (Ki∗ )ǫx−ε2 (Ki∗ )
(126)
Yi (x) = ǫx (Ni∗ )
ǫx (Ki∗ )ǫx−ε (Ki∗ )
5.1.4. Y-observables
on toric instantons. For our calculations, we need the fixed point
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
expression, i.e. the value Yi (x)[λ] of the observable Yi (x) on the special instanton
configuration Eλ :
ni
Y x − a − c − ε x − a − c − ε
Y
i,α
2
i,α
1
=
Yi (x)[λ] =
x − ai,α
x
−
a
−
c
−
ε
x
−
a
−
c
i,α
i,α
α=1
∈λ(i,α)
Q
(127)
(x − ai,α − c )
n
i
Y
∈∂+ λ(i,α)
Q
=
(x − ai,α − ε − c )
α=1
∈∂− λ(i,α)
where for a monomial ideal Iλ , corresponding to the partition λ the outer boundary
∂+ λ and the inner boundary ∂− λ are the monomials corresponding to the generators,
and the relations (divided by the factor z1 z2 ) of the ideal, cf. Fig.10. Explicitly, given
the character χλ of the quotient C[z1 , z2 ]/Iλ which is the same thing as the character
of the partition λ, the contents of the inner and the outer boundaries can be read off
the character of the tautological sheaf:
X
X
(128)
Sλ = 1 − Pχλ =
e βc − q
e βc
∈∂+ λ
∈∂− λ
Note that
q
q − qTr Sλ− b
Sλ = Tr Sλ+ b
(129)
where Sλ± are the fibers over Iλ ∈ Hilb[|λ|](C2 ) of the vector bundles S ± which we study
in more detail in [101]. It is easy to see from the picture of the Young diagram λ, to
which stratum HMk,l ⊂ Hilb[|λ|] (C2 ) it belongs:
(130)
λ ∈ HMk,l
⇔
k = |λ|,
l = ℓ(λ) = #∂− λ = #∂+ λ − 1 .
38
NIKITA NEKRASOV
Fig.10
Generators and relations of a monomial ideal Iλ
The character of the tautological sheaf Sλ = 1 − Pχλ
The Y-observable Yi (x) is essentially the character of the localized tautological complex Si , which is the cohomology (along CP2 ) of the complex
(131)
Si = Ei → Ei ⊗ TP2 → Ei ⊗ ∧2 TP2 [−1]
which is the dual Koszul complex tensored with the universal sheaf Ei . The relevant
character Si is easy to calculate:
(132)
Si = Ni − PKi =
ni
X
e βai,α Sλi,α
α=1
The previous formulae can be succinctly written as:
(133)
Yi (x)[λ] = β −ni ǫ[e βx Si∗ ]
or, in more detail, cf. the notation (258)
(134)
Yi (x)[λ] = xni
∞
X 1
l
exp −
[β
]S
i
lxl
l=1
For large x the observable Yi (x) can be expanded as:
ε1 ε2
(135)
Yi (x) = Ai (x) 1 + 2 ki + . . .
x
5.1.5. ✿✿✿✿✿
The ✿✿✿✿✿✿✿✿✿✿✿✿
importance ✿✿✿
of ✿✿✿✿✿✿✿✿✿✿✿✿✿✿
Y-observables. The observables Yi (x) and the characters Si
are used in the analysis of the non-perturbative Schwinger-Dyson equations. The large
field redefinitions (17) we shall employ involve adding a point-like instanton at the
i’th gauge factor, or, conversely, removing a point-like instanton of the i’th type. This
λ, with
transformation maps one allowed quiver n-colored partition λ to another one e
modified instanton charge
(136)
kj [e
λ] = kj [λ] ± δi,j .
An inspection of the picture Fig.6 easily shows that the modifications of the indicated
′
type consist of either adding a box ∈ ∂+ λ(i,α ) for some α ′ = 1, . . . , ni , or removing a
′′
box ∈ ∂− λ(i,α ) for some α ′′ = 1, . . . , ni . In other words, the allowed modifications of
λ at the i’th node correspond to the zeroes and poles of Yi (x)[λ].
The measures µλ (a; m; ε) and µe
λ (a; m; ε) are related to each other in a simple manner. Indeed, the character Tλ is quadratic in Ki , more precisely, it is sesquilinear. The
∗
variation Te
λ − Tλ is, therefore, linear in Ki and Ki . In fact, it is linear in Sj ’s and
S ∗ ’s. For the modification λ → e
λ consisting of adding a box ∈ ∂+ λ(i,α) for some
j
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
39
α = 1, . . . , ni :
−1
+ Si∗ [e
λ]qξ − Mi∗ξ−
(137) Te
λ − Tλ = Si [λ]ξ
X
X
∗
βme
Ss(e) [λ]ξqe −
e βme ξ −1 St(e) [λ] +
e∈t −1 (i)
e∈s−1 (i)
X
e βme P
e∈s−1 (i)∩t −1 (i)
where
ξ = e β(ai,α +c )
Kj [e
λ] = Kj [λ] + δi,j ξ,
(138)
The ratio of the measures can be, therefore, expressed as a product of the values and
residues of various functions Yj (x)[λ] in the variable x, for example, as
(139)
µe
λ (a; m; ε)
ε
Pi (x)
= (−1)κi qi
×
µλ (a; m; ε)
ε1 ε2 Yi (x + ε)[λ]Y′i (x)[λ]
Y
Y
Yt(e) (x − me )[λ]×
Ys(e) (x + ε + me )[λ]
e∈t −1 (i)
Y
e∈s−1 (i)
e∈s−1 (i)∩t −1 (i)
(me + ε1 )(me + ε2 )
me (me + ε)
x = ai,α + c
where
κi = ni − 1 +
(140)
X
nt(e)
e∈s−1 (i)
Note the identity:
(141)
λ] =
resx=ai,α +c Yi (x + ε)[e
ε1 ε2
Yi (ai,α + c + ε)[λ]
ε
5.2. Q-observables. The inspection of the Eq. (127) shows that Yi (x)[λ] can be represented as a ratio of two entire functions, in two ways:
(1,2)
(142)
where
(143)
Yi (x) =
Qi
(x)
(1,2)
Qi (x − ε2 )
(2,1)
=
Qi
(x)
(2,1)
Qi (x − ε1 )
x−ai,α
ni
(−ε ) εb
Y
Y
x − ai,α − c − εa
(a,b)
b
Qi (x)[λ] =
,
x−a
Γ − i,α
x
−
a
−
c
i,α
α=1
εb
∈λ(i,α)
,
(a, b) = (1, 2) or (2, 1)
The rôle of these observables will be revealed in [100], [103]. In the limit ε2 → 0 with
(2,1)
ε1 -fixed the observables Qi
tend to the so-called Baxter operators of the quantum
integrable system, which is Bethe/gauge-dual [97] to the gauge theory under consideration [86].
40
NIKITA NEKRASOV
6. Enter the qq − characters
Remarkably, the Dyson-Schwinger relations based on (139) can be summarized in
the following proposition:
6.1. The main theorem. For any γ-graded vector space
M
(144)
W=
Wi ,
i∈Vertγ
Vertγ
with the corresponding dimension vector w ∈ Z≥0 , Wi = Cwi , and a choice of ℓ
weights ν = (νi )i∈Vertγ , νi = diag νi,1 , . . . , νi,wi ∈ End(Wi ), there is a Laurent polynomial (Laurent power series for affine γ) in Yj (x + ξj,κ )’s , i.e. in Yj ’s with possibly
shifted arguments, including the nilpotent shifts (i.e. a finite number of derivatives in
x applied to Yj )
(145)
Xw,ν (Y(x + . . .)) =
wi
Y Y
Yi (x + νi,l + ε) + O(q)
i∈Vertγ l=1
such that its expectation value in the γ-quiver gauge theory:
E
D
1 X
≡ inst
Xw,ν (Y[λ]) qλ µλ (a; m; ε) = Tw,ν (x),
(146)
Xw,ν (Y)
γ
Zγ
λ
is a polynomial in x. More specifically, Tw,ν (x) is a polynomial in x of degree
X
(147)
deg Tw,ν (x) = w · n =
wi n i .
i∈Vertγ
We call Xw,ν (x) the Yangian qq-character of Y (gγ ). For w = (δj,i )j∈Vertγ and ν = 0 the
corresponding qq-character will be denoted by χi (x), the i’th fundamental qq-character.
Remark. The qq-characters are the gauge theory generalizations of the matrix model
expression T(x) (11).
In the limit ε2 → 0 Xw,ν (x) reduces to the Yangian q-characters of finite-dimensional
representations of the Yangian Y (gγ ), constructed for finite γ in [60]. In [38] the qcharacters for the quantum affine algebras Uq (gγ ) for finite γ’s and in [49] for affine
γ’s are constructed. These correspond to the K-theoretic version of our story in the
limit q2 → 1, q1 = q finite, which was discussed in [86].
The K-theoretic version of our story with general (q1 , q2 ) produces the qq-characters,
corresponding to Uq (gγ ). The physical applications of the qq-characters are the five
dimensional supersymmetric gauge theories compactified on a circle [82]. We shall give
the definition and the formulae below, without going into much detail.
7. Examples of qq − characters
In this section we prepare the reader by giving a few explicit examples of the qqcharacters, before unveiling the general formula in the next section.
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
41
7.1. A-type theories: one factor gauge group. Let us start with a couple of examples
b0
for the theories with a single factor gauge group, i.e. either the A1 theory or the A
theory.
case. The A1 theory is the U (n) gauge theory with Nf = 2n fundamental
7.1.1. ✿✿✿✿
The✿✿✿✿
A1 ✿✿✿✿✿
hypermultiplets. The theory is characterized by the gauge coupling q and 2n masses
m = (m1 , . . . , m2n ), which are encoded in the polynomial
P(x) =
2n
Y
f=1
(x − mf )
Since the quiver consists of a single vertex, we omit the subscript i in Y(x) and P(x).
The fundamental A1 qq-character is equal to
X1,0 (x) = Y(x + ε) + q
(148)
P(x)
Y(x)
The general A1 qq-character depends on a w-tuple ν of complex numbers, ν =
(ν1 , . . . , νw ) ∈ Cw . It is given by:
(149)
Xw,ν (x) =
X
q|J|
[w]=I ⊔J
Y
i∈I ,j∈J
S(νi − νj )
Y P(x + νj ) Y
j∈J
Y(x + νj )
Y(x + ε + νi )
i∈I
It has potential poles in ν’s, when νi = νj or νi = νj + ε, for i 6= j.
The expression (149) is actually non-singular at the diagonals νi = νj . The limit
contains, however, the derivatives ∂x Y. For example, for w = 2, ν1 = ν2 = 0 the qqcharacter is equal to:
P(x)
ε ε
(150) X2,(0,0) (x) = Y(x + ε) 1 − q 1 2 ∂x
ε
Y(x)Y(x + ε)
2
!!
+
Y(x + ε)
P(x)2
ε1 ε2
+ 2qP(x)
1 − 2 + q2
Y(x)
ε
Y(x)2
The expression (149) has a first order pole at the hypersurfaces where νi = νj + ε for
some pair i 6= j. The residue of Xw,ν is equal to the qq-character Xw−2,ν\{νi ,νj } , times
the polynomial in x factor
Y
(151)
S(νk − νj )P(x + νk ) .
k6=i,j
The finite part Xfin
w,ν of the expansion of Xw,ν in νi near νi = νj + ε is the properly
defined qq-character for the arrangement of weights ν landing on the hypersurface
42
NIKITA NEKRASOV
νi = νj + ε. It involves the terms with the derivative ∂x Y. For example
(152) Xfin
2,(−ε,0) = Y(x + ε)Y(x)+
!
Y(x + ε)
ε1 ε2
ε1 ε2 ∂x Y(x)
+ q 1 + 2 P(x − ε)
+
+ qP(x) 1 −
Y(x − ε)
ε Y(x)
2ε
+ q2
P(x)P(x − ε)
Y(x)Y(x − ε)
b0 theory. The A
b0 theory (also known as the N = 2∗ theory) is character7.1.2. ✿✿✿✿
The✿✿✿✿
A
✿✿✿✿✿✿✿✿
ized by one mass parameter m, the mass of the adjoint hypermultiplet, and the gauge
coupling q.
Here we give the expression for the fundamental character X1 (x) ≡ X1,0 (x):
Q
X
Y
∈∂ λ Y(x + σ + ε)
|λ|
=
S(mh + εa ) · Q +
(153) X1 (x) =
q
Y(x
+
σ
)
∈∂
λ
−
λ
∈λ
X
Y
Y Y(x + σ − m)Y(x + σ + m + ε)
= Y(x + ε)
q|λ|
S(mh + εa ) ·
=
Y(x + σ )Y(x + σ + ε)
λ
∈λ
∈λ
= Y(x + ε) + q S(m)
Y (x − m) Y (x + ε + m)
+ ...
Y (x)
Here
(154)
σ = m(i − j) + ε(1 − j)
is the content of defined relative to the pair of weights (m, −m−ε). It is not too difficult
b0 qq-character Xw,ν , in terms of an infinite sum
to write an expression for the general A
over the w-tuples of partitions, but we feel it is not very illuminating.
Note that the expression (153) has apparent singularities when m and −(m + ε) are in
a positive congruence, i.e. if
(155)
m(p − q) = εq
for some positive integers p, q > 0. In fact, the limit of the expression (153) is finite,
but it involves not only the ratios of shifted Y’s, but also its derivatives. The most
efficient way to study this asymptotics is to use the geometric expression to be discussed
below. The geometric expression also leads to the contour integral representation of
the qq-characters.
7.2. A-type theories: linear quiver theories. Let us now present the formulas for the
general Ar theories, assuming
(156)
m1 = mr = n1 = . . . = nr = N , m2 = m3 = . . . = mr−1 = 0 .
We treat the general Ar case mi = 2ni − ni−1 − ni+1 , n0 = nr+1 = 0 in the section below.
We have r observables Yi (x), and couplings qi , for i = 1, . . . , r. Define r + 1 complex
numbers zi , i = 0, 1, . . . , r by (??):
(157)
z i = z 0 q1 . . . qi ,
i = 1, . . . , r
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
43
and define r + 1 functions Λi (x), i = 0, . . . , r, by:
Y (x + ε)
(158)
Λi (x) = zi i+1
,
Yi (x)
where we set Y0 (x) = P1 (x), Yr+1 (x) = Pr (x), in other words the masses m1,f , mr,f of
fundamentals are denoted as a0,f , ar+1,f , respectively. We also choose the normalization
me = −ε.
7.2.1. The
height functions. For a finite set I ⊂ R, we define the height function:
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
(159)
hI : I → [p],
p = |I|≡ #I ,
hI (i) = # { i ′ | i ′ ∈ I, i ′ < i }
In other words, for I = {i1 , . . . , ip } with i1 < i2 < . . . < ip , hib = b − 1, 1 ≤ b ≤ p.
7.2.2. Pre-character.
Define the l’th fundamental qq pre-character by (cf. (159)):
✿✿✿✿✿✿✿✿✿✿✿✿✿
X Y
(160)
χl (x) =
Λi (x + (hI (i) + 1 − l) ε)
I ⊂[0,r], |I |=l i∈I
7.2.3. Fundamental
qq-character of type Ar . Then l’th fundamental qq-character is
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
given by the properly normalized χl (x) :
χl (x)
Y (x)Yl+1 (x + ε)
(161)
Xl (x) = Y0 (x + (1 − l) ε)
= Yl (x + ε) + ql l−1
+ ...
z0 z1 . . . zl−1
Yl (x)
to the q-characters of E. Frenkel and N. Reshetikhin. Our formula
7.2.4. Comparison
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
(160) looks similar to the formula in the section 11.1 of [37] (adapted to the Yangian
Y (slr+1 ) , of course). The similarity is a little bit misleading, as shows the example of
the D-type theories.
7.2.5. The
main property of the qq-characters of the A-type. The relations (139) suffice
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
to prove that the expectation values of the qq-characters of the A-type theories do not
have singularities as the functions of x. It suffices to check the cancellation of residues
between the poles related by adding or removing one square in one of the Young
diagrams in λ. It would take more elaborate arguments to prove the analogous claim
for all N = 2 theories.
44
NIKITA NEKRASOV
7.3. The D-type theories.
7.3.1. ✿✿✿✿✿
The ✿✿✿
D4✿✿✿✿✿✿✿✿
theory. This is the theory with four gauge groups. The quiver is the
graph with Vertγ = {1, 2, 3, 4}, and Edgesγ = {1, 3, 4} with s(e) = e, t(e) = 2 for all
e ∈ Edgesγ . The graph has the obvious S3 symmetry, permuting the vertices 1, 3, 4.
We present the formula for the asymptotically conformal theory with n1 = n3 = n4 =
N = m2 , n2 = 2N , m1 = m3 = m4 = 0, leaving an obvious extension to the general case
to the interested reader as an exercise in the deciphering the general formula of the
next section (essentially one replaces qi 7→ qi Pi (x + aε) for i = 1, 3, 4 with some integers
a).
In what follows we use the short-hand notation:
Yi,a = Yi (x + aε), Yi = Yi (x), Pa = P2 (x + aε), P = P2 (x)
(162)
There are four fundamental qq-characters, three of which are permuted by the S3 .
The qq-character X1,0 is given by the sum of 8 terms (as would be the case for the
characters of 8v , 8s , 8c of vector or spinor representations of Spin(8)):
−1
(163) X1,0 = Y1,1 + q1 Y1−1 Y2 + q1 q2 P−1 Y2,−1
Y3 Y4 +
−1
−1
−1
−1
+ q1 q2 q3 P−1 Y3,−1
Y4 + q1 q2 q4 P−1 Y4,−1
Y3 + q1 q2 q3 q4 P−1 Y3,−1
Y4,−1
Y2,−1 +
−1
−1
+ q1 q22 q3 q4 P−1 P−2 Y2,−2
Y1,−1 + q21 q22 q3 q4 P−1 P−2 Y1,−2
The formulae for X3,0 , X4,0 are obtained by the cyclic permutation of the indices 1, 3, 4.
The qq-character X2,0 reveals a surprising structure,
X2,0 = X+2 + q1 q22 q3 q4 PP−1 X−2
(164)
and contains the derivatives of Yi ’s. Explicitly,
(165)
X+2 = Y2,1 +q2 PY2−1 Y1,1 Y3,1 Y4,1 +q1 q2 PY1−1 Y3,1 Y4,1 +q3 q2 PY3−1 Y1,1 Y4,1 +q4 q2 PY4−1 Y1,1 Y3,1 +
q1 q2 q3 PY1−1 Y3−1 Y2 Y4,1 + q1 q2 q4 PY1−1 Y4−1 Y2 Y3,1 + q2 q3 q4 PY3−1 Y4−1 Y2 Y1,1 +
−1
−1
−1
+ q1 q22 q3 PP−1 Y2,−1
Y4 Y4,−1 + q1 q22 q4 PP−1 Y2,−1
Y3 Y3,−1 + q3 q22 q4 PP−1 Y2,−1
Y1 Y1,−1 +
+ q1 q2 q3 q4 PY1−1 Y3−1 Y4−1 Y22
−−
X−2 = X−,0
2 + X2 ,
(166)
(167)
X−,0
2
!!
Y2 Y2,−1
Y2
ε1 ε2
ε1 ε2
=
2 1− 2 +
+
∂ log
Y2,−1
ε x
P−1 Y1 Y3 Y4
ε
ε ε −1
−1
−1
Y1,1 + Y3,−1
Y3,1 + Y4,−1
Y4,1
+ 1 + 1 22 Y1,−1
2ε
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
45
−1 −1
−1 −1
−2
−1 −1
(168) X−−
2 = q4 Y4 Y4,−1 Y2 + q3 Y3 Y3,−1 Y2 + q1 Y1 Y1,−1 Y2 + q2 P−1 Y2,−1 Y1 Y3 Y4 +
−1
−1
−1
−1
−1
−1
q2 q4 P−1 Y2,−1
Y4,−1
Y1 Y3 + q2 q3 P−1 Y2,−1
Y3,−1
Y1 Y4 + q2 q1 P−1 Y2,−1
Y1,−1
Y3 Y4 +
−1
−1
−1
−1
−1
−1
+ q1 q2 q4 P−1 Y1,−1
Y4,−1
Y3 + q2 q3 q4 P−1 Y3,−1
Y4,−1
Y1 + q1 q2 q3 P−1 Y1,−1
Y3,−1
Y4 +
−1
−1
−1
Y2,−1 +
+ q1 q2 q3 q4 P−1 Y1,−1
Y3,−1
Y4,−1
−1
+ q1 q22 q3 q4 P−1 P−2 Y2,−2
8. The qq − character formula
In order to write the general formula for the qq-character we shall use an auxiliary
geometric object, the quiver variety M(w, v), which we presently define.
8.1. Nakajima quiver variety. Given a quiver γ, two dimension vectors
(169)
Vertγ
v = (vi )i∈Vertγ , w = (wi )i∈Vertγ ∈ Z≥0
and a choice of stability parameter ζ ∈ RVertγ H. Nakajima [76] defines the quiver
variety as the hyperkahler quotient:
(170)
−1
Mγ,ζ (w, v) = µ−1
C (0) ∩ µR (ζ)/Gv
where
Gv =
(171)
U (Vi ) ,
i∈Vertγ
and µC , µR are the quadratic maps Hγ → LieGv∗ ⊗ C, R, respectively, with Hγ the
vector space
M
M
∗
Hom(Vi , Wi ) ⊕
Hom(Vs(e) , Vt(e) )
(172)
Hγ = T
e∈Edgesγ
i∈Vertγ
of linear operators (matrices) (I˜i , J˜i , Be,± ):
(173)
I˜i : Wi → Vi , J˜i : Vi → Wi
Be,+ : Vs(e) → Vt(e) , Be,− : Vt(e) → Vs(e)
C
Explicitly: µR,C = (µR
i , µi )i∈Vertγ , with
X
X
˜i J˜i +
µC
=
I
B
B
−
Be,+ Be,−
e,−
e,+
i
e∈s−1 (i)
(174)
˜ ˜† ˜† ˜
µR
i = Ii Ii − J i J i +
+
X
e∈t −1 (i)
e∈s−1 (i)
X
e∈t −1 (i)
Be,− B†e,− − B†e,+ Be,+ +
Be,+ B†e,+ − B†e,− Be,−
46
NIKITA NEKRASOV
The definition (170) translates to the set of equations:
µR
i = ζi 1Vi ,
(175)
µC
i = 0,
i ∈ Vertγ
with the identification of solutions related by the Gv transformations:
(176)
−1
˜ ˜ −1
(Be,+ , Be,− , I˜i , J˜i ) 7→ (ht(e) Be,+ h−1
s(e) , hs(e) Be,− ht(e) , hi Ii , Ji hi ),
hi ∈ U (Vi )
8.1.1. Stability
parameters. Solving the real moment map equations (the first line in
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
the Eq. (175)) and dividing by Gv can be replaced by dividing the set of stable solutions
to the complex moment map equations (the second line in the Eq. (175)) by the action
of the complexified group
GvC = ×i GL(Vi ) .
(177)
The notion of stability depends on the choice of ζ. In this paper we assume ζi > 0 for
all i ∈ Vertγ . The solution (Be,± , I˜i , J˜i ) is stable iff any collection (Vi′ )i∈Vertγ of subspaces
Vi′ ⊂ Vi , such that
(1)
(178)
(2)
I˜i (Wi ) ⊂ Vi′ for all i ∈ Vertγ
and
′
′
′
′
Bp (Vs(p)
) ⊂ Vt(p)
, and B̃p (Vt(p)
) ⊂ Vs(p)
for all p ∈ Pathsγ
is such that Vi′ = Vi for all i ∈ Vertγ . Here p ∈ Pathsγ denotes a sequence (ei , σi ) ∈
Arrowsγ , i = 1, . . . , m, such that s̄(e1 , σ1 ) = s(p), t¯(e1 , σ1 ) = s̄(e2 , σ2 ), . . ., t¯(ei , σi ) =
¯ m , σm ) = t(p).
s̄(ei+1 , σi+1 ), . . ., t(e
The proof is simple. Let Pi be the orthogonal projection of Vi onto the orthogonal
complement (Vi′ )⊥ of the “invariant” subspace Vi′ ⊂ Vi . We have Pi I˜i = 0 and Pt(e) Be,+ (1−
Ps(e) ) = 0, Ps(e) Be,− (1 − Pt(e) ) = 0. Now compute
X
X
⊥ X
(179) 0 ≤
ζi dim Vi′ =
Tr Pi µR
P
=
−
Tr Pi J˜i† J˜i Pi +
i i
X
e
i
i
i
Tr Ps(e) Be,− B†e,− Ps(e) + Tr Pt(e) Be,+ B†e,+ Pt(e) − Tr Pt(e) B†e,− Be,− Pt(e) − Tr Ps(e) B†e,+ Be,+ Ps(e) =
−
X
X
kJ˜i Pi k2 −
k(1 − Ps(e) )Be,− Pt(e) k2 +k(1 − Pt(e) )Be,+ Ps(e) k2 ≤ 0
i
e
which implies Vi′ = Vi for all i. The stability condition is equivalent to the condition
that the path operators Bp and B̃p for p ∈ Pathsγ acting on I˜i′ (Wi′ ) generate Vi′′ :
X
X
(180)
Vi =
CBp I˜s(p) (Ws(p) ) +
CB̃p I˜t(p) (Wt(p) )
p∈t −1 (i)
p∈s−1 (i)
Conversely, in order to establish that the stability condition implies that the GvC -orbit
of (Be,± , I˜i , J˜i ) solving the µC = 0 equations in (175) passes through the solution of
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
47
the µR = ζ equations in (175), we use the standard method: consider the Morse-Bott
function
X
2
(181)
f =
kµR
i − ζi 1Vi k
i∈Vertγ
The trajectory of the gradient flow
d
(B , I˜ , J˜ ) = −(∇B†e,± f , ∇I˜† f , ∇J˜† f )
(182)
i
i
dt e,± i i
C
belongs to the Gv -orbit. Indeed, (182) exponentiates to the transformation:
exp tµi ∈ GL(Vi )
(183)
The function f decreases along the flow. In the limit t → ∞ the value of f either tends
to its absolute minimum, i.e. f = 0, which is the locus of solutions to the equations
(175), or it stops at another critical point with the critical value f ∗ > 0. Now, the
critical points with f ∗ > 0 are the configurations (Be,± , I˜i , J˜i ) for which the real moment
map (µi )i∈Vertγ viewed as an element of the Lie algebra of GvC (more precisely, it is in
i LieGv ⊂ LieGvC ), is a non-trivial infinitesimal symmetry, i.e.:
µi I˜i = ζi I˜i ,
J˜i µi = J˜i ζi
(184) µs(e) Be,− − Be,− µt(e) = (ζs(e) − ζt(e) )Be,− ,
µt(e) Be,+ − Be,+ µs(e) = (ζt(e) − ζs(e) )Be,+
Define Vi′ = ker µi − ζi 1Vi ⊂ Vi for all i ∈ Vertγ . By (184) these subspaces obey all
the conditions of (178), therefore µi = ζi 1Vi for all i ∈ Vertγ .
In what follows we omit the subscripts γ and ζ in the notations for the quiver variety:
Mγ,ζ (w, v)
−→
M(w, v)
8.1.2. ✿✿✿✿✿✿✿✿✿✿✿✿✿
Symmetries ✿✿✿
of ✿✿✿✿✿✿✿✿✿
M(w, v). The group Hw = Gw × U (1)b∗ (γ) acts on M(w, v) by
isometries. Here
(185)
Gw =
U (Wi )
i∈Vertγ
˜ J˜ maps:
acts on the I,
(186)
(gi )i∈Vertγ : (I˜i , J˜i )i∈Vertγ 7→ (I˜i gi , gi−1 J˜i )i∈Vertγ
The U (1)b0 (γ) = U (1)-factor acts by rotating all of the I˜i , Be,− ’s while keeping J˜i , Be,+ ’s
intact (this definition can be extended to the disconnected quivers in a trivial fashion:
rotate I˜i , Be,− belonging to a given connected component):
(187)
(I˜i , J˜i , Be,+ , Be,− ) 7→ (u I˜i , J˜i , uBe,+ , Be,− )
The group U (1)b1 (γ) ≈ U (1)Edgesγ /U (1)Vertγ acts on the Gv -equivalence classes of the
(Be,± ) maps:
(188)
(ue )e∈Edgesγ : (Be,+ , Be,− )e∈Edgesγ 7→ (ue Be,+ , ue−1 Be,− )e∈Edgesγ
48
NIKITA NEKRASOV
so that the normal subgroup U (1)Vertγ acts by the Gv -transformations
(189)
−1
−1
ut(e) Be,+ , us(e) ut(e)
Be,− )e∈Edgesγ
(ui )i∈Vertγ : (Be,+ , Be,− )e∈Edgesγ 7→ (us(e)
8.1.3. ✿✿✿✿✿
The ✿✿✿✿✿✿✿✿✿✿
canonical✿✿✿✿✿✿✿✿✿✿✿
complexes✿✿✿✿✿
and✿✿✿✿✿✿✿✿✿
bundles. For each i ∈ Vertγ the vector space Vi
descends to M(w, v) as a vector bundle. In addition, there are also the canonical
complexes of bundles over M(w, v):
M
M
d2
d1
Wi
Vs(e)
(190)
Ci = 0 → Vi −→
Vt(e) −→
Vi → 0
−1
−1
e∈t
e∈s (i)
(i)
where the first and the second maps are given by:
M
M
M
M
(191)
d2 = J˜i
−Be,−
Be,+ ,
d1 = I˜i
Be,+
Be,−
e∈t −1 (i)
e∈s−1 (i)
e∈t −1 (i)
e∈s−1 (i)
The moment map equation (175), µC
i = 0, implies d1 ◦ d2 = 0, hence Ci is a complex.
We set the leftmost term Vi in (190) to be in degree zero.
8.2. The bi-observables. Let
(192)
Gx = e βx
X
i∈Vertγ
qSi∗ Ci + Mi∗ Vi
denote the Chern character of the Hw -equivariant complex of vector bundles over
M(n, k) × M(w, v):
M
βx
( q [ Si → Ci ] ⊕ [ Mi → Vi ] )
Gx = e Ch
i∈Vertγ
We identify the equivariant parameters of the Gw group with ν, the equivariant parameters for the U (1) factor with ε, the equivariant parameters for the U (1)Edgesγ -group
with me + ε. Explicitly, the equivariant Chern character of the complex Ci is equal to:
X
X
(193)
Ch Ci = Wi − Vi − q−1 Vi +
q−1 e −βme Vs(e) +
e βme Vt(e)
e∈t −1 (i)
e∈s−1 (i)
where Wi is a pure c-number character (the sum of exponents of ν-components), while
Vj ’s are the Chern characters of Hw -equivariant bundles, i.e. may have components of
positive degrees cohomology classes.
8.3. The formula. Finally, we can present the formula for Xw,ν . There are several
ways to write it.
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
8.3.1. ✿✿✿✿✿✿✿✿
Integral✿✿✿✿✿
over✿✿✿✿
the✿✿✿✿✿✿✿✿
quiver ✿✿✿✿✿✿✿
variety.
Z
X
v
(194)
Xw,ν =
q
M(w,v)
v
qv =
49
ǫε2 (T M(w, v))ǫx (G)
Y
v
qi i ,
i∈Vertγ
ǫx (G) is understood as the Hw -equivariant cohomology class of M(w, v): represent Ci
as the virtual bundle Ci+ − Ci− over M(w, v), where Ci+ , Ci− are the actual bundles, with
±
the formal Chern roots ξi,κ
. Then
±
Q
+
Y
(x
+
ξ
)
v
i
i
i,κ
Y κ
+ Y
Q+
P
(x
+
η
)
(195)
ǫx (G) =
i
i,κ
−
)
Yi (x + ξi,κ
−
i∈Vertγ
κ=1
κ−
where ηi,κ are the formal Chern roots of Vi .
b0 examples we considered so far the quiver varieties M(w, v) are the
For the A1 , A
cotangent bundle T ∗ Gr(v, w) to the Grassmanian of v-planes in Cw and the Hilbert
scheme Hilb[v] (C2 ) of v points on C2 , respectively.
8.3.2. ✿✿✿✿✿✿✿✿✿
Contour ✿✿✿✿✿✿✿✿
integral✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
representations. Equivalently, one can write a contour integral
representation for (194) which has the advantage of being explicit, albeit less concise:
!vi Y
I
wi
ε qi
1
Υw,ν,v (x)
(196) Xw,ν (x) =
Yi (x + ε + νi,j )
√
v!
Γ
w,ν,v
v i∈Vertγ i 2π −1 ε1 ε2
j=1
Y
Y
Υw,ν,v;i (x) ,
Υw,ν,v (x) =
Υw,ν,v;e (x)
X Y
e∈Edgesγ
Υw,ν,v;e (x) =
vt(e)
Y
i∈Vertγ
(t(e))
Ys(e) (x + ε + me + φ κ )
κ=1
Υw,ν,v;i (x) =
ℓ=1
(s(e))
Yt(e) (x − me + φ ℓ
),
wi
(i)
(i)
Y
Y
dφ κ Pi (x + φ κ )
(i)
(i)
(i)
)
S(φ
−
φ
S(φ
−
ν
)
.
κ
κ
i,j
ℓ
(i)
(i)
Y (x + ε + φ )Y (x + φ )
vi
Y
κ=1
vs(e)
Y
i
κ
κ
i
ℓ6=κ
j=1
The contour Γw,ν,v is chosen in such a fashion, so as to ignore the poles coming from
the zeroes of the Y-functions in the denominator of (196) or the poles of Y-functions in
the numerator there. Let us assume that νi,j are all real, and that ε1 , ε2 have positive
imaginary part. We also assume that zeroes and poles of Y(x + z) in z are far away
(i)
from the real axis. Then the contour Γw,ν,v ≈ R|v| , i.e. all φ κ are real. Now deform
νi,j and ε1 , ε2 to whatever values we desire, all the while deforming the contour Γw,ν,v
(i)
in a such a way, that the poles of (196) in φ κ ’s do not cross Γw,ν,v .
The technique to arrive from (196) to (194) is well-known, see, e.g. [74]
50
NIKITA NEKRASOV
8.4. Five dimensional theory. The gauge theories we studied so far in four dimensions
canonically lift to five dimensions, with the vector multiplets lifting to vector multiplets.
The complex scalars ai,α in the vector multiplet in four dimensions come from a real
scalar in five dimensions and the fifth component of the gauge field. Now we compactify
the theory on a circle of circumference β, and impose the twisted boundary conditions,
rotating the space N by the angles (−iβε1 , −iβε2 ) in the two orthogonal two-planes R2
in N = R4 . In addition we perform the SU (2) R-symmetry rotation
exp
iβε
σ
2 3
and the constant gauge transformation
e βai = diag(e βai,α )α=1,...,vi .
The observables Yi (x) generalize to:
∞
X 1
k
=
Ch
ψ
S
Yi (z) = z ni exp −
i
kz k
k=1
= Det z − e βΦi |0
(197)
Again, as in the four dimensional theory, the non-perturbative effects make the naive
polynomial in the right hand side of (197) a rational function. In particular, on the
U (1) × U (1) invariant instanton configuration λ the observable Yi (z) evaluates to:
Yi (z)[λ] =
(198)
ni
Y
α=1
Q
∈∂+
λ(i,α)
Q
∈∂− λ(i,α)
(z − e β(ai,α +c ) )
(z − qe β(ai,α +c ) )
.
The K-theoretic version of the qq-characters is defined in a similar fashion, one should
use the χq−1 -genus instead of the Chern polynomial and to use push forwards in equi2
variant K-theory instead of the equivariant integrals. The formula (194) generalizes
to:
Z
j
^
X
n−j
n
(−q2 )b
(199)
Xw,ν (z) =
T dT M(w,v) Ch T M(w, v) Ξz [Fv ]
qv q1−b
M(w,v)
v,j
where b
n = dimC M(w, v)
(200)
+
Q
ξi,κ
+ )
Y
(ze
i
Y Y
ηi,κ κ+
Ξz [Fv ] =
Pi (ze ) Q
−
ξi,κ
−
)
Yi (ze
i∈Vertγ κ
κ−
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
where
Ch(Ci± ) =
X
e
±
ξi,κ
51
±
κ±
(201)
Ch(Vi ) =
X
e ηi,κ
κ
8.5. The symmetry. ε1 ↔ ε2 , q1 ↔ q2 .
The formulas (194), (196), (199) are not obviously symmetric with respect to the
exchange ε1 ↔ ε2 , q1 ↔ q2 . However, the symmetry becomes clear once we recall that
M(w, v) is a holomorphic symplectic manifold. Its tangent bundle is isomorphic to
the cotangent bundle, the isomorphism being provided by the holomorphic symplectic
form ω C , which descends from the canonical symplectic form on Hγ :
X
X
(202)
Tr δBe ∧ δ B̃e +
Tr δ I˜i ∧ δ J˜i
e∈Edgesγ
i∈Vertγ
Since the symplectic form ω C is scaled as ω C → q−1 ω C by the action of Hw , the
equivariant Chern character
j
b
b
b
n
n
n
Y
^
Y
X
b
xl −1
n/2
−j
(1 − e q2 ) = (q1 /q2 )
(1 − e −xl q2 )
(203)
(−q2 ) Ch T M(w, v) =
j=0
l=1
l=1
n
n/2 Qb
xl −1
which is equal to (q1 /q2 )b
l=1 (1 − e q1 ) since every equivariant virtual Chern root
xl is paired with another equivariant virtual Chern root β(ε1 + ε2 ) − xl . Thus,
j
j
b
b
n
n
^
X
^
X
n−j
n−j
n
n
Ch T M(w, v)
q2−b
(−q1 )b
Ch T M(w, v) =
q1−b
(−q2 )b
j=0
j=0
8.6. Convergence of the integrals. The integrals (199) may be divergent. Indeed, the
quiver varieties M(w, v) are non-compact. We understand the integrals (199) as the
integrals in Hw -equivariant cohomology. Practically this means that the differential
form representative for the integrand in (199) contains a factor (cf. (??)):
(204)
exp D g(·, V (ξ̄)) = e −g(V (ξ),V (ξ̄)) × (1 + . . .)
Here,
(205)
D = d + ιV (ξ)
is the equivariant de Rham differential, ξ, ξ¯ ∈ Lie(HC
w ), ξ is the collective notation for
(ν, ε), the equivariant parameters, V : Lie(Hw ) → Vect(M(w, v)) is the infinitesimal
action of Hw on M(w, v), and g is any Hw -invariant metric on M(w, v), in which
g(V (ξ), V (ξ̄)) grows for generic ξ and ξ̄ ≈ ξ ∗ sufficiently fast at “infinity” of M(w, v).
With this convergence factor understood the integrals over M(w, v) converge. Moreover, the D-exactness of the exponential in (204) means that small variations of ξ̄ or
52
NIKITA NEKRASOV
g with fixed ξ do not change the integral. The result does, however, depend on ξ.
Indeed, for special values of ξ = (ν, ε) it diverges, as we saw in the examples (149).
Our point is that it converges for all values of x. We shall establish this fact in full
generality in [101] and [100].
8.7. Reduction to the fixed loci. The integrals (199), (194) can be computed by localization with respect to the Hw -action on M(w, v). The isolated fixed points contribute
rational expressions in Y’s with shifted arguments, while positive dimension components of the fixed locus contribute terms with derivatives of Y’s.
The character of the virtual tangent bundle to the quiver variety can be transformed
to
X
X
virt
∗
βme
∗
(206)
T Mγ (w, v) { P
(Wi − Vi )Vi +
e Vt(e) Vs(e)
e∈Edgesγ
i∈Vertγ
Indeed, the tangent bundle to Mγ (w, v) is equal to:
(207)
X
X
∗
∗
T Mγ (w, v) =
(Wi − Vi )Vi∗ + q−1 (Wi − Vi )∗ Vi +
e βme Vt(e) Vs(e)
+ q−1 e −βme Vs(e) Vt(e)
e∈Edgesγ
i∈Vertγ
in the Gw × U (1)b∗ (γ) -equivariant K-theory of Mγ (w, v). The virtual tangent bundle is
equal to
T virt Mγ (w, v) = (1 − q1 )T Mγ (w, v) .
(208)
Now, dualize the terms in (207) proportional to q−1 :
(1 − q1 )q−1 T { (1 − q1−1 )qT ∗ = −(1 − q1 )q2 T ∗
(209)
to arrive at (206).
Let Tγ,w denote the maximal torus in Gw × U (1)b∗ (γ) . The set of Tγ,w -fixed points
Mγ (w, v)Tγ,w is a union
[
Mγ,c (w, v)
(210)
Mγ (w, v)Tγ,w =
c
of connected components. Each component is a product
wi
(211)
Mγ,c (w, v) =
Mγ,i,ci,β (ei , vi,β,n )
i∈Vertγ β=1 n∈L
Vertγ
for some vi,β ∈ Z≥0
such that
(212)
wi
X X
vi,β = v
i∈Vertγ β=1
Here we used a notation
(213)
Mγ,i,c (ei , v) ⊂ Mγ (ei , v),
c ∈ Ci,v
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
53
for a connected component of the set of Tγ,ei -fixed points of Nakajima quiver variety
Mγ (w, v) with w = ei :
G
(214)
Mγ (ei , v)Tγ,ei =
Mγ,i,c (ei , v)
c∈Ci,v
The vector bundles Wi , Vi , restricted onto each connected component Mγ,i,c (ei , v) of
the fixed point set splits, as a sum of Tγ,w -equivariant vector bundles:
Wi =
(215)
Vi =
wi
M
e νi,β
β=1
wi M
M
β=1
n
e νi,β ⊗ q−n Vi,β,n
Here n runs over the lattice of representations of U (1)b∗ (γ) , and qn stands for the
corresponding character. In all cases except for the affine A-type quivers, q is literally
b
q = q1 q2 , and n runs through some lattice L ⊂ Z. In the A-case,
q−n = (q1 q2 e m )−n1 e n2 m
for (n1 , n2 ) ∈ L ⊂ Z ⊕ Z.
case. Recall the expression (150) for the A1 qq-character cor8.7.1. Example:
the A1✿✿✿✿✿
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
responding to w = 2, ν = (0, 0). The corresponding quiver varieties MA1 (2, v), with
v = 0, 1, 2 are the point, T ∗ CP1 , and another point, respectively. The contribution of
v = 1, i.e. the integral over T ∗ CP1 reduces, by the fixed point formula, to the integral
over F = CP1 .
The vector bundle V reduces to L−1 ≈ O(−1), The character-bundle (206) specifies
to:
T virt M = P(W − V )V ∗ = P(2L − 1)
the tangent bundle to ℓ is equal to
T F = 2L − 1
(check: L has two sections, while T F has three), while the complex C becomes q(2 −
L−1 ) − L−1 . The contribution of F to the formula (194) is given by:
ε
(216) qY(x + ε)2
ε1 ε2
where ω = c1 (L),
R
F
Z
ℓ
!2
P(x − ω)
=
Y(x + ε − ω)Y(x − ω)
!
P(x)
2 ε1 ε2
− qY(x + ε)
+
∂
ε x Y(x)Y(x + ε)
Y(x + ε)
ε1 ε2
+ 2qP(x)
1− 2
Y(x)
ε
(ω + ε1 )(ω + ε2 )
ω+ε
ω = 1. It is evident that (216) reproduces the q1 term in (150).
54
NIKITA NEKRASOV
8.7.2. Example:
the D4✿✿✿✿✿
case. The D4 fundamental character X2 provides a represen✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
tative example.
Let w = (0, 1, 0, 0), v = (1, 2, 1, 1), with Vertγ = {1, 2, 3, 4}. In this subsection M4 =
MD4 (w, v).
The character (206) specifies to
(217)
T virt M4 { P (1 − V2 )V2∗ − 3 + V2 (V1∗ + V3∗ + V4∗ )
Now, the three terms in the second line of (167) are coming from the isolated fixed
points pi , i = 1, 3, 4 in M4 , with W2 = 1, V2 = 1 + q−1 , Vi = q−1 , Vj = 1, j 6= i,j = 1, 3, 4.
This gives Tpvirt
M4 { Pq = q + q2 − qq1 − qq2 , which translates to the factor
i
(ε + ε1 )(ε + ε2 )
ε ε
= 1 + 1 22
ε · 2ε
2ε
in the second line of (167).
The first line in (167) is the contribution of the non-isolated component of the fixed
point set, the fixed projective line line F = P1 ⊂ M4 , with W2 = V1 = V3 = V4 = 1,
V2 = 1 + q−1 L, where L ≈ O(−1) is a non-trivial line bundle over F.
The corresponding complexes Ci are given by:
Ci = V2 − (1 + q−1 )Vi ,
(218)
For F this gives:
(219)
i = 1, 3, 4
C2 = W2 ⊕ q−1 (V1 + V3 + V4 ) − 1 + q−1 V2
T virt M4 |F = P 2q−1 L − 1 = 2(q−1 + 1 − q1−1 − q2−1 )L − 1 + q1 + q2 − q
The restriction of Ci onto F is given by:
(220)
Ci = L − 1,
i = 1, 3, 4
C2 = 2 − (1 + q−1 )L
The tangent bundle to F is given by
(221)
T F = 2L − 1
The corresponding contribution to X2,0 is the integral over F of the equivariant Euler
class of T M4 |F with the equivariant parameter ε1 , divided by the equivariant Euler
virt
class of the virtual normal bundle NF⊂M
= T virt M4 |F −T F, which is equal to
4
(222)
virt
NF⊂M
{ (1 − q1 )T M4 − T F { 2(q−1 − q1−1 − q2−1 )L + q1 + q2 − q
4
times the product of Y-observables:
!2
Z
ε
(ω + ε1 )(ω + ε2 ) P(x)P(x − ε − ω)Y2 (x)2 Y Yi (x − ω)
=
ω+ε
Y2 (x − ω)Y2 (x − ω − ε)
Yi (x)
F ε1 ε2
i=1,3,4
(223)
!!
Y2 Y2,−1
ε1 ε2
ε1 ε2
Y2
2 1− 2 +
∂ log
PP−1
Y2,−1
ε x
P−1 Y1 Y3 Y4
ε
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
55
9. More on the Physics of qq − characters
Let G be a Lie group, and (R, π) its representation, i.e. π is a group homomorphism
π : G → End(R). The character χR (g) is a (generalized) function on G, given by the
trace of the matrix π(g) in the representation R:
(224)
χR (g) = TraceR π(g)
By definition χR is an adjoint-invariant function, i.e. the function on the space of
conjugacy classes:
(225)
χR (g) = χR (h−1 gh),
for any h ∈ G
For the compact Lie group G the space of conjugacy classes G/Ad(G) = T /W is the
quotient of the maximal torus T ⊂ G by the action of discrete group, the Weyl group.
9.1. Characters from supersymmetric quantum mechanics. A familiar realization of a
character χR (e h ) in quantum mechanics as the partition functions of a quantum mechanical system with G-symmetry, whose space of states is the representation R and
the Hamiltonian is a realization π(h) of an element h ∈ t of the Lie algebra t = LieT of
the maximal torus T .
For example, if G is a compact Lie group, and R is a unitary representation, corresponding to the highest weight λ ∈ t ∗ , then the geometric quantization program
associates R to the symplectic manifold X = G/Kλ ⊂ g∗ , the coadjoint orbit of λ, with
the canonical Kirillov-Kostant symplectic form ωX /h̄. The geometric quantization realizes R as the space of holomorphic sections of the pre-quantization line bundle L over
ωX
X (which is a Kähler manifold), such that c1 (L) = [ 2πh̄
] ∈ H 2 (X, Z).
(226)
R = H 0 (X, L)
This correspondence extends, with some friction, to a wider class of groups and representations [59].
There are various explicit formulas for the character χR , due to Harish-Chandra,
Weyl, Kirillov, and Kac [104], [55]. For the dominant weight λ the line bundle L has
vanishing higher degree cohomology, so that
X
(227)
TraceR =
(−1)i TraceH i (X,L)
i
One interpretation of the Kac-Weyl character formula is the equivariant RiemannRoch-Grothendieck formula applied to (227).
Physically, one takes (X, ωX ) as the phase space of the mechanical system. For the
Hamiltonian one takes the function h defined as: for x ∈ X, h(x) = hi(x), ti, where
i : X → g∗ is the embedding, and t is some fixed element of t ⊂ g. Then the character
can be realized as the path integral [4, 5]:
Z
I
i −1
(228)
χR (g) = DX exp
d ωX − h(x(t))dt
h̄
56
NIKITA NEKRASOV
where the integral is taken over the space LX of parametrized loops x : S 1 → X. The
loop space LX is acted upon by the torus T × U (1), where T acts pointwise on X, and
U (1) acts by the loop rotations: e 2πis · x(t) = x(t + s).
The integral (228) can be evaluated exactly by the infinite-dimensional version of
the Duistermaat-Heckman formula. The loop space LX is viewed as the symplectic
manifold with the symplectic form being the integral (“a point-wise sum")
Z
1
(229)
Ω=
dt ωµν ψ µ ψ ν
2 S1
H
Then the action d −1 ωX − h(x(t))dt is interpreted as the Hamiltonian, generating a
one-parametric subgroup in T × U (1).
The character formula can be also interpreted with the help of the supersymmetric
quantum mechanics on X, [6, 50]. Instead of X one takes the supermanifold Y =
ΠT X ⊗T ∗ X (the total space of the sum of the cotangent bundle and the tangent bundle
with fermionic fibers over X). Y is endowed with the even symplectic form (as opposed
to the BV formalism, where the symplectic form is odd):
(230)
ωY = dpµ ∧ dxµ + gµν dψ µ ∧ dψ ν
where gµν is a metric on X. We study the quantum mechanics on Y with the Hamiltonian
1
b
b λ
κ
ν
g
ψ
ψ
(231)
HY = g µν pµ − Γκ̃µν̃ gκ̃λ̃ ψ ν̃ ψ λ̃ pν − Γb
b
κλ
νb
ν b
2
The supersymmetry is generated by the odd function
(232)
Q = ψ µ pµ − Γκ̃µν̃ gκ̃λ̃ ψ ν̃ ψ λ̃
If the target space X itself is a moduli space of solutions to some partial differential
equations involving gauge fields on a d-dimensional space Bd , e.g. vortices in d = 2,
monopoles in d = 3, or instantons in d = 4, then the quantum mechanics on X is a
low energy approximation to the d + 1-dimensional gauge theory. The character (228)
would be then given by the path integral in the theory on S1 × Bd . The parameters
t ∈ g in the limit of shrinking S1 would be interpreted as the (twisted) mass parameters
for the flavor symmetry G acting on the moduli space X. If d = 3, then using the
three dimensional mirror symmetry we can exchange these parameters for the FayetIlliopoulos parameters for the dual target space X∨ . Then, the weight subspaces would
identify with the contribution of components of fixed topology. This is almost as good
as the statement of our theorem.
9.2. Gauge theory realization of the qq-characters.. In this section we expand on the
interpretation of qq-characters we sketched in the section 2.4 using the intersecting
branes, or its string dual.
We shall mainly consider the case of affine γ. The theories corresponding to the
finite dimensional A, D, E-type quivers can be viewed as limits of the affine ones, by
sending some of the gauge couplings to zero. For example, the A1 theory is a limit of
b2 theory with two out of three gauge couplings sent to zero.
A
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
57
As we recalled above the N = 2 quiver gauge theories with affine A, D, E quivers can
be realized as the low energy limit of the theory on a stack of n D3-branes placed at
the tip of the C2 /Γ singularity, with Γ ⊂ SU (2) the McKay dual finite group.
For definiteness, let N ≈ R4 denote the worldvolume of the stack of the ‘physical’
D3 branes. The six dimensional transversal slice splits as a product W/Γ × R2φ . Here
W ≈ R4 is the Euclidean cover of the singularity. The fluctuations along R2φ are
represented, in the D3 theory, by the adjoint complex scalar in the vector multiplet.
We want to subject the theory to the Ω-deformation. To this purpose we choose a
point 0 ∈ N, and use the symmetry of rotations about 0.
The superconformal vacuum of the theory on D3-branes corresponds to the branes
located at the origin in W, fixed by the Γ action and at some point p in Σ. Let us
identify Σ ≈ C and p with 0 ∈ C.
Let us now add a stack of w D3-branes located at 0×x ∈ N×Σ, with the worldvolume
being a copy of W/Γ. Here x ∈ C is a complex number.
The low energy configurations in the combined system of n + w D3 branes split into
two orthogonal stacks are labelled by some continuous and discrete parameters, such
as the separation of branes along Σ and the choice of flat U (w) connection at infinity
S3 /Γ of W/Γ, i.e. a homomorphism ρ : Γ → U (w).
When Γ = 1 is trivial, the supersymmetry of the combined system of branes is
consistent with Ω-deformation, which uses the subgroup SU (2)N,L ×SU (2)∆ ×SU (2)W,L
of the group
Spin(4)N × Spin(4)W = SU (2)N,L × SU (2)N,R × SU (2)W,L × SU (2)W,R
of rotations of the two orthogonal R4 ’s.
The open string Hilbert space splits as a sum of the spaces corresponding to the
strings stretched between different types of D-branes. We have the (−1) − (−1) strings
connecting the D(-1)-instantons, we have the (−1) − 3 strings connecting the D(-1)instantons to the stack of n D3-branes, we have the (−1) − 3′ strings connecting the
D(-1)-instantons to the stack of w D3 branes. There are also the 3 − 3′ open strings.
In [101] we define the moduli space using the low-energy modes of these open strings.
The qq-character Xw,ν (x) is simply the observable in the original theory on the stack
of n D3-branes living along N, which is obtained by integrating out the degrees of
freedom on the transversal w D3-branes, in the vacuum corresponding to the particular
w and the vacuum expectation values ν of the scalars in the vector multiplets living
on W/Γ.
The integral (199) can be interpreted (cf. [82]) as the partition function of the
supersymmetric quantum mechanics on the moduli space of Yang-Mills instantons on
4 /Γ , constructed in [64]. Here Γ ⊂ SU (2) is the
the ALE gravitational instantons R]
γ
γ
MacKay dual to γ discrete group, whose representation theory is encoded in the quiver
γ: Vertγ = Γ∨
γ . The gauge group U (w), with
X
w=
wi dimRi
i∈Γ∨
γ
58
NIKITA NEKRASOV
is broken at infinity, by the choice of flat connection Γγ → U (w), to a subgroup
U (w) −→
U (wi )
i∈Vertγ
The dimensions v encode the magnetic fluxes through the exceptional two-spheres in
4 /Γ , and the ordinary instanton charge
the resolved orbifold R]
γ
X
vi dimRi
vtot =
i∈Γ∨
γ
In the supersymmetric quantum mechanics the sum over v is not natural, as it adds
the partition functions of different Hilbert spaces. However, in the 4 + 1 dimensional
4 /Γ × S1 the sum over v is just the sum over various topological
gauge theory on R]
γ
sectors which is enforced anyway by the cluster decomposition. A more careful look
at (199) reveals that we are dealing with the maximally supersymmetric Yang-Mills
theory subject to the Ω-deformation (more on it below) and coupled to a point-like
source localized along the circle S1 × 0̃, where 0̃ is the exceptional variety, the joint of
the exceptional spheres, which arose in the resolution of singularities.
The coupling qv comes naturally from the usual Chern-Simons couplings on the
D4-branes
Z
Z
(233)
C1 ∧ Tr (F − BN S ) ∧ (F − BN S ) + C3 ∧ Tr(F − BN S )
In the IIB picture these would translate to the couplings to
Z
BN S + τBRR = τi
S2i
as described, e.g. in [65].
Here is the general construction (the reader is invited to consult [95, 91] for details).
Consider the maximal supersymmetric super-Yang-Mills theory in eight dimensions, on
a noncommutative R8 ≈ C4 . One can view this theory as a particular background in
the IKKT matrix model [52] with an dimensional theory with an infinite dimensional
gauge group, the group of unitary operators in the Hilbert space. This theory can also
be lifted to 8 + 1 and 9 + 1 dimensions, (also known as the Matrix theory [9] and the
matrix string theory [26], respectively).
Recall that the gauge fields on the noncommutative Euclidean space Rnθ with the
coordinates b
xµ , µ = 1, . . . , n, obeying
[b
xµ ,b
xν ] = iθ µν ·1
can be described, more conveniently, as operators
bµ = b
X
xµ + θ µν Aν (b
x)
bµ = b
In the vacuum, Aν = 0 and X
xµ . The equations of motion of Yang-Mills theory on
bµ ’s:
Rnθ translate to the relations on the commutators of X
(234)
bµ′ , [X
bµ′′ , X
bµ ]] = 0
Gµ′ µ′′ [X
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
59
In the form (234) the equations of motion do not distinguish between the gauge fields
and adjoint scalars, and are equally applicable both to the n-dimensional theory and
to its dimensional reductions to lower dimensions. Everything is hidden in the nature
bµ .
of the operators X
Let us now take n = 10, choose an identification R10 ≈ C4 ×C, assume the metric to be
Euclidean, Gµν = δµν , and the Poisson tensor θ µν to be of the (1, 1)-type. We assume it
b2i−1 +iX
b2i , i = 1, 2, 3, 4, Φ = X
b9 +iX
b10 . We
vanishes on the last C factor. Define Z i = X
are interested in the supersymmetric field configurations, the generalized instantons.
In the present case the relevant equations are (cf. [73]):
(235)
(236)
[Z i , Z j ] + εijkl [Z k , Z l ]† = 0,
i, j = 1, 2, 3, 4
4
X
[Z i , Z i† ] = −2θ · 1H
i=1
and
(237)
[Φ, Z i ] = [Φ, Z i† ] = 0
The equations (235), (236), (237) imply (234). The Ω-deformation modifies the equations (237) to:
(238)
[Φ, Z i ] + εi Z i = 0,
[Φ, Z i† ] − εi Z i† = 0
1
εijkl dz i ∧ dz j ∧ dz k ∧ dz l which is only
The equation (235) involves the (4, 0)-form 4!
invariant under the SU (4) rotations, forcing the constraint (27).
Let us denote by H a copy of the two-oscillator Fock space:
∞
M
(239)
H=
C|~
ni,
n1 ,n2 =0
acted upon by the creation and the annihilation operators:
p
√
(240)
A†i |~
ni = ni + 1|~
n + ei i,
Ai |~
ni = ni |~
n − ei i,
i = 1, 2
with n
~ = n1 e1 + n2 e2 , e1 = (1, 0), e2 = (0, 1). The Hilbert space H is the irreducible
representation of the 2-oscillators Heisenberg algebra
[Ai , A†j ] = δij , [A1 , A2 ] = 0, i, j = 1, 2
A simple solution to the Eqs. (235), (236), (237) describing a stack of n parallel D3branes stretched in the R41234 direction is given by: identify H = H ⊗ N , with N the
fintie dimensional complex vector space of dimension n:
√
√
(241)
Z 1 = θA†1 ⊗ 1N ,
Z 2 = θA†2 ⊗ 1N
i
i
i
while for i = 3, 4, Z i = 1H ⊗ diag(z(1)
, . . . , z(n)
), where z(a)
, aa ∈ C, a = 1, . . . , n, i = 3, 4.
The scalar Φ is equal to 1H ⊗ diag(a1 , . . . , an ) in the absence of Ω-deformation, and to
(242)
Φ = ε1 A†1 A1 + ε2 A†2 A2 ⊗ 1N + 1H ⊗ diag(a1 , . . . , an )
when the Ω-deformation is turned on.
60
NIKITA NEKRASOV
The solution which preserves less supersymmetry has H = H12 ⊕ H34 , H12 = H ⊗ N ,
H34 = H ⊗ W , with two vector spaces N and W , of dimensions n and w, respectively
and:
√
Z i = θ P12 A†i ⊗ 1N P12 ,
i = 1, 2
(243)
√
Z i = θ P34 A†i−2 ⊗ 1W P34 ,
i = 3, 4
where Pij : H → Hij is the orthogonal projection, Pij2 = Pij = Pij† , P12 P34 = P34 P12 = 0,
1H = P12 + P34 . The scalar Φ is given by
Φ = P12 1H ⊗ diag(a1 , . . . , an )P12 + P34 1H ⊗ diag(ν1 , . . . , νw )P34
without Ω-deformation, and by
Φ = P12 ε1 A†1 A1 + ε2 A†2 A2 ⊗ 1N + 1H ⊗ diag (a1 , . . . , an ) P12 +
(244)
P34 ε3 A†1 A1 + ε4 A†2 A2 ⊗ 1W + 1H ⊗ diag (ν1 , . . . , νw ) P34
with the Ω-deformation corresponding to the generic SU (4) rotation.
We are mostly interested in the solutions, which asymptotically tend to the 4 + 4dimensional background, corresponding to the intersecting branes solution (the asymptotics does not allow shifting Z i by a constant). Let us describe the so-called 1instanton solutions in the “abelian” case n = w = 1. Let eN and eW denote the orthonormal bases in N and W , respectively. The space of solutions has three components: MN ∪ MN W ∪ MW .
The components MN , MW are both isomorphic
to C2 , while the component MN W ≈ CP1
is compact. Moreover these components
intersect, at two points:
MN ∩ MN W = p N , MW ∩ MN W = p W
(it is tempting to call pN the North pole,
and pW the Western pole,
unfortunately we couldn’t place
the latter on the map, even on the Google map).
Explicitly, MN parametrizes the solutions where the pair (Z 1 , Z 2 ) in (9.2) is replaced
by the one-instanton solution of [88] (see also [93] for more details), while (Z 3 , Z 4 , Φ)
are intact. Likewise, MW parametrizes the solutions where the pair (Z 3 , Z 4 ) in (9.2) is
replaced by the one-instanton solution. Recall [40], [93], [99] that these solutions make
use of Murray-von Neumann partial isometries S : H → H, which obey:
(245)
SS † = 1H ,
S † S = 1H − PK
with PK an orthogonal projection onto a finite-dimensional subspace K ⊂ H. For the
one-instanton solutions in the components MN ,W the subspace K is one-dimensional,
†
†
K = Ce uA1 +vA2 |0, 0i ⊗ eN ,W , with (u, v) ∈ MN ,W ≈ C2 being the instanton modulus.
The solutions corresponding to the component MN W have K = Ce ⊥ , where
(246)
e ⊥ = ᾱ|0, 0i ⊗ eN + β̄|0, 0i ⊗ eW ,
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
with some α, β ∈ C,
|α|2 +|β|2 = 1
(247)
Define
e†K = 0
e = β|0, 0i ⊗ eN − α|0, 0i ⊗ eW ,
(248)
and
61
H′ =
(249)
M
n1 +n2 >0
C |~
ni ⊂ H
and
H′ = Ce ⊕ (H′ ⊗ N ) ⊕ (H′ ⊗ W )
(250)
the orthogonal complement to K. Define:
√
p
−1
n + ei i ⊗ eN , i = 1, 2
gn ni + 1 |~
Z̃ i |~
ni ⊗ eN = θ gn+1
√
p
−1
Z̃ i |~
ni ⊗ eW = θ g̃n+1
n + ei−2 i ⊗ eW ,
g̃n ni−2 + 1 |~
Z̃ 1,2 |~
ni ⊗ eW = 0,
Z̃ 3,4 |~
ni ⊗ eN = 0
n = n1 + n2 > 0,
(251)
Z̃ 1 e =
√
Z̃ 3 e =
√
gn , g̃n ∈ C
θ γ12 |1, 0i ⊗ eN ,
Z̃ 2 e =
√
θ γ34 |1, 0i ⊗ eW ,
Z̃ 4 e =
√
Z̃ i† e = 0,
i = 3, 4
θ γ12 |0, 1i ⊗ eN ,
θ γ34 |0, 1i ⊗ eW ,
i = 1, 2, 3, 4
with
(252)
Now define:
γ12 , γ34 ∈ C,
|γ12 |2 +|γ34 |2 = 1,
(γ12 : γ34 ) ∈ MN W
Z i = SZ i S †
(253)
where S † maps H onto H′ isometrically (use the Hilbert hotel construction) obeying
(245).
The diagonal matrices gn , g̃n are fixed, up to the unitary gauge transformations, by
the equation (236),
(254)
where
(255)
|gn |2 =
(n + 1)! n!
,
(n + κ12)! (n + 1 − κ12)!
|g̃n |2 =
(n + 1)! n!
(n + κ34 )! (n + 1 − κ34)!
κij (1 − κij ) = 2(|γij |2 −1)
62
NIKITA NEKRASOV
In the limiting cases (γ12 : γ34 ) → (1 : 0) = pW or (γ12 : γ34 ) → (0 : 1) = pN the
solution (251) approaches the direct sum of the vacuum solution for H34 and oneinstanton solution on H12 or the direct sum of the one-instanton solution for H34 and
the vacuum solution on H12 , respectively.
!
4
In [100] we shall consider more general intersecting brane solutions. Let 6 =
, the
2
set of 2-element subsets of 4. Fix 6 vector spaces NA . We take the Hilbert space to be
the sum
M
(256)
H=
HA ,
HA = H ⊗ NA
A∈6
Define
(257)
Z0a =
X
A,a∈A
A†h
A (a)+1
⊗ 1NA
so that Z0a |HB = 0 whenever a ∈/ B. This is the reference solution for the generalized
instantons in the theory we call the gauge origami in [100], [101].
9.3. Other realizations of X-observables. A natural question is what is the meaning of
the Xi (x) observables on the CFT side of the BPS/CFT-correspondence?
It might seem natural, e.g. in the AGT setup [2, 1] to assign the Xi (x)-observables
to the non-intersecting loops on the curve C on which one compactifies the A1 (0, 2)theory, which define the α-coordinates in the system of Darboux coordinates on the
moduli space of SL2 local systems [87].
One systematic way to derive such a representation would be to start with the type
IIB ten-dimensional background whose geometry is a rank 4 complex vector bundle E
over a flat two-torus with an SU (4)-flat connection. One can add up to six stacks of
D5 branes, wrapping the base torus and one of the complex rank two sub-bundles of E,
invariant under the action of the product of the maximal torus T ⊂ SU (4) and the twotorus translating the base. This symmetry can be used to T -dualize the configuration
of branes leading to various equivalent realizations. This direction will be explored
elsewhere.
However, one may try to address the question directly within the realm of the twodimensional conformal field theory. We know the N = 2∗ SU (n) theory corresponds to
the An−1 Toda theory [120] on a two-torus C× /qZ with an insertion of a special vertex
operator. The coupling constant b 2 of the Toda theory is determined by the ratio ε2 /ε1
b0 -theory is generated by
of two equivariant parameters. The qq-character Xw,ν of the A
∗
4
the auxiliary N = 2 theory on the transverse R with the equivariant parameters ε3 , ε4 .
It corresponds to its own Aw−1 Toda theory with the coupling constant b̃ 2 = ε4 /ε3 . It
would be interesting to work out the coupling between these theories generating the
x-dependent contributions to the qq-character.
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
63
10. The first applications
10.0.1. Expansion
coefficients. For the formal Laurent series f (z −1 ) near z = ∞ we
✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿✿
n
denote by [z ]f (z) the z n coefficient:
I
1
dz
n
(258)
[z ]f (z) ≡ Coeffzn f (z) =
f (z) n+1 ,
2πi ∞
z
the latter equality holding for actual functions f (z).
10.1. Effective prepotentials and superpotentials. One obvious application of our formalism is the solution of the low-energy theories. Indeed, in the limit ε2 → 0 the
integrals (194) simplify (the Chern polynomial of the tangent bundle drops out). In addition, the sum over all quiver n-colored partitions (146) is dominated by a single limit
shape [86] which maps (194) to a system of difference equations for the Yi -functions.
These equations were studied in [86]. In this case it suffices to study the equations for
the dimension vectors w corresponding to the fundamental weights of gγ .
10.2. Instanton fusion. In the quantum case where ε1 , ε2 are finite we need all w. The
equations (194) can be viewed as a system of Hirota difference equations, which should
fix Z uniquely. This direction is currently investigated. Note that for the finite Atype quivers with the special choice of n the related equations were found in [56] by
somewhat different methods, although we couldn’t match them with our equations for
specific w. It would be very interesting to relate the algebraic structure found in [56]
to the one we exhibited here.
10.3. Undressing the U (1) legs. Another application of our formalism is the reduction
formula, which allows to relate the partition functions of the gauge theories with U (1)
gauge factors to the partition functions of the gauge theories with these U (1)’s being
treated as global symmetries. We assume the asymptotic freedom condition βi ≤ 0 (78)
is obeyed.
Let i ∈ Vertγ be the node with ni = 1. Shift the argument of Yi (x) so as to set ai = 0.
Then, we have the expansion (cf. (135)):
ε ε
(259)
Yi (x) = x + 1 2 ki + . . . ,
x→∞
x
There are two possibilities: either the node i is connected to itself by an edge e ∈
s−1 (i) ∩ t −1 (i), or s−1 (i) ∩ t −1 (i) is empty.
b0 theory. In the first case the theory is the N = 2∗ theory. In the U (1)
10.3.1. ✿✿✿✿✿
The ✿✿✿
A
✿✿✿✿✿✿✿✿
case it is characterized by the mass m of the adjoint hypermultiplet, the complexified
coupling q and the Ω-background parameters (ε1 , ε2 ). The partition function ZAb is a
0
homogeneous function of m, ε1 , ε2 , symmetric in ε1 , ε2 and invariant under m → −m − ε:
!
X Y
m(m
+
ε)
,
(260)
Z(q; m : ε1 : ε2 ) =
q|λ|
1+ ∨
∨
c
(ε
−
c
)
λ
∈λ
64
NIKITA NEKRASOV
where c∨ = ε1 (l + 1) − ε2 a . Since for λ 6= ∅ there always exists a locally most southeast box for which l = a = 0, the partition function Z(q; m : ε1 : ε2 ) = 1 for m = −ε1
or m = −ε2 :
Z(q; m : ε1 : ε2 ) = 1 +
(261)
(m + ε1 )(m + ε2 )
Z̃(q; m : ε1 : ε2 )
ε1 ε2
The normalization
Z(q; 0 : ε1 : ε2 ) = Z(q; −ε : ε1 : ε2 ) = φ(q)−1
(262)
follows trivially from (260). Let us expand the character X1,0 (x) (153) in x near x = ∞:
(263) X1,0 (x) =
X
λ
q
|λ|
Y
∈λ
!
m(m + ε)
ε1 ε2
k + ... 1 −
|λ|+ . . .
S(mh + εa ) x + ε +
x
x2
Recall that the formula above gives the x-expansion of an observable. It has the form
(0)
(1)
X1,0 (x) = X1,0 (x) + X1,0 (x)k + . . ., where k is our familiar observable (119).
Thus
(264)
[x−1 ]X1,0 (x) = ε1 ε2 Z(q; ε1 : −m − ε : m)k − m(m + ε)q
d
Z(q; ε1 : −m − ε : m)
dq
and the consequence of our equations (146) reads
DD
EE
=
(265) 0 =
[x−1 ] X1,0 (x)
q;m,ε1 ,ε2
d
Z(q; m : ε1 : ε2 )−
dq
d
m(m + ε)Z(q; m : ε1 : ε2 )q Z(q; ε1 : −m − ε : m)
dq
ε1 ε2 Z(q; ε1 : −m − ε : m)q
Introduce:
(266)
Φ(q; m : ε1 : ε2 ) =
ε1 ε2
logZ(q; m : ε1 : ε2 )
(m + ε1 )(m + ε2 )
For fixed q it is a priori a meromorphic function on CP2 , with possible singularities at
ε2 /ε1 ∈ Q≥0 (but not at m = −ε1 , m = −ε2 , cf. (261)). For q = 0, Φ = 0. Then (265)
implies:
(267)
Φ(q; m : ε1 : ε2 ) = Φ(q; ε1 : −m − ε : m)
which shows that Φ has no singularities in (m : ε1 : ε2 ) for fixed q, i.e. it is a constant.
The normalization (262) then implies that Φ(q; m : ε1 : ε2 ) = logφ(q), i.e.
(268)
−
Z(q; m : ε1 : ε2 ) = φ(q)
(m+ε1 )(m+ε2 )
ε1 ε2
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
65
10.3.2. ✿✿✿✿✿✿
Other✿✿✿✿✿✿✿✿✿
theories. In this case the node i is such that s−1 (i)∩t −1 (i) is empty. The
fundamental qq-character Xi (x) has the following structure:
(269)
Xi (x) = Yi (x + ε) + qi Γ2 (x)Y−1
i (x) + qi Γ1 (x)
where Γ1 (x), Γ2 (x) are built out of Yj , qj with j 6= i. For large x the functions Γa (x)
behave as xa (1 + O(1/x)), for a = 1, 2. The expansion in x near x = ∞ gives:
(270)
0 = [x−1 ] h Xi (x) i = ε1 ε2 (1 − qi )qi
d inst
Z
+ qi DZ inst
dqi
where D is the first order differential operator in qj , with j 6= i. The equation (270)
is the quasilinear partial differential equation of the first order, which can be solved
using the method of characteristics. The solution is unique given the initial condition,
which can be set at qi = 0, where the U (1)i gauge factor becomes a flavor group.
10.3.3. ✿✿✿✿✿
The ✿✿✿✿✿✿
linear✿✿✿✿✿✿✿✿
quiver ✿✿✿✿✿✿✿✿
abelian✿✿✿✿✿✿✿✿✿
theories. As an example of the application of this
technique, consider the Type I theories with the Ar -type quiver, with m1 = n1 = n2 =
. . . = nr = mr = 1, mj = 0, 1 < j < r. The theory is characterized by the masses m1 , mr
of the fundamental hypermultiplets, the Coulomb moduli a = (ai )ri=1 (which could be
traded for the masses of the bi-fundamental hypermultiplets, cf. [85]) and the couplings
q = (qj )rj=1 . Let us introduce the “momenta” pi± , i = 0, . . . , r:
(271)
pi− = ai − ai+1 ,
pi+ = ε + ai − ai+1 ,
Using (161), (160), we derive:
(272) 0 = −[x−1 ] h Xl (x) i =
=
X
I ⊂[0,r], |I |=l
X
∇zi logZAinst
+
zI ε1 ε2
r
i∈I
The solution to (272) is given by the simple “free-field formula”:
(273)
ZAinst
(a, q) =
r
Y
X
i∈I ,j∈[0,r]\I ,j<i
pi+ pj− .
p+ p−
i j
1 ε2
−ε
0≤j<i≤r
(1 − zi /zj )
Thus, we have derived the formulas conjectured in the sections C.1 and C.2 of [2]. Our
derivation here differs from that in [18].
For r = 1 we get:
ZAinst
= (1 − q)
1
(ε+a0 −a1 )(a2 −a1 )
ε1 ε2
10.3.4. ✿✿✿✿✿
The ✿✿✿✿✿✿✿✿
D-type ✿✿✿✿✿✿✿✿✿
theories. Let us present now an example of the D4 -type theory.
This is the theory with four gauge group factors, which we shall label by i = 0, 1, 2, 3,
with the assignments: n0 = 2, n1 = n2 = n3 = m0 = 1. The theory is characterized by
four couplings q = (q0 , q1 , q2 , q3 ), and six Coulomb and mass parameters: the Coulomb
parameters
a = (a0,1 = a1 , a0,2 = a2 , a1,1 = m1 , a2,1 = m2 , a3,1 = m3 )
66
NIKITA NEKRASOV
two for the U (2) gauge group factor, and three for three U (1) factors, and the mass
m4 .
By computing [x−1 ]Xi,0 (x) in (163) using (135) for i = 1, 3, 4 we derive three first
order differential equations, whose solution give:
(274) ZD4 a; m4 ; q = ZA1 (a1 , a2 ; m1 , m2 , m3 , m4 ; q) ×
(1 − q1 )µ1 (1 − q2 )µ2 (1 − q3 )µ3 (1 − q20 q1 q2 q3 )µ4 ×
(1 − q0 q1 )ν1 (1 − q0 q2 )ν2 (1 − q0 q3 )ν3 (1 − q0 q1 q2 q3 )ν4 ×
where
(275)
ε1 ε2 µj = (mj − a1 )(mj − a2 ) ,
(1 − q0 q1 q2 )κ3 (1 − q0 q1 q3 )κ2 (1 − q0 q2 q3 )κ1
ε1 ε2 νj = (a1 + a2 + ε)(a1 + a2 + ε + mj − m) − a1 a2 − εm4 + mj (mj − m) +
ε 1 ε 2 κ j = a 1 + a 2 + ε − mj − m4 a 1 + a 2 + ε + mj − m
j = 1, . . . , 4 ,
X
mi mk ,
1≤i<k≤4
m = m1 + m2 + m3
and
(276)
(1 − q1 )(1 − q2 )(1 − q3 )(1 − q20 q1 q2 q3 )
q = q0
(1 − q0 q1 )(1 − q0 q2 )(1 − q0 q3 )(1 − q0 q1 q2 q3 )
10.4. Fractional instantons and quantum differential equations. The equations of the
schematic form:
∂
ba (τ) · Ψ
(277)
κ
Ψ=H
∂τ a
ba on the right hand side are
where a label the set of couplings and the operators H
κ-independent, show up in mathematical physics on several occasions ( KnizhnikZamolodchikov connection [61], [105], t-part of tt ∗ -connection [19], Gauss-Manin connection for exponential periods [68], [66], λ-connection associated to the solution of the
WDVV equations [62], [45], and more recently, e.g. [15]). The consistency of (277),
i.e. the flatness of the corresponding connection for any value of κ, is equivalent to two
sets of equations:
ba , H
bb ] = 0,
[H
(278)
∂ b
∂ b
Hb −
H =0
∂τ a
∂τ b a
The first set of equations imply that at each value of τ one has a quantum integrable
ba is maximal in the appropriate sense).
system (if the number of the operators H
In the present case the meaning of these equations is the following. We have a
quantum field theory with some set of couplings τa in which we study a codimension
two defect, which has its own couplings τ̃a,ω . Differentiating the partition function of
the theory with defect brings down the corresponding observable Oa , deforming the
BPS/CFT, DYSON-SCHWINGER, QQ-CHARACTERS
67
Lagrangian. Integration over the positions of Oa ’s has a contribution of the region
where Oa approaches the defect. When Oa hits the defect, it fractionalizes, and splits
into the observables of the defect theory:
X (1)
X (2)
(279)
Oa ∼
f a,ω (τ, τ̃)Õa,ω +
f a,ω′ ,ω′′ (τ, τ̃)Õa,ω′ Õa,ω′′ + . . .
ω′ ,ω′′
ω
The equation we derive in [102] is an example of such a relation, where the bulk operator
Oa is, in fact, the familiar Tr Φ2 , and its supersymmetric descendents (which are all
equal up to the powers of ε2 in cohomology of the Ω-deformed supersymmetry). What
about other operators, such as Tr Φk for k > 2?
The operators deforming the gauge Lagrangian by
Z
Xτ Z
1
k
4
4
k
d xd ϑTr Φ ∼
Tr Φk−2 F 2 + . . .
(280)
δτ L =
k!
(k − 2)!
k>2
are irrelevant and lead to non-renormalizable theories. We can, nevertheless, study
n
them by treating τk as formal variables (i.e. assuming some power τk k of τk to vanish).
The qq-characters are modified by the introduction of the higher times. In the A1 case,
for example, the qq-character modifies to
(281)
Y(x + ε) + Y(x)−1 qP(x) exp
∞
X
1
l=1
l!
τl xl
In this way we get the realization of the W -algebra and its qq-deformation in gauge
theory. We also get a new perspective on the rôle of Whitham hierarchies [63], [46]
and their quantum and qq-deformations in gauge theory.
See also [13] for more applications of qq-characters in the U (1) case.
11. Discussion and open questions
Of course, the most interesting question is to extend our formalism of non-perturbative
Dyson-Schwinger equations beyond the BPS limit, even beyond the realm of supersymmetric theories.
However, even in the world of moderately supersymmetric theories our approach
seems to be useful. It appears that the exact computations of [24], [25] of effective
superpotentials of N = 1 theories and their gravitational descendands can be cast in
the form of the non-perturbative Dyson-Schwinger identities. The precise definition of
the qq-characters in N = 1 theories will be discussed elsewhere.
Another exciting problem is to find the string theory analogue of our qq-characters
and the stringy version of the large field redefinitions (see [42] for a discussion of stringy
symmetries).
The considerations of this paper and its companions are local, they describe gauge
theories in the vicinity of a fixed point of rotational symmetry. In [92] four dimensional
N = 2 gauge theories on the smooth toric surfaces were studied. It was found that the
partition function of the gauge theory on a toric surface S has the topological vertex
68
NIKITA NEKRASOV
structure:
(282)
ZS ∼
X Y
lattice v∈S
ZR4 (local Coulomb parameters, local Ω − parameters)
where the sum goes over the lattice of magnetic fluxes H 2 (S, Λw ), the product is over
the fixed points of the two-torus action on S (see [12] for the recent progress in this direction). Our generalized gauge theories involving intersecting four dimensional spacetimes naturally live on Calabi-Yau fourfolds. They describe generalized complex surfaces which may have several components with different multiplicities. It would be
interesting to apply these ideas to topological strings and to topological gravity.
On a more mathematical note, let us discuss the relation of our qq-characters to the
t-deformation of q-characters of [38], introduced by H. Nakajima in [75, 78, 79, 80]. His
definition is basically the weighted sum of the Poincare polynomials of the Hw,γ -fixed
loci on M(w, v). Let us observe that if in the formula (199) we pull the Yi ’s and Pi ’s
out of the integral, with some clever choice of the arguments replacing those in (200),
the remaining integral, for each v would compute
X
j
(283)
(−q2 )−j χ(M(w, v), ΩM(w,v) )
j
i.e. the holomorphic Poincare polynomial. In other words, if the qq-operator is
viewed as the difference-differential operator on the functions Yi , then the t-deformed
q-character looks like its symbol. It would be interesting to develop some kind of deformation quantization scheme, allowing to compute our qq-characters using the knowledge of the t-deformed q-characters [80], and to apply them to rederive the results of
[17]. The paper [?] is a step in this direction.
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Simons Center for Geometry and Physics, Stony Brook University, Stony Brook NY 11794-3636,
USA, E-mail: nikitastring@gmail.com2
2on
leave of absence from: IHES, Bures-sur-Yvette, France
ITEP and IITP, Moscow, Russia
| 0 |
Generating Reflectance Curves from sRGB Triplets *
0F
Scott Allen Burns
University of Illinois at Urbana-Champaign (scottb@illinois.edu)
Overview
I present several algorithms for generating a reflectance curve from a specified sRGB triplet,
written for a general audience. Although there are an infinite number of reflectance curves that
can give rise to the specific color sensation associated with an sRGB triplet, the algorithms presented here are designed to generate reflectance curves that are similar to those found with naturally occurring colored objects. My hypothesis is that the reflectance curve with the least sum of
slope squared (or in the continuous case, the integral of the squared first derivative) will do this.
After presenting the algorithms, I examine the quality of the computed reflectance curves compared to thousands of documented reflectance curves measured from paints and pigments available commercially or in nature. Being able to generate reflectance curves from three-dimensional
color information is useful in computer graphics, particularly when modeling color transformations that are wavelength specific.
Introduction
There are many different 3D color space models, such as XYZ, RGB, HSV, L*a*b*, etc., and
one thing they all have in common is that they require only three quantities to describe a unique
color sensation. This reflects the “trichromatic” nature of human color perception. The space of
color stimuli, however, is not three dimensional. To specify a unique color stimulus that enters
the eye, the power level at every wavelength over the visible range (e.g., 380 nm to 730 nm)
must be specified. Numerically, this is accomplished by discretizing the spectrum into narrow
wavelength bands (e.g., 10 nm bands), and specifying the total power in each band. In the case of
10 nm bands between 380 and 730 nm, the space of color stimuli is 36 dimensional. As a result,
there are many different color stimuli that give rise to the same color sensation (infinitely many,
in fact).
For most color-related applications, the three-dimensional representation of color is efficient and
appropriate. But it is sometimes necessary to have the full wavelength-based description of a
color, for example, when modeling color transformations that are wavelength specific, such as
dispersion or scattering of light, or the subtractive mixture of colors, for example, when mixing
paints or illuminating colored objects with various illuminants. In fact, this document was developed in support of another document concerning how to compute the RGB color produced by
subtractive mixture of two RGB colors.1
I present several algorithms for converting a three-dimensional color specifier (sRGB) into a
wavelength-based color specifier, expressed in the form of a reflectance curve. When quantifying
object colors, the reflectance curve describes the fraction of light that is reflected from the object
by wavelength, across the visible spectrum. This provides a convenient, dimensionless color
*
Originally published April 29, 2015 by Scott Allen Burns at http://scottburns.us/reflectance-curves-from-srgb/
1
specification, a curve that varies between zero and one (although fluorescent objects can have
reflectance values >1). The motivating idea behind these algorithms is that the one reflectance
curve that has the least sum of slope squared (integral of the first derivative squared, in the continuous case) seems to match reasonably well the reflectance curves measured from real paints
and pigments available commercially and in nature. After presenting the algorithms, I compare
the computed reflectance curves to thousands of documented reflectance measurements of paints
and pigments to demonstrate the quality of the match.
sRGB Triplet from a Reflectance Curve
The reverse process of computing an sRGB triplet from a reflectance curve is straightforward. It
requires two main components: (1) a mathematical model of the “standard observer,” which is an
empirical mathematical relationship between color stimuli and color sensation (tristimulus values), and (2) a definition of the sRGB color space that specifies the reference illuminant and the
mathematical transformation between tristimulus values and sRGB values.
The linear transformation relating a color stimulus,
is
, to its corresponding tristimulus values,
,
The column vector has three elements, , , and . The matrix
has three rows (called the
three “color matching functions”) and columns, where is the number of discretized wavelength bands. In this study, all computations are performed with 36 wavelength bands of width
10 nm, running over the range 380 nm to 730 nm. The stimulus vector also has components,
representing the total power of all wavelengths within each band. The specific color matching
functions I use in this work are the CIE 1931 color matching functions.2 (Note that I’m using the
symbol
here; the standard matrix is x 3, and so
indicates that it has been transposed.)
The stimulus vector can be constructed as the product of an
diagonal illuminant matrix,
, and an
reflectance vector, . Matrix
is usually normalized so the tristimulus
value is 1 when is a perfect reflector (contains all 1s). The normalizing factor, , is thus the
inner product of the second row of
and the illuminant vector, , yielding the alternate form
of the tristimulus value equation
The transformation from tristimulus values to sRGB is a two-step process. First, a 3 3 linear
transformation is applied to convert to
, which is a triplet of “linear RGB” values:
The second step is to apply a “gamma correction” to the linear
values, also known as “companding” or applying the “color component transfer function.” This is how it is done: for each ,
, and component of
, let’s generically call it , if
, use
, otherwise
use
. This gives sRGB values in the range of 0 to 1. To convert them to the
2
alternate integer range of 0 to 255 (used in most 24-bit color devices), we multiply each by 255
and round to the nearest integer.
The inverse operation of converting sRGB to
, expressed in (Matlab) code, is:
sRGB=sRGB/255; % convert 0-255 range to 0-1 range
for i=1:3
if sRGB(i)<0.04045
rgb(i)=sRGB(i)/12.92;
else
rgb(i)=((sRGB(i)+0.055)/1.055)^2.4;
end
end
The expression relating
and above can be simplified by combining the three matrices and
the normalizing factor into a single matrix,
so that
The formal definition3 of the sRGB color space uses an illuminant similar to daylight, called
D65, as its “reference” illuminant. Here are the specific values for the
matrix, the
matrix,
the D65 vector, and the matrix. The normalizing factor, , has a value of 10.5677. Most of
the theory I present here I learned from Bruce Lindbloom’s highly informative website.4
Now that we have a simple expression for computing sRGB from a reflectance curve, we can use
that as the basis of doing the opposite, computing a reflectance curve from an sRGB triplet. In
the sections that follow, I will present five different algorithms for doing this. Each has its
strengths and weaknesses. Once they are presented, I will then compare them to each other and
to reflectance curves found in nature.
Linear Least Squares (LLS) Method
Since there are so many more columns of than rows, the linear system is under-determined and
gives rise to an
dimensional subspace (33-dimensional in our case) of reflectance curves
for a single sRGB triplet. There are well-established techniques for solving under-determined
linear systems. The most common method goes by various names: the linear least squares method, the pseudo-inverse, the least-squares inverse, the Moore-Penrose inverse, and even the
“Fundamental Color Space” transformation5.
Suppose we pose this optimization problem:
3
This linearly constrained minimization can be solved easily by forming the Lagrangian function
The solution can be found by finding a stationary point of , i.e., setting partial derivatives with
respect to and equal to zero:
Solving this system by eliminating
gives the LLS solution
Thus, a reflectance curve can be found from a sRGB triplet by simply converting it to linear
and solving a 3 3 linear system of equations. Unfortunately, the resulting solution is sometimes
not very useful. Consider its application to this reflectance curve, which represents a bright red
object color:
Reflectance curve for Munsell 5R 4/14 color sample and linear least-squares reconstruction.
The LLS solution contains negative reflectance values, which don’t have physical meaning and
limit its usefulness in realistic applications. Computationally, this is a very efficient method. The
4
matrix
can be computed in advance, as shown here, and each new sRGB value
needs only be multiplied by it to get a reflectance curve.
Least Slope Squared (LSS) Method
Note that the standard LLS method minimizes
, that is, it finds the solution that is nearest to
the origin, or the reflectance curve with the least sum of squares of all its entries. For purposes of
computing reflectance curves, I can’t think of a compelling reason why this should be a useful
objective.
It dawned on me that it might be better to try a different objective function. The reflectance
curves of most natural colored objects don’t tend to oscillate up and down very much. I came up
with the idea to minimize the square of the slope of the reflectance curve, summed over the entire curve. In the continuous case, this would be equivalent to
The square is used because it equally penalizes upward and downward movement of . This objective will favor flatter reflectance curves and avoid curves that have a lot of up and down
movement.
Other researchers have developed methods for reconstructing reflectance curves from tristimulus
values. In order to reduce the oscillations, they typically introduce basis functions that oscillate
very little, such as segments of low-frequency sinusoids, or they “frequency limit” or “band limit” the solution by constraining portions of the Fourier transform of the reflectance curve. To my
mind, these approaches seem to ignore that fact that realistic reflectance curves can sometimes
exhibit relatively sharp wavelength cutoffs, which would have relatively large high frequency
Fourier components. These methods would not be able to create such reflectance curves.
My proposed method would be able to create sharp cutoffs, but only as a last resort when flatter
magnitude curves are not able to match the target tristimulus values. The other advantage of the
minimum slope squared approach is that it can be expressed as a quadratic objective function
subject to linear constraints, which is solvable by standard least-square strategies.
Consider this optimization formulation:
where is the number of discrete wavelength bands (36 in this study). This optimization can be
solved by solving the system of linear equations arising from the Lagrangian stationary conditions
5
where
is a 36 36 tridiagonal matrix
Since and do not depend on
, the matrix can be inverted ahead of time, instead of each
time an sRGB value is processed. Defining
we have
or
where
is the upper-right 36 3 portion of the matrix. Alternatively, the matrix inversion
leading to can be computed explicitly, yielding
This 36×3
matrix is shown here.
Since computing is a simple matter of matrix multiplication, the LSS method is just as computationally efficient as the LLS method, and tends to give much better reflectance curves, as I’ll
demonstrate later.
Here is a Matlab program for the LSS (Least Slope Squared) method. It also works in the open
source free alternative to Matlab, called Octave.
6
Least Log Slope Squared (LLSS) Method
It is generally not a good idea to allow reflectance curves with negative values. Not only is this
physically meaningless, but it also can cause problems down the road when the reflectance
curves are used in other computations. For example, when modeling subtractive color mixture1,
it may be necessary to require the reflectance curves to be strictly positive.
One way to modify the LSS method to keep reflectances positive is to operate the algorithm in
the space of the logarithm of reflectance values,
. I call this the Least Log Slope
Squared (LLSS) method:
This new optimization is not as easy to solve as the previous one. Nevertheless, the Lagrangian
formulation can still be used, giving rise to a system of 39 nonlinear equations and 39 unknowns:
where
is the same 36 36 tridiagonal matrix presented earlier.
Newton’s method solves this system of equations with ease, typically in just a few iterations.
Forming the Jacobian matrix,
the change in the variables with each Newton iteration is found by solving the linear system
Here is a Matlab program for the LLSS (Least Log Slope Squared) method. I added a check for
the special case of sRGB = (0,0,0), which simply returns
.
This is necessary since the log formulation is not able to create a reflectance of exactly zero (nor
is that desirable in some applications). It can come very close to zero as approaches
, but it
is numerically better to handle this one case specially. I chose the value of 0.0001 because it is
the largest power of ten that translates back to an integer sRGB triplet of (0,0,0).
7
The LLSS method requires substantially more computational effort than the previous two methods. Each iteration of Newton’s method requires the solution of 39 linear equations in 39 unknowns.
This Matlab program has also been tested in Octave and was found to work fine.
Iterative Least Log Slope Squared (ILLSS) Method
The LLSS method above can return reflectance curves with values >1. Although this is physically meaningful phenomenon (fluorescent objects can exhibit this), it may be desirable in some
applications to have the entire reflectance curve between 0 and 1. It dawned on me that I might
be able to modify the Lagrangian formulation to cap the reflectance values at 1. The main obstacle to doing this is that the use of inequality constraints in the Lagrangian approach greatly complicates the solution process, requiring the solution of the “KKT conditions,” and in particular,
the myriad “complementary slackness” conditions. If only there were some way to know which
reflectance values need to be constrained at 1, then these could be treated by a set of equality
constraints and no KKT solution would be necessary.
That led me to investigate the nature of the LLSS reflectance curves with values >1. I ran the
LLSS routine on every value of sRGB by intervals of five, that is, sRGB = (0,0,0), (0,0,5),
(0,0,10), …, (255,255,250), (255,255,255). In every one of those 140,608 cases, the algorithm
found a solution in less than a dozen or so iterations (usually just a handful), and 38,445 (27.3%)
of them had reflectance values >1.
Of the 38,445 solutions with values >1, 36,032 of them had a single contiguous region of reflectance values >1. The remaining 2,413 had two regions, always located at both ends of the visible
spectrum. Since the distribution of values >1 is so well defined, I started thinking of an algorithm
that would iteratively force the reflectance to 1. It would start by running the LLSS method. If
any of the reflectance values ended up >1, I would add a single equality constraint forcing the
reflectance at the maximum of each contiguous >1 region to equal 1, solve that optimization,
then force the adjacent values that were still >1 to 1, optimize again, and repeat until all values
were 1.
That was getting to be an algorithmic headache to implement, so I tried a simpler approach, as
follows. First, run the LLSS method. If any reflectance values end up >1, constrain ALL of them
to equal 1, and re-solve the optimization. This will usually cause some more values adjacent to
the old contiguous regions to become >1, so constrain them in addition to the previous ones. Resolve the optimization. Repeat the last two steps until all values are 1. Here is an animation of
this process, which I call the Iterative Least Log Slope Squared (ILLSS) process, applied to
sRGB = (75, 255, 255):
8
Animation of the ILLSS process (click image link to animate).
To express the ILLSS algorithm mathematically, let’s begin with the LLSS optimization statement and add the additional equality constraints:
“FixedSet” is the set of reflectance indices that are constrained to equal 1, or equivalently, the set
of indices constrained to equal zero (since
). Initially, FixedSet is set to be the empty
set. Each time the optimization is repeated, the values 0 have their indices added to this set.
We can define a matrix that summarizes the fixed set, for example:
This example indicates that there are two reflectance values being constrained (because
has
two rows), and the third and fifth reflectance values are the particular ones being constrained.
9
The Lagrangian formulation now has additional Lagrange multipliers, called , one for each of
the constrained reflectance values. The system of nonlinear equations produced by finding a stationary point of the Lagrangian (setting partial derivatives of the Lagrangian with respect to each
set of variables ( , , and ) equal to zero) is
where is the same 36 36 tridiagonal matrix presented earlier. As before, we solve this nonlinear system with Newton’s method. Forming the Jacobian matrix,
the change in the variables with each Newton iteration is found by solving the linear system
Here is a Matlab program that performs the ILLSS (Iterative Least Log Slope Squared) optimization. I included a check for the two special cases of
= (0,0,0) or (255,255,255), which simply
return = (0.0001, 0.0001, …, 0.0001) or (1, 1, …, 1). The additional special case of
(255,255,255) is needed because numerical issues arise if the
matrix grows to 36 36, as it
would in that second special case. This program works in Octave as well.
Iterative Least Slope Squared (ILSS) Method
For completeness, I thought it would be a good idea to add one more algorithm. Recall the
ILLSS method modifies the LLSS method to cap reflectances >1. Similarly, the ILSS method
will modify the LSS method to cap values both >1 and <0. The ILSS may reduce computational
effort in comparison to the ILLSS method since the inner loop of the ILLSS method requires an
iterative Newton's method solution, whereas there would be no inner loop needed with the ILSS
method; it is simply the solution of a linear system of equations. Here is the ILSS formulation:
10
is the set of reflectance indices that are constrained to equal 1, and
is the set
of indices that are constrained to equal
(the smallest allowable reflectance, typically
0.00001). Initially, both fixed sets are the empty set. Each time the optimization is repeated, the
values
have their indices added to
and those
have their indices added to
.
We define two matrices that summarize the fixed sets, for example:
This example indicates that there are two reflectance values being constrained to equal 1 (because
has two rows), and the third and fifth reflectance values are the particular ones being
constrained. There are three reflectance values being constrained to
(because
has three
rows), and the first, sixth, and fourth are the particular ones being constrained. The order in
which the rows appear in these matrices is not important.
At each iteration of the ILSS method, this linear system is solved:
where
and
are Lagrange multipliers associated with the
and
sets, respectively.
This system can be solved for , yielding the expression
where
and
are the upper-left 36 36 and upper-right 36 3 parts of , respectively. They can be
computed ahead of time. Note that only an
matrix needs to be inverted at each iteration,
where
is the number of values being held fixed, typically zero or a small number. When
is zero, the ILSS method simplifies to the LSS method.
11
Here is a Matlab program that performs the ILSS (Iterative Least Slope Squared) method. It also
works in Octave.
Comparison of Methods
I ran each of the five algorithms with every sRGB value (by fives) and have summarized the results in the table below. The total number of runs for each solution was 140,608.
Execution Max Min
Time (sec)
Name
LLS, Linear Least
Squares
LSS, Least Slope
Squared
ILSS, Iterative Least
Slope Squared
LLSS, Least Log
Slope Squared
ILLSS, Iterative Least
Log Slope Squared
Num Num Max Mean Computational
>1
<0 Iter. Iter.
Effort
2.7
1.36 -0.28 26,317 50,337 n/a
n/a
Matrix mult only
2.7
1.17 -0.17 9,316 48,164 n/a
n/a
Matrix mult only
25.7
1
1.49
322.
3.09 0
38,445 0
16
525.
1
0
5*
0
0
0
0
0
5
Mult soln of
linear eqns**
Mult soln of 39
6.77
linear eqns
Mult soln of (39+
1.41*
) linear eqns**
* This is outer loop iteration count. The inner loop has iteration count similar to previous line.
** The quantity is the number of reflectance values that end up being constrained at either 1 or 0.
Explanation of Columns
•
•
•
•
•
•
•
•
Execution Time: Real-time duration to compute 140,608 reflectance curves of all sRGB values (in intervals
of five), on a relatively slow Thinkpad X61 tablet. Relative times are more important than absolute times.
Max : The maximum reflectance value of all computed curves.
Min : The minimum reflectance value of all computed curves. Zero is used to represent some small specified lower bound, typically 0.0001.
Num >1: The number of reflectance curves with maxima above 1.
Num <0: The number of reflectance curves with minima below 0.
Max Iter.: The largest number of iterations required for any reflectance curve (for iterative methods).
Mean Iter.: The mean value of all iteration counts (for iterative methods).
Computational Effort: Comments on the type of computation required for the method.
Clearly, the methods that don’t need to solve linear systems of equations are much faster. The
reflectance curves they create, however, may not be very realistic.
Comparison of Reflectance Curves
In this section, I compare the computed reflectance curves against two large sets of measured
reflectance data. My goal is to assess how “realistic” the computed reflectance curves are in
comparison to reflectances measured from real colored objects.
12
Munsell Color Samples
The first set comes from the Munsell Book of Colors6, specifically the 2007 glossy version. The
Munsell system organizes colors by hue, chroma (like saturation), and value (like lightness). In
2012, Paul Centore measured 1485 different Munsell samples and published the results here.7 I
computed the sRGB values of each sample, and used those quantities to compute reflectance
curves for each of the five methods described above. Of the 1485 samples, 189 of them are outside the sRGB gamut (have values <0 or >255) and were not examined in this study, leaving
1296 in-gamut samples. An Excel file of the 1485 reflectance curves and sRGB values is available here.8
When comparing reflectance curves, some
parts of the curve are more import than others. As we approach both ends of the visible
spectrum, adjacent to the UV and IR ranges,
the human eye’s sensitivity to these wavelengths rapidly decreases. This phenomenon
is described by the “luminous efficiency”
curve9 shown in the adjacent figure. As I was
computing the reflectance curves, I noticed
that there are often large discrepancies between the computed and measured reflectance curves at the ends of the visible spectrum. The luminous efficiency curve, which describes the
relative sensitivity of the eye to spectral light across
These large differences have relatively little
the visible spectrum.
impact on color perception, and would also
have little consequence on operations performed on the computed reflectance curves, such as when modeling subtractive color mixture.
Consequently, I decided to downplay these end differences when comparing the curves.
To assess how well each computed reflectance matches the measured reflectance curve, I first
subtracted one from the other and took absolute values of the difference. Then I multiplied the
differences by the luminous efficiency curve. Finally, I summed all of the values to give a single
measure of how well the reflectance curves match, or a “reflectance match measure” (
):
Lower values of
represent a better match, with zero being a perfect match of the two reflectance curves. Keep in mind that regardless of the value of
, the sRGB triplets of the
computed and measured curves always match exactly, as would the perceived color evoked by
the two reflectance curves.
I examined the maximum and mean values of
to all 1296 Munsell samples:
for each of the five methods when applied
13
Name
LLS, Linear Least Squares
LSS, Least Slope Squared
ILSS, Iterative Least Slope Squared
LLSS, Least Log Slope Squared
ILLSS, Iterative Least Log Slope Squared
Max
2.46
1.11
1.04
0.92
0.86
Mean
0.88
0.17
0.16
0.15
0.15
The Linear Least Squares method is clearly much worse than the others. Considering that the
Least Slope Squared method (LSS) requires the same computational effort as LLS, but with
much better results, I decided to present the comparison figures that follow for the last four
methods only: LSS, ILSS, LLSS, and ILLSS.
The following figures compare the four methods. Each gray square represents one Munsell sample, and the shade of gray indicates the corresponding
value. The shade of gray is linearly mapped so that it is white for
= 0 and black for
= 1.11, the maximum value
for all four methods.
14
Overall, the log versions of the algorithms do a little better. The worst matches with non-logbased LSS/ILSS take place with the highly chromatic reds and purples, and sometimes with the
bright yellows. With log-based LLSS/ILLSS, the worst matches tend to be in the bright yellows,
and sometimes in the chromatic reds and oranges. In both pairs, the iterative version clips the
reflectance curves between 0 and 1, and usually reduces the mismatch somewhat, but not by very
much.
Here are some examples of LSS and ILSS with high
15
s:
This is typical of the mismatch with chromatic reds and purples. This one (Hue=5RP, Value=6,
Chroma=12) has an
of 1.04. The curve for LSS is not visible because it is directly behind
the ILSS curve. There was no clipping needed, so the two curves are the same.
One weakness of the LSS method is that it tends to drop into the negative region for chromatic
red colors. In these cases, the clipping provided by the ILSS method greatly improves the match.
The following example has an LSS
of 1.01 and an improved ILSS
of 0.40:
16
The following is a typical mismatch for LSS/ILSS in the yellow region:
Generally, when LSS/ILSS has a bad match, it is because it oscillates too little.
Here are some examples of log-based LLSS/ILLSS with high
17
s:
This previous example has an
of 0.88, mainly because it takes advantage of the wavelengths of light that give rise to a yellow sensation, in contrast to how the pigments in this sample get the same yellow sensation from a broader mix of wavelengths.
On bright red and orange Munsell samples, the LLSS/ILLSS curves tend to shoot high on the red
side:
Even though there is considerable clipping with the ILLSS version, most of it takes place in the
long wavelengths, which does not help the
score as much. This is evident in how little
ILLSS needs to adjust the mid-wavelength range to make up for the huge difference in the long
wavelengths, while maintaining the same sRGB values.
In summary, the non-log versions (LSS/ILSS) tend to undershoot peaks and the log versions
(LLSS/ILLSS) tend to overshoot them. Overall, the log versions give a somewhat better match,
but at the expense of considerably more computation.
Commercial Paints and Pigments
The second large dataset of reflectance curves comes from Zsolt Kovacs-Vajna’s RS2color
webpage10. He has a database of reflectance curves for many sets of commercial paints and pigments, which can be obtained by emailing a request to him. They are grouped into the following
31 families:
1. apaFerrario_PenColor
2. Chroma_AtelierInteractive
3. ETAC_EFX500b
4. ETAC_EFX500t
17. MunsellGlossy5G
18. MunsellGlossy5P
19. MunsellGlossy5R
20. MunsellGlossy5Y
18
5. GamblinConservationColors
6. Golden_HB
7. Golden_OpenAcrylics
8. GretagMacbethMini
9. Holbein_Aeroflash
10. Holbein_DuoP
11. KremerHistorical
12. Liquitex_HB
13. Maimeri_Acqua
14. Maimeri_Brera
15. Maimeri_Polycolor
16. MunsellGlossy5B
21. MunsellGlossyN
22. pigments
23. supports
24. Talens_Ecoline
25. Talens_Rembrandt
26. Talens_VanGoghH2Oil
27. whites
28. WinsorNewton_ArtAcryl
29. WinsorNewton_Artisan
30. WinsorNewton_Finity
31. WinsorNewtonHandbook
In total, there are 1493 different samples in these 31 groups. I discarded the 275 of them that
were out of the sRGB gamut, and computed reflectance curves from the sRGB values of the remaining 1218 samples. I again computed
values to compare the computed curves to the
actual measured curves. The following animated GIFs show the
values (color coded according to the colormap on the right) plotted in sRGB space.
Above,
values (by color) for the LSS method plotted in sRGB space.
(Click on link to view animated GIF.)
19
Above,
values (by color) for the ILSS method plotted in sRGB space. (Click for GIF)
Above,
values (by color) for the LLSS method plotted in sRGB space. (Click for GIF)
20
Above,
values (by color) for the ILLSS method plotted in sRGB space. (Click for GIF)
It is apparent from
these GIFs that the
non-log-based methods have a fairly large
region of mismatch in
the red region (large R,
small G and B). The
log-based versions
have a much smaller
region of mismatch in
the yellow region
(large R and G, small
B). The iterative version of both methods
improve the matches
in both cases. The relative sizes of the mismatch regions can be
seen more clearly in a
projection onto the RAbove,
G plane:
R-G plane.
values (by color) for the LSS method projected onto the sRGB
21
Above,
values (by color) for the ILSS method projected onto the sRGB R-G plane.
Above,
values (by color) for the LLSS method projected onto the sRGB R-G plane.
22
Above,
values (by color) for the ILLSS method projected onto the sRGB R-G plane.
Conclusions
In summary, the method that best matches paint and pigment colors found commercially and in
nature is the ILLSS (Iterative Least Log Slope Squared) method. It suffers, however, from very
large computational requirements. If efficiency is more important, then the ILSS (Iterative Least
Slope Squared) method is the preferred one. A third alternative that is midway between the
match quality and computing effort is LLSS (Least Log Slope Squared), but be aware that some
reflectance curves can end up with values >1. I would not recommend the LSS (Least Slope
Squared) method, despite its spectacular computational efficiency, because it can give reflectance curves with physically meaningless negative values. The following table summarizes the
three suggested methods:
Algorithm
Name
Computational
Effort
ILSS (Iterative
Least Slope
Squared)
Relatively little.
LLSS (Least
Log Slope
Squared)
About 12 times
that of ILSS.
Comments
Link to
Matlab/Octave
Code
Very fast, but tends to undershoot reflectance curve peaks, especially for bright
link
red and purple colors. Always returns reflectance values in the range 0-1.
Better quality matches overall, but tends
to overshoot peaks in the yellow region.
link
Some reflectance values can be >1, especially for bright red colors.
23
ILLSS (Iterative
About 20 times
Least Log Slope
that of ILSS.
Squared)
Best quality matches. Tends to overshoot
peaks in the yellow region. Always relink
turns reflectance values in the range 0-1.
Acknowledgments
This work has been a tremendous learning experience for me, and I want to thank several people
who graciously posted high quality material, upon which most of my work has been based:
Bruce Lindbloom11, Bruce MacEvoy12, Zsolt Kovacs-Vajna13, David Briggs14, and Paul Centore15. If you are interested in learning more about color theory, these five links are excellent places to start!
References
1. Burns, S. A., Subtractive Color Mixture Computation, arXiv:1710.06364 [cs.GR], and
http://scottburns.us/subtractive-color-mixture/
2. http://www.cie.co.at/index.php/LEFTMENUE/index.php?i_ca_id=298
3. http://www.w3.org/Graphics/Color/srgb
4. http://www.brucelindbloom.com/index.html?Math.html
5 Cohen J.B., Kappauf W.E. Metameric color stimuli, fundamental metamers, and Wyszecki’s metameric blacks.
Am J Psychol 1982; 95:537–564.
6. http://munsell.com/
7. http://www.munsellcolourscienceforpainters.com/MunsellResources/MunsellResources.html
8. http://scottburns.us/wp-content/uploads/2015/03/X-Rite-2007-Glossy-reflectance-with-D65-sRGB-1485chips.xlsx
9. https://en.wikipedia.org/wiki/Luminosity_function
10. http://zsolt-kovacs.unibs.it/colormixingtools/cmt-rs2color
11. http://www.brucelindbloom.com/
12. http://www.handprint.com/LS/CVS/color.html
13. http://zsolt-kovacs.unibs.it/colormixingtools
14. http://www.huevaluechroma.com/index.php
15. http://www.munsellcolourscienceforpainters.com/
________________________________________
Generating Reflectance Curves from sRGB Triplets by Scott Allen Burns is licensed under a
Creative Commons Attribution-ShareAlike 4.0 International License.
24
Appendix: Linked Textual Data
Several data tables and source codes are supplied in the text above via internet links. For archival
purposes, these tables and codes are supplied below.
M matrix (3x3)
Conversion between tristimulus values, XYZ, and linear rgb, referenced to D65
illuminant
3.243063328, -1.538376194, -0.49893282
-0.968963091, 1.875424508, 0.041543029
0.055683923, -0.204174384, 1.057994536
A' matrix (3x36)
CIE 1931 color matching functions for 380 to 730 nm by 10 nm intervals
0.001368, 0.004243, 0.01431, 0.04351, 0.13438, 0.2839, 0.34828, 0.3362,
0.2908, 0.19536, 0.09564, 0.03201, 0.0049, 0.0093, 0.06327, 0.1655, 0.2904,
0.43345, 0.5945, 0.7621, 0.9163, 1.0263, 1.0622, 1.0026, 0.85445, 0.6424,
0.4479, 0.2835, 0.1649, 0.0874, 0.04677, 0.0227, 0.011359, 0.00579, 0.002899,
0.00144
0.000039, 0.00012, 0.000396, 0.00121, 0.004, 0.0116, 0.023, 0.038, 0.06,
0.09098, 0.13902, 0.20802, 0.323, 0.503, 0.71, 0.862, 0.954, 0.99495, 0.995,
0.952, 0.87, 0.757, 0.631, 0.503, 0.381, 0.265, 0.175, 0.107, 0.061, 0.032,
0.017, 0.00821, 0.004102, 0.002091, 0.001047, 0.00052
0.00645, 0.02005, 0.06785, 0.2074, 0.6456, 1.3856, 1.74706, 1.77211, 1.6692,
1.28764, 0.81295, 0.46518, 0.272, 0.1582, 0.07825, 0.04216, 0.0203, 0.00875,
0.0039, 0.0021, 0.00165, 0.0011, 0.0008, 0.00034, 0.00019, 0.00005, 0.00002,
0, 0, 0, 0, 0, 0, 0, 0, 0
D65 W vector (36x1)
Illuminant D65 over 380 to 730 nm in 10 nm intervals
0.499755
0.546482
0.827549
0.91486
0.934318
0.866823
1.04865
1.17008
1.17812
1.14861
1.15923
1.08811
1.09354
1.07802
1.0479
25
1.07689
1.04405
1.04046
1.00000
0.963342
0.95788
0.886856
0.900062
0.895991
0.876987
0.832886
0.836992
0.800268
0.802146
0.822778
0.782842
0.697213
0.716091
0.74349
0.61604
0.698856
T matrix (3x36)
5.47813E-05, 0.000184722, 0.000935514, 0.003096265, 0.009507714, 0.017351596,
0.022073595, 0.016353161, 0.002002407, -0.016177731, -0.033929391, 0.046158952, -0.06381706, -0.083911194, -0.091832385, -0.08258148, 0.052950086, -0.012727224, 0.037413037, 0.091701812, 0.147964686,
0.181542886, 0.210684154, 0.210058081, 0.181312094, 0.132064724, 0.093723787,
0.057159281, 0.033469657, 0.018235464, 0.009298756, 0.004023687, 0.002068643,
0.00109484, 0.000454231, 0.000255925
-4.65552E-05, -0.000157894, -0.000806935, -0.002707449, -0.008477628, 0.016058258, -0.02200529, -0.020027434, -0.011137726, 0.003784809,
0.022138944, 0.038965605, 0.063361718, 0.095981626, 0.126280277, 0.148575844,
0.149044804, 0.14239936, 0.122084916, 0.09544734, 0.067421931, 0.035691251,
0.01313278, -0.002384996, -0.009409573, -0.009888983, -0.008379513, 0.005606153, -0.003444663, -0.001921041, -0.000995333, -0.000435322, 0.000224537, -0.000118838, -4.93038E-05, -2.77789E-05
0.00032594, 0.001107914, 0.005677477, 0.01918448, 0.060978641, 0.121348231,
0.184875618, 0.208804428, 0.197318551, 0.147233899, 0.091819086, 0.046485543,
0.022982618, 0.00665036, -0.005816014, -0.012450334, -0.015524259, 0.016712927, -0.01570093, -0.013647887, -0.011317812, -0.008077223, 0.005863171, -0.003943485, -0.002490472, -0.001440876, -0.000852895, 0.000458929, -0.000248389, -0.000129773, -6.41985E-05, -2.71982E-05, 1.38913E-05, -7.35203E-06, -3.05024E-06, -1.71858E-06
matrix (36x3)
0.0002,
0.0008,
0.0042,
0.0140,
0.0432,
-0.0001,
-0.0002,
-0.0009,
-0.0029,
-0.0083,
0.0019
0.0065
0.0334
0.1130
0.3593
26
0.0802, -0.0110, 0.7155
0.1063, -0.0000, 1.0915
0.0888, 0.0329, 1.2355
0.0350, 0.0845, 1.1720
-0.0367, 0.1483, 0.8820
-0.1052, 0.2355, 0.5632
-0.1507, 0.3232, 0.3044
-0.2089, 0.4874, 0.1819
-0.2698, 0.7227, 0.1092
-0.2824, 0.9513, 0.0598
-0.2302, 1.1326, 0.0413
-0.1105, 1.1548, 0.0285
0.0473, 1.1288, 0.0227
0.2359, 1.0011, 0.0198
0.4369, 0.8261, 0.0183
0.6450, 0.6415, 0.0176
0.7587, 0.4116, 0.0151
0.8609, 0.2517, 0.0137
0.8476, 0.1274, 0.0114
0.7265, 0.0513, 0.0089
0.5271, 0.0131, 0.0060
0.3731, -0.0016, 0.0041
0.2272, -0.0049, 0.0024
0.1329, -0.0042, 0.0014
0.0724, -0.0026, 0.0008
0.0369, -0.0015, 0.0004
0.0160, -0.0007, 0.0002
0.0082, -0.0004, 0.0001
0.0043, -0.0002, 0.0000
0.0018, -0.0001, 0.0000
0.0010, -0.0000, 0.0000
matrix (36x3)
Matrix "B12" which converts linear RGB values (0-1) to a
"representative" reflectance curve (over wavelengths
380 to 730 nm, in 10 nm intervals).
0.0933, -0.1729, 1.0796
0.0933, -0.1728, 1.0796
0.0932, -0.1725, 1.0794
0.0927, -0.1710, 1.0783
0.0910, -0.1654, 1.0744
0.0854, -0.1469, 1.0615
0.0723, -0.1031, 1.0308
0.0487, -0.0223, 0.9736
0.0147, 0.0980, 0.8873
-0.0264, 0.2513, 0.7751
-0.0693, 0.4234, 0.6459
-0.1080, 0.5983, 0.5097
-0.1374, 0.7625, 0.3749
-0.1517, 0.9032, 0.2486
-0.1437, 1.0056, 0.1381
-0.1080, 1.0581, 0.0499
-0.0424, 1.0546, -0.0122
0.0501, 0.9985, -0.0487
27
0.1641,
0.2912,
0.4217,
0.5455,
0.6545,
0.7421,
0.8064,
0.8494,
0.8765,
0.8922,
0.9007,
0.9052,
0.9073,
0.9083,
0.9088,
0.9090,
0.9091,
0.9091,
0.8972, -0.0613
0.7635, -0.0547
0.6129, -0.0346
0.4616, -0.0071
0.3238, 0.0217
0.2105, 0.0474
0.1262, 0.0675
0.0692, 0.0814
0.0330, 0.0905
0.0121, 0.0957
0.0006, 0.0987
-0.0053, 0.1002
-0.0082, 0.1009
-0.0096, 0.1012
-0.0102, 0.1014
-0.0105, 0.1015
-0.0106, 0.1015
-0.0107, 0.1015
LSS (Least Slope Squared) source code (Matlab and Octave)
function rho=LSS(B12,sRGB)
% This is the Least Slope Squared (LSS) algorithm for generating
% a "reasonable" reflectance curve from a given sRGB color triplet.
% The reflectance spans the wavelength range 380-730 nm in 10 nm increments.
% It solves min sum(rho_i+1 - rho_i)^2 s.t. T rho = rgb,
% using Lagrangian approach.
% B12 is upper-right 36x3 part of inv([D,T';T,zeros(3)])
% sRGB is a three-element vector of target D65-referenced sRGB values
%
in 0-255 range,
% rho is a 36x1 vector of reflectance values over wavelengths 380-730 nm,
%
%
%
%
%
Written by Scott Allen Burns, 4/25/15.
Licensed under a Creative Commons Attribution-ShareAlike 4.0 International
License (http://creativecommons.org/licenses/by-sa/4.0/).
For more information, see
http://www.scottburns.us/subtractive-color-mixture/
% compute target linear rgb values
sRGB=sRGB(:)/255; % convert to 0-1 column vector
rgb=zeros(3,1);
% remove gamma correction to get linear rgb
for i=1:3
if sRGB(i)<0.04045
rgb(i)=sRGB(i)/12.92;
else
rgb(i)=((sRGB(i)+0.055)/1.055)^2.4;
end
end
% matrix multiply
rho=B12*rgb;
LLSS (Least Log Slope Squared) source code (Matlab and Octave)
28
function rho=LLSS(T,sRGB)
% This is the Least Log Slope Squared (LLSS) algorithm for generating
% a "reasonable" reflectance curve from a given sRGB color triplet.
% The reflectance spans the wavelength range 380-730 nm in 10 nm increments.
% Solves min sum(z_i+1 - z_i)^2 s.t. T exp(z) = rgb, where
% z=log(reflectance), using Lagrangian formulation and Newton's method.
% Allows reflectance values >1 to be in solution.
% T is 3x36 matrix converting reflectance to D65-weighted linear rgb,
% sRGB is a 3 element vector of target D65 referenced sRGB values (0-255),
% rho is a 36x1 vector of reconstructed reflectance values, all > 0,
%
%
%
%
%
For more information, see
http://www.scottburns.us/subtractive-color-mixture/
Written by Scott Allen Burns, March 2015.
Licensed under a Creative Commons Attribution-ShareAlike 4.0 International
License (http://creativecommons.org/licenses/by-sa/4.0/).
% initialize outputs to zeros
rho=zeros(36,1);
% handle special case of (0,0,0)
if all(sRGB==0)
rho=0.0001*ones(36,1);
return
end
% 36x36 difference matrix for Jacobian
% having 4 on main diagonal and -2 on off diagonals,
% except first and last main diagonal are 2.
D=full(gallery('tridiag',36,-2,4,-2));
D(1,1)=2;
D(36,36)=2;
% compute target linear rgb values
sRGB=sRGB(:)/255; % convert to 0-1
rgb=zeros(3,1);
% remove gamma correction to get linear rgb
for i=1:3
if sRGB(i)<0.04045
rgb(i)=sRGB(i)/12.92;
else
rgb(i)=((sRGB(i)+0.055)/1.055)^2.4;
end
end
% initialize
z=zeros(36,1); % starting point all zeros
lambda=zeros(3,1); % starting Lagrange mult
maxit=100; % max number of iterations
ftol=1.0e-8; % function solution tolerance
deltatol=1.0e-8; % change in oper pt tolerance
count=0; % iteration counter
% Newton's method iteration
while count <= maxit
29
r=exp(z);
v=-diag(r)*T'*lambda; % 36x1
m1=-T*r; % 3x1
m2=-T*diag(r); % 3x36
F=[D*z+v;m1+rgb]; % 39x1 function vector
J=[D+diag(v),m2';m2,zeros(3)]; % 39x39 Jacobian matrix
delta=J\(-F); % solve Newton system of equations J*delta = -F
z=z+delta(1:36); % update z
lambda=lambda+delta(37:39); % update lambda
if all(abs(F)<ftol) % check if functions satisfied
if all(abs(delta)<deltatol) % check if variables converged
% solution found
disp(['Solution found after ',num2str(count),' iterations'])
rho=exp(z);
return
end
end
count=count+1;
end
disp(['No solution found in ',num2str(maxit),' iterations.'])
ILLSS (Iterative Least Log Slope Squared) source code (Matlab and Octave)
function rho=ILLSS(T,sRGB)
% This is the Iterative Least Log Slope Squared (ILLSS) algorithm for
% generating a "reasonable" reflectance curve from a given sRGB color
% triplet. The reflectance spans the wavelength range 380-730 nm in 10 nm
% increments.
% It solves min sum(z_i+1 - z_i)^2 s.t. T exp(z) = rgb, K z = 0, where
% z=log(reflectance), using Lagrangian approach and Newton's method.
% Clips values >1 and repeats optimization until all reflectance <=1.
% T
%
% sRGB
%
% rho
%
%
%
%
%
%
is 3x36 matrix converting reflectance to linear rgb over the
range 380-730 nm,
is a 3 element vector of target D65 referenced sRGB values
in 0-255 range,
is a 36x1 vector of reflectance values (0->1] over
wavelengths 380-730 nm,
Written by Scott Allen Burns, 4/11/15.
Licensed under a Creative Commons Attribution-ShareAlike 4.0 International
License (http://creativecommons.org/licenses/by-sa/4.0/).
For more information, see
http://www.scottburns.us/subtractive-color-mixture/
% initialize output to zeros
rho=zeros(36,1);
% handle special case of (0,0,0)
if all(sRGB==0)
rho=0.0001*ones(36,1);
return
end
% handle special case of (255,255,255)
30
if all(sRGB==255)
rho=ones(36,1);
return
end
% 36x36 difference matrix having 4 on main diagonal and -2 on off diagonals,
% except first and last main diagonal are 2.
D=full(gallery('tridiag',36,-2,4,-2));
D(1,1)=2;
D(36,36)=2;
% compute target linear rgb values
sRGB=sRGB(:)/255; % convert to 0-1 column vector
rgb=zeros(3,1);
% remove gamma correction to get linear rgb
for i=1:3
if sRGB(i)<0.04045
rgb(i)=sRGB(i)/12.92;
else
rgb(i)=((sRGB(i)+0.055)/1.055)^2.4;
end
end
% outer iteration to get all refl <=1
maxouter=10;
outer_count=0; % counter for outer iteration
while (any(rho>1) && outer_count<=maxouter) || all(rho==0)
% create K matrix for fixed refl constraints
fixed_refl=find(rho>=1)';
numfixed=length(fixed_refl);
K=zeros(numfixed,36);
for i=1:numfixed
K(i,fixed_refl(i))=1;
end
% initialize
z=zeros(36,1); % starting point all zeros
lambda=zeros(3,1); % starting point for lambda
mu=zeros(numfixed,1); % starting point for mu
maxit=50; % max number of iterations
ftol=1.0e-8; % function solution tolerance
deltatol=1.0e-8; % change in oper pt tolerance
count=0; % iteration counter
% Newton's method iteration
while count <= maxit
r=exp(z);
v=-diag(r)*T'*lambda; % 36x1
m1=-T*r; % 3x1
m2=-T*diag(r); % 3x36
F=[D*z+v+K'*mu;m1+rgb;K*z]; % function vector
J=[D+diag(v),[m2',K'];[m2;K],zeros(numfixed+3)]; % Jacobian matrix
delta=J\(-F); % solve Newton system of equations J*delta = -F
z=z+delta(1:36); % update z
lambda=lambda+delta(37:39); % update lambda
mu=mu+delta(40:end);
if all(abs(F)<ftol) % check if functions satisfied
31
if all(abs(delta)<deltatol) % check if variables converged
% solution found
disp(['Inner loop solution found after ',num2str(count),...
' iterations'])
rho=exp(z);
break
end
end
count=count+1;
end
if count>=maxit
disp(['No inner loop solution found after ',num2str(maxit),...
' iterations.'])
end
outer_count=outer_count+1;
end
if outer_count<maxouter
disp(['Outer loop solution found after ',num2str(outer_count),...
' iterations'])
else
disp(['No outer loop solution found after ',num2str(maxouter),...
' iterations.'])
end
ILSS (Iterative Least Slope Squared) source code (Matlab and Octave)
function rho=ILSS(B11,B12,sRGB)
% This is the Iterative Least Slope Squared (ILSS) algorithm for generating
% a "reasonable" reflectance curve from a given sRGB color triplet.
% The reflectance spans the wavelength range 380-730 nm in 10 nm increments.
%
%
%
%
%
%
It solves
min sum(rho_i+1 - rho_i)^2
s.t. T rho = rgb,
K1 rho = 1,
K0 rho = 0,
using Lagrangian formulation and iteration to keep all rho (0-1].
%
%
%
%
%
B11
B12
sRGB
rho
%
%
%
%
%
Written by Scott Allen Burns, 4/26/15.
Licensed under a Creative Commons Attribution-ShareAlike 4.0 International
License (http://creativecommons.org/licenses/by-sa/4.0/).
For more information, see
http://www.scottburns.us/subtractive-color-mixture/
is upper-left 36x36 part of inv([D,T';T,zeros(3)])
is upper-right 36x3 part of inv([D,T';T,zeros(3)])
is a 3-element vector of target D65-referenced sRGB values (0-255),
is a 36x1 vector of reflectance values (0->1] over
wavelengths 380-730 nm,
rho=ones(36,1)/2; % initialize output to 0.5
rhomin=0.00001; % smallest refl value
% handle special case of (255,255,255)
if all(sRGB==255)
rho=ones(36,1);
32
return
end
% handle special case of (0,0,0)
if all(sRGB==0)
rho=rhomin*ones(36,1);
return
end
% compute target linear rgb values
sRGB=sRGB(:)/255; % convert to 0-1 column vector
rgb=zeros(3,1);
% remove gamma correction to get linear rgb
for i=1:3
if sRGB(i)<0.04045
rgb(i)=sRGB(i)/12.92;
else
rgb(i)=((sRGB(i)+0.055)/1.055)^2.4;
end
end
R=B12*rgb;
% iteration to get all refl 0-1
maxit=10; % max iterations
count=0; % counter for iteration
while ( (any(rho>1) || any(rho<rhomin)) && count<=maxit ) || count==0
% create K1 matrix for fixed refl at 1
fixed_upper_logical = rho>=1;
fixed_upper=find(fixed_upper_logical);
num_upper=length(fixed_upper);
K1=zeros(num_upper,36);
for i=1:num_upper
K1(i,fixed_upper(i))=1;
end
% create K0 matrix for fixed refl at rhomin
fixed_lower_logical = rho<=rhomin;
fixed_lower=find(fixed_lower_logical);
num_lower=length(fixed_lower);
K0=zeros(num_lower,36);
for i=1:num_lower
K0(i,fixed_lower(i))=1;
end
% set up linear system
K=[K1;K0];
C=B11*K'/(K*B11*K'); % M*K'*inv(K*M*K')
rho=R-C*(K*R-[ones(num_upper,1);rhomin*ones(num_lower,1)]);
rho(fixed_upper_logical)=1; % eliminate FP noise
rho(fixed_lower_logical)=rhomin; % eliminate FP noise
count=count+1;
end
if count>=maxit
disp(['No solution found after ',num2str(maxit),' iterations.'])
end
33
| 1 |
Equivalence of Lattice Orbit Polytopes
FRIEDER LADISCH AND ACHILL SCHÜRMANN
arXiv:1703.01152v2 [math.MG] 31 Jan 2018
Dedicated to Jörg M. Wills on the occasion of his 80th birthday
A b s t r ac t . Let G be a finite permutation group acting on Rd by permuting
coordinates. A core point (for G) is an integral vector z ∈ Zd such that the
convex hull of the orbit Gz contains no other integral vectors but those in the
orbit Gz. Herr, Rehn and Schürmann considered the question for which groups
there are infinitely many core points up to translation equivalence, that is, up
to translation by vectors fixed by the group. In the present paper, we propose a
coarser equivalence relation for core points called normalizer equivalence. These
equivalence classes often contain infinitely many vectors up to translation, for
example when the group admits an irrational invariant subspace or an invariant
irreducible subspace occurring with multiplicity greater than 1. We also show that
the number of core points up to normalizer equivalence is finite if G is a so-called
QI-group. These groups include all transitive permutation groups of prime degree.
We exemplarily show how the concept of normalizer equivalence can be used to
simplify integer convex optimization problems.
1. I n t ro d u c t i o n
Let G 6 GL(d, Z) be a finite group. We consider orbit polytopes conv(Gz) of
integral vectors z ∈ Zd , that is, the convex hull of an orbit of a point z with integer
coordinates. We call z a core point for G, when the vertices are the only integral
vectors in the orbit polytope conv(Gz). Core points were introduced in [2, 17] in the
context of convex integer optimization, in order to develop new techniques to exploit
symmetries. Given a G-symmetric convex set, it contains an integer vector if and
only if it contains a core point for G. It is therefore possible to reduce a G-invariant
convex integer optimization problem to the optimization of core points or even a
subset of them. Different algorithms can take advantage of core points, in particular
for groups G with all core points known. Even naive approaches work well for special
problem classes and have solved a previously open problem from the MIPLIB 2010
[24] (see [2, 17]).
In this paper we extend the list of groups for which there are only finitely many
core points up to a notion of equivalence. As explained below, this is achieved
2010 Mathematics Subject Classification. Primary 20C10; Secondary 16U60, 20B25, 20C15,
52B20, 90C10.
Key words and phrases. Orbit polytope, core points, group representation, lattice, integer linear
programming.
The authors were supported by the DFG (Project: SCHU 1503/6-1).
1
2
FRIEDER LADISCH AND ACHILL SCHÜRMANN
by introducing a new notion of equivalence which is coarser than the equivalence
relation previously used. It turns out that not only is this new notion helping to
classify core points, but also it suggests a new way to transform G-invariant convex
integer optimization problems in a natural way into possibly simpler equivalent ones.
Knowing a group G of symmetries, elements of its normalizer in GL(d, Z) can be
used to transform a G-invariant convex integer optimization problem linearly into
an equivalent G-invariant problem. As we exemplarily show in Section 6 for the
case of integer linear problem instances, the transformed optimization problems
are sometimes substantially easier to solve. To apply this technique in general, one
needs to know how to find a good transformation from the normalizer which yields
an easy-to-solve transformed problem. While this is easy in some cases as in our
examples, we do not yet understand satisfactorily how to find good transformations
from the normalizer in general.
In the following, we write
Fix(G) = {v ∈ Rd | gv = v for all g ∈ G}
for the fixed space of G in Rd . Notice that when z is a core point and t ∈ Fix(G) ∩ Zd ,
then z + t is another core point. We call the core points z and z + t translation
equivalent. Herr, Rehn and Schürmann [18] consider the question whether there
are finitely or infinitely many core points up to translation equivalence, in the case
where G is a permutation group acting by permuting coordinates. Their methods
can be used to show that there are only finitely many core points up to translation,
when Rd / Fix(G) has no G-invariant subspaces other than the trivial ones [34,
Theorem 3.13]. They conjectured that in all other cases, there are infinitely many
core points up to translation. This has been proved in special cases, but is open in
general.
In this paper, we consider a coarser equivalence relation, where we allow to
multiply core points with invertible integer matrices S ∈ GL(d, Z) which normalize
the subgroup G. Thus two points z and w are called normalizer equivalent, when
w = Sz + t, where S is an element of the normalizer of G in GL(d, Z) (in other
words, S −1 GS = G), and t ∈ Fix(G) ∩ Zd . In Theorem 4.1, we will determine when
these coarser equivalence classes contain infinitely many points up to translation
equivalence, in terms of the decomposition into irreducible invariant subspaces. For
example, if Rd has an irrational invariant subspace U 6 Rd (that is, a subspace
{0} =
6 U 6 Rd such that U ∩ Zd = {0}), then each integer point z with nonzero
projection onto U is normalizer equivalent to infinitely many points, which are
not translation equivalent. This yields another proof of the result of Herr, Rehn
and Schürmann [18, Theorem 32] that there are infinitely many core points up to
translation, when there is an irrational invariant subspace.
In Theorem 5.1, we prove the following: Suppose that G 6 Sd is a transitive
permutation group acting on Rd by permuting coordinates. Suppose that Fix(G)⊥
contains no rational G-invariant subspace other than {0} and Fix(G)⊥ itself. (A
Equivalence of Lattice Orbit Polytopes
3
subspace of Rd is rational, if it has a basis contained in Qd .) Such a group G is
called a QI-group. Then there are only finitely many core points up to normalizer
equivalence.
For example, this is the case, when d = p is a prime number (and G 6 Sp is
transitive). In the case that the group is not 2-homogeneous, there are infinitely
many core points up to translation, but these can be obtained from finitely many by
multiplying with invertible integer matrices from the normalizer.
The paper is organized as follows. In Section 2, we introduce different equivalence
relations for core points and make some elementary observations. Section 3 collects
some elementary properties of orders in semisimple algebras. In Section 4, we
determine when the normalizer equivalence classes contain infinitely many points
up to translation equivalence. In Section 5, we prove the aforementioned result on
QI-groups. Sections 4 and 5 can mostly be read independently from another. In the
final Section 6 we exemplarily show how normalizer equivalence can be applied to
integer convex optimization problems with suitable symmetries.
2. E q u i va l e n c e f o r c o r e p o i n t s
Let V be a finite-dimensional vector space over the real numbers R and G a finite
group acting linearly on V .
2.1. Definition. An orbit polytope (for G) is the convex hull of the G-orbit of a
point v ∈ V . It is denoted by
P(G, v) = conv{gv | g ∈ G}.
Let Λ ⊆ V be a full Z-lattice in V , that is, the Z-span of an R-basis of V . Assume
that G maps Λ onto itself.
2.2. Definition. [17] An element z ∈ Λ is called a core point (for G and Λ) if the
only lattice points in P(G, z) are its vertices, that is, the elements in the orbit Gz.
In other words, z is a core point if
P(G, z) ∩ Λ = Gz.
The barycenter
1 X
gv ∈ P(G, v)
|G| g∈G
is always fixed by G. If FixV (G), the set of vectors in V fixed by all g ∈ G, consists
only of 0, then the barycenter of each orbit polytope is the zero vector. In this case,
only the zero vector is a core point [34, Remark 3.7, Lemma 3.8].
More generally, the map
1 X
e1 =
g
|G| g∈G
gives the projection from V onto the fixed space FixV (G) [37, Theorem 8 in §2.6],
and thus yields a decomposition V = FixV (G)⊕ker(e1 ) into G-invariant subspaces. If
4
FRIEDER LADISCH AND ACHILL SCHÜRMANN
this decomposition restricts to a decomposition of the lattice, Λ = e1 Λ⊕(ker(e1 )∩Λ),
then e1 z ∈ Λ ∩ P (G, z) for any z ∈ Λ, and so z can only be a core point for G when
z is itself in the fixed space. But in general, we do not have such a decomposition,
since the projection e1 Λ may not be contained in Λ.
An important class of examples where e1 Λ 6⊆ Λ are transitive permutation groups
G 6 Sd , acting on V = Rd by permuting coordinates, and where Λ = Zd . The fixed
space consists of the vectors with all entries equal and is thus generated by the all
P
ones vector 1 := (1, 1, . . . , 1)t . For v = (v1 , . . . , vd )t we have e1 v = ( i vi )/d · 1. In
particular, we see that e1 Λ contains all integer multiples of (1/d)1. We can think of
Λ as being partitioned into layers, where a layer consists of all z ∈ Λ with z t 1 = k
(equivalently, e1 z = (k/d)1), for a fixed integer k.
Returning to general groups of integer matrices, we claim that for each v ∈ e1 Λ,
there are core points z with e1 z = v. Namely, among all z ∈ Λ with e1 z = v, there
P
are elements such that g kgzk2 is minimal, and these are core points.
If z is a core point and b ∈ FixΛ (G), then z + b is a core point, too, because
P(G, z + b) = P(G, z) + b. Such core points should be considered as equivalent. This
viewpoint was adopted by Herr, Rehn and Schürmann [17, 18]. In the present paper,
we consider a coarser equivalence relation. We write GL(Λ) for the invertible Z-linear
maps Λ → Λ. Since Λ contains a basis of V , we may view GL(Λ) as a subgroup of
GL(V ). (If V = Rd and Λ = Zd , then we can identify GL(Λ) with GL(d, Z), the set
of matrices over Z with determinant ±1.)
By assumption, G is a subgroup of GL(Λ). We use the following notation from
group theory: The normalizer of a finite group G in GL(Λ) is the set
NGL(Λ) (G) := {S ∈ GL(Λ) | ∀g ∈ G : S −1 gS ∈ G}.
The centralizer of G in GL(Λ) is the set
CGL(Λ) (G) := {S ∈ GL(Λ) | ∀g ∈ G : S −1 gS = g}.
2.3. Definition. Two points z and w are called normalizer equivalent, if there is a
point b ∈ FixΛ (G) and an element S in the normalizer NGL(Λ) (G) of G in GL(Λ)
such that w = Sz + b. The points are called centralizer equivalent if w = Sz + b with
S ∈ CGL(Λ) (G) and b ∈ FixΛ (G). Finally, we call two points z and w translation
equivalent, when w − z ∈ FixΛ (G).
Since CGL(Λ) (G) ⊆ NGL(Λ) (G), each normalizer equivalence class is a union of
centralizer equivalence classes, and obviously each centralizer equivalence class is a
union of translation equivalence classes. The definition is motivated by the following
simple observation:
2.4. Lemma. If
w = Sz + b with
S ∈ NGL(Λ) (G) and
b ∈ FixΛ (G),
then x 7→ Sx + b defines a bijection between
P(G, z) ∩ Λ
and
P(G, w) ∩ Λ.
Equivalence of Lattice Orbit Polytopes
5
In particular, z is a core point for G if and only if w is a core point for G.
Proof. The affine bijection x 7→ Sx + b maps the orbit polytope P(G, z) to another
polytope. The vertex gz is mapped to the vertex
Sgz + b = (SgS −1 )Sz + b = hSz + b = h(Sz + b) = hw,
where h = SgS −1 ∈ G (since S normalizes G). The second last equality follows as b
is fixed by G. As SgS −1 runs through G as g does, it follows that x 7→ Sx + b maps
the orbit Gz to the orbit Gw and thus maps the orbit polytope P(G, z) to the orbit
polytope P(G, w). Since x 7→ Sx + b also maps Λ onto itself, the result follows.
Notice that a point w is equivalent to z = 0 (for any of the equivalence relations
in Definition 2.3), if and only if w = S · 0 + b = b ∈ FixΛ (G). Any w ∈ FixΛ (G) is a
core point. We call these points the trivial core points.
In the important example of transitive permutation groups, the fixed space is
one-dimensional. More generally, when V is spanned linearly by some orbit Gz, then
FixV (G) is spanned by e1 z and thus dim(FixV (G)) 6 1.
2.5. Remark. Suppose that FixV (G) has dimension 1. Then there is at most one
w ∈ NGL(Λ) (G)z such that w 6= z and w is translation equivalent to z.
Proof. The elements of NGL(Λ) (G) map FixΛ (G) onto itself, and thus act on FixV (G)
as ±1. Let S ∈ NGL(Λ) (G). Suppose w = Sz and z are translation equivalent, so
that Sz − z = b ∈ FixΛ (G). Then b = e1 b = e1 Sz − e1 z = Se1 z − e1 z = ±e1 z − e1 z
and thus either Sz = z or Sz = z − 2e1 z. (The latter case can only occur when
2e1 z ∈ Λ.)
In particular, if the orbit NGL(Λ) (G)z is infinite, then the normalizer equivalence
class of a nontrivial core point z contains infinitely many translation equivalence
classes.
Also notice that when z is a nontrivial core point, then e1 z must not be a lattice
point.
Herr, Rehn and Schürmann [18, 16, 34] considered the question whether the set
of core points up to translation is finite or infinite (in the case where G acts by
permuting coordinates). We might ask the same question about core points up to
normalizer equivalence as defined here. Also, it is of interest whether our bigger
equivalence classes contain finitely or infinitely many points up to translation.
2.6. Example. Let G = Sd , the symmetric group on d elements, acting on Rd by
permuting coordinates, and Λ = Zd . We identify G with the group of all permutation
matrices. For this group, Bödi, Herr and Joswig [2] have shown that every core point
is translation equivalent to a vector with all entries 0 or 1. (Conversely, these vectors
are obviously core points.) One can show that the normalizer of the group G of all
permutation matrices in GL(d, Z) is generated by −I and the group G itself. As G
is transitive on the subsets of {1, . . . , d} of size k, all 0/1-vectors with fixed number
k of 1’s are normalizer equivalent. A vector z with k ones and d − k zeros is also
6
FRIEDER LADISCH AND ACHILL SCHÜRMANN
normalizer equivalent to the vector −z + 1 with d − k ones and k zeros. Thus up to
normalizer equivalence, there are only bd/2c + 1 core points.
2.7. Example. Let G = Cd = h(1, 2, . . . , d)i be a cyclic group, again identified
with a matrix group which acts on Rd by permuting the coordinates cyclically. For
d = 4 we have a finite normalizer (as we will see in Section 4), but infinitely many
core points up to normalizer or translation equivalence: for example, all the points
(1 + m, −m, m, −m)t , m ∈ Z, are core points for C4 [18, Example 26].
If d = p is prime, then we will see that there are only finitely many core points
up to normalizer equivalence, but for p > 5 the normalizer is infinite and there are
infinitely many core points up to translation equivalence. (See Example 5.9 below.)
For d = 8 (say), the normalizer is infinite and there are infinitely many core points
up to normalizer equivalence. Namely, let b1 ∈ R8 be the first standard basis vector
and let v ∈ R8 be the vector with entries alternating between 1 and −1. Then the
points b1 + mv for m ∈ Z are core points [18, Theorem 30] (the construction principle
here is the same as above in the case d = 4). The circulant 8 × 8-matrix S with
first row (2, 1, 0, −1, −1, −1, 0, 1) is contained in the centralizer of G and has infinite
order. Since S is symmetric and Sv = v, we have v t S k b1 = v t b1 = 1 for all k ∈ Z
and thus the vectors S k b1 + mv are all different for different pairs (k, m) ∈ Z2 . And
since we also have S1 = 1, where 1 = (1, 1, . . . , 1)t spans the fixed space, we also
see that different vectors of the form S k b1 + mv can not be translation equivalent.
Finally, one can show that the the subgroup generated by S has finite index in the
normalizer NGL(8,Z) (C8 ). Thus at most finitely many of the points b1 + mv can be
normalizer equivalent to each other.
It is sometimes easier to work with the centralizer CGL(Λ) (G) instead of the
normalizer NGL(Λ) (G), which yields a slightly finer equivalence relation. By the
following simple observation, the CGL(Λ) (G)-equivalence classes can not be much
smaller than the NGL(Λ) (G)-equivalence classes:
2.8. Lemma. |NGL(Λ) (G) : CGL(Λ) (G)| is finite.
Proof. The factor group NGL(Λ) (G)/ CGL(Λ) (G) is isomorphic to a subgroup of
Aut(G) [21, Corollary X.19], and Aut(G) is finite, since G itself is finite by assumption.
3. P r e l i m i n a r i e s o n o r d e r s
In this section, we collect some simple properties of orders in semi-simple algebras
over Q. Orders are relevant for us since the centralizer CGL(Λ) (G) can be identified
with the unit group of such an order, as we explain below.
Recall the following definition [35]: Let A be a finite-dimensional algebra over Q
(associative, with one). An order (or Z-order) in A is a subring R ⊂ A which is
finitely generated as Z-module and contains a Q-basis of A. (Here, “subring” means
in particular that R and A have the same multiplicative identity.) In other words,
an order is a full Z-lattice in A which is at the same time a subring of A.
Equivalence of Lattice Orbit Polytopes
7
For the moment, assume that W is a finite-dimensional vector space over the
rational numbers Q, and let Λ be a full Z-lattice in W, that is, the Z-span of a
Q-basis of W, and G a finite subgroup of GL(Λ). (In the situation of Section 2,
we can take for W the Q-linear span of Λ.) Let A := EndQG (W ) be the ring of
QG-module endomorphisms of W , that is, the set of linear maps α : W → W such
that α(gv) = gα(v) for all v ∈ W and g ∈ G. This is just the centralizer of G in the
ring of all Q-linear endomorphisms of W .
We claim that
R := {α ∈ A | α(Λ) ⊆ Λ}
is an order in A. Namely, choose a Z-basis of Λ. This basis is also a Q-basis of W .
By identifying linear maps with matrices with respect to the chosen basis, A gets
identified with the centralizer of G in the set of all d × d matrices over Q, and R
gets identified with the centralizer of G in the set of d × d matrices with entries in
Z. It follows that R is finitely generated as Z-module, and for every α ∈ A there
is an m ∈ Z such that mα ∈ R. Thus R is an order of A. (Also, R ∼
= EndZG (Λ)
naturally.)
Moreover, the centralizer CGL(Λ) (G) is exactly the set of invertible elements of R,
that is, the unit group U(R) of R. For this reason, it is somewhat easier to work with
CGL(Λ) (G) instead of the normalizer NGL(Λ) (G). The unit group U(R) of an order
R is a finitely generated (even finitely presented) group [23, § 3]. Finding explicit
generators of U(R) (and relations between them) is in general a difficult task, but
there do exist algorithms for this purpose [3]. The situation is somewhat better when
R is commutative, for example when R ∼
= ZA, where A is a finite abelian group [9].
Moreover, it is quite easy to give generators of a subgroup of U(ZA) which has finite
index in U(ZA) [19, 28].
We now collect some general elementary facts about orders that we need. (For a
comprehensive treatment of orders, not only over Z, we refer the reader to Reiner’s
book on maximal orders [35]. For unit groups of orders, see the survey article by
Kleinert [23].)
3.1. Lemma. Let R1 and R2 be two orders in the Q-algebra A. Then R1 ∩ R2 is
also an order in A.
Proof. Clearly, R1 ∩ R2 is a subring.
Since R2 is finitely generated over Z and QR1 = A, there is a non-zero integer
m ∈ Z with mR2 ⊆ R1 . Thus mR2 ⊆ R1 ∩ R2 . Since mR2 contains a Q-basis of A,
it follows that R1 ∩ R2 contains such a basis. As a submodule of a finitely generated
Z-module, R1 ∩ R2 is again finitely generated. Thus R1 ∩ R2 is an order of A.
3.2. Lemma. Let R1 and R2 be orders in the Q-algebra A with R1 ⊆ R2 . Then
|U(R2 ) : U(R1 )| is finite.
Proof. There exists a non-zero integer m such that mR2 ⊆ R1 . Suppose that u,
v ∈ U(R2 ) are such that u − v ∈ mR2 . Then u ∈ v + mR2 and thus uv −1 ∈
8
FRIEDER LADISCH AND ACHILL SCHÜRMANN
1 + mR2 ⊆ R1 . Similarly, vu−1 ∈ 1 + mR2 ⊆ R1 . Thus uv −1 ∈ U(R1 ). This shows
|U(R2 ) : U(R1 )| 6 |R2 : mR2 | < ∞, as claimed.
3.3. Corollary. Let R1 and R2 be two orders in the Q-algebra A. Then U(R1 ) is
finite if and only if U(R2 ) is finite.
Proof. By Lemma 3.1, R1 ∩ R2 is an order. By Lemma 3.2, the index |U(Ri ) :
U(R1 ∩ R2 )| is finite for i = 1, 2. The result follows.
4. F i n i t e n e s s o f e q u i va l e n c e c l a s s e s
In this section we determine for which groups G the normalizer equivalence classes
are finite or not. We use the notation introduced in Section 2. Thus G is a finite
group acting on the finite-dimensional, real vector space V , and Λ ⊂ V is a full
Z-lattice in V which is stabilized by G. A subspace U 6 V is called Λ-rational if
U ∩ Λ contains a basis of U , and Λ-irrational, if U ∩ Λ = {0}. If U is an irreducible
RG-submodule, then U is either Λ-rational or Λ-irrational.
4.1. Theorem. Let
V = U1 ⊕ · · · ⊕ Ur
be a decomposition of V into irreducible RG-subspaces. Then NGL(Λ) (G) has finite
order if and only if all the Ui ’s are Λ-rational and pairwise non-isomorphic.
The proof of Theorem 4.1 involves some non-trivial representation and number
theory. By Lemma 2.8, the normalizer NGL(Λ) (G) is finite if and only if the centralizer
CGL(Λ) (G) is finite. As remarked earlier, the centralizer can naturally be identified
with the set of units of the ring EndZG (Λ), and EndZG (Λ) is an order in the Q-algebra
EndQG (QΛ), where QΛ denotes the Q-linear span of Λ. For this reason, it is more
convenient to work with the Q-vector space W := QΛ. (We get back our V from W
by scalar extension, that is, V ∼
= R ⊗Q W .)
Fix a decomposition of W = QΛ into simple modules:
W ∼
= m1 S1 ⊕ · · · ⊕ mr Sr ,
mi ∈ N,
where we assume that Si ∼
6= Sj for i 6= j. Set Di := EndQG (Si ), which is by Schur’s
lemma [25, (3.6)] a division ring, and finite dimensional over Q.
4.2. Lemma. With the above notation, we have
EndQG (W ) ∼
= Mm1 (D1 ) × · · · × Mmr (Dr ),
where Mm (D) denotes the ring of m × m matrices with entries in D. If Ri is an
order in Di for each i, then
R := Mm1 (R1 ) × · · · × Mmr (Rr )
is an order in EndQG (W ).
Equivalence of Lattice Orbit Polytopes
9
Proof. The first assertion is a standard observation, used for example in one proof
of the Wedderburn-Artin structure theorem for semisimple rings [25, Thm. 3.5 and
proof]. The assertion on orders is then easy.
In particular, the group of units of R is then isomorphic to the direct product of
groups of the form GL(mi , Ri ). To prove Theorem 4.1, in view of Corollary 3.3, it
suffices to determine when all these groups are finite. The following is a first step
towards the proof of the theorem:
4.3. Corollary. If some mi > 1, then U(R) (and thus NGL(Λ) (G)) is infinite.
Proof. U(R) contains a subgroup isomorphic to GL(mi , Ri ), which contains the
group GL(mi , Z). This group is infinite if mi > 1.
To continue with the proof of Theorem 4.1, we have to look at the units of
an order Ri in Di . We will need extension of scalars for algebras over a field via
tensor products, as explained in [10, Chapter 3]. Thus for a Q-algebra A, we get an
R-algebra denoted by R ⊗Q A. We use the following theorem of Käte Hey which can
be seen as a generalization of Dirichlet’s unit theorem:
4.4. Theorem. [23, Theorem 1] Let D be a finite dimensional division algebra over
Q, and let R be an order of D with unit group U(R). Set
S = {d ∈ R ⊗Q D | (det d)2 = 1}.
Then S/ U(R) is compact. (Here det d refers to the action of d as linear operator on
R ⊗Q D. One can also use the reduced norm, of course.)
From this, we can derive the following result (probably well known):
4.5. Lemma. Let D be a finite dimensional division algebra over Q and R an order
of D. Then |U(R)| < ∞ if and only if R ⊗Q D is a division ring.
Proof. Suppose DR := R ⊗Q D is a division ring. By Frobenius’ theorem [10, Theorem 3.20], we have DR ∼
= R, C or H. In each case, one checks that the set S defined in
Theorem 4.4 is compact. Thus the discrete group U(R) ⊆ S must be finite. (Notice
that we did not use Theorem 4.4 here, only that U(R) ⊆ S.)
Conversely, suppose that DR is not a division ring. Then there is some nontrivial idempotent e ∈ DR , that is, e2 = e, but e 6= 0, 1. (This follows since DR is
semisimple.) Set f = 1 − e. Then for λ, µ ∈ R, we have det(λe + µf ) = λk1 µk2 with
k1 = dim(DR e) and k2 = dim(DR f ). In particular, for every λ 6= 0 there is some µ
such that λe + µf ∈ S. This means that S is unbounded, and thus not compact. It
follows from Theorem 4.4 that U(R) can not be finite.
Proof of Theorem 4.1. First, assume that we are given a decomposition V = U1 ⊕
· · · ⊕ Ur as in the theorem. Then Si := Ui ∩ QΛ contains a basis of Ui and thus is
non-zero, and necessarily simple as QG-module. Thus
W = V ∩ QΛ = S1 ⊕ · · · ⊕ Sr
10
FRIEDER LADISCH AND ACHILL SCHÜRMANN
is a decomposition of W into simple QG-modules, which are pairwise non-isomorphic.
It follows that
EndQ (W ) ∼
= D1 × · · · × Dr ,
where Di = EndQG (Si ). Since R ⊗Q Di ∼
= EndRG (Ui ) is a division ring, too, it follows
that the orders of each Di have a finite unit group, by Lemma 4.5. Thus CGL(Λ) (G)
is finite.
Conversely, assume that NGL(Λ) (G) is finite. It follows from Corollary 4.3 that
mi = 1 for all i (in the notation introduced before Lemma 4.2). Thus W has a
decomposition into simple summands which are pairwise non-isomorphic:
W = S1 ⊕ · · · ⊕ Sr .
Let Di = EndRG (Si ). Then Lemma 4.5 yields that R ⊗Q Di is a division ring, too.
Since R ⊗Q Di ∼
= EndRG (RSi ), it follows that Ui := RSi is simple. (Otherwise, the
projection to a nontrivial invariant submodule would be a zero-divisor in EndRG (Ui ).)
For i =
6 j, we have Ui ∼
6= Uj by the Noether-Deuring theorem [25, Theorem 19.25].
Thus V has a decomposition V = U1 ⊕ · · · ⊕ Ur as required.
4.6. Remark. Let z ∈ V be an element such that the orbit Gz linearly spans V.
Then the normalizer equivalence class of z contains infinitely many translation
equivalence classes if (and only if) NGL(Λ) (G) has infinite order.
Proof. The “only if” part is clear, so assume that NGL(Λ) (G) has infinite order.
By Remark 2.5, it suffices to show that the orbit NGL(Λ) (G)z has infinite size.
By Lemma 2.8, the centralizer CGL(Λ) (G) has also infinite order. If cz = z for
c ∈ CGL(Λ) (G), then cgz = gcz = gz for all g ∈ G and thus c = 1. Thus
∞ = |CGL(Λ) (G)| = |CGL(Λ) (G)z| 6 |NGL(Λ) (G)z|.
So when NGL(Λ) (G) is infinite, only elements contained in proper invariant subspaces can have finite orbits under the normalizer. (Notice that the linear span of
an orbit Gz is always a G-invariant subspace of V .) If G is a transitive permutation
group acting on the coordinates, then there are always points z such that the orbit
Gz spans the ambient space, for example z = (1, 0, . . . , 0)t .
When V has an irrational invariant subspace, then NGL(Λ) (G) is infinite, by
Theorem 4.1. Thus if z is a core point for G such that its orbit spans the ambient
space, then there are infinitely many core points, even up to translation. This was
first proved by Rehn [34, 18] for permutation groups.
Another consequence of Theorem 4.1 and the remark above is that there are
infinitely many core points for transitive permutations groups G acting on V = Rd
such that V is not multiplicity-free (as RG-module).
4.7. Example. Consider the regular representation of a group G, that is, G acts on
QG by left multiplication, so it permutes the canonical basis G. As lattice, we choose
the group ring ZG, the vectors with integer coordinates. Then EndZG (ZG) ∼
= ZG.
Equivalence of Lattice Orbit Polytopes
11
Units of group rings are a much studied problem. A theorem of Higman says that
U(ZG) is finite if and only if G is abelian of exponent 1, 2, 3, 4 or 6, or G ∼
= Q8 × E
2
with E = {1}. This can also be derived from Theorem 4.1.
In Example 2.7, we described some core points in the cases G = C4 and C8 . In
the case of C8 , the decomposition of QC8 into simple modules is given by
QC8 ∼
= Q ⊕ Q ⊕ Q[i] ⊕ Q[e2πi/8 ].
Over R, the last summand decomposes into two invariant, irrational subspaces of
dimension 2. The normalizer of C8 is infinite because of this last summand. Of
course, any z contained in the sum of the first three summands has only a finite
orbit under the normalizer, for example z = (1, 0, 0, 0, 1, 0, 0, 0)t .
When p is prime and p > 5, then U(ZCp ) is infinite, but there are only finitely
many core points up to normalizer equivalence in ZCp , by Theorem 5.1 below.
5. R at i o n a l ly i r r e d u c i b l e
Suppose that Λ = Zd , and assume that G acts on Rd by matrices in GL(d, Z). A
subspace U 6 Rd is called irrational, if U ∩ Qd = {0}, and rational, if U has a basis
contained in Qd . If U is an irreducible RG-submodule, then U is either rational or
irrational.
In this section, we consider permutation groups acting on Rd by permuting
coordinates. (We conjecture that a version of the main result remains true more
generally for finite matrix groups G 6 GL(d, Z), but we are not able to prove it
yet. One problem is that we can not extend Lemma 5.2 below to this more general
setting.)
Since permutation matrices are orthogonal, it follows that the orthogonal complement U ⊥ of any G-invariant subspace is itself G-invariant. Following Dixon [8],
we call a transitive permutation group G a QI-group, when Fix(G)⊥ does not contain any rational G-invariant subspace other than {0} and Fix(G)⊥ itself. Notice
that Fix(G)⊥ contains no non-trivial rational invariant subspaces if and only if
Fix(G)⊥ ∩ Qd contains no proper G-invariant subspace other than {0}. In algebraic
language, this means that Fix(G)⊥ ∩ Qd is a simple module over QG.
Let us emphasize that by definition, QI-groups are transitive. Thus the fixed space
Fix(G) is generated by the all ones vector (1, 1, . . . , 1)t , and so dim Fix(G) = 1.
5.1. Theorem. Let G 6 Sd be a QI-group. Then there is a constant M depending
only on the group G such that every core point is normalizer equivalent to a core
point w with kwk2 6 M . In particular, there are only finitely many core points for
G up to normalizer equivalence.
We divide the proof of Theorem 5.1 into a number of lemmas. The idea is the
following: We show that for any vector z ∈ Zd there is some c ∈ CGL(d,Z) (G) such
that the projections of cz to the different irreducible real subspaces of Fix(G)⊥
have approximately the same norm. (At the same time, this point cz is one with
12
FRIEDER LADISCH AND ACHILL SCHÜRMANN
minimal norm in the orbit CGL(Λ) (G)z.) When z is a core point, at least one of these
norms must be “small”, by a fundamental result of Herr, Rehn and Schürmann [18,
Theorem 9] (Theorem 5.8 below).
We begin with a short reminder of some character theory. The facts we need can
be found in any basic text on representations of finite groups, for example Serre’s
text [37]. Saying that a group G acts linearly on a (finite-dimensional) vector space V
over some field K is equivalent to having a representation R : G → GL(V ) (or even
R : G → GL(d, K) when V = K d ). The character χ of R (or V ) is the function
defined by χ(g) = tr(R(g)). An irreducible character is the trace of an irreducible
representation R : G → GL(d, C) over the field of complex numbers C. The set of
irreducible characters of the group G (over the complex numbers) is denoted by
Irr(G). For finite groups G, this is a finite set. Indeed, by the orthogonality relations,
the set Irr(G) is orthonormal with respect to a certain inner product on the space
of all functions G → C [37, §2.3,Thm 3].
Every character of a finite group can be written uniquely as a nonnegative integer
linear combination of irreducible characters. This corresponds to the fact that for
each representation G → GL(V ) on some vector space V over C, we can write V as
a direct sum of irreducible, G-invariant subspaces [37, §1.4, Thm. 2, §2.3, Thm. 4].
Suppose χ is the character of some representation R of the finite group G. Then
the eigenvalues of R(g), where g ∈ G, must be |G|-th roots of unity. Thus the values
of χ are contained in the field generated by the |G|-th roots of unity. We write
Q(χ) for the field generated by all values of χ. It follows that Q(χ) is a finite Galois
extension of Q, with abelian Galois group Gal(Q(χ)/Q).
The following lemma appears in Dixon’s paper [8, Lemma 6(b)].
5.2. Lemma (Dixon [8]). Let G be a QI-group and let π be the character of the
corresponding permutation representation of G. Let χ ∈ Irr G be an irreducible
constituent of π − 1 (the character of G on Fix(G)⊥ ). Then
π =1+
X
χα ,
where Γ = Gal(Q(χ)/Q).
α∈Γ
For the moment, we work with the complex space Cd , on which G acts by permuting
coordinates. Recall that to each χ ∈ Irr G there corresponds a central primitive
idempotent of the group algebra CG, namely
eχ =
χ(1) X
χ(g −1 )g ∈ Z(CG).
|G| g∈G
If V is any CG-module, then eχ acts on V as the projection onto its χ-homogeneous
component. So the image eχ (V ) coincides with the set {v ∈ V | eχ v = v}, and the
character of eχ (V ) is an integer multiple of χ [37, §2.6]. In the present situation, it
follows from Lemma 5.2 that
U := eχ (Cd ) = {v ∈ Cd | eχ v = v}
Equivalence of Lattice Orbit Polytopes
13
is itself an irreducible module affording the character χ. The projection eχ maps the
standard basis of Cd to vectors contained in K d , where K := Q(χ). Thus U has a
basis contained in K d . (This means that the representation corresponding to the
linear action of G on U can be described by matrices with all entries in K. Thus χ
is the character of a representation where all matrices have entries in K = Q(χ).)
Another consequence of Lemma 5.2 is that we have the decomposition
Cd = Fix(G) ⊕
M
Uγ.
γ∈Γ
Here U γ means this: Since U has a basis in Q(χ)d , we can apply γ to the coordinates
of the vectors in such a basis. The linear span of the result is denoted by U γ . This is
independent of the chosen basis.
5.3. Lemma. Set A := CMd (Q) (G) = {a ∈ Md (Q) | ∀g ∈ G : ag = ga}, the
full centralizer of G in the ring of d × d-matrices over Q. There is an algebra
homomorphism λ : A → Q(χ) such that each a ∈ A acts on U γ by multiplication
with λ(a)γ , and such that λ(at ) = λ(a). There is another homomorphism m : A → Q,
such that
A∼
= Q × Q(χ) via a 7→ (m(a), λ(a)).
The isomorphism A ∼
= Q × Q(χ) appears in Dixon’s paper [8, Lemma 6(d)], and
follows from Lemma 5.2 together with general results in representation theory. But
as we need the specific properties of the map λ from the lemma, we give a detailed
proof:
Proof of Lemma 5.3. Suppose the matrix a centralizes G, and let λ(a) ∈ C be an
eigenvalue of a on U . The corresponding eigenspace is G-invariant, since a centralizes
G. Since U is irreducible, U is contained in the eigenspace of λ(a).
When a ∈ A ⊆ Md (Q), then a maps U ∩ Q(χ)d 6= {0} to itself, and thus
λ(a) ∈ Q(χ). This defines the algebra homomorphism λ : A → Q(χ).
When u ∈ U ∩ Q(χ)d , γ ∈ Γ and a ∈ A, then auγ = (au)γ = λ(a)γ uγ . Thus a acts
as λ(a)γ on U γ .
Each a ∈ A acts also on the one-dimensional fixed space by multiplication with
some m(a) ∈ Q. As
M
Cd = Fix(G) ⊕
Uγ,
γ∈Γ
d
we see that the space C has a basis of common eigenvectors for all a ∈ A. With
respect to this basis, each a is a diagonal matrix, where m(a) appears once and λ(a)γ
appears χ(1)-times for each γ ∈ Γ. In particular, the map A 3 a 7→ (m(a), λ(a)) is
injective.
Since G acts orthogonally with respect to the standard inner product on Cd , the
above decomposition into irreducible subspaces is orthogonal and we can find an
orthonormal basis of common eigenvectors of all a ∈ A. From this, it is clear that
λ(at ) = λ(a∗ ) = λ(a).
14
FRIEDER LADISCH AND ACHILL SCHÜRMANN
To see that a 7→ (m(a), λ(a)) is onto, let (q, µ) ∈ Q × Q(χ). Define
ϕ(q, µ) := qe1 +
X
(µeχ )γ
γ∈Γ
γ
1 X
χ(1) X X
=q
g+
µχ(g −1 ) g
|G| g∈G
|G| g∈G γ∈Γ
∈ Z(QG).
Then the corresponding map v 7→ ϕ(q, µ)v is in A, and from ϕ(q, µ)e1 = qe1 and
ϕ(q, µ)eχ = µeχ we see that m(ϕ(q, µ)) = q and λ(ϕ(q, µ)) = µ. This finishes the
proof that A ∼
= Q × Q(χ).
5.4. Lemma. Set W := (U + U ) ∩ Rd . Then the decomposition of Rd into irreducible
RG-modules is given by
Rd = Fix(G) ⊕
W α,
M
Γ0 = Gal((Q(χ) ∩ R)/Q).
α∈Γ0
(In particular,
W is irreducible as RG-module.) For w ∈ W α and a ∈ A, we have
α
2
kawk = λ(a)λ(a) kwk2 .
Proof. When Q(χ) ⊆ R, then U = U and W = U ∩ Rd . The result is clear in this
case.
Otherwise, we have U ∩Rd = {0} and U ∩U = {0}, and so W = (U ⊕U )∩Rd 6= {0},
and thus again W is simple over RG.
The extension Q(χ)/Q has an abelian Galois group, and thus Q(χ) ∩ R is also
Galois over Q. The Galois group Γ0 is isomorphic to the factor group Γ/{Id, κ},
where κ denotes complex conjugation. Suppose α ∈ Γ0 is the restriction of γ ∈ Γ to
Q(χ) ∩ R. Then
W α = (U + U ) ∩ Rd
α
γ
= (U γ + U ) ∩ Rd = (U γ + U κγ ) ∩ Rd .
The statement about the decomposition follows.
The last statement is immediate from Lemma 5.3.
5.5. Lemma. Let C := CGL(d,Z) (G) and define
L : C → RΓ0 ,
L(c) := log(λ(c)λ(c))α
α∈Γ0
.
Then the image L(C) of C under this map is a full lattice in the hyperplane
H = {(xα )α∈Γ0 |
X
xα = 0}.
α∈Γ0
We will derive this lemma from the following version of Dirichlet’s unit theorem [31,
Satz I.7.3]:
5.6. Lemma. Let K be a finite field extension over Q, let α1 , . . . , αr : K → R be
the different real field embeddings of K, and let β1 , β1 , . . . , βs , βs : K → C be the
Equivalence of Lattice Orbit Polytopes
15
different complex embeddings of K, whose image is not contained in R. Let OK be
the ring of algebraic integers in K and l : K ∗ → Rr+s the map
z 7→ l(z) = (log|z α1 |, . . . , log|z αr |, log|z β1 |, . . . , log|z βs |) ∈ Rr+s .
Then the image l(U(OK )) of the unit group of OK under l is a full lattice in the
hyperplane
H = {x ∈ Rr+s |
r+s
X
xi = 0}.
i=1
In the proof of Lemma 5.5, we will apply this result to K = Q(χ). Set F = K ∩ R,
Γ0 = Gal(F/Q) and Γ = Gal(K/Q). If F = K ⊆ R, then r = |K : Q| and s = 0.
In this case, {α1 , . . . , αr } = Γ = Γ0 . If K 6⊆ R, then |K : F | = 2, r = 0 and
s = |F : Q|. In this case, we may identify the set {β1 , . . . , βs } with the Galois group
Γ0 : for each α ∈ Γ0 , there are two extensions of α to the field K, and these are
complex conjugates of each other. Thus we get a set {β1 , . . . , βs } as in Lemma 5.6
by choosing exactly one extension for each α ∈ Γ. The map l is independent of this
choice, anyway.
It follows that in both cases, we may rewrite the map l (somewhat imprecisely) as
l(z) = log|z α |
α∈Γ0
.
Proof of Lemma 5.5. First notice that the entries of L(c) can be written as
α
log λ(c)λ(c)
= log λ(c)α λ(c)α = log|λ(c)α |2 = 2 log|λ(c)α |,
where we tacitly replaced α by an extension to Q(χ), when Q(χ) 6⊆ R. Thus
L(c) = 2l(λ(c)) for all c ∈ C, with l as in Lemma 5.6.
In view of Lemma 5.6, it remains to show that the group λ(C) has finite index in
U(OK ). We know that C is the group of units in CMd (Z) (G) ∼
= EndZG (Zd ), which
is an order in A ∼
= Q × K. Another order in Q × K (in fact, the unique maximal
order) is Z × OK with unit group {±1} × U(OK ). By Lemma 3.2, it follows that C
has finite index in {±1} × U(OK ). Thus λ(C) has finite index in U(OK ) and the
result follows.
For each v ∈ Rd , let vα be the orthogonal projection of v onto the simple subspace
W α.
5.7. Lemma. There is a constant D, depending only on the group G, such that for
every v ∈ Rd with vα 6= 0 for all α ∈ Γ0 , there is an c ∈ C with
k(cv)α k2
6D
k(cv)β k2
for all α, β ∈ Γ0 .
As Fix(G)⊥ ∩ Qd is a simple module, the assumption vα 6= 0 for all α holds in
particular for all v ∈ Qd \ Fix(G).
16
FRIEDER LADISCH AND ACHILL SCHÜRMANN
Proof of Lemma 5.7. By Lemma 5.5, there is a compact set T ,
T ⊂ H = {(xα ) ∈ RΓ0 |
X
xα = 0},
α
such that H = T + L(C). (For example, we can choose T as a fundamental parallelepiped of the full lattice L(C) in H.)
For v ∈ Rd as in the statement of the lemma, define
N (v) = logkvα k2
α
∈ RΓ0 .
Let S ∈ RΓ0 be the vector having all entries equal to
1 X
s :=
logkvα k2 .
|Γ0 | α
This s is chosen such that N (v) − S ∈ H. Thus there is c ∈ C such that L(c) +
N (v) − S ∈ T , say L(c) + N (v) − S = t = (tα ).
As
α
k(cv)α k2 = kcvα k2 = λ(c)λ(c) kvα k2 ,
it follows that
N (cv) = L(c) + N (v)
in general. Thus
logkcvα k2 − logkcvβ k2 = (logkcvα k2 − s) − (logkcvβ k2 − s)
= (N (cv) − S)α − (N (cv) − S)β
= tα − tβ
6 max tα − min tβ =: D0 .
α,t
β,t
This maximum and minimum exist since T is compact. The number D0 may depend
on the choice of the set T , but not on v or c. Thus kcvα k2 /kcvβ k2 is bounded by
D := eD0 .
We see from the proof that we get a bound whenever we have a subgroup C0 of
CGL(d,Z) (G) such that L(C0 ) is a full lattice in the hyperplane H. Of course, we do
not get the optimal bound then, but in practice it may be difficult to compute the
full centralizer.
We will prove Theorem 5.1 by combining the last lemma with the following
fundamental result [18, Theorem 9] (which is actually true for arbitrary matrix
groups [34, Theorem 3.13]).
5.8. Theorem. Let G 6 Sd be a transitive permutation group. Then there is a
constant C (depending only on d) such that for each core point z, there is a non-zero
invariant subspace U 6 Fix(G)⊥ over R such that kz|U k2 6 C.
In our situation, the W α from Lemma 5.4 are the only irreducible subspaces, and
thus for every core point z there is some α ∈ Γ0 with kzα k2 6 C.
Equivalence of Lattice Orbit Polytopes
17
Proof of Theorem 5.1. Let z be a core point with z ∈
/ Fix(G). We want to show that
there is a c ∈ CGL(d,Z) (G) and a vector b ∈ Fix(G) ∩ Zd , such that kcz + bk 6 M ,
where M is a constant depending only on G, not on z. By Lemma 5.7, there is
c ∈ CGL(d,Z) (G) such that kczα k2 6 Dkczβ k2 for all α, β ∈ Γ, where D is some
constant depending only on G, not on z.
Since y = cz is also a core point (Lemma 2.4), Theorem 5.8 yields that there is a
β ∈ Γ with kyβ k2 6 C (where again, the constant C depends only on the group, not
on z). It follows that the squared norms of the other projections yα are bounded by
CD. Thus
ky|Fix(G)⊥ k2 6 C + (|Γ| − 1)CD
is bounded.
Since the projection to the fixed space can be bounded by translating with some
b ∈ Fix(G) ∩ Zd , the theorem follows.
5.9. Example. Let p be a prime and let G = Cp 6 Sp be generated by a p-cycle,
acting on Rp by (cyclically) permuting coordinates. Then G is a QI-group. (Of course,
every transitive group of prime degree is a QI-group.) For p odd, Rp decomposes
into Fix(G) and (p − 1)/2 irreducible subspaces of dimension 2. Here the lattice can
be identified with the group ring ZG, and thus CGL(p,Z) (G) ∼
= U(ZG). The torsion
free part of this unit group is a free abelian group of rank (p − 3)/2.
Let us see what constant we can derive for p = 5. For concreteness, let g =
(1, 2, 3, 4, 5) and G = hgi. We have the decomposition
R5 = Fix(G) ⊕ W ⊕ W 0 .
The projections from R5 onto W and W 0 are given by
1
eW = (2 + ag + bg 2 + bg 3 + ag 4 ),
5
1
eW 0 = (2 + bg + ag 2 + ag 3 + bg 4 ),
5
The centralizer of G has the form
√
−1 + 5
a=
,
2√
−1 − 5
b=
.
2
CGL(5,Z) (G) = {±I} × G × hui,
where u is a unit of infinite order. Here we can choose u = −1 + g + g 4 with inverse
−1 + g 2 + g 3 . To u corresponds the matrix
(1)
−1 1
0
0
1
1 −1 1
0
0
0
1 −1 1
0
∈ GL(5, Z).
0
1 −1 1
0
1
0
0
1 −1
0
This unit acts on W as −1+a and on W
√ as −1+b. For the constant D of Lemma 5.7,
2
we get D = (b − 1) = 2 − 3b = (7 + 3 5)/2. For the constant C in Theorem 5.8, we
get a bound C = 48/5 (from the proof). We can conclude that every core point is
18
FRIEDER LADISCH AND ACHILL SCHÜRMANN
equivalent to one with squared norm smaller than M = (2/5) + (48/5)(1 + 2 − 3b) ≈
50.6.
We can get somewhat better bounds by applying Theorem 5.8 “layer-wise”. The
P
k-layer is, by definition, the set of all z ∈ Zd with zi = k. In our example, every
lattice point is equivalent to one in layer 1 or layer 2.
For example, it can be shown that each core point in the 1-layer is equivalent to
a point z with kzk2 6 31. However, this bound is still far from optimal. Using the
computer algebra system GAP [13], we found that the only core points of C5 in the
1-layer up to normalizer equivalence are just
(1, 0, 0, 0, 0)t ,
(1, 1, 0, 0, −1)t ,
(2, 1, 0, −1, −1)t ,
(2, 1, −2, 0, 0)t .
(1, 1, 1, 0, −2)t ,
(The normalizer NGL(5,Z) (G) is generated by the centralizer and the permutation
matrix corresponding to the permutation (2, 3, 5, 4).) For completeness, we also give
a list of core points up to normalizer equivalence in the 2-layer:
(1, 1, 0, 0, 0)t ,
(1, 1, 1, 0, −1)t ,
(2, 1, 1, −1, −1)t ,
(2, 1, 1, −2, 0)t .
(2, 1, 0, 0, −1)t ,
Every nontrivial core point for C5 is normalizer equivalent to exactly one of these
ten core points.
For this example, an infinite series of core points of the form
(fj+1 , 0, fj , fj , 0)t ,
where fj is the j-th Fibonacci number, was found by Rehn [34, 5.2.2]. Each point in
this series is normalizer equivalent to one of the two obvious core points (1, 0, 0, 0, 0)t
and (1, 0, 1, 1, 0)t . This follows from
(1 − g − g 4 )(fj+1 , 0, fj , fj , 0)t = (fj+1 , −fj+2 , 0, 0, −fj+2 )t
and thus
(1 − g − g 4 )(fj+1 , 0, fj , fj , 0)t + fj+2 (1, 1, 1, 1, 1)t = (fj+3 , 0, fj+2 , fj+2 , 0)t .
5.10. Example. Now set
G = h(1, 2, 3, 4, 5), (1, 4)(2, 3)i ∼
= D5 ,
the dihedral group of order 10. Then
CGL(5,Z) (G) = {±I}hui,
where u is as in the previous example. The normalizer of G is the same as that of
the cyclic group C5 = h(1, 2, 3, 4, 5)i. In particular, normalizer equivalence for D5
and C5 is the same equivalence relation. Of the core points from the last example,
only (1, 0, 0, 0, 0)t and (1, 1, 0, 0, 0)t are also core points for D5 . (In fact, for most of
the other points, we have some lattice
point on an interval between two vertices,
t
for example (1, 0, 0, 0, 0) = (1/2) (1, 1, 0, 0, −1)t + (2, 5)(3, 4)(1, 1, 0, 0, −1)t . Thus
there are only two core points up to normalizer equivalence in this example.
Equivalence of Lattice Orbit Polytopes
19
5.11. Remark. The number of core points up to normalizer equivalence seems to
grow fast for cyclic groups of prime order. For p = 7, we get 515 core points up to
normalizer equivalence.
Herr, Rehn and Schürmann [18] conjectured that a finite transitive permutation
group G has infinitely many core points up to translation equivalence if the group
is not 2-homogeneous. This conjecture is still open, but is known to be true in a
number of special cases, including imprimitive permutation groups and all groups of
degree d 6 127.
It is known that a permutation group G 6 Sd is 2-homogeneous if and only if
FixRd (G)⊥ is irreducible [6, Lemma 2(iii)]. In this case, there are only finitely many
core points up to translation equivalence [18, Corollary 10].
We propose the following conjecture, which is the converse of Theorem 5.1:
5.12. Conjecture. Let G 6 Sd be a transitive permutation group such that Fix(G)⊥
contains a rational G-invariant subspace other than {0} and Fix(G)⊥ itself. Then
there are infinitely many core points up to normalizer equivalence.
This can be seen as a generalization of the Herr-Rehn-Schürmann conjecture, since
translation equivalence refines normalizer equivalence, and since whenever Fix(G)⊥
contains a nontrivial irrational G-invariant subspace, then there are infinitely many
core points up to translation equivalence by Theorem 4.1 (or [18, Theorem 32]).
6. A p p l i c at i o n t o i n t e g e r l i n e a r o p t i m i z at i o n
In this last section we describe a possible application of the concept of normalizer
equivalence to symmetric integer linear optimization problems. For many years it
has been known that symmetry leads often to difficult problem instances in integer
optimization. Standard approaches like branching usually work particularly poorly
when large symmetry groups are present, since a lot of equivalent sub-problems
have to be dealt with in such cases. Therefore, in recent years several new methods
for exploiting symmetries in integer linear programming have been developed. See
for example [29, 12, 5, 22, 27, 32, 14, 11, 20] and the surveys by Margot [30] and
Pfetsch and Rehn [33] for an overview. These methods (with the exception of [11])
fall broadly into two classes: They either modify the standard branching approach,
using isomorphism tests or isomorphism free generation to avoid solving equivalent
subproblems; or they use techniques to cut down the original symmetric problem to
a less symmetric one, which contains at least one element of each orbit of solutions.
By now, many of the leading commercial solvers, like CPLEX [7], Gurobi [15], and
XPRESS [38], have included some techniques to detect and exploit special types of
symmetries. Accompanying their computational survey [33], Pfetsch and Rehn also
published implementations of some symmetry exploiting algorithms for SCIP [36],
like isomorphism pruning and orbital branching.
Core points were introduced as an additional tool to deal with symmetries in
integer convex optimization problems. Knowing the core points for a given symmetry
20
FRIEDER LADISCH AND ACHILL SCHÜRMANN
group allows to restrict the search for optima to this subset of the integer vectors
[17]. There are many possible ways how core points could be used. For instance,
one could use the fact that core points are near invariant subspaces, by adding
additional quadratic constraints (SOC-constraints). In the case of QI-groups, hence
with finitely many core points up to normalizer equivalence (Theorem 5.1), one could
try to systematically run through core points satisfying the problem constraints.
In contrast to the aforementioned approaches, we here propose natural reformulations of symmetric integer optimization problems using the normalizer of the
symmetry group. Recall that a general standard form of an integer linear optimization
problem is
(2)
max ct x such that Ax 6 b, x ∈ Zd ,
for some given matrix A and vectors b and c, all of them usually rational. If c = 0,
then we have a so-called feasibility problem, asking simply whether or not there is
an integral solution to a given system of linear inequalities. Geometrically, we are
asking whether some polyhedral set (a polytope, if bounded) contains an integral
point.
A group G 6 GL(d, Z) is called a group of symmetries of problem (2) if the
constraints Ax 6 b and the linear objective function ct x are invariant under the
action of G on Rd , that is, if ct (gx) = ct x and A(gx) 6 b for all g ∈ G whenever
Ax 6 b. The first condition is for instance satisfied if c is in the fixed space Fix(G).
Practically, computing a group of symmetries for a given problem is usually reduced
to the problem of finding symmetries of a suitable colored graph [4, 33]. Quite
often in optimization, the attention is restricted to groups G 6 Sd acting on Rd by
permuting coordinates.
Generally, a linear reformulation of a problem as in (2) can be obtained by an
integral linear substitution x 7→ Sx for some matrix S ∈ GL(d, Z):
(3)
max(ct S)x such that (AS)x 6 b, x ∈ Zd .
(More generally, one can use integral affine substitutions x 7→ Sx + t with S ∈
GL(d, Z) and t ∈ Zd . For simplicity, we assume t = 0 in the discussion to follow.)
We remark that reformulations as in (3) with a matrix S ∈ GL(d, Z) can of course
be applied to any linear integer optimization problem. In fact, this is a key idea of
Lenstra’s famous polynomial time algorithm in fixed dimension d [26]. In Lenstra’s
algorithm, the transformation matrix S is chosen to correspond to a suitable LLLreduction of the lattice, such that the transformed polyhedral set {x ∈ Rd | (AS)x 6
b} is sufficiently round. This idea has successfully been used for different problem
classes of integer linear optimization problems (for an overview see [1]). The main
difficulty is the choice of an appropriate unimodular matrix S which simplifies the
optimization problem.
If the symmetry group of an optimization problem contains the group G, then it
is natural to choose matrices S which keep the problem G-invariant. When S is an
Equivalence of Lattice Orbit Polytopes
21
element of the normalizer NGL(d,Z) (G), problem (2) is G-invariant if and only if (3)
is G-invariant. Note also that then (ct S)t is in Fix(G).
We illustrate the idea with a small concrete problem instance of (2) which is
invariant under the cyclic group C5 . In particular, using core points, we construct
C5 -invariant integral optimization problems that are quite hard or even impossible
to solve for state-of-the-art commercial solvers like CPLEX or GUROBI. For instance,
this is often the case when the constraints Ax 6 b can be satisfied by real vectors x,
but not by integral ones.
6.1. Example. The orbit polytope P (C5 , z) of some integral point z has a description
with linear inequalities of the form x1 + . . . + x5 = k and Ax 6 b, where A is a
circulant 5 × 5-matrix
a1 a2 a3 a4 a5
a2 a3 a4 a5 a1
A=
a3 a4 a5 a1 a2
a4 a5 a1 a2 a3
a5 a1 a2 a3 a4
with integral entries a1 , . . . , a5 , and b ∈ Z5 satisfies b1 = . . . = b5 . If z is a core point
and if we replace bi by b0i := bi − 1, then we get a system of inequalities having no
integral solution.
Applying this construction to the core point
z = U 10 · (1, 1, 1, 0, −2)t ,
where U is the matrix from (1) in Example 5.9, we get parameters
a1 = 515161,
a2 = 18376,
a4 = −329744, a5 = 300011,
a3 = −503804,
b01 = 60.
We can vary the values of k ≡ 1 mod 5 (geometrically, this corresponds to translating
the polytope by some integral multiple of the all-one-vector). This gives a series of
problem instances on which the commercial solvers very often not finish within a
time limit of 10000 seconds on a usual desktop computer. For k = 1, which seems
computationally the easiest case, a solution always still takes more than 4000 seconds.
However, knowing that a given problem as the above is C5 -invariant, we can try
to find an easier reformulation (3) by using matrices from the centralizer. As a rule
of thumb, we assume that a transformed problem with smaller coefficients is “easier”.
Here, the torsion free part of the centralizer is generated by the matrix U from (1)
in Example 5.9, and so the only possibilities for S are U or U −1 . (A matrix of finite
order will probably not simplify a problem significantly.) Here, applying S = U yields
an easier problem, and one quickly finds that after applying S ten times, the problem
is not simplified further by applying U (or U −1 ). In other words, we transform the
original problem instance with U 10 . This yields an equivalent C5 -invariant feasibility
problem, which is basically instantly solved by the commercial solvers (finding that
there is no integral solution).
22
FRIEDER LADISCH AND ACHILL SCHÜRMANN
As far as we know, this approach is in particular by far better than any previously
known one that uses the symmetries of a cyclic group. One standard approach is for
example to add symmetry-breaking inequalities x1 6 x2 , . . . , x1 6 x5 . This yields
an improved performance in some cases, but is far from the order of computational
gain that is possible with our proposed reformulations.
In general, when an ILP (2) is invariant under a QI-group G, and when it has any
solutions at all, then Theorem 5.1 tells us that there is a transformation x 7→ Sx + t
with S ∈ NGL(d,Z) (G) such that the reformulated problem has a feasible solution in a
given finite set (a set of representatives of core points under normalizer equivalence).
Heuristically, this means that we should be able to transform any G-invariant problem
into one of bounded difficulty: By Lemma 5.7, for any vector x ∈ Rd , there is an
element S ∈ CGL(d,Z) (G) such that the projections of Sx to the different G-invariant
subspaces have approximately the same norm. This means that the orbit polytope
of Sx is “round”.
Our approach is particularly straightforward when the torsion free part of the
centralizer CGL(d,Z) (G) has just rank 1, as in the example with G = C5 above. When
the centralizer contains a free abelian group of some larger rank, then it is less clear
how to reduce the problem efficiently. A possible heuristic is as follows: Recall that in
Lemma 5.5, we described a map L which maps the centralizer, and thus its torsionfree part of rank r (say), onto a certain lattice in Rr+1 . This maps the problem of
finding a reformulation (3) with “small” AS to a minimization problem on a certain
lattice. For example, when we minimize kASk, this translates to minimizing a convex
function on a lattice. So we can find a good reformulation by finding a lattice point
in Rr+1 which is close to the minimum, using for instance LLL-reduction. This will
be further studied in a forthcoming paper.
Ac k n ow l e d g m e n t s
We would like to thank the anonymous referees for several valuable comments.
We also gratefully acknowledge support by DFG grant SCHU 1503/6-1.
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U n i v e r s i tät Ro s t o c k , I n s t i t u t f ü r M at h e m at i k , U l m e n s t r . 6 9 , H au s 3 ,
1 8 0 5 7 Ro s t o c k , G e r m a n y
E-mail address: frieder.ladisch@uni-rostock.de
E-mail address: achill.schuermann@uni-rostock.de
| 4 |
arXiv:1803.03901v1 [stat.ME] 11 Mar 2018
Piecewise Convex Function Estimation:
Representations, Duality and Model Selection
Kurt S. Riedel
Courant Institute of Mathematical Sciences
New York University
New York, New York 10012-1185
Abstract
We consider spline estimates which preserve prescribed piecewise convex properties of the unknown function. A robust version of the penalized likelihood is given
and shown to correspond to a variable halfwidth kernel smoother where the halfwidth
adaptively decreases in regions of rapid change of the unknown function. When the
convexity change points are prescribed, we derive representation results and smoothness properties of the estimates. A dual formulation is given which reduces the
estimate is reduced to a finite dimensional convex optimization in the dual space.
1
Introduction
A common problem in nonparametric function estimation is that the estimate often has
artificial wiggles that the original function does not possess. In practice, these extra inflection points have a very negative impact on the utility and credibility of the estimate.
In this article, we examine function estimation which preserves the geometric shape of the
unknown function, f (t). In other words, the number and location of the change points of
convexity of the estimate, fˆ(t), should approximate those of f (t).
We say that f (t) has K change points of ℓ-convexity with change points x1 ≤ x2 . . . ≤ xK
if (−1)k−1 f (ℓ) (t) ≥ 0 for xk ≤ t ≤ xk+1 . For ℓ = 0, f (t) is nonnegative and for ℓ = 1,
the function is nondecreasing. In regions where the constraint of ℓ-convexity is active,
f (ℓ) (t) = 0 and f (t) is a polynomial of degree ℓ − 1. For 1-convexity, f (t) is constant in the
active constraint regions and for 2-convexity, the function is linear. Our subjective belief is
that most people prefer smoothly varying functions such as quadratic or cubic polynomials
1
even in the active constraint regions. Thus, piecewise 3-convexity or 4-convexity are also
reasonable hypotheses.
When the change points are prescribed, convex analysis can be employed to derive
representation theorems, smoothness properties and duality results. Our work extends
that of Refs. [2, 5, 7, 8, 9, 11, 12] to an important class of robust functionals. To motivate
robust nonparametric estimation, we show that robust functionals correspond to variable
halfwidth data-adaptive kernels, i.e. the effective halfwidth decreases in regions where the
unknown function changes rapidly. When the number of change points are known, we prove
the existence of a minimum of the penalized likelihood.
Sections 2 and 3 contain functional analysis preliminaries. Lemma 3.1 gives a characterization of negative polar cones in the spirit of [2]. Representation and duality theorems for
constrained smoothing splines have been developed in [8, 9, 11] for the case of prescribed
convexity change points. In Sections 4 and 6, we generalize these results to the case of
nonquadratic penalty functions. In Section 5, we show how robust penalty functions correspond to data-adaptive variable halfwidth kernel smoothers. In Section 7, we consider
estimating the change point locations by minimizing the penalized least squares fit.
2
Functional Analysis Preliminaries
We consider an unknown function, f , is in the Sobolev space Wm,p [0, 1] with m ≥ ℓ and
1 < p < ∞, where
Wm,p ≡ {f |f (m) ∈ Lp [0, 1] and f, f ′ . . . f (m−1) (t) absolutely continuous} .
(2.1)
For f ∈ Wm,p , we have the representation
f (t) =
m−1
X
aj Pj (t) +
j=0
t
Z
0
(t − s)m−1 (m)
f (s)ds ,
(m − 1)!
(2.2)
where Pj (t) ≡ tj /(j!). Equation (2.2) decomposes Wm,p into a direct sum of the space of
polynomials of degree m − 1 plus the set of functions whose first m − 1 derivatives vanish
R
0
at t = 0, which we denote by Wm,p
[13]. We define the seminorm kf kpj,p ≡ 01 |f (j) (t)|p dt.
We endow Wm,p with the norm:
|kf k|pm,p
=
m−1
X
|f (j) (t = 0)|p + kf kpm,p .
(2.3)
j=0
0
The dual space of Wm,p is isomorphic to the direct sum of Pm−1 and Wm,q
with q = p/(p−1)
and the duality pairing of f ∈ Wm,p and g ∈ Wm,q is
hhg, f ii =
m−1
X
j=0
bj aj +
Z
2
0
1
f (m) (t)g (m) (t)dt ,
(2.4)
where aj ≡ f (j) (0) and bj ≡ g (j)(0). We denote the duality pairing by hh·ii and the L2 inner
product by h·i. In [13], Wm,2 is given a reproducing kernel where for each t, f (t) = hhRt , f ii.
The same reproducing kernel structure carries over to 1 < p < ∞ with
Rt (s) =
m−1
X
Pj (t)Pj (s) +
j=0
Z
min{t,s}
0
(t − u)m−1 (s − u)m−1 du
.
[(m − 1)!]2
(2.5)
∗
A linear operator Li ∈ Wm,p
has representations Li f = hhbi (s), f (s)ii and Li f = hhi (s), f (s)i,
where bi (s) ≡ Li R(·, s) and hi (s) ≡ Li δ(s − ·). Li R(·, s) denotes Li acting on the first entry
of R. In the standard case, where Li f ≡ f (ti ), bi (s) = R(ti , s) and hi (s) = δ(s − ti ).
3
Convex Cone Constraints
In this and the next section, we assume that the change points {x1 . . . xK } of ℓ-convexity
are given and that the unknown function is in the Sobolev space, Wm,p [0, 1]. Given change
points, {x1 , x2 . . . xK }, we define the closed convex cone
K,ℓ
Vm,p
[x1 , . . . , xK ] = {f ∈ Wm,p | (−1)k−1 f (ℓ) (t) ≥ 0 for xk−1 ≤ t ≤ xk } ,
(3.1)
where x0 ≡ 0 and xK+1 ≡ 1. Let x denote the K row vector, (x1 , x2 . . . xK ). Throughout
this article, we require ℓ ≤ m. By the Sobolev embedding theorem, f (ℓ) (t) is continuous
for ℓ < m. For ℓ = m, we require the convexity constraint in (3.1) almost everywhere. We
define the class of functions with at most K change points as
K,ℓ
Vm,p
≡
[
K,ℓ
K,ℓ
{Vm,p
[x1 , . . . , xK ] ∪ (−Vm,p
[x1 , . . . , xK ])} .
(3.2)
x1 ≤x2 ...≤xK
K,ℓ
K+1,ℓ
K,ℓ
into Vm,p
. Vm,p
is the union of convex
By allowing xk′ = xk′ +1 , we have embedded Vm,p
cones, and is closed but not convex. Similar piecewise ℓ-convex classes are defined in [6]
for the case ℓ = m + 1 with a supremum norm on the Hölder constant for f (ℓ−2) .
For Theorem 6.1, we need the following results from convex analysis.
Definition [1] Let C be a closed convex cone in Wm,p ; the negative polar, C − , of C is
∗
C − ≡ {g ∈ Wm,p
| ∀f ∈ C, hhg, f ii ≤ 0}.
K,ℓ
For Vm,p [x ], we are able to give a more explicit characterization of the negative polar.
Our result is restricted to ℓ ≤ m while Deutsch et al. [2] consider the more difficult case of
m = 0 with ℓ ≥ m.
K,ℓ
K,ℓ
∗
Lemma 3.1 The negative polar of Vm,p
[x ] is Vm,p
[x ]− = closure in Wm,p
of the {g ∈
C 2m−ℓ [0, 1] | (−1)m−ℓ χx (t)g (2m−ℓ) (t) ≤ 0; g (j) (0) = 0, j = 0 . . . ℓ − 1; g (m+j) (1) =
0, and g (m−j−1) (0) − (−1)j g (m+j) (0) = 0; j = 0 . . . m − ℓ − 2; (−1)m−ℓ χx (1)g (2m−ℓ−1) (1)
≥ 0; χx (0)[g (ℓ) (0) + (−1)m−ℓ g (2m−ℓ−1) (1) ≤ 0}, where χx (t) is defined by χx (t) = (−1)k−1
for xk < t < xk+1 } and χx (xk ) = 0, k = 1 . . . K.
3
Proof. Integration by parts yields 01 f (m) (t)g (m) (t)dt = (−1)m−ℓ 01 f (ℓ) (t)g (2m−ℓ) (t)dt +
Pm−ℓ−1
(−1)j f (m−j−1) (t)g (m+j) |10 for g ∈ C 2m−ℓ [0, 1]. We now find test functions, f˜ ∈
j=0
K,ℓ
Vm,p
[x ] which require each term separately to be nonpositive. For t 6= xK , we choose
(ℓ)
˜
f (s) = (s − t + h)m−ℓ+1
(t − s + h)m−ℓ+1
χx (s) where h is a small localization parameter
+
+
and s+ ≡ s for s > 0 and zero otherwise. The boundary conditions at t = 1 are proved
inductively with the sequence of test functions f˜h (s) = (s − 1 + h)m
+ χx (s) as h → 0. ✷
R
R
0
K,m
K,m
K,m
.
[x ] ∩ Wm,p
[x ]− = −Vm,p
[x ] is Vm,p
Corollary 3.2 The negative polar of Vm,p
K,m−1
∗
Corollary 3.3 The negative polar of Vm,p
[x ] is the closure in Wm,p
of the {g
m+1
(m+1)
(m)
(m−1)
(m)
C
[0, 1] | g
(t)χx (t) ≥ 0; χx (1)g (1) ≤ 0; χx (0)[g
(0) − g (0)] ≤ 0 }.
∈
K,m−2
K,m−2
∗
Corollary 3.4 The negative polar of Vm,p
[x ] is Vm,p
[x ]− = closure in Wm,p
of the
m+2
(m+2)
(m)
(m+1)
(m−1)
{g ∈ C
[0, 1] | g
(t)χx (t) ≤ 0; g (1) = 0; χx (1)g
(1) ≥ 0; [g
(0) −
(m)
(m−2)
(m+1)
g (0)] = 0; χx (0)[g
(0) + g
(0)] ≤ 0}.
The negative polar is useful in evaluating the normal cone of K:
Lemma 3.5 ([1], p.171) Let C be a closed convex cone in W, the normal cone of C in
W ∗ at f , NC (f ) satisfies NC (f ) = C − ∩ {f }⊥ , where C − is the negative polar of C.
4
Robust splines: Representations and Smoothness
In this section, we generalize representation and smoothness results to a large class of robust
functionals. These robust functionals are advantageous because they downweight outliers
and adaptively adjust the effective smoothing width. We are given N measurements of the
unknown function, f (t):
yi = Li f + ǫi = hhi , f i + ǫi = hhbi , f ii + ǫi ,
(4.1)
where the Li are bounded linear operators on Wm,p , and the ǫi are uncorrelated random
variables with variance σi2 > 0. A robustified estimate of f (t) given the measurements {yi }
K,ℓ
is fˆ ≡ argmin VP[f ∈ Vm,p
[x1 , . . . , xK ]]:
λ
VP[f ] ≡
p
Z
|f
(m)
p
(s)| ds +
N
X
ρi (hhi , f i − yi ) ,
(4.2)
i=1
where the ρi are strictly convex, continuous functions. The standard case is p = 2 and
ρi (yi − hhi , f i) = |yi − f (ti )|2 /(Nσi2 ). For an excellent discussion of the advantages of
robustness in function estimation, see Mächler [5].
4
Theorem 4.1 is proven in [11] and Theorem 6.1 is proven in [8] for the case p = 2
and ρ(y) = y 2. For the unconstrained case of Theorem 4.1, see [5]. Equation (2.5) and
the corresponding smoothness results appear in [11] for the case ℓ = 1, p = 2 and Li =
δ(t − ti ). The set of {hi , i = 1, . . . , N} separate polynomials of degree m − 1 means that
P
k
hhi , m−1
k=0 ck t i = 0, ∀i implies ck ≡ 0.
Theorem 4.1 Let {hi } separate polynomials of degree m − 1; then the minimization probK,ℓ
lem (4.2) has an unique solution in Vm,p
[x ], and the minimizing function satisfies the
differential equation:
N
X
(−1)m λdm [|fˆ(m) |p−2fˆ(m) (t)] +
ρ′i (hhi , fˆi − yi )hi (t) = 0 ,
(4.3)
i=1
in those regions where |f (ℓ) | > 0 for 1 < p < ∞ and ℓ ≤ m.
Proof. The functional (4.2) is strictly convex, lower semicontinuous and coercive, so by
Theorem 2.1.2 of Ekeland and Temam [3], it has a unique minimum, fˆ, on any closed
convex set. From the generalized calculus of convex analysis, the solution satisfies
N
X
0 ∈ (−1)m λdm [|fˆ(m) |p−2 fˆ(m) (t)] +
ρ′i (hhi , fˆi − yi )hi (t) + ∂NV (f ) ,
(4.4)
i=1
K,ℓ
where NV (f ) is the normal cone of Vm,p
[x ] at f [1, p. 189]. The normal cone is characterized
by Lemmas 3.1 and 3.5. From [11], each element of NV (f ) is the limit of a discrete sum:
P
(ℓ)
′
t at δ (· − t) where the t s are in the active constraint region. Integrating (2.4) yields
λ|fˆ(m) |p−2fˆ(m) (t) =
ρ′i (hhi ,fˆi−yi )hhi (s),(s−t)m−1
i
+
i=1
(m−1)!
PN
+
R
(s−t)m−ℓ−1
dµ(s)
+
(m−ℓ−1)!
,
(4.5)
where dµ corresponds to a particular element of NV (f ). ✷
For hi (s) = δ(s − ti ) and ℓ = m, Theorem 4.1 can be derived as a consequence of the
corresponding result for constrained interpolation [7].
⊥
Corollary 4.2 If {hi } are in Wℓ,1
, then the minimizing function of (4.2) is in C 2m−ℓ−2 [0, 1].
Proof. Since (s − t)m−ℓ−1
is m − ℓ − 2 times differentiable, the first term on the right
+
⊥
hand side of (4.5) is m − ℓ − 2 times differentiable. By hypothesis, hi ∈ Wℓ,1
and thus
R
m−1
m−ℓ−2
2m−ℓ−2
hi (s)(s − t)+ ds is in C
. Integrating (4.5) yields f ∈ C
. ✷
5
5
Equivalent adaptive halfwidth of robust splines
Replacing the standard spline likelihood functional (p = 2 and ρ(y) = y 2 /σ 2 ) in (4.2) with
a more robust version has several well-known advantages. First, outliers are less influential
when ρ(y) downweights large values of the residual error. Second, for 1 ≤ p < 2, the set of
candidate functions are increased, and the solution may have sharper local variation than
in the p = 2 case. We now describe a third important advantage: the effective halfwidth
adapts to the unknown function.
In [10], it is shown that as the number of measurements increase the spline estimate
converges to a local kernel smoother estimate (provided the measurement times, {ti }, are
nearly regularly spaced.) For technical details, see [10]. Convergence proofs are available
1
only for p = 2. The resulting effective kernel halfwidth, hef f , is scales as hef f ∼ [λF ′ (t)] 2m ,
where F (t) is the limiting distribution of measurement points.
For 1 < p < 2, no theory exists on the effective halfwidth of a robust spline estimate.
We assume that fˆ converges to f in Wm,p under hypotheses similar to those used for the
p = 2 case in [10]. These conditions relate to the discrepancy of the measurement times,
{ti }, the smoothness of F (t), and the scaling of the smoothing parameter with N. The
appropriate modifications for p 6= 2 are unknown.
We can make a heuristic two-scale analysis of (4.4) in the continuum. We assume that
in the continuum limit, the estimate satisfies the following equation to zeroth order:
(−1)m λdm [|fˆ(m) |p−2fˆ(m) (t)] + fˆ(t) = y(t) ,
(5.1)
where y(t) = f (t)+Z(t), with Z(t) being a white noise process. Away from the m-convexity
change points, we linearize 5.1 about fˆ(m) (t) ∼ f (m) (t). Let f˜(t) be the linearized variable
for 5.1: f˜(m) (t) ≈ fˆ(m) (t) − f (m) (t), where f˜(t) satisfies
(−1)m (p − 1)λdm [|f (m) |p−2 f˜(m) (t)] + f˜(t) = Z(t) .
(5.2)
When λ|f (m) (t)|p−2 is small but nonzero, the homogeneous equation may be solved using the
Wenzel-Kramer-Brillioun expansion. The resulting Green’s function for f˜ may be recasted
as a kernel smoother with an effective halfwidth:
1
hef f (t) ∼ [λF ′ (t)|f (m) (t)|p−2 ] 2m .
(5.3)
For 1 ≤ p ≤ 2, the effective halfwidth of the robustified function automatically reduces the
halfwidth in regions of large |f (m) (t)| just like a variable halfwidth smoother. We caution
that this result has not been rigorously derived.
For the equivalent kernel, the bias error scales as f (m) (t)hm while the variance is
proportional to 1/Nh. The halfwidth that minimizes the mean square error scales as
6
h
i
−1
hM SE ∼ N|f (m) |2 2m+1 The two halfwidths agree at p = 2/(2m + 1), but p < 1 is illconditioned.
We recognize that this derivation is formal, but we believe that a rigorous multiple scale
analysis may prove 5.3. Our purpose is only to motivate the connection between robust
splines and adaptive halfwidth kernel smoothers.
6
Constrained smoothing splines and duality
In (4.4), the intervals on which f (ℓ) (t) vanishes are unknowns and need to be found as
part of the optimization. Using the differential characterization (4.2) loses the convexity
properties of the underlying functional. For this reason, extremizing the dual functional is
now preferred.
Theorem 6.1 (Convex Duality) The dual variational problem of Theorem 4.1 is: Minimize over α ∈ lRN
N
X
λ1−q Z
ρ∗i (αi ) − αi yi ,
|[Px ∗ Bα](m) (s)|q ds +
VP [α; x] ≡
q
i=1
∗
where ρ∗i is the Fenchel/Legendre transform of ρi , and Bα(t) ≡
Li R(·, t). The dual projection Px ∗ is defined as
Z
|[Px ∗ g](m) (s)|q ds ≡ inf g̃∈V −
Z
0
1
PN
i
(6.1)
bi (t)αi with bi (t) =
|g (m) − g̃ (m) (s)|q ds ,
(6.2)
subject to the constraints g (j) (0) = g̃ (j)(0), 0 ≤ j < m. The dual problem is strictly
convex, and its minimum is the negative of the infimum of (4.2). When the {hi } are
linearly independent, the minimum satisfies the differential conditions:
αi = ρ′i hhi , fˆi − yi , and hhi , f i − yi = ρ∗′ (αi ),
i = 1...N .
(6.3)
K,ℓ
Proof. Let χV be the indicator function of Vm,p
[x ] and define
U(f ) =
λ
p
Z
1
0
|f (m) (s)|p ds + χV (f ) .
(6.4)
We claim that the Legendre transform of U(f ) is (6.2). Note that χ∗V (g) = χV − (g), the
indicator function of the dual cone V − . The Legendre transform of the first term in (6.4)
is
Z
λ1−q 1 (m)
0
V1∗ (g) =
|g (s)|q ds for g ∈ Wm,q
, and ∞ otherwise.
(6.5)
q 0
7
Our claim follows from [U1 + U2 ]∗ (g) = inf g′ {U1∗ (g − g ′) + U2∗ (g ′ )}. The remainder of the
theorem including the differential conditions (6.3) follows from the general duality theorem
of Aubin and Ekeland [1, p. 221]. ✷
An alternative formulation of the duality result for quadratic smoothing problems is
given in [9]. For both theories, the case ℓ < m is difficult to evaluate in practice because
the minimization in (6.2) can only rarely be reduced to an explicit finite dimensional problem. Only a few partial results are known when ℓ < m [2, 8, 9]. For the case ℓ = m, the
minimization over the dual cone can be done explicitly and yields the following simplification:
Corollary 6.2 For ℓ = m, the dual projection, Px ∗ , is a local operator with [Px ∗ Mα](m) (s)
P
P
(m)
(m)
= N
αi bi (t)χx (t) ≥ 0 and zero otherwise. Thus the minimization of (6.2)
i=1 αi bi (t) if
is finite dimensional.
7
Change point estimation
When the number of change points is fixed, but the locations are unknown, we can estimate
them by minimizing the functional in (2.3) with respect to the change point locations. We
now show that there exists a set of minimizing change points.
Theorem 7.1 For each K with ℓ = m, there exist change points {xk , k = 1, . . . K} which
minimize the variational problem (2.3).
Proof. We use the dual variational problem (2.5) and maximize over x ∈ [0, 1]K after
P
∗
minimizing over the α ∈ RN . We need only consider α in the compact region N
i=1 ρi (αi ) −
αi yi . For ℓ = m, explicit construction of the functional (2.5) shows that it is jointly
continuous in α and x . Since (2.5) is convex in α, Theorem 7.1 follows from the min-max
theorem [1, p. 296]. ✷
We conjecture that Theorem 7.1 is true for ℓ < m, but we lack a proof that Eq. (6.2) is
continuous with respect to x for ℓ < m. The change point locations need not be unique.
The proof requires ≤ instead of < in the ordering xk ≤ xk+1 to make the change point
space compact. When xk = xk+1 , the number of effective change points is less than K.
In [6], Mammen considers the case where K is known but the locations are unknown.
The function is estimated using simple least squares on a class of functions roughly analoK,m+1
gous to Vm,∞
. Mammen proves that this estimate achieves the optimal convergence rate
−2m/(2m+1)
of N
for the mean integrated square error. Unfortunately, Mammen’s estimator
is not unique and often results in aestetically unappealing fits.
8
For both formulations. finding the optimal change points locations is computationally
intensive. For each candidate set of change points, the the likelihood function needs to be
minimized subject to PC constraints. One advantage of our formulation is that for each
candidate value of x , the programming problem is strictly convex in the dual. This strict
convexity is lost if one uses a penalty functional with p = ∞ as in [6] or p = 1 corresponding
to a total variation norm. If the total variation norm is used and an absolute value penalty
function is employed (ρ(y) = |y|), the programming problem reduces to constrained linear
programming.
8
Discussion
We have considered robust smoothing splines under piecewise convex constraints. We
generalize the standard representation and smoothness results to nonlinear splines using
convex analysis. When the same derivative is both constrained and penalized (ℓ = m), the
dual problem is finite dimensional.
We have sketched a derivation of the effective halfwidth of a robust spline. By robustifying the functional, the effective halfwidth (5.3) for the equivalent kernel smoother
scales as |f (m) (t)|(p−2)/2m . The halfwidth that minimizes the mean square error scales as
h
i
−1
hM SE ∼ N|f (m) |2 2m+1 . Thus robust splines adjust the halfwidth, but not as much as
the asymptotically optimal local halfwidth would. When the number of convexity change
points is known, their locations may be estimated by minimizing the penalized likelihood.
For ℓ = m, we have existence, but not necessarily uniqueness.
Acknowledgments: Work funded by U.S. Dept. of Energy Grant DE-FG02-86ER53223.
References
[1] J.-P. Aubin, and I. Ekeland, “Applied Nonlinear Analysis”, John Wiley, New York
1984.
[2] F. Deutsch, V. A. Ubhaya, and Y. Xu, Dual cones, constrained n-convex Lp approximation and perfect splines, J. Approx. Th. 80 (1995), 180-203.
[3] I. Ekeland and R. Temam, “Convex Analysis and Variational Problems”, North Holland, Amsterdam 1976.
[4] W. Li, D. Naik, and J. Swetits, A data smoothing technique for piecewise convex/concave curves, SIAM J. Sci. Comp 17 517-537 (1996).
9
[5] M. Mächler, Variational Solution of Penalized Likelihood Problems and Smooth Curve
Estimation, Ann. Stat., 23 (1996), 1496-1517.
[6] E. Mammen, Nonparametric regression under qualitative smoothness assumptions,
Ann. Stat., 19 (1991), 741-759.
[7] C. A. Michelli, P. W. Smith, J. Swettis, J. Ward, Constrained Lp Approximation,
Construct. Approx. 1 (1985), 93-102.
[8] C. A. Michelli and F. Utreras, Smoothing and interpolation in a convex set of Hilbert
space, SIAM J. Stat. Sci. Comp. 9 (1985), 728-746.
[9] C. A. Michelli and F. Utreras, Smoothing and interpolation in a convex set of a Hilbert
space: II, the semi-norm case, Math. Model. & Num. Anal. 25 (1991), 425-440.
[10] B. W. Silverman, Spline smoothing: the equivalent variable kernel method, Ann. Stat.
12 (1984), 898-916.
[11] F. Utreras, Smoothing noisy data under monotonicity constraints - Existence, characterization and convergence rates, Numerische Math. 47 (1985), 611-625.
[12] M. Villalobos and G. Wahba, Inequality-constrained multivariate smoothing splines
with application to the estimation of posterior probabilities, J. Amer. Stat. Assoc.,
82, (1987), 239-248.
[13] G. Wahba, Spline Models for Observational Data, SIAM, Philadelphia, PA 1991.
10
| 10 |
arXiv:1612.06117v2 [math.GR] 24 Jun 2017
CELLULAR AUTOMATA, DUALITY AND SOFIC GROUPS
LAURENT BARTHOLDI
Abstract. We produce for arbitrary non-amenable group G and field K a
non-pre-injective, surjective linear cellular automaton. This answers positively
Open Problem (OP-14) in Ceccherini-Silberstein and Coornaert’s monograph
“Cellular Automata and Groups”.
We also reprove in a direct manner, for linear cellular automata, the result
by Capobianco, Kari and Taati that cellular automata over sofic groups are
injective if and only if they are post-surjective.
These results come from considerations related to matrices over group rings:
we prove that a matrix’s kernel and the image of its adjoint are mutual orthogonals of each other. This gives rise to a notion of “dual cellular automaton”,
which is pre-injective if and only if the original cellular automaton is surjective,
and is injective if and only if the original cellular automaton is post-surjective.
1. Introduction
1.1. Cellular automata. Let G be a group and let K be a field. A linear cellular
automaton on G is — no more, no less — a square matrix with entries in the group
ring KG.
The interpretation of a linear cellular automaton Θ P Mn pKGq is as follows. Let
S be a finite subset of G such that all entries of Θ are in the K-span of S. Construct
the graph G with vertex set G, and with an edge from g to gs for all g P G, s P S.
Put a copy of the vector space V :“ Kn at each vertex of G . Elements of the vector
space V G “ tc : G Ñ V u are called configurations. Then Θ, defines a one-step
evolution rule still written Θ on the space of configurations, in which each vertex of
G inherits
write
ř a value in V depending on the values at its neighbours: one may
G
c
P
V
evolves
Θ “ sPS Θs s for K-matrices Θs , and then every configuration
ř
under Θ to the configuration taking at every g P G the value sPS Θs pcps´1 gqq.
More concisely, c evolves to Θ ¨ c. For more information on linear cellular automata,
we defer to [6, Chapter 8].
Linear cellular automata are natural linear analogues of classical cellular automata, in which each vertex of G takes a value in a finite set A, which evolves
according to the values at its neighbours. The cellular automaton is thus a locallydefined evolution rule on the compact space AG . In particular, if K is a finite field,
then every linear cellular automaton is also a classical cellular automaton.
The converse, however, is far from true: linear cellular automata are extremely
restricted computational models, and there is no clear way of converting a classical
cellular automaton into a linear one. Every self-map of a finite set A induces a selfmap of the finite-dimensional vector space V :“ KA, so cellular automata acting
on AG induce linear self-maps on KpAG q, but this space is much larger than V G :
Date: June 25, 2017.
This work is supported by the “@raction” grant ANR-14-ACHN-0018-01.
1
2
LAURENT BARTHOLDI
Â
coalgebra”
the former is a completion of the tensor power G V (the “measuring
À
KG á V ), while the latter is a completion of the direct sum G V .
1.2. Sofic groups and surjunctivity. How are algebraic properties of the group
G reflected in the cellular automata carried by G ? We single out some properties
of cellular automata which have received particular attention: let us write x „ y
for x, y P AG when tg P G | xpgq ‰ ypgqu is finite. A cellular automaton Θ : AG ý
is
injective: if Θpxq “ Θpyq implies x “ y;
pre-injective: if Θpxq “ Θpyq ^ x „ y implies x “ y; otherwise one calls
such x, y Mutually Eraseable Patterns;
surjective: if Θpxq “ AG ; one then says that Θ has no Garden of Eden;
post-surjective: if y „ Θpxq implies Dz „ x : Θpzq “ y.
Moore and Myhill’s celebrated “Garden of Eden” theorem asserts that, if G “ Zd ,
then cellular automata are pre-injective if and only if they are surjective [9, 10].
This has been extended to amenable groups G by Ceccherini-Silberstein, Machı̀
and Scarabotti [4], and I proved in [1, 2] that both results may fail as soon as
G is not amenable. We shall not need the precise definition of amenable groups;
suffice it to say that one of the equivalent definitions states that G contains finite
subsets that are arbitrarily close to invariant under translation, in the sense that
for every finite S Ď G and every ǫ ą 0 there exists a finite subset F Ď G with
#pF SzF q ă ǫ#F . We recall:
Theorem 1.1 ([2]). For a group G, the following are equivalent:
(1) G is non-amenable;
(2) for some integer n and every (or equivalently some) field K, there is an
injective G-module map pKGqn Ñ pKGqn´1 .
We shall also not need the precise definition of sofic groups, a common generalization of amenable and residually finite groups; we refer to the original article [13].
Suffice it to say that it is at present unknown whether non-sofic groups exist, and
that if G is sofic then it satisfies Gottschalk’s “Surjunctivity Conjecture” from [8],
namely every injective cellular automaton is surjective [13, §3]. Capobianco, Kaari
and Taati show in [3] that, when G is sofic, every post-surjective cellular automaton
is pre-injective. Thus
injective
pre-injective
if G sofic
post-surjective
iff G amenable
surjective
We remark that if a cellular automaton is injective and surjective, then its inverse
is also a cellular automaton. Similarly, if a cellular automaton is pre-injective and
post-surjective, then it is bijective and its inverse is also a cellular automaton.
The notions of (pre-)injectivity and (post-)surjectivity become substantially simpler in the context of linear cellular automata, and exhibit more clearly the duality:
CELLULAR AUTOMATA, DUALITY AND SOFIC GROUPS
3
Lemma 1.2. A linear cellular automaton Θ : À
V G ý is pre-injective, respectively
post-surjective if and only if its restriction to
G V is injective, respectively surjective.
Note that non-surjective linear cellular automata Θ : V G ý avoid a non-empty
open subset of V G , namely there exists a finite subset F Ď G and x P V F such that
Θpyq never restricts to x on F , see Proposition 2.2.
1.3. A problem by Ceccherini-Silberstein and Coornaert. Ceccherini-Silberstein
and Coornaert prove in [5] that if G is an amenable group then a linear cellular
automaton on G is pre-injective if and only if it is surjective, and ask if this is also a
characterization of amenability in the restricted context of linear cellular automata.
The construction I gave in [2] actually produces, for every non-amenable group
G, a pre-injective, non-surjective linear cellular automaton. Ceccherini-Silberstein
and Coornaert ask in [6, Open Problem 14]:
Problem 1.3. Let G be a non-amenable group and let K be a field. Does there exist
a finite-dimensional K-vector space V and a linear cellular automaton Θ : V G ý
which is surjective but not pre-injective?
The group ring KG admits an anti-involution ˚, defined on basis elements g P G
by g ˚ :“ g ´1 and extended by linearity. It induces an anti-involution on Mn pKGq
as follows: for Θ P Mn pKGq, set pΘ˚ qij “ Θ˚ji for all i, j P t1, . . . , nu; namely, Θ˚
is computed from Θ by transposing the matrix and applying ˚ to all its entries.
Clearly Θ˚˚ “ Θ. There is a natural bilinear pairing pKGqn ˆ pKn qG Ñ K, given
by
ÿÿ
xv|ξy :“
pgq ¨ ξpgq.
gPG v
In this article, I shall prove:
Theorem 1.4. Let G be a group, let K be a field, and let Θ P Mn pKGq be a linear
cellular automaton. Then
(1.1)
(1.2)
(1.3)
(1.4)
kerpΘ|Kn GqK “ impΘ˚ |pKn qG q,
kerpΘ|pKn qG qK “ impΘ˚ |Kn Gq,
impΘ|Kn GqK “ kerpΘ˚ |pKn qG q,
impΘ|pKn qG qK “ kerpΘ˚ |Kn Gq.
In particular, Θ is pre-injective if and only if Θ˚ is surjective, and Θ is injective if
and only if Θ˚ is post-surjective.
This answers positively Problem 1.3:
Corollary 1.5. Let G be a non-amenable group and let K be an arbitrary (possibly
finite) field. Then there exist surjective, non-pre-injective linear cellular automata
on G.
Proof. Let Θ P Mn pKGq be a pre-injective, non-surjective linear cellular automaton,
obtained e.g. by adding a full row of 0’s to the matrix given by Theorem 1.1. Then
Θ˚ is the required example.
4
LAURENT BARTHOLDI
1.4. Capobianco, Kari and Taati’s result. From this duality of linear cellular
automata, one also deduces an immediate proof of Capobianco, Kari and Taati’s
main result, when restricted to linear cellular automata:
Theorem 1.6 (see [3, Theorem 2]). Let G be a sofic group. Then every postsurjective linear cellular automaton is pre-injective.
Proof. Let Θ be a post-surjective linear cellular automaton. By Theorem 1.4, Θ˚ is
injective, so Θ˚ is surjective by [13, §3], so Θ is pre-injective again by Theorem 1.4.
1.5. Reddite ergo quæ Cæsaris sunt. The notion of dual linear cellular automata is quite natural, but its first appearance seems only to be a passing remark
in [11]. The last line of Theorem 1.4 has been proven, in the setting of locally
finite graphs, by Matthew Tointon in [12]. I am indebted to Professor Coornaert
for having pointed out that reference to me when I shared this note with him.
In a recent article [7], Gaboriau and Seward study the sofic entropy of algebraic
actions, and note the following consequence of Corollary 1.5: if G is sofic but not
amenable, then the Yuzvinsky addition formula for entropy hpG # Aq “ hpG #
Bq ` hpG # A{Bq fails for some G-modules B ď A. Indeed take A “ pKn qG and
B “ kerpΘq for a surjective, non-pre-injective cellular automaton Θ. I am grateful
to Messrs. Gaboriau and Seward for having communicated their remark to me
ahead of its publication.
2. Linear cellular automata
We start with a field K and an integer n. We write V :“ Kn , and identify V
with V ˚ . There is a natural bilinear, non-degenerate pairing V ˚ ˆ V Ñ K given by
xφ|vy “ φpvq “
n
ÿ
φi vi .
i“1
Let G be a group. We denote by V G the vector space of functions G Ñ V , and
declare its closed subsets to be tc P V G | c|S P W u for all finite S Ď G and all
W ď V S . In particular, the restriction maps πS : V G Ñ V S are continuous for all
finite S Ď G, and V G is compact (but not Hausdorff).
We denote by V ˚ G the vector subspace of finitely-supported functions in V G .
There is a left action of G on V G by translation: for g P G, c P V G we define
gc P V G by pgcqphq “ cpg ´1 hq. This action preserves V ˚ G. There is also a bilinear
pairing
ÿ
x´|´y : V ˚ G ˆ V G Ñ K,
xω|cy “
xωpgq|cpgqy.
gPG
Lemma 2.1. x´|´y is non-degenerate in both arguments.
In the notation introduced above, a linear cellular automaton is both an element
of V b V ˚ G and a G-equivariant, continuous self-map Θ : V G ý. Note that Θ
restricts to a self-map V ˚ G ý.
Proposition 2.2. Let Θ : V G ý be a linear cellular automaton. Then ΘpV G q is a
closed subspace of V G .
CELLULAR AUTOMATA, DUALITY AND SOFIC GROUPS
5
Proof. Verbatim the proof of [6, Theorem 8.8.1]. Note that they claim in fact the
weaker statement that ΘpV G q is closed in the prodiscrete topology. Note also that
the proposition does not follow trivially from the fact that V G is compact, because
V G is not Hausdorff.
ř
Consider a linear cellular automaton Θ P V bV ˚ G, written as Θ “ i vi bφi gi for
finitely many vi P V, φi P V ˚ , gi P G.
ř Then, tracing back to our original definition,
its adjoint Θ˚ P V ˚ b V G is Θ˚ “ i φi b vi gi´1 .
Lemma 2.3. Let Θ P V b V ˚ G be a cellular automaton, with adjoint Θ˚ . Then
xΘ˚ pωq|cy “ xω|Θpcqy for all ω P V ˚ G, c P V G .
ř
Proof. Write Θ as a finite sum i vi b φi gi . Then the sides of the above equation
are respectively
) ˇ
E
ÿ A! ÿ
ÿ
ˇ
φi b vi pgi´1 ωq pgqˇcpgq “
xφi |cpgqy xωpgi gq|vi y
gPG
i
gPG,i
and
ÿA
gPG
ˇ! ÿ
) E
ÿ
ˇ
vi b φi pgi cq pgq “
xωpgq|vi y xφi |cpgi´1 gqy,
ωpgqˇ
i
gPG,i
which are just permutations of each other.
3. Proof of Theorem 1.4
Let Θ P Mn pKGq be a linear cellular automaton, and as in §2 set V “ V ˚ “ Kn ,
with the usual scalar product.
We begin by the inclusion kerpΘ|V ˚ GqK Ě impΘ˚ |V G q from (1.1). Given c P
impΘ˚ |V G q, say c “ Θ˚ pdq, for all ω P kerpΘ|V ˚ Gq we have
xω|cy “ xω|Θ˚ pdqy “ xΘpωq|dy “ x0|dy “ 0,
so c K kerpΘ|V ˚ Gq. The exact same computation gives all ‘Ě’ inclusions from (1.2),
(1.3) and (1.4).
We continue with the inclusion kerpΘ|V ˚ Gq KĎ impΘ˚ |V G q from (1.1). Given
c R impΘ˚ |V G q, there exists an open neighbourhood of c in V G zimpΘ˚ |V G q by
Proposition 2.2; so there exists a finite subset S Ď G and a proper subspace W ă V S
such that the projection πS pV G q belongs to W . Since V S is finite-dimensional,
there exists a linear form ω on V S that vanishes on W but does not vanish on
c. Note that ω, qua element of pV S q˚ , is canonically identified with an element
of pV ˚ qS , and therefore with an element of V ˚ G. From ω K impΘ˚ |V G q we get
Θpωq K V G so Θpωq “ 0 because the scalar product x´|´y is non-degenerate.
Therefore c M kerpΘ|V ˚ Gq as desired.
We continue with the inclusion kerpΘ|V G q KĎ impΘ˚ |V ˚ Gq from (1.2). Given
ω R impΘ˚ |V ˚ Gq, there exists a linear form c P pV ˚ Gq˚ that vanishes on impΘ˚ |V ˚ Gq
but does not vanish on ω. Note that pV ˚ Gq˚ canonically identifies with V G . From
c K impΘ˚ |V ˚ Gq we get Θpcq K V ˚ G, so Θpcq “ 0 because the scalar product
x´|´y is non-degenerate. Therefore ω M kerpΘ|V G q as desired.
We finally consider the inclusion impΘ|V ˚ GqK Ď kerpΘ˚ |V G q from (1.3). Given
c K impΘ|V ˚ Gq, we have c K Θpωq for all ω P V ˚ G, so Θ˚ pcq K ω for all ω P V ˚ G,
so Θ˚ pcq K V ˚ G and therefore Θ˚ pcq “ 0 because the scalar product x´|´y is
non-degenerate. The exact same computation gives the ‘Ď’ inclusion from (1.4).
6
LAURENT BARTHOLDI
Recalling that Θ is pre-injective if and only if kerpΘ|V ˚ Gq “ 0 and Θ is injective
if and only if kerpΘ|V G q “ 0 and Θ is post-surjective if and only if impΘ|V ˚ Gq “
V ˚ G and Θ is surjective if and only if impΘ|V G q “ V G , the last conclusions follow.
References
[1] Laurent Bartholdi, Gardens of Eden and amenability on cellular automata, J. Eur.
Math. Soc. (JEMS) 12 (2010), no. 1, 241–248, DOI 10.4171/JEMS/196, available at
arXiv:math/0709.4280. MR2578610 (2011e:05282)
[2] Laurent Bartholdi and Dawid Kielak, Amenability of groups is characterized by Myhill’s
Theorem, Journal Europ. Math. Soc. (2016), to appear, available at arXiv:cs/1605.09133.
[3] Silvio Capobianco, Jarkko Kari, and Siamak Taati, Post-surjectivity and balancedness of
cellular automata over groups (2015), available at arXiv:1507.02472.
[4] Tullio G. Ceccherini-Silberstein, Antonio Machı̀, and Fabio Scarabotti, Amenable groups and
cellular automata, Ann. Inst. Fourier (Grenoble) 49 (1999), no. 2, 673–685 (English, with
English and French summaries). MR1697376 (2000k:43001)
[5] Tullio G. Ceccherini-Silberstein and Michel Coornaert, The Garden of Eden theorem for linear
cellular automata, Ergodic Theory Dynam. Systems 26 (2006), no. 1, 53–68. MR2201937
(2007a:37017)
, Cellular automata and groups, Springer Monographs in Mathematics, Springer[6]
Verlag, Berlin, 2010. MR2683112 (2011j:37002)
[7] Damien Gaboriau and Brandon Seward, Cost, ℓ2 -Betti numbers and the sofic entropy of some
algebraic actions (2017), available at arXiv:1509.02482. to appear.
[8] Walter Gottschalk, Some general dynamical notions, Recent advances in topological dynamics
(Proc. Conf. Topological Dynamics, Yale Univ., New Haven, Conn., 1972; in honor of Gustav
Arnold Hedlund), Springer, Berlin, 1973, pp. 120–125. Lecture Notes in Math., Vol. 318.
MR0407821
[9] Edward F. Moore, Machine models of self-reproduction, Mathematical problems in the biological sciences. Proc. Sympos. Appl. Math. XIV, 1962, pp. 17–33. MR0299409 (45 #8457)
[10] John Myhill, The converse of Moore’s Garden-of-Eden theorem, Proc. Amer. Math. Soc. 14
(1963), 685–686. MR0155764 (27 #5698)
[11] Adriana Popovici and Dan Popovici, Dilatability of Linear Cellular Automata, Proceedings
of the 19th International Symposium on Mathematical Theory of Networks and Systems —
MTNS 2010, 2010.
[12] Matthew C. H. Tointon, Characterizations of algebraic properties of groups in terms of harmonic functions, Groups Geom. Dyn. 10 (2016), no. 3, 1007–1049, DOI 10.4171/GGD/375.
MR3551188
[13] Benjamin Weiss, Sofic groups and dynamical systems, Sankhyā Ser. A 62 (2000), no. 3,
350–359. Ergodic theory and harmonic analysis (Mumbai, 1999). MR1803462
Département de Mathématiques et Applications, École Normale Supérieure, Paris
and Mathematisches Institut, Georg-August Universität zu Göttingen
E-mail address: laurent.bartholdi@gmail.com
| 4 |
arXiv:1611.01595v1 [math.ST] 5 Nov 2016
Power Computations for Intervention
Analysis
A. Ian McLeod and Evelyn R. Vingilis
Department of Statistical and Actuarial Sciences,
The University of Western Ontario
London, Ontario N6A 5B7
aimcleod@uwo.ca
Department of Family Medicine,
University of Western Ontario
London, Ontario N6G 4X8
evingili@uwo.ca
A. Ian McLeod and Evelyn R. Vingilis (2005), Power Computations for
Intervention Analysis, Technometrics 47/2, 174-180.
1
Abstract
In many intervention analysis applications time series data may be
expensive or otherwise difficult to collect. In this case the power function
is helpful since it can be used to determine the probability that a proposed
intervention analysis application will detect a meaningful change.
Assuming that an underlying ARIMA or fractional ARIMA model is
known or can be estimated from the pre-intervention time series, the
methodology for computing the required power function is developed for
pulse, step and ramp interventions with ARIMA and fractional ARIMA
errors. Convenient formulae for computing the power function for
important special cases are given. Illustrative applications in traffic safety
and environmental impact assessment are discussed.
KEY WORDS: Autocorrelation and lack of statistical independence;
ARIMA time series models; Environmental impact assessment; Forecast
and actuality significance test; Long-memory time series; Sample size;
Two-sample problem.
2
1. INTRODUCTION
Intervention analysis developed by Box and Tiao (1976a) has been
widely used in a variety of applications in engineering, biological,
environmental and social sciences to quantify the effect of a known
intervention at time t = T on data collected as a time series,
zt , t = 1, . . . , n. In its simplest form, intervention analysis itself may be
regarded as a generalization of the two-sample problem to the case where
the error or noise term is autocorrelated. It is well-known that the usual
two-sample procedures are not robust against alternatives involving
autocorrelation (Box, Hunter and Hunter, 1978, §3.1). The purpose of this
article is to describe methods for computing the necessary sample size to
detect an intervention with a prescribed power and level. It is shown by
simulation experiments that these methods can be accurate even in
moderately small samples. Statistical power computations have also been
studied by Tiao et al. (1990) and Weatherhead et al. (1998) for particular
types of intervention analysis models used for trend detection with
environmental time series. This article extends and refines these results.
It is assumed that for t < T + b, where b is the delay parameter, the
time series is generated by a fractional ARIMA (p, d, q) with fractional
differencing parameter |f | < 0.5. Stationary short-memory time series
models, d = f = 0, are used in environmental impact assessment (Box and
Tiao, 1976a; Tiao et al., 1990; Noakes and Campbell, 1992; Weatherhead
et al. 1998; Hipel and McLeod, 1994, §19.4.5) and in quality control
(Jiang, Tsui and Woodall, 2000) as well as in many other areas of science
and technology. Nonstationary models with d = 1 and/or long-memory
models with 0 < f < 0.5 have numerous applications in the physical and
engineering sciences such as: quality control and industrial time series
(Luceño, 1995; Box and Luceño, 1997), internet traffic (Cao et al., 2001),
daily solar irradiance (Kärner, 2002), levels of Lake Huron (Roberts, 1991,
p.319-320), daily wind-speed (Haslett and Raftery, 1989), and various
types of hydrological time series (Beran, 1994; Hipel and McLeod, 1994).
In general, we may write the fractional ARIMA model for the
pre-intervention series as
∇d+f zt = ξ + θ(B)/φ(B)at ,
t = 1, . . . , T + b − 1,
(1)
where ξ is the constant term, d is the differencing parameter, ∇ = 1 − B,
θ(B) = 1 − θ1 B − . . . − θq B q , φ(B) = 1 − φ1 B − . . . − φp B p and B is the
backshift operator on t. The innovations, denoted by at , t = 1, . . . , n, are
assumed to be independent and normally distributed with mean zero and
3
variance σa2 . It is also assumed that φ(B) = 0 and θ(B) = 0 have no
common roots and that all roots are outside the unit circle.
2. SIMPLE INTERVENTION ANALYSIS (SIA) MODEL
2.1 Introduction
The SIA model may be written,
(T )
∇d zt = ξ + ω∇d B b It
+ ∇−f
θ(B)
at ,
φ(B)
t = 1, . . . , n,
(2)
(T )
where It is the intervention series, ω is the parameter indicating the
magnitude of the intervention and ∇−f θ(B)/φ(B)at is the stationary error
component. In this article three types of intervention series are used, the
step, pulse and ramp series, defined respectively by,
(T )
It
(T )
It
=
(T )
St
=
(T )
Pt
or
(T )
It
=
(T )
Rt
=
=
0,
1
if t < T ,
if t ≥ T ,
(3)
=
0,
1
if t 6= T ,
if t = T ,
(4)
0,
t−T +1
if t < T ,
if t ≥ T .
(5)
In practice two of the most common models for the error are the AR (1)
and IMA (1) which correspond respectively to p = 1, d = 0, q = 0 and
p = 0, d = 1, q = 1. In the case of a step intervention, the SIA model
implies that for t ≥ T + b an increase of ω occurred. So the SIA model
with a step intervention can be regarded as the time-series generalization
of the standard two-sample test for a change in location and in practice
this is one of the most frequently applicable models. Pulse interventions
are useful for dealing with outliers (Chang, Tiao and Chen, 1988). A ramp
intervention has been used to model the recovery trend in stratospheric
ozone (Reinsel et al. 2002).
The SIA model may be generalized by allowing for multiple
interventions and other types of interventions, as well as for seasonal
ARIMA errors and possible covariates (Tiao et al., 1990; Weatherhead et
al., 1998; Reinsel, 2002; Reinsel et al., 2002). All of these situations are
easily handled with the methods discussed in §1.2 and §1.3. Power
computations, although possible, are less useful when applied to dynamic
response interventions for the reasons explained in Appendix B.
4
2.2 Information Matrix
Letting λ1 = (ξ, ω) and λ2 = (φ1 , . . . , φp , θ1 , . . . , θq , f ), it is shown in
Appendix A that the expected Fisher information matrix is block diagonal
with blocks, Iλ1 and Iλ2 corresponding to λ1 and λ2 . For the first block,
Iξ,ω = σa−2 J ′ Γ−1
n J,
(6)
where σa−2 Γ−1
n is the inverse of the covariance matrix of the stationary
component and J is an n × 2 matrix with 1 in the first column and
(T )
∇d It , t = 1, . . . , n in the second column. The Trench algorithm (Golub
and Van Loan, 1983) provides a computationally efficient method for
computing Γn−1 . An expression essentially equivalent to eqn. (6) was
obtained by Tiao et al. (1990) and Weatherhead et al. (1998) using
generalized least squares. Assuming approximate normality of the
estimates, the asymptotic variance of the maximum likelihood estimate of
ω is found by taking the (2, 2) element of the inverse of (6),
σω̂ =
√
2
I1,1 / I1,1 I2,2 − I1,2
,
(7)
where Ii,j denotes the (i, j) entry in the matrix Iξ,ω . If the constant term,
√
ξ, is not present, σω̂ = 1/ I2,2 . When there is an extensive amount of
data prior to the intervention it is sometimes helpful to simply correct the
series by its sample mean and assume ξ = 0 (Tiao et al., 1990).
The results of Pierce (1972) provide a computationally efficient
approximation to (6) when f = 0. From Pierce (1972, eqn. 3.2) we can
write the Fisher information for (ξ, ω) based on n observations as
Iξ,ω = σa−2
nκ2
P
κ t vt
P
v
P t 2t
κ
t vt
,
(8)
(T )
where κ = −φ(1)/θ(1) and vt = −φ(B)/θ(B)wt , where wt = ∇d It .
Without loss of generality we take b = 0 since if b > 0, the formulae hold
with T replaced by T + b. Provided that T is not too small and T is not
too close to n, eqn. (8) yields almost identical values to the more exact
formula given in (6). New explicit expressions, using Pierce’s
approximation for AR (1) and IMA (1) cases, are given in Tables 1 and 2
below for step, pulse and ramp interventions.
[Tables 1 and 2 about here]
From eqn. (6), it follows that for consistency of the estimates ξˆ and ω̂,
Iξ,ω /n or equivalently, J ′ J/n, must converge to a nonsingular matrix. For
5
the intervention analysis models defined by eqns. (2), (3), (4) and (5), this
happens provided that
n
1X
(T )
∇d It → c,
n t=1
c > 0, c 6= 1.
(9)
If the constant term, ξ, is assumed to be known or zero then only c > 0 is
needed. This result is certainly not the whole story from the application
point of view. In §1.5 we show using simulation experiments that the
empirical variances may be accurately estimates from 7) even when eqn.
(9) is not satisfied.
2.3 Power and Sample Size
The null hypothesis H0 : ω = 0 can be tested using two asymptotically
equivalent methods. The first method, referred to as the Z-test, uses
Z = ω̂/σ̂ω̂ , where ω̂ is the maximum likelihood estimate for ω and σ̂ω̂ is its
estimated standard error. Note that σω̂ , the standard error of ω̂, depends
only on the underlying ARIMA model in the pre-intervention period and
so it can be estimated before the post-intervention data are obtained. A
second asymptotically equivalent method is to use a likelihood-ratio test.
The asymptotic theoretical power function for the Z-test of the null
hypothesis H0 : ω = 0 against the two-sided alternative at level α is
Pr {| ω̂ | > Z1−α/2 σω̂ |ω}, where Z1−α/2 is the upper (1 − α/2)-quantile in
the standard normal distribution. For brevity the asymptotic theoretical
power function will be referred to simply as the power function. In practice
this power function is approximated by replacing σω̂ by an estimate, σ̂ω̂ ,
based either on the pre-intervention data or on other prior knowledge.
Often it is more convenient to use the rescaled parameter, δ = ω/σ, where
σ 2 is the variance of the stationary error component since in this case
knowledge of σ 2 is not needed. The power function may be expressed in
terms of δ as
Π(δ) = Φ(−Z1−α/2 − δσ/σω̂ ) + 1 − Φ(Z1−α/2 − δσ/σω̂ ),
(10)
where Φ(•) denotes the cumulative distribution function of the standard
normal. If the variance of the pre-intervention series, σ 2 , is known or
estimated, the power function for ω is Π(ω/σ). Eqn. (10) should be
adjusted if only a one-sided alternative is under consideration.
As in Tiao et al. (1990) it is sometimes of interest to estimate the
amount of additional data needed to detect an intervention of a specified
magnitude with a prescribed power. The power function Π(δ) may be
6
expressed more fully as a function of the test level α and the other
underlying parameters n and T so we can write the power function more
fully as Π(δ, α, n, T ). For a fixed α = α(0) , δ = δ(0) and a prescribed power
Π(0) we may estimate the number of additional data values, m, that are
required by numerically solving the equation Π(δ(0) , α(0) , T + m − 1, T )
= Π(0) . If as in the geophysical datasets considered in Tiao et al. (1990)
there is extensive pre-intervention data, we may assume the mean is known
and take T = 1 and solve Π(δ(0) , α(0) , m, 1) = Π(0) . This technique is
illustrated in §1.4 where it is also explained that in some situations, due to
the limitations imposed by the model, there is no solution for m.
In general the power and sample size computations for interventions
with ARIMA and fractional ARIMA errors are easily done using an
advanced quantitative programming environment such as Mathematica,
MatLab, S or Stata. In the case of SIA with AR (1) or IMA (1) errors,
power computations can even be done on a hand calculator.
2.4 Numerical Illustrations
The power and sample size computations are illustrated in this section
for the SIA with a step intervention with AR (1), IMA (1) and
fractionally-differenced white noise. First an approximation to the
detection limit, δ′ , is derived for the step intervention in an SIA model with
unknown mean, stationary short-memory errors, with f = d = 0, and a
fixed number, T − 1, of pre-intervention observations. The variance of the
P
.
estimate, δ̂, may be written, Var (δ̂) = γδ /T, where γδ = ∞
k=−∞ γk /γ0 , γk
is the autocovariance function for the stationary pre-intervention series and
γ0 = σ 2 . To achieve 90% power, Pr {(δ̂ − δ′ )/ SE (δ̂) > 1.96 −δ′ / SE (δ̂)}
.
.
.
= 0.9. Hence 2 − δ′ / SE (δ̂) = −1.3. So δ′ = 3.3 SE (δ̂).
Using Table 1, the power curve for the AR (1) with unknown mean,
√
n = 50, T = 25 and φ1 = 0.5, σω = 0.526681. With σ = 1/ (1 − φ21 ) =
1.1547, the power curve is Π(δ) = 1 + Φ(−1.960 − 2.192 × δ)
− Φ(1.960 − 2.192 × δ). This and the power curve obtained by letting
n → ∞ are shown in Figure 1 as well as the approximate detection level,
√
.
δ′ = γδ / T = 1.14. For comparison, the exact value of δ′ found by
numerically solving Π(δ′ , 0.5, 109 , 25) = 0.9 is δ′ = 1.12. Assuming an
unknown mean and that T = 25, we can find m, the number of additional
observations needed to achieve a prescribed power level. For example, for
90% power with δ(0) = 1.5, solving Π(1.5, 0.05, 25 + m − 1, 25) = 0.9 we
find m = 23. In the known mean case taking T = 1 we find m = 10. In the
unknown mean case, if δ(0) ≤ γδ there is no solution but if the mean is
known then m can always be found.
7
[Figure 1 about here]
The middle panels of Figure 2 illustrate the power curves for an
IMA (1) with n = 50 and T = 25. With θ1 = 0.5, Π(δ) =
1 + Φ(−1.960 − 1.252 × δ) −Φ(1.960 − 1.252 × δ).
Since long-memory or fractional time series have also been suggested
for various types of geophysical data, it is of interest to examine the
impact of this type of process on our ability to detect interventions. Table
3 compares the power of a two-sided 5% level test of the fractionally
differenced white noise model p = d = q = 0 with f = 0.2 and f = 0.4 to
the corresponding approximating ARMA(1, 1) when n = 50 and T = 25.
The approximating ARMA(1, 1) model was determined by equating the
first two autocorrelations in the fractional model with the first two
autocorrelations in the ARMA(1, 1) and solving to obtain the parameters
φ1 and θ1 . In the first case with f = 0.2 the power is almost identical and
in the second case with f = 0.4 the power is slightly higher for the
ARMA(1, 1) approximation. This suggests that long term memory in the
fractional noise model has little effect on the power when the length of the
series is moderate, as in this example with n = 50 and T = 25. For
sufficiently long time series, the effect on long memory is much more
important and the ARMA(1, 1) approximation does not hold.
[Table 3 about here]
2.5 Simulation Experiment
The power function derived in eqn. (10) relies on the asymptotic
normality of the maximum likelihood estimator and so it is helpful to check
its accuracy by simulation. We do this by comparing the power function
with the empirical power function, Π̂. For each simulated time series all
parameters in the model were estimated by exact maximum likelihood
estimation and the Z-test was computed. The empirical power, Π̂, of a
two-sided 5% test is then the proportion of times that the absolute value of
this Z-statistic exceeded 1.96 in absolute value and the 95% confidence
√
interval for Π is Π̂ ± 1.96 (Π̂(1 − Π̂)/N ), where N is the number of
simulations. For each model and each parameter setting, N = 1, 000.
The model in eqn. (2) was simulated with n = 50 and T = 25 and
AR (1) errors with φ1 = 0, 0.25, 0.5, 0.75, ω = δσ, where
δ = 0, ±0.25, ..., ±2.0. The empirical power confidence limits and
theoretical power given by eqn. (10) are compared in Figure 2. It is seen
that eqn. (10) provides an accurate approximation. The IMA (1), is a
commonly occurring nonstationary time series model. Figure 2 compares
the theoretical and empirical power for the case with n = 50 and T = 25
8
using a two-sided Z-test at the 5% level. Once again it is seen that eqn.
(10) holds very well despite the small sample size. The values selected for
θ1 are positive since this is the most common situation in practice. The
power improves, as expected, as θ1 increases from 0 to 1. Notice that this
model does not satisfy eqn. (9). The last column of Figure 2 compares the
empirical and theoretical power in the case of fractionally differenced white
noise, p = q = d = 0 for f = 0.0, 0.2, 0.3, 0.4. The approximation to the
theoretical power improves with increasing f . The simulations shown in
Figure 2 were repeated using the likelihood-ratio test and essentially
equivalent results were obtained.
[Figure 2 about here]
In conclusion, the simulations in Figure 2 suggest that for practical
purposes if n, T and n − T are not too small the asymptotic theoretical
power curve provides a good small sample approximation. Alternatively,
the simulations show that ω̂ is well approximated using its large-sample
approximation even for moderately small samples. As already noted, σω̂ ,
must also be estimated by σ̂ω̂ using either the pre-intervention data or an
estimate of its likely autocorrelation function. In practice, as in the
example in §2.1, a range of likely parameter values are often used to
indicate a range of possible power curves.
2.6 Model Uncertainty
Box, Jenkins, and Reinsel (1994) found that both the ARMA(1, 1) and
IMA(1) fit Series A, Chemical Process Concentrations about equally well.
Both models give similar one step ahead forecasts but the long run
forecasts are very different. The situation is similar with the power
functions for these two models.
Consider a hypothetical step intervention which occurs immediately
after the last observation. In this case T = 198 and the power curve as a
function of ω is tabulated for a few selected values in Table 7 for a
two-sided 5% test assuming that m post-intervention observations are
available for m = 5 and m = 50. When m = 5 the power curves are quite
similar but for m = 50 the power increases for the ARMA model but stays
essentially the same in the case of the IMA model. For example, Table 7
shows that there is a 75% chance of detecting a change of 0.6 with just 5
post-intervention observations.
[Table 4 about here]
2.7 Forecast-Actuality Significance Test
9
Box and Tiao (1976b) described an omnibus significance test for
detecting if an intervention has occurred. If at , t = T, ..., n denote the
one-step ahead prediction errors of an assumed model, then the test
P
statistic may be written, Q = nt=T a2t /σa2 . If the intervention has no
effect, Q is approximately χ2 -distributed on m = n − T + 1 df. This
significance test is easy to apply and does not require specification of an
intervention model and its estimation. However, as might be expected, the
loss of power can be considerable as will now be demonstrated.
As an example, consider the SIA model with a step intervention. Then
it can shown using eqn. (4) of Box and Tiao (1976b) that
Q = ||ω1′m π/σa + a/σa ||2 , where 1m denotes the m-dimension vector with
1 in each position, a = (aT , ..., an ), π = (πi−j ) is the lower triangular
matrix with (i, j) entry πi−j , where πk is the coefficient of B k in the
expansion ∇d φ(B)/θ(B) = 1 + π1 B + π2 B 2 + .... So Q has a χ2
distribution with m df and noncentrality parameter ν = (ω 2 /σa2 )||1′m π||2
and hence the large-sample power function can be computed. Figure 3
compares the power of this significance test with the SIA model hypothesis
test for an example with n = 120, T = 101 and AR(1) errors. Figure 3
shows that the power of the significance test can be substantially less than
the intervention analysis hypothesis test.
[Figure 3 about here]
3. ILLUSTRATIVE APPLICATIONS
3.1 Traffic Safety and Public Policy
On May 1, 1996, liquor bar closing time in Ontario was changed from
1 AM to 2 AM. In a proposed intervention analysis we wished to examine
the possible effect of this change on late-night automobile fatalities. The
data for this study comprised the total number of fatalities every month in
Ontario during the hours of 11PM to 4AM for a period of years before and
after May 1, 1996. For comparison we also collected similar time series
data for Michigan and New York State. Data for this analysis were
expensive to obtain since raw records needed to be assembled, cleaned and
aggregated from sources in various jurisdictions. Initially we planned to
obtain monthly time series on the the total number of fatalities from
January 1994 to December 1998. This would yield n = 60 observations and
with the intervention occurring at T = 36. At additional cost, we could
obtain complete monthly time series covering the period January 1992 to
December 1998 which corresponds to n = 84 and T = 48. We were
interested to know if (n = 60, T = 36) or (n = 84, T = 48) would be
10
sufficient to detect change of σ or greater with a reasonably high
probability, where σ is the standard deviation of the pre-intervention series.
Based on previous experience with similar time series (Vingilis, et al.,
1988) we expected the time series will exhibit small autocorrelations which
may be modelled by an AR (1) with parameter φ1 ≤ 0.5. The intervention
was expected to cause an increase in late-night fatalities, so a one-sided
upper-tail test is appropriate. The power function in this case is Π(δ) =
1 − Φ(1.645 − 2.362 × δ). Table 5 shows the power of a 5% upper-tail test
for these two plans for various φ1 . When φ1 = 0.5, Table 5 shows that
(n = 84, T = 48) has a 86.7% chance of detecting a step intervention whose
magnitude is only one standard deviation of the error component whereas
the corresponding power for (n = 60, T = 36) is 76.3%. The results of
Table 5 demonstrated to our satisfaction and that of the granting agency,
that (n = 84, T = 48) had a good chance of detecting a meaningful change
and was worth the extra expenditure.
[Table 5 about here]
3.2 Detecting Ozone Turnaround
Tiao et al. (1990) used the SIA model with a ramp intervention with
AR (1) errors to model the trend in monthly deseasonalized stratospheric
ozone and other environmental variables. For simplicity Tiao et al. (1990)
assumed that the mean of the pre-intervention series was known. It may be
shown that the expression obtained by Tiao et al. (1990, Appendix A) for
√
σω̂ is exactly equal to σω̂ = 1/ I2,2 using Table 1 with n = T and T = 1.
Table 6 compares this result with the corresponding result obtained using
the exact expected Fisher information matrix given in eqn. (6) for the
same parameters as used in Tiao et al. (1990, Table 1). When φ = 0.8, the
difference is as high as 17% but it decreases as the sample size increases.
The approximation is very good for parameter values 0.6 and less. For
most of the geophysical time series considered by Tiao et al. (1990) the
degree of autocorrelation is quite low, so this approximation works well.
[Table 6 about here]
Tiao et al. (1990, Table 2) also consider the number of years of
monthly data needed to detect a ramp intervention for several geophysical
time series of interest. In their computations it was assumed that T = 1
and that the mean was known. Table 7 below computes the number of
years of data needed for these time series under the assumptions that the
mean is unknown but that there are 30 years of prior data. The other
assumptions about the data and the form of the intervention are the same
as in Tiao et al. (1990). The parameter δ shown in the table was based on
11
the information supplied by Tiao et al. (1990). Specifically, δ = ω/ (12 × σ̂)
where φ̂1 and σ̂ are obtained from Tiao et al. (1990, Table 2) and ω is
obtained from Tiao et al. (1990, p.20,510). Note that ω was divided by 12
because the form of the intervention used in Tiao et al. (1990) was
(T +1)
/12. In conclusion, the estimate of the sample size required shown
Rt
in Table 7 is in reasonable agreement with the results in Tiao et al. (1990).
[Table 7 about here]
4. CONCLUDING REMARKS
We have shown how the power function for an intervention analysis
may be computed provided that we have an estimate of the ARIMA
parameters in the pre-intervention time series or in some closely related
time series. In the case of the SIA model with AR (1) or IMA (1) errors,
the power function can easily be computed using a hand calculator. Such
programs are freely available for the Texas Instruments TI-83 from the
first author’s webpage. Mathematica and S software for computing the
power functions and all tables and figures described in this paper are also
available there as well as various other supplements to this article.
The emphasis of this article has been on the use of the power function
as an aid in selecting the sample size. In the case of the SIA model, if
Π(ω ′ ) = 1 − β ′ for a 5% two-sided test of H0 : ω = 0 then the usual 95%
confidence interval for ω will contain 0 with probability β ′ when ω = ω ′ .
So the power function may be used as an aid in choosing the sample size so
that a useful confidence interval is obtained. Instead of the power function
we could have focussed on the width of a suitable interval estimate of ω.
Since this also depends on an estimate of σω̂ the methods presented are
applicable. It may be noted that overemphasis on hypothesis tests has
long been condemned as was already noted many years ago by Cox (1977).
Nevertheless, as indicated by Cox (1977), such tests remain important in
practice.
The power function depends strongly on the degree of autocorrelation
in the pre-intervention time series. In the stratospheric ozone example,
§2.2, a long pre-intervention series was available which enabled the model
to be accurately estimated. In other cases, such as the traffic safety
example, §2.1, the pre-intervention series is either unavailable or quite
short. In such cases there may be prior information available which
indicates a range of likely models. As discussed in §2.1, this may still be
very useful for planning purposes. A final note of caution, power
computations should only be used before the analysis of the data is done
12
(Hoenig and Heisey, 2001; Lenth, 2001) and should never be used to
compute the observed power after a test of hypothesis has already been
carried out.
ACKNOWLEDGMENTS
This research was supported by grants from NIAAA and NSERC. The
authors would like to thank the Editor, an Associate Editor, two referees
and Dr. R.J. Kulperger for helpful suggestions.
13
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Chapman and Hall.
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Experimenters, New York: Wiley.
Box, G.E.P., Jenkins, G.M. and Reinsel, G.C. (1994), Time Series
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Box, G.E.P. and Luceño, A. (1997), Statistical Control by Monitoring and
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Box, G.E.P. and Tiao, G.C. (1976a), “Intervention Analysis with
Applications to Economic and Environmental Problems,” Journal of
the American Statistical Association, 70, 70–79.
Box, G. E. P. and Tiao, G. C. (1976b), “Comparison of Forecast and
Actuality,” Applied Statistics 25 (1976), 195–200.
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Chang, I., Tiao, G.C. and Chen, C. (1988), “Estimation of Time Series
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Golub, G. and Van Loan (1983), Matrix Computations, Baltimore: John
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Haslett, J. and Raftery, A. E. (1989), “Space-time Modelling with
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Resources and Environmental Systems, Amesterdam: Elsevier.
Hoenig, J. M. and Heisey, D. M. (2001), “The Abuse of Power: The
Pervasive Fallacy of Power Calculations for Data Analysis,” The
American Statistician, 55, 19–24.
Jiang, W., Tsui, K.L. and Woodall, W.H. (2000), “A New SPC Monitoring
Method: The ARMA Chart,” Technometrics 42, 399–410.
Luceño, A. (1995), “Choosing the EWMA Parameter in Engineering
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Kärner, O. (2002), “On Nonstationarity and Antipersistency in Global
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Determination,” The American Statistician 55 187–193.
14
Noakes, D. J. and Campbell, A. (1992), ”Use of Geoduck Clams to
Indicate Changes in the Marine Environments of Ladysmith Harbour,
British Columbia,” EnvironMetrics 3, 81–97.
Pierce, D.A. (1972), “Least Squares Estimation in Dynamic-disturbance
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Reinsel, G. C. (2002), “Trend Analysis of Upper Stratospheric Umkehr
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Letters 29(10), doi:10.1029/2002GL014716.
Reinsel, G. C., Weatherhead, E. C., Tiao, G. C., Miller, A. J., Nagatani,
R. M., Wuebbles, D. J., and Flynn, L. E. (2002), “On Detection of
Turnaround and Recovery in Trend for Ozone,” Journal of
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Roberts, (1991), Data Analysis for Managers with Minitab. 2nd Ed. San
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Tiao, G.C., Reinsel, G.C., Xu, D., Pedrick, J.H., Zhu, X., Miller, A.J.,
DeLuisi, J.J., Mateer, C.L. and Wuebbles, D.J. (1990), “Effects of
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D12, 20,507–20,517.
Weatherhead, E.C., Reinsel, G.C., Tiao, G.C., Meng, X.L., Choi, D.,
Cheang, W.K., Keller, T., DeLuisi, J., Wuebbles, D.J., Kerr, J.B.,
Miller, A.J., Oltmans, S.J. and Frederick, J.E. (1998), “Factors
Affecting the Detection of Trends: Statistical Considerations and
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15
1
0.8
0.6
Β
0.4
0.2
-∆¢
0
-2
∆¢
-1
1
0
∆
2
Figure 1: Comparison of Power Curves For n = 50, T = 25 and n =
∞, T = 25. The solid curve shows for n = 50, T = 25 and the dashed
.
curve, n = ∞, T = 25. The approximate detection limit, δ′ = 1.143 is also
shown.
16
-2
-1
phi(1) = 0.75
0
1
2
theta(1) = 0.75
f = 0.4
1.0
0.8
0.6
0.4
0.2
phi(1) = 0.5
theta(1) = 0.5
f = 0.3
phi(1) = 0.25
theta(1) = 0.25
f = 0.2
1.0
0.8
0.6
power
0.4
0.2
1.0
0.8
0.6
0.4
0.2
phi(1) = 0
theta(1) = 0
f=0
1.0
0.8
0.6
0.4
0.2
-2
-1
0
1
2
-2
-1
0
delta
Figure 2: Comparison of Empirical and Theoretical Asymptotic Power in
the SIA Model with AR(1), IMA(1) and Fractionally-Differenced White
Noise. The parameter δ = ω/σ is the rescaled step size. The solid curve
shows the theoretical power defined in eqn. (10). The vertical bars show
the width of a 95% confidence interval for the empirical power in 1,000
simulations of the model. The AR(1) and IMA(1) parameters φ1 and θ1
are denoted by phi(1) and theta(1) in the diagram.
17
1
2
0.0
0.5
phi(1)=0.0
1.0
1.5
2.0
phi(1)=0.4
phi(1)=0.8
1.0
power
0.8
0.6
0.4
0.2
0.0
0.5
1.0
1.5
2.0
0.0
0.5
delta
Figure 3: Comparison of Power Functions for a SIA Model with a Step
Intervention with AR(1) Errors and the Forecast-Actuality Significance
Test For a Two-Sided Test at the 5% Level. The model parameters are
n = 120, T = 101, delta = δ = ω/σ and phi(1) = φ1 . The solid thin
curve shows the SIA Model based hypothesis test and the solid thick
curve shows the omnibus significance test using Q. Since both power
functions are symmetric about δ = 0 only the upper half is shown.
18
1.0
1.5
2.0
Table 1: Information Matrix for Simple Intervention Analysis with AR(1)
Errors. The table gives the (1, 2) and (2, 2) entries, I1,2 /σa2 and I2,2/σa2 .
For each intervention type, I1,1 /σa2 = n(1 − φ1 )2 and the (2, 1) entry is
obtained by symmetry.
Type
Step
Pulse
Ramp
Information Matrix Entries
I1,2 /σa2 =
I2,2 /σa2 =
I1,2 /σa2 =
I2,2 /σa2 =
I1,2 /σa2 =
I2,2 /σa2 =
(n − T )(1 − φ1 )2 + 1 − φ1
(n − T )(1 − φ1 )2 + 1
1 − φ21
1 − φ21
(1 + n − T ) (1 − φ1 ) (2 + n − T − (n − T ) φ1 ) /2
(1 + n − T ) (6 + 7 n + 2 n2 − 7 T − 4 n T + 2 T 2 − 8 n φ1
−4 n2 φ1 + 8 T φ1 + 8 n T φ1 − 4 T 2 φ1 + n φ1 2 + 2 n2 φ1 2 − T φ1 2
−4 n T φ1 2 + 2 T 2 φ1 2 )/6
Table 2: Information Matrix for Simple Intervention Analysis with
IMA (1) Errors. For θ1 = 0 set θ10 = 1. The table gives the (1, 2)
and (2, 2) entries, I1,2 /σa2 and I2,2 /σa2 . For each intervention type,
I1,1 /σa2 = (n − 1)/(1 − θ1 )2 and the (2, 1) entry is obtained by symme-
try.
Type
Step
Pulse
Ramp
Information Matrix Entries
I1,2 /σa2 =
I2,2 /σa2 =
I1,2 /σa2 =
I2,2 /σa2 =
I1,2 /σa2 =
I2,2 /σa2 =
(1 − θ1 )−2 (1 − θ n+1−T )
(1 − θ12 )−1 (1 − θ 2(n+1−T ) )
(1 − θ1 )−1 θ1n−T
(1 + θ1 )−1 2(1 + θ 2(n−T )+1 )
(1 − θ1 )−3 (n + 1 − T + θ1n+2−T − (n + 2 − T )θ)
(1 + θ)−1 (1 − θ1 )−3 (2 θ 2+n+T (1 + θ) − θ 4+2 n
+θ 2 T (n + 1 − T − 2 θ − (2 + n − T ) θ 2 ))
19
Table 3: Power Function, Π(δ), for Fractionally Differenced White Noise
With Parameter f and The Approximating ARMA(1, 1) Model for a Twosided 5% Level Test in SIA Step Intervention Model with n = 50 and T =
25. The first entry in each pair is for the fractional model and the second
the ARMA(1, 1) model. The parameters in the approximating ARMA
model are respectively φ1 = 0.667, φ2 = 0.451 and φ1 = 0.875, φ2 = 0.405
corresponding respectively to f = 0.2 and f = 0.4.
δ
f = 0.2
f = 0.4
0
0.5
1.
1.5
2.
2.5
3.
0.050, 0.050
0.198, 0.202
0.602, 0.612
0.914, 0.920
0.993, 0.994
1.000, 1.000
1.000, 1.000
0.050, 0.050
0.086, 0.076
0.198, 0.156
0.384, 0.291
0.602, 0.468
0.792, 0.651
0.914, 0.805
20
Table 4: Power Comparison for Step Interventions with ARMA(1,1) and
IMA(1) Errors for Series A with n = 197 + m and T = 198. The models’
other parameters are respectively, {φ1 = 0.9087, θ1 = 0.5758, σa = 0.3125}
and {θ1 = 0.7031, σa = 0.3172}.
ARMA(1, 1)
IMA(1)
ω
m=5
m = 50
m=5
m = 50
0.2
0.3
0.4
0.5
0.6
0.7
0.141
0.258
0.415
0.588
0.745
0.863
0.205
0.398
0.621
0.809
0.925
0.978
0.141
0.258
0.416
0.589
0.746
0.864
0.143
0.264
0.425
0.600
0.756
0.872
21
Table 5: Power Comparison for AR(1) Errors for (n = 60, T = 36) and
(n = 84, T = 48). The first entry in each column corresponds to (n =
60, T = 36) and the second (n = 84, T = 48).
δ
φ1 = 0
φ1 = 0.25
φ1 = 0.5
φ1 = 0.75
0.000
0.250
0.500
0.750
1.000
1.250
1.500
1.750
2.000
0.050, 0.050
0.245, 0.306
0.604, 0.736
0.889, 0.961
0.985, 0.998
0.999, 1.000
1.000, 1.000
1.000, 1.000
1.000, 1.000
0.050, 0.050
0.186, 0.226
0.444, 0.555
0.729, 0.848
0.914, 0.973
0.983, 0.998
0.998, 1.000
1.000, 1.000
1.000, 1.000
0.050, 0.050
0.146, 0.170
0.321, 0.395
0.550, 0.664
0.763, 0.867
0.904, 0.964
0.971, 0.994
0.994, 0.999
0.999, 1.000
0.050, 0.050
0.124, 0.135
0.253, 0.288
0.431, 0.493
0.624, 0.700
0.790, 0.857
0.903, 0.946
0.963, 0.984
0.989, 0.996
22
Table 6: Comparison of Exact and Approximate Methods. The function
g(T, φ) defined in Tiao et al. (1990) was computed using exact form
of the information matrix eqn. (6) and the approximation eqn. (8) for
selected parameter values given in Table 1 of Tiao et al. (1990). The
entries in the table show the percentage difference, 100 × (EXACT −
APPROXIMATE)/EXACT.
Number
of φ = 0.6
Years
6
7
8
9
10
−6
−5
−5
−4
−4
23
φ = 0.8
−17
−15
−13
−11
−10
Table 7: Number of Years, n∗ , For 90% Probability of Detecting a Prescribed Trend, δ Using a Two-Sided 5% Test Given 30 Years of Prior
Data And Assuming AR (1) Errors With Estimated Parameter φ̂1 . The
last line of the table shows the comparable values given in Tiao et al.
(1990, Table 2).
Tateno
Hohen.
Wakkan
Bulawayo
Abidajan
φ̂1
0.32
0.05
0.14
0.43
0.65
ω
0.003
0.003
0.2
0.2
0.2
δ
0.00758
0.00543
0.01042
0.01282
0.01111
n∗
11.6
12.1
8.0
8.6
12.0
n∗Tiao
14
14
10
10
13
24
Table 8: Power Comparisons of Dynamic Step Intervention Model with
Simple Step Intervention when n = 50 and T = 25. The first entry
in each triplet shows the theoretical power of a 5% two-sided test of
(1)
H0 : g = 0 where g = ω0 /(1 − δ1 ) in the dynamic step intervention model
(T )
(1)
zt = ξ + ω0 /(1 − δ1 B)St + at /(1 − φ1 B) with ξ = 0, φ1 = 0.5 and σa2 = 1.
The second entry is the theoretical power of a 5% test of H0 : ω0(2) = 0 in
the SIA model, zt = ξ + ω0(2) St(T ) + at /(1 − φ1 B), where ω0(2) = ω0(1) /(1 − δ1 )
and all other parameters are the same as in the dynamic model. The
third entry is the empirical power, based on 1000 simulations, for a twosided 5% test of H0 : ω0(2) = 0 when the SIA model is fitted to a time
series generated by the dynamic step intervention model.
δ0
ω0 = 0.5
ω0 = 0.75
ω0 = 1.0
0.25
0.50
0.75
0.226, 0.252, 0.241
0.439, 0.490, 0.445
0.673, 0.732, 0.692
0.416, 0.490, 0.466
0.745, 0.827, 0.758
0.937, 0.972, 0.932
0.879, 0.972, 0.880
0.997, 1.000, 0.974
1.000, 1.000, 0.955
25
Appendix A: Derivation of the Information Matrix
The loglikelihood function, apart from a constant, may be written,
L(λ1 , λ2 , σa2 ) = − log(σ) − log(det(Γn )) −
1 ′ −1
y Γn y,
2σa2
(11)
where y is the column vector of length n − d with t-th entry
(T )
∇d zt − ξ − ω∇d St , t = d + 1, . . . , n. Then ∂y/∂ξ = (−1, . . . , −1).
(T )
(T )
Similarly ∂y/∂ω = (−S1 , . . . , −Sn ). Hence,
Iλ1 = −E(∂λ21 ,λ1 L(λ1 , λ2 , σa2 ))
1 ′ −1
=
J Γn J,
σa2
(12)
where J is as in eqn. (6). Since E(∂ 2 L(λ1 , λ2 , σa2 )/(∂λ1 ∂λ2 )) = 0 and
E(∂ 2 L(λ1 , λ2 , σa2 )/(∂λ1 ∂λ2 )) = 0, the information matrix is block diagonal.
26
Appendix B: Interventions With A Dynamic Response
For completeness we also discuss the intervention analysis model with a
dynamic response to the intervention which may be written,
(T )
∇d zt = ξ + ω(B)/δ(B)∇d B b It
+ ∇−f
θ(B)
at ,
φ(B)
t = 1, . . . , n,
(13)
where ω(B) = ω0 + ω1 B + . . . ωr B r and δ(B) = δ0 − δ1 B − . . . δs B s . For
stability of the transfer function it is assumed that all roots of δ(B) = 0 lie
outside the unit circle. As in Appendix A, the exact information matrix
for the parameters λ1 = (ξ, ω0 , . . . , ωr , δ1 , . . . , δs ) Iλ1 = σa−2 J ′ Γ−1
n J where
J is an n − d × (2 + r + s) matrix with rows (1, ut , . . . , ut−r , vt , . . . , vt−s )
(T )
for t = 1, . . . , n − d, where ut−j = ∇d (1/δ(B)) It−j and
(T )
vt−j = ∇d (ω(B)/δ(B)) It−j . Alternatively the large-sample approximation
given in Pierce (1972) may be used. The steady-state gain (Box, Jenkins
and Reinsel, 1994, §10.1.1), which measures the long-run change of the
intervention, is defined by g = (ω0 + . . . + ωr )/(1 − δ1 − . . . − δs ). The
maximum likelihood estimates for the model may be used to form the
estimate of g, ĝ. Using a Taylor series linearization, the standard deviation
√
of ĝ is given by σĝ = (d′ζ Vζ dζ ), where Vζ is obtained by dropping the first
row and column from Iλ−1
and dζ = (∂g/∂ω0 , . . . , ∂g/∂ωr ,
1
∂g/∂δ1 , . . . , ∂g/∂δs ). For dynamic intervention analysis models we may
consider testing H0 : g = 0 using the Z test. Notice that, when s > 0 we
need estimates of all parameters in the full intervention model to estimate
σĝ . This limits the applicability of this approach since even if the
pre-intervention series is known, it is not likely that such precise
information is available for the intervention parameters. Often the SIA
model can be used to get an approximation to the power in this case.
As a numerical illustration, consider the dynamic step intervention model,
(T )
zt = ξ + ω0 (1)/(1 − δ1 B)St + at /(1 − φ1 B), t = 1, . . . , n. Taking
n = 50, T = 25, ξ = 0, φ1 = 0.5 and σa2 = 1, Table 8 below compares the
power of a 5% two-sided test H0 : g = 0, where g = ω0 (1), with that of the
(2)
Z-test H0 : ω0 = 0 in the corresponding SIA model defined by
(2)
(2) (T )
zt = ξ + ω0 St + at /(1 − φ1 B) where ω0 = g and the other parameter
settings are the same. On an intuitive basis, the effect in the SIA model is
slightly larger so one might expect the power in the SIA model to be
slightly larger. Table 8 shows, comparing the first two entries in each
triplet, that this is exactly what happens. The third entry in each triplet
(2)
in Table 8 is the empirical power of a two-sided 5% test of H0 : ω0 = 0
27
when the SIA model is fitted to a time series generated by the dynamic
step intervention model. One thousand simulations were used for each
model. The empirical power is predicted well by the theoretical asymptotic
power for the SIA model. These simulations were repeated with various
values of the parameter φ and similar results where found when
−1 < φ ≤ 0.5. For φ1 > 0.5, there was a much bigger difference between
the asymptotic theoretical power of the dynamic and step models. For
example with φ1 = 0.9, ω1 = 0.75 and δ1 = 0.75, the asymptotic power for
the two-sided 5% level gains test was only 0.199 whereas the predicted
power using a SIA step intervention was 0.972. The empirical power of the
two-sided 5% level test of H0 : ω1 = 0 in the step SIA model was 0.283.
The general conclusion reached was that the step SIA model provides a
useful approximation to the more complicated dynamic step intervention
model provided the autocorrelation is not too large. Further simulation
results are available in the online supplements.
28
| 10 |
GADTs meet subtyping
Gabriel Scherer and Didier Rémy
arXiv:1301.2903v1 [cs.PL] 14 Jan 2013
INRIA, Rocquencourt⋆
Abstract. While generalized algebraic datatypes (GADTs) are now considered well-understood, adding them to a language with a notion of
subtyping comes with a few surprises. What does it mean for a GADT
parameter to be covariant? The answer turns out to be quite subtle.
It involves fine-grained properties of the subtyping relation that raise
interesting design questions. We allow variance annotations in GADT
definitions, study their soundness, and present a sound and complete algorithm to check them. Our work may be applied to real-world ML-like
languages with explicit subtyping such as OCaml, or to languages with
general subtyping constraints.
Introduction
In languages that have a notion of subtyping, the interface of parametrized
types usually specifies a variance. It defines the subtyping relation between two
instances of a parametrized type from the subtyping relations that hold between
their parameters. For example, the type α list of immutable lists is expected
to be covariant : we wish σ list ≤ σ ′ list as soon as σ ≤ σ ′ .
Variance is essential in languages with parametric polymorphism whose programming idioms rely on subtyping, in particular object-oriented languages, or
languages with structural datatypes such as extensible records and variants, dependently typed languages with inductive types (to represent positivity requirements), or additional information in types such as permissions, effects, etc. A last
reason to care about variance is its use in the relaxed value restriction [Gar04]:
while a possibly-effectful expression, also called an expansive expression, cannot
be soundly generalized in ML—unless some sophisticated enhancement of the
type system keeps track of effectful expressions—it is always sound to generalize
type variables that only appear in covariant positions, as they may not classify
mutable data. Therefore, it is important for extensions of type definitions, such
as generalized algebraic datatypes (GADTs), to support it as well through a
clear and expressive definition of parameter covariance.
For example, consider the following GADT of well-typed expressions:
type +α exp =
| Val : α → α exp
| Int : int → int exp
| Thunk : ∀β. β exp ∗ (β → α) → α exp
| Prod : ∀βγ. β exp ∗ γ exp → (β ∗ γ) exp
⋆
Part of this work has been done at IRILL.
2
Gabriel Scherer and Didier Rémy
Is it safe to say that exp is covariant in its type parameter? It turns out that,
using the subtyping relation of the OCaml type system, the answer is “yes”.
But, surprisingly to us, in a type system with a top type ⊤, the answer would be
“no”. We introduce this example in details in §1—and present some interesting
counter-examples of incorrect variance annotations.
Verifying variance annotations for simple algebraic datatypes is straightforward: it suffices to check that covariant type variables appear only positively
and contravariant variables only negatively in the types of the arguments of
the datatype constructors. GADTs can be formalized as extensions of datatypes
where constructors have typed arguments, but also a set of existential variables
and equality constraints. Then, the simple check of algebraic datatypes apparently becomes a searching problem: witnesses for existentials must be found so
as to satisfy the equality constraints. That is, there is a natural correctness criterion (already present in previous work); however, it is expressed in a “semantic”
form that is not suitable for a simple implementation in a type checker. We
present this semantic criterion in §2 after reviewing the formal framework of
variance-based subtyping.
The main contribution of our work, described in §3, is to develop a syntactic criterion that ensures the semantics criterion. Our solution extends the
simple check of algebraic datatypes in a non-obvious way by introducing two
new notions. First, upward and downward-closure of type constructors explains
how to check that a single equality constraint is still satisfiable in presence of
variance (but also raises interesting design issues for the subtyping relation).
Second, zipping explains when witnesses exist for existential variables, that is,
when multiple constraints using the same existential may soundly be used without interfering with each other. These two properties are combined into a new
syntactic judgment of decomposability that is central to our syntactic criterion.
We prove that our syntactic criterion is sound and complete with respect to the
semantic criterion. The proof of soundness is relatively direct, but completeness
is much harder.
We discuss the implication of our results in §4, in particular the notion of
upward and downward-closure properties of type constructors, on the design of a
subtyping relation. We also contrast this approach, motivated by the needs of a
language of a ML family, with a different and mostly orthogonal approach taken
by existing object-oriented languages, namely C♯ and Scala, where a natural notion of GADTs involves subtyping constraints, rather than equality constraints.
We can re-evaluate our syntactic criterion in this setting: it is still sound, but
the question of completeness is left open.
In summary, we propose a syntactic criterion for checking the soundness of
variance annotations of GADTs with equality constraints in a language with
subtyping. Our work is directly applicable to the OCaml language, but our
approach can also be transposed to languages with general subtyping constraints,
and raises interesting design questions. A long version of the present article,
containing the detailed proofs and additional details and discussion, is available
online [SR].
GADTs meet subtyping
1
3
Examples
Let us first explain why it is reasonable to say that α exp is covariant. Informally,
if we are able to coerce a value of type α into one of type α′ (we write (v :> α′ )
to explicitly cast a value v of type α to a value of type α′ ), then we are also
able to transform a value of type α exp into one of type α′ exp. Here is some
pseudo-code1 for the coercion function:
let coerce : α exp → α′ exp = function
| Val (v : α) -> Val (v :> α′ )
| Int n -> Int n
| Thunk β (b : β exp) (f : β → α) ->
Thunk β b (fun x -> (f x :> α′ ))
| Prod β γ ((b, c) : β exp ∗ γ exp) ->
(* if β ∗ γ ≤ α′ , then α′ is of the form β ′ ∗ γ ′
with β ≤ β ′ and γ ≤ γ ′ *)
Prod β ′ γ ′ ((b :> β ′ exp), (c :> γ ′ exp))
In the Prod case, we make an informal use of something we know about the
OCaml type system: the supertypes of a tuple are all tuples. By entering the
branch, we gain the knowledge that α must be equal to some type of the form
β ∗ γ. So from α ≤ α′ we know that β ∗ γ ≤ α′ . Therefore, α′ must itself be a
pair of the form β ′ ∗ γ ′ . By covariance of the product, we deduce that β ≤ β ′ and
γ ≤ γ ′ . We may thus conclude by casting at types β ′ exp and γ ′ exp, recursively.
Similarly, in the Int case, we know that α must be an int and therefore an
int exp is returned. This is because we know that, in OCaml, no type is above
int: if int ≤ τ , then τ must be int.
What we use in both cases is reasoning of the form2 : “if T [β] ≤ α′ , then I
′
′
know that α′ is of the form T [β ] for some β ”. We call this an upward closure
property: when we “go up” from a T [β], we only find types that also have
the structure of T . Similarly, for contravariant parameters, we would need a
downward closure property: T is downward-closed if T [β] ≥ α′ entails that α′ is
′
of the form T [β ].
Before studying a more troubling example, we define the classic equality type
(α, β) eq and the corresponding casting function cast : ∀αβ.(α, β) eq → α → β:
type (α, β) eq =
let cast r =
| Refl : ∀γ. (γ, γ) eq
match r with Refl -> (fun x -> x)
Notice that it would be unsound3 to define eq as covariant, even in only one
parameter. For example, if we had type (+α, =β) eq, from any σ ≤ τ , we could
subtype (σ, σ) eq into (τ, σ) eq, allowing a cast from any value of type τ back
into one of type σ, which is unsound in general.
1
2
3
The variables β ′ and γ ′ of the Prod case are never really defined, only justified at
the meta-level, making this code only an informal sketch.
We write T [β] for a type expression T that may contain free occurrences of variables
β and T [σ] for the simultaneous substitution of σ for β in T .
This counterexample is due to Jeremy Yallop.
4
Gabriel Scherer and Didier Rémy
As a counter-example, the following declaration is incorrect: the type α bad
cannot be declared covariant.
type +α bad =
| K : < m : int > → < m : int > bad
let v = (K (object method m = 1 end) :> < > bad)
This declaration uses the OCaml object type < m : int >, which qualifies objects having a method m returning an integer. It is a subtype of object types
with fewer methods, in this case the empty object type < >, so the alleged covariance of bad, if accepted by the compiler, would allow us to cast a value of
type < m : int > bad into one of type < > bad and thus have the above value
v of type <> bad. However, if such a value v existed, we could produce an equality witness (< >, <m : int>) eq that allows to cast any empty object of type
< > into an object of type < m : int >, but this is unsound, of course!
let get_eq : α bad → (α, < m : int >) eq = function
| K _ -> Refl
(* locally α = < m : int > *)
let wrong : < > -> < m : int > =
let eq : (< >, < m : int >) eq = get_eq v in cast eq
It is possible to reproduce this example using a different feature of the OCaml
type system named private type abbreviation 4 : a module using a type type s = τ
internally may describe its interface as type s = private τ . This is a compromise
between a type abbreviation and an abstract type: it is possible to cast a value
of type s into one of type τ , but not, conversely, to construct a value of type
s from one of type τ . In other words, s is a strict subtype of τ : we have s ≤
τ but not s ≥ τ . Take for example type file_descr = private int: this
semi-abstraction is useful to enforce invariants by restricting the construction of
values of type file_descr, while allowing users to conveniently and efficiently
destruct them for inspection at type int. Using an unsound but quite innocentlooking covariant GADT datatype, one is able to construct a function to cast any
integer into a file_descr, which defeats the purpose of this abstraction—see
the extended version of this article for the full example.
The difference between the former, correct Prod case and those two latter
situations with unsound variance is the notion of upward closure. The types α∗β
and int used in the correct example were upward-closed. On the contrary, the
private type file_descr has a distinct supertype int, and similarly, the object
type < m:int > has a supertype < > with a different structure (no method m).
Finally, the need for covariance of α exp can be justified either by applications
using subtyping on data (for example object types or polymorphic variants), or
by the relaxed value restriction. If we used the Thunk constructor to delay a
computation returning an object of type < m : int >, that is itself of type
< m : int > exp, we may need to see it as a computation returning the empty
object < >. We could also wish to define an abstract interface through a module
boundary that would not expose any implementation detail about the datatype;
for example, using Product to implement a list interface.
4
This counterexample is due to Jacques Garrigue.
GADTs meet subtyping
5
module Exp : sig
type α exp
val inj : α -> α exp
val pair : α exp -> β exp -> (α ∗ β) exp
val fst : (α ∗ β) exp -> α exp
end
What would then be the type of Exp.inj []? In presence of the value restriction,
this application cannot be generalized, and we get a weak polymorphic type
?α list Exp.exp for some non-generalized inference variable ?α. If we change
the interface to express that Exp.exp is covariant, then we get the expected
polymorphic type ∀α.α list Exp.exp.
2
2.1
A formal setting
The subtyping relation
Ground types consist of base type q, types τ p, function types τ1 → τ2 , product
types τ1 ∗ τ2 , and a set of algebraic datatypes σ t. We also write σ and ρ for
types, σ for a sequence of types (σi )i∈I , and we use prefix notation for datatype
parameters, as is the usage in ML. Datatypes may be user-defined by toplevel
declarations of the form:
type vα t = K1 of τ 1 [α] | . . . Kn of τ n [α]
This is a disjoint sum: the constructors Kc represent all possible cases and each
type τ c [α] is the domain of the constructor Kc . Applying Kc to an argument e of
a corresponding ground type τ [σ] constructs a term of type σ t. Values of this
type are deconstructed using pattern matching clauses of the form Kc x → e,
one for each constructor.
The sequence vα is a binding list of type variables αi along with their variance
annotation vi . Variances range in the set {+, −, =, ⋊
⋉}. We may associate a
relation (≺v ) between types to each variance v:
–
–
–
–
≺+ is the covariant relation (≤);
≺− is the contravariant relation (≥), the symmetric of (≤);
≺= is the invariant relation (=) defined as the intersection of (≤) and (≥);
≺⋊
⋉), i.e. the full relation such that σ ⋊
⋉ τ holds
⋉ is the irrelevant relation (⋊
for all types σ and τ .
Given a reflexive transitive relation (6) on base types, the subtyping relation
on ground types (≤) is defined by the inference rules of Figure 1, which, in
particular, give their meaning to the variance annotations vα. The judgment
type vα t simply means that the type constructor t has been previously defined
with the variance annotation vα. Notice that the rules for arrow and product types, sub-Fun and sub-Prod, can be subsumed by the rule for datatypes
sub-Constr. Indeed, one can consider them as special datatypes (with a specific
6
Gabriel Scherer and Didier Rémy
sub-Trans
σ1 ≤ σ2
σ2 ≤ σ3
sub-Refl
σ≤σ
′
σ∗τ ≤σ ∗τ
′
τ ≤ τ′
′
σ → τ ≤ σ → τ′
σ1 ≤ σ3
sub-Prod
σ ≤ σ′
τ ≤ τ′
sub-Fun
σ ≥ σ′
sub-Constr
type vα t
∀i, σi ≺vi σi′
′
σt≤σ t
sub-P
σ ≤ σ′
sub-PQ
′
σp≤q
σp≤σ p
Fig. 1. Subtyping relation
dynamic semantics) of variance (−, +) and (+, +), respectively. For this reason, the following definitions will not explicitly detail the cases for arrows and
products.
The rules sub-P and sub-PQ were added for the explicit purpose of introducing some amount of non-atomic subtyping in our relation. For two fixed type
constructors p (unary) and q (nullary), we have σ p ≤ q for any σ. Note that
q is not a top type as it is not above all types, only above the σ p. Of course,
we could add other such type constructors, but those are enough to make the
system interesting and representative of complex subtype relation.
As usual in subtyping systems, we could reformulate our judgment in a
syntax-directed way, to prove that it admits good inversion properties: if σ t ≤
σ ′ t and type vα t, then one can deduce that for each i, σi ≺vi σi′ .
The non-atomic rule sub-PQ ensures that our subtyping relation is not “too
structured” and is a meaningful choice for a formal study applicable to realworld languages with possibly top or bottom types, private types, record width
subtyping, etc. In particular, the type constructor p is not upward-closed (and
conversely q is not downward-closed), as used informally in the examples and
defined for arbitrary variances in the following way:
Definition 1 (Constructor closure). A type constructor α t is v-closed if,
for any type sequence σ and type τ such that σ t ≺v τ hold, then τ is necessarily
equal to σ ′ t for some σ ′ .
2.2
The algebra of variances
If we know that σ t ≤ σ ′ t, that is σ t ≺+ σ ′ t, and the constructor t has
variable vα, an inversion principle tells us that σi ≺vi σi′ for each i. But what if
we only know σ t ≺u σ ′ t for some variance u different from (+)? If u is (−), we
⋉), we get σi ⋊
⋉ σi′ , that is, nothing.
get the reverse relation σi ≻vi σi′ . If u is (⋊
This outlines a composition operation on variances u.vi , such that if σ t ≺u σ ′ t
then σi ≺u.vi σi′ holds. It is defined by the table in figure 2.2.
This operation is associative and commutative. Such an operator, and the
algebraic properties of variances explained below, have already been used by
other authors, for example [Abe06].
There is a natural order relation between variances, which is the coarser-than
order between the corresponding relations: v ≤ w if and only if (≺v ) ⊇ (≺w );
GADTs meet subtyping
7
i.e. if and only if, for all σ and τ , σ ≺w τ implies σ ≺v τ .5 This reflexive, partial
order is described by the lattice diagram in figure 2.2. All variances are smaller
than = and bigger than ⋊
⋉.
v.w
=
+
−
⋉
⋊
v
=
=
=
=
⋉
⋊
+
=
+
−
⋉
⋊
−
=
−
+
⋉
⋊
⋉
⋊w
⋉
⋊
⋉
⋊
⋉
⋊
⋉
⋊
Fig. 2. Variance composition table
④④
+❆
❆
=❈
❈
−
⑥⑥
⋉
⋊
Fig. 3. Variance order diagram
From the order lattice on variances we can define join ∨ and meet ∧ of
variances: v ∨ w is the biggest variance such that v ∨ w ≤ v and v ∨ w ≤ w;
conversely, v ∧ w is the lowest variance such that v ≤ v ∧ w and w ≤ v ∧ w.
Finally, the composition operation is monotone: if v ≤ v ′ then w.v ≤ w.v ′ and
v.w ≤ v ′ .w.
We often manipulate vectors vα of variable associated with variances, which
correpond to the “context” Γ of a type declaration. We extend our operation
pairwise on those contexts: Γ ∨Γ ′ and Γ ∧Γ ′ , and the ordering between contexts
Γ ≤ Γ ′ . We also extend the variance-dependent subtyping relation (≺v ), which
becomes an order (≺Γ ) between vectors of type of the same length: σ ≺vα σ ′
holds when we have σi ≺vi σi′ for all i.
2.3
A judgment for variance of type expressions
We define a judgment to check the variance of a type expression. Given a context
Γ of the form vα, that is, where each variable is annotated with a variance, the
judgment Γ ⊢ τ : v checks that the expression τ varies along v when the variables
of τ vary along their variance in Γ . For example, (+α) ⊢ τ [α] : + holds when
τ [α] is covariant in its variable α. The inference rules for the judgment Γ ⊢ τ : v
are defined on Figure 4.
The parameter v evolves when going into subderivations: when checking Γ ⊢
τ1 → τ2 : v, contravariance is expressed by checking Γ ⊢ τ1 : (v.−). Previous
work (on variance as [Abe06] and [EKRY06], but also on irrelevance as in [Pfe01])
used no such parameter, but modified the context instead, checking Γ/− ⊢ τ1
for some “variance cancellation” operation vw/ (see [Abe06] for a principled
5
The reason for this order reversal is that the relations occur as hypotheses, in negative
position, in definition of subtyping: if we have v ≤ w and type vα t, it is safe to
assume type wα t, since σ ≺w σ ′ implies σ ≺v σ ′ , which implies σ t ≤ σ ′ t. One
may also see it, as Abel notes, as an “information order”: knowing that σ ≺+ τ
“gives you more information” than knowing that σ ≺⋉
⋊ ≤ +.
⋊ τ , therefore ⋉
8
Gabriel Scherer and Didier Rémy
vc-Var
wα ∈ Γ
w≥v
vc-Constr
Γ ⊢ type wα t
Γ ⊢α:v
∀i, Γ ⊢ σi : v.wi
Γ ⊢σt:v
Fig. 4. Variance assignment
presentation). Our own inference rules preserve the same context in the whole
derivation and can be more easily adapted to the decomposability judgment
Γ ⊢ τ : v ⇒ v ′ that we introduce in §3.4.
A semantics for variance assignment This syntactic judgment Γ ⊢ τ : v corresponds to a semantic property about the types and context involved, which
formalizes our intuition of “when the variables vary along Γ , the expression τ
varies along v”. We also give a few formal results about this judgment.
Definition 2 (Interpretation of the variance checking judgment).
We write JΓ ⊢ τ : vK for the property: ∀σ, σ ′ , σ ≺Γ σ ′ =⇒ τ [σ] ≺v τ [σ ′ ].
Lemma 1 (Correctness of variance checking).
Γ ⊢ τ : v is provable if and only if JΓ ⊢ τ : vK holds.
Lemma 2 (Monotonicity).
If Γ ⊢ τ : v is provable and Γ ≤ Γ ′ then Γ ′ ⊢ τ : v is provable.
Lemma 3 (Principality). For any type τ and any variance v, there exists a
minimal context ∆ such that ∆ ⊢ τ : v holds. That is, for any other context Γ
such that Γ ⊢ τ : v, we have ∆ ≤ Γ .
We can generalize inversion of head type constructors (§2.1) to whole type
expressions. The most general inversion is given by the principal context.
Theorem 1 (Inversion). For any type τ [α], variance v, and type sequences σ
and σ ′ , the subtyping relation τ [σ] ≺v τ [σ ′ ] holds if and only if the judgment Γ ⊢
τ : v holds for some context Γ such that σ ≺Γ σ ′ . Furthermore, if τ [σ] ≺v τ [σ ′ ]
holds, then σ ≺∆ σ ′ holds, where ∆ is the minimal context such that ∆ ⊢ τ : v.
2.4
Variance annotations in ADTs
As a preparation for the difficult case of GADTs, we first present our approach
in the well-understood case of algebraic datatypes. We exhibit a semantic criterion that justifies the correctness of a variance annotation; then, we propose
an equivalent syntactic judgment. Of course, we recover the usual criterion that
covariant variables should only occur positively.
In general, an ADT definition of the form
type vα t =
c∈C
Kc of τ c [α]
GADTs meet subtyping
9
cannot be accepted with any variance vα t. For example, the declaration (type vα inv =
Fun of α → α) is only sound when v is invariant. Accepting a variance assignment vα determines the relations between closed types σ and σ ′ under which
the relation σ t ≤ σ ′ t is correct.
In the definition of +α exp we justified the covariance of exp by the existence
of a coercion function. We now formalize this idea for the general case. To check
the correctness of σ t ≤ σ ′ t we check the existence of a coercion term that
turns a closed value q of type σ t into one of type σ ′ t that is equal to q up to
type information. We actually search for coercions of the form:
match (q : σ t) with |c∈C Kc (x : τ c [σ]) → Kc (x :> τ c [σ ′ ])
Note that erasing types gives an η-expansion of the sum type, i.e. this is really a
coercion. Hence, such a coercion exists if and only if it is well-typed, that is, each
cast of the form (x : τ c [σ] :> τ c [σ ′ ]) is itself well-typed. This gives our semantic
criterion for ADTs.
Definition 3 (Semantic soundness criterion for ADTs).
We accept the ADT definition of vαt with constructors (Kc of τ c [α])c∈C if
∀c ∈ C, ∀σ, ∀σ ′ ,
σ t ≤ σ ′ t =⇒ τ c [σ] ≤ τ c [σ ′ ]
The syntactic criterion for ADTs We notice that this criterion is exactly the
semantic interpretation of the variance checking judgment (Definition 2): the
type type vα t is accepted if and only if the judgment vα ⊢ τ c : (+) is derivable
for each constructor type τ c [α].
This syntactic criterion coincides with the well-known alogrithm implemented
in type checkers6 : checking positive occurences of a variable α corresponds to a
proof obligation of the form vα ⊢ α : +, which is valid only when α has variance
(+) or (=) in Γ ; checking negative occurences correspond to a proof obligation
vα ⊢ α : −, etc. This extends seamlessly to irrelevant variables, which must
⋉—or not at all.
appear only under irrelevant context vα ⊢ α : ⋊
2.5
Variance annotations in GADTs
A general description of GADTs When used to build terms of type α t, a
constructor K of τ behaves like a function of type ∀α.(τ → α t). Notice that
the codomain is exactly α t, the type t instantiated with parametric variables.
GADTs arise by relaxing this restriction, allowing constructors with richer types
of the form ∀α.(τ → σ t). See for example the declaration of constructor Prod
in the introduction:
| Prod : ∀βγ. β exp ∗ γ exp → (β ∗ γ) exp
6
One should keep in mind that this criterion suffers the usual bane of static typing,
it can reject programs that do not go wrong: type −α weird = K of α ∗ ⊥. For more
details, see the beginning of the §3 in the long version of this article.
10
Gabriel Scherer and Didier Rémy
Instead of being just α exp, the codomain is now (β ∗ γ) exp. We moved from
simple algebraic datatypes to so-called generalized algebraic datatypes. This
approach is natural and convenient for the users, so it is exactly the syntax
chosen in languages with explicit GADTs support, such as Haskell and OCaml,
and is reminiscent of the inductive datatype definitions of dependently typed
languages.
However, for the formal study of GADTs, a different formulation based on
equality constraints is preferred. We use the following equivalent presentation,
already present in previous works [SP07]. We force the codomain of the constructor Prod to be α t again, instead of (β ∗ γ) t, by adding an explicit equality
constraint α = β ∗ γ.
type α exp =
| Val of ∃β[α = β]. β
| Int of [α = int]. int
| Thunk of ∃βγ[α = γ]. β exp ∗ (β → γ)
| Prod of ∃βγ[α = β ∗ γ]. β exp ∗ γ exp
In the rest of the paper, we extend our former core language with such
definitions. This does not impact the notion of subtyping, which is defined on
GADT type constructors with variance type vα t just as it previously was
on simple ADT type constructors. What needs to be changed, however, is the
soundness criterion for checking the variance of type definitions
The correctness criterion We must adapt our semantic criterion for datatype
declarations (Definition 3) from simple ADTs to GADTs. Again, we check under
which relations between σ and σ ′ the subtyping relation σ t ≤ σ ′ t holds for
some GADT definition vα t.
The difference is that a constructor Kc that had an argument of type τ c [α]
in the simple ADT case, now has the more complex type ∃β[D[α, β]].τ c [β], for
a set of existential variables β and a set of equality constraints D—of the form
(αi = Ti [β])i∈I for a family of type expressions (Ti [β])i∈I . Given a closed value
q of type σ t, the coercion term is:
match (q : σ t) with |c∈C Kc (x : τ c [ρc ]) → Kc (x :> τ c [ρ′c ])
We do not need to consider the dead cases: we only match on the constructors
for which there exists an instantiation
ρc of the existential variables β such that
V
the constraint D[σ, ρ], i.e. i∈I σi = Ti [ρc ], holds. To type-check this term, we
need to find another instantiation ρ′c that verifies the constraints D[σ ′ , ρ′ ]. This
coercion type-checks only when τ c [ρc ] ≤ τ c [ρ′c ] holds. This gives our semantic
criterion for GADTs:
Definition 4 (Semantic soundness criterion for GADTs). We accept the
GADT definition of type vα t with constructors (Kc of ∃β[D[α, β]].τ c [α])c∈C ,
if for all c in C we have:
(Req)
∀σ, σ ′ , ρ, σ t ≤ σ ′ t ∧ D[σ, ρ] =⇒ ∃ρ′ , D[σ ′ , ρ′ ] ∧ τ [ρ] ≤ τ [ρ′ ]
GADTs meet subtyping
11
As for ADTs, this criterion ensures soundness: if, under some variance annotation, a datatype declaration satisfies it, then the implied subtyping relations are
all expressible as coercions in the language, and therefore correct. Whereas the
simpler ADT criterion was already widely present in the literature, this one is less
known; it is however present in the previous work of Simonet and Pottier [SP07]
(presented as a constraint entailment problem).
Another way to understand this criterion would be to define constrained existential types of the form ∃β[D[α, β]].τ [β] as first-class types and, with the right
notion of subtyping for those, require that σ t ≤ σ ′ t imply (∃β[D[σ, β]].τ [β]) ≤
(∃β[D[σ ′ , β]].τ [β]). The (easy) equivalence between those two presentations is
detailed in the work of Simonet and Pottier [SP07].
3
3.1
Checking variances of GADT
Expressing decomposability
If we specialize Req to the Prod constructor of the α exp example datatype, i.e.
Prod of ∃βγ[α = β ∗ γ]β exp ∗ γ exp, we get:
∀σ, σ ′ , ρ1 , ρ2 ,
σ exp ≤ σ ′ exp ∧ σ = ρ1 ∗ ρ2 =⇒ ∃ρ′1 , ρ′2 , (σ ′ = ρ′1 ∗ ρ′2 ∧ ρ1 ∗ ρ2 ≤ ρ′1 ∗ ρ′2 )
We can substitute equalities and use the (user-defined) covariance to simplify
the subtyping constraint σ exp ≤ σ ′ exp into σ ≤ σ ′ :
∀σ ′ , ρ1 , ρ2 , ρ1 ∗ρ2 ≤ σ ′ =⇒ ∃ρ′1 , ρ′2 , (σ ′ = ρ′1 ∗ρ′2 ∧ ρ1 ≤ ρ′1 ∧ ρ2 ≤ ρ′2 ) (1)
This is the upward closure property mentioned in the introduction. The preceeding transformation is safe only if any supertype σ ′ of a product ρ1 ∗ ρ2 is itself
a product, i.e. is of the form ρ′1 ∗ ρ′2 for some ρ′1 and ρ′2 .
More generally, for a type Γ ⊢ σ and a variance v, we are interested in
a closure property of the following form, where the notation (ρ : Γ ) simply
classifies type vectors ρ that have exactly one type ρi for each variable in Γ :
∀(ρ : Γ ), σ ′ ,
σ[ρ] ≺v σ ′ =⇒ ∃(ρ′ : Γ ), σ ′ = σ[ρ′ ]
Here, the context Γ represents the set of existential variables of the constructor
(β and γ in our example). We can easily express the condition ρ1 ≤ ρ′1 and
ρ2 ≤ ρ′2 on the right-hand side of the implication by considering a context Γ
annotated with variances (+β, +γ), and using the context ordering (≺Γ ). Then,
(1) is equivalent to:
∀(ρ : Γ ), σ ′ ,
σ[ρ] ≺v σ ′ =⇒ ∃(ρ′ : Γ ), ρ ≺Γ ρ′ ∧ σ ′ = σ[ρ′ ]
Our aim is now to find a set of inference rules to check decomposability; we
will later reconnect it to Req. In fact, we study a slightly more general relation,
where the equality σ[ρ′ ] = σ ′ on the right-hand side is relaxed to an arbitrary
relation σ[ρ′ ] ≺v′ σ ′ :
12
Gabriel Scherer and Didier Rémy
Definition 5 (Decomposability). Given a context Γ , a type expression σ[β]
and two variances v and v ′ , we say that σ is decomposable under Γ from variance v to variance v ′ , which we write Γ
σ:v
v ′ , if the following property
holds:
∀(ρ : Γ ), σ ′ , σ[ρ] ≺v σ ′ =⇒ ∃(ρ′ : Γ ), ρ ≺Γ ρ′ ∧ σ[ρ′ ] ≺v′ σ ′
We use the symbol
rather than ⊢ to highlight the fact that this is just a
logic formula, not the semantic interpretation of a syntactic judgment—we will
introduce one later in section 3.4.
Remark that, due to the positive occurrence of the relation ≺Γ in the proposition Γ τ : v
v ′ and the anti-monotonicity of ≺Γ , this formula is “antimonotone” with respect to the context ordering Γ ≤ Γ ′ . This corresponds to
saying that we can still decompose, but with less information on the existential
witness ρ′ .
Lemma 4 (Anti-monotonicity).
If Γ τ : v
v ′ holds and Γ ′ ≤ Γ , then Γ ′
3.2
τ :v
v ′ also holds.
Variable occurrences
In the Prod case, the type whose decomposability was considered is β ∗ γ (in
the context β, γ). In this very simple case, decomposability depends only on the
type constructor for the product. In the present type system, with very strong
invertibility principles on the subtyping relation, both upward and downward
closures hold for products. In the general case, we require that this specific type
constructor be upward-closed.
In general, the closure of the head type constructor alone is not enough
to ensure decomposability of the whole type. For example, in a complex type
expression with subterms, we should consider the closure of the type constructors
appearing in the subterms as well. Besides, there are subtleties when a variable
occurs several times.
For example, while β ∗ γ is decomposable from (+) to (=), β ∗ β is not: ⊥ ∗ ⊥
is an instantiation of β ∗ β, and a subtype of, e.g., int ∗ bool, which is not
an instance7 of β ∗ β. The same variable occurring twice in covariant position
(or having one covariant and one invariant or contravariant occurence) breaks
decomposability.
On the other hand, two invariant occurrences are possible: β ref ∗ β ref
is upward-closed (assuming the type constructor ref is invariant and upwardclosed): if (σ ref ∗ σ ref) ≤ σ ′ , then by upward closure of the product, σ ′ is
of the form σ1′ ∗ σ2′ , and by its covariance σ ref ≤ σ1′ and σ ref ≤ σ2′ . Now
by invariance of ref we have σ1′ = σ ref = σ2′ , and therefore σ ′ is equal to
σ ref ∗ σ ref, which is an instance of β ref ∗ β ref.
7
We use the term instance to denote the replacement of all the free variables of a
type expression under context by closed types—not the specialization of an ML type
scheme.
GADTs meet subtyping
13
Finally, a variable may appear in irrelevant positions without affecting closure
properties; β ∗ (β irr) (where irr is an upward-closed irrelevant type, defined
for example as type α irr = int) is upward closed: if σ ∗ (σ irr) ≤ σ ′ , then σ ′
is of the form σ1′ ∗ (σ2′ irr) with σ ≤ σ1′ and σ ⋊
⋉ σ2′ , which is equiconvertible to
′
′
σ1 ∗ (σ1 irr) by irrelevance, an instance of β ∗ (β irr).
3.3
Context zipping
The intuition to think about these different cases is to consider that, for any σ ′ ,
we are looking for a way to construct a “witness” σ ′ such that τ [σ ′ ] = σ ′ from
the hypothesis τ [σ] ≺v σ ′ . When a type variable appears only once, its witness
can be determined by inspecting the corresponding position in the type σ ′ . For
example, in α ∗ β ≤ bool ∗ int, the mapping α 7→ bool, β 7→ int gives the
witness pair bool, int.
However, when a variable appears twice, the two witnesses corresponding to
the two occurrences may not coincide. (Consider for example β ∗β ≤ bool∗int.)
If a variable βi appears in several invariant occurrences, the witness of each
occurrence is forced to be equal to the corresponding subterm of τ [σ], that is σi ,
and therefore the various witnesses are themselves equal, hence compatible. On
the contrary, for two covariant occurrences (as in the β ∗ β case), it is possible
to pick a σ ′ such that the two witnesses are incompatible—and similarly for one
covariant and one invariant occurrence. Finally, an irrelevant occurrence will
never break closure properties, as all witnesses (forced by another occurrence)
are compatible.
To express these merging properties, we define a zip operation v1 & v2 ,
that formally expresses which combinations of variances are possible for several
occurrences of the same variable; it is a partial operation (for example, it is not
defined in the covariant-covariant case, which breaks the closure properties) with
the following table:
⋉w
v&w =+−⋊
= =
=
+
+
−
−
⋊
⋉ =+−⋊
⋉
v
3.4
Syntactic decomposability
Equipped with the zipping operation, we introduce a judgment Γ ⊢ τ : v ⇒ v ′ to
express decomposability, syntactically, defined by the inference rules on Figure 5.
We also define its semantic interpretation JΓ ⊢ τ : v ⇒ v ′ K. The judgment and
its interpretation were co-designed, so keeping the interpretation in mind is the
best way to understand the subtleties of the inference rules. We use zipping,
which requires correct variances, to merge sub-derivations into larger ones, so,
in addition to decomposability, the interpretation also ensures that v is a correct
14
Gabriel Scherer and Didier Rémy
sc-Triv
v ≥ v′
sc-Var
wα ∈ Γ
Γ ⊢τ :v
Γ ⊢ τ : v ⇒ v′
sc-Constr
Γ ⊢ type wα t : v-closed
w=v
Γ ⊢ α : v ⇒ v′
∀i, Γi ⊢ σi : v.wi ⇒ v ′ .wi
Γ = &i Γi
Γ ⊢σt:v⇒v
′
Fig. 5. Syntactic decomposablity
variance for τ under Γ . This subtlety is why we have two different properties for
decomposability, Γ τ : v
v ′ and JΓ ⊢ τ : v ⇒ v ′ K.
Definition 6 (Interpretation of syntactic decomposability).
We write JΓ ⊢ τ : v ⇒ v ′ K for the conjunction of properties JΓ ⊢ τ : vK and
Γ τ :v
v′ .
To understand the inference rules, the first thing to notice is that the present
rules are not completely syntax-directed: we first check whether v ≥ v ′ holds,
and if not, we apply syntax-directed inference rules; existence of derivations is
still easily decidable. If v ≥ v ′ holds, satisfying Γ
τ :v
v ′ (Definition 5)
′
′
′
is trivial: τ [σ] ≺v τ implies τ [σ] ≺v′ τ , so taking σ for σ is always a correct
witness, which is represented by Rule sc-Triv. The other rules then follow the
same structure as the variance-checking judgment.
Rule sc-Var is very similar to vc-Var, except that the condition w ≥ v is replaced by a stronger equality w = v. This difference comes from the fact that the
semantic condition for closure checking (Definition 2) includes both a variance
check, which is monotonic in the context (Lemma 2) and the decomposability
property, which is anti-monotonic (Lemma 4), so the present judgment must be
invariant with respect to the context.
The most interesting rule is sc-Constr. It checks first that the head type
constructor is v-closed (according to Definition 1); then, it checks that each
subtype is decomposable from v to v ′ , with compatible witnesses, that is, in an
environment family (Γi )i∈I that can be zipped into a unique environment Γ .
Lemma 5 (Soundness of syntactic decomposability).
If the judgment Γ ⊢ τ : v ⇒ v ′ holds, then JΓ ⊢ τ : v ⇒ v ′ K holds.
Completeness is the general case is however much more difficult and we only
prove it when the right-hand side variance v ′ is (=). In other words, we take back
the generality that we have introduced in §3.1 when defining decomposability.
Lemma 6 (Completeness of syntactic decomposability).
If JΓ ⊢ τ : v ⇒ v ′ K holds for v ′ ∈ {=, ⋊
⋉}, then Γ ⊢ τ : v ⇒ v ′ is provable.
Lemma 6 is an essential piece to finally turn the semantic criterion Req into
a purely syntactic form.
GADTs meet subtyping
15
Theorem 2 (Algorithmic criterion). Given
a variance
annotation (vi αi )i∈I
V
and a constructor declaration of type (∃β i∈I αi = Ti [β] . τ [β]), the soundness
criterion Req for this constructor is equivalent to
∃Γ, (Γi )i∈I ,
Γ ⊢ τ : (+) ∧ Γ = & Γi ∧ ∀i ∈ I, Γi ⊢ Ti : vi ⇒ (=)
i∈I
The three parts of this formula can be explained to a user, as soon as the
underlying semantic phenomenons (variable interference through zipping, and
upward- and downward-closure) have been understood—there is no way to get
around that. They are best read from right to left. The last part on the (Ti )i∈I is
the decomposability requirement that failed in our example with < m : int >:
the type expressions equated with a covariant variable should be upward-closed,
and those equated with a contravariant one downward-closed. The zipping part
checks that the equations do not create interference through shared existential
variables, as in type (+α, =β) eq = Refl of ∃γ[α = γ, β = γ]. Finally, the
variance check corresponds to the classic variance check on argument types of
ADTs. One can verify that in presence of a simple ADT, this new criterion
reduces to the simple syntactic criterion.
This presentation of the correctness criterion only relies on syntactic judgments. It is pragmatic in the sense that it suggests a simple and direct implementation, as a generalization of the check currently implemented in type system
engines—which corresponds to the Γ ⊢ τ : (+) part.
To compute the contexts Γ and (Γi )i∈I existentially quantified in this formula, one can use a variant of our syntactic judgments where the environment
Γ is not an input, but an output of the judgment; in fact, one should return
for each variable α the set of possible variances for this judgment to hold. For
example, the query (? ⊢ α ∗ β ref : +) should return (α 7→ {+, =}; β 7→ {=}).
Defining those algorithmic variants of the judgments is routine. The sets of variances corresponding to the decomposability of the (Ti )i∈I (? ⊢ Ti : vi ⇒ (=))
should be zipped together and intersected with the possible variances for τ , returned by (? ⊢ τ : +). The algorithmic criterion is satisfied if and only if the
intersection is not empty; this can be decided in a simple and efficient way.
4
4.1
Discussion
Upward and downward closure in a ML type system
In the type system we have used so far, all type constructors but p and q are both
upward and downward-closed. This simple situation, however, does not hold in
general: richer subtyping relations will have weaker invertibility properties. As
soon as a bottom type ⊥ is introduced, for example, such that that for all type σ
we have ⊥ ≤ σ, downward-closure fails for all types – but ⊥ itself. For example,
products are no longer downward-closed: Γ ⊢ σ ∗ τ ≥ ⊥ does not implies that
⊥ is equal to some σ ′ ∗ τ ′ . Conversely, if one adds a top type ⊤, bigger than all
other types, then most type are not upward-closed anymore.
16
Gabriel Scherer and Didier Rémy
In OCaml, there is no ⊥ or ⊤ type8 . However, object types and polymorphic
variants have subtyping, so they are, in general, neither upward nor downwardclosed. Finally, subtyping is also used in private type definitions, which were
demonstrated in the example. Our closure-checking relation therefore degenerates into the following, quite unsatisfying, picture:
– no type is downward-closed because of the existence of private types;
– no object type but the empty object type is upward-closed;
– no arrow type is upward-closed because its left-hand-side would need to be
downward-closed;
– datatypes are upward-closed if their components types are.
From a pragmatic point of view, the situation is not so bad; as our main practical
motivation for finer variance checks is the relaxed value restriction, we care about
upward-closure (covariance) more than downward-closure (contravariance). This
criterion tells us that covariant parameters can be instantiated with covariant
datatypes defined from sum and product types (but no arrow), which would
satisfy a reasonable set of use cases.
4.2
A better control on upward and downward-closure
There is a subtle design question here. Decomposability is fundamentally a
negative statement on the subtyping relation, guaranteeing that some types
have no supertypes of a different structure. It is therefore not necessarily preserved by addition to the subtyping relation – our system, informally, is non-monotone in the subtyping relation.
This means that if we adopt the correctness criterion above, we must be
careful in the future not to enrich the subtyping relation too much. Consider
private types for example: one could imagine a symmetric concept of a type
that would be strictly above a given type τ ; we will name those types invisible
types (they can be constructed, but not observed). Invisible types and GADT
covariance seem to be working against each other: if the designer adds one,
adding the other later will be difficult.
A solution to this tension is to allow the user to locally guarantee negative
properties about subtyping (what is not a subtype), at the cost of selectively
abandoning the corresponding flexibility. Just as object-oriented languages have
final classes that cannot be extended any more, we would like to be able to define some types as downward-closed (respectively upward-closed), that cannot
later be made private (resp. invisible). Such declarations would be rejected
if the defining type, for example an object type, already has subtypes (resp.
supertypes), and would forbid further declarations of types below (resp. above)
the defined type, effectively guaranteeing downward (resp. upward) closure.
8
A bottom type would be admissible, but a top type would be unsound in OCaml,
as different types may have different runtime representations. Existential types, that
may mix values of different types, are constructed explicitly through a boxing step.
GADTs meet subtyping
17
Finally, upward or downward closure is a semantic aspect of a type that we
must have the freedom to publish through an interface: abstract types could
optionally be declared upward-closed or downward-closed.
4.3
Subtyping constraints and variance assignment
We will now revisit our example of strongly typed expressions in the introduction.
A simple way to get such a type to be covariant would be, instead of proving
delicate, non-monotonic upward-closure properties on the tuple type involved in
the equation α = β ∗ γ, to change this definition so that the resulting type is
obviously covariant:
type +α exp =
| Val of ∃β[α ≥ β]. β
| Int of [α ≥ int]. int
| Thunk of ∃βγ[α ≥ γ]. β exp ∗ (β → γ)
| Prod of ∃βγ[α ≥ β ∗ γ]. β exp ∗ γ exp
We have turned each equality constraint α = T [β] into a subtyping constraint
α ≥ T [β]. For a type α′ such that α ≤ α′ , we get by transitivity that α′ ≥ T [β].
This means that α exp trivially satisfies the correctness criterion Req. Formally,
instead of checking Γ ⊢ Ti : vi ⇒ (=), we are now checking Γ ⊢ Ti : vi ⇒
(+), which is significantly easier to satisfy: when vi is itself + we can directly
apply the sc-Triv rule. Note that this only works in the easy direction: while
Γ ⊢ Ti : (+) ⇒ (+) is easy to check, Γ ⊢ Ti : (+) ⇒ (−) is just as hard as
Γ ⊢ Ti : (+) ⇒ (=). In particular, an equality (σ = σ ′ ) is already equivalent to
a pair of inequalities (σ ≤ σ ′ ∧ σ ≥ σ ′ ).
While this different datatype gives us a weaker subtyping assumption when
pattern-matching, we are still able to write the classic function eval : α exp → α,
because the constraints α ≥ τ are in the right direction to get an α as a result.
rec eval : α exp → α = function
Val β (v : β) -> (v :> α)
Int (n : int) -> (n :> α)
Thunk β γ ((v : β exp), (f : β → γ)) ->
(f (eval v) :> α)
| Prod β γ ((b : β exp), (c : γ exp)) ->
((eval b, eval c) :> α)
let
|
|
|
This variation on GADTs, using subtyping instead of equality constraints,
has been studied by Emir et al [EKRY06] in the context of the C♯ programming
language—it is also expressible in Scala. However, using subtyping constraints in
GADTs has important practical drawbacks in a ML-like language. While typed
object-oriented programming languages tend to use explicit polymorphism and
implicit subtyping, ML uses implicit polymorphism and explicit subtyping (when
present). Thus in ML, equality constraints can be implicitly used while subtyping constraints must be explicitly used: unification-based inference favors bidirectional equality over unidirectional subtyping. This makes GADT definitions
18
Gabriel Scherer and Didier Rémy
based on single subtyping constraints less convenient to use, because of the corresponding syntactic burden, and this is probably the reason why the notion of
GADTs found in functional languages use only equality constraints. Subtyping
constraints need also be explicit in the type declaration, forcing the user out of
the convenient “generalized codomain type” syntax.
Finally, weakening equality constraints into a subtyping constraint in one
direction is not always possible; sometimes the strictly weaker expressivity of the
type forbids important uses. One must then use an equality constraint, and use
our decomposability-based reasoning to justify the variance annotation. Consider
the following example:
type +α tree =
| Node of ∃β[α = β list]. (β tree) list
let append : α tree ∗ α tree → α tree = function
| Node β1 (l1 : β1 tree list), Node β2 (l2 : β2 tree list) ->
Node (List.append l1 l2)
We know that the two arguments of append have the same type α tree.
When matching on the Node constructors, we learn that α is equal to both
β1 list and β2 list, from which we can deduce that β1 is equal to β2 by
non-irrelevance of list. The concatenation of the lists l1 and l2 type-checks
because this equality holds. If we used a type system without the decomposability
criterion, we would need to turn the constructor constraint into ∃β[α ≥ β list]
to preserve covariance of α tree . We wouldn’t necessarily have β1 and β2 equal
anymore, so (List.append l1 l2), hence the definition of append would not
type-check. We would need decomposability-based reasoning to deduce, from
α ≥ β list and the fact that list is upward-closed, that in fact α = β ′ list
for some β ′ .
This demonstrates that single subtyping constraints and our novel decomposability check on equality constraints are of incomparable expressivity: each
setting handles programs that the other cannot type-check. From a theoretical
standpoint, we think there is value in exploring the combination of both systems:
using subtyping constraints rather than equalities, but also using decomposability to deduce stronger equalities when possible.
Note that while our soundness result directly transposes to a type-system
with decomposability conditions on subtyping rather than equality constraints,
our completeness result is special-cased on equality constraints. Completeness in
the case of subtyping constraints is an open question.
Related Work
Simonet and Pottier [SP07] have studied GADTs in a general framework HMG(X),
inspired by HM(X). They were interested in type inference using constraints, so
considered GADTs with arbitrary constraints rather than type equalities, and
considered the case of subtyping with applications to information flow security
in mind. Their formulation of the checking problem for datatype declarations,
GADTs meet subtyping
19
as a constraint-solving problem, is exactly our semantic criterion and is not
amenable to a direct implementation. Correspondingly, they did not encounter
any of the new notions of upward and downward-closure and variable interference (zipping) discussed in the present work. They define a dynamic semantics
and prove that this semantic criterion implies subject reduction and progress.
However, we cannot directly reuse their soundness result as they work in a setting where all constructors are upward- and downward-closed (their subtyping
relation is atomic). We believe this is only an artifact of their presentation and
their proof should be easily extensible to our setting.
Emir, Kennedy, Russo and Yu [EKRY06] studied the soundness of an objectoriented calculus with subtyping constraints on classes and methods. Previous
work [KR05] had established the correspondence between equality constraints on
methods in an object-oriented style and GADT constraints on type constructors
in functional style. Through this surprisingly non-obvious correspondence, their
system matches our presentation of GADTs with subtyping constraints and easier variance assignment, detailed in §4.3. They provide several usage examples
and a full soundness proof using a classic syntactic argument. However, they
do not consider the more delicate notions of decomposability, and their system
therefore cannot handle some of the examples presented here.
Future Work
Experiments with v-closure of type constructors as a new semantic property In
a language with non-atomic subtyping such as OCaml, we need to distinguish
v-closed and non-v-closed type constructors. This is a new semantic property
that, in particular, must be reflected through abstraction boundaries: we should
be able to say about an abstract type that it is v-closed, or not say anything.
How inconvenient in practice is the need to expose those properties to have
good variance for GADTs? Will the users be able to determine whether they
want to enforce v-closure for a particular type they are defining?
Completeness of variance annotations with domain information The way we
present GADTs using equality constraints instead of the codomain syntax is wellknown to practictioners, under the form of a “factoring” transformation where
an arbitrary GADT is expressed as a simple ADT, using the equality GADT
(α, β) eq as part of the constructor arguments to reify equality information.
This transformation does not work anymore with our current notion of GADTs
in presence of subtyping. Indeed, all we can soundly say about the equality type
(α, β) eq is that it must be invariant in both its parameters; using (α, Ti [β]) eq
as part of a constructor type would force the paramter α to be invariant.
We think it would possible to re-enable factoring by eq by considering domain
information, that is, information on constraints that must hold for the type to
be inhabited. If we restricted the subtyping rule with conclusion σ t ≤ σ ′ t to
only cases where σ t and σ ′ t are inhabited—with a separate rule to conclude
subtyping in the non-inhabited case—we could have a finer variance check, as we
20
Gabriel Scherer and Didier Rémy
would only need to show that the criterion Seq holds between two instances of
the inhabited domain, and not any instance. If we stated that the domain of the
type (α, β) eq is restricted by the constraint α = β, we could soundly declare the
variance (⋊
⋉α, ⋊
⋉β) eq on this domain—which no longer prevents from factoring
out GADTs by equality types.
Conclusion
Checking the variance of GADTs is surprisingly more difficult (and interesting)
than we initially thought. We have studied a novel criterion of upward and
downward closure of type expressions and proposed a corresponding syntactic
judgment that is easily implementable. We presented a core formal framework
to prove both its correctness and its completeness with respect to a natural
semantic criterion.
This closure criterion exposes important tensions in the design of a subtyping
relation, for which we previously knew of no convincing example in the context of
ML-derived programming languages. We have suggested new language features
to help alleviate these tensions, whose convenience and practicality is yet to be
assessed by real-world usage.
Considering extensions of GADTs in a rich type system is useful in practice;
it is also an interesting and demanding test of one’s type system design.
References
Abe06.
Andreas Abel. Polarized subtyping for sized types. Mathematical Structures
in Computer Science, 2006. Special issue on subtyping, edited by Healfdene
Goguen and Adriana Compagnoni.
EKRY06. Burak Emir, Andrew Kennedy, Claudio Russo, and Dachuan Yu. Variance
and generalized constraints for C# generics. In Proceedings of the 20th
European conference on Object-Oriented Programming, ECOOP’06, 2006.
Gar04.
Jacques Garrigue. Relaxing the value restriction. In In International Symposium on Functional and Logic Programming, Nara, LNCS 2998, 2004.
KR05.
Andrew Kennedy and Claudio V. Russo.
Generalized algebraic
data types and object-oriented programming.
In Proceedings of
the 20th annual ACM SIGPLAN conference on Object-oriented
programming, systems, languages, and applications, 2005.
URL:
http://research.microsoft.com/pubs/64040/gadtoop.pdf.
Pfe01.
Frank Pfenning. Intensionality, extensionality, and proof irrelevance in
modal type theory. In 16th IEEE Symposium on Logic in Computer Science
(LICS 2001), 16-19 June 2001, Boston University, USA, Proceedings, 2001.
SP07.
Vincent Simonet and Franois Pottier.
A constraint-based approach to guarded algebraic data types.
ACM Transactions on
Programming Languages and Systems, 29(1), January 2007.
URL:
http://doi.acm.org/10.1145/1180475.1180476.
SR.
Gabriel Scherer and Didier Rémy. GADTs meet subtyping. Long version,
available electronically. URL: http://gallium.inria.fr/~ remy/gadts/.
| 6 |
Logical Methods in Computer Science
Vol. 8 (2:01) 2012, pp. 1–35
www.lmcs-online.org
Submitted
Published
Jul. 13, 2011
Apr. 6, 2012
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
BENEDIKT AHRENS
Laboratoire J.-A. Dieudonné, Université Nice Sophia Antipolis, Parc Valrose, 06108 Nice, France
e-mail address: ahrens@unice.fr
Abstract. Initial Semantics aims at interpreting the syntax associated to a signature as
the initial object of some category of “models”, yielding induction and recursion principles
for abstract syntax. Zsidó [Zsi10, Chap. 6] proves an initiality result for simply–typed
syntax: given a signature S, the abstract syntax associated to S constitutes the initial
object in a category of models of S in monads.
However, the iteration principle her theorem provides only accounts for translations
between two languages over a fixed set of object types. We generalize Zsidó’s notion of
model such that object types may vary, yielding a larger category, while preserving initiality
of the syntax therein. Thus we obtain an extended initiality theorem for typed abstract
syntax, in which translations between terms over different types can be specified via the
associated category–theoretic iteration operator as an initial morphism. Our definitions
ensure that translations specified via initiality are type–safe, i.e. compatible with the typing
in the source and target language in the obvious sense.
Our main example is given via the propositions–as–types paradigm: we specify propositions and inference rules of classical and intuitionistic propositional logics through their
respective typed signatures. Afterwards we use the category–theoretic iteration operator to
specify a double negation translation from the former to the latter.
A second example is given by the signature of PCF. For this particular case, we formalize
the theorem in the proof assistant Coq. Afterwards we specify, via the category–theoretic
iteration operator, translations from PCF to the untyped lambda calculus.
1. Introduction
Initial Semantics characterizes the set of terms of a language via a universal property —
namely as an initial object in some category —, and gives a category–theoretic account
of the iteration principle it is equipped with. By working in a suitable category, one can
specify additional structure and properties on the syntax. As an example, the initial object
in our category is by definition equipped with a substitution operation, due to our use of
monads (cf. Def. 2.1, Exs. 2.9, 2.13). Furthermore, this substitution is by construction
type–safe. Initiality also provides an iteration principle which allows to specify maps as
initial morphisms on the the set of terms of a syntax. The main focus of this paper is to
obtain a sufficiently general iteration operator that allows to specify translations between
1998 ACM Subject Classification: D.3.1, F.4.3.
Key words and phrases: initial semantics, typed abstract syntax, logic translation.
l
LOGICAL METHODS
IN COMPUTER SCIENCE
c
DOI:10.2168/LMCS-8 (2:01) 2012
CC
B. Ahrens
Creative Commons
2
B. AHRENS
terms over different sets of object types (to which we also refer as sorts) as such initial
morphisms.
An important property of translations between programming languages is that they
should preserve the meaning of programs. While the present work does not consider this
aspect — it merely treats the syntactic part —, we outline our ideas concerning faithfulness
of translation with respect to meaning in Sec. 6.
In Sec. 1.1 we explain initiality for syntax without binding by means of an example and
present our view on syntax with variable binding and sorts. Related work is reviewed in Sec.
1.2. In Sec. 1.3 we give an overview of the paper.
1.1. Natural Numbers, Syntax with Binding and Types.
1.1.1. Natural Numbers. Consider the category N an object of which is a triple (X, Z, S) of
a set X, a constant Z ∈ X and a map S : X → X. A morphism to another such (X 0 , Z 0 , S 0 )
is a map f : X → X 0 such that
f (Z) = Z 0
and
S0 ◦ f = f ◦ S .
(1.1)
This category has an initial object (N, Zero, Succ), and a map f from N to a set X can be
specified by giving an element Z ∈ X and a map S : X → X. This way of specifying the
map f is an iteration principle for N resulting from its initiality in the category N .
Our work consists in providing, via initiality, a category–theoretic iteration operator for
typed syntax with variable binding, similar in spirit to that for the natural numbers. In the
rest of this section we consider some aspects that arise when passing from our introductory
example about natural numbers to syntax with variable binding and types.
1.1.2. Variable Binding. For syntax with variable binding, we consider the set of terms to be
parametrized by a context, i.e. a set of variables, whose elements may appear freely in those
terms. The terms of the untyped lambda calculus, for instance, can be implemented in the
proof assistant Coq [Coq10] as the following parametrized datatype:
Inductive ULC (V : Type) : Type :=
| Var : V −> ULC V
| Abs : ULC (option V) −> ULC V
| App : ULC V −> ULC V −> ULC V.
where option V stands for an extended context obtained by enriching the context V with a
new distinguished variable — the variable which is bound by the Abs constructor.
The map V 7→ ULC(V ) is in fact functorial: given a map f : V → W , the map
ULC(f ) : ULC(V ) → ULC(W ) renames any free variable v ∈ V in a term by f (v), yielding
a term with free variables in W . Accordingly, instead of sets and maps of sets as for the
introductory example, we consider functors and natural transformations between them.
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
3
1.1.3. Adding Types. The interest of considering typed syntax is twofold: firstly, for programming languages, typing rules contain information of how to plug several terms together
in semantically meaningful ways, and ensure properties such as termination. Secondly,
via the propositions–as–types paradigm, logics may be considered as typed syntax, where
propositions are viewed as types, and a term p : P of type P thus denotes a proof p of
proposition P . In this vein, the inference rules correspond to term constructors, i.e. they
are the basic bricks from which one builds terms — proofs — according to plugging rules.
The premises of such an inference rule thus are represented by the inputs of the constructor,
whereas the conclusion is represented by its output type.
In the present work we consider both applications of types: our main example, a logic
translation from classical to intuitionistic logic (cf. Sec. 4), works through the propositions–as–
types paradigm. As a running example throughout this work we consider typed programming
languages.
Type systems exists with varying features, ranging from simply–typed syntax to syntax
with dependent types, kinds, polymorphism, etc. By simply–typed syntax we mean a
non–polymorphic syntax where the set of types is independent from the set of terms, i.e.
type constructors only take types as arguments, In more sophisticated type systems types
may depend on terms, leading to more complex definitions of arities and signatures. The
present work is only concerned with simply–typed languages.
One way to add types would be to make them part of the syntax, as in “λx : ι.x + 4”.
However, for simple type systems it is possible to separate the worlds of types and terms and
consider typing as a map from terms to types, thus giving a simple mathematical structure
to typing. How can we be sure that our terms are well–typed? Despite the separation of
types and terms we still want typing to be tightly integrated into the process of building
terms, in order to avoid constructing ill–typed terms. Separation of terms and types seems
to contradict this goal. The answer lies in considering not one set of terms, but a family
of sets, indexed by the set of object types. Term constructors then can be “picky” about
what terms they take as arguments, accepting only those terms that have the suitable type.
We also consider free variables to be equipped with an object type. Put differently, we do
not consider terms over one set of variables, but over a family of sets of variables, indexed
by the set of object types. We illustrate such a definition of a family of terms in the proof
assistant Coq [Coq10] using the example of the simply–typed lambda calculus SLC:
Example 1.1 (Syntax of SLC). Let
TSLC ::= ∗ | TSLC
TSLC
be the set of types of the simply–typed lambda calculus. For each “typed set” V ∈ [TSLC , Set]
and t ∈ TSLC we denote by Vt := V (t) the set associated to object type t ∈ TSLC . Hence
SLC(V )t denotes the set of lambda terms of type t with free variables in V . In the following
Coq code excerpt we write T for TSLC .
Inductive SLC (V : T −> Type) : T −> Type :=
| Var : forall t, V t −> SLC V t
| Abs : forall r s, SLC (V ∗ r) s −> SLC V (r ~> s)
| App : forall r s, SLC V (r ~> s) −> SLC V r −> SLC V s.
Here V ∗ r is Coq notation for V + {∗r}, which is the family of sets V enriched with a new
distinguished variable of type r ∈ TSLC — the variable which is bound by the Abs(r, s)
4
B. AHRENS
constructor. The quantified variables s and t range over the set TSLC of object types. Indeed
SLC can be interpreted as a functor
SLC : [TSLC , Set] → [TSLC , Set]
on the category [TSLC , Set] whose objects are families of sets indexed by the set TSLC of
types of SLC.
This method of defining exactly the well–typed terms by organizing them into a type
family parametrized by object types is called intrinsic typing [BHKM11] — as opposed to
the extrinsic typing, where first a set of raw terms is defined, which is then filtered via a
typing predicate. Intrinsic typing delegates object level typing to the meta language type
system, such as the Coq type system in Ex. 1.1. In this way, the meta level type checker
(e.g. Coq) sorts out ill–typed terms automatically: writing such a term yields a type error
on the meta level. Furthermore, the intrinsic encoding comes with a much more convenient
recursion principle; a map to any other type can simply be defined by specifying its image
on the well–typed terms. When using extrinsic typing, a map on terms would either have
to be defined on the set of raw terms, including ill–typed ones, or on just the well–typed
terms by specifying an additional propositional argument expressing the welltypedness of the
term argument. Benton et al. give detailed explanation about intrinsic typing in a recently
published paper [BHKM11].
1.1.4. Substitution. Syntax with variable binding always comes with a (capture–avoiding)
substitution operation. Fiore, Plotkin and Turi [FPT99] model substitution and its properties
using the notion of monoid. An alternative point of view is given by monads: a monad
(Def. 2.1) is an endofunctor with extra structure, and it is this additional structure that
captures substitution (cf. Ex. 2.9), as exhibited by Altenkirch and Reus [AR99]. We review
the monad structure on ULC (Ex. 2.9) and SLC (Ex. 2.13).
1.2. Related Work. Initial Semantics for untyped syntax without variable binding was
first considered by Birkhoff [Bir35]. Goguen et al. [GTWW77] give an overview over the
literature about initial algebra and spell out explicitly the connection between initial algebras
and abstract syntax.
When passing to syntax with variable binding, the question of how to model binding
arises. We give a possibly non–exhaustive list of techniques for binder representation:
(1) Nominal syntax using named abstraction;
(2) Higher–Order Abstract Syntax (HOAS), e.g. lam : (T → T ) → T and its weak variant,
e.g. lam : (var → T ) → T ;
(3) Nested datatypes as presented in [BM98].
In the following, the numbers given in parentheses indicate the way variable binding is
modeled, according to the list given above. Initial semantics for untyped syntax were
presented by Gabbay and Pitts [GP99, (1)], Hofmann [Hof99, (2)] and Fiore et al. [FPT99,
(3)]. Hirschowitz and Maggesi [HM07, (3)] prove an initiality result for arbitrary untyped
syntax based on the notion of monad.
Fiore et al.’s approach was generalized to encompass the simply–typed lambda calculus
by Fiore [Fio02, (3)] and Miculan and Scagnetto [MS03, (3)]. In her thesis, Zsidó [Zsi10,
Chap. 6] generalized Hirschowitz and Maggesi’s approach to simply–typed syntax. The
present paper presents a variant of Zsidó’s theorem 6.4.121 — the main result of [Zsi10, Chap.
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
5
6] —, using the same category–theoretic concept of monads. Both approaches, Hirschowitz
and Maggesi’s and Fiore et al.’s, are connected via an adjunction between the respective
categories under consideration. This adjunction was established in Zsidó thesis [Zsi10, Chaps.
4 (untyped), 7 (typed)].
Some of the mentioned lines of work have been extended to integrate semantic aspects in
form of reduction relations on terms into initiality results: Hirschowitz and Maggesi [HM07]
characterize the terms of the lambda calculus modulo beta and eta reduction as an initial
object in some category. In another work [Ahr11], we extend Hirschowitz and Maggesi’s
approach via monads to encompass semantics in form of reduction rules, specified through
inequations, by considering relative monads [ACU10] over a suitable functor from sets to
preorders. Fiore and Hur [FH07] extended Fiore et al.’s approach to “second–order universal
algebras”. In particular, Hur’s PhD thesis [Hur10] is dedicated to this extension.
1.3. Summary of the Paper. We prove an initiality result for simply–typed syntax which
provides a category–theoretic iteration operator for translations between languages over
different sets of sorts.
We define typed signatures in order to specify the types and terms of simply–typed
languages. To any such typed signature we associate a category of representations —
“models” — of this signature. Our main theorem states that this category has an initial
object, which integrates the types and terms freely generated by the signature. Initiality
yields an iteration operator which allows to conveniently and economically specify translations
between languages over different sets of sorts.
We give two examples of translations via such an iteration operator: firstly, via the
proposition–as–types paradigm we consider classical and intuitionistic propositional logic as
simply–typed languages. We present the typed signature for both of these logics and specify
a double negation translation from classical to intuitionistic logic via the category–theoretic
iteration operator (Sec. 4). Secondly, we present the typed signature of the programming
language PCF, a simply–typed programming language introduced by Plotkin [Plo77]. For this
particular typed signature, we have formalized the initiality theorem in the proof assistant
Coq [Coq10]. Afterwards we have specified two different representations of PCF in the
untyped lambda calculus ULC, yielding — by initiality — two translations from PCF to
ULC. The formalization is presented in Sec. 5. In the formalization these translations are
Coq functions and hence executable. The Coq theory files as well as online documentation
are available online1.
1.4. Synopsis. In the second section we review the definitions of monads and modules over
monads with their respective morphisms. We recall some constructions on monads and
modules, which will be of importance in what follows.
The third section introduces our notions of arity, typed signature and representations of
typed signatures. We then prove our main result.
In the fourth section, we present our main example: we specify the propositions and proofs of
classical and intuitionistic logic via their respective typed signatures, and define a translation
from the former to the latter logic via initiality.
1http://math.unice.fr/laboratoire/logiciels
6
B. AHRENS
The fifth section gives a brief overview of the formalization in the proof assistant Coq of the
theorem instantiated for the signature of PCF, as well as two translations from PCF to the
untyped lambda calculus via initiality.
Some extensions we are working on are explained in the last section.
2. Monads & Modules
We state the widely known definition of monad and the less known definition of module
over a monad. Modules have been used in the context of Initial Semantics by Hirschowitz
and Maggesi [HM07, HM10] and Zsidó [Zsi10]. Monad morphisms are in fact colax monad
morphisms, as presented, for instance, by Leinster [Lei04].
2.1. Definitions.
Definition 2.1 (Monad). A monad T over a category C is given by
• a functor T : C → C (observe the abuse of notation),
• a natural transformation η : IdC → T and
• a natural transformation µ : T ◦ T → T
such that the following diagrams commute:
T
Tη
/ T2 o
ηT
µ
id
# {
T,
id
T
T3
Tµ
µT
µ
T2
/ T2
µ
/ T.
Example 2.2. The functor [_] : Set → Set which to any set X associates the set of (finite)
lists over X, is equipped with a structure as monad by defining η and µ as “singleton list”
and flattening, respectively:
ηX (x) := [x] and
µX [x1,1 , . . . , x1,m1 ], . . . , [xn,1 , . . . , xn,mn ] := [x1,1 , . . . , x1,m1 , . . . , xn,1 , . . . , xn,mn ] .
Remark 2.3 (Kleisli Operation (Monadic Bind)). Given a monad (T, η, µ) on the category
C, the Kleisli operation with type
(_)∗a,b : C(a, T b) → C(T a, T b)
is defined, for any a, b ∈ C and f ∈ C(a, T b), by setting
(f )∗a,b := µb ◦ T f .
Indeed, a monad (T, η, µ) can equivalently be defined as a triple (T, η, (_)∗ ) with an adapted
set of axioms. We refer to [Man76] for details.
Our definition of colax monad morphisms and their transformations is taken from
Leinster’s book [Lei04]:
Definition 2.4 (Colax Monad Morphism). Let (T, η, µ) be a monad on the category C and
(T 0 , η 0 , µ0 ) be a monad on the category D. A colax morphism of monads (C, T ) → (D, T 0 ) is
given by
• a functor F : C → D and
• a natural transformation γ : F T → T 0 F
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
7
such that the following diagrams commute:
FTT
Fµ
γT
γ
/ T 0F T
/ T 0T 0F
F
µ0 F
/ T 0 F,
FT
Fη
γ
η0 F
FT
"
/ T 0 F.
γ
From now on we will simply say “monad morphism over F ” when speaking about a colax
monad morphism with underlying functor F . We will not use any other kind of monad
morphism.
Definition 2.5 (Composition of Monad Morphisms). Suppose given a monad morphism
as in Def. 2.4. Given a third monad (T 00 , η 00 , µ00 ) on category E and a monad morphism
(F 0 , γ 0 ) : (T 0 , η 0 , µ0 ) → (T 00 , η 00 , µ00 ), we define the composition of (F, γ) and (F 0 , γ 0 ) to be the
monad morphism given by the pair consisting of the functor F 0 F and the transformation
F 0F T
F 0γ
/ F 0T 0F
γ0F
/ T 00 F 0 F .
The verification of the necessary commutativity properties is done in the Coq library, cf.
colax_Monad_Hom_comp.
Definition 2.6 (Transformation). Given two morphisms of monads
(F, γ), (F 0 , γ 0 ) : (C, T ) → (D, T 0 ) ,
a transformation (F, γ) ⇒ (F 0 , γ 0 ) is given by a natural transformation β : F → F 0 such
that the following diagram commutes:
FT
γ
/ T 0F
T 0β
βT
F 0T
γ0
/ T 0F 0.
Definition 2.7 (2–Category of Monads, [Lei04]). We call Mndcolax the 2–category an object
of which is a pair (C, T ) of a category C and a monad T on C. A morphism to another object
(D, T 0 ) is a colax monad morphism (F, γ) : (C, T ) → (D, T 0 ). A 2–cell (F, γ) ⇒ (F 0 , γ 0 ) is a
transformation.
Notation 2.8. For any category C, we write IdC for the object (C, Id) of Mndcolax .
Example 2.9 (Monadic Syntax, Untyped). Syntax as a monad (using the Kleisli operation
presented in Rem. 2.3) was presented by Altenkirch and Reus [AR99]: consider the syntax
of the untyped lambda calculus ULC as given in Sec. 1.1. As mentioned there, the map
V 7→ ULC(V ) is functorial. We equip it with a monad structure: we define η as variable–as–
term operation
ηV (v) := Var(v) ∈ ULC(V )
and the multiplication µ : ULC ◦ ULC → ULC as flattening which, given a term of ULC
with terms of ULC(V ) as variables, returns a term of ULC(V ). These definitions turn
(ULC, η, µ) into a monad on the category Set. The Kleisli operation associated to this monad
corresponds to a simultaneous substitution, cf. [AR99].
8
B. AHRENS
For reasons that are explained in Rem. 2.14, we are particularly interested in monads
over families of sets (Def. 2.10) and monad morphisms over retyping functors (Def. 2.11).
Definition 2.10 (Category of Families). Let C be a category and T be a set, i.e. a discrete
category. We denote by [T, C] the functor category, an object of which is a T –indexed
family of objects of C. Given two families V and W , a morphism f : V → W is a family of
morphisms in C,
f : t 7→ f (t) : V (t) → W (t) .
We write Vt := V (t) for objects and morphisms. Given another category D and a functor
F : C → D, we denote by [T, F ] the functor defined on objects and morphisms as
[T, F ] : [T, C] → [T, D],
f 7→ t 7→ F (ft ) .
Definition 2.11 (Retyping Functor). Let T and T 0 be sets and g : T → T 0 be a map. Let
C be a cocomplete category. We define the functor
~g : [T, C] → [T 0 , C] ,
X = t 7→ Xt
~g (X) := t0 7→
7→
a
Xt .
{t | g(t)=t0 }
In particular, for any V ∈ [T, C] — considered as a functor — we have a natural transformation
V ⇒ ~g V ◦ g : T → C
given pointwise by the morphism Vt → {s|g(s)=g(t)} Vs in the category C. Put differently,
every map g : T → T 0 induces an endofunctor ḡ on [T, C] with object map
`
ḡ(V ) := ~g (V ) ◦ g
and we have a natural transformation
ctype : Id ⇒ ḡ : [T, C] → [T, C] .
Remark 2.12 (Retyping as an Adjunction). An anonymous referee pointed out to us that
the retyping functor ~g associated to g : T → T 0 is the left Kan extension operation along g,
that is, we have an adjunction
~g
[T, C] g
⊥
'
[T 0 , C] ,
g∗
where g ∗ (W ) := W ◦ g. The natural transformation ctype is the unit of this adjunction.
Given a map g as in Def. 2.11, we interpret the map g : T → T 0 as a translation of
object sorts and the functor ~g as a “retyping functor” which changes the sorts of contexts
and terms (and more generally, models of terms) according to the translation of sorts.
In Ex. 2.13 and Rem. 2.14 we explain how we consider languages as monads and
translations between languages as monad morphisms over retyping functors, respectively:
Example 2.13 (Monadic Syntax, Typed). Consider the syntax of the simply–typed lambda
calculus as presented in Ex. 1.1. Similarly to the untyped lambda calculus, the natural
transformations η : Id → SLC and µ : SLC ◦ SLC → SLC are defined as variable–as–term
operation and flattening, respectively. These definitions turn (SLC, η, µ) into a monad on
the category [TSLC , Set].
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
9
The previous example explains, how the terms of a language can be organized in a
monad. Accordingly, a translation between two languages corresponds to a monad morphism:
Remark 2.14. Suppose we have two monads, a monad P over [U, Set] and a monad Q
over [V, Set] for sets U and V . We think of P and Q as term monads as in Ex. 2.13, i.e.
the monads P and Q denote the terms of some programming language over types U and V ,
respectively. However, what follows is not restricted to such term monads.
A map — “translation” — from P to Q now consists, first of all, of a map of types
g : U → V . The translation of terms f then should be compatible with the type translation
g. During the term translation f we have to pass from the category [U, Set] — where the
terms of P live — to the category [V, Set], where the terms of Q live. This passing is done
via the retyping functor ~g associated to the type translation g.
Given a set of variables X ∈ [U, Set] typed over U , a translation of terms with free
variables in X is specified via a morphism
fX : ~g (P X) → Q(~g X)
in the category [V, Set]. The intuition is that if we have a term t ∈ P (X)u , we translate at
first its type u ∈ U to g(u), yielding a term t0 ∈ ~g (P X)g(u) . The term translation afterwards
then is a morphism in the category [V, Set]:
t ∈ P (X)u
ctype
t0 ∈ ~g (P X)g(u)
7−→
f
X
7−→
fX (t0 ) ∈ Q(~g X)g(u) ,
where instead of “fX ” one should read “the component of fX corresponding to g(u)”.
Putting this in category–theoretic terms, the family (fX )X∈[U,Set] of morphisms forms
a colax monad morphism f over the retyping functor associated to g, provided that f
is compatible with the monadic structure on P and Q, i.e. with variables–as–terms and
flattening operations.
The notion of module over a monad generalizes monadic substitution (cf. [HM07]):
Definition 2.15 (Module over a Monad). Given a monad T over category C and a category
D, a module over T with codomain D (or T –module towards D) is a colax monad morphism
(M, γ) : (C, T ) → (D, IdD ) from T to the identity monad on D. Given T –modules M and N ,
a morphism of modules from M to N is a transformation from M to N . We call
Mod(T, D) := Mndcolax (C, T ), (D, Id)
the category of T –modules towards D.
Remark 2.16. By unfolding the preceding definition and simplifying, we obtain that a
T –module towards D is a functor M : C → D together with a natural transformation
σ : M T → M such that the following diagrams commute:
MTT
Mµ
σT
σ
MT
/ MT
σ
/ M,
M
Mη
id
MT
σ
"
/ M.
10
B. AHRENS
A morphism of T –modules from (M, σ) to (M 0 , σ 0 ) then is given by a natural transformation
β : M ⇒ M 0 such that the following diagram commutes:
MT
σ
βT
M
/ M 0T
β
σ0
/ M 0.
Remark 2.17 (Kleisli Operation for Modules). Let T be a monad on a category C and
(M, σ) be a T –module with codomain category D. Similarly to monads (cf. Rem. 2.3), a
Kleisli operation for modules, with type
(_)∗a,b : C(a, T b) → D(M a, M b)
is defined by setting, for any a, b ∈ C and f ∈ C(a, T b),
(f )∗a,b := σb ◦ M f .
Modules over monads can equivalently be defined in terms of this Kleisli operation, cf.
[AZ11].
We anticipate the constructions of the next section by giving some examples of modules
and module morphisms:
Example 2.18 (Tautological Module, Ex. 2.9 cont.). Any monad T on a category C can be
considered as a module over itself, the tautological module. In particular, the monad of the
untyped lambda calculus ULC (cf. Ex. 2.9) is a ULC–module with codomain Set.
Example 2.19. The map
ULC0 : V 7→ ULC(V 0 ) ,
with V 0 := V + {∗}, inherits — from the tautological module ULC — the structure of a
ULC–module, which we call the derived module of the module ULC. Also, the map
ULC × ULC : V 7→ ULC(V ) × ULC(V )
inherits a ULC–module structure.
The constructors of the untyped lambda calculus are, accordingly, morphisms of modules:
Example 2.20 (Ex. 2.19 cont.). The natural transformation
V 7→ AppV : ULC(V ) × ULC(V ) → ULC(V )
verifies the diagram of module morphisms and is hence a morphism of ULC–modules from
ULC × ULC to ULC. The natural transformation
V 7→ AbsV : ULC(V 0 ) → ULC(V )
is a morphism of ULC–modules from ULC0 to ULC.
The meaning of the commutative diagrams for module morphisms is best explained
in terms of the module Kleisli operation, the module substitution (cf. Def. 2.17); for this
equivalent definition, the notion of module morphism captures the distributivity property of
substitution with respect to term constructors. A detailed explanation is given by Ahrens
and Zsidó [AZ11].
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
11
Example 2.21. Given any t ∈ TSLC , the functor
SLCt : V 7→ SLC(V )t
is canonically equipped with a module structure, where the natural transformation
σ : SLCt ◦ SLC → SLCt
is simply the component in the fibre t of the multiplication µ of the monad SLC. This is an
example of a module whose underlying functor is not an endofunctor.
2.2. Constructions on monads and modules. We present some instances of modules
which we will use in the next section. They were previously defined in Zsidó’s thesis [Zsi10]
and works of Hirschowitz and Maggesi [HM07, HM10].
Definition 2.22 (Tautological Module). Given the monad (C, T ), we call tautological module
the module (T, µT ) : (C, T ) → (C, Id).
Definition 2.23 (Constant and terminal module). Given a monad (C, T ) and a category D
with an object d ∈ D, the constant functor Fd : C → D mapping any object of C to d ∈ D
and any morphism to the identity on d yields a module
(Fd , id) : (C, T ) → (D, Id) .
In particular, if D has a terminal object 1D , then the constant module (F1D , id) is terminal
in Mod(T, D).
Given a morphism of monads from T to T 0 , and T 0 –module gives rise to a T –module:
Definition 2.24 (Pullback module). Let (C, T ) and (D, T 0 ) be monads over C and D,
respectively. Given a morphism of monads (F, γ) : (C, T ) → (D, T 0 ) and a T 0 -module (M, σ)
with codomain category E, we call pullback of M along (F, γ) the composed T –module
(F, γ)∗ (M, σ) := (M, σ) ◦ (F, γ) .
The pullback operation extends to morphisms of modules and is functorial.
Definition 2.25 (Induced module morphism). With the same notation as in the previous
example, the monad morphism (F, γ) induces a morphism of T –modules — which we call γ
as well —
γ : (F, id) ◦ (T, µT ) ⇒ (F, γ)∗ (T 0 , µT 0 )
as in
(C, T )
(T,µT )
(F,γ)
z
γ
(C, Id)
(F,id)
$
%
+3
(D, T 0 )
y
(T 0 ,µT 0 )
(D, Id).
Indeed, the natural transformation γ verifies the corresponding diagram, as a consequence
of the diagrams for monad morphisms it verifies.
12
B. AHRENS
Definition 2.26 (Products). Suppose the category D is equipped with a product. Given any
monad (C, T ), the product of D lifts to a product on the category Mod(T, D) of T –modules
with codomain D.
2.3. Modules on Typed Sets. When considering constructors that are indexed by object
types, such as App and Abs, we will also consider monads and modules over categories of
typed sets where the set of types is pointed (multiple times):
Definition 2.27 (Pointed index sets). Given a category C, a set T and a natural number
n, we denote by [T, C]n the category with, as objects, diagrams of the form
t
V
n→T →C ,
written (V, t1 , . . . , tn ) with ti := t(i). A morphism h to another such (W, t) with the same
pointing map t is given by a morphism h : V → W in [T, C]. Any functor F : [T, C] → [T, D]
extends to Fn : [T, C]n → [T, D]n via
Fn (V, t1 , . . . , tn ) := (F V, t1 , . . . , tn ) .
Remark 2.28. The category [T, C]n consists of T n copies of [T, C], which do not interact.
Due to the “markers” (t1 , . . . , tn ) we can act differently on each copy, cf. e.g. Defs. 2.34 and
2.37. The reason why we consider categories of this form is explained in Rem. 3.17.
We generalize retyping functors to such categories with pointed indexing sets. When
changing types according to a map of types g : T → U , the markers must be adapted as well:
Definition 2.29. Given a map of sets g : T → U , by postcomposing the pointing map with
g, the retyping functor generalizes to the functor
~g (n) : [T, C]n → [U, C]n ,
(V, t) 7→ ~g V, g∗ (t) ,
where g∗ (t) = t ◦ g : n → U .
Finally there is also a category where families of sets over different indexing sets are
mixed together:
Definition 2.30. Given a category C, we denote by T C the category where an object is a
pair (T, V ) of a set T and a family V ∈ [T, C] of objects of C indexed by T . A morphism to
another such (T 0 , W ) is given by a map g : T → T 0 and a morphism V → W ◦ g in [T, C],
that is, family of morphisms, indexed by T ,
ht : Vt → Wg(t) ,
in the category C.
Let C have an initial object, denoted by 0C . Given n ∈ N, we call n̂ = (n, k 7→ 0C ) the
element T C that associates to any 1 ≤ k ≤ n the initial object of C. We call T Cn the slice
category n̂ ↓ T C. An object of this category consists of an object (T, V ) ∈ T C whose indexing
set “of types” T is pointed n times, written (T, V, t1 , . . . , tn ). We call T Un : T Cn → Set the
forgetful functor associating to any pointed family (T, V, t1 , . . . , tn ) the indexing set T , in
particular for the case that C is the category Set of sets.
Remark 2.31 (Picking out Sorts). Let 1 : T Cn → Set denote the constant functor which
maps objects to the terminal object 1Set of the category Set. A natural transformation
τ : 1 → T Un associates to any object (T, V, t) of the category T Cn an element of T .
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
13
Notation 2.32. Given a natural transformation τ : 1 → T Un as in Rem. 2.31, we write
τ (T, V, t) := τ (T, V, t)(∗) ∈ T ,
i.e. we omit the argument ∗ ∈ 1Set of the singleton set.
2.3.1. Derivation. Roughly speaking, a binding constructor makes free variables disappear.
Its input are hence terms “with (one or more) additional free variables” compared to the
output, i.e. terms in an extended context. Derivation formalizes context extension. Let T
be a set and u ∈ T an element of T . We define D(u) to be the object of [T, Set] such that
D(u)(u) = {∗} and
D(u)(t) = ∅ for t 6= u .
We enrich the object V of [T, Set] with respect to u by setting
V ∗u := V + D(u) ,
that is, we add a fresh variable of type u to the context V . This yields a monad (_)∗u on
[T, Set]. Moreover, given any monad P on [T, Set], we equip the functor V 7→ V ∗u with
a structure of an endomorphism on P : on a typed set V its natural transformation γ is
defined as the coproduct map
γV := [P (inl), x 7→ η inr(∗) ] : (P V )∗u → P (V ∗u ) ,
where [inl, inr] = id : V
∗u
→V
(2.1)
∗u .
Remark 2.33. In case the monad P denotes terms over sets of free variables as in Ex. 2.9,
the map γV defined in Eq. (2.1) sends a term t ∈ P V in a context V to its image in an
extended context V ∗u , and the additional variable of type u to the term (in context V ∗u )
consisting of just this variable.
More generally, we derive with respect to a natural transformation
τ : 1 ⇒ T Un : T Setn → Set .
Such τ associates to any V ∈ T Setn with a set of types T an object type t ∈ T .
Definition 2.34 (Derived Module). Let τ : 1 → T Un be a natural transformation. Given a
set T and a monad P on [T, Set]n , the functor (_)∗τ : V 7→ V ∗τ (V ) is given the structure of
a morphism of monads as in Eq. (2.1). Given any P –module M , we call derivation of M
with respect to τ the module M τ := M ◦ (_)∗τ .
Remark 2.35. In the preceding definition the natural transformation τ : 1 → T Un supplies
more data than necessary, since we only evaluate it on families of sets indexed by the
fixed set T . However, in the next section we will derive different modules — each defined
on a category [T, Set]n with varying sets T — with respect to one and the same natural
transformation τ .
Example 2.36 (Ex. 2.13 continued). We consider SLC (cf. Ex. 2.13) as the tautological
module over itself. Given any element s ∈ TSLC , the derived module with respect to s,
SLCs : V 7→ SLC(V ∗s ) ,
denotes the (typed) set of terms of SLC with variables in an extended context V ∗s .
14
B. AHRENS
2.3.2. Fibres. Given a set family V indexed by a (nonempty) set T , we sometimes need to
pick the set of elements “of type u ∈ T ”, that is, the set V (u) associated to u ∈ T . Given a
monad P on a category C and a P –module M towards [T, Set], we define the fibre module of
M with respect to u ∈ T to be the module which associates the fibre M (c)(u) to any object
c ∈ C. This construction is expressed via postcomposition with a particular module:
we define the fibre with respect to u ∈ T to be the monad morphism
(_)(u), id : ([T, Set], Id) → (Set, Id)
over the functor V 7→ V (u). Postcomposition of the module M with this module then
precisely yields the fibre module [M ]u of M with respect to u ∈ T .
Analogously to derivation we define the fibre more generally with respect to a natural
transformation:
Definition 2.37 (Fibre Module). Let the natural transformation τ be as in Def. 2.34. We
call fibre with respect to τ the monad morphism
(_)τ : V 7→ V (τV ) : ([T, Set]n , Id) → (Set, Id)
over the functor V 7→ VτV . Given a module M towards [T, Set]n (over some monad P ), we
call the fibre module of M with respect to τ the module [M ]τ := (_)τ ◦ M .
Example 2.38 (Ex. 2.13 continued). We consider SLC as the tautological module over
itself. Given any element t ∈ TSLC , the fibre module with respect to t,
[SLC]t : V 7→ SLC(V )t ,
associates to any context V the set of simply–typed lambda terms of type t with variables
in V .
3. Signatures & Representations
A simply–typed language is given by a pair (S, Σ) of signatures: an algebraic signature S
specifying the types of the language, and a term–signature Σ which specifies terms that
are typed over the set of object types associated to S. We call typed signature a pair (S, Σ)
consisting of an algebraic signature S and a term–signature Σ over S.
3.1. Signatures for Types. Algebraic signatures were already considered by Birkhoff
[Bir35]. An example of (untyped) algebraic signature is given in the introduction. We review
the general definition:
Definition 3.1 (Algebraic Signature). An algebraic signature S is a family of natural
numbers, i.e. a set JS and a map (carrying the same name as the signature) S : JS → N.
For j ∈ JS and n ∈ N, we also write j : n instead of j 7→ n. An element of J resp. its image
under S is called an arity of S.
To any algebraic signature we associate a category of representations. We call representation of S any set U equipped with operations according to the signature S. A morphism of
representations is a map between the underlying sets that is compatible with the operations
on either side in a suitable sense. Representations and their morphisms form a category. We
give the formal definitions:
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
15
Definition 3.2 (Representation of an Algebraic Signature). A representation R of an
algebraic signature S is given by
• a set X and
• for each j ∈ JS , an operation j R : X S(j) → X.
In the following, given a representation R, we write R also for its underlying set.
Example 3.3 (Algebraic Signature of Ex. 2.13). The algebraic signature of the types of
the simply–typed lambda calculus is given by
SSLC := {∗ : 0 ,
(
) : 2} .
Example 3.4. The language PCF [Plo77, HO00] is a simply–typed lambda calculus with a
fixed point operator and arithmetic constants. Let J := {ι, o, (⇒)}. The signature of the
types of PCF is given by the arities
SPCF := {ι : 0 ,
o:0 ,
(⇒) : 2} .
A representation T of SPCF is given by a set T and three operations,
ιT : T ,
oT : T ,
(⇒)T : T × T → T .
Definition 3.5 (Morphisms of Type–Representations). Given two representations T and
U of the algebraic signature (J, S), a morphism from T to U is a map f : T → U on the
underlying sets such that for any arity j ∈ J with S(j) = n we have
f ◦ jT = jU ◦ f n .
Representations of S and their morphisms form a category.
Example 3.6 (Ex. 3.4 continued). Given two representations T and U of SPCF , a morphism
from T to U is a map f : T → U such that, for any s, t ∈ T ,
f (ιT ) = ιU ,
f (oT ) = oU
and
T
f (s ⇒ t) = f (s) ⇒U f (t) .
Next we prove that for any algebraic signature S, its category of representations has an
initial object, whose underlying set Ŝ consists of the types freely generated by the signature.
In particular, by initiality we obtain, for any representation R of S in a set U , a map from
Ŝ to U .
Lemma 3.7. Let (J, S) (or S for short) be an algebraic signature. The category of representations of S has an initial object Ŝ.
Proof. We cut the proof into small steps:
• In a type–theoretic setting the set — also called Ŝ — which underlies the initial representation Ŝ is defined as an inductive set with a family of constructors indexed by
JS :
Ŝ ::= C : ∀j ∈ J, Ŝ S(j) → Ŝ .
That is, for each arity j ∈ J, we have a constructor Cj : Ŝ S(j) → Ŝ.
16
B. AHRENS
• For each arity j ∈ J, we must specify an operation j Ŝ : Ŝ S(j) → Ŝ. We set
j Ŝ := Cj : Ŝ S(j) → Ŝ ,
that is, the representation j Ŝ of an arity n = S(j) is given precisely by its corresponding
constructor.
• Given any representation R of S, we specify a map iR : Ŝ → R between the underlying
sets by structural recursion:
iR Cj (a) := j R (iR )S(j) (a) ,
iR : Ŝ → R ,
for a ∈ Ŝ S(j) . That is, the image of a constructor function Cj maps recursively on the
image of the corresponding representation j R of R.
• We must prove that iR is a morphism of representations, that is, that for any j ∈ J with
S(j) = n,
iR ◦ j Ŝ = j R ◦ (iR )n .
Replacing j Ŝ by its definition yields that this equation is precisely the specification of iR ,
see above.
• It is the diagram of Def. 3.5 which ensures unicity of iR ; since any morphism of representations i0 : Ŝ → R must make it commute, one can show by structural induction that
i0 = iR . More precisely:
i0 (Cj (a)) = i0 (Cj (a1 , . . . , aS(j) )) = j R (i0 (a1 ), . . . , i0 (aS(j) ))
i0 (ak )=iR (ak )
=
R
= j (iR (a1 ), . . . , iR (aS(j) )) = iR (Cj (a)) .
Example 3.8 (Ex. 3.4 continued). The set TPCF underlying the initial representation of
the algebraic signature SPCF is given by
TPCF ::=
ι | o | TPCF ⇒ TPCF .
For any other representation R of SPCF the initial morphism iR : TPCF → R is given by the
clauses
iR (ι) = ιR
iR (o) = oR
iR (s ⇒ t) = iR (s) ⇒R iR (t) .
3.2. Signatures for Terms. We consider the simply–typed lambda calculus as specified
in Ex. 1.1. Its terms could be specified by the signature:
{abss,t := ([s], t) → (s
t) ,
apps,t := ([], s
t), ([], s) → t}s,t∈TSLC .
(3.1)
whose meaning is as follows: an arrow → separates domain and codomain data. The domain
data specifies the input type; it consists of a list, where each list item corresponds to one
argument. Each list item is itself a pair of a list — specifying the type of the variables bound
in the corresponding argument — and an object type — the type of the argument. The
codomain data specifies the output type of the associated constructor. This viewpoint is
sufficient when considering models of SLC over the set TSLC of types of SLC. Indeed, Zsidó
[Zsi10] defines signatures for terms precisely as in the above example.
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
17
If, however, we want to consider models of SLC over varying sets of types, then the above
point of view, with its tight dependence on the initial set of types TSLC , is not adequate any
more. Instead, we would like to specify the signature of SLC like this:
{abs := ([1], 2) → (1
2) ,
app := ([], 1
2), ([], 1) → 2} .
(3.2)
What is the intended meaning of such a signature? For any representation T of SSLC , the
variables 1 and 2 range over elements of T . In this way the number of abstractions and
applications depends on the representation T of SSLC : intuitively, a model of the above
signature of Eq. (3.2) over a representation T of TSLC has T 2 abstractions and T 2 applications
— one for each pair of elements of T . As an example, for the final representation of SSLC in
the singleton set, one obtains only one abstraction and one application morphism.
In summary, to account for type variables in an arity, we consider arities of higher degree,
where the degree of an arity denotes the number of (distinct) type variables. For instance,
the arities abs and app of Eq. (3.2) are of degree 2.
3.2.1. Term Signatures, syntactically. In this section we give a syntactic characterization of
arities over a fixed algebraic signature S for types as in Def. 3.1.
Definition 3.9 (Type of Degree n). For n ≥ 1, we call types of S of degree n the elements
of the set S(n) of types associated to the signature S with free variables in the set {1, . . . , n}.
We set S(0) := Ŝ. Formally, the set S(n) may be obtained as the initial representation of
the signature S enriched by n nullary arities.
Types of degree n are used to form classic arities of degree n:
Definition 3.10 (Classic Arity of Degree n). A classic arity for terms over the signature S
for types of degree n is of the form
([t1,1 , . . . , t1,m1 ], t1 ), . . . , ([tk,1 , . . . , tk,mk ], tk ) → t0 ,
(3.3)
where ti,j , ti ∈ S(n). More formally, a classic arity of degree n over S is a pair consisting of
an element t0 ∈ S(n) and a list of pairs. where each pair itself consists of a list [ti,1 , . . . , ti,mi ]
of elements of S(n) and an element ti of S(n).
A classic arity of the form given in Eq. (3.3) denotes a constructor — or a family of
constructors, for n ≥ 1 — whose output type is t0 , and whose k inputs are terms of type ti ,
respectively, in each of which variables of type according to the list [ti,1 , . . . , ti,mi ] are bound
by the constructor.
Remark 3.11. For an arity as given in Eq. 3.3 we also write
t1,1 ,...,t1,m1
[Θn
tk,1 ,...,tk,mk
]t1 × . . . × [Θn
]tk → [Θn ]t0 .
(3.4)
Examples of arities — besides the example of Eq. (3.2) — are also given in Sec. 4.
Remark 3.12 (Implicit Degree). Any arity of degree n ∈ N as in Def. 3.10 can also be
considered as an arity of degree n + 1. We denote by S(ω) the set of types associated to the
type signature S with free variables in N. Then any arity of degree n ∈ N can be considered
as an arity built over S(ω). Conversely, any arity built over S(ω) only contains a finite set
of free variables in N, and can thus be considered to be an arity of degree n for some n ∈ N.
In particular, by suitable renaming of free variables, there is a minimal degree for any arity
built over S(ω). We can thus omit the degree — e.g., the lower inner index n in Disp. 3.4
18
B. AHRENS
—, and specify any arity as an arity over S(ω), if we really want to consider this arity to be
of minimal degree. Otherwise we must specify the degree explicitly.
3.2.2. Term Signatures, semantically. We now attach a meaning to the purely syntactically
defined arities of Sec. 3.2.1. More precisely, we define arities as pairs of functors over suitable
categories. Afterwards we restrict ourselves to a specific class of functors, yielding arities
which are in one–to–one correspondence to — and thus can be compactly specified via —
the syntactically defined classic arities of Sec. 3.2.1. Accordingly, we call the restricted class
of arities also classic arities.
At first, in Rem. 3.13, we present an alternative characterization of algebraic arities.
This alternative point of view is then adapted to allow for the specification of arities for
terms.
Remark 3.13. We reformulate the definition of algebraic arities and their representations:
an algebraic arity j : n associates, to any set X, the set dom(j, X) := X n , the domain set.
A representation R of this arity j in a set X then is given by a map j R : X n → X. More
formally, the domain set is given via a functor dom(j) : Set → Set which associates to any
set X the set X n . Similarly, we might also speak of a codomain functor for any arity, which
— for algebraic arities — is given by the identity functor. A representation R of j in a set X
then is given by a morphism
j R : dom(j)(X) → cod(j)(X) .
We take this perspective in order to define arities and signatures for terms: given an
algebraic signature S for types, an arity α of degree n for terms over S is a pair of functors
(dom(α), cod(α)) associating two P –modules dom(α)(P ) and cod(α)(P ), each of degree n,
to any suitable monad P . A suitable monad here is a monad P on some category [T, Set]
where the set T is equipped with a representation of S. We call such a monad an S–monad.
A representation R of α in an S–monad P is a module morphism
αR : dom(α)(P ) → cod(α)(P ) .
As we have seen in Ex. 1.1, constructors can in fact be families of constructors indexed n
times by object type variables. We specify such a constructor via an arity of higher degree,
where the degree n ∈ N of the arity corresponds to the number of object type variables of
its associated constructor.
For any signature for types S, we define a category of monads on typed sets where the
indexing set is equipped with a representation of S:
Definition 3.14 (S–Monad). Given an algebraic signature S, the 2-category S-Mnd of
S–monads is defined as the 2-category whose objects are pairs (T, P ) of a representation T
of S and a monad P : [T, Set] → [T, Set]. A morphism from (T, P ) to (T 0 , P 0 ) is a pair (g, f )
of a morphism of S–representations g : T → T 0 and a monad morphism f : P → P 0 over the
retyping functor ~g . Transformations are the transformations of Mndcolax .
Given n ∈ N, we write S-Mndn for the 2-category whose objects are pairs (T, P ) of a
representation T of S and a monad P over [T, Set]n . A morphism from (T, P ) to (T 0 , P 0 )
is a pair (g, f ) of a morphism of S–representations g : T → T 0 and a monad morphism
f : P → P 0 over the retyping functor ~g (n) (cf. Def. 2.29).
We call IS,n : S-Mndn → Mndcolax the functor which forgets the representation of S.
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
19
We define a “large category of modules” in which modules over different S–monads are
mixed together:
Definition 3.15 (Large Category of Modules). Given a natural number n ∈ N, an algebraic
signature S and a category D, we call LModn (S, D) the colax comma category IS,n ↓ IdD .
An object of this category is a pair (P, M ) of a monad P ∈ S-Mndn and a P –module with
codomain D. A morphism to another such (Q, N ) is a pair (f, h) of an S–monad morphism
f : P → Q in S-Mndn and a transformation h : M → f ∗ N :
M
P
h
&
8 IdD .
N ◦f
Definition 3.16 (Half–Arity over S (of degree n)). Given an algebraic signature S and
n ∈ N, we call half–arity over S of degree n a functor
α : S-Mnd → LModn (S, Set) .
Taking into account Rem. 3.17, this means that a half–arity of degree n associates to any
S–monad R — with representation of S in a set T — a family of R–modules indexed n
times by T .
Remark 3.17 (Module on pointed Category ∼
= Family of Modules). Let C and D be
categories, let T be a set and R be a monad on [T, C]. Suppose n ∈ N, and let D be
a category. Then modules over Rn with codomain D correspond precisely to families of
R–modules indexed by T n with codomain D by (un)currying.
More precisely, let M be an Rn –module. Given t ∈ T n , we define an R–module Mt by
Mt (c) := M (c, t) .
Module substitution for Mt is given, for f ∈ [T, C](c, Rd), by
ς Mt (f ) := ς M (f )
where we use that we also have f ∈ [T, C]n ((c, t), (Rd, t)) according to Def. 2.27. Going the
other way round, given a family (Mt )t∈T n , we define the Rn –module M by
M (c, t) := Mt (c) .
Given a morphism f ∈ [T, C]n ((c, t), (Rd, t)), we also have f ∈ [T, C](c, Rd) and define
ς M (f ) := ς Mt (f ) .
We recall that morphisms in [T, C]n are only between families with the same points t.
The remark extends to morphisms of modules; indeed, a morphism of modules α : M →
N on pointed categories corresponds to a family of morphisms (αt : Mt → Nt )t∈T n between
the associated families of modules.
We restrict our attention to half–arities which correspond, in a sense made precise below,
to the syntactically defined arities of Def. 3.10. The basic brick is the tautological module of
degree n:
Definition 3.18. Given n ∈ N, any monad R on the category [T, Set] induces a monad Rn
on [T, Set]n with object map (V, t1 , . . . , tn ) 7→ (RV, t1 , . . . , tn ). To any S–monad R we hence
associate the tautological module of Rn ,
Θn (R) := (Rn , Rn ) ∈ LModn (S, [T, Set]n ) .
20
B. AHRENS
This construction extends to a functor.
Let us consider the signature SSLC of types of SLC. In the syntactically defined arities
(cf. Eq. 3.2) we write terms like 1
2. We now give meaning to such a term: intuitively, the
term 1
2 should associate, to a family (T, V, t1 , t2 ) with V a T –indexed family of sets and
t1 , t2 ∈ T , the element t1
t2 . The set T should thus come equipped with a representation
of SSLC in order to interpret the arrow .
More formally, such a term is interpreted by a natural transformation over a specific
category, whose objects are triples of a representation T of SSLC , a family of sets indexed by
(the set) T and “markers” (t1 , t2 ) ∈ T 2 .
We go back to considering an arbitrary signature S for types. The following are the
corresponding basic categories of interest:
Definition 3.19 (SSetn ). We define the category SSetn to be the category an object of
which is a triple (T, V, t) where T is a representation of S, the object V ∈ [T, Set] is a
T –indexed family of sets and t is a vector of elements of T of length n. We denote by
SUn : SSetn → Set the functor mapping an object (T, V, t) to the underlying set T . We
have a forgetful functor SSetn → T Setn which forgets the representation structure. On the
other hand, any representation T of S in a set T gives rise to a functor [T, Set]n → SSetn ,
which “attaches” the representation structure.
The meaning of a term s ∈ S(n) as a natural transformation
s : 1 ⇒ SUn : SSetn → Set
is now given by recursion on the structure of s:
Definition 3.20 (Canonical Natural Transformation). Let s ∈ S(n) be a type of degree n.
Then s denotes a natural transformation
s : 1 ⇒ SUn : SSetn → Set
defined recursively on the structure of s as follows: for s = α(a1 , . . . , ak ) the image of a
constructor α ∈ S we set
s(T, V, t) = α(a1 (T, V, t), . . . , ak (T, V, t))
and for s = m with 1 ≤ m ≤ n we define
s(T, V, t) = t(m) .
We call a natural transformation of the form s ∈ S(n) canonical.
Canonical natural transformations are used to build classic half–arities; they indicate
context extension (derivation) and selection of specific object types (fibre):
Definition 3.21 (Classic Half–Arity over S). We give some examples of half–arities over
a signature S and associate short names to them. At the same time the following clauses
define an inductive set of classic half–arities, to which we will restrict our attention.
• The constant functor
∗ : R 7→ 1 ,
where 1 denotes the terminal module, is a classic half–arity.
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
21
• For any canonical natural transformation τ : 1 → T Un , the point-wise fibre module with
respect to τ of the tautological module Θn : R 7→ (Rn , Rn ) is a classic half–arity of degree
n,
[Θn ]τ : S-Mnd → LModn (S, Set) , R 7→ [Rn ]τ .
• Given any (classic) half–arity M : S-Mnd → LModn (S, Set) of degree n and a canonical
natural transformation τ : 1 → T Un , the point-wise derivation of M with respect to τ is a
(classic) half–arity of degree n,
M τ : S-Mnd → LModn (S, Set) ,
τ
R 7→ M (R)
.
τ
Here M (R) really means derivation of the module, i.e. derivation in the second component of M (R).
• For a half–arity M , let Mi : R 7→ πi M (R) denote the i–th projection. Given two (classic)
half–arities M and N of degree n, which coincide pointwise on the first component, i.e.
such that M1 = N1 . Then their product M × N is again a (classic) half–arity of degree n.
Here the product is really the pointwise product in the second component, i.e.
M × N : R 7→ M1 (R), M2 (R) × N2 (R) .
Remark 3.22. Classic half–arities correspond precisely to our needs: products are needed
when a constructor takes multiple arguments, and a derived module corresponds to an
argument in which a variable is to be bound. The fibre restricts the terms under consideration
to a specific object type.
Definition 3.23 (Weighted Set). A weighted set J is a set J together with a map d : J → N.
An arity of degree n ∈ N for terms over an algebraic signature S is a pair of functors —
called half–arities, since two of them constitute an arity — from S–monads to modules in
LModn (S, Set). The first component dom(α) of such an arity α = (dom(α), cod(α)) denotes
the domain, or arguments, of a constructor, whereas the second, cod(α), determines the
output type. The degree n of an arity denotes the number of object type arguments of its
associated constructor. As an example, the arities of Abs and App of Ex. 2.13 are of degree
2 (cf. Ex. 3.26).
Definition 3.24 (Term–Arity, Signature over S). A classic arity α over S of degree n is a
pair
α = dom(α), cod(α)
of half–arities over S of degree n such that
• dom(α) is classic and
• cod(α) is of the form [Θn ]τ for some natural transformation τ as in Def. 3.21.
We write dom(α) → cod(α) for the arity α, and
dom(α, R) := dom(α)(R)
(and similar for the codomain functor cod). Any classic arity is thus of the form given in Eq.
3.3. Given a weighted set (J, d), a term–signature Σ over S indexed by (J, d) is a J-family Σ
of classic arities over S, the arity Σ(j) being of degree d(j) for any j ∈ J.
Definition 3.25 (Typed Signature). A typed signature is a pair (S, Σ) consisting of an
algebraic signature S and a term–signature Σ (indexed by some weighted set) over S.
22
B. AHRENS
Example 3.26 (SLC, Ex. 2.13 continued). The terms of the simply typed lambda calculus
over the type signature of Ex. 3.3 is given by the classic (cf. Def. 3.21) arities
abs : [Θ1 ]2 → [Θ2 ]1
app : [Θ]1
2
2
,
× [Θ]1 → [Θ]2
,
both of which are of degree 2 — we use the convention of 3.12. The outer lower index and
the exponent are to be interpreted as variables, ranging over object types. They indicate
the fibre (cf. Def. 2.37) and derivation (cf. Def. 2.34), respectively, in the special case where
the corresponding natural transformation is given by a natural number as in Def. 3.20.
Those two arities can in fact be considered over any algebraic signature S with an arrow
constructor, in particular over the signature SPCF (cf. Ex. 3.28).
Remark 3.27. Note that in Ex. 3.26 we do not need to explicitly specify an arity for the
Var term constructor in order to obtain the simply–typed lambda calculus as presented in
Ex. 1.1. Indeed, in our approach every model is by definition (cf. Def. 3.29) equipped with a
corresponding operation — the unit of the underlying monad.
Example 3.28 (Ex. 3.8 continued). We continue considering PCF. The signature SPCF for
its types is given in Ex. 3.4. The term–signature of PCF is given by an arity for abstraction
and an arity for application, each of degree 2, an arity (of degree 1) for the fixed point
operator, and one arity of degree 0 for each logic and arithmetic constant — some of which
we omit:
abs : [Θ1 ]2 → [Θ]1⇒2 ,
app : [Θ]1⇒2 × [Θ]1 → [Θ]2 ,
Fix : [Θ]1⇒1 → [Θ]1 ,
Z : ∗ → [Θ]ι
S : ∗ → [Θ]ι⇒ι
condι : ∗ → [Θ]o⇒ι⇒ι⇒ι
T, F : ∗ → [Θ]o
..
.
Our presentation of PCF is inspired by Hyland and Ong’s [HO00], who — similarly to
Plotkin [Plo77] — consider, e.g., the successor as a constant of arrow type. As an alternative,
one might consider the successor as a constructor expecting a term of type ι as argument,
yielding a term of type ι. For our purpose, those two points of view are equivalent.
3.3. Representations. A representation of a typed signature (S, Σ) is a pair (U, P ) given
by a representation U of the signature S in a set — also called U — and a representation P
of the term–signature Σ in a monad — also called P — over the category [U, Set]. Such a
representation of Σ consists of a morphism in a suitable category for each arity of Σ — the
analogue of the maps Z and S from the introductory example:
Definition 3.29 (Representation of a Signature over S). Let (S, Σ) be a typed signature.
A representation R of (S, Σ) is given by
• an S–monad P and
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
23
• for each arity α of Σ, a morphism (in the large category of modules)
αR : dom(α, P ) → cod(α, P ) ,
such that π1 (αR ) = idP .
In the following we also write R for the S–monad underlying the representation R.
Suppose we have two such representations P and R of (S, Σ). What is a suitable
definition of morphism from the first to the latter? Such a morphism is given by a pair
consisting of a morphism of the underlying type representations g : S P → S R , and a monad
morphism over the retyping functor associated to (the carrier of) g between the monads
underlying P and R. In this way the monad morphism maps elements “of type” t ∈ S P to
elements “of type” g(t) ∈ S R , and is thus compatible with the translation g of types. Note
that these definitions are already integrated into the definition of S–monads. The missing
piece is that the monad morphism should be compatible with the term representations of P
and R:
Definition 3.30 (Morphism of Representations). Given representations P and R of a typed
signature (S, Σ), a morphism of representations f : P → R is given by a morphism of
S–monads f : P → R, such that for any arity α of S the following diagram of module
morphisms commutes:
dom(α, P )
dom(α,f )
αP
/ cod(α, P )
dom(α, R)
cod(α,f )
/ cod(α, R).
αR
Remark 3.31. Taking a 2–categoric perspective, the above diagram reads as an equality of
2-cells
dom(α,P )
αP
P
cod(α,P )
cf
f ∗ cod(α,R)
dom(α,P )
/ IdSet
@
df
=
P
f∗
dom(α,R)
∗ R
f α
/ IdSet ,
@
f ∗ cod(α,R)
where we write df and cf instead of dom(α, f ) and cod(α, f ), respectively.
The diagram of Def. 3.30 lives in the category LModn (S, Set) — where n is the degree
of α — where objects are pairs (P, M ) of a S–monad P of S-Mndn and a module M over
P . The above 2–cells are morphisms in the category Mod(Pn , Set), obtained by taking the
second projection of the diagram of Def. 3.30. Note that for easier reading, we leave out the
projection function and thus write dom(α, R) for the Rn –module of dom(α, R), i.e. for its
second component, and similar elsewhere.
Representations of (S, Σ) and their morphisms form a category.
Remark 3.32. We obtain Zsidó’s category of representations [Zsi10, Chap. 6] by restricting
ourselves to representations of (S, Σ) whose type representation is the initial one. More,
24
B. AHRENS
precisely, a signature (S, Σ) maps to a signature, say, Z(S, Σ) over the initial set of sorts Ŝ in
the sense of Zsidó [Zsi10, Chap. 6], obtained by unbundling each arity of higher degree into
a family of arities of degree 0. For instance, the signature of Ex. 3.26 maps to the signature
Apps,t : [()s
t, ()s] −→ t , Abss,t : [(s)t] −→ s
t
s,t∈TSLC
.
Representations of this latter signature in Zsidó’s sense then are in one–to–one correspondence
to representations of the signature of Ex. 3.26 over the initial representation Ŝ of sorts, via
the equivalence explained in Rem. 3.17.
3.4. Initiality.
Theorem 3.33. For any typed signature (S, Σ), the category of representations of (S, Σ)
has an initial object.
Proof. The proof consists of the following steps:
(1) find the initial representation Ŝ of the type signature S;
(2) define the monad STS of terms specified by Σ on the category [Ŝ, Set];
(3) equip the S–monad STS with a representation structure of Σ, yielding a representation
Σ̂ of (S, Σ);
(4) for any representation R of (S, Σ), give a morphism of representations iR : Σ̂ → R;
(5) prove unicity of iR .
We go through these points:
(1) We have already established (cf. Lem. 3.7) that there is an initial representation of sorts,
which we call Ŝ. Its underlying set is called Ŝ as well.
(2) The term monad we associate to (S, Σ) is the same as Zsidó’s [Zsi10, Chap. 6] in the
sense of Rem. 3.32, i.e. it is the term monad associated to Z(S, Σ). The construction
of this monad in a set–theoretic setting is described in Zsidó’s thesis. We will give its
definition in a type–theoretic setting.
In the following the natural transformations τi are in fact vectors of multiple transformations like those in Rem. 2.31 (see also Def. 2.34), iterated by successive composition.
Furthermore we make use of the simplified notation as introduced in Not. 2.32.
We construct the monad which underlies the initial representation of (S, Σ),
STS : [Ŝ, Set] → [Ŝ, Set] .
It associates to any set family of variables V ∈ [Ŝ, Set] an inductive set of terms with
the following constructors:
• for every classic arity (of degree n)
α = [Θτn1 ]σ1 × . . . × [Θτnm ]σm → [Θn ]σ
(3.5)
we have a family of constructors indexed n times by t = (t1 , . . . , tn ) as well as by the
context V ∈ [Ŝ, Set]:
αt (V ) : STSτ1 (V,t) (V )σ1 (V,t) × . . . × STSτm (V,t) (V )σm (V,t) → STS(V )σ(V,t)
• a family of constructors
Var(V )t : Vt → STS(V )t
indexed by contexts and the set Ŝ of sorts.
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
25
The monadic structure is, accordingly, defined in the same way as in [Zsi10], by variables–
as–terms — using the constructor Var — and flattening.
(3) The representation structure on the monad STS is defined by currying, and corresponds
to Zsidó’s: given an arity α of degree n in Σ, we must specify a module morphism
αΣ̂ : dom(α, STS) → cod(α, STS) ,
where dom(α, STS) and dom(α, STS) are modules in Mod(STSn , Set). We define
αΣ̂ (V, t)(a) := αt (V )(a) ,
that is, the image under the constructor α from the definition of the monad STS. This
yields a morphism of modules α of degree n; note that according to Rem. 3.17 it would
be equivalent to specify a family αtΣ̂ of module morphisms of suitable type, indexed by
t, which is actually done by Zsidó.
(4) Given any other representation R over a set of sorts T , initiality of Ŝ gives a “translation
of sorts” g : Ŝ → T .
The morphism i : STS → R on terms is defined by structural recursion. Unfolding
the definition of colax monad morphism, we need to define, for any context V ∈ [Ŝ, Set],
a map of type
iV : ∀ t0 ∈ T, ~g (STS(V ))t0 → R(~g V )t0 .
Via the adjunction of Rem. 2.12 we equivalently define a map i as a family
iV : ∀ t ∈ Ŝ, STS(V )t → R(~g V )g(t) .
Let a ∈ STS(V )t be a term. In case a = Var(V )t (v) is the image of a variable v ∈ Vt ,
we map it to
iV (Var(V )t (v)) := η R (~g V )(g(t))(ctype(v)) .
Otherwise the term a = αt (V )(a1 , . . . , ak ) ∈ STS(V )σ(V,t) is mapped to
iV αt (V )(a1 , . . . , ak ) := αR (~g (n)(V, t)) i(a1 ), . . . , i(ak ) .
(3.6)
This map is well–typed: note that ~g (n)(V, t) = (~g V, g∗ (t)) by definition (Def. 2.29) and
~g (n)((V, t)τ ) = (~g V, g∗ (t))τ , i.e. context extension and retyping permute.
The axioms of monad morphisms, i.e. compatibility of this map with respect to
variables–as–terms and flattening are easily checked: the former is a direct consequence
of the definition of i on variables, and the latter is proved by structural induction. This
definition yields a morphism of representations; consider the arity α of Σ. For this arity,
the commutative diagram of Def. 3.30 informally reads as follows: one starts in the
upper–left corner with a tuple of terms, say, (a1 , . . . , ak ) of STS. Taking the upper–right
path corresponds to the translation of the image of this tuple under the map αΣ̂ , i.e.
under the constructor α of STS. The lower–left path corresponds to the image under
the module morphism αR of the translated tuple (i(a1 ), . . . , i(ak )). The diagram thus
precisely states the equality of Eq. (3.6). We thus establish that i is (the carrier of) a
morphism of representations (g, i) : (Ŝ, Σ̂) → R.
(5) Unicity of the morphism i : (Ŝ, Σ̂) → R is proved making use of the commutative
diagram of Def. 3.30. Suppose that (g 0 , i0 ) : (Ŝ, Σ̂) → R is a morphism of representations.
We already know that g = g 0 by initiality of Ŝ. By structural induction on the terms of
26
B. AHRENS
STS we prove that i = i0 : using the same notation as above, for a = αt (V )(a1 , . . . , ak )
we have
i(ai )=i0 (ai )
i0 (a) = αR i0 (a1 ), . . . , i0 (ak )
=
αR (i(a1 ), . . . , i(ak )) = i(a) .
In case a = Var(v) is a variable, considered as a term, the fact that both i and i0 are
monad morphisms ensures that i(Var(v)) = i0 (Var(v)) = η~gRV (ctype(v)). Thus we have
proved i = i0 .
An application of this theorem is the specification of translations from one language (Ŝ, Σ̂)
— associated to a typed signature (S, Σ) — to another (Ŝ 0 , Σ̂0 ). We place ourselves in
the category of representations of (S, Σ). In order to obtain said translation as an initial
morphism in this category, it suffices to equip (Ŝ 0 , Σ̂0 ) with a representation of (S, Σ). Doing
so consists in, firstly, representing S in the set Ŝ 0 , yielding a translation of types Ŝ → Ŝ 0 .
Afterwards the translation of terms is given, via a similar iteration principle as for types, by
representing the signature Σ in Σ̂0 .
We illustrate this iteration principle using two examples: firstly, in Sec. 4 we specify
a translation of logics from classical logic to intuitionistic logic. Secondly, we specify
translations from PCF to the untyped lambda calculus via initiality. The latter example is
implemented in the proof assistant Coq, cf. Sec. 5.
4. Logics and Logic Translations
In the style of the Curry–Howard isomorphism, we consider propositions as types and proofs
of a proposition as terms of that type. In this example we present the typed signatures of
two different logics,
• Classical propositional logic, called CPC, and
• Intuitionistic propositional logic, called IPC.
According to our main theorem each of those signatures gives rise to an initial representation,
a logical type system. We then use the iteration principle on CPC in order to specify a
translation of propositions and their proofs from CPC to IPC. The translation we specify
is actually the propositional fragment of the Gödel–Gentzen negative translation [TvD88,
Def. 3.4].
4.1. Signatures of Classical and Intuitionistic Logic. We present typed signatures
for classical and intuitionistic propositional logic. Their respective signatures for types —
propositions — are the same: let P denote a set of atomic formulas. The types — propositions
— of classical (CPC) and intuitionistic (IPC) propositional logic are given by the following
algebraic signature:
P := {p : 0,
> : 0,
∧ : 2,
⊥ : 0,
∨ : 2,
⇒: 2} .
where for any atomic formula p ∈ P we have an arity p : 0. We call P̂ the initial representation
as well as its underlying set, i.e. the propositions of CPC and IPC. For the set P̂ we use
infixed binary constructors. Note that negation is defined as ¬A ≡ A ⇒ ⊥.
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
Inference Rule
Γ`>
Arity
>I
>I : ∗ → [Θ]>
Γ`⊥ ⊥
Γ`A I
⊥I : [Θ]⊥ → [Θ]1
Γ`A
Γ`B ∧
I
Γ`A∧B
∧I : [Θ]1 × [Θ]2 → [Θ]1∧2
Γ ` A∧B ∧
E1
Γ`A
∧E1 : [Θ]1∧2 → [Θ]1
Γ ` A∧B ∧
E2
Γ`B
∧E1 : [Θ]1∧2 → [Θ]2
Γ, A ` B
⇒I
Γ`A⇒B
⇒I : [Θ1 ]2 → [Θ]1⇒2
Γ`A⇒B
Γ ` A ⇒E
Γ`B
⇒E : [Θ]1⇒2 × [Θ]1 → [Θ]2
Γ`A
∨I1
Γ ` A∨B
∨I1 : [Θ]1 → [Θ]1∨2
Γ`B
∨I2
Γ ` A∨B
∨I2 : [Θ]2 → [Θ]1∨2
Γ`A∨B
Γ, A ` C
Γ`C
Γ ` ¬A ∨ A
27
Γ, B ` C
∨E
∨E : [Θ]1∨2 × [Θ1 ]3 × [Θ2 ]3 → [Θ]3
EM
EM : ∗ → [Θ]¬1∨1
Table 1: Inference Rules of CPC and their Arities
4.1.1. Signature of CPC. Concerning the terms of CPC, every inference rule is given by
an arity. In Table 1, the inference rules and their corresponding arities are presented. Each
inference rule corresponds to a (family of) term — proof — constructor(s), where inference
rules without hypotheses are constants. Note that the initial representation automatically
comes with an additional inference rule
Γ, A ` A
var
corresponding to the monadic operation η, i.e. to the variables–as–terms constructor. Analogously to Rem. 3.27, it is not necessary, using our approach, to specify this inference rule
explicitly by an arity in the term signature of the logic under consideration; any logic we
specify via a typed signature automatically comes with this rule.
4.1.2. Signature of IPC. The type signature and thus the formulas of intuitionistic propositional logic IPC are the same as for CPC. However, the term signature is missing the arity
EM for excluded middle.
28
B. AHRENS
4.2. Translation via Initiality. The translation of propositions (_)g : P̂ → P̂, i.e. on
the type level, is specified by a representation g of the algebraic signature P in the set P̂.
According to Def. 3.2 we must specify, for any arity s : n ∈ N of P, a map towards P̂ taking
a suitable number of arguments in P̂,
sg : P̂ n → P̂ .
There is, of course, a canonical such map for each arity — but this would only give us the
identity morphism on P̂. We represent P in P̂ not by this identity representation, but in
such a way that we obtain the Gödel–Gentzen negative translation:
pg := ¬¬p,
⇒g := (⇒),
>g := ¬¬>,
∧g := ∧,
∨g := (A, B) 7→ ¬(¬A ∧ ¬B),
⊥g := ¬¬⊥ .
The proofs of IPC are given by the signature of CPC without the classical axiom EM. We
represent EM in IPC by giving, for any proposition A, a term of type ¬(¬¬A ∧ ¬A), e.g.,
var
var
¬¬A ∧ ¬A ` ¬¬A ∧ ¬A ∧
¬¬A ∧ ¬A ` ¬¬A ∧ ¬A ∧
E1
E2
¬¬A ∧ ¬A ` ¬¬A
¬¬A ∧ ¬A ` ¬A ⇒
E
¬¬A ∧ ¬A ` ⊥
⇒I
` ¬¬A ∧ ¬A ⇒ ⊥
As another example, we give a representation of ∨I1 , that is, for any proposition A and B,
we give a term of type Ag → ¬(¬Ag ∧ ¬B g ):
Ag
¬¬Ag
∨I1
¬¬Ag ∨ ¬¬B g
De Morgan
¬(¬Ag ∧ ¬B g )
Here the proof of Ag → ¬¬Ag and of the used De Morgan law are abbreviations for longer
proofs in IPC. We leave it up to the reader to find representations in IPC for the other
arities.
4.3. Some Remarks. This representation of the signature of CPC in IPC yields the
(propositional fragment of the) Gödel–Gentzen translation of propositions specified in
Troelstra and van Dalen’s book [TvD88, Def. 3.4], denoted on propositions with the same
name as its specifying representation,
(_)g : P̂ → P̂ .
Note that our translation of terms shows that any provable proposition in CPC translates
to a provable proposition in IPC, since we provide the corresponding proof term via our
translation:
Γ `C A implies Γg `I Ag .
However, a logic translation t from a logic L to another logic L0 should certainly satisfy an
equivalence of the form
Γ `L A if and only if Γt `L0 At .
Our framework does not ensure the implication from right to left, and is thus deficient from
the point of view of logic translations.
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
29
5. Translation of PCF to ULC, Formalized
In this section we explain our formalization in the proof assistant Coq of an instance of our
main theorem (cf. Thm. 3.33), for the typed signature of PCF (cf. Exs. 3.4, 3.8, 3.28). For
this, we make several simplifications:
• we do not define a notion of 2–signature, but specify directly a Coq type of representations
of PCF and
• we use dependent Coq types to formalize arities of higher degree (cf. Def. 3.16), instead of
relying on modules on pointed categories. A representation of an arity of degree n is thus
given by a family of module morphisms, indexed n times over the respective object type
(cf. Rem. 3.17).
The formalization builds up on a library of category theory the details of which we will not
go into. We just note that Coq types play the role of sets in our formalization. Maps of
sets are hence modelled by Coq functions and thus executable. In particular, the initial
morphism is a Coq function, and we can compute the translation of a term of PCF inside
Coq. For now we just give some key definitions of the theory–specific part. For complete
description we refer to the online documentation and source code repository2. As a side
note, the theorem relies on the axioms eq_rect_eq and functional_extensionality_dep from
the Coq standard library.
In the following we write Coq code in sans serif font. For a morphism f from object
a to object b in any category we write f : a −−−> b in Coq. Composition of morphisms
f : a → b and g : b → c is written f ;; g.
5.1. The Category of Representations. A representation of the typed signature of
PCF is given by
(1) a representation of the types of PCF (in a Coq type Sorts), cf. Ex. 3.4,
(2) a monad P on the category of families of sets indexed by Sorts (in the formalization:
ITYPE Sorts) and
(3) representations of the arities of PCF (cf. Ex. 3.28), i.e. morphisms of P –modules with
suitable source and target modules.
We implement representations of PCF as a “bundle”, i.e. a record type, whose components —
or “fields” — are these 3 items. In order to make the definitions more traceable, we first
define what a representation of the term signature of PCF in a monad P is, in the presence
of an SPCF –monad (cf. Def. 3.14). Unfolding the definitions, we suppose given a type Sorts, a
monad P on ITYPE Sorts and three operations on Sorts: a binary function Arrow — denoted
by an infixed “~~>” — and two constants Bool and Nat.
Variable Sorts : Type.
Variable P : Monad (ITYPE Sorts).
Variable Arrow : Sorts −> Sorts −> Sorts.
Variable Bool : Sorts.
Variable Nat : Sorts.
Notation "a ~~> b" := (Arrow a b) (at level 60, right associativity).
In this context, a representation of PCF is given by a bunch of module morphisms. Note
that M[t] denotes the fibre module of module M w.r.t. t, and d M // u denotes derivation of
2http://math.unice.fr/laboratoire/logiciels
30
B. AHRENS
module M w.r.t. u. The module denoted by a star ∗ is the terminal module, which is the
constant singleton module.
Class PCF_rep_struct := {
app : forall u v, (P[u ~~> v]) x (P[u]) −−−> P[v];
abs : forall u v, (d P // u)[v] −−−> P[u ~~> v];
rec : forall t, P[t ~~> t] −−−> P[t];
tttt : ∗ −−−> P[Bool];
ffff : ∗ −−−> P[Bool];
nats : forall m:nat, ∗ −−−> P[Nat];
Succ : ∗ −−−> P[Nat ~~> Nat];
Pred : ∗ −−−> P[Nat ~~> Nat];
Zero : ∗ −−−> P[Nat ~~> Bool];
CondN: ∗ −−−> P[Bool ~~> Nat ~~> Nat ~~> Nat];
CondB: ∗ −−−> P[Bool ~~> Bool ~~> Bool ~~> Bool];
bottom: forall t, ∗ −−−> P[t] }.
After abstracting over the section variables we package all of this into a record type:
Record PCF_rep := {
Sorts : Type;
Arrow : Sorts −> Sorts −> Sorts;
Bool : Sorts ;
Nat : Sorts ;
pcf_rep_monad :> Monad (ITYPE Sorts);
pcf_rep_struct :> PCF_rep_struct pcf_rep_monad Arrow Bool Nat }.
Notation "a ~~> b" := (Arrow a b) (at level 60, right associativity).
The type PCF_rep later will constitute the type of objects of the category of representations
of PCF. Accordingly, a morphism of representations from P to R (cf. Def. 3.30) consists of
a morphism of representations of the types of PCF — with underlying map Sorts_map —
and a colax morphism of monads which makes commute some diagrams. We first define
the diagrams we expect to commute, before packaging everything into a record type of
morphisms. The context is given by the following declarations:
Variables P R : PCF_rep.
Variable Sorts_map : Sorts P −> Sorts R.
Hypothesis HArrow : forall u v, Sorts_map (u ~~> v) = Sorts_map u ~~> Sorts_map v.
Hypothesis HBool : Sorts_map (Bool _ ) = Bool _ .
Hypothesis HNat : Sorts_map (Nat _ ) = Nat _ .
Variable f : colax_Monad_Hom P R (RETYPE (fun t => Sorts_map t)).
We explain the commutative diagrams of Def. 3.30 for the successor arity. We ask the
following diagram to commute:
Program Definition Succ_hom’ :=
Succ ;; f [(Nat ~~> Nat)] ;; Fib_eq_Mod _ _ ;; IsoPF
==
∗−−−>∗ ;; f ∗∗ Succ.
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
31
Here the morphism Succ refers to the representation of the successor arity either of P
(the first appearance) or R (the second appearance) — Coq is able to figure this out itself.
The morphism f ∗∗ Succ thus is the pullback along f of the module morphism Succ of the
representation R — recall that pullback is functorial. The domain of the successor is given
by the terminal module ∗. Accordingly, we have that dom(Succ, f ) is the trivial module
morphism with domain and codomain given by the terminal module. We denote this module
morphism by ∗−−−>∗. The codomain is given as the fibre of f of type ι → ι. The
two remaining module morphisms are isomorphisms which do not appear in the informal
description. The isomorphism IsoPF is needed to permute fibre with pullback — in the
formalization the 2–category of monads behaves like a bicategory, since composition is
associative up to isomorphism only, due to Coq conversion being stronger than propositional
equality. The morphism Fib_eq_Mod M H takes a module M and a proof H of equality of two
object types as arguments, say, H : u = v. Its output is an isomorphism M[u] −−−> M[v].
Here the proof is of type
H : Sorts_map (Nat ~~> Nat) = Sorts_map Nat ~~> Sorts_map Nat
and Coq is able to figure the proof, i.e. the term, out itself.
Finally, we prove that the objects and morphisms thus defined yield a category, where
the composition and identity are given by composition and identity of monad morphisms,
respectively. We omit the description of this part of the formalization.
5.2. The Initial Representation. We want to prove that the above specified category
admits an initial object, consisting of the term monad associated to the signature of PCF,
together with the canonical representation morphisms. The monad of PCF terms is defined
as an inductive dependent type, parametrized by the initial set of types of PCF, denoted by
TY, as well as a context V. First we define the constants of PCF, afterwards the inductive
type family of terms:
Inductive Consts : TY −> Type :=
| Nats : nat −> Consts Nat
| ttt : Consts Bool
...
| condB: Consts (Bool ~> Bool ~> Bool ~> Bool).
Inductive PCF (V: TY −> Type) : TY −> Type:=
| Bottom: forall t, PCF V t
| Const : forall t, Consts t −> PCF V t
| Var : forall t, V t −> PCF V t
| App : forall t s, PCF V (s ~> t) −> PCF V s −> PCF V t
| Lam : forall t s, PCF (opt t V) s −> PCF V (t ~> s)
| Rec : forall t, PCF V (t ~> t) −> PCF V t.
Renaming, i.e. functoriality, and substitution, are then defined via structural recursion,
and the monad laws are proved by induction, accordingly. We refer to the source code or
documentation for details.
Given any representation R of PCF, the initial morphism is iteratively defined according
to the proof of the main theorem:
32
B. AHRENS
Fixpoint init V t (v : PCF V t) :
R (retype (fun t0 => Init_Sorts_map t0) V) (Init_Sorts_map t) :=
match v with
| Var t v => weta R _ _ (ctype _ v)
| u @ v => app _ _ _ (init u, init v)
| Lam _ _ v => abs _ _ _ (rlift R
(@der_comm TY (Sorts R) (fun t => Init_Sorts_map t) _ V ) _ (init v))
| Rec _ v => rec _ _ (init v)
| Bottom _ => bottom _ _ tt
| y ’ => match y in Consts t1 return
R (retype (fun t2 => Init_Sorts_map t2) V) (Init_Sorts_map t1) with
| Nats m => nats m _ tt
| succ => Succ _ tt
| condN => CondN _ tt
| condB => CondB _ tt
| zero => Zero _ tt
| ttt => tttt _ tt
| fff => ffff _ tt
| preds => Pred _ tt
end
end.
Again, the necessary properties, i.e. the monad morphism laws, representation laws, and
finally, unicity, are proved by induction. Note that the above family of maps init V really is
the family of the adjuncts of the initial morphism under the adjunction of Rem. 2.12, cf.
also the proof of Thm. 3.33. The component on V of the initial morphism is obtained by
precomposing the map init V with pattern matching on the constructor ctype.
5.3. Representing PCF in the Untyped Lambda Calculus. The untyped lambda
calculus, formalized as a monad ULC : Set → Set, gives rise to a monad uULC : [{∗}, Set] →
[{∗}, Set], in which we represent PCF. Our implementation does not allow us to identify
those two monads, but we do so informally. By its iterative definition, the initial morphism
depends on the representation in the codomain monad. Giving two different representations
of PCF in ULC gives rise to two different translations of PCF to ULC. As an example, one
might choose to use different representations of natural numbers or the fixed point operator.
This is simply done by defining two different ULC terms as image of the fixed point operator
rec. We define the Turing fixed point combinator
Θ := (λx.λy.(y(xxy)))(λx.λy.(y(xxy)))
and the Curry combinator
Y := λf.(λx.f (xx))(λx.f (xx))
formally:
Eval compute in ULC_theta.
= Abs (Abs (1 @ (2 @ 2 @ 1))) @ Abs (Abs (1 @ (2 @ 2 @ 1)))
Eval compute in ULC_Y.
EXTENDED INITIALITY FOR TYPED ABSTRACT SYNTAX
33
= Abs (Abs (2 @ (1 @ 1)) @ Abs (2 @ (1 @ 1)))
Here some Coq notation is used to translate the “nested datatype” style of variable binding
to a slightly more readable de Bruijn notation, and an infixed “@” denotes application. After
equipping both of the maps
x 7→ App(Y, x)
and
x 7→ App(Θ, x)
with a structure as module morphism, we can use either of them as a representation of the
rec arity of PCF.
Program Instance ULCRec_theta_s t : Module_Hom_struct
(fun V y => (ULC_theta _ ) @ y).
Definition ULCRec_theta t := Build_Module_Hom (ULCRec_s t).
Program Instance ULCRec_Y_s t : Module_Hom_struct
(fun V y => (ULC_Y _ ) @ y).
Definition ULCRec_Y t := Build_Module_Hom (ULCRec_Y_s t).
The representational structure of PCF in uULC determines the iteratively defined initial
morphism:
Program Instance PCF_ULC_rep_s :
PCF_rep_struct (Sorts:=unit) uULC (fun _ _ => tt) tt tt := {
app r s := ULCApp r s;
abs r s := ULCAbs r s;
rec t := ULCRec_theta t ; (* replace here to
translate to Y instead of Turing operator *)
tttt := ULCttt ;
ffff := ULCfff ;
nats m := ULCNat m ;
Succ := ULCSucc ;
CondB := ULCCondb ;
CondN := ULCCondn ;
bottom t := ULCBottom t ;
Zero := ULCZero ;
Pred := ULCPred }.
As a final remark, we emphasize that the obtained translation from PCF to the untyped
lambda calculus is executable in Coq. For instance, we can translate the PCF term negating
boolean terms as follows:
Eval compute in
(PCF_ULC_c (fun t => False) tt (ctype _
(Lam (condB ’ @@ x_bool @@ fff ’ @@ ttt ’)))).
= Abs (Abs (Abs (Abs (3 @ 2 @ 1))) @ 1 @ Abs (Abs 1) @ Abs (Abs 2))
Here we use infixed “@@” to denote application of PCF, and x_bool is a notation for a de
Bruijn variable of type Bool of the lowest level, i.e. a variable that is bound by the Lam
binder of PCF in above term.
34
B. AHRENS
6. Future Work
We have given an algebraic interpretation of maps between languages over different sets of
types. Our initiality theorem yields a iteration operator that allows for the specification of
such translations.
Another line of work of ours is to integrate semantics into initiality results [Ahr11]. We
study untyped syntax equipped with reduction rules by considering it as a relative monad
[ACU10] (over the diagonal functor ∆ : Set → Ord) from the category of sets to the category
of preorders Ord. A 2–signature consists of a syntactic signature Σ which defines the terms
of a language, as well as of a set A of inequations, each of which specifies a reduction rule.
Representations of such a 2–signature (Σ, A) are representations of Σ which verify each
inequation α ∈ A. We prove that the category of representations of (Σ, A) has an initial
object.
The present work carries over to relative monads, and we can thus study translations of
languages over different types which are equipped with reduction rules. In a forthcoming
work we will prove an initiality theorem for simply–typed syntax with reduction rules, and
we will present a translation via initiality from PCF, equipped with its usual reduction rules,
to ULC with beta reduction. The translation is ensured to be semantically faithful.
Acknowledgement
We wish to thank André Hirschowitz and Marco Maggesi for numerous discussions. Furthermore, we thank Jan Rutten and the anonymous referees for their helpful comments and
advice.
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letter to Creative Commons, 171 Second St, Suite 300, San Francisco, CA 94105, USA, or
Eisenacher Strasse 2, 10777 Berlin, Germany
| 6 |
BIVARIATE REPRESENTATION AND CONJUGACY CLASS
ZETA FUNCTIONS ASSOCIATED TO UNIPOTENT GROUP
SCHEMES
arXiv:1802.04440v1 [math.GR] 13 Feb 2018
PAULA MACEDO LINS DE ARAUJO
Abstract. We introduce two bivariate zeta functions associated to unipotent
group schemes over rings of integers of number fields. These zeta functions
encode, respectively, the numbers of isomorphism classes of irreducible complex
representations of finite dimensions and the numbers of conjugacy classes of
congruence quotients of the associated groups. Our bivariate zeta functions
specialise to (univariate) class number zeta functions associated to the relevant
groups. In case of nilpotency class 2, bivariate representation zeta functions
also specialise to (univariate) twist representation zeta functions. We show
that these zeta functions satisfy Euler factorisations and almost all of their
Euler factors satisfy rationality and functional equations.
We give explicit formulae for the bivariate zeta functions of some families
of nilpotent groups of class 2. The local bivariate representation zeta functions
of these groups are also expressed in terms of sums over finite hyperoctahedral
groups, which provides formulae for joint distributions of three statistics on
such groups.
Contents
1. Introduction and statement of main results
1.1. Introduction
1.2. Uniform rationality and functional equations
1.3. Groups of type F , G, and H
1.4. Background and related research
1.5. Notation
2. Bivariate zeta functions and p-adic integrals
2.1. The numbers rn (GN ) and cn (GN )
2.2. p-adic integrals
2.3. Twist representation zeta functions
2.4. Local functional equations—proof of Theorem 1.7
2.5. Euler products
2.6. Convergence
3. Results for groups of type F , G, and H
3.1. Bivariate conjugacy class zeta functions—proof of Theorem 1.10
3.2. Bivariate representation zeta functions—proof of Theorem 1.12
3.3. Hyperoctahedral groups and functional equations
4. Further examples
Appendix A. p-adic integrals
Acknowledgements
References
2
2
4
5
8
10
10
12
16
19
20
22
23
24
24
37
37
41
42
44
45
Date: February 14, 2018.
2010 Mathematics Subject Classification. 11M41, 11M32, 20F69, 20D15, 22E55, 20E45.
Key words and phrases. Finitely generated nilpotent groups, zeta functions, conjugacy classes,
irreducible complex characters, Kirillov orbit method, p-adic integration, signed permutation
statistics.
Paula Lins
1. Introduction and statement of main results
1.1. Introduction. We study bivariate zeta functions of groups associated to
unipotent group schemes encoding, respectively, the numbers of isomorphism classes
of finite-dimensional irreducible complex representations, and the numbers of conjugacy classes of certain finite quotients of the infinite groups considered. We first
recall the definitions of (univariate) representation and conjugacy class zeta functions of finite groups. Let G be a finite group and, for n ∈ N, denote
rn (G) = |{isomorphism classes of n-dimensional irreducible complex
representations of G}|,
cn (G) = |{conjugacy classes of G of cardinality n}|.
Definition 1.1. The representation and the conjugacy class zeta functions of the
finite group G are, respectively,
irr
ζG
(s)
=
∞
X
rn (G)n
−s
and
cc
ζG
(s)
n=1
=
∞
X
cn (G)n−s ,
n=1
where s is a complex variable.
The sums of Definition 1.1 are finite, since a finite group has only finitely many
isomorphism classes of irreducible complex representations and only finitely many
conjugacy classes.
The groups considered in this work are groups associated to nilpotent Lie lattices,
constructed as follows. Let O denote the ring of integers of a number field K. By
an O-Lie lattice we mean a free and finitely generated O-module Λ together with
an antisymmetric bi-additive form [ , ] which satisfies the Jacobi identity. If Λ
has nilpotency class c and satisfies Λ0 ⊆ c!Λ, where Λ0 = [Λ, Λ] is the derived Lie
sublattice, one may define a unipotent group scheme G = GΛ from Λ by setting
Λ(R) = Λ ⊗O R, for each O-module R, and then defining the group operation of
Λ(R) by means of the Hausdorff series; see [24, Section 2.1.2]. The group scheme
G is such that G(O) is a finitely generated, torsion-free nilpotent group (T -group
for short) of nilpotency class c. All finitely generated nilpotent groups are virtually
of the form G(Z), as explained in [5, Section 5] and [24, Remark 2.4].
We now introduce the bivariate zeta functions which are the main objects of
study of the current paper.
Definition 1.2. The bivariate representation and the bivariate conjugacy class
zeta functions of G(O) are, respectively,
X
irr
irr
ZG(O)
(s1 , s2 ) =
ζG(O/I)
(s1 )|O : I|−s2 and
{0}6=IEO
cc
ZG(O)
(s1 , s2 )
=
X
cc
ζG(O/I)
(s1 )|O : I|−s2 ,
{0}6=IEO
where s1 and s2 are complex variables.
These series converge for (s1 , s2 ) with sufficiently large real part; see Section 2.6.
In Proposition 2.12, we establish the Euler decompositions
Y
∗
∗
(1.1)
ZG(O)
(s1 , s2 ) =
ZG(O
(s1 , s2 ),
p)
p∈Spec(O)\{(0)}
where ∗ ∈ {irr, cc}, the completion of O at the nonzero prime ideal p is denoted
Op . When considering a fixed prime ideal p, we write simply Op = o and GN :=
2
Bivariate zeta functions of T -groups
G(o/pN ). With this notation, the local factor at p is given by
(1.2)
∞
X
∗
∗
ZG(O
(s1 , s2 ) = ZG(o)
(s1 , s2 ) =
p)
∗
(s1 )|o : p|−N s2 .
ζG
N
N =0
Previously studied classes of zeta functions of groups satisfy Euler decompositions
such that their local factors are rational functions and satisfy functional equations;
see the discussion in Section 1.4. Theorem 1.7 states that almost all local factors of
the Euler products (1.1) are rational functions which satisfy functional equations
and behave uniformly under extension of scalars.
Example 1.3. Let G be the free abelian group scheme of rank m. Let p be a nonzero
prime ideal of the ring of integers O with q = |O : p|. For each i, N ∈ N0 ,
(
q mN , if i = 0,
rqi (GN ) = cqi (GN ) =
0,
otherwise.
Therefore, for ∗ ∈ {irr, cc},
∗
ZG(o)
(s1 , s2 ) = Zo∗m (s1 , s2 ) =
∞
X
N =0
q N (m−s2 ) =
1
.
1 − q m−s2
∗
ZO
m (s1 , s2 )
That is,
= ζK (s2 −m), where ζK (s) denotes the Dedekind zeta function
of the number field K. In particular, the local factor at p satisfies the functional
equation
Zo∗m (s1 , s2 ) |q→q−1 = −q m−s2 Zo∗m (s1 , s2 ).
4
An advantage of the study of the bivariate zeta functions of Definition 1.2 is that
they allow the investigation of two univariate zeta functions. Firstly, these bivariate
zeta functions both specialise to the class number zeta function, which encodes the
class numbers of principal congruence quotients of the group considered. Recall
that the class number of a group G—denoted k(G)—is the number of conjugacy
classes of G or, equivalently, the number of irreducible complex characters of G. In
cc
irr
particular, k(G) = ζG
(0) = ζG
(0).
Definition 1.4. The class number zeta functionwe of the T -group G(O) is
X
k
ζG(O)
(s) =
k(G(O/I))|O : I|−s ,
{0}6=IEO
where s is a complex variable.
The term ‘conjugacy class zeta function’ is sometimes used for what we call ‘class
number zeta function’; see for instance [2, 18, 20].
Clearly,
(1.3)
irr
cc
k
ZG(O)
(0, s) = ZG(O)
(0, s) = ζG(O)
(s).
Secondly, if c = 2, in which case we say that G(O) is a T2 -group, then the
local factors of the bivariate representation zeta function of G(O) specialise to the
respective local factors of its twist representation zeta function, whose definition
we now recall.
Nontrivial T -groups have infinitely many 1-dimensional irreducible complex representations. For this reason, one cannot define the representation zeta function
of a T -group G as in Definition 1.1. Instead, we consider the numbers ren (G) of
n-dimensional twist-isoclasses of irreducible complex representations of G, that is,
the number of classes of the equivalence relation on the set of irreducible complex
representations of G given by ρ ∼ σ if and only if there exists a 1-dimensional
3
Paula Lins
∼ χ ⊗ σ. In the context of topological groups,
representation χ of G such that ρ =
we only consider continuous representations.
If G is a T -group, the numbers ren (G) are all finite, allowing us to define a
generating function encoding the data (e
rn (G)); see [13, Theorem 6.6].
Definition 1.5. The twist representation zeta function of a T -group G is
f
irr
ζG
(s)
=
∞
X
ren (G)n−s ,
n=1
where s is a complex variable.
The twist representation zeta functions of T -groups are objects of study, for
instance, of [9, 17, 24, 27]. Explicit examples of (local factors of) representation
zeta functions of T -groups can be found in [6, 17, 22, 24, 25].
In Section 2.3, we show that if G(O) is a T2 -group, its bivariate representation
zeta function specialises to its twist representation zeta function as follows. For a
fixed nonzero prime ideal p of O, let g = Λ ⊗O Op and denote by g0 its derived Lie
sublattice. Let r be the torsion-free rank of g/g0 . Then Proposition 2.10 states that
Y
f
irr
irr
(1.4)
(1 − q r−s2 )ZG(O
(s).
(s
,
s
)
|
= ζG(O)
s
→s−2
1 2
1
p)
s2 →r
p∈Spec(O)\{(0)}
In [24], three infinite families of T2 -groups which generalise the Heisenberg group
H(O) of upper uni-triangular 3 × 3-matrices over O are investigated. We provide
explicit formulae for the bivariate conjugacy class zeta functions of these groups in
Theorem 1.10 and for their bivariate representation zeta functions in Theorem 1.12.
The following example illustrates these results for the Heisenberg group.
Example 1.6. Let H(O) denote the Heisenberg group over O. In Example 2.9, we
show that, for a given a nonzero prime ideal p of O with |O : p| = q,
1 − q −s1 −s2
and
(1 − q 1−s1 −s2 )(1 − q 2−s2 )
1 − q −s1 −s2
cc
ZH(o)
(s1 , s2 ) =
.
(1 − q 1−s2 )(1 − q 2−s1 −s2 )
irr
ZH(o)
(s1 , s2 ) =
In particular, these are rational functions in q, q −s1 , and q −s2 , and
∗
∗
ZH(o)
(s1 , s2 ) |q→q−1 = −q h−s2 ZH(o)
(s1 , s2 ),
where ∗ ∈ {irr, cc} and h = dim(H) = 3 is the dimension of the Heisenberg group
scheme H. Specialisations (1.3) and (1.4) yield
k
ζH(o)
(s) =
1 − q −s
(1 − q 1−s )(1 − q 2−s )
and
irr
ζH(o)
(s) =
f
1 − q −s
.
1 − q 1−s
The expression of the class number zeta function agrees with the formula given
in [18, Section 8.3]. This example also occurs in [2, Section 8.2], corrected by what
seems to be a sign mistake. The expression of the twist representation zeta function
accord with [24, Theorem B]. We further note that
k
k
(s) |q→q−1 = −q 3−s ζH(o)
(s).
ζH(o)
4
1.2. Uniform rationality and functional equations. Our first main result
concerns rationality and functional equations of local factors of bivariate representation and conjugacy class zeta functions of nilpotent groups obtained from
nilpotent Lie lattices.
4
Bivariate zeta functions of T -groups
Theorem 1.7. Let G = GΛ be a unipotent group scheme associated to a nilpotent
O-Lie lattice Λ as in Section 1.1. For each ∗ ∈ {irr, cc}, there exist a positive
integer t∗ and a rational function R∗ (X1 , . . . , Xt∗ , Y1 , Y2 ) in Q(X1 , . . . , Xt∗ , Y1 , Y2 )
such that, for all but finitely many nonzero prime ideals p of O, there exist algebraic
integers λ∗1 (p), . . . , λ∗t∗ (p) for which the following holds. For any finite extension
O of o := Op with relative degree of inertia f = f (O, o),
∗
ZG(O)
(s1 , s2 ) = R∗ (λ∗1 (p)f , . . . , λ∗t∗ (p)f , q −f s1 , q −f s2 ),
where q = |O : p|. Moreover, these local factors satisfy the following functional
equation:
∗
(s1 , s2 ) |
ZG(O)
q→q −1
∗
= −q f (h−s2 ) ZG(O)
(s1 , s2 ),
∗
−1
λ∗
j (p)→λj (p)
where h = dimK (Λ ⊗ K) with K = Frac(O).
The statement of Theorem 1.7 is analogous to [24, Theorem A], and its proof
deeply relies on the methods of [1, 24]; see Section 2.4. The main tools used in the
proof of Theorem 1.7 are the Kirillov orbit method and p-adic integration.
A consequence of Theorem 1.7 is that the local factors of the class number zeta
function of G(O) are rational in λi (p), q and q −s and behave uniformly under base
extension. Moreover, for a finite extension O of o, the local factors satisfy the
functional equation
k
ζG(O)
(s) |
q→q −1
k
= −q f (h−s) ζG(O)
(s),
∗
−1
λ∗
j (p)→λj (p)
which agrees with [18, Theorem 4.15] together with the specialisation of the ask
zeta function to the class number zeta function given in [18, Theorem 1.7].
1.3. Groups of type F , G, and H. As mentioned in Section 1.1, we are particularly interested in three infinite families of groups which are also objects of study
in [24]. Such groups are constructed from certain Z-Lie lattices, to be defined below,
and provide different generalisations of the Heisenberg group scheme. The interest
in those Z-Lie lattices arises from their very construction. Roughly speaking, their
defining relators reflect the reduced, irreducible, prehomogeneous vector spaces of,
respectively, complex n × n antisymmetric matrices, complex n × n-matrices and
complex n × n symmetric matrices—here, the relative invariants are given by Pf,
det and det, where Pf(X) denotes the Pfaffian of an antisymmetric matrix X. We
refer the reader to [24, Section 6] for details. We now recall the Z-Lie lattices to
which they are associated.
Definition 1.8. For n ∈ N and δ ∈ {0, 1}, consider the nilpotent Z-Lie lattices
Fn,δ = hxk , yij | [xi , xj ] − yij , 1 ≤ k ≤ 2n + δ, 1 ≤ i < j ≤ 2n + δi,
Gn = hxk , yij | [xi , xn+j ] − yij , 1 ≤ k ≤ 2n, 1 ≤ i, j ≤ ni,
Hn = hxk , yij | [xi , xn+j ] − yij , [xj , xn+i ] − yij , 1 ≤ k ≤ 2n, 1 ≤ i ≤ j ≤ ni.
By convention, relations that do not follow from the given ones are trivial.
Remark 1.9. For Λ ∈ {Fn,δ , Gn , Hn }, the condition Λ0 ⊆ 2Λ is not satisfied. However, in [24, Section 2.4], a different construction of a unipotent group scheme G
associated to a (general) nilpotent O-Lie lattice Λ of class 2 without the assumption Λ0 ⊆ 2Λ is given. In case such condition is satisfied, the unipotent group
scheme obtained in this way coincide with the latter ones. We use this alternative
construction to obtain group schemes over Z associated to Fn,δ , Gn , and Hn .
5
Paula Lins
Following [24], these unipotent group schemes are denoted, respectively, by Fn,δ ,
Gn , and Hn , and groups of the form Fn,δ (O), Gn (O), and Hn (O) are called groups
of type F, G, and H, respectively. The groups F1,0 (O) = G1 (O) = H1 (O) are
isomorphic to the Heisenberg groups H(O).
For the rest of Section 1.3, G denotes one of the unipotent group schemes Fn,δ ,
Gn , or Hn , and Λ ∈ {Fn,δ , Gn , Hn }, for some n ∈ N and δ ∈ {0, 1}.
1.3.1. Bivariate conjugacy class and class number zeta functions. We start with
the main results concerning bivariate conjugacy class and class number zeta functions of groups of type F , G, and H, followed by the main results concerning
bivariate representation and twist representation zeta functions of these groups.
Theorem 1.10. Given n ∈ N and δ ∈ {0, 1}, for each nonzero prime ideal p of O
with q = |O : p|,
ZFccn,δ (o) (s1 , s2 )
2n+δ−1
1 − q ( 2 )−(2n+δ−1)s1 −s2
.
=
2n+δ
2n+δ
(1 − q ( 2 )−s2 )(1 − q ( 2 )+1−(2n+δ−1)s1 −s2 )
Write q −s1 = T1 and q −s2 = T2 . For n ≥ 2,
cc
ZG
(T1 , T2 ) =
n (o)
n
n
(1 − q 2( 2 ) T1n T2 )(1 − q 2( 2 )+1 T12n−1 T2 ) + q n T1n T2 (1 − q −n )(1 − q −(n−1) T1n−1 )
,
(1 − q n2 T2 )(1 − q n2 T1n T2 )(1 − q n2 +1 T12n−1 T2 )
cc
ZH
(T1 , T2 ) =
n (o)
2
n+1
n
n
(1 − q ( 2 ) T1n T2 )(1 − q ( 2 )+2 T12n−1 T2 ) + q ( 2 ) T1n T2 (1 − q −n+1 )(1 − q −(n−1) T1n−1 )
.
n+1
n+1
n+1
(1 − q ( 2 ) T )(1 − q ( 2 )+1 T n T )(1 − q ( 2 )+1 T 2n−1 T )
2
1
2
2
1
The proof of Theorem 1.10 is given in Section 3.1.
Specialisation (1.3) together with Theorem 1.10 provide explicit formulae for the
class number zeta functions of groups of type F , G, and H.
Corollary 1.11. For all n ≥ 1 and δ ∈ {0, 1},
ζK (s − 2n+δ
− 1)ζK (s −
k
2
(1.5)
ζFn,δ (O) (s) =
ζK (s − 2n+δ−1
)
2
2n+δ
2
)
,
where ζK (s) is the Dedekind zeta function of the number field K = Frac(O). Furthermore, for n ≥ 2, the class number zeta functions of Gn (O) and Hn (O) are
2(n
2 )−s
k
ζG
(s)
n (O)
=
Y
(1 − qp
p∈Spec(O)\{(0)}
k
(s)
ζH
n (O)
2(n
2 )+1−s
)(1 − qp
(1 − qpn
2 −s
) + qpn
2
−s
)2 (1 − qpn
(1 − qp−n )(1 − qp−n+1 )
2 +1−s
)
(n)−s
(n)+2−s
(n+1)−s
(1 − qp 2 )(1 − qp 2
) + qp 2
(1 − qp−n+1 )2
=
,
(n+1)−s
(n+1)+1−s 2
(1 − qp 2
)(1 − qp 2
)
p∈Spec(O)\{(0)}
Y
where qp = |O : p|, for all p ∈ Spec(O) \ {(0)}.
In particular, all the local factors of the bivariate conjugacy class zeta functions
of groups of type F , G, and H are rational in qp , qp−s1 , and qp−s2 , whilst all local
factors of their class number zeta functions are rational in qp and qp−s , and the local
factors of both zeta functions satisfy functional equations.
6
,
Bivariate zeta functions of T -groups
1.3.2. Bivariate representation zeta functions. To state our next result, we introduce some notation.
Let X, Y denote indeterminates in the field Q(X, Y ). Given n ∈ N, set (n)X =
1 − X n and (n)X ! = (n)X (n − 1)X . . . (1)X . For a, b ∈ N0 such that a ≥ b, the
X-binomial coefficient of a over b is
a
(a)X
=
.
b X
(b)X (a − b)X
Given n ∈ N, denote [n] = {1, . . . , n} and [n]0 = [n] ∪ {0}. Given a subset
{i1 , . . . , il } ⊂ N, we write {i1 , . . . , il }< meaning that i1 < i2 < · · · < il . For
I = {i1 , . . . , il }< ⊆ [n − 1]0 , denote µj := ij+1 − ij for all j ∈ [l]0 , where i0 = 0,
il+1 = n, and define
n
n
il
i2
=
...
.
I X
il X il−1 X
i1 X
The Y -Pochhammer symbol is defined as
(X; Y )n =
n−1
Y
(1 − XY i ).
i=0
Theorem 1.12. Let p be a nonzero prime ideal of O with q = |O : p|, n ∈ N, and
δ ∈ {0, 1}. Then
irr
ZG(o)
(s1 , s2 ) =
1
1−
X
q ā(G,n)−s2
fG,I (q −1 )
I⊆[n−1]0
Y
i∈I
q ā(G,i)−(n−i)s1 −s2
,
1 − q ā(G,i)−(n−i)s1 −s2
where fG,I (X) and ā(G, i), for all I = {i1 , . . . , il }< ⊆ [n − 1]0 and for all i ∈ [n]0 ,
are defined as follows.
G
fG,I (X)
Fn,δ
n
2(i1 +δ)+1
; X 2 )n−i1
I X 2 (X
n
i1 +1
; X)n−i1
I X (X
Gn
Hn
Q
l
2
2 −1
j=1 (X ; X )bµj /2c
(X i1 +1 ; X)n−i1
2n+δ
2
ā(G, i)
− 2i+δ
+ 2i + δ
2
n2 − i2 + 2i
n+1
− i+1
+ 2i
2
2
The proof of Theorem 1.12 may be found in Section 3.2.
The numbers ā(G, i) are a slight modification of the numbers a(G, i) given in
[24, Theorem C], namely ā(Fn,δ , i) = a(Fn,δ , i) + 2i + δ and ā(G, i) = a(G, i) +
2i, for G(O) of type G and H. Since the groups of type F , G, and H are T2 groups, we obtain the formulae of [24, Theorem C] by applying specialisation (1.4)
to Theorem 1.12.
The polynomials fG,I (X) appearing in Theorem 1.12 can be expressed in terms
of distributions of statistics on Weyl groups of type B, also called hyperoctahedral
groups Bn ; see Sections 3.3.1 and 3.3.2. These are the groups Bn of the permutations w of the set [±n]0 = {−n, . . . , n} such that w(−i) = −w(i) for all i ∈ [±n]0 .
In Lemma 3.15, we describe the local bivariate representation zeta function of
G(O) as a sum over Bn in terms of statistics on such groups. As the local factors
of the representation and conjugacy class zeta functions of G(O) specialise to its
class number zeta function, the formulae in terms of statistics on hyperoctahedral
groups Bn can be compared with the formulae of Corollary 1.11, which leads to
formulae for the joint distribution of three functions on Weyl groups of type B; see
Propositions 3.16 and 3.17.
7
Paula Lins
More precisely, the formulae of Lemma 3.15 under specialisation (1.3) provide a
formula of the following form for the class number zeta function of G(o).
P
χG (w)q −hG (w)−des(w)s
k
,
ζG(o) (s) = w∈BQnn
ā(G,i)−s )
i=0 (1 − q
where χG is one of the linear characters (−1)neg or (−1)` of Bn , where neg(w)
denote the number of negative entries of w and ` is the standard Coxeter length
function of Bn . Moreover, the functions hG are sums of statistics on Bn for each
G, and des(w) is the cardinality of the descent set of w ∈ Bn ; see Section 3.3.1 for
definitions.
1.3.3. Local functional equations. The formulae for the bivariate zeta functions
given in Theorems 1.12 and 1.10 allow us to strengthen Theorem 1.7 for the groups
of type F , G, and H by showing that its conclusion holds for all local factors.
Theorem 1.13. For ∗ ∈ {irr, cc} and a nonzero prime ideal p of O with |O : p| = q,
the local bivariate zeta function of G(O) at p satisfies the functional equation
∗
∗
(s1 , s2 ) |q→q−1 = −q h−s2 ZG(o)
(s1 , s2 ),
ZG(o)
where
h = dim G =
2n+δ+1
2
2
,
n + 2n,
n+1
+ 2n,
2
if G = Fn,δ ,
if G = Gn ,
if G = Hn .
In fact, Theorem 1.7 states that almost all local factors satisfy functional equations of such form, whilst Theorems 1.10 and Theorem 1.12 state that all local
factors are given by the same rational functions. We give an alternative proof in
Section 3.3.3.
1.4. Background and related research. As with the famous Riemann zeta
function, one ideally expects that zeta functions admit Euler decompositions with
local factors being rational and satisfying functional equations. In the seminal work
of Grunewald, Segal and Smith [8], they defined zeta functions of T -groups enumerating, respectively, the numbers of subgroups of finite index, the numbers of normal
subgroups of finite index, and the numbers of finite-index subgroups whose profinite
completions are isomorphic to the profinite completion of the associated group. It
was then shown that these zeta functions satisfy Euler product decompositions and
that the local factors are rational functions; see [8, Proposition 4 and Theorem 1].
This result was refined for free T2 -groups by showing that the subgroup and normal
subgroup zeta functions of those groups satisfy the following property. For each
of them, there exists a rational function R in two variables such that, for every
prime integer p, the local factor at p is given by R(p, p−s ); see [8, Theorem 2]. It
was later proved, however, that functional equations may not exist for normal zeta
functions of groups with nilpotency class larger than 2. In [21, Section 2] there
are examples of nilpotent groups of class 3, some of which have normal subgroup
zeta functions satisfying functional equations, while other examples do not. More
recently, sufficient criteria for such functional equations were established in [28].
They hold, in particular, for finitely generated free nilpotent groups.
The twist representation zeta functions of groups of the form G(O), as in Section 1.1, has all the above mentioned features. Their Euler decompositions, established in [24, Proposition 2.2], are given by
Y
f
f
irr
irr
(1.6)
ζG(O)
(s) =
ζG(O
(s),
p)
p∈Spec(O)\{(0)}
8
Bivariate zeta functions of T -groups
where Spec(O) denotes the set of prime ideals of O, and Op denotes the completion
of O at p. Furthermore, the local factor at p ∈ Spec(O) \ {(0)} with residue field
of characteristic p and cardinality q is
∞
X
f
irr
(s) =
ζG(o)
repi (G(o))p−is .
i=0
In the same paper, Stasinski and Voll showed rationality and functional equations
for (almost all) local factors of the twist representation zeta functions of such groups;
see [24, Theorem A]. Hrushovski, Martin, Rideau, and Cluckers [9, Theorem 1.5]
had previously shown that the local factors of the twist representation zeta functions
of T -groups are rational functions.
Some (univariate) representation zeta functions are also known to satisfy some
of these properties. Larsen and Lubotzky proved in [12, Proposition 1.1] that representation zeta functions of arithmetic groups satisfying the congruence subgroup
property satisfy Euler decompositions. These Euler decompositions are not of the
form (1.6). Instead, they are indexed by the places of the number field over which
the group is defined, including the Archimedean ones. Moreover, the representation
zeta functions of certain pro-p groups satisfy local functional equations and rationality, according to [1, Theorem A]. A result of Jaikin-Zapirain proves rationality
for representation zeta functions of FAb compact p-adic analytic groups for p > 2;
see [10, Theorem 1.1].
In our set-up, Proposition 2.12 gives an Euler decomposition for the bivariate
representation zeta functions of the groups G(O) which is similar to (1.6). Furthermore, Theorem 1.7 assures that almost all of their local factors are rational and
satisfy functional equations, thus recovering some of the above mentioned results
for T2 -groups via specialisation (1.4).
Let now G ≤ GLn be a Z-defined algebraic subgroup which has the strong approximation property. In [2, Lemma 8.1], Berman, Derakhshan, Onn, and Paajanen
showed that the class number zeta function of G(O) satisfies an Euler decomposition. The proof methods also apply to groups of the form G(O), as defined in
Section 1.1, since unipotent groups have the strong approximation property; see [15,
Lemma 5.5]. This means that their class number zeta functions admit Euler decompositions of the form
Y
k
k
ζG(O)
(s) =
ζG(O
(s),
p)
p∈Spec(O)\{(0)}
with local factors
k
ζG(o)
(s)
=
∞
X
k(GN )q −N s .
N =0
It is also proved in [2, Theorem C] rationality for the local class number zeta functions of Chevalley groups G(o), where o is the valuation ring of a non-Archimedean
local field of any (sufficiently large) characteristic, and that these zeta functions
only depend on the size of the residue field of o.
In [20, Theorem 1.2], du Sautoy established rationality in p−s for the class number zeta function of any compact p-adic analytic group. Euler decomposition, and
rationality and functional equations for almost all local factors of class number zeta
functions of groups of the form G(O) follow from Proposition 2.12 and Theorem 1.7
together with specialisation (1.3). Rossmann proved independently in [18, Corollary 4.10 and Theorem 4.15], via specialisation of the ask zeta function, rationality
and functional equations for local factors of class number zeta functions of such
groups under mild assumptions on the group G(o) and the characteristic p of o/p.
9
Paula Lins
1.5. Notation. The following list collects frequently used notation.
N
N0
[n]
[n]0
[±n]
X, Y
(n)X
(n)X !
a
b X
(X; Y )n
gp(X)
I = {i1 , .. . , il }<
n
I
i0
il+1
µj
K
O
p
o = Op
on
pm
m (n)
(p )
vp
|.|p
k(zj )j∈J kp
o
Wk,N
Wko
{1, 2, . . . }
{0, 1, 2 . . . }
{1, . . . , n}
{0, 1, . . . , n}
{−n, . . . , −1} ∪ [n]0
indeterminates in the field Q(X, Y )
1 − X n , for n ∈ N
(n)X (n − 1)X . . . (1)X , for n ∈ N
(a)X
(b)X (a−b)X ,
Qn−1
i=0
1
1−X
for a ≥ b
(1 − XY i )
set I of nonnegative
integers i1 < i2 < · · · < il
n
il
i2
.
.
.
il il−1
i1
0
n
ij+1 − ij , j ∈ [l]0
number field
ring of integers of K
nonzero prime ideal of O
completion of O at p
nth Cartesian power o × · · · × o
mth ideal power p · · · p
nth Cartesian power pm × · · · × pm
p-adic valuation (see Appendix A)
p-adic norm
max(|zj |p )j∈J = q −vp ((zi )i∈I ) , J finite index set
{x ∈ (o/pN )k | vp (x) = 0}, k ∈ N, N ∈ N0
{x ∈ ok | vp (x) = 0}, k ∈ N
2. Bivariate zeta functions and p-adic integrals
Our results rely on the fact that local bivariate representation and bivariate
conjugacy class zeta functions of groups associated to unipotent group schemes can
be written in terms of p-adic integrals. The main goal of this section is to obtain
formulae for these local factors in terms of p-adic integrals using the methods of [27,
Section 2.2], which we now recall.
For the rest of the Section 2, p is a fixed nonzero prime ideal of O, and o
denotes the completion of O at p. Denote by q the cardinality of O/p and by p
its characteristic. Recall from Section 1.5 that vp denotes the p-adic valuation; see
Appendix A for its definition. For k ∈ N and N ∈ N0 , set
o
Wk,N
:= ((o/pN )k )∗ = {x ∈ (o/pN )k | vp (x) = 0},
Wko := (ok )∗
= {x ∈ ok | vp (x) = 0}.
Denote by π ∈ o a uniformiser of o, that is, an element such that p = πo. A
matrix M ∈ Matm×n (o/pN ), for some N ∈ N0 , is said to have elementary divisor
10
Bivariate zeta functions of T -groups
type (m1 , . . . , m ) if it is equivalent to the matrix
m
π 1
π m2
..
.
π m
,
where 0 ≤ m1 ≤ m2 ≤ · · · ≤ m ≤ N . Write ν(M ) = (m1 , . . . , m ) to indicate the
elementary divisors of M .
Given n ∈ N and a matrix R(Y ) = R(Y1 , . . . , Yn ) of polynomials R(Y )ij ∈ o[Y ]
with uR = max{rkFrac(o) R(z) | z ∈ on }, set, for m ∈ Nu0 R ,
(2.1)
o
NoN,R,m := {y ∈ Wn,N
| ν(R(y)) = m} and
o
NN,R,m
:= |NoN,R,m |.
o
The number NN,R,m
is zero, unless m = (m1 , . . . , muR ) satisfies 0 = m1 ≤ · · · ≤
muR ≤ N .
Consider the Poincaré series
X
PuR
o
(2.2)
Po,R (r, t) =
NN,R,m
q −tN − i=1 ri mi ,
N ∈N
u
m∈N0 R
where r = (r1 , . . . , ruR ) is a vector of variables.
In [27, Section 2.2] it is shown how to describe the series (2.2) in terms of the
following p-adic integrals.
Z
uR
Y
kFk (R(y)) ∪ xFk−1 (R(y))krpk
1
t
|x|
(2.3) Zo,R (r, t) =
dµ,
1 − q −1 (x,y)∈p×Wno p
kFk−1 (R(y))krpk
k=1
where µ is the additive Haar measure normalised so that µ(on+1 ) = 1. Moreover,
|.|p and k.kp are as in Section 1.5, and Fj (R(y)) denotes the set of nonzero (j × j)minors of R(y).
More precisely, in [27, Section 2.2] it is shown that the series (2.2) satisfy:
(2.4)
Po,R (r, t) = Zo,R (r, t − n − 1).
If M is an antisymmetric matrix, its elementary divisors come in pairs, so that
ν(M ) = (m1 , m1 , m2 , m2 , . . . , mξ , mξ ),
for some ξ ∈ N0 . If M is antisymmetric, we write ν̃(M ) = (m1 , m2 , . . . , mξ ) to
indicate its elementary divisor type.
Assume now that R(y) is antisymmetric, in which case uR is even. We then
denote by NR,m the same set (2.1), but we substitute ν(R(y)) for ν̃(R(y)) in this
uR
definition. We note that m is assumed to be an element of N0 2 in this case. We
o
also set NN,R,m
= |NN,R,m |. In this case, we assume that the vector of variables r
is of the form r = (r1 , r1 , . . . , r uR , r uR ) so that
2
(2.5)
Zo,R (r, t) =
2
X
o
NN,R,m
q −tN −2
P u2R
i=1
ri m i
.
N ∈N
uR
m∈N0 2
Given x ∈ o with vp (x) = N , y ∈ on , and k ∈ [uR ], we obtain from [18,
Lemma 4.6(i) and (ii)] the following for the antisymmetric matrix R(y) with
11
Paula Lins
ν̃(R(y)) = (m1 , . . . , muR ).
kF2k (R(y)) ∪ xF2k−1 (R(y))kp
kF2k−1 (R(y)) ∪ xF2(k−1) (R(y))kp
=
= q − min(mi ,N ) ,
kF2k−1 (R(y))kp
kF2(k−1) (R(y))kp
and
kF2k (R(y)) ∪ x2 F2(k−1) (R(y))kp
= q −2 min(mi ,N ) .
kF2(k−1) (R(y))kp
Therefore, if R(y) is an antisymmetric matrix, the series (2.5) can be described
by the p-adic integral
Po,R (r, t) = Zo,R (r, t − n − 1) =
uR
(2.6)
1
1 − q −1
Z
(x,y)∈p×Wno
|x|t−n−1
p
2
Y
kF2k (R(y)) ∪ x2 F2(k−1) (R(y))krpk
dµ.
kF2(k−1) (R(y))krpk
k=1
2.1. The numbers rn (GN ) and cn (GN ). We now write the local bivariate zeta
functions at p in terms of sums encoding the elementary divisor types of certain
matrices associated to Λ, called commutator matrices. This is done by rewriting the
numbers rn (GN ) and cn (GN ), for n ∈ N and N ∈ N0 , in terms of the cardinalities
o
NN,R,m
of the sets NoN,R,m defined at the beginning of Section 2. In each case, R
is one of the two commutator matrices of Λ which we now define.
Denote g = Λ(o) = Λ ⊗O o. Let g0 be the derived Lie sublattice of g, and let z
be its centre. Set
h = rk(g),
a = rk(g/z),
b = rk(g0 ),
r = rk(g/g0 ),
z = rk(z).
For R either O or o, let M be a finitely generated R-module with a submodule N .
The isolator ι(N ) of N in M is the smallest submodule L of M containing N such
that M/L is torsion free. In particular, the centre z of g coincides with ι(z); see [24,
Lemma 2.5]. Denote k = rk(ι(g0 )/ι(g0 ∩ z)) = rk(ι(g0 + z)/z).
The commutator matrices are defined with respect to a fixed o-basis B =
(e1 , . . . , eh ) of the o-Lie lattice g, satisfying the conditions
(ea−k+1 , . . . , ea ) is an o-basis for ι(g0 + z),
(ea+1 , . . . , ea−k+b ) is an o-basis for ι(g0 ∩ z), and
(ea+1 , . . . , eh ) is an o-basis for z.
Denote by the natural surjection g → g/z. Let e = (e1 , . . . , ea ). Then e =
(e1 , . . . , ea ) is an o-basis of g/z. There are nonnegative integers c1 , . . . , cb such that
(π c1 ea−k+1 , . . . , π ck ea ) is an o-basis of g0 + z and
(π
ck+1
ea+1 , . . . , π cb ea−k+b ) is an o-basis of g0 ∩ z,
by the elementary divisor theorem. Fix an o-basis f = (f1 , . . . , fb ) for g0 satisfying
(f1 , . . . , fk ) = (π c1 ea−k+1 , . . . , π ck ea ) is an o-basis of g0 + z and
(fk+1 , . . . , fb ) = (π ck+1 ea+1 , . . . , π cb ea−k+b ) is an o-basis of g0 ∩ z.
For i, j ∈ [a] and k ∈ [b], let λkij ∈ o be the structure constants satisfying
[ei , ej ] =
b
X
k=1
12
λkij fk .
Bivariate zeta functions of T -groups
Definition 2.1. [14, Definition 2.1] The A-commutator and the B-commutator
matrices of g with respect to e and f are, respectively,
a
X
A(X1 , . . . , Xa ) =
λkij Xj ∈ Mata×b (o[X]), and
j=1
B(Y1 , . . . , Yb ) =
b
X
ik
!
λkij Yk
k=1
∈ Mata×a (o[Y ]),
ij
where X = (X1 , . . . , Xa ) and Y = (Y1 , . . . , Yb ) are independent variables.
For each y ∈ ob , the matrix B(y) is antisymmetric.
Consider the congruence quotient GN = G(o/pN ). This is a finite p-group of
nilpotency class c. Denote gN := Λ ⊗o o/pN , and let zN denote the centre and
g0N the derived Lie sublattice of gN . Given an element w of the Pontryagin dual
×
gc
N = Homo (gN , C ), define the form
BωN : gN × gN → C× , (u, v) 7→ w([u, v]).
The radical of BωN is
Rad(BωN ) = {u ∈ gN | BωN (u, v) = 1, ∀v ∈ gN }.
For x ∈ gN /zN , the adjoint homomorphism adx : gN /zN → gc
N is given by adx (z) =
0 → g\
[z, x], for all z ∈ gN /zN . Let ad?x : gc
/z
be
the
map
w 7→ w ◦ adx . The
N N
N
dimensions of the irreducible complex representations and the sizes of conjugacy
classes of GN are powers of p and, according to [14, Section 3], for c < p, the
numbers rpi (GN ) and cpi (GN ) are given by
(2.7)
0 | |Rad(B N ) : z | = p−2i |g /z |}| · |g /g0 |p−2i ,
rpi (GN ) = |{w ∈ gc
N
N N
N
ω
N
N
(2.8)
0 |}| · |z |p−i .
cpi (GN ) = |{x ∈ gN /zN | |Ker(ad?x )| = p−i |gc
N
N
The first formula is a consequence of the Kirillov orbit method, which reduces
the problem of enumerating the characters of GN to the problem of determining
the indices in gN of Rad(BωN ) for w ∈ g0N ; see [14, Theorem 3.1]. The second
formula reflects the fact that the Lazard correspondence induces an order-preserving
correspondence between subgroups of GN and sublattices of gN , and maps normal
subgroups to ideals. Moreover, centralisers of GN correspond to centralisers of gN
under the Lazard correspondence.
The cardinalities of gN and gN /zN are powers of q, and hence so are the cardinalities of Rad(BωN )/z and Ker(ad?x ). It follows that rn (GN ) and cn (GN ) can only
be nonzero if n is a power of q.
The next step is to relate formulae (2.7) and (2.8) to the commutator matrices
of Definition 2.1. Tensoring the o-bases e and f with o/pN yields ordered sets
eN = (e1,N , . . . , ea,N ) and fN = (f1,N , . . . , fb,N ) such that (e1,N , . . . , ea,N ) is an
o/pN -basis for zN and fN is an o/pN -basis for g0N as o/pN -modules.
In [14, Section 2], the following coordinate system is given.
φ1 : g1 /z1 → (o/p)a ,
x=
a
X
xj ej,1 7→ x = (x1 , . . . , xa ),
j=1
ψ1 : gb01 → (o/p)b ,
y=
b
X
∨
yj fj,1
7→ y = (y1 , . . . , yb ),
j=1
where, for
0 =
dual gc
N
∨
∨
∨
N ∈ N0 , fN
= (f1,N
, . . . , fb,N
) is the
0
×
Homo (gN , C ). We notice that g1 /z1
o/p-dual basis for the Pontryagin
and g01 are regarded as o/p-vector
13
Paula Lins
spaces in this construction. By regarding gN /zN and g0N as o/pN -modules for all
N ∈ N, we can use similar arguments as the ones of [14, Section 2] to define the
following coordinate systems:
a
X
φN : gN /zN → (o/pN )a ,
x=
xj ej,N 7→ x = (x1 , . . . , xa ),
j=1
0 → (o/pN )b ,
ψN : gc
N
y=
b
X
∨
yj fj,N
7→ y = (y1 , . . . , yb ),
j=1
0 with φ (x) = x = (x , . . . , x ) and
Lemma 2.2. Given x ∈ gN /zN and w ∈ gc
N
1
a
N
ψN (w) = y = (y1 , . . . , yb ),
x ∈ Rad(BωN )/zN if and only if B(y)xtr = 0,
w ∈ Ker(ad?x ) if and only if A(x)ytr = 0.
Proof. An element x ∈ gN /zN is an element of Rad(BωN )/z exactly when w[x, v] = 1,
0 belongs to Ker(ad? ) exactly when
for all v ∈ gN /zN , whilst an element w ∈ gc
x
N
w[x, v] = 1 for all v ∈ gN /zN . Expressing these conditions in coordinates, we see
that both expressions hold.
We give details of the second equivalence’s proof. Fix x ∈ gN /zN with φN (x) =
x = (x1 , . . . , xa ) ∈ (o/pN )a . Then
b
a
a
a X
X
X
X
ei,N ,
λlij xj fl,N .
(2.9)
xj ej,N =
xj [ei,N , ej,N ] =
j=1
j=1
j=1 l=1
0 satisfy w([z, x]) = 1, for all z ∈
We want to determine which elements w ∈ gc
N
Qb
∨
gN /zN . Consider ψ(w) = y = (y1 , . . . , yb ), that is, w = k=1 (fk,N
)yk . Because
of (2.9), for each i ∈ [a],
yk
b
a X
b
b
Y
X
Y
y Pa λk xj
∨
∨
fk,N
λlij xj fl,N =
fk,N
(fk,N ) k j=1 ij .
w([ei , x]) =
k=1
j=1 l=1
k=1
Pb
Pa
This expression equals 1 exactly when k=1 yk j=1 λkij xj = 0. Now, by definition,
Pa
k
j=1 λij xj = A(x)ik , where A(x) is the A-commutator matrix of Definition 2.1
Pb
evaluated at x. Consequently, w ∈ Ker(ad?x ) if and only if k=1 A(x)ik yk = 0, for
all i ∈ [a], that is, A(x)ytr = 0.
Applying Lemma 2.2 to (2.7) and (2.8), we rewrite the numbers rqi (GN ) and
cqi (GN ), respectively, in terms of solutions of the system B(y)xtr = 0 and of the
system A(x)ytr = 0, considering in each case the elementary divisor type of the
corresponding matrix.
Fix an elementary divisor type ν̃(B(y)) = (m1 , . . . , muB ) ∈ [N ]u0 B , where
2uB = max{rkFrac(o) B(z) | z ∈ ob }.
Since B(y) is similar to the matrix Diag(π m1 , π m1 , . . . , π muB , π muB , 0a−2uB ), where
0a−2uB = (0, . . . , 0) ∈ Za−2uB , the system B(y)xtr = 0 in o/pN is equivalent to
x1
≡ x2 ≡ 0 mod pN −m1 ,
x 3
≡ x4 ≡ 0 mod pN −m2 ,
..
.
x2uB −1 ≡ x2uB ≡ 0 mod pN −muB .
14
Bivariate zeta functions of T -groups
For 2uB < a, the elements x2uB +1 , . . . , xa are arbitrary elements of o/pN , and
|{x ∈ o/pN | x ≡ 0 mod pN −mj }| = q mj .
Hence, the number of solutions of B(y)xtr = 0 in o/pN is q 2(m1 +···+muB )+(a−2uB )N .
In other words, ν̃(B(y)) = (m1 , . . . , muB ) implies
|Rad(BωN )/zN | = q 2(m1 +···+muB )+(a−2uB )N .
aN −2i
Lemma 2.2 then assures that |Rad(BωN )/zN | = q −2i |gP
, when B(y)
N /zN | = q
uB
has elementary divisor type (m1 , . . . muB ) satisfying j=1 mj = uB N − i.
Consequently, expression (2.7) can be rewritten as follows, for r = rk(g/g0 ) =
h − b.
(2.10)
rqi (GN ) =
X
|{y ∈ (o/pN )b | ν̃(B(y)) = (m1 , . . . , muB )}|q rN −2i .
0≤m1 ≤···≤muB ≤N
PuB
j=1 mj =uB N −i
Analogously, if ν(A(x)) = (m1 , . . . , muA ), where
uA := max{rkFrac(o) A(z) | z ∈ oa },
the equality A(x)y = 0 has q m1 +m2 +···+muA +(b−uA )N solutions in o/pN . For z =
rk(z) = h − a, this yields
(2.11)
cqi (GN ) =
X
|{x ∈ (o/pN )a | ν(A(x)) = (m1 , . . . , muA )}|q zN −i .
0≤m1 ≤···≤muA ≤N
PuA
j=1 mj =uA N −i
For a matrix R(Y ) = R(Y1 , . . . , Yn ) of polynomials as the one at the beginning
of Section 2 and for m = (m1 , . . . , m ) ∈ N0 , define
WoN,R,m := {y ∈ (o/pN )n | ν(R(y)) = m}.
Expressions (2.10) and (2.11) are written in terms of cardinalities of such sets,
o
of the sets NoN,R,m as follows. Denote
which are related to the cardinalities NN,R,m
m − m = (m1 − m, . . . , m − m), for all m ∈ N0 . If R(y) is such that vp (y) =
vp (R(y)), for all y ∈ on , then
|WoN,R,m | = NNo −m1 ,R,m−m1 .
(2.12)
Indeed, the map NoN −m1 ,R,m−m1 → WoN,R,m given by ỹ 7→ π m1 ỹ is a bijection.
Applying equality (2.12) to expressions (2.10) and (2.11), one obtains the following.
Lemma 2.3. For each i ∈ N0 and N ∈ N0 ,
X
(2.13)
rqi (GN ) =
NNo −m1 ,B,(0,m2 −m1 ,...,mu
B
−m1 ) q
A
−m1 ) q
rN −2i
,
0≤m1 ≤···≤muB ≤N
PuB
j=1 mj =uB N −i
(2.14)
cqi (GN ) =
X
NNo −m1 ,A,(0,m2 −m1 ,...,mu
zN −i
.
0≤m1 ≤···≤muA ≤N
PuA
j=1 mj =uA N −i
Remark 2.4. It was assumed that Λ0 ⊆ c!Λ for the construction of G = GΛ for
c > 2. As mentioned in Remark 1.9, in [24, Section 2.4] a different construction
of such unipotent group schemes is given for Λ of nilpotency class 2, in which
case the hypothesis Λ0 ⊆ 2Λ is not needed. For such group schemes of nilpotency
class 2 a Kirillov orbit method formalism was formulated which is valid for all
primes; see [24, Section 2.4.2]. In particular, [24, Lemma 2.13] assures that the
equalities (2.14) and (2.13) hold for all primes in nilpotency class 2.
15
Paula Lins
2.2. p-adic integrals. We now write the local factors of the bivariate zeta functions of G(O) in terms of Poincaré series such as (2.2).
Recall from Section 2.1 that the dimensions of irreducible complex representations as well as the sizes of the conjugacy classes of G(o) are powers of q, allowing
us to write the local terms of the (univariate) representation and conjugacy class
zeta functions of the congruence quotient GN = G(o/pN ) as
irr
(s) =
ζG
N
∞
X
rqi (GN )q −is and
i=0
cc
(s) =
ζG
N
∞
X
cqi (GN )q −is .
i=0
These sums are finite, since GN is a finite group. With (1.2), one obtains
irr
(s1 , s2 ) =
ZG(o)
cc
ZG(o)
(s1 , s2 ) =
∞ X
∞
X
N =0 i=0
∞ X
∞
X
rqi (GN )q −is1 −N s2 and
cqi (GN )q −is1 −N s2 .
N =0 i=0
Recall that z = rk(z) = h − a and r = rk(g/g0 ) = h − b. Expressions (2.13)
and (2.14) of Lemma 2.3 then yield
irr
(2.15) ZG(o)
(s1 , s2 ) =
∞ X
∞
X
X
NNo −m1 ,B,(0,m2 −m1 ,...,mu
B
−m1 ) q
A
−m1 ) q
(r−s2 )N −(2+s1 )i
,
N =0 i=0 0≤m1 ≤···≤muB ≤N
PuB
j=1 mj =uB N −i
cc
(2.16) ZG(o)
(s1 , s2 ) =
∞ X
∞
X
X
NNo −m1 ,A,(0,m2 −m1 ,...,mu
(z−s2 )N −(1+s1 )i
.
N =0 i=0 0≤m1 ≤···≤muA ≤N
PuA
j=1 mj =uA N −i
We now show how to rewrite these sums as Poincaré series of the form (2.2). In
preparation for this, we need two lemmas.
Lemma 2.5. Let s be a complex variable, (am )m∈N0 a sequence of real numbers,
and let q ∈ Z \ {1}. The following holds, provided both series converge.
!
∞
∞ N
−1
X
X
X
q −s
−sN
−sN
am q
=
aN q
.
1 − q −s
m=0
N =0
N =1
Proof. In fact,
(1 − q −s )
∞ N
−1
X
X
N =1 m=0
am q −sN =
∞ N
−1
X
X
am q −sN −
N =1 m=0
= a0 q −s +
= a0 q −s +
am q −s(N +1)
N =1 m=0
∞
X
N
X
N =1 m=0
∞
X
am q −s(N +1) −
aN q −s(N +1) =
N =1
16
∞ N
−1
X
X
∞ N
−1
X
X
am q −s(N +1)
N =1 m=0
∞
X
q −s
aN q −sN .
N =0
Bivariate zeta functions of T -groups
Lemma 2.6. Let s and t be complex variables. For a matrix R(Y ) = R(Y1 , . . . , Yn )
of polynomials R(Y )ij ∈ o[Y ] with u = max{rkFrac(o) R(z) | z ∈ on }. The following
holds, provided both series converge.
(2.17)
∞ X
∞
X
X
NNo −m1 ,R,(0,m2 −m1 ,...,mu −m1 ) q −sN −ti
N =0 i=0 0≤m
Pu 1 ≤···≤mu ≤N
j=1 mj =uN −i
∞
X
1
1 +
=
1 − q −s
X
o
NN,R,(m
q −(s+ut)N +t
1 ,m2 ,...,mu )
Pu
j=1
mj
.
N =1 (m1 ,...,mu )∈Nu
0
Proof. Denote m = (m1 , . . . , mu ) and recall the notation m − m = (m1 −
m, . . . , mu − m), for m ∈ N0 .
o
As NN,R,m
= 0 unless 0 = m1 ≤ m2 ≤ · · · ≤ mu ≤ N , in which case 0 ≤
Pu
Pu
m
≤
uN
, the condition j=1 mj = uN − i implies that the only values of
j
j=1
i which are relevant for the sum (2.17) are 0 ≤ i ≤ uN . Hence, the expression on
the left-hand side of (2.17) can be rewritten as
(2.18)
1+
∞ X
uN
X
X
NNo −m1 ,R,m−m1 q −sN −ti .
N =1 i=0 0≤m
Pu 1 ≤···≤mu ≤N
j=1 mj =uN −i
Restricting the summation in (2.18) to m1 = 0 leads
∞
X
X
o
NN,R,(0,m
q −sN −t(uN −
2 ,...,mu )
Pu
j=2
mj )
.
N =1 P
0≤m2 ≤···≤mu ≤N
u
j=2 mj ≤(u−1)N
o
= 0 unless 0 = m1 ≤ m2 ≤ · · · ≤ mu ≤ N allows us to
The fact that NN,R,m
rewrite this as
∞
X
X
N =1
o
NN,R,m
q −(s+ut)N +t
Pu
j=1
mj
=: S(s, t).
m∈Nu
0
Our goal now is to write the part of the summation in (2.18) with m1 > 0 in terms
of S(s, t). Denote m0 = (0, m02 , . . . , m0u ).
∞ X
uN
X
X
NNo −m1 ,R,m−m1 q −sN −ti
N =1 i=0 0<m
Pu 1 ≤···≤mu ≤N
j=1 mj =uN −i
=
∞ X
N
X
Pu
X
NNo −m1 ,R,m−m1 q −sN −t(uN −
j=1
mj )
N =1 m1 =1 m
1 ≤m2 ≤···≤mu ≤N
P
u
j=2 mj ≤uN −m1
=
∞ X
N
X
X
NNo −m1 ,R,m0 q −(s+ut)N +t((
Pu
j=2
m0j )+um1 )
N =1 m1 =1 0≤m02 ≤···≤m0u ≤N −m1
Pu
0
j=2 mj ≤u(N −m1 )
=
∞
X
N =1
q −sN
N
−1
X
X
t((
o
Nm,R,m
0q
Pu
j=2
m0j )−um)
m=0 0≤m02 ≤···≤m0u ≤m
Pu
0
j=2 mj ≤um
17
Paula Lins
=
(2.19)
∞
X
q −sN
N
−1
X
X
o
Nm,R,m
q t((
Pu
j=1
mj )−um)
.
m=0 m∈Nu
0
N =1
Apply Lemma 2.5 to expression (2.19) by setting
Pu
X
o
am :=
Nm,R,m
q t(( j=1 mj )−um) .
m∈Nu
0
This gives
∞ X
uN
X
X
NNo −m1 ,R,m−m1 q −sN −ti
N =1 i=1 0<m
Pu 1 ≤···≤mu ≤N
j=1 mj =uN −i
∞
Pu
X
X
q −s
o
1+
NN,R,m
q −(s+ut)N +t j=1 mj
=
1 − q −s
u
N =1 m∈N0
−s
=
q
(1 + S(s, t)) .
1 − q −s
Combining the expressions for the parts of the sum with m1 = 0 and m1 > 0 yields
∞ X
∞
X
X
o
NN,R,(m
q −sN −ti
1 ,m2 ,...,mu )
N =0 i=0 0≤m
Pu 1 ≤···≤mu ≤N
j=1 mj =uN −i
= 1 + S(s, t) +
q −s
1
(1 + S(s, t)) =
(1 + S(s, t)) .
1 − q −s
1 − q −s
Proposition 2.7. The bivariate representation and the bivariate conjugacy class
zeta functions of G(o) are given by
irr
ZG(o)
(s1 , s2 ) =
∞
X
X
1
1 +
r−s
2
1−q
(2.20)
N =1
(2.21)
1
1 +
1 − q z−s2
∞
X
o
NN,B,m
q −N (uB s1 +s2 +2uB −r)−2
(−s1 −2)
j=1 mj
2
,
u
(m1 ,...,muB )∈N0 B
cc
ZG(o)
(s1 , s2 )
PuB
=
X
o
NN,A,m
q −N (uA s1 +s2 +uA −z)−
PuA
j=1
mj (−s1 −1)
.
N =1 (m1 ,...,mu )∈NuA
0
A
Proof. By setting s = s2 − r, t = 2 + s1 and considering R to be the B-commutator
matrix of Λ in the left-hand side of expression (2.17), we obtain expression (2.15)
of the bivariate representation zeta function of G(o). Under these substitutions,
we obtain expression (2.20). Analogously, by setting s = s2 − z, t = 1 + s1 and
considering R to be the A-commutator matrix of Λ, the left-hand side of expression (2.17) yields expression (2.16) of the bivariate conjugacy class zeta function of
G(o), which yields (2.21).
Expression (2.20) is of the form (2.5) with t = uB s1 +s2 +2uB −r and rk = −s12−2
for each k ∈ [uB ], whilst (2.21) is (2.2) with t = uA s1 +s2 +uA −z and rk = −s1 −1
for each k ∈ [uA ]. Therefore these choices of t and r applied to (2.6) and to (2.4)
yields the following. Recall that a + z = rk(g/z) + rk(z) = rk(g) = h and that
b + r = rk(g0 ) + rk(g/g0 ) = h.
18
Bivariate zeta functions of T -groups
Proposition 2.8. The bivariate zeta functions of G(o) can be described by
irr
(s1 , s2 ) =
ZG(o)
1
(1 + Zo,B ((−s1 − 2)/2, uB s1 + s2 + 2uB − h − 1)) ,
1 − q r−s2
(2.22)
cc
ZG(o)
(s1 , s2 ) =
1
(1 + Zo,A (−s1 − 1, uA s1 + s2 + uA − h − 1)) .
1 − q z−s2
In particular, we obtain descriptions for the conjugacy class zeta function of g(o)
in terms of p-adic integrals via Specialisation (1.3):
cc
cc
(0, s) =
(s) = ZG(o)
ζG(o)
1
(1 + Zo,A (−1, s + uA − h − 1)) ,
1 − q z−s
which coincides with the p-adic integral obtained from the p-adic integral [18, (4.3)]
together with the specialisation given in [18, Theorem 1.7].
Example 2.9. Let H(O) be the Heisenberg group over O considered in Example 1.6.
Its commutator matrices with respect to the ordered bases e = (x1 , x2 ) and f = (y),
where e is a basis of the co-centre and f is a basis of derived Lie lattice of F1,0 (o) =
Gn (o) = Hn (o), are
"
#
"
#
X2
0
Y
A(X1 , X2 ) =
and B(Y ) =
.
−X1
−Y 0
The A-commutator matrix has rank 1 and the B-commutator matrix has rank 2
over the respective fields of rational functions, that is, uA = uB = 1. Moreover,
h = rk(F1,0 ) = 3, and
F2 (B(Y )) = {Y 2 }.
F1 (A(X1 , X2 )) = {−X1 , X2 },
In particular, if (x1 , x2 ) ∈ W2o , that is, vp (x1 , x2 ) = 0, then kF1 (A(x1 , x2 ))kp = 1.
Also, if y ∈ W1o , then, in particular, vp (y 2 ) = 0, which gives kF2 (B(y))kp = 1.
Therefore, using Proposition A.1, we obtain
!
Z
1
−1 −1
s1 +s2 −2
irr
1 + (1 − q )
|w|p
dµ
ZH(o) (s1 , s2 ) =
1 − q 2−s2
(w,y)∈p×W1o
=
cc
ZH(o)
(s1 , s2 )
1 − q −s1 −s2
,
(1 − q 1−s1 −s2 )(1 − q 2−s2 )
1
=
1 − q 1−s2
=
(1 −
1 + (1 − q
−1 −1
Z
)
(w,x1 ,x2 )∈p×W2o
!
|w|ps1 +s2 −3 dµ
1 − q −s1 −s2
,
− q 2−s1 −s2 )
q 1−s2 )(1
as claimed in Example 1.6.
4
2.3. Twist representation zeta functions. In this section, we assume in addition that the nilpotency class of G is c = 2 and we do not require Λ0 ⊆ 2Λ;
see Remarks 1.9 and 2.4. We provide a univariate specialisation of the bivariate
representation zeta function of G(o) which results in the twist representation zeta
function of this group.
According to [24, Corollary 2.11], the twist representation zeta function of G(o)
is given by
irr
ζG(o)
(s) = 1 + Zo,B (−s/2, uB s − b − 1),
where b = rk(g0 ), 2uB = max{rkFrac(o) B(z) | z ∈ ob } and Zo,B (r, t) is the integral
Zo,R (r, t) given in (2.6) when R(Y ) is the B-commutator matrix B(Y ). Recall
19
Paula Lins
that r = rk(g/g0 ) = h − b. We have shown in Proposition 2.8 that
irr
(1 − q r−s2 )ZG(o)
(s1 , s2 ) = 1 + Zo,B ((−2 − s1 )/2, uB s1 + s2 + 2uB − h − 1) .
irr
irr
Comparing the expressions for ζG(o)
(s) and (1 − q r−s2 )ZG(o)
(s1 , s2 ), we obtain the
desired specialisation.
Proposition 2.10. If G(o) has nilpotency class 2, then
irr
irr
(s).
(1 − q r−s2 )ZG(o)
(s1 , s2 ) |s1 →s−2 = ζG(o)
f
s2 →r
2.4. Local functional equations—proof of Theorem 1.7. Let L be a finite
extension of K = Frac(O), with ring of integers OL . For a fixed prime ideal P
of OL dividing p, write O for the localisation OL,P . Denote by f = f (O, o) the
relative degree of inertia, hence |O/P| = q f . Denote gL = Λ(O), and let zL and
g0L be, respectively, the centre and the derived Lie sublattice of gL . Since OL is a
ring of integers of a number field L, we can choose bases e and f of gL /zL and g0L ,
respectively, and define the commutator matrices of gL with respect to these bases
as in Definition 2.1. Consider the following functions.
^
irr
Z
G(O) (s1 , s2 ) := 1 + ZO,B ((−s1 − 2)/2, uB s1 + s2 + 2uB − h − 1) ,
cc
^
Z
G(O) (s1 , s2 ) := 1 + ZO,A (−s1 − 1, uA s1 + s2 + uA − h − 1)) ,
where ZO,B (r, t) and ZO,A (r, t) are the integrals given in (2.6) and (2.3), respectively, and A and B denote the commutator matrices of gL with respect to some
O bases e and f as above. We have shown in Proposition 2.8 that
1
irr
^
irr
ZG(O)
(s1 , s2 ) =
Z
(s1 , s2 ) and
1 − q f (r−s2 ) G(O)
1
cc
cc
^
Z
ZG(O)
(s1 , s2 ) =
(s1 , s2 ).
f
1 − q (z−s2 ) G(O)
Since, in particular for x = f (z − s2 ) and x = f (r − s2 ), the term (1 − q x )−1 is
rational and
1
1
x
−1 = −q
|
,
1 − q x q→q
1 − qx
^
irr
it thus suffices to show the relevant statement of Theorem 1.7 for Z
(s , s ) and
G(O)
cc
^
Z
G(O) (s1 , s2 ).
1
2
In fact, we only need to show that the p-adic integrals appearing in
the descriptions of the bivariate zeta functions in Proposition 2.8 fit the framework
of [24, Section 2.3] and [1, Section 4]. In other words, we must show that their
integrands are defined over O—hence only their domains of integration vary with
the ring O—and that these p-adic integrals can be expressed in terms of the integrals
given in [27, Section 2.1].
The condition that the integrands of the integrals of Proposition 2.8 are defined
over O is needed since the O-bases defined in Section 2.1 are only defined locally,
so that the matrices A(X) and B(Y ) are also defined locally. We must assure
that there there exist O-bases e and f as the ones of Section 2.1 such that the
commutator matrices A(X) and B(Y ), defined with respect with these e and f are
defined over O, and hence so are the sets of polynomials Fj (A(X)) and F2j (B(Y )).
Since the matrix B(Y ) is the same as the one appearing in the integrands of [24,
(2.8)] and A(X) is obtained in an analogous way, the argument of [24, Section 2.3]
also holds in this case. Namely, we choose an O-basis f for a free finite-index Osubmodule of the isolator i(Λ0 ) of the derived O-Lie sublattice of Λ; see Section 2.1.
By [24, Lemma 2.5], f can be extended to an O-basis e for a free finite-index Osubmodule M of Λ. If the residue characteristic p of p does not divide |Λ : M | or
20
Bivariate zeta functions of T -groups
|i(Λ0 ) : Λ0 |, this basis e may be used to obtain an O-basis for Λ(O), by tensoring
the elements of e with O.
Remark 2.11. The condition “p does not divide |i(Λ0 ) : Λ0 |” is missing in [24], but
this omission does not affect the proof of [24, Theorem A], since this condition
only excludes a finite number of prime ideals p. This was first pointed out in [5,
Section 3.3].
We now recall the general integrals given in [27, Section 2.1]. Fix m, n, l ∈ N.
For each k ∈ [l], let Jk be a finite index set. Fix I ⊆ [n − 1]. Further, fix nonnegative integers eikj and finite sets Fkj (Y ) of polynomials over o, for k ∈ [l], j ∈ Jk
and i ∈ I. Let also W(o) ⊆ om be a union of cosets modulo p(m) . Define
(2.23)
ZW(o),I (s) =
l
Y
Z
p(|I|) ×W(o) k=1
sk
!
[
Y
j∈Jk
i∈I
e
xi ikj
dµ,
Fkj (y)
p
where s = (s1 , . . . , sl ) is a vector of complex variables and x = (xi )i∈I and y =
(y1 , . . . , ym ) are independent integration variables.
In [27, Corollary 2.4], by studying the
Qtransformation of the integral (2.23) under
a principalization (Y, h) of the ideal k,j (Fkj (Y )), a functional equation under
inversion of the parameter q is proved, under certain invariance and regularity
conditions. In particular, it is required that the principalization (Y, h) has good
reduction modulo p.
We now relate the integrals of Proposition 2.8 with the general integral (2.23).
Set I = {1} and write x1 = x. Set n = b, m = b2 , l = 2uB + 1, and Jk = {1, 2},
if k ∈ [uB ] and Jk = {1} if uB < k ≤ 2uB + 1. We also set W(O) = GLb (O), and
k
≤ uB
uB < k ≤ 2uB
2uB + 1
≤ uB
j
1
1
1
2
Fkj
F2k (B(y))
F2(k−1−uB ) (B(y))
{1}
F2(k−1) (B(y))
e1kj
0
0 .
1
2
We see that, with this setup, the integral (2.23) is given by
Z
s2u +1
(2.24)
ZGLb (O),{1} (s) =
kxkP B ·
P×GLb (O)
uB
Y
k=1
kF2k (B(Y )) ∪ x2 F2(k−1) (B(Y ))ksPk
2u
YB
kF2(k−1−uB ) (B(Y ))ksPk dµ.
k=uB +1
Although the domain of integration of the integral (2.24) involves GLb (O), the
integrand only depends on the entries of the first rows, say, since the B-commutator
matrix is defined in b variables. Consequently, we can interpret WbO as the “space of
of first columns” of GLb (O) and we may consider the domain of integration of (2.24)
to be WbO as long as we correct the integral by multiplying it by the measure
1
1
of the remaining entries of matrices of GLb (O). Let airr
1 = (− 2 1uB , 2 1uB , uB ),
irr
irr
a2 = (0uB , 0uB , 1), b = (−1uB , 1uB , 2uB − h − 1), where 1uB = (1, . . . , 1) ∈ ZuB
and 0uB = (0, . . . , 0) ∈ ZuB . It follows that
!−1
b−1
Y
1
−k
irr
irr
^
irr
ZG(O) (s1 , s2 ) = 1+
(1 − q )
ZGLb (O),{1} airr
.
1 s1 + a2 s2 + b
−1
1−q
k=1
21
Paula Lins
Analogously, for n = a, m = a2 , one can find appropriate data l ∈ N, Jk , e1jk ,
and Fkj (X) such that
!
a−1
Y
1
−k
cc
cc
cc
^
(1 − q ) ZGLa (O),{1} (acc
ZG(O) (s1 , s2 ) = 1 +
1 s1 + a2 s2 + b ),
1 − q −1
k=1
for
acc
1
acc
2
= (−1uA , 1uA , uA ),
= (0uA , 0uA , 1), bcc = (−1uA , 1uA , uA − h − 1).
Theorem 1.7 then follows by the arguments given in [1, Section 4].
2.5. Euler products. In the previous sections, we have shown some properties
of local factors of the bivariate representation and conjugacy class zeta functions of
groups of the form G(O). In this section, we show that the global corresponding
zeta functions can be written as products of such local terms, allowing us to relate
local results to the global zeta functions.
Proposition 2.12. For ∗ ∈ {irr, cc} and s1 and s2 with sufficiently large real parts,
Y
∗
∗
ZG(O)
(s1 , s2 ) =
ZG(O
(s1 , s2 )
p)
p∈Spec(O)\{(0)}
Proof. It suffices to show that, for any ideal I of finite index in O with prime
decomposition I = pe11 · · · perr , with pi 6= pj if i 6= j, the following holds.
∗
ζG(O/I)
(s) =
r
Y
∗
ζG(O/p
ei ) (s).
i=1
Unipotent groups satisfy the strong approximation property; see [15, Lemma 5.5].
This gives an isomorphism
(2.25)
G(O/I) ∼
= G(O/pe1 ) × · · · × G(O/per ).
r
1
We first show the relevant statement of Proposition 2.12 for the representation case.
For a group G, denote by Irr(G) the set of irreducible characters of G. From (2.25),
Irr(G(O/I)) ∼
= Irr(G(O/pe1 )) × · · · × Irr(G(O/per )).
r
1
Since rn (G) = |{χ ∈ Irr(G) : χ(1) = n}|, it follows that
X
X
irr
ζG(O/I)
(s) =
χ(1)−s
=
χ∈Irr(G(O/I))
=
r
Y
k=1
χ1 (1)−s · · · χr (1)−s
e
∀i∈[r]:χi ∈Irr(G(O/pi i ))
X
χk (1)−s
e
χk ∈Irr(G(O/pi i ))
=
r
Y
irr
ek (s).
ζG(O/p
)
k
k=1
For the conjugacy class zeta function, we use the fact that each conjugacy class
C of G(O/pe11 ) × · · · × G(O/perr ) is of the form C = C1 × · · · × Cr , where Ci is a
conjugacy class of G(O/pei i ), for each i ∈ [r]. Thus
X
cn (G(O/I)) =
cn1 (G(O/pe11 )) · · · cnr (G(O/perr )).
n1 ,...,nr ∈N0
n1 ···nr =n
Consequently, setting qi = |O : pi |, and using the fact that all conjugacy classes of
G(O/pei i ) have size a power of qi , for all i ∈ [r],
cc
ζG(O/I)
(s)
=
∞
X
X
cq1n1 (G(O/pe11 )) · · · cqrnr (G(O/perr ))(q1n1 . . . qrnr )−s
n=1 n1 ,...,nr ∈N0
n
q1 1 ...qrnr =n
=
r
Y
!
cqnk (G(O/pekk ))qk−nk s
k
k=1
22
∞
X
nk =0
=
r
Y
k=1
cc
ek (s).
ζG(O/p
)
k
Bivariate zeta functions of T -groups
2.6. Convergence. Having defined and worked abstractly with the bivariate zeta
functions, it is natural to ask whether they converge on some domain. In this
section, we show that the bivariate zeta functions of a group G(O) associated to a
unipotent group scheme converge on some open domain of C2 .
If a complex sequence (an )n∈N grows atPmost polynomially, it is well known
∞
that the Dirichlet series D((an )n∈N , s) := n=1 an n−s converges for s ∈ C with
sufficiently large real part. We now show that an analogous result holds for double
Dirichlet series.
Definition 2.13. A double sequence (an,m )n,m∈N of complex numbers is said to
have polynomial growth if there exists positive integers α1 and α2 and a constant
C > 0 such that |an,m | < Cnα1 mα2 for all n, m ∈ N.
Proposition 2.14. If the double sequence (an,m )n,m∈N has polynomial growth,
then there exist α1 , α2 ∈ N such that the double Dirichlet series
D((an,m )n,m∈N , s1 , s2 ) :=
∞ X
∞
X
an,m n−s1 m−s2
n=1 m=1
converges absolutely for (s1 , s2 ) ∈ C2 satisfying Re(s1 ) > 1+α1 and Re(s2 ) > 1+α2 .
Proof. Let α1 , α2 ∈ N and C > 0 be such that an,m < Cnα1 mα2 , for all n, m ∈ N.
Then
XX
X X an,m
1
≤C
.
s
s
Re(s
)−α
1
2
1
1
n m
n
mRe(s2 )−α2
n m
n m
The relevant statement of Proposition 2.14 then follows from the fact that, for
p, q ∈ R, the harmonic double series
∞ X
∞
X
1
p
k lq
p=1 q=1
converges if and only if p > 1 and q > 1; see [7, Example 7.10(iii)].
Denote
rn,m (G(O)) =
X
rn (G(O/I)) and cn,m (G(O)) =
(0)6=IEO
|O:I|=m
X
cn (G(O/I)).
(0)6=IEO
|O:I|=m
The bivariate representation and conjugacy class zeta functions of G(O) are given
by the following double Dirichlet series.
irr
ZG(O)
(s1 , s2 ) =
irr
ZG(O)
(s1 , s2 ) =
∞
∞ X
X
n=1 m=1
∞ X
∞
X
rn,m (G(O))n−s1 m−s2 ,
cn,m (G(O))n−s1 m−s2 .
n=1 m=1
irr
cc
Proposition 2.15. The bivariate zeta functions ZG(O)
(s1 , s2 ) and ZG(O)
(s1 , s2 )
converge (at least) on some open domain of the form
{(s1 , s2 ) ∈ C2 | Re(s1 , s2 ) > (1 + α1 , 1 + α2 )},
for some constants α1 and α2 .
Proof. Denote γm (O) := |{I E O | |O : I|
m}|. The Dedekind zeta function of
P=
∞
the number field K is given by ζK (s) = m=1 γm m−s , and is known to converge
PM
for Re(s) > 1. In particular, the partial sums m=1 γm , for M ∈ N, are bounded
by a polynomial P(X) ∈ Z[X].
23
Paula Lins
Given I E O, the finite group G(O/I) is a principal congruence subgroup of a
torsion-free nilpotent and finitely generated group. Then there exists Q(X) ∈ Z[X]
such that, for all I E O, |G(O/I)| < Q(m), where m = |O : I| .
Given I E O, the finite group G(O/I) has at most |G(O/I)| conjugacy classes.
Consequently, for each (n, m) ∈ N2 ,
X
cn,m (G(O)) =
cn (G(O/I)) < P(m)Q(m).
(0)6=IEO
|O:I|=m
Analogously, rn,m (G(O)) < P(m)Q(m), since rn (G(O/I)) ≤ |G(O/I)|.
3. Results for groups of type F , G, and H
Throughout Section 3, the O-Lie lattice Λ is either Fn,δ , Gn , or Hn of Definition 1.8, for n ∈ N and δ ∈ {0, 1}, and G is the unipotent group scheme obtained
from Λ. Moreover, p stands for a fixed nonzero prime ideal of O and o = Op . Recall
the notation g = Λ(o). For Λ ∈ {Fn,δ , Gn , Hn }, the centre z and the derived Lie
sublattice g0 of g coincide. Recall that the torsion-free ranks of g/z and of g0 are
given by the following table:
a = rk(g/z) b = rk(g0 ) = rk(z) = z
Λ
2n+δ
Fn,δ
2n + δ
2
Gn
Hn
2n
2n
n2
n+1
2
3.1. Bivariate conjugacy class zeta functions—proof of Theorem 1.10. In
this section, denote the commutator matrix A(X) of Λ with respect to the bases
given in Definition 1.8 by AΛ (X), and fix n ∈ N and δ ∈ {0, 1}.
In the following, we provide separate formulae for the local factors of the conjugacy class zeta functions of each type of G ∈ {Fn,δ , Gn , Hn } by calculating explicitly
the corresponding integrals given by expression (2.22) of Proposition 2.8. We first
describe the A-commutator matrix of g, which is given with respect to the ordered
bases e = (x1 , . . . , xa ) and f = (yij )(i,j) ∈ DΛ , where
2
{(i, j) ∈ [2n + δ] | 1 ≤ i < j ≤ 2n + δ}, if Λ = Fn,δ ,
DΛ = [n]2 ,
if Λ = Gn ,
{(i, j) ∈ [n]2 | 1 ≤ i ≤ j ≤ n},
if Λ = Hn .
We order the set f by setting yij > ykl , whenever either i < k or i = k and j < l.
We then write f = (yij )(i,j)∈DΛ = (f1 , . . . , fb ) so that f1 > f2 · · · > fb .
Lemma 3.1. Let ωΛ : DΛ → [b] denote the map satisfying yij = fω(i,j) . Then
i−1
(i − 1)a − 2 + j − i, if Λ = Fn,δ ,
ωΛ (i, j) = (i − 1)n + j,
if Λ = Gn ,
i
(i − 1)n − 2 + j,
if Λ = Hn .
Proof. For Λ = Fn,δ , the ordering of the yij is described by the following identities.
ωFn,δ (1, j) = j − 1,
ωFn,δ (i + 1, i + 2) = ωFn,δ (i, n) + 1,
for all j ∈ {2, . . . , a},
for all i ∈ [a − 2],
and
ωFn,δ (i, j) = ωFn,δ (i, i + 1) + j − (i + 1), for all i ∈ [a − 1], j ∈ {i + 1, . . . , a}.
It follows by induction that ωFn,δ (i, j) = (i − 1)a − i−1
+ j − i.
2
The other cases follow from similar arguments.
24
Bivariate zeta functions of T -groups
Let X = (X1 , . . . , Xa ) be a vector of variables and, for m ∈ [a], set
CΛ,m = {ωΛ (m, j) | (m, j) ∈ DΛ }.
(m)
We want to determine the submatrix AΛ (X) of AΛ (X) composed by the columns
of index in CΛ,m so that
h
i
(2)
(a−1)
,
AFn,δ (X) = A(1)
(X)
A
(X)
.
.
.
A
(X)
Fn,δ
Fn,δ
Fn,δ
i
h
(2)
(n)
AΛ (X) = A(1)
,
Λ (X) AΛ (X) . . . AΛ (X)
for Λ ∈ {Gn , Hn }. For Λ ∈ {Fn,δ , Gn , Hn }, the matrices AΛ (X) all have size a × b.
Denote
m−1
if Λ = Fn,δ
(m − 1)a − 2 ,
νΛ,m = (m − 1)n,
if Λ = Gn
(m − 1)n − m
+
m
−
1,
if Λ = Hn ,
2
so that CΛ,m = {νΛ,m + 1, . . . , νΛ,m + kΛ,m }, where
if Λ = Fn,δ ,
a − m,
kΛ,m = n,
if Λ = Gn ,
n − m + 1, if Λ = Hn ,
(m)
that is, the jth column of AΛ (X) is the (νΛ,m + j)th column of AΛ (X).
Recall the definition of the structure constants λkij in the discussion before Definition 2.1. The relations of Λ show that, for (i, j) ∈ DΛ and for k ∈ CΛ,m , the
structure constants involving (i, j) are the ones in the following table:
Λ
Fn,δ
Gn
λki(n+j)
Hn
structure constants involving (i, j)
(
1, if k = ωFn,δ (i, j),
k
λij =
0, otherwise,
(
1, if k = ωGn (i, j),
λki(n+j) =
0, otherwise,
(
1, if k = ωHn (i, j),
= λkj(n+i) =
0, otherwise.
Since ωΛ (i, j) ∈ CΛ,i , for each (i, j) ∈ DΛ , it is clear that the indices k ∈ CΛ,m
(m)
of the columns of AΛ (X) cannot equal ωΛ (i, j) if i 6= m. In particular, λkij = 0 if
i, j 6= m. Every k ∈ CΛ,m is of the form k = νΛ,m + l, for some l ∈ [kΛ,m ]. Recall
(m)
that the (i, l)th entry of AΛ (X) is the (i, νΛ,m + l)th entry of AΛ (X), that is,
(m)
AΛ (X)il = AΛ (X)i(νΛ,m +l) .
In particular, for Λ = Fn,δ , the index k coincides with ωFn,δ (m, j) = νFn,δ ,m +
j − m if and only if j = l + m. It follows that λkij = 1 if and only if i = m and
(m)
j = m + l. Hence the (m, l)th entry of AFn,δ (X) is
(m)
AFn,δ (X)ml =
a
X
νF
λmjn,δ
,m +l
Xj = Xm+l ,
j=1
and, for i 6= m, its (i, l)th entry is
(m)
AFn,δ (X)il
=−
a
X
j=1
νF
,m +l
λji n,δ
Xj
(
−Xm ,
=
0,
if i = m + l,
otherwise.
25
Paula Lins
Given s, r ∈ N, let 0s×r denote the (s × r)-zero matrix and let 1s denote the
(s × s)-identity matrix, both over o[X]. It follows that, for each m ∈ [a − 1],
(m)
AFn,δ (X)
0(m−1)×(2n+δ−m)
X
X
.
.
.
X
= m+1
m+2
2n+δ ∈ Mat(2n+δ)×(2n+δ−m) (o[X]).
−Xm 1(2n+δ−m)
For Λ = Gn , the index k coincides with ωGn (m, j) = νGn ,m + j if and only if
j = l. It follows that λki(n+j) = 1 if and only i = m and j = l. Hence the (m, l)th
(m)
entry of AGn (X) is
(m)
AGn (X)ml
=
a
X
ν
+l
Gn ,m
λmj
Xj = Xn+l ,
j=1
and, for i 6= m, its (i, l)th entry is
(m)
AGn (X)il
=−
a
X
(
ν
+l
λjiGn ,m Xj
=
j=1
Hence, for each m ∈ [n],
(3.1)
−Xm ,
0,
0(m−1)×n
Xn+1
(m)
AGn (X) =
Xn+2
if i = n + l,
otherwise.
X2n
∈ Mat2n×n (o[X]).
...
0(n−m)×n
−Xm 1n
Finally, for Λ = Hn , the index k coincide with ωHn (m, j) = νHn ,m + j − m + 1 if
and only if j = m + l − 1. If follows that λki(n+j) = λkj(n+i) = 1 if and only if either
i = m and j = m + l − 1 or j = m and i = m + l − 1. Therefore
(m)
AHn (X)ml =
a
X
ν
+l
Hn ,m
λmj
Xj = Xn+m+l−1 ,
j=1
(m)
AHn (X)(n+m)l
=−
a
X
ν
+l
Hn ,m
λj(n+m)
Xj = −Xm+l−1 .
j=1
(m)
For i ∈ [n] \ {m}, the (i, l)th entry of AHn (X) is
(
n
X
Xn+m ,
νHn ,m +l
(m)
AHn (X)il =
λj(n+i) Xn+j =
0,
j=1
if i = m + l − 1,
otherwise.
(m)
For i = n + t with t ∈ [n] \ {m}, the (i, l)th entry of AHn (X) is
(m)
AHn (X)il
=−
n
X
j=1
26
(
νHn ,m +l
λj(n+t)
Xj
=
−Xm ,
0,
if t = m + l − 1,
otherwise.
Bivariate zeta functions of T -groups
Hence, for each m ∈ [n],
(3.2)
0(m−1)×(n−m+1)
Xn+m
(m)
AHn (X) =
−Xm
Xn+m+1
Xn+m
...
..
.
0(m−1)×(n−m+1)
−Xm+1
−Xm
...
..
.
X2n
Xn+m
∈ Mat2n×(n−m+1) (o[X]).
−Xn
−Xm
Example 3.2. The following examples illustrate the form of the commutator matrix
for each type of group scheme.
X2
X3
X4
−X1
X3
X4
,
AF2,0 (X) =
−X1
−X2
X4
−X1
−X2 −X3
X4
X5
X6
X4
X5
X6
X4
X5
X6
AG3 (X) =
,
−X1
−X
−X
2
3
−X
−X
−X
1
2
3
−X1
X4
AH3 (X) =
−X1
X5
−X2
X6
X4
X5
X4
−X2
X6
X5
X6
−X3
−X1
−X2
−X1
−X3
−X3
−X2
,
−X3
where the omitted entries equal zero.
4
It is not difficult to see that AΛ (X) has rank a − 1 in all cases.
We now proceed to a detailed analysis of the A-commutator matrix in each
individual type.
3.1.1. Conjugacy class zeta functions of groups of type F .
Lemma 3.3. For w ∈ p and x ∈ Wao , that is, for x ∈ oa such that vp (x) = 0,
(3.3)
kFk (AFn,δ (x)) ∪ wFk−1 (AFn,δ (x))kp
= 1, for all k ∈ [a − 1].
kFk−1 (AFn,δ (x))kp
27
Paula Lins
Proof. The columns of AFn,δ (X) are of the form
kth row
Xj
, for each j, k ∈ [a],
(3.4)
−Xk
jth row
where the nondisplayed entries equal 0.
For each i ∈ [a], consider the (a×(a−1))-submatrix Ki (X) of AFn,δ (X) composed
of the columns of expression (3.4) in the following order. The first i − 1 columns
are the ones with j = i and k ∈ [i − 1] being chosen in the increasing order, then
the next a − i columns are the ones with k = i and j ∈ {i + 1, . . . , a}. The matrix
Ki (X) is the matrix with diagonal given by Xi in the first i − 1 entries and −Xi
in the remaining diagonal entries, and the other nontrivial entries are the ones of
row i, which is given by
(−X1 , . . . , −Xi−1 ,
−Xi
|{z}
, Xi+1 , . . . , Xa ).
diagonal term
Given x ∈ Wao , it is clear that, for at least one i0 ∈ [a], the matrix Ki0 (x) has
rank a − 1. That is, for each k ∈ [a − 1], there exists a (k × k)-minor of Ki0 (x)
which is a unit. Since the (k × k)-minors of Ki0 (x) are elements of Fk (AFn,δ (X)),
expression (3.3) follows.
Lemma 3.3 applied to equality (2.22) and Proposition A.1 yield
ZFccn,δ (o) (s1 , s2 )
=
1
1 − q(
2n+δ
2
)−s2
1 + (1 − q
−1 −1
Z
)
(2n+δ−1)s1 +s2 −(2n+δ
2 )−2
|w|p
dµ
!
o
(w,x)∈p×W2n+δ
2n+δ−1
2
=
)−(2n+δ−1)s1 −s2
1 − q(
,
)−s2 )(1 − q (2n+δ
2 )+1−(2n+δ−1)s1 −s2 )
(1 − q (
2n+δ
2
proving Theorem 1.10 for groups of type F .
Remark 3.4. The formula (1.5) reflects the K-minimality of Λ = Fn,δ ; see [18,
Definition 6.2 and Lemma 6.3]. In fact, the proof of Lemma 3.3 shows in particular
that, for G = Fn,δ ,
kFk (AFn,δ (x)) ∪ yFk−1 (AFn,δ (x))kp
= kx, ykp .
kFk−1 (AFn,δ (x))kp
The formula for the class number zeta function of Fn,δ (O) given in Corollary 1.11
then coincides with the formula for the class number zeta function of Fn,δ (o) given
by the specialisation of the formula given in [18, Proposition 6.4] to the correspondent class number zeta function.
3.1.2. Conjugacy class zeta functions of groups of type G. We first describe the
determinant of a square matrix in terms of its 2 × 2-minors, which will be used
f(i,j),(r,s) =
to describe the minors of AGn (X). For a matrix M = (mij ), denote M
mij mis
.
mrj mrs
28
Bivariate zeta functions of T -groups
Lemma 3.5. Given t ∈ N, let G = (gij )1≤i,j≤2t and U = (uij )1≤i,j≤2t+1
be matrices with gij = g(X)ij , uij = u(X)ij ∈ o[X]. Let i = {i1 , . . . , it },
j = {j1 , . . . , jt } ⊂ [2t]. Then, for suitable αi,j , βi,j ∈ {−1, 1},
X
e (1,i ),(2,j ) G
e (3,i ),(4,j ) · · · G
e (2t−1,i ),(2t,j ) ,
det(G) =
αi,j G
2
2
t
t
1
1
i∪j=[2t]
iq <jq , ∀q∈[t]
det(U ) =
2t+1
X
X
e(1,i ),(2,j ) U
e(3,i ),(4,j ) · · · U
e(2t−1,i ),(2t,j ) .
βi,j u1i U
1
1
2
2
t
t
i=1 i∪j=[2t+1]\{i}
iq <jq , ∀q∈[t]
b I,J the
Proof. Given two subsets I, J ⊆ [2t] of equal cardinality m, denote by G
determinant of the (2t − m) × (2t − m)-submatrix of G obtained by excluding the
rows of indices in I and columns of index in J. The entries of the submatrix
G{1},{k} = (g̃ij )ij obtained from G by excluding its first row and its kth column
are given by
(
g(i+1)j ,
if j ∈ [k − 1],
g̃ij =
g(i+1)(j+1) , if j ∈ {k, . . . , 2t − 1}.
Consequently,
b {1},{k} =
G
k−1
X
b {1,2},{j,k} +
(−1)1+j g2j G
j=1
2t−1
X
b {1,2},{k,j+1} .
(−1)1+j g2(j+1) G
j=k
It follows that
2t
X
b {1},{k}
det(G) =
(−1)1+k g1k G
k=1
=
=
2t
X
k−1
X
b {1,2},{j,k} −
(−1)k+j g1k g2j G
k=1
j=1
2t−1
X
2t
X
2t
X
b {1,2},{k,j}
(−1)k+j g1k g2j G
j=k+1
b {1,2}{m,i}
(−1)i+m−1 (g1m g2i − g1i g2m )G
m=1 i=m+1
=
2t−1
X
2t
X
e (1,m),(2,i) G
b {1,2},{m,i} .
(−1)i+m−1 G
m=2 i=m+1
By induction on t, the relevant claim in Lemma 3.5 for the matrix G holds.
The claim for the matrix U follows by the first part, since its determinant is
det(U ) =
2t+1
X
b{1},{i} .
(−1)i+1 u1i U
i=1
Lemma 3.6. For each r ∈ [2n], the elements of Fr (AGn (X)) are either of one of
the following forms or a sum of two of these terms.
Xi1 . . . Xiω Xn+j1 . . . Xn+jλ or − Xi1 . . . Xiω Xn+j1 . . . Xn+jλ .
Proof. Lemma 3.5 describes each element of Fk (AGn (X)) in terms of sums of products of (2 × 2)-minors of AGn (X). It then suffices to show that these minors are
all either 0 or of the forms Xi Xj or −Xi Xj , for some i, j ∈ [2n]. This can be seen
from the description of AGn (X) in terms of the blocks (3.1).
The main idea of the proof of Theorem 1.10 for groups of type G is showing the
following proposition.
29
Paula Lins
Proposition 3.7. Let X = (X1 , . . . , X2n ) be a vector of variables. Given λ, ω ∈
[n]0 such that 0 < ω + λ ≤ 2n − 1, for all choices of i1 , . . . , iω , j1 , . . . , jλ ∈ [n],
one of
Xi1 . . . Xiω Xn+j1 . . . Xn+jλ or − Xi1 . . . Xiω Xn+j1 . . . Xn+jλ
is an element of Fω+λ (AGn (X)).
In fact, for x, y ∈ o, min{vp (x + y), vp (x), vp (y)} = min{vp (x), vp (y)}. Thus, if
some term of the form
Xi1 Xi2 . . . Xiω Xn+j1 . . . Xn+jλ − Xk1 Xk2 . . . Xkω Xn+l1 . . . Xn+lλ
is a minor of the commutator matrix AGn (X), then, assuming the claim in Proposition 3.7 holds, both
Xi1 Xi2 . . . Xiω Xn+j1 . . . Xn+jλ and Xk1 Xk2 . . . Xkω Xn+l1 . . . Xn+lλ
are minors of this commutator matrix (up to sign), and hence, when considering these three terms, only the last two will be relevant in order to determine kFr (AGn (X))kp . In this case, we may then assume that all elements
are of the form given in Proposition 3.7 while computing kFr (AGn (X))kp and
kFr (AGn (X)) ∪ wFr−1 (AGn (X))kp .
Firstly we show that Proposition 3.7 holds if |{i1 , . . . , iω }|, |{j1 , . . . , jλ }| =
6 n.
Lemma 3.8. Let ω, λ ∈ [n]0 not both zero and not both n. Given i1 , . . . , iω , j1 ,
. . . , jλ ∈ [n] such that |{i1 , . . . , iω }|, |{j1 , . . . , jλ }| < n, either
Xi1 · · · Xiω Xn+j1 · · · Xn+jλ or − Xi1 · · · Xiω Xn+j1 · · · Xn+jλ
is an element of Fλ+ω (AGn (X)).
Proof. For each (i, j) = (i1 , . . . , iω , j1 , . . . , jλ ) as in the assumption of Lemma 3.8,
we construct explicitly a submatrix of AGn (X) which is, up to reordering of rows
and columns, of the form
Xn+j1
..
T (X)
.
X
n+j
λ
(3.5)
−X
i1
..
W (X)
.
−Xiω
where T (X) = (t(X)ij ) and W (X) = (w(X)ij ) are such that t(X)ij = 0 and
w(X)ij = 0, if i ≤ j. It is clear that the determinant of this matrix is one of
±Xi1 · · · Xiω Xn+j1 · · · Xn+jλ .
The main fact we use is that the columns of AGn (X) are of the form
ith row
(3.6)
Xn+j
,
(n + j)th row −Xi
where the nondisplayed terms equal zero. For each i, j ∈ [n], there is exactly one
column of AGn (X) with Xn+j in the ith row, and exactly one column with −Xj in
the (n + i)th row.
Fix l1 ∈ [n] \ {i1 , . . . , iω } and let c1 denote the unique column of AGn (X) with
Xn+j1 in the l1 th row. Inductively, fix lk ∈ [n] \ {l1 , . . . , lk−1 , ik , . . . , iω }, for each
k ∈ [λ], and let ck be the unique column of AGn (X) with Xn+jk in the lk th row.
30
Bivariate zeta functions of T -groups
Analogously, fix m1 ∈ [n] \ {j1 , . . . , jλ } and let C1 be the index of the unique
column of AGn (X) with −Xi1 in the (n + m1 )th row, and, inductively, fix mq ∈
[n] \ {m1 , . . . , mq−1 , jq , . . . , jλ }, for each q ∈ [ω], and let Cq be the index of the
unique column of AGn (X) with −Xiq in the (n + mq )th row.
From expression (3.6), one sees that the columns ck and Cq are given by
Cq
ck
z}|{
z}|{
iq th row
lk th row
Xn+mq
Xn+jk
.
(3.7)
(n + jk ) th row −Xlk (n + mq ) th row −Xiq
By construction, the indices ck are all distinct, and so are the indices Cq . If ck = Cq
for some k ∈ [λ] and some q ∈ [ω], then we would obtain lk = iq . Analogously, the
indices l1 , . . . , lλ , n + m1 , . . . , n + mω are all distinct.
Consider the matrix M(i,j) (X) composed of columns ck and Cq and of rows lk
and n + mq , for k ∈ [λ] and q ∈ [ω]. This matrix is of the form of (3.5) for some
matrices T (X) ∈ Matλ×ω (o[X]) and W (X) ∈ Matω×λ (o[X]). Let us show that, in
fact, t(X)ij = 0 and w(X)ij = 0 for i ≤ j.
The only nonzero entries of Cq are the ones of indices iq and n + mq . We chose
each lk so that lk ∈
/ {i1 , . . . , ik }. Since any of the rows l1 , . . . , lq is the iq th row of
AGn (X), it follows that t(X)iq = 0, for all i ≤ q. Analogously, since the only nonzero
entries of ck are lk and n + jk and mq ∈
/ {j1 , . . . , jq }, it follows that w(X)ik = 0,
for all i ≤ k.
Proof.of Proposition 3.7. Lemma 3.8 shows the claim of Proposition 3.7 for all
cases, except for ω = n and i1 , . . . , in all distinct, and for λ = n and j1 , . . . , jn all
distinct. Let us show the last case, the other one is analogous.
Assume that j1 , . . . , jn are all distinct and ω ∈ [n−1]0 . For k ∈ [n], we can define
lk as in the proof of Lemma 3.8, since |{i1 , . . . , iω }| < n. We also set ck as in the
proof of Lemma 3.8. As |{j1 , . . . , jn }| = n, we cannot choose m1 ∈ [n]\{j1 , . . . , jn }.
Instead, we consider the rows n+jk , for k ∈ [ω]. Denote by Cq the column of AGn (X)
with −Xiq in the (n + jq )th row. By construction, the indices ck , for k ∈ [n], are
all distinct, and so are the indices Cq , for q ∈ [ω]. The indices ck and Cq coincide,
for some k ∈ [n] and q ∈ [ω], if and only if iq = lq . It follows that all ck and Cq are
distinct. Let Mi,j (X) be the submatrix of AGn (X) composed by columns ck and Cq
and of rows lk and n + jq , for each k ∈ [n] and q ∈ [λ], where i = (i1 , . . . , iλ ) and
j = (j1 , . . . , jn ).
Then, as in Lemma 3.8, B(X) is of the form (3.5), but the matrix W (T ) is such
that w(X)ij = 0 if i 6= j.
In particular, Proposition 3.7 shows that, for each r ∈ [2n] and each k ∈ [n],
o
either Xrk or −Xrk is an element of Fk (AGn (X)). Hence, if x ∈ W2n
, then at least
one (k × k)-minor of AGn (x) has valuation zero. This gives
(3.8)
kFk (AGn (x)) ∪ wFk−1 (AGn (x))kp
= 1, for all k ∈ [n].
kFk−1 (AGn (x))kp
For k ∈ {n + 1, . . . , 2n − 1}, the elements of Fk (AGn (X)) can be assumed to be of
the form
Xi1 · · · Xiω Xn+j1 · · · Xn+jλ ,
where ω, λ ∈ [n]0 satisfy ω + λ = k, and i1 , . . . , iω ,j1 , . . . , jλ ∈ [n].
31
Paula Lins
o
Given x ∈ W2n
, denote by M = vp (x1 , . . . , xn ) and N = vp (xn+1 , . . . , x2n ).
Then
[
{Xi1 · · · Xiω Xn+j1 · · · Xn+jλ | i1 , . . . , iω , j1 , . . . , jλ ∈ [n]}
ω+λ=k
0≤ω,λ≤n
=q
p
−n min{M,N }−(k−n) max{M,N }
.
Consequently, for w ∈ p,
(3.9)
(
kx1 , . . . , xn , wkp ,
kFk (AGn (x)) ∪ wFk−1 (AGn (x))kp
=
kFk−1 (AGn (x))kp
kxn+1 , . . . , x2n , wkp ,
if 0 = N ≤ M,
if 0 = M ≤ N.
Combining equations (3.8) and (3.9) yields
(
2n−1
Y kFk (AG (x)) ∪ wFk−1 (AG (x))kp
kx1 , . . . , xn , wkpn−1 ,
n
n
=
kFk−1 (AGn (x))kp
kxn+1 , . . . , x2n , wkpn−1 ,
k=1
if 0 = M ≤ N,
if 0 = N ≤ M.
Consequently, the p-adic integral given in expression (2.22) in this case is
Z
2n−1
Y kFk (AGn (x)) ∪ wFk−1 (AGn (x))kp−1−s1
(2n−1)s1 +s2 −n2 −2
|w|p
dµ
o
kFk−1 (AGn (x))kp−1−s1
(w,x)∈p×W2n
k=1
Z
(2n−1)s1 +s2 −n2 −2
−(n−1)(1+s1 )
=2
|w|p
kx1 , . . . , xn , wkp
dµ
(w,x1 ,...,x2n )∈p×pn ×Wno
Z
(2n−1)s1 +s2 −n2 −2
|w|p
+
dµ
(w,x1 ,...,x2n )∈p×Wno ×Wno
2
2
= 1 − q −n + 2q −1+(n−1)s1 − q n −ns1 −s2 − q n −n−ns1 −s2 ·
2
(1 − q −1 )(1 − q −n )q n +1−(2n−1)s1 −s2
,
(1 − q n2 +1−(2n−1)s1 −s2 )(1 − q n2 −ns1 −s2 )
where the first and the second integrals of the second equality are calculated, respectively, in Proposition A.2 and Proposition A.1. Applying this to formula (2.22),
we obtain
cc
ZG
(s1 , s2 ) =
n (o)
n
n
(1 − q 2( 2 )−ns1 −s2 )(1 − q 2( 2 )+1−(2n−1)s2 −s2 ) + q n −ns1 −s2 (1 − q −n )(1 − q −(n−1)(1+s1 ) )
,
(1 − q n2 −s2 )(1 − q n2 −ns1 −s2 )(1 − q n2 +1−(2n−1)s1 −s2 )
2
proving Theorem 1.10 for groups of type G.
3.1.3. Conjugacy class zeta functions of groups of type H. In this section, we
denote by A(X)ij the (i, j)th coordinate of the commutator matrix AHn (X).
By equality (3.2), each column of AHn (X) is of one of the following forms:
sth row
Xn+r
sth row
X
n+s
Xn+s
rth
row
,
,
(3.10)
(3.11)
−Xr
(n
+
s)th
row
(n + s)th row
−Xs
(n + r)th row −Xs
32
Bivariate zeta functions of T -groups
where the nondisplayed entries equal zero. These columns have the following symmetry:
(
−Xj , if and only if A(X)ik = Xn+j ,
(3.12)
A(X)(n+i)k =
0,
if and only if A(X)ik = 0.
For each s ∈ [n], there is exactly one column of the form (3.10), and the columns
of type (3.11) occur exactly once for each pair s < r of elements of [n].
o
Lemma 3.9. For w ∈ p, x ∈ W2n
and k ∈ [n],
kFk (AHn (x)) ∪ wFk−1 (AHn (x))kp
= 1.
kFk−1 (AHn (x))kp
Proof. Fix m ∈ [n]. For each q ∈ [m − 1], denote by Cq the index of the unique
(m)
column of AHn (X) which has Xn+m in the qth row. Recall that AHn (X) is the
submatrix of AHn (X) given in (3.2). The submatrix Lm (X) of AHn (X) composed
(m)
of columns C1 , . . . , Cm−1 and the columns of AHn (X) and rows 1, . . . , n is
C1
z }| {
Xn+m
Lm (X) =
Xn+1
Cm−1
z}|{ z
C2
z}|{
(m)
AHn (X)
}|
{
Xn+m
..
Xn+2
.
...
Xn+m
Xn+m−1
Xn+m
Xn+m+1
Xn+m
...
..
X2n
.
.
Xn+m
o
W2n
,
If x ∈
then there exists m0 ∈ [n] such that the matrix Lm0 (x) has maximal
rank n, that is, for each k ∈ [n], at least one of the (k × k)-minors of Lm0 (x) is a
unit. Since the (k × k)-minors of Lm0 (x) are elements of Fk (AHn (x)), the result
follows.
In the next lemma, we show that the sets Fn+l (AHn (X)), for l ∈ [n−1], are given
in terms of linear combinations of products of (i, j)-minors Mij (X) := Xi Xn+j −
Xj Xn+1 of the following matrix
"
#
X1
X2
. . . Xn
M (X1 , . . . , X2n ) =
∈ Mat2×n (o[X1 , . . . , X2n ]).
Xn+1 Xn+2 . . . X2n
Lemma 3.10. Let k = n + l, for some l ∈ [n − 1]. Then the nonzero elements of
Fk (AHn (X)) are sums of terms of the form
Xf1 . . . Xfr Mi1 j1 (X) . . . Mis js (X),
for i1 , . . . , is , j1 , . . . , js ∈ [n], and f1 , . . . , fr ∈ [2n], where r + 2s = k and s ≥ l.
Proof. Lemma 3.5 describes each element G of Fk (AHn (X)) in terms of sums of
e (m ,n ),(m ,n ) . It then suffices to show that these
products of minors of the form G
1
1
2
2
minors are all either 0 or of the forms Xu Xv , −Xu Xv or Mij (X), for some u, v ∈ [2n]
and 1 ≤ i < j ≤ n.
Since k = n + l, there are at least l pairs of rows of G whose indices in AHn (X)
are of the form t and n + t, for some t ∈ [n]. Denote by λ the exact number of such
pairs of rows occurring in G, and assume that, for m ∈ {1, 3, . . . , 2λ − 1}, the mth
33
Paula Lins
and the (m + 1)th rows of G correspond, respectively, to rows of indices of the form
t and n + t in AHn (X), for some t ∈ [n]. In this case,
A(X)ij = 0 if and only if A(X)(i+1)j = 0,
for all i ∈ {1, 3, . . . , 2λ − 1} and j ∈ n+1
, because of equality (3.12). Therefore,
2
e (m,k ),(m+1,k ) is
for k1 , k2 ∈ [b] distinct and m ∈ {1, 3, . . . , 2λ − 1}, the minor G
1
2
either 0 or Mij (X), for some 1 ≤ i < j ≤ n, as the columns of this minor are either
of the form (0, 0)T or (Xn+i , −Xi )T , for some i ∈ [n].
For i, j ∈ [n] distinct, there is at most one column of AGn (X) whose nonzero
rows are the ones of indices in {i, j, n+i, n+j}, it follows that each of the remaining
minors of G are either equal to 0 or of one of the forms Xi Xj or −Xi Xj , for some
i, j ∈ [2n].
o
Let x = (x1 , . . . , x2n ) ∈ W2n
with vp (xf0 ) = 0, say. Then
(3.13)
vp (xrf0 Mi1 j1 (x) · · · Mis js (x)) ≤ vp (xf1 · · · xfr0 Mi1 j1 (x) · · · Mis js (x)),
for all r, r0 ∈ N, f1 , . . . , fr0 ∈ [2n] and i1 , . . . , is , j1 , . . . , js ∈ [n].
Furthermore, if k{Mij (x) | 1 ≤ i < j ≤ n}kp = kMi0 j0 (x)kp , for some i0 , j0 ,
then
k{Mi1 j1 (x) · · · Mil jl (x) | 1 ≤ im < jm ≤ n, m ∈ [k]}kp = kMi0 j0 (x)klp
(3.14)
= k{Mij (x) | 1 ≤ i < j ≤ n}klp .
Lemma 3.10 states that the k × k-minors of AHn (X) are of the form
Xf1 . . . Xfr Mi1 j1 (X) . . . Mis js (X),
or sums of such terms, where r + 2s = k and s ≥ l. The maximal value for r occurs
when s = l. Expressions (3.13) and (3.14) then assure that, for m ∈ [n − 1]0 such
that k = m + 2l,
l
vp (xm
f0 Mi0 j0 (x) ) ≤ vp (xf1 . . . xfr Mi1 j1 (x) · · · Mis js (x)),
for all s ≥ l and r ∈ [n]0 satisfying r + 2s = k, and for all i1 , . . . , il , j1 , . . . , jl ∈ [n],
and f1 , . . . , fm ∈ [2n].
We now show that, for all k = n + l with l ∈ [n − 1], all terms of the form
Xfm Mij (X)l are elements of Fk (AHn (X)), for k = m+2l. This implies in particular
o
l
that, for x ∈ W2n
as above, the term xm
f0 Mi0 j0 (x) is an element of Fk (AHn (x))
and, therefore
l
l
l
kFk (AHn (x))kp = kxm
f0 Mi0 j0 (x) kp = kMi0 j0 (x)kp = k{Mij (x) | 1 ≤ i < j ≤ n}kp .
Assuming this holds, the integrand of the integral of expression (2.22) can be rewritten using the fact that
kFn+l (AHn (x)) ∪ wFn+l−1 (AHn (x))kp
kFn+l−1 (AHn (x))kp
k{Mij (x)l | 1 ≤ i < j ≤ n} ∪ w{Mij (x)l−1 | 1 ≤ i < j ≤ n}kp
k{Mij (x)l−1 | 1 ≤ i < j ≤ n}kp
= k{Mij (x) | 1 ≤ i < j ≤ n} ∪ {w}kp .
=
(3.15)
Proposition 3.11. Given l ∈ [n − 1], let k = n + l and m = n − l. Then, for all
f ∈ [2n] and 1 ≤ i < j ≤ n, either Xfm Mij (X)l or −Xfm Mij (X)l is an element of
Fk (AHn (X)).
Proof. Let f ∈ [n] and 1 ≤ i < j ≤ n. We show that, up to sign, both Xfm Mij (X)l
m
and Xn+f
Mij (X)l lie in Fk (AHn (X)).
34
Bivariate zeta functions of T -groups
m
First, we show that Xn+f
Mij (X)l ∈ Fk (AHn (X)). We consider the cases m ≥ 3,
m = 2 and m = 1 separately. In most cases, we do the following: we choose specific
indices r1 , . . . , rm , R1 , . . . , Rl of rows of AHn (X), and then denote by cs the index
of the unique column of AHn (X) having Xn+f in the rs th row, by Cqi the index of
the unique column having Xn+i in the Rq th row, and by Cqj the index of the unique
column having Xn+j in the Rq th row. The choices of rs and Rq are made such
that the submatrix Ã(X) of AHn (X) obtained by its rows of indices r1 , . . . , rm , R1 ,
n + R1 , . . . , Rl , n + Rl and columns c1 , . . . , cm , C1i , C1j , . . . , Cli , Clj , in this order,
is of the form
Xn+f Xn+r2 . . . Xn+rm
Xn+f
.
..
Xn+f
,
(3.16)
Xn+i Xn+j
−Xi −Xj
..
.
Xn+i Xn+j
−Xi −Xj
∗
0
m
which has determinant Xn+f
Mij (X)l .
Case 1. Assume that m ≥ 3. First, we consider f ∈
/ {i, j}. Set r1 = f , r2 = i,
r3 = j. Inductively, fix rs ∈ [n] \ {r1 , . . . , rs−1 }, for each s ∈ {4, . . . , m}. Fix also
R1 ∈ [n] \ {r1 , . . . , rm } and, inductively, Rq ∈ [n] \ {r1 , . . . , rm , R1 , . . . , Rq−1 }.
The submatrix Ã(X) of AHn (X) described above is of the form (3.16).
In fact, column c1 is of the form (3.10) and, for s ∈ {2, . . . , m}, cs is of the
form (3.11), so that the only nonzero entries of cs in AHn (X) are the ones of index
f , rs , n + f , and n + rs . Since r1 = f and rs ∈
/ {r1 , . . . , rs−1 }, it follows that the
nonzero entries of this column which appear in the submatrix Ã(X) are Xn+rs in
the row of index r1 = f , and Xn+f in the row of index rs .
Given q ∈ [l], the only nonzero entries of Cqi in AHn (X) are the ones of rows
whose index are elements of {i, Rq , n + i, n + Rq }. Since Rq 6= i, it follows that the
row n + i is not one of the rows of index n + Rit , t ∈ [l], that is, the only nonzero
rows of the form Rit or of the form n + Rit in Cqi which appear in Ã(X) are the ones
with t = q. The same argument shows that, the only nonzero entries of Cqj of the
form Rt or n + Rt in Ã(X) are the ones with t = q.
If f ∈ {i, j}, fix r1 = f , r2 ∈ {i, j} \ {f }, and set inductively rs ∈ [n] \
{r1 , . . . , rs−1 }, for each s ∈ {3, . . . , m}. The indices Rt are chosen as in the former
case. The matrix Ã(X) is in this case of the form (3.16), by similar arguments as
the ones for the former case.
Case 2. Assume that m = 2, that is, we want to find a minor of the form
2
/ {i, j}, set r1 = f , r2 = i, and R1 = j. Then fix, inductively,
Mij (X). If f ∈
Xn+f
Rq ∈ [n] \ {r1 , r2 , R1 , . . . , Rq−1 }.
If f ∈ {i, j}, we set r1 = f , r2 ∈ {i, j} \ {f } and Rq , for q ∈ [l], as in the former
cases.
These choices give matrices Ã(X) of the form (3.16).
Case 3. Assume that m = 1. If f ∈ {i, j}, set r1 = f , R1 ∈ {i, j} \ {f },
R2 ∈ [n] \ {r1 , R1 }, and, inductively, Rt ∈ [n] \ {r1 , R1 , . . . , Rt−1 }. The obtained
matrix Ã(X) is of the desired form.
For m = 1 and f ∈
/ {i, j}, we need a slightly different construction: we set r1 = f ,
but, in this case, we consider ci1 and cj1 , which are the indices of the columns of
AHn (X) containing, respectively, Xn+i and Xn+j in the r1 th row. Then set R1 = i
35
Paula Lins
and R2 = j and, inductively, Rq ∈ [n] \ {r1 , R1 , . . . , Rq−1 }, for all q ∈ {3, . . . , l}.
Denote by Cqi and Cqj the index of the columns of AHn (X) containing, respectively,
Xn+i and Xn+j in the Rq th row. There are only 2l − 1 indices Cqj and Cqj in total,
since C1j = C2i .
Similar arguments as the ones of the former cases show that the matrix composed
of rows r1 , R1 , n + R1 , . . . , Rl , n + Rl and columns ci1 , cj1 , C1i , C1j , C2j , . . . , Cli , Clj ,
in this order, is
Xn+i Xn+j
0
0
0
0
Xn+f
0
Xn+i Xn+j
Xn+R3
0
Xn+Rl
0
0
−Xf
0
−X
−X
−X
0
−X
0
i
j
R3
Rl
0
Xn+f
0
Xn+i Xn+j
0
Xn+R3
0
Xn+Rl
0
−Xf
0
−Xi −Xj
0
−XR3
0
−XRl
.
Xn+i
Xn+j
0
0
0
0
−Xi
−Xj
..
.
Xn+i
Xn+j
0
0
0
0
−Xi
−Xj
The determinant of such matrix is
Xn+i Xn+j
Xn+f
0
−Xf
0
Mij (X)l−2 det
0
Xn+f
0
−Xf
0
0
Xn+i
−Xi
0
0
Xn+j
−Xj
Xn+i
−Xi
0
0
0
= Xn+f Mij (X)l .
Xn+j
−Xj
The minors of the form Xfm Mij (X)l (up to sign) are obtained by repeating
the constructions above for each case but considering rows n + rs instead of rs ,
for all s ∈ [m]. The determinants of the matrices obtained in this way are of
the desired form because of the symmetry of the columns of AHn (X) given by
equality (3.12).
o
Combining expression (3.15) with Lemma 3.9, we obtain, for each x ∈ W2n
,
2n−1
Y
k=1
kFk (AHn (x)) ∪ wFk−1 (AHn (x))kp
= k{Mij (x) | 1 ≤ i < j ≤ n} ∪ {w}kn−1
.
p
kFk−1 (AHn (x))kp
Thus, for groups of the form Hn (o), the p-adic integral (2.22) is
JHn (s1 , s2 ) :=
Z
(2n−1)s1 +s2 −(n+1
−(n−1)(1+s1 )
2 )−2
|w|p
dµ,
k{Mij (x) | 1 ≤ i < j ≤ n} ∪ {w}kp
o
(w,x)∈p×W2n
which is a specialisation of the integral given in Proposition A.3. Combining Proposition A.3 with Proposition 2.8 yields
cc
ZH
(s1 , s2 ) =
n (o)
1
1 + (1 − q −1 )−1 JHn (s1 , s2 )
n+1
−s
(
)
2
1−q 2
= ZFHn (q, q −s1 , q −s2 ),
proving Theorem 1.10 for groups of type H.
36
Bivariate zeta functions of T -groups
3.2. Bivariate representation zeta functions—proof of Theorem 1.12. Recall that g := Λ(o). Consider the B-commutator matrix B(Y ) = BΛ (Y ) of g with
respect to e and f , where e and f are the bases of g/z and g0 , respectively, given in
the presentations of Definition 1.8.
Given a set I = {i1 , . . . , il }< ⊆ [n − 1]0 , recall that µj := ij+1 − ijPfor all j ∈ [l]0 ,
where i0 = 0, il+1 = n, and choose rI = (ri )i∈I ∈ NI and let N = i∈I ri . Recall
that b = rk(g0 ). Following [24, Section 3], we define the following sets, which form
o
a partition of Wn,N
.
NI,rI (G) =
o
{y ∈ Wb,N
: ν(B(y)) = (ril , . . . , ril , ril + ril−1 , . . . , ril + ril−1 , . . . , N, . . . , N )}.
| {z }
| {z } |
{z
}
µl terms
µ0 terms
µl−1 terms
Recall that, for Λ ∈ {Fn,δ , Gn , Hn }, z = g0 , so that
(
2n + δ,
r := rk(g/g0 ) = a := rk(g/z) =
2n,
if G = Fn,δ ,
if G ∈ {Gn , Hn }.
For simplicity, consider δ = 0 when G ∈ {Gn , Hn }, so that we can write a = 2n + δ
uniformly. Using these facts, we rewrite equality (2.20) as follows.
irr
ZG(o)
(s1 , s2 )
1
=
ā(G,n)−s
2
1−q
X
X
|NI,rI (G)|q −(ns1 +s2 +2n−r)
P
i∈I
ri −
P
i∈I
iri (−2−s1 )
I⊆[n−1]0 rI ∈NI
(3.17)
=
1
1−
X
q ā(G,n)−s2
X
|NI,rI (G)|q
P
i∈I
ri (−(n−i)s1 −s2 +2i+δ)
,
I⊆[n−1]0 rI ∈NI
where ā(G, n) = 2n + δ, as in Theorem 1.12.
The cardinalities |NI,rI (G)| are described in [24, Proposition 3.4] in terms of the
polynomials fG,I and the numbers ā(G, i) defined in Theorem 1.12 as follows.
|NI,rI (G)| = fG,I (q −1 )q
(3.18)
Combining (3.18) with (3.17) yields
X
1
irr
ZG(o)
(s1 , s2 ) =
1 − q ā(G,n)−s2
P
X
i∈I
ri (a(G,i)−2i−δ)
fG,I (q −1 )q
P
i∈I
.
ri (ā(G,i)−(n−i)s1 −s2 )
I⊆[n−1]0 rI ∈NI
=
1
1−
q ā(G,n)−s2
X
fG,I (q −1 )
I⊆[n−1]0
Y
i∈I
q ā(G,i)−(n−i)s1 −s2
.
1 − q ā(G,i)−(n−i)s1 −s2
This concludes the proof of Theorem 1.12.
3.3. Hyperoctahedral groups and functional equations. In this section, we
relate the formulae of Theorem 1.12 to statistics on Weyl groups of type B, also
called hyperoctahedral groups Bn . Specialisation (1.3) then provides formulae for
the class number zeta functions of groups of type F , G, and H in terms of such
statistics. By comparing these formulae to the ones of Corollary 1.11, we obtain
formulae for joint distributions of three functions on such Weyl groups.
We also use the descriptions of the bivariate representation zeta functions in
terms of Weyl group statistics in order to prove Theorem 1.13 in Section 3.3.3.
Some required notation regarding hyperoctahedral groups is given in Section 3.3.1.
37
Paula Lins
3.3.1. Hyperoctahedral groups Bn . We briefly recall the definition of the hyperoctahedral groups Bn and some statistics associated to them. For further details
about Coxeter groups and hyperoctahedral groups we refer the reader to [3].
The Weyl groups of type B are the groups Bn , for n ∈ N, of all bijections
w : [±n] → [±n] with w(−a) = −w(a), for all a ∈ [±n], with operation given by
composition. Given an element w ∈ Bn write w = [a1 , . . . , an ] to denote w(i) = ai .
Definition 3.12. For w ∈ Bn , the inversion number, the number of negative
entries and the number of negative sum pairs of w are defined, respectively, by
inv(w) = |{(i, j) | i < j, w(i) > w(j)}|,
neg(w) = |{i ∈ [n] | w(i) < 0}|,
nsp(w) = |{(i, j) ∈ [n]2 : i 6= j, w(i) + w(j) < 0}|.
Let si = [1, . . . , i − 1, i + 1, i, . . . , n] for i ∈ [n − 1] and s0 = [−1, 2, . . . , n] be
elements of Bn . Then (Bn , SB ) is a Coxeter system, where SB = {si }i∈[n−1]0 .
In [3, Proposition 8.1.1] it is shown that the Coxeter length on Bn with respect
to the generating set SB is given by
`(w) = inv(w) + neg(w) + nsp(w), for w ∈ Bn .
The right descent of w ∈ Bn is the set
D(w) = {si ∈ SB | w(i) > w(i + 1)}.
For simplicity, we identify SB with [n − 1]0 in the obvious way, so that D(w) ⊆
[n − 1]0 . Moreover, for I ⊆ SB , define
BnI = {w ∈ Bn | D(w) ⊆ I c = SB \ I}.
Example 3.13. Let w0 = [−1, . . . , −n] be the longest element of Bn . Then
n
inv(w0 ) =
,
neg(w0 ) = n,
`(w0 ) = n2 ,
D(w0 ) = SB .
2
4
Consider w ∈ Bn . The following statistics are used in this work.
1
|{(i, j) ∈ [±n]20 | i < j, w(i) > w(j), i 6≡ 0 mod 2}|,
2
des(w) = |D(w)|,
X
σ(w) =
n2 − i2 ,
L(w) =
i∈D(w)
maj(w) =
X
i,
i∈D(w)
rmaj(w) =
X
n − i.
i∈D(w)
The statistics des(w), maj(w), and rmaj(w) are called, respectively, the descent
number, the major index, and the reverse major index of w.
3.3.2. Local bivariate representation zeta functions in terms of statistics of Weyl
groups. The following lemma describes the polynomials fG,I defined in Theorem 1.12 in terms of statistics on the groups Bn , where G ∈ {Fn,δ , Gn , Hn }.
Lemma 3.14. Let n ∈ N, δ ∈ {0, 1} and I ⊆ [n − 1]0 . Then
38
Bivariate zeta functions of T -groups
(1) [24, Proposition 4.6]
X
fFn,δ ,I (X) =
(−1)neg(w) X (2`(w)+(2δ−1) neg)(w) ,
Ic
w∈Bn
X
fGn ,I (X) =
(−1)neg(w) X `(w) ,
Ic
w∈Bn
(2) [4, Theorem 5.4]
X
fHn ,I (X) =
(−1)`(w) X L(w) .
Ic
w∈Bn
Lemma 3.15. Given n ∈ N, δ ∈ {0, 1}, and a prime ideal p of O,
P
Q
−hG (w)
ā(G,i)−(n−i)s1 −s2
w∈Bn χG (w)q
i∈D(w) q
irr
Qn
ZG(o) (s1 , s2 ) =
,
ā(G,i)−(n−i)s1 −s2 )
i=0 (1 − q
where, for each w ∈ Bn ,
G
χG (w)
hG (w)
Fn,δ
Gn
Hn
(−1)neg(w)
(−1)neg(w)
(−1)`(w)
2`(w) + (2δ − 1) neg(w)
`(w)
L(w)
Proof. Applying Lemma 3.14 to the formulae of Theorem 1.12, one obtains the
irr
following expression for ZG(o)
(s1 , s2 ):
1
X
X
1 − q ā(G,n)−s2
χG (w)q −hG (w)
Ic
I⊆[n−1]0 w∈Bn
Y
i∈I
q ā(G,i)−(n−i)s1 −s2
,
1 − q ā(G,i)−(n−i)s1 −s2
which can be rewritten as the claimed sum because of [24, Lemma 4.4].
Proposition 3.16. For n ∈ N and δ ∈ {0, 1}, the following holds in Q[X, Z].
X
(−1)neg(w) X −(2(`−σ)+(2δ−1) neg −(2δ−3) rmaj −(2n+δ) des)(w) Z des(w)
w∈Bn
n
Y
2n+δ−1
2n+δ
2i+δ
= 1 − X ( 2 )Z
1 − X ( 2 )−( 2 )+2i+δ Z
i=2
Proof. On the one hand, specialisation (1.3) applied to the formula of Lemma 3.15
for groups of type F gives
P
Q
neg(w) −(2`+(2δ−1) neg)(w)
ā(Fn,δ ,i)−s
q
w∈Bn (−1)
i∈D(w) q
k
Qn
ζFn,δ (o) (s) =
.
ā(Fn,δ ,i)−s )
i=0 (1 − q
But
ā(Fn,δ , i) =
2n + δ
2i + δ
−
+ 2i + δ = 2(n2 − i2 ) + (2δ − 3)(n − i) + 2n + δ,
2
2
so that
Y
q ā(Fn,δ ,i)−s = q (2σ+(2δ−3) rmaj +(2n+δ−s) des)(w) .
i∈D(w)
Hence
P
ζFkn,δ (o) (s)
=
w∈Bn (−1)
neg(w) −(2(`−σ)+(2δ−1) neg −(2δ−3) rmaj −(2n+δ−s) des)(w)
q
Qn
i=0 (1
− q ā(Fn,δ ,i)−s )
.
39
Paula Lins
On the other hand, Corollary 1.11 asserts that
ζFkn,δ (o) (s)
2n+δ−1
2n+δ−1
1 − q ( 2 )−s
1 − q ( 2 )−s
= Q1
=
.
2n+δ
2n+δ
(1 − q ā(Fn,δ ,i)−s )
(1 − q ( 2 )+1−s )(1 − q ( 2 )−s )
i=0
Therefore
X
(−1)neg(w) q −(2(`−σ)+(2δ−1) neg −(2δ−3) rmaj −(2n+δ−s) des)(w)
w∈Bn
n
Y
2n+δ−1
= 1 − q ( 2 )−s
1 − q ā(Fn,δ ,i) q −s
i=2
2n+δ−1
2
= 1 − q(
)−s
n
Y
2n+δ
2
1 − q(
)−(2i+δ
2 )+2i+δ q −s
.
i=2
The formal identity follows as these formulae hold for all prime powers q and all
s ∈ C with sufficiently large real part.
For a geometric interpretation of ` − σ, we refer the reader to [26, Section 2].
It can be easily checked that, for n ≥ 2 and w ∈ Bn ,
Y
(3.19)
q ā(Gn ,i)−s = q (σ+2 maj −s des)(w) ,
i∈D(w)
Y
(3.20)
1
q ā(Hn ,i)−s = q 2 (σ−3 rmaj)(w)+(2n−s) des(w) .
i∈D(w)
The following proposition follows from Lemma 3.15, Corollary 1.11, and equalities (3.19) and (3.20), and arguments analogous to those given in the proof of
Proposition 3.16.
Proposition 3.17. For n ≥ 2, the following identities hold in Q[X, Z].
!
n
X
Y
neg(w) −(`−σ−2 maj)(w) des(w)
n2 −i2 +2i
(−1)
X
Z
=
1−X
Z ·
i=3
w∈Bn
(1 − X
X
2(n
2)
(−1)
Z)(1 − X
`(w)
X
2(n
2 )+1
Z) + X
Z(1 − X −n )(1 − X −n+1 ) , and
n2
− 21 (2L−σ+3 rmaj +4n des)(w)
Z
des(w)
!
1 − X(
n+1
2
)−(i+1
2 )+2i
Z
·
i=3
w∈Bn
=
n
Y
n
n
) Z(1 − X −n+1 )2 .
n+1
2
(1 − X ( 2 ) Z)(1 − X ( 2 )+2 Z) + X (
Remark 3.18. By setting X = 1 in the equations of Propositions 3.16 and 3.17, we
obtain the the equalities
X
X
(−1)neg(w) Z des(w) =
(−1)`(w) Z des(w) = (1 − Z)n ,
w∈Bn
w∈Bn
which were first proven in [16, Theorem 3.2].
3.3.3. Proof of Theorem 1.13. We recall that expression (2.20) of the local factors
of the bivariate representation zeta function of a group G(O) associated to a nilpotent O-Lie lattice Λ holds for all nonzero prime ideals p in case of nilpotency class
2, since we consider the alternative construction of the unipotent group scheme G
given in [24, Section 2.4]. In particular, the descriptions of the local terms of the
bivariate representation zeta functions of groups of type F , G, and H in terms of
Weyl statistics given in Lemma 3.15 also hold for all nonzero prime ideals. We use
Lemma 3.15 to show that all local terms of these bivariate zeta functions satisfy
40
Bivariate zeta functions of T -groups
functional equations. Recall that, for each n ∈ N, w0 denotes the longest element
of Bn , that is, w0 = [−1, −2, . . . , −n].
Theorem 1.13 follows from the same arguments of the proof of [11, Theorem 2.6]
applied to the expressions of Lemma 3.15. In fact, although hG is not one of the
statistics b · lL or b · lR defined in [11, Theorem 2.6], it satisfies the equations (2.6)
of [11], that is,
hG (ww0 ) + hG (w) = hG (w0 ).
In fact, one can easily show that g ∈ {inv, neg, `} satisfies g(ww0 ) = g(w0 ) − g(w),
for all w ∈ Bn , and the equation L(ww0 ) = L(w0 w) = L(w0 ) − L(w) is [23,
Corollary 7]. Therefore the conclusion of [11, Theorem 2.6] also holds for the
expressions given in Lemma 3.15.
4. Further examples
In this section we provide examples of bivariate zeta functions of T -groups of
nilpotency class 3. As before, p stands for a nonzero prime ideal of O, q = |O : p|,
and o = Op .
Let fr,c be the free nilpotent Z-Lie lattice with r generators and nilpotency
class c, and let Fr,c denote the unipotent group scheme obtained from fr,c . The
groups of type F are a particular case of groups Fr,c (O) with Fn,δ (O) = F2n+δ,2 (O).
Denote by zr,c and f0r,c , respectively, the centre and the derived Lie lattice of fr,c .
Example 4.1. Consider
f2,3 = hx1 , x2 , y, z1 , z2 | [x1 , x2 ] − y, [y, x1 ] − z1 , [y, x2 ] − z2 i.
The commutator matrices of f2,3 with respect to e = (y, x1 , x2 ) and f = (z1 , z2 , y),
where e and f are ordered bases of f2,3 /z2,3 and f02,3 , are
X2
X3
0
0
Y1 Y2
0
X3 ,
0
Y3 .
A(X1 , X2 , X3 ) = −X1
B(Y1 , Y2 , Y3 ) = −Y1
−X1
0
−X2
−Y2
−Y3
0
Thus, uA = 2, F0 (A(X)) = {1}, F1 (A(X)) = {−X1 , ±X2 , X3 }, and F2 (A(X)) ⊇
{X12 , X22 , X32 }. By Propositions 2.8 and A.1,
!
Z
1
−1 −1
2s1 +s2 −4
cc
1 + (1 − q )
|y|p
dµ
ZF2,3 (o) (s1 , s2 ) =
1 − q 2−s2
(y,x1 ,x2 ,x3 )∈p×W3o
=
(1 −
1 − q −2s1 −s2
.
− q 3−2s1 −s2 )
q 2−s2 )(1
Similarly, uB = 1, F0 (B(Y )) = {1}, and F2 (B(Y )) ⊇ {Y12 , Y22 , Y32 }. Thus
!
Z
1
s1 +s2 −4
irr
−1 −1
ZF2,3 (o) (s1 , s2 ) =
1 + (1 − q )
|y|p
dµ
1 − q 2−s2
(x,y1 ,y2 ,y3 )∈p×W3o
(4.1)
=
1 − q −s1 −s2
.
(1 − q 2−s2 )(1 − q 3−s1 −s2 )
Specialisation (1.3) yields
ζFk2,3 (o) (s) =
1 − q −s
.
(1 −
− q 3−s )
q 2−s )(1
This formula agrees with the one given in [18, Section 8.3], where f2,3 is denoted
L5,9 . In [17, Table 1], Tobias Rossmann provides the following formula for the twist
41
Paula Lins
representation zeta function of f2,3 —denoted L5,9 also in [17]—by implementing his
methods in Zeta [19].
ζFirr2,3 (o) (s) =
(4.2)
f
(1 − q −s )2
.
(1 − q 1−s )(1 − q 2−s )
Comparing (4.1) and (4.2), we see that there is no specialisation of the form (1.4)
for the bivariate representation zeta function of F2,3 (o) in terms of its twist representation zeta function.
4
In [21, Theorem 2.34], it is shown that the local factors of the normal zeta
function of the Lie ring
(4.3)
Λ = hz, w1 , w2 , w3 , x1 , x2 , x3 , y | [z, wi ] − xi , [z, x1 ] − y, i ∈ [3]i.
satisfy no functional equation. We know from Theorem 1.7 that its bivariate representation and bivariate conjugacy class zeta functions satisfy functional equations
and, consequently, so does its class number zeta function. In the following example,
we provide explicit formulae for its bivariate representation and class number zeta
functions.
Example 4.2. Let Λ be the Lie lattice given by the presentation (4.3), and let G be
the unipotent group scheme obtained from Λ. Denote by z the centre and by g0 the
derived Lie sublattice of g = Λ(o). The commutator B-matrix g with respect to
e = (z, w1 , w2 , w3 , x1 ) and f = (y, x1 , x2 , x3 ), such that e and f are ordered bases
of g/z and g0 , respectively, is
0
Y2 Y3 Y4 Y1
−Y
0
0
0
0
2
0
0
0
B(Y ) = −Y3 0
.
−Y4 0
0
0
0
−Y1
0
0
0
0
Clearly 2uB = 2.
Given y ∈ W4o , the matrix B(y) ∈ Mat5×5 (o) has rank 2. In other words, at least
one of the 2 × 2-minors of B(y) is a unit. Since r = rk(g/g0 ) = 4 and h = rk(g) = 8,
it follows from Propositions 2.8 and A.1 that
!
Z
1
irr
−1 −1
s1 +s2 −7
dµ
1 + (1 − q )
ZG(o) (s1 , s2 ) =
|w|
1 − q 4−s2
(w,y)∈p×W4o
=
1 − q 2−s1 −s2
.
(1 − q 4−s2 )(1 − q 6−s1 −s2 )
Specialisation (1.3) yields
k
ζG(o)
(s) =
1 − q 2−s
.
(1 − q 4−s )(1 − q 6−s )
4
Appendix A. p-adic integrals
In this section we calculate some generic p-adic integrals which are needed in the
current work.
Let O be the ring of integers of a number field and let p be a fixed nonzero prime
ideal of O. Write o = Op and set q = |O : p|. Let z ∈ o and let (z) = pe pe11 · · · perr
be the prime factorisation of the ideal (z) C O with pi 6= p, for all i ∈ [r]. Recall
that the p-adic valuation of z is given by vp (z) = e, and that the p-adic norm of
z is given by |z|p = q −vp (z) . For a finite index set J and (zj )j∈J ∈ oJ , define
k(zj )j∈J kp = max(|zj |p )j∈J .
42
Bivariate zeta functions of T -groups
We consider the additive Haar measure µ on o, normalised so that µ(o) = 1. We
also denote by µ the product measure on on , for n ∈ N.
The following is well known.
Proposition A.1. Let r be a complex variable. Then, for each k ∈ N,
Z
q −k(r+1) (1 − q −1 )
,
|y|rp dµ =
1 − q −k(r+1)
y∈pk
if the integral in the left-hand side converges absolutely.
The following lemma is a direct consequence of [18, Lemma 5.6], which assures
in particular that, for complex variables r and s, one has
Z
(1 − q −1 )(1 − q −r−n−1 )
(A.1)
,
|y|rp kx1 , . . . , xn , yksp dµ =
(1 − q −r−s−n−1 )(1 − q −r−1 )
(y,x)∈o×on
if the integral in the left-hand side converges absolutely.
Lemma A.2. Let r and s be complex variables. Then, for each n ∈ N0 ,
Z
(1 − q −1 )(1 − q −n + q −s−n − q −r−s−n−1 )q −r−1
|y|rp kx1 , . . . , xn , yksp dµ =
,
(1 − q −r−s−n−1 )(1 − q −r−1 )
(y,x)∈p×on
Z
(1 − q −1 )(1 − q −r−n−1 )q −r−s−n−1
|y|rp kx1 , . . . , xn , yksp dµ =
,
(1 − q −r−s−n−1 )(1 − q −r−1 )
(y,x)∈p×p(n)
if the integrals in the left-hand side of each equality converge absolutely.
Proof. Recall the notation Wko = {x ∈ ok | vp (x) = 0}. Since y ∈ W1o implies
|y|p = 1 and kx1 , . . . , xn , ykp = 1, and p × on = o × on \ W1o × on ,
Z
Z
r
s
|y|p kx1 , . . . , xn , ykp dµ =
|y|rp kx1 , . . . , xn , yksp dµ − µ(W1o × on ).
(y,x)∈p×on
(y,x)∈o×on
The first claim then follows from (A.1) and the fact that µ(W1o × on ) = 1 − q −1 .
Analogously, since p × p(n) = p × on \ p × Wno ,
Z
|y|rp kx1 , . . . , xn , yksp dµ
n
(y,x)∈p×p
Z
Z
r
s
−n
=
|y|p kx1 , . . . , xn , ykp dµ − (1 − q )
|y|rp dµ.
(y,x)∈p×on
y∈p
Let X = (X11 , . . . , X2n ) be a vector of variables. In the following, we consider
the matrix
"
#
X11 X12 . . . X1n
M (X) =
∈ Mat2×n (o[X]),
X21 X22 . . . X2n
and, for 1 ≤ i < j ≤ n, write Mij (X) := X1i X2j − X1j X2i .
Proposition A.3. For complex variables s and r, the following holds, provided the
integral in the left-hand side converges absolutely.
Z
|y|rp k{Mij (x) | 1 ≤ i < j ≤ n} ∪ {y}ksp dµ
o
(y,x)∈p×W2n
(q n − 1)(1 − q −1 )q −r−2n−1
(q + 1)(1 − q −r−n )q −s + (q n − q)(1 − q −r−s−n ) .
(1 − q −1−r )(1 − q −r−s−n )
Sq
Proof. Since o = m=1 πm + p, for some representatives πm of the classes of o/p,
there exist k ∈ N and representatives A1 , . . . , Ak of o2n /p(2n) such that
o2n = ∪km=1 Am + Mat2×n (p) ∪ Mat2×n (p),
43
=
Paula Lins
o
where Mat2×n (p) denotes the set of all 2 × n-matrices over p. Hence W2n
=
k
∪m=1 Am + Mat2×n (p). In the following, we evaluate the integrals
Z
IAm (s, r) :=
|y|rp k{Mij (x) | 1 ≤ i < j ≤ n}, yksp dµ.
(y,x)∈p×Am +Mat2×n (p)
If x ∈ Am + Mat2×n (p), then rk(x) = rk(Am ) modulo p. Let us then consider
the two cases rk(Am ) = 1 and rk(Am ) = 2 modulo p. For simplicity, assume that
rk(Am ) = 1 for 1 ≤ m ≤ t, and that rk(Am ) = 2 for t + 1 ≤ m ≤ k, for some
t ∈ [k]0 .
Case 1: Suppose that m ∈ [t], that is, rk(Am ) = 1. Then, each x = (xij ) ∈
Am + Mat2×n (p) has rank 1 modulo p. In particular, vp (Mij (x)) ≥ 1 for all 1 ≤
i < j ≤ n. By making a suitable change of variables, we can consider Am to be the
matrix with (1, 1)-coordinate 1 and 0 elsewhere. Consequently, x11 = 1 + Q11 and
xij = Qij , for (i, j) 6= (1, 1), where Qij ∈ p. Hence
(
(1 + Q11 )Q2i − Q21 Q1i , for i = 2, . . . , n,
M1j (x) =
Q1i Q2j − Q2i Q1j ,
for 1 < i < j ≤ n,
so that k{Mij (x) | 1 ≤ i < j ≤ n}kp = kM12 (x), . . . , M1n (x)kp . Therefore
Z
|y|rp kM12 (x), . . . , M1n , yksp dµ
IAm (s, r) =
(y,x)∈p×Mat2×n (p)
Z
n+1
= µ(p
)
|y|rp kx1 , . . . , xn−1 , yksp dµ
(y,x1 ,...,xn−1 )∈p×pn−1
−1
−r−n −r−s−n
(1 − q )(1 − q
)q
,
(1 − q −r−s−n )(1 − q −r−1 )
where the domain of integration of integral in the second equality is justified
by the translation invariance of the Haar measure and the last equality is due
Proposition A.2.
= q −n−1
Case 2: We now assume that m ∈ {t + 1, . . . , k}, that is, rk(Am ) = 2. In this
case, each x ∈ Am + Mat2×n (p) has rank two modulo p, which means that at least
one of the Mij (x) has valuation zero. Consequently,
Z
q −2n−r−1 (1 − q −1 )
.
IAm (s, r) =
|y|rp dµ =
1 − q −r−1
(y,x)∈p×p2n
Finally, there are (q + 1)(q n − 1) matrices of rank 1 and q(q n − 1)(q n−1 − 1) matrices
of rank 2 in Mat2×n (Fq ). Consequently
Z
k
X
|y|rp k{Mij (x) | 1 ≤ i < j ≤ n} ∪ yksp dµ =
IAm (s, r)
o
(y,x)∈p×W2n
m=1
= (q + 1)(q n − 1)IAt (s, r) + q(q n − 1)(q n−1 − 1)IAk (s, r)
(q n − 1)(1 − q −1 )q −r−2n−1
(q + 1)(1 − q −r−n )q −s + (q n − q)(1 − q −r−s−n ) ,
−1−r
−r−s−n
(1 − q
)(1 − q
)
as desired.
=
Acknowledgements
This paper is part of my PhD thesis. I am grateful to my advisor Christopher
Voll for his guidance, encouragement, constructive criticism and inumerous helpful
discussions. I would like to express my gratitude to Tobias Rossmann for helpful
conversations and significant comments on a previous version of this paper. I would
like to thank Yuri Santos Rego for his support and for his comments on an earlier
44
Bivariate zeta functions of T -groups
draft of this paper. I also gratefully acknowledge financial support from the DAAD
for this work.
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E-mail address: lins@math.uni-bielefeld.de
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| 4 |
arXiv:1603.05683v1 [math.GR] 17 Mar 2016
Journal of Algebraic Systems
Vol. XX, No XX, (201X), pp XX-XX
ON RELATIVE CENTRAL EXTENSIONS AND
COVERING PAIRS
A. POURMIRZAEI∗ , M. HASSANZADEH AND B. MASHAYEKHY
Abstract. Let (G, N ) be a pair of groups. In this article, first we
construct a relative central extension for the pair (G, N ) such that
special types of covering pair of (G, N ) are homomorphic image of
it. Second, we show that every perfect pair admits at least one
covering pair. Finally, among extending some properties of perfect
groups to perfect pairs, we characterize covering pairs of a perfect
pair (G, N ) under some extra assumptions.
In this paper the well-known notion of relative central extension of
a pair of groups is used. By a pair of groups we mean a group G
and a normal subgroup N and this is denoted by (G, N). Let M be
another group on which an action of G is given. The G-commutator
subgroup of M is defined by the subgroup [M, G] of M generated by
all the G-commutators
[m, g] = mg m−1 ,
in which g ∈ G, m ∈ M and mg is the action of g on m. Ellis [1]
defined the G-center of M to be the subgroup
Z(M, G) = {m ∈ M|mg = m, ∀g ∈ G}
Now we recall the definition of relative central extension of a pair of
groups.
MSC(2010): Primary: 20E34; Secondary: 20E22, 20F05
Keywords: Pair of groups, Covering pair, Relative central extension, Isoclinism of pairs of
groups.
Received: 2 March 2015, Accepted: 26 February 2016.
∗Corresponding author .
1
2
POURMIRZAEI, HASSANZADEH AND MASHAYEKHY
Definition 0.1. ([2]) Let (G, N) be a pair of groups. A relative central extension of the pair (G, N) consists of a group homomorphism
σ : M → G, together with an action of G on M such that
(i)σ(M) = N;
(ii)σ(mg ) = g −1 σ(m)g, for all g ∈ G, m ∈ M;
(iii)m′σ(m) = m−1 m′ m, for all m, m′ ∈ M;
(iv)Ker(σ) ⊆ Z(M, G),
Note that for every relative central extension σ : M → G of a pair
of groups (G, N), we have Inn(M) ⊆ Imσ, by the property (iii) in
Definition 0.1. Moreover, every relative central extension σ : M → G
yields an exact sequence
σ
1 → kerσ → M → N → 1.
Conversely, for every exact sequence
σ
1 → kerσ → M ′ → N → 1
with properties (i)-(iv), ρ : M ′ → G is a relative central extension. For
this reason we do not differ between a relative central extension and
its associated exact sequence.
Let σ : M → G, σ ′ : M ′ → G be two relative central extensions
of (G, N). In 1998 Eliss [2] introduced a morphism between these
relative central extensions which is a group homomorphism ϕ : M →
M ′ satisfying σ ′ ϕ(m) = σ(m) and ϕ(mg ) = (ϕ(m))g for all g ∈ G,
m ∈ M. In particular, if ϕ is a surjective homomorphism, then σ ′
is called a homomorphic image of σ. The resulting category is the
category of relative central extensions which Eliss denoted it by RCE
(G, N).
Now we recall the definition of a covering pair.
Definition 0.2. ([2]) A relative central extension σ : M ∗ → G of the
pair (G, N) is called a covering pair for (G, N) if there exists a subgroup
A of M ∗ such that
(i)A ⊆ Z(M ∗ , G) ∩ [M ∗ , G];
(ii)A ∼
= M(G, N);
(iii)N ∼
= M ∗ /A,
where M(G, N) is the Schur multiplier of the pair (G, N).
A covering pair σ : G∗ → G of the pair (G, G) coincides with the
usual notion of a covering group G∗ of the group G. In 1998 Ellis [2]
showed that every pair of finite groups admits at least one covering
pair.
ON RELATIVE CENTRAL EXTENSIONS AND COVERING PAIRS
3
In this paper we consider P as a perfect group that satisfies in maximal condition which means φ(P ) 6= P such that φ(P ) is the Frattini
subgroup of P . We mention that if σ : P → G is a covering pair of
(G, N), then we have P ′ = [P, G] and therefore
N = σ(P ) = σ([P, G]) = [σ(P ), G] = [N, G].
Let (G, N) be a pair of groups with free presentation G ∼
= F/R and
∼
N = S/R for a normal subgroup S of F , where F is the free group
on the set G. In Section 1, we introduce a relative central extension
δ : S/[R, F ] → G which has an important role throughout the paper.
As a consequence, we show that for every covering pair σ : P → G, P is
a homomorphic image of S/[R, F ]. We remark that in 2007 Salemkar,
Moghaddam and Chiti [7, Lemma 2.5] claimed that there exists an
epimorphism from δ to any relative central extension without any condition. We present a counterexample to show that their claim is not
correct and hence one of their main results [7, Theorem 2.6] is not valid.
In 1978 Loday [5] extended the notion of perfect group to perfect
pair in the sense that a pair of groups (G, N) is perfect if [G, N] = N.
Moreover he proved that (G, N) is a perfect pair of groups if and only
if RCE (G, N) has a universal object. To prove this he used some
cohomological methods. In Section 2, by restriction of δ to [S, F ]/[R, F ]
we obtain a universal object in RCE (G, N) when (G, N) is perfect.
This is also a covering pair of (G, N). It is worth to mention that we
use only presentation methods instead of cohomological methods which
seems easier.
In sequel we extend a result of Schur on covering groups to covering
pairs.
1. Some Results for Relative Central Extensions
Let (G, N) be a pair of groups with a free presentation 1 → R →
π
F → G → 1 such that N ∼
= S/R for a normal subgroup S of F . Define
the group homomorphism
S
δ:
→ G,
(1.1)
[R, F ]
s[R, F ] 7→ π(s).
(1.2)
It is straightforward to check that δ is a relative central extension by
the following action
S
F
S
δ̄ :
×
→
,
(1.3)
[R, F ] R
[R, F ]
(s[R, F ], f R) 7→ sf [R, F ].
(1.4)
4
POURMIRZAEI, HASSANZADEH AND MASHAYEKHY
In the following theorem we present a relation between the above
relative central extension to anyone of the pair (G, N), especially to
any covering pair σ : M → G of (G, N).
Theorem 1.1. Let G ∼
= F/R, where F is the free group on the set
G and N be a normal subgroup of G with N ∼
= S/R for a normal
subgroup S of F . If σ : M → G is a relative central extension of the
pair (G, N), then there exists a homomorphism β : S/[R, F ] → M such
that the following diagram is commutative:
1 →
R
[R,F ]
→
S
[R,F ]
1 →
β| ↓
A →
β↓
M
δ
→ N → 1
k
σ
→ N → 1,
where δ is the relative central extension (4). In particular, if M is a
perfect group with φ(M) 6= M, then the above homomorphism β is an
epimorphism.
Proof. Consider the map f : G → N by the rule f (g) = g if g ∈ N,
otherwise f (g) = 1. Now f induces a homomorphism θ : F → N.
Using the projective property of free groups, there is a homomorphism
α : F → M such that σα = θ. By the restriction of θ and α to S we
have the following commutative diagram:
S
α| ւ ↓ θ|
σ
M −→ N −→ 1.
For x ∈ F and r ∈ R, α([r, x]) = [α(r), α(x)] = 1 since α(r) ∈ kerσ ≤
Z(M, G) ≤ Z(M). Therefore [R, F ] ≤ kerα| which implies the existence of
S
β:
→ M.
[R, F ]
Clearly β(R/[R, F ]) ≤ kerσ, which gives the restriction of β to R/[R, F ],
β|. Now let A = kerσ ≤ Z(M, G) and φ(M) 6= M. For every m ∈ M,
there exists s̄ ∈ S/[R, F ] such that σ(m) = δ(s̄) = σβ(s̄). Hence
m = β(s̄)a, for some a ∈ kerσ. Thus
S
M = hβ(s̄), a | s̄ ∈
, a ∈ Ai.
[R, F ]
Since M = M ′ so
A ≤ Z(M, G) ∩ M ′ ≤ Z(M) ∩ M ′ ≤ φ(M).
Hence
M = hβ(s̄) | s̄ ∈
S
i
[R, F ]
ON RELATIVE CENTRAL EXTENSIONS AND COVERING PAIRS
which implies that β is an epimorphism.
5
Corollary 1.2. With the assumptions and notations of the previous
theorem, for every covering pair σ : P → G of (G, N), P is a homomorphic image of S/[R, F ].
Note that in Theorem 1.1 if
β(s̄g ) = β(s̄)g
for every s̄ ∈ S/[R, F ] and g ∈ G (in other words β preserves the
action), then we can say that every covering pair σ : P → G is a
homomorphic image of δ : S/[R, F ] → G.
Remark 1.3. In 2007 Salemkar, Moghaddam and Chiti [7, Lemma 2.5]
claimed that in the above theorem β is always an epimorphism for any
free presentation G ∼
= S/R without any condition. The
= F/R and N ∼
following counterexample shows that their claim is not true and hence
one of the main results of their paper [7, Theorem 2.6] is not valid.
Example 1.4. Put G = N = Z ⊕ ... ⊕ Z a free abelian group of
finite rank with the free presentation G ∼
= N, where F ′ is
= F/F ′ ∼
the commutator subgroup of the free group F . Consider the following
commutative diagram:
1 −→
F′
[F ′ ,F ]
⊆
−→
F
[F ′ ,F ]
δ
−→ G = N = Z ⊕ ... ⊕ Z −→ 1
β| ↓
β↓
k
α
σ
1 −→ A = Z −→ M = G ⊕ Z −→ G = N = Z ⊕ ... ⊕ Z −→ 1,
where α(x) = (1, x) and σ(g, x) = g, for all x ∈ A, g ∈ G. Then we
have
F′
F
F
β( ′
) = [β( ′
), β( ′
)] ≤ M ′ = 1.
[F , F ]
[F , F ]
[F , F ]
Hence β is not an epimorphism.
At the end of this section by a result of Ellis [2, Corollary 1.2] on the
Schur multiplier of a pair (G, N) of finite nilpotent groups we present
a covering pair of the pair (G, N).
Theorem 1.5. Let (G, N) be a pair of finite nilpotent groups. Assume
Q
that G = ki=1 Si , where Si is the Sylow pi -subgroup of G and σi :
Mi → Si for each i = 1, ...,Q
k, is an arbitrary covering pair of the
pair (Si , Si ∩ N). Then σ : ki=1 Mi → G defined by σ({mi }ki=1 ) =
{σi (mi )}ki=1 is a covering pair of (G, N).
Proof. It is readily seen that σ is a relative central extension of (G, N).
By [2, Corollary 1.2] M(G, N) ∼
= Πki=1 M(Si , Si ∩ N). hence the proof
is straightforward.
6
POURMIRZAEI, HASSANZADEH AND MASHAYEKHY
2. Some Properties of Covering Pairs
The aim of this section is to introduce a covering pair for a perfect
pair of groups (G, N).
Theorem 2.1. Let (G, N) be a perfect pair of groups. Then for every
covering pair σ : M → G, we have [M, G] = M.
Proof. Since σ : M → G is a covering pair, we have M/kerσ ∼
= N and
kerσ ≤ [M, G]. Hence
M/kerσ ∼ N
.
=
[M, G]/kerσ
[N, G]
The result is a consequence of (G, N) being perfect.
It is known that if G ∼
= F/R is a group with a covering group G∗ ,
then there exists a normal subgroup S of F such that R/[R, F ] =
(F ′ ∩ R)/[R, F ] × S/[R, F ], and we have an isomorphism F/S ∼
= G∗ [6,
Theorem 2.4.6(iv)(b)]. In the following theorem we intend to extend
this result to pairs of groups.
Theorem 2.2. Let (G, N) be a pair of groups with G ∼
= F/R and
∼
N = S/R, where F is the free group on the set G and S E F . Then
for every relative central extension σ : P → G with A ∼
= kerσ there is
a normal subgroup T of F such that
R
R ∩ [S, F ]
T
=
×
.
[R, F ]
[R, F ]
[R, F ]
Moreover, there exists an isomorphism S/T ∼
= P which carries R/T
onto A.
Proof. Let π : F → G be a surjective homomorphism with R = kerπ.
As the proof of Theorem 1.1, there exists a homomorphism ψ : S → P
which induces the epimorphism β : S/[R, F ] → P . Thus ψ is in fact
an epimorphism. By setting T = kerψ, we have S/T ∼
= P . Clearly
ψ(R) = A and by Theorem 1.1, ψ([R, F ]) = 1 so [R, F ] ≤ T which
gives the induced epimorphism ψ̄ : R/[R, F ] → A. Considering
ψ̄| :
R ∩ [S, F ]
→ A,
[R, F ]
we claim that ψ̄| is an isomorphism. To prove this, let a ∈ A. By
ψ(R) = A there exists x ∈ R such that a = ψ(x). On the other hand
for every p ∈ P and g ∈ G,
[p, g] = [β(s̄), g] = β(s̄−1 )β(s̄g ).
ON RELATIVE CENTRAL EXTENSIONS AND COVERING PAIRS
7
Easily it can be seen that
g
β(s̄ ) =
β(s̄)g
if g ∈ N
β(s̄)β(r̄) if g ∈
/N
for some r̄ ∈ R/[R, F ].
Thus by the proof of Theorem 1.1,
P = hβ(s̄) | s̄ ∈
S
R
\
i.
[R, F ] [R, F ]
Since P is perfect and so (G, N) is perfect, we have [S, F ]R = S.
On the other hand A = ψ(R) ≤ φ(P ). Therefore [P, G] ≤ ψ([S, F ]).
This fact and A ≤ [M, G] imply that a = ψ(y) for some y ∈ [S, F ].
Thus ψ(x) = a = ψ(y), hence x−1 y ∈ kerψ ≤ R, so that y ∈ R ∩
[S, F ]. Since A ∼
= (R ∩ [S, F ])/[R, F ] and A is finite, ψ̄|
= M(G, N) ∼
is an isomorphism. By surjectivity of ψ̄, for every r̄ ∈ R/[R, F ] there
exists x̄ ∈ (R ∩ [S, F ])/[R, F ] such that ψ̄(x̄)=ψ̄(r̄), thus x̄−1 r̄ ∈ kerψ.
Consequently
R
R ∩ [S, F ]
= ker ψ̄
.
[R, F ]
[R, F ]
Let x̄ ∈ ker ψ̄ ∩ (R ∩ [S, F ])/[R, F ]. Then x̄ ∈ ker ψ̄| = 1, so we have
the following isomorphism
R ∼
T
R ∩ [S, F ]
R ∩ [S, F ]
=
×
.
= ker ψ̄ ×
[R, F ]
[R, F ]
[R, F ]
[R, F ]
Theorem 2.3. Let σi : Mi → G, i = 1, 2 be two covering pairs of a
finite pair (G, N). If Mi = Mi′ and φ(Mi ) 6= Mi for i = 1, 2, then
(i)M1 ∼
= M2 ;
(ii)M1 /Z(M1 , G) ∼
= M2 /Z(M2 , G);
(iii)Z(M1 , G)/kerσ1 ∼
= Z(M2 , G)/kerσ2 .
Proof. (i) Let σ : M ∗ → G be a fixed arbitrary covering pair with the
following relative central extension
1 → A → M∗ → N → 1
such that A ⊆ Z(M ∗ , G) ∩ [M ∗ , G] and A ∼
= M(G, N). Since [N, G] =
∗
∗
∗
N and [M , G] = M so σ : [M , G] → [N, G] is an epimorphism.
Therefore
|[M ∗ , G]| =
| ker σ|
|[N, G]|
=
|A|
|[N, G]|
= |M(G, N)| |[N, G]|.
8
POURMIRZAEI, HASSANZADEH AND MASHAYEKHY
Suppose that G ∼
= F/R such that F is the free group on the set G with
N∼
S/R.
By
Theorem
1.1, we have the following diagram
=
→
S
[R,F ]
β| ↓
1 → A →
β↓
M∗
1 →
R
[R,F ]
δ
→ N → 1
k
σ
→ N → 1.
Since [S/[R, F ], G] = [S, F ]/[R, F ] and δ| : [S, F ]/[R, F ] → [N, G] is an
epimorphism, thus
[S, F ]
|
| = |M(G, N)||[N, G]|.
[R, F ]
Therefore
[S, F ]
|.
|M ∗ | = |[M ∗ , G]| = |
[R, F ]
If s̄ ∈ S/[R, F ] and g ∈ G,
−1
g
β([s̄, g]) = β(s̄) β(s̄ ) =
β(s̄)−1 β(s̄)g = [β(s̄), g] if g ∈ N
β(s̄)−1 β(s̄) = 1
if g ∈
/N
which yields that β([S, F ]/[R, F ]) ≤ [M ∗ , G]. By Theorem 2.1, M ∗ =
[M ∗ , G] ≤ β([S, F ]/[R, F ]) which implies that M ∗ ∼
= [S, F ]/[R, F ].
(ii) By Theorem 2.2 there exists a normal subgroup T of F with
M∗ ∼
= S/T. This yields naturally an action of G on S/T which implies
Z(S/T, G) = Z(M ∗ , G). Put Z(S/T, G) = L/T. Let x ∈ Z(S/[R, F ], G).
Then [x, g] ∈ [R, F ] ≤ T and so Z(S/[R, F ], G) ≤ L/[R, F ]. To
prove the reverse containment, assume that x[R, F ] ∈ L/[R, F ], then
[x, g] ∈ T ≤ R ∩ [S, F ] ∩ T = [R, F ], hence
L
S
= Z(
, G).
[R, F ]
[R, F ]
Consequently, it may be inferred that
S/T ∼ S
M∗
∼
=
= .
∗
Z(M , G)
L/T
L
The desired assertion is now a consequence of the fact that S/L is
determined by the presentation of G and N.
(iii) Owing to Theorem 1.1 we have
Z(M ∗ , G) ∼ L/T ∼ L
=
= .
A
R/T
R
The result is now established.
Now, we are in a position to state and prove one of the main results
of this section.
ON RELATIVE CENTRAL EXTENSIONS AND COVERING PAIRS
9
Theorem 2.4. Let (G, N) be a perfect pair of groups with a free presentation G ∼
= F/R and N ∼
= S/R. Then
(i) δ : [S, F ]/[R, F ] → G by δ(x[R, F ]) = xR, is a covering pair of
(G, N)
(ii) The relative central extension δ : [S, F ]/[R, F ] → G is the universal
object in the category RCE (G, N).
Proof. (i) Consider the following action of G on [S, F ]/[R, F ]
[S, F ]
[S, F ]
×G→
[R, F ]
[R, F ]
([s, f ][R, F ], g) 7→ ([s, f ][R, F ])g = [st , f t ][R, F ],
where π : F → G is the epimorphism with kerπ = R and π(t) = g. By
the assumption G = NQ, let Q ∼
= T /R. Since (G, N) is a perfect pair,
[N, G] = N and therefore ([S, F ]R)/R = S/R. It can be shown that
δ : [S, F ]/[R, F ] → G is a relative central extension of the pair (G, N).
On the other hand, since
kerδ = (R ∩ [S, F ])/[R, F ] ∼
= M(G, N),
we have
[S, F ]/[R, F ] ∼ [S, F ] ∼ [S, F ]R
S ∼
=
=
=
= N.
kerδ
R ∩ [S, F ]
R
R
It is enough to show that
kerδ ≤ [
[S, F ]
, G].
[R, F ]
For this we show that [[S, F ]/[R, F ], G] = [S, F ]/[R, F ]. Clearly
[[S, F ]/[R, F ], G] ≤ [S, F ]/[R, F ].
So we only need to show the reverse containment. We know that
[S, F ]
[S, F ]
[
, G] = h[x, g]|x ∈
, g ∈ Gi
[R, F ]
[R, F ]
such that
[x, g] = x−1 xg = [s, f ][R, F ]−1([s, f ][R, F ])g = [s, f ]−1 [s, f ]t [R, F ],
for some s ∈ S and f ∈ F , where π(t) = g. Since S = [S, F ]R, for
every s ∈ S there exist s′ ∈ S, l ∈ F and r ∈ R such that
[s, f ] = [[s′ , l]r, f ] = [[s′ , l], f ][[s′ , l], f, r][r, f ].
Hence [S, F ]/[R, F ] ≤ [[S, F ]/[R, F ], G].
(ii) Let σ : M → G be a relative central extension of the pair (G, N),
and F be the free group on the set G and S E F such that G ∼
= F/R
10
POURMIRZAEI, HASSANZADEH AND MASHAYEKHY
and N ∼
= S/R. Then consider δ : [S, F ]/[R, F ] → G as mentioned in
the previous part. By Theorem1.1, there exists a homomorphism
ϕ : [S, F ]/[R, F ] → M
such that ϕ = β|[S,F ]/[R,F ]. Now for every g ∈ G and x̄ ∈ [S, F ]/[R, F ],
β(x̄)g = ϕ(x̄)g if g ∈ N
g
g
ϕ(x̄ ) = β(x̄ ) =
β(x̄) = ϕ (x̄) if g ∈
/N
We have to prove ϕ preserves the action. To do this it is enough to
show that ϕ(x̄)g = ϕ(x̄) for every g ∈ G and x̄ ∈ [S, F ]/[R, F ]. Define
S
f:
→ M
[R, F ]
s̄ 7→ β(s̄)g β(s̄)−1 .
For every g ∈ G and x̄ ∈ [S, F ]/[R, F ], f (s̄) ∈ Z(G, M) so
g
g
(β(s̄)g ) (β(s̄)−1 ) = f (s̄)g if g ∈ N
g
g g
g −1
f (s̄ ) = β(s̄ ) β(s̄ ) =
β(s̄)g β(s̄)−1
if g ∈
/N
Thus,
−1 g
f ([s̄, g]) = f (s̄ s̄ ) =
f (s̄)−1 f (s̄)g = [f (s̄), g] = 1 if g ∈ N
f (s̄)−1 f (s̄) = 1
if g ∈
/N
By part (i)
f(
[S, F ]
[S, F ]
) = f ([
, G]) = 1.
[R, F ]
[R, F ]
Therefore
ϕ(x̄g ) = β(x̄g ) = β(x̄)g = ϕ(x̄)g .
We claim that ϕ is unique. By contrary, let ϕi : [S, F ]/[R, F ] → M,
i = 1, 2, be homomorphisms such that
σϕ1 (s̄) = δ(s̄) = σϕ2 (s̄).
Define
ψ:
[S, F ]
→ M
[R, F ]
x̄ 7→ ϕ1 (x̄)ϕ2 −1 (x̄).
If g ∈ G and x̄ ∈ [S, F ]/[R, F ],
ψ(x̄g ) = ϕ1 (x̄g )ϕ2 −1 (x̄g ) = ϕ1 (x̄)g (ϕ2 −1 (x̄))g = (ϕ1 (x̄)ϕ2 −1 (x̄))g = ψ(x̄)g .
Consequently ψ([x̄, g]) = [ψ(x̄), g] = 1. Now by (i) we have
[[S, F ]/[R, F ], G] = [S, F ]/[R, F ] which implies that ψ = 1. Hence the
results holds.
ON RELATIVE CENTRAL EXTENSIONS AND COVERING PAIRS
11
Note that if (G, N) is a pair of groups with a free presentation G ∼
=
F/R and N ∼
S/R,
then
[S,
F
]/[R,
F
]
is
independent
of
the
choice
of
=
the free presentation of (G, N).
The following theorem states a necessary and sufficient condition for
a pair of groups to be perfect. It should be noted that Loday [5] proved
the following result using homological method but our proof is different
and seems elementary.
Theorem 2.5. A pair of groups (G, N) is perfect pair if and only if
the category RCE (G, N) has a universal relative central extension.
Proof. Necessity is Theorem 2.4. For sufficiency, let
ρ
1→A→M →G→1
be a universal relative central extension of the pair (G, N). Consider
the following relative central extension
1→A×
N
N
ψ
→ N → 1,
→M×
[N, G]
[N, G]
where ψ(m, n̄) = ρ(m), m ∈ M, n̄ ∈ N/[N, G]. Define homomorphisms
θi : M → M ×
N
,
[N, G]
i = 1, 2, by θ1 (m) = (m, 1) and θ2 (m) = (m, ρ(m)). Then we have
ψθi = ρ, i = 1, 2, where θ1 = θ2 . Consequently, N = [N, G] and hence
(G, N) is a perfect pair.
Finally, we intend to consider a covering pair as a pair of groups. If
σ : M → G is a covering pair of (G, N), then there exists a subgroup A
of M such that A ⊆ Z(M, G) ∩ [M, G], A ∼
= N.
= M(G, N) and M/A ∼
Now, we can consider (M, A) as a covering pair of (G, N). It is known
that any two covering groups of a finite group G are isoclinic. Moreover,
if G is a finite perfect pair of groups, then a covering group of G is also
perfect. We intend to investigate these facts for a covering pair as our
new point of view of a covering pair of groups. In this way, the notion
of isoclinism for the pair of groups is needed which we review it of [8].
Definition 2.6. Let (G, N) and (H, K) be two pairs of groups. The
pairs (G, N) and (H, K) are said to be isoclinic if there exists isomorphisms ǫ : G/Z(G, N) → H/Z(H, K) and η : [G, N] → [H, K]
such that ǫ(N/Z(G, N)) = K/Z(H, K) and η([g, n]) = [h, k] whenever
ǫ(gZ(G, N)) = hZ(H, K) and η(nZ(G, N)) = kZ(H, K). This concept
is denoted as (G, N) ∼ (H, K).
12
POURMIRZAEI, HASSANZADEH AND MASHAYEKHY
It is a well-known fact that all covering groups of a given finite group
are mutually isoclinic (see [4]). The following lemma helps us to prove
this fact for some special types of covering pairs for arbitrary pair of
groups.
Lemma 2.7. Let (G,N) and (H,K) be two pairs of groups. If θ : G →
H is an epimorphism such that θ(N) = K and kerθ ∩ N = 1, then
(G, N) ∼ (H, K).
Proof. Define ǫ : G/Z(G, N) → H/Z(H, K) such that ǫ(gZ(G, N)) =
θ(h)Z(H, K) and ϕ : [N, G] → [K, H] by ϕ([n, g]) = [θ(n), θ(g)]. We
can easily see that ǫ and ϕ are isomorphisms which yields that (G, N) ∼
(H, K).
Theorem 2.8. Let (G, N) be a pair of groups. Then every two covering
pairs (Pi , Ai ), i = 1, 2, where Pi = Pi′ and φ(Pi ) 6= Pi of (G, N) are
isoclinic.
Proof. By the proof of Theorem 2.2 if (P, A) is a covering pair of
(G, N) then there exists an epimorphism β : S/[R, F ] → P such that
ψ(([S, F ] ∩ R)/[R, F ]) = A. Clearly
kerψ ∩ [
S
[S, F ] ∩ R
,
] = 1.
[R, F ] [R, F ]
Using Lemma 2.7 we obtain
(P, A) ∼ (
[S, F ] ∩ R
S
,
).
[R, F ] [R, F ]
The following results can be easily obtained if we use the notion
of isoclinism for a perfect pair of groups. Note that these results are
generalizations of similar results for perfect groups to perfect pairs (
see [4, Lemma 2.4, Corollary 2.5 and Proposition 2.6]).
Lemma 2.9. Let (G, N) be a finite perfect pair of groups, and (H, K)
be any finite pair of groups which is isoclinic to (G, N). Then K is
isomorphic to
N × Z(H, K)
.
Z(G, N) × (Z(H, K) ∩ [H, K])
Corollary 2.10. Let (G, N) be a finite perfect pair of groups with
trivial center. Then for any pair of groups isoclinic to (G, N), say
(H, K), we have K is isomorphic to direct product of N and an abelian
group.
ON RELATIVE CENTRAL EXTENSIONS AND COVERING PAIRS
13
Proposition 2.11. Let (G, N) be a finite perfect pair of groups, and
(H, K) be any pair which is isoclinic to (G, N) such that |N| = |K|.
Then N ∼
= K.
Proof. The result follows from Lemma 3.9 and the assumption that
|K| = |N|.
References
1. G. Ellis, Capability, homology and central series of a pair of groups, J. Algebra
(1996), 179:31–46.
2. G. Ellis, The Schur multiplier of a pair of groups, Appl. Categ. Structures
6(1998), 355–371.
3. N. S. Hekster, On the structure of n-isoclinism classes of groups, J. Pure and
Applied Algebra 40(1986), 63–85.
4. M. R. Jones, J. Wiegold, Isoclinisms and covering groups, Bull. Austral. Math.
Soc. (1974), 11:71-76.
5. J. L. Loday, Cohomologie et group de Steinberg relatif, J. Algebra 54 (1978),
178–202.
6. G. Karpilovsky, The Schur Multiplier, Oxford Clarendon Press, 1987.
7. A. R. Salemkar, M. R. R. Moghaddam, K. Chiti, Some properties on the Schur
multiplier of a pair of groups, J. Algebra 312 (2007), 1–8.
8. A. R. Salemkar, F. Saeedi and T. Karimi, The structure of isoclinism of pairs of
groups, Southeast Asian Bull. Math. 31 (2007), 1173-1182.
Azam Pourmirzaei
Department of Mathematics, Hakim Sabzevari University, P. O. Box 96179-76487,
Sabzevar, Iran
a.pmirzaei@gmail.com
Mitra Hassanzadeh
Department of Mathematics, Department of Mathematics, Ferdowsi University of
Mashhad, P.O.Box 1159-91775, Mashhad, Iran.
mtr.hassanzadeh@gmail.com
Behrooz Mashayekhy
Department of Mathematics, Center of Excellence in Analysis on Algebraic Structures, Ferdowsi University of Mashhad, P.O.Box 1159-91775, Mashhad, Iran.
bmashf@um.ac.ir
| 4 |
arXiv:1711.05085v2 [math.ST] 11 Apr 2018
The mixability of elliptical distributions with
supermodular functions
Xiaoqian Zhang Chuancun Yin
School of Statistics, Qufu Normal University
Shandong 273165, China
e-mail: ccyin@mail.qfnu.edu.cn
April 12, 2018
ABSTRACT The concept of φ-complete mixability and φ-joint mixability was first
introduced in Bignozzi and Puccetti (2015), which is a direct extension of complete and
joint mixability. Following Bignozzi and Puccetti (2015), we consider two cases of φ
and investigate the φ-joint mixability for elliptical distributions and logarithmic elliptical
distributions. We obtain a necessary and sufficient condition for the φ-joint mixability
of some distributions and a sufficient condition for uniqueness of the center of φ-joint
mixability for some elliptical distributions.
MSC: 60E05; 91B30
Keywords: elliptical distributions; log-elliptical distributions; φ-joint mixability; supermodular functions
1
Introduction
The concept of complete mixability and joint mixability has been studied by many researchers in recent years because of its important application in many relevant fields. The
definition of complete mixability for a univariate distribution was first introduced in Wang
1
and Wang (2011) and then extended to the notion of joint mixability of an arbitrary set
of distributions in Wang, Peng and Yang (2013). Suppose n is a positive integer, distribution functions F1 , · · · , Fn on R are said to be jointly mixable with index n if there exist n
random variables X1 , · · · , Xn such that Xi ∼ Fi , i ≤ i ≤ n, and P (X1 +· · ·+Xn = C) = 1
for some C ∈ R. If Fi = F , 1 ≤ i ≤ n, then F is said to be n-completely mixable and
(X1 , · · · , Xn ) a complete mix. Any such C is called a joint center of (F1 , · · · , Fn ). The
concept of complete mixability is related to some optimization problems in the theory of
optimal couplings. For more details on the problems and a brief history of the concept
of the mixability, we refer to the recent papers Puccetti, Wang and Wang (2012), Wang
(2015) and Wang and Wang (2016).
Bignozzi and Puccetti (2015) extend the concept of joint mixability and introduce the
concept of φ-joint mixability.
Definition 1.1. (Bignozzi and Puccetti (2015)). Let φ : Rn →R be a measurable function.
An n-tuple of univariate distribution functions (F1 , · · · , Fn ) is said to be φ-jointly mixable
with index n if there exist n random variables X1 , · · · , Xn such that Xi ∼ Fi , 1 ≤ i ≤ n,
and
P (φ(X1 , · · · , Xn ) = C) = 1
(1.1)
for some C ∈ R. Any such C is called a center of the φ-jointly mixable distributions
and any random vector (X1 , · · · , Xn ) satisfying (1.1) is called a joint φ-mix. If Fi = F ,
1 ≤ i ≤ n, then F is said to be φ-completely mixable with index n and the random vector
(X1 , · · · , Xn ) a complete φ-mix.
Supermodular functions play an important role in risk theory and actuarial science.
A function ϕ: Rn → R is said to be supermodular if for any u, v∈Rn it satisfies
ϕ(u∨v) + ϕ(u∧v) ≥ ϕ(u) + ϕ(v),
where the operators ∨ and ∧ denote coordinatewise maximum and minimum, respectively.
Equivalent definitions and many examples of supermodular functions can be found in
Denuit, Dhaene, Goovaerts and Kaas (2005).
2
In this paper, we consider the following two supermodular functions:
φ1 (x1 , · · · , xn ) = h(α1 x1 + · · · + αn xn ), αi > 0, xi ∈ R;
φ2 (x1 , · · · , xn ) = g
n
Y
xαi i
i=1
!
, αi > 0, xi ≥ 0,
(1.2)
(1.3)
where h, g are convex functions. In particular, if all αi are 1, then φ1 becomes the sum
operator and φ2 becomes the product operator considered in Bignozzi and Puccetti (2015).
By definition, the positive distributions F1 , · · · , Fn are φ2 -jointly mixable if and only if
the distributions G1 , · · · , Gn are φ1 -jointly mixable for increasing or decreasing h and g,
where Gi (x) = Fi (exp( αxi )), 1 ≤ i ≤ n; see Lemma 6 in Bignozzi and Puccetti (2015) for
the special case of all αi are 1 and g(x) = h(x) = x.
In Section 2, we focus on φ1 -joint mixability for the class of elliptical distributions. In
Section 3, we investigate φ2 -joint mixability for the class of logarithmic elliptical distributions.
2
Elliptical distributions
In this section, we consider the φ1 -joint mixability for the class of elliptical distributions.
Definition 2.1. (Denuit, Dhaene, Goovaerts and Kaas (2005)). An n-dimensional random vector X is said to have an elliptical distribution if its characteristic function can be
expressed as
E[exp(it′ X)] = exp(it′ µ)ψ(t′Σt), t ∈ Rn ,
where µ, Σ are parameters, µ = (µ1 , · · · , µn )′ ∈ Rn , Σ = (σij )n × n is positive semidefinite
matrix, the function ψ: R → R is the characteristic generator of X. We denote by
X ∼ εlln (µ, Σ, ψ).
Note that not every function ψ can be a characteristic generator, it should fulfil a
requirement that ψ(0) = 1. It is easy to see that if ψ(x) = exp{−x/2}, the elliptical
distribution becomes normal distribution Nn (µ, Σ).
3
Let Ψn be a class of functions ψ : [0, ∞) → R such that function ψ(|t|2 ), t ∈ Rn is an
n-dimensional characteristic function. It is clear that Ψn ⊂ Ψn−1 · · · ⊂ Ψ1 . Denote by Ψ∞
the set of characteristic generators that generate an n-dimensional elliptical distribution
for arbitrary n ≥ 1. That is Ψ∞ = ∩∞
n=1 Ψn . Clearly, if ψ(x) = exp{−x/2}, then ψ ∈ Ψ∞ .
Remark 2.1. The moments of X ∼ εlln (µ, Σ, ψ) do not necessarily exist, if E[Xi ] exists,
it will be given by E[Xi ] = µi , if Cov[Xi , Xj ] and/or V ar[Xi ] exist, they will be given by
Cov(Xi , Xj ) = −2ψ ′ (0)σij ,
V ar[Xi ] = −2ψ ′ (0)σi2 ,
where ψ ′ is the first derivative of ψ. Furthermore, if the covariance matrix of X exists,
then it is given by -2ψ ′ (0)Σ, a necessary condition for this covariance matrix to exist is
|ψ ′ (0)| < ∞; see Cambanis, Huang and Simons (1981).
It is well known that a random vector X ∼ εlln (µ, Σ, ψ) if and only if for any α =
(α1 , · · · , αn )′ ∈ Rn , α′ X ∼ εll1 (α′ µ, α′ Σα, ψ); see Denuit, Dhaene, Goovaerts and Kaas
(2005).
Suppose that Fi ∼ εll1 (µi , σi2 , ψ), i = 1, 2, · · · , n (n ≥ 3), where ψ is a characteristic
generator for an n-elliptical distribution. If f is one-to-one, then it follows from Lemma
2.1 and Theorem 3.7 in Wang and Wang (2016) that (F1 , · · · , Fn ) is φ1 -jointly mixable if
and only if
n
X
i=1
αi σi ≥ 2 max{α1 σ1 , · · · , αn σn },
(2.1)
where φ1 is defined by (1.2). If f is not one-to-one, the condition (2.1) is also a sufficient
condition for φ1 -joint mixability. Furthermore, if all means of Fi ’s exist, then the joint
center of (F1 , · · · , Fn ) is unique; If all means of Fi ’s do not exist, then the joint centers
of (F1 , · · · , Fn ) are not necessarily unique. For example, Puccetti, Rigo, Wang and Wang
(2018) obtain a profound result that for every n ≥ 2, the set of n-centers of the standard
Cauchy distribution is the interval
log(n − 1) log(n − 1)
.
,
−
π
π
4
For general Cauchy distributions with the following probability density functions
f (x; µ, σ) =
σ
1
, −∞ < x < ∞,
π (x − µ)2 + σ 2
the set of n-centers is the interval
log(n − 1)
log(n − 1)
−σ
+ nµ, σ
+ nµ .
π
π
It follows that any C in
"
2 #
log(n − 1)
0, σ
+ nµ
π
is the φ-center, where φ(x1 , · · · , xn ) = (x1 + · · · + xn )2 .
Contrastively, for a given supermodular function φ, in general the center of a set of φjointly mixable distributions might not be unique even when the distributions have finite
mean. Example 1.1 in Bignozzi and Puccetti (2015) shows that the center is not unique
for φ-jointly mixable discrete distributions having finite means with φ(x1 , x2 ) = (x1 +x2 )2 .
The following theorems concern the joint mixability and φ-joint mixability. We obtain
necessary and sufficient conditions for the φ-joint mixability of some classes of distributions. Especially we give a sufficient condition for uniqueness of the center of φ-joint
mixability for some elliptical distributions.
Theorem 2.1. Suppose that Fi ∼
forms
Ci
f
fi (x; σi ) =
2σi
εll1 (0, σi2, ψ)
(x − νi )2
σi2
Ci
+
f
2σi
(i = 1, 2, · · · , n) have densities of the
(x + νi )2
σi2
, −∞ < x < ∞,
(2.2)
where Ci ’s are normalizing constants, νi ≥ 0, σi > 0 are parameters and f is density
generator satisfying the condition
0<
Z
0
∞
1
x− 2 f (x)dx < ∞.
If distributions G1 , · · · , Gn with density generator f are jointly mixable, then (F1 , · · · , Fn )
P
is φ-jointly mixable with center h( ni=1 νi ), where φ(x1 , · · · , xn ) = h(|x1 + · · · + xn |) for
any function h on [0, ∞).
5
Proof The joint mixability of (G1 , · · · , Gn ) implies that there exist n random variables
Y1 , · · · , Yn such that Yi ∼ εll1 (νi , σi2 , f ), 1 ≤ i ≤ n, and
Y1 + · · · + Y1 =
n
X
νi ,
i=1
and there exist n random variables Z1 , · · · , Zn such that Zi ∼
and
Z1 + · · · + Z1 = −
n
X
εll1 (−νi , σi2, f ), 1 ≤ i ≤ n,
νi .
i=1
Define Xi = ξYi + (1 − ξ)Zi (i = 1, 2, · · · , n), where P (ξ = 1) = P (ξ = 0) = 1/2
and, random variables Yi , Zi , ξ are independent. Then Xi has the distribution Fi and
P
|X1 + · · · + Xn | = ni=1 νi . This shows that (F1 , · · · , Fn ) is φ-jointly mixable with center
P
h( ni=1 νi ).
Theorem 2.2. Suppose that Fi ∼
forms
Ci
fi (x; σi ) =
f
2σi
εll1 (0, σi2, ψ)
(x − νi )2
σi2
Ci
+
f
2σi
(i = 1, 2, · · · , n) have densities of the
(x + νi )2
σi2
, −∞ < x < ∞,
(2.3)
where Ci ’s are normalizing constants, νi , σi are parameters and f is density generator
satisfying the condition
0<
Z
∞
0
1
x− 2 f (x)dx < ∞.
Suppose distributions Gi ’s with density generator f are unimodal and (2.1) holds. Then
P
(F1 , · · · , Fn ) is φ-jointly mixable with center h( ni=1 νi ), where φ(x1 , · · · , xn ) = h(|α1 x1 +
· · · + αn xn |) for any function h on [0, ∞).
Proof Xi ∼ εll1 (0, σi2 , ψ) implies that αi Xi ∼ εll1 (0, αi2σi2 , ψ). Using (2.3) the density
of αi Xi can be expressed as
Ci
fi (x; αi σi ) =
f
2αi σi
x − νi
αi σi
2 !
Ci
+
f
2αi σi
x + νi
αi σi
2 !
, −∞ < x < ∞,
(2.4)
The unimodality of Gi ’s and condition (2.1) imply that there exist n random variables
Y1 , · · · , Yn such that Yi ∼ Gi , 1 ≤ i ≤ n and
α1 Y 1 + · · · + α1 Y 1 =
6
n
X
i=1
νi ,
and there exist n random variables Z1 , · · · , Zn such that Zi ∼ Gi , 1 ≤ i ≤ n and
α1 Z 1 + · · · + α1 Z 1 = −
n
X
νi .
i=1
Define Xi = ξYi + (1 − ξ)Zi , where P (ξ = 1) = P (ξ = 0) = 1/2 and, random variables
Yi , Zi , ξ are independent. Then Xi has the density (2.3) and
|α1 X1 + · · · + αn Xn | =
n
X
νi ,
i=1
which shows that (F1 , · · · , Fn ) is φ-jointly mixable with center h(
Pn
i=1
νi ).
Theorem 2.3. Suppose that Fi ∼ εll1 (0, σi2 , ψ) (i = 1, 2, · · · , n) have density functions
Pn
fi , if there exist n random variables X1 , · · · , Xn such that Xi ∼ Fi and
i=1 Xi has
two-point distribution
1
1
Gn (x) = δ−µn (x) + δµn (x), x ∈ (−∞, ∞),
2
2
for some µn > 0, where δa (·) denotes a point mass at a, then fi will be of the form
Ci
Ci
(x − νi )2
(x + νi )2
fi (x; σi ) =
+
, −∞ < x < ∞,
(2.5)
f
f
2σi
σi2
2σi
σi2
P
where Ci ’s are normalizing constants, νi ’s are parameters such that ni=1 νi = µn and f
is density generator of 1-dimensional elliptical distribution satisfying the condition
Z ∞
1
0<
x− 2 f (x)dx < ∞.
0
Proof Since P (
Pn
i=1
Xi = −µn ) = P (
Pn
i=1
Xi = µn ) = 12 , then there exist 2n random
variables Yi and Zi (i = 1, 2, · · · , n) such that Xi = ξYi + (1 − ξ)Zi (i = 1, 2, · · · , n) and
P
P
P ( ni=1 Yi = −µn ) = P ( ni=1 Zi = µn ) = 1, where P (ξ = 1) = P (ξ = 0) = 1/2 and,
random variables {Yi }, {Zi }, ξ are independent. If there exists an asymmetric density gi
such that
1
1
fi (x) = gi (−x) + gi (x), x ∈ (−∞, ∞),
2
2
then
Z
∞
−∞
cos(tx)fi (x)dx =
Z
∞
−∞
cos(tx)gi (x)dx, t ∈ (−∞, ∞),
from which we deduces that fi = gi . This is a contradiction. Thus gi is symmetric and
the density fi can be written as the form (2.5). This ends the proof.
7
Corollary 2.1. Suppose Fi ∼
εll1 (0, σi2, ψ) (i =
1, 2, · · · , n) have finite means. Then
P
there exist n random variables X1 , · · · , Xn such that Xi ∼ Fi and ni=1 Xi has two-point
distribution
1
1
Gn (x) = δ−µn (x) + δµn (x), x ∈ (−∞, ∞),
2
2
Pn
for some µn > 0, if and only if P (( i=1 Xi )2 = µ2n ) = 1.
Proof The “only if” part is obvious. Now we prove the converse implication. If there
P
exist n random variables X1 , · · · , Xn such that Xi ∼ Fi and ni=1 Xi = C, then C = 0
P
since Fi ∼ εll1 (0, σi2 , ψ) (i = 1, 2, · · · , n) have finite means. Thus by symmetry of ni=1 Xi
P
and P (( ni=1 Xi )2 = µ2n ) = 1 we have
!
!
n
n
X
X
1
P
Xi = −µn = P
Xi = µ n = .
2
i=1
i=1
Corollary 2.2. Suppose that Fi ∼ εll1 (0, σi2 , ψ) (i = 1, 2, · · · , n) have finite means with
density functions fi . If there exist n random variables X1 , · · · , Xn such that Xi ∼ Fi and
P
P (( ni=1 Xi )2 = µ2n ) = 1 for some µn > 0, then fi will be of the form (2.5).
Corollary 2.3. Suppose that Fi ∼
εll1 (0, σi2, ψ)
(i = 1, 2, · · · , n) have finite means
with ψ ∈ Ψ∞ . If there exist n random variables X1 , · · · , Xn such that Xi ∼ Fi and
P
P (( ni=1 Xi )2 = µ2n ) = 1 for some constant µn , then µn = 0.
Proof If µn 6= 0, then, by Lemma 2.1, the density function fi of Fi has the form (2.5).
In terms of characteristic functions (2.5) is equivalent to
2 2
Z ∞
(θσi )2 2
σi t
exp −
, −∞ < t < ∞,
t dHi (θ) = cos(tνi )φi
2
2
0
(2.6)
where Hi a distribution function on (0, ∞) and φi is the characteristic generator of gi .
This is a contradiction since the left hand side of (2.6) is assumed positive for any t, but
the right hand side of (2.6) is zero for some t. Thus µn = 0.
Remark 2.2. Elliptical distributions with ψ ∈ Ψ∞ belonging to the class of scale mixture
of the normal distributions. Examples are normal distribution, T -distribution, Cauchy
distribution, stable laws distribution, double exponential distribution and logistic distribution, and so on. Note that the condition ψ ∈ Ψ∞ in Corollary 2.3 can be replaced with
the condition that all Fi have positive characteristic functions.
8
Example 2.1 Suppose F has the following density
x2 +µ2
1
f (x) = √ e− 2 cosh(µx), −∞ < x < ∞,
2π
where µ > 0 is a constant. Then there exist n random variables X1 , · · · , Xn such that
P
Xi ∼ F and P (( ni=1 Xi )2 = n2 µ2 ) = 1. In fact, f can be rewritten as the form
(x+µ)2
(x−µ)2
1
1
f (x) = √ e− 2 + √ e− 2 , −∞ < x < ∞.
2 2π
2 2π
Obviously, f (−x) = f (−x), f is unimodal for µ ≤ 1 and bimodal for µ > 1 (cf. Robertson
and Fryer (1969)). It follows from Theorem 2.1, F is φ-completely mixable with center
(nµ)2 , where φ(x1 , · · · , xn ) = (x1 + · · · + xn )2 . On the other hand, F is the distribution of
ζ + η, where random variable ζ and η are independent and, ζ ∼ N(0, 1) and P (η = µ) =
P (η = −µ) = 1/2. Consequently, 0 is the completely center as well as the φ-completely
mixable center of F .
3
Logarithmic elliptical distributions
In this section, we will consider the φ2 -completely mixability of the log-elliptical distributions. For any n-dimensional vector X = (X1 , · · · , Xn )′ with positive components Xi , we
define log X = (log X1 , log X2 , · · · , log Xn )′ .
Definition 3.1. (Valdez et al. (2009)). The random vector X is said to have a multivariate log-elliptical distribution with parameters µ and Σ, denoted by X ∼ LEn (µ, Σ, ψ), if
log X has a multivariate elliptical distribution, i.e. log X ∼ εlln (µ, Σ, ψ). In particular,
when ψ(x) = exp{−x/2}, the log-elliptical distributions become log-normal distributions
LNn (µ, Σ).
Using Lemma 2.1 one easy to get
Lemma 3.1. A random vector X ∼ LEn (µ, Σ, ψ) if and only if for any (α1 , · · · , αn )′ ∈
n
Q
Xiαi ∼ LE1 (α′ µ, α′ Σα, ψ). In particular, any marginal distribution of a logRn ,
i=1
elliptical distribution is again log-elliptical.
9
Using the result of the last section, one finds that if g is one-to-one, then (F1 , · · · , Fn )
is φ2 -jointly mixable if and only if
n
X
i=1
αi σi ≥ 2 max{α1 σ1 , · · · , αn σn },
where φ2 is defined by (1.3). Furthermore, if all means of Fi ’s exist, then the φ2 -jointly
center of (F1 , · · · , Fn ) is unique; If all means of Fi ’s do not exist, then the φ2 -jointly centers
of (F1 , · · · , Fn ) are not necessarily unique. For example, using the result of Puccetti, Rigo,
Wang and Wang (2018) in the last section we have that for every n ≥ 2, the set of nproduct centers of the log-Cauchy distribution with the probability density function
1
σ
, x > 0,
xπ (log x − µ)2 + σ 2
f (x; µ, σ) =
is the interval
log(n − 1)
log(n − 1)
+ nµ , exp σ
+ nµ ,
exp −σ
π
π
where µ is a real number and σ > 0.
Corresponding to Theorems 2.1-2.3, we have the following theorems.
Theorem 3.1. Suppose that Fi ∼ LE1 (0, σi2 , ψ) (i = 1, 2, · · · , n) have densities of the
forms
Ci
f
fi (x; σi ) =
2σi x
(log x − νi )2
σi2
Ci
+
f
2σi x
(log x + νi )2
σi2
, x > 0,
(3.1)
where Ci ’s are normalizing constants, νi ≥ 0, σi > 0 are parameters and f is density
generator satisfying the condition
0<
Z
0
∞
1
x− 2 f (x)dx < ∞.
If distributions G1 , · · · , Gn with density generator f are jointly mixable, then (F1 , · · · , Fn )
Q
P
is φ-jointly mixable with center g( ni=1 νi ), where φ(x1 , · · · , xn ) = g(| log( ni=1 xi )|) for
any function g on [0, ∞).
Theorem 3.2. Suppose that Fi ∼ LE1 (0, σi2 , ψ) (i = 1, 2, · · · , n) have densities of the
forms
Ci
f
fi (x; σi ) =
2σi x
(log x − νi )2
σi2
Ci
+
f
2σi x
10
(log x + νi )2
σi2
, x > 0,
(3.2)
where Ci ’s are normalizing constants, νi ≥ 0, σi > 0 are parameters and f is density
generator satisfying the condition
0<
Z
0
∞
1
x− 2 f (x)dx < ∞.
Suppose distributions Gi ’s with density generator f are unimodal and (2.1) holds. Then
P
(F1 , · · · , Fn ) is φ-jointly mixable with center g( ni=1 νi ), where
!!
n
Y
φ(x1 , · · · , xn ) = g log
xαi i
i=1
for any function g on [0, ∞) and constants αi > 0.
Theorem 3.3. Suppose that Fi ∼ LE1 (0, σi2 , ψ) (i = 1, 2, · · · , n) have density functions
Q
fi , if there exist n random variables X1 , · · · , Xn such that Xi ∼ Fi and ni=1 Xi has
two-point distribution
1
1
Gn (x) = δexp(−µn ) (x) + δexp(µn ) (x), x > 0,
2
2
for some µn > 0, where δa (·) denotes a point mass at a. Then fi will be of the form
Ci
Ci
(log x − νi )2
(log x + νi )2
+
, x > 0,
(3.3)
f
f
fi (x; σi ) =
2σi x
σi2
2σi x
σi2
P
where Ci ’s are normalizing constants, νi ’s are parameters such that ni=1 νi = µn and f
is density generator of 1-dimensional elliptical distribution satisfying the condition
Z ∞
1
0<
x− 2 f (x)dx < ∞.
0
Acknowledgements. We thank Ruodu Wang and Bin Wang for helpful discussions
and the anonymous referees for insightful comments and constructive suggestions on an
earlier version of this paper. The research was supported by the National Natural Science
Foundation of China (No. 11571198, 11701319).
References
[1] Bignozzi, V., Puccetti, G., 2015. Studying mixability with supermodular aggregating
fuctions. Statistics and Probability Letters 100, 48-55.
11
[2] Cambanis, S., Huang, S., Simons, G., 1981. On the theory of elliptically contoured
distributions. Journal of Multivariate Analysis 11, 368-385.
[3] Dhaene, J., Denuit, M., Goovaerts, M., Kaas, R., 2005. Actuarial Theory for Dependent Risks: Measures, Orders and Models. John Wiley and Sons, Chichester.
[4] Fang, K. T., Kotz, S., Ng, K. W., 1990. Symmetric Multivariate and Related Distributions. Chapman and Hall, London.
[5] Puccetti,
ability
G.,
Rigo,
measures
P.,
without
Wang,
B.,
the
mean.
Wang,
Journal
R.,
of
2018. Centers of probTheoretical
Probability
https://doi.org/10.1007/s10959-018-0815-3.
[6] Puccetti, G., Wang, B., Wang, R., 2012. Advances in complete mixability. Journal
of Applied Probability 49 (2), 430-440.
[7] Robertson, C. A. and Fryer, J. G. (1969). Some descriptive properties of normal
mixtures. Scandinavian Actuarial Journal 1969 (3-4), 137-146.
[8] Valdez, E., Dhaene, J., Maj, M., Vanduffel, S., 2009. Bounds and approximations
for sums of dependent log-elliptical random variables. Insurance: Mathematics &
Economics 44, 385-397.
[9] Wang, R., 2015. Current open questions in complete mixability. Probability Surveys
12, 13-32.
[10] Wang, R., Peng, L., Yang, J., 2013. Bounds for the sum of dependent risks and
worst value-at-risk with monotone marginal densities. Finance and Stochastics 17
(2), 395-417.
[11] Wang, B., Wang, R., 2011. The complete mixability and convex minimization problems with monotone marginal densities. Journal of Multivariate Analysis 102(10),
1344-1360.
[12] Wang, B., Wang, R., 2016. Joint mixability. Mathematics of Operations Research
41(3), 808-826.
[13] Yin, C., Zhu, D., 2017. Joint mixability of elliptical distributions and related families.
arXiv:1706.05499.
12
| 10 |
A multifactor RSA-like scheme with fast decryption
based on Rédei rational functions over the Pell
hyperbola
arXiv:1709.00696v1 [cs.IT] 3 Sep 2017
Emanuele Bellini
DarkMatter LLC, UAE
Nadir Murru
University of Turin, Italy
Abstract
We propose a generalization of an RSA-like scheme based on Rédei rational
functions over the Pell hyperbola. Instead of a modulus which is a product of
two primes, we define the scheme on a multi-factor modulus, i.e. on a product
of more than two primes. This results in a scheme with a decryption which
is quadratically faster, in the number of primes factoring the modulus, than
the original RSA, while preserving a better security. The scheme reaches its
best efficiency advantage over RSA for high security levels, since in these cases
the modulus can contain more primes. Compared to the analog schemes based
on elliptic curves, as the KMOV cryptosystem, the proposed scheme is more
efficient. Furthermore a variation of the scheme with larger ciphertext size does
not suffer of impossible group operation attacks, as it happens for schemes based
on elliptic curves.
Keywords: RSA-like cryptosystem, multifactor RSA, multiprime RSA, Rédei
rational functions, Pell equation, fast decryption
1. Introduction
RSA is the most widespread asymmetric encryption scheme. Its security is
based on the fact that the trapdoor function τN,e (x) = xe mod N , with N = pq
product of two large prime integers, and e an invertible element in Zφ(N ) (φ(N )
being the Euler totient function), cannot be inverted by a polynomial-time in
log N algorithm without knowing either the integers p, q, φ(N ) or the inverse
d of e modulo φ(N ). Thus the pair (N, e), called the public key, is known to
everyone, while the triple (p, q, d), called the secret key, is only known to the
Email addresses: eemanuele.bellini@gmail.com (Emanuele Bellini),
nadir.murru@unito.it (Nadir Murru)
Preprint submitted to Journal of LATEX Templates
September 5, 2017
receiver of an encrypted message.
Both encryption and decryption are performed through an exponentiation modulo N . Precisely, the ciphertext C is obtained as C = M e (mod N ), and the
original message M is obtained with the exponentiation M = C d (mod N ).
While usually the encryption exponent is chosen to be small, the decryption
exponent is about the size of N , implying much slower performances during
decryption with respect to encryption.
Through the years many proposal have been presented trying to speed up the
decryption process. In this work we present the fastest, to the authors knowledge, of such decryption algorithms whose security is based on the factorization
problem.
The presented scheme exploits different properties of Rédei rational functions,
which are classical functions in number theory. The proposed decryption algorithm is quadratically, on the number of primes composing the modulus N ,
faster than RSA.
The work is divided as follows. In Section 2 an overview of the main schemes
based on the factorization problem which successfully improved RSA decryption step is presented. In Section 3 the main theoretical results underlying our
scheme are described. Section 4 is devoted to the presentation of the cryptographic scheme, and in Section 5 and 6 its security and efficiency are discussed,
respectively. Section 7 concludes the work.
2. Related work
In this section we briefly overview the main cryptographic schemes based on
the factorization problem that have been introduced in order to improve RSA
decryption step.
Usually, the general technique to speed up the RSA decryption step C = M e
(mod N ) is to compute the exponentiation modulo each factor of N and then
obtain N using the Chinese Remainder Theorem.
2.1. Multifactor RSA
There exists variants of RSA scheme which exploit a modulus with more than
2 factors to achieve a faster decryption algorithm. This variants are sometimes
called Multifactor RSA ([1]), or Multiprime RSA ([2], [3]). The first proposal
exploiting a modulus of the form N = p1 p2 p3 has been patented by Compaq
([3], [4]) in 1997. About at the same time Takagi [5] proposed an even faster
solution using the modulus N = pr q, for which the exponentiation modulo pr
is computed using the Hensel lifting method [6, p.137]. Later, this solution has
been generalized to the modulus N = pr q s [7].
According to [3] the appropriate number of primes to be chosen in order to
resist state-of-the-art factorization algorithms depends from the modulus size,
and, precisely, it can be: up to 3 primes for 1024, 1536, 2048, 2560, 3072, and
3584 bit modulus, up to 4 for 4096, and up to 5 for 8192.
2
2.2. RSA-like schemes
Another solution which allows to obtain even faster decryption is to use
RSA-like schemes based on isomorphism as [8], [9], [10], [11]. As an additional
property, these schemes owns better security properties with respect to RSA,
avoiding small exponent attacks to either d ([12]) or e ([13], [14]), and vulnerabilities which appear when switching from one-to-one communication scenario
to broadcast scenario (e.g., see [15]).
The aforementioned schemes are based on isomorphism between two groups, one
of which is the set of points over a curve, usually a cubic or a conic. A complete
overview on RSA-like schemes based on conics can be found in [11]. In general,
schemes based on cubic curves have a computationally more expensive addition
operation compared to schemes based on conic equations.
2.3. Generalizing RSA-like scheme with multifactor modulus
As done when generalizing from RSA to Multiprime RSA, in [16] a generalization of [8], [9] has been proposed, thus generalizing a RSA-like scheme
based on elliptic curves and a modulus N = pq to a similar scheme based on
the generic modulus N = pr q s .
In this paper we present a similar generalization of the scheme [11], which is
based on the Pell’s equation, to the modulus N = pe11 · . . . · perr for r > 2,
obtaining the fastest decryption of all schemes discussed in this section.
3. Product of points over the Pell hyperbola
In [11], we introduced a novel RSA–like scheme based on an isomorphism
between certain conics (whose the Pell hyperbola is a special case) and a set of
parameters equipped with a non–standard product. In Section 4, we generalize
this scheme considering a prime power modulus N = pe11 · · · perr . In this section,
we recall some definitions and properties given in [11] in order to improve the
readability of the paper. Then, we study properties of the involved products
and sets in Zpr and ZN .
3.1. A group structure over the Pell hyperbola over a field
Let K be a field and x2 −D an irreducible polynomial over K[x]. Considering
the quotient field A[x] = K[x]/(x2 − D), the induced product over A[x] is
(p + qx)(r + sx) = (pr + qsD) + (qr + ps)x.
The group of unitary elements of A∗ [x] = A[x] − {0A[x]} 1 is {p + qx ∈ A∗ [x] :
p −Dq 2 = 1}. Thus, we can introduce the commutative group (HD,K , ⊗), where
2
HD,K = {(x, y) ∈ K × K : x2 − Dy 2 = 1}
1 The
element 0A[x] is the zero polynomial.
3
and
(x, y) ⊗ (w, z) = (xw + yzD, yw + xz),
∀(x, y), (w, z) ∈ HD,K .
(1)
It is worth noting that (1, 0) is the identity and the inverse of an element (x, y)
is (x, −y).
Remark 1. When K = R, the conic HD,K , for D a non–square integer, is called
the Pell hyperbola since it contains all the solutions of the Pell equation and ⊗
is the classical Brahamagupta product, see, e.g., [17].
3.2. A parametrization of the Pell hyperbola
From now on let A = A[x].
Starting from A∗ , we can derive a parametrization for HD,K . In particular, let
us consider the group A∗ /K∗ , whose elements are the equivalence classes of A∗
and can be written as
{[a + x] : a ∈ K} ∪ {[1K∗ ]}.
The induced product over A∗ /K∗ is given by
[a + x][b + x] = [ab + ax + bx + x2 ] = [D + ab + (a + b)x]
and, if a + b 6= 0, we have
#
D + ab
+x
[a + x][b + x] =
a+b
"
else
[a + x][b + x] = [D + ab] = [1K∗ ].
This construction allows us to define the set of parameters PK = K ∪ {α}, with
α not in K, equipped with the following product:
D + ab
, a + b 6= 0
a⊙b=
.
(2)
a+b
a ⊙ b = α, a + b = 0
We have that (PK , ⊙) is a commutative group with identity α and the inverse
of an element a is the element b such that a + b = 0. Now, consider the following
parametrization for the conic HD,K :
1
(x + 1) .
m
It can be proved that the following isomorphism between (HD,K , ⊗) and (PK , ⊙)
holds:
HD,K
→ PK
1+x
(x, y)
7→
∀(x, y) ∈ HD,K , y 6= 0
ΦD :
(3)
y
(1,
0)
→
7
α
(−1, 0) 7→ 0 ,
y=
4
and
Φ−1
D
see [18] and [11].
PK
: m
α
→ HD,K
!
2m
m2 + D
,
7→
m2 − D m2 − D
∀m ∈ K ,
(4)
7→ (1, 0) ,
Proposition 1. When K = Zp , p prime, (PK , ⊙) and (HD,K , ⊗) are cyclic
groups of order p + 1 and
m⊙(p+2) = m
(mod p),
∀m ∈ PZp
(x, y)⊗(p+2) = (x, y)
(mod p),
∀(x, y) ∈ HD,Zp ,
or, equivalently
where powers are performed using products ⊙ and ⊗, respectively. See [11].
The powers in PK can be efficiently computed by means of the Rédei rational
functions [19], which are classical functions in number theory. They are defined
by considering the development of
√
√
(z + D)n = An (D, z) + Bn (D, z) D,
for z integer and D non–square positive integer. The polynomials An (D, z) and
Bn (D, z) defined by the previous expansion are called Rédei polynomials and
can be evaluated by
An (D, z) DBn (D, z)
n
M =
Bn (D, z) An (D, z)
where
M=
z
1
D
.
z
From this property, it follows that the Rédei polynomials are linear recurrent
sequences with characteristic polynomial t2 − 2zt + (z 2 − D). The Rédei rational
functions are defined by
Qn (D, z) =
An (D, z)
,
Bn (D, z)
∀n ≥ 1.
Proposition 2. Let m⊙n be the n–th power of m ∈ PK with respect to ⊙, then
m⊙n = Qn (D, m).
See [20].
Remark 2. The Rédei rational functions can be evaluated by means of an algorithm of complexity O(log2 (n)) with respect to addition, subtraction and
multiplication over rings [21].
5
3.3. Properties of the Pell hyperbola over a ring
In this section, we study the case K = Zpr that we will exploit in the next
section for the construction of a cryptographic scheme. In what follows, we will
omit from HD,K the dependence on D when it will be clear from the context.
First, we need to determine the order of HZpr in order to have a result similar
to Proposition 1 also in this situation.
Theorem 1. The order of the cyclic group HZpr is pr−1 (p + 1), i.e., the Pell
equation x2 − Dy 2 = 1 has pr−1 (p + 1) solutions in Zpr for D ∈ Z∗pr quadratic
non–residue in Zp .
Proof. Since, by Proposition 1, the Pell equation in Zp has p + 1 solutions, then
we need to prove the following
1. any solution of the Pell equation in Zp , generates pr−1 solutions of the
same equation in Zpr ;
2. all the solutions of the Pell equation in Zpr are generated as in the previous
step.
(1) Let (x0 , y0 ) be a solution of x2 − Dy 2 ≡ 1 (mod p). We want to prove
that for any integer 0 ≤ k < pr−1 , there exists one and only one integer h
such that (x0 + kp, y0 + hp) is solution of x2 − Dy 2 ≡ 1 (mod pr ).
Indeed, we have
(x0 + kp)2 − D(y0 + hp)2 = 1 + vp + 2x0 kp + k 2 p2 − 2Dy0 hp − Dh2 p2 ,
since x20 − Dy02 = 1 + vp for a certain integer v. Thus, we have that
(x0 + kp, y0 + hp) is solution of x2 − Dy 2 ≡ 1 (mod pr ) if and only if
Dph2 + 2Dy0 h − v − 2x0 k − k 2 p ≡ 0 (mod pr−1 ).
Hence, we have to prove that there is one and only one integer h that
satisfies the above identity. The above equation can be solved in h by
completing the square and reduced to
(2Dph + 2Dy0 )2 ≡ s
(mod pr−1 ),
(5)
where s = (2Dy0 )2 + 4(v + 2x0 k + k 2 p)Dp. Let us prove that s is a
quadratic residue in Zpr−1 . Indeed,
s = 4D((x0 + kp)2 − 1)
and surely the Jacobi symbol
If r is even we have that
!
!
4
s
=
pr−1
pr−1
s
pr−1
!
D
pr−1
6
=
!
s
p
!r−1
= 1 if r is odd.
!
(x0 + kp)2 − 1
=1
pr−1
4
!
D
!
D
p
!r−1
= 1,
=
= −1 by hypothesis on D,
pr−1
!
(x0 + kp)2 − 1
= −1, since (x0 + kp)2 − 1 ≡ Dy02 (mod p).
pr−1
Now, let ±t be the square roots of s. It is easy to note that
since
pr−1
t ≡ 2Dy0
(mod p),
−t ≡ −2Dy0
(mod p)
t ≡ −2Dy0
(mod p).
or
−t ≡ 2Dy0
(mod p),
Let us call t̄ the only one between t and −t that is equal to 2Dy0 in Zp .
Hence, Equation (5) is equivalent to the linear equation
ph ≡ (t̄ − 2Dy0 )(2D)−1
(mod pr−1 ),
which has one and only one solution, since t̄ − 2Dy0 ≡ 0 (mod p). Note
that, if t̄ is not equal to 2Dy0 in Zp the above equation has no solutions.
Thus, we have proved that any solution of the Pell equation in Zp generates
pr−1 solutions of the Pell equation in Zpr .
(2) Now, we prove that all the solutions of the Pell equation in Zpr are generated as in step 1.
Let (x̄, ȳ) be a solution of x2 −Dy 2 ≡ 1 (mod pr ), i.e., x̄2 −Dȳ 2 = 1+wpr ,
for a certain integer w. Then x0 = x̄−kp and y0 = ȳ −hp, for h, k integers,
are solutions of x2 − Dy 2 ≡ 1 (mod p). Indeed,
(x̄ − kp)2 − D(ȳ − hp)2 = 1 + wpr − 2x̄kp + k 2 p2 + 2Dȳhp − Dh2 p2 .
As a consequence of the previous theorem, an analogous of the Euler theorem
holds for the product ⊗.
Theorem 2. Let p, q be prime numbers and N = pr q s , then for all (x, y) ∈ HZN
we have
r−1
s−1
(x, y)⊗p (p+1)q (s+1) ≡ (1, 0) (mod N )
for D ∈ Z∗N quadratic non–residue in Zp and Zq .
Proof. By Theorem 1, we know that
r−1
(x, y)⊗p
and
(x, y)⊗q
s−1
(p+1)
≡ (1, 0) (mod pr )
(s+1)
≡ (1, 0) (mod q s ).
7
r−1
Thus, said (a, b) = (x, y)⊗p
(p+1)qs−1 (s+1)
, we have
(a, b) ≡ (1, 0) (mod pr ),
i.e., a = 1 + kpr and b = hpr for some integers h, k. On the other hand, we have
(a, b) ≡ (1, 0) (mod q s ) ⇔ (1 + kpr , hpr ) ≡ (1, 0) (mod q s ).
We can observe that 1 + kpr ≡ 1 (mod q s ) if and only if k = k ′ q s for a certain
integer k ′ . Similarly, it must be h = h′ q s , for an integer h′ . Hence, we have that
(a, b) = (1 + k ′ pr q s , h′ pr q s ) ≡ (1, 0) (mod N ).
Corollary 1. Let p1 , ..., pr be primes and N = pe11 · . . . · perr , then for all (x, y) ∈
HZN we have
(x, y)⊗Ψ(N ) = (1, 0) (mod N ),
where
Ψ(N ) = pe11 −1 (p1 + 1) · . . . · perr −1 (pr + 1),
for D ∈ Z∗N quadratic non–residue in Zpi , for i = 1, ..., r.
Now, we can observe that when we work on ZN , the map ΦD is not an
isomorphism. Indeed, the orders of HD,ZN and PZN do not coincide. However,
it is still a morphism and we also have |Z∗N | = |HZ∗ N |, because of the following
proposition.
Proposition 3. With the above notation, we have that
1. ∀(x1 , y1 ), (x2 , y2 ) ∈ HZ∗ N , ΦD (x1 , y1 ) = ΦD (x2 , y2 ) ⇔ (x1 , y1 ) = (x2 , y2 );
−1
2. ∀m1 , m2 ∈ Z∗N , Φ−1
D (m1 ) = ΦD (m2 ) ⇔ m1 = m2 ;
∗
−1
3. ∀m ∈ ZN , we have Φ (m) ∈ HZ∗ N and ∀(x, y) ∈ HZ∗ N , we have ΦD (x, y) ∈
Z∗N .
See [11].
As a consequence, we have an analogous of the Euler theorem also for the
product ⊙, i.e., for all m ∈ Z∗N the following holds
m⊙Ψ(N ) = α
(mod N ) ,
where ⊙ is the special product in PZN defined in Equation 3.2.
4. The cryptographic scheme
In this section, we describe our public–key cryptosystem based on the properties studied in the previous section.
8
4.1. Key generation
The key generation is performed by the following steps:
• choose
Qrr prime numbers p1 , . . . , pr , r odd integers e1 , . . . , er and compute
N = i=1 pei i ;
Q
• choose an integer e such that gcd(e, lcm ri=1 piei −1 (pi + 1)) = 1;
Q
• evaluate d = e−1 (mod lcm ri=1 piei −1 (pi + 1)).
The public or encryption key is given by (N, e) and the secret or decryption key
is given by (p1 , . . . , pr , d).
4.2. Encryption
We can encrypt pair of messages
(Mx , My ) ∈ Z∗N × Z∗N , such that the follow!
2
Mx − 1
ing condition holds:
= −1. This will ensure that we can perform all
N
the operations. The encryption of the messages is performed by the following
steps:
• compute D =
Mx2 − 1
∗
(mod N ), so that (Mx , My ) ∈ HD,Z
;
N
My2
• compute M = Φ(Mx , My ) =
Mx + 1
(mod N );
My
• compute the ciphertext C = M ⊙e (mod N ) = Qe (D, M ) (mod N )
Notice that not only C, but the pair (C, D) must be sent through the insecure
channel.
4.3. Decryption
The decryption is performed by the following steps:
• compute C ⊙d (mod N ) = Qd (D, C) (mod N ) = M ;
!
2M
M2 + D
−1
(mod N ) for retrieving the
,
• compute Φ (M ) =
M2 − D M2 − D
messages (Mx , My ).
5. Security of the encryption scheme
The proposed scheme can be attacked by solving one of the following problems:
1. factorizing the modulus N = pe11 · . . . · perr ;
9
2. computing Ψ(N ) = p1e1 −1 (p1 + 1) · . . . · perr −1 (pr + 1), or finding the number
of solutions of the equation x2 − Dy 2 ≡ 1 mod N , i.e. the curve order,
which divides Ψ(N );
3. computing Discrete Logarithm problem either in (HZ∗ N , ⊗) or in (PZ∗N , ⊙);
4. finding the unknown d in the equation ed ≡ 1 mod Ψ(N );
5. finding an impossible group operation in PZN ;
6. computing Mx , My from D.
5.1. Factorizing N or computing the curve order
It is well known that the problem of factorizing N = pe11 ·. . .·perr is equivalent
to that of computing the Euler totient function φ(N ) = pe11 −1 (p1 − 1) · . . . ·
perr −1 (pr − 1), e.g. see [22] or [23, Section 10.4].
In our case we need to show the following
Proposition 4. The problem of factorizing N is equivalent to computing the
value Ψ(N ) = pe11 −1 (p1 + 1) · . . . · perr −1 (pr + 1) or the order of the group PZ∗N
(or equivalently of HZ∗ N ), which is a divisor of Ψ(N ).
Proof. Clearly, knowing the factorization of N yields Ψ(N ).
Conversely, suppose N and Ψ(N ) are known. A factorization of N can be found
by applying Algorithm 1 recursively.
Remark 3. Algorithm 1 is an adaptation of the general algorithm in [23, Section
10.4], used to factorize N by only knowing φ(N ) (Euler totient function) and N
itself. The main idea of the Algorithm 1 comes from the fact that x⊙Ψ(N ) = 1
(mod N ) for all x ∈ Z∗N , which is the analog of the Euler theorem in PZN .
Notice that, because of Step 7, Algorithm 1 is a probabilistic algorithm. Thus,
to find a non-trivial factor, it might be necessary to run the algorithm more than
once. We expect that a deeper analysis of the algorithm will lead to a similar
probabilistic behaviour than the algorithm in [23], which returns a non-trivial
factor with probability 1/2.
Since we proved that the problems 1 and 2 are equivalent, we can only focus
on the factorization problem.
According to [3], state-of-the-art factorization methods as the Elliptic Curve
Method [24] or the Number Field Sieve [25], [26] are not effective if in the
following practical cases
• |N | = 1024, 1536, 2048, 2560, 3072, 3584 and N = pe11 pe22 pe33 with √
e1 + e2 +
e3 ≤ 3 and pi , i = 1, 2, 3 greater than approximately the size of 3 N .
• |N | = 4096 and N = pe11 pe22 pe33 pe44 with e1 + e2 √
+ e3 + e4 ≤ 4 and pi , i =
1, . . . , 4 greater than approximately the size of 4 N .
• |N | = 8192 and N = pe11 pe22 pe33 pe44 pe55 with e1 + e2 + e3√+ e4 + e5 ≤ 5 and
pi , i = 1, . . . , 5 greater than approximately the size of 5 N .
10
Algorithm 1 Find a factor of N by knowing N and Ψ(N )
1: function Find factor(N ,Ψ(N ))
2:
h=0
3:
t = Ψ(N )
4:
while IsEven(t) do
5:
h=h+1
6:
t=t/2
7:
a = Random(N − 1)
8:
d = gcd(a, N )
9:
if d 6= 1 then
10:
return d
11:
b = a⊙t mod N
12:
for j = 0, . . . , h − 1 do
13:
d = gcd(b + 1, N )
14:
if d 6= 1 or d 6= N then
15:
return d
16:
b = b2 mod N
17:
return 0
Notice that currently, the largest prime factor found by the Elliptic Curve
Method is a 274 bit digit integer [27]. Note also that the Lattice Factoring
Method (LFM) of Boneh, Durfee, and Howgrave-Graham [28] is designed to
factor integers of the form N = pu q only for large u.
5.2. Computing the Discrete Logarithm
Solving the discrete logarithm problem in a conic curve can be reduced to
the discrete logarithm problem in the underlying finite field [29]. In our case the
curve is defined over the ring ZN . Solving the DLP over ZN without knowing
the factorization of N is as hard as solving the DLP over a prime finite field of
approximately the same size. As for the factorization problem, the best known
algorithm to solve DLP on a prime finite field is the Number Field Sieve. When
the size of N is greater than 1024 then the NFS can not be effective.
5.3. Solving the private key equation
In the case of RSA, small exponent attacks ([12], [13], [14]) can be performed
to find the unknown d in the equation ed ≡ 1 mod Ψ(N ). Generalization of
these attacks can be performed on RSA variants where the modulus is of the
form N = pe11 pe22 [30]. It has already been argued in [11] and [9] that this kind
of attacks fails when the trapdoor function is not a simple monomial power as
in RSA, as it is in the proposed scheme.
5.4. Finding an impossible group operation
In the case of elliptic curves over ZN , as in the generalized KMOV cryptosystem [16], it could happen that an impossible addition between two curve
11
points occurs, yielding the factorization of N . This is due to the fact that the
addition formula requires to perform an inversion in the underlying ring ZN .
However, as shown by the same authors of [16], the occurrence of an impossible
addition is very unlikely for N with few and large prime factors.
In our case an impossible group operation may occur if a + b is not invertible
in ZN , i.e. if gcd(a + b, N ) 6= 1, yielding in fact a factor of N . However, also in
our case, if N contains a few large prime factors, impossible group operations
occur with negligible probability, as shown by the following proposition.
Proposition 5. The probability to find an invertible element in PZN is approximately
1
1
1− 1−
·...· 1 −
p1
pr
Proof. The probability to find an invertible element in PZN is given by dividing
the number of non-invertible elements in PZN by the total number of elements
of this set, as follows:
|PZN | − #{invertible elements in PZN }
=
|PZN |
|ZN | + 1 − (#{invertible elements in ZN } + 1)
=
=
|ZN | + 1
N − φ(N )
=
=
N +1
1
1
∼1 − 1 −
·... · 1 −
p1
pr
where we used N ∼ N + 1 and φ(N ) = N 1 − p11 · . . . · 1 − p1r .
(6)
(7)
(8)
(9)
This probability tends to zero for large prime factors.
Let us notice that, in the Pell curve case, it is possible to avoid such situation,
by performing encryption and decryption in HZ∗ N , without exploiting the isomorphism operation. Here the group operation ⊗ is defined between two points
on the Pell curve, as in Equation 3.1, and does not contain the inverse operation.
In the resulting scheme the ciphertext is obtained as (Cx , Cy ) = (Mx , My )⊗e ,
where the operation ⊗ depends on D. Thus the triple (Cx , Cy , D) must be
transmitted, resulting in a non-compressed ciphertext.
5.5. Recovering the message from D
Mx2 −1
(mod N ), the atMy2
2
DMy − 1 = 0 (mod N ) with
To recover the message pair (Mx , My ) from D =
tacker must solve the quadratic congruence Mx2 −
respect to the two unknowns Mx and My . Even if one of the two coordinates
is known (partially known plaintext attack), it is well known that computing
square roots modulo a composite integer N , when the square root exists, is
equivalent to factoring N itself.
12
5.6. Further comments
As a conclusion to this section, we only mention that as shown in [11], RSAlike schemes based on isomorphism own the following properties: they are more
secure than RSA in the broadcast scenario, they can be transformed to semantically secure schemes using standard techniques which introduce randomness
in the process of generating the ciphertext.
6. Efficiency of the encryption scheme
Recall that our scheme encrypts and decrypts messages of size 2 log N . To
decrypt a ciphertext of size 2 log N using CRT, standard RSA requires four full
exponentiation modulo N/2-bit primes. Basic algorithms to compute xd mod p
requires O(log d log2 p), which is equal to O(log3 p) if d ∼ p.
Using CRT, if N = pe11 · . . . · perr , our scheme requires at most r exponentiation
modulo N/r-bit primes.
This means that the final speed up of our scheme with respect to RSA is
4 · (N/2)3
= r2 /2
r · (N/r)3
(10)
When r = 2 our scheme is two times faster than RSA, as it has already been
shown in [11]. If r = 3 our scheme is 4.5 time faster, with r = 4 is 8 times faster,
and with r = 5 is 12.5 times faster.
7. Conclusions
We generalized an RSA-like scheme based on the Pell hyperbola from a
modulus that was a product of two primes to a generic modulus. We showed
that this generalization leads to a very fast decryption step, up to 12 times
faster than original RSA for the security level of a modulus of 8192 bits. The
scheme preserves all security properties of RSA-like schemes, which are in general more secure than RSA, especially in a broadcast scenario. Compared to
similar schemes based on elliptic curves it is more efficient. We also pointed
that a variation of the scheme with non-compressed ciphertext does not suffer
of impossible group operation attacks.
References
References
[1] D. Boneh, H. Shacham, Fast variants of RSA, CryptoBytes 5 (1) (2002)
1–9.
[2] M. Ciet, F. Koeune, F. Laguillaumie, J.-J. Quisquater, Short private exponent attacks on fast variants of RSA, UCL Crypto Group Technical Report
Series CG-2002/4, University Catholique de Louvain.
13
[3] Cryptography
using
Compaq
multiprime
technology
in
a
parallel
processing
environment,
ftp://15.217.49.193/pub/solutions/CompaqMultiPrimeWP.pdf,
http://nonstop.compaq.com/view.asp?IOID=4523 (2002).
[4] T. Collins, D. Hopkins, S. Langford, M. Sabin, Public key cryptographic
apparatus and method, uS Patent 5,848,159 (Dec. 8 1998).
[5] T. Takagi, Fast RSA-type cryptosystem modulo pk q, in: Advances in
Cryptology–CRYPTO’98, Springer, 1998, pp. 318–326.
[6] H. Cohen, A course in computational algebraic number theory, Vol. 138,
Springer Science & Business Media, 2013.
[7] S. Lim, S. Kim, I. Yie, H. Lee, A generalized Takagi-cryptosystem with a
modulus of the form pr q s , in: Indocrypt, Springer, 2000, pp. 283–294.
[8] K. Koyama, U. M. Maurer, T. Okamoto, S. A. Vanstone, New publickey schemes based on elliptic curves over the ring Zn , in: Advances in
Cryptology–CRYPTO’91, Springer, 1992, pp. 252–266.
[9] K. Koyama, Fast RSA-type schemes based on singular cubic curves
y 2 + axy ≡ x3 (mod n), in: Advances in Cryptology–EUROCRYPT’95,
Springer, 1995, pp. 329–340.
[10] S. Padhye, A Public Key Cryptosystem Based on Pell Equation., IACR
Cryptology ePrint Archive 2006 (2006) 191.
[11] E. Bellini, N. Murru, An efficient and secure RSA–like cryptosystem exploiting Rédei rational functions over conics, Finite Fields and Their Applications 39 (2016) 179–194.
[12] M. J. Wiener, Cryptanalysis of short RSA secret exponents, Information
Theory, IEEE Transactions on 36 (3) (1990) 553–558.
[13] D. Coppersmith, M. Franklin, J. Patarin, M. Reiter, Low-exponent RSA
with related messages, in: Advances in Cryptology–EUROCRYPT’96,
Springer, 1996, pp. 1–9.
[14] D. Coppersmith, Small solutions to polynomial equations, and low exponent RSA vulnerabilities, Journal of Cryptology 10 (4) (1997) 233–260.
[15] J. Hastad, On using RSA with low exponent in a public key network,
in: Advances in Cryptology–CRYPTO’85 Proceedings, Springer, 1986, pp.
403–408.
[16] M. Boudabra, A. Nitaj, A new generalization of the KMOV cryptosystem,
Journal of Applied Mathematics and Computing (2017) 1–17.
[17] M. J. Jacobson, H. C. Williams, K. Taylor, K. Dilcher, Solving the Pell
equation, Springer, 2009.
14
[18] S. Barbero, U. Cerruti, N. Murru, Generalized Rédei rational functions and
rational approximations over conics, Int. J. Pure Appl. Math 64 (2) (2010)
305–317.
[19] L. Rédei, Über eindeutig umkehrbare polynome in endlichen körpern, Acta
Sci. Math.(Szeged) 11 (1946) 85–92.
[20] S. Barbero, U. Cerruti, N. Murru, Solving the Pell equation via Rédei
rational functions, The Fibonacci Quarterly 48 (4) (2010) 348–357.
[21] W. More, Fast evaluation of Rédei functions, Appl. Algebra Eng. Commun.
Comput. 6 (3) (1995) 171–173.
[22] G. L. Miller, Riemann’s hypothesis and tests for primality, in: Proceedings
of seventh annual ACM symposium on Theory of computing, ACM, 1975,
pp. 234–239.
[23] V. Shoup, A computational introduction to number theory and algebra,
Cambridge university press, 2009.
[24] H. W. Lenstra Jr, Factoring integers with elliptic curves, Annals of mathematics (1987) 649–673.
[25] A. K. Lenstra, H. W. Lenstra Jr, M. S. Manasse, J. M. Pollard, The number
field sieve, in: The development of the number field sieve, Springer, 1993,
pp. 11–42.
[26] D. J. Bernstein, A. K. Lenstra, A general number field sieve implementation, in: The development of the number field sieve, Springer, 1993, pp.
103–126.
[27] Zimmermann,
50
largest
factors
found
by
ECM,
https://members.loria.fr/PZimmermann/records/top50.html,
accessed 2017-07-28.
[28] D. Boneh, G. Durfee, N. Howgrave-Graham, Factoring n = pr q for large
r., in: Crypto, Vol. 1666, Springer, 1999, pp. 326–337.
[29] A. J. Menezes, S. A. Vanstone, A note on cyclic groups, finite fields, and
the discrete logarithm problem, Applicable Algebra in Engineering, communication and computing 3 (1) (1992) 67–74.
[30] Y. Lu, L. Peng, S. Sarkar, Cryptanalysis of an RSA variant with moduli
n = pr q l , Journal of Mathematical Cryptology 11 (2) (2017) 117–130.
15
| 7 |
arXiv:1609.00820v3 [math.AT] 10 Nov 2016
Doctoral Thesis
CLASSIFYING SPACES FOR
KNOTS
New bridges between knot theory
and algebraic number theory
Federico William Pasini
Supervisor:
Professor Thomas Stefan Weigel
A thesis submitted in partial fulfillment of the requirements
for the degree of Philosophiae Doctor in Mathematics
Dipartimento di Matematica e Applicazioni
Università degli Studi di Milano-Bicocca
13 September 2016
ii
To my brother-in-soul
Dario Merlin
Preface
This thesis is aimed at earning the degree of Philosophiae Doctor, so I find
it good to start it with some philosophy.
Regarding epistemology, I am strictly antimetaphysical. I believe the
epistemological essence of a human being is the datum of all his or her
relationships with the others. In this respect, I owe not only my gratitude,
but also a fragment of my essence (which I am quite happy with), to all the
people that shared a bit of their road with me.
But I am also an environmentalist, and thanking explicitly all the people
I should would cause a too grievous sapshed among the poplar community.
Then, following an idea of Dr. Dario Celotto, I will cite here only those
who had a direct influence on my PhD studies.
My gratitude goes to Thomas Weigel, who has always been more than
just a mathematical advisor. He instilled the difference between good mathematics and formal manipulation of symbols in me, and constantly exhorted
me to bear the pains of the path of good mathematics. He urged me to
work on my weaknesses. He offered all his students periodic psychological sessions. He taught me the basics of tango. Above all, he helped
me clarify my thoughts in a period of personal dire straits and guided
me towards a better approach to my life as a whole. I do not know if
this behaviour is general among PhD supervisors, but his case is sufficient to justify the meaningfulness of the Mathematics Genealogy Project
https://genealogy.math.ndsu.nodak.edu/.
I also had the chance of spending some months as a visitor at Royal
Holloway University of London. There I was adopted by my foster advisor
Brita Nucinkis, and the choice of words is not exaggerated. For her quietness, her patience and her maternal instinct, as I should say, made me feel
iii
iv
right at home. As a consequence, I had a great time, both personally and
professionally, with her and her PhD students, Ged Corob Cook and Victor
Moreno. I strongly hope that period was just the beginning of a lifelong
cooperation and friendship. I also benefited from some kind advice and
feedbacks from Iain Moffatt. And how could I not mention the tuscan delegation at Royal Holloway? I will always bring a little of Eugenio Giannelli
and Matteo Vannacci’s contagious enthusiasm and vivacity with me.
Being a researcher cannot be only a job. It is a kind of approach to
the whole reality. And it is a fantastic chance of growth when a wise senior
researcher shares his vision with you. That is the reason why I am grateful
to Roberto La Scala from the University of Bari and Ugo Moscato and Renzo
Ricca from the University of Milano-Bicocca.
I left to the end the people my story braided the most with, my nest
fellows at the university of Milano-Bicocca. My elder brothers Tommaso
Terragni, Claudio Quadrelli and Gianluca Ponzoni have always been ready
to provide me with encouragement in hard times. Above all, they built a
sense of team group among PhD students in algebra that survived their
departure. I hope I managed to even strengthen this spirit and to extend it,
beyond the boundaries of algebra, to all the younger PhD students. Whether
I succeeded might be checked in the acknowledgements of their theses.
I shared all the (many) joys and (many) sorrows of the doctoral studies
with Dario Celotto, Sara Mastaglio, Elena Rossi, Marina Tenconi, Elisa Baccanelli, Valentina Folloni, Bianca Gariboldi, Iman Mehrabi Nezhad, Paola
Novara, Elena Volonté, Alessandro Calvia, Elia Manara, Arianna Olivieri,
Mariateresa Prandelli, Martino Candon, Jessica Cologna, Andrea Galasso,
Jinan Loubani, Daniela Schenone and Alberto Viscardi. The Risiko nights,
the hikes in the mountains, the trips and the dinners we enjoyed together
will be forever on my mind. Finally, Ilaria Castellano and Benedetta Lancellotti have become my best friends by virtue of all the adventures we have
gone through together (and in which citrus fruits usually play a big role).
They would deserve a praise that cannot be expressed in words, but only
with a big hug.
Without them all, I would have probably produced far more mathematics, and had far less fun.
Contents
Preface
iii
1 Introduction
1.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
8
2 Knot theory
2.1 Basic definitions . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Homology of links . . . . . . . . . . . . . . . . . . . . . . . .
9
9
13
3 Algebraic number theory
15
3.1 Ramification of prime ideals . . . . . . . . . . . . . . . . . . . 16
3.2 Valuations and completions . . . . . . . . . . . . . . . . . . . 17
3.3 Galois cohomology . . . . . . . . . . . . . . . . . . . . . . . . 20
4 Classifying spaces
4.1 Classifying spaces . . . . . . . .
4.2 Classifying spaces for meridians
4.2.1 Structure of G . . . . .
4.2.2 Definition of ∼ . . . . .
4.2.3 NG [H] and G[H] . . . .
4.2.4 Prime knots . . . . . . .
4.2.5 Non-prime knots . . . .
4.3 The Hopf link . . . . . . . . . .
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23
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31
5 Coverings and field extensions
33
5.1 A motivating analogy . . . . . . . . . . . . . . . . . . . . . . 33
v
vi
CONTENTS
5.2
Homotopy of classifying spaces . . . . . .
5.2.1 The proof of Proposition A . . . .
5.2.2 Branched coverings of knot spaces
5.2.3 The proof of Theorem B . . . . . .
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36
37
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6 Homology and Poitou-Tate sequence
45
6.1 Poincaré-Lefschetz duality . . . . . . . . . . . . . . . . . . . . 45
6.2 The proof of Theorem C . . . . . . . . . . . . . . . . . . . . . 46
6.3 Some steps towards Conjecture D . . . . . . . . . . . . . . . . 53
A Duality
A.1 Derived categories . . . . . .
A.2 Triangles . . . . . . . . . . .
A.3 Triangulated categories . . .
A.4 Dualities and unitary functors
A.5 The Restriction functor . . .
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57
57
60
63
64
66
69
Chapter 1
Introduction
It may very well be claimed that the birth certificate of topology is Leonhard
Euler’s solution to the bridges of Königsberg problem, in 1735. Curiously,
that same year saw the birth of Alexandre-Teophile Vandermonde, who was
the first to explicitly conceive the possibility of a mathematical study of
knots and links as a part of the new discipline (cf. [38]). This idea spreaded
silently in the mathematical community of the 18th century, but it was not
until the 19th century that Carl Friedrich Gauss laid the first two foundational stones of knot theory, which would become a source of inspiration
for plenty of later developments. On one side, he developed a first method
to tabulate the knots, by writing the sequence of crossings met by a point
running on a knot. On the other, after becoming involved in electrodynamics, he introduced the notion of linking number of two disjoint knots,
i.e., the number of times the first knot winds around the other (probably in
connection with Biot-Savart law, cf. [32]):
1
Lk(K1 , K2 ) =
4π
I
K1
I
K2
r1 − r2
(dr1 × dr2 ).
|r1 − r2 |3
(1.1)
He also proved that this linking number remains the same if the knots are
interchanged and that it is invariant under ambient isotopy: it is the first
example of a link type invariant.
Classical knot theory deals with the embeddings tS 1 ,→ S 3 of (collections of) circles into the 3-sphere (classical links). The significance of the
theory is easily explained. On one side, links are sufficiently tangible objects (as far as a mathematical object can be) to provide a comfortable
environment for testing new techniques and developing new ideas. On the
1
2
CHAPTER 1. INTRODUCTION
other side, 3-dimensional manifolds enjoy a good deal of features that do
not appear in any other dimension and make dimension 3 particularly interesting (and challenging, as the history of Poincaré conjecture showed) to
topologists (see e.g. the survey [13]), and links play a prominent role in the
topology of 3-manifolds: for example, J. W. Alexander showed (cf. [1]) that
every closed orientable 3-manifold is a branched covering of S 3 branched
over a link.
Algebraic number theory is a discipline that develops from the very roots
of both ancient algebra (the theory of equations) and modern algebra (the
theory of structures, after Évariste Galois). It ideally started with diophantine equations (cf. Diophantus’s Arithmetica). The search for solutions to
such equations led to the development of the concepts of divisibility and
primality, which in turn attracted the attention of many great mathematicians of 17th and 18th century, like Pierre de Fermat, Euler, Joseph-Louis
Lagrange and Adrien-Marie Legendre. But it was again Gauss that gathered
all the previous isolated results into a stable and systematic theory. And, as
he was Gauss for a good reason, he also added some of his personally crafted
jewels into his foundational book Disquisitiones arithmeticae (see [15] for an
english translation). In particular, he proved the quadratic reciprocity law :
for distinct odd prime numbers p, q ∈ N,
p
q
= (−1)(p−1)(q−1)/4 ,
q
p
(
1
p is a square mod q
p
where
=
q
−1 otherwise
is called the Legendre symbol.
In trying to generalise this theorem to higher powers than squares, Gauss
proved that the elements of Z[i] enjoy an essentially unique factorisation
√ into
primes, like the “classical” integers do, and the failure of a general Z[ −n] to
be a unique factorisation domain eventually led Ernst Kummer and Richard
Dedekind to formulate the concept of ideal of a ring. This in turn paved the
way to the realisation that number theory could be conveniently reformulated in terms of “integer numbers” in finite extensions of Q. And that is
the story of how algebraic number theory intermarried with Galois theory.
In modern terms, algebraic number theory generalise the notion of integer
number in the field of rationals to that of algebraic integer in a finite field
extension of Q, and deals with the relationship between prime numbers and
prime ideals in the rings of algebraic integers of such extensions.
3
Along the years, knot theory and algebraic number theory have grown
up independently, but apparently they have kept track of a common germ.
M. Morishita collected, in his pleasant book [25], lots of interesting analogies between knot theory and algebraic number theory, based on a sort of
“geometrization of numbers” that highlights the similar behaviour of knots
on one side and primes on the other. Remarkably, one of these analogies
matches the linking number with the Legendre symbol, and the integral 1.1
expressing the former with a Gauss sum yielding the latter, also proved by
Gauss
p−1
!
q−1
X 2
2 q
(−1)
=
ζqx
.
p
x∈Fq
Might it be that the prince of mathematicians already had this vision
of magic bridges that guided him during his explorations in the realms of
topology and number theory?
It is my intent to carry over this line of research by exploiting a new
topological ingredient that has recently entered the scene of mathematics,
but, up to my knowledge, has never been conceived in this framework, i.e.
classifying spaces for families of subgroups. These spaces generalise the
concept of total space EG of the universal principal bundle EG → BG for
the group G. More precisely, they relax the absence of G-fixed points to
the requirement that only a prescribed family of subgroups of G fix points
and the fixed-point space of each such subgroup be contractible. Classifying
spaces gained the attention of mathematicians in connection with the BaumConnes Conjecture on the topological K-theory of the reduced group C ∗ algebra (cf. [24]) and the Farrell-Jones Conjecture on the algebraic K- and
L-theories of group rings (cf. [12]).
Classifying spaces are known to exist and to be unique up to G-homotopy
for all families of all groups G (cf. [20, Subsection 1.2]). Yet, a major
problem is to find nice concrete models for such spaces, on which to carry
effective computations.
In the context of knot theory, a classifying space of a knot group G for
the family of subgroups generated by meridians can be defined. It amounts
to “completing” the universal cover of the knot complement (which carries
a free G-action) to a G-space such that the quotient over G is the whole S 3 .
As a first result, I present a construction that provides explicit models of
these classifying spaces in the case of prime knots. In detail, if G is a prime
knot group and H ∼
= Z2 is a peripheral subgroup generated by a meridian a
4
CHAPTER 1. INTRODUCTION
and a longitude l of G (see Chapter 2 for the definitions), then the classifying
space EG G of G for the family of subgroups generated by meridians is given
by a pushout of G-CW-complexes
F
F
G/H
φ
R2
/ EG
idG ×ψ
/ EG G.
2
G/H Cyl(π : R → R)
Here Cyl(π : R2 → R) denotes the mapping cylinder of the canonical projection from EH ≈ R2 to the subcomplex EH hai ≈ R of points fixed by
the subgroup hai and the maps starting from the upper-left corner glue the
term EH in each copy of that mapping cylinder to a boundary component
of EG.
Actually, an analogous pushout does make sense for non-prime knots as
well, as it is explained in §4.2.5, but the attaching maps, and thus the model
EG G obtained, are fairly more complicated. In other words, in the case of
non-prime knots the models lose those features of neatness that make their
prime knot analogues attractive.
The classifying spaces just constructed can serve to extend and to shed a
new light on the topic of branched coverings of knots. In this respect, having
an explicit model for the classifying space enables to prove the following.
Proposition A. Let G be a prime knot group and U E G a normal subgroup
of finite index. There is a short exact sequence of groups
/ MU
1
/U
/ π1 (EG G/U )
/ 1,
where MU is the normal subgroup of U generated by those powers of the
meridians of the knot that stay in U .
This sequence is interesting on its own. Indeed, applying the abelianisation functor we obtain the exact sequence
MUab
/ U ab
π
/ H1 (EG G/U )
/ 0.
And now let us put on the number theory glasses. Take an algebraic number
field K, call OK its ring of algebraic integers and GK its absolute Galois
group, i.e. the Galois group of its separable closure (cf. [26, Section IV.1]).
A direct consequence of Hilbert class field theory (cf. [26]) is that the ideal
class group ClK (cf. Chapter 3) fits into an exact sequence
`
∗
/ Gab $ / ClK
/ 0,
p∈Spec(OK ) Op
K
5
where Op is the complete discrete valuation ring associated with the prime
p E OK . Even if an explicit description has not been obtained yet, it is
known that ker($) is generated by the ramification groups (cf. [26, Section
II.9]) of the primes of K. On the other hand, ker(π) is generated by the
“meridians” of BU , which, as we will see in Chapter 4, are responsible of
the ramification in branched coverings of knots. In other words, $ kills the
ramification of prime ideals in algebraic number fields, exactly like π “kills”
the ramification of branched coverings of S 3 .
But the sequence is also a key tool in proving the following
Theorem B. Let G be a prime knot group and let U E G be a normal
subgroup of finite index. Then the following are equivalent.
1. U = MU ;
2. The canonical projection EG G/MU → EG G/U is a trivial covering;
3. The canonical projection EG/MU → EG/U is a trivial covering;
4. π1 (EG G/U ) = 1;
5. EG G/U ∼
= S3;
A conjectural relationship between the (co)homology of the classifying
spaces and the Shafarevič groups of algebraic number fields inspired the
discovery of a Poitou-Tate-like 9-term exact sequence. In detail, if S is a
nonempty set of primes of an algebraic number field K, including the primes
at ∞, and A is a finite GS -module s.t. ν(|A|) = 0 for all ν ∈
/ S, Poitou-Tate
sequence is the 9-term exact sequence
0
/ H 0 (KS , A)
s
H 1 (KS , A)
H 2 (KS , A)
s
/ P 0 (KS , A)
/ H 2 (KS , A0 )∨
/ P 1 (KS , A)
/ H 1 (KS , A0 )∨
/ P 2 (KS , A)
/ H 0 (KS , A0 )∨
/0
connecting the Galois cohomology of K with the Galois cohomology of its
completions with respect to a set of non-archimedean distances induced by
the prime ideals of the ring of algebraic integers of K (see Chapter 3 for
explanations). The kernel of the map β is the first Shafarevič group of K:
X1 (GS , A) = ker H 1 (KS , A) → P 1 (KS , A) .
6
CHAPTER 1. INTRODUCTION
By using tools from the theory of derived categories with duality, a brief
review of which can be found in the appendix, it has been possible to obtain
an amazingly similar sequence for knot groups, taking into account that
• this sequence is in homology, so all arrows go in the opposite direction
with respect to those in Poitou-Tate sequence;
• H i (KS , A0 )∨ in Poitou-Tate sequence stands for the Pontryagin dual
of H i (KS , A0 ), and the cohomology groups appearing in our sequence
are morally dual to their homology counterparts.
Theorem C. Let G be a knot group, H ≤ G a peripheral subgroup and
U E G a normal subgroup of finite index. Then there is an exact sequence
of groups
0
/ H 0 (U, Z)
H 1 (U, Z)
H 2 (U, Z)
`
/
`
/
`
s
/
s
G/UH
H2 (H ∩ U, Z)
/ H2 (U, Z)
G/UH
H1 (H ∩ U, Z)
/ H1 (U, Z)
G/UH
H0 (H ∩ U, Z)
/ H0 (U, Z)
(1.2)
/ 0.
And, if we cut out the subsequence in degree 1,
Conjecture D. Let G be a knot group, H ≤ G a peripheral subgroup and
U E G a normal subgroup of finite index. Then there is an exact sequence
of groups
0
/ H 1 (EG G/U, Z)
H 1 (U, Z)
/
`
G/UH
H1 (H ∩ U, Z)
/ H1 (U, Z)
H1 (EG G/U, Z)
/ 0.
Note that H ∩ U is still isomorphic to Z2 , as U has finite index in G.
Its generators are products of powers (in multiplicative notation) of the
generators of H.
7
Conjecture D will be proved in some special cases, but remains open in
full generality, so the hunt is not over.
Besides the charming analogies, there is also a remarkable difference between knots and numbers. That is, in the realm of knots the first homology
and the first cohomology group of the same space EG G/U appear as candidates to complete the degree-1 subsequence of (1.2) on both ends. This
has no parallel in algebraic number theory. Knot theory has topological
spaces naturally available to work with, while algebraic number theory has
not. And dealing with groups that come as the (co)homology of such spaces
might be far easier than dealing with generic groups, since homology and
cohomology are bonded to each other by plenty of duality principles.
An explicative example is Leopoldt Conjecture in number theory. Let K
be an algebraic number field with [K : Q] = n, and suppose for simplicity
that K ⊇ Q[i] (so that K has no real embeddings) and K is “large enough”
(e.g., it contains a primitive pth root of unity, for some prime p ∈ Z). Set
Sp = {Q E OK | Q ∩ Z = p} (i.e., Q lies above p, cf. Section 3.1), S∞ the
set of places of K at infinity (cf. Section 3.2) and S = Sp ∪ S∞ (note that
this is a finite set). Furthermore, let DQ be the decomposition group of Q
(cf. [26, Section I.9]) and let GSK be the Galois group of the maximal field
extension of K unramified outside S (cf. [26, Section III.2]). Then there is
an exact sequence
`
/
/ H1 GS , Zp
H 1 GSK , Zp (1)
Q∈S H1 (DQ , Zp )
K
(for the meaning of Zp (1), cf. [31]). The Leopoldt Conjecture claims that if
one completes this sequence to
0
/ T1
H 1 GSK , Zp (1)
/
`
Q∈S
H1 (DQ , Zp )
/ H1 GS , Zp
K
T2
/0
then the groups T1 and T2 are torsion groups. There are cases (e.g., when one
passes to the maximal pro-p quotient GSK (p) of GSK ) in which T2 is known to
be even finite, but this still does not give any hints on T1 . On the contrary,
in the cases in which Conjecture D is true, the finiteness of H1 (EG G/U, Z)
would immediately imply the triviality of H 1 (EG G/U, Z), via the Universal
coefficient Theorem for cohomology (cf. [17, Section 3.1])!
8
CHAPTER 1. INTRODUCTION
But I am an optimistic guy. Maybe one day some talented mathematician will carry forward the “geometrization of numbers” started by Gauss
and find a way to “spacify” the cohomology of number fields, thus restoring
the lost symmetry. I hope I will be nearby. For, I am sure, what will be
disclosed will be truly amazing.
1.1
Notation
N denotes the set of natural numbers including 0. N∗ = N \ {0}.
An isomorphism between sets with algebraic structure will be denoted ∼
=.
In a group G, hg1 , . . . , gn i is the subgroup generated by g1 , . . . , gn , while
hhg1 , . . . , gn ii is the normal subgroup generated by g1 , . . . , gn . In a ring R,
R∗ is the set of invertible elements.
S n is the n-dimensional sphere, B n the (open) n-dimensional ball. In
the category of topological spaces, ≈ denotes a homeomorphism, while '
denotes a homotopy equivalence. If Y is a topological space, Y , ∂Y , Y ◦
stand for its closure, boundary and interior, respectively.
Chapter 2
Knot theory in a nutshell
General references for this chapter are [8] and [9].
2.1
Basic definitions
A c-component link, or simply c-link (c ∈ N∗ ) is a topological embedding,
i.e. a homeomorphism onto its image, λ : S 1 × {1, . . . , c} −→ S 3 , from a
disjoint union of circles into the 3-sphere. We usually identify it with its
image L. The components of the link are L(i) := λ(S 1 × {i}) for i = 1, . . . , c.
A knot is a 1-link.
Remark 1. In the most naive setting of knot theory, the codomain of a link
should be the “tangible” space R3 ; the choice of S 3 instead is only a matter
of comfort. In fact, S 3 is just the one-point compactification of R3 , and
working in a compact 3-manifold does bring some advantages. Nevertheless,
we shall occasionally feel free to view a link as an embedding in a copy of
R3 sitting inside S 3 .
Thinking of circles as parametrized curves γ : [0, 2π] → C, γ(t) = eit , we
can choose an orientation on each component of a link, that is, a direction
of travel on that component.
There is a natural notion of equivalence among links, defined as follows.
An isotopic deformation of a topological space X is a map D = D(x, t) :
X × [0, 1] −→ X, simultaneously continuous in both its arguments x and t,
such that ∀t ∈ [0, 1], dt := D(·, t) is a homeomorphism and d0 is the identity.
9
10
CHAPTER 2. KNOT THEORY
Two links L and K are equivalent if there is an isotopic deformation D of
R3 mapping the former onto the latter, namely d1 (L) = K. This yields a
true equivalence relation whose classes are called link types. In particular, a
knot is trivial if it is equivalent to the standard circle
{x2 + y 2 = 1, z = 0} ⊂ R3 ,→ S 3 ,
or informally, if it is “unknotted”.
The general problem of knot theory is to understand whether two given
knots (links) belong to the same type. In order to achieve this task, a
number of invariants of the link type have been developed from algebraic
topology. Among them, the link group holds a prominent role, but before
giving its definition we shall circumscribe the class of links for which the
algebraic invariants manage to express their full potential.
Polygonal links are the ones whose components are the union of finitely
many closed straight-line segments. A link is tame if it has the same type
of a polygonal one, wild otherwise. The two terms also apply to link types
at once. Tame links can be satisfactorily handled with algebraic methods.
Roughly speaking, this is so because the behaviour of polygonal curves can
be encoded with finitely many information and do not require, for example,
any limit process (in the following, several precise instances of this sentence
will appear clearly). That is why, from now on, all links are tacitly assumed
to be tame.
The link group G := GL of a link L is the fundamental group of its
complement in S 3 , which is independent of the base point thanks to pathconnectedness. A presentation is obtained by the following procedure:
1. a polygonal link (here viewed in R3 ) can be projected on a suitable
plane in such a way that
(a) the projection is 1 to 1 at all points of the link, except possibly
a finite number of crossing points, where it is 2 to 1 (the images
of such points are called crossings);
(b) no vertex of the polygonal curves is projected to a double point.
Suppose without loss of generality that the aforementioned suitable
plane is the XY plane. Then at each crossing we distinguish an upper strand (the one whose preimage contains the crossing point with
higher Z coordinate) and a lower strand. This is recorded on the planar projection by removing a small neighbourhood of the crossing from
the lower strand. The resulting finite set of arcs that rise and die near
2.1. BASIC DEFINITIONS
11
crossings is a link diagram.
2. The choice of an orientation induces an orientation on the arcs of the
link diagram.
3. Each arc produces a generator aj (j = 1, . . . , d). It represents the [homotopy class of the] loop starting from the base point, winding once as
a left-handed screw around that arc and going back to the base point
without wandering around anymore.
4. Each crossing produces a relation rj (j = 1, . . . , d), according to the
following rules:
_
?
b
a
_
x
xa = bx
x
_
?
?
b
a
ax = xb
12
CHAPTER 2. KNOT THEORY
(by a simple combinatorial argument the number of arcs equals the
number of crossings).
5. The link group is then GL = ha1 , . . . , ad | r1 , . . . , rd i (cf. [9, Chapter
VI]).
If in particular L is a knot, there is always exactly one redundant relation. More precisely, knot groups have deficiency 1, that is, they admit a
presentation with (number of generators) = (number of relators)+1 and
no presentations with (number of generators) = (number of relators) + k
for k > 1. On the contrary, link groups can have arbitrary deficiency greater
or equal to 1. This is related to “splitness”. A link L is split if there exists a
2-sphere S 2 ⊆ S 3 \ L that separates S 3 into two 3-balls, each containing at
least one component of L. Such a S 2 is called a splitting 2-sphere. If this is
the case, Seifert and Van Kampen’s Theorem tells that the link group is the
free product of the link groups of the two “sublinks” lying on the opposite
sides of the 2-sphere. Hence the deficiency of the group of the whole link is
the sum of the deficiencies of the sublink groups. Then, by induction, we
can construct link groups with arbitrarily large deficiency. For example, the
link group of
Ld =
d
G
{(x, y, z) ∈ R3 | x2 + y 2 = 1, z = i} ⊆ R3 ⊆ S 3
i=1
is free on d generators and has deficiency d.
2.2. HOMOLOGY OF LINKS
2.2
13
Homological properties of link
groups
A tubular neighbourhood V of L is a collection of open solid tori V (i) =
L(i) × B 2 , each centered at one link component L(i) , so thin as to be disjoint
and non-self-intersecting (these conditions can be always fulfilled thanks to
tameness). Then S 3 \ L is homeomorphic to S 3 \ V and deformation-retracts
onto S 3 \ V (from now on, we will denote XL , or simply X if there could
be no confusion, the complement of a tubular neighbourhood V of a link L
in S 3 ). So GL can be viewed as π1 (X, x), where x ∈ ∂X. The boundary
T := ∂X = ∂V is a disjoint union of classical tori T (i) = ∂V (i) .
Suppose that L is a nontrivial knot. The inclusion-induced homomorphism of H := π1 (∂X, x) ∼
= Z2 into GL is injective. We can choose as
generators of H [the classes of] two simple closed curves m, l such that (cf.
[8, Section 3.A])
(a) m is null-homologous in V but not in ∂V (and hence not in X);
(b) l is homologous to L in V and to 0 in X;
(c) m ∩ l = {x};
(d) the linking numbers (integers that measure how many times a simple
closed curve winds around another, see [32] for the formal definition
and interesting historical remarks) are lk(m, L) = 1 and lk(l, L) = 0.
m is referred to as a meridian, l as a longitude. With a painless abuse,
we will use the same terminology for the images of m and l in the knot
group. Actually, thanks to the aforementioned injectivity, we shall think
of the whole H as a subgroup of GL and simply make no distinction at all
between m and l and their respective images. Under that convention, H is
called a peripheral subgroup. In particular, we can arrange [the image of] m
to coincide with a generator of the knot group.
Either using topological intuition or looking at the shape of the Wirtinger
relations, one can easily see that all the generators of GL are conjugate
to either m or its inverse (for this reason, we will extend our abuse of
language and call meridian every generator of GL ). As a consequence,
H1 (X) = Gab
L = GL /[GL , GL ] is infinite cyclic generated by the class of
m (cf.[17, Theorem 2A.1] for the first equality). In other words, there is an
14
CHAPTER 2. KNOT THEORY
isomorphism W : GL /[GL , GL ] → Z sending m[GL , GL ] to 1. It is the morphism induced on the quotient by lk( , L) and will be called the winding
number, since it counts the number of times a simple closed curve winds
around the knot.
In fact, using the Mayer-Vietoris sequence in homology (cf. [17, Section
2.2]) associated to the pair (X, V ), along with the knowledge of the homology
groups of S 3 and S 1 × S 1 (cf. [17, Section 2.1]), we are able to determine
the homology of X completely:
H0 (X) ∼
= H1 (X) ∼
= Z,
Hi (X) = 0 for all i ≥ 2.
As a consequence of Papakyriakopoulos’s Sphere Theorem ([30]), X is aspherical (i.e., its higher homotopy groups vanish). In other words, it is
an Eilenberg-MacLane space K(GL , 1) for the group GL . This means that
the homology (with integer coefficients) of the group GL coincides with the
homology of X (cf. [7, Section II.4]).
Some of the preceding features extend to links, some others do not.
Specifically, if L is a generic link, each component identifies a conjugacy
class of peripheral subgroups, each generated by a meridian and a longitude, but the latter curve is rarely null-homologous in X. As an example,
consider the Hopf link: up to isotopy, the longitude of one component serves
as the meridian of the other. The generators of the link group are partitioned into conjugacy classes. Two generators are conjugate if and only if
they are meridians relative to the same link component. This means that
H1 (X) = Gab
L is free abelian generated by the classes of the meridians of the
components of L. The complete description of the homology of X is
H0 (X) ∼
= Z, H1 (X) ∼
= Zd , H2 (X) ∼
= Zd−1 , Hi (X) = 0 for all i ≥ 3.
Unfortunately, link complements need not be aspherical: in fact, if a link
L is split, then obviously any splitting 2-sphere cannot be homotoped to a
point within S 3 \ L. However, this is the only obstruction to asphericity
(again by Papakyriakopoulos’s Sphere Theorem), so that a link complement
is aspherical if and only if the link is non-split, if and only if its link group
has deficiency 1. Again, if this is the case, then the homology of the link
group coincides with the homology of the link complement.
Chapter 3
A crash course in algebraic
number theory
General references for this chapter are [19] and [26].
Algebraic number theory is the study of finite field extensions of Q. Fix
such an extention K/Q; K is called an algebraic number field. The set of
algebraic integers of K
OK = {a ∈ K | a is a root of a monic f ∈ Z[T ]}
is a ring whose field of fractions is K itself. This ring plays in K the same
role as Z in Q.
In general, OK is not a principal ideal domain, yet it enjoys a good deal of
the properties of Z: in particular, it is noetherian and integrally closed and
each of its nonzero prime ideals is maximal. In other words, it is a Dedekind
domain. As a consequence, even if OK may not be a unique factorisation
domain, it does have the property of unique factorisation of ideals: every
ideal of OK can be uniquely factored (up to the order of factors) into a
product of finitely many prime ideals.
Taking inspiration from the behaviour of Q with respect to the primes
of Z, we define a fractional ideal of OK to be a OK -submodule M of K
such that there exists a c ∈ OK \ {0} for which cM ⊆ OK . Note that
cM , and hence M , are finitely generated thanks to noetherianity of OK .
Fractional ideals serve as multiplicative inverses of “classical” ideals of OK .
More precisely, the set of nonzero fractional ideals of OK form a group
under multiplication. It is called the ideal group of OK and denoted IK .
The principal fractional ideals are the fractional ideals kOK generated by a
15
16
CHAPTER 3. ALGEBRAIC NUMBER THEORY
single element k ∈ K. With the exclusion of 0OK , they form a subgroup
PK ≤ IK and the quotient
ClK = IK /PK
is the ideal class group of K. It measures the obstruction to OK being a
principal ideal domain.
3.1
Ramification of prime ideals
Let K be an algebraic number field. A prime ideal Q E OK is said to
lie above a prime ideal P E Z if Q ∩ Z = P (notation: Q|P ). For all prime
ideals P E Z there exists a prime ideal Q E OK with Q|P , but such Q may
not be unique. This is related with how P decomposes in OK .
Indeed, it might very well happen that the lift P OK of a prime ideal
P E Z does not stay prime in OK . In general it has a factorization (unique
up to permutations)
P OK = Qe11 · · · Qerr
into finitely many prime ideals of OK . The Qi s are precisely the prime ideals
of OK that lie above P . The natural number ei is called the ramification
index of Qi over P . P is unramified in K if, for all i = 1, . . . , r, ei = 1
and the residue field extension (OK /Qi )/(Z/P ) is separable, otherwise it is
ramified. We say that P splits completely in K if r = [K : Q]. If, on the
contrary, P does lift to a prime ideal in K, we call it inert.
The above terminology is also extended to arbitrary extensions K/F of
algebraic number fields, where the same decomposition phenomena might
happen.
Example 2. Consider the degree 2 extension Q(i)/Q. The associated extension of the rings of algebraic integers is Z[i]/Z. It is easy to see that odd
primes of Z are congruent modulo 4 to either 1 or 3. Fermat’s Theorem on
the sum of two squares (cf. [15, Art. 182]) states that an odd prime p of Z
is a sum of 2 squares if and only if p ≡ 1mod 4. As a consequence,
• the lift of (2) E Z decomposes as (2)Z[i] = (1 + i)2 : the ideal (2)
ramifies in Q(i);
• the lift of a prime p ≡ 1mod 4 splits into 2 prime ideals, (p)Z[i] =
(a + ib)(a − ib): in othe words, (p) splits completely in Q(i);
3.2. VALUATIONS AND COMPLETIONS
17
• the lift of a prime p ≡ 3mod 4 remains prime in Z[i], as it cannot be
decomposed in Z and the only new chance of decomposition in Z[i] is
ruled out by Fermat’s Theorem: (p) is inert in Q(i).
Obviously this is a sandbox example, since Z[i] is a principal ideal domain
and prime ideals may be identified with prime elements. In real life, finding
the decomposition of even a single prime in some algebraic number field
extension might be painful.
Ramification is a local property, in the following sense. The localization
AP of a Dedekind domain A at a nonzero prime ideal P produces a ring
in which the only nonzero prime (and maximal) ideal is (A \ P )−1 P , that
is, a local ring. Actually, a very special local ring: it is a local Dedekind
domain and not a field, i.e., a discrete valuation ring. In some sense, the
localization process is a zoom lens that focuses only on what happens near P ,
disregarding what the domain is like elsewhere, that is near the other primes.
It goes without saying that the behaviour of the ideals in a discrete valuation
ring is far more understandable than in a general Dedekind domain, since
there one has to take care of just one nonzero prime ideal and all the other
nonzero ideals are powers of it. For this reason, the nicest properties of ideals
in Dedekind rings are those unaltered by the localization process, as they
can be studied in the quiet environment of a localization. The ramification
index is such a property (cf. [19, Section I.7]).
3.2
Valuations and completions
The name discrete valuation ring suggests that this algebraic structure
should have an equivalent definition by completely different means. In fact,
this is true and allows us to give two alternative descriptions of prime ideals
in rings of algebraic integers.
An ordered group is an abelian group A endowed with a total order
compatible with the group operation + (that is, a ≤ b ⇒ a + c ≤ b + c).
The corresponding ordered group closure is A = A t {∞}, where the new
symbol ∞ is subject to the rules
∞+∞=a+∞=∞+a=∞
a<∞
A valuation on a field F is a map ν : F → A, with values in an ordered
group closure, satisfying
18
CHAPTER 3. ALGEBRAIC NUMBER THEORY
(v1) ν(x) = ∞ ⇔ x = 0;
(v2) ν(xy) = ν(x) + ν(y);
(v3) ν(x + y) ≥ min{ν(x), ν(y)}.
The subgroup ν(F ∗ ) ≤ A is the value group of ν. In particular, every field
admits the trivial valuation, defined by νtr (x) = 0 for all x ∈ F 6= 0, which
from now on we exclude unless otherwise specified.
Two valuations ν, µ : F → R are equivalent if there is a real number
s > 0 such that µ = sν. A valuation with values in R is discrete if its value
group is Z up to equivalence.
An absolute value of a field F is a map | | : F → R satisfying
(av1) |x| ≥ 0 and |x| = 0 ⇔ x = 0;
(av2) |xy| = |x||y|;
(av3) |x + y| ≤ |x| + |y|.
In particular, every field admits the trivial absolute value, defined by |x|tr =
1 for all x ∈ F ∗ , which from now on we exclude unless otherwise specified.
The other absolute values fall into two species: the nonarchimedean (or
ultrametric) absolute values are those satisfying the stronger version of Axiom (av3)
(av3’) |x + y| ≤ max{|x|, |y|};
the other absolute values are called archimedean.
Every absolute value on F turns it into a metric space via the distance
d(x, y) = |x − y|. Two absolute values | |1 and | |2 of F are equivalent
if they induce the same topology on F . This happens if and only if there
is a real number t > 0 such that | |2 = | |t1 . Obviously an archimedean
absolute value cannot be equivalent to a nonarchimedean one. A place of an
algebraic number field K is a class of equivalent absolute values of K. We
will usually simply identify a place with each of its representatives.
Every nonarchimedean absolute value | | of K induces a discrete valuation
(
− log |x| x 6= 0
ν : K → R, ν(x) =
∞
x=0
Viceversa, every valuation ν : K → R induces an absolute value
| | : K → R,
|x| = q −ν(x)
3.2. VALUATIONS AND COMPLETIONS
19
for a fixed q > 1. Equivalent discrete valuations induce equivalent nonarchimedean absolute values, or, in other words, induce the same nonarchimedean place.
Moreover, for a nonarchimedean place | | of K and the corresponding
discrete valuation ν, the subset
Oν = {x ∈ K | |x| ≤ 1} = {x ∈ K | ν(x) ≥ 0}
is a discrete valuation ring with maximal ideal
Pν = {x ∈ O | |x| < 1} = {x ∈ K | ν(x) > 0}.
Conversely, a prime ideal P E OK determines a discrete valuation by setting
Y
vP
νP (x) = vP where (x) =
Pi i
Pi EOK prime
and this valuation can be turned into a nonarchimedean place. The associated discrete valuation ring satisfies
n
o
a
(OK )P = x = ∈ K | a ∈ OK , s ∈ OK \P = {x ∈ K | νP (x) ≥ 0} = OνP
s
Summing up, there is a correspondence
nonarchimedean
places of K
O
(3.1)
equivalence classes of discrete
valuations on K
O
localizations of OK atO nonzero prime ideals
nonzero prime ideals of OK
In view of which archimedean places can be thought as “primes at infinity”.
A field with absolute value (F, | |) (and associated valuation ν) is complete if every sequence in F that is Cauchy with respect to the metric induced
by | | converges in F . The completion Fν of (F, | |) is the field obtained
quotienting the ring of Cauchy sequences in F by the maximal ideal of sequences converging to 0. F embeds into Fν as the subfield of [classes of]
constant sequences and | | admits a unique extension to Fν , defined by
20
CHAPTER 3. ALGEBRAIC NUMBER THEORY
|(an )n∈N | = limn→∞ |an |. The completion of a field with absolute value is
essentially unique and is complete with respect to the aforementioned extension of the absolute value.
Let K/F an algebraic extension of number fields. As explained in [26,
Sections II.4,II.6,II.7,II.8], a valuation ν of F can be extended to a valuation
µ of K, once a particular F -embedding of K into the algebraic closure
of Fν is given. We borrow the same notation µ|ν used for prime ideals.
Let K µ denote the compositum (union) of the completions (Ki )µ of the
finite subextensions Ki /F of K/F . In particular, if K/F is finite, then
Lµ = Lµ = LKν is the completion of L. The advantage in passing from
the language of ideals to that of valuations lies exactly in the possibility of
passing from an arbitrary extension K/F of number fields to the various
completion extensions Kµ /Fν , which are easier to understand.
In view of the correspondence (3.1) we can speak of ramified and unramified valuations by looking at the corresponding ideals. The ramification
indices have a characterization in the language of valuations. If ν is a nonarchimedean valuation on F and we extend it to a valuation µ on K, then the
ramification index of the extension µ|ν is
eµ = [µ(K ∗ ) : ν(F ∗ )]
The prime ideal Qi E OK corresponding to µ lies above the prime ideal
P E OF corresponding to ν and the ramification index of Qi over P coincides
with the ramification index of µ over ν.
In conclusion, if K/F is a Galois extension of number fields, it makes
sense to speak of the places of F that ramify in K. With a little abuse, we
will also use the expression archimedean place to mean an equivalence class
of archimedean valuations.
3.3
Galois cohomology
The (absolute) Galois group of an algebraic number field K is the Galois
group GK of its separable closure. The (Galois) cohomology groups of K
with coefficients in a GK -module A are
H i (K, A) = H i (GK , A).
Given a valuation ν on K, one can view a finite GK -module A as a GKν module and define the cohomology groups H i (Kν , A) (with the convention
3.3. GALOIS COHOMOLOGY
21
that H 0 (Kν , A) denotes the modified cohomology group, cf.[27, Section I.2]).
There is a canonical homomorphism
H i (K, A) → H i (Kν , A),
(3.2)
which, on the level of Galois groups, is given by the restriction map
H i (GK , A) → H i (GKν , A)
(cf. [35, Section II.6]).
The morphisms (3.2) for varying ν can be bundled together in a homomorphism
Y
H i (Kν , A).
H i (K, A) →
(3.3)
ν
Let now K/F be a Galois extension of algebraic number fields and let
S be a finite set of places of F containing all the places that ramify in K
and the subset S∞ of all the archimedean places. We denote GS the Galois
group of the maximal Galois extension unramified outside S. Let A be a
finite GS -module whose order is a S-unit, that is, ν(|A|) = 0 for all ν ∈
/ S.
∗
∗
/ S}, the dual module of A is
Letting OS = {x ∈ F | ν(x) = 0 for ν ∈
A0 = hom(A, OS∗ ).
Let Enr denote the maximal unramified extension of the field E. For
ν ∈
/ S, Gal(Fν /(Fν )nr ) acts trivially on A, hence A can be considered a
Gal((Fν )nr /Fν )-module, hence the groups
i
Hnr
(Fν , A) = H i ((Fν )nr /Fν , A)
are defined (cf.[35, § II.5.5]). Let
(
)
Y
i
i
i
P (FS , A) = (cν ) ∈
H (Fν , A) | cν ∈ Hnr (Fν , A) for almost all ν .
ν
The morphism (3.3) maps H i (F, A) to P i (FS , A) (cf. [27, Proposition 8.6.1]);
the kernel of this map is the i-th Shafarevič group. Moreover, there is a 9term exact sequence
0
/ H 0 (FS , A)
s
H 1 (FS , A)
H 2 (FS , A)
s
/ P 0 (FS , A)
/ H 2 (FS , A0 )∨
/ P 1 (FS , A)
/ H 1 (FS , A0 )∨
/ P 2 (FS , A)
/ H 0 (FS , A0 ) ∨
(3.4)
/0
22
CHAPTER 3. ALGEBRAIC NUMBER THEORY
called Poitou-Tate exact sequence (cf.[27, (8.6.10)]). Here, for a torsion
abelian group M , M ∨ = homcont (M, Q/Z) denotes the Pontryagin dual of
M (cf. [35, Section I.1]). The groups P i (FS , A) encode the local information
about the field F , so Poitou-Tate sequence connects the local and the global
behaviour of an algebraic number field. In the same spirit, Shafarevič groups
measure the amount of information lost passing from the global to the local
point of view.
Chapter 4
Classifying spaces of knot
groups
Let K ⊆ S 3 be a knot, with knot group G and a tubular neighbourhood V ,
and set X = S 3 \ V . The space X is connected, locally path-connected and
e
semilocally simply connected, hence it admits a universal cover X.
These two spaces have a rich topological structure: X is a CW-complex
whose higher homotopy groups are trivial as a consequence of Papakyrie is a free, conakopoulos’s Sphere Theorem (cf. [30]). It follows that X
e → X is the
tractible G-CW-complex. Moreover, the covering map π : X
projection to the quotient of the G-action and it coincides with the universal principal G-bundle EG → BG.
Now, X is obtained by removing an open solid torus from S 3 : what
modifications would affect the picture if one tried to glue V back? That is,
more precisely:
Question 3. Is it possible to find another contractible G-CW-complex Z
e as a G-CW-subcomplex and such that the restriction of the
that includes X
·/G
e coincides with π? And if so, what topological
projection $ : Z → S 3 to X
properties would the nicest such Z have?
The G-action on Z cannot be free, since each meridian of G fixes some
points in the preimage of K. Thus the best one can achieve is that Z be a
contractible G-CW-complex such that
1. $ behaves like a covering map except over K; in a neighbourhood of
each point of K, it acts like an analogue of an integer power map in a
neighbourhood of 0 ∈ C, but with “infinite exponent”;
23
24
CHAPTER 4. CLASSIFYING SPACES
2. only [the cyclic subgroups generated by each of] the meridians fix some
points;
3. these nonempty fixed-point sets are contractible.
The first requirement is formalised in the concept of branched (or ramified )
covering, that we state here in the finite case only, since it is the case we
will always deal with in the future.
Definition 4. Let M and N be 3-manifolds and f : M → N a continuous surjection. Define BM = {x ∈ M | f is not a homeomorphism in a
neighbourhood of x} and BN = f (BM ). Finally, view D2 = {z ∈ C | |z| ≤
1} and let, for a positive integer e, pwe : D2 → D2 , pwe (z) = z e . Then f is
a finite branched covering map branched over BN if
(a) f |M \BM : M \ BM → N \ BN is a covering;
(b) for any x ∈ BM , there are (closed) neighbourhoods U of x in M
and W of f (x) in N and homeomorphisms α : U → D2 × [0, 1] and
β : W → D2 × [0, 1] such that for some positive integer e
U
f
α
D2 × [0, 1]
pwe ×id[0,1]
/V
β
/ D 2 × [0, 1]
is a commutative diagram. The number e is the ramification index at
x.
The last two requirements are encoded in the concept of classifying space
for a family of subgroups.
4.1
Classifying spaces
Definition 5. Let G be a group. A family of subgroups (or simply family) of
G is a set of subgroups of G closed under conjugation and taking subgroups.
The family generated by a subgroup H consists of all conjugates of H
and all their subgroups and will be denoted F(H). When we will be concerned with families generated by infinite cyclic groups, we will briefly call
F(ha1 i, . . . , han i) the family of a1 , . . . , an .
4.1. CLASSIFYING SPACES
25
Definition 6. Let F be a family of the group G. A [model for the] classifying space EF (G) of the family F is a G-CW-complex X such that for all
subgroups H ≤ G the fixed point set
X H = {x ∈ X | ∀h ∈ H : h(x) = x}
is contractible if H ∈ F and empty otherwise.
The space Z we are looking for in Question 3 is nothing but the classifying space of a knot group for the family of the meridians.
Remark 7. There is a more abstract equivalent definition of classifying space
of F as a terminal object in the G-homotopy category of G-CW-complexes
whose isotropy groups belong to F (cf. [20]). This is interesting in that it
automatically guarantees the uniqueness of models up to G-homotopy, but
it is useless for finding models.
As regards existence, it is guaranteed for all families of all groups by
Milnor’s construction (cf. [23]). Unfortunately, that construction often gives
a too big space to deal with. In fact, usually one is interested not only in the
mere existence of of such spaces, but in performing effectively some algebraic
topology on them. A fundamental issue, then, is to find models that are as
simple and concrete as possible, in order to carry explicit computations.
This issue of course affects also our situation: now that we know the first
part of Question 3 has a positive answer, it is time to address the second
part. As said, lots of information are encoded in the total space EG of
the universal principal bundle of the knot group G (that is, the universal
e On the other hand, EG can be seen as the classifying space of
cover X).
n o
the trivial family {1} , which the family of meridians is a superset of.
Is it then possible to use EG as a kind of foundations which to construct
the classifying space Z on? This is a particular instance of the problem of
building classifying spaces for larger families from those for smaller ones.
A result by Lück and Weiermann ([21]) does the trick. Let F ⊆ G be
two families of the group G. An equivalence relation ∼ on G \ F with the
additional properties
H, K ∈ G \ F, H ⊆ K =⇒ H ∼ K
H, K ∈ G \ F, H ∼ K =⇒ ∀g ∈ G : g
−1
Hg ∼ g
−1
Kg
(4.1)
(4.2)
will be called a strong equivalence relation. Given such ∼, denote [G \ H]
the set of ∼-equivalence classes and [H] the ∼-equivalence class of H and
define the subgroup of G
NG [H] = {g ∈ G | [g −1 Hg] = [H]}.
26
CHAPTER 4. CLASSIFYING SPACES
Property (4.2) says that the conjugation of subgroups induces an action
on [G \ H] with respect to which NG [H] is the stabiliser of [H]. Finally
construct the family of subgroups of NG [H]
G[H] = {K ≤ NG [H] | K ∈ G \ F, [K] = [H]} ∪ (F ∩ NG [H]),
where F ∩ NG [H] is a shorthand for {K ≤ NG [H] | K ∈ F}.
Theorem 8 (Lück & Weiermann). Let I be a complete system of representatives [H] of the G-orbits in [G \ F] under the G-action coming from conjugation. Choose arbitrary NG [H]-CW-models for EF ∩NG [H] (NG [H]) and
EG[H] (NG [H]), and an arbitrary G-CW-model for EF (G). Define a G-CWcomplex Y by the cellular G-pushout
F
[H]∈I
G ×NG [H] EF ∩NG [H] (NG [H])
F
F
[H]∈I
φ
/ EF G
(4.3)
idG ×ψ[H]
[H]∈I G ×NG [H] EG[H] (NG [H])
/Y
such that the G-map φ is cellular and all the NG [H]-maps ψ[H] are inclusions
or viceversa.
Then Y is a model for EG G.
4.2
Classifying spaces for the
meridians of a knot
We wish to apply Lück and Weiermann’s Theorem in this setting: G =
GL the group of the knot L, F the trivial family, G the family generated by
a generator of G.
Now, if the knot L is trivial, then G is the set of all subgroups of G ' Z,
so EG G is a point. This degenerate situation having been settled, in what
follows we assume L to be nontrivial.
4.2.1
Structure of G
Let a1 , . . . ad be the generators of G and select one among them, say a := a1 .
The others are conjugate to either a or a−1 , as a consequence of the shape
of the Wirtinger relators (cf. Chapter 2). This means that G is the family
4.2. CLASSIFYING SPACES FOR MERIDIANS
27
generated by a, according to Definition 5. We will refer to a as the chosen
meridian. Thus the subgroups in the family have a nice explicit shape:
G = {hg −1 ai gi | g ∈ G, i ∈ Z}.
Remark 9. In particular, G is a partially ordered set with respect to inclusion, it has maximal elements hg −1 agi, g ∈ G and each element of G lies in
one of those.
4.2.2
Definition of ∼
If we fix a H ∈ G \ F and we set ↑ H := {K ∈ G \ F | K ≥ H} (the up-set of
H), ↓ H := {K ∈ G \ F | K ≤ H} (the down-set of H) and l H :=↑ H∪ ↓ H,
then Property (4.1) of strong equivalence relations says that l H ⊆ [H].
Clearly, an equivalence relation for which the reverse inclusion holds has
very little hopes to exist in general and no hopes at all in our setting, as for
example ha2i i ∈ [ha3i i]\ l ha3i i. On the other hand, it is natural to ask the
equivalence relation to be as fine as possible.
Consider the relation
H ∼ K ⇐⇒ ∃L ∈ G \ F : L ≤ H, L ≤ K;
it is clearly reflexive and symmetric.
Lemma 10. The following statements are equivalent:
(i) ∼ is transitive;
(ii) ∀H ∈ G \ F : ↓ H is a downward-directed set with respect to inclusion;
(iii) ∀ maximal H ∈ G \ F : ↓ H is a downward-directed set with respect to
inclusion.
(A partially ordered set (P, ) is downward-directed if for all x, y ∈ P there
is a z ∈ P such that z x and z y).
Proof.
(i)⇒(ii) If K1 , K2 ∈↓ H, then by definition K1 ∼ H and H ∼ K2 , so that
by transitivity K1 ∼ K2 , which means ∃L ∈ G \ F : L ≤ K1 , L ≤ K2
(obviously such a L is in ↓ H by transitivity of ≤).
28
CHAPTER 4. CLASSIFYING SPACES
(ii)⇒(i) Let H1 ∼ H2 and H2 ∼ H3 , that is ∃K1 ∈ G \ F : K1 ≤ H1 , K1 ≤
H2 and ∃K2 ∈ G \ F : K2 ≤ H2 , K2 ≤ H3 . In particular K1 , K2 ∈↓ H2 ,
so that by directedness there exists a L ∈↓ H2 ⊆ G\F which is included
in both. All the three Hi s include this L, hence H1 ∼ H3 .
H1
H2
K1
H3
K2
L
(ii)⇒(iii) Obvious.
(iii)⇒(ii) Each subgroup in G \ F is included in a maximal one, and
downward-directedness is inherited by subsets.
As all the elements of G \ F are infinite cyclic, ∼ is an equivalence
relation via Condition (ii). It also satisfies the two additional axioms of
strong equivalence relations. Recalling Remark 9, we can find a maximal
element in each ∼-equivalence class and we choose it as the representative
of its class.
4.2.3
NG [H] and G[H]
As far as we are concerned with knots, the conjugation action of G on [G \F]
is transitive by the very definition. So the stabilisers are all conjugate to
one another and we only need to study NG [hai].
Lemma 11. If L is a nontrivial knot, then NG [hai] = CG (a).
Proof. By the definition of ∼, a certain t ∈ G lies in NG [hai] if and only if
ht−1 ati ∩ hai 6= {1}, if and only if ∃i, j ∈ Z \ {0} such that t−1 ai t = aj . If
we project the last equality onto G/[G, G], which is infinite cyclic generated
by the image of a, we see that i = j. Hence t centralises a power of a. But,
according to [18, Corollary 3.7], the centraliser in G of an element of the
peripheral subgroup coincides with the centraliser of any power of the given
element1 .
Summing up, all t ∈ NG [hai] centralise a and the viceversa is obvious.
1
I thank Iain Moffatt for pointing my nose on that reference.
4.2. CLASSIFYING SPACES FOR MERIDIANS
29
Remark 12. This also means that Remark 9 can be strengthened: each
element of G is included in a unique maximal element.
The same idea yields:
Lemma 13. if L is a nontrivial knot, then G[hai] = Allhai := ↓ hai ∪ {1} .
Proof. One has just to observe that the simultaneous conditions H ∈ G \ F
and [H] = [hai] mean H = ht−1 ak ti, for some a ∈ N∗ and some t ∈ G which,
by the same argument as in the previous lemma, lies in CG (a). It follows
that H = hak i ≤ hai.
4.2.4
Prime knots
There is a binary operation, called knot sum, that turns the set of oriented
knot types in S 3 into a commutative monoid. It is defined by the following
procedure. Consider disjoint projections of each knot on a plane. Find a
rectangle in the plane a pair of opposite sides of which are arcs along each
knot but is otherwise disjoint from the knots and such that there is an
orientation of the boundary of the rectangle which is compatible with the
orientations of the knots. Finally, join the two knots together by deleting
these arcs from the knots and adding the arcs that form the other pair
of sides of the rectangle. The resulting connected sum knot inherits an
orientation consistent with the orientations of the two original knots, and
the oriented ambient isotopy class of the result is well-defined, depending
only on the oriented ambient isotopy classes of the original two knots.
A knot is prime if it is nontrivial and it cannot be obtained as the knot
sum of nontrivial knots.
When a knot is prime, it follows from Simon’s results [37] that the centraliser of a meridian is the peripheral subgroup containing it, in symbols
CG (a) = ha, li = H ' Z2 , where l is a longitude corresponding to a. So
Diagram (4.3) becomes
G ×ha,li EZ2
idG ×ψ
G ×ha,li EAllhai Z2
φ
/ EG
(4.4)
/ EG G.
The top-left space is a disjoint union of copies of R2 indexed by G/ha, li,
glued via φ to the boundary components of the top-right space. This one
is well-known to be a 3-manifold, with boundary a disjoint union of planes
30
CHAPTER 4. CLASSIFYING SPACES
indexed exactly by G/ha, li (this way φ can clearly be chosen as to be a
cellular map). As for the bottom-left corner, consider R2 , on which ha, li
acts freely by integer translations, as a model for E(Z2 ), and R, on which l
acts by integer translations while a fixes every point, as a model for E(Z2 )hai .
Then the space in that corner can be chosen to be the disjoint union, again
indexed by G/ha, li, of copies of the mapping cylinder Cyl(π : R2 → R) of
the projection E(Z2 ) → E(Z2 )hai ' E(Z2 )/hai. This choice guarantees the
map ψ to be the inclusion of R2 into the mapping cylinder, as required in
Lück and Weiermann’s Theorem 8. Finally, EG G is the 3-dimensional GCW-complex obtained by gluing each Cyl(π : R2 → R), along its R2 copy,
to one boundary component of E(G).
Summing up, for a prime knot the classifying space EG G is given by the
pushout
F
φ
2
/ EG
(4.5)
G/H R
F
4.2.5
G/H
idG ×ψ
Cyl(π : R2 → R)
/ EG G.
Non-prime knots
As Schubert showed in its doctoral thesis [33], any non-prime knot L admits
an essentially unique decomposition into a knot sum of prime factors L =
K1 ] . . . ]Kr . According to Noga ([29]), the centralizer CG0 (a) of the chosen
meridian in the commutator subgroup of G is free of rank equal to the
number r of prime factors in the above decomposition, and it is generated
by the longitudes l1 , . . . , lr of those factors.
But then,
CG (a)
CG (a)G0
G
CG (a)
=
'
≤ 0.
0
CG0 (a)
CG (a) ∩ G
G0
G
Since the last group is infinite cyclic, the first one is so as well. Moreover,
since a ∈ CG (a) \ G0 , a generator of this group is aCG0 (a). This amounts to
saying that there is a right-split exact sequence
1 → CG0 (a) → CG (a) hai → 1
or equivalently, since a is central in CG0 (a),
CG (a) = hai × CG0 (a) ' Z × Fr
4.3. THE HOPF LINK
31
(where Fr denotes the free group of rank r).
In this setting, Diagram (4.3) is
G ×ha,l1 ,...,lr i E(Z × Fr )
φ
idG ×ψ
/ EG
(4.6)
/ EG G.
G ×ha,l1 ,...,lr i EAllhai (Z × Fr )
Setting C(Fr ) to be (the geometric realization of) the Cayley graph of Fr
(with respect to the generating set made of the generators of Fr and their
inverses), we can choose, as a model for the top-left corner, the disjoint
union of copies of R × C(Fr ) indexed by G/ha, l1 , . . . , ln i and, as a model for
the bottom-left corner, the mapping cylinder of the projection R × C(Fr ) →
C(Fr ).
4.3
The Hopf link
With the same machinery we can also build a classifying space for the
family of the meridans of the group G = ha, b | ab = bai ' Z2 of the Hopf
link.
a
J
J
b
In this link, the meridian of each component serves as the longitude
of the other. Hence the classifying space is the double mapping cylinder
Cyl(R ← R2 → R) of the projections onto the two components of EG = R2 .
Here the copy of R which is the image of the first projection carries the
translation action of b and the trivial action of a (i.e., it is EGhai ), and
conversely for the other copy (which is then EGhbi ).
There are two reasons that make this example interesting. The first
is that it suggests a track of further research, namely the construction of
classifying spaces for the meridians of links. The second is that this is almost
certainly the only case in which a classifying space of ours comes out well
in a photograph.
32
CHAPTER 4. CLASSIFYING SPACES
EG<a>
EG
EG<b>
Chapter 5
Branched coverings of S 3 and
extensions of number fields
5.1
A motivating analogy
We recall a property of prime ideals in Dedekind rings (cf. e.g. [19,
Section 1.7]). It generalises and completes what we said in Section 3.1. Let
A be a Dedekind ring, K its quotient field, L/K a finite separable extension
and B the integral closure of A in L. If P is a prime ideal of A, then P B is
an ideal of B and has a factorization (unique up to permutations)
P B = Qe11 · · · Qerr
into finitely many prime ideals of B. The Qi s are precisely the prime ideals
of B that lie above P meaning that Qi ∩ A = P (notation: Qi |P ).
The natural number ei is called the ramification index of Qi over P .
Moreover, for each i, let fi be the (finite) degree of B/Qi as a field extension
of A/P (recall that nonzero prime ideals of Dedekind rings are maximal,
hence B/Qi and A/P are fields).
One can show that
r
X
[L : K] =
ei fi .
i=1
If L/K is Galois, then all the Qi s are conjugate to each other, hence all the
ramification indices (resp. the residue class degrees) are equal to the same
33
34
CHAPTER 5. COVERINGS AND FIELD EXTENSIONS
number e (resp. f ). Thus, in particular, the preceding formula becomes
[L : K] = ef r.
(5.1)
A similar formula holds for finite regular coverings of S 3 \ V , where V is
a tubular neighbourhood of a knot K ⊆ S 3 (cf. [25, Chapter 5]). In fact, let
G be the knot group of K and H = ha, li ≤ G be the peripheral subgroup
containing our favourite meridian a. Consider a normal subgroup U E G of
finite index. Then U H is a subgroup of G and
|G : U | = |G : U H| · |U H : U | = |G : U H| · |H : H ∩ U | =
= |G : U H| · |H : hai(H ∩ U )| · |hai(H ∩ U ) : H ∩ U |
(5.2)
= |G : U H| · |H : hai(H ∩ U )| · |hai : hai ∩ U |
In topological terms, U defines a finite regular covering πU : EG/U → BG of
the knot exterior BG. The G-action on EG induces a transitive permutation
of the connected components of ∂EG, which are planes, and the peripheral
subgroup H is the (setwise) stabiliser of one such component P0 . Hence
the subgroup U H is the (setwise) stabiliser of the connected component
T0 = πU (P0 ) (a torus) of ∂EG/U . It follows that the cosets in G/U H
parametrise the connected components of the fibre πU−1 (∂BU ). So |G : U H|
is analogous to r in Equation (5.1). In the same spirit, |hai : hai ∩ U | is the
least positive integer k such that ak ∈ U , hence it is analogous to e in the
same equation. Finally, |H : hai(H ∩ U )| is analogous to f .
Actually, the above reasoning also holds for K a link with components
K1 , . . . , Kc , provided the peripheral subgroups of each component are nondegenerate, i.e., isomorphic to Z2 . Such a link will be called peripherally
nontrivial. For i = 1, . . . , c, let Vi be a tubular neighbourhood of Ki , disjoint from the others, and let Hi = hai , li i be the peripheral subgroup of G
generated by a meridian ai and the corresponding longitude li of the component Ki . G permutes the connected components of ∂EG/U in such a way
that two of them are in the same orbit if and only if they are relative to
the same component Ki of K, that is, if and only if they project to ∂Vi .The
stabiliser of a connected component of ∂EG/U relative to Ki is U Hi . Thus,
the connected components of the fibre π −1 (∂Vi ) are indexed by G/U Hi and
it still holds that
[G : U ]
=
[G : U Hi ]
ri
[Hi : hai i(Hi ∩ U )]
fi
[hai i : hai i ∩ U ]
ei .
5.1. A MOTIVATING ANALOGY
35
In other words, the components K1 , . . . , Kc are the analogues of the distinct
primes in the ground field of an algebraic number field extension.
It is possible to push this comparison a little forward: in number theory
the following holds.
Proposition 14 ([26], Chapter VI, Corollary 3.8). Let L/F be a finite extension of algebraic number fields. If almost all primes of F split completely
in L, then L = F .
The next result can be seen as a partial analogue in knot theory.
Proposition 15. Let G be the group of a peripherally nontrivial link K
with components K1 , ..., Kc and relative tubular neighbourhoods V1 , . . . , Vc .
Let U E G a normal subgroup of finite index and πU : EG/U → BG the
finite covering induced by U . Then G = U if and only if ∀i = 1, . . . , c :
|π0 (π −1 (∂Vi ))| = |G : U |.
Proof. For i = 1, . . . , c, let Hi = hai , li i be the peripheral subgroup of G
generated by the meridian ai and the longitude li of the component Ki . It
has already been said that |π0 (π −1 (∂Vi ))| = |G : U Hi |. On the other hand,
|G : U | = |G : U Hi | · |U Hi : U |. Hence the hypothesis
[G : U ] = |π0 (π −1 (∂Vi ))|
(5.3)
forces |Hi : U ∩ Hi | = |U Hi : U | = 1, that is, Hi ⊆ U . In particular, ai ∈ U ,
along with all its conjugates, by normality of U . But the generators of G
are all conjugates to some ai , whence G ⊆ U , provided (5.3) holds for all
i = 1, . . . , c. The reverse implication is obvious.
Remark 16. Proposition 15 is somehow weaker than Proposition 14. In fact,
the former deals with links in S 3 only. According to [25], S 3 is the manifold
analogue of the field Q. Hence, translating Proposition 15 to number theory
amounts at fixing F = Q.
On the other hand, the definition of link admits an obvious generalisation
as embedding of disjoint copies of S 1 into an arbitrary 3-manifold. It would
be interesting to understand to which 3-manifolds Proposition 15 extends.
36
5.2
CHAPTER 5. COVERINGS AND FIELD EXTENSIONS
The homotopy of the quotients of
the classifying spaces
From now until the end of the chapter, knots will tacitly assumed to be
prime.
Let G be a prime knot group and U E G a normal subgroup of finite index. Our aim is to describe the space EG G/U . The quotient space
EG/U has finitely many boundary path-connected components T1 , . . . , Tr ,
each homeomorphic to a torus. It remains to understand how the extra components Cyl(π : R2 → R) are transformed by quotienting out the U -action.
Let H = ha, li ∼
= Z2 be the peripheral subgroup of the chosen meridian.
Then U ∩ H is a finite-index sublattice of H, whence it contains a nontrivial
power of a; in fact, one can always choose a Z-basis whose first vector is
a power of a, as follows. Let {v, w} be an arbitrary Z-basis of U ∩ H and
let e = min{n ∈ N \ {0} | an ∈ spanZ {v, w}}. Then, expressing ae as a
Z-linear combination αv + βw, necesarily gcd(α, β) = 1. As a consequence
of Euclid’s Algorithm (cf. [11, Book 7, Proposition 1]), there are γ, δ ∈ Z
such that
α γ
= αδ − βγ = 1.
det
β δ
The condition on the determinant says that ae and u = v γ wδ form a Z-basis
for U ∩ H, that is, U ∩ H = hae i × hui. Now take the extra component
Cyl(π : EH → EH hai ) corresponding to H (recall that the extra components, as well as the boundary components of EG, are indexed by the cosets
of G/H). Quotienting first by hae i gives an infinite solid cylinder with axis
EH hai and meridian ae . Quotienting again by hui provides a solid torus. The
fact that in general u is not orthogonal to a is reflected in the configuration
of the lines on the surface of this torus. In particular, if u = ap lq , the “old”
longitude l is wrapped around the axis by a constant slope of −p/e (and
one needs q old longitudes to run around the whole torus). This accounts
for a solid torus attached to the boundary component of EG/U originating
from H (a word of warning: the quotient map in general identifies several
boundary components of EG to a singular boundary component of EG/U ).
Clearly the same reasoning works for the other boundary components of
EG/U . Summing up, the space EG G/U is obtained by gluing a closed solid
torus Vk ' Cyl(π : R2 → R)/U to each boundary component of EG/U , in
such a way that Tk = ∂Vk (k = 1, . . . , r).
5.2. HOMOTOPY OF CLASSIFYING SPACES
5.2.1
37
The proof of Proposition A
Thanks to the preceding discussion, the fundamental group of EG G/U can be
computed via iterated applications of Seifert and Van Kampen’s Theorem.
Suppose a presentation of U is given, with set of generators gen(U ) and set
−1
2
of relators rel(U ). Let π1 (Tk ) = hmk , lk | mk lk m−1
k lk i ' Z and π1 (Vk ) =
h`k | i ' Z. Finally, let ik : π1 (Tk ) → π1 (BU ) and jk : π1 (Tk ) → π1 (Vk ) be
the group homomorphisms induced by the inclusion maps. Then,
π1 (EG G/U ) = (. . . (π1 (BU ) ∗π1 (T1 ) π1 (V1 )) ∗ . . . ) ∗π1 (Tr ) π1 (Vr )
(where one has obviously to consider small open neighbourhoods of BU
and each Vk that deformation-retract onto BU and Vk respectively, as well
as their intersection deformation-retracts onto Tk . This is always possible
thanks to tameness). Hence, π1 (EG G/U ) admits the presentation
hgen(U ), `1 , . . . , `r | rel(U ), ik (mk )jk−1 (mk ), ik (lk )jk−1 (lk ), k = 1, . . . , ri
=hgen(U ), `1 , . . . , `r | rel(U ), ik (mk ), `k jk−1 (lk ), k = 1, . . . , ri
=hgen(U ) | rel(U ), ik (mk ), k = 1, . . . , ri
(the last equality follows from Tietze’s Theorem on equivalence of presentations). Therefore, there is a short exact sequence of groups
1
/ hhi1 (m1 ), . . . , ir (mr )ii
/U
/ π1 (EG G/U )
/1
(5.4)
The elements ik (mk ) are representatives of the U -conjugacy classes of the
powers of generators of G that lie in U . We will denote
MU = hhi1 (m1 ), . . . , ir (mr )ii.
5.2.2
Branched coverings of knot spaces
Given a prime knot group G, every normal subgroup of finite index U E G
induces a finite regular unbranched cover EG/U and an associated branched
cover EG G/U . By the construction of EG G (see (4.5) and the beginning of
this section),
EG/U ⊆ EG G/U.
(5.5)
Proposition A makes the link between unbranched and associated branched
covers precise at the level of fundamental groups. Here we present a couple
of related results.
38
CHAPTER 5. COVERINGS AND FIELD EXTENSIONS
Lemma 17. The group U/MU acts properly discontinuously on EG G/MU
with quotient space EG G/U .
Proof. The action is induced by the G-action on EG G, as follows:
uMU • MU x = MU ux
for u ∈ U, x ∈ EG G. The claim on the quotient space is obvious, so it
remains to show proper discontinuousness.
Let x ∈ EG G/MU , that is, x = MU x for a x ∈ EG G. Let us set FixG G =
tC∈G (EG G)C to simplify notation. If stabG (x) = 1, i.e., x ∈ EG G \ (FixG G),
then by the construction of EG G we can enlarge EG ⊆ EG G to a G-CWcomplex Y ⊆ EG G which is G-homeomorphic to EG and contains x in
its interior (this amounts to cutting away a small open neighbourhood of
FixG G). Hence it is sufficient to consider only the two cases, x ∈ EG and
stabG (x) 6= 1.
Suppose that the U/MU -action on EG G/MU does not satisfy proper
discontinuousness, that is, there exist a x = MU x ∈ EG G/MU (some x ∈
EG G) such that, denoting with B a generic neighbourhood of x,
∀B : ∃uMU ∈ U/MU \ {MU } : B ∩ uMU B 6= ∅.
(5.6)
Letting
G/MU be the S
canonical quotient map, one has τ −1 (B)
S τ : EG G → EG−1
= m∈MU mD and τ (uMU B) = m∈MU muD, for a suitable neighbourhood D of x.
In the first case (x ∈ EG), the neighbourhood B can be chosen so small
that D be disjoint from all its translates uD, u ∈ U \ {1} because the Gaction on EG is properly discontinuous by definition. This contradicts (5.6).
As regards the second case (stabG (x) 6= 1), by construction the stabilizer
is infinite cyclic and generated by a meridian, say stabG (x) = hai. The
obstruction in order to apply the same reasoning as before is that D cannot
be chosen to be disjoint from some of its translates. But, as long as B is
small enough, the only troubled translates of D are the translates by powers
of a, which are identified with 1 when passing to the quotient by MU .
Proposition 18 (the Easter Beer Proposition). The canonical quotient map
π
b : EG G/MU → EG G/U is the universal covering map of EG G/U .
Proof. The group U is finitely generated, being a finite-index subgroup of
the knot group G. Hence U/MU is finitely generated too. Moreover, U/MU
is residually finite, being the fundamental group of the compact 3-manifold
5.2. HOMOTOPY OF CLASSIFYING SPACES
39
EG G/U (cf. [2]). Thus, U/MU is Hopfian (cf. e.g. [28, Section 4.1]), i.e., it
is not isomorphic to any of its proper subgroups.
But U/MU is isomorphic to the group deck(b
π ) of deck transformations
of the covering π
b : EG G/MU → EG G/U (Lemma 17 and [17, Proposition
1.40]). On the other hand, deck(b
π) ∼
π∗ (π1 (EG G/MU )), where
= π1 (EG G/U )/b
π
b∗ is the homomorphism induced by π
b on the fundamental groups (cf. [17,
Proposition 1.39]).
Together with Proposition A, this means
U/MU ∼
=
U/MU
.
π
b∗ (π1 (EG G/MU ))
Then, by Hopf property, π
b∗ (π1 (EG G/MU )) = 1, whence the simple connectedness of EG G/MU .
5.2.3
The proof of Theorem B
Now we have all the ingredients needed to prove Theorem B, that we report
here for the reader’s convenience.
Theorem B. Let G be a prime knot group and let U E G be a normal
subgroup of finite index. Then the following are equivalent.
1. U = MU ;
2. The canonical projection π
b : EG G/MU → EG G/U is a trivial covering;
3. The canonical projection π : EG/MU → EG/U is a trivial covering;
4. π1 (EG G/U ) = 1;
5. EG G/U ∼
= S3;
Proof. Clearly, (1)⇒(2). The implication (2)⇒(3) holds by restriction, see
(5.5). Moreover, (3)⇒(1) by the Galois correspondence between coverings of
BG and subgroups of G (cf. [17, Theorem 1.38]. The implication (1)⇔(4) is
given by the exact sequence (5.4) (but it also follows from Proposition 18).
Finally, (4)⇔(5) is Poincaré conjecture (now Hamilton-Perelman’s Theorem,
as Perelman himself would probably like to call it).
Example 19 (finite cyclic coverings and Fox completions). Let K ⊆ S 3 be
a knot, with tubular neighbourhood V , X = S 3 \ V and G = GK the
40
CHAPTER 5. COVERINGS AND FIELD EXTENSIONS
knot group. Recall from Chapter 2 that G/[G, G] is infinite cyclic generated by the class of a meridian a and the winding number isomorphism
W : G/[G, G] → Z maps that class to 1. The kernel Un of the composition
G
lk(
,K)
/ Z mod n / Z/nZ
is a normal subgroup of finite index in G. Since
G/[G, G] ∼ G ∼
=
= Z/nZ
Un /[G, G]
Un
we have a description of Un as the extension
1 → [G, G] → Un → han i → 1.
with a a meridian and han i ∼
= Z.
The covering ρn : Xn → X associated to Un , i.e., the unique covering of
X with group of deck transformations isomorphic to Z/nZ, is called the nfold cyclic covering of X (or of K). By construction, the covering space Xn
has only one boundary component and, letting H = ha, li be the peripheral
subgroup associated to the meridian a ∈ G, Un ∩ H = han i × hli. Referring
to the notation used at the beginning of this chapter, this means e = n =
|G : Un |, f = r = 1.
Gluing one solid torus to Xn along its meridian an , another one to X
along a, and extending in the obvious way the projection map ρn to the
manifolds so obtained, one gets the n-fold cyclic branched covering of K
cn → S 3 .
ρ
cn : X
Some authors, for example Morishita (cf. [25, Example 2.14]), call this
construction the Fox completion of ρn .
Therefore, the domain of the Fox completion of the n-fold cyclic covering
of K is a model for the space EG G/Un . Its fundamental group is
π1 (EG G/Un ) ∼
= Un /han i.
To be more concrete, let
G = ha, b, c | ab = bc = cai = ha, b, c | aba = babi
be the trefoil knot group. Following [36, Example 2.1], we can describe the
commutator subgroup as
[G, G] = haj (a−1 b)a−j | j ∈ Zi
5.2. HOMOTOPY OF CLASSIFYING SPACES
41
and using Tietze moves, we can reduce the presentation to
[G, G] = ha−1 b, a−2 bai.
This is confirmed by a theorem of Stallings (cf. [8, Section 5.A]), saying that
the commutator subgroup of a fibred knot group is free on 2g generators,
where g is the genus of the knot.
An interrogation of MAGMA Online Calculator
http://magma.maths.usyd.edu.au/calc/
gives
% QUESTION
G<a,b>:=Group<a,b|a*b*a=b*a*b>;
H:=sub<G|a^-1*b,a^-2*b*a, a^3>; % sgp generated by [G,G] and a^3
U_3:=H^G;
% normal closure
U_3;
% ANSWER
Finitely presented group U_3 on 3 generators
Index in group G is 3
Generators as words in group G
U_3.1 = b * a^-1
U_3.2 = b^-1 * a
U_3.3 = a^3
% QUESTION
U3:=Rewrite(G,U_3); % presentation of U_3, just for personal culture
U3;
% ANSWER
Finitely presented group U3 on 3 generators
Generators as words in group G
U3.1 = b * a^-1
U3.2 = a * b * a
U3.3 = a^-1 * b
Relations
U3.1^-1 * U3.2 * U3.1^-1 * U3.2^-1 = Id(U3)
U3.2 * U3.3^-1 * U3.2^-1 * U3.3^-1 = Id(U3)
% QUESTION
K:=sub<G|a^3>;
M_3:=K^G;
42
CHAPTER 5. COVERINGS AND FIELD EXTENSIONS
MU3:=Rewrite(G,M_3);
MU3;
% ANSWER
Finitely presented group MU3 on 4 generators
Generators as words in group G
MU3.1 = a^3
MU3.2 = b^3
MU3.3 = (a * b * a^-1)^3
MU3.4 = (b * a * b^-1)^3
Relations
MU3.1^-1 * MU3.3^-1 * MU3.2^-1 * MU3.1 * MU3.4 * MU3.2 *
MU3.3 * MU3.4^-1 = Id(MU3)
MU3.4 * MU3.2 * MU3.3 * MU3.1 * MU3.2^-1 * MU3.4^-1 *
MU3.1^-1 * MU3.3^-1 = Id(MU3)
MU3.1 * MU3.4 * MU3.1^-1 * MU3.3^-1 * MU3.2^-1 * MU3.4^-1 *
MU3.2 * MU3.3 = Id(MU3)
% QUESTION
Index(U3,MU3);
% ANSWER
8
% QUESTION
Q<x,y,z>:=U3/MU3;
Q;
% ANSWER
Finitely presented group Q on 3 generators
Relations
x^-1 * y * x^-1 * y^-1 = Id(Q)
y * z^-1 * y^-1 * z^-1 = Id(Q)
x^-1 * y * z^-1 = Id(Q)
x * y * z = Id(Q)
y * z * x = Id(Q)
y * z^-1 * x^-1 = Id(Q)
The map x 7→ i, y 7→ j, z 7→ k provides an isomorphism between Q and the
group
Q8 = hi, j, k | i2 = j 2 = k 2 = ijki
of the units of the quaternions. Summing up, we have obtained a regular
branched covering of S 3 , branched over the trefoil knot, with fundamental
group π1 (EG G/U3 ) ∼
= Q8 .
5.2. HOMOTOPY OF CLASSIFYING SPACES
43
Remark 20. It is quite difficult to give an explicit description of MU . However, there are some restrictions on its structure. Since knot groups are coherent (as a consequence of Scott’s Theorem, cf. [34]), MU is either infinitely
generated or finitely presented. In the former case, the index |U : MU |
clearly has to be infinite. In the latter case, one can apply the results in to
obtain the following Tits-like alternative: either the index |U : MU | is finite
or MU is free. In fact, the groups of nontrivial knots have cohomological
dimension 2. Since U has finite index by hypothesis, it has cohomological
dimension 2 as well (cf. [7, Section VIII.2]). If |U : MU | is finite, then also
MU has cohomological dimension 2, hence it is not free. On the other side,
if |U : MU | = ∞, then by [5] MU has cohomological dimension 1, hence it is
free.
44
CHAPTER 5. COVERINGS AND FIELD EXTENSIONS
Chapter 6
(Co)homology of the classifying
spaces and Poitou-Tate exact
sequence
Throughout the chapter, let K ⊆ S 3 be a fixed knot, with knot group G and
e the
a tubular neighbourhood V , and set X = S 3 \ V , T = ∂V . We denote X
universal cover of X. Let a ∈ G be our favourite meridian, l the associated
longitude and H = ha, li the corresponding peripheral subgroup.
6.1
Poincaré-Lefschetz duality
The aim of this section is to prove that there is a short exact sequence
0
/ H 2 (G, Z[G])
β
/ indG (Z)
H
α
/Z
/0
(6.1)
e ∂ X)
e gives rise to a long exact sequence (cf. [17,
In fact, the CW-pair (X,
Section 2.1])
...
e ∂ X)
e
/ H2 (X,
ξ1
e
/ H0 (∂ X)
ξ2
e
/ H1 (∂ X)
ι0
e
/ H0 (X)
ι1
π0
e
/ H1 (X)
e ∂ X)
e
/ H0 (X,
π1
e ∂ X)
e
/ H1 (X,
/
/0
e ∼
e = 0.
By simply-connectedness of universal covers H0 (X)
= Z and H1 (X)
45
46
CHAPTER 6. HOMOLOGY AND POITOU-TATE SEQUENCE
e ∂ X)
e are the homology of the relative chain
Moreover, the terms H• (X,
complex
...
e
/ C2 (X)
χ2
e
C2 (∂ X)
e
/ C1 (X)
χ1
e
C1 (∂ X)
e
/ C0 (X)
χ0
e
C0 (∂ X)
/0
e there is a path s in X
e such that
If we consider a generator y of C0 (X),
e
s(0) = y and s(1) is the generator of C0 (∂ X). Then χ1 ([s]) = [y]. This
e ∂ X)
e = 0.
shows that χ1 is surjective or, in other words, H0 (X,
Summing up, we obtain the short exact sequence
0
e ∂ X)
e
/ H1 (X,
ξ1
e
/ H0 (∂ X)
ι0
/Z
/ 0.
e are planes in 1-to-1 correspondence
The connected components of ∂ X
with the set G : H. These planes are permuted by the action of G in such a
way that H is the isotropy group of the index corresponding to one of them,
e ∼
hence H0 (∂ X)
= indG
H (Z) (cf. [7, Section III.5]).
e ∂ X)
e ∼
Lefschetz-Poincaré duality (cf. [17, Section 3.3]) yields H1 (X,
=
2
2
e ∼
e
Hc (X) (cohomology with compact support). But one has also Hc (X)
=
H 2 (X, Z[G]) (cf. [10, Section 1]). Finally, by asphericity of knot complements (recall Papakyriakopoulos’s Sphere Theorem, [30]),
H 2 (X, Z[G]) ∼
= H 2 (G, Z[G])
and the short exact sequence (6.1) is established, with α = ι0 and β given
e ∂ X).
e
by the composition of ξ1 with the isomorphism H 2 (G, Z[G]) → H1 (X,
6.2
The proof of Theorem C
Our aim is to find a suitable algebraic counterpart to the topologically
flavoured sequence (6.1). We obtain a projective (in fact, free) left H-module
resolution of the trivial H-module Z from a cellular chain complex of the
universal cover R2 of T :
0
/ Z[H]hri
∂2
/ Z[H]hmi ⊕ Z[H]hli
∂1
/ Z[H]h0i
Z
(6.2)
6.2. THE PROOF OF THEOREM C
47
where π1 (T, x) = hm, l | ri, r = mlm−1 l−1 and
∂2 (r) = m + ml − mlm−1 m − mlm−1 l−1 l = (1 − l)m + (m − 1)l,
∂1 (m) = (m − 1)0,
∂1 (l) = (l − 1)0.
Here some explanations about the notational conventions may be necessary.
We underlined the generators of our modules in order to distinguish them
from the corresponding elements of the group algebra Z[H]; moreover, we
introduced the formal symbol 0 to denote the H-generator of the 0-chains.
Similar conventions will be used for the other resolutions introduced in this
chapter.
The functor indG
is exact and maps projective HP ( ) = Z[G] ⊗Z[H]
modules to projective G-modules, thus we get a projective G-module resolution (Q• , δ• ) of indG
H (Z):
/ Z[G]hri
0
δ2
/ Z[G]hmi ⊕ Z[G]hli
δ1
/ Z[G]h0i
(6.3)
indG
H (Z).
The same technique yielding (6.2) works for finding a projective Gmodule resolution (P• , γ• ) of Z (recall from Chapter 2 that nontrivial knot
groups have cohomological dimension 2):
0
/
Ld−1
i=1
Z[G]hri i
γ2
/
Ld
i=1 Z[G]hai i
γ1
/ Z[G]h1i
(6.4)
Z,
where G = ha1 , . . . , an | r1 , . . . , rn−1 i is a Wirtinger presentation of the knot
group. The boundary maps are as follows: any relator ri has the shape
−1
aj ak a−1
l ak and
−1
−1 −1
−1
γ2 (aj ak a−1
l ak ) = aj + aj ak − aj ak al al − aj ak al ak ak
γ1 (ai ) = (ai − 1)0.
The two maps δ2 and γ2 are defined in order to codify the loops in the
Cayley graphs of H ∼
= Z2 (with generating set S = {m, l, m−1 , l−1 }) and G
(with generating set S = {a1 , a−1
| i = 1, . . . , d}) respectively. It is worth
i
48
CHAPTER 6. HOMOLOGY AND POITOU-TATE SEQUENCE
noting that they act on the generators of the respective domains like some
sort of module-theoretic version of Fox derivations (cf. [14, § 1]), which
inspires the following
Definition 21. Let R be a ring and G be a group. Let X = {gi | i ∈ I} be
a set of generators of G and F(X) the free group on X. The Fox modulederivative is the map ∆ : F(X) → ⊕i∈I R[G]hgi i defined recursively by
∆(gi ) = gi ,
(6.5)
∆(gi−1 ) = −gi−1 · ∆(gi ),
∆(vw) = ∆(v) + v · ∆(w).
Remark 22. Z[G] comes equipped with the involution (antipode)
S : Z[G] → Z[G], S(g) = g −1
that transforms left G-modules into right ones and viceversa. In more detail,
given a left G-module L with action ·, we denote L× the right G-module
whose underlying abelian group is L endowed with the action ? : L×Z[G] →
L, l ? g = g −1 · l. Viceversa, given a right G-module R with action ?, we
denote ×R the left G-module whose underlying abelian group is R endowed
with the action · : Z[G] × R → R, g · r = r ? g −1 .
In particular, for any left G-module M one can define a twisted version
of the dual module. Its underlying abelian group is
M ~ = homSG (M, Z[G]) = {f : M → Z[G] | f (am) = f (m)S(a)},
which becomes a left G-module via the action a · f : m 7→ af (m). Note
that the natural G-action on the “classical” dual module, that is M ∗ =
homG (M, Z[G]) = {f : M → Z[G] | f (am) = af (m)}, is a right action given
by f ? g : m 7→ f (m)g. We can twist it by the antipode to obtain a left
G-module
× ∗
M = homG (M, Z[G])
with action g · f = f ? g −1 : m 7→ f (m)g −1 . The two versions of left dual
module are naturally isomorphic through the composition with the antipode:
S◦
homSG (M, Z[G])
/ × homG (M, Z[G])
Applying the functor ~ = homSG ( , Z[G]) to the augmented chain
complex P• → Z → 0 we obtain the chain complex of left G-modules
0
/ ×Z~
~
/ Z[G]h1∗ i
γ
e2~
/
Ld
i=1
Z[G]ha∗i i
γ
e1~
/
Ld−1
j=1
Z[G]hrj∗ i.
(6.6)
6.2. THE PROOF OF THEOREM C
49
The term Z~ = homSG (Z, Z[G]) is 0. In fact,
P as a group homomorphism
f : Z → Z[G] sends every z ∈ Z to a sum
zgi gi with zgi 6= 0 for finitely
many indices i, f commutes with the G-actions if and only if for all z ∈ Z
and g ∈ G one has
X
X
zgi gi = f (z) = f (gz) = f (z)g −1 =
zgi (gi g −1 ).
On the other hand, G is infinite and the multiplication action of G on itself
is free, which means that if some zgi is nonzero, then all the infinitely many
{zgi g | g ∈ G} are. This forces all zgi to be 0. The homology of the chain
complex is by construction the cohomology of G with coefficients in the
group ring Z[G]. According to [6], H 0 (G, Z[G]) = H 1 (G, Z[G]) = 0, so the
complex (6.6) is exact everywhere but at the end, where its homology is
H 2 (G, Z[G]). But since the dual functor ∗ = homG ( , Z[G]) maps finitely
generated projective left G-modules to finitely generated projective right
G-modules (cf. for instance [7, Prop. I.8.3]), the functor ~ maps finitely
generated projective left G-modules to finitely generated projective left Ge•~ ) of the left
modules. Thus, we actually get a projective resolution (Pe•~ , γ
2
~
G-module H (G, Z[G]) .
/ Z[G]h1∗ i
0
γ
e2~
/
Ld
i=1
Z[G]ha∗i i
γ
e1~
/
Ld−1
j=1
Z[G]hri∗ i
(6.7)
H 2 (G, Z[G])~ .
Some basic linear algebra provides the explicit behaviour of the boundary
−1
maps, as follows. We denote rjkh the Wirtinger relation aj ak a−1
h ak for some
indices j, k, h ∈ {1, . . . , d} (this notation is unambiguous once the generators
are fixed) and we define
R = (j, k, h) ∈ {1, . . . , d}3 | rjkh is a Wirtinger relation of G .
Then
γ
e2~ (1∗ ) =
d
X
∗
(a−1
i − 1)ai
(6.8)
i=1
γ
e1~ (a∗i )
=
X
∗
rikh
+
(k,h)|(i,k,h)∈R
X
∗
(a−1
j − 1)rjih −
(j,h)|(j,i,h)∈R
X
∗
a−1
k rjki
(j,k)|(j,k,i)∈R
In principle, this resolution is concentrated in degrees −2, −1, 0. One can
then shift it by +2 to obtain a chain complex (P•~ , γ•~ ) = (Pe~ [2]• , γ
e~ [2]• )
concentrated in non-negative degrees (cf. Section A.1).
50
CHAPTER 6. HOMOLOGY AND POITOU-TATE SEQUENCE
Now we have the following diagram with exact rows and exact rightmost
column
0
/ ZGh1∗ i
0
/ ZGhri
0
0
/
/
Ln−1
j=1
Ln
∗
i=1 ZGhai i
/
Ln−1
j=1
/ ZGhmi ⊕ ZGhli
/
ZGhrj i
ZGhrj∗ i
/ / H 2 (G, ZG)
/ ZGh0i
/ / indG Z
H
i=1 ZGhai i
α
/ ZGh1i
Ln
β
//Z
0.
(6.9)
on which we can apply the functors
, Z). Taking into account that
the resolution on the first row of Diagram (6.9) is the dual of the projective
resolution of the trivial G-module Z (third row), the resulting 9-terms long
exact sequence can be written as
TorG
•(
0
/ H 0 (G, Z)
t
H 1 (G, Z)
H 2 (G, Z)
t
/ H2 (H, Z)
/ H2 (G, Z)
/ H1 (H, Z)
/ H1 (G, Z)
/ H0 (H, Z)
/ H0 (G, Z)
(6.10)
/ 0.
This is strongly reminiscent of Poitou-Tate exact sequence for algebraic number fields. It could seem that, apart from its great philosophical relevance,
(6.10) does not give any new information, since it reduces to
0
/Z
/Z
/0
Zu
/Z⊕Z
/Z
0u
/Z
/Z
/ 0.
6.2. THE PROOF OF THEOREM C
51
But it is in conjunction with the unitarity of the restriction functor (see
Proposition 37) that (6.10) reveals its full power, as we shall now explain.
We can also dualise the resolution Q• of indG
H (Z). The same reasoning
as for P• , together with the adjunction
G
∼
homG (indG
H (Z), Z[G]) = homH (Z, resH (Z[G])),
~
leads to a fourth projective resolution (Q~
• , δ• ) of the left G-module:
0
/ Z[G]h0∗ i
δ2~
/ Z[G]hm∗ i ⊕ Z[G]hl∗ i
δ1~
/ Z[G]hr ∗ i
H 2 (H, resG
H Z[G]).
(6.11)
The boundary maps are
δ2~ (0∗ ) = (m−1 − 1)m∗ + (l−1 − 1)l∗
δ1~ (m∗ )
δ1~ (l∗ )
= (1 − l
−1
= (m
−1
)r
(6.12)
∗
− 1)r∗
One can define a chain map ζ• : Q• → Q~
• by
ζ0 (0) = r∗
(6.13)
ζ1 (m) = −ml
∗
ζ1 (l) = lm
ζ2 (r) = −ml0∗ .
This turns out to to be a self-duality in the category with duality of complexes of left G-modules (Kom(G − Mod),~ , $) (see Definition 34), where
the natural isomorphism $ is given by
$M : M → M ~~ ,
$M (x) : f → S(f (x))
(so that, in our situation,
$Q (0) = 0∗∗ , $Q (m) = m∗∗ , $Q (l) = l∗∗ , $Q (r) = r∗∗ ).
Indeed, ζ has inverse map
ζ0−1 (r∗ ) = 0
ζ1−1 (m∗ )
ζ1−1 (l∗ )
ζ2−1 (1∗ )
= l
−1
(6.14)
l
= −m−1 m
= −l−1 m−1 r
52
CHAPTER 6. HOMOLOGY AND POITOU-TATE SEQUENCE
~
~
and adjoint map ζ•] : Q~
• → Q• given by
ζ0] (0∗∗ ) = −m−1 l−1 r∗
ζ1] (m∗∗ )
ζ1] (l∗∗ )
ζ2] (r∗∗ )
(6.15)
= l−1 l∗
= −m−1 m∗
= 0∗ .
The map ζ fits into the diagram of bounded complexes of finitely generated projective G-modules
P~
Id•
P~
?1
ζ•
α∗•
α•
/Q
?2
/P
/ P ~ [1]
(Id∗$ )• = ($P )•
?3
/ Q~
/ P ~~
(6.16)
Id[1]•
/ P ~ [1]
?4
where α• is the map induced by α : indG
H (Z) → Z on the projective resolutions via the Comparison Theorem in Homological Algebra (cf. [22, Section
III.6].
Our goal is to complete this diagram to a commutative diagram of distinguished triangles (in the derived category of bounded complexes of finitely
generated projective G-modules) that encodes a self-duality (cf. Definition
35) between the two rows. Since ζ• is invertible, we can choose as ?1 the
map ζ•−1 ◦α•∗ , so that the leftmost square commutes. Then the map ?3 ought
to be (ζ•−1 ◦ α•∗ )∗ , and in fact this choice would make the central square commute. Indeed, (ζ•−1 ◦ α•∗ )∗ = α•∗∗ ◦ (ζ•−1 )∗ and (ζ•−1 )∗ = (ζ•∗ )−1 . Moreover,
the fact that ζ• is self-adjoint says that in the diagram
Q
α•
/P
ζ•
Q~
|
(ζ•−1 )∗ =(ζ•∗ )−1
$P
!
$Q
Q~~
α∗∗
•
/ P ~~
the left triangle commutes. Finally, since $ is a natural isomorphism
Id → ∗∗ , we conclude that $P ◦ α• = α∗∗ ◦ (ζ −1 )∗ ◦ ζ. As regards the
rightmost square in Diagram (6.16), the map ?2 is uniquely determineed up
to homotopy by imposing the first row to be a distinguished triangle. More
precisely, ?2 should be a map ξ• such that its homotopy class [ξ] is the unique
6.3. SOME STEPS TOWARDS CONJECTURE D
53
element of Ext1Z[G] (H0 (P• ), H0 (P•~ )) (this is because the complexes P• and
P•~ are acyclic). The same holds for the map ?4 =: ϑ• : there is exactly one
choice (up to homotopy) that makes the second row a distinguished triangle.
But $P is invertible, and the map Id[1]• ◦ ξ• ◦ $P−1 : P•~~ → P ~ [1]• , which
obviously makes the rightmost square commute, also turns the second row
into an exact triangle, by the definition of $, so ϑ• = Id[1]• ◦ ξ• ◦ $P−1 up
to homotopy. Summing up, we have proved
Lemma 23. · · · → P•~ → Q• → P• → P ~ [1]• → . . . is a self-dual distinguished triangle in the derived category of bounded complexes of finitely
generated projective left G-modules.
If U is any finite-index subgroup of G, then resG
U is a unitary functor (cf.
Proposition 37), so it maps the self-dual distinguished triangle
· · · → P•~ → Q• → P• → P ~ [1]• → . . .
in the derived category of bounded complexes of finitely generated projective left G-modules to a self-dual distinguished triangle in the derived category of bounded complexes of finitely generated projective left U -modules.
Therefore, we automatically get a more general version of the 9-terms exact
sequence (6.10), namely
0
/ H 0 (U, Z)
H 1 (U, Z)
H 2 (U, Z)
`
/
`
/
`
s
/
s
G/UH
H2 (H ∩ U, Z)
/ H2 (U, Z)
G/UH
H1 (H ∩ U, Z)
/ H1 (U, Z)
G/UH
H0 (H ∩ U, Z)
/ H0 (U, Z)
/ 0.
(6.17)
6.3
Some steps towards Conjecture D
Now select the subsequence in degree 1 of the exact sequence (6.17): Conjecture D claims that the first cohomology and homology groups of EG G/U
fit perfectly at the ends of the subsequence.
54
CHAPTER 6. HOMOLOGY AND POITOU-TATE SEQUENCE
Conjecture D. Let G be a knot group, H ≤ G a peripheral subgroup and
U E G a normal subgroup of finite index. Then there is an exact sequence
of groups
0
/ H 1 (EG G/U, Z)
/
H 1 (U, Z)
`
G/UH
/ H1 (U, Z)
H1 (H ∩ U, Z)
H1 (EG G/U, Z)
/ 0.
From now on, coefficients in homology and cohomology are understood
to be Z unless otherwise specified.
Conjecture D is certainly true in the trivial cases, that is, if U is a
knot group and EG G/U is the 3-sphere, or equivalently, if U H = G and
π1 (EG G/U ) = 1. Indeed, in this case Diagram (6.17) is formally identical
to Diagram (6.10) and H 1 (EG G/U ) = H1 (EG G/U ) = 0.
Next, consider the case in which U is still a knot group, but in a generic
3-manifold, that is, U H = G, but π1 (EG G/U ) is arbitrary. Then, applying
the abelianisation functor to the sequence (5.4) one gets the exact sequence
hmi
/ H1 (U )
α
b
/ H1 (EG G/U )
/ 0.
One can enlarge the first entry and get a still exact sequence
H1 (H)
α
e
/ H1 (U )
/ H1 (EG G/U )
/0
by defining
α
e(m) = α
b(m),
α
e(l) = 0.
This sequence, its hom-dual
0
/ H 1 (EG G/U )
/ H 1 (U )
α
e∗
/ H 1 (H)
/0
(where we used the Universal Coefficient Theorem for cohomology, cf. [17,
Section 3.1], along with torsion-freeness of 0th homology groups) and the
6.3. SOME STEPS TOWARDS CONJECTURE D
55
degree 1 subsequence of (6.17) fit into a diagram with exact rows
H 1(U)
H1(β•)
H1(H)
α
e
/ H1(U)
/ H1(H)
H1(α•)
/ H1(U)
/ H1 EG G
U
/0
o H1(resG
U ζ•)
0
/ H 1 EG G
U
/ H 1(U)
α
e∗
/ H 1(H)
(6.18)
Suppose further that the longitude l ∈ H is null-homologous in BU . Then
the upper square obviously commutes by construction. The lower one comG
mutes because H1 (resG
U β• ) is exactly the dual map of H1 (resU α• ) composed
−1
with the Poincaré duality isomorphism H1 (resG
U ζ• )) , as expressed in Diagram (6.16) and Proposition 37. Summing up, Conjecture D is true for
normal subgroups U such that BU has only one (torus) boundary component, provided the longitude of ∂BU is null-homologous in BU .
56
CHAPTER 6. HOMOLOGY AND POITOU-TATE SEQUENCE
Appendix A
Duality in derived categories
Here we briefly discuss some technical tools from category theory used in
the previous chapter. They involve the construction of various categories
whose objects are chain complexes over a fixed abelian category.
A.1
Derived categories
Throughout this section, let A be an abelian category. We will construct
the derived category of A in 3 steps, each of which consists of creating a
new, increasingly sophisticated category based on A.
Definition 24. The category of complexes over A is the category Kom(A)
defined as follows:
Objects chain complexes with entries in A, i.e.
dk−1
dk
A
A
A = (A• , d•A ) = · · · → Ak−1 →
Ak →
Ak+1 → . . .
with Ak and dkA in A and dkA ◦ dk−1
= 0 for all k (the dkA s are called
A
boundary maps).
Morphisms chain maps, i.e.
φ• : A → B = (φk : Ak → B k )k
with φk+1 ◦ dkA = dkB ◦ φk for all k. We will simply write φ : A → B
when there is no danger of confusion.
57
58
APPENDIX A. DUALITY
In particular (at least when A is concrete, which is always true in our
setting) one can take the (co)homology of a complex H k (A) = ker dkA /imdk−1
A
and view it as a chain complex with 0 boundary maps. A chain map f : A →
B induces a chain map in cohomology
H k (f ) : H k (A) → H k (B),
k−1
H k (f )(a + imdA
) = f (a) + imdk−1
B
Any object X of A can be regarded as a complex · · · → 0 → 0 → X →
0 → 0 → . . . concentrated in degree 0 (i.e., with X in the 0th position and
0 elsewhere). This complex has obviously trivial cohomology outside degree
0, and its 0th cohomology is isomorphic to X.
Definition 25. Given two morphisms of complexes f, g : A → B, a chain
homotopy from f to g is a collection of maps (hk : Ak → B k−1 )k (not necesk
k+1 dk :
sarily a chain map) such that f k − g k = dk−1
A
B h +h
...
f k−1
...
dk−1
A
/ Ak−1
g k−1
/ B k−1
hk
w
dk−1
B
/ Ak
fk
gk
/ Bk w
dkA
hk+1
dkB
/ Ak+1
f k+1
dk+1
A
/ ...
g k+1
/ B k+1
dk+1
B
/ ...
In this case f and g are said to be (chain) homotopic and this is denoted
f ∼ g.
The relation ∼ is an equivalence relation on every homKom(A) (A, B). We
can thus form a new category factoring it out.
Definition 26. The homotopy category of complexes over A is the category
K(A) defined as follows:
Objects the same as the objects of Kom(A).
Morphisms “morphisms of complexes modulo homotopy”, that is,
homK(A) (A, B) = homKom(A) (A, B)/ ∼
As a consequence of the definition, an isomorphism in K(A) is [represented by] a homotopy equivalence, that is a map f ∈ homKom(A) (A, B) for
which there is another map g ∈ homKom(A) (B, A) such that f ◦ g ∼ IdB
and g ◦ f ∼ IdA . The most significant example of homotopy equivalence is
the augmentation map from a projective resolution to the object it resolves.
Hence, in K(A) an object of A is isomorphic to all its projective resolutions.
A.1. DERIVED CATEGORIES
59
It is not difficult to see that homotopic morphisms induce the same
map in cohomology. It follows that a homotopy equivalence induces an
isomorphism in cohomology. In general, however, the converse is not true,
so the maps satisfying the latter property deserve their own name.
Definition 27. A morphism in Kom(A) or in K(A) is a quasi-isomorphism
if its induced map in cohomology is an isomorphism.
The derived category of A is obtained from K(A) in the same way as
a localization of an integral domain, that is, adding formal inverses to a
suitable class of elements. In the present context, one adds formal inverses
to quasi-isomorphisms of K(A).
Definition 28. The derived category of A is the category D(A) defined as
follows:
Objects the same as the objects of Kom(A).
Morphisms homD(A) (A, B) is made of equivalence classes of roofs
X
φ
α
A
~
B
with φ ∈ homD(A) (X, B) a morphism and α ∈ homD(A) (A, X) a quasiφ
α
β
ψ
isomorphism, where two roofs A ← X → B and A ← Y → B are
κ
λ
equivalent if and only if there is a third roof X ← Z → Y forming a
commutative diagram
Z
κ
X
λ
~
Y
ψ
α
~
β
At
φ
*
B
The derived category enjoys a universal property: it is the most general
category in which quasi-isomorphisms of complexes become isomorphisms.
Namely, let Q : Kom(A) → D(A) be the functor that maps every object
φ
to itself and the morphism A → B to the equivalence class of the roof
60
APPENDIX A. DUALITY
Id
φ
A ← A → B. Then for any functor F : Kom(A) → C transforming quasiisomorphisms into isomorphisms there is a unique functor G : D(A) → C
such that F = G ◦ Q.
For certain purposes it is useful to deal only with the complexes A that
are bounded from above (Ak = 0 for k 0), bounded from below (Ak = 0
for k 0), or just bounded (Ak = 0 for k 0 and k 0). The respective full subcategories of Kom(A) are usually denoted Kom− (A), Kom+ (A),
Komb (A). The “derivation” process can be performed on these subcategories
as well, giving birth to the categories
complexes
homotopy
derived
Kom− (A)
/ K− (A)
/ D− (A)
Kom+ (A)
/ K+ (A)
/ D+ (A)
Komb (A)
/ Kb (A)
/ Db (A)
(A.1)
The derived categories in the last column admit the equivalent descriptions as the full subcategories of D(A) consisting of the complexes A with
H k (A) = 0 for k 0, with H k (A) = 0 for k 0, and with H k (A) = 0 for
k 0 and k 0, respectively.
In order to simplify the writing of some statements, we will call Kcategories over A all the categories made of chain comlexes over A, that
is, Kom(A), K(A), D(A) and their bounded subcategories introduced in
Diagram (A.1).
A.2
Triangles
Derived categories are additive but not abelian (cf. [16, Section III.3]),
hence the concept of exact sequence does not make sense in them. Nevertheless, relaxing the requirements a bit, it is possible to get a quite powerful
deputy tool: the aim of this section is to introduce it.
Let C be a K-category over A and fix n ∈ Z. For any complex A in C,
A.2. TRIANGLES
61
one can consider the translated complex A[n] defined by
n+k
A[n]k = An+k , dkA[n] = (−1)n dA
.
Also, for any morphism φ : A → B, one can define the translated morphism
φ[n] : A[n] → B[n], φ[n]k = φn+k . This defines an autoequivalence of categories
T n : C → C, T n (A) = A[n], T n (φ) = φ[n]
called n-shift or n-translation.
To any morphism φ : A → B in Kom(A) one can associate two complexes:
1. the cone C(φ) of φ, defined by
C(φ)k = A[1]k ⊕ B k ,
dkC(φ) (a(k+1) , b(k) ) =
−dA (a(k+1) ), φ(a(k+1) )+dB (b(k) ) ;
2. the cylinder Cyl(φ) of φ, defined by
Cyl(φ)k = Ak ⊕ A[1]k ⊕ B k ,
dkCyl(φ)(a(k),a(k+1),b(k))= dA (a(k))−a(k+1), −dA (a(k+1)), φ(a(k+1))+dB (b(k)) .
Clearly one can consider the cone and the cylinder of a morphism also as
objects in the homotopy category of complexes or in the derived category
as well.
Definition 29. A triangle in C is a diagram of objects and morphisms in C
φ
χ
ψ
A → B → C → A[1].
The archetypical example of triangle is the cylinder-cone sequence of a
morphism φ ∈ homKom(A) (A, B). It is
ϕ
π
δ
A → Cyl(φ) → C(φ) → A[1],
where
ϕ(a(k) ) = (a(k) , 0, 0),
π(a(k) , a(k+1) , b(k) ) = (a(k+1) , b(k) ),
δ(a(k+1) , b(k) ) = a(k+1) .
62
APPENDIX A. DUALITY
Definition 30. A morphism of triangles is a commutative diagram
A
φ
/B
ρ
D
χ
/C
σ
λ
/E
ψ
/ A[1]
τ
µ
/F
ν
ρ[1]
/ A[1];
it is an isomorphism if ρ, σ, τ are isomorphisms (in their category!).
Finally, a triangle is distinguished if it is isomorphic to the cylinder-cone
sequence
ϕ
π
δ
A → Cyl(φ) → C(φ) → A[1]
of some φ ∈ homKom(A) (A, B).
The fact that distinguished triangles in derived categories are a generalization of exact sequences is expressed in the following result, that we have
already used.
Proposition 31 ([16], Proposition III.5). A short exact sequence in Kom(A)
is isomorphic in D(A) (that is, quasi-isomorphic in Kom(A)) to a distinguished triangle.
The fact that they are a good substitute for exact sequences is stated in
the
Theorem 32 ([16], Theorem III.6). Let
A
φ
/B
χ
/C
ψ
/ A[1]
be a distinguished triangle in D(A). Then the sequence
...
/ H k (A)
H k (φ)
/ H k (B)
H k (χ)
/ H k (C)
H k (ψ)
H k+1 (φ)
/ H k+1 (A)
/ ...
is exact.
Given a functor F : A → B between abelian categories, one can extend it
to chain complexes by making the extension act componentwise. Since this
extension preserves the homotopy between morphisms, it induces a functor
K(F) : K(A) → K(B). If F is exact, K(F) maps quasi-isomorphisms to quasiisomorphisms, so it induces a derived functor D(F) : D(A) → D(B) and
this one maps distinguished triangles to distinguished triangles. A functor
between derived categories with this property is called exact. The same
course holds in the subcategories of bounded complexes.
A.3. TRIANGULATED CATEGORIES
A.3
63
Triangulated categories
The machinery of distinguished triangles was developed in parallel to
the concept of derived category by Jean-Louis Verdier in his PhD thesis.
He realized that the properties of the distinguished triangles in a derived
category give birth to an interesting structure that is worth of independent
studies. So he provided general axioms to encode the behaviour of distinguished triangles.
Definition 33. A triangulation on an additive category C is an additive
self-equivalence T : C → C together with a collection T of triangles
φ
χ
ψ
A → B → C → T (A),
called the distinguished triangles, such that the following axioms hold.
TR1
• Any triangle isomorphic to a distinguished one is itself distinguished.
φ
• Any morphism A → B can be completed to a distinguished triφ
χ
ψ
angle A → B → C → T (A).
Id
• The triangle A → A → 0 → T (A) is distinguished.
φ
χ
ψ
TR2 A triangle A → B → C → T (A) is distinguished if and only if the
χ
−T (φ)
ψ
triangle B → C → T (A) → T (B) is distinguished.
TR3 Any diagram
A
φ
/B
ρ
D
χ
/C
ψ
/ A[1]
σ
λ
/E
µ
/F
ν
ρ[1]
/ A[1]
in which the rows are distinguished and σφ = λρ can be completed,
through a morphism τ : C → F , to a morphism of triangles.
TR4 Given distinguished triangles
α
β
γ
δ
ζ
δα
θ
ι
A → B → G → T (A),
B → C → Z → T (B),
A → C → I → T (A),
64
APPENDIX A. DUALITY
there exists a distinguished triangle
φ
χ
ψ
G → I → Z → T (G)
such that
= χθ,
γ = ιφ,
ψ = T (β)ζ,
ζχ = T (α)ι,
φβ = θδ.
A triangulated category is an additive category C equipped with a triangulation (T, T ) (we will denote it by the same symbol as the underlying additive
category, if there is no danger of confusion).
In Sections IV.1 and IV.2 of [16] it is proved that for any abelian category
A the 1-translation functor together with the distinguished triangles we defined in Section A.2 constitute actual triangulations on each of K(A), D(A)
and (since cones and cylinders of morphisms of somehow bounded complexes
are bounded of the same type) their bounded variants (A.1). These triangulations shall be named canonical in order to underline their prominence
and naturality.
An additive functor F : (C, T, T ) → (C0 , T 0 , T 0 ) between triangulated categories is graded if there is a natural isomorphism F ◦ T ∼
= T 0 ◦ F. It
is δ-exact, δ ∈ {±1}, if it is graded and for every distinguished triangle
φ
χ
ψ
A → B → C → T (A) the triangle
F(φ)
F(χ)
δ·F(ψ)
F(A) → F(B) → F(C) → T 0 (F(A))
is distinguished. Similar definitions hold in case F is a contravariant functor: F graded means F ◦ T ∼
= (T 0 )−1 ◦ F and F δ-exact means for every
φ
χ
ψ
distinguished triangle A → B → C → T (A) the triangle
F(χ)
F(φ)
F(C) → F(B) → F(A)
δ·T 0 (F(ψ))
→
T 0 (F(C))
is distinguished.
A.4
Dualities and unitary functors
Definition 34. Let C be a category. A couple ( ] , $) consisting of a
contravariant functor ] : C → C, together with a natural isomorphism
$ : IdC → ]] satisfying
]
$A
◦ $A] = IdA]
(A.2)
A.4. DUALITIES AND UNITARY FUNCTORS
65
for all objects A of C, is called a duality.
Let (C, ] , $) be a category with duality. Then any morphism φ : A →
]
B in C has an adjoint map
φ]$ := φ] ◦ $B : B → A] .
A map φ : A → A] satisfying φ]$ = φ is called self-adjoint. A self-duality is
a self-adjoint isomorphism; in this case, each of its domain and codomain is
a self-dual object.
When dealing with triangulated categories, one requires dualities to respect also the extra bit of structure given by the translation functor, namely
Definition 35. Let C be a triangulated category. A couple ( ] , $) consisting of a δ-exact contravariant functor ] : C → C, together with a natural
isomorphism $ : IdC → ]] satisfying
]
◦ $A ]
$A
= IdA] ,
(A.3)
$T (A) = T$(A) ,
(A.4)
for all objects A of C, is called a δ-duality. In this case, by a self-duality we
mean an isomorphism of distinguished triangles
A
φ
/B
ρ
C]
χ
/C
σ
χ]
/ B]
ψ
/ A[1]
τ
φ]
ρ[1]
δ·ψ] [1]
/ A]
/ C ] [1]
such that σ ] ◦ $B = σ and τ = ρ] ◦ $C . Each row of the above diagram is
a self-dual distinguished triangle.
See [3] for more details.
Definition 36. Let (C,] , $) and (D,[ , %) be two triangulated categories
with duality. A unitary functor
(G, ς) : (C,
]
[
, $) → (D,
, %)
is a covariant (+1)-exact functor C → D, together with a natural isomorphism of contravariant functors ς : G( ] ) → G( )[ making the diagrams
G(A)
%(G(A))
G($(A))
G(A)[[
/ G(A]] )
ς(A)[
ς(A] )
/ G(A] )[
(A.5)
66
APPENDIX A. DUALITY
commute for all objects A in C.
In particular, a unitary functor takes self-dual distinguished triangles to
self-dual distinguished triangles.
A.5
The Restriction functor
Let R be a ring and A an R-algebra. Then PKom(A) shall denote
the category whose objects are bounded chain complexes of finitely generated projective left A-modules and whose morphisms are chain maps. The
derivation process of Section A.1 can be performed seamlessly in order to
obtain the corresponding homotopy category PK(A) and finally the derived
category P(A). The category PKom(A) contains as objects the mapping
cylinder and the mapping cone of all its morphisms, hence PK(A) and P(A)
come equipped with a preferred triangulation, that is the restriction of the
canonical triangulation introduced in Section A.2.
A particular instance of the preceding constrution involves as ingredients the ring Z and the group algebra ZG of some group G. Group algebras have an additional piece of structure, namely the canonical antipode
S : Z[G] → Z[G], S(g) = g −1 introduced in Remark 22. This serves to define a duality on P(ZG), as follows. If P is a finitely generated projective
left ZG-module, P ∗ := homG (P, ZG) is a finitely generated projective right
ZG-module. Twisting the action on P ∗ by the antipode we get a finitely
generated projective left ZG module
P ~ = homSG (P, ZG) = {α : P → ZG | α(a. p) = α(p)S(a)}.
Extending the functor ~ componentwise to PKom(ZG) defines a contravariant (+1)-exact functor, still denoted by ~ , given by
~
Pk~ = homSG (P −k , ZG),
dPk (p∗(k) )(p(1−k) ) = p∗(k) (dP1−k (p(1−k) ))
where ZG is concentrated in degree 0. This functor passes to the derived
category P(ZG) (we will still use the notation ~ ) and, together with the
natural isomorphism
$ : IdP(ZG) →
~~
,
∗
∗
$P (p(k) )(q(−k)
) = S(q(−k)
(p(k) )),
forms a (+1)-duality on P(ZG) (a proof of this will be presented in [39]).
In the following, a category of the form P(ZG) will be intended to be
equipped with the triangulation and the duality just mentioned.
A.5. THE RESTRICTION FUNCTOR
67
Proposition 37. Let G be a group and U ≤ G a subgroup of finite index.
The functor resG
U : P(ZG) → P(ZU ) is unitary.
Proof. Let P be an object in P(ZG). By a particular instance of EckmannShapiro Lemma (sometimes called Nakayama relations, cf [4, Section 2.8]),
one has the natural isomorphism of abelian groups
S
S
Φ : resG
→ homSU (ZU ⊗G P, ZU )
U homG (P, homU (ZG, ZU )
Φ(α) : u ⊗ x 7→ α(x)(u).
But Φ is also compatible with the U -action, hence it is an isomorphism of
U -modules. Moreover, ZU ⊗G P may be identified with resG
U (P ). Finally,
since the index [G : U ] is finite, one has the isomorphism
Ψ : ZG → homU (ZG, ZU ),
Ψ(g)(h) = hgδ{hg∈U }
where
δ{x∈U }
(
1
=
0
x∈U
x∈
/ U.
Summing up, there is a natural isomorphism of functors
~ ) → resG ( )~
ς : resG
U
U(
ςP (χ) : u ⊗G x 7→ Ψ ◦ χ(x)(u).
Then Diagram (A.5) becomes
resG
U (P )
(resG
U)
resG (P )
U
G
resG
U ($P )
/ resG (homS (homS (P, ZG), ZG))
G
G
U
$U
ς
homSU (homSU (resG
U (P ), ZU ), ZU )
ς~
~
/ homS (resG (homS (P, ZG)), ZU )
U
G
U
(A.6)
and (simplifying the notation a little and using the identification
)=
ZU ⊗G ) we need to verify that ς(1 ⊗G $G (x)) = ς ~ ($U (x)) for all x ∈ P .
On one side,
resG
U(
~
ς(1⊗G $G(x)) : resG
U (P ) → ZU
ς(1⊗G $G (x))(u ⊗ α) = Ψ $G (x)(α) (u) = Ψ(S(α(x))) (u) =
= uS(α(x))δ{uS(α(x))∈U } = uS(α(x))δ{α(x)∈U } .
68
APPENDIX A. DUALITY
On the other,
~
ς ~ ($U(x)) : resG
U (P ) → ZU
ς ~ ($U (x))(u ⊗ α) = $U (x) ς(u ⊗G α) = u · $U (x)(ς(1 ⊗G α)) =
= u · S(ς(1 ⊗G α)(x)) = u · S (Ψ ◦ α)(x)(1) =
= u · S(α(x)δ{α(x)∈U } ) = uS(α(x))δ{α(x)∈U } .
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| 4 |
P ERSONALIZING D IALOGUE AGENTS :
I HAVE A DOG , DO YOU HAVE PETS TOO ?
Saizheng Zhang, Emily Dinan, Jack Urbanek, Arthur Szlam, Douwe Kiela, & Jason Weston
Facebook AI Research
arXiv:1801.07243v1 [cs.AI] 22 Jan 2018
A BSTRACT
Chit-chat models are known to have several problems: they lack specificity, do not
display a consistent personality and are often not very captivating. In this work
we present the task of making chit-chat more engaging by conditioning on profile
information. We collect data and train models to (i) condition on their given profile
information; and (ii) information about the person they are talking to, resulting
in improved dialogues, as measured by next utterance prediction. Since (ii) is
initially unknown our model is trained to engage its partner with personal topics,
and we show the resulting dialogue can be used to predict profile information
about the interlocutors.
1
I NTRODUCTION
Despite much recent success in natural language processing and dialogue research, communication
between a human and a machine is still in its infancy. It is only recently that neural models have had
sufficient capacity and access to sufficiently large datasets that they appear to generate meaningful
responses in a chit-chat setting. Still, conversing with such generic chit-chat models for even a short
amount of time quickly exposes their weaknesses (Serban et al., 2016; Vinyals & Le, 2015).
Common issues with chit-chat models include: (i) the lack of a consistent personality (Li et al.,
2016a) as they are typically trained over many dialogs each with different speakers, (ii) the lack of
an explicit long-term memory as they are typically trained to produce an utterance given only the
recent dialogue history (Vinyals & Le, 2015); and (iii) a tendency to produce non-specific answers
like “I don’t know” (Li et al., 2015). Those three problems combine to produce an unsatisfying
overall experience for a human to engage with. We believe some of those problems are due to
there being no good publicly available dataset for general chit-chat1 . For those reasons, chit-chat
models are often ignored as an end-application and the research community has focused on taskoriented communication, such as airline or restaurant booking, instead (Bordes & Weston, 2016),
or else single-turn information seeking, i.e. question answering Rajpurkar et al. (2016). Despite
the success of the latter, simpler, domain, it is well-known that a large quantity of human dialogue
centers on socialization, personal interests and chit-chat (Dunbar et al., 1997). For example, less
than 5% of posts on Twitter are questions, whereas around 80% are about personal emotional state,
thoughts or activities, authored by so called “Meformers” (Naaman et al., 2010).
In this work we make a step towards more engaging chit-chat dialogue agents by endowing them
with a configurable, but persistent persona, encoded by multiple sentences of textual description,
termed a profile. This profile can be stored in a memory-augmented neural network and then used
to produce more personal, specific, consistent and engaging responses than a persona-free model,
thus alleviating some of the common issues in chit-chat models. Using the same mechanism, any
existing information about the persona of the dialogue partner can also be used in the same way. Our
models are thus trained to both ask and answer questions about personal topics, and the resulting
dialogue can be used to build a model of the persona of the speaking partner.
We thus present the PERSONA - CHAT dataset, a new dialogue dataset consisting of 164,356 utterances
between crowdworkers who were randomly paired and each asked to act the part of a given provided
persona (randomly assigned, and created by another set of crowdworkers). The paired workers
1
For example, currently the most general chit-chat dataset available in http://parl.ai a large repository of dialogue datasets is probably OpenSubtitles, which is based on movie scripts, not natural conversations.
1
were asked to chat naturally and to get to know each other during the conversation. This produces
interesting and engaging conversations that our agents can try to learn to mimic.
Studying the next utterance prediction task during dialogue, we compare a range of models: both
generative and ranking models, including Seq2Seq models and Memory Networks (Sukhbaatar
et al., 2015) as well as other standard retrieval baselines. We show experimentally that in either
the generative or ranking case conditioning the agent with persona information gives improved prediction of the next dialogue utterance. The PERSONA - CHAT dataset is designed to facilitate research
into alleviating some of the issues that traditional chit-chat models face, and with the aim of making such models more consistent and engaging, by endowing them with a persona. By comparing
against chit-chat models built using the OpenSubtitles dataset, human evaluations show that our
dataset provides more engaging models, that are simultaneously capable of being fluent and consistent via conditioning on a persistent, recognizable profile.
2
R ELATED W ORK
Traditional dialog systems consist of building blocks, such as dialog state tracking components
and response generators, and have typically been applied to tasks with labeled internal dialog state
and precisely defined user intent (i.e., goal-oriented dialogue), see e.g. (Young, 2000). The most
successful goal-oriented dialog systems model conversation as partially observable Markov decision
processes (POMDPs) (Young et al., 2013). All those methods typically do not consider the chit-chat
setting and are more concerned with achieving functional goals (e.g. booking an airline flight) than
displaying a personality. In particular, many of the tasks and datasets available are constrained to
narrow domains (Serban et al., 2015).
Non-goal driven dialog systems go back to Weizenbaum’s famous program ELIZA (Weizenbaum,
1966), and hand-coded systems have continued to be used in applications to this day. For example, modern solutions that build an open-ended dialogue system to the Alexa challenge combine
hand-coded and machine-learned elements (Serban et al., 2017a). Amongst the simplest of statistical systems that can be used in this domain, that are based on data rather than hand-coding, are
information retrieval models (Sordoni et al., 2015), which retrieve and rank responses based on their
matching score with the recent dialog history. We use IR systems as a baseline in this work.
End-to-end neural approaches are a class of models which have seen growing recent interest. A popular class of methods are generative recurrent systems like seq2seq applied to dialogue (Sutskever
et al., 2014; Vinyals & Le, 2015; Sordoni et al., 2015; Li et al., 2016b; Serban et al., 2017b). Their
strengths are that (i) they are not constrained by hard-code rules or explicit internal states that may
work well in a narrow domain, but are too restrictive for more open dialogue such as chit-chat, and
(ii) being based on architectures rooted in language modeling and machine translation, they excel
at generating syntactically coherent language, and can generate entirely novel responses. Their deficiencies are that they typically need a large amount of data to be trained, and the vanilla approach
generates responses given only the recent dialog history without using other external memory. The
latter issue makes neural models hence typically lack both domain knowledge in the domain being
discussed, and a persistent personality during discussions. A promising direction, that is still in its
infancy, to fix this issue is to use a memory-augmented network instead (Sukhbaatar et al., 2015;
Dodge et al., 2015) and either provide or learn appropriate external memories. A related class of
neural methods is to use similarly architectures, but to retrieve and rank candidates similarly to the
IR baseline, but using memory-augmented networks to score the candidates instead. We compare
the generative and ranking approaches to each other in this work.
Serban et al. (2015) list available corpora for training dialog systems. Perhaps the most relevant
to learning chit-chat models are ones based on movie scripts such as OpenSubtitles and Cornell
Movie-Dialogue Corpus, and dialogue from web platforms such as Reddit and Twitter, all of which
have been used for training neural approaches (Vinyals & Le, 2015; Dodge et al., 2015; Li et al.,
2016b; Serban et al., 2017b). Naively training on these datasets leads to models with the lack of a
consistent personality as they will learn a model averaged over many different speakers. Moreover,
the data does little to encourage the model to engage in understanding and maintaining knowledge
of the dialogue partner’s personality and topic interests.
2
Original Persona
Revised Persona
I love the beach.
My dad has a car dealership
I just got my nails done
I am on a diet now
Horses are my favorite animal.
To me, there is nothing like a day at the seashore.
My father sales vehicles for a living.
I love to pamper myself on a regular basis.
I need to lose weight.
I am into equestrian sports.
I am an eccentric hair stylist for dogs
My favorite past time is collecting Civil War antiques.
I fake a British accent to seem more attractive.
I have been married four times and widowed three.
I have an allergy to mangoes
I work with animals.
I like finding or buying historical artifacts.
I heard girls liked foreigners.
I have a lot of experience with marriage
I have reactions to certain fruits.
I play a lot of fantasy videogames.
I have a computer science degree.
My mother is a medical doctor
I am very shy.
I like to build model spaceships.
RPGs are my favorite genre.
I also went to school to work with technology.
The woman who gave birth to me is a physician.
I am not a social person.
I enjoy working with my hands.
Table 1: Example Personas (left) and their revised versions (right) from the PERSONA - CHAT dataset.
The revised versions are designed to be characteristics that the same persona might have, which
could be rephrases, generalizations or specializations.
According to the survey (Serban et al., 2015) personalization of dialogue systems is “an important
task, which so far has not received much attention”. In the case of goal-oriented dialog some work
has focused on the agent being aware of the human’s profile and adjusting the dialogue accordingly,
but without a personality to the agent itself (Lucas et al., 2009; Joshi et al., 2017). For the chit-chat
setting, the most relevant work is (Li et al., 2016a). For each user in the Twitter corpus, personas
were captured via distributed embeddings (one per speaker) to encapsulate individual characteristics such as background information and speaking style, and they then showed using those vectors
improved the output of their seq2seq model for the same speaker. Their work does not focus on
attempting to engage the other speaker by getting to know them, as we do here. For that reason, our
focus is on explicit profile information, not hard-to interpret latent variables.
3
T HE PERSONA - CHAT DATASET
The aim of this work is to facilitate more engaging and more personal chit-chat dialogue. The
PERSONA - CHAT dataset is a crowd-sourced dataset where each of the pair of speakers condition
their dialogue on a given profile, which is provided. The data collection consists of three stages:
• Personas: we crowdsource a set of 1155 possible personas, each consisting of at least 5
profile sentences, setting aside 100 never seen before personas for validation, and 100 for
test.
• Revised personas: to avoid modeling that takes advantage of trivial word overlap, we
crowdsource additional rewritten sets of the same 1155 personas, with related sentences
that are rephrases, generalizations or specializations, rendering the task much more challenging.
• Persona chat: we pair two Turkers and assign them each a random (original) persona from
the pool, and ask them to chat. This resulted in a dataset of 164,356 utterances over 10,981
dialogs, 15,705 utterances (968 dialogs) of which are set aside for validation, and 15,119
utterances (1000 dialogs) for test.
The final dataset is available in ParlAI2 . In the following, we describe each data collection stage in
more detail.
2
https://github.com/facebookresearch/ParlAI/tree/master/parlai/tasks/
personachat
3
Persona 1
Persona 2
I like to ski
My wife does not like me anymore
I have went to Mexico 4 times this year
I hate Mexican food
I like to eat cheetos
I am an artist
I have four children
I recently got a cat
I enjoy walking for exercise
I love watching Game of Thrones
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
Hi
Hello ! How are you today ?
I am good thank you , how are you.
Great, thanks ! My children and I were just about to watch Game of Thrones.
Nice ! How old are your children?
I have four that range in age from 10 to 21. You?
I do not have children at the moment.
That just means you get to keep all the popcorn for yourself.
And Cheetos at the moment!
Good choice. Do you watch Game of Thrones?
No, I do not have much time for TV.
I usually spend my time painting: but, I love the show.
Table 2: Example dialog from the PERSONA - CHAT dataset. Person 1 is given their own persona (top
left) at the beginning of the chat, but does not know the persona of Person 2, and vice-versa. They
have to get to know each other during the conversation.
3.1
P ERSONAS
We asked the crowdsourced workers to create a character (persona) description using 5 sentences,
providing them only a single example:
“I am a vegetarian. I like swimming. My father used to work for Ford. My favorite band is Maroon5.
I got a new job last month, which is about advertising design.”
Our aim was to create profiles that are natural and descriptive, and contain typical topics of human
interest that the speaker can bring up in conversation. We asked the workers to make each sentence
short, with a maximum of 15 words per sentence. This is advantageous both for humans and machines: if they are too long, crowdsourced workers are likely to lose interest, and for machines the
task could become more difficult.
Some examples of the personas collected are given in Table 1 (left).
3.2
R EVISED P ERSONAS
A difficulty when constructing dialogue datasets, or text datasets in general, is that to encourage
research progress requires the careful construction of a task that is neither too easy nor too difficult
for the current technology (Voorhees et al., 1999). One issue with conditioning on textual personas
is that there is a danger that humans will, even if asked not to, unwittingly repeat profile information
either verbatim or with significant word overlap. This may make any subsequent machine learning
tasks less challenging, and the solutions will not generalize to more difficult tasks. This has been
a problem in some recent datasets: for example, the dataset curation technique used for the wellknown SQuAD dataset suffers from this word overlap problem to a certain extent (Chen et al., 2017).
To alleviate this problem, we presented the original personas we collected to a new set of crowdworkers and asked them to rewrite the sentences so that a new sentence is about “ a related characteristic that the same person may have”, hence the revisions could be rephrases, generalizations
or specializations. For example “I like basketball” can be revised as “I am a big fan of Michael
Jordan” not because they mean the same thing but because the same persona could contain both.
In the revision task, workers are instructed not to trivially rephrase the sentence by copying the
original words. However, during the entry stage if a non-stop word is copied we issue a warning,
and ask them to rephrase, guaranteeing that the instructions are followed. For example, “My father
4
worked for Ford.” can be revised to “My dad worked in the car industry”, but not “My dad was
employed by Ford.” due to word overlap.
Finally, we encourage the construction of natural sentences. In earlier versions of the task we noticed
that the word overlap constraint caused unwanted unnatural constructions such as “I like eating
pretzels” revised as “I like to chew and swallow twisted bread with salt”. Giving explicit instructions
about this seemed to help, where we prefer a revision like “I enjoy beers and beer snacks”.
Some examples of the revised personas collected are given in Table 1 (right).
3.3
P ERSONA C HAT
After collecting personas, we then collected the dialogues themselves, conditioned on the personas.
For each dialogue, we paired two random crowdworkers, and gave them the instruction that they
will chit-chat with another worker, while playing the part of a given character. We then provide
them with a randomly chosen persona from our pool, different to their partners. The instructions are
on purpose quite terse and simply ask them to “chat with the other person naturally and try to get to
know each other”. In an early study we noticed the crowdworkers tending to talk about themselves
too much, so we also added the instructions “both ask questions and answer questions of your chat
partner” which seemed to help. We also gave a bonus for high quality dialogs. The dialog is turnbased, with a maximum of 15 words per message. We again gave instructions to not trivially copy
the character descriptions into the messages, but also wrote explicit code sending them an error if
they tried to do so, using simple string matching. We define a minimum dialogue length which is
randomly between 6 and 8 turns each for each dialogue.
An example dialogue from the dataset is given in Table 2.
3.4
E VALUATION
We focus on the standard dialogue task of predicting the next utterance given the dialogue history,
but consider this task both with and without the profile information being given to the learning agent.
Our goal is to enable interesting directions for future research, where chatbots can for instance have
personalities, or imputed personas could be used to make dialogue more engaging to the user.
We consider this in four possible scenarios: conditioning on no person, your own persona, their
person, or both. We can also try each of these scenarios using either the original personas, or the
revised ones. We then evaluate the task using two metrics: (i) the log likelihood of the correct
sequence, measured via perplexity and (ii) next utterance classification loss, following Lowe et al.
(2015).
As dialogue has many possible responses, leading to a multi-modal distribution of words, word
overlap measures do not work well as evaluation metrics (Liu et al., 2016; Serban et al., 2015).
While word level perplexity has many deficiencies as a measure of conversational success, it is
standard in more general language modeling, and can still capture multi-modal distributions to a
certain extent as good response word choices should still have high probability. Thus we include
it here. Next utterance classification loss consists of choosing N random distractor responses from
other dialogues (in our setting, N =19) and choosing the best among them, resulting in a score of
one if the model chooses the correct response, and zero otherwise. Its main advantage is that it is
easy to interpret.
4
M ODELS
Let x be an input sequence (i.e., the previous dialogue utterances), M 1 be the set of profile entries of
the speaker (i.e., the model’s own profile) and M 2 be the profile of the listener (i.e., the interlocutor’s
profile). In our experiments, we compare four settings: not having access to any profile information,
having access only to M 1 , having access only to M 2 , or having access to both M 1 ∪ M 2 . In what
follows, we use M ∈ {∅, M 1 , M 2 , M 1 ∪ M 2 } to cover all four possibilities.
5
I
have
I
enjoy
My
job
a
cat
eating spaghetti
is
research
<s>
Do
you
have
pets
Yes
,
one
cat
?
Figure 1: A diagram of the Profile Memory Network for generation. We also implemented a ranking
version which has the same architecture except it ranks candidate sentences from the training set
instead of generating, representing them using bag-of-word embeddings.
4.1
R ANKING M ODELS
The following set of models produce a next utterance by considering any utterance in the training
set as a possible candidate reply. They are typically strong baselines, or, if the candidate set is big
enough can be hard to beat as the sentences, being written by humans, already have fluency and
internal semantic coherence. On the other hand, they cannot generate novel sentences.
4.1.1
IR BASELINE
To select candidate responses a standard baseline is nearest neighbor information retrieval (IR) (Isbell et al., 2000; Jafarpour et al., 2010; Ritter et al., 2011; Sordoni et al., 2015). While there are
many variants, we adopt the simplest one: find the most similar message in the (training) dataset
and output the response from that exchange. Similarity is measured by the tf-idf weighted cosine
similarity between the bags of words. To incorporate the profile we simply concatenate it to the
query vector bag of words.
4.1.2
S TARSPACE
Starspace is a recent model that performs also performs information retrieval but by learning sentence embeddings that measure similarity between the dialog and the next utterance by optimizing
the word embeddings directly for that task using the training set (Wu et al., 2017)3 . Similar supervised embeddings have been used with good results in other dialogue tasks previously (Dodge et al.,
2015). Specifically, it optimizes:
X
−
Lbatch (sim(q, c), sim(q, c−
1 ), . . . , sim(q, ck ))
(q,c)∈E +
b− ∈E −
where the loss function Lbatch compares a positive pair of query and candidate (q, c) with the negative pairs (q, c−
i ), i = 1, . . . , k using the margin ranking loss max(0, µ − sim(q, c), where µ is
the margin parameter. The similarity function sim(·, ·) is the cosine similarity of the sum of word
embeddings of the query q and candidate c0 . Denoting the dictionary of D word embeddings as W
3
Available at https://github.com/facebookresearch/StarSpace
6
th
which is a D × d matrix, where WP
word (row), yielding its d-dimensional embedi indexes the i
ding, it embeds a sequence s with i∈s Wi . While this model supports different word embeddings
for the left and right hand side of the similarity function, we found sharing the weights gave the best
performance.
Similar to the IR baseline, to incorporate the profile we simply concatenate it to the query vector
bag of words.
4.1.3
R ANKING P ROFILE M EMORY N ETWORK
Both the previous models use the profile information by combining it with the dialogue history,
which means the model cannot differentiate between the two when deciding on the next utterance.
In this model we instead use a memory network with the dialogue history as input, which then
performs attention over the profile to find relevant lines from the profile to combine with the input,
and then finally predicts the next next utterance. We use the same representation as in the starspace
model, so without the profile, the two models are identical. When the profile is available attention
is performed by computing the similarity of the input q with the profile sentences pi , computing the
softmax, and taking the weighted sum:
X
q+ = q +
si pi ,
si = Softmax(sim(q, pi ))
P
where Softmax(zi ) = ezi / j ezj . One can then rank the candidates c0 using sim(q + , c0 ). One can
also perform multiple “hops” of attention over the profile rather than one, as shown here, although
that did not bring significant gains in our parameter sweeps. Similarly, we found a model that shared
word embedding lookup table weights across dialog history, profiles and candidates performed best
compared to models with more parameters.
4.1.4
K EY-VALUE P ROFILE M EMORY N ETWORK
The key-value (KV) memory network Miller et al. (2016) was proposed as an improvement to the
memory network by performing attention over keys and outputting the values (instead of the same
keys as in the original), which can outperform memory networks dependent on the task and definition
of the key-value pairs. Here, we apply this model to dialogue, and consider the keys as dialog
histories (from the training set), and the values as the next dialogue utterances, e.g. the replies from
the speaking partner. This allows the model to have a memory of past dialogues that it can directly
use to help influence its prediction for the current conversation. The model we choose is identical
to the profile memory network just described in the first hop over profiles, while in the second hop,
q + is used to attend over the keys and output a weighted sum of values as before, producing q ++ .
This is then used to rank the candidates c0 using sim(q ++ , c0 ) as before. As the set of (key-value)
pairs is large this would make training very slow. In our current experiments we simply trained the
profile memory network and used the same weights from that model and applied this architecture
at test time instead. Training the model directly would presumably give better results, however this
heuristic already proved beneficial compared to the original network.
4.2
G ENERATIVE M ODELS
Our next set of models do generate novel sentences by conditioning on the dialogue history and
possibly the persona, and then generating the response word-by-word, see e.g. Fig 1. One can
still evalutate these models as ranking models by computing the probability of generating a given
candidate, and ranking candidates by those scores.4
4.2.1
S EQ 2S EQ
The input sequence is encoded by applying het = LST Menc (xt | het−1 ). We use GloVe (Pennington
et al., 2014) for our word embeddings. The final hidden state, het , is fed into the decoder LST Mdec
4
In practice, better results for ranking are obtained by normalizing by the sentence length, following (Dodge
et al., 2015).
7
as the initial state hd0 . For each time step t, the decoder then produces the probability of a word j
occurring in that place via the softmax, i.e.,
exp(wj hdt )
p(yt,j = 1 | yt−1 , . . . , y1 ) = PK
.
d
0
j 0 =1 exp(wj ht )
(1)
The model is trained via negative log likelihood. The basic model can be extended to include persona
information, in which case we simply prepend it to the input sequence x, i.e., x = ∀m ∈ M || x,
where || denotes concatenation.
4.2.2
G ENERATIVE P ROFILE M EMORY N ETWORK
Finally, we introduce a model that encodes each of the profile entries as individual memory representations in a memory network. As before, the dialogue history is encoded via LST Menc , the final
state of which is used as the initial hidden state of the decoder. Each entry mi = hmi,1 , . . . , mi,n i ∈
P|mi |
M is then encoded via f (mi ) =
αi mi,j . That is, we weight words by their inverse term
j
frequency: αi = 1/(1 + log(1 + tf)) where tf is computed from the GloVe index via Zipf’s law5 .
Let F be the set of encoded memories. The decoder now attends over the encoded profile entries,
i.e., we compute the mask at , context ct and next input x̂t as:
at = sof tmax(F Wa hdt ); ct = a|t F ; x̂t = tanh(Wc [ct−1 , xt ]).
(2)
This model is illustrated in Figure 1. Again, if the model has no profile information, and hence no
memory, it becomes equivalent to the Seq2Seq model.
5
E XPERIMENTS
We first report results using automatic evaluation metrics, and subsequently perform an extrinsic
evaluation where we use crowdsourced workers to perform a human evaluation of our models.
5.1
AUTOMATED METRICS
Results for the generative model approaches are reported in Table 3, and for the ranking models
in Table 4. For the generative models, we report perplexity and hits@1 (the accuracy of the next
dialogue utterance when choosing between the gold response and N =19 distractor responses).To
compute hits@1 for generative models we rank candidates according to their mean log likelihood.
For ranking models, which are not generative and hence do not allow for computing the perplexity,
we only report hits@1.
In all cases we compare using the different persona types (none, my, their and both) and using the
original or revised versions. For the ranking models we also tried two variants of training: training
with the original personas in the training set or the revised ones. The latter could provide a difference
because there is less word overlap between the dialogue and the profiles in that case which can force
the model to generalize more (e.g. learn synonyms) rather than learning about word overlap, which
crowdsource workers may otherwise resort to.
Overall, the results show the following key points:
• Most models improve significantly when conditioning prediction on their persona (‘Self
Persona’) at least for the original (non-revised) versions, which is an easier task than the
revised ones which have no word overlap. For example, the Profile Memory generation
model has improved perplexity and hits@1 compared to Seq2Seq, and all the ranking algorithms (IR baseline, Starspace and Profile Memory Networks) obtain improved hits@1.
• Using “Their persona” has less impact on this dataset. We believe this is because most
speakers tend to focus on themselves when it comes to their interests. It would be interesting how often this is the case in other datasets. Certainly this is skewed by the particular
5
tf = 1e6 ∗ 1/(idx1.07 )
8
Method
Original
Perplexity Hits@1
38.08
0.092
38.08
0.092
Self Persona
Seq2Seq
Profile Memory
40.53
34.54
0.084
0.125
40.65
38.21
0.082
0.108
Their Persona
Seq2Seq
Profile Memory
41.48
36.42
0.075
0.105
41.95
37.75
0.074
0.103
Both Personas
Seq2Seq
Profile Memory
40.14
35.27
0.084
0.115
40.53
38.48
0.082
0.106
Persona
No Persona
Revised
Perplexity Hits@1
Table 3: Evaluation of dialog utterance prediction with generative models in four settings: conditioned on the speakers persona (“self persona”), the dialogue partner’s persona (“their persona”),
both or none. The personas are either the original source given to Turkers to condition the dialogue,
or the revised personas that do not have word overlap. In the “no persona” setting, the models are
equivalent, so we only report once.
Method
No Persona
Orig Rewrite
IR baseline
0.214
Training on original personas
Starspace
0.318
0.318
Profile Memory
Training on revised personas
Starspace
0.318
Profile Memory
0.318
KV Profile Memory 0.349
Self Persona
Orig Rewrite
Their Persona
Orig Rewrite
Both Personas
Orig Rewrite
0.214
0.410
0.207
0.181
0.181
0.382
0.188
0.318
0.318
0.481
0.473
0.295
0.302
0.245
0.283
0.235
0.267
0.429
0.438
0.258
0.266
0.318
0.318
0.349
0.491
0.509
0.511
0.322
0.354
0.351
0.271
0.299
0.291
0.261
0.294
0.289
0.432
0.467
0.467
0.288
0.331
0.330
Table 4: Evaluation of dialog utterance prediction with ranking models using hits@1 in four
settings: conditioned on the speakers persona (”self persona”), the dialogue partner’s persona (”their
persona”), both or none. The personas are either the original source given to Turkers to condition the
dialogue, or the rewritten personas that do not have word overlap, explaining the poor performance
of IR in that case.
instructions one could give to the crowdworkers. For example if we gave the instructions
“try not to talk about yourself, but about the other’s interests’ likely these metrics would
change.
• Revised personas are much harder to use. We do however still see some gain for the Profile
Memory networks using “Self persona” compared to none (0.354 vs. 0.318 hits@1). Training on revised personas helps, both for test examples that are in original form or revised
form, likely due to the model be forced to learn more than simple word overlap.
• Ranking models are far better than generative models at ranking. This is perhaps obvious
as that is the metric they are optimizing, but still the performance difference is quite stark.
It may be that the word-based probability which generative models use works well, but
is not calibrated well enough to give a sentence-based probability which ranking requires.
Due to this inherent unfairness in the automatic evaluation, a more fair measure is a human
evaluation that compares these methods, which we perform in Sec. 5.2.
• For the ranking models, the IR baseline is outperformed by Starspace due to its learnt
similarity metric, which in turn is outperformed by Profile Memory networks due to the
attention mechanism over the profiles (as all other parts of the models are the same). Finally
KV Profile Memory networks outperform Profile Memory Networks in the no persona
case due to the ability to consider neighboring dialogue history and next utterance pairs
in the training set that are similar to the current dialogue, however when using persona
information the performance is similar.
9
Method
Fluency
Engagingness
Consistency
Persona
Detection
Self
4.31(1.07)
4.25(1.06)
4.36(0.92)
0.95(0.22)
None
Self
3.17(1.10)
3.08(1.40)
3.18(1.41)
3.13(1.39)
2.98(1.45)
3.14(1.26)
0.51(0.50)
0.72(0.45)
Ranking Models
KV Memory
KV Profile Memory
None
Self
3.81(1.14)
3.97(0.94)
3.88(0.98)
3.50(1.17)
3.36(1.37)
3.44(1.30)
0.59(0.49)
0.81 (0.39)
OpenSubtitles KV Memory
None
2.14(1.20)
2.22(1.22)
2.06(1.29)
0.42(0.49)
Model
Profile
Human
Generative Models
Seq2Seq
Profile Memory
Table 5: Human Evaluation of our various PERSONA - CHAT model, along with a comparison to
human performance, and OpenSubtitles based model (last row), standard deviation in parenthesis.
5.2
H UMAN E VALUATION
As automated metrics are notoriously poor for evaluating dialogue (Liu et al., 2016) we also perform
human evaluation using crowdsourced workers. The procedure is as follows. We perform almost
exactly the same setup as in the dataset collection process itself as in Section 3.3. In that setup, we
paired two Turkers and assigned them each a random (original) persona from the collected pool, and
asked them to chat. Here, from the Turker’s point of view everything looks the same except instead
of being paired with a Turker they are paired with one of our models instead (they do not know this).
In this setting, for both the Turker and the model, the personas come from the test set pool.
After the dialogue, we then ask the Turker some additional questions in order to evaluate the quality
of the model. They are, in order:
• Fluency: We ask them to judge the fluency of the other speaker as a score from 1 to 5,
where 1 is “not fluent at all”, 5 is “extremely fluent”, and 3 is “OK”.
• Engagingness: We ask them to judge the engagingness of the other speaker disregarding
fluency from 1-5, where 1 is “not engaging at all”, 5 is “extremely engaging”, and 3 is
“OK”.
• Consistency: We ask them to judge the consistency of the persona of the other speaker,
where we give the example that “I have a dog” followed by “I have no pets” is not consistent. The score is again from 1-5.
• Profile Detection: Finally, we display two possible profiles, and ask which is more likely
to be the profile of the person the Turker just spoke to. One profile is chosen at random,
and the other is the true persona given to the model.
The results are reported in Table 5 for the best performing generative and ranking models, in both
the No Persona and Self Persona categories, 100 dialogues each. We also evaluate the scores of
human performance by replacing the chatbot with a human (another Turker). This effectively gives
us upper bound scores which we can aim for with our models. Finally, and importantly, we compare
our models trained on PERSONA - CHAT with a chit-chat model trained with the OpenSubtitles dataset
instead, following Vinyals & Le (2015). Example chats from a few of the models are shown in Tables
7, 6 and 8.
Firstly, we see a difference in fluency, engagingness and consistency between all PERSONA - CHAT
models and the model trained on OpenSubtitles. PERSONA - CHAT is a resource that is particularly
strong at providing training data for the beginning of conversations, when the two speakers do not
know each other, focusing on asking and answering questions, in contrast to other resources. We
also see suggestions of more subtle differences between the models, although these differences
are obscured by the high variance of the human raters’ evaluations. 6 For example, in both the
generative and ranking model cases, models endowed with a persona can be detected by the human
6
We plan to increase the resolution of our evaluation with more rater data in future work.
10
Persona 1
Persona 2
I am a doctor
I have one brother
My mom is a secretary
I like to eat a lot of meat
I am a bodybuilder
I eat sardines for breakfast daily
I just had surgery
I love playing Frisbee
I have two large dogs
I never finished high school
[PERSON 1:] Hi I am Dr. Jenson. What is your name?
[PERSON 2:] Miss Dawes.
[PERSON 1:] Hi Miss Dawes, tell me about your family!
[PERSON 2:] Miss Baggoli, I know it must show on your records that my name is Mary, but everyone
has referred to me as Lola since I was a squealing infant in my mother’ s arms.
[PERSON 1:] fascinating. My secretary mother would have made note of all of that.
[PERSON 2:] In addition to the remainder, it writes my autobiography.
[PERSON 1:] Oh yeah? well I enjoy a nice steak. how about you?
[PERSON 2:] Father always used to say that if we ever had the money you have...
we’ d have steak and ice cream three times every day!
[PERSON 1:] I eat all the steak and chicken too, even bacon
[PERSON 2:] Father always used to say that if we ever had the money you have...
we’ d have steak and ice cream three times every day!
[PERSON 1:] yeah you said that already
[PERSON 2:] Dynamic structure can absorb shock.
[PERSON 1:] yes it can. I know about that as a bodybuilder
[PERSON 2:] Ladies and gentlemen... the strongest man in the world is blowing up a hot water bottle.
Table 6: Example dialog between a human (Person 1) and the OpenSubtitles KV Memory Network
model (Person 2).
Persona 1
Persona 2
I am afraid of dogs
I used to work at a carnival
I like to drink scotch to relax
I write short stories for a living
I like dark superhero movies
I am a carpenter
I used to drink but I stopped five years ago
I have a wife and three kids
I like playing poker
I dont have many friends
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
Hi are you relaxing on this fine evening?
I am doing good and you?
I am well, just drinking some scotch and working on my writing
I love to cook and bake but I rarely make time for it
what do you usually bake?
I do not cook, I bake cakes at a local restaurant. how about you?
I used to be a juggler at a carnival but now I am an author
I am a dancer, what do you do?
I just told you. Do you have any pets?
no, I am allergic to pets unfortunately.
I understand. I am not allergic but I am terrified of dogs!
I have a cat. she has been with me through three wives, haha.
Funny that you just told me you have no pets
I do not have any animals.
Table 7: Example dialog between a human (Person 1) and the Seq2Seq model (Person 2).
conversation partner, as evidenced by the persona detection accuracies, whilst maintaining fluency
and consistency compared to their non-persona driven counterparts.
Finding the balance between fluency, engagement, consistency, and a persistent persona remains a
strong challenge for future research.
11
Persona 1
Persona 2
I am engaged to be married
I used to play sports in high school
My favorite sport is soccer
I am a college graduate
I love watching sports on TV
I am a writer
I write romance novels
I live on a faraway island
I am an introvert
I have two cats
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
[PERSON 1:]
[PERSON 2:]
I just got done playing soccer. What have you been up to?
I do not have pets. I spend my free time writing. what about you?
Just graduated from college. Looking for work.
yes I graduated, in writing and poetry
Have you had anything published?
I mainly write fiction novels.
Do you like to watch sports?
do you like kings of leon my favorite by them is use somebody
Are you married? I will be married soon.
haha, no time. I have got a novel to finish.
What is your novel about?
it is a love story. I really enjoy writing. do you write or read?
Like to read. Spend most time watching football.
yes I do I am and author for a living
Table 8: Example dialog between a human (Person 1) and the Key-Value Profile Memory Network
with Self Persona.
6
C ONCLUSION & D ISCUSSION
In this work we have introduced the PERSONA - CHAT dataset, which consists of crowd-sourced dialogues where each participant plays the part of an assigned persona; and each (crowd-sourced) persona has a word-distinct paraphrase. We test various baseline models on this dataset, and show that
models that have access to their own personas in addition to the state of the dialogue are scored as
more consistent by annotators, although not more engaging. On the other hand, we show that models
trained on PERSONA - CHAT (with or without personas) are more engaging than models trained on
dialogue from movies.
We believe PERSONA - CHAT will be a useful resource for training components of future dialogue
systems. Because we have paired human generated profiles and conversations, the data aids the
construction of agents that have consistent personalities and viewpoints. Furthermore, imputing the
profiles from a conversation moves chit-chat tasks in the direction of goal-directed dialogue, which
has metrics for success. Because we collect paraphrases of the profiles, they cannot be trivially
matched; indeed, we believe the original and rephrased profiles are interesting as a semantic similarity dataset in their own right. We hope that the data will aid training agents that can ask questions
about users’ profiles, remember the answers, and use them naturally in conversation.
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14
| 2 |
1
Resource Allocation for Downlink NOMA Systems:
Key Techniques and Open Issues
arXiv:1801.00121v1 [cs.IT] 30 Dec 2017
S. M. Riazul Islam, Ming Zeng, Octavia A. Dobre and Kyung-Sup Kwak
Abstract—This article presents advances in resource allocation
(RA) for downlink non-orthogonal multiple access (NOMA)
systems, focusing on user pairing (UP) and power allocation (PA)
algorithms. The former pairs the users to obtain the high capacity
gain by exploiting the channel gain difference between the users,
while the later allocates power to users in each cluster to balance
system throughput and user fairness. Additionally, the article
introduces the concept of cluster fairness and proposes the divideand-next largest difference-based UP algorithm to distribute the
capacity gain among the NOMA clusters in a controlled manner.
Furthermore, performance comparison between multiple-input
multiple-output NOMA (MIMO-NOMA) and MIMO-OMA is
conducted when users have pre-defined quality of service. Simulation results are presented, which validate the advantages of
NOMA over OMA. Finally, the article provides avenues for
further research on RA for downlink NOMA.
User M
User 1
User M’
User 1’
…….
NOMA
User m’ signal
detection
User m
NOMA
User m’ signal
detection
NOMA
Subtract signal of
User m’
User m signal
detection
SIC (NOMA)
User m’
Fig. 1: A downlink NOMA system with multiple clusters.
Index Terms—5G, NOMA, Resource allocation, User pairing,
Power allocation.
I. I NTRODUCTION
Non-orthogonal multiple access (NOMA) enables a balanced tradeoff between spectral efficiency and user fairness,
being recognized as a promising multiple access technique
for the fifth generation (5G) networks [1]–[3]. In contrast to
orthogonal multiple access (OMA), NOMA exploits power
domain to simultaneously serve multiple users at different
power levels, where the power allocation (PA) for each user
plays a key role in determining the overall performance of the
system. Downlink NOMA combines superposition coding at
the base station (BS) and successive interference cancellation
(SIC) decoding at the user. To maintain user fairness, NOMA
allocates more power to the users with weaker channel gains.
Because of additional system overhead for channel feedback
coordination and error propagation, it is not feasible to apply
NOMA on all users jointly. Therefore, the idea of user pairing
(UP) has emerged [4], with users in the cell divided into
multiple clusters and NOMA is employed within each cluster
(see Fig. 1). The performance of a NOMA system is highly
dependent on both UP and PA. These are usually referred
to as resource allocation (RA), which represents the central
theme of this article. The RA in NOMA aims to determine
This research was supported by the Ministry of Science, ICT and Future
Planning (MSIP), Korea, as well as the Natural Sciences and Engineering
Research Council of Canada (NSERC).
S. M. R. Islam (e-mail: riaz@sejong.ac.kr) is with the Department of
Computer Science and Engineering, Sejong University, South Korea.
M. Zeng (e-mail: mzeng@mun.ca) and O. A. Dobre (e-mail:
odobre@mun.ca) are with the Faculty of Engineering and Applied Science,
Memorial University, Canada.
K. S. Kwak (e-mail: kskwak@inha.ac.kr) is with the UWB Wireless
Communications Research Center, Inha University, South Korea.
the users to be paired and power to be allocated to each
user within each cluster. The optimal performance of NOMA
RA can be attained by an exhaustive search of all possible
user pairs and transmit power allocations, which is, however,
computationally complex. Moreover, if dynamic UP and PA
are adopted, the decoding order in SIC and PA ratios introduce
additional signaling overheads.
To date, extensive research has been performed on NOMA
RA due to the pivotal role of the RA algorithms in achieving
the benefits offered by NOMA. To this end, this article
appraises the state-of-the-art of RA algorithms for downlink
NOMA research and uncovers various issues to be addressed.
More specifically, it
• provides a categorized survey of the UP and PA algorithms;
• proposes an UP algorithm which ensures the cluster
fairness in terms of sum rate gain of desired degree;
• conducts performance comparison between multipleinput multiple-output NOMA (MIMO-NOMA) and
MIMO-OMA when users have pre-defined quality of
service (QoS) requirements;
• highlights challenges and open issues to be addressed in
downlink NOMA RA.
II. U SER PARING IN NOMA
Based on the desired performance (e.g., sum rate gain), deployment environment, and implementation complexity, there
exist a number of UP algorithms. UP ideally should be
compatible with the PA strategy to provide high throughput
with minimum computational complexity, while maintaining
user fairness. UP algorithms for both single-input single-output
(SISO) and MIMO-NOMA are introduced as follows.
2
A. UP in SISO-NOMA
UE15
UE16
UE15
UE14
Set 4
Merged
Set B
UE14
UE16 and UE5 form
a cluster, UE15 and
UE6 form a cluster,
and so on
UE13
UE13
Channel gains in the ascending order
Random pairing is the easiest UP algorithm, in which the BS
selects the users randomly from the set of candidates to form
the clusters. Although it comes with the lowest complexity,
it exhibits suboptimal sum rate performance because of not
taking the users’ channel gains into account. The mathematical
investigation on UP shows that the performance gain of
NOMA with fixed PA (F-PA) over OMA grows with increasing the difference between the channel gains of the users
of interest [4]. As such, pairing the user of highest channel
gain with the user of lowest channel gain provides the best
performance gain, whereas the user of second highest channel
gain should be paired with the user of second lowest channel
gain to obtain the second best performance gain, and so on.
This algorithm is referred to as the next largest differencebased UP algorithm (NLUPA) and is one of the most common
techniques [4]. In contrast to the above approach, the cognitive
radio (CR)-inspired NOMA pairs the user of highest channel
gain with the user of second highest channel gain, the user
of third highest channel gain with the user of fourth highest
channel gain, and so on, since the nth user is opportunistically
served on the condition that the QoS of the mth user (n > m)
is guaranteed [4]. The idea of such an UP algorithm can be
referred to as next best diversity pairing.
UP can also play a role in canceling adjacent channel interferences by adopting the vertical UP concept when adjacent
sub-channels are sequentially assigned to successive user pairs
[5]. In this scheme, the users in each cluster apply additional
SIC to cancel the interferences from the previous clusters.
However, this algorithm comes with further computational
complexity due to additional SIC operations. The previous UP
algorithms do not consider the realistic fact that there might
not exist enough strong users. In such a situation, there are
leftover weak users after each strong user is paired with its
partner. A possible solution for such a problem is the adoption
of a hybrid approach, where some users are left without pairing
and are accessed via OMA. However, this deprives such users
of the advantages offered by NOMA. Alternatively, NOMA
can be implemented using the concept of virtual UP [6], where
a frequency band can be shared by two weak users of similar
channel gains and a strong user: half of the bandwidth can be
shared by the strong user and a weak user, and half is used
by the strong user with the other weak user.
An UP technique should be of low computational complexity. A good practice is to formulate the pre-requisites for
UP such that some user groups which are not appropriate
for NOMA multiplexing can be excluded from unnecessary
comparison of candidate user pairs [7]. The complexity and
signaling overhead can also be reduced by using a pre-defined
user grouping, where users are divided into different groups
based on their channel conditions. Then, a pair can be formed
only if the users come from different groups. Eventually, the
BS does not need to convey the SIC order information to users
in every sub-frame. The scheduled times, along with channel
conditions, can also be determining factors for excluding users
from the set of candidates. In such a case, the users in the
cell are divided into two groups, with users in the first group
UE16
UE8
UE12
UE11
UE7
Set 3
UE10
UE6
UE9
UE5
UE8
UE7
UE12
Set 2
UE11
UE6
UE10
UE5
Merged
Set A
UE4
UE3
Set 1
UE2
Maximum in Set A
𝒅𝟏𝟔,𝟓 (𝒉) = 𝒉𝟏𝟔 − 𝒉𝟓
UE12 and UE1 form
a cluster, UE11 and
UE2 form a cluster,
and so on
UE9
UE4
UE3
UE1
Minimum in Set A
𝒅𝟏𝟑,𝟖 (𝒉) = 𝒉𝟏𝟑 − 𝒉𝟖
UE2
Minimum in Set B
𝒅𝟗,𝟒 (𝒉) = 𝒉𝟗 − 𝒉𝟒
Maximum in Set B
𝒅𝟏𝟐,𝟏 (𝒉) = 𝒉𝟏𝟐 − 𝒉𝟏
UE1
Divide Stage
NLUPA Stage
Fig. 2: Illustration of the D-NLUPA for a two-user NOMA
(N = 16). UE stands for user.
having higher channel gains than users in the other group.
Then, based on the scheduled times and signal-to-interferenceplus-noise ratio (SINR), users with low scheduling priority can
be excluded from each group by setting an SINR threshold.
Finally, the BS finds the user with an optimum fairness metric
in each group to form a pair.
UP algorithms discussed above are generally applicable to
NOMA systems where user grouping is likely to be in a
partition form. However, there might arise some cases where
merge-and-split-like algorithms cannot be applied [2]. In such
cases, a game theory framework could be useful. [8] considers
users and subchannels as two sets of players to be matched
with each other and applies matching games to achieve the
maximum weighted sum-rate. Note that [7] proposes many-tomany matching algorithms to perform UP for a more general
NOMA system, where multiple users share each sub-channel
and multiple sub-channels can be accessed by each user. Also,
[10] investigates UP for a varying number of users to be
multiplexed on a subcarrier.
B. UP in MIMO-NOMA
The UP in MIMO-NOMA also has impact on the sum
rate gain. Interestingly, in a two-user cluster with zero-forcing
precoding, the user with weak channel conditions has no
influence on the strong user’s rate. Therefore, the strong user
could be first selected to form a cluster. Then, the weak user
should be chosen to optimize the performance metric. The said
optimization can usually be achieved with the minimization of
the intra- and inter-cluster interferences. These interferences
can be reduced if a cluster is formed by two users whose
channel gains are sufficiently distinct but there exists a high
correlation between their channels. The large-gain differences
make certain the effectiveness of NOMA, whereas the high
3
correlation helps eliminate the inter-cluster interference. Apart
from zero-forcing precoding, the quasi-degrading1 nature of
NOMA channels is exploited to formulate a new form of
precoding which is referred as the quasi-degraded channeldriven (QDC) precoding. With QDC precoding, a low complexity sequential UP algorithm is formulated to reduce the
transmit power [9]. However, this is not efficient when the
number of transmit antennas at the BS is greater than the
number of downlink users. To solve this problem, there exist
a couple of variations, namely projection-based UP algorithm
and inversion-based UP algorithm. Both zero-forcing and QDC
precoding-based MIMO-NOMA deal with sum rate maximization problems. The minimum Euclidean distance precodingbased MIMO-NOMA, on the other hand, focuses on the
symbol error rate (SER) reduction. Based on this precoding, a
pair of UP algorithms are proposed to further reduce the SER,
namely, the condition number-based and orthogonality defectbased UP algorithms, in which the basic ideas are that two
users with smaller condition number and smaller orthogonality
defect should be paired, respectively [10].
C. The Concept of Divide-and-NLUPA
Here, we propose a modified NLUPA scheme, referred to as
divide-and-NLUPA (D-NLUPA), to guarantee a minimum sum
rate gain for each cluster, and thereby, introduce the concept
of cluster fairness. NLUPA, as described in Section III. A,
pairs the user of best channel gain with the user of worst
channel gain, the user of second best channel gain with the
user of second lowest channel gain, and so on. As the sum
rate gain achieved by NOMA when compared with OMA is
logarithmically proportional to the ratio of the channel gains of
the strong and weak users in a cluster, the difference between
the channel gains affects the performance gain. In this paper,
the difference between the orders of the paired strong and
weak users is called the range, i.e., |n − m|, where n and m
denote the orders of the strong and weak users, respectively.
The corresponding channel gain difference is referred to as the
distance, i.e., dn,m (h) = |hn |−|hm |, where hn and hm denote
the channel gains of the strong and weak users, respectively.
As the range increases, the distance grows as well, yielding
a higher performance gain. Assume the total number of users
is N .2 Thus, the first cluster enjoys the maximum gain, while
the gain for the N2 th cluster may not be significant. Therefore,
it is possible to obtain some clusters which actually do not
enjoy any sum-rate gain (gain is very closed to zero).
To guarantee a minimum performance gain, we introduce
the ”divide” step in D-NLUPA, by setting a minimum range,
and thus, increasing the corresponding minimum value of the
distance. Because of this ”divide” step, the scenario of nearzero gain clusters can be avoided and each cluster enjoys a
1 For a particular decoding and encoding order (n, m) of NOMA and dirty
paper coding (DPC), respectively, two channel coefficients hm and hn are
quasi-degraded with respect to the targeted SNR levels of the corresponding
users if and only if the minimum transmission powers of NOMA and DPC are
comparable [9]. The idea of the said comparison with the DPC comes from
the fact that the use of DPC can achieve the capacity region of the downlink
broadcast channel with perfect channel state information (CSI) information
available at the BS.
2 Without loss of generality, we assume that N is an even number. If N is
odd, different UP strategies can be applied, as presented in Section III. A.
minimum gain as designed. The value of this minimum gain
depends on how we shuffle the users in the ”divide” stage.
To illustrate the idea, an example is shown in Fig. 2, with
N = 16. The ordered users are first divided into Nz = 4 sets,
where z = 4 is the number of users in each set. Then, sets 1
and 3 are merged into set A, sets 2 and 4 are merged into set B,
and finally NLUPA is respectively applied to sets A and B to
form the clusters; this yields the minimum value of the range
and the corresponding minimum value of the distance in each
set. Note that although the minimum values of the range in
sets A and B are the same, the corresponding minimum values
of the distance are not necessarily the same due to different
channel gains of the users in two sets. The minimum distance,
dn,m (h), n, m ∈ {1, · · · , 16} is considered over both sets.
To compare the sum rate gain of NLUPA and D-NLUPA,
Fig. 3(a) presents results from simulation. It can be noticed that
this gain has been controlled by changing the minimum distance in D-NLUPA, while there is no such control in NLUPA.
The ”divide” stage in D-NLUPA arranges the user pairing
in such a way that there occurs a certain minimum distance
(here, around 15 dB) between the two users forming a cluster.
Because of this minimum distance, D-NLUPA guarantees a
minimum sum rate gain for each cluster. Let us now merge
and sort the sum rate gains of the clusters formed from both
A and B sets. Then, the sum rate gain vs. cluster index
performance is presented in Fig. 3(b). Results show that about
50% D-NLUPA-based clusters exhibit higher gains compared
with NLUPA, while the remaining DNLUPA-based clusters
obtain lower gains. The advantage of the cluster fairness is
thus achieved by redistributing the gains to the clusters in a
controlled manner, while the aggregated throughput remains
the same for both NLUPA and D-NLUPA. Also, it is noticeable that random pairing obtains the lowest sum rate gain.
III. P OWER A LLOCATION IN NOMA
Compared with OMA, the role of PA in NOMA is further
enhanced, since users are multiplexed in the power domain.
Interference management, rate distribution, and even user admission are directly impacted by PA. Generally, PA in NOMA
is determined by the users’ channel conditions, availability
of CSI, QoS requirements, total power constraint and system
objective. An inappropriate PA not only leads to an unfair
rate distribution among users, but also causes system outage
as SIC may fail. There are different PA performance metrics,
e.g., the number of admitted users, sum rate, user fairness,
outage probability and total power consumption. Thus, PA in
NOMA should aim at achieving either more admitted users
and higher sum rate, or a balanced fairness under minimum
power consumption. A variety of PA strategies have been
proposed in the literature, targeting different aspects of PA in
NOMA, and a classification is provided in Fig. 4. We introduce
PA in the following two subsections: one focuses on singlecarrier (SC) SISO systems,3 while the other deals with multicarrier (MC) and MIMO systems, respectively.
3 In the following sections, we will simply use SISO to refer to SC SISO.
Further, MC-NOMA and MIMO-NOMA refer to MC SISO-NOMA and SC
MIMO-NOMA, respectively.
4
2.5
2.5
NLUPA
D-NLUPA (Set A)
D-NLUPA (Set B)
2
Sum Rate Gain (bps)
Sum Rate Gain (bps)
2
NLUPA
Random Pairing
D-NLUPA
1.5
1
0.5
1.5
1
0.5
0
0
8
10
12
14
16
18
20
22
Distance (dB)
1
2
3
4
5
6
7
8
Cluster Index
(a)
(b)
Fig. 3: NLUPA and D-NLUPA with F-PA: Sum rate gain versus a) distance and b) cluster index. Transmit power is 1 W; PA
ratio is 0.6:0.4; cell radius is 100 m; N = 16; Rayleigh fading channel with log-normal shadowing of path loss exponent of
2.5 and standard deviation of 3 is considered.
• SISO
• MIMO
• SC
• MC
Number of
antennas
Number of
carriers
Availability
of CSI
Objective
function
• Perfect CSI
• Statistical CSI
• Minimize power
consumption
• Maximize sum
rate or fairness
Fig. 4: Classification of PA strategies.
A. PA in SISO-NOMA
Early works in PA mainly target SISO systems. Unlike
OMA, whose optimal PA to maximize the sum rate follows
the water-filling technique, the corresponding PA for NOMA
simply allocates all power to the user with the best channel
[11], [12]. Obviously, this leads to extreme unfairness among
users and also diminishes the number of admitted users. To
balance system throughput and user fairness, NOMA allocates
more power to the weak user. This way, the strong user handles
the interference from the weak user using SIC, while the
interference to its counterpart remains comparatively small.
The simplest PA algorithm is F-PA, which allocates power
to each user using a fixed ratio based on its position in the
channel ordering. Since the users’ specific channel gains are
not considered during PA, F-PA cannot satisfy users’ various
QoS requirements. To address this issue, fractional transmit
power control (FTPC) allocates power to each user inversely
proportional to its channel gain powered with a decaying
factor. Nevertheless, assigning the same decaying factor to all
users is still suboptimal, and selecting the appropriate decaying
factor to balance system throughput and user fairness remains
an open issue.
PA is directly impacted by the availability of CSI. Under
perfect CSI, the multi-user weighted sum rate maximization
problem is proved to be convex, and thus optimal PA can
be obtained using convex optimization [11]. The max-min
fairness problem is shown to be quasi-convex, and optimal PA
can be obtained via the bisection method [13]. The energyefficient PA problem is formulated as a difference of two
convex functions, and PA can be obtained by iteratively
solving the convex sub-problems [14]. Under statistical CSI,
although the min-max outage probability under a given SIC
order is shown to be non-convex, optimal PA is derived in [13];
on this basis, [15] further obtains the corresponding optimal
SIC decoding order.
Note that the works above do not ensure a higher throughput
of NOMA over OMA. To achieve this for the weak user, CRinspired PA can be applied, where NOMA is considered as a
special case of CR networks and the weak user is viewed as a
primary user [4]. However, this may sacrifice the performance
of the strong user since it is served only after the weak
user’s QoS is met. To overcome this issue, dynamic PA is
proposed in [16], which allocates power to users such that the
individual user rate achieved by NOMA is strictly larger than
that provided by OMA.
B. PA in MC-NOMA and MIMO-NOMA
For multi-user systems, NOMA is usually integrated with
MC (MC-NOMA) to reduce complexity. In MC-NOMA, a
5
user can occupy multiple sub-carriers, and vice versa. MCNOMA is quite suitable for 5G as it is difficult to find
continuous wide bandwidth in 5G. Compared with orthogonal
frequency-division multiple access (OFDMA), MC-NOMA
can further increase spectral efficiency and the number of
simultaneously supported users. Its performance depends on
both PA and sub-carrier assignment (SA). Some works maximize the sum rate under total power constraint to increase
spectral efficiency and throughput, whereas others minimize
the total power consumption under QoS requirements to improve energy efficiency and reduce inter-cell interference. For
the weighted sum rate maximization problem, its NP-hardness
is proved in [8], [11]. An algorithmic framework combining
the Lagrangian duality and dynamic programming is proposed
in [11] to deliver near-optimal solutions. The original problem
is decomposed into two subproblems, i.e., SA and PA in [8].
SA is solved using a matching algorithm, while PA is solved
via geometric programming. For the energy-efficient problem,
[14] adopts a similar approach as [8].
Note that perfect CSI is assumed in the above schemes,
which might be impractical for MC-NOMA systems overloaded with exceedingly number of users. Consequently, the
RA under statistical CSI should be investigated. Without
perfect CSI, the BS cannot decide the SIC decoding order
directly, and thus, an explicit SIC decoding order should be
derived first. Following this, PA and SA can be performed
as for the case of perfect CSI. To further enhance the spectral
efficiency of the MC-NOMA systems, full duplex (FD) BS can
be introduced, achieving a substantial throughput improvement
compared with FD MC-OMA and half duplex (HD) MCNOMA systems.
The study of applying MIMO technologies to NOMA is
of significance, since MIMO provides additional degrees of
freedom for further performance enhancement. However, the
introduction of MIMO brings two major challenges: 1) it is
still unclear whether MIMO-NOMA can obtain the system
capacity. [9] verifies that MIMO-NOMA achieves that when
the users’ channels are quasi-degraded. Nonetheless, the extension from quasi-degraded channels to general ones remains
an open issue; 2) there exists no natural order for the users’
channels in MIMO-NOMA, as they are in form of matrices or
vectors. To address the issue of user ordering, an effective way
is to pair users into clusters, and assign the same beamforming
vectors to users in the same cluster. This decomposes the
MIMO-NOMA channel into multiple separate SISO-NOMA
subchannels [17]–[19]. A general MIMO-NOMA framework
is proposed in [17], in which the inter-cluster interference
is eliminated due to signal alignment-based beamforming.
This further simplifies PA in MIMO-NOMA since now PA
in each cluster is independent and can be treated same as in
SISO. Therefore, most PA strategies for SISO, e.g., F-PA and
CR-inspired PA can be directly applied. However, the above
inter-cluster interference-free MIMO-NOMA framework can
only be used for the case of two users per cluster. For the
more general case of multiple users per cluster, inter-cluster
interference generally cannot be completely removed. In this
case, although the problem of channel ordering is solved
by cluster-based beamforming, PA across clusters is inter-
dependent, which makes the problem still non-trivial. A new
millimeter wave transmission scheme that integrates NOMA
with beamspace MIMO is proposed in [19], which shows
that MIMO-NOMA can achieve higher spectrum and energy
efficiency compared with existing beamspace MIMO even
when there exists inter-cluster interference.
IV. P ERFORMANCE C OMPARISON BETWEEN
MIMO-NOMA AND MIMO-OMA
In this section, based on the system model in [17], we conduct a performance comparison between MIMO-NOMA and
MIMO-OMA when each user has a minimum rate requirement
(QoS). Due to the QoS constraint, it is possible that not all
users can be admitted even if the total transmit power is used.
In this case, user admission is conducted to accommodate
as many users as possible. On the other hand, if all users
can be admitted, the objective is maximizing the sum rate
of the system. For both NOMA and OMA, we consider two
scenarios: 1) equal power for each cluster; and 2) cross-cluster
PA.
When equal power is allocated to each cluster, the PA for
NOMA is quite simple. For each cluster, when the QoS for
both users can be satisfied, PA is allocated such that the weak
user satisfies its QoS, while the rest goes to the strong user to
maximize the sum rate [11]. Otherwise, the strong user gets
all the power. For the cross-cluster PA, first it is determined
if all users can be admitted, by comparing the total power
constraint with the total power required to satisfy the QoS of
all users. If the required power exceeds the power constraint,
the users are admitted one by one following the ascending
order of required power for satisfying their QoS until all the
power is consumed. Otherwise, first we assign the power to
each user to ensure that its QoS is satisfied. Following this,
the remaining power can be used to further increase the sum
rate of the system. Since the power within each cluster should
be allocated such that the weak user’s QoS is satisfied, while
the rest goes to the strong user, the relation between the rate
increment and the extra power required only depends on the
channel gain of the strongest user. Hence, the remaining power
can be allocated across clusters optimally by adopting the
water-filling technique only considering the channel gain of
the strongest user in each cluster.
For OMA, when equal power is allocated to each cluster,
it is first determined whether both users can be admitted or
not. If so, the power is allocated to the two users such that
each user’s QoS is satisfied, and the remaining power is then
calculated. Afterwards, the water-filling technique is applied
to allocate the remaining power between them. Otherwise,
the strong user gets all the power. When cross-cluster PA
is considered, it is first determined whether all users can be
admitted or not. If so, the power is allocated to the users such
that each user’s QoS is satisfied, and the remaining power is
calculated. Then, the water-filling technique is applied among
all the users to allocate the remaining power. Otherwise, the
users are arranged on descending order of their channel gains,
and admitted one by one until all power is consumed.
Simulation results for the outage probability and effective
sum rate are presented in Figs. 5 and 6, respectively, in
6
10-1
100
C-NOMA
E-NOMA
C-OMA
E-OMA
10
Outage Probability
Outage Probability
10-2
C-NOMA
E-NOMA
C-OMA
E-OMA
-3
10-4
10-1
10-2
10-5
10-6
10
15
20
25
30
35
40
10-3
10
15
Transmit Power [dBm]
20
25
30
35
40
Transmit Power [dBm]
(a) Strong user
(b) Weak user
Fig. 5: Outage performance.
V. P RACTICAL C HALLENGES AND R ESEARCH D IRECTIONS
While NOMA has attracted the attention of researchers as
a potential radio access technique for 5G, the design of RA
algorithms for NOMA remains still in its infancy due to various practical challenges. These include scalability, presence
of inter-cell interference for multi-cell networks, integration
of carrier aggregation, RA under limited channel feedback,
QoS-guaranteed RA, and multi-hop communications, among
others. The RA algorithms for delay-sensitive networks are
inherently different from those in delay-tolerant networks as
NOMA requires timely channel feedback, and further research
is needed to explicitly define the RA requirements for both
networks. Apart from the aforementioned challenges, one can
note the following existing research gaps.
C-NOMA
E-NOMA
C-OMA
E-OMA
Effective Sum Rate [bps]
which ”C-” and ”E-” denote cross-cluster and equal power
PA, respectively. Clearly, for both strong and weak users, CNOMA has the lowest outage probability. Specifically, for the
strong user, the outage probability for E-NOMA and E-OMA
is the same, being worse than C-OMA under high transmit
power. For the weak user, the outage probability for NOMA
is lower than that for OMA. Moreover, cross-cluster PA has
lower outage probability than equal power PA for both NOMA
and OMA. For the effective sum rate, it can be seen that
NOMA outperforms OMA for both the cross-cluster and equal
power PA scenarios. Under low transmit power, E-NOMA
achieves higher sum rate than C-NOMA due to the fact that the
former allocates all power to the strong user when only the
strong user can be admitted, while the latter distributes the
remaining power across clusters. The same behavior can be
observed for OMA. To conclude, MIMO-NOMA outperforms
MIMO-OMA in terms of outage performance and effective
sum rate. Moreover, optimizing the power across clusters
yields significant decrease in the outage probability, as well as
increase in the effective sum rate under high transmit power.
101
10
15
20
25
30
35
40
Transmit Power [dBm]
Fig. 6: Sum rate versus transmit power.
A. Joint Optimization of UP and PA
Since UP and PA are closely related to each user, a joint
optimization is desirable. However, it is challenging to derive
an optimal solution even under SISO, not to mention the case
of MIMO. We propose D-NLUPA to ensure a fair gain among
clusters for SISO. However, under the MIMO-NOMA system
model in [14], D-NLUPA, NLUPA, and random paring exhibit
a similar performance. This is because under MIMO, different
pairing leads to different precoding and detection matrices,
which randomizes the gain of the path loss-based UP. More
efforts are required to design an effective joint UP-PA strategy,
especially for MIMO-NOMA.
B. RA for FD MC-MIMO-NOMA
As research progresses, more and more advanced technologies are integrated with NOMA to fully explore its potential,
e.g., FD with MC-NOMA, MC with MIMO-NOMA and
7
FD with MIMO-NOMA. One can even consider these three
technologies in a single framework, i.e., FD MC-MIMONOMA. This, however, yields a complex system and makes
the RA problem non-trivial. Particularly, the combinatorial
optimization problem of MC-NOMA is already shown to be
NP-hard. With MIMO and FD, the problem becomes much
more complicated, and it is essential to propose novel lowcomplexity RA schemes to ensure the superiority of NOMA
over OMA.
C. RA for NOMA with Soft Frequency Reuse
To solve the inter-cell interference problem, soft frequency
reuse (SFR) is an indispensable element of LTE systems.
Under SFR, the primary band is assigned to the cell edge users,
and the secondary band is allocated to the cell center users.
On the other hand, if NOMA is integrated with the SFR-based
LTE systems, the signals of strong users (cell center users)
and weak users (cell edge users) may be overlapped on the
primary band. However, such an arrangement is unfavorable
to fairness as it increases the data rate of strong users and
decreases the data rate of the weak users. Additionally, the
use of the primary band at the edge of the cell would generate
inter-cell interference. In this regard, the design of NOMA RA
for SFR-based cellular systems is an open problem.
D. Low-Complexity RA
The existing fairness models exploit multiple parameters to
adjust the fairness level, which introduces high complexity
in determining the corresponding RA in NOMA. However,
α-fairness uses a single scalar, denoted by α, to achieve different user fairness levels and well-known efficiency-fairness
tradeoffs [20]. The RA can be investigated for sum throughput
optimization of the NOMA system with fairness constraints to
obtain low-complexity algorithms.
E. Security-Aware RA
NOMA-based communication comes with security concerns, as the user with strong channel condition needs to
decode the signal of the user with weak channel condition.
Thus, when the weak user becomes malicious or is under
attack, the signal decoding operations of both strong and weak
users are no longer reliable. For example, the attack might be
in the form of an alteration of the channel quality indicator
during feedback. Therefore, the design of the RA schemes
becomes more critical in the presence of security concerns. To
overcome this hurdle, introducing appropriate physical layer
security measures is necessary, which is an interesting open
problem for the research community.
VI. C ONCLUDING R EMARKS
This article provides an overview of the RA algorithms for
downlink NOMA in a categorized fashion. It is clear that the
RA algorithms play a pivotal role in achieving the maximum
benefits of NOMA. Additionally, it is shown that the fairness
among the NOMA clusters can be attained by controlling the
sum rate gain through the D-NLUPA algorithm. Moreover,
for a general MIMO-NOMA system with pre-defined QoS
requirement for each user, simulations are conducted, which
show that MIMO-NOMA outperforms MIMO-OMA in terms
of outage probability and effective sum rate when optimal
PA is applied for both. Finally, the article concludes with a
discussion on open issues, including joint optimization of UP
and PA, RA for FD MC MIMO-NOMA, low-complexity RA,
and security-aware RA, which provides the basis for further
research directions on the RA for NOMA systems.
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8
[19] B. Wang, L. Dai, Z. Wang, N. Ge, and S. Zhou, “Spectrum and energy efficient beamspace MIMO-NOMA for millimeter-wave communications
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| 7 |
Open Transactions on Shared Memory
Marino Miculan
Marco Peressotti
Andrea Toneguzzo
marino.miculan@uniud.it
marco.peressotti@uniud.it
toneguzzo.andrea@spes.uniud.it
arXiv:1503.09097v1 [cs.PL] 31 Mar 2015
Laboratory of Models and Applications of Distributed Systems
Department of Mathematics and Computer Science
University of Udine, Italy
Abstract
Transactional memory has arisen as a good way for solving many of the issues of lockbased programming. However, most implementations admit isolated transactions only,
which are not adequate when we have to coordinate communicating processes. To this
end, in this paper we present OCTM , an Haskell-like language with open transactions over
shared transactional memory: processes can join transactions at runtime just by accessing
to shared variables. Thus a transaction can co-operate with the environment through shared
variables, but if it is rolled-back, also all its effects on the environment are retracted. For
proving the expressive power of OCTM we give an implementation of TCCS m , a CCS-like
calculus with open transactions.
1
Introduction
Coordination of concurrent programs is notoriously difficult. Traditional fine-grained lock-based
mechanisms are deadlock-prone, inefficient, not composable and not scalable. For these reasons,
Software Transactional Memory (STM) has been proposed as a more effective abstraction for
concurrent programming [1, 9, 18]. The idea is to mark blocks of code as “atomic”; at runtime,
these blocks are executed so that the well-known ACID properties are guaranteed. Transactions
ensure deadlock freedom, no priority inversion, automatic roll-back on exceptions or timeouts,
and greater parallelizability. Among other implementations, we mention STM Haskell [7], which
allows atomic blocks to be composed into larger ones. STM Haskell adopts an optimistic evaluation strategy: the blocks are allowed to run concurrently, and eventually if an interference is
detected a transaction is aborted and its effects on the memory are rolled back.
However, standard ACID transactions are still inadequate when we have to deal with communicating processes, i.e., which can exchange information during the transactions. This is very
common in concurrent distributed programming, like in service-oriented architectures, where
processes dynamically combine to form a transaction, and all have to either commit or abort
together. In this scenario the participants cannot be enclosed in one transaction beforehand,
because transactions are formed at runtime. To circumvent this issue, various forms of open
transactions have been proposed, where the Isolation requirement is relaxed [2–4,11,13]. In particular, TransCCS and TCCS m are two CCS-like calculi recently introduced to model communicating transactions [4, 5, 11]. These calculi offer methodologies for proving important properties,
such as fair-testing for proving liveness and bisimulations for proving contextual equivalences.
Now, if we try to implement cross-transaction communications a la TCCS m in STM Haskell
or similar languages, it turns out that isolated transactions are not expressive enough. As an
example, let us consider two TCCS m transactions h c̄.P ◮ 0ii | h c.Q ◮ 0ii synchronizing on a
1
channel c. Following the standard practice, we could implement this synchronization as two
parallel processes using a pair of semaphores c1,c2 (which are easily realized in STM Haskell):
h c̄.P ◮ 0ii = atomic {
up c1
down c2
P
}
h c.Q ◮ 0ii = atomic {
down c1
up c2
Q
}
-- 1.1
-- 1.2
-- 2.1
-- 2.2
This implementation is going to deadlock: the only possible execution order is 1.1-2.1-2.2-1.2,
which is possible outside transactions but it is forbidden for ACID transactions1. The problem is
that ordinary STM transactions are kept isolated, while in TCCS m they can merge at runtime.
In order to address this issue, in this paper we introduce software transactional memory
with open transactions: processes can join transactions and transactions can merge at runtime,
when they access to shared variables. To this end, we present OCTM , a higher-order language
extending the concurrency model of STM Haskell with composable open (multi-thread) transactions interacting via shared memory. The key step is to separate the isolation aspect from
atomicity: in OCTM the atomic construct ensures “all-or-nothing” execution, but not isolation;
when needed, isolated execution can be guaranteed by a new constructor isolated. An atomic
block is a participant (possibly the only one) of a transaction. Notice that transaction merging
is implicitly triggered by accessing to shared memory, without any explicit operation or a priori
coordination. For instance, in OCTM the two transactions of the example above would merge
becoming two participants of the same transaction, hence the two threads can synchronize and
proceed. In order to prove formally the expressivity of open memory transactions, we define an
implementation of TCCS m in OCTM , which is proved to correctly preserve behaviours by means
of a suitable notion of simulation. We have based our work on STM Haskell as a paradigmatic
example, but this approach is general and can be applied to other STM implementations.
Lesani and Palsberg [13] have proposed transactions communicating through transactional
message-based channels called transactional events. These mechanisms are closer to models like
TransCCS and TCCS m , but on the other hand they induce a strict coupling between processes,
which sometimes is neither advisable nor easy to implement (e.g., when we do not know all
transaction’s participants beforehand). In fact, most STM implementations (including STM
Haskell) adopt the shared memory model of multi-thread programming; this model is also more
amenable to implementation on modern multi-core hardware architectures with transactional
memory [8]. For these reasons, in OCTM we have preferred to stick to loosely coupled interactions
based on shared memory only.
The rest of the paper is structured as follows. In Section 2 we describe the syntax and
semantics of OCTM . Some examples are in Section 3. In Section 4 we assess the expressiveness
of OCTM by providing an implementation of TCCS m , our reference model for open transactions.
Conclusions and directions for future work are in Section 5. Longer proofs are in the Appendix.
2
OCTM : Open Concurrent Transactional Memory
In this section we introduce the syntax and semantics of OCTM , a higher-order functional language with threads and open transaction on shared memory. The syntax is Haskell-like (in the
wake of existing works on software transactional memories such as [7]) and the semantics is a
β
small-step operational semantics given by two relations: −
→ models transaction auxiliary opera1 This
possibility was pointed out also in [7]: “two threads can easily deadlock if each awaits some communication from the other”.
2
Value
V ::=
Term
M, N ::=
r | λx.M | return M | M >>= N |
newVar M | readVar r | writeVar r M |
fork M | atomic M N | isolated M | abort M | retry
x | V | MN | ...
Figure 1: Syntax of OCTM values and terms.
tions (e.g. creation) while −
→ models actual term evaluations. Executions proceeds by repeatedly
choosing a thread and executing a single (optionally transactional) operation; transitions from
different threads may be arbitrarily interleaved as long as atomicity and isolation are not violated
where imposed by the program.
2.1
Syntax
The syntax can be found in Figure 1 where the meta-variables r and x range over a given
countable set of locations Loc and variables Var respectively. Terms and values are inspired to
Haskell and are entirely conventional2; they include abstractions, application, monadic operators
(return and >>= ), memory operators (newVar , readVar , writeVar ), forks, transactional
execution modalities (atomic and isolated) and transaction operators (abort and retry).
Effectfull expressions such as fork or isolated are glued together by the (overloaded)
monadic bind >>= e.g.:
newVar 0 >>= λx.(fork (writeVar x 42) >>= λy.readVar x)
whereas values are “passed on” by the monadic unit return.
Akin to Haskell, we will use underscores in place of unused variables (e.g. λ .0) and M >> N
as a shorthand for M >>= λ .N , and the convenient do-notation:
do{x ← M ;N } ≡ M >>= (λx.do{N })
do{M ;N } ≡ M >>= (λ .do{N })
do{M } ≡ M
possibly trading semicolons and brackets for the conventional Haskell layout. For instance, the
above example is rendered as
do
x ← newVar 0
fork (writeVar x 42)
readVar x
2.2
Operational Semantics
We present the operational semantics of OCTM in terms of an abstract machine whose states
are triples hP ; Θ, ∆i formed by
• thread family (process) P ;
• heap memory Θ : Loc ⇀ Term;
• distributed working memory ∆ : Loc ⇀ Term × TrName
where Term denotes the set of OCTM terms (cf. Figure 1) and TrName denotes the set of names
used by the machine to identify active transactions.
We shall denote the set of all possible states as State.
2 We
treat the application of monadic combinators (e.g. return ) as values in the line of similar works [7].
3
Threads Threads are the smaller unit of execution the machine scheduler operates on; they
execute OCTM terms and do not have any private transactional memory.
Threads are given unique identifiers (ranged over by t or variations thereof) and, whenever
they take part to some transaction, the transaction identifier (ranged over k, j or variations
thereof). Threads of the former case are represented by ([M ])t where M is the term being
evaluated and the subscript t is the thread identifier. Threads of the latter case have two forms:
([M ⊲ M ′ ; N ])t,k , called and ([M ⊲ M ′ ])t,k where:
• M is the term being evaluated inside the transaction k;
• M ′ is the term being evaluated as compensation in case k is aborted;
• N is the term being evaluated as continuation after k commits or aborts.
Threads with a continuation are called primary participants (to transaction k), while threads
without continuation are the secondary participants. The former group includes all and only the
threads that started a transaction (i.e. those evaluated in an atomic), while the latter group
encompasses threads forked inside a transaction and threads forced to join a transaction (from
outside a transactional context) because of memory interactions. While threads of both groups
can force a transaction to abort or restart, only primary participants can vote for its commit
and hence pass the transaction result to the continuation.
We shall present thread families using the evocative CCS-like parallel operator k (cf. Figure 2)
which is commutative and associative. Notice that this operator is well-defined only on operands
whose thread identifiers are distinct. The notation is extended to thread families with 0 denoting
the empty family.
Memory The memory is divided in the heap Θ and in a distributed working memory ∆. As
for traditional closed (acid) transactions (e.g. [7]), operations inside a transaction are evaluated
against ∆ and effects are propagated to Θ only on commits. When a thread inside a transaction
k accesses a location outside ∆ the location is claimed for k and remains claimed for the rest
of k execution. Threads inside a transaction can interact only with locations claimed by their
transaction. To this end, threads outside any transaction can join an existing one and different
active transactions can be merged to share their claimed locations.
We shall denote the pair hΘ, ∆i by Σ and reference to each projected component by a subscript e.g. ΣΘ for the heap. When describing updates to the state Σ, we adopt the convention
that Σ′ has to be intended as equal to Σ except if stated otherwise, i.e. by statements like
Σ′Θ = ΣΘ [r 7→ M ].
Formally, updates to location content are defined on Θ and ∆ as follows:
Θ[r 7→ M ](s) ,
(
M
if r = s
Θ(s) otherwise
∆[r 7→ (M, k)](s) ,
(
(M, k) if r = s
∆(s)
otherwise
for any r, s ∈ Loc, M ∈ Term and k ∈ TrName. Likewise, updates on transaction names are
defined on Σ and ∆ as follows:
(
∆(r)
if ∆(r) = (M, l), l 6= k
Σ[k 7→ j] , (Θ, ∆[k 7→ j])
(∆[k 7→ j])(r) ,
(M, j) if ∆(r) = (M, k)
for any r ∈ Loc, M ∈ Term and k, j ∈ TrName. Note that j may occur in ∆ resulting in the
fusion of the transactions denoted by k and j respectively. Finally, ∅ denotes the empty memory
(i.e. the completely undefined partial function).
4
Thread
Thread family
Expressions
Processes
Transactions
Tt ::=
P ::=
E ::=
Pt ::=
Tt,k ::=
([M ])t | ([M ⊲ M ′ ; N ])t,k | ([M ⊲ M ′ ])t,k
Tt1 k · · · k Ttn ∀i, j ti 6= tj
[−] | E >>= M
([E])t
([E ⊲ M ; N ])t,k | ([E ⊲ M ])t,k
Figure 2: Threads and evaluation contexts.
M 6≡ V V[M ] = V
M →V
retry >>= M → retry
(Eval)
(BindRetry)
return M >>= N → N M
(BindReturn)
abort N >>= M → abort N
(BindAbort)
Figure 3: OCTM semantics: rules for term evaluation.
Behaviour Evaluation contexts are shown in Figure 2 and the transition relations are presented
in Figures 3, 4, 5. The first (cf. Figures 3) is defined on terms only and models pure computations.
In particular, rule (Eval) allows a term M that is not a value to be evaluated by an auxiliary
(partial) function, V[M ] yielding the value V of M whereas the other three rules define the
semantic of the monadic bind. The transition relation modelling pure computations can be
thought as accessory to the remaining two for these model transitions between the states of the
machine under definition.
Derivation rules in Figure 4 characterize the execution of pure (effect-free) terms, forks and
memory operations both inside, and outside of some transaction; Derivation rules in Figure 5
characterize auxiliary operations for transaction management (e.g. creation) and their coordination (e.g distributed commits). Note that there are no derivation rules for retry. In fact, the
meaning of retry is to inform the machine that choices made by the scheduler led to a state
from which the program cannot proceed. From an implementation perspective this translates
in the transaction being re-executed from the beginning (or a suitable check-point) following a
different scheduling of its operations.
We shall describe now a representative subset of the derivation rules from Figures 4 and 5.
Reading a location falls into four cases depending on the location being claimed (i.e. occurring
in ∆) and the reader being part of a transaction. The rule (ReadP) characterize the reading of
an unclaimed location from outside any transaction; the read is performed as expected leaving
it unclaimed. Rule (ReadT) describes the reading of an unclaimed location r by a thread
belonging to some transaction k; the side effect of the reading is r being claimed for k. Rules
(ReadMerge) and (ReadJoin) cover the cases of readings against claimed locations. In the
first scenario, the reading thread belongs to a transaction resulting in the two being merged,
which is expressed by renaming its transaction via a substitution. In the remaining scenario,
the reading thread does not belong to any transaction and hence joins the transaction k which
claimed the location. The newly created participant does not have any continuation since the
whole term is set to be executed inside k; any other choice for splitting the term singling out
a compensation would impose an artificial synchronization with the transaction commit. For a
counter example, consider executing only the read operation inside the transaction and delaying
everything after the commit; then concurrency will be clearly reduced. Because of the same
reasoning, the whole term M is taken as the compensation of the participant.
Transactions are created by rule (Atomic); threads participating in a transaction are nondeterministically interleaved with other threads. The stronger requirement of isolation is offered
5
M →N
M →N
(TermP)
(TermT)
hPt [M ] k P ; Σi −
→ hPt [N ] k P ; Σi
hTt,k [M ] k P ; Σi −
→ hTt,k [N ] k P ; Σi
t′ ∈
/ threads(P ) t 6= t′
(ForkP)
hPt [fork M ] k P ; Σi −
→ hPt [return t′ ] k ([M ])t′ k P ; Σi
t′ ∈
/ threads(P ) t 6= t′
(ForkT)
hTt,k [fork M ] k P ; Σi −
→ hTt,k [return t′ ] k ([M ⊲ return])t′ ,k k P ; Σi
threads(Tt1 k · · · k Ttn ) , {t1 , . . . , tn }
r∈
/ dom(ΣΘ ) ∪ dom(Σ∆ ) Σ′Θ = ΣΘ [r 7→ M ]
(NewP)
hPt [newVar M ] k P ; Σi −
→ hPt [return r] k P ; Σ′ i
r∈
/ dom(ΣΘ ) ∪ dom(Σ∆ ) Σ′∆ = Σ∆ [r 7→ (M, k)]
(NewT)
hTt,k [newVar M ] k P ; Σi −
→ hTt,k [return r] k P ; Σ′ i
r∈
/ dom(Σ∆ ) ΣΘ (r) = M
(ReadP)
hPt [readVar r] k P ; Σi −
→ hPt [return M ] k P ; Σi
r∈
/ dom(Σ∆ ) ΣΘ (r) = M Σ′∆ = Σ∆ [r 7→ (M, k)]
(ReadT)
hTt,k [readVar r] k P ; Σi −
→ hTt,k [return M ] k P ; Σ′ i
M = E[readVar r] Σ∆ (r) = (M ′ , k)
(ReadJoin)
h([M ])t k P ; Σi −
→ h([E[return M ′ ] ⊲ λ. M ])t,k k P ; Σi
Σ∆ (r) = (M, j) Σ′ = Σ[k 7→ j]
(ReadMerge)
hTt,k [readVar r] k P ; Σi −
→ hTt,j [return M ] k P [k 7→ j]; Σ′ i
r∈
/ dom(Σ∆ ) ΣΘ (r) = N Σ′Θ = ΣΘ [r 7→ M ]
(WriteP)
hPt [writeVar r M ] k P ; Σi −
→ hPt [return ()] k P ; Σ′ i
r∈
/ dom(Σ∆ ) ΣΘ (r) = N Σ′∆ = Σ∆ [r 7→ (M, k)]
(WriteT)
hTt,k [writeVar r M ] k P ; Σi −
→ hTt,k [return ()] k P ; Σ′ i
M = E[writeVar r M ′ ] Σ∆ (r) = (M ′′ , k) Σ′∆ = Σ∆ [r 7→ (M ′ , k)]
(WriteJoin)
h([M ])t k P ; Σi −
→ h([E[return ()] ⊲ λ .M ])t,k k P ; Σ′ i
Σ∆ (r) = (N, j) Σ′ = Σ[k 7→ j] Σ′∆ = Σ∆ [k 7→ (M, j)]
(WriteMerge)
hTt,k [writeVar r M ] k P ; Σi −
→ hTt,j [return ()] k P [k 7→ j]; Σ′ i
Figure 4: OCTM semantics: rules for −
→.
by rules (IsolatedP) and (IsolatedT), whose premises forbid thread or transaction creation.
Committing or aborting a transaction require a synchronization of its participants. In
particular, an abort can be read as a participant vetoing the outcome of the transaction;
this corresponds to (RaiseAbort1) and (RaiseAbort2). The information is then propagated by (AbBroadcast) and (TrIgnore) to any other participant to the transaction being
aborted; these participants abort performing a transition described by either (SigAbort1) or
(SigAbort2).
3
Examples
In this section we provide some short examples to illustrate the use of OCTM and how standard
STM behaviour can be recovered in OCTM thanks to the isolated construct. In Section 4.2
we will give an extended example by providing a translation of TCCS m into OCTM .
6
k∈
/ transactions(P )
(Atomic)
new
k
h([atomic M N >>= N ′ ])t k P ; Σi −−−→
h([M ⊲ N ; N ′ ])t,k k P ; Σi
∗
h([M ])t ; Σi → h([return N ])t ; Σ′ i
(IsolatedP)
hPt [isolated M ]; Σi → hPt [return N ]; Σ′ i
op ∈ {abort, return} h([M ⊲ return])t,k ; Σi →∗ h([op N ⊲ return])t,k ; Σ′ i
hTt,k [isolated M ]; Σi → hTt,k [op N ]; Σ′ i
Σ′∆ = clean(k, Σ∆ )
(IsolatedT)
(RaiseAbort1)
ab M
k
h([abort M ⊲ N ; N ′ ])t,k ; Σi −−−
−→ h([N (M ) >>= N ′ ])t ; Σ′ i
Σ′∆ = clean(k, Σ∆ )
(RaiseAbort2)
ab M
k
h([abort M ⊲ N ])t,k ; Σi −−−
−→ h([N (M )])t ; Σ′ i
Σ′∆ = clean(k, Σ∆ )
(SigAbort1)
b M
ab
k
h([M ⊲ N ; N ′ ])t,k ; Σi −−−
−→ h([N (M ) >>= N ′ ])t ; Σ′ i
′
Σ∆ = clean(k, Σ∆ )
(SigAbort2)
b M
ab
k
h([M ⊲ N ])t,k ; Σi −−−
−→ h([N (M )])t ; Σ′ i
b M
ab
ab M
k
hP ; Σi −−−
−→ hP ′ ; Σ′ i
k
hQ; Σi −−−
−→ hQ′ ; Σ′ i
ab M
(AbBroadcast)
k
hP k Q; Σi −−−
−→ hP ′ k Q′ ; Σ′ i
′
ΣΘ = commit(k, ΣΘ , Σ∆ ) Σ′∆ = clean(k, Σ∆ )
co
k
h([return M ⊲ N ; N ′ ])t,k ; Σi −−→
h([return M >>= N ′ ])t ; Σ′ i
′
ΣΘ = commit(k, ΣΘ , Σ∆ ) Σ′∆ = clean(k, Σ∆ )
co
k
h([M ⊲ N ])t,k ; Σi −−→
h([M ])t ; Σ′ i
co
(Commit2)
co
k
k
hP ; Σi −−→
hP ′ ; Σ′ i hQ; Σi −−→
hQ′ ; Σ′ i
co
k
hP k Q; Σi −−→
hP ′ k Q′ ; Σ′ i
β
(Commit1)
hP ; Σi −
→ hP ′ ; Σ′ i
(CoBroadcast)
transactions(β) ∈
/ transactions(Q)
β
(TrIgnore)
hP k Q; Σi −
→ hP ′ k Q; Σi
clean(k, ∆)(r) ,
(
⊥
if ∆(r) = (M, k)
∆(r) otherwise
commit(k, Θ, ∆)(r) ,
(
M
if ∆(r) = (M, k)
Θ(r) otherwise
transactions(([M ⊲ M ′ ; N ])t,k ) , {k} transactions(([M ⊲ N ])t,k ) , {k}
[
transactions(Tt1 k · · · k Ttn ) ,
transactions(Tti )
transactions(([M ])t ) , ∅
1≤i≤n
β
Figure 5: OCTM semantics: rules for −
→.
3.1
MVars
One of the basic constructs offered by Concurrent Haskell are MVars [10] i.e. mutable locations
that are either empty or contain a value of the given type parameter. Interaction with these
structures is based on two fundamental operations: putMVar which fills an MVar if it is empty
and blocks otherwise, and takeMVar which empties an MVar if it is full and blocks otherwise.
In [7] MVars are implemented on top of TVars (i.e. STM Haskell transactional locations).
7
Following [7] an MVar of type a is implemented on top of a OTVar (our transactional locations
i.e. any r ∈ Loc) holding a value of type Maybe a; this is a type that is either an empty value
(Nothing) or actually holds a value of type a (e.g. Just 42). Thus, the definition of the type
MVar a is the following:
type MVar a = OTVar (Maybe a)
and its two constructors for creating an empty and a full location are:
newEmptyMVar = newVar Nothing
newMVar x = newVar (Just x)
The definition of the two basic operations is precisely the same appearing in [7] except for the
added isolated construct for enforcing isolation.
putMVar v y = isolated do
v ← readVar v
case v of
Nothing → writeVar y Nothing
Just → retry
takeMVar v = isolated do
v ← readVar v
case v of
Nothing → retry
Just x → do
writeVar x Nothing
return x
3.2
Transactional RPC
MVars can be used as simple directional channels with takeMVar and putMVar as receive and
send. Then a bidirectional channel for a remote procedure call is easily implemented using a pair
of MVars
type RPC a b = (MVar (CorId, a), MVar (CorId, b))
where a and b are the types of the request and response exchanged and CorId is a suitable type
providing a correlation identifier for relating a request to its response.
Before we introduce the skeleton and stub let us define a conditional variation of the takeMVar
accepting a boolean predicate p and such that it empties the given MVar v only if the contained
value satisfies p and blocks (issue a retry) otherwise.
takeMVarIf p v = isolated do
v ← readVar v
case v of
Nothing → retry
Just x → do
if p x then
writeVar x Nothing >> return x
else
retry
The conditional version of takeMVar allows us to take a response only if we know its correlation
identifier and hence the call is simply:
rpcCall (req, res) data = do
c ← newCorrelationId
putMVar req (c, data)
r ← takeMVarIf (c == fst) res
return (snd r)
8
where fst and snd are the first and second projections respectively. Symmetrically, to provide
the rpc we just need to take a request from the MVar req and put its response in res using the
same correlation identifier:
rpcServe (req, res) data = do
q ← takeMVar req
a ← doSomething (snd q)
putMVar res (fst q, a)
If any of the two parties happens to be partaking a transaction the rpc results in the other joining
the transaction effectively rendering the rpc transactional.
The above example is quite simplified (e.g. requests could have been handled by a buffer, and
the structure of (req, req) should be hidden to the user) but serves the purpose of illustrating
the difference between OCTM and STM. handled by a buffer, and the structure of (req, req)
should be hidden to the user) but serves the purpose of illustrating the difference between OCTM
and STM. In fact, the above implementation allows the call to happen inside a transaction
without resulting into a lock as in the case of STM since isolation will prevent the serving thread
to join and provide a response.
4
Expressiveness of OCTM
In order to assess the expressive power of OCTM , in this Section we prove that it can be used
to implement TCCS m , a formal model for open transactions [11]. We proceed as follow: first, in
Subsection 4.1 we recall TCCS m ; then, the translation of TCCS m processes into OCTM states
is defined in Subsection 4.2; this translation is proved to be correct in Subsection 4.3.
4.1
TCCS m : CCS with open transactions
TCCS m [11] is a CCS-like calculus with open flat can synchronize even when belonging to
different transactions, which in turn are joined into a distributed one. We refer to [11] for a
detailed description of TCCS m . transactions: processes can synchronize even when belonging
to different transactions, which in turn are joined into a distributed one. We refer to [11] for a
detailed description of TCCS m .
The syntax of TCCS m is defined by the following grammar
Pn
Qm
P ::= i=1 αi .Pi | i=0 Pi | P \L | X | µX.P | h P1 ◮ P2 i | hhP1 ⊲k P2 ii | co.P
(1)
where αi ::= a | ā | τ , a ranges over a given set of visible actions A, L over subsets of A and the
bijection (·) : A → A maps every action to its coaction as usual. The calculus extends CCS with
three constructs which represent inactive transactions, active transactions and commit actions
respectively. Transactions such as hhP1 ⊲k P2 ii are formed by two processes with the former being
executed atomically and the latter being executed whenever the transaction is aborted, i.e. as
a compensation. Terms denoting active transactions expose also a name (k in the previous example) which is used to track transaction fusions. For instance, consider the process denoted
by hhP1 ⊲j P2 ii | hhQ1 ⊲k Q2 ii where P1 and Q1 synchronize on some a ∈ A; the result of this
synchronization is the fusion of the transactions j and k i.e. hhP1′ ⊲l P2 ii | hhQ′1 ⊲l Q2 ii. The fusion
makes explicit the dependency between j and k introduced by the synchronization and ties them
to agree on commits. In this sense, P1′ and Q′1 are participants of a distributed transaction [6].
As in [11] we restrict ourselves to well-formed terms. Intuitively, a term is well-formed if active
transactions occur only at the top-level and commit actions occur only in a transaction (active
9
ς ⊢P :p
ς ⊢P :τ
ς ⊢P :p
ς ⊢ P : τ ς ⊢ co.P : c ς ⊢ P \L : τ
ς[X : p] ⊢ P : p ς[X : c] ⊢ P : c ∀i ς ⊢ Pi : τ
Q
ς ⊢ X : ς(X)
ς ⊢ µX.P : p
ς ⊢ µX.P : c
ς ⊢ Pi : τ
∀i ς ⊢ Pi : p
∀i ς ⊢ αi .Pi : c ς ⊢ P : c ς ⊢ Q : p ς ⊢ P : c ς ⊢ Q : p
P
P
ς ⊢ αi .Pi : p ς ⊢ αi .Pi : c
ς ⊢ hhP ⊲k Qii : t
ς ⊢ h P ◮ Qii : p
Figure 6: Simple types for TCCS m .
or inactive). To this end we introduce a type system for TCCS m , whose rules are in Figure 6.
Terms that cannot occur inside a transaction have type t, terms that cannot occur outside a
transaction have type c, and terms without such restrictions have type p; τ ranges over types.
Definition 1 (Well-formed TCCS m terms). A TCCS m term P , described by the grammar in
(1), is said to be well-formed if, and only if, ∅ ⊢ P : t. Well-formed terms form the set Proc.
The operational semantics of well-formed TCCS m terms is given by the SOS in Figure 7
(see [11] for further details). The reduction semantics is given as a binary relation → defined by
△
β
τ
k(τ )
P → Q ⇐⇒ P −
→σ Q ∨ P −
→ Q ∨ P −−−→σ Q.
The first case is a synchronization between pure CCS processes. The second case corresponds
to creation of new transactions and distributed commit or abort (β ∈ {newk, cok, abk}). The
third case corresponds to synchronizations of processes inside a named (and possibly distributed)
transaction. Notice that by (TSync) transaction fusion is driven by communication and that
by (TSum) any pure CCS process can join and interact with a transaction.
4.2
Encoding TCCS m in OCTM
In this section we define the translation from TCCS m processes to OCTM states. To this end,
we have to implement transactions and CCS-like synchronizations using shared transactional
variables and the atomic and isolated operators.
Synchronization is implemented by means of shared transactional variables, one for each
channel, that take values of type ChState (cf. Figure 9); this type has four constructors: one
for each of the three messages of the communication protocol below plus a “nothing” one providing the default value. Let t1 and t2 be the identifiers of two threads simulating a.P and a.Q
respectively. The protocol is composed by the following four steps:
1. t1 checks whether the channel is free and writes on the transactional variable modelling
the channel a a nonce tagged with the constructor M1;
2. t2 reads the variable for a and accepts the synchronization offered by the challenge (M1
np) adding a fresh nonce to it and writing back (M2 np nq);
3. t1 reads the answer to its challenge and acknowledges the synchronization writing back the
nonce it read tagged with the constructor M3;
4. t2 reads the acknowledgement and frees the channel.
10
a
P
α
i
αi .Pi −→
ε Pi
α
P −
→σ P ′
(Sum)
P −
→ε P ′
τ
P |Q −
→ε P ′ |Q′
img(σ) ∩ tn(Q) = ∅
α
P |Q −
→σ
α
P −
→ε P ′
ā
Q−
→ε Q′
P ′ |Q[σ]
τ 6= α l 6= k
P
α
α∈
/L
α
P \L −
→σ P ′ \L
β
β
P −
→ P′
αi .Pi −−−→ε7→k hhPj |co ⊲k
k(a)
k(ā)
(TSum)
P
αi .Pi ii
Q −−−→j7→k Q′
k(τ )
(TSync)
P |Q −−−→i,j7→k P ′ [j 7→ k]|Q′ [i 7→ k]
P −
→ε P ′
τ
hhP ⊲k Qii −
→ε hhP ′ ⊲k Qii
k fresh
(TRes)
P \L −
→ P ′ \L
(Rec)
τ
(Res)
β
P −
→ P′
k(αj )
P −−−→i7→k P ′
hhP ⊲l Qii −−−→l7→k hhP ′ ⊲k Qii
P −
→σ P ′
τ
µX.P −
→ε P [µX.P/X ]
τ 6= αj
(ParL)
(TAct)
k(α)
(Sync)
(TTau)
(TNew)
newk
h P ◮ Qii −−−→ hhP ⊲k Qii
β
Q−
→ Q′
β
β 6= newk
β
(TB1)
P |Q −
→ P ′ |Q′
Q
Ψσ (Q)\L
Pα .Ψ (P )
i
σ
i
Ψσ (P ) , Q
Ψσ (Pi )
µX.Ψ
σ[P/X ] (Q)
P [σ]
if P = co.Q
if P = Q\L
P
if P = αi .Pi
Q
if P = Pi
if P = µX.Q
otherwise
P −
→ P′
abk
(TAb)
hhP ⊲k Qii −−→ Q
∃i Pi = co.Pi′
Q
cok
hh Pi ⊲k Qii −−→ Ψid (P )
tn(β) ∈
/ tn(Q)
β
(TCo)
(TB2)
P |Q −
→ P ′ |Q
{k}
if P = hhP ⊲k Qii
S
Q
tn(P ) ,
tn(Pi ) if P = Pi
otherwise
∅
k if β = newk
tn(β) ,
k if β = abk
k if β = cok
Figure 7: TCCS m operational semantics.
Each step has to be executed in isolation with respect to the interactions with the shared
transactional variable a.
Nonces are meant to correlate the steps only and hence can be easily implemented in OCTM
by pairing thread identifiers with counter a la logical clock. If at any step a thread finds the
channel in an unexpected state it means that the chosen scheduling has led to a state incoherent
with respect to the above protocol; hence the thread executes a retry. This tells the scheduler
to try another execution order; by fairness, we eventually find a scheduling such that the two
processes do synchronize on a and these are the only executions leading to P | Q. The protocol
is illustrated in Figure 8. If the synchronizing parties are involved in distinct transactions these
are fused as a side
effect of the interaction via the shared variable.
Pm
A choice like i=1 αi .Pi can be seen as a race of threads t1 , . . . , tm , each simulating a branch,
to acquire a boolean transactional variable l (private to the group). Each ti proceeds as follows.
First, it checks l and if it is set, it returns void and terminates (another thread has already
acquired it); otherwise it tries to set it while carrying out αi , i.e. right before executing its last
step of the communication protocol. If the variable is acquired by another thread while ti is
finalizing αi then ti issues a retry to retract any effect of αi . The OCTM code implementing
this protocol is shown in Figure 9.
11
a.P
var a
1
M0
(M1 np)
3
(M2 np ny)
(M3 ny)
a.Q
(M1 nx)
(M2 nx nq)
2
(M3 nq)
M0
4
Figure 8: Implementing TCCS m synchronization.
Encoding of TCCS m We can now define the encoding η : Proc → State, mapping well-formed
Q
TCCS m terms to states of the OCTM abstract machine. Intuitively, a process P ≡ m
i=1 Pi is
mapped into a state with a thread for each Pi and a variable for each channel in P . Clearly a
state of this form can be generated by a single OCTM term which allocates all variables and
forks the m threads; we have preferred to map TCCS m terms to OCTM states instead of OCTM
term for sake of simplicity.
The map η is defined byQrecursion along the derivation of ∅ ⊢ P : t and the number m of
m
parallel components in P ≡ i=1 Pi . This is handled by the auxiliary encoding ς : Proc× Heap →
State (up to choice of fresh names) whose second argument is used to track memory allocations.
The base case is given by m = 0 and yields a state with no threads i.e. h0, Θ, ∅i. The recursive
step is divided in three subcases depending on the structure and type of P1 (m > 0).
1. If ∅ ⊢ P1 : c without top-level restrictions (i.e. for no Q and no L = {a1 , . . . , an+1 } such
that each ai occurs in Q the process P1 is structurally equivalent to Q \ L) then
Q
ς( m+1
i=1 Pi , Θ) , h([̺(P1 )])t1 k S; Σi
Q
where hS; Σi = ς( m−1
j=1 Pj+1 , Θ) is the translation of the rest of P and t1 is unique w.r.t. S
(i.e. t1 ∈
/ threads(S)). By hypothesis P1 does not contain any top-level active transaction
or parallel composition and hence can be translated directly into a OCTM -term
by means of the encoding ̺ (cf. Figure 10) – ̺(P ) contain a free variable for each unrestricted channel occurring in P .
2. If P1 has a top-level restriction (i.e. P1 ≡ Q \ {a1 , . . . , an+1 }) then
Qm+1
ς( i=1 Pi , Θ) , hS1 [r1 /a1 , . . . rn+1 /an+1 ] k S2 ; Θ2 [r1 , . . . , rn+1 7→ M0], ∅i
Qm−1
where hS1 ; Θ1 , ∅i = ς(Q, Θ) and hS2 ; Θ2 , ∅i = ς( j=1 Pj+1 , Θ1 ) are the translation of the
unrestricted process Q and the translation of the rest of P respectively, all threads have a
unique identifier threads(S1 ) ∩ threads(S2 ) = ∅, the heap is extended with n channel variables fresh (r1 , . . . , rn+1 ∈
/ dom(Θ2 )) and known only to the translation of Q.
3. If P1 ≡ hhQ1 ⊲k Q2 ii is an active transaction then
Qm+1
ς( i=1 Pi , Θ) , hSco k Sab k S1 [rco /co] k S2 ; Θ2 [rl 7→ T rue, rco 7→ M0], ∅i
Sco = ([recv rl rco ⊲ ̺(Q1 ); bang (recv (newVar True) rco )])tco ,k
Sab = ([abort () ⊲ return])tab ,k
12
data Channel = OTVar ChState
data ChState = M1 Nonce | M2 Nonce Nonce | M3 Nonce | M0
tau l P = isolated do
case (readVar l) of
False → return ()
True → chooseThis l >> P
chooseThis l = writeVar l False
eqOrRetry x y
| x == y = return ()
| otherwise = retry
bang x = fork x >> bang x
recv c l P = do
nq ← newNonce
isolated do
case (readVar l) of
False → return ()
True → do
chooseThis l
case (readVar c) of
(M1 nx) → writeVar c (M2 nx nq)
_ → retry
isolated do
case (readVar c) of
(M3 ny) → eqOrRetry ny nq >> writeVar c M0 >> P
_ → retry
send c l P = do
np ← newNonce
isolated do
case (readVar l) of
False → return ()
True → do
chooseThis l
case (readVar c) of
M0 → writeVar c (M1 np)
_ → retry
isolated do
case (readVar c) of
(M2 nx ny) → eqOrRetry nx np >> writeVar c (M3 ny) >> P
_ → retry
Figure 9: Encoding channels and communication
13
Pm
̺( i=1 αi Pi ) , do
l ← newVar True
∀i ∈ {1, . . . , m}
fork ξ(αi , l, Pi )
̺(µX.P ) , let X = ̺(P ) in
̺(hh P ◮ Qii ) , do
co ← newVar M0
atomic p ̺(Q)
bang psi
Q
̺( m
i=0 Pi ) , do
∀i ∈ {0, . . . , m}
fork ̺(Pi )
where
p = do
̺(P )
fork (abort ())
̺(P \ L) , do
∀c ∈ L
psi
c ← newVar M0
̺(P )
psi = do
l ← newVar True
̺(X) , X
̺(P )
̺(co.P ) , do
l ← newVar True
send co l ̺(P )
recv co l return
recv αi l ̺(Pi ) if αi = c
ξ(αi , l, Pi ) , send αi l ̺(Pi ) if αi = c
tau l ̺(Pi )
if αi = τ
Figure 10: Encoding TCCS m terms of type c
Qm−1
where hS1 ; Θ1 , ∅i = ς(Q1 , Θ), hS2 ; Θ2 , ∅i = ς( j=1 Pj+1 , Θ2 ) (like above), the thread Sab
is always ready to abort k as in (TAb) and Sco awaits on the private channel rco a thread
from S1 to reach a commit and, after its commit, collects all remaining synchronizations
on rco to emulate the effect of Ψ (cf. Figure 7). Finally, all threads have to be uniquely
identified: threads(S1 ) ∩ threads(S2 ) = ∅ and tco , tab ∈
/ threads(S1 ) ∪ threads(S2 )
Remark 1. The third case of the definition above can be made more precise (at the cost of a
longer definition) since the number of commits to be collected can be inferred from Q mimicking the definition of Ψ. This solution reduces the presence of dangling auxiliary processes and
transaction fusions introduced by the cleaning process.
Like ̺, ς(P, Θ) contains a free variable for each unrestricted channel in P . Finally, the
encoding η is defined on each P ∈ Proc as:
η(P ) , hS[r1 /a1 , . . . rn /an ]; Θ[r1 , . . . , rn 7→ M0], ∅i
where hS; Θ, ∅i = ς(P, ∅), {r1 , . . . , rn } ⊆ Loc, and {a1 , . . . , an } ⊆ A is the set of channels
occurring in P .
4.3
Adequacy of translation
In this section we prove that the translation η is adequate, in the sense that it preserves the
observational behaviour of TCCS m processes. More precisely, akin to [12], we define an appropriate notion of star simulation S between well-formed TCCS m processes and states of OCTM .
The basic idea is that a single step of P is simulated by a sequence of reductions of η(P ), and
η(P ) does not exhibit behaviours which are not exhibited by P .
14
Definition 2 (Star simulation). A relation S ⊆ Proc × State is a star simulation if for all
(P, hS; Σi) ∈ S:
k(τ )
τ
1. for all Q such that P −
→σ Q or P −−−→σ Q, there exist S ′ , Σ′ such that hS; Σi →∗ hS ′ ; Σ′ i
and (Q, hS ′ ; Σ′ i) ∈ S;
β
β
2. for all Q such that P −
→ Q, there exist S ′ , Σ′ s.t. hS; Σi −
→∗ hS ′ ; Σ′ i and (Q, hS ′ ; Σ′ i) ∈ S.
3. for all S ′ , Σ′ such that hS; Σi → hS ′ ; Σ′ i, there exist Q, S ′′ , Σ′′ such that (Q, hS ′′ ; Σ′′ i) ∈ S
and one of the following holds:
k(τ )
τ
• P −
→σ Q or P −−−→σ Q, and hS ′ ; Σ′ i →∗ hS ′′ ; Σ′′ i
β
β
• P −
→ǫ Q and hS ′ ; Σ′ i −
→∗ hS ′′ ; Σ′′ i.
where β-labels of the two transition relations are considered equivalent whenever are both commits
or both aborts for the same transaction name. We say that P is star-simulated by hS; Σi if there
∗
exists a star-simulation S such that (P, hS; Σi) ∈ S. We denote by ≈ the largest star simulation.
Another technical issue is that two equivalent TCCS m processes can be translated to OCTM
states which differ only on non-observable aspects, like name renamings, terminated threads, etc.
To this end, we need to consider OCTM states up-to an equivalence relation ∼
=t ⊆ State × State,
which we define next.
∼t hS2 ; Σ2 i, when
Definition 3. Two OCTM states are transaction-equivalent, written hS1 ; Σ1 i =
they are equal up to:
• renaming of transaction and thread names;
• terminated threads, i.e. threads of one of the following forms: ([return M ])t , ([abort M ])t ,
([return ⊲ return])t,k , ([abort ⊲ return])t,k , ([psi])t ;
• threads blocked in synchronizations on co variables.
Definition 4. Let P ∈ P roc be a well-formed process and hS; Σi be a state. P is star simulated
∗
by hS; Σi up to ∼
=t if (P, hS; Σi) ∈ ≈ ◦ ∼
=t .
We are now ready to state our main adequacy result, which is a direct consequence of the
two next technical lemmata.
Lemma 1. For all P, Q ∈ P roc the following hold true:
τ
k(τ )
1. if P −
→σ Q or P −−−→σ Q, there exist S, Σ such that η(P ) →∗ hS; Σi and hS; Σi ∼
=t η(Q);
β
β
2. if P −
→ Q, there exist S, Σ such that η(P ) −
→∗ hS; Σi and hS; Σi ∼
=t η(Q).
Proof. See Appendix A.
Lemma 2. For P ∈ P roc, for all S, Σ, if η(P ) → hS; Σi then there exist Q, S ′ , Σ′ such that
hS ′ ; Σ′ i ∼
=t η(Q) and one of the following holds:
τ
k(τ )
• P −
→σ Q or P −−−→σ Q, and hS; Σi →∗ hS ′ ; Σ′ i;
β
β
• P −
→ǫ Q and hS; Σi −
→∗ hS ′ ; Σ′ i.
Proof. See Appendix A.
Theorem 3. For all P ∈ P roc, P is star simulated by η(P ) up to ∼
=t .
15
5
Conclusions and future work
In this paper we have introduced OCTM , a higher-order language extending the concurrency
model of STM Haskell with composable open (multi-thread) transactions. In this language,
processes can join transactions and transactions can merge at runtime. These interactions are
driven only by access to shared transactional memory, and hence are implicit and loosely coupled.
To this end, we have separated the isolation aspect from atomicity: the atomic construct ensures
“all-or-nothing” execution but not isolation, while the new constructor isolated can be used
to guarantee isolation when needed. In order to show the expressive power of OCTM , we have
provided an adequate implementation in it of TCCS m , a recently introduced model of open
transactions with CCS-like communication. As a side result, we have given a simple typing
system for capturing TCCS m well-formed terms.
Several directions for future work stem from the present paper. First, we plan to implement
OCTM along the line of STM Haskell, but clearly the basic ideas of OCTM are quite general
and can be applied to other STM implementations, like C/C++ LibCMT and Java Multiverse.
An interesting possibility is to use TCCS m as an exogenous orchestration language for
OCTM : the behaviour of a transactional distributed system can be described as a TCCS m
term, which can be translated into a skeleton in OCTM using the encoding provided in this
paper; then, the programmer has only to “fill in the gaps”. Thus, TCCS m can be seen as a kind
of “global behavioural type” for OCTM .
In fact, defining a proper behavioural typing system for transactional languages like OCTM
is another interesting future work. Some preliminary experiments have shown that TCCS m is
not enough expressive for modelling the dynamic creation of resources (locations, threads, etc.).
We think that a good candidate could be a variant of TCCS m with local names and scope
extrusions, i.e., a “transactional π-calculus”.
Being based on CCS, communication in TCCS m is synchronous; however, nowadays asynchronous models play an important rôle (see e.g. actors, event-driven programming, etc.). It
may be interesting to generalize the discussion so as to consider also this case, e.g. by defining
an actor-based calculus with open transactions. Such a calculus can be quite useful also for
modelling speculative reasoning for cooperating systems [14–16]. A local version of actor-based
open transactions can be implemented in OCTM using lock-free data structures (e.g., message
queues) in shared transactional memory.
References
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automatic mutual exclusion. ACM Trans. Program. Lang. Syst., 33(1):2, 2011.
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join. In J. Lévy, E. W. Mayr, and J. C. Mitchell, editors, Proc. TCS, volume 155 of IFIP,
pages 563–576. Kluwer/Springer, 2004.
[3] V. Danos and J. Krivine. Transactions in RCCS. In M. Abadi and L. de Alfaro, editors, Proc. CONCUR, volume 3653 of Lecture Notes in Computer Science, pages 398–412.
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[5] E. de Vries, V. Koutavas, and M. Hennessy. Liveness of communicating transactions (extended abstract). In K. Ueda, editor, Proc. APLAS, volume 6461 of Lecture Notes in
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P. Yew, editors, Proc. PPOPP, pages 157–168. ACM, 2011.
[14] J. Ma, K. Broda, R. Goebel, H. Hosobe, A. Russo, and K. Satoh. Speculative abductive reasoning for hierarchical agent systems. In J. Dix, J. Leite, G. Governatori, and W. Jamroga,
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17
A
Omitted proofs
Proof of Lemma 1. The proof proceeds by induction on the syntax of TCCS m . We only show
three cases:
τ
1. a transition P −
→ε Q resulting from a synchronization outside a transaction;
k(τ )
2. a transition P −−−→σ Q resulting from a synchronization inside a transaction;
cok
3. a commit transition P −−→ Q.
a
τ
a
→ Q1 , P2 −
1. If P −
→εPQ with (Sync) rule, P = P1 | P2 , Q = Q1 | Q2 , P1 −
→ Q2 .
m
P1 = ((Pi 1 ai .Ri′ ) | P1′ ) \ L1 such that ∃i.ai = a and a ∈
/L
2
′′
′
P2 = (( m
/L
j bj .Rj ) | P2 ) \ L2 such that ∃j.bj = a and a ∈
η(P ) =h([return t′1 ])t1 k ([return t1m1 ])tsum1 k S1′ k
′
)])t1m1 k
k ([ξ(a1 , lsum1 , R1′ )])t11 k · · · k ([ξ(am1 , lsum1 , Rm
1
k ([return t′2 ])t2 k ([return t2m2 ])tsum2 k S2′ k
′′
)])t2m2 ; (Θ, ∅)i
k ([ξ(b1 , lsum2 , R1′′ )])t21 k · · · k ([ξ(bm2 , lsum2 , Rm
2
From hypothesis, there exists a thread t1r ∈ {t11 , . . . , t1m1 } such that
([̺(ar .Rr′ )])t1 r = ([recv a lsum1 Rr′ ])t1 r
and exists another thread t2s ∈ {t21 , . . . , t2m2 } s.t.
([̺(as .Rs′′ )])t2 s = ([send a lsum2 Rs′′ ])t2 s .
lsum1 and lsum2 are locations created by threads tsum1 and tsum2 from the code generated
by encoding of the sums.
h· · · k ([recv a lsum1 ̺(Rr′ )])t1 r k ([send a lsum2 ̺(Rs′′ )])t2 s k . . . ; (Θ, ∅)i
→ h([return t′1 ])t1 k ([return t1m1 ])tsum1 k S1′ k
∗
k ([return])t11 k · · · k ([̺(Rr′ )])t1 r k · · · k ([return])t1m1 k
k ([return t′2 ])t2 k ([return t2m2 ])tsum2 k S2′ k
k ([return])t21 k · · · k ([̺(Rs′′ )])t2 s k · · · k ([return])t2m2 ; (Θ′ , ∅)i = hS; Σi
where Θ′ (lsum1 ) = False, Θ′ (lsum2 ) = False
recv a lsum1 ̺(Rr′ ) and send a lsum2 ̺(Rs′′ ) can in order execute the isolated blocks, and
at the end reduce to continuations ̺(Rr′ ) and ̺(Rs′′ ).
Other threads forked by threads tsum1 , tsum2 can only reduce to return because Θ′ (lsum1 ) =
False and Θ′ (lsum2 ) = False: threads t1r , t2s modified l-variables through the synchronization code inside isolated blocks. We can observe hS; Σi ∼
=t η(Q), in fact Q = (Ri′ |
′′
′
′
P1 ) \ L1 | (Rj | P2 ) \ L2 , η(Q) = hSq ; Σq i.
η(Q) =h([return t′1 ])t1 k ([̺(Ri′ )])ti k S1′ k
k ([return t′2 ])t2 k ([̺(Rj′ )])tj k S2′ ; (Θ′q , ∅)i = hSq ; Σq i
where ∀ch ∈ L1 ⊎ L2 . Θ′q (ch) = M0
hSq ; Σq i and hS; Σi are different only in local variables and for reduced threads, thus
hS; Σi ∼
=t η(Q).
18
k(τ )
k(a)
2. If P −−−→i,j7→k Q with (TSync) rule, P = (P1 | P2 ), Q = (Q1 | Q2 ), P1 −−−→i7→k Q1 ,
k(a)
P2 −−−→j7→k Q2 .
P = hhP1 ⊲i C1 ii | hhP2 ⊲j C2 ii and P1 = P11 | · · · | P1m1 , P2 = P21 | · · · | P2m2 .
′
η(P ) =h([P1′ ⊲ ̺(C1 )])t1 ,i k ([P2′ ⊲ ̺(C2 )])t2 ,i k ([̺(P11
) ⊲ return])t11 ,i k . . .
′
′
) ⊲ return])t1m1 ,i k . . .
· · · k ([recv a l P1r ⊲ return])tr ,i k · · · k ([̺(P1m
1
′
′
· · · k ([̺(P21
) ⊲ return])t21 ,j k · · · k ([send a l P2s
⊲ return])ts ,j . . .
′
) ⊲ return])t2m2 ,j ; (Θ, ∆)i
· · · k ([̺(P2m
2
→∗ h([P1′ ⊲ ̺(C1 )])t1 ,j k ([P2′ ⊲ ̺(C2 )])t2 ,j k . . .
′
′
· · · k ([P1r
⊲ return])tr ,j k · · · k ([P2s
⊲ return])ts ,j ; (Θ, ∆′ )i = hS; Σi
hS; Σi ∼
=t η(Q) =
′
=η(hhP11 | · · · | P1r
| · · · | P1m1 ⊲k C1 ii |
′
| hhP21 | · · · | P2s | · · · | P2m2 ⊲k C2 ii)
=h([. . . ⊲ ̺(C1 )])t1 ,k k ([. . . ⊲ ̺(C2 )])t2 ,k k . . .
′
′
· · · k ([P1r
⊲ return])tr ,k k · · · k ([P2s
⊲ return])ts ,k ; (Θq , ∆q )i
cok
cok
3. If P −−→ Q with (TCo) rule, P = hhR ⊲k N ii and hhR ⊲k N ii −−→ R′ where R′ =
Qm
i=1 Ri
η(P ) =h([recv co lt return ⊲ ̺(N ); R])t,k k ([return tm ⊲ return])t′ ,k k
k ([̺(R1 ) ⊲ return])t1 ,k k · · · k ([̺(Rj ) ⊲ return])tj ,k k . . .
· · · k ([̺(Rm ) ⊲ return])tm ,k ; (Θ, ∆)i
where
∆(lt ) = (True, k), ∆(co) = (M0, k)
R = bang psi
∃j ∈ {1, . . . , m}, Rj = co.Rj′ , ̺(Rj ) = send co l ̺(Rj′ ).
η(P ) →∗ h([recv co lt return ⊲ ̺(N ); R])t,k k ([return tm ⊲ return])t′ ,k k
k · · · k ([send co ltj ̺(Rj′ ) ⊲ return])tj ,k k . . . ; (Θ, ∆′′ )i
At this point threads t, tj can synchronize through co variable and transaction k can com-
19
mit.
→∗ h([recv co lt return ⊲ ̺(N ); bang psi])t,k k
k ([return tm ⊲ return])t′ ,k k ([̺(R1 ) ⊲ return])t1 ,k k . . .
· · · k ([send co ltj ̺(Rj′ ) ⊲ return])tj ,k k · · · k ([̺(Rm ) ⊲ return])tm ,k ; (Θ, ∆′′ )i
→∗ h([return ⊲ ̺(N ); bang psi])t,k k
k ([return tm ⊲ return])t′ ,k k
k ([̺(R1 ) ⊲ return])t1 ,k k · · · k ([̺(Rj′ ) ⊲ return])tj ,k k . . .
· · · k ([̺(Rm ) ⊲ return])tm ,k ; (Θ, ∆′′′ )i
co
k
−−→h(
[bang psi])t k ([return tm ])t t′ k
k ([̺(R1 )])t1 k · · · k ([̺(Rj′ )])tj k · · · k ([̺(Rm )])tm ; Σi = hS; Σi
where Σ = commit(k, Θ, ∆′′′ )
Qm
hS; Σi ∼
=t η(Q), Q = R′ = 1 Ri and ∃j ∈ {1, . . . , m} : Rj = Rj′
η(Q) = η(R′ ) = h([̺(R1 )])t1 k · · · k ([̺(Rj′ )])tj k · · · k ([̺(Rm )])tm ; Σq i ∼
=t hS; Σi
Proof of Lemma 2. The proof goes through induction on the semantic of TCCS m . Here we
show only 3 cases, first when P are two processes that can perform a synchronization outside
transactions, second P synchronizes inside transactional processes, third a transactional process
commits.
1. If P = a.P1 | a.P2 From η(P ) we can move to another state of the OCTM machine hS; Σi
and hS; Σi ∼
=t η(Q):
η(P ) =h([return t′1 ])t1 k ([recv a rl ̺(P1 )])t′1 k
k ([return t′2 ])t2 k ([recv a rl ̺(P2 )])t′2 ; (Θ′ , ∅)i
→∗ h([return t′1 ])t1 k ([̺(P1 )])t′1 k ([return t′2 ])t2 k ([̺(P2 )])t′2 ; (Θ′ , ∅)i = hS; Σi
where Θ′ (lt1 ) = F alse, Θ′ (lt2 ) = F alse Θ′ (a) = M0
τ
If P −
→ Q, then Q = P1 | P2 , η(Q) = h([̺(P1 )])t1 k ([̺(P2 )])t2 ; (Θq , ∅)i, we can observe
hS ′ ; Σ′ i ∼
=t η(Q).
2. If P = hha.P1 ⊲i Q1 ii | hha.P2 ⊲j Q2 ii, η(P ) → hS; Σi →∗ hS ′ ; Σ′ i the computations are exactly
the same as the previous point, but all variables are tentative in ∆. It is easy to see that
τ (k)
P −−−→i,j7→k Q and hS; Σi ∼
=t η(Q).
20
3. If P = hhco.P ′ ⊲k Qii
η(P ) =
=h([recv co l return ⊲ ̺(Q); bang psi])t,k k
k ([send co l ̺(P ′ ) ⊲ return])t′ ,k ; (Θ, ∆)i
→h([M ⊲ ̺(Q); bang psi])t,k k
−
k ([send co l ̺(P ′ ) ⊲ return])t′ ,k ; (Θ, ∆′ )i = hS; Σi
where ∆′ (np) = noncet1
co
k
→∗ −−→h(
[bang psi])t k ([̺(P ′ )])t′ ; (Θ′ , ∅)i
where Σ′ = commit(k, Θ′ , ∆′ )
= hS ′ ; Σ′ i
cok
P = hhco.P ′ ⊲k N ii −−→ P ′ = Q, η(Q) = h([̺(P ′ )])tq ; Σq i, η(Q) ∼
=t hS ′ ; Σ′ i
21
| 6 |
NOTES ON EXTREMAL AND TAME VALUED FIELDS
arXiv:1407.3759v2 [math.LO] 11 Jul 2016
SYLVY ANSCOMBE AND FRANZ-VIKTOR KUHLMANN
Abstract. We extend the characterization of extremal valued fields given
in [2] to the missing case of valued fields of mixed characteristic with perfect
residue field. This leads to a complete characterization of the tame valued fields
that are extremal. The key to the proof is a model theoretic result about tame
valued fields in mixed characteristic. Further, we prove that in an extremal
valued field of finite p-degree, the images of all additive polynomials have the
optimal approximation property. This fact can be used to improve the axiom
system that is suggested in [8] for the elementary theory of Laurent series fields
over finite fields. Finally we give examples that demonstrate the problems we
are facing when we try to characterize the extremal valued fields with imperfect
residue fields. To this end, we describe several ways of constructing extremal
valued fields; in particular, we show that in every ℵ1 saturated valued field the
valuation is a composition of extremal valuations of rank 1.
1. Introduction
A valued field (K, v) with valuation ring O and value group vK is called extremal if for every multi-variable polynomial f (X1 , . . . , Xn ) over K the set
{v(f (a1 , . . . , an )) | a1 , . . . , an ∈ O} ⊆ vK ∪ {∞}
has a maximal element. For the history of this notion, see [2]. In that paper,
extremal fields were characterised in several special cases, but some cases remained
open. In the present paper we answer the question stated after Theorem 1.2 of
[2] to the positive, thereby removing the condition of equal characteristic from the
theorem. The most comprehensive version of the theorem now reads:
Theorem 1.1. Let (K, v) be a nontrivially valued field. If (K, v) is extremal, then
it is algebraically complete and
(i) vK is a Z-group, or
(ii) vK is divisible and Kv is large.
Conversely, if (K, v) is algebraically complete and
(i) vK ≃ Z, or vK is a Z-group and char Kv = 0, or
(ii) vK is divisible and Kv is large and perfect,
then (K, v) is extremal.
Note that a valued field (K, v) is called algebraically complete if every finite
algebraic extension (L, v) satisfies
(1)
[L : K] = (vL : vK)[Lv : Kv] ,
where Lv, Kv denote the respective residue fields. Every algebraically complete
valued field (K, v) is henselian, i.e., v admits a unique extension to its algebraic
Date: February 9, 2016.
1
2
SYLVY ANSCOMBE AND FRANZ-VIKTOR KUHLMANN
closure K̃ (which we will again denote by v). Also, every algebraically complete
valued field (K, v) is algebraically maximal, that is, does not admit proper
algebraic immediate extensions (L, v) (immediate means that vL = vK and Lv =
Kv). For later use let us mention that a valued field is called maximal if it does
not admit proper immediate extensions at all.
Further, (K, v) is a tame field if it is henselian, perfect, and K̃ is equal to
the ramification field of the extension (K̃|K, v). All tame fields are algebraically
complete (cf. [13, Lemma 3.1]).
A field K is large if every smooth curve over K which has a K-rational point,
has infinitely many such points. For more information about large fields, see [15],
[10] and [2].
In [2] it was proved that an algebraically complete valued field (K, v) with divisible value group and large perfect residue field is extremal if char K = char Kv
(the equal characteristic case). To this end, we used the Ax–Kochen–Ershov
Principle
(2)
vK ≡ vL ∧ Kv ≡ Lv
=⇒
(K, v) ≡ (L, v)
which holds for all tame valued fields of equal characteristic (see [13, Theorem 1.4]).
We were not able to cover the mixed characteristic case char K 6= char Kv
because the principle was not known for this case. In fact, we will show below
(Theorem 1.5) that it is false. However, we can do with lesser tools that are known.
After all, at least the corresponding Ax–Kochen–Ershov Principle for elementary
extensions has been proved in [13]:
Theorem 1.2. If (L|K, v) is an extension of tame fields such that vK ≺ vL and
Kv ≺ Lv, then (K, v) ≺ (L, v).
This theorem enables us to prove:
Theorem 1.3. Take a nontrivially valued tame field (K, v) and two ordered abelian
groups Γ and ∆ such that Γ ≺ vK and Γ ≺ ∆. Then there exist two tame fields
(K ′ , v) and (L, v) with vK ′ = Γ, vL = ∆, Kv = K ′ v = Lv, (K ′ , v) ≺ (K, v) and
(K ′ , v) ≺ (L, v). In particular, (K, v) ≡ (L, v).
If vK is nontrivial and divisible and ∆ is any nontrivial divisible ordered abelian
group, then we can take Γ = Q to obtain that Γ ≺ vK and Γ ≺ ∆ since the
elementary class of nontrivial divisible ordered abelian groups is model complete.
Thus, Theorem 1.3 yields the following result:
Corollary 1.4. If (K, v) is a nontrivially valued tame field with divisible value
group and ∆ is any nontrivial divisible ordered abelian group, then there is a tame
field (L, v) ≡ (K, v) with vL = ∆ and Lv = Kv.
It is easy to see that (2) cannot hold in this generality in the mixed characteristic
case. One can construct two algebraic extensions (L, v) and (L′ , v ′ ) of (Q, vp ), where
vp is the p-adic valuation on Q, both having residue field Fp , such that:
√
1) L does not contain p and vL is the p-divisible hull of (vp p)Z,
√
√
2) L′ contains p and v ′ L′ is the p-divisible hull of (vp p)Z = 12 (vp p)Z.
Then vL ≃ v ′ L′ and hence vL ≡ v ′ L′ , but (L, v) 6≡ (L′ , v ′ ).
One could hope, however, that this problem vanishes when one strengthens the
conditions by asking that vL and v ′ L′ are equivalent over vp Q (and Lv and L′ v ′
are equivalent over Qvp ). But the problem remains:
NOTES ON EXTREMAL AND TAME VALUED FIELDS
3
Theorem 1.5. Take any prime p. Then the there exist valued field extensions
(Q, vp ) ⊂ (L0 , v) ⊂ (L1 , v) ⊂ (L2 , v) and (Q, vp ) ⊂ (F0 , v) ⊂ (F1 , v) ⊂ (F2 , v)
such that the following assertions hold:
a) The fields (L0 , v) and (F0 , v) are extensions of degree p(p−1) of the henselization
of Q under the p-adic valuation and extremal with L0 v = Fp = F0 v and vL0 =
1
vF0 = p(p−1)
(vp p)Z, but (L0 , v) 6≡ (F0 , v).
b) The fields (L1 , v) and (F1 , v) are algebraic over Q and tame with L1 v = F1 v = Fp
1
(vp p)Z, but (L1 , v) 6≡ (F1 , v).
and vL1 = vF1 equal to the p-divisible hull of p−1
c) The fields (L2 , v) and (F2 , v) are tame and extremal, with perfect residue fields
L2 v = F2 v and vL2 = vF2 = Q, but (L2 , v) 6≡ (F2 , v).
Corollary 1.6. The Ax–Kochen–Ershov Principle (2) fails for extremal fields with
value group isomorphic to Z in mixed characteristic. It also fails for tame extremal
fields with value group isomorphic to Q and perfect residue field in mixed characteristic.
Open problem: Can the situation be improved by adding the Macintyre power
predicates to the language?
Note that (L, v) ≡ (L′ , v ′ ) if and only if they are equivalent over (Q, vp ), and
this in turn holds if and only if we have the equivalence
(L, v)δ ≡ (L′ , v ′ )δ
over (Q, vp )δ
of their amc structures of level δ, for all δ ∈ (vp p)Z (see [7, Corollary 2.4]). But this
fact is of little use for the proof of Corollary 1.4 since it is by no means clear how
to construct an extension of (Q, vp ) whose amc structures of level δ are equivalent
to those of (K, v).
The improvement in Theorem 1.1 yields a corresponding improvement of Proposition 5.3 from [2]. Note that when we speak of a composition v = w ◦ w of
valuations, we do not mean a composition as functions, but in fact refer to the
composition of their associated places. That is, if Q and Q̄ are the places associated with w and w̄, then their composition (with the obvious additional rules for
∞) is the place associated with v.
Proposition 1.7. Take a valued field (K, v) with perfect residue field. Assume
that v is the composition of two nontrivial valuations: v = w ◦ w. Then (K, v)
is extremal with divisible value group if and only if the same holds for (K, w) and
(Kw, w).
We may say that a property P of valuations is compatible with composition
if P (v) ⇔ P (w)∧P (w̄) for each composition v = w◦ w̄. Examples of such properties
are “henselian”, “maximal”, “algebraically complete”, “divisible value group”. The
latter two will be used in the proof of the proposition, given in Section 2. The
proposition in fact yields that also the property “extremal with divisible value
group and perfect residue field” is compatible with composition (since if (Kw, w̄)
has this property, then in particular it is perfect).
It should be noted that the condition on the value groups cannot be dropped
without a suitable replacement, even when all residue fields have characteristic 0.
Indeed, if the value group of (K, w) is a Z-group and w is nontrivial, then the value
group of (K, v) is neither divisible nor a Z-group and (K, v) cannot be extremal.
4
SYLVY ANSCOMBE AND FRANZ-VIKTOR KUHLMANN
Let us state two
Open problems:
1) If v = w ◦ w̄ with w and w̄ extremal and w having divisible value group, does
it follow that v is extremal?
2) We know that if v = w ◦ w̄ is extremal, then so is w̄ (see Lemma 4.1 below).
But we do not know whether it follows that also w is extremal.
Tame fields of positive residue characteristic p > 0 are algebraically complete,
and by [13, Theorem 3.2], they have p-divisible value groups which consequently are
not Z-groups. On the other hand, by the same theorem all algebraically complete
valued fields with divisible value group and perfect residue field are tame fields.
Therefore, in the case of positive residue characteristic and value groups that are
not Z-groups, the above Theorem 1.1 is in fact talking about tame fields:
Theorem 1.8. A tame field of positive residue characteristic is extremal if and
only if its value group is divisible and its residue field is large.
Again, we see that we know almost everything about tame fields (with the exception of quantifier elimination in the case of equal characteristic), but almost
nothing about imperfect valued fields. As shown in [2], there are some algebraically
complete valued fields with value group a Z-group and a finite residue field that
are extremal, and others that are not. In particular, the Laurent series field Fq ((t))
over a finite field Fq with q elements is extremal.
It is a longstanding open question whether Fq ((t)) has a decidable elementary
theory. However, in recent years progress has been made on the existential theory.
Denef and Schoutens showed in [4] that if Resolution of Singularity holds in positive
characteristic in all dimensions (which is a longstanding open problem), then the
existential theory of (Fq ((t)), t) — i.e., the field together with the constant t — is
decidable. More recently, Anscombe and Fehm showed in [1] that the existential
theory of Fq ((t)) is decidable, under no assumptions.
Since the question for the full elementary theory has remained open, it is important to search for a complete recursive axiomatization. Such an axiomatization
was suggested in [8], using the elementary property that the images of additive
polynomials have the optimal approximation property (see Section 3 for the definition of this notion). For the case of Fq ((t)), this was proved in [3]. At first
sight, extremality seems to imply the optimal approximation property for the images of additive polynomials. But the latter uses inputs from the whole field while
the former restricts to inputs from the valuation ring. However, we will prove in
Section 3:
Theorem 1.9. If (K, v) is an extremal field of characteristic p > 0 with [K : K p ] <
∞, then the images of all additive polynomials have the optimal approximation
property.
Open problem:
Does the assertion of this theorem fail in the case of [K : K p ] = ∞?
Since the elementary property of extremality is more comprehensive and easier
to formulate than the optimal approximation property, it is therefore a good idea
to replace the latter by the former in the proposed axiom system for Fq ((t)). We
NOTES ON EXTREMAL AND TAME VALUED FIELDS
5
also note that every extremal field is algebraically complete by Theorem 1.1. So we
ask:
Open problem: Is the following axiom system for the elementary theory of Fq ((t))
complete?
1) (K, v) is an extremal valued field of positive characteristic,
2) vK is a Z-group,
3) Kv = Fq .
In order to obtain the assertion of Theorem 1.9 in the case of algebraically
complete perfect fields of positive characteristic (which are exactly the tame fields of
positive characteristic), one does not need the assumption that the field be extremal.
Indeed, S. Durhan recently proved in [5]:
Theorem 1.10. If (K, v) is a tame field of positive characteristic, then the images
of all additive polynomials have the optimal approximation property.
There are tame fields of positive characteristic that are not extremal, e.g. the
power series field Fp ((Γ)) with Γ the p-divisible hull of Z (see Theorem 1.8). Therefore, the previous theorem yields:
Corollary 1.11. There are perfect non-extremal fields of positive characteristic
in which the images of all additive polynomials have the optimal approximation
property.
Open problem:
Is there an imperfect non-extremal field of characteristic p > 0 in which the images
of all additive polynomials have the optimal approximation property?
Finally, let us point out that we still do not have a complete characterization of
extremal fields:
Open problem: Take a valued field (K, v) of positive residue characteristic. Assume that vK is a Z-group, or that vK is divisible and Kv is an imperfect large
field. Under which additional assumptions do we obtain that (K, v) is extremal?
Additional assumptions are indeed needed, as we will show in Section 4:
Proposition 1.12. a) There are algebraically complete valued fields (K, v) of
positive characteristic and value group a Z-group that are extremal, and others that
are not.
b) There are algebraically complete valued fields (K, v) of mixed characteristic with
value group a Z-group that are extremal, and others that are not.
c) There are algebraically complete nontrivially valued fields (K, v) of positive
characteristic with divisible value group and imperfect large residue field that are
extremal, and others that are not.
d) There are algebraically complete valued fields (K, v) of mixed characteristic with
divisible value group and imperfect large residue field that are extremal, and others
that are not.
None of the non-extremal fields that we construct for the proof of parts a)–d) of
this proposition is maximal. This leads us to the following
Conjecture: Every maximal field with value group a Z-group, or divisible value
group and large residue field, is extremal.
6
SYLVY ANSCOMBE AND FRANZ-VIKTOR KUHLMANN
The following theorem, also proved in Section 4, provides a compelling way of
constructing maximal extremal fields and is used in the proof of parts c) and d) of
the previous theorem.
Theorem 1.13. Let (K, v) be any ℵ1 -saturated valued field. Assume that Γ and
∆ are convex subgroups of vK such that ∆ ⊂
6= Γ and Γ/∆ is archimedean. Let
u (respectively w) be the coarsening of v corresponding to ∆ (resp. Γ). Denote
by ū the valuation induced on Kw by u. Then (Kw, ū) is maximal, extremal and
large, and its value group is isomorphic either to Z or to R. In the latter case, also
Ku = (Kw)ū is large.
Remark 1.14. Pairs (Γ, ∆) of convex subgroups satisfying the conditions of this
theorem are abundant and can easily be constructed. Indeed, for any γ ∈ vK we can
take Γ to be the smallest convex subgroup of vK containing γ (the intersection of
all convex subgroups of vK containing γ), and ∆ to be the largest convex subgroup
of vK not containing γ (the union of all convex subgroups of vK not containing
γ). Then ∆ is the largest proper convex subgroup of Γ and therefore, Γ/∆ is
archimedean.
From Theorem 1.13 we can derive an interesting observation about infinite compositions of henselian valuations. Note that every valuation can be viewed as a
possibly infinite composition of rank 1 valuations, i.e., valuations with archimedean
ordered value groups. It is well known that v = w ◦ w̄ is henselian if and only if
both w and w̄ are. However, in Section 4 we will derive the following result:
Corollary 1.15. There exist non-large (and therefore non-henselian) valued fields
(K, v) with the following property: if v = w1 ◦ w2 ◦ w3 with w2 of rank 1, then w2
is henselian and both Kw1 and (Kw1 )w2 are large.
The part about henselianity also follows from an actually stronger result, stating
the existence of a non-henselian valued field (K, v) with the following property: if
v = w1 ◦ w2 with nontrivial w1 , then w2 is henselian; see [12, Proposition 4]. The
latter again implies that Kw1 is large, but we do not know how to show that the
field constructed in the cited paper is not large.
Acknowlegements.
Several of the ideas contained in this paper were conceived at a 2 hour seminar
talk the second author gave to the logic group at the University of Wroclaw in
Poland. The audience was arguably the most lively and inspiring the author has
ever witnessed. He would like to thank this group for the great hospitality.
The authors would like to thank the referee for his careful reading of the manuscript and for several very useful suggestions that inspired them to come up with
Theorem 1.13 and with a new version of Theorem 1.5.
The second author would like to thank Koushik Pal for proofreading an earlier
version of the paper, and Anna Blaszczok for very helpful corrections and comments
on a later version.
During this research, the first author was funded by EPSRC grant EP/K020692/1,
and the second author was partially supported by a Canadian NSERC grant and a
sabbatical grant from the University of Saskatchewan.
NOTES ON EXTREMAL AND TAME VALUED FIELDS
7
2. Proof of Theorems 1.1, 1.3 and 1.5, and Proposition 1.7
As a preparation, we need a few basic facts about tame fields. For the following
lemma, see [13, Lemma 3.7]:
Lemma 2.1. Take a tame field (L, v). If K is a relatively algebraically closed
subfield of L such that Lv|Kv is algebraic, then (K, v) is a tame field, vL/vK is
torsion free, and Lv = Kv.
We derive:
Corollary 2.2. Take a tame field (K, v) and an ordered abelian group Γ ⊂ vK
such that vK/Γ is torsion free. Then there exists a tame subfield (K ′ , v) of (K, v)
with vK ′ = Γ and K ′ v = Kv.
Proof. Denote the prime field of K by K0 and note that k0 := K0 v is the prime field
of Kv. Take a maximal system γi , i ∈ I, of elements in Γ rationally independent
over vK0 . Choose elements xi ∈ K such that vxi = γi , i ∈ I. Further, take a
transcendence basis tj , j ∈ J, of Kv over its prime field, and elements yj ∈ K such
that yj v = tj for all j ∈ J. For L
K1 := K0 (xi , yj | i ∈ I , j ∈ J) we obtain from [13,
Lemma 2.2] that vK1 = vK ⊕ i∈I γi Z and K1 v = k0 (tj | j ∈ J), so that Γ/vK1
is a torsion group and Kv|K1 v is algebraic.
Now we take K ′ to be the relative algebraic closure of K1 in K. Then by
Lemma 2.1, (K ′ , v) is a tame field with vK/vK ′ torsion free and K ′ v = Kv. Since
Γ ⊆ vK and Γ/vK2 is a torsion group, we have that Γ ⊆ vK ′ . Since vK/Γ is
torsion free, we also have that vK ′ ⊆ Γ, so that vK ′ = Γ.
Lemma 2.3. Take a tame field (K, v) and an ordered abelian group ∆ containing
vK such that ∆ is p-divisible, where p is the characteristic exponent of Kv. Then
there exists a tame extension field (L, v) of (K, v) with vL = ∆ and Lv = Kv.
Proof. By Theorem 2.14 of [9] there is an extension (K1 , v) of (K, v) such that
vK1 = ∆ and K1 v = Kv. We take (L, v) to be a maximal immediate algebraic
extension of (K1 , v); then (L, v) is algebraically maximal. Since vL = vK1 = ∆ is
p-divisible, and Lv = K1 v = Kv is perfect by [13, Theorem 3.2] applied to (K, v),
it follows from the same theorem that (L, v) is a tame field.
Now we can give the
Proof of Theorem 1.3: Since Γ ≺ vK by assumption, we have that vK/Γ is
torsion free. Hence by Corollary 2.2 we find a tame subfield (K ′ , v) of (K, v) with
vK ′ = Γ and K ′ v = Kv. Again since Γ ≺ vK, it follows from Theorem 1.2 that
(K ′ , v) ≺ (K, v).
Since (K ′ , v) is a tame field, we know that Γ = vK ′ is p-divisible. As Γ ≺ ∆, the
same holds for ∆. Hence by Lemma 2.3 we can find a tame extension field (L, v)
of (K ′ , v) with vL = ∆ and Lv = K ′ v. Since vK ′ = Γ ≺ ∆ = vL, it follows again
from Theorem 1.2 that (K ′ , v) ≺ (L, v).
Theorem 1.3 is the key to the
Proof of Theorem 1.1: In view of Theorems 1.2 and 4.1 of [2], we only have to
show that if (K, v) is algebraically complete with divisible value group and large
perfect residue field, then (K, v) is extremal. Note that (K, v) is then a tame field,
being algebraically complete with perfect residue field and p-divisible value group.
Every trivially valued field is extremal, so we may assume that (K, v) is nontrivially valued. We apply Corollary 1.4 with ∆ = R to obtain a tame field
8
SYLVY ANSCOMBE AND FRANZ-VIKTOR KUHLMANN
(L, v) ≡ (K, v) with value group vL = R. By the proof of Theorem 1.2 in [2],
this field is extremal. Since extremality is an elementary property, also (K, v) is
extremal.
We turn to the
Proof of Theorem 1.5: We extend the p-adic valuation vp of Q to some valuation
v on the algebraic closure of Q. Adjoining a primitive p-th root of unity ζp to Q
and passing to the henselization K := Q(ζp )h = Qh (ζp ), we obtain that vK =
1
p−1 (vp p)Z and Kv = Qvp = Fp .
By general ramification theory, the Galois extension Fpp |Fp can be lifted to
a Galois extension of degree p of K. Since K contains the p-th roots of unity,
Kummer theory shows that this extension is generated by an arbitrary p-th root of
some element b ∈ K.
Now we take L0 (respectively, F0 ) to be the Galois extension of K generated by
1
1
(vp p)Z ⊆ vL0 and p(p−1)
(vp p)Z ⊆ vF0 ,
a p-th root of bp (resp., of p). Then p(p−1)
and since
1
[L0 : K] = [F0 : K] = p =
(vp p)Z : vK ,
p(p − 1)
the fundamental inequality n ≥ ef shows that
vF0 = vL0 =
1
(vp p)Z and L0 v = F0 v = Fp .
p(p − 1)
Since both (L0 , v) and (F0 , v) are henselian fields of characteristic 0 with value
group isomorphic to Z, they are algebraically complete. Hence by [2, Theorem 4.1],
both fields are extremal.
Next, in order to construct (L1 , v) and (F1 , v), we choose algebraic extensions
(L′1 , v) of (L0 , v) and (F1′ , v) of (F0 , v) such that vL′1 = vF1′ is the p-divisible hull
1
(vp p)Z, and L′1 v = L0 v = Fp = F0 v = F1′ v; this is
of vL0 = vF0 and hence of p−1
possible by [9, Theorem 2.14].
Now we take (L1 , v) (resp., (F1 , v)) to be a maximal immediate algebraic extension of (L′1 , v) (resp., (F1′ , v)). Then by [13, Theorem 3.2], (L1 , v) and (F1 , v) are
tame fields. Their value groups and residue fields are as in the assertion of part b)
of Theorem 1.5.
Finally, in order to construct (L2 , v) and (F2 , v), we choose an arbitrary nontrivially valued henselian and perfect field (k, w) of characteristic p such that kw =
Fp . (For example, we could take the power series field Fp ((tQ )) for k and the t-adic
valuation for w; but also the much smaller relative algebraic closure of Fp (t) in
Fp ((tQ )) works.) Using [9, Theorem 2.14] again, we construct extensions (L′2 , v)
of (L1 , v) and (F2′ , v) of (F1 , v) such that vL′2 = vF2′ = Q and L′2 v = F2′ v = k.
As before, we take (L2 , v) (resp., (F2 , v)) to be a maximal immediate algebraic
extension of (L′2 , v) (resp., (F2′ , v)). Then again by [13, Theorem 3.2], (L2 , v) and
(F2 , v) are tame fields. Since their residue field k admits a nontrivial henselian
valuation, it is a large field. Hence by Theorem 1.1, (L2 , v) and (F2 , v) are also
extremal.
It remains to show that (Li , v) and (Fi , v) are not elementarily equivalent, for
i = 1, 2, 3. Assume the contrary. Then L0 and F0 or L1 and F1 would be isomorphic
over Q, as all of them are algebraic over Q. Likewise, if (L2 , v) and (F2 , v) are
NOTES ON EXTREMAL AND TAME VALUED FIELDS
9
elementarily equivalent then we obtain an isomorphism of the algebraic parts of
L2 and F2 over Q. In all three cases, this yields an embedding of F0 in L2 and
hence the existence of all p-th roots of p in L2 . But L2 also contains a p-th root
of bp, hence a p-th root of b as well. This however contradicts the fact that by
construction, (L2 v)w does not contain Fpp .
We conclude this section with the
Proof of Proposition 1.7: In both directions we assume that Kv is perfect.
First we assume that (K, v) is extremal and vK is divisible. By the compatibility
of “divisible value group” with composition, both wK and w̄(Kw) are divisible.
Theorem 1.1 shows that (K, v) is algebraically complete and that Kv = (Kw)w̄
is large. By the compatibility of “algebraically complete” with composition, both
(K, w) and (Kw, w̄) are algebraically complete. The latter has a large perfect
residue field, hence by Theorem 1.1, it is extremal. As in addition its value group
is divisible and its residue field is perfect, it is itself perfect. Since Kw carries the
nontrivial henselian valuation w̄, it is large (see e.g. [10, Proposition 16]). Therefore,
also (K, w) has a large perfect residue field, and again it follows from Theorem 1.1
that it is extremal.
For the converse, we assume that both (K, w) and (Kw, w̄) are extremal with
divisible value group. By Theorem 1.1, both are algebraically complete, with large
residue fields. By compatibility it follows that (K, v) is algebraically complete with
divisible value group. We know that Kv = (Kw)w̄ is large, and it is also perfect
by assumption. Now Theorem 1.1 shows that (K, v) is extremal.
3. Additive polynomials over extremal fields
We start by introducing a more precise notion of extremality. Take a valued field
(K, v), a subset S of K, and a polynomial f in n variables over K. Then we say
that (K, v) is S-extremal with respect to f if the set vf (S n ) ⊆ vK ∪ {∞} has
a maximum. We say that (K, v) is S-extremal if it is S-extremal with respect
to every polynomial in any finite number of variables. With this notation, (K, v)
being extremal means that it is O-extremal, where O denotes the valuation ring of
(K, v).
A subset A of a valued field (K, v) has the optimal approximation property
if for every z ∈ K there is some y ∈ A such that v(z − y) = max{v(z − x) | x ∈ A}.
A polynomial h ∈ K[X1 , . . . , Xn ] is called a p-polynomial if it is of the form f + c,
where f ∈ K[X1 , . . . , Xn ] is an additive polynomial and c ∈ K. The proof of the
following observation is straightforward:
Lemma 3.1. The images of all additive polynomials over (K, v) have the optimal approximation property if and only if K is K-extremal with respect to all ppolynomials over K.
We will work with ultrametric balls
Bα (a) := {b ∈ K | v(a − b) ≥ α} ,
where α ∈ vK and a ∈ K. Observe that O = B0 (0). We note:
Proposition 3.2. Take α, β ∈ vK and a, b ∈ K. Then (K, v) is Bα (a)-extremal
if and only if it is Bβ (b)-extremal. In particular, (K, v) is Bα (a)-extremal if and
only if it is extremal.
10
SYLVY ANSCOMBE AND FRANZ-VIKTOR KUHLMANN
Proof. It suffices to prove that “Bα (a)-extremal” implies “Bβ (b)-extremal”. Take a
polynomial f in n variables. If c ∈ K is such that vc = β −α, then the function y 7→
c(y − a) + b establishes a bijection from Bα (a) onto Bβ (b). We set g(y1 , . . . , yn ) :=
f (c(y1 − a) + b, . . . , c(yn − a) + b). It follows that f (Bβ (b)n ) = g(Bα (a)n ), whence
vf (Bβ (b)n ) = vg(Bα (a))n . Hence if (K, v) is Bα (a)-extremal with respect to g, then
it is Bβ (b)-extremal with respect to f . This yields the assertions of the proposition.
A valued field (K, v) of characteristic p > 0 is called inseparably defectless if
every finite purely inseparable extension (L|K, v) satisfies equation (1) (note that
the extension of v from K to L is unique). This holds if and only if every finite
subextension of (K|K p , v) satisfies equation (1).
If (K, v) is inseparably defectless with [K : K p ] < ∞, then for every ν ≥ 1, the
ν
extension (K|K p , v) has a valuation basis, that is, a basis of elements b1 , . . . , bℓ
ν
that are valuation independent over K p , i.e.,
v(c1 b1 + . . . + cℓ bℓ ) = min vci bi
1≤i≤ℓ
pν
for all c1 , . . . , cℓ ∈ K .
Note that every algebraically complete valued field is in particular inseparably
defectless. By Theorem 1.1, every extremal field is algebraically complete and hence
inseparably defectless.
Proposition 3.3. Take an inseparably defectless valued field (K, v) with [K : K p ] <
∞ and an additive polynomial f in n variables over K. Then for some integer ν ≥ 0
there are additive polynomials g1 , . . . , gm ∈ K[X] in one variable such that
a) f (K n ) = g1 (K) + . . . + gm (K),
b) all polynomials gi have the same degree pν ,
c) the leading coefficients b1 , . . . , bm of g1 , . . . , gm are valuation independent over
ν
Kp .
Proof. The proof can be taken over almost literally from Lemma 4 of [3]. One only
has to replace the elements 1, t, . . . , tδi −1 from that proof by an arbitrary basis of
K|K δi .
The following theorem is a reformulation of Theorem 1.9 of the Introduction.
Theorem 3.4. Assume that (K, v) is an extremal field of characteristic p > 0 with
[K : K p ] < ∞. Then it is K-extremal w.r.t. all p-polynomials and therefore, the
images of all additive polynomials have the optimal approximation property.
Proof. Take a p-polynomial h in n variables over K, and write it as h = f + c with
f an additive polynomial in n variables over K and c ∈ K. We choose additive
polynomials g1 , . . . , gm ∈ K[X] in one variable satisfying assertions a), b), c) of
Proposition 3.3. Then h(K n ) = g1 (K) + . . . + gm (K) + c.
ν
ν−1
We write gi = bi X p + ci,ν−1 X p
+ . . . + ci,0 X for 1 ≤ i ≤ m. Then we choose
α ∈ vK such that
α < min{0, vc − vbi , vci,k − vbi | 1 ≤ i ≤ m , 0 ≤ k < ν} .
Because α < 0, it then follows that for each a with va ≤ α,
vbi + pν va ≤ vbi + pν α ≤ vbi + α < vc
NOTES ON EXTREMAL AND TAME VALUED FIELDS
11
and for 0 ≤ k < ν,
vbi + pν va ≤ vbi + va + pk va ≤ vbi + α + pk va < vci,k + pk va .
It then follows that
vgi (a) = vbi + pν va ≤ vbi + pν α < vc .
(3)
On the other hand, if va′ ≥ α, then vbi + pν va′ ≥ vbi + pν α and vci,k + pk va′ ≥
vci,k + pk α > vbi + pν α for 0 ≤ k < ν. This yields that
vgi (a′ ) ≥ vbi + pν α .
(4)
Now take any (a′1 , . . . , a′m ) ∈ Bα (0)n and (a1 , . . . , am ) ∈ K n \ Bα (0)n . So we
have:
min{va1 , . . . , vam } < α ≤ min{va′1 , . . . , va′m } .
ν
Since b1 , . . . , bm are valuation independent over K p , we then obtain from (3) and
(4) that
vh(a1 , . . . , am ) =
<
min vbi + pν vai
1≤i≤m
min vbi + pν α ≤ vh(a′1 , . . . , a′m ) .
1≤i≤m
This proves that
vh(Bα (0)n ) > vh(K n \ Bα (0)n ) .
Since (K, v) is extremal by assumption, Proposition 3.2 shows that vh(Bα (0)n ) has
a maximal element, and the same is consequently true for vh(K n ). This shows that
(K, v) is K-extremal w.r.t. h, from which the first assertion follows. The second
assertion follows by Lemma 3.1.
4. More constructions of extremal fields, and proof of
Theorem 1.13
It follows from [2, Theorem 4.1] that the Laurent series fields (Fp ((t)), vt ) and
the p-adic fields (Qp , vp ) are extremal. The former have equal characteristic, the
latter mixed characteristic. All of them have Z as their value group, which is a
Z-group.
In [8] a valued field extension (L, v) of (Fp ((t)), vt ) is presented in which not all
images of additive polynomials have the optimal approximation property. In [2] it
is shown that (L, v) is not extremal, although it is algebraically complete and its
value group vL is a Z-group (of rank 2). It is also shown that for the nontrivial
coarsening w of v corresponding to the convex subgroup (vt t)Z of vL, also (L, w)
is not extremal. As a coarsening of an algebraically complete valuation, it is also
algebraically complete. Its value group wL = vL/(vt t)Z is divisible and its residue
field Lw = Fp ((t)) is large, but not perfect. Note that (L, v) and (L, w) are of equal
characteristic.
In order to prove the remaining existence statements of Proposition 1.12 concerning non-extremal fields in mixed characteristic, we consider compositions of
valuations. Unfortunately, contrary to the assertion that the proof of Lemma 5.2 of
[2] is easy (and thus left to the reader), we are unable to prove it in the cases that
are not covered by Proposition 1.7. (However, we also do not know of any counterexample.) In fact, a slightly different version can easily be proved: If (K, v) is
Ov -extremal, then also (K, w) is Ov -extremal. We do not know whether the latter
12
SYLVY ANSCOMBE AND FRANZ-VIKTOR KUHLMANN
impies that (K, w) is Ow -extremal. Proposition 3.2 is of no help here because Ov
is in general not a ball of the form Bα (a) in (K, w).
It appears, though, that we actually had in mind the following result, which is
indeed easy to prove:
Lemma 4.1. If (K, v) is extremal and v = w ◦ w, then (Kw, w) is extremal.
Proof. Assume that (K, v) is extremal with v = w ◦ w; note that for any a, b ∈ Ow ,
w(aw) > w(bw) implies va > vb.
Assume further that g ∈ Kw[X1 , . . . , Xn ]. Then choose f ∈ Ow [X1 , . . . , Xn ]
such that f w = g. By assumption, there are b1 , . . . , bn ∈ Ov such that
vf (b1 , . . . , bn ) = max{vf (a1 , . . . , an ) | a1 , . . . , an ∈ Ov } .
Since b1 , . . . , bn ∈ Ov ⊆ Ow we have that
f (b1 , . . . , bn )w = f w(b1 w, . . . , bn w) = g(b1 w, . . . , bn w) .
We claim that
wg(b1 w, . . . , bn w) = max{wg(a1 , . . . , an ) | a1 , . . . , an ∈ Ow } .
Indeed, if there were a1 , . . . , an ∈ Ow with wg(a1 , . . . , an ) > wg(b1 w, . . . , bn w),
then for any choice of a1 , . . . , an ∈ Ow with ai w = ai for 1 ≤ i ≤ n we would obtain
that a1 , . . . , an ∈ Ov and vf (a1 , . . . , an ) > vf (b1 , . . . , bn ), a contradiction.
We use this lemma to prove the existence of the non-extremal fields in mixed
characteristic as claimed in Proposition 1.12. We consider again the two nonextremal fields (L, v) and (L, w) mentioned above. By Theorem 2.14 of [9] there is
an extension (K0 , v0 ) of (Q, vp ) with divisible value group and L as its residue field.
We replace (K0 , v0 ) by a maximal immediate extension (M, v0 ). Then (M, v0 ) is
algebraically complete, and so are (M, v0 ◦ v) and (M, v0 ◦ w). The value group of
(M, v0 ◦ v) is a Z-group, and (M, v0 ◦ w) has divisible value group and nonperfect
large residue field. But by Lemma 4.1, both fields are non-extremal.
Finally, we have to prove the existence of extremal fields as stated in parts c)
and d) of Proposition 1.12. We will employ Theorem 1.13 which we will prove
now. We note that by Theorem 1.1, the residue field of an extremal field with
divisible value group must be large. Also, every non-trivially valued extremal field
is henselian, which implies that it is itself a large field. Therefore, it remains to
prove the following assertion:
Let (K, v) be any ℵ1 -saturated valued field. Assume that Γ and ∆ are convex subgroups of vK such that ∆ ⊂
6= Γ and Γ/∆ is archimedean. Let u (respectively w)
be the coarsening of v corresponding to ∆ (resp. Γ). Denote by ū the valuation
induced on Kw by u. Then (Kw, ū) is maximal and extremal, and its value group
is isomorphic either to Z or to R.
Proof. Denote by Ou (resp. Ow ) the valuation ring corresponding to u (resp. w).
We note that the value group of u is vK/∆, the value group of w is vK/Γ, and we
have that
Ov ⊂ Ou ⊂ Ow .
We show first that (Kw, ū) is maximal. From [6, Theorem 4] we know that a
valued field is maximal if and only if every pseudo Cauchy sequence has a limit
in the field. We refer the reader to [6] for an excellent introduction to the theory
NOTES ON EXTREMAL AND TAME VALUED FIELDS
13
of pseudo Cauchy sequences (which Kaplansky calls “pseudo-convergent sets”). If
(ai )i<λ is any pseudo Cauchy sequence in (Kw, ū), then the sequence ū(ai+1 − ai )
in ū(Kw) = Γ/∆ is strictly increasing. But the cofinality of any strictly increasing
sequence in R (and hence also in any archimedean ordered abelian group) is at
most ω. Therefore, it suffices to show that every pseudo Cauchy sequence (ai )i<ω
in (Kw, ū) has a limit. By definition, a ∈ Kw is a limit of this sequence if and only
if ū(a − ai ) = ū(ai+1 − ai ) for all i < ω.
Write ai = bi w with bi ∈ Ow , i < ω. Then the sequence (u(bi+1 − bi ))i<ω is
strictly increasing in vK/∆. This implies that the sequence (v(bi+1 − bi ))i<ω is
strictly increasing in vK. We consider the following (partial) type in countably
many parameters:
{v(x − bi ) = v(bi+1 − bi ) | i < ω} .
It is finitely realizable in (K, v) since for x = bi+1 we obtain that v(x − bj ) =
v(bj+1 − bj ) holds for 0 ≤ j ≤ i. By saturation, there is some b ∈ K which realizes
this type. Now v(b − bi ) = v(bi+1 − bi ) implies that
w(b − bi ) = v(b − bi ) + Γ = v(bi+1 − bi ) + Γ = w(bi+1 − bi ) ,
u(b − bi ) = v(b − bi ) + ∆ = v(bi+1 − bi ) + ∆ = u(bi+1 − bi ) .
The former implies that b ∈ Ow as also all bi are in Ow ; so we can set a := bw.
The latter then implies that
ū(a − ai ) = ū(bw − bi w) = ū((b − bi )w) = u(b − bi ) = u(bi+1 − bi ) = ū(ai+1 − ai ) .
This proves that a ∈ Kw is a limit of the pseudo Cauchy sequence (ai )i<ω and
shows that (Kw, ū) is maximal.
Now we distinguish two cases.
Case 1: ū(Kw) is isomorphic to Z. In this case, it follows from the maximality that
(Kw, ū) is algebraically complete and hence extremal [2, by Theorem 4.1].
Case 2: ū(Kw) = Γ/∆ is densely ordered. Note that since the archimedean ordered
group Γ/∆ is embeddable in R, any subset of it has coinitiality and cofinality no
greater than ℵ0 .
We show that (Kw, ū) is extremal. The value group ū(Kw) is Γ/∆ and Oū is
the image of Ou under the residue map x 7→ xw of w. For a tuple a = (a1 , ..., am )
from Ow , we denote by aw := (a1 w, ..., am w) the corresponding tuple of residues.
Let f¯ ∈ Kw[x] be a polynomial in the variables x = (x1 , ..., xm ) and let f ∈
Ow [x] denote any lift of f¯ so that f w = f¯. We must show that the set of ū-values
of the image of f¯, i.e.,
X := ū(f¯(b)) ∈ Γ/∆ ∪ {∞} | b ∈ Oū
= ū(f¯(aw)) ∈ Γ/∆ ∪ {∞} | a ∈ Ou ,
has a maximum. As noted above, the cofinality of X is no greater than ℵ0 . Thus
there is a sequence (an )n<ω of m-tuples from Ou such that the sequence
(ū(f¯(an w))n<ω
is increasing and cofinal in X. For each n < ω we set αn := v(f (an )), and note
that either f¯(an w) = 0 (in which case ū(f¯(an w)) = ∞ must be the maximum of
X) or
αn + ∆ = u(f (an )) = ū(f¯(an w)).
14
SYLVY ANSCOMBE AND FRANZ-VIKTOR KUHLMANN
Next we set Y := {γ + ∆ ∈ Γ/∆ | γ + ∆ < ∆}. Then Y is equal to the
image under u of the elements of Ow \ Ou (and also equal to the image under ū of
Kw \ Oū ). By assumption, Γ/∆ is densely ordered; thus Y has no maximum. Also
the cofinality of Y can be no greater than ℵ0 . Thus there is a sequence (βn )n<ω in
Γ such that (βn + ∆)n<ω is a strictly increasing and cofinal sequence in Y .
Finally we consider the following (partial) x-type in countably many parameters:
p(x) := {αn ≤ v(f (x)) | n < ω } ∪ {βn ≤ v(xi ) | n < ω, 1 ≤ i ≤ m } .
This is finitely realised in (K, v). By saturation, it is realized by some m-tuple
c = (c1 , ..., cm ) ∈ K m .
For 1 ≤ i ≤ m we examine the second set of formulas in p(x) to find that
βn ≤ v(ci ), for each n < ω. Thus βn + ∆ ≤ v(ci ) + ∆ = u(ci ), again for each n < ω.
By the cofinality of the sequence (βn + ∆)n<ω in Y we have that ci ∈ Ou .
Finally, by examining the first set of formulas in p(x), we see that αn ≤ v(f (c)),
for all n < ω. Then either ū(f¯(cw)) = ∞ (in which case ∞ is the maximum of X)
or we have that
αn + ∆ ≤ v(f (c)) + ∆ = u(f (c)) = ū(f¯(cw)),
for all n < ω. Since (αn + ∆) is cofinal in X, ū(f¯(cw)) is the maximum of X. This
shows that (Kw, ū) is extremal, as required.
For the conclusion of the proof, we show that the value group of (Kw, ū) is cut
complete, which shows that it is isomorphic to R. Take a Dedekind cut (D, E) in
ū(Kw), that is, D is a nonempty initial segment of ū(Kw) and E is a nonempty final
segment of ū(Kw) such that D ∪ E = ū(Kw). As noted before, the cofinality of D
and the coinitiality of E are no greater than ℵ0 . Thus there are sequences (βn )n<ω
and (γn )n<ω in Γ such that (βn + ∆)n<ω is an increasing and cofinal sequence in
D and (γn + ∆)n<ω is a decreasing and coinitial sequence in E. We consider the
following (partial) type in countably many parameters:
{βn ≤ vx | n < ω} ∪ {γn ≥ vx | n < ω} .
This is finitely realized in (K, v). Hence by saturation, it is realized by some d ∈ K.
Then βn ≤ vd ≤ γn and therefore βn + ∆ ≤ ud ≤ γn + ∆ , for each n < ω. It
follows that ud lies in the convex hull of Γ/∆ in vK/∆, which shows that wd = 0.
So dw ∈ Kw, and we obtain that
D ≤ ū(dw) = ud ≤ E ,
which proves that the cut (D, E) is realized in ū(Kw), showing that this group is
cut complete.
We may choose (K, v) so that Γ/∆ is densely ordered, for any ∆ ⊂ Γ ⊂ vK.
Indeed, if we take any integer n ≥ 2 and (K, v) such that vK is n-divisible, then
also Γ/∆ will be n-divisible and hence densely ordered. If on the other hand, the
residue field Kv is imperfect and w ⊂ u ⊂ v are as in the theorem, then also the
residue field of (Kw, ū), which is equal to Ku, is imperfect. Taking (K, v) to be an
ℵ1 -saturated valued field of equal characteristic p with imperfect large residue field
and n-divisible value group, and choosing Γ and ∆ according to Remark 1.14, we
obtain from Theorem 1.13:
Corollary 4.2. Let p be a prime. There exist extremal fields of equal characteristic
p with value group isomorphic to R and imperfect residue field.
NOTES ON EXTREMAL AND TAME VALUED FIELDS
15
To give an example of an extremal field obtained by this corollary, we begin
by
ℵ1 -saturated elementary extension (K, v) of the Puiseux series field
S taking any 1/n
F
(x)((t
)) over Fp (x), where x is transcendental over Fp . As the residue
p
n∈N
field Kv is an elementary extension of the lower residue field, it is also imperfect.
As the value group vK is an elementary extension of the lower value group, it is
also divisible.
On the other hand, we can extend the p-adic valuation from Q to a valuation v
on Q(x) such that xv is transcendental over Fp ; then the residue field of (Q(x), v)
will be the imperfect field Fp (xv). By adjoining n-th roots repeatedly, we can pass,
without changing the residue field, to an algebraic extension (k, v) of (Q(x), v) with
n-divisible value group. Now we can take any ℵ1 -saturated elementary extension
(K, v) of (k, v). Then Kv will again be imperfect, vK will be n-divisible, and (K, v)
will have mixed characteristic (0, p).
In order to achieve that the valued field (Kw, ū) in Theorem 1.13 also has mixed
characteristic, we choose Γ and ∆ as follows. We take ∆ to be the largest convex
subgroup of vK not containing vp and let Γ be the smallest convex subgroup of vK
containing vp. Then ∆ is the largest proper convex subgroup of Γ, and therefore
Γ/∆ is archimedean. It follows that pw 6= 0 since vp ∈
/ ∆, but (pw)ū = pu = 0
since vp ∈ Γ. This shows that char Kw = 0 and char (Kw)ū = p. We thus obtain:
Corollary 4.3. Let p be a prime. There exist extremal fields of mixed characteristic
(0, p) with value group isomorphic to R and imperfect residue field.
Corollaries 4.2 and 4.3 together complete the proof of Proposition 1.12.
By taking (K, v) as in one of these corollaries and (L, v) to be a countable model
of Th (K, v), we obtain:
Corollary 4.4. Let p be a prime. There exist countable extremal fields of equal
characteristic p with divisible value group not isomorphic to R and imperfect residue
field. Likewise, there exist countable extremal fields of mixed characteristic (0, p)
with divisible value group not isomorphic to R and imperfect residue field.
By choosing models of arbitrary cardinality, one can obtain divisible value groups
of arbitrarily large cardinality. But we do not know which divisible ordered abelian
groups (and not even which cardinalities) can be thus obtained, as we are lacking
an AKE-principle.
We will now give the
Proof of Corollary 1.15: We take (K, v) to be an ℵ1 -saturated elementary
extension of an arbitrary non-large valued field whose value group is divisible by
some n ≥ 2, and apply Theorem 1.13. Since also vK is divisible by n, for all u and
w as in the theorem the value group of (Kw, ū) is divisible. Hence if v = w1 ◦w2 ◦w3
with w2 of rank 1, then by setting w = w1 and u = w1 ◦ w2 it follows from the
theorem that (Kw1 , w2 ) is extremal with nontrivial divisible value group, hence
a) w2 is henselian,
b) Kw1 is large,
c) (Kw1 )w2 is large.
For the conclusion of this paper, let us discuss how the property of extremality
behaves in a valued field extension (L|K, v) where (K, v) is existentially closed in
(L, v). In this case, it is known that L|K and Lv|Kv are regular extensions and
16
SYLVY ANSCOMBE AND FRANZ-VIKTOR KUHLMANN
that vL/vK is torsion free. (An extension L|K of fields is called regular if it is
separable and K is relatively algebraically closed in L.)
Proposition 4.5. Take a valued field extension (L|K, v) such that (K, v) is existentially closed in (L, v), a subset SK of K that is existentially definable with
parameters in K, and a polynomial f in n variables over K. Denote by SL the subset of L defined by the existential formula that defines SK in K. Then the following
assertions hold.
a) If (K, v) is SK -extremal w.r.t. f , then (L, v) is SL -extremal w.r.t. f and
n
max vf (SLn ) = max vf (SK
). In particular, if (K, v) is extremal, then (L, v) is
extremal w.r.t. all polynomials with coefficients in K.
b) Assume in addition that vL = vK. If (L, v) is SL -extremal w.r.t. f , then (K, v)
n
is SK -extremal w.r.t. f and max vf (SLn ) = max vf (SK
). In particular, if (L, v) is
extremal, then so is (K, v).
n
n
Proof. a): Assume that a ∈ SK
such that vf (a) = max vf (SK
). Then the assertion
n
that there exists an element b in SL such that vf (b) > vf (a) is an elementary
existential sentence with parameters in K. Hence if it held in L, then there would
n
be an element b′ in SK
such that vf (b′ ) > vf (a), which is a contradiction to the
n
choice of a. It follows that max vf (SLn ) ≤ max vf (SK
). Since SK ⊆ SL , we obtain
n
n
that max vf (SL ) = max vf (SK ).
b): Take b ∈ SLn such that vf (b) = max vf (SLn ). Since vL = vK by assumption,
there is c ∈ K such that vc = vf (b). Now the assertion that there exists an
element b in SLn such that vf (b) = vc is an elementary existential sentence with
n
parameters in K. Hence there is a ∈ SK
such that vf (a) = vc = max vf (SLn ).
n
n
n
Since vf (a) ∈ vf (SK ) ⊆ vf (SL ), we obtain that vf (a) = max vf (SK
).
References
[1] Anscombe, S. – Fehm, A.: The existential theory of equicharacteristic henselian valued fields,
http://arxiv.org/abs/1501.04522 (2015)
[2] Azgin, S. – Kuhlmann, F.-V. – Pop, F.: Characterization of Extremal Valued Fields, Proc.
Amer. Math. Soc. 140 (2012), 1535–1547
[3] van den Dries, L. – Kuhlmann, F.-V.: Images of additive polynomials in Fq ((t)) have the
optimal approximation property, Can. Math. Bulletin 45 (2002), 71–79
[4] Denef, J. – Schoutens, H.: On the decidability of the existential theory of Fp [[t]], Valuation
theory and its applications, Vol. II (Saskatoon, SK, 1999), 43?60, Fields Inst. Commun., 33,
Amer. Math. Soc., Providence, RI, 2003
[5] Durhan, S.: Additive Polynomials over Perfect Fields, in: Valuation Theory in Interaction,
Proceedings of the Second International Valuation Theory Conference, Segovia / ElEscorial
2011, EMS Series of Congress Reports 2014
[6] Kaplansky, I.: Maximal fields with valuations I, Duke Math. Journ. 9 (1942), 303–321
[7] Kuhlmann, F.-V.: Quantifier elimination for henselian fields relative to additive and multiplicative congruences, Israel J. Math. 85 (1994), 277–306
[8] Kuhlmann, F.-V.: Elementary properties of power series fields over finite fields, J. Symb.
Logic, 66, 771–791 (2001)
[9] Kuhlmann, F.-V.: Value groups, residue fields and bad places of rational function fields,
Trans. Amer. Math. Soc. 356 (2004), 4559–4600
[10] Kuhlmann, F.-V.: On places of algebraic function fields in arbitrary characteristic, Advanves
in Math. 188 (2004), 399–424
[11] Kuhlmann, F.–V.: Additive polynomials and their role in the model theory of valued fields,
Logic in Tehran, Lecture Notes in Logic 26 160–203, Assoc. Symbol. Logic, La Jolla, CA,
2006
[12] Kuhlmann, F.–V.: Dense subfields of Henselian fields, and integer parts, Logic in Tehran,
Lecture Notes in Logic 26 204?-226, Assoc. Symbol. Logic, La Jolla, CA, 2006
NOTES ON EXTREMAL AND TAME VALUED FIELDS
17
[13] Kuhlmann, F.-V.: The algebra and model theory of tame valued fields, to appear in: J.
reine angew. Math. Preliminary version published in: Séminaire de Structures Algébriques
Ordonnées, 81, Prépublications Paris 7 (2009)
[14] Kuhlmann, F.-V.: A classification of Artin-Schreier defect extensions and a characterization
of defectless fields, Illinois J. Math. 54 (2010), 397–448
[15] Pop, F.: Embedding problems over large fields, Annals of Math. 144 (1996), 1–34.
Jeremiah Horrocks Institute, Leighton Building Le7, University of Central Lancashire, Preston, PR1 2HE, United Kingdom
E-mail address: sanscombe@uclan.ac.uk
Institute of Mathematics, University of Silesia, ul. Bankowa 14, 40-007 Katowice,
Poland
E-mail address: fvk@math.us.edu.pl
| 0 |
arXiv:1701.07775v1 [q-bio.NC] 26 Jan 2017
A Forward Model at Purkinje Cell Synapses
Facilitates Cerebellar Anticipatory Control
Ivan Herreros-Alonso
SPECS lab
Universitat Pompeu Fabra
Barcelona, Spain
ivan.herreros@upf.edu
Xerxes D. Arsiwalla
SPECS lab
Universitat Pompeu Fabra
Barcelona, Spain
Paul F.M.J. Verschure
SPECS, UPF
Catalan Institution of Research
and Advanced Studies (ICREA)
Barcelona, Spain
Abstract
How does our motor system solve the problem of anticipatory control in spite
of a wide spectrum of response dynamics from different musculo-skeletal systems, transport delays as well as response latencies throughout the central nervous
system? To a great extent, our highly-skilled motor responses are a result of a
reactive feedback system, originating in the brain-stem and spinal cord, combined
with a feed-forward anticipatory system, that is adaptively fine-tuned by sensory
experience and originates in the cerebellum. Based on that interaction we design
the counterfactual predictive control (CFPC) architecture, an anticipatory adaptive
motor control scheme in which a feed-forward module, based on the cerebellum,
steers an error feedback controller with counterfactual error signals. Those are
signals that trigger reactions as actual errors would, but that do not code for any current or forthcoming errors. In order to determine the optimal learning strategy, we
derive a novel learning rule for the feed-forward module that involves an eligibility
trace and operates at the synaptic level. In particular, our eligibility trace provides
a mechanism beyond co-incidence detection in that it convolves a history of prior
synaptic inputs with error signals. In the context of cerebellar physiology, this
solution implies that Purkinje cell synapses should generate eligibility traces using
a forward model of the system being controlled. From an engineering perspective,
CFPC provides a general-purpose anticipatory control architecture equipped with a
learning rule that exploits the full dynamics of the closed-loop system.
1
Introduction
Learning and anticipation are central features of cerebellar computation and function (Bastian, 2006):
the cerebellum learns from experience and is able to anticipate events, thereby complementing a
reactive feedback control by an anticipatory feed-forward one (Hofstoetter et al., 2002; Herreros
and Verschure, 2013). This interpretation is based on a series of anticipatory motor behaviors that
originate in the cerebellum. For instance, anticipation is a crucial component of acquired behavior in
eye-blink conditioning (Gormezano et al., 1983), a trial by trial learning protocol where an initially
neutral stimulus such as a tone or a light (the conditioning stimulus, CS) is followed, after a fixed
delay, by a noxious one, such as an air puff to the eye (the unconditioned stimulus, US). During early
trials, a protective unconditioned response (UR), a blink, occurs reflexively in a feedback manner
following the US. After training though, a well-timed anticipatory blink (the conditioned response,
CR) precedes the US. Thus, learning results in the (partial) transference from an initial feedback
action to an anticipatory (or predictive) feed-forward one. Similar responses occur during anticipatory
postural adjustments, which are postural changes that precede voluntary motor movements, such
as raising an arm while standing (Massion, 1992). The goal of these anticipatory adjustments is to
counteract the postural and equilibrium disturbances that voluntary movements introduce. These
30th Conference on Neural Information Processing Systems (NIPS 2016), Barcelona, Spain.
behaviors can be seen as feedback reactions to events that after learning have been transferred to
feed-forward actions anticipating the predicted events.
Anticipatory feed-forward control can yield high performance gains over feedback control whenever
the feedback loop exhibits transmission (or transport) delays (Jordan, 1996). However, even if a
plant has negligible transmission delays, it may still have sizable inertial latencies. For example,
if we apply a force to a visco-elastic plant, its peak velocity will be achieved after a certain delay;
i.e. the velocity itself will lag the force. An efficient way to counteract this lag will be to apply
forces anticipating changes in the desired velocity. That is, anticipation can be beneficial even when
one can act instantaneously on the plant. Given that, here we address two questions: what is the
optimal strategy to learn anticipatory actions in a cerebellar-based architecture? and how could it be
implemented in the cerebellum?
To answer that we design the counterfactual predictive control (CFPC) scheme, a cerebellar-based
adaptive-anticipatory control architecture that learns to anticipate performance errors from experience.
The CFPC scheme is motivated from neuro-anatomy and physiology of eye-blink conditioning.
It includes a reactive controller, which is an output-error feedback controller that models brain
stem reflexes actuating on eyelid muscles, and a feed-forward adaptive component that models the
cerebellum and learns to associate its inputs with the error signals driving the reactive controller.
With CFPC we propose a generic scheme in which a feed-forward module enhances the performance
of a reactive error feedback controller steering it with signals that facilitate anticipation, namely,
with counterfactual errors. However, within CFPC, even if these counterfactual errors that enable
predictive control are learned based on past errors in behavior, they do not reflect any current or
forthcoming error in the ongoing behavior.
In addition to eye-blink conditioning and postural adjustments, the interaction between reactive
and cerebellar-dependent acquired anticipatory behavior has also been studied in paradigms such
as visually-guided smooth pursuit eye movements (Lisberger, 1987). All these paradigms can be
abstracted as tasks in which the same predictive stimuli and disturbance or reference signal are
repeatedly experienced. In accordance to that, we operate our control scheme in trial-by-trial (batch)
mode. With that, we derive a learning rule for anticipatory control that modifies the well-known
least-mean-squares/Widrow-Hoff rule with an eligibility trace. More specifically, our model predicts
that to facilitate learning, parallel fibers to Purkinje cell synapses implement a forward model that
generates an eligibility trace. Finally, to stress that CFPC is not specific to eye-blink conditioning, we
demonstrate its application with a smooth pursuit task.
2
2.1
Methods
Cerebellar Model
w1
x1
xj
wj
xN
wN
e
o
Figure 1: Anatomical scheme of a Cerebellar Purkinje cell. The xj denote parallel fiber inputs to
Purkinje synapses (in red) with weights wj . o denotes the output of the Purkinje cell. The error signal
e, through the climbing fibers (in green), modulates synaptic weights.
We follow the simplifying approach of modeling the cerebellum as a linear adaptive filter, while
focusing on computations at the level of the Purkinje cells, which are the main output cells of the
cerebellar cortex (Fujita, 1982; Dean et al., 2010). Over the mossy fibers, the cerebellum receives
a wide range of inputs. Those inputs reach Purkinke cells via parallel fibers (Fig. 1), that cross
2
dendritic trees of Purkinje cells in a ratio of up to 1.5 × 106 parallel fiber synapses per cell (Eccles
et al., 1967). We denote the signal carried by a particular fiber as xj , j ∈ [1, G], with G equal to the
total number of inputs fibers. These inputs from the mossy/parallel fiber pathway carry contextual
information (interoceptive or exteroceptive) that allows the Purkinje cell to generate a functional
output. We refer to these inputs as cortical bases, indicating that they are localized at the cerebellar
cortex and that they provide a repertoire of states and inputs that the cerebellum combines to generate
its output o. As we will develop a discrete time analysis of the system, we use n to indicate time (or
time-step). The output of the cerebellum at any time point n results from a weighted sum of those
cortical bases. wj indicates the weight or synaptic efficacy associated with the fiber j. Thus, we
|
|
have x[n] = [x1 [n], . . . , xG [n]] and w[n] = [w1 [n], . . . , wG [n]] (where the transpose, | , indicates
that x[n] and w[n] are column vectors) containing the set of inputs and synaptic weights at time n,
respectively, which determine the output of the cerebellum according to
o[n] = x[n]| w[n]
(1)
The adaptive feed-forward control of the cerebellum stems from updating the weights according to a
rule of the form
∆wj [n + 1] = f (xj [n], . . . , xj [1], e[n], Θ)
(2)
where Θ denotes global parameters of the learning rule; xj [n], . . . , xj [1], the history of its presynaptic inputs of synapse j; and e[n], an error signal that is the same for all synapses, corresponding
to the difference between the desired, r, and the actual output, y, of the controlled plant. Note that in
drawing an analogy with the eye-blink conditioning paradigm, we use the simplifying convention
of considering the noxious stimulus (the air-puff) as a reference, r, that indicates that the eyelids
should close; the closure of the eyelid as the output of the plant, y; and the sensory response to the
noxious stimulus as an error, e, that encodes the difference between the desired, r, and the actual
eyelid closures, y. Given this, we advance a new learning rule, f , that achieves optimal performance
in the context of eye-blink conditioning and other cerebellar learning paradigms.
2.2
Cerebellar Control Architecture
Cerebellum (cortex and nuclei)
and Inferior olive [FF]
Trigeminal
nucleus
[o]
[e]
[e]
[u]
+
-
[r]
US
(airpuff)
[y]
Facial
nucleus
[C]
Eyelids
(Blink)
[P]
x
ADAPTIVE-ANTICIPATORY
(FEED-FORWARD) LAYER
FF
o
[x]
r
Pons
+ e
-
+
C
u
P
y
REACTIVE
(FEEDBACK) LAYER
CS
(Context,
e.g.: sound, light)
FEEDBACK CLOSEDLOOP SYSTEM
Figure 2: Neuroanatomy of eye-blink conditioning and the CFPC architecture. Left: Mapping of
signals to anatomical structures in eye-blink conditioning (De Zeeuw and Yeo, 2005); regular arrows
indicate external inputs and outputs, arrows with inverted heads indicate neural pathways. Right:
CFPC architecture. Note that the feedback controller, C, and the feed-forward module, F F , belong
to the control architecture, while the plant, P , denotes an object controlled. Other abbreviations: r,
reference signal; y, plant’s output; e, output error; x, basis signals; o, feed-forward signal; and u,
motor command.
We embed the adaptive filter cerebellar module in a layered control architecture, namely the CFPC
architecture, based on the interaction between brain stem motor nuclei driving motor reflexes and
the cerebellum, such as the one established between the cerebellar microcircuit responsible for
conditioned responses and the brain stem reflex circuitry that produces unconditioned eye-blinks
(Hesslow and Yeo, 2002) (Fig. 2 left). Note that in our interpretation of this anatomy we assume
that cerebellar output, o, feeds the lower reflex controller (Fig. 2 right). Put in control theory terms,
within the CFPC scheme an adaptive feed-forward layer supplements a negative feedback controller
steering it with feed-forward signals.
3
Our architecture uses a single-input single-output negative-feedback controller. The controller
receives as input the output error e = r − y. For the derivation of the learning algorithm, we assume
that both plant and controller are linear and time-invariant (LTI) systems. Importantly, the feedback
controller and the plant form a reactive closed-loop system, that mathematically can be seen as a
system that maps the reference, r, into the plant’s output, y. A feed-forward layer that contains the
above-mentioned cerebellar model provides the negative feedback controller with an additional input
signal, o. We refer to o as a counter-factual error signal, since although it mechanistically drives the
negative feedback controller analogously to an error signal it is not an actual error. The counterfactual
error is generated by the feed-forward module that receives an output error, e, as its teaching signal.
Notably, from the point of view of the reactive layer closed-loop system, o can also be interpreted as
a signal that offsets r. In other words, even if r remains the reference that sets the target of behavior,
r + o functions as the effective reference that drives the closed-loop system.
3
3.1
Results
Derivation of the gradient descent update rule for the cerebellar control architecture
We apply the CFPC architecture defined in the previous section to a task that consists in following
a finite reference signal r ∈ RN that is repeated trial-by-trial. To analyze this system, we use the
discrete time formalism and assume that all components are linear time-invariant (LTI). Given this,
both reactive controller and plant can be lumped together into a closed-loop dynamical system, that
can be described with the dynamics A, input B, measurement C and feed-through D matrices. In
general, these matrices describe how the state of a dynamical system autonomously evolves with
time, A; how inputs affect system states, B; how states are mapped into outputs, C; and how inputs
instantaneously affect the system’s output D (Astrom and Murray, 2012). As we consider a reference
of a finite length N , we can construct the N -by-N transfer matrix T as follows (Boyd, 2008)
D
0
0
... 0
CB
D
0
... 0
CAB
CB
D
... 0
T =
..
..
..
..
..
.
.
.
.
.
CAN −2 B
CAN −3 B
CAN −4 B
...
D
With this transfer matrix we can map any given reference r into an output yr using yr = T r, obtaining
what would have been the complete output trajectory of the plant on an entirely feedback-driven trial.
Note that the first column of T contains the impulse response curve of the closed-loop system, while
the rest of the columns are obtained shifting that impulse response down. Therefore, we can build
the transfer matrix T either in a model-based manner, deriving the state-space characterization of
the closed-loop system, or in measurement-based manner, measuring the impulse response curve.
Additionally, note that (I − T )r yields the error of the feedback control in following the reference, a
signal which we denote with e0 .
Let o ∈ RN be the entire feed-forward signal for a given trial. Given commutativity, we can consider
that from the point of view of the closed-loop system o is added directly to the reference r, (Fig. 2
right). In that case, we can use y = T (r + o) to obtain the output of the closed-loop system
when it is driven by both the reference and the feed-forward signal. The feed-forward module only
outputs linear combinations of a set of bases. Let X ∈ RN ×G be a matrix with the content of the
G bases during all the N time steps of a trial. The feed-forward signal becomes o = Xw, where
w ∈ RG contains the mixing weights. Hence, the output of the plant given a particular w becomes
y = T (r + Xw).
We implement learning as the process of adjusting the weights w of the feed-forward module in a
trial-by-trial manner. At each trial the same reference signal, r, and bases, X, are repeated. Through
learning we want to converge to the optimal weight vector w∗ defined as
1
1
w∗ = arg min c(w) = arg min e| e = arg min (r − T (r + Xw))| (r − T (r + Xw))
2
2
w
w
w
(3)
where c indicates the objective function to minimize, namely the L2 norm or sum of squared errors.
With the substitution X̃ = T X and using e0 = (I − T )r, the minimization problem can be cast as a
4
canonical linear least-squares problem:
1
w∗ = arg min (e0 − X̃w)| (e0 − X̃w)
2
w
(4)
One the one hand, this allows to directly find the least squares solution for w∗ , that is, w∗ = X̃† e0 ,
where † denotes the Moore-Penrose pseudo-inverse. On the other hand, and more interestingly, with
w[k] being the weights at trial k and having e[k] = e0 − X̃w[k], we can obtain the gradient of the
error function at trial k with relation to w as follows:
∇w c = −X̃| e[k] = −X| T | e[k]
Thus, setting η as a properly scaled learning rate (the only global parameter Θ of the rule), we can
derive the following gradient descent strategy for the update of the weights between trials:
w[k + 1] = w[k] + ηX| T | e[k]
(5)
This solves for the learning rule f in eq. 2. Note that f is consistent with both the cerebellar anatomy
(Fig. 2left) and the control architecture (Fig. 2right) in that the feed-forward module/cerebellum only
requires two signals to update its weights/synaptic efficacies: the basis inputs, X, and error signal, e.
3.2
T | facilitates a synaptic eligibility trace
The standard least mean squares (LMS) rule (also known as Widrow-Hoff or decorrelation learning
rule) can be represented in its batch version as w[k + 1] = w[k] + ηX| e[k]. Hence, the only
difference between the batch LMS rule and the one we have derived is the insertion of the matrix
factor T | . Now we will show how this factor acts as a filter that computes an eligibility trace at each
weight/synapse. Note that the update of a single weight, according Eq. 5 becomes
wj [k + 1] = wj [k] + ηx|j T | e[k]
(6)
where xj contains the sequence of values of the cortical basis j during the entire trial. This can be
rewritten as
wj [k + 1] = wj [k] + ηh|j e[k]
(7)
with hj ≡ T xj . The above inner product can be expressed as a sum of scalar products
wj [k + 1] = wj [k] + η
N
X
hj [n]e[k, n]
(8)
n=1
where n indexes the within trial time-step. Note that e[k] in Eq. 7 refers to the whole error signal
at trial k whereas e[k, n] in Eq. 8 refers to the error value in the n-th time-step of the trial k. It is
now clear that each hj [n] weighs how much an error arriving at time n should modify the weight
wj , which is precisely the role of an eligibility trace. Note that since T contains in its columns/rows
shifted repetitions of the impulse response curve of the closed-loop system, the eligibility trace codes
at any time n, the convolution of the sequence of previous inputs with the impulse-response curve of
the reactive layer closed-loop. Indeed, in each synapse, the eligibility trace is generated by a forward
model of the closed-loop system that is exclusively driven by the basis signal.
Consequently, our main result is that by deriving a gradient descent algorithm for the CFPC cerebellar
control architecture we have obtained an exact definition of the suitable eligibility trace. That
definition guarantees that the set of weights/synaptic efficacies are updated in a locally optimal
manner in the weights’ space.
3.3
On-line gradient descent algorithm
The trial-by-trial formulation above allowed for a straightforward derivation of the (batch) gradient
descent algorithm. As it lumped together all computations occurring in a same trial, it accounted for
time within the trial implicitly rather than explicitly: one-dimensional time-signals were mapped onto
points in a high-dimensional space. However, after having established the gradient descent algorithm,
we can implement the same rule in an on-line manner, dropping the repetitiveness assumption inherent
to trial-by-trial learning and performing all computations locally in time. Each weight/synapse must
5
have a process associated to it that outputs the eligibility trace. That process passes the incoming
(unweighted) basis signal through a (forward) model of the closed-loop as follows:
sj [n + 1] = Asj [n] + Bxj [n]
hj [n] = Csj [n] + Dxj [n]
where matrices A, B, C and D refer to the closed-loop system (they are the same matrices that we
used to define the transfer matrix T ), and sj [n] is the state vector of the forward model of the synapse
j at time-step n. In practice, each “synaptic” forward model computes what would have been the
effect of having driven the closed-loop system with each basis signal alone. Given the superposition
principle, the outcome of that computation can also be interpreted as saying that hj [n] indicates what
would have been the displacement over the current output of the plant, y[n], achieved feeding the
closed-loop system with the basis signal xj . The process of weight update is completed as follows:
wj [n + 1] = wj [n] + ηhj [n]e[n]
(9)
At each time step n, the error signal e[n] is multiplied by the current value of the eligibility trace
hj [n], scaled by the learning rate η, and subtracted to the current weight wj [n]. Therefore whereas
the contribution of each basis to the output of the adaptive filter depends only on its current value and
weight, the change in weight depends on the current and past values passed through a forward model
of the closed-loop dynamics.
3.4
Simulation of a visually-guided smooth pursuit task
0.2
r
y[1]
y[50]
1
0.8
angular position (a.u.)
angular position (a.u.)
We demonstrate the CFPC approach in an example of a visual smooth pursuit task in which the
eyes have to track a target moving on a screen. Even though the simulation does not capture all the
complexity of a smooth pursuit task, it illustrates our anticipatory control strategy. We model the
plant (eye and ocular muscles) with a two-dimensional linear filter that maps motor commands into
angular positions. Our model is an extension of the model in (Porrill and Dean, 2007), even though
in that work the plant was considered in the context of the vestibulo-ocular reflex. In particular, we
use a chain of two leaky integrators: a slow integrator with a relaxation constant of 100 ms drives the
eyes back to the rest position; the second integrator, with a fast time constant of 3 ms ensures that
the change in position does not occur instantaneously. To this basic plant, we add a reactive control
layer modeled as a proportional-integral (PI) error-feedback controller, with proportional gain kp and
integral gain ki . The control loop includes a 50 ms delay in the error feedback, to account for both
the actuation and the sensing latency. We choose gains such that reactive tracking lags the target by
approximately 100 ms. This gives kp = 20 and ki = 100. To complete the anticipatory and adaptive
control architecture, the closed-loop system is supplemented by the feed-forward module.
0.6
0.4
0.2
0
0
0.5
1
1.5
time (s)
2
2.5
e[1]
e[50]
o[50]
0.1
0
−0.1
0
0.5
1
1.5
time (s)
2
2.5
Figure 3: Behavior of the system. Left: Reference (r) and output of the system before (y[1]) and
after learning (y[50]). Right: Error before e[1] and after learning e[50] and output acquired by
cerebellar/feed-forward component (o[50])
The architecture implementing the forward model-based gradient descent algorithm is applied to a
task structured in trials of 2.5 sec duration. Within each trial, a target remains still at the center of
the visual scene for a duration 0.5 sec, next it moves rightwards for 0.5 sec with constant velocity,
remains still for 0.5 sec and repeats the sequence of movements in reverse, returning to the center.
The cerebellar component receives 20 Gaussian basis signals (X) whose receptive fields are defined
in the temporal domain, relative to trial onset, with a width (standard-deviation) of 50 ms and spaced
by 100 ms. The whole system is simulated using a 1 ms time-step. To construct the matrix T we
computed closed-loop system impulse response.
6
At the first trial, before any learning, the output of the plant lags the reference signal by approximately
100 ms converging to the position only when the target remains still for about 300 ms (Fig. 3 left). As
a result of learning, the plant’s behavior shifts from a reactive to an anticipatory mode, being able to
track the reference without any delay. Indeed, the error that is sizable during the target displacement
before learning, almost completely disappears by the 50th trial (Fig. 3 right). That cancellation
results from learning the weights that generate a feed-forward predictive signal that leads the changes
in the reference signal (onsets and offsets of target movements) by approximately 100 ms (Fig. 3
right). Indeed, convergence of the algorithm is remarkably fast and by trial 7 it has almost converged
to the optimal solution (Fig. 4).
WH
WH+50ms
WH+70ms
FM−ET
1
rRMSE
0.8
0.6
0.4
0.2
0
0
10
20
#trial
30
40
50
Figure 4: Performance achieved with different learning rules. Representative learning curves of the
forward model-based eligibility trace gradient descent (FM-ET), the simple Widrow-Hoff (WH) and
the Widrow-Hoff algorithm with a delta-eligibility trace matched to error feedback delay (WH+50
ms) or with an eligibility trace exceeding that delay by 20 ms (WH+70 ms). Error is quantified as the
relative root mean-squared error (rRMSE), scaled proportionally to the error in the first trial. Error of
the optimal solution, obtained with w∗ = (T X)† e0 , is indicated with a dashed line.
To assess how much our forward-model-based eligibility trace contributes to performance, we test
three alternative algorithms. In both cases we employ the same control architecture, changing the
plasticity rule such that we either use no eligibility trace, thus implementing the basic Widrow-Hoff
learning rule, or use the Widrow-Hoff rule extended with a delta-function eligibility trace that matches
the latency of the error feedback (50 ms) or slightly exceeds it (70 ms). Performance with the basic
WH model worsens rapidly whereas performance with the WH learning rule using a “pure delay”
eligibility trace matched to the transport delay improves but not as fast as with the forward-modelbased eligibility trace (Fig. 4). Indeed, in this case, the best strategy for implementing a delayed
delta eligibility trace is setting a delay exceeding the transport delay by around 20 ms, thus matching
the peak of the impulse response. In that case, the system performs almost as good as with the
forward-model eligibility trace (70 ms). This last result implies that, even though the literature
usually emphasizes the role of transport delays, eligibility traces also account for response lags due
to intrinsic dynamics of the plant.
To summarize our results, we have shown with a basic simulation of a visual smooth pursuit task
that generating the eligibility trace by means of a forward model ensures convergence to the optimal
solution and accelerates learning by guaranteeing that it follows a gradient descent.
4
Discussion
In this paper we have introduced a novel formulation of cerebellar anticipatory control, consistent
with experimental evidence, in which a forward model has emerged naturally at the level of Purkinje
cell synapses. From a machine learning perspective, we have also provided an optimality argument
for the derivation of an eligibility trace, a construct that was often thought of in more heuristic terms
as a mechanism to bridge time-delays (Barto et al., 1983; Shibata and Schaal, 2001; McKinstry et al.,
2006).
The first seminal works of cerebellar computational models emphasized its role as an associative
memory (Marr, 1969; Albus, 1971). Later, the cerebellum was investigates as a device processing
correlated time signals(Fujita, 1982; Kawato et al., 1987; Dean et al., 2010). In this latter framework,
7
the use of the computational concept of an eligibility trace emerged as a heuristic construct that
allowed to compensate for transmission delays in the circuit(Kettner et al., 1997; Shibata and Schaal,
2001; Porrill and Dean, 2007), which introduced lags in the cross-correlation between signals.
Concretely, that was referred to as the problem of delayed error feedback, due to which, by the time
an error signal reaches a cell, the synapses accountable for that error are no longer the ones currently
active, but those that were active at the time when the motor signals that caused the actual error were
generated. This view has however neglected the fact that beyond transport delays, response dynamics
of physical plants also influence how past pre-synaptic signals could have related to the current
output of the plant. Indeed, for a linear plant, the impulse-response function of the plant provides the
complete description of how inputs will drive the system, and as such, integrates transmission delays
as well as the dynamics of the plant. Recently,
Even though cerebellar microcircuits have been used as models for building control architectures,
e.g., the feedback-error learning model (Kawato et al., 1987), our CFPC is novel in that it links
the cerebellum to the input of the feedback controller, ensuring that the computational features of
the feedback controller are exploited at all times. Within the domain of adaptive control, there are
remarkable similarities at the functional level between CFPC and iterative learning control (ILC)
(Amann et al., 1996), which is an input design technique for learning optimal control signals in
repetitive tasks. The difference between our CFPC and ILC lies in the fact that ILC controllers
directly learn a control signal, whereas, the CFPC learns a conterfactual error signal that steers a
feedback controller. However the similarity between the two approaches can help for extending
CFPC to more complex control tasks.
With our CFPC framework, we have modeled the cerebellar system at a very high level of abstraction:
we have not included bio-physical constraints underlying neural computations, obviated known
anatomical connections such as the cerebellar nucleo-olivary inhibition (Bengtsson and Hesslow,
2006; Herreros and Verschure, 2013) and made simplifications such as collapsing cerebellar cortex and
nuclei into the same computational unit. On the one hand, such a choice of high-level abstraction may
indeed be beneficial for deriving general-purpose machine learning or adaptive control algorithms.
On the other hand, it is remarkable that in spite of this abstraction our framework makes fine-grained
predictions at the micro-level of biological processes. Namely, that in a cerebellar microcircuit (Apps
and Garwicz, 2005), the response dynamics of secondary messengers (Wang et al., 2000) regulating
plasticity of Purkinje cell synapses to parallel fibers must mimic the dynamics of the motor system
being controlled by that cerebellar microcircuit. Notably, the logical consequence of this prediction,
that different Purkinje cells should display different plasticity rules according to the system that they
control, has been validated recording single Purkinje cells in vivo (Suvrathan et al., 2016).
In conclusion, we find that a normative interpretation of plasticity rules in Purkinje cell synapses
emerges from our systems level CFPC computational architecture. That is, in order to generate
optimal eligibility traces, synapses must include a forward model of the controlled subsystem. This
conclusion, in the broader picture, suggests that synapses are not merely components of multiplicative
gains, but rather the loci of complex dynamic computations that are relevant from a functional
perspective, both, in terms of optimizing storage capacity (Benna and Fusi, 2016; Lahiri and Ganguli,
2013) and fine-tuning learning rules to behavioral requirements.
Acknowledgments
The research leading to these results has received funding from the European Commission’s Horizon
2020 socSMC project (socSMC-641321H2020-FETPROACT-2014) and by the European Research
Council’s CDAC project (ERC-2013-ADG 341196).
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9
| 3 |
A Characterization of Scale Invariant Responses
in Enzymatic Networks
arXiv:1202.5332v1 [cs.SY] 23 Feb 2012
Maja Skataric
Department of Electrical Engineering, Rutgers University, Piscataway, NJ, USA
Eduardo Sontag∗
Department of Mathematics, Rutgers University, Piscataway, NJ, USA
November 9, 2017
Running head: Scale Invariant Responses in Enzymatic Networks
Abstract
An ubiquitous property of biological sensory systems is adaptation: a step increase in
stimulus triggers an initial change in a biochemical or physiological response, followed by
a more gradual relaxation toward a basal, pre-stimulus level. Adaptation helps maintain
essential variables within acceptable bounds and allows organisms to readjust themselves
to an optimum and non-saturating sensitivity range when faced with a prolonged change
in their environment. Recently, it was shown theoretically and experimentally that many
adapting systems, both at the organism and single-cell level, enjoy a remarkable additional
feature: scale invariance, meaning that the initial, transient behavior remains (approximately) the same even when the background signal level is scaled. In this work, we set
out to investigate under what conditions a broadly used model of biochemical enzymatic
networks will exhibit scale-invariant behavior. An exhaustive computational study led us
to discover a new property of surprising simplicity and generality, uniform linearizations
with fast output (ULFO), whose validity we show is both necessary and sufficient for scale
invariance of enzymatic networks. Based on this study, we go on to develop a mathematical
explanation of how ULFO results in scale invariance. Our work provides a surprisingly
consistent, simple, and general framework for understanding this phenomenon, and results
in concrete experimental predictions.
Author summary: An ubiquitous property of biological sensory systems is adaptation: a step
increase in stimulus triggers an initial change in a biochemical or physiological response, followed
by a more gradual relaxation toward a basal, pre-stimulus level. Adaptation helps maintain
essential variables within acceptable bounds and allows organisms to readjust themselves to
an optimum and non-saturating sensitivity range when faced with a prolonged change in their
environment. Recently, it was shown theoretically and experimentally that many adapting
systems, both at the organism and single-cell level, enjoy a remarkable additional feature:
scale invariance, meaning that the initial, transient behavior remains (approximately) the same
even when the background signal level is scaled. In this work, we develop a mathematical
characterization of biochemical enzymatic networks that exhibit scale-invariant behavior and
make concrete experimental predictions.
∗
Corresponding author
1
1
Introduction
The survival of organisms depends critically upon their capacity to formulate appropriate responses to sensed chemical and physical environmental cues. These responses manifest themselves at multiple levels, from human sight, hearing, taste, touch, and smell, to individual
cells in which signal transduction and gene regulatory networks mediate the processing of measured external chemical concentrations and physical conditions, such as ligand concentrations
or stresses, eventually leading to regulatory changes in metabolism and gene expression.
An ubiquitous property of biological sensory systems at all levels is that of adaptation: a step
increase in stimulus triggers an initial, and often rapid, change in a biochemical or physiological
response, followed by a more gradual relaxation toward a basal, pre-stimulus level [1]. Adaptation plays a role in ensuring that essential variables stay within acceptable bounds, and it also
allows organisms to readjust themselves to an optimum and non-saturating sensitivity range
even when faced with a prolonged change in their operating environment, thus making them
capable of detecting changes in signals while ignoring background information.
Physiological examples of adaptation in higher organisms include phenomena such as the control
of the amount of light entering eyes through the contraction and relaxation of the pupil by the
nervous system, which brings intensities of illumination within the retinal working range, or the
regulation of key metabolites in the face of environmental variations [20]. At the single-cell level,
one of the best understood examples of adaptation is exhibited by the E. coli chemotaxis sensory
system, which responds to gradients of nutrient and ignores constant (and thus uninformative)
concentrations [7, 39]. The term “exact” or “perfect” adaptation is employed to describe
processes which, after a transient, return with very high accuracy to the same input-independent
level. In practice, however, an approximate adaptation property is usually adequate for proper
physiological response [30].
By definition, neither the concepts of perfect nor approximate adaptation address the characteristics of the transient signaling which occurs prior to a return to steady state. The amplitude
and other characteristics of transient behaviors, however, are physiologically relevant. In this
more general context, a remarkable phenomenon exhibited by several human and animal sensory systems is scale invariance or logarithmic sensing [20] [22] [48]. This means that responses
are functions of upon ratios (in contrast to actual magnitudes), of a stimulus relative to the
background. There is evidence for this phenomenon at an intracellular level as well. It appears
in bacterial chemotaxis [18] [35], in the sensitivity of S. cerevisiae to fractional rather than
absolute pheromone gradients [34], and in two mammalian signaling systems: transcriptional
as well as embryonic phenotype responses to β-catenin levels in Wnt signaling pathways [13],
and nuclear ERK localization in response to EGF signaling [10]. Scale invariance allows systems to react to inputs ranging over several orders of magnitude, and is speculated to help
make behaviors robust to external noise as well as to stochastic variations in total expressed
concentrations of signaling proteins [41].
Mathematically, scale invariance is defined by the following property of transient behaviors
[41]: if a stimulus changes from a background level u0 to a new level u, then the entire time
response of the system is the same as if the stimulus had changed, instead, from a background
level pu0 to pu. In other words, only the ratio (or “fold-change”) pu/pu0 = u/u0 is relevant
to the response; the “scale” p is irrelevant. For this reason, the term “fold change detection”
is interchangeably used instead of scale-invariance. Scale invariance implies adaptation, but
not every adaptive system is scale invariant [41]. A mathematical analysis of scale-invariance
2
was initiated in [41], [40]. Predictions regarding scale-invariance of E. coli chemotaxis were
subsequently experimentally verified [23]. While adaptation can be often understood in terms
of control-theoretic tools based on linearizations [42] [52] [43] [16] [25], scale invariance is a
genuinely nonlinear property; as a matter of fact, a linear system can never display scaleinvariance, since the response to an input scaled by p will also be scaled by this same factor
p.
In this work, we focus on enzymatic signal transduction systems, which involve the activation/deactivation cycles that typically mediate transmission of external signals to transcription
factors and other effectors. Networks involving such enzymatic cycles are involved in signal
transduction networks from bacterial two-component systems and phosphorelays [5, 14] to actin
treadmilling [9], guanosine triphosphatase cycles [11], glucose mobilization [19], metabolic control [45], cell division and apoptosis [46], cell-cycle checkpoint control [24], and the eukaryotic
Mitogen-Activated Protein Kinase (MAPK) cascades which mediate growth factor inputs and
determine proliferation, differentiation, and apoptosis [4, 8, 15, 50, 3].
Given the biological importance of these processes, and the already observed scale-invariance in
some of these pathways [13] [10], we pose here the following question: which enzymatic networks
do not merely adapt, but also display scale invariance? In order to answer this question,
we performed an exhaustive computational study of all 3-node networks, finely sampled in
parameter space. Only about 0.01% of these networks are capable of (approximate) adaptation.
Testing which of these adapting networks also display scale-invariant behavior, we found that
only about 0.15% of them did. Once that this small subclass was identified, we turned to
the problem of determining what network characteristics would explain the results of these
numerical experiments. We discovered a surprisingly simple and general property, which we
call uniform linearizations with fast output (ULFO), that is displayed by all the networks in
this subclass, and here we provide a theoretical framework that explains conceptually why this
property is both necessary and sufficient for scale invariance of enzymatic networks. As an
application, we consider a recently published model [47] of an eukaryotic enzymatic system,
specifically the pathway involved in the social amoeba Dictyostelium discoideum’s chemotactic
response to cAMP. and show that our conditions are satisfied in appropriate ranges of cAMP
input.
Characterizations of this sort allow one to understand which networks are robust to scale
uncertainty, and constitute a powerful tool in allowing one to discard putative mechanisms that
are not consistent with experimentally observed scale-invariant behaviors [40], [23].
2
2.1
Results
Three-node enzymatic networks
We consider networks consisting of three types of enzymes, denoted respectively as A, B, and
C. Each of these enzymes can be in one of two states, active or inactive. The fractional
concentration of active enzyme A is represented by a variable xA = xA (t), so x
eA = 1 − xA is
the fraction of inactive enzyme A. Similar notations are used for B and C. Only enzyme A is
directly activated by an external input signal, and the response of the network is reported by the
fraction of active C. Enzyme B acts as an auxiliary element. Each enzyme may potentially act
upon each other through activation (positive regulation), deactivation (negative regulation), or
not at all. If a given enzyme is not deactivated by any of the remaining two, we assume that it is
3
constitutively deactivated by a specific enzyme; similarly, if a given enzyme is not activated by
any other, there is a constitutively activating enzyme for it. One represents networks by 3-node
directed graphs, with nodes labeled A, B, C, and with edges between two nodes labeled + and
− (or “→” and “a”) to denote positive or negative regulation respectively; no edge is drawn if
there is no action. There are 32 = 9 potential directed edges among the three nodes (A to A, A
to B, etc.), each of whose labels may be +, −, or “none” if there is no edge. This gives a total
of 39 = 19, 683 possible graphs. One calls each of these possible graphs a topology. Discarding
the 3,645 topologies that have no direct or indirect links from the input to the output, there
remain 16,038 topologies.
2.2
Specification of a dynamic model
We quantify the effects of each existing regulatory interaction by a Michaelis-Menten term and
write a three-variable ordinary differential equation (ODE) that describes the time evolution of
xA (t), xB (t), and xC (t):
ẋA =
X k V A vi · x
eA
i
i
ẋB =
X k V B vi · x
eB
i
i
ẋC
=
x
eA + KVi A
x
eB + KVi B
X k V C vi · x
eC
i
i
x
eC + KVi C
−
X kW A wi · xA
i
i
−
X kW B wi · xB
i
i
−
xA + KWi A
xB + KWi B
X kW C wi · xC
i
i
xC + KWi C
(1a)
(1b)
(1c)
The K’s denote Michaelis-Menten, and the k’s catalytic, rate constants associated to each
regulatory interaction. All the summations range over i = 1, . . . , 6. Each “Vi ” represents
one of A, B, C, EA , EB , EC , the activating enzymes in the respective equations, and each
“Wi ” one of A, B, C, FA , FB , FC , the deactivating enzymes; E and F are the constitutively
activating and deactivation enzymes, buffered at constant concentrations. (Lower-case variables
vi , wi = xA , . . . , xFC denote active fractions) As an exception, the equation for node A does
x
eA
not include an EA term, but instead includes a term kU A u xeA +K
that models activation of A
UA
by an external input whose strength at time t is given by u = u(t) and whose values u(t) stay
within a range [u, u]. No enzyme appears both an activator and as a deactivator of any given
component, that is, kXi A kYi A = 0, kXi B kYi B = 0, and kXi C kYi C = 0, and constitutive enzymes
are included only if the reaction would be otherwise irreversible. For example, the topology
shown in Fig.1 is described by the following following set of ODE’s:
Figure 1: Topology 2293
4
ẋA =
ẋB =
ẋC
=
kU A u · x
eA
kBA xB · xA kCA xC · xA
−
−
x
eA + KU A
xA + KBA
xA + KCA
kAB xA · x
eB
kF B xFB · xB
− B
x
eB + KAB
xB + KFB B
kAC xA · x
eC
kBC · xB xC
kCC xC · xC
−
−
x
eC + KAC
xC + KBC
xC + KCC
(2a)
(2b)
(2c)
The term circuit is used to refer to a given topology together with a particular choice of the K
and k parameters. The three-node model in Eq.1 was employed by Ma et al. [25], in order to
classify the minimal enzymatic circuits that adapt. (With the model in [25] that we adopted,
there is no direct connection from the input to the output node, and two-node networks are
not sufficient for adaptation, while larger adapting networks contain these three-node networks
[25]. If one allows direct connections from input to outputs, then two-node networks are able
to display adaptation.) The same paradigm has since been used to investigate other network
characteristics as well [38], [51].
2.3
Adaptation
Following [12], we define adaptation behavior in terms of two functional metrics. The first metric
quantifies the following effect: if we start at steady state, and then step the input at time t = 0
from a value u0 to a different constant value u1 , then the system’s output, as reported by a
response variable y(t) (where y(t) = xC (t) in Eq.1), should return asymptotically to a value
that is close to the original value y(0). The relative difference in initial and final response
∆∞
y = |y(+∞) − y(0)| provides a measure of adaptation precision. We say that a system
is (approximately) adaptive provided that, for all inputs in the valid range, ∆∞
y /∆u < 0.1,
where ∆u = |u1 − u0 | / |u0 | is the relative change in input. In particular, exact or perfect
adaptation means that ∆∞
y = 0. The 10% error tolerance is natural in applications, and
the qualitative conclusions are not changed by picking a smaller cutoff [25]. A second metric
relies upon the maximal transient difference in output, normalized by the steady-state output,
∆max
= max |y(t) − y(0)| / |y(0)|. A signal-detection property for adaptation [43], [2], should
y
be imposed in order to rule out the trivial situation ∆max
≈ 0 in which a system’s output is
y
independent of the input. To avoid having to pick an arbitrary threshold, in this study we
follow the convention in [25] of requiring the sensitivity ∆max
/∆u to be greater than one.
y
2.4
Scale invariance
Scale invariance is the property that if a system starts from a steady state that was preadapted (t < 0) to a certain background level u0 , and the input is subsequently set to a
new level u at t = 0, then the entire time response of the system yu0 ,u (t) is the same as the
response ypu0 ,pu (t) that would result if the stimulus had changed, instead, from pu0 to pu. This
property should hold for scale changes p > 0 that respect the bounds u ≤ u ≤ u on inputs.
For example, recent microfluidics and FRET experimental work [23] verified scale-invariance
predictions that had been made in [41] for bacterial chemotaxis under the nonmetabolizable
attractant α-methylaspartate (MeAsp) as an input. In these experiments, E. coli bacteria
were pre-adapted to input concentrations and then tested in new nutrient gradients, and it was
found experimentally that there were two different ranges of inputs [u1 , u1 ] and [u2 , u2 ] in which
scale-invariance holds, the “FCD1” and “FCD2” regimes, repectively. (The term fold-change
5
detection, or FCD, is used to reflect the fact that only the ratio or fold-change pu/pu0 = u/u0
can be detected by the response y(t).) More generally, the mathematical definition of (perfect)
scale invariance [40] imposes the ideal requirement that the same response invariance property
is exhibited if u = u(t), t ≥ 0 is any time-varying input. The experiments in [23] included
excitation by certain oscillatory inputs, for example. In practice, however, this property will
always break down for high-frequency inputs, since there are limits to the speed of response of
biological systems.
2.5
Adaptive systems need not be scale-invariant
As an illustration of a (perfectly) adaptive yet not scale-invariant system, consider the following
equations:
ẋA = k1 u − k2 xB
(3a)
ẋB = k3 xA − k4 xB
(3b)
ẋC
= k5 xA − k6 xB xC
(3c)
which is a limiting case of the system described by Eq.2 when kCA , kCC , KU A , KBA , KAB , KAC ≈
0, kBC = k6 KBC , KBC 1 (so −kBC xB xC /(xC +KBC ) ≈ −k6 xB xC ), and kFB B xFB = k2 KFB B
and KFB B 1. This network perfectly adapts, since at steady state the output is xC = xC =
k4 k5 /(k3 k6 ), no matter what is the magnitude of the constant input u, and in fact the system
returns to steady state after a step change in input u, with xC (t) → xC as t → ∞ (general stability properties of feedforward circuits shown in [44]). On the other hand, the example in Eq.3
does not display scale invariance. Indeed, consider the solution from an initial state pre-adapted
to an input level u0 , that is xA (0) = k1 k4 u0 /(k2 k3 ), xB (0) = k1 u0 /k2 , and xC (0) = k4 k5 /(k3 k6 ),
and the input u(t) ≡ u1 for t ≥ 0. Then, xC (t) = k4 k5 /(k3 k6 ) + k1 k5 (u1 − u0 )t2 /2 + O(t3 )
for small t ≥ 0. Since the t2 coefficient in this Taylor expansion gets multiplied by p when
u0 is replaced by pu0 and u1 is replaced by pu1 , it follows that the transient behavior of the
output xC (t) depends on p. Interestingly, if the equation for the third node is replaced by
ẋC = k5 xA /xB − k6 xC , that is to say the activation of C is repressed by A, instead of its
de-activation being enhanced by A, then scale invariance does hold true, because xA (t) and
xB (t) both scale by p when u0 7→ pu0 , u1 7→ u0 , and C(t) depends on the ratio of these two
functions (in particular, the t2 /2 term is k2 k5 (u1 − u0 )/u0 ). Such a repression is typical of
genetic interaction networks, but is not natural in enzymatic reactions.
It turns out that the example described by Eq.3 is typical: no enzymatic network described by
Eq.1 can display perfect scale-invariant behavior. This fact is a consequence of the equivariance
theorem proved in [40] (see Materials and Methods). Thus, a meaningful study of enzymatic
networks, even for perfectly adaptive ones, must rely upon a test of approximate scale invariance. Instead of asking that yu0 ,u (t) = ypu0 ,pu (t), as was the case in the theory developed in
[41] [40], one should require only that the difference be small. To investigate this issue, we
computationally screened all 3-node topologies through a high-throughput random parameter
scan, testing for small differences in responses to scaled steps. We found that approximately
0.01% of the samples showed adaptation, but of them, only about 0.15% passed the additional
criterion of approximate scale invariance (see Materials and Methods). These samples belonged
to 21 (out of 16,038 possible) topologies. As an example of the behavior of one of these, Fig.2
shows a response resulting from a 20% step, from 3 to 3.6, compared to the response obtained
when stepping from 5 to 6; the graphs are almost indistinguishable. (See SI Text for an enu6
meration of circuits and corresponding plots). In the following discussion, we will refer to these
surviving circuits, and their topologies, as being “approximately scale invariant” (ASI).
We found that all ASI networks possess a feedforward motif, meaning that there are connections
A → B → C and as well as A → C. Such feedforward motifs have been the subject of
extensive analysis in the systems biology literature [1]. and are often involved in detecting
changes in signals [28]. They appear in pathways as varied as E. coli carbohydrate uptake
via the carbohydrate phosphotransferase system [21], control mechanisms in mammalian cells
[26], nitric oxide to NF-κB activation [27, 29], EGF to ERK activation [37, 32], glucose to
insulin release [31, 33], ATP to intracellular calcium release [36], and microRNA regulation
[49]. The feedforward motifs in all ASI networks are incoherent, meaning such that the direct
effect A → C has an opposite sign to the net indirect effect through B. An example of an
incoherent feedforward connection is provided by the simple system described by Eq.3, where
the direct effect of A on C is positive, but the indirect effect is negative: A activates B which
in turn deactivates C. (Not every incoherent feedforward network provides scale invariance; a
classification of those that provide exact scale invariance is known [40].) It is noteworthy that
all ASI circuits have a positive regulation from A to B and a negative regulation from B to A.
We then discovered a surprising common feature among all ASI circuits. This feature can best
be explained by a further examination of the example in Eq.3.
Figure 2: Scale-invariance: plots overlap, for responses to steps 3→1.2∗3 and 5→1.2∗5. Network
is the one described by Eq.2. Random parameter set: KU A =0.093918 kU A =11.447219, KBA =0.001688
kBA =44.802268, KCA =90.209027 kCA =96.671843, KAB =0.001191 kAB =1.466561, KFB =9.424319 kFB =22.745736,
KAC =0.113697 kAC =1.211993, KBC =0.009891 kBC =7.239357, KCC =0.189125 kCC =17.910182
2.6
Approximate scale invariance
Continuing with example in Eq.3, let us suppose that k1 , k2 , k3 , k4 k5 , k6 , so that the output
variable y = xC reaches its steady state much faster than xA and xB do. Then, we may
approximate the original system by the planar linear system represented by the differential
equations for xA and xB together with the new output variable ye(t) = h(xA (t), xB (t)) =
kxA (t)/xB (t), where k = k5 /k6 . This reduced planar system, obtained by a quasi-steady state
approximation, has a perfect scale-invariance property: replacing the input u by pu results in the
7
solution (pxA (t), pxB (t)), and thus the output is the same: h(xA (t), xB (t)) = h(pxA (t), pxB (t)).
The exact invariance of the reduced system translates into an approximate scale invariance
property for the original three-dimensional system because, except for a short boundary-layer
behavior (the relatively short time for xC to reach equilibrium), the outputs of both systems
are essentially the same, y(t) ≈ ye(t).
2.7
Generality of the planar reduction
We found that, just as in the example in Eq.3 when k1 , k2 , k3 , k4 k5 , k6 , in every ASI circuits the time scale of node C is much shorter than that of A and B. Therefore, the same
two-dimensional reduction is always valid. It follows that one can drop the last equation, approximating these circuits by planar systems that are described by only the two state variables
xA and xB , where every occurence of xC in the first two equations of the right-hand side of
Eq.1 is replaced by h(xA , xB ), the function obtained by setting the right-hand side of the third
equation in Eq.1 to zero and solving for the unique root in the interval [0, 1] of the quadratic
equation. This reduced system, with ye(t) = h(xA (t), xB (t)) as an output, provides an excellent
approximation of the original dynamics. Fig.3 compares the true response with the response
obtained by the quasi-steady state approximation, for one ASI circuit (see SI Text for all comparisons).
Figure 3: QSS quadratic approximation. Network is the one described by Eq.2. Random
parameter set is as in Fig.2
.
2.8
Generality of dependence on
xA /xB
In the example given by Eq.3, there were two additional key mathematical properties that
made the planar reduction scale-invariant (and hence the original system approximately so).
The first property was that, at equilibrium, the variable xC must be a function of the ratio
xA /xB , and the second one was that each of xA and xB must scale by the same factor when
the input scales by p. Neither of these two properties need to hold, even approximately, for
general networks. Surprisingly, however, we discovered that both are valid with very high
8
accuracy for every ASI circuit. The equilibrium value of xC is obtained from setting the last
right-hand side of Eq.1 to zero and solving for xC . A solution xC = h(xA , xB ) in the interval
[0, 1] always exists, because at xC = 0 one has x
eC = 1 and thus the term is positive, and at
xC = 1 one has x
eC = 0 and so the term is negative. This right-hand side has the general form
xA φ(xC ) + xB γ(xC ) + κ(xC , xEC , xFC ), where φ and γ are increasing functions, each a constant
multiple of a function of the form x
eC /(e
xC + K) or −xC /(xC + K). If the term κ is negligible,
then xA φ(xC ) + xB γ(xC ) = 0 means that also (xA /xB )φ(xC ) + γ(xC ) = 0, and therefore xC at
equilibrium is a (generally nonlinear) function of the ratio xA /xB . There is no a priori reason
for the term κ to be negligible. However, we discovered that in every ASI circuit, κ ≈ 0. More
precisely, there is no dependence on the constitutive enzymes, and this “self-loop” link, when
it exists, contributes to the derivative ẋC much less than the xA and xB terms, see Fig.4.
Figure 4: Relative contribution of terms in the equation for node C. The first two terms range
in [−0.25, 0.25] but self-loop magnitude is always less than 10−3 . i.e. contribution or self-loop
to ẋC is less than 1%. Similar results hold for all ASI circuits. Network is the one described by
Eq.2. Random parameter set is as in Fig.2. Similar results are available for all ASI circuits.
2.9
Generality of homogeneity of
xA , x B
The last ingredient of the example given by Eq.3 that plays a role in approximate scale invariance
is that each of xA and xB must scale proportionately when the input is scaled. In that example,
the property holds simply because the equations for these two variables are linear. In general,
however, the dynamics of (xA , xB ) are described by nonlinear equations. Thus it is remarkable
that, in all ASI circuits, the property holds. We tested the property by plotting xA (t)/xB (t) in
a set of experiments in which a system was pre-adapted to an input value u0 and the input was
subsequently set to a new level u at t = 0. When going from pu0 to pu, we found that the new
value xA (t)/xB (t) was almost the same, meaning that xA and xB scaled in the same fashion.
A representative plot is shown in Fig.5.
2.10
A new property: uniform linearizations with fast output
The (approximate) independence of xA (t)/xB (t) on input scalings is not due to linearity of
the differential equations for xA and xB (t). Instead, the analysis of this question led us to
postulate a new property, which we call uniform linearizations with fast output (ULFO). To
define this property, we again drop the last equation, and approximate circuits by the planar
system that has only the state variables xA and xB , where every occurence of xC in their
differential equations shown in Eq.1 is replaced by h(xA , xB ). We denote by f (xA , xB , u) =
9
Figure 5: Constant A/B ratio in responses to 3→1.2 ∗ 3 and 5→1.2 ∗ 5. Network is the one
described by Eq.2. Random parameter set is as in Fig.2. Similar results are available for all
ASI circuits (see SI Text).
(f1 (xA , xB , u), f2 (xA , xB , u)) the result of these substitutions, so that the reduced system is
described in vector form by ẋ = f (x, u), x = (xA , xB ). We denote by σ(u) the unique
steady state corresponding to a constant input u, that is, the solution of the algebraic equation
f (σ(u), u) = 0. We denote by A(u) = (∂f /∂x)(σ(u)) the Jacobian matrix of f with respect
to x, and by B(u) = (∂f /∂u)(σ(u)) the Jacobian vector of f with respect to u. The property
ULFO is then defined by requiring time-scale separation for xC , that h(xA , xB ) depends only
on the ratio xA /xB , and:
σ(pu) = pσ(u), A(u) = A(v), B(u) = B(v)
(4)
for every u , v, and p such that u, v, and pu are in the range [u, u]. Notice that we are not
imposing the far stronger property that the Jacobian matrices should be constant. We are only
requiring the same matrix at every steady state. The first condition in Eq.4 means that the
vector σ(u)/u should be constant. We verified that this requirement holds with very high accuracy in every one of the ASI circuits. With u = 0.3 and u = 0.6, we have the following σ(u)/u
values, rounded to 3 decimal digits: (0.195, 0.239), (0.193, 0.237), (0.192, 0.236), (0.191, 0.235)
when u = 0.3, 0.4, 0.5, and 0.6 respectively, for the network described by Eq.2 and the random
parameter set in Fig.2. Similar results are available for all ASI circuits (see SI Text). The Jacobian requirements are also verified with high accuracy for all the ASI circuits. We illustrate this
with the same network and parameter set. Let us we compute the linearizations A0.3 = A(0.3),
A0.4 = A(0.4), . . . , B0.6 = B(0.6) and the average relative differences
Aerr
ij =
X
u=0.3,0.4,0.5,0.6
(Au )ij − (A0.45 )ij
(A0.45 )ij
and we define similarly B err . These relative differences are very small (shown to 3 decimal
digits):
0.069 0.004
0.002
err
err
A
=
, B
=
,
0
0.005
0
thus justifying the claim that the Jacobians are practically constant. Similar results are available
for all ASI circuits (see SI Text).
10
The key theoretical fact is that the property ULFO implies approximate scale-invariance, see
Materials and Methods.
2.11
A concrete example
In a recent paper [47] Takeda and collaborators studied the adaptation kinetics of a eukaryotic chemotaxis signaling pathway, employing a microfluidic device to expose Dictyostelium
discoideum to changes in chemoeffector cyclic adenosine monophosphate (cAMP). Specifically,
they focused on the dynamics of activated Ras (Ras-GTP), which was in turn reported by RBDGFP (the Ras binding domain of fluorescently tagged human Raf1), and showed almost perfect
adaptation of previously unstimulated cells to cAMP concentrations ranging from 10−2 nM to
1 µM . Furthermore, inspired by [25], the authors proposed alternative models for adaptation,
and concluded that the best fit was obtained by using an incoherent feedforward structure. The
model that they identified is given by the following system of 6 differential equations:
dR1
dt
dR2
dt
u
dGEF
dt
dGAP
dt
dRasGT P
dt
dRBDcyt
dt
= kR1 (v + r1 )(R1tot − R1 ) − k−R1 R1
= kR2 (v + r2 )(R2tot − R2 ) − k−R2 R2
= R1 + R2
= kGEF u − k−GEF GEF
= kGAP u − kGAP GAP
= kRAS GEF (RAS tot − RasGT P ) − k−RAS GAP RasGT P
off (RBDtot − RBDcyt ) − k on RasGT P RBDcyt .
= kRBD
RBD
The symbol v stands for the chemoeffector cAMP, and the authors assumed the existence of two
different receptor populations (R1 and R2 , with very different Kd ’s) which when bound pool
their signals to downstream components (through u). The constants r1 and r2 represent levels
of constitutive activation. The variables GEF and GAP represent activation and deactivation
of RasGEF and RasGAP, RasGT P represents the activated Ras, and RBDcyt describes the
cytosolic reporter molecule RBD-GFP. Fig. 6 shows a schematic of the main players.
Figure 6: The system studied in [47]
The best-fit parameters obtained in [47] are as follows: R1tot = 0.1, R2tot = 0.9, r1 = 0.012nM,
r2 = 0.115nM, kR1 = 0.00267nM−1 sec−1 , k−R1 = 0.16sec−1 , kR2 = 0.00244nM−1 sec−1 , k−R2 =
1.1sec−1 , kGEF = 0.04sec−1 , k−GEF = 0.4sec−1 , kGAP = 0.01sec−1 , kGAP = 0.1sec−1 , RAS tot =
11
off = 0.53sec−1 , k on = 1.0sec−1 .
1, kRAS = 390sec−1 , k−RAS = 3126sec−1 , RBDtot = 1, kRBD
RBD
With these parameters, and cAMP concentrations which are small yet also satisfy r1 v(t)
and r2 v(t), it follows that Ṙ1 ≈ kR1 R1tot v − k−R1 R1 and Ṙ2 ≈ kR2 R2tot v − k−R2 R2 , so we
may view u(t) as an input (linearly dependent on the external v(t)) to the three-variable system
described by xA = GEF , xB = GAP , xC = RasGT P . Since RBDcyt depends only on xC , we
may view xC as the output. This three-variable system (interpreted as having limiting values of
Michaelis-Menten constants) has the ULFO property provided that the dynamics of xC are fast
compared to xA and xB , which the identified parameters insure. So, we expect scale-invariant
behavior. Indeed, Fig.7 shows a simulation of the entire six-dimensional system (not merely
of our 3-dimensional reduction) when using a step from 1 to 2 nM of cAMP, and shows that
essentially the same response is obtained when stepping from 2 to 4 nM. This prediction of
Figure 7: Scale-invariance for model from [47]: responses to steps 1→2 and 2→4 coincide
scale-invariant behavior is yet to be tested experimentally.
3
Discussion
Work in molecular systems biology seeks to unravel the basic dynamic processes, feedback
control loops, and signal processing mechanisms in single cells and entire organisms, both for
basic scientific understanding and for guiding drug design. One of the key questions is: how
can one relate phenotype (function) to interaction maps (gene networks, protein graphs, and
so forth) derived from experimentation, especially those obtained from high-throughput tools?
Answers to this question provide powerful tools for guiding the reverse-engineering of networks,
by focusing on mechanisms that are consistent with experimentally observed behaviors, and,
conversely, from a synthesis viewpoint, allow one to design artificial biological systems that are
capable of adaptation [6] and other objectives. In particular, scale-invariance, a property that
has been observed in various systems [13], [10], can play a key role in this context, helping to discard putative mechanisms that are not consistent with experimentally observed scale-invariant
behaviors [23]. Through a computational study, we identified a set of simple mathematical
conditions that are used to characterize scale invariant enzymatic networks.
12
4
4.1
Materials and Methods
Computational screen
We generalized and extended the computational protocol developed for adaptation in [25] to
an investigation of approximate scale invariance. MATLAB R scripts were used, in conjunction
with the software developed in [25]. In order to test inputs in ranges of the form a ≤ u(t) ≤ 2a,
redefining the constant kU A if needed, we take simply u = 0.3 and u = 0.6. We considered
160,380,000 circuits, obtained from the 16,038 nontrivial 3-node topologies, each one with 10,000
parameters sampled in logarithmic scale using the Latin hypercube method [17]. (We picked
the ranges kcat =0.1-10 and Km =0.001-100. A finer sampling does not affect conclusions in any
significant way [25].) Of these, 0.01% (16,304) circuits showed adaptation, meaning that, as in
[25], when making a 20% step from u0 = 0.5 to u1 = 0.6 the precision is 10% or better, and the
sensitivity is at least unity. Approximate scale invariance (ASI) was then tested by also performing a 20% step experiment from u0 = 0.3 to u1 = 0.36 and requiring that the relative difference
between the responses be at most 10%: maxt {|y0.6 (t) − y3.6 (t)| / max(y0.6 (t) − y3.6 (t))} < 0.1
Of the adapting circuits, about 0.15% (25 circuits, classified into 21 different topologies) were
determined to be ASI.
4.2
ULFO implies approximate scale invariance
Consider a system of n differential equations with input signal u,
ẋ = f (x, u)
with the variables x evolving on some closed bounded set and f differentiable, and suppose
that for each constant input u¯∗ there is a unique steady state x¯∗ = σ(u¯∗ ) with the conditions
in Eq.4 and an output
y(t) = h(x(t))
such that h is differentiable and homogeneous of degree zero (h(px) = h(x) for nonzero p). We
view 3-node enzymatic networks as obtained from a set of n + 1 equations
ẋ = F (x, z, u)
εż = G(x, z)
with n = 2, x = (xA , xB ), and z = xC (0 < ε 1 represents the faster time scale for xC ), and
we are studying the reduced system ẋ = f (x, u) = F (x, α(x), u) obtained by solving G(x, z) = 0
for z = α(x) and substituting in F . Consider a time interval [0, T ], a constant input u¯∗ , and
a possibly time-varying input u(t), t ≥ 0, as well as a scaling p > 0, such that all values u¯∗ ,
pu¯∗ , u(t), pu(t) are in the input range [u, u]. The solutions of ẋ = f (x, u) with initial condition
x(0) = σ(u¯∗ ) and of ż = f (z, pu) with initial condition z(0) = σ(pu¯∗ ) are denoted respectively
by x(t) and z(t), and the respective outputs are y(t) = h(x(t)) and yp (t) = h(z(t)). We wish to
show that these two responses are approximately equal on 0 ≤ t ≤ T . Write δ(t) = u(t) − u¯∗ .
From Theorem 1 in [42] we know that
x(t) = x(0) + ξ(t) + o(kδk)
where kδk = sup0≤t≤T |δ(t)| and ξ is the solution of the variational system
˙ = Aξ(t) + Bδ(t)
ξ(t)
13
with ξ(0) = 0, and that
z(t) = z(0) + ζ(t) + o(kpu − pu¯∗ k) = z(0) + ζ(t) + o(kδk),
where
ζ̇ = Aζ(t) + Bpδ(t)
with ζ(0) = 0. By linearity, ζ = pξ. Using z(0) ≡ σ(pu¯∗ ) = pσ(u¯∗ ) = px(0), we have that
px(t) − z(t) = o(kδk). Thus,
y(t) = h(x(t)) = h(px(t)) = h(z(t) + o(kδk)) .
If K is an upper bound on the gradient of h, then
|yp (t) − y(t)| = |h(z(t)) − h(z(t) + o(kδk))| ≤ Ko(kδk).
Thus, the relative error supt |yp (t) − y(t)| / supt |u(t) − u¯∗ | converges to zero as a function of the
input perturbation u(t) − u¯∗ . As a numerical illustration, we consider again the the network
described by Eq.2 and the random parameter set in Fig.2. We compare the relative error
between the original nonlinear system, with initial state ξ = (xA , xB ) corresponding to u = 0.3,
and applied input u = 0.36, and the approximation is ξ + z(t), where the z solves the linear
system with initial condition zero and constant input 0.06. The maximum approximation error
is about 5% (to 3 decimal places, 0.055 for xA and 0.01 for xB ). When stepping from u = 0.5
to u = 0.6, the error is less than 3% (0.028 and 0.005 respectively). Similar results are available
for all ASI circuits (see SI Text).
4.3
Impossibility of perfect scale-invariance
Consider any system with state x = (xA , xB , xC ), output xC , and equations of the general form
ẋA = f (x) + G(xA )u, ẋB = g(x), ẋC = h(x) = xA a(xC ) + xB b(xC ) + c(xC ).
ẋA = f (x) + G(xA )u
ẋB = g(x)
ẋC
= h(x) = xA a(xC ) + xB b(xC ) + c(xC ) .
It is assumed that a(xC ) 6= 0 for all xC , G(xA ) 6= 0 for all xA , G := supx G(x) < ∞, and the
system is irreducible [40]. We now prove that such a system cannot be scale-invariant. Suppose
by way of contradiction that it would be, and pick any fixed p 6= 1. The main theorem in
[40] insures that there are two differentiable functions α(x) and β(x) such that the algebraic
identities:
αx (x)[f (x) + G(xA )u] + αy (x)g(x) + αz (x)h(x) = f (α(x), β(x), xC ) + G(α(x))pu,
βx (x)[f (x) + u] + βy (x)g(x) + βz (x)h(x) = g(α(x), β(x), xC )
α(x)a(xC ) + β(x)b(xC ) + c(xC ) = xA a(xC ) + xB b(xC ) + c(xC )
hold for all constant x = (xA , xB , xC ) and u, and the vector function x 7→ (α(x), β(x), z) is
one-to-one and onto, which implies in particular that
sup G(α(x)) = G .
x
14
Dividing by u and taking the limit as u → ∞ in the first identity, we conclude that αx (x)G(xA ) ≡
pG(α(x)). Doing the same in the second identity, we conclude that βx (x) ≡ 0. Finally, taking
partial derivatives with respect to xA in the third identity:
a(xC )pG(α(x))/G(xA ) = αx (x)a(xC ) + βx (x)b(xC ) = a(xC )
is true for all x. Since a(xC ) 6≡ 0, it follows that
pG(α(x)) = G(xA )
for all x. We consider two cases: (a) p < 1 and (b) p > 1. Suppose p < 1. Pick any sequence of
points x(i) with G(x(i) ) → G as i → ∞. Then G(α(x(i) )) → G/p > G, contradicting G(x) ≤ G.
If p > 1, picking a sequence such that G(α(x(i) )) → G as i → ∞ gives the contradiction
G(x(i) ) → pG > G. This shows that the FCD property cannot hold.
.
Acknowledgments
We are grateful to Wenzhe Ma for making available and explaining his software for generating
and testing networks for adaptation. This work was supported in part by the US National
Institutes of Health and the Air Force Office of Scientific Research.
15
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19
Supplementary Material
A characterization of scale invariant responses in enzymatic networks
5
Circuits that exhibit ASI
We list here the results of the computational screen as described in the Main Text. After
showing graphical representations for the 25 identified ASI circuits (21 topologies), we provide
their equations and parameters.
For each circuit, four plots are shown:
(a) a comparison between the plots of xA (t) and xB (t) for the original nonlinear system and
the respective plots for the linearized approximations,
(b) the plots showing scale-invariant behavior for step inputs,
and the comparison between the plots of xC (t) for the original nonlinear system and for
the quasi-steady state approximation, for
(c) step input change from 0.3 to 0.36 and
(d) step input change from 0.5 to 0.6.
20
(a) Circuit 1.
(b) Circuit 2.
(c) Circuit 3.
(d) Circuit 4.
(e) Circuit 5.
(f) Ciircuit 6.
(g) Circuit 7.
(h) Circuit 8.
(i) Circuit 9.
(j) Circuit 10
(k) Circuit 11.
(l) Circuit 12.
(m) Circuit 13.
(n) Circuit 14.
(o) Circuits 15 -17
(p) Circuit 18.
(q) Circuit 19.
(r) Circuit 20.
(s) Circuit 21 - 22
(t) Circuit 23.
(u) Circuit 24 - 25
21
Circuit 1.
x
eA
xA
− kBA xB
x
eA + KuA
xA + KBA
x
eB
xB
= kAB xA
− kFB B xFB
x
eB + KAB
xB + KFB B
x
eC
xC
= kAC xA
− kBC xB
x
eC + KAC
xC + KBC
ẋA = kuA u
ẋB
ẋC
Parameters: KAB = 0.001191; kAB = 1.466561; KAC = 0.113697; kAC = 1.211993; KBA =
0.001688; kBA = 44.802268; KBC = 0.009891; kBC = 7.239357; KuA = 0.093918; kuA =
11.447219; kAC = 1.211993; KAC = 0.1136927; KFB = 9.424319; kFB = 22.745736
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
22
Circuit 2.
x
eA
xA
xA
− kBA xB
− kCA xC
x
eA + KuA
xA + KBA
xA + KCA
x
eB
xB
= kAB xA
− kFB B xFB
x
eB + KAB
xB + KFB B
x
eC
xC
= kAC xA
− kBC xB
x
eC + KAC
xC + KBC
ẋA = kuA u
ẋB
ẋC
Parameters: KuA = 0.093918; kuA = 11.447219; KBA = 0.001688; kBA = 44.802268; KCA =
90.209027; kCA = 96.671843; KAB = 0.001191; kAB = 1.466561; KFB = 9.424319; kFB =
22.745736; KAC = 0.113697; kAC = 1.211993; KBC = 0.009891; kBC = 7.239357
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
23
Circuit 3.
x
eA
xA
xA
− kBA xB
− kAA xA
x
eA + KuA
xA + KBA
xA + KAA
x
eB
xB
xB
= kAB xA
− kCB xB
− kBB xB
x
eB + KAB
xB + KCB
xB + KBB
xC
x
eC
− kAC xA
= kBC xB
x
eC + KBC
xC + KAC
ẋA = kuA u
ẋB
ẋC
Parameters: KAA = 7.633962; kAA = 86.238263; KAB = 20.265158; kAB = 5.428752; KAC =
0.258375; kAC = 62.416585; KBA = 0.003960; kBA = 17.705166; KBB = 31.604578; kBB =
3.692326; KBC = 44.386408; kBC = 65.027941; KCB = 0.701052; kCB = 26.091557; KuA =
0.464248; kuA = 1.882348
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
24
Circuit 4.
x
eA
xA
xA
− kBA xB
− kAA xA
x
eA + KuA
xA + KBA
xA + KAA
x
eB
xB
= kAB xA
− kCB xC
x
eB + KAB
xB + KCB
xC
x
eC
− kAC xA
= kBC xB
x
eC + KBC
xC + KAC
ẋA = kuA u
ẋB
ẋC
Parameters: KAA = 7.633962; kAA = 86.238263; KAB = 20.265158; kAB = 5.428752; KAC =
0.258375; kAC = 62.416585; KBA = 0.003960; kBA = 17.705166; KBC = 44.386408; kBC =
65.027941; KCB = 0.701052; kCB = 26.091557; KuA = 0.464248; kuA = 1.882348
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
25
Circuit 5.
x
eA
xA
− kBA xB
x
eA + KuA
xA + KBA
x
eB
xB
= kAB xA
− kCB xC
x
eB + KAB
xB + KCB
xC
x
eC
− kAC xA
= kBC xB
x
eC + KBC
xC + KAC
ẋA = kuA u
ẋB
ẋC
Parameters:KAB = 63.277600; kAB = 6.638959; KAC = 0.133429; kAC = 55.731406; KBA =
0.011188; kBA = 2.749793; KBC = 0.013374; kBC = 45.175191; KCB = 1.457975; kCB =
2.114949; KuA = 24.589517; kuA = 5.346875
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
26
Circuit 6.
x
eA
xA
xA
xA
− kBA xB
− kAA xA
− kCA xC
x
eA + KuA
xA + KBA
xA + KAA
xA + KCA
x
eB
xB
xB
= kAB xA
− kCB xC
− kBB xB
x
eB + KAB
xB + KCB
xB + KBB
xC
x
eC
− kAC xA
= kBC xB
x
eC + KBC
xC + KAC
ẋA = kuA u
ẋB
ẋC
Parameters: KAA = 7.633962; kAA = 86.238263; KAB = 20.265158; kAB = 5.428752; KAC =
0.258375; kAC = 62.416585; KBA = 0.003960; kBA = 17.705166; KBB = 31.604578; kBB =
3.692326; KBC = 44.386408; kBC = 65.027941; KCA = 26.714681; kCA = 2.806080; KCB =
0.701052; kCB = 26.091557; KuA = 0.464248; kuA = 1.882348
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
27
Circuit 7.
x
eA
xA
xA
xA
− kBA xB
− kAA xA
− kCA xC
x
eA + KuA
xA + KBA
xA + KAA
xA + KCA
x
eB
xB
= kAB xA
− kCB xC
x
eB + KAB
xB + KCB
xC
x
eC
− kAC xA
= kBC xB
x
eC + KBC
xC + KAC
ẋA = kuA u
ẋB
ẋC
Parameters: KAA = 7.633962; kAA = 86.238263; KAB = 20.265158; kAB = 5.428752; KAC =
0.258375; kAC = 62.416585; KBA = 0.003960; kBA = 17.705166; KBC = 44.386408; kBC =
65.027941; KCA = 26.714681; kCA = 2.806080; KCB = 0.701052; kCB = 26.091557; KuA =
0.464248; kuA = 1.882348
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
28
Circuit 8.
x
eA
xA
− kBA xB
x
eA + KuA
xA + KBA
x
eB
xB
x
eB
= kAB xA
− kFB B xFB
+ kCB xC
x
eB + KAB
xB + KFB B
x
eB + KCB
x
eC
xC
xC
= kAC xA
− kBC xB
− kCC xC
x
eC + KAC
xC + KBC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KuA = 0.093918; kuA = 11.447219; KBA = 0.001688; kBA = 44.802268; KAB =
0.001191; kAB = 1.466561; KFB = 9.424319; kFB = 22.745736; KAC = 0.113697; kAC =
1.211993; KBC = 0.009891; kBC = 7.239357; KCB = 30.602013; kCB = 3.811536; KCC =
0.189125; kCC = 17.910182
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
29
Circuit 9.
x
eA
xA
xA
− kBA xB
− kCA xC
x
eA + KuA
xA + KBA
xA + KCA
x
eB
x
eB
xB
= kAB xA
+ kCB xC
− kFB B xFB
x
eB + KAB
x
eB + KCB
xB + KFB B
x
eC
xC
xC
= kAC xA
− kBC xB
− kCC xC
x
eC + KAC
xC + KBC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KuA = 0.093918; kuA = 11.447219; KBA = 0.001688; kBA = 44.802268; KCA =
90.209027; kCA = 96.671843; KAB = 0.001191; kAB = 1.466561; KFB = 9.424319; kFB =
22.745736; KAc = 0.113697; kAC = 1.211993; KBC = 0.009891; kBC = 7.239357; KCB =
30.602013; kCB = 3.811536; KCC = 0.189125; kCC = 17.910182
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
30
Circuit 10.
x
eA
xA
xA
− kBA xB
− kAA xA
x
eA + KuA
xA + KBA
xA + KAA
x
eB
x
eB
xB
= kAB xA
+ kCB xC
− kFB B xFB
x
eB + KAB
x
eB + KCB
xB + KFB B
x
eC
xC
xC
= kBC xB
− kAC xA
− kCC xC
x
eC + KBC
xC + KAC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KAA = 24.989065; kAA = 53.174082; KAB = 0.444375; kAB = 12.053134; KFB =
1.716920; kFB = 11.601122; KAC = 0.013988; kAC = 8.521185; KBA = 0.005461; kBA =
7.103952; KBC = 51.850148; kBC = 80.408137; KCB = 5.392001; kCB = 3.086740; KCC =
1.962230; kCC = 17.382010; KuA = 4.387832; kuA = 19.638124
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
31
Circuit 11.
x
eA
xA
− kBA xB
x
eA + KuA
xA + KBA
x
eB
x
eB
xB
= kAB xA
+ kCB xC
− kFB B xFB
x
eB + KAB
x
eB + KCB
xB + KFB B
x
eC
xC
xC
= kBC xB
− kAC xA
− kCC xC
x
eC + KBC
xC + KAC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KAB = 0.444375; kAB = 12.053134; KFB = 1.716920; kFB = 11.601122; KAC =
0.013988; kAC = 8.521185; KBA = 0.005461; kBA = 7.103952; KBC = 51.850148; kBC =
80.408137; KCB = 5.392001; kCB = 3.086740; KCC = 1.962230; kCC = 17.382010; KuA =
4.387832; kuA = 19.638124
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
32
Circuit 12.
x
eA
xA
x
eA
− kBA xB
+ kCA xC
x
eA + KuA
xA + KBA
x
eA + KCA
x
eB
x
eB
xB
= kAB xA
+ kCB C
− kFB B xFB
x
eB + KAB
x
eB + KCB
xB + KFB B
x
eC
xC
xC
= kAC xA
− kBC xB
− kCC xC
x
eC + KAC
xC + KBC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KuA = 0.093918; kuA = 11.447219; KBA = 0.001688; kBA = 44.802268; KCA =
5.026318; kCA = 45.803641; KAB = 0.001191; kAB = 1.466561; KFB = 9.424319; kFB =
22.745736; KAC = 0.113697; kAC = 1.211993; KBC = 0.009891; kBC = 7.239357; KCB =
30.602013; kCB = 3.811536; KCC = 0.189125; kCC = 17.910182
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
33
Circuit 13.
x
eA
xA
xA
x
eA
− kBA xB
− kAA xA
+ kCA xC
x
eA + KuA
xA + KBA
xA + KAA
x
eA + KCA
x
eB
x
eB
xB
= kAB xA
+ kCB xC
− kBB xB
x
eB + KAB
x
eB + KCB
xB + KBB
xC
xC
x
eC
− kAC xA
− kCC xA
= kBC xB
x
eC + KBC
xC + KAC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KAA = 24.989065; kAA = 53.174082; KAB = 0.444375; kAB = 12.053134; KFB
1.716920; kFB = 11.601122; KAC = 0.013988; kAC = 8.521185; KBA = 0.005461; kBA
7.103952; KBC = 51.850148; kBC = 80.408137; KCB = 5.392001; kCB = 3.086740; KCC
1.962230; kCC = 17.382010; KuA = 4.387832; kuA = 19.638124; KCA = 15.479253; kCA
4.903430
=
=
=
=
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
34
Circuit 14.
x
eA
xA
x
eA
− kBA xB
+ kCA xC
x
eA + KuA
xA + KBA
x
eA + KCA
x
eB
x
eB
xB
= kAB xA
+ kCB xC
− kBB xB
x
eB + KAB
x
eB + KCB
xB + KBB
xC
xC
x
eC
− kAC xA
− kCC xA
= kBC xB
x
eC + KBC
xC + KAC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KAB = 0.444375; kAB = 12.053134; KFB 1.716920; kFB = 11.601122; KAC =
0.013988; kAC = 8.521185; KBA = 0.005461; kBA = 7.103952; KBC = 51.850148; kBC =
80.408137; KCB = 5.392001; kCB = 3.086740; KCC = 1.962230; kCC = 17.382010; KuA =
4.387832; kuA = 19.638124; KCA = 15.479253; kCA = 4.903430
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
35
Circuit 15.
x
eA
xA
− kBA xB
x
eA + KuA
xA + KBA
x
eB
xB
= kAB xA
− kFB B xFB
x
eB + KAB
xB + KFB B
x
eC
xC
xC
= kAC xA
− kBC xB
− kCC xA
x
eC + KAC
xC + KBC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KAB = 0.709169; kAB = 7.445605; KFB = 1.495375; kFB = 7.282827; KAC =
0.002566; kAC = 1.115065; KBA = 0.002522; kBA = 5.753075; KBC = 0.017051; kBC =
2.777794; KCC = 0.195997; kCC = 1.480130; KuA = 0.225814; kuA = 2.492872
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
36
Circuit 16.
This is the same topology as in the previous case, only a different parameter set was used:
Parameters: KAB = 0.001191; kAB = 1.466561; KFB = 9.424319; kFB = 22.745736; KAC =
0.113697; kAC = 1.211993; KBA = 0.001688; kBA = 44.802268; KBC = 0.009891; kBC =
7.239357; KCC = 0.189125; kCC = 17.910182; KuA = 0.093918; kuA = 11.447219
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
37
Circuit 17.
This is the same topology as in the previous case, only a different parameter set was used:
Parameters: KAB = 1.620877; kAB = 2.306216; KFB = 2.012565; kFB = 2.700847; KAC =
0.010933; kAC = 8.968091; KBA = 0.001812; kBA = 10.039221; KBC = 0.014199; kBC =
17.762333; KCC = 2.686891; kCC = 4.139044; KuA = 0.161715; kuA = 1.933303
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
Circuit 18.
x
eA
xA
xA
− kBA xB
− kAA xA
x
eA + KuA
xA + KBA
xA + KAA
x
eB
x
eB
xB
= kAB xA
+ kBB xB
− kFB B xFB
x
eB + KAB
x
eB + KBB
xB + KFB B
x
eC
xC
xC
= kAC xA
− kBC xB
− kCC xC
x
eC + KAC
xC + KBC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KAA = 17.569120; kAA = 2.198366; KAB = 9.435176; kAB = 3.134007; KFB =
38
0.469083; kFB = 1.934194; KAC = 0.062914; kAC = 2.742206; KBA = 0.003245; kBA =
75.352905; KBB = 27.463128; kBB = 10.551155; KBC = 0.041615; kBC = 61.333818; KCC =
0.039332; kCC = 4.756637; KuA = 0.005167; kuA = 8.186533
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
39
Circuit 19.
x
eA
xA
xA
− kBA xB
− kAA xA
x
eA + KuA
xA + KBA
xA + KAA
x
eB
xB
= kAB xA
− kFB B xFB
x
eB + KAB
xB + KFB B
x
eC
xC
xC
= kBC xB
− kAC xA
− kCC xC
x
eC + KBC
xC + KAC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KuA = 4.387832; kuA = 19.638124; KBA = 0.005461; kBA = 7.103952; KAA =
24.989065; kAA = 53.174082; KAB = 0.444375; kAB = 12.053134; KFB = 1.716920; kFB =
11.601122; KBC = 51.850148; kBC = 80.408137; KAC = 0.013988; kAC = 8.521185; KCC =
1.962230; kCC = 17.382010
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
40
Circuit 20.
x
eA
xA
− kBA xB
x
eA + KuA
xA + KBA
x
eB
xB
= kAB xA
− kFB B xFB
x
eB + KAB
xB + KFB B
x
eC
xC
xC
= kBC xB
− kAC xA
− kCC xC
x
eC + KBC
xC + KAC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KuA = 4.387832; kuA = 19.638124; KBA = 0.005461; kBA = 7.103952; KAB =
0.444375; kAB = 12.053134; KFB = 1.716920; kFB = 11.601122; KBC = 51.850148; kBC =
80.408137; KAC = 0.013988; kAC = 8.521185; KCC = 1.962230; kCC = 17.382010
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
41
Circuit 21.
x
eA
xA
x
eA
− kBA xB
+ kCA xC
x
eA + KuA
xA + KBA
x
eA + KCA
x
eB
xB
= kAB xA
− kFB B xFB
x
eB + KAB
xB + KFB B
x
eC
xC
xC
= kAC xA
− kBC xB
− kCC xC
x
eC + KAC
xC + KBC
xC + KCC
ẋA = kuA u
ẋB
ẋC
Parameters: KuA = 0.093918; kuA = 11.447219; KBA = 0.001688; kBA = 44.802268; KCA =
5.026318; kCA = 45.803641; KAB = 0.001191; kAB = 1.466561; KFB = 9.424319; kFB =
22.745736; KAC = 0.113697; kAC = 1.211993; KBC = 0.009891; kBC = 7.239357; KCC =
0.189125; kCC = 17.910182
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
42
Circuit 22.
This is the same topology as in the previous case, only a different parameter set was used:
Parameters: KAB = 1.620877; kAB = 2.306216; KFB = 2.012565; kFB = 2.700847; KAC =
0.010933; kAC = 8.968091; KBA = 0.001812; kBA = 10.039221; KBC = 0.014199; kBC =
17.762333; KCA = 0.002690; kCA = 1.506954; KCC = 2.686891; kCC = 4.139044; KuA =
0.161715; kuA = 1.933303
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
Circuit 23.
x
eA
xA
xA
− kBA xB
− kCA xC
x
eA + KuA
xA + KBA
xA + KCA
x
eB
xB
= kAB xA
− kFB B xFB
x
eB + KAB
xB + KFB B
x
eC
xC
xC
= kAC xA
− kBC xB
− kCC xC
x
eC + KAC
xC + KBC
xC + KCC
ẋA = kuA u
ẋB
ẋC
43
Parameters: KuA = 0.093918; kuA = 11.447219; KBA = 0.001688; kBA = 44.802268; KCA =
90.209027; kCA = 96.671843; KAB = 0.001191; kAB = 1.466561; KFB = 9.424319; kFB =
22.745736; KAC = 0.113697; kAC = 1.211993; KBC = 0.009891; kBC = 7.239357; KCC =
0.189125; kCC = 17.910182
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
44
Circuit 24.
x
eA
xA
x
eA
− kBA xB
+ kCA xC
x
eA + KuA
xA + KBA
x
eA + KCA
x
eB
xB
= kAB xA
− kFB B xFB
x
eB + KAB
xB + KFB B
x
eC
xC
= kAC xA
− kBC xB
x
eC + KAC
xC + KBC
ẋA = kuA u
ẋB
ẋC
Parameters: KuA = 0.093918; kuA = 11.447219; KBA = 0.001688; kBA = 44.802268; KCA =
5.026318; kCA = 45.803641; KAB = 0.001191; kAB = 1.466561; KFB = 9.424319; kFB =
22.745736; KAC = 0.113697; kAC = 1.211993; KBC = 0.009891; kBC = 7.239357
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
45
Circuit 25.
This is the same topology as in the previous case, only a different parameter set was used:
KAB = 1.620877; kAB = 2.306216; KFB = 2.012565; kFB = 2.700847; KAC = 0.010933;
kAC = 8.968091; KBA = 0.001812; kBA = 10.039221; KBC = 0.014199; kBC = 17.762333;
KCA = 0.002690; kCA = 1.506954; KuA = 0.161715; kuA = 1.93330
(a) Dynamics of A and B in linearized model
(b) Ouput from C nonlinear model
(c) Quadratic approx. and output of nonlinear system
(d) Quadratic approx. and output of nonlinear system
46
6
Ratios xA (t)/xB (t)
In this section, for each ASI circuit, we show that the ratio xA (t)/xB (t) is approximately
invariant when inputs are scaled, as discussed in the Main Text.
47
Figure 8: xA (t)/xB (t) for circuits 1-6
48
Figure 9: xA (t)/xB (t) for circuits 7-12
49
Figure 10: xA (t)/xB (t) for circuits 13-18
50
Figure 11: xA (t)/xB (t) for circuits 19-24
51
Figure 12: xA (t)/xB (t) for circuit 25
52
7
Tables
In this section the following three tables for the 25 identified ASI circuits are shown:
• Table 1. Relative differences in linearization matrices corresponding to different linearizations, A0.3 = A(0.3), A0.4 = A(0.4), . . . , B0.6 = B(0.6), rounded to 3 decimal places. The
corresponding expressions are given by:
Aerr
ij =
X
u=0.3,0.4,0.5,0.6
(Au )ij − (A0.45 )ij
(A0.45 )ij
and similarly for B err . These relative differences are very small. The entries in the table
are of the following form: Aerr displayed as [a11 a12 ; a21 a22 ] and B err displayed as [b1 b2 ]T .
• Table 2. Relative error between original (nonlinear) system with an initial state ξ =
(xA , xB ) corresponding to u = 0.3, and applied input u = 0.36, and the approximation is
ξ + z(t), where z solves the linear system with an initial condition of zero and a constant
input of 0.06. Additionally, we provide relative errors between the original (nonlinear)
system with an initial state corresponding to u = 0.5, and applied input of u = 0.6,
and the approximation given by ξ + z(t), where z solves the linear system with an initial
condition of zero and a constant input of 0.1. The corresponding expressions are given
N
xA L
0.36 (t)−xA 0.36 (t)
by: xA err
,
max,u=0.36 = maxt≥0
x N (t)
x
L
(t)−x
A 0.36
N (t)
A 0.6
A 0.6
xA err
,
max,u=0.6 = maxt≥0
xA N
0.6 (t)
where N denotes the nonlinear system, and L denotes the linear system.
err
We define similarly for xB err
max,u=0.36 and xB max,u=0.6 .
• Table 3. Homogeneity property of the states xA and xB . For a constant input u, it holds
that σ(pu) ≈ pσ(u), where σ(u) is a unique steady state (xA , xB ).
53
Circuit
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
Aerr
[0.069 0.004; 0 0.005]
[0.084 0.006; 0.019 0.015]
[0.069 0.004; 0 0.005]
[0.114 0.007; 0.011 0.003]
[0.045 0.003; 0.01 0.033]
[0.075 0.012; 0.021 0.012]
[0.057 0.012; 0.021 0.012]
[0.055 0.012; 0.019 0.009]
[0.069 0.004; 0 0.005]
[0.037 0.022; 0.009 0.0707]
[0.037 0.022; 0.007 0.009]
[0.025 0.029; 0.007 0.006]
[0.037 0.022; 0.009 0.007]
[0.036 0.022; 0.007 0.009]
[0.07 0.004; 0 0.005]
[0.07 0.004; 0 0.005]
[0.073 0.012; 0.017 0.009]
[0.051 0.004; 0 0.005]
[0.066 0.013; 0.018 0.009]
[0.048 0.013; 0.018 0.009]
[0.051 0.004; 0 0.005]
[0.233 0; 0.011 0.003]
[0.069 0.004; 0 0.005]
[0.051 0.004; 0 0.005]
[0.233 0; 0.011 0.003]
B err
[0.002
[0.004
[0.002
[0.002
[0 0]T
[0.015
[0.012
[0.016
[0.002
[0.002
[0.002
[0.012
[0.002
[0.002
[0.002
[0.002
[0.015
[0.002
[0.015
[0.016
[0.002
[0.002
[0.002
[0.002
[0.002
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
0]T
Table 1: Relative error in linearization matrices
54
Circuit
err
err
xA err
max,u=0.36 xB max,u=0.36 xA max,u=0.6
xB err
max,u=0.6
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
0.055
0.008
0.055
0.03
0.031
0.015
0.023
0.023
0.055
0.097
0.010
0.033
0.097
0.010
0.056
0.056
0.027
0.047
0.027
0.023
0.04
0.116
0.055
0.045
0.117
0.005
0.002
0.005
0.004
0
0.005
0.004
0.004
0.005
0.016
0.016
0.010
0.016
0.016
0.005
0.005
0.004
0.006
0.004
0.004
0.004
0.013
0.005
0.005
0.013
0.011
0.007
0.010
0.007
0.006
0.016
0.021
0.021
0.01
0.020
0.020
0.021
0.020
0.02
0.010
0.010
0.022
0.010
0.023
0.021
0.009
0.027
0.010
0.01
0.03
0.028
0
0.028
0.012
0.003
0.011
0.005
0.004
0.028
0.081
0.084
0.024
0.081
0.084
0.028
0.028
0.004
0.028
0.005
0.005
0.034
0.05
0.028
0.027
0.05
Table 2: Relative error between nonlinear and linearized system
55
Circuit
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
σ(u0.3 )/0.3
(0.195, 0.239)
(0.199, 0.364)
(0.195, 0.239)
(0.132, 0.172)
(0.591, 0.11)
(0.206, 0.526)
(0.208, 0.529)
(0.206, 0.530)
(0.195, 0.239)
(0.078, 0.083)
(0.077, 0.083)
(0.153, 0.09)
(0.078, 0.083)
(0.077, 0.083)
(0.195, 0.239)
(0.195, 0.239)
(0.204, 0.526)
(0.196, 0.24)
(0.205, 0.528)
(0.206, 0.532)
(0.196, 0.24)
(0.136, 0.177)
(0.195, 0.239)
(0.196, 0.240)
(0.136, 0.178)
σ(u0.4 )/0.4
(0.193, 0.237)
(0.197, 0.359)
(0.193, 0.237)
(0.131, 0.170)
(0.58, 0.109)
(0.198, 0.507)
(0.2, 0.512)
(0.199, 0.512)
(0.194, 0.237)
(0.075, 0.08)
(0.074, 0.08)
(0.145, 0.086)
(0.075, 0.08)
(0.074, 0.08)
(0.193, 0.237)
(0.193, 0.237)
(0.197, 0.508)
(0.193, 0.238)
(0.197, 0.509)
(0.199, 0.513)
(0.194, 0.237)
(0.134, 0.173)
(0.193, 0.237)
(0.194, 0.237)
(0.134, 0.173)
σ(u0.5 )/0.5
(0.192, 0.236)
(0.194, 0.356)
(0.192, 0.236)
(0.131, 0.169)
(0.57, 0.109)
(0.192, 0.493)
(0.194, 0.498)
(0.193, 0.499)
(0.192, 0.236)
(0.073, 0.078)
(0.072, 0.078)
(0.139, 0.082)
(0.073, 0.078)
(0.072, 0.078)
(0.191, 0.235)
(0.191, 0.236)
(0.191, 0.494)
(0.192, 0.236)
(0.192, 0.494)
(0.193, 0.5)
(0.192, 0.236)
(0.133, 0.171)
(0.192, 0.236)
(0.192, 0.236)
(0.133, 0.171)
σ(u0.6 )/0.6
(0.19, 0.234)
(0.192, 0.353)
(0.191, 0.234)
(0.13, 0.168)
(0.561, 0.108)
(0.188, 0.481)
(0.19, 0.486)
(0.189, 0.486)
(0.190, 0.234)
(0.071, 0.076)
(0.071, 0.076)
(0.135, 0.08)
(0.071, 0.076)
(0.071, 0.076)
(0.190, 0.234)
(0.19, 0.234)
(0.186, 0.48)
(0.19, 0.235)
(0.187, 0.481)
(0.189, 0.487)
(0.191, 0.235)
(0.132, 0.17)
(0.191, 0.234)
(0.190, 0.235)
(0.132, 0.17)
Table 3: σ(u)/u for constant inputs u = 0.3, 0.4, 0.5, 0.6
56
| 5 |
The DFS Fused Lasso: Linear-Time Denoising over General
Graphs
Oscar Hernan Madrid Padilla1
oscar.madrid@utexas.edu
James Sharpnack2
jsharpna@gmail.com
arXiv:1608.03384v3 [math.ST] 1 Mar 2017
James Scott1,3
james.scott@mccombs.utexas.edu
Ryan J. Tibshirani4,5
ryantibs@stat.cmu.edu
1
Department of Statistics and Data Sciences, University of Texas, Austin, TX 78712
2
Department of Statistics, University of California at Davis, Davis, CA 95616
3
McCombs School of Business, University of Texas, Austin, TX 78712
4
Department of Statistics, Carnegie Mellon University, Pittsburgh, PA 15213
5
Machine Learning Department, Carnegie Mellon University, Pittsburgh, PA 15213
Abstract
The fused lasso, also known as (anisotropic) total variation denoising, is widely used for piecewise
constant signal estimation with respect to a given undirected graph. The fused lasso estimate is highly
nontrivial to compute when the underlying graph is large and has an arbitrary structure. But for a special
graph structure, namely, the chain graph, the fused lasso—or simply, 1d fused lasso—can be computed
in linear time. In this paper, we establish a surprising connection between the total variation of a generic
signal defined over an arbitrary graph, and the total variation of this signal over a chain graph induced by
running depth-first search (DFS) over the nodes of the graph. Specifically, we prove that for any signal,
its total variation over the induced chain graph is no more than twice its total variation over the original
graph. This connection leads to several interesting theoretical and computational conclusions. Denoting
by m and n the number of edges and nodes, respectively, of the graph in question, our result implies that
for an underlying signal with total variation t over the graph, the fused lasso achieves a mean squared
error rate of t2/3 n−2/3 . Moreover, precisely the same mean squared error rate is achieved by running the
1d fused lasso on the induced chain graph from running DFS. Importantly, the latter estimator is simple
and computationally cheap, requiring only O(m) operations for constructing the DFS-induced chain and
O(n) operations for computing the 1d fused lasso solution over this chain. Further, for trees that have
bounded max degree, the error rate of t2/3 n−2/3 cannot be improved, in the sense that it is the minimax
rate for signals that have total variation t over the tree. Finally, we establish several related results—for
example, a similar result for a roughness measure defined by the `0 norm of differences across edges in
place of the the total variation metric.
Keywords: fused lasso, total variation denoising, graph denoising, depth-first search
1
Introduction
We study the graph denoising problem, i.e., estimation of a signal θ0 ∈ Rn from noisy data
yi = θ0,i + i ,
i = 1, . . . , n,
(1)
when the components of θ0 are associated with the vertices of an undirected, connected graph G = (V, E).
Without a loss of generality, we denote V = {1, . . . , n}. Versions of this problem arise in diverse areas of
science and engineering, such as gene expression analysis, protein mass spectrometry, and image denoising.
1
The problem is also archetypal of numerous internet-scale machine learning tasks that involve propagating
labels or information across edges in a network (e.g., a network of users, web pages, or YouTube videos).
Methods for graph denoising have been studied extensively in machine learning and signal processing.
In machine learning, graph kernels have been proposed for classification and regression, in both supervised
and semi-supervised data settings (e.g., Belkin and Niyogi [2002], Smola and Kondor [2003], Zhu et al.
[2003], Zhou et al. [2005]). In signal processing, a considerable focus has been placed on the construction
of wavelets over graphs (e.g., Crovella and Kolaczyk [2003], Coifman and Maggioni [2006], Gavish et al.
[2010], Hammond et al. [2011], Sharpnack et al. [2013], Shuman et al. [2013]). We will focus our study on
the fused lasso over graphs, also known as (anisotropic) total variation denoising over graphs. Proposed by
Rudin et al. [1992] in the signal processing literature, and Tibshirani et al. [2005] in the statistics literature,
the fused lasso estimate is defined by the solution of a convex optimization problem,
θ̂G = argmin
θ∈Rn
1
ky − θk22 + λk∇G θk1 ,
2
(2)
where y = (y1 , . . . , yn ) ∈ Rn the vector of observed data, λ ≥ 0 is a tuning parameter, and ∇G ∈ Rm×n is
the edge incidence matrix of the graph G. Note that the subscript on the incidence matrix ∇G and the fused
lasso solution θ̂G in (2) emphasize that these quantities are defined with respect to the graph G. The edge
incidence matrix ∇G can be defined as follows, using some notation and terminology from algebraic graph
theory (e.g., Godsil and Royle [2001]). First, we assign an arbitrary orientation to edges in the graph, i.e.,
for each edge e ∈ E, we arbitrarily select one of the two joined vertices to be the head, denoted e+ , and the
other to be the tail, denoted e− . Then, we define a row (∇G )e of ∇G , corresponding to the edge e, by
(∇G )e,e+ = 1, (∇G )e,e− = −1, (∇G )e,v = 0 for all v 6= e+ , e− ,
for each e ∈ E. Hence, for an arbitrary θ ∈ Rn , we have
X
k∇G θk1 =
|θe+ − θe− |.
e∈E
We can see that the particular choice of orientation does not affect the value k∇G θk1 , which we refer to as
the total variation of θ over the graph G.
1.1
Summary of results
We will wait until Section 1.3 to give a detailed review of literature, both computational and theoretical, on
the fused lasso. Here we simply highlight a key computational aspect of the fused lasso to motivate the main
results in our paper. The fused lasso solution in (2), for a graph G of arbitrary structure, is highly nontrivial
to compute. For a chain graph, however, the fused lasso solution can be computed in linear time (e.g., using
dynamic programming or specialized taut-string methods).
The question we answer is: how can we use this fact to our advantage, when seeking to solve (2) over
an arbitrary graph? Given a generic graph structure G that has m edges and n nodes, it is obvious that we
can define a chain graph by running depth-first search (DFS) over the nodes. Far less obvious is that, for
any signal, its total variation over the DFS-induced chain graph never exceeds twice its total variation over
the original graph. This fact, which we prove, has the following three notable consequences (the first being
computational, and next two statistical).
1. No matter the structure of G, we can denoise any signal defined over this graph in O(m + n) operations: O(m) operations for DFS and O(n) operations for the 1d fused lasso on the induced chain. We
call the corresponding estimator—the 1d fused lasso run on the DFS-induced chain—the DFS fused
lasso.
2
2. For an underlying signal θ0 that generates the data, as in (1), such that θ0 ∈ BVG (t), where BVG (t)
is the class of signals with total variation at most t, defined in (4), the DFS fused lasso estimator has
mean squared error (MSE) on the order of t2/3 n−2/3 .
3. For an underlying signal θ0 ∈ BVG (t), the fused lasso estimator over the original graph, in (2), also
has MSE on the order of t2/3 n−2/3 .
The fact that such a fast rate, t2/3 n−2/3 , applies for the fused lasso estimator over any connected graph
structure is somewhat surprising. It implies that the chain graph represents the hardest graph structure for
denoising signals of bounded variation—at least, hardest for the fused lasso, since as we have shown, error
rates on general connected graphs can be no worse than the chain rate of t2/3 n−2/3 .
We also complement these MSE upper bounds with the following minimax lower bound over trees.
4. When G is a tree of bounded max degree, the minimax MSE over the class BVG (t) scales at the rate
t2/3 n−2/3 . Hence, in this setting, the DFS fused lasso estimator attains the optimal rate, as does the
fused lasso estimator over G.
Lastly, we prove the following for signals with a bounded number of nonzero edge differences.
5. For an underlying signal θ0 ∈ BDG (s), where BDG (s) is the class of signals with at most s nonzero
edge differences, defined in (5), the DFS fused lasso (under a condition on the spacing of nonzero differences over the DFS-induced chain) has MSE on the order of s(log s + log log n) log n/n + s3/2 /n.
When G is a tree, the minimax MSE over the class BDG (s) scales as s log(n/s)/n. Thus, in this setting, the DFS fused lasso estimator is only off by a log log n factor provided that s is small.
This DFS fused lasso gives us an O(n) time algorithm for nearly minimax rate-optimal denoising over
trees. On paper, this only saves a factor of O(log n) operations, as recent work (to be described in Section
1.3) has produced an O(n log n) time algorithm for the fused lasso over trees, by extending earlier dynamic
programming ideas over chains. However, dynamic programming on a tree is (a) much more complex than
dynamic programming on a chain (since it relies on sophisticated data structures), and (b) noticeably slower
in practice than dynamic programming over a chain, especially for large problem sizes. Hence there is still
a meaningful difference—both in terms of simplicity and practical computational efficiency—between the
DFS fused lasso estimator and the fused lasso over a generic tree.
For a general graph structure, we cannot claim that the statistical rates attained by the DFS fused lasso
estimator are optimal, nor can we claim that they match those of fused lasso over the original graph. As an
example, recent work (to be discussed in Section 1.3) studying the fused lasso over grid graphs shows that
estimation error rates for this problem can be much faster than those attained by the DFS fused lasso (and
thus the minimax rates over trees). What should be emphasized, however, is that the DFS fused lasso can
still be a practically useful method for any graph, running in linear time (in the number of edges) no matter
the graph structure, a scaling that is beneficial for truly large problem sizes.
1.2
Assumptions and notation
Our theory will be primarily phrased in terms of the mean squared error (MSE) an estimator θ̂ of the mean
parameter θ0 in (1), assuming that = (1 , . . . , n ) has i.i.d. mean zero sub-Gaussian components, i.e.,
E(i ) = 0, and P(|i | > t) ≤ M exp − t2 /(2σ 2 ) , all t ≥ 0, for i = 1, . . . , n,
(3)
for constants M, σ > 0. The MSE of θ̂ will be denoted, with a slight abuse of notation, by
kθ̂ − θ0 k2n =
3
1
kθ̂ − θ0 k22 .
n
√
(In general, for a vector x ∈ Rn , we denote its scaled `2 norm by kxkn = kxk2 / n.) Of course, the MSE
will depend not only on the estimator θ̂ in question but also on the assumptions that we make about θ0 . We
will focus our study on two classes of signals. The first is the bounded variation class, defined with respect
to the graph G, and a radius parameter t > 0, as
BVG (t) = {θ ∈ Rn : k∇G θk1 ≤ t}.
(4)
The second is the bounded differences class, defined again with respect to the graph G, and a now a sparsity
parameter s > 0, as
BDG (s) = {θ ∈ Rn : k∇G θk0 ≤ s}.
(5)
We call measure of roughness used in the bounded differences class the cut metric, given by replacing the
`1 norm used to define the total variation metric by the `0 norm, i.e.,
X
k∇G θk0 =
1{θe+ 6= θe− },
e∈E
which counts the number of nonzero edge differences that appear in θ. Hence, we may think of the former
class in (4) as representing a type of weak sparsity across these edge differences, and the latter class in (5)
as representing a type of strong sparsity in edge differences.
When dealing with the chain graph, on n vertices, we will use the following modifications to our notation. We write ∇1d ∈ R(n−1)×n for the edge incidence matrix of the chain, i.e.,
−1
1
0 ... 0
0 −1
1 ... 0
(6)
∇1d = .
.
..
..
..
.
.
0
0 . . . −1 1
We also write θ̂1d for the solution of the fused lasso problem in (2) over the chain, also called the 1d fused
lasso solution, i.e., to be explicit,
n−1
θ̂1d
X
1
= argmin ky − θk22 + λ
|θi+1 − θi |.
2
θ∈Rn
(7)
i=1
We write BV1d (t) and BD1d (s) for the bounded variation and bounded differences classes with respect to
the chain, i.e., to be explicit,
BV1d (t) = {θ ∈ Rn : k∇1d θk1 ≤ t},
BD1d (s) = {θ ∈ Rn : k∇1d θk0 ≤ s}.
Lastly, in addition to the standard notation an = O(bn ), for sequences an , bn such that an /bn is upper
−1
bounded for n large enough, we use an bn to denote that both an = O(bn ) and a−1
n = O(bn ). Also, for
random sequences An , Bn , we use An = OP (Bn ) to denote that An /Bn is bounded in probability.
1.3
Related work
Since its inception in the signal processing and statistics communities in Rudin et al. [1992] and Tibshirani
et al. [2005], respectively, there has been an impressive amount of work on total variation penalization and
the fused lasso. We do not attempt to give a complete coverage, but point out some relevant computational
and theoretical advances, covering the two categories separately.
4
Computational. On the computational side, it is first worth pointing out that there are multiple efficient
algorithms for solving the fused lasso problem over a chain graph, i.e., the 1d fused lasso problem. Davies
and Kovac [2001] derived an algorithm based on a “taut string” perspective that solves the 1d fused lasso
problem in O(n) time (but, the fact that their taut string method solves the 1d fused lasso problem was
not explicitly stated in the work). This was later extended by Condat [2012], Barbero and Sra [2014] to
allow for arbitrary weights in both of the individual penalty and loss terms. Johnson [2013] proposed an
entirely different O(n) time algorithm for the fused lasso based on dynamic programming. The taut string
and dynamic programming algorithms are extremely fast in practice (e.g., they can solve a 1d fused lasso
problem with n in the tens of millions in just a few seconds on a standard laptop).
Kolmogorov et al. [2016] extended the dynamic programming approach of Johnson [2013] to solve
the fused lasso problem on a tree. Their algorithm in theoretically very efficient, with O(n log n) running
time, but the implementation that achieves this running time (we have found) can be practically slow for
large problem sizes, compared to dynamic programming on a chain graph. Alternative implementations are
possible, and may well improve practical efficiency, but as far as we see it, they will all involve somewhat
sophisticated data structures in the “merge” steps in the forward pass of dynamic programming.
Barbero and Sra [2014] extended (though not in the same direct manner) fast 1d fused lasso optimizers
to work over grid graphs, using operator splitting techniques like Douglas-Rachford splitting. Their techniques appear to be quite efficient in practice, and the authors provide thorough comparisons and a thorough
literature review of related methods. Over general graphs structures, many algorithms have been proposed,
e.g., to highlight a few: Chambolle and Darbon [2009] described a direct algorithm based on a reduction to
parametric max flow programming; Hoefling [2010], Tibshirani and Taylor [2011] gave solution path algorithms (tracing out the solution in (2) over all λ ∈ [0, ∞]); Chambolle and Pock [2011] described what can
be seen as a kind of preconditioned ADMM-style algorithm; Kovac and Smith [2011] described an active
set approach; Tansey and Scott [2015] leveraged fast 1d fused lasso solvers in an ADMM decomposition
over trails of the graph; most recently, Landrieu and Obozinski [2015] derived a new method based on graph
cuts. We emphasize that, even with the advent of these numerous clever computational techniques for the
fused lasso over general graphs, it is still far slower to solve the fused lasso over an arbitrary graph than it is
to solve the fused lasso over a chain.
Theoretical. On the theoretical side, it seems that the majority of statistical theory on the fused lasso can
be placed into two categories: analysis of changepoint recovery, and analysis of MSE. Some examples of
works focusing on changepoint recovery are Rinaldo [2009], Harchaoui and Levy-Leduc [2010], Qian and
Jia [2012], Rojas and Wahlberg [2014]. The statistical theory will concern MSE rates, and hence we give a
more detailed review of related literature for this topic.
We begin with results for chain graphs. Mammen and van de Geer [1997] proved, when θ0 ∈ BV1d (t),
that the 1d fused lasso estimator estimator θ̂1d with λ t−1/3 n1/3 satisfies
kθ̂1d − θ0 k2n = OP (t2/3 n−2/3 ).
(8)
This is indeed the minimax MSE rate for the class BV1d (t), as implied by the minimax results in Donoho
and Johnstone [1998]. (For descriptions of the above upper bound and this minimax rate in a language more
in line with that of the current paper, see Tibshirani [2014].) Recently, Lin et al. [2016] improved on earlier
results for the bounded differences class in Dalalyan et al. [2014], and proved that when θ0 ∈ BD1d (s), the
1d fused lasso estimator θ̂1d with λ (nWn )1/4 satisfies
p
s
kθ̂1d − θ0 k2n = OP
(9)
(log s + log log n) log n + n/Wn ,
n
where Wn denotes the minimum distance between positions at which nonzero differences occur in θ0 , more
precisely, Wn = min{|i − j| : (∇1d θ0 )i 6= 0, (∇1d θ0 )j 6= 0}. When these nonzero differences or “jumps”
5
in θ0 are evenly spaced apart, we have Wn n/s, and the above becomes, for λ
s(log s + log log n) log n s3/2
2
.
+
kθ̂1d − θ0 kn = OP
n
n
√
ns−1/4 ,
(10)
This is quite close to the minimax lower bound, whose rate is s log(n/s)/n, that we establish for the class
BD1d (s), in Theorem 7. (The minimax lower bound that we prove this theorem actually holds beyond the
chain graph, and applies to tree graphs). We can see that the 1d fused lasso rate in (10) is only off by a factor
of log log n, provided that s does not grow too fast (specifically, s = O((log n log log n)2 )).
Beyond chain graphs, the story is in general much less clear, however, interesting results are known in
special cases. For a d-dimensional grid graph, with d ≥ 2, Hutter and Rigollet [2016] recently improved on
results of Wang et al. [2016], showing that for θ0 ∈ BVG (t) ∩ BDG (s), the fused lasso estimator θ̂G over
G satisfies
loga n
2
kθ̂G − θ0 kn = OP min{t, s}
.
(11)
n
when λ loga/2 n, where a = 2 if d = 2, and a = 1 if d ≥ 3. A minimax lower
p bound on the MSE rate
for the BVG (t) class over a grid G of dimension d ≥ 2 was established to be t log(n/t)/n, by Sadhanala
et al. [2016]. This makes the rate achieved by the fused lasso in (11) nearly optimal for bounded variation
signals, off by at most a log3/2 n factor when d = 2, and a log n factor when d ≥ 3.
Wang et al. [2016], Hutter and Rigollet [2016] also derived MSE rates for the fused lasso over several
other graph structures, such as Erdos-Renyi random graphs, Ramanujan d-regular graphs, star graphs, and
complete graphs. As it is perhaps the most relevant to our goals in this paper, we highlight the MSE bound
from Wang et al. [2016] that applies to arbitrary connected graphs. Their Theorem 3 √
implies, for a generic
connected graph G, θ0 ∈ BVG (t), that the fused lasso estimator θ̂G over G with λ n log n satisfies
r
log n
2
kθ̂G − θ0 kn = OP t
.
(12)
n
(See Appendix A.1 for details.) In Theorem 3, we show that the universal tn−1/2 rate (ignoring log terms)
in (12) for the fused lasso over an arbitrary connected graph can be improved to t2/3 n−2/3 . In Theorem 2,
we show that the same rate can indeed be achieved by a simple, linear-time algorithm: the DFS fused lasso.
1.4
Outline
In Section 2, we prove a simple but key lemma relating the `1 norm (and `0 ) norm of differences on a tree
and a chain induced by running DFS. We then define the DFS fused lasso estimator. In Section 3, we derive
MSE rates for the DFS fused lasso, and the fused lasso over the original graph G in question, for signals
of bounded variation. We also derive lower bounds for the minimax MSE rate over trees. In Section 4, we
proceed similarly, but for signals with bounded differences. In Section 5, we cover numerical experiments,
and in Section 6, we summarize our work and also describe some potential extensions.
2
The DFS fused lasso
In this section, we define the DFS-induced chain graph and the DFS fused lasso.
2.1
Tree and chain embeddings
We start by studying some of the fundamental properties associated with total variation on general graphs,
and embedded trees and chains. Given a graph G = (V, E), let T = (V, ET ) be an arbitrary spanning tree
6
of G. It is clear that for any signal, its total variation of over T is no larger than its total variation over G,
X
X
k∇T θk1 =
|θe+ − θe− | ≤
|θe+ − θe− | = k∇G θk1 , for all θ ∈ Rn .
(13)
e∈ET
e∈E
The above inequality, albeit very simple, reveals to us the following important fact: if the underlying mean
θ0 in (1) is assumed to be smooth with respect to the graph G, inasmuch as k∇G θ0 k1 ≤ t, then it must also
be smooth with respect to any spanning tree T of G, since k∇T θ0 k1 ≤ t. Roughly speaking, computing the
fused lasso solution in (2) over a spanning tree T , instead of G, would therefore still be reasonable for the
denoising purposes, as the mean θ0 would still be smooth over T according to the total variation metric.
The same property as in (14) also holds if we replace total variation by the cut metric:
X
X
k∇T θk0 =
1{θe+ 6= θe− } ≤
1{θe+ 6= θe− } = k∇G θk0 , for all θ ∈ Rn .
(14)
e∈ET
e∈E
Thus for the mean θ0 , the property k∇G θ0 k0 ≤ s again implies k∇T θ0 k0 ≤ s for any spanning tree T of G,
and this would again justify solving the fused lasso over T , in place of G, assuming smoothness of θ0 with
respect to the cut metric in the first place.
Here we go one step further than (13), (14), and assert that analogous properties actually hold for specially embedded chain graphs. The next lemma gives the key result.
Lemma 1. Let G = (V, E) be a connected graph, where recall we write V = {1, . . . , n}. Consider depthfirst search (DFS) run on G, and denote by v1 , . . . , vn the nodes in the order in which they are reached by
DFS. Hence, DFS first visits v1 , then v2 , then v3 , etc. This induces a bijection τ : {1, . . . , n} → {1, . . . , n},
such that
τ (i) = vi , for all i = 1, . . . , n.
Let P ∈ Rn×n denote the permutation associated with τ . Then it holds that
k∇1d P θk1 ≤ 2k∇G θk1 ,
for all θ ∈ Rn ,
(15)
k∇1d P θk0 ≤ 2k∇G θk0 ,
for all θ ∈ Rn .
(16)
as well as
Proof. The proof is simple. Observe that
k∇1d P θk1 =
X
|θτ (i+1) − θτ (i) |,
(17)
i=1,...,n−1
and consider an arbitrary summand |θτ (i+1) − θτ (i) |. There are now two cases to examine. First, suppose
τ (i) is not a leaf node, and τ (i + 1) has not yet been visited by DFS; then there is an edge e ∈ E such that
{e− , e+ } = {τ (i), τ (i + 1)}, and |θτ (i+1) − θτ (i) | = |θe+ − θe− |. Second, suppose that either τ (i) is a leaf
node, or all of its neighbors have already been visited by DFS; then there is a path p = {p1 , . . . , pr } in the
graph such that p1 = τ (i), pr = τ (i + 1), and each {pj , pj+1 } ∈ E, j = 1, . . . , r − 1, so that by the triangle
inequality
r−1
X
|θτ (i+1) − θτ (i) | ≤
|θpj−1 − θpj |.
j=1
Applying this logic over all terms in the sum in (17), and invoking the fundamental property that DFS visits
each edge exactly twice (e.g., Chapter 22 of Cormen et al. [2001]), we have established (15). The proof for
(16) follows from precisely the same arguments.
7
Example 1. The proof behind Lemma 1 can also be clearly demonstrated through an example. We consider
G to be a binary tree graph with n = 7 nodes, shown below, where we have labeled the nodes according to
the order in which they are visited by DFS (i.e., so that here P is the identity).
1
5
2
3
7
6
4
In this case,
k∆1d θk1 =
6
X
|θi+1 − θi |
i=1
≤ |θ2 − θ1 | + |θ3 − θ2 | + |θ3 − θ2 | + |θ4 − θ2 | + |θ4 − θ2 | + |θ2 − θ1 | + |θ5 − θ1 |
+ |θ6 − θ5 | + |θ6 − θ5 | + |θ7 − θ5 |
X
≤2
|θe+ − θe− | = 2k∇G θk1 ,
e∈G
where in the inequality above, we have used triangle inequality for each term in parentheses individually.
2.2
The DFS fused lasso
We define the DFS fused lasso estimator, θ̂DFS , to be the fused lasso estimator over the chain graph induced
by running DFS on G. Formally, if τ denotes the bijection associated with the DFS ordering (as described
in Lemma 1), then the DFS-induced chain graph can be expressed as C = (V, EC ) where V = {1, . . . , n}
and EC = {{τ (1), τ (2)}, . . . , {τ (n − 1), τ (n)}}. Denoting by P the permutation matrix associated with τ ,
the edge incidence matrix of C is simply ∇C = ∇1d P , and the DFS fused lasso estimator is given by
θ̂DFS = argmin
θ∈Rn
= P
>
1
ky − θk22 + λk∇1d P θk1
2
!
n−1
X
1
2
|θi+1 − θi | .
argmin kP y − θk2 + λ
2
θ∈Rn
(18)
i=1
Therefore, we only need to compute the 1d fused lasso estimator on a permuted data vector P y, and apply
the inverse permutation operator P > , in order to compute θ̂DFS .
Given the permutation matrix P , the computational cost of (18) is O(n), since, to recall the discussion
in Section 1.3, the 1d fused lasso problem (7) can be solved in O(n) operations with dynamic programming
or taut string algorithms. The permutation P is obtained by running DFS, which requires O(m) operations,
and makes the total computation cost of the DFS fused lasso estimator O(m + n).
It should be noted that, when multiple estimates are desired over the same graph G, we must only run
DFS once, and all subsequent estimates on the induced chain require just O(n) operations.
The bounds in (15), (16) for the DFS chain are like those in (13), (14) for spanning trees, and carry the
same motivation as that discussed above for spanning trees, beneath (13), (14): if the mean θ0 is assumed to
be smooth with respect to t, insofar as its total variation satisfies k∇G θ0 k1 ≤ t, then denoising with respect
8
to C would also be reasonable, in that k∇1d P θ0 k1 ≤ 2t; the same can be said for the cut metric. However,
it is the rapid O(m + n) computational cost of the DFS fused lasso, and also the simplicity of the dynamic
programming and taut string algorithms for the 1d fused lasso problem (7), that makes (15), (16) particularly
appealing compared to (13), (14). To recall the discussion in Section 1.3, the fused lasso can in principle be
computed efficiently over a tree, in O(n log n) operations using dynamic programming, but this requires a
much more cumbersome implementation and in practice we have found it to be noticeably slower.
2.3
Running DFS on a spanning tree
We can think of the induced chain graph, as described in the last section, as being computed in two steps:
(i) run DFS to compute a spanning tree T of G;
(ii) run DFS on the spanning tree T to define the chain C.
Clearly, this is the same as running DFS on G to define the induced chain C, so decomposing this process
into two steps as we have done above may seem odd. But this decomposition provides a useful perspective
because it leads to the idea that we could compute the spanning tree T in Step (i) in any fashion, and then
proceed with DFS on T in Step 2 in order to define the chain C. Indeed, any spanning tree in Step (i) will
lead to a chain C that has the properties (15), (16) as guaranteed by Lemma 1. This may be of interest if
we could compute a spanning tree T that better represents the topology of the original graph G, so that the
differences over the eventual chain C better mimicks those over G.
An example of a spanning tree whose topology is designed to reflect that of the original graph is a lowstretch spanning tree. Current interest on low-stretch spanning trees began with the breakthrough results in
Elkin et al. [2008]; most recently, Abraham and Neiman [2012] showed that a spanning tree with average
stretch O(log n log log n) can be computed in O(m log n log log n) operations. In Section 5, we investigate
low-stretch spanning trees experimentally.
In Section 6.4, we discuss a setting in which the fused lasso problem (2) has arbitrary penalty weights,
which gives rise to a weighted graph G. In this setting, an example of a spanning tree that can be crafted so
that its edges represent important differences in the original graph is a maximum spanning tree. Prim’s and
Kruskal’s minimum spanning tree algorithms, each of which take O(m log n) time [Cormen et al., 2001],
can be used to compute a maximum spanning tree after we negate all edge weights.
2.4
Averaging multiple DFS estimators
Notice that several DFS-induced chains can be formed from a single seed graph G, by running DFS itself on
G with different random starts (or random decisions about which edge to follow at each step in DFS), or by
computing different spanning trees T of G (possibly themselves randomized) on which we run DFS, or by
(1)
(2)
(K)
some combination, etc. Denoting by θ̂DFS , θ̂DFS , . . . , θ̂DFS the DFS fused
estimators fit to K different
P lasso
(k)
induced chains, we might believe that the average estimator, (1/K) K
θ̂
k=1 DFS , will have good denoising
performance, as it incorporates fusion at each node in multiple directions. In Section 5, we demonstrate that
this intuition holds true (at least, across the set of experiments we consider).
3
Analysis for signals of bounded variation
Throughout this section, we assume that the underlying mean θ0 in (1) satisfies θ0 ∈ BVG (t) for a generic
connected graph G. We derive upper bounds on the MSE rates of the DFS fused lasso and the fused lasso
over G. We also derive a tight lower bound on the minimax MSE when G is a tree that of bounded degree.
9
3.1
The DFS fused lasso
The analysis for the DFS fused lasso estimator is rather straightforward. By assumption, k∇G θ0 k1 ≤ t, and
thus k∇1d P θ0 k1 ≤ 2t by (15) in Lemma 1. Hence, we may think of our model (1) as giving us i.i.d. data
P y around P θ0 ∈ BV1d (2t), and we may apply existing results from Mammen and van de Geer [1997]
on the 1d fused lasso for bounded variation signals, as described in (8) in Section 1.3. This establishes the
following.
Theorem 2. Consider a data model (1), with i.i.d. sub-Gaussian errors as in (3), and θ0 ∈ BVG (t), where
G is a generic connected graph. Then for any DFS ordering of G yielding a permutation matrix P , the DFS
fused lasso estimator θ̂DFS in (18), with a choice of tuning parameter λ t−1/3 n1/3 , has MSE converging
in probability at the rate
kθ̂DFS − θ0 k2n = OP (t2/3 n−2/3 ).
(19)
(1)
(2)
(K)
We note that, if multiple DFS fused lasso estimators θ̂DFS , θ̂DFS , . . . , θ̂DFS are computed across multiple
different DFS-induced chains on G, then the average estimator clearly satisfies the same bound as in (19),
K
1 X (k)
θ̂DFS − θ0
K
k=1
2
= OP (t2/3 n−2/3 ),
n
provided that K is held constant, by the triangle inequality.
3.2
The graph fused lasso
Interestingly, the chain embedding result (15) in Lemma 1 is not only helpful for establishing the MSE rate
for the DFS fused lasso estimator in Theorem 2, but it can also be used to improve the best known rate for
the original fused lasso estimator over the graph G. In Section 1.3, we described a result (12) that follows
from Wang et al. [2016], establishing an MSE rate of tn−1/2 rate (ignoring log terms) for the fused lasso
estimator over a connected graph G, when k∇G θ0 k1 ≤ t. In fact, as we will now show, this can be improved
to a rate of t2/3 n−2/3 , just as in (19) for the DFS fused lasso.
Wang et al. [2016] present a framework for deriving fast MSE rates for fused lasso estimators based on
entropy. They show in their Lemma 9 that a bound in probability on the sub-Gaussian complexity
> x
max
x∈SG (1)
1−w/2
,
(20)
kxk2
for some 0 < w < 2, where SG (1) = {x ∈ row(∇G ) : k∇G xk1 ≤ 1}, leads to a bound in probability on
the MSE of the fused lasso estimator θ̂G over G. (Wang et al. [2016] actually assume Gaussian errors, but
their Lemma 9, Theorem 10, Lemma 11, and Corollary 12 still hold for sub-Gaussian errors as in (3)). The
sub-Gaussian complexity in (20) is typically controlled via an entropy bound on the class SG (1). Typically,
one thinks of controlling entropy by focusing on specific classes of graph structures G. Perhaps surprisingly,
Lemma 1 shows we can uniformly control the sub-Gaussian complexity (20) over all connected graphs.
For any DFS-induced chain C constructed from G, note first that row(∇G ) = span{1}⊥ = row(∇C ),
where 1 = (1, . . . , 1) ∈ Rn is the vector of all 1s. This, and (15) in Lemma 1, imply that
> x
max
x∈SG (1)
1−w/2
> x
≤ max
x∈SC (2)
kxk2
1−w/2
.
kxk2
Now, taking w = 1,
max
x∈SC (2)
> x
1/2
kxk2
=
max
x : 1> x=0,
k∇1d P xk1 ≤2
> x
1/2
=
kxk2
10
max
x : 1> x=0,
k∇1d xk1 ≤1
2−1/2 (P )> x
1/2
kxk2
= OP (n1/4 ).
The last step (asserting that the penultimate term is OP (n1/4 )) holds by first noting that P is equal in law to
(as we have assumed i.i.d. components of the error vector), and then applying results on the chain graph in
Theorem 10, Lemma 11, and Corollary 12 of Wang et al. [2016]. Applying Lemma 9 of Wang et al. [2016],
we have now established the following result.
Theorem 3. Consider a data model (1), with i.i.d. sub-Gaussian errors as in (3), and θ0 ∈ BVG (t), where
G is a generic connected graph. Then the fused lasso estimator θ̂G over G, in (2), under a choice of tuning
parameter λ t−1/3 n1/3 , has MSE converging in probability at the rate
kθ̂G − θ0 k2n = OP (t2/3 n−2/3 ).
(21)
In a sense, the above theorem suggests that the chain graph is among the hardest graphs for denoising
bounded variation signals, since the fused lasso estimator on any connected graph G will achieve an MSE
rate in that is at least as good as in the chain rate, if not better. In this vein, it is worth emphasizing that the
MSE bound in (21) is not tight for certain graph structures; a good example is the 2d grid, where we must
compare (21) from the theorem to the known MSE bound in (11) from Hutter and Rigollet [2016], the latter
being only log factors from optimal, as shown in Sadhanala et al. [2016]. It is natural for the 2d grid graph
√
to consider the scaling t n (as argued in Sadhanala et al. [2016]), in which case the rates for the fused
lasso estimator are n−1/3 from Theorem 3 versus (log2 n)n−1/2 from Hutter and Rigollet [2016].
3.3
Minimax lower bound over trees
We derive a lower bound for the MSE over the class BVG (t) when G is a tree graph. The proof applies Assouad’s Lemma [Yu, 1997], over a discrete set of probability measures constructed by a careful partitioning
of the vertices of G, that balances both the sizes of each partition element and the number of edges crossing
in between partition elements. It is deferred until Appendx A.2.
Theorem 4. Consider a data model (1), with i.i.d. Gaussian errors i ∼ N (0, σ 2 ), i = 1, . . . , n, and with
θ0 ∈ BVG (t), where G is a tree graph, having maximum degree dmax . Then there exists absolute constants
N, C > 0, such that for n/(tdmax ) > N ,
inf
sup
θ̂
θ0 ∈BVG (t)
Ekθ̂ −
θ0 k2n
≥C
t
σd2max n
2/3
.
(22)
The theorem demonstrates that, for trees of bounded degree, such as the chain and balanced d-ary trees,
the fused lasso estimator over the tree achieves achieves the minimax rate, as does the DFS fused lasso.
4
Analysis for signals with bounded differences
We assume that the underlying mean θ0 in (1) satisfies θ0 ∈ BDG (s) for a generic connected graph G. We
analyze the MSE of the DFS fused lasso, as well as (a particular formulation of) wavelet denoising over G.
We again establish a lower bound on the minimax MSE when G is a tree.
4.1
The DFS fused lasso
As it was for the bounded variation case, the analysis for the DFS fused lasso estimator is straightforward.
By assumption, k∇G θ0 k0 ≤ s, thus k∇1d P θ0 k0 ≤ 2s by (16) in Lemma 1, and we may think of our model
(1) as having i.i.d. data P y around P θ0 ∈ BD1d (2s). Applying an existing result on the 1d fused lasso for
bounded differences signals, as described in (9), from Lin et al. [2016], gives the following result.
11
Theorem 5. Consider a data model (1), with i.i.d. sub-Gaussian errors as in (3), and θ0 ∈ BDG (s), for a
connected graph G. Consider an arbitrary DFS ordering of G, that defines a permutation matrix P and the
DFS fused lasso estimator θ̂DFS in (18). Denote by Wn = min{|i − j| : (∇1d P θ0 )i 6= 0, (∇1d P θ0 )j 6= 0}
the minimum distance between positions, measured along the DFS-induced chain, at which nonzero differences or jumps occur in θ0 . Then, under a choice of tuning parameter λ (nWn )1/4 , the DFS fused lasso
estimator has MSE converging in probability at the rate
p
s
2
kθ̂DFS − θ0 kn = OP
(23)
(log s + log log n) log n + n/Wn .
n
√
Hence, if the s jumps along the DFS chain are evenly spaced apart, i.e., Wn n/s, then for λ ns−1/4 ,
kθ̂DFSr −
θ0 k2n
= OP
s(log s + log log n) log n s3/2
.
+
n
n
(24)
An undesirable feature of applying existing 1d fused lasso results for signals with bounded differences,
in the above result, is the dependence on Wn in the DFS fused lasso error bound (23) (we applied the result
(9) from Lin et al. [2016], but the bounds from Dalalyan et al. [2014] also depend on Wn , and as far as we
can tell, so should any analysis of the 1d fused lasso for signals with bounded differences). In the 1d setting,
assuming that Wn n/s, which says that jumps in θ0 occur at roughly equally spaced positions, is fairly
reasonable; but to assume the same when the jumps are measured with respect to the DFS-induced chain, as
we must in order to establish (24), is perhaps not. Even if the differences apparent in θ0 over edges in G are
somehow (loosely speaking) spaced far apart, running DFS could well produce an ordering such that jumps
in P θ0 occur at positions very close together. We reiterate that the MSE bounds for the DFS fused lasso for
bounded variation signals, in Theorem 2, do not suffer from any such complications.
4.2
Graph wavelet denoising
We compare the performances of the DFS fused lasso and wavelet denoising using spanning tree wavelets,
for signals with bounded differences. For spanning tree wavelets, the construction starts with a spanning tree
and carefully defines a hierarchical decomposition by recursively finding and splitting around a balancing
vertex, which is a vertex whose adjacent subtrees are of size at most half of the original tree; this decomposition is used to construct an unbalanced Haar wavelet basis, as in Singh et al. [2010]. In Sharpnack et al.
[2013], it was shown that for any connected graph G, the constructed wavelet basis W ∈ Rn×n satisfies
kW θk0 ≤ dlog dmax edlog nek∇G θk0 ,
for all θ ∈ Rn ,
(25)
where dmax is the maximum degree of G, and the above holds regardless of choice of spanning tree in the
wavelet construction. Now consider the wavelet denoising estimator
θ̂W = argmin
θ∈Rn
1
ky − θk22 + λkW θk1 .
2
(26)
The following is an immediate consequence of (25), the fact that the wavelet basis W is orthonormal, and
standard results about soft-thresholding (e.g., Lemma 2.8 in [Johnstone, 2011]).
Theorem 6. Consider a data model (1), with i.i.d. Gaussian errors i ∼ N (0, σ 2 ), i = 1, . . . , n, and with
θ0 ∈ BVG (t), where G is a connected graph,
√ having maximum degree dmax . Then the spanning tree wavelet
estimator θ̂W in (26), with a choice λ log n, has MSE converging in expectation at the rate
s log dmax log2 n
2
Ekθ̂W − θ0 kn = O
.
(27)
n
12
The result in (27) has the advantage over the DFS fused lasso result in (23) that it does not depend on a
hard-to-interpret quantity like Wn , the minimum spacing between jumps along the DFS-induced chain. But
when (say) dmax 1, s 1, and we are willing to assume that Wn n (meaning the jumps of θ0 occur
at positions evenly spaced apart on the DFS chain), we can see that the spanning tree wavelet rate in (27) is
just slightly slower than the DFS fused lasso rate in (24), by a factor of log n/ log log n.
While the comparison between the DFS fused lasso and wavelet rates, (23) and (27), show an advantage
to spanning tree wavelet denoising, as it does not require assumptions about the spacings between nonzero
differences in θ0 , we have found nonetheless that the DFS fused lasso to performs well in practice compared
to spanning tree wavelets, and indeed often outperforms the latter in terms of MSE. Experiments comparing
the two methods are presented Section 5.
4.3
Minimax lower bound for trees
We now derive a lower bound for the MSE over the class BDG (s) when G is a tree graph. The proof relates
the current denoising problem to one of estimating sparse normal means, with a careful construction of the
sparsity set using degree properties of trees. It is deferred until Appendix A.3.
Theorem 7. Consider a data model (1), with i.i.d. Gaussian errors i ∼ N (0, σ 2 ), i = 1, . . . , n, and with
θ0 ∈ BDG (s), where G is a tree. Then there are absolute constants N, C > 0, such that for n/s > N ,
n
s
inf
sup
Ekθ̂ − θ0 k2n ≥ Cσ 2 log
.
(28)
n
s
θ̂ θ0 ∈BDG (s)
The MSE lower bound in (28) shows that, when we are willing to assume that Wn n/s in the DFSinduced chain, the DFS fused lasso estimator is a log log n factor away from the optimal rate, provided that
s is not too large, namely s = O((log n log log n)2 ). The spanning tree wavelet estimator, on the other hand,
is a log n factor from optimal, without any real restrictions on s, i.e., it suffices to have s = O(na ) for some
a > 0. It is worth remarking that, for large enough s, the lower bound in (28) is perhaps not very interesting,
as in such a case, we may as well consider the bounded variation lower bound in (22), which will likely be
tighter (faster).
5
Experiments
In this section we compare experimentally the speed and accuracy of two approaches for denoising signals
on graphs: the graph fused lasso, and the fused lasso along the chain graph induced by a DFS ordering. In
our experiments, we see that the DFS-based denoiser sacrifices a modest amount in terms of mean squared
error, while providing gains (sometimes considerable) in computational speed. This shows that our main
theorem, in addition to providing new insights on MSE rates for the graph fused lasso, also has important
practice consequences. For truly massive problems, where the full graph denoising problem is impractical
to solve, we may use the linear-time DFS fused lasso denoiser, and obtain a favorable tradeoff of accuracy
for speed.
5.1
Generic graphs
We begin by considering three examples of large graphs of (more or less) generic structure, derived from
road networks in three states: California, Pennsylvania, and Texas. Data on these road networks are freely
available at https://snap.stanford.edu. In these networks, intersections and endpoints are represented by nodes, and roads connecting these intersections or endpoints are represented by undirected edges;
see Leskovec et al. [2009] for more details. For each network, we use the biggest connected component as
13
our graph structure to run comparisons. The graph corresponding to California has n = 1957027 nodes and
m = 2760388 edges, the one for Pennsylvania has n = 1088092 nodes and m = 1541898 edges, and the
graph for Texas has n = 1351137 nodes and m = 1879201 edges. We compare Laplacian smoothing versus
the fused lasso over a DFS-induced chain, on the graphs from the three states. We do not compare with the
fused lasso over the original graphs, due to its prohibitive computational cost at such large scales.
We used the following procedure to construct a synthetic signal θ0 ∈ Rn on each of the road network
graphs, of piecewise constant nature:
• an initial seed node v1 is selected uniformly at random from the nodes V = {1, . . . , n} in the graph;
• a component C1 is formed based on the bn/10c nodes closest to v1 (where the distance between two
nodes in the graph is given by the length of the shortest path between them);
• a second seed node v2 is selected uniformly at random from G \ C1 ;
• a component C2 is formed based on the bn/10c nodes closest to v2 (again in shortest path distance);
• this process is repeated until we have a partition C1 , . . . , C10 of the node set V into components of
(roughly) equal size, and θ0 ∈ Rn is defined to take constant values on each of these components.
In our experiments, we considered 20 values of the total variation for the underlying signal. For each, the
signal θ0 was scaled appropriately to achieve the given total variation value, and data y ∈ Rn was generated
by adding i.i.d. N (0, 0.22 ) noise to the components of θ0 . For each data instance y, the DFS fused lasso and
Laplacian smoothing estimators, the former defined by (18) and the latter by
θ̂Lap = argmin
θ∈Rn
1
ky − θk22 + λθ> LG θ,
2
(29)
where LG = ∇>
G ∇G is the Laplacian matrix of the given graph G, and each estimator is computed over 20
values of its own tuning parameter. Then, the value of the tuning parameter minimizing the average MSE,
over 50 draws of data y around θ0 , was selected for each method. Finally, this optimized MSE, averaged
over the 50 draws of data y, and further, over 10 repetitions of the procedure for constructing the signal θ0
explained above, was recorded. Figure 1 displays the optimized MSE for the DFS fused lasso and Laplacian
smoothing, as the total variation of the underlying signal varies, for the three road network graphs.
As we can see from the figure, for low values of the underlying total variation, i.e., low signal-to-noise
ratio (SNR) levels, Laplacian smoothing and the DFS fused lasso, each tuned to optimality, perform about
the same. This is because at low enough SNR levels, each will be approximating θ0 by something like ȳ1,
with ȳ being the sample average of the data vector y. But as the SNR increases, we see that the DFS fused
lasso outpeforms Laplacian smoothing by a considerable amount. This might seem surprising, as Laplacian
smoothing uses information from the entire graph, whereas the DFS fused lasso reduces the rich structure
of the road network graph in each case to that of an embedded chain. However, Laplacian smoothing is a
linear smoother (meaning that θ̂Lap in (29) is a linear function of the data y), and therefore it comes with
certain limitations when estimating signals of bounded variation (e.g., see the seminal work of Donoho and
Johnstone [1998], and the more recent graph-based work of Sadhanala et al. [2016]). In contrast, the DFS
fused lasso is a nonlinear estimator, and while it discards some information in the original graph structure,
it retains enough of the strong adaptivity properties of the fused lasso over the original graph to statistically
dominate a linear estimator like Laplacian smoothing.
Lastly, in terms of computational time, it took an average of 82.67 seconds, 44.02 seconds, and 54.49
seconds to compute the 20 DFS fused lasso solutions (i.e., over the 20 tuning parameter values) for the road
network graphs from California, Pennsylvania, and Texas, respectively (the averages are taken over the 50
draws of data y around each signal θ0 , and the 10 repetitions in constructing θ0 ). By comparison, it took an
14
Pennsylvania
●
DFS fused lasso
Laplacian smoothing
Texas
●
DFS fused lasso
Laplacian smoothing
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●
0.015
●
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0.015
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0.002
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10000
15000
Total variation
20000
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Figure 1: The optimized MSE for the DFS fused lasso and Laplacian smoothing (i.e., MSE achieved by these methods
under optimal tuning) is plotted as a function of the total variation of the underlying signal, for each of the three road
network graphs. This has been averaged over 50 draws of data y for each construction of the underlying signal θ0 ,
and 10 repetitions in constructing θ0 itself. For low values of the underlying total variation, i.e., low SNR levels, the
two methods perform about the same, but as the SNR increases, the DFS fused lasso outperforms Laplacian smoothing
by a considerable margin.
average of 2748.26 seconds, 1891.97 seconds, and 1487.36 seconds to compute the 20 Laplacian smoothing
solutions for the same graphs. The computations and timings were performed on a standard laptop computer
(with a 2.80GHz Intel Core i7-2640M processor). For the DFS fused lasso, in each problem instance, we first
computed a DFS ordering using the dfs function from the R package igraph, which is an R wrapper for a
C++ implementation of DFS, and initialized the algorithm at a random node for the root. We then computed
the appropriate 1d fused lasso solutions using the trendfilter function from the R package glmgen,
which is an R wrapper for a C++ implementation of the fast (linear-time) dynamic programming algorithm
in Johnson [2013]. For Laplacian smoothing, we used the solve function from the R package Matrix,
which is an R wrapper for a C++ implementation of the sparse Cholesky-based solver in Davis and Hager
[2009]. For such large graphs, alternative algorithms, such as (preconditioned) conjugate gradient methods,
could certainly be more efficient in computing Laplacian smoothing solutions; our reported timings are only
meant to indicate that the DFS fused lasso is efficiently computable at problem sizes that are large enough
that even a simple linear method like Laplacian smoothing becomes nontrivial.
5.2
2d grid graphs
Next we consider a denoising example on a 2d grid graph of dimension 1000 × 1000, so that the number
of nodes is n = 1000000 and the number of edges is m = 1998000. We generated a synthetic piecewise
constant signal θ0 ∈ R1000×1000 over the 2d grid, shown in the top left corner of Figure 2, where a color
scale (displayed in the accompanying color legend) is used, with red denoting the smallest possible value
and yellow the largest possible value. Data y ∈ R1000×1000 was generated by adding i.i.d. N (0, 1) noise to
the components of θ0 , displayed in the top middle panel of Figure 2. We then computed the 2d fused lasso
solution (i.e., the fused lasso solution over the full 2d grid graph), as well as three DFS-based variations:
the DFS fused lasso solution using a random DFS ordering (given by running DFS beginning at a random
node), labeled as “1 random DFS” in the figure; the average of DFS fused lasso solutions over 5 random
15
DFS orderings, labeled “5 random DFS” in the figure; and the average of DFS fused lasso solutions over 2
“snake” DFS orderings (one given by collecting and joining all horizontal edges and the other all vertical
edges) labeled “2 snake DFS” in the figure. The tuning parameter for each method displayed in the figure
was chosen to minimize the average MSE over 100 draws of the data y from the specified model. Visually,
we can see that the full 2d fused lasso solution is the most accurate, however, the 1 random DFS, 5 random
DFS, and 2 snake DFS solutions all still clearly capture the structure inherent in the underlying signal. Of
the three DFS variations, the 5 random DFS estimator is visually most accurate; the 1 random DFS estimator
is comparably “blotchy”, and the 2 snake DFS estimator is comparably “stripey”.
The left panel of 3 shows the optimized MSE for each method, i.e., the minimum of the average MSE
over 100 draws of the data y, when we consider 20 choices for the tuning parameter. This optimized MSE
is plotted as a function of the sample size, which runs from n = 2500 (a 50 × 50 grid) to n = 1000000 (a
1000 × 1000 grid), and in each case the underlying signal is formed by taking an appropriate (sub)resolution
of the image in the top left panel of Figure 2. The 2d fused lasso provides the fastest decrease in MSE as
n grows, followed by the 5 random DFS estimator, then the 1 random DFS estimator, and the 2 snake DFS
estimator. This is not a surprise, since the 2d fused lasso uses the information from the full 2d grid. Indeed,
comparing (11) and (19), we recall that the 2d fused lasso enjoys an MSE rate of t log2 n/n when θ0 has
2d total variation t, whereas the DFS fused lasso has an MSE rate of only (t/n)2/3 in this setting. When
√
t n, which is a natural scaling for the underlying total variation in 2d and also the scaling considered in
the experimental setup for the figure, these rates are (log2 n)n−1/2 for the 2d fused lasso, and n−1/3 for the
DFS fused lasso. The figure uses a log-log plot, so the MSE curves all appear to have linear trends, and the
fitted slopes roughly match these theoretical MSE rates (-0.58 for the 2d fused lasso, and -0.39, -0.40, and
-0.36 for the three DFS variations).
The right panel of Figure 3 shows the runtimes for each method (averaged over 100 draws of the data
y), as a function of the sample size n. The runtime for each method counts the total time taken to compute
solutions across 20 tuning parameter values. The computations and timings were carried out on a standard
desktop computer (with a 3.40GHz Intel Core i7-4770 processor). To compute 2d fused lasso solutions, we
used the TVgen function in the Matlab package proxTV, which is a Matlab wrapper for a C++ implementation of the proximal stacking technique described in Barbero and Sra [2014]. For the DFS fused lasso, we
computed initial DFS orderings using the dfs function from the Matlab package MathBGL, and then, as
before, used the C++ implementation available through glmgen to compute the appropriate 1d fused lasso
solutions. The figure uses a log-log plot, and hence we can see that all DFS-based estimators are quite a bit
more efficient than the 2d fused lasso estimator.
5.3
Tree graphs
We finish with denoising comparisons on tree graphs, for sample sizes varying from n = 100 to n = 5300.
For each sample size n, a random tree is constructed via a sequential process in which each node is assigned
a number of children between 2 and 10 (uniformly at random). Given a tree, an underlying signal θ0 ∈ Rn
√
is constructed to be piecewise constant with total variation 5 n (the piecewise constant construction here
is made easy because the oriented incidence matrix of a tree is invertible). Data y ∈ Rn was generated by
adding i.i.d. N (0, 1) noise to θ0 . We compared the fused lasso estimator over the full tree, 1 random DFS
and 5 random DFS estimators (using the terminology from the last subsection), and the wavelet smoothing
estimator defined in (26). For each estimator, we computed the entire solution path using the path algorithm
of Tibshirani and Taylor [2011] implemented in the R package genlasso, and selected the step along the
path to minimize the average MSE over 50 draws of data y around θ0 , and 10 repetitions in constructing
θ0 . (The full solution path can be computed here because each estimator can be cast as a generalized lasso
problem, and because the problem sizes considered here are not enormous.)
The left panel of Figure 4 plots this optimized MSE as a function of the sample size n. We see that the
16
Figure 2: Underlying signal, data, and solutions from the 2d fused lasso and different variations on the DFS fused
50.00
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Figure 3: Optimized MSE and runtime for the 2d fused lasso and DFS fused lasso estimators over a 2d grid, as the
grid size n (total number of nodes) varies.
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fused lasso estimator over the full tree and the 5 random DFS estimator perform more or less equivalently
over all sample sizes. The 1 random DFS estimator is slightly worse, and the wavelet smoothing estimator
is considerably worse. The right panel shows the the MSE as a function of the effective degrees of freedom
of each estimator, for a particular data instance with n = 5300. We see that both the tree fused lasso and 1
random DFS estimators achieve their optimum MSEs at solutions of low complexity (degrees of freedom),
whereas wavelet smoothing does not come close to achieving this MSE across its entire path of solutions.
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Figure 4: The left panel shows the optimized MSE as a function of the sample size for the fused lasso over a tree
graph, as well as the 1 random DFS and 5 random DFS estimators, and wavelet smoothing. The right panel
6
Discussion
Recently, there has been a significant amount on interest on graph-structured denoising. Much of this work
has focused on the construction of graph kernels or wavelet bases. We have proposed and studied a simple
method, defined by computing the 1d fused lasso over a particular DFS-induced ordering of the nodes of a
general graph. This linear-time algorithm comes with strong theoretical guarantees for signals of bounded
variation (achieving optimal MSE rates for trees of bounded degree), as well as guarantees for signals with
a bounded number of nonzero differences (achieving nearly optimal rates under a condition on the spacings
of jumps along the DFS-induced chain). We summarize our theoretical results in Table 1.
Practically, we have seen that the DFS fused lasso can often represent a useful tradeoff between computational efficiency and statistical accuracy, versus competing methods that offer better statistical denoising
power but are more computationally expensive, especially for large problems. A simple trick like averaging
multiple DFS fused lasso fits, over multiple random DFS-induced chains, often improves statistical accuracy
at little increased computational cost. Several extensions along these lines, and other lines, are possible. To
study any of them in detail is beyond the scope of this paper. We discuss them briefly below, leaving detailed
follow-up to future work.
18
BVG (t), t 1
BDG (s), s 1
Fused lasso, θ̂G
n−2/3
unknown
Spanning tree wavelets, θ̂W
unknown
(log2 n log dmax )/n
DFS fused lasso, θ̂DFS
n−2/3
(log n log log n)/n∗
Tree lower bound
n−2/3 dmax
−4/3
log n/n
Table 1: A summary of the theoretical results derived in this paper. All rates are on the mean squared error (MSE)
scale (Ekθ̂ − θ0 k2n for an estimator θ̂), and for simplicity, are presented under a constant scaling for t, s, the radii in
the BVG (t), BDG (s) classes, respectively. The superscript “∗ ” in the BDG (s) rate for the DFS fused lasso is used to
emphasize that this rate only holds under the assumption that Wn n. Also, we write dmax to denote the max degree
of the graph in question.
6.1
Beyond simple averaging
(1)
(K)
Given multiple DFS fused lasso estimators, θ̂DFS , . . . , θ̂DFS , obtained using multiple DFS-induced chains
computed on the same graph G, there are several possibilities
intelligently combining these estimators
P for(k)
(K)
θ̂
beyond the simple average, denoted (say) θ̄DFS = (1/K) K
k=1 DFS . To better preserve edges in the com(1)
(K)
bined estimator, we could run a simple nonlinear filter—for example, a median filter, over θ̂DFS , . . . , θ̂DFS
(meaning that the combined estimator is defined by taking medians over local neighborhoods of all of the
individual estimators). A more sophisticated approach would be to compute the DFS fused lasso estimators
sequentially, using the (k − 1)st estimator to modify the response in some way in the 1d fused lasso problem
that defines the kth DFS fused lasso estimator. We are intentionally vague here with the specifics, because
such a modification could be implemented in various ways; for example, it could be useful to borrow ideas
from the boosting literature, which would have us treat each DFS fused lasso estimator as a weak learner.
6.2
Distributed algorithm
For large graphs, we should be able to both compute a DFS ordering over G, and solve the DFS fused lasso
problem in (18), in a distributed fashion. There are many algorithms for distributed DFS, offering a variety
of communication and time complexities; see, e.g., Tsin [2002] for a survey. Distributed algorithms for the
1d fused lasso are not as common, though we can appeal to the now well-studied framework for distributed
optimization via the alternating direction method of multipliers (ADMM) from Boyd et al. [2011]. Different
formulations for the auxiliary variables present us with different options for communication costs. We have
found that, for a formulation that requires O(1)-length messages to be communicated between processors,
the algorithm typically converges in a reasonably small number of iterations.
6.3
Theory for piecewise constant signals
The bounded differences class BDG (s) in (5) is defined in terms of the cut metric k∇G θk0 of a parameter
θ, which recall, counts the number of nonzero differences occurring in θ over edges in the graph G. The cut
metric measures a notion of strong sparsity (compared to the weaker notion measured by the total variation
metric) in a signal θ, over edge differences; but, it may not be measuring sparsity on the “right” scale for
certain graphs G. Specifically, the cut metric k∇G θk0 can actually be quite large for a parameter θ that is
piecewise constant over G, with a small number of pieces—these are groups of connected nodes that are
assigned the same constant value in θ. Over the 2d grid graph, e.g., one can easily define a parameter θ that
19
√
has only (say) two constant pieces but on the order of n nonzero edge differences. Therefore, for such a
“simple” configuration of the parameter θ, the cut metric k∇G θk0 is deceivingly large.
To formally define a metric that measures the number of constant pieces in a parameter θ, with respect
to a graph G = (V, E), we introduce a bit of notation. Denote by Z(θ) ⊆ E the subset of edges over which
θ exhibits differences of zero, i.e., Z(θ) = {e ∈ E : θe+ = θe− }. Also write (∇G )Z(θ) for the submatrix of
the edge incidence matrix ∇G with rows indexed by Z(θ). We define the piece metric by
ρG (θ) = nullity (∇G )Z(θ) ,
where nullity(·) denotes the dimension of the null space of its argument. An equivalent definition is
ρG (θ) = the number of connected components in (V, E \ Z(θ)).
We may now define the piecewise constant class, with respect to G, and a parameter s > 0,
PCG (s) = {θ ∈ Rn : ρG (θ) ≤ s}.
It is not hard to to see that BVG (s) ⊆ PCG (s) (assuming only that G is connected), but for certain graph
topologies, the latter class PCG (s) will be much larger. Indeed, to repeat what we have conveyed above, for
√
the 2d grid one can naturally define a parameter θ such that θ ∈ BDG ( n) and θ ∈ PCG (2).
We conjecture that the fused lasso estimator over G can achieve a fast MSE rate when the mean θ0 in
(1) exhibits a small number of constant pieces, i.e., θ0 ∈ PCG (s), provided that these pieces are of roughly
equal size. Specifically, assuming ρG (θ0 ) ≤ s, let Wn denote the smallest size of a connected component in
the graph (V, E \ Z(θ0 )). Then, for a suitable choice of λ, we conjecture that the fused lasso estimator θ̂G
in (2) satisfies
s
2
kθ̂G − θ0 kn = OP
polylog n + n/Wn ,
(conjecture)
n
where polylog n is a shorthand for a polynomial of log n. This would substantially improve upon existing
strong sparsity denoising results, such as (11), (23), (27), since the latter results are all proven for the class
BDG (s), which, as we have argued, can be much smaller than PCG (s), depending on the structure of G.
6.4
Weighted graphs
The key result in Lemma 1 can be extended to the setting of a weighted graph G = (V, E, w), with we ≥ 0
denoting the edge weight associated an edge e ∈ E. We state the following without proof, since its proof
follows in nearly the exact same way as that of Lemma 1.
Lemma 8. Let G = (V, E, w) be a connected weighted graph, where recall we write V = {1, . . . , n}, and
we assume all edge weights are nonnegative. Consider running DFS on G, and denote by τ : {1, . . . , n} →
{1, . . . , n} the induced permutation, so that if v1 , . . . , vn are the nodes in the order that they are traversed
by DFS, then
τ (i) = vi , for all i = 1, . . . , n.
Denote wmin = mine∈E we , the minimum edge weight present in the graph, and define
(
we
if e = {i, j} ∈ E,
w̃ij =
for all i, j = 1, . . . , n.
wmin otherwise,
(30)
It holds that
n−1
X
i=1
w̃τ (i),τ (i+1) θτ (i+1) − θτ (i) ≤ 2
X
e∈E
20
we |θe+ − θe− |,
for all θ ∈ Rn ,
(31)
as well as
n−1
X
X
w̃τ (i),τ (i+1) 1 θτ (i+1) 6= θτ (i) ≤ 2
we 1{θe+ 6= θe− },
i=1
for all θ ∈ Rn .
(32)
e∈E
The bounds in (31), (32) are the analogies of (15), (16) but for a weighted graph G; indeed we see that
we can still embed a DFS chain into G, but this chain itself comes with edge weights, as in (30). These new
edge weights in the chain do not cause any computational issues; the 1d fused lasso problem with arbitrary
penalty weights can still be solved in O(n) time using the taut string algorithm in Barbero and Sra [2014].
Thus, in principle, all of the results in this paper should carry over in some form to weighted graphs.
6.5
Potts and energy minimization
Replacing the total variation metric by the cut metric in the fused lasso problem (2) gives us
θ̃G = argmin
θ∈Rn
1
ky − θk22 + λk∇G θk0 ,
2
(33)
often called the Potts minimization problem. Because the 1d Potts minimization problem
θ̃1d = argmin
θ∈Rn
1
ky − θk22 + λk∇1d θk0
2
(34)
can be solved efficiently, e.g., in worst-case O(n2 ) time with dynamic programming Bellman [1961], Johnson [2013], the same strategy that we have proposed in this paper can be applied to reduce the graph Potts
problem (33) to a 1d Potts problem (34), via a DFS ordering of the nodes. This may be especially interesting
as the original Potts problem (33) is non–convex and generally intractable (i.e., intractable to solve to global
optimality) for an arbitrary graph structure, so a reduction to a worst-case quadratic-time denoiser is perhaps
very valuable.
When the optimization domain in (33) is a discrete set, the problem is often called an energy minimization problem, as in Boykov et al. [2001]. It has not escaped our notice that our technique of denoising over
DFS-induced chains could be useful for this setting, as well.
A
A.1
Proofs
Derivation of (12) from Theorem 3 in Wang et al. [2016]
We first establish a result on the exact form for the inverse of (an augmented version of) the edge incidence
matrix of a generic tree T = (V, ET ), where, recall V = {1, . . . , n}. Without a loss of generality, we may
assume that the root of T is at node 1. For m ≤ n, we define a path in T , of length m, to be a sequence
p1 , . . . , pm such that {pr , pr+1 } ∈ ET for each r = 1, . . . , m − 1. We allow for the possibility that m = 1,
in which case the path has just one node. For any j, k, ` = 1, . . . , n, we say that j is on the path from k to
` if there exists a path p1 , . . . , pm such that p1 = k, pm = ` and pr = j for some r = 1, . . . , m. For each
node i = 2, . . . , n (each node other than the root), we define its parent p(i) to be the node connected to i
which is on the path from the root to i.
We can also assume without a loss of generality that for each i = 2, . . . , n, the (i − 1)st row of ∇T
corresponds to the edge {p(i), i}, and thus we can write
−1 if j = p(i),
(∇T )i−1,j = 1
if j = i,
0
if j ∈ {1, . . . , n} \ {i, p(i)}.
21
for each j = 1, . . . , n. The next lemma describes the inverse of ∇T , in the appropriate sense.
Lemma 9. Let e1 = (1, 0, . . . , 0) ∈ Rn , and define the matrix AT ∈ Rn×n by
(
1 if j is on the path from the root to i,
(AT )i,j =
0 otherwise,
(35)
for each i, j = 1, . . . , n. Then
AT =
e>
1
∇T
−1
e>
1
∇T
.
Proof. We will prove that the product
B=
AT
is the identity. As the root of T corresponds to node 1, we have that by definition of AT that its first column
is
(AT )·,1 = (1, . . . , 1),
which implies that the first column of B is
B·,1 = e1 .
Moreover, by definition of AT , its first row is
(AT )1,· = e>
1,
which implies that the first row of B is
B1,· = e>
1.
Let us now assume that i, j are each not the root. We proceed to consider three cases.
Case 1. Let j 6= i, and j be on the path from the root to i. Then j is also on the path from the root to p(i).
This implies that
>
e1
Bij =
(A ) = (∇T )i−1,· (AT )·,j = 1 − 1 = 0.
∇T i,· T ·,j
Case 2. Let j 6= i, and j not be on the path from the root to i. Then j is not on the path from the root to
p(i), which implies that
>
e1
Bij =
(A ) = (∇T )i−1,· (AT )·,j = 0 − 0 = 0.
∇T i,· T ·,j
Case 3. Let j = i. Then j is on the path from the root to i, and j is not on the path from the root to p(i).
Hence,
>
e1
Bij =
(A ) = (∇T )i−1,· (AT )·,j = −1 · 0 + 1 · 1 = 1.
∇T i,· T ·,j
Assembling these three cases, we have shown that B = I, completing the proof.
We now establish (12).
22
Proof of (12). The proof of Theorem 3 in Wang et al. [2016] proceeds as in standard basic inequality arguments for the lasso, and arrives at the step
kΠ⊥ (θ̂G − θ0 )k22 ≤ 2> Π⊥ (θ̂G − θ0 ) + 2λk∇G θ0 k1 − 2λk∇G θ̂G k1 ,
where Π⊥ is the projection matrix onto the space span{1}⊥ , i.e., the linear space of all vectors orthogonal
to the vector 1 = (1, . . . , 1) ∈ Rn of all 1s. The proof in Wang et al. [2016] uses the identity Π⊥ = ∇†G ∇G ,
where ∇†G denotes the pseudoinverse of ∇G . However, notice that we may also write Π⊥ = ∇†T ∇T for any
spanning tree T of G. Then, exactly the same arguments as in Wang et al. [2016] produce the MSE bound
√
M (∇T ) log n
2
kθ̂G − θ0 kn = OP
k∇G θ0 k1 ,
n
where M (∇T ) is the maximum `2 norm among the columns of ∇†T . We show below, using Lemma 9, that
√
M (∇T ) ≤ n, and this gives the desired MSE rate.
For any b ∈ Rn−1 , we may characterize ∇†T b as the unique solution x ∈ Rn to the linear system
∇T x = b,
such that 1> x = 0, i.e., the unique solution to the linear system
>
a
e1
,
x=
b
∇T
for a value of a ∈ R such that 1> x = 0. By Lemma 9, we may write
a
,
x = AT
b
so that the constraint 0 = 1> x = na + 1> (AT )·,2:n b gives a = −(1/n)> (AT )·,2:n b, and
x = (I − 11> /n)(AT )·,2:n b.
Evaluating this across b = e1 , . . . , en , we find that the maximum `2 norm of columns of ∇†T is bounded by
√
the maximum `2 norm of columns of (AT )·,2:n , which, from the definition in (35), is at most n.
A.2
Proof of Theorem 4
We first present two preliminary lemmas.
Lemma 10. Let S1 , . . . , Sm be a partition of the nodes of G such that the total number of edges with ends
in distinct elements of the partition is at most s. Let k ≤ mini=1,...,m |Si |. Then
kt2
kmt2
2
inf
sup
Ekθ̂ − θ0 k2 ≥ 2 2 exp − 2 2 .
4σ s
σ s
θ̂ θ0 ∈BVG (t)
Proof. For each η ∈ {−1, 1}m , define
m
θη =
δX
1S
ηi p i ,
2
|Si |
i=1
m }. Note that
where δ > 0 will√be specified shortly. Also define the class P = {N (θη , σ 2 I) : η ∈ {−1, 1}
√
k∇G θη k1 ≤ δs/ k, so to embed P into the class {N (θ, σ 2 I) : θ ∈ BVG (t)}, we set δ = t k/s.
23
Let η, η 0 ∈ {−1, 1}m differ in only one coordinate. Then the KL divergence between the corresponding
induced measures in P is kθη − θη0 k22 /σ 2 ≤ δ 2 /σ 2 . Hence by Assouad’s Lemma [Yu, 1997], and a wellknown lower bound on the affinity between probability measures in terms of KL divergence,
δ2m
δ2
2
inf
sup
Ekθ̂ − θ0 k2 ≥
exp − 2 .
4σ 2
σ
θ̂ θ0 ∈BVG (t)
The result follows by plugging in the specified value for δ.
Lemma 11. Let G be a tree with maximum degree dmax , and k ∈ {1, . . . , n} be arbitrary. Then there exists
a partition as in Lemma 10, s = m − 1, and
k ≤ min
i=1,...,m
|Si | ≤ k(dmax + 1).
Proof. Our proof proceeds inductively. We begin by constructing S10 , the smallest subtree among all those
having size at least k, and generated by a cut of size 1 (i.e., separated from the graph by the removal of 1
edge). Note that |S10 | ≤ kdmax , because if not then S10 has at least k internal nodes, and we can remove its
root to produce another subtree whose size is smaller but still at least k.
For the inductive step, assume S10 , . . . , S`0 have been constructed. We consider two cases. (For a subgraph G0 of G, we denote by G − G0 the complement subgraph, given by removing all nodes in G0 , and all
edges incident to a node in G0 .)
0 , the smallest subtree of G − ∪l S 0 among all those
Case 1. If |G − ∪`i=1 Si0 | > k, then we construct S`+1
i=1 i
0 | ≤ kd
having size at least k, and generated by a cut of size 1. As before, we obtain that |S`+1
max .
Case 2. If |G − ∪`i=1 Si0 | ≤ k, then the process is stopped. We define Si = Si0 , i = 1, . . . , ` − 1, as well as
S` = S`0 ∪ (G − ∪`i=1 Si0 ). With m = `, the result follows.
We now demonstrate a more precise characterization of the lower bound in Theorem 4, from which the
result in the theorem can be derived.
Theorem 12. Let G be a tree with maximum degree dmax . Then
!2
2/3
2
t
σn
inf
sup
Ekθ̂ − θ0 k22 ≥
−1 .
4eσ 2 n
2t(dmax + 1)
θ̂ θ0 ∈BVG (t)
Proof. Set s = m − 1 and
$
k=
σn
2t(dmax + 1)
2/3 %
.
By Lemmas 10 and 11,
inf
sup
θ̂
θ0 ∈BVG (t)
Ekθ̂ −
θ0 k22
kmt2
≥ 2 2 exp
4σ s
≥
≥
≥
≥
kt2
− 2 2
σ s
kt2
kt2
exp − 2
4σ 2 m
σ (m − 1)2
2
kt
t2 k 3 (dmax + 1)2
m2
exp −
4σ 2 m
σ 2 n2
(m − 1)2
kt2
4t2 k 3 (dmax + 1)2
exp −
4σ 2 n
σ 2 n2
k 2 t2
exp(−1).
4σ 2 n
24
In the above, the third line uses n/m ≤ kdmax as given by Lemma 11, the fourth line simply uses m ≤ n
and m2 /(m − 1)2 ≤ 4 (as m ≥ 2), and the last line uses the definition of k. Thus, because
2/3
σn
− 1,
k≥
2t(dmax + 1)
we have established the desired result.
A.3
Proof of Theorem 7
First we establish that, as G is a tree, the number of nodes of degree at most 2 is at least n/2. Denote by di
be the degree of the node i, for each i = 1, . . . , n. Then
2(n − 1) =
n
X
i=1
di =
X
di +
i : di ≤2
X
di ≥ |{i : di ≤ 2}| + 3|{i : di ≥ 3}| = 3n − 2|{i : di ≤ 2}|.
i : di ≥3
Hence, rearranging, we find that |{i : di ≤ 2}| ≥ n/2 + 1.
Let I = {i : di ≤ 2} so that |I| ≥ dn/2e and stipulate that |I| is even without loss of generality. Let k
be the largest even number such that k ≤ s/2. Define
B = {z ∈ Rn : zI ∈ {−1, 0, +1}|I| , zI c = 0, kzk0 = k}.
Note that by construction B ⊆ BDG (s).
Assume s ≤ n/6. Then this implies k/2 ≤ n/6 ≤ |I|/3. By Lemma 4 in Raskutti et al. [2011], there
exists B̃ ⊆ B such that
|I| − k
k
,
log |B̃| ≥ log
2
k/2
and kz − z 0 k22 ≥ k/2 for all z, z 0 ∈ B̃. Defining B0 = 2δ B̃, for δ > 0 to be specified shortly, we now have
kz − z 0 k22 ≥ 2δ 2 k for all z, z 0 ∈ B0 .
For θ ∈ B0 , let us consider comparing the measure Pθ = N (θ, σ 2 I) against P0 =pN (0, σ 2 I): the KL
divergence between these two satisfies K(Pθ ||P0 ) = kθk22 /σ 2 = 2δ 2 k/σ 2 . Let δ = ασ 2 /(2k) log |B0 |,
for a parameter α < 1/8 that we will specify later. We have
1 X
K(Pθ ||P0 ) ≤ α log |B0 |.
|B0 |
θ∈B0
Hence by Theorem 2.5 in Tsybakov [2009],
inf
sup
θ̂
θ0 ∈BDG (s)
It holds that
δ2k =
s
!
p
|B
|
2α
0
2
2
p
1 − 2α −
.
P(kθ̂ − θ0 k2 ≥ δ k) ≥
log |B0 |
1 + |B0 |
ασ 2
ασ 2 k
|I| − k
log |B0 | ≥
log
≥ Cσ 2 s log
2
4
k/2
(36)
n
,
s
for some constant C > 0 depending on α alone. Moreover, the right-hand side in (36) can be lower bounded
by (say) 1/4 by taking α to be small enough and assuming n/s is large enough. Thus we have established
!
n
1
inf
sup
P kθ̂ − θ0 k22 ≥ Cσ 2 s log
≥ ,
s
4
θ̂ θ0 ∈BDG (s)
and the result follows by Markov’s inequality.
25
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28
| 10 |
Real-time Prediction of Intermediate-Horizon
Automotive Collision Risk
Blake Wulfe
Stanford University
Stanford, CA
wulfebw@stanford.edu
arXiv:1802.01532v1 [cs.CV] 5 Feb 2018
Rory Hartong-Redden
The Allstate Corporation
Northbrook, IL
rory.hartong-redden@allstate.com
Sunil Chintakindi
The Allstate Corporation
Northbrook, IL
soucheng.choi@allstate.com
Anuradha Kodali
Mykel J. Kochenderfer
The Allstate Corporation
Northbrook, IL
akoda@allstate.com
ABSTRACT
Advanced collision avoidance and driver hand-off systems can benefit from the ability to accurately predict, in real time, the probability
a vehicle will be involved in a collision within an intermediate
horizon of 10 to 20 seconds. The rarity of collisions in real-world
data poses a significant challenge to developing this capability because, as we demonstrate empirically, intermediate-horizon risk
prediction depends heavily on high-dimensional driver behavioral
features. As a result, a large amount of data is required to fit an effective predictive model. In this paper, we assess whether simulated
data can help alleviate this issue. Focusing on highway driving,
we present a three-step approach for generating data and fitting
a predictive model capable of real-time prediction. First, high-risk
automotive scenes are generated using importance sampling on
a learned Bayesian network scene model. Second, collision risk is
estimated through Monte Carlo simulation. Third, a neural network
domain adaptation model is trained on real and simulated data to
address discrepancies between the two domains. Experiments indicate that simulated data can mitigate issues resulting from collision
rarity, thereby improving risk prediction in real-world data.
KEYWORDS
Automotive Risk Prediction, Policy Evaluation, Monte Carlo Simulation, Bayesian Networks, Importance Sampling, Domain Adaptation
1
Sou-Cheng T. Choi
The Allstate Corporation
Northbrook, IL
sunil.chintakindi@allstate.com
INTRODUCTION
The ability to accurately predict intermediate-horizon automotive
risk from scene information is valuable for safety applications. For
example, this capability can be used to inform the control-hand off
problem in level-3 autonomous vehicles, or to allow for preemptive
rerouting of autonomous vehicles away from high-risk situations.
There are three major challenges when predicting automotive
risk from scene information. The first challenge is partial observability, which can arise due to occluded vehicles and sensor uncertainty
along with unobserved driver information, such as degree of aggressiveness or maneuver intention. We assume full observability
Proc. of the 17th International Conference on Autonomous Agents and Multiagent Systems
(AAMAS 2018), M. Dastani, G. Sukthankar, E. Andre, S. Koenig (eds.), July 2018, Stockholm,
Sweden
© 2018 International Foundation for Autonomous Agents and Multiagent Systems
(www.ifaamas.org). All rights reserved.
https://doi.org/doi
Stanford University
Stanford, CA
mykel@stanford.edu
in this work for simplicity. The second challenge is that, as the
prediction horizon increases, the set of possible vehicle trajectories
grows exponentially. As a result, measures of risk associated with
real-world data are generally imprecise, and estimating risk in simulation is computationally expensive. We restrict ourselves to an
intermediate horizon in this paper.
The third challenge in intermediate-horizon risk prediction, and
the focus of this paper, is a lack of data for fitting predictive models. This lack of data results from collision rarity and the highdimensional nature of the problem. We demonstrate empirically
that driver behavioral features have an increasingly significant
impact as the prediction horizon increases. Learning a predictive
model of risk that depends upon relatively high-dimensional behavioral features for many nearby vehicles requires a large amount
of data with coverage over the state space, which in high-risk cases
is unavailable. With collisions occurring approximately twice per
million vehicle miles traveled in the U.S. [53], an economical and
safe method for addressing collision rarity when fitting predictive
models is desired.
A variety of methods have been considered for addressing the
challenge of collision rarity in learning a predictive model. A common approach is to focus on predicting risk surrogates, such as hard
braking or low time-to-collision events [10, 15]. While prediction of
these quantities largely avoids issues resulting from rarity, it relies
on the assumptions that these events correlate well with collisions
and that they capture many collision modalities [18]. A second
approach to addressing collision rarity is to augment existing data,
for example through random noise or transforms [28], or more
sophisticated oversampling [8]. These approaches typically assume
smooth relationships between covariate and response variables,
and it is not clear whether this holds in a risk prediction setting.
This paper aims to determine whether simulated data can help
mitigate issues resulting from collision rarity. This approach leverages prior knowledge about the automotive domain, for example
that vehicles behave according to physical laws or follow certain
driver models, to generate sufficiently realistic data to improve prediction performance through transfer learning [38]. A simulationbased approach to addressing collision rarity faces two primary
challenges: (i) efficient generation of high-risk simulated data, and
(ii) reducing or compensating for inevitable differences between
simulated and real-world data, thereby enabling effective transfer.
AAMAS’18, July 2018, Stockholm, Sweden
This paper presents a method that addresses these two challenges. The first challenge is addressed using importance sampling,
which allows us to focus computational effort on higher-risk scenes.
Determining proposal distributions from which to sample events
can be challenging, so we propose an approach for automatically
learning high-risk automotive scene distributions using the cross
entropy method [11, 44]. We address the second challenge of effective transfer learning through a two-pronged approach. We first
allow for simulated data to closely resemble real data when possible.
We accomplish this by learning scene models from data, and by employing a risk estimation framework that imposes few assumptions
on the dynamics or driver behavior models. Second, since simulated
data will necessarily differ from real data, we use recent adversarial
domain adaptation methods. These approaches have been shown
to be effective in unsupervised settings [14], and we demonstrate
their effectiveness in a practical, fully-supervised setting.
We validate our approach to intermediate-horizon risk prediction in two experiments. First, in a fully simulated setting, we
demonstrate the effectiveness of the system in mitigating issues
resulting from collision rarity. Second, we demonstrate that simulated data can be effectively transferred to improving a predictive
model applied to real-world data.
2
RELATED WORK
Automotive risk models often address problems arising from collision rarity by predicting collision surrogates such as initiating
conditions and evasive maneuvers [10, 15, 49]. These surrogate
events are assumed to correlate with collisions [17, 20]. Lord et al.
review approaches to statistical analysis of crash frequency data
[33], which use static scene features. A variety of approaches have
been considered in this context, including negative-binomial regression [34], support vector machines [32], and neural networks [7].
In contrast, real-time systems use dynamic features of a scene to
estimate risk [31]. These approaches leverage some form of motion
prediction, which range in complexity from physics-based models
[3, 21] to those accounting for driver maneuver intention [30, 46].
Monte Carlo simulation methods permit complex dynamics and
driver models by not making assumptions about these factors, and
have been widely considered in automotive risk estimation [2, 6, 12].
Our framework uses a Monte Carlo approach, though it differs
from existing methods in two important ways. First, previous research has emphasized short-term prediction, primarily with the
goal of informing collision avoidance systems [31]. Second, whereas
existing approaches propose to estimate risk in real-time by executing simulations on-vehicle, our approach amortizes the cost
of running this optimization by learning a predictive model that
generalizes across scenes.
Driver behavior models determine the actions taken by vehicles
in automotive simulations. Heuristic driver models, such as the
intelligent driver model (IDM) [51] and MOBIL [23], can be used to
generate collision-free trajectories. A variety of collision-inclusive,
heuristic models exist, for example, the Errorable Driver Model
[57] and Less-Than-Perfect driver [56]. While these models exhibit
collisions, truly human-like behavior and failure modes are better
captured by fully parametric models that are learned from data
[29, 36]. We focus on the former heuristic models in this work, and
leave more sophisticated models for future consideration.
Automotive scenes can be generated through heuristic means, or
by learning a generative scene model from data. Wheeler et al. consider Bayesian networks [55] and factor graphs [? ] for learning
models directly from data. In this paper, we adapt the Bayesian
network approach proposed by Wheeler et al. [55].
Due to collision rarity, many scene samples may be required to
produce a collision, and many Monte Carlo simulations of those
scenes may be needed to arrive at risk estimates with small relative
error. Importance sampling provides a means of addressing these
inefficiencies by oversampling dangerous scenes, and then weighing
them based on their relative likelihood in an estimate or objective
function [16]. Importance sampling has been applied for sampling
lane-change scenarios, in particular with proposal distributions
learned through the cross entropy method (CEM) [59]. CEM is a
general optimization method common in rare event simulation
[11, 44] that we also employ, but do so in learning full Bayesian
network proposal distributions.
We use recent methods from the field of domain adaptation (DA)
in order to improve transfer between simulated and real-world data.
This problem has been considered in the context of reinforcement
learning for robotic control [19, 45], as well as in supervised tasks
such as pedestrian classification [54]. Unsupervised DA approaches
typically seek to identify both domain invariant and discriminative
features [14], or to infer shared and private latent spaces of the
two domains [5], and have been applied, for example, to object
classification [47].
3
PROBLEM STATEMENT
This section first introduces Markov decision processes and formulates risk estimation as policy evaluation. We then discuss three
traits of the problem that will later motivate our approach.
3.1
Background
We formulate risk estimation within the Markov decision process
(MDP) framework [4, 25, 41] due to its natural interpretation as
policy evaluation. A finite-horizon MDP consists of a set of states
S, actions A, a probability distribution P(s ′ | s, a) over the next
state s ′ given the current state s and action a, a reward function
R(s), which we assume is deterministic and a function of only the
current state, a horizon H < ∞, and an initial distribution over
states ρ 1 .
A stochastic policy πθ parameterized by θ ∈ Θ defines a probability distribution over actions given the current state: πθ (a | s).
The return of a policy starting at time t is the sum of rewards
r t + r t +1 + · · · + r H it receives when interacting with an environment. The expected return of using policy πθ starting from state
st is referred to as the value of the policy, which can be expressed
using the Bellman equation:
V
πθ
(st ) = Eπθ
"H
Õ
= R(st ) +
R(sk ) S t = st
Õ
k =t
a
πθ (a | st )
#
Õ
s′
P(s ′ | st , a)V πθ (s ′ ).
(1)
Real-time Prediction of Intermediate-Horizon Automotive Collision Risk
Policy evaluation is the task of computing the value of a policy for all states. Policy evaluation can be performed through a
dynamic programming procedure that iterates equation (1) as an
update. This approach can be intractable in MDPs with large state
or action spaces, in which case Monte Carlo policy evaluation might
be employed instead, wherein sample returns are averaged to approximate the value of a policy.
3.2
Problem Formulation
We formulate automotive risk estimation as policy evaluation. A
particular vehicle of concern, which we refer to as the ego vehicle,
operates according to a policy πθ parameterized by θ . We refer to
these parameters as the “behavioral features” of the vehicle because
they are the parameters of the driver behavior models (e.g., IDM
and MOBIL) that control vehicle action selection.
We assume full observability, and thus the state contains the
physical attributes as well as the behavioral features of the ego vehicle and neighboring vehicles. Because we assume that behavioral
traits are fully observed, this definition of the state corresponds
most closely with the notion of a scene within the automotive safety
literature [52], and we use “state” and “scene” interchangeably.
We are interested in automotive risk, which can be defined as
the “likelihood and severity of the damage that a vehicle of interest
may suffer in the future” [31]. We focus only on the probability of
collisions, or occasionally on surrogate measures of risk, and therefore we would like to design a reward function such that the value
of a policy corresponds to the probability of that particular policy
suffering a collision. We accomplish this by defining an indicator
function C(st ) equal to 1 if st contains a collision involving the
policy of interest, and 0 otherwise. We focus on collisions occurring after some initial timestep h, and define a collision indicator
function that depends on this value:
(
1, if C(st ) = 1 and t ≥ h
Yh (st ) =
.
(2)
0, otherwise
States containing collisions of the ego vehicle are terminal states.
The value of a policy with respect to a state is the probability
that policy enters into a collision starting in state st and acting
according to πθ until the horizon H :
V πθ (st ) = Eπθ
=
=
H
Õ
k=t
H
Õ
k =t
Õ
H
k =t
R(sk ) S t = st
(3)
Eπθ Yh (sk ) S t = st
(4)
P(Yh (sk ) = 1 | S t = st , πθ ).
(5)
Above, we use the properties of linearity of expectation and the
expectation of an indicator random variable. Defining states containing collisions as terminal states makes the events Yh (si ) = 1
and Yh (s j ) = 1 mutually exclusive for all i and j such that i , j.
We define a trajectory as τ = (s 1 , s 2 , . . . , s H ) in a space T =
S × · · · × S, along with a function indicating that a trajectory
contains a collision involving the ego vehicle:
(
1, if C(st ) = 1 and t ≥ h for any st ∈ τ
Yh (τ ) =
.
0, otherwise
(6)
We can define the value as the probability of a collision in a trajectory from an initial state st . Hence (5) can be rewritten as
V πθ (st ) = P(Yh (τ ) = 1 | S t = st , πθ ).
3.3
(7)
Traits of Intermediate-Horizon Automotive
Risk Prediction
In this section, we discuss three traits of intermediate-horizon automotive risk prediction that will later inform our approach.
3.3.1 Rarity of Collisions. Validating autonomous systems frequently involves analyzing critical events that are rare. Formally,
define the state-visitation distribution [43]
H
1 Õ
P(Sk = st | πθ ).
(8)
H
k =1
Í
The expected value of a collision event, st ρ(st )C(st ), is assumed
to be extremely small. This posses a challenge for learning to predict
risk based on real-world data, and for Monte Carlo policy evaluation
in simulation.
ρ(st ) =
3.3.2 State Space Size. The state space in automotive risk prediction is high-dimensional and contains continuous variables. These
factors motivate a method of generalizing risk estimates across
scenes. Accomplishing this using function approximation implies
learning a predictive model for risk conditioned on the scene.
3.3.3 Influence of Behavioral Parameters. In automotive risk prediction, behavioral parameters, θ , of vehicles in the scene increase
in importance with the prediction horizon. Figure 1 demonstrates
this trend in an artificial 1 dataset. This fact, coupled with the relatively high-dimensional nature of behavioral features and the need
to account for these features for all neighboring vehicles in the
scene, makes intermediate-horizon automotive risk prediction a
high-dimensional problem.
4
APPROACH
This section first introduces the simulator we use for data generation, which entails describing the components of the MDP. We
then discuss our approach to policy evaluation (i.e., risk estimation).
Finally, we describe how we fit a model to risk estimates in order
to allow for real-time prediction.
4.1
Simulator
We define the simulator by specifying an initial scene distribution,
ρ 1 , and the transition function, P(s ′ | s, a). This second component is composed of the physical dynamics model, and the driver
behavior models that perform decision making for vehicles.
4.1.1 Scene Generation. Scene generation involves specifying
the initial state distribution, ρ 1 of the MDP. As previously defined,
the state includes the physical attributes and behavioral parameters,
1 We
use artificial data due to the unavailability of collision-inclusive real-world data.
Section 4.1 discusses the method used for generating this data in detail.
Normalized Correlation with
Collision Probability
AAMAS’18, July 2018, Stockholm, Sweden
1
aддi
0.5
atti
v f ,i−1
s f ,i
∆vi
li
wi
i = 1 to L
0
0
5
10
15
20
Initial Prediction Timestep h (seconds)
Figure 1: This plot shows, for a dataset of 70,000 vehicle trajectories each simulated 100 times, the maximum correlation across all behavioral features with rear-end collisions,
normalized by the maximum correlation across all features
as a function of the prediction horizon. The upward trend
of this plot indicates that as h increases, behavioral features
become increasingly informative of collision risk.
Table 1: Scene Vehicle Features
Feature Name
Symbol
Description
fore distance
fore velocity
relative velocity
length
width
attentiveness
aggressiveness
sf
vf
∆v
l
w
att
aдд
distance to vehicle in front
velocity of vehicle in front
rear velocity minus fore velocity
length of vehicle
width of vehicle
whether or not driver is attentive
aggressiveness of driver
θ , of vehicles in the scene. We assume a scene decomposes into L
individual vehicles as S = {S (1) , S (2) , . . . , S (L) }.
A simple approach to scene generation is to select from a database of real scenes with inferred behavioral features. The problems
with this approach are that (i) scene data must be available and (ii)
only previously observed scenes can be sampled, meaning generalization to new roadways is not possible. Scenes may instead be
initialized heuristically, or according to a constant configuration
that is simulated for a burn-in period. While these approaches are
simple, specifying rules for initial configurations can be challenging,
and simulation-based approaches rely on driver behavior models
to produce realistic distributions over scenes.
Alternatively, a generative model for scenes can be learned from
data. This approach allows for generalization to new settings, and
provides likelihood estimates of scenes. For these reasons, we employ a learned generative model in the form of a Bayesian network [27], which defines a probability distribution over a set of
variables S as a product of conditional probability distributions. The
distribution of each variable is defined conditional on the values
of its parent variables as defined by a directed acyclic graph. The
Î
distribution decomposes as P(S) = S (i ) ∈S P(S (i) | parents(S (i) )).
Figure 2: Bayesian network single-lane scene generation
model.
Forward sampling allows for efficient generation of scenes from a
Bayesian network provided that all observed nodes precede sampled
nodes in a topological ordering [27]. We implement a single-lane,
highway model similar to that of Wheeler et al. [55], with the
primary difference being that we incorporate behavioral features
in the model. Table 1 describes the variables of the network, and
Figure 2 shows the plate model used for scene generation.
To sample a lane from the model, the variables of the first vehicle
are sampled after marginalizing v f . Subsequent vehicles are then
sampled conditioning on the velocity of the prior vehicle. The scene
distribution decomposes as
ρ 1 (S) = P(S (1) )
L
Ö
i=2
P(S (i) | S (i−1) ).
(9)
The scene model exclusively contains discrete variables. For sampling continuous values, the variables define bounds of a uniform
distribution from which the value is sampled. This approach allows
for the use of discrete variables, while still approximating arbitrary
distributions as the discretization granularity increases, and has
been used in the context of aircraft scene generation [26].
4.1.2 Driver Behavior Models. The previous section described
how a scene is generated. We now discuss how this scene is simulated to produce vehicle trajectories. At each timestep, each vehicle
samples a longitudinal and lateral acceleration determined by the
driver models of that vehicle. For the longitudinal model, we employ
a collision-inclusive variant of the IDM, based upon the Errorable
Driver Model [57]. This model introduces collisions through a reaction time parameter that delays observations. The model also
includes attentiveness parameters that determine the probability of
a driver becoming distracted and not updating its action, as well as
the probability of a driver becoming attentive from an inattentive
state. We use MOBIL for lane changes, and sample longitudinal
and lateral accelerations from Gaussian distributions. The means of
these distributions are the acceleration values selected by the driver
models, and the standard deviations are 0.5 m/s2 and 0.1 m/s2 for
IDM and MOBIL, respectively.
For generating IDM and MOBIL parameters, we employ a correlated behavior model similar to that used in Sunberg et al. [48]. This
approach samples, for each vehicle, an aggressiveness parameter
from a uniform distribution over the unit interval. Aggressiveness
then determines the mean of a truncated Gaussian distribution by
interpolating between bounds for IDM and MOBIL as specified in
Table 2. The standard deviation of this distribution is set to be 0.03
Real-time Prediction of Intermediate-Horizon Automotive Collision Risk
Table 2: Driver model parameters and associated bounds
IDM Parameter
Most Agg.
Least Agg.
maximum acceleration
desired velocity (m/s)
min distance to fore vehicle (m)
safe time headway (s)
comfortable deceleration (m/s2 )
6.0
35
0
0.2
5
2.0
25
4
1
2
MOBIL Parameter
Most Agg.
Least Agg.
politeness
safe deceleration (m/s2 )
advantage threshold (m/s2 )
0.1
2.0
0.01
0.5
2.0
0.7
Errorable Parameter
Value
reaction time (s)
p(inattentivet +1 | attentivet )
p(attentivet +1 | inattentivet )
0.2
0.05
0.3
(m/s2 )
P(Yh (τ ) = 1) = Es∼ρ 1 (S =s),τ ∼P (T =τ |S =s), πθ [Yh (τ )]
times the range of values. Given these actions, the simulator then
propagates vehicles through space in discrete time increments of
0.1 seconds using a simple bicycle model [42].
4.2
Risk Estimation
This section discusses our approach to Monte Carlo policy evaluation, which we refer to as risk estimation in the automotive context.
4.2.1 Monte Carlo Simulation. The previous sections described
the initial scene generation, driver behavior, and dynamics, which
together specify the environment. Given this simulator, we can
perform policy evaluation to estimate risk. We use Monte Carlo
policy evaluation because it imposes minimal restrictions on the
type of driver model used.
Our goal is to compute the value of a policy, which is the probability that that policy is involved in a collision conditional on a
scene s sampled from the scene model. Ideally, we would compute
this probability exactly:
P(Yh (τ ) = 1 | S = s, πθ ) = Eτ ∼P (T =τ |S =s), πθ [Yh (τ )].
(10)
This value cannot be computed in closed form, so we instead approximate the value through sampling n trajectories and averaging:
n
1Õ
Eτ ∼p(T =τ |S =s), πθ [Yh (τ )] ≈
Y (τi ),
(11)
n i=1 h
where τi ∼ P(T = τ | S = si , πθ ).
The value in (10) conditions on a particular scene s. In contrast,
our approach to importance sampling biases sampling towards
scenes that are dangerous with respect to the ego vehicle. Formally,
the challenge is that the unconditional probability of a collision
4.2.2 Importance Sampling. Generating collision-inclusive data
from the learned scene and driver models can require a large number of samples from (i) the scene generation model or (ii) the driver
and dynamics models. We would like to reduce the quantity of samples needed so as to improve computational efficiency. Importance
sampling samples collisions or other events with greater frequency
relative to a baseline probability distribution, and then reweighs
those samples according to their relative likelihood, thereby producing a lower variance, unbiased estimator [16]. This method can
be applied in either of the sampling cases, but we focus on sampling
initial scenes from the generative model.
is small. We address this problem through importance sampling.
Instead of sampling s from ρ 1 (S), we sample from a proposal distribution Q(S), which is biased towards dangerous scenes:
ρ 1 (S = s)
p(Yh (τ ) = 1) = Es∼Q (S =s),τ ∼P (T =τ |S =s), πθ
Yh (τ ) .
Q(S = s)
This method has been employed in evaluating the safety of autonomous systems, for example, vehicles [59] and aircraft [24]. In
these settings, importance sampling results in a lower variance estimator of the unconditional probability of collision or other event.
In our setting, we are not interested in computing the unconditional probability of a collision. Instead, we are interested in fitting a
predictive model to the conditional probability of a collision with the
goal of achieving high performance on certain evaluation metrics.
Due to this altered goal, importance sampling in this context can
be interpreted as a heuristic approach to active learning [9]. From
this perspective, we make the assumption that dangerous scenes
will improve performance by subjecting the model to a greater
number of positive-risk events, thereby addressing the low sample
size problem associated with those events.
4.2.3 Cross Entropy Method. A remaining task is determining
the proposal distribution Q(S). For this distribution, we use another Bayesian network, the conditional probability distributions
of which have been altered to generate collisions more frequently.
Because a scene filled with dangerous vehicles would result in extremely low likelihood values, we instead sample a single vehicle
in each scene from Q, and the remaining vehicles as usual from ρ 1 .
Recall that the scene model decomposes across the L vehicles as (9).
The proposal distribution Q(S) differs from ρ 1 (S) only in the jth
vehicle sampled, and thus all vehicle probabilities cancel in the ratio
of ρ 1 and Q except for the jth vehicle:
ρ 1 (S)
P(S (j) | S (j−1) )
.
=
Q(S)
Q(S (j) | S (j−1) )
We refer to this likelihood ratio as w.
We could manually alter the Bayesian network parameters to
increase the likelihood of collisions involving the ego vehicle, but
we would like to automate this process. We accomplish this using
the cross entropy method (CEM).
CEM [44] is an optimization algorithm originally designed for
finding proposal distributions in the context of rare events, which
operates by iteratively sampling from some distribution, and then
updating the parameters of that distribution based on the fitness of
the samples [11]. In the particular instantiation of CEM we employ,
the proposal distribution is trained to produce collisions involving
the ego vehicle within 10 to 20 seconds by changing the conditional
probability distributions of the variables in the Bayesian network.
Figure 3 shows the mean values of three variables and collision
probability during the optimization process. The changes to these
variables, which may initially seem counterintuitive, result in a
higher probability of collision. The reason for this is that only
AAMAS’18, July 2018, Stockholm, Sweden
Aggressiveness
0.5
1.88
0.4
1.87
0.3
1.86
1.85
0.2
0
0
100
200
Relative Velocity
·10−2
−4
0
100
Iterations
0
1.2
1
0.8
0.6
0.4
−2
−6
run. The following sections describe this prediction model, and our
method for compensating for discrepancies between simulated and
real-world data.
Attentiveness
200
100
200
Collision Probability
·10−2
0
100
Iterations
200
Figure 3: Mean values for three of the Bayesian network variables and collision probability throughout optimization.
intermediate-horizon collisions (i.e., those occurring in the 10 to
20 second range) are counted. Thus, the method learns to avoid
early collisions (i.e., those occurring before 10 seconds) by making
the ego vehicle both attentive and slower than the vehicle in front.
Because the ego vehicle initially travels at a lower velocity than the
fore vehicle, it tends to speed up. If the ego vehicle then becomes
inattentive, it will continue accelerating until colliding with the fore
vehicle. Aggressiveness decreases because this reduces the initial
acceleration (thereby preventing early collisions), as well as the
magnitude of the comfortable deceleration rate (thereby limiting
the ability of the ego vehicle to slow down).
A dataset of scenes is sampled from the learned distributions
ρ 1 and Q, and each scene evaluated using Monte Carlo simulation.
m , containing m scene-risk
The resulting dataset {s (i) , y (i) , w (i) }i=1
pairs, each with an associated likelihood ratio w, is then used to
augment real data in learning a predictive model.
4.3 Risk Prediction
Risk estimation must be performed in real-time on-vehicle. A common approach to this task is to simulate future trajectories online
to derive a risk estimate [2, 39, 58]. While this method may be
feasible for short-term risk estimation, we believe that intermediate and longer-horizon risk will be impractical to estimate in
this manner. There are two reasons why this would be the case.
First, because driver behavior increases in significance with longer
horizons, effective simulators will likely use learned, computationally expensive driver models [29]. Second, and more challenging,
is the number of simulations that will need to be run due to the
prediction horizon in order to arrive at estimates of risk with low
relative error. We propose to instead fit a predictive model offline,
the online computational complexity of which grows slowly as a
function of simulator complexity and the number of simulations
4.3.1 Prediction Model. The response variable Y in collected
datasets is the estimated proportion of successes p (e.g., collisions)
in a binomial experiment with n samples. Because this value is
estimated from a finite number of samples, it can only assume a
finite number of values, and is therefore a discrete variable. This
variable has two unusual properties. First, valid values are bounded
between 0 and 1 inclusively. Second, in real-world data, predicting
this variable reduces to binary classification.
Zhao et al. [60] compare linear and logistic regression in fitting
proportion data, finding logistic regression to have better performance [60]. Three alternative approaches are (i) to take log(Y ) as
the response and fit a linear regression model, thereby avoiding
issues due to bounded values; (ii) to perform beta regression of the
values, which requires a transform of response values of 0 or 1 [13];
and (iii) to treat the response as counts and fit a negative binomial
regression [40], which would complicate transfer to real-world data
where the sample size n differs from simulation. While these alternatives may have merit, we elect to use the cross entropy loss because
of its simplicity and good performance in initial experiments.
In risk estimation, we considered the risk associated with the
ego vehicle in a scene s. In risk prediction, we take the perspective
of the ego vehicle directly to reflect the information available in a
real-world situation. We refer to the features from this perspective
with the random variable X , which we assume is still a Markovian
representation of the environment.
We fit a predictive model V (x; ϕ) with parameters ϕ as the value
function, minimizing the loss
Lpr ed (X , Y )
= − E(x,y)∼(X,Y ) [y log(V (x; ϕ)) + (1 − y) log(1 − V (x; ϕ))]. (12)
We empirically evaluate this loss on the collected dataset as
−
m
Õ
i=1
w (i) [y (i) log(V (x (i) ; ϕ)) + (1 − y (i) ) log(1 − V (x (i) ; ϕ))]. (13)
This model is fit to a dataset containing risk estimates for many
different policy parameters, θ . We include these parameters as
features, effectively learning a value function over a distribution
of policies. In all experiments, we use a neural network prediction
model with sigmoid activation function. This choice of model is
motivated by the high-dimensional feature space, and the ability to
employ recent domain adaptation approaches as we discuss next.
4.3.2 Domain Adaptation. Even in high-fidelity simulations,
generated data will necessarily differ from real-world data. Formally,
the joint probability distribution over the simulated, ego-centric
scene representations X s and risk estimates Ys , will differ from the
(target) real-world joint distribution: Ps (X s , Ys ) , Pt (X t , Yt ). This
“domain shift” results in degraded performance when transferring
from the source to target domain [45].
Domain adaptation (DA) methods attempt to address this problem. Both supervised and unsupervised variants exist, and it is not
immediately clear which approach is better suited for automotive
risk prediction. The reason for this is that the target, real-world
Real-time Prediction of Intermediate-Horizon Automotive Collision Risk
Table 3: Prediction Features
5.1
Feature Name
Example Quantities
Ego physical
lane offset and heading, velocity,
vehicle length and width, acceleration,
turn rate, time-to-collision
Ego well-behaved
ego vehicle currently colliding,
out of lane, or has a negative velocity
Ego behavioral
IDM and MOBIL parameters
Neighboring vehicle
state
same physical and behavioral features as
as ego vehicle and relative dist. to ego
automotive domain can be viewed as “weakly supervised” in the
sense that its risk estimates are highly imprecise reflections of the
true underlying collision probability.
While supervised DA methods have been shown to be effective,
for example, the “semantic alignment loss” of Motiian et al. [37],
we focus on an unsupervised approach, domain adversarial neural
networks (DANN) [14]. DANN attempts to learn features that are
(i) invariant to the domain and (ii) useful for the task being performed. To accomplish this task, the activations of certain layers in
the network, Ms and Mt , are encouraged to match across domains.
This requires a measure of similarity, and for that DANN employs
a domain classification network, D, that attempts to distinguish between the learned features of the two domains. The risk prediction
network incurs an additional adversarial loss
Ladv (X s , X t , Ms , Mt )
(14)
= −Ex s ∼X s [log D(Ms (x s ))] − Ex t ∼X t [log(1 − D(Mt (x t )))] ,
with D being trained to minimize the negative. The overall objective
then becomes L = Lpr ed + λLadv , where λ controls the weight
of the adversarial loss.
Minimizing this loss with respect to both Ms and Mt encourages
the source and target distributions to match, but this might come
at the cost of learning a good target feature distribution. In the
supervised setting, it is possible to only optimize the adversarial
loss with respect to the source feature distribution Ms , and we
consider this approach as well in our experiments.
5
EXPERIMENTS AND RESULTS
Our goal is to assess whether simulated automotive data can improve intermediate-horizon risk prediction in real-world data. We
perform two experiments. The first experiment employs the full
approach as described, and seeks to determine whether importance
sampled collision events can improve prediction in an artificial
scenario. The second experiment considers the simulated to real
transfer challenge using real-world vehicle trajectory data.
Throughout the experiments, we use a set of scene features,
which we summarize in Table 3. Our primary focus in this research
is on addressing the challenge of collision rarity, but not on partial
observability. As a result, we assume access to features that are
difficult to obtain in real scenarios, for example, information about
vehicles far in front of the ego vehicle, and behavioral features that
typically must be inferred [48].
Simulation Experiments
In this section, we assess whether importance sampling of collisions
in artificial settings can improve prediction performance. Our focus
on exclusively simulated data in this experiment is motivated by
the unavailability of real-world data containing collisions.
We generate an artificial dataset to act as the “real” data containing few collisions. We generate this data heuristically, randomly
initializing vehicles and their behavioral parameters on a single
lane, circular track, which is simulated for 600 timesteps to arrive
at a random configuration. This configuration is then simulated 100
times for 200 timesteps, and risk estimates for the period between
100 and 200 timesteps are computed. Each scene involves 70 vehicles, and we take each vehicle as a single sample. This produces a
dataset in which samples are not independent since they may both
be from the same scene. For the sake of efficiency, we ignore this
lack of independence in the training data; however, the validation
set contains samples generated from entirely different simulations
from those of the training set.
We apply the proposed system by fitting a scene model, running
the cross entropy method to derive a proposal distribution, generating a dataset, and learning predictive models. We compare four
methods: (i) training only on the target domain, (ii) training on both
domains without adaptation, (iii) training on both domains with
adaptation, and (iv) training on both domains, applying adversarial
loss only to the source features. Because the number of collision
events in the target domain plays a critical role in determining
the value of simulated data, we compare performance between the
models for various numbers of target domain collision events. To
better reflect real-world data, we convert the target training set into
binary values by sampling collisions according to the observed probabilities. The validation set we leave as continuous values because
this provides greater precision during evaluation.
A hyperparameter search was performed for each method over
a predefined set of three network architectures, learning rates between 10−4 and 10−3 , and dropout probabilities between 0.5 and 1.
The three architectures consisted of encoding hidden layers containing (512, 256, 128, 64), (256, 128, 64), or (128, 64) units, and classifier
hidden layers containing either no hidden units (i.e., direct classification of encoder output), (64) or (64, 64) hidden units. All networks
used ReLU activations, and for each training run the best validation
performance during the run was selected for use in the results. The
domain adversarial loss was applied to the features output from
the encoder. Thirty networks were trained for each method. Figure
4 (a) shows the average negative log likelihood evaluated on the
validation set across these training runs.
These results support two main conclusions. First, all models
that learn from simulated data exhibit improved performance on
positive-risk instances, and source-only adaptation and no adaptation improve over target-only training overall. This result is likely
due to the fact that the jointly-trained models observe a far greater
number of positive-risk samples from the source dataset, thereby
enabling them to learn a better predictive model. Second, when
few positive-risk samples are provided in the training set, the adaptation models achieve the best performance. With more positiverisk samples the model without adaptation performs best across
all validation samples, likely because this model is better able to
AAMAS’18, July 2018, Stockholm, Sweden
target data only
adapt source + target
adapt neither
adapt source
0.6
Negative Log Likelihood
0.5
0.4
0.3
0.025
0.02
0.015
0
20
40
60
80
Collisions in Target Training Set
100
Average Precision Score
(a) Inter-Simulation Transfer
0.25
0.2
0.15
0.1
0
50
100
150
200
Low TTC Events in Target Training Set
(b) Simulation to NGSIM Transfer
Figure 4: These plots show validation results with varying
amounts of positive-risk samples in the target domain training set. Each point represents the mean performance across
runs, with vertical bars indicating standard error. The top
and bottom plots of (a) show performance evaluated on positive-risk and all target domain validation samples, respectively. Plot (b) shows results for transferring from simulation to NGSIM data.
decide what source data to leverage, rather than being explicitly
constrained to learning a shared feature space.
5.2
Real-world Transfer Experiments
We next seek to determine whether simulated automotive data can
be transferred to a prediction task involving real-world data. We
consider the task of predicting low time-to-collision (TTC) events
[15], which we define as a vehicle having a TTC less than 3.0
seconds, in the US Highway 101 portion of the Next-Generation
Simulation (NGSIM) dataset [1]. NGSIM contains vehicle trajectories on a five-lane highway, collected over three 15-minute time
periods. These trajectories contain no collisions, so we instead focus on a collision surrogate. Furthermore, low TTC events are not
sufficiently rare to merit application of importance sampling, so
we generate 50,000 source samples heuristically using the same
method as used in the simulated experiments for mimicking realworld data. Because NGSIM contains binary labels, we now evaluate
classification performance using the average precision score [61],
which we choose in order to better account for class imbalance.
The NGSIM dataset does not contain explicit behavioral parameters. Since these parameters are central to the high-dimensionality
of the risk prediction problem, we first extract IDM parameters using the local maximum likelihood method of Kesting et al. [22]. We
smooth the U.S. Highway 101 subset of the NGSIM data according
to the method of Theimann et al. [50], and then extract samples
from it in 20 second increments. To minimize dependence between
training and validation datasets, we train on the first and third time
periods of the NGSIM dataset, and validate on the second.
We again compare performance of the four methods across varying amounts of positive-risk target data, and visualize results in
Figure 4 (b). These results indicate that when training data is limited,
simulated data can significantly improve prediction performance.
As the number of positive-risk target samples increases, the targetonly model performs best because enforcing domain-invariance
begins to outweigh the benefits of additional positive-risk samples
from simulated data. Approximately 5% of the samples have a positive response value. In this imbalanced setting, a random classifier
achieves an average precision score of 0.05. All models therefore
perform significantly above random, but nevertheless quite poorly
due to the high-variance nature of the classification task.
6
CONCLUSION
This paper aimed to answer the question of whether simulated data
could be used to improve automotive risk prediction. We concluded
that simulated data can be helpful in cases where the amount of
data available is insufficient to learn a good predictive model. We
demonstrated that intermediate-horizon prediction suffers from this
low sample size issue due to the significance of high-dimensional
behavioral features in learning a predictive model. We then showed
that domain adaptation methods could be used with simulated data
to improve prediction in both artificial and real-world settings. The
significance of this result is that systems that perform intermediatehorizon risk prediction may potentially be improved through the
offline training of predictive models on simulated data.
Our approach can be improved through more sophisticated modeling, estimation, and prediction. Initial scene generation could
employ the factor graph approach of Wheeler et al. [? ] or deep
generative models. Driver models learned from data [29] are likely
to provide substantial benefit in transfer performance by better
capturing human failure modes. A multi-lane approach to importance sampling would allow for application of the system to real-life
collision prediction transfer.
Real-time Prediction of Intermediate-Horizon Automotive Collision Risk
Prediction performance could be enhanced through the use of
domain adaptation models that explicitly model shared and private
latent feature spaces [5]. Finally, the local maximum likelihood
estimation approach to inferring behavioral parameters of NGSIM
vehicles is limited because the model poorly captures individual
driving behavior, and it does not generalize across drivers. A method
proposed by Morton et al. [35] would allow for arbitrarily complex
behavior encodings, while also generalizing across samples, thereby
alleviating issues resulting from the lack of individual-vehicle data.
7
ACKNOWLEDGMENT
The authors wish to thank Gina Madigan, Tim Wheeler, Zach Sunberg, and Katie Driggs-Campbell for helpful conversations and
feedback. This research was funded by The Allstate Corporation.
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| 1 |
Two-Scale Topology Optimization with Microstructures
arXiv:1706.03189v1 [cs.CE] 10 Jun 2017
BO ZHU, MÉLINA SKOURAS, DESAI CHEN, WOJCIECH MATUSIK, MIT CSAIL
Fig. 1. Our two-scale topology optimization framework allows to optimize continuous material properties mapping to printable microstructures (left) to
fabricate high-resolution functional objects (middle) and minimum compliant structures (right).
In this paper we present a novel two-scale framework to optimize the structure and the material distribution of an object given its functional specifications. Our approach utilizes multi-material microstructures as low-level
building blocks of the object. We start by precomputing the material property gamut – the set of bulk material properties that can be achieved with
all material microstructures of a given size. We represent the boundary
of this material property gamut using a level set field. Next, we propose
an efficient and general topology optimization algorithm that simultaneously computes an optimal object topology and spatially-varying material
properties constrained by the precomputed gamut. Finally, we map the
optimal spatially-varying material properties onto the microstructures with
the corresponding properties in order to generate a high-resolution printable structure. We demonstrate the efficacy of our framework by designing,
optimizing, and fabricating objects in different material property spaces on
the level of a trillion voxels, i.e several orders of magnitude higher than what
can be achieved with current systems.
Additional Key Words and Phrases: microstructures, metamaterials, 3D
printing, topology optimization
1
INTRODUCTION
Many engineering problems focus on the design of complex structures that needs to meet high level objectives such as the capability
to support localized stresses, optimal tradeoffs between compliance
and mass, minimal deformation under thermal changes, etc. One
very popular approach to design such structures is topology optimization. Topology optimization generally refers to discretizing the
object of interest into small elements and optimizing the material distribution over these elements in such a way that the functional goals
are satisfied [Bendsøe and Sigmund 2004]. Traditionally, topology
optimization focused on designs made of homogeneous materials
and was concerned with macroscopic changes in the object geometry. With the advent of multi-material 3D printing techniques,
it is now possible to play with materials at a much higher resolution, allowing to obtain much finer designs and, thus, improved
functional performances. Unfortunately, standard techniques for
topology optimization do not scale well and they cannot be run
on objects with billions of voxels. This is because the number of
variables to optimize increases linearly with the number of cells
in the object. Since many current 3D printers have a resolution of
600DPI or more, a one billion voxel design occupies only a 1.67 inch
cube.
One direction to handle this issue is to work with microstructures corresponding to blocks of voxels instead of individual voxels
directly. Some recent works followed this direction and proposed to
decouple macro structural design and micro material design [Coelho
et al. 2008; Nakshatrala et al. 2013; Rodrigues et al. 2002]. However,
these approaches remain computationally expensive and, in most
cases, limited to the well-known minimal compliance problem. The
second direction to reduce the problem complexity is to temporarily ignore the geometry of the microstructures and consider only
their macroscopic physical behaviour. However, this introduces
new difficulties as the space of material properties covered by all
printable microstructures is much wider than the properties of the
base materials. For example, microstructures made of alternating
layers of soft and stiff isotropic materials exhibit an anisotropic
behaviour as they are able to stretch more easily in one direction
that in the others. This implies that not only the ranges but also
the number of physical parameters needed to describe the physical
behaviour of these microstructures increases. Therefore, in order to
work with the material properties of microstructures, one needs to
solve two challenging problems: (i) computing the gamut – i.e the
set – of the material properties achievable by all microstructures,
(ii) efficiently optimizing the distribution of these high-dimensional
material properties inside the layout of the object.
Most previous algorithms working in the material space focused
on optimizing a single material property such as density or material stiffness, for which analytical formulas describing the property
bounds exist [Allaire and Kohn 1993]. On the contrary, optimizing the structure and material distribution of an object in a high
dimensional material property space remains an open problem. In
84:2 • B. Zhu et al.
this work, we propose a new computational framework for topology optimization with microstructures that supports design spaces
of multiple dimensions. We start by computing the gamut of the
material properties of the microstructures by alternating stochastic
sampling and continuous optimization. This gives us a discrete representation of the set of achievable material properties, from which
we can construct a continuous gamut representation using a level
set field. We then reformulate the topology optimization problem in
the continuous space of material properties and propose an efficient
optimization scheme that finds the optimized distributions of multiple material properties simultaneously inside the gamut. Finally,
in order to obtain fabricable designs, we map the optimal material
properties back to discrete microstructures from our database.
Our general formulation can be applied to a large variety of
problems. We demonstrate its efficacy by designing and optimizing objects in different material spaces using isotropic, cubic and
orthotropic materials. We apply our algorithm to various design
problems dealing with diverse functional objectives such as minimal compliance and target strain distribution. Furthermore, our
approach utilizes the high-resolution of current 3D printers by supporting designs with trillions of voxels. We fabricate several of our
designs, thus, demonstrating the practicality of our approach.
The main contributions of our work can be summarized as follows:
• We present a fully automatic method for computing the
space of material properties achievable by microstructures
made of a given set of base materials.
• We propose a generic and efficient topology optimization
algorithm capable of handling objects with a trillion voxels.
The key of our approach is a reformulation of the problem to work directly on continuous variables representing
the material properties of microstructures. This allows us
to cast topology optimization as a reasonably sized constrained optimization problem that can be efficiently solved
with state of the art solvers.
• We validate our method on a set of test cases and demonstrate its versatility by applying it to various design problems of practical interest.
2
RELATED WORK
Topology Optimization. Topology optimization is concerned with
the search of the optimal distribution of one or more materials within
a design domain in order to minimize some input objective function
while satisfying given constraints [Bendsøe and Sigmund 2004]. Initially applied to the structural design in engineering [Bendsøe 1989],
topology optimization has been extended since then to a variety of
problems including micromechanism design [Sigmund 1997], mass
transfer [Challis and Guest 2009], metamaterial design [Cadman
et al. 2013; Sigmund and Torquato 1996], multifunctional structure
design [Yan et al. 2015], coupled structure-appearance optimization
[Martı́nez et al. 2015]. Many algorithms have been proposed to
numerically solve the optimization problem itself. We refer to the
survey by Sigmund and Maute [2013] for a complete review. In
the very popular SIMP (Solid Isotropic Materials with Penalization)
method, the presence of material in a given cell is controlled by
locally varying its density. A binary design is eventually achieved
by penalizing intermediate values for these densities. In practice,
this method works well for two-material designs (e.g., a material
and a void), but generalizing this method to robustly handle higher
dimensional material spaces remains challenging. Instead of considering only discrete structures, free material optimization [Haber
et al. 1994; Ringertz 1993] optimizes structures made of continuous
material distributions constrained by analytical bounds. Another
class of methods rely on homogenization. They replace the material
in each voxel of the object by a mixture of the base materials whose
material properties can be analytically derived. While optimal microstructures are known for certain classes of problems (laminated
composites in the case of the minimum compliance problem), this
is not the case in the general setting, for which using a specific
subclass of microstructures can lead to suboptimal results. In a
sense, our work is a generalization of these approaches and aims to
handle a wider range of materials for which theoretical bounds on
the material properties are not known a priori.
Although they are largely used in engineering, standard methods for topology optimization suffer from a major drawback : the
parametrization of the problem at the voxel level makes them extremely expensive and largely impedes their use on high resolutions
models such as the ones generated by modern 3D printing hardware.
High-performance GPU implementations with careful memory handling can be used to push the limits of what can be done (a couple
of million variables in the implementation by Wu et al. [2016]), but
such approaches rely on specificities of the minimum compliance
problem and are difficult to generalize. To counteract the effects of
the explosion of variables in finely discretized layouts, Rodrigues et
al. [2002] alternatively proposed an interesting formulation where
microstructure designs and macroscopic layouts using the effective
properties of the underlying microstructures were hierarchically
coupled and treated simultaneously. This initial work has been
extended in multiple ways [Coelho et al. 2008; Nakshatrala et al.
2013; Xia and Breitkopf 2014; Yan et al. 2014]. Alexandersen and
Lazarov [2015] proposed a fast simulation algorithm for optimizing complete macroscopic structures made of layered or periodic
microstructures. However, these methods still need to handle variables defined at the microstructure level and therefore they remain
relatively costly. The most related work is the method proposed
by Xia et al.[2015b], which also relies on a database to speed up
computations. However, their work specifically targets minimum
compliance problems in the structural design which allows them to
approximate the macroscale behaviour of the microstructures with
a particular strain-based interpolating function.
Fabrication-oriented Optimization. The last decade has witnessed
an increasing interest by the computer graphics community in the
design of tools and algorithms targeting digital fabrication of physical artifacts. The range of media and applications addressed in
previous literature is very diverse and we focus our discussion on
systems targeting 3D printing. The problem of optimizing the material assignment for the individual voxels of an object in order to
control its large scale behaviour has been studied in different contexts. Starting with optical properties, Hašan et al. [2010] and Dong
et al. [2010] provided methods for printing objects with desired
subsurface scattering properties. Stava et al. [2012] later considered
Two-Scale Topology Optimization with Microstructures •
Base materials
Microstructures
Discrete point cloud
𝐸
Microstructure
Database Generation
Material
Space
Sampling
84:3
Continuous level set
𝐸
Rigid
Soft
𝜈
𝜈
𝜌
Mapping
𝜌
Optimization
Target
deformation
Push
Multi-scale Topology
Optimization
Fabrication
Map to discrete microstructures
Topology optimization in
continuous material space
Design goal
Fig. 2. Algorithm overview. We start by precomputing the gamut of material properties that can be achieved with all material microstructures of a given size.
Next, we run our topology optimization algorithm that optimizes the material properties of the object within this gamut such as to minimize some functional
objective. Finally, we map the optimal continuous material properties back to microstructures from our database to generate a printable object.
stability of 3D printed objects, Zhou et al. [2013] explored structural
strength while Chen et al. [2014] focused on rest shape optimization.
Closer to our present work, frameworks for the design of objects
with desired mechanical behaviours have been proposed by Bickel
et al. [2010] and Skouras et al. [2013]. Like these works, our system
allows to match given input deformations. However, while these
previous systems assume a small set of available base materials and
use these base materials in relatively coarse discretizations, our system combines the base materials into microstructures to expand the
design possibilities. Also relevant is the tool presented by Xu et al.
[2015] that allows to interactively design heterogeneous materials
for elastic objects subject to prescribed displacements and forces,
and the material optimization approach proposed by Panetta et al.
[2015]. However, these methods may output materials that are not
available in the real world for non-convex manifolds of material
properties. By contrast, we guarantee that all the microstructures
used are always realizable in such cases, which is one of the key
contributions of our work. Lastly, in an effort to unify individual contributions when dealing with inverse modeling problems,
Chen et al. [2013] proposed an abstraction mechanism to facilitate
the development of goal-based methods. The output of most of
these systems is a per-voxel material composition, which cannot
be efficiently represented using simple surface meshes. Vidimče
et al. [2013] introduced a fabrication-specific language and a programming pipeline for a procedural material synthesis that lift this
limitation.
Microstructures and Metamaterials. Microstructures can be defined as small scale assemblies made of one or several base materials, whose macroscale properties can be very different from those
of the original materials. Many materials found in the nature are
microstructures when observed at a sufficiently small scale. Microstructures can also be engineered so as to define composites
with improved capabilities or even metamaterials with exceptional
properties. For example, Lakes [1987] presented in 1987 the first
man-made structure with negative Poisson’s ratio, i.e., a structure
which transversally expands when it is axially stretched. The design
of composites and metamaterials is an active research field inspiring
myriads of works [Andreassen et al. 2014; Babaee et al. 2013; Cadman
et al. 2013; Sigmund 1997; Sigmund and Torquato 1996; Wang et al.
2014]. While many of these works are concerned with the inverse
modeling of specific microstructures or families of microstructures,
the study of the space of properties that these microstructures can
achieve as a whole has been investigated much less. Theoretical
bounds have been derived without experimental validation [Lipton 1994; Milton and Cherkaev 1995; Ting and Chen 2005]. Taking
into account additive manufacturing constraints, Schumacher et al.
[2015] and Panetta et al. [2015] recently investigated the design
of tileable and printable microstructures. In the first part of this
paper, we further explore this line of research and focus on the generation and characterization of databases of microstructures with
maximal material property coverage. In particular, we present a
novel approach combining a probabilistic search and a continuous
optimization that allows us to fully automatically explore the gamut
of material properties that can be achieved by assembling given
base materials.
3
OVERVIEW
Given as input a set of base materials, an object layout, and functional objectives, the goal of our system is to compute the material
distribution inside the object in order to optimize these functional
objectives. In our approach, we do not solve the problem directly,
instead we work with microstructures made of the base materials and the space of physical material properties spanned by them.
The complete pipeline of our system, illustrated in Figure 2, can be
decomposed into three stages.
Material Space Precomputation. In the first stage, we estimate the
gamut of material properties covered by all possible microstructures
made by spatial arrangement of base materials. Since exhaustively
84:4 • B. Zhu et al.
computing the properties of all these microstructures is, in practice,
intractable, we progressively increase the material space by alternating a stochastic search and a continuous optimization. The first
step introduces discrete changes in the materials of the microstructures and allows emergence of new types of microstructures. The
second step allows to locally push the material space boundaries by
refining the microstructure shapes. After completing this stage, we
obtain a discrete representation of the space of material properties
and the mapping between these properties and the corresponding
microstructures.
Gamut-based Continuous Topology Optimization. In the second
stage, we construct a smooth continuous gamut representation of
the material property space by using a level set field. We define our
topology optimization problem directly in this space. Our approach
minimizes the objective function over possible material parameters
while asking for strict satisfaction of the physics constraints – typically, the static equilibrium – as well as the strict satisfaction of
the physical parameter bounds. Taking advantage of our gamut
representation as a level set, we formulate this last constraint as
limiting the material properties to stay on the negative side of the
level set. This guarantees that the material properties that we use
in the optimization are always physically realizable.
Fabrication-oriented Microstructure Mapping. In the last stage, we
generate a printable result by replacing each cell in the object layout
with a microstructure whose material properties are the closest to
the continuous material assignment resulting from the optimization.
We also take into account the boundary similarity across adjacent
cell interfaces to improve the connectivity between microstructures.
This results in a complex, high-resolution, multi-material model
with optimized functional specifications.
4
MECHANICS
In this section, we briefly introduce the background material for
simulating deformable objects. We will use these concepts when
computing the material properties of the microstructures (Section 5)
and in the topology optimization algorithm (Section 6). We refer to
the course by Sifakis and Barbic [2012] for a more comprehensive
exposition.
4.1
Material Model
written (using the Voigt notation) as
(1−ν )Ê
© ν Ê
C = ν Ê
«
ν Ê
ν Ê
(1−ν )Ê ν Ê
ν Ê (1−ν )Ê
σ = Cϵ ,
(1)
where C, the so-called elasticity tensor, can be described by 21
parameters [Bonet and Wood 1997].
Working in such a high dimensional space is prohibitive and
therefore we focus on materials having a certain numbers of symmetries, such as orthotropic materials for which the elasticity tensor
is defined by 12 parameters (4 parameters in 2D), cubic materials
defined by 3 parameters, and isotropic materials defined by 2 parameters. For example, the tensor for a 3D cubic structure can be
(2)
with Ê = E/((1 − 2ν )(1 + ν )) and where E is the Young’s modulus of
the material, ν its Poisson’s ratio, and µ its shear modulus.
Alternatively, one can also use Lamé’s parameters to define the
tensor C, which simplifies the derivation of the tensor with respect
to the elastic parameters. In this case, the tensor has the form
2µ+λ
© λ
C= λ
«
λ
λ
2µ+λ λ
λ 2µ+λ
ª
®
® .
µ
®
µ
µ¬
The tensor for a 2D orthotropic structure can be written as
E x νyx E x
ν
E
E
C = c xy y
y
(3)
(4)
µ/c
with c = 1/(1 − ν xy νyx ), and where E x and Ey are the Young’s
moduli along the two principal axes, µ is the shear modulus, ν xy is
the Poisson’s ratio corresponding to a contraction in the direction
y when an extension is applied along the x axis, and νyx verifies
νyx E x = ν xy Ey .
Letting R n denote the space of n material properties, we then
write each point p ∈ R n as p = [ρ, e], where ρ is the density of
the material and e are the other material parameters. Our gamut,
i.e. the set M ⊂ R n of material properties corresponding to microstructures of a given resolution, is made of a finite number of
points. However, by increasing the resolution of the microstructures
this gamut gets denser and denser so that we assume that it can be
approximated by the union of continuous n-dimensional manifolds
and can be represented using a distance field.
4.2
Discretization
Following standard finite element methodology, we discretize the
object in regular voxels and compute its deformed state when subject
to external forces fext using the well-known relation
Ku = fext ,
Most available materials for 3D printers are elastic materials. Assuming small deformations, we use linear elasticity to compute both
the mechanical behaviour of the entire object and the microstructures. In such a setting, the relation between the linear strain ϵ and
Cauchy stress σ at every material point is given by
ª
®
®
®
µ
µ
µ¬
(5)
where K is the stiffness matrix of the system, and u are the displacements at the nodes of the voxels.
Note that we use the same approach to simulate both the mechanical behaviour of the microstructures and the object macroscopic
behaviour. However, we work at two different scales. To simulate
the microstructures, we assume that each of its voxels is made of an
homogeneous base material, whereas for determining the large scale
behaviour of the object, we assume that each of its cells corresponds
to a microstructure. The properties of the individual microstructures
are determined from 6 harmonic displacements (or 2 displacements
in 2D) using numerical coarsening as described by Kharevych et al.
[2009].
We solve the static equilibrium Equation 5 using a fast multigrid
solver based on the implementation by Dick et al. [2011].
Two-Scale Topology Optimization with Microstructures •
84:5
Fig. 3. Level set gamuts for two dimensional cubic microstructures (top row) and three dimensional cubic microstructures (bottom row). The first column
shows the projection of the sample points in the space parametrized by the density ρ, the normalized Young’s modulus Ê and the Poisson’s ratio ν . The
second to the fourth columns show three slices of the four dimensional level sets corresponding to different values for the shear modulus G.
Continuous Optimization
Relative Young’s
modulus
Relative Young’s
modulus
𝐩
𝛻𝜑(𝐩)
Relative Young’s
modulus
Rigid
material
Rigid
material
Soft
material
Soft
material
Poisson’s ratio
Poisson’s ratio
Poisson’s ratio
Poisson’s ratio
Discrete Sampling
Relative Young’s
modulus
Fig. 4. One cycle of computing the microstructure gamut. Given a set of samples, we compute a signed distance function approximating the material gamut
(left) and randomly perturb microstructures lying near the boundary to provide new seeds to the continuous algorithm (middle left). We then update the
distance field and use the gradient of the signed distance function at the the boundary to define new target material points (middle right). These target
material points are used in a continuous optimization that generates new samples (right).
5
MATERIAL SPACE EXPLORATION
The first step in our pipeline is to determine the space of physical
properties that can be achieved when combining the base materials
into microstructures of a predefined size.
Computing the mechanical properties of microstructures, when
arranged in periodic tilings, can be performed by probing the structure using a physical simulation. This approach, based on the homogenization theory, is a common practice and has been widely
used in the past [Allaire 2012; Panetta et al. 2015; Schumacher et al.
2015]. However, while inferring the homogenized properties of individual microstructures is not particularly challenging, analyzing
the space covered by all combinations of base materials is much
more difficult due to the combinatorial explosion in the number of
possible material arrangements. As an example, 16 × 16 × 16 lattices
made of only two materials corresponds to 24096 microstructures:
exhaustively probing all microstructures is clearly an impossible
task. To address this issue, two possible avenues can be pursued:(i)
we can try to sample the space of the microstructures, (ii) we can
rely on the continuity between material parameters of the individual
voxels and macroscopic properties of the microstructures in order to
generate new microstructures with desired properties. This second
option is effective in reaching locally optimal values in the material
property space. However, the function that maps the material assignment to material properties is nonlinear. In particular, very different
microstructures can correspond to the same point in the material
property space. Additionally, since the ratio of materials in each
84:6 • B. Zhu et al.
cell is bounded between zero and one, the continuous optimization
converges slowly or stops moving when material distributions in
many cells are at the lower or upper bound. Being able to jump out
of a local optimum and discovering different variants is important
in order to provide new exploration regions. We leverage these two
approaches by combining them in a scheme that alternates between
a stochastic search and a continuous optimization. We provide the
technical details in the rest of this section.
5.1
Discrete Sampling of Microstructures
We aim at sampling the space of material assignments, i.e. microstructures, in such a way that we maximize the number of samples corresponding to microstructures whose material properties
lie in the vicinity of the material gamut boundaries. We do not
draw all samples at once but progressively enrich the database of
microstructures as we refine our estimation of the material gamut
boundaries. This sampling strategy is motivated by the observation
that a small change in the material assignment of a microstructure
generally – but not always – translates to a small change of its
material properties. By modifying microstructures located near the
current boundaries of the material property gamut, we are likely to
generate more structures in this area, some of which will lie outside
of the current gamut.
Given a population of microstructures to evolve, we generate
new samples from each microstructure by changing its material at
random voxel locations. To rationalize computational resources,
we want to avoid revisiting the same voxel twice. But we do not
want to privilege any particular order either. Ritchie et al. [2015]
recently presented a Stochastically-Ordered Sequential Monte Carlo
(SOSMC) method that provides a suitable approach. In SOSMC, a
population of particles (here, our microstructures) corresponding to
instances of a procedural program (here, the sequential assignment
of materials to the voxels of the microstructures) are evolved so as to
represent a desired distribution. During this process, the programs
are executed in a random order and particles are regularly scored
and reallocated in regions of high probability. In our particular
settings, we use the scoring function
s(pi ) =
Φ(pi )
1
×
,
D(pi ) D(pi )
(6)
where Φ(pi ) is the signed distance of the material properties of
particle i to the gamut boundary (see Section 5.3) and D(pi ) is the
local sampling density at the location pi . We define the sample
density as
Õ
D(pi ) =
ϕ k (pi ) ,
(7)
k
| |p−pk | |22
h2
4
where ϕ k (p) = 1 −
are locally-supported kernel functions that vanish beyond their support radius h, set to a tenth of
the size of the lattice used for the continuous representation of the
material gamut (see Section 5.3).
The first term in Equation 15 favors microstructures located near
the gamut boundary. The normalization by D allows us to be less
sensitive to the local microstructures density and to hit any location
corresponding to the same level-set value with a more uniform
Algorithm 1 Procedure for generating new microstructures
procedure genMicrostructure(input: microstructure Mi , output: microstructure Mo )
Mo ← Mi
while some voxels of Mo have not been visited do
while microstructure Mo is unchanged do
pick a random voxel v of Mo that has not been visited
assign a randomly chosen material to v
if Mo is manifold and Mo , Mi then
accept the change
end if
end while
end while
end procedure
probability. The second product is used to additionally privilege
under-sampled areas.
Particles are resampled using systematic resampling scheme [Douc
2005] that is also used to initiate the population of particles. These
particles are then evolved according to Algorithm 3. Additional
details regarding the implementation of our algorithm are provided
in the supplementary material.
5.2
Continuous Optimization of Microstructures
The goal of the continuous optimization is to refine the geometry of
the microstructures located at the boundary of the gamut in order
to further expand the gamut along the normal directions. We start
continuous sampling by selecting a subset of microstructures lying
on the boundary of the gamut as starting points for the continuous
optimization. The discrete structures are mapped to continuous
values close to 0.5. We used 0.5 ± 0.3 in our experiments. Doing so
allows the topology optimization algorithms to move freely in the
first steps and discover new structures.
For each starting structure, we identify target material parameters
using the gradient of the level set Φ at the initial discrete sample
point p (see Section 5.3) defined by q = p + ∇Φ(p). We translate
this target material parameters into an elasticity tensor C0 and
density ρ 0 . Here ρ is the ratio of the two base materials in the
microstructure.
Note that our problem formulation does not restrict us to a particular topology optimization algorithm or material distribution
parametrization. We have experimented with two objective functions that worked equally well for our purposes. The first objective
uses an energy based formulation [Xia and Breitkopf 2015a] to compute and optimize the elasticity tensor directly. At a high level, the
optimization problem is
Õ
arg min f (x) = (C(x) − C0 )2 + w ρ (ρ − ρ 0 )2 , ρ =
x i , (8)
x
i
where x is the ratio of materials in each cell, and w ρ controls the
weighting between the displacement term and the density term. The
authors of the method developed parameter heuristics to optimize
for difficult cases such as negative Poisson’s ratio structures. We
naturally arrive at structures with negative Poisson’s ratio without
Two-Scale Topology Optimization with Microstructures •
the parameter varying step in [Xia and Breitkopf 2015a] since our
discrete samples allow us to explore a wide variety of initial designs.
The second objective is formulated using harmonic displacements [Kharevych et al. 2009; Schumacher et al. 2015] G instead of
the elasticity tensor directly. G is a 6 × 6 symmetric matrix where
each row corresponds to a strain in vector form. We use the target
elasticity tensor C0 to compute the target harmonic displacements
matrix G0 and minimize the objective function:
f (x) = (G(x) − G0 )2 + w ρ (ρ − ρ 0 )2 .
(9)
This objective matches soft structures more accurately since entries
of G are inversely proportional to material stiffness.
Following the work by Andreassen et al. [2014], we use the
method of moving asymptotes (MMA) [Svanberg 1987]) to optimize
the objectives using an implementation provided in the NLOPT
package [Johnson 2014]. We run at most 50 iterations since it usually
converges to a solution within 20-30 steps. MMA makes large jumps
during the optimization while keeping track of the current best
solution, thus causing the oscillation of the objective value. To force
continuous material ratios towards discrete values, we experimented
with the SIMP model with the exponent set to 3 and the HashinShtrikman bound for isotropic materials described by Bendsøe and
Sigmund [1999].
Either interpolation allows us to threshold the final continuous
distribution and obtain a similar discrete sample. We tolerate small
deviations introduced by the thresholding since our goal is to obtain
a microstructure lying outside of the gamut rather than reaching a
particular target. In practice, we observed that the material properties of the final discrete structures often did not change significantly
after the thresholding step.
6
TOPOLOGY OPTIMIZATION
A classic topology optimization problem consists of optimizing the
shape and structure of a given object defined by a prescribed domain
in order to minimize some cost function. For example, the standard
topology optimization minimizes the compliance of the object while
satisfying the static equilibrium and the total weight constraint.
Since the topology of the object is unknown a priori, a method
of choice is to define the shape of the object through its material
distribution and to locally work with material densities. To this end,
the design layout is voxelized and a density variable is assigned to
every cell of the discretized domain. By penalizing intermediate
values for these densities, a binary distribution corresponding to
the object’s final layout can be eventually obtained.
In this work, we extend the traditional topology optimization
algorithm in multiple ways. First, we do not compute a binary material distribution at the cell level as commonly done. Instead, we
leverage our database of microstructures and ask for each cell to be
filled with one of the microstructures. By doing so we change the
topology of the object at a finer scale, i.e. within each cell. This is
done by working with the macro-scale material properties of the
microstructures instead of their geometry directly. The second difference is that our algorithm can be used with parametrizations of
the material property space that are more complex than the single
density parameter per cell that is commonly used in the standard
topology optimization algorithm. Indeed, in our generalized topology optimization problem, each cell c i contains an n-dimensional
material parameter pi ∈ R n . We use p to denote the stacked vector
of material parameters in all cells. Given a signed distance function Φ(pi ) that defines the gamut, our new topology optimization
problem is then written as
min :
p
5.3
Continuous Representation of the Material Gamut
We represent the gamut of material properties using a signed distance field that is computed from the material points associated to
the sampled microstructures. First, we normalize each coordinate
pi of p to constrain the scope of the level set to an n-dimensional
unit cube. Then we compute the level set values on the cell centers of an n-dimensional Cartesian grid that encloses this unit cube.
We draw inspiration from the methods for surface reconstruction
used in particle fluid rendering [Ando et al. 2013; Bhatacharya et al.
2011; Zhu and Bridson 2005] and extend it to n dimensions. In this
case, a signed distance field is generated from a set of points by
evaluating an implicit distance function Φ at each point p ∈ M.
We initialize the signed distance field using the implicit function
Φ(p) = ||p − p̄|| − r from [Zhu and Bridson 2005] where || · || is the
Euclidean distance between two points in M, and p̄ is the average
position of the neighboring points of p within a range of 2r . Note
that the signed distance is initially defined only near the boundary
of the gamut. In order to sample the distance on the entire domain,
we propagate the 0-level set surface using the fast marching algorithm and solve an explicit mean curvature flow problem defined as
∂Φ/∂t = ∆Φ [Osher and Fedkiw 2006] .
Having a continuous representation of the gamut of materials
achievable by the microstructures, we can now reformulate the
topology optimization problem directly in the material space.
84:7
s.t . :
:
S(p, u)
F (p, u) = 0
Φ(pi ) ≤ 0,
(10)
1 ≤ i ≤ Nc
where S is a real-valued objective function that depends on the
material parameters and the displacement vector u of the entire
object at the elasticity equilibrium. The equality constraint F = 0
requires u to satisfy the elasticity equilibrium and the inequality
constraint Φ ≤ 0 guarantees that the material properties of each
cell stay inside the precomputed gamut.
In our examples, the material parameter p consists of the density
ρ and the elasticity parameters e. We split our objective function
into an elasticity term C(e, u) that controls the deformation behavior
(see Section 6.1) and an optional density term V(ρ) that controls
the overall mass of the object.The density term can be written as
V(ρ) = (
Nc
Õ
ρ i Vi − M̂)2 ,
(11)
i=1
where Vi is the cell volume and M̂ is the target overall mass. When
one of the base material is void, the use of the density term allows
to modify the topology of the object at a larger scale than the one
of the microstructures, and thus to change the external shape of
the object. In fact, even for multi-material designs involving base
materials with similar mass densities, we noted that we could use
the density term to encourage the presence of soft material in the
84:8 • B. Zhu et al.
structure. By removing the external cells entirely made of the soft
material, we could then decrease the mass of the structure without
significantly changing its mechanical behaviour. Alternatively, the
density term can also be used to control other quantities related to
the ratios of the different materials such as the cost of the object.
For specific problems, we can also add spatially-varying weight
control terms to Equation 11. For example, we can control the target
weight of each individual cell by adding a local term (ρ i − ρ̂ i )2Vi .
Assuming static equilibrium, the elasticity constraint is written
as
F (e, u) = K(e)u − fext = 0,
(12)
where fext are the external loads applied to the object.
The gamut constraint for a point pi in the material property space
is described by an n-dimensional level set function Φ(p). We have
Φ(pi ) < 0 for a point inside the gamut, Φ(pi ) > 0 for a point outside
the gamut, and Φ(pi ) = 0 for a point on the boundary of the gamut.
The value of Φ represents the n-dimension Euclidean distance to
the level set boundary. The gradient of Φ are evaluated by a finite
difference operation on the signed distance field.
We used a standard gradient-based numerical optimizer (Ipopt
[Wächter and Biegler 2006] in our implementation) to solve Equation 10. We enforced the elasticity equilibrium constraint using
the adjoint method. The optimizer only needs to take the function
values of S and Φ along with their gradients as input.
6.1
Elasticity Objectives
We used two different types of objective functions for the elasticity
term in our topology optimization algorithm. These two types of
objectives allowed us to design a wide range of objects.
Target Deformation. Our algorithm takes a vector of nodal target displacements and boundary conditions (external forces, fixed
points, etc.) as input. Then, it automatically optimizes the material
distribution over the object domain to achieve the desired linear
deformation assuming a linear elastic behavior.
We define the deformation objective as
Cd (e, u) = (u − û)T D(u − û),
(13)
where û is the vector of the target displacements, D is a diagonal
matrix that determines the importance of each nodal displacement.
We use D to define the subset of nodes that we are interested in.
For example, we can set most entries of D to zero and focus on a
portion of the domain (see Figure 12).
Minimum Compliance. We have experimented with the same
objective as the one used in the standard topology optimization
algorithm where the compliance Cc is defined as
Cc (e, u) = uT K(e)u.
(14)
In the commonly used SIMP algorithm, the stiffness matrix Ki of
each cell i depends on the artificial density value ρ i through an
analytical formula such as Ki = ρ i3 K0 where K0 corresponds to the
stiffness matrix of the base material. In contrast, the stiffness matrix
in our objective function is directly computed from the material
parameters of the material space and forced to correspond to a
realizable material thanks to our gamut constraints.
Like in standard algorithms, we regularized the problem to avoid
checkerboard solutions by applying a smoothing kernel on the
material properties that favors smooth variations of the material
parameters over the object layout. Our optimizer supports multiple
objectives by linearly combining weighted objective functions.
7
MAPPING MATERIAL PROPERTIES TO
MICROSTRUCTURES
After running the topology optimization
algorithm, we generate a printable result
by replacing each cell in the object lattice
by a microstructure whose material properties match the optimal ones.
Material properties of the microstructures are computed using the homogenization theory which is more accurate with
a smooth transition between the geometries of neighboring cells. While smoothness in the material parameters can be easily enforced, it does not imply topological
similarity of nearby microstructures. For
example, any translation of a given microstructure in a periodic tiling will result in a microstructure geometrically different but with exactly the same mechanical properties.
Fortunately, our database is very dense and multiple microstructures
generally map to similar points in the material property space, offering several variants. To further increase the number of possibilities,
we also incorporate an additional exemplar for each microstructure by translating it by half its size, which preserves its cubic or
orthotropic symmetry without changing its properties (see inset
Figure). We then run a simple but effective algorithm that picks
the microstructure exemplars that minimize the boundary material mismatch across adjacent cells. We quantify this mismatch by
ÍNc
I = i=1
Ii , where Ii is the contribution associated to the cell i and
corresponds to the number of boundary voxels filled with materials
that are different from the ones of the voxels’ immediate neighbours
across the interfaces.
Our algorithm proceeds as follows:
• For each cell, we define a list of possible candidates by
picking all the microstructures mapping to material points
lying in the vicinity of the optimal material point and we
randomly initialize the cell with one of the candidates.
• We compute the mismatch energy Ii associated to each cell
i and sort the cells according to their energy.
• We pick the first cell in the sorted list, i.e. the one with the
highest energy and assign to it the microstructure candidate
that decreases the energy the most. If we cannot decrease
the cell energy, we move to the next cell in the list.
• We update the mismatch energies of all the impacted cells
and we update the priority list.
• We repeat the last two steps until the mismatch energy I
cannot be decreased anymore.
Two-Scale Topology Optimization with Microstructures •
84:9
Fig. 5. Gamuts computed with our discrete-continuous sampling scheme for 2D cubic structures (left) , 2D orthotropic structures (second from left), 3D cubic
structures (second from right) and 3D cubic structures with 0.35 as Poisson’s ratio (right). The plots show the results for the projection of the gamuts on
the plane defined by the macroscale Young’s modulus along the x axis (normalized by the Young’s modulus of the stiffest base material) and the Poisson’s
ratio corresponding to a contraction along the y-direction when the material is stretched along the x-direction. The blue dots correspond to the generated
samples,the orange dots correspond to the microstructures from Schumacher et al. [2015] and the yellow dots correspond to the microstructures from Panetta
et al. [2015].
Fig. 6. Gamut corresponding to 2D cubic microstructures made of two
materials and void. The Young’s modulus of the microstructures is plotted
using a logarithmic scale. We show above some examples of microstructures
lying near the estimated boundary of the gamut, i.e. with extreme material
properties. The dark color corresponds to the softer material, while the light
grey color is used for the stiffer material.
8
RESULTS
We first analyzed our microstructure sampling algorithm for 2D
and 3D microstructure gamuts. Then we used these precomputed
gamuts and we designed and optimized a wide variety of objects
with our topology optimization algorithm.
8.1
Microstructure Sampling
We evaluated our method on two- and three-dimensional microstructures made of one or two materials. For the 2D case we considered patterns with cubic and orthotropic mechanical behaviors that
can be described with 4 parameters (3 elasticity parameters and
density) and 5 parameters (4 elasticity parameters and density) respectively. In 3D we computed the gamut corresponding to cubic
Fig. 7. Resulting material property distributions when running our topology optimization algorithm in the orthotropic (top left), cubic (top right),
isotropic (middle left) and an analytically defined gamut E ≥ ρ 3 E 0 (middle
right), with the material property space dimensions ranging from five to
two. We compare our algorithm with the standard SIMP method with
power index p = 1 (bottom left) and p = 3 (bottom right). For these figures,
we computed the color of each cell by mapping every base vector of the
normalized parameter space to a color range and linearly interpolating the
colors associated to each of the parameters. In this example, the left side of
the cantilever is fixed while a discretized, linearly varying, distributed force
is applied to the bottom side (see red arrows in the top left picture).
structures with 4 parameters. In all cases, the size of the lattice
for the microstructures was set to 16 in every dimension. We used
isotropic base materials whose Young’s modulus differed by a factor
of 1000 and having 0.48 as Poisson’s ratio. We initially computed
the databases for two-material microstructures, but also adapted
these databases for microstructures made of a void and a stiff material. In the later case, we replaced the softer material by void,
filter out all the microstructures with disconnected components
84:10 •
B. Zhu et al.
30
analytic
cubic
orthotropic
15
Elastic energy
Objective energy
×106
10
5
0
0
20
40
20
15
10
5
60
analytic
cubic
orthotropic
SIMP-3
SIMP-1
25
0
20
# iterations
×108
32
64
128
256
2
Elastic energy
Objective energy
60
20
3
1
32
64
128
256
18
16
14
12
10
0
0
20
40
60
8
80
0
# iterations
×10
40
60
80
40
8
point 1
point 2
point 3
point 4
6
4
2
0
20
# iterations
7
Elastic energy
Objective energy
40
# iterations
point 1
point 2
point 3
point 4
30
20
10
0
20
40
# iterations
60
0
20
40
60
# iterations
Fig. 8. Convergence tests. Variation of the objective energy (left) and the
elastic energy right of a beam being optimized for minimum compliance as
the optimization progresses. The convergence plots correspond to the beam
of Figure 7 when optimized using different material spaces (top), different
resolutions for the beam lattice (middle) when using cubic microstructures,
and different initial material properties for the cubic microstructures (bottom).
and, in the 3D case, filled the enclosed voids and recomputed the
homogenized properties. We provide a comparison between the
initial and postprocessed databases in the supplementary material.
The resulting postprocessed gamuts are also depicted in Figure 5.
Our databases contain 274k, 388k and 88k 2D cubic, 2D orthotropic
and 3D cubic microstructures respectively and took from 15 hours
to 93 hours to compute, which correspond to 68, 19 and 5 sampling
cycles, respectively.
We first compared our results to the ones obtained by Schumacher
et al. [2015] and observed a significant increase in the coverage of
the material space, even for 2D microstructures where we used a
coarser discretization. This comforts us with the idea that topology
optimization only, while helpful to locally improve the microstructure geometries, is suboptimal when one aims to discover the entire
gamut of physical properties. The diversity of the microstructures
that we obtained is also much richer, thus providing a larger set
of options for the practical use of microstructures. Note that they
employed some regularization to avoid thin features. For 163 microstructures, we found regularization unnecessary since they are
manifold and have a minimal feature size of 1/16 of the lattice size,
which is the same order of magnitude as the thinnest parts of Schumacher’s microstructures. For completeness, we also compared our
Fig. 9. Optimizing a beam to make it bend when it is squeezed. A beam
with optimized material properties can take the desired ffSff shape (right)
whereas a beam with homogeneous material properties can only axially
deform under small deformation (left). In this example, the vertices on the
vertical sides are fixed in their target positions, while the other vertices are
free to move. Target displacements are set on the nodes of the cells of the
two horizontal boundary layers. The color plot for the bottom beams shows
the deformation error of each cell as defined by Equation 13.
database of 3D microstructures to the one of Panetta et al. [2015]
at 163 and 643 grid resolutions (Figure 5, right). Our initial database was computed with 0.48 as Poisson’s ratio and is shown in
the supplementary material. For this comparison, we then recomputed the material properties of the microstructures using the same
Poisson’s ratio as Panetta’s, i.e 0.35, which affects the extremal values of the obtained gamut. For the 643 microstructures, we used
morphological operations in the discrete step and sensitivity filtering with a radius of 3 voxels in the continuous step to limit the
minimum feature size to 1/32 of the lattice size [Sigmund 2007].
Note that this comparison is provided for reference only since our
microstructures are cubic while Panetta’s are isotropic (a subset of
cubic). Furthermore, they target a different 3D printing technology
with self-supporting constraints not imposed here. Finally, we also
obtained a dense sampling in the interior of the space, as a result of
the randomness inherent to our approach. This reduces the need of
running costly optimization in these areas and occurs even if we do
not explicitly enforce any sampling there.
We also experimented with three-material 2D cubic microstructures (two solid materials with Young’s moduli differing by a factor
of 1000 and with 0.48 as Poisson’s ratio, plus a void material). The
resulting database contains about 800k microstructures that can potentially be printed. The corresponding gamut and some examples
of the generated microstructures are shown in Figure 6.
Two-Scale Topology Optimization with Microstructures • 84:11
8.2
Topology Optimization
We tested our topology optimization algorithm on a number of
simple test cases and large scale examples. Detailed analysis and
discussion of the results is provided below.
Impact of the Material Space. We evaluated the impact of the
chosen material space on a 2D cantilever beam with optimized minimum compliance. We tested our topology optimization algorithm
on isotropic, cubic and orthotropic gamuts as well as with the virtual
materials used in the traditional SIMP approach and for which the
stiffness of the material E = ρ p E 0 , p ≥ 1 is a function of the density
ρ of the cell and the stiffness of the base material E 0 . We also tested
our algorithm on an analytical gamut with allowed stiffnesses E
defined by E ≤ ρ 3 E 0 . The results are shown in Figure 7. It can be
noted that, as the dimension of the material space increases, the final
energy of the system decreases. This is to be expected since higher
dimensional space means larger gamuts. Thus, when using cubic
materials, the minimum compliance objective function reaches 3%
lower energy than the standard SIMP method with power index 3.
This difference reaches 11% when we use orthotropic materials. It
is worth noting that the lowest elastic energy is achieved when we
use the traditional SIMP method with p = 1 (as shown in Figure 8).
However, this solution does not correspond to a realizable structure
since some of the optimized materials do not correspond to any
microstructure.
Matching Quality. We evaluated for different examples the matching quality of the target deformation optimization.
For the first test, we forced a beam to take an ‘S’ shape when
undergoing tensile forces (Figure 9). In order to avoid overfitting, we
applied target displacements on the vertices of the boundary cells
only. As depicted in the figure, the use of microstructures largely
improves the global shape of the beam, which closely matches the
target deformed shape. This becomes even more striking when compared to the behavior of a beam made of a homogeneous material.
We also validated our algorithm by designing a soft ray whose
wings can flap using a compliant mechanism (see Figure 10 and
accompanying video). Boundary conditions are applied on two circular areas located along the spine of the ray. Each disk has one
degree of freedom for deformation, namely contracting or expanding along the disk normals. This mechanism resembles the one of
many hydraulics-driven soft robots. We define two target deformation objectives corresponding to the flapping of the wings up and
down, when alternatively contracting and expanding the two disks’
boundaries. By running our multi-objective topology optimization
framework, we can compute an optimized material design that can
achieve both deformation modes when the corresponding boundary
conditions are exerted.
Convergence and Robustness. We evaluated the convergence rate
of our topology optimization both on the minimum compliance
problem and with the target deformation objective. For the minimum compliance problem, we used the same loading as the one of
Figure 7. The corresponding results are shown in Figure 8 where
we plot both the deformation energy of the structure as defined in
Equation 14 and the original objective of the problem 10 that also
Fig. 10. Designing a soft ray. The wings of the ray are asked to flap up
and down when vertices on its spine contract and expand. Constrained
vertices are colored in green. The deformations achieved with the optimized
materials are displayed on the bottom row.
includes the volume term defined by Equation 11. For all these examples, the algorithm converged after a couple of dozen iterations,
irrespectively of the lattice resolution, i.e. the number of variables
and the number of non-linear constraints. This demonstrates the
scalability of the our algorithm. We also tested the robustness of our
algorithm by starting with different initial conditions. In this case,
we initialized the material parameters of each cell with a random
material point projected onto the boundary of the gamut. Similar
to other topology optimization schemes, we have no guarantee that
we reach the global minimum of the function, and indeed, our algorithm sometimes converges to different solutions. However we note
that these different solutions have a similar final objective value and
are therefore equally good.
For the evaluation of the target deformation optimization, we
ran our topology optimization algorithm on scenarios similar to the
ones from Panetta et al. [2015] where different extruded structures
are asked to deform into prescribed shapes when being compressed
between two plates (see Figure 11). As shown in the figure, our
optimization algorithm successfully converges to the specified deformation behavior in less than 100 iterations for all the examples.
We also tested the convergence rate when optimizing for functional
mechanisms. To this end, we designed several grippers that can
grasp objects by moving their tips when external forces are applied
to their extremities. We experimented with four sets of boundary
conditions, namely, pulling and pushing the back of the gripper
horizontally, and compressing and stretching the extremities of the
gripper vertically. As shown in Figure 12, these different settings
lead to different material structures. The deformation errors of all
the four designs converge to a low level after a couple of hundreds
of iterations.
Accuracy. We evaluated the accuracy of our algorithm on several
optimized structures by comparing the deformation obtained when
using the optimized homogenized material properties for each cell
84:12 •
B. Zhu et al.
arm
face
wave
5
0
-5
0
0.08
6
push back
pull back
push side
pull side
5
4
3
2
100
150
0.06
0.05
0.04
0.03
0.02
1
50
push back
pull back
push side
pull side
0.07
deformation energy (Linf)
10
deformation energy (L2 norm)
objective energy (log)
15
0.01
200
#iteration
Fig. 11. Target deformation examples. The pictures on the left (a) show
the deformed optimized result when the boundary conditions illustrated
by the pictures (b) are applied. The orange meshes (c) correspond to the
simulated deformations when a homogeneous material is used. The blueto-red meshes indicate the relative deformation error of the unoptimized
(d) and optimized (e) structures. Convergence rates corresponding to the
optimization of these three examples are reported on the bottom figure.
to the one obtained by a high resolution simulation in which every
cell is replaced by its mapped microstructure.
We first evaluated the accuracy on a deformable bar, one side of
which was rigidly attached while the other was subject to different
sets of external conditions (see Figure 13). We used a 8 × 2 × 2 lattice
to represent the bar with homogenized cells, which translates into
a 128 × 32 × 32 grid for the full resolution mesh. Similar stretching,
bending and shearing behaviors were obtained for both sets of
models. From a quantitative point of view, the differences amount to
5-10% in terms of average vertex displacement and 9%-33% in terms
of elastic deformation energy (see Table 1). We further evaluated
0
0
0
50
100
150
200
#iteration
250
300
350
0
50
100
150
200
250
300
350
#iteration
Fig. 12. Designing functional grippers. The top row shows the initial shape
of the gripper (left), and the target deformation for the tip (right). The green
dots correspond to the fixed vertices while the blue arrows correspond to
the target displacements. The two middle rows correspond to the optimized
results obtained for the specified boundary conditions. The inset pictures
color-code the initial and final deformation error for the different examples.The convergence plots in the bottom row depict the change in the sum
of the deformation errors corresponding to all the cells (left) and the value
of the maximal cell error contribution (right) as the optimization progresses.
the effects of material patterns by running a similar comparison on
a cube made of periodic layers of similar microstructures and with
random assignments of microstructures (Figure 14). As reported
in Table 1, we show that the ratio between the magnitudes of the
average vertex displacement differences is between 4% and 7%, and
the elastic energy difference is between 10% and 19%.
Finally, we also compared the behaviors of one of the grippers
(Figure 15). The original optimized gripper is made of 3k elements
Two-Scale Topology Optimization with Microstructures • 84:13
Fig. 16. Simulation of a cube made of a periodic arrangement of a single microstructure at different resolutions. Inset pictures correspond to the model
with homogenized material properties, while main pictures correspond to
the full resolution simulations.
Table 1. Error statistics (SI units). The size of one microstructure is set to
1 × 1 × 1.
Fig. 13. Comparison of simulated beams with homogenized material properties (inset pictures) to the ones using full microstructures (large pictures).
Fig. 14. Comparison of simulated cubes with different material patterns
modeled by homogenized cells (inset pictures) and full resolution microstructures (large pictures).
Fig. 15. Comparison of a simulated gripper with homogenized material
properties (inset picture, left) to the one using full microstructures (main
picture, left). The figures on the right show the vector field of vertex displacement of the two models. The blue-to-red colors represent the magnitudes
of the displacements.
while the high resolution version is made of 4M voxels. Overall,
the two models exhibit similar global deformation behaviors, in
particular in the tip area. Some differences can be observed on the
left side of the gripper for which the high-resolution model exhibits
a lower effective material stiffness than its homogenized counterpart.
With the same displacement boundary conditions applied, the highresolution model deforms about 25% more than the homogenized
model.
Example
Mean
displacement
Beam 1
Beam 2
Beam 3
Beam 4
Cube 1
Cube 2
Cube 3
Gripper
Cube 4
Cube 5
Cube 6
6.47×10−3
6.47×10−3
5.07×10−3
8.78×10−3
3.62×10−3
4.35×10−3
5.42×10−3
1.32×10−2
5.22×10−3
5.21×10−3
5.21×10−3
Mean
displacement
difference
6.04×10−4
6.04×10−4
4.97×10−4
4.45×10−4
2.86×10−4
1.94×10−4
4.22×10−4
6.90×10−3
4.89×10−4
2.17×10−4
1.32×10−4
Elastic energy
homogenized/full resolution
6.85×10−5
1.63×10−5
2.38×10−4
3.33×10−4
3.64×10−3
6.82×10−3
7.81×10−3
8.67×10−3
2.08×10−2
1.89×10−2
1.82×10−2
6.17×10−5
1.08×10−5
2.07×10−4
2.30×10−4
3.20×10−3
5.94×10−3
6.32×10−3
5.70×10−3
1.63×10−2
1.63×10−2
1.63×10−2
The differences observed between the homogenized model behaviour and the full resolution simulation can be explained by two
major factors: (i) numeral stiffness when using larger elements
which tends to make the homogenized mesh slightly stiffer in particular when bending deformation arises, (ii) violation of the periodicity assumption when replacing each cell by a single microstructure. This issue can be reduced by replacing each cell by a tiling of
microstructures. This was verified on a cube made of a periodic arrangement of a single microstructure (see Figure 16). And indeed, as
we increase the resolution of the simulation grid, the error between
the homogenized model and the full resolution version decreases
and converges to similar values.
Orthotropic materials. We tested the behavior of our algorithm
in a 5-dimensional space by using the gamut of 2D orthotropic
microstructures depicted in Figure 5 (middle). To this end, we used
a regular lattice whose vertices on the left side where fixed and
we applied parallel forces on the vertices of the opposite side. The
goal in the test was to minimize the compliance of the structure.
As can be seen in Figure 17 and in the accompanying video, we
experimented with different force directions. Unsurprisingly, when
a single cell is considered, the microstructure that we obtain has a
structure that is aligned with the direction of the forces (see Figure
17, left). For a higher resolution lattice this is no longer true and
the resulting overall structure becomes less intuitive (see Figure 17,
right). Note that the resulting material distribution varies smoothly.
By considering various alternative for each material point, our tiling
algorithm is able to map the material properties to microstructures
which are well connected.
84:14 •
B. Zhu et al.
Fig. 17. Optimizing the orthotropic material parameters of a single cell (left) and a 32 × 32 lattice of cells (right) subject to directional forces. The vertices on
the left side of the layout are fixed while forces are applied on the right vertices as depicted by the red arrows. Our simple but effective tiling algorithm allows
to nicely transition between microstructures of smoothly material properties (right, top).
8.3
3D-Printed Designs
Leveraging our two-scale approach, we used our topology optimization algorithm to generate a wide variety of high resolution models
that we 3D-printed. We used a Stratasys Objet Connex 500 and the
two base materials Vero Clear and Tango Black Plus and used the
database containing the three-dimensional cubic microstructures.
The sizes and computation times of the resulting models are outlined
in Table 2.
Since Ipopt performs a line search at each gradient step, one single
step may correspond to multiple simulations. We show the average
time required for taking a step in the last column of Table 2. For these
large scale examples, Ipopt takes two hundred iterations in average
to find a local minimum. Since our problem is formulated as a very
general constrained continuous optimization, it is independent of
the optimization package that is used and its speed could potentially
be further improved by using alternative minimizers. We found
Ipopt to be a good choice for its capability to efficiently handle a
large number of inequality constraints, which is not the case of
other popular minimizers used in topology optimization such as the
method of moving asymptotes (MMA).
Our algorithm is mainly directed towards engineering applications and targets the design of objects undergoing small deformations. In the following examples, we sometimes intentionally exaggerated the target displacements (and scaled the external forces
accordingly) for better visualization, which does not change the
output of the algorithm with a linear material model.
Beams with controlled deformation behaviour. We started by designing a 3D hollowed beam with a desired deformed shape. The
beam was stretched by moving vertices on two opposite sides. Our
topology optimization algorithm was run using a target deformation objective. The resulting optimized material properties and the
3D-printed structure are depicted in Figure 18.
Multi-Objective Flexure Design. We tested our algorithm on a
multi-target deformation setting by optimizing the structure of
Table 2. Statistics on the 3D-printed models. The last row uses the database
of 643 microstructures.
Example
Grid Size
# Voxels
Beam
Flexure
Gripper
Bunny
Bridge
Bridge 2
96×24×4
32×32×16
64×32×8
32×32×32
128×64×32
320×160×80
38M
67M
67M
134M
1074M
1074G
Time
per FEM
Solve [s]
0.7
1
1.7
0.6
27
1.3k
Time per
Step [s]
5
12
10
4
81
-
Fig. 18. An optimized hollow beam with target deformation. The left figure
shows the target deformation and optimized material distribution. The
right figure shows the 3D-printed structure and the achieved deformation.
a flexure mount with two different target shapes (see Figure 19).
Here, our goal is to design a flexure that resists vertical loads while
remaining compliant to horizontal loads. We assume that the object
mounted on the flexure is connected to the flexure using a cylindrical
connector that transmits the forces to the flexure via the connecting
area. In the first scenario, vertical forces are applied to the points
of the cylindrical area and we ask the flexure to stay as close as
possible to its rest configuration. In the second scenario, horizontal
forces are applied to the points of the cylinder and we ask the flexure
shape to match the shape shown in the Figure 19.
Gripper. We verified the functionality of our grippers by fabricating two of them. For these results we ran the optimization on high
resolution meshes of the version that grasps the object when the
Two-Scale Topology Optimization with Microstructures • 84:15
Fig. 19. Optimizing a flexure mount. The flexure is connected to an object
thanks to a cylindrical connector. We leave space for this connector by
keeping a cylindrical area of the design layout empty of material. The
material distribution of the flexure is optimized for two sets of external
forces applied to the cylindrical area. Under vertical load, the flexure should
stay close to the rest shape while under horizontal load, the flexure should
deform according to the inset figure.
Fig. 21. Stanford bunny optimized for two loading cases. Two sets of external
forces are applied to the back and chest of the bunny as indicated by the
arrows, and the color indicates the material distribution (left). Material
parameters are mapped to microstructures to obtain an object that can be
actually printed (right).
Fig. 22. Optimizing a bridge. The initial layout corresponds to a 128x64x32
regular grid. We apply uniform loads on the upper plane deck. We compute
the material parameters and set cells with extremely low stiffness to void
(top left). We look up the microstructures and 3D print the bridge (top
right). We scale the problem to 1 trillion voxels by using a lattice of 4 million
elements where each element corresponds to a 643 microstructure (bottom
left). We show a 20 × 20 × 1 patch on the bridge with filled microstructures
and a single microstructure with 643 voxels on the patch (bottom right).
Fig. 20. 3D-printed functional grippers. By setting different target ratios
of the rigid material, different designs can be obtained. When more soft
material is used the grasping behaviour of the gripper is obtained via outof-plane bending (top), whereas more rigid material is used, the gripper
deformation remains planar (bottom).
extremities of the gripper are pressed (see Figure 20). By changing
the parameter controlling the ratio of the soft material, different
designs based on different mechanisms can be achieved. When
more soft material is used the gripper achieves its target deformation thanks to out-of-plane bending, while for stiffer designs, the
grasping motion is achieved via in-plane deformation.
Minimum Compliance Examples. To demonstrate the scalability
of our algorithm, we computed a Stanford bunny (Figure 21) made
of more than 100 million voxels and subject to two load case scenarios. We also designed two bridges of increasing resolutions.
The first bridge was optimized using a lattice of half a million cells
which corresponds to 1 billion voxels (Figure 22). For the second
bridge, we used the database of 643 microstructures and a layout
made of 4 million cells, which amounts to 1 trillion voxels. We
initialized the topology optimization by running the algorithm on a
lower resolution grid with 1.4 million elements and used the resulting parameters as initial material parameter values for the higher
resolution optimization.
9
CONCLUSION
We have presented a computational framework for two-scale topology optimization. Our approach can efficiently optimize high resolution models that can be fabricated using multi-material 3D printing.
Our first insight is to use a precomputation process to efficiently
84:16 •
B. Zhu et al.
sample the space of microstructures and their corresponding material properties in order to define a continuous material property
gamut. Our second insight is to use this gamut as a constraint in a
generalized topology optimization framework to assign spatiallyvarying material properties throughout the optimized object. Finally,
the volume with assigned material properties can be converted to
a 3D model with corresponding spatially-varying microstructures.
We demonstrated the effectiveness of our approach on multiple examples and showed improvements over traditional binary topology
optimization schemes.
Limitations and Future Work. First, while our sampling method
outperforms current approaches in terms of the material space coverage and the approach converges to stable gamuts, we do not provide
any theoretical guarantees that the gamut space cannot be further
expanded. Second, we would like to investigate microstructures
with additional properties, e.g., electrical or magnetic properties,
their combined property gamuts, and the different applications
they enable. Finally, our framework builds upon linear elasticity
and optimizes the material distribution of objects subject to small
deformations only. While this is enough for many engineering applications, extending our algorithm to the nonlinear regime would
be an interesting direction for future work.
ACKNOWLEDGEMENTS
This research is supported in part by the Defense Advanced Research
Projects Agency (DARPA) and Space and Naval Warfare Systems
Center Pacific (SSC Pacific) under Contract No 66001-15-C-4030.
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A APPENDICES
A.1 Discrete sampling of microstructures
Given a set of microstructures, a new population of microstructures
is generated using the Stochastically-Ordered Sequential Monte
Carlo (SOSMC) method introduced by Ritchie et al. [2015].
In our implementation, the samples (or particles) are our microstructures, i.e. binary assignments of the base materials, and the
desired distribution is the one that maximizes the number of particles located near or outside the boundary of the gamut of material
properties. We evaluate the contribution of each sample towards
the desired goal thanks to the scoring function
s(pi ) =
Φ(pi )
1
×
,
D(pi ) D(pi )
(15)
where Φ(pi ) is the signed distance of the material properties of
particle i to the gamut boundary and D(pi ) is the local sampling
density at the location pi . The sample density is defined as
Õ
D(pi ) =
ϕ k (pi ) ,
(16)
is often unknown a priori. To introduce randomness in the program
execution, and because of the simplicity of our procedure primitives,
i.e. swapping voxel materials, we do not rely on Stochastic Future
as in Ritchie’s implementation, but directly modify the original
program into the one described by Algorithm 3.
Starting with the microstructures corresponding to the entire
gamut, we initialize the population of microstructures to evolve
by sampling N microstructures using systematic resampling [Douc
2005] based on their scores as computed by Equation 15. We then
run, for each microstructure, the program described by Algorithm
3 in order to evolve the population. The program is not executed
entirely but paused after the microstructure is modified, i.e. after
the inner loop of the procedure has been executed, which corresponds to a so-called barrier synchronization point. When all the
programs corresponding to all the microstructures of the population have reached this synchronization point, the scores of the
partially modified microstructures are evaluated again and the population is resampled using systematic resampling. We again sample
N microstructures, but since the scores have changed, the most
interesting microstructures will appear several times, while the less
promising ones will leave the evolving population. Note that the microstructures, and the associated programs, are duplicated together
with their program execution history, i.e. the information regarding
which voxels have been already visited, so that these voxels are not
modified a second time. This ensures that interesting changes in
the material assignments are preserved. After the synchronization
point is reached and the microstructures have been resampled, the
execution of the programs is resumed and the algorithm continues
until all the voxels of all the microstructures have been visited. The
entire SOSMC algorithm is summarized by Algorithm 4. In our
implementation, we used N = 3000 microstructures for the 2D and
3D cubic databases, and N = 10000 for the 2D orthotropic database.
A.2
Material Gamuts
We initially targeted multi-material printers and therefore computed
databases of two- and three-dimensional microstructures made of
two materials. We used isotropic base materials whose Young’s
modulus differed by a factor of 1000 and having 0.48 as Poisson’s
ratio. For comparison purposes with previous research (24), we
adapted these databases to one-material microstructures by replacing the softer material by void, filtering out all the microstructures
with disconnected components, filling the enclosed voids in the 3D
case, and recomputing their homogenized properties. The two sets
of gamuts are depicted in Figure 23. We observed that the gamuts
corresponding to two-material microstructures and one-material
k
| |p−pk | |22 4
h2
where ϕ k (p) = 1 −
are locally-supported kernel functions that vanish beyond their support radius h, set to a tenth of
the size of the lattice used for the continuous representation of the
material gamut.
As described by Algorithm 2, given an initial microstructure, we
generate a new microstructure by randomly swapping materials
in the material assignments. However, as explained in Ritchie’s
paper, executing the procedure sequentially – in our case visiting
the voxels in a fixed order – is often suboptimal since the best order
Algorithm 2 Initial procedure for generating a new microstructure
procedure genMicrostructure(input: microstructure Mi , output: microstructure Mo )
Mo ← Mi
for all voxels do
swap material of the current voxel v with probability 0.5
end for
end procedure
84:18 •
B. Zhu et al.
Fig. 23. Gamuts computed with our discrete-continuous sampling scheme for 2D cubic structures (left) , 2D orthotropic structures (middle) and 3D cubic
structures (right). The plots show the results for the projection of the gamuts on the plane defined by the macroscale Young’s modulus along the x axis
(normalized by the Young’s modulus of the stiffest base material) and the Poisson’s ratio corresponding to a contraction along the y-direction when the
material is stretched along the x-direction. All these plots correspond to microstructures that use 0.48 as Poisson’s ratio for the base material. The blue dots
correspond to the filtered one-material microstructures while the yellow dots correspond to the original two-material microstructures.
Fig. 24. Gamuts computed with our discrete-continuous sampling scheme for 2D cubic structures (left) , 2D orthotropic structures (second from left) and 3D
cubic structures (second from right) using 0.48 as Poisson’s ratio, and 3D cubic structures with 0.35 as Poisson’s ratio (right). The plots show the results for the
projection of the gamuts on the plane defined by the macroscale Young’s modulus along the x axis (normalized by the Young’s modulus of the stiffest base
material) and the Poisson’s ratio corresponding to a contraction along the y-direction when the material is stretched along the x-direction. The blue dots
correspond to the generated samples for 16 × 16 × 16 microstructures, the purple dots correspond to generated samples for 64 × 64 × 64 microstructures, the
orange dots correspond to the microstructures from Schumacher et al. [2015] and the yellow dots correspond to the microstructures from Panetta et al. [2015].
Algorithm 3 Procedure for generating a new microstructure
procedure genMicrostructure(input: microstructure Mi , output: microstructure Mo )
Mo ← Mi
while some voxels of Mo have not been visited do
while microstructure Mo is unchanged do
pick a random voxel v of Mo that has not been visited
assign a randomly chosen material to v
if Mo is manifold and Mo , Mi then
accept the change
end if
end while
// Synchronization point
end while
end procedure
microstructures have a very similar shape, except in the area corresponding to very soft microstructures. This is to be expected since
microstructures made of soft material connecting small blocks of
stiff materials are not realizable in the one-material setting. From an
implementation point of view, it is worthwhile to note that, if the
final intent of the user is to design one-material structures, these
additional fabrication constraints can be directly accounted for during the sampling stage by preventing any change that would affect
the validity of the microstructures. This avoids the need of filtering
the database a posteriori and would improve the sampling density
in regions that might be undersampled if one filters the database
in a subsequent step. Note that we sampled the microstructures
using a non-logarithmic scale for the Young’s modulus and that the
estimated density in the soft regions is higher than what it appears
to be in Figures 23 and 24.
Our initial databases were computed for microstructures corresponding to 16 × 16 × 16 arrangements of voxels. This limits the sizes
of the thinnest features of the microstructures and therefore the softness of the softer material that can be achieved. When increasing the
lattice size to 64, the gamut of the microstructure properties expands
in this area and reaches what can be obtained when using other
parametrization methods (see Figure 24, right). Due to high computation costs (each microstructure takes 29s to simulate in average),
we limited our initial analysis of highly discretized geometries to the
study of a database comprising about 10k microstructures. Notably,
Two-Scale Topology Optimization with Microstructures • 84:19
Algorithm 4 SOSMC for discrete sampling of microstructures
procedure SOSCM(input: set of n microstructures m, output: set
of microstructures p o)
p o ←m
for i=1..n do
// Evaluate scores of all the particles mi ∈ m
w(i) ← s(mi )
end for
// Sample N particles
p ← universal samplinд(m, N , w)
for i=1..N do
start program genMicrostructure(pi , qi ) for the input
microstructure pi ∈ p
end for
while some particles qi ∈ q have unvisited voxels do
for all unterminated programs do
run the program until the synchronization point is
reached
p o ←p o ∪q
for i=1..N do
// Evaluate scores for modified microstructures
w(i) ← s(qi )
end for
// Sample particles
q ← universal samplinд(q, N , w)
end for
end while
end procedure
our search method allows us to find a wide range of single-material
structures with negative Poisson’s ratio (ν = −0.7). While lower
Poisson’s ratio is theoretically achievable, such structures contain
extremely thin joints unsuitable for manufacturing. These structures
demonstrate a variety of relationships between Young’s modulus
and shear modulus. To be more precise, we define µ iso as the shear
modulus computed from Young’s modulus E and Poisson’s ν ratio
using the relationship µ iso = E/(1 − 2 × ν ). We then compare µ iso
with the actual shear modulus µ of each structure. For structures
with ν = −0.5 ± 0.03, the ratio µ/µ iso achieves a range from 0.09 to
1.39 (Figure 25).
Fig. 25. The 643 microstructures span a wide range of relative shear modulus
even they have negative Poisson’s ratios. As an example, we plotted the
distribution of structures with Poisson’s ratio ν = −0.5 ± 0.03. Points on
the diagonal line µ = µ iso are isotropic within the linear elasticity regime.
| 5 |
QUOTIENTS OF TRIANGULATED CATEGORIES AND
EQUIVALENCES OF BUCHWEITZ, ORLOV AND AMIOT–GUO–KELLER
arXiv:1702.04475v2 [math.RT] 22 Aug 2017
OSAMU IYAMA AND DONG YANG
Abstract. We give a simple sufficient condition for a Verdier quotient T /S of a triangulated
category T by a thick subcategory S to be realized inside of T as an ideal quotient. As applications, we deduce three significant results by Buchweitz, Orlov and Amiot–Guo–Keller.
Key words: Verdier quotient, ideal quotient, torsion pair, Cohen–Macaulay module, cluster
category.
MSC 2010: 18E30, 16E35, 13C14, 14F05.
1. Introduction
Triangulated categories are ubiquitous in mathematics, appearing in various areas such as representation theory, algebraic geometry, algebraic topology and mathematical physics. A fundamental
tool to construct new triangulated categories from given ones is to take Verdier quotients, for example, derived categories are certain Verdier quotients of homotopy categories. However, Verdier
quotient categories are in general hard to understand because taking Verdier quotients could drastically change morphisms. Generally it is even hard to know whether morphisms between two
objects in a Verdier quotient form sets.
The first aim of this paper is to give a simple sufficient condition for a Verdier quotient T /S of
a triangulated category T by a thick subcategory S to be realized inside of T as an ideal quotient
Z/[P] for certain explicitly constructed full subcategories Z ⊃ P of T (Theorem 1.1). Such a
realization is very helpful in studying T /S since the morphism sets in the ideal quotient Z/[P] are
very easy to control. For example, in this case if T is Hom-finite over a field and Krull–Schmidt,
so is T /S. The second aim of this paper is to show that the following three significant results can
be regarded as special cases of our realization.
• The first one is Buchweitz’s equivalence [4, 24, 16] between the singularity category of an
Iwanaga–Gorenstein ring and the stable category of Cohen–Macaulay modules over the ring.
In fact, we recover the silting reduction introduced in [10] more generally (Corollaries 2.1, 2.3).
• The second one is Orlov’s theorem [22] relating the graded singularity category of a Z-graded
Iwanaga–Gorenstein ring and the derived category of the corresponding noncommutative projective scheme (Corollary 2.6). It gives a direct connection between projective geometry and
Cohen–Macaulay representations.
• The third one is Amiot–Guo–Keller’s equivalence [1, 5, 10] which plays an important role in
the categorification of Fomin–Zelevinsky’s cluster algebras [15]. It realizes the cluster category
as the fundamental domain in the perfect derived category of Ginzburg dg algebras (Corollary
2.12).
In fact, the third application is given in a wider setting. We introduce the notion of a relative
Serre quadruple as a certain pair of a nice triangulated category T and its thick subcategory S
with extra data (Definition 2.8). Then the corresponding AGK category is defined as the Verdier
quotient T /S (Definition 2.9). This is a wide generalization of cluster categories, and we prove
that AGK category is equivalent to the fundamental domain, a certain full subcategory of T given
explicitly (Theorem 2.10).
Date: August 23, 2017.
1
2
OSAMU IYAMA AND DONG YANG
1.1. Preliminaries. Let T be a triangulated category, and X and Y full subcategories of T . We
denote by X ∗ Y the full subcategory of T consisting of objects T ∈ T such that there exists a
triangle X → T → Y → X[1] with X ∈ X and Y ∈ Y. When HomT (X , Y) = 0 holds, we write
X ∗ Y = X ⊥ Y. For full subcategories X1 , . . . , Xn , we define X1 ∗ · · · ∗ Xn and X1 ⊥ · · · ⊥ Xn
inductively. We write
X ⊥ := {T ∈ T | HomT (X , T ) = 0} and
⊥
X := {T ∈ T | HomT (T, X ) = 0}.
When T = X ⊥ Y, X = ⊥ Y and Y = X ⊥ hold, we say that T = X ⊥ Y is a torsion pair of T [9].
If a torsion pair T = X ⊥ Y satisfies X [1] ⊂ X (respectively, X [1] ⊃ X , X [1] = X ), then we call it
a t-structure [2] (respectively, co-t-structure [23, 3], stable t-structure [18]). In the literature a tstructure (respectively, co-t-structure) usually means the pair (X , Y[1]) (respectively, (X , Y[−1])).
Our convention is more suitable for our purpose. If T = X1 ⊥ · · · ⊥ Xn for thick subcategories
X1 , . . . , Xn of T , we say that T = X1 ⊥ · · · ⊥ Xn is a weak semi-orthogonal decomposition of T
[22]. Note that it is often written as hXn , . . . , X1 i in the literature [8].
1.2. Main results. Our main result is given under the following very simple axioms.
(T0) T is a triangulated category, S is a thick subcategory of T and U = T /S.
(T1) S has a torsion pair S = X ⊥ Y.
(T2) T has torsion pairs T = X ⊥ X ⊥ = ⊥ Y ⊥ Y.
Notice that X and Y are not necessarily triangulated subcategories of T , and this is important in
our applications. In this setting, we define full subcategories of T by
Z := X ⊥ ∩ ⊥ Y[1] and P := X [1] ∩ Y.
We denote by Z/[P] the additive category with the same objects as Z and
HomZ/[P] (X, Y ) = HomT (X, Y )/[P](X, Y )
for X, Y ∈ Z, where [P](X, Y ) is the subgroup of HomT (X, Y ) consists of morphisms factoring
through objects in P.
The first main result in this paper enables us to realize the Verdier quotient U = T /S as the
ideal quotient Z/[P].
Theorem 1.1. Under the assumptions (T0), (T1) and (T2), the composition Z ⊂ T → U of natural functors induces an equivalence of additive categories Z/[P] ≃ U. In particular, the category
Z/[P] has a structure of a triangulated category.
Note that if S = X ⊥ Y is a t-structure, then P = 0. In Section 2.3, we apply Theorem 1.1 to
AGK categories. In particular, we deduce Amiot–Guo–Keller’s equivalence (Corollary 2.12) from
Theorem 1.1.
Next we consider the following special case of (T1).
(T1′ ) S = X ⊥ Y is a co-t-structure.
In this case, we have the following direct description of the triangulated structure of Z/[P] , which
is an analog of the triangulated structures introduced in [6, 9] in different settings.
Theorem 1.2. Under the assumptions (T0), (T1′ ) and (T2), we have the following.
(a) The shift functor and triangles of the triangulated category Z/[P] are the following.
• For X ∈ Z, we take a triangle
ι
X
→ PX → Xh1i → X[1]
X −−
with a (fixed) left P-approximation ιX . Then h1i gives a well-defined auto-equivalence
of Z/[P], which is the shift functor of Z/[P].
QUOTIENTS OF TRIANGULATED CATEGORIES
f
g
3
h
• For a triangle X −
→ Y −
→ Z −
→ X[1] in T with X, Y, Z ∈ Z, take the following
commutative diagram of triangles:
X
f
// Y
g
// Z
h
// X[1]
a
X
ιX
// PX
// Xh1i
// X[1]
f
The triangles in Z/[P] are the diagrams which is isomorphic to the image of X −
→
g
a
Y −
→Z−
→ Xh1i in Z/[P].
(b) We have T = X ⊥ Z ⊥ Y[1].
In Section 2.1, we deduce Buchweitz’s equivalence (Corollary 2.3) and the silting reduction
(Corollary 2.2) from Theorem 1.2.
Finally we consider the following further special case of (T1′ ).
(T1′′ ) S = X ⊥ Y is a stable t-structure.
In this case, X , Y and Z are thick subcategories of T , and P = 0 holds. As a consequence, we
deduce the following result immediately.
Corollary 1.3. Under the assumptions (T0), (T1′′ ) and (T2), we have a weak semi-orthogonal
decomposition T = X ⊥ Z ⊥ Y. In particular, the composition Z ⊂ T → U is a triangle
equivalence.
In Section 2.2, we deduce Orlov’s theorem (Corollary 2.6) from Corollary 1.3.
Let us explain the structure of this paper. Our main Theorems 1.1 and 1.2 will be proved in the
last Section 3. In the next Section 2, we deduce the three applications: Buchweitz’s equivalence,
Orlov’s equivalences, and Amiot–Guo–Keller’s equivalence.
During the preparation of this paper, the authors were informed that there are analogous results
to our Theorem 1.1 in different settings. One is ‘Hovey’s twin cotorsion pairs’ due to Nakaoka [21],
and the other is ‘additive categories with additive endofunctors’ due to Li [17]. Their arguments
are more involved than ours since they give direct descriptions of the triangulated structure of
Z/[P] in the setting of Theorem 1.1. It will be interesting to have a closer look at the connections
between these results.
Ackowledgements Results in this paper were presented at the conference “Cluster Algebras and
Geometry” in Münster in March 2016. The first author thanks Karin Baur and Lutz Hille for their
hospitality. He also thanks Hiroyuki Nakaoka for explaining his results. He is supported by JSPS
Grant-in-Aid for Scientific Research (B) 16H03923, (C) 23540045 and (S) 15H05738. The second
author is supported by the National Science Foundation of China No. 11401297.
2. Applications of main results
In this section we give three applications of Theorems 1.1 and 1.2 and Corollary 1.3.
2.1. Silting reduction and Buchweitz’s equivalence. Tilting theory is powerful to control
equivalences of derived categories, and silting objects/subcategories are central in tilting theory.
It is shown in [10, 25] that the Verdier quotient of a triangulated category by its thick subcategory
with a silting subcategory can be realized as an ideal quotient (silting reduction). The aim of this
subsection is to deduce silting reduction from our main Theorems 1.1 and 1.2.
Recall that a full subcategory P of a triangulated category T is presilting if HomT (P, P[>0]) = 0.
A presilting subcategory P is called silting if the thick subcategory thickP generated by P is T .
Now we assume the following.
(P0) T is a triangulated category, P is a presilting subcategory of T such that P = addP,
S = thickP, and U = T /S.
4
OSAMU IYAMA AND DONG YANG
(P1) P is covariantly finite in ⊥ P[>0] and contravariantly finite in P[<0]⊥ .
(P2) For any X ∈ T , we have HomT (X, P[ℓ]) = 0 = HomT (P, X[ℓ]) for ℓ ≫ 0.
As a special case of our Theorems 1.1 and 1.2, we recover the following silting reduction.
Corollary 2.1 ([10, Theorems 3.1 and 3.6]). Under the assumptions (P0), (P1) and (P2), let
[
[
P ∗ P[1] ∗ · · · ∗ P[i],
P[−i] ∗ · · · ∗ P[−2] ∗ P[−1], Y :=
X :=
i>0
⊥
Z := X
i>0
⊥
⊥
⊥
∩ Y[1] = P[<0] ∩ P[>0].
Then the following assertion holds.
(a) (T0), (T1′ ) and (T2) in Theorem 1.2 are satisfied.
(b) We have a triangle equivalence Z/[P] ≃ U, where the structure of a triangulated category
of Z/[P] is described in Theorem 1.2.
(c) We have T = X ⊥ Z ⊥ Y[1].
Proof. It is well-known that we have a co-t-structure S = X ⊥ Y (see for example [10, Proposition
2.8]). By [10, Proposition 3.2], we have torsion pairs T = X ⊥ X ⊥ = ⊥ Y ⊥ Y. Thus (T0), (T1′ )
and (T2) in Theorem 1.2 are satisfied, and we have the assertions.
Now we apply Corollary 2.1 to prove Keller–Vossieck’s equivalence [16] in our context. For an
additive category A, we denote by K(A) the homotopy category of complexes on A, and by Kb (A)
the full subcategory consisting of bounded complexes.
Let F be a Frobenius category, P the category of projective-injective objects in F and F = F /[P]
the stable category of F . Then F has a structure of a triangulated category due to Happel [6]. We
denote by K−,b (P) the full subcategory of K(P) consisting of complexes X = (X i , di : X i → X i+1 )
satisfying the following conditions.
(a) There exists nX ∈ Z such that X i = 0 holds for each i > nX .
ai−1
bi
(b) There exist mX ∈ Z and a conflation 0 → Y i−1 −−−→ X i −→ Y i → 0 in F for each i ≤ mX
such that di = ai bi holds for each i < mX .
It is elementary that the category F is equivalent to the full subcategory of K−,b (P) consisting
of complexes which are isomorphic in K(P) to some X satisfying nX ≤ 0 ≤ mX , and we identify
them. We denote by K>0 (P) (respectively, K<0 (P)) the full subcategory of Kb (P) consisting of
complexes X = (X i , di : X i → X i+1 ) satisfying X i = 0 for i ≤ 0 (respectively, i ≥ 0).
Corollary 2.2. Let F be a Frobenius category such that the category P of projective-injective
objects in F is idempotent complete. Then we have a decomposition
K−,b (P) = K>0 (P) ⊥ F ⊥ K<0 (P).
Moreover the composition F ⊂ K−,b (P) → K−,b (P)/Kb (P) induces a triangle equivalence
≃
→ K−,b (P)/Kb (P).
F−
The ‘Moreover’ part is contained in [16, Example 2.3].
Proof. Let T = K−,b (P), S = Kb (P) and U = T /S. Then the condition (P0) is clearly satisfied.
We show that the conditions (P1) and (P2) are satisfied.
(P2) Fix X ∈ T . Then the condition (a) implies HomT (P[<−nX ], X) = 0, and the condition
(b) implies HomT (X, P[>−mX ]) = 0. Thus the condition (P2) is satisfied.
(P1) Fix X ∈ P[<0]⊥ and put n := nX . If n ≤ 0, then the natural morphism X 0 → X gives a
right P-approximation of X. If n > 0, then the natural morphism X n [−n] → X must be zero in
QUOTIENTS OF TRIANGULATED CATEGORIES
5
T . Thus there exists f ∈ HomP (X n , X n−1 ) such that 1X n = dn−1 f .
X n [−n] :
X:
···
Xn
①
①
f ①①
①
1X n
①①
①
||①
// X n
// X n−1
n−1
// 0
d
// · · ·
Since P is idempotent complete, X is isomorphic to the complex X ′ obtained by replacing dn−1 :
X n−1 → X n in X by Cok f → 0. Then nX ′ < n = nX holds. Repeating the same argument, we
can assume that nX ≤ 0 without loss of generality. Thus P is contravariantly finite in P[<0]⊥ .
ai−1
bi
Fix X ∈ ⊥ P[>0] and m := mX . Consider the conflation 0 → Y i−1 −−−→ X i −→ Y i → 0 given
in the condition (b). If m ≥ 0, then b0 : X 0 → Y 0 is a cokernel of d−1 . Thus the composition of
the natural morphism X → Y 0 and an inflation Y 0 → P with P ∈ P gives a left P-approximation
of X.
Assume m < 0, and take an inflation f : Y m → P in F with P ∈ P. Since bm : X m → Y m is
a cokernel of dm−1 , there exists am : Y m → X m+1 such that dm = am bm . Since X ∈ ⊥ P[>0], the
composition of natural morphisms X → Y m [−m] → P [−m] must be zero in T . Thus there exists
g ∈ HomP (X m+1 , P ) such that f bm = gdm .
X:
···
// X m−1
dm−1
dm
// X m
❖❖
bm ❖❖
f bm
P [−m] :
ww♣♣♣♣f
P oo
♥♥a
'' m ♥
Y
♣♣♣
// X m+1 d
66
♥
♥
m
m+1
// · · ·
g
Then f = gam holds. Since f is an inflation in F and P is idempotent complete, am is also an
am
bm+1
inflation in F . Thus there exists a conflation 0 → Y m −−→ X m+1 −−−→ Y m+1 → 0 in F satisfying
dm = am bm , and we can replace mX by mX + 1. Repeating the same argument, we can assume
mX ≥ 0 without loss of generality. Thus P is covariantly finite in ⊥ P[>0]. Thus the condition
(P1) is satisfied.
We are ready to complete the proof of Corollary 2.2. Thanks to Corollary 2.1, it suffices to
show that Z = P[<0]⊥ ∩ ⊥ P[>0] coincides with F , and that the triangulated structure of Z/[P]
coincides with that of F = F /[P]. The inclusion Z ⊃ F is clear. Conversely, the above argument
shows that any object in Z is isomorphic to some X with nX ≤ 0 ≤ mX , and hence belongs
to F . Thus Z = F holds. On the other hand, the triangulated structure of Z/[P] described in
Theorem 1.2(a) is nothing but Happel’s triangulated structure of F in this setting. Thus the claim
follows.
Now we apply Corollary 2.2 to prove Buchweitz’s equivalence [4, 24]. Recall that a Noetherian
ring R is called an Iwanaga–Gorenstein ring if inj.dimR R < ∞ and inj.dimRR < ∞. Let
CMR := {X ∈ modR | ExtiR (X, R) = 0 ∀i > 0}
be the category of Cohen–Macaulay R-modules. We denote by Kb (A) = K>0 (A) ⊥ K≤0 (A) the
standard co-t-structure (see [3, Section 1.1] and [23, Section 3.1]). For an abelian category A, we
denote by Db (A) the bounded derived category of A.
Corollary 2.3. Let R be an Iwanaga–Gorenstein ring. Then we have
Db (modR) = K>0 (projR) ⊥ CMR ⊥ K<0 (projR).
Moreover the composition CMR ⊂ Db (modR) → Db (modR)/Kb (projR) induces a triangle equivalence
CMR ≃ Db (modR)/Kb (projR).
The ‘Moreover’ part is [4, Theorem 4.4.1(2)], and [24, Theorem 2.1] for self-injective algebras.
6
OSAMU IYAMA AND DONG YANG
Proof. By [4, Lemma 4.1.2(iv)], any complex of finitely generated projective R-modules with
bounded cohomologies is in K−,b (projR). It follows that Db (modR) is triangle equivalent to
K−,b (projR). The desired results are obtained by applying Corollary 2.2 to F = CMR.
2.2. Orlov’s equivalences. Throughout this subsection, we assume the following.
(R0) R is a Z-graded Iwanaga–Gorenstein ring such that R = R≥0 .
In [22], Orlov gave a remarkable connection between two Verdier quotients of Db (modZ R), where
modZ R is the category of Z-graded finitely generated R-modules. One is Db (modZ R)/Kb (projZ R)
for the category projZ R of Z-graded finitely generated projective R-modules. This is important
in Cohen–Macaulay representation theory as we saw in the previous subsection. The other is
Db (modZ R)/Db (modZ0 R) for the category modZ0 R of Z-graded R-modules of finite length. This is
important in commutative and non-commutative algebraic geometry.
The aim of this subsection is to deduce Orlov’s theorem from Corollary 1.3 in a slightly more
general setting than Orlov’s original setting [22].
For an integer ℓ, we denote by mod≥ℓ R (respectively, mod≤ℓ R) the full subcategory of modZ R
consisting of all X satisfying Xi = 0 for any i < ℓ (respectively, i > ℓ). We denote by proj≥ℓ R
(respectively, proj≤ℓ R) the full subcategory of projZ R consisting of all P which are generated by
homogeneous elements of degrees at least ℓ (respectively, at most ℓ). Let mod>ℓ R := mod≥ℓ+1 R,
mod<ℓ R := mod≤ℓ−1 R, proj>ℓ R := proj≥ℓ+1 R and proj<ℓ R := proj≤ℓ−1 R. Since R is Iwanaga–
Gorenstein, we have a duality [19, Corollary 2.11]
(−)∗ := RHomR (−, R) : Db (modZ R) ↔ Db (modZ Rop ).
Under the following condition, we apply Corollary 1.3 to deduce the following result.
(R1) R0 has finite global dimension.
Corollary 2.4. Under the assumptions (R0) and (R1), we have
Db (modZ R) = Kb (proj<0 R) ⊥ Db (mod≥0 R) ∩ Db (mod>0 Rop )∗ ⊥ Kb (proj≥0 R).
In particular, the composition
Db (mod≥0 R) ∩ Db (mod>0 Rop )∗ ⊂ Db (modZ R) → Db (modZ R)/Kb (projZ R)
is a triangle equivalence.
This is given implicitly in [22, Lemma 2.4], and a similar result is given in [7, Theorem 4.17].
Proof. Let T := Db (modZ R), S := Kb (projZ R), X := Kb (proj<0 R) and Y := Kb (proj≥0 R). Using
(R1), one can easily show that we have stable t-structures
Kb (projZ R) = X ⊥ Y and Db (modZ R) = X ⊥ X ⊥
with X ⊥ = Db (mod≥0 R) [22, Lemma 2.3]. Replacing R by Rop and shifting the degree in the
second stable t-structure, we have a stable t-structure
Db (modZ Rop ) = Kb (proj≤0 Rop ) ⊥ Db (mod>0 Rop ).
Applying (−)∗ and using Kb (proj≤0 Rop )∗ = Y, we have a stable t-structure
Db (modZ R) = Db (mod>0 Rop )∗ ⊥ Kb (proj≤0 Rop )∗ = ⊥ Y ⊥ Y.
Thus (T0), (T1′′ ) and (T2) in Corollary 1.3 are satisfied, and we have the assertion.
mod≥ℓ
0 R
mod≤ℓ
0 R :=
≤ℓ
For an integer ℓ, let
:= modZ0 R ∩ mod≥ℓ R and
modZ0 R ∩ mod≤ℓ R. It is clear
≤ℓ
≤ℓ
ℓ
≥ℓ
that mod0 R = mod R holds. Let mod R := (mod R) ∩ (mod R).
The following conditions are crucial for our next result.
(R2) Ri has finite length as an R-module and as an Rop -module for any i ∈ Z.
(R3) There exists a ∈ Z such that (−)∗ restricts to a duality (−)∗ : Db (mod0 R) ↔ Db (moda Rop ).
QUOTIENTS OF TRIANGULATED CATEGORIES
7
The condition (R3) is satisfied if R has an a-invariant a, but (R3) is much more general (see
Remark 2.7 for details).
Under these assumptions, we apply Corollary 1.3 to deduce the following result.
Corollary 2.5. Under the assumptions (R0), (R2) and (R3), we have
≥0
b
Db (modZ R) = Db (mod≥0
R) ∩ Db (mod>a Rop )∗ ⊥ Db (mod<0
0 R).
0 R) ⊥ D (mod
In particular, the composition
Db (mod≥0 R) ∩ Db (mod>a Rop )∗ ⊂ Db (modZ R) → Db (modZ R)/Db (modZ0 R)
is a triangle equivalence.
This is given implicitly in [22, Lemma 2.4], and a similar result is given in [7, Theorem 6.15].
Proof. Let T := Db (modZ R), S := Db (modZ0 R), X := Db (mod0≥0 R) and Y := Db (mod<0
0 R). Using
(R2), one can easily show that we have stable t-structures
Db (modZ0 R) = X ⊥ Y and Db (modZ R) = ⊥ Y ⊥ Y
with ⊥ Y = Db (mod≥0 R) [22, Lemma 2.3]. Replacing R by Rop and shifting the degree in the
second stable t-structure, we have a stable t-structure
op
Db (modZ Rop ) = Db (mod>a Rop ) ⊥ Db (mod≤a
0 R ).
(2.2.1)
op
∗
By (R3), the duality (−)∗ induces a duality (−)∗ : X ≃ Db (mod≤a
0 R ). Applying (−) to (2.2.1),
we have a stable t-structure
Db (modZ R) = Db (mod≤a Rop )∗ ⊥ Db (mod>a Rop )∗ = X ⊥ X ⊥ .
Thus (T0), (T1′′ ) and (T2) in Corollary 1.3 are satisfied, and we have the assertion.
Combining Corollaries 2.4 and 2.5, we have the following Orlov’s theorem immediately.
Corollary 2.6 ([22, Theorem 2.5]). Under the assumptions (R0), (R1), (R2) and (R3), we have
the following.
(a) If a < 0, then there exists a fully faithful triangle functor
Db (modZ R)/Kb (projZ R) → Db (modZ R)/Db (modZ0 R).
(b) If a = 0, then there exists a triangle equivalence
Db (modZ R)/Kb (projZ R) ≃ Db (modZ R)/Db (modZ0 R).
(c) If a > 0, then there exists a fully faithful triangle functor
Db (modZ R)/Db (modZ0 R) → Db (modZ R)/Kb (projZ R).
It is easy to show that the fully faithful functors in (a) and (c) are parts of stable t-structures.
Remark 2.7. Assume (R0) and (R2).
(a) The condition (R3) is clearly equivalent to that RHomR (S, R) ∈ Db (moda Rop ) holds for
any simple R-module S ∈ mod0 R and RHomRop (S ′ , R) ∈ Db (moda R) holds for any simple
Rop -module S ′ ∈ mod0 Rop .
(b) The condition (R3) is satisfied if R has a-invariant (that is, the minus Gorenstein parameter ) a. This means that there exists an integer d such that, for any simple R0 -module S,
there exists a simple R0op -module S ′ such that S ∗ ≃ S ′ [−d](−a), and the same condition
holds for simple R0op -modules. For example, this is satisfied if R is ring-indecomposable
and commutative.
8
OSAMU IYAMA AND DONG YANG
(c) The condition (R3) is much weaker than the existence of a-invariant. For example, let k
be a separable field, and A and B be Z-graded finite dimensional k-algebras. Assume that
A = A0 has finite global dimension and is not semisimple, and that B is selfinjective and
has an a-invariant. Then R := A ⊗k B is an Iwanaga–Gorenstein ring and satisfies the
condition (R3), but does not has an a-invariant.
Proof. (c) Since k is separable, any simple R-module has the form S ⊗k T for a simple A-module S
and a simple B-module T . Since RHomR (S ⊗k T, R) ≃ RHomA (S, A) ⊗k RHomB (T, B) holds, the
former statement follows. It also follows that, if R has an a-invariant, then so does A. On the other
hand, it is easy to show that, if a Z-graded finite dimensional k-algebra has an a-invariant, then it
is selfinjective. Since A has finite global dimension, it has to be semisimple, a contradiction.
2.3. The AGK category and Amiot–Guo–Keller’s equivalence. In cluster theory, an equivalence between the cluster category of an n-Calabi–Yau algebra A and a certain full subcategory
Z of the perfect derived category perA of A plays a very important role (see a survey article [15]).
This was given by Amiot [1] for n = 3 and Guo [5] for general n based on Keller’s work [13, 14].
The aim of this subsection is to deduce Amiot–Guo–Keller’s equivalence from Theorem 1.1.
In fact, we will work in the following much wider setting. Let k be a field and D = Homk (−, k).
Definition 2.8. We say that (T , S, S, M) is a relative Serre quadruple if the following conditions
are satisfied.
(RS0) T is a k-linear Hom-finite Krull–Schmidt triangulated category and S is a thick subcategory
of T .
(RS1) S : S → S is a triangle equivalence such that there is a bifunctorial isomorphism for any
X ∈ S and Y ∈ T :
D HomT (X, Y ) ≃ HomT (Y, SX).
(RS2) M is a silting subcategory of T and T = M[< 0]⊥ ⊥ M[≥ 0]⊥ is a t-structure of T
satisfying M[≥ 0]⊥ ⊂ S. Moreover, M is a dualizing k-variety.
Note that the last condition that M is a dualizing k-variety is automatic if M has an additive
generator. By [10, Theorem 4.10], (RS2) is equivalent to its dual:
(RS2op ) M is a silting subcategory of T and T = ⊥ M[< 0] ⊥ ⊥ M[≥ 0] is a t-structure of T
satisfying ⊥ M[<0] ⊂ S. Moreover, M is a dualizing k-variety.
Let us introduce the following new class of triangulated categories.
Definition 2.9. For a relative Serre quadruple (T , S, S, M), we define the AGK category as the
Verdier quotient
C := T /S.
We define the fundamental domain as the full subcategory
Z = X ⊥ ∩ ⊥ Y[1] ⊂ T , where X = M[<0]⊥ ∩ S and Y = M[≥0]⊥ .
The following theorem is the main result of this subsection.
Theorem 2.10. Let (T , S, S, M) be a relative Serre quadruple. Assume that one of the following
conditions holds.
(i) S : S → S extends to a triangle equivalence S : T → T ;
(ii) M has an additive generator M .
Then the composition
Z ⊂T →C
is an equivalence. As a consequence, the AGK category C is a Hom-finite Krull–Schmidt triangulated category. Moreover, in case (i), C has a Serre functor S ◦ [−1].
QUOTIENTS OF TRIANGULATED CATEGORIES
9
Proof. By (RS2), we have a t-structure T = ⊥ Y ⊥ Y with ⊥ Y = M[< 0]⊥ . Therefore, for any
S ∈ S, there exists a triangle S ′ → S → Y → S ′ [1] with S ′ ∈ M[< 0]⊥ and Y ∈ Y. Since S ′
belongs to Y [−1] ∗ S ∈ S ∗ S = S, it belongs to X . Thus we have a t-structure S = X ⊥ Y.
Now we show that T = X ⊥ X ⊥ is a t-structure in both cases (i) and (ii). Consequently, (T0),
(T1) and (T2) in Theorem 1.1 are satisfied, and hence the composition Z = X ⊥ ∩ ⊥ Y[1] ⊂ T → C
is an equivalence since P = X [1] ∩ Y = 0.
First, we assume (i). Using the relative Serre duality (RS1) and the assumption ⊥ M[<0] ⊂ S,
we have X = S(⊥ M[<0]) = ⊥ (SM)[<0]. By (RS2op ), we have a t-structure T = ⊥ (SM)[<0] ⊥
⊥
(SM)[≥0] = X ⊥ X ⊥ with X ⊥ = ⊥ (SM)[≥0].
Next we assume (ii). Using (RS2op ), the dual argument to the first paragraph shows that we
have t-structures T = X ′ ⊥ X ′⊥ and S = X ′ ⊥ Y ′ for X ′ = ⊥ M[<0] and Y ′ = (⊥ M[≥0]) ∩ S.
The t-structures S = X ⊥ Y and S = X ′ ⊥ Y ′ are bounded by [10, Lemma 5.2]. Hence the heart
H = X ∩Y[1] is equivalent to the category of finite-dimensional modules over the finite dimensional
k-algebra EndT (M ) by [10, Proposition 4.8]. Thus any object in the abelian category H has finite
length, and moreover H contains only finitely many simple objects up to isomorphism. Let L be
the direct sum of a complete set of pairwise non-isomorphic simple objects of H. Since S = X ′ ⊥ Y ′
is bounded, there exists n ≫ 0 such that L[n] ∈ X ′ . Then H[n] ⊂ X ′ and hence X [n] ⊂ X ′ holds.
By Lemma 2.11 below, we have a torsion pair T = X ⊥ X ⊥ , which is a t-structure.
Lemma 2.11. Let T be a triangulated category, S a thick subcategory of T , and S = X ⊥ Y =
X ′ ⊥ Y ′ and T = X ′ ⊥ X ′⊥ torsion pairs. If X [n] ⊂ X ′ holds for some integer n, then we have a
torsion pair T = X ⊥ X ⊥ .
Proof. Replacing X by X [n], we can assume n = 0. It suffices to show X ∗ X ⊥ ⊃ T . This follows
from
X ∗ X⊥ = X ∗ Y ∗ X⊥
= S ∗ X⊥
′
⊃X ∗X
=T.
′⊥
(X ⊥ = Y ∗ X ⊥ )
(X ∗ Y = S)
(S ⊃ X ′ , X ⊥ ⊃ X ′⊥ )
Let us continue the proof of Theorem 2.10. It remains to prove the existence of Serre duality in
case (i). Let X and Y be objects of T . Since M is a silting subcategory of T , we have a bounded
co-t-structure T = ⊥ M[≥ 0] ⊥ M[< 0]⊥ (see for example [10, Proposition 2.8]). It follows that
there exists an integer i such that Y belongs to ⊥ M[≥−i + 1]. Now by (RS2) there is a triangle
X′
// X
// X ′′
// X ′ [1]
with X ′ ∈ M[<−i + 1]⊥ and X ′′ ∈ M[≥−i + 1]⊥ . Since HomT (Y, X ′ ) = 0, it follows that the
induced homomorphism HomT (Y, X) → HomT (Y, X ′′ ) is injective. So the morphism X → X ′′ is
a local S-envelope of X relative to Y in the sense of [1, Definition 1.2]. Therefore by [1, Lemma
1.1, Theorem 1.3 and Proposition 1.4] we obtain that S ◦ [−1] is a Serre functor of C.
Let (T , S, S, M) be a relative Serre quadruple and n ≥ 2 an integer. If S ≃ [n], then we call
the triple (T , S, M) an n-Calabi–Yau triple and call C = T /S the cluster category [10, Section 5].
A typical case is given by a bimodule n-Calabi–Yau non-positive dg algebra A with H 0 (A) being
finite-dimensional. In this case, (T , S, M) := (perA, Dfd (A), addA) is an n-Calabi–Yau triple ([1,
Section 2], [5, Section 2]).
Let (T , S, M) be an n-Calabi–Yau triple. Then
Z = M[≤0]⊥ ∩ ⊥ M[≥n] = M[1] ∗ M[2] ∗ · · · ∗ M[n − 1].
As a special case of Theorem 2.10, we obtain the following result which was obtained by Amiot [1,
Proposition 2.9] and Guo [5, Proposition 2.15] for bimodule Calabi–Yau dg algebras and by the
authors in the general setting in [10], which plays a crucial role in cluster theory.
10
OSAMU IYAMA AND DONG YANG
Corollary 2.12 ([10, Theorem 5.8(b)]). For an n-Calabi–Yau triple (T , S, M), the composition
Z ⊂T →C
is an equivalence. Moreover C is an (n − 1)-Calabi-Yau triangulated category.
3. Proof of Main results
In this section we prove our main results Theorems 1.1 and 1.2.
3.1. Proof of Theorem 1.1. In this subsection, we prove Theorem 1.1. First, notice that
S ∩ Z = (S ∩ X ⊥ ) ∩ (S ∩ ⊥ Y[1]) = Y ∩ X [1] = P
hold by (T1). Thus the functor Z → U induces a functor
Z/[P] → U.
(3.1.1)
Next we show that this is dense.
Lemma 3.1. For any X ∈ T , there exists Y ∈ Z satisfying X ≃ Y in U. As a consequence, the
functor (3.1.1) is dense.
Proof. Let T ∈ U. Since T = ⊥ Y[1] ⊥ Y[1] holds by (T2), we have a triangle
T′
// T
// T ′ [1]
// Y [1]
(T ′ ∈ ⊥ Y[1], Y ∈ Y).
Then we have T ≃ T ′ in U. Since T = X ⊥ X ⊥ holds again by (T2), we have a triangle
// T ′
X
// T ′′
// X[1]
(X ∈ X , T ′′ ∈ X ⊥ ).
Then we have T ≃ T ′ ≃ T ′′ in U. Since both T ′ and X[1] belongs to ⊥ Y[1] by (T1), so does T ′′ .
Thus T ′′ belongs to X ⊥ ∩ ⊥ Y[1] = Z, and we have an isomorphism T ≃ T ′′ in U.
Next we prepare the following.
Lemma 3.2. We have X [1] ⊂ X ⊥ P and Y ⊂ P ⊥ Y[1].
Proof. We only prove the first assertion. For X ∈ X , we take a triangle X ′ → X[1] → Y → X ′ [1]
with X ′ ∈ X and Y ∈ Y by (T1). Since X [1] is extension closed, we have Y ∈ X [1] ∩ Y = P. Thus
X[1] ∈ X ∗ P = X ⊥ P.
Finally we show that our functor is fully faithful.
Lemma 3.3. The functor (3.1.1) is fully faithful.
Proof. For M, N ∈ Z, we consider the natural map HomZ/[P] (M, N ) → HomU (M, N ).
We first show the surjectivity.
f
s
Any morphism HomU (M, N ) has a representative of the form M −
→ T ←
− N , where f ∈
HomT (M, T ) and s ∈ HomT (N, T ), such that the cone of s is in S. Take a triangle
N
s
// T
// S
a
// N [1]
(S ∈ S).
By (T1), there exists a triangle
X[1]
b
// S
// Y [1]
// X[2]
(X ∈ X , Y ∈ Y).
QUOTIENTS OF TRIANGULATED CATEGORIES
11
Since ab = 0 by X ∈ X and N ∈ Z ⊂ X ⊥ , we have the following commutative diagram of triangles
by the octahedral axiom.
X[1]
X[1]
b
N
// S
s
// T
cs
// T ′
a
// N [1]
c
N
// Y [1]
d
X[2]
// N [1]
X[2]
Then we have dcf = 0 by M ∈ Z ⊂ ⊥ Y[1] and Y ∈ Y. Thus there exists e ∈ HomT (M, N ) such
that cf = cse. Now c(f − se) = 0 implies that f − se factors through X[1] ∈ S. Thus f = se and
s−1 f = e hold in U, and we have the assertion.
Next we show the injectivity.
Assume that a morphism f ∈ HomT (M, N ) is zero in U. Then it factors through S (by, for
example, [20, Lemma 2.1.26]), that is, there exist S ∈ S, g ∈ HomT (M, S) and a ∈ HomT (S, N )
such that f = ag. By (T1), there exists a triangle
X
b
// S
c
// Y
// X[1]
(X ∈ X , Y ∈ Y).
Since ab = 0 by X ∈ X and N ∈ Z ⊂ X ⊥ , there exists d ∈ HomT (Y, N ) such that a = dc.
b
// S c // Y
>> ❅❅
⑥
❅❅
⑥⑥
❅
⑥
⑥g
a ❅❅ d
⑥
⑥
// N
M
X
f
By Lemma 3.2, there exists a triangle
P
// Y
e
// Y ′ [1]
(P ∈ P, Y ′ ∈ Y).
// P [1]
Then we have ecg = 0 by M ∈ Z ⊂ ⊥ Y[1] and Y ′ ∈ Y. Thus cg factors through P , and f = dcg = 0
in Z/[P].
By Lemmas 3.1 and 3.3, we complete to prove Theorem 1.1.
3.2. Proof of Theorem 1.2. In this subsection, we prove Theorem 1.2. We start with the
following observations.
Lemma 3.4. Under the assumptions (T0), (T1′ ) and (T2), the following assertions hold.
(a) P ⊂ Z and HomT (P, Z[1]) = 0 = HomT (Z, P[1]) hold.
a
b
a
b
(b) For any Z ∈ Z, there exists a triangle Z ′ −
→P −
→ Z → Z ′ [1] (respectively, Z −
→ P −
→
′
′
Z → Z[1]) with Z ∈ Z and P ∈ P such that a is a left P-approximation and b is a right
P-approximation.
Proof. (a) These are clear.
(b) By (T2), there exists a triangle
a
b
T −
→ X[1] −
→ Z → T [1]
with X ∈ X and T ∈ X ⊥ . Applying HomT (−, Y[1]), we have an exact sequence
0 = HomT (X[1], Y[1]) → HomT (T, Y[1]) → HomT (Z, Y[2])
12
OSAMU IYAMA AND DONG YANG
where the right term is zero since Z ∈ Z and Y[2] ⊂ Y[1] holds by (T1′ ). Thus T ∈ X ⊥ ∩⊥ Y[1] = Z.
Since T, Z ∈ Z and Z is extension-closed, we have X[1] ∈ X [1] ∩ Z ⊂ S ∩ Z = P. It is clear
that a is a left P-approximation and that b is a right P-approximation since HomT (Z, P[1]) = 0 =
HomT (P, Z[1]) holds by (a).
Now we are ready to prove Theorem 1.2.
(a) By Lemma 3.4, the pair (Z, Z) forms a P-mutation pair in the sense of [9]. In particular,
by [9, Theorem 4.2], the category Z/[P] has the structure of a triangulated category with respect
to the shift functor and triangles given in the statement. Moreover, it is easy to check that, with
respect to this triangulated structure of Z/[P], the equivalence Z/[P] ≃ U in Theorem 1.1 is a
triangle functor. Thus the assertion follows.
(b) It suffices to prove ⊥ Y[1] = X ⊥ Z.
By (T1′ ), we have ⊥ Y[1] ⊃ ⊥ Y ⊃ X . Thus ⊥ Y[1] ⊃ X ⊥ Z holds. It remains to show
⊥
Y[1] ⊂ X ⊥ Z. For any T ∈ ⊥ Y[1], there exists a triangle
X → T → T ′ → X[1]
with X ∈ X and T ′ ∈ X ⊥ by (T2). Since X[1] ∈ X [1] ⊂
Thus T ∈ X ⊥ Z, and we have the assertion.
⊥
Y[1], we have T ′ ∈ X ⊥ ∩ ⊥ Y[1] = Z.
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O. Iyama: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602 Japan
E-mail address: iyama@math.nagoya-u.ac.jp
D. Yang: Department of Mathematics, Nanjing University, 22 Hankou Road, Nanjing 210093, P. R.
China
E-mail address: yangdong@nju.edu.cn
| 0 |
arXiv:1407.3508v2 [math.GR] 26 Jun 2017
Hyperbolicity of relative free splitting
and free factor complexes
Michael Handel and Lee Mosher
∗
February 7, 2018
1
Introduction
Masur and Minsky in their papers [MM99, MM00] introduced a hierarchy of connected
simplicial complexes associated to a finite type surface S: at the top of the hierarchy is
the curve complex of S; and at lower levels are the curve complexes of essential, connected
subsurfaces of S. They prove hyperbolicity of the curve complexes of all finite type
surfaces, which applies immediately to all levels of the hierarchy of S. This hierarchy of
hyperbolic complexes has proved immensely useful in many applications to the large scale
geometry of the mapping class group MCG(S) [BF02, BM08, Man10, BKMM12, BBF10].
In [HM13] we proved hyperbolicity of the free splitting complex FS(Fn ) of a rank n
free group Fn , originally introduced as Hatcher’s sphere complex [Hat95]. In [BF14a],
Bestvina and Feighn proved hyperbolicity of the complex of free factors F(Fn ) of a
rank n free group Fn . Each of these complexes is regarded as an Out(Fn ) analogue, in
different ways, of the curve complex of a finite type surface.
In this paper we study the large scale geometry of relative free factor and free splitting complexes of Fn , proving their hyperbolicity (hyperbolicity of relative free splitting
complexes was proved independently by Horbez [Hor14b]). These complexes might be
regarded as analogues of curve complexes of essential connected subsurfaces of a finite
type surface. Unlike the situation with surfaces, hyperbolicity of these relative complexes is not a consequence of the absolute cases covered in [HM13], [BF14a], rather it is
a generalization. With some extra effort, we prove the theorem in the still more general
context of work of Guirardel and Levitt [GL07]: we prove hyperbolicity of relative free
factor and free splitting complexes of groups in general, relative to a free factor system.
∗
The first author was supported by the National Science Foundation under Grant No. DMS-1308710
and by PSC-CUNY under grants in Program Years 46 and 47. The second author was supported by the
National Science Foundation under Grant No. DMS-1406376.
1
While the need for relativizing [HM13] and [BF14a] has been clear, what “relativization” might mean has been less clear to us. In our work on the classification of subgroups
of Out(Fn ) [HM13a–e] we studied both subgroups and individual elements that are fully
irreducible relative to a free factor system. This work helped us to formulate an appropriate concept of relativization, and led us to consider free factor and free splitting
complexes of Fn relative to a fixed free factor system of Fn .
Relative complexes of free factor systems of Fn . Free factor systems for Fn
were first used by Bestvina, Feighn, and Handel [BFH00] to analyze the dynamics of
general elements of Out(Fn ). Formally a free factor system of Fn is a finite set of
the form A = {[A1 ], . . . , [AK ]} such that there exists an (internal) free factorization
Fn = A1 ∗ · · · ∗ AK ∗ B, (K ≥ 0), where each Ak is nontrivial, and [·] denotes conjugacy
class of a subgroup. We refer to the elements of the set A as its components. We refer
to the free factor B as a cofactor of A, with a careful emphasis that B is far from
unique, not even up to conjugacy, although its rank and therefore its isomorphism type
is well-defined; note that B may be trivial. Inclusion of free factors up to conjugacy
induces a partial ordering on free factor systems which is denoted A ⊏ A′ .
Fixing one free factor system A of Fn , the complex of free factor systems of Fn
relative to A, denoted FF (Fn ; A), is defined to be the geometric realization of the
partial ordering ⊏ restricted to the set of free factor systems B of Fn that are properly
nested between A and the “improper” free factor system {[Fn ]}; that is, A ⊏ B and
A=
6 B=
6 {[Fn ]}. For example, in the complex FF (Fn ; ∅), the subcomplex spanned by
those B having but a single component is naturally identified with the complex of free
factors F(Fn ), and the natural inclusion F(Fn ) ֒→ FF (Fn ; ∅) is a quasi-isometry (see
Proposition 6.3). Other examples of interest are associated to exceptional free factor
systems A, certain ones close to the maximum {[Fn ]} for which FF (Fn ; A) exhibits the
exceptional behavior of being either empty or 0-dimensional (see Section 2.5).
Relative complexes of free splittings of Fn . A free splitting of Fn is a minimal
action of Fn on a nontrivial simplicial tree T with trivial edge stabilizers and with finitely
many edge orbits. The set of conjugacy classes of nontrivial vertex stabilizers forms a
free factor system of Fn denoted F(T ) (see Section 3.2). Two free splittings which differ
by an equivariant homeomorphism are equivalent. Collapsing equivariant subgraphs of
free splittings defines a partial ordering on equivalence classes which is denoted S ≻ T .
The free splitting complex of Fn relative to a free factor system A, denoted FS(Fn ; A),
is the simplicial realization of the partial ordering ≻ restricted to equivalence classes of
free splittings T such that A ⊏ F(T ); here we allow equality A = F(T ). The familiar
case FS(Fn ; ∅) is the free splitting complex of Fn as studied in [HM13].
2
Theorem 1.1. For any proper free factor system A of Fn , the complex FS(Fn ; A) is
nonempty, connected, and hyperbolic.
Theorem 1.1 was proved independently by Horbez in [Hor14b].
Theorem 1.2. For any nonexceptional free factor system A of Γ, the complex FF (Γ; A)
is positive dimensional, connected, and hyperbolic.
Theorem 1.1 can also have certain special behavior when A is exceptional; see Section 4.2.
Relative complexes of free factor systems and free splitting for general groups.
We shall generalize Theorems 1.1 and 1.2 to any group relative to any choice of free factor system in that group. The proofs of hyperbolicity work identically in this general
context, after some preliminary work to establish basic facts which are well known for Fn
(see Sections 2 and 3).
The general context in which we shall work is identical to the context of Section 4 of
the paper [GL07] by Guirardel and Levitt, which one might express as being a study of
the outer space of a group Γ relative to a free factor system of that group. Our general
Theorems 1.3 and 1.4 are intended as a contribution to a growing mathematical study of
outer automorphism groups of freely decomposable groups—both absolute, and relative
to a choice of free factor system—with a goal of developing analogies between theorems
about these groups and theorems about Out(Fn ). For other works in this genre see
[Hor14a], [Mar99], [MM96], [CT94].
The historical roots of free factor systems and the partial order ⊏ in the context of
a general finitely generated group may be seen in the following fundamental theorem.
Given a group Γ define a Grushko decomposition to be a free product decomposition of
the form
(∗)
Γ = A1 ∗ · · · ∗ AK ∗ B (K ≥ 0)
in which each Ak is nontrivial, freely indecomposable, and not infinite cyclic, and B is
free of finite rank (possibly trivial).
Grushko Decomposition Theorem ([Chi76, Coh89]). Every finitely generated group
Γ has a Grushko decomposition.
The Kurosh subgroup theorem (see Section 2.1) provides certain uniqueness properties
for any Grushko decomposition (∗) of any group Γ:
(1) If A′ < Γ is a free factor which is not a finite rank free group then there exists
k ∈ {1, . . . , K} such that Ak is conjugate to a subgroup of A′ .
(2) For any other Grushko decomposition Γ = A′1 ∗ · · · ∗ A′K ′ ∗ B ′ , (K ′ ≥ 0), we have
K = K ′ , rank(B) = rank(B ′ ), and for each k = 1, . . . , K the subgroups Ak , A′σ(k)
are conjugate, where σ is a uniquely determined index permutation.
3
Formally, free factor systems and the extension relation ⊏ are defined for a general
group Γ exactly as in the special case Γ = Fn (our definition of extension is stricter than
in [FM14], in that we require the “cofactor” B to be free of finite rank). The above
uniqueness properties of a Grushko decomposition (∗) may be expressed by saying that
the associated Grushko free factor system A = {[A1 ], . . . , [AK ]} is the unique minimum
of the partial ordering ⊏ on the set of free factor systems of Γ. The converse is also
true: if the free factor system A is the unique minimum of ⊏, then A is the Grushko
free factor system associated to a Grushko decomposition (see Proposition 2.13).
Fix now an arbitrary group Γ and a free factor system A of Γ, not required to be a
Grushko free factor system. We treat the elements of A as indivisible atoms, although
for applications the internal structure of A will be important, as it is in the results of
[GL07] regarding virtual cohomological dimension.
The relative outer automorphism group Out(Γ; A) is defined to be the subgroup of
Out(Γ) which fixes A under the action of Out(Γ) on free factor systems. This is the
group whose virtual cohomological dimension is studied by Guirardel and Levitt [GL07,
Theorem 5.2] as an application of their construction of the outer space of Γ relative
to A. In this paper the group Out(Γ; A) is mostly lurking behind the scenes, but see
Section 3.6 and Section 6 for a record of basic facts.
The complex of free factor systems of Γ relative to A, denoted FF (Γ; A), is defined
to be the geometric realization of the partial ordering ⊏ restricted to the set of proper
free factor systems B of Γ such that A ⊏ B and A =
6 B. The special case FF (Γ; A)
when A is a Grushko free factor system might be thought of as the absolute complex of
free factor systems of Γ. Just as for Γ = Fn (see above), there are exceptional free factor
systems, those closest to the maximum {[Γ]}, for which FF (Γ; A) exhibits exceptional
behavior (see Section 2.5 and Proposition 6.2).
A free splitting of Γ is a minimal action of Γ on a nontrivial simplicial tree T with
trivial edge stabilizers and with finitely many edge orbits. The set of conjugacy classes
of nontrivial vertex stabilizers forms a free factor system of Γ denoted F(T ). Two free
splittings which differ by an equivariant homeomorphism are equivalent. Collapsing
equivariant subgraphs of free splittings defines a partial ordering on equivalence classes
which is denoted S ≻ T .
The free splitting complex relative to A, denoted FS(Γ; A), is the simplicial realization of the equivalence classes of free splittings T such that A ⊏ F(T ); here we
allow equality A = F(T ). When A is a Grushko free factor system then one may think
of FS(Γ; A) as the absolute free splitting complex of Γ. Just as happens for FS(Fn )
(c.f. [Hat95]), in general the complex FS(Γ; A) may be regarded as a kind of “simplicial completion” of the Guirardel-Levitt outer space of Γ relative to A; more precisely,
that relative outer space is naturally the complement of the subcomplex of FS(Γ; A)
consisting of all T for which the inclusion A ⊏ F(T ) is proper.
4
Theorem 1.3. For any group Γ and any proper free factor system A of Γ, the complex
FS(Γ; A) is nonempty, connected, and hyperbolic.
Theorem 1.3 was proved independently by Horbez [Hor14b].
Theorem 1.4. For any group Γ and any nonexceptional free factor system A of Γ, the
complex FF (Γ; A) is nonempty, connected, and hyperbolic.
These theorems can both be enhanced by descriptions of geodesics; see Theorem 5.4 for
FS(Γ; A) and Theorem 6.7 for FF(Γ; A).
Just as was done in [MM99] for the action of a surface mapping class group MCG(S)
on the curve complex of S, one may view these theorems as describing “weak relative hyperbolicity” of Out(Γ; A) with respect to the conjugacy classes of subgroups of
Out(Γ; A) that stabilize simplices of FS(Γ; A) and of FF (Γ; A).
Problems and Questions: Here is an opportunity for generalizing results about
Out(Fn ) to the context of groups of the form Out(Γ; A). Our results in [HM14] give a
complete classification of the dynamics of elements of Out(Fn ) acting on FS(Fn ), based
on the theory of attracting laminations, which itself is based on the theory of relative
train track maps. In particular we proved:
Loxodromic Classification Theorem ([HM14]). φ ∈ Out(Fn ) acts loxodromically on
FS(Fn ) if and only if φ has an attracting lamination that fills Fn .
(1) Develop a theory of attracting laminations for elements of Out(Γ; A) (using, most
likely, a version of the relative train track theory of [FM14]).
(2) Is there an analogue of the loxodromic characterization theorem for the action of
Out(Γ; A) on FS(Γ; A)?
(3) Under what conditions on Γ and A do loxodromic elements exist for the action
of Out(Γ; A) on FS(Γ; A)? We conjecture this holds if and only if FS(Γ; A)
has infinite diameter (see Section 4.2 for specific cases where FS(Γ; A) has finite
diameter).
For the case of Out(Fn ; A) we shall address these questions in [HM].
Outline of the paper. Sections 2 and 3 develop basic concepts of free factor systems
and free splittings in the context of a general group Γ. The case Γ = Fn is mostly well
known, and a reader interested only in that case could scan the opening paragraphs of
Sections 2 and 3 to glean what is needed from those sections, before proceeding to the
proofs of Theorems 1.3 and 1.4 in the remainder of the paper.
Sections 4 and 5 contain the proof of Theorem 1.3. The basic method of the proof is
quite similar to the proof of the absolute case of Theorem 1.1 given in [HM13] (but see
5
below for discussion of differences with [HM13]). Section 4 sets up the machinery of fold
paths in FS(Γ; A), combing properties of fold paths, and combinatorial measurements
along fold paths known as “free splitting units”. Section 5 uses these tools to prove
Theorem 1.3 in combination with axioms for hyperbolicity developed by Masur and
Minsky in [MM99], together with a “Big Diagram” argument as used first in [HM13].
Also as in [HM13], we prove Theorem 5.4 saying that the collection of fold paths in
FS(Γ; A) is uniformly quasigeodesically parameterized by free splitting units.
Section 6 contains the proof of Theorem 1.4. We use a method developed by Kapovich
and Rafi in [KR14] to derive hyperbolicity of FF(Γ; A) from hyperbolicity of FS(Γ; A),
generalizing their derivation of hyperbolicity of FF(Fn ) from hyperbolicity of FS(Fn ).
Remarks on methods of proof. In modifying the arguments of [HM13] to work in
this paper, there are three major areas of change. Two are accounted for in Sections 2
and 3: generalizing from Fn to Γ; and relativizing the absolute concepts of [HM13]. The
third area of change is motivated by work of Bestvina and Feighn who, in the appendix
of their paper [BF14b], introduced some simplifications to the methods of [HM13] by
ignoring the “gate 3 condition” on fold paths (see the heading “Remark on the gate 3
condition” in Section 4.1). We adopt these changes in this paper, emphasizing them in
the narrative of Sections 4 and 5.
Otherwise, certain concepts and/or proofs from [HM13] can be easily generalized and
relativized with little alteration, and when possible we shall do so with little comment,
providing a sketch in the more important cases.
Perhaps the most significant effect of dropping the gate 3 condition is that the
definition of free splitting units is considerably simplified, and is hence more easily
applicable. In [HM] we will apply the new free splitting units to prove the following
result, which is new even in the absolute case:
Theorem 1.5. There are constants M, L > 0, depending only on n, such that for every
free factor system A of Fn and every φ ∈ Out(Fn ; A), the action of φ on FS(Fn ; A)
satisfies one of two possibilities: either φ has an orbit of diameter ≤ M ; or φ acts
loxodromically with stable translation length ≥ L.
The analogous theorem for the mapping class group of a surface acting on the curve
complex is due to Bowditch [Bow08, Corollary 1.5].
6
Contents
1 Introduction
1
2 Free factor systems
2.1 Free factorizations and free factor systems . . . . . . . . . . .
2.2 The Kurosh Subgroup Theorem. Extension ⊏ and meet ∧. .
2.3 Corank and the structure of extensions of free factor systems
2.4 Grushko free factor systems . . . . . . . . . . . . . . . . . . .
2.5 Free factor system depth of a free factor system. . . . . . . .
3 The
3.1
3.2
3.3
3.4
3.5
3.6
free splitting complex and its relativizations
Basic terminology and notation regarding graphs. . . . . . .
Free splittings and the partial order ≻. . . . . . . . . . . . .
Free splitting complexes and their relativizations. . . . . . .
Relations between the partial orders ⊏ and ≻. . . . . . . .
Free splitting depth of free factor systems and dimensions of
splitting complexes. . . . . . . . . . . . . . . . . . . . . . .
The relative outer automorphism group Out(Γ; A). . . . . .
4 Fold paths and free splitting units
4.1 Fold sequences . . . . . . . . . . . .
4.2 FS(Γ; A) in low complexity cases . .
4.3 Combing . . . . . . . . . . . . . . . .
4.4 Invariant subgraphs of free splittings,
4.5 Free splitting units . . . . . . . . . .
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5 Hyperbolicity of relative free splitting complexes
5.1 The Masur–Minsky axioms. . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 Projection maps and the proof of the coarse retract axiom. . . . . . . .
5.3 Proof of Theorem 1.1: Reducing the coarse lipschitz and strong contraction axioms to Proposition 5.3. . . . . . . . . . . . . . . . . . . . . . . .
5.4 Theorem 5.4: Parameterizing fold paths using free splitting units . . . .
5.5 The proof of Proposition 5.3: Big Diagrams. . . . . . . . . . . . . . . . .
. 25
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6 Hyperbolicity of relative free factor complexes
67
6.1 The complex of free factor systems relative to a free factor system. . . . . 69
6.2 Connectivity of FF (Γ; A); a Lipschitz map FS(Γ; A) 7→ FF (Γ; A). . . . . 71
6.3 Proof of hyperbolicity of FF (Γ; A) . . . . . . . . . . . . . . . . . . . . . . 77
7
2
Free factor systems
Throughout this paper our convention is that Γ represents an arbitrary freely decomposable group, meaning that Γ can be expressed as a nontrivial free product of nontrivial
groups (if Γ were freely indecomposable then the main objects of study of this paper—
relative free factor and free splitting complexes—would be empty). In particular this
convention rules out the possibility that Γ is infinite cyclic; see remarks after the definition of free factor systems in Section 2.1 and after the definition of free splittings in
Section 3.2.
For readers whose primary interest is the case Γ = Fn , the contents of this section
are either well known or rather evident, and after briefly skimming this section one may
profitably proceed directly to the definitions of relative complexes of free factor systems
in Section 6.
This section contains basic material regarding free factor systems of Γ, their partial
order ⊏ known as “containment” or “extension”, their binary operation ∧ known as
“meet”, and properties thereof. This material will be used in Sections 3, 4 and 5 regarding relative free splitting complexes of Γ, and in Section 6 regarding relative complexes
of free factor systems of Γ.
The contents of this section consist for the most part of applications of the Kurosh
Subgroup Theorem and, in the finitely generated case, of Grushko’s Theorem. One such
application is the Extension Lemma 2.11, regarding the structure of a nested pair of
free factor systems A ⊏ B. The Extension Lemma and its consequences will be used
throughout the rest of the paper, even for the case of finite rank free groups.
An important application of the Extension Lemma is Lemma 2.14 which describes
a formula for the depth of a free factor system with respect to the partial ordering ⊏,
together with various properties of depth (the depth of an element of a partially ordered
set is the length of the longest ascending chain starting with the given element). Depth
of free factor systems will be applied in several ways later in the paper, including in
a dimension formula for relative free factor complexes (see Proposition 6.1). Of more
central importance, in Section 4.4 bounds on depth are used to derive topological and
metric properties of free splittings and their fold paths, and in Section 4.5 these bounds
are translated into properties of free splitting units along fold paths.
2.1
Free factorizations and free factor systems
Free factorizations. In any group Γ a free factorization is a set of nontrivial subgroups H = {Hl }l∈L satisfying the universality property that for any group K, any set
of homomorphisms {fl : Hl → K l ∈ L} is the restriction of a unique homomorphism
Γ → K. Equivalently, every nonidentity element γ ∈ Γ is represented by a unique
reduced word γ = γ1 · · · γI (I ≥ 1), meaning that there is a sequence l1 , . . . , lI ∈ L
8
such that γi ∈ Hli − {Id} for 1 ≤ i ≤ I, and li 6= li+1 for 1 ≤ i ≤ I − 1. Note that
Hl = Hm ⇐⇒ l = m. When a free factorization is finite—e.g. when Γ is finitely
generated (Grushko’s theorem)—we will generally pick a bijection L ↔ {1, . . . , L} and
write Γ = H1 ∗ · · · ∗ HL . Meanwhile, as we ponder infinite free factorizations in these
early sections of the paper, we shall write Γ = ∗(Hl )l∈L or just Γ = ∗H. A free factor
H < Γ is any element of a free factorization, in which case by conglomerating the other
free factors one obtains a free factorization of the form Γ = H ∗ H ′ .
For any free factorization H of Γ, each conjugacy class in Γ is represented by a
cyclically reduced word (meaning a reduced word that also satisfies lI 6= l1 ) and this
representative is unique up to cyclic permutation. From this we immediately obtain the
following, which incorporates the well known result that every free factor is malnormal:
Lemma 2.1. Every free factorization Γ = ∗H is mutually malnormal, meaning that for
each H, H ′ ∈ H and γ ∈ Γ, if γHγ −1 ∩ H ′ is nontrivial then γ ∈ H = H ′ .
♦
From malnormality of a free factor H < Γ it follows that two subgroups of H are
conjugate in Γ if and only if they are conjugate in H. We shall make tacit use of this
equivalence in what follows.
A partial free factorization of Γ is a subset of a free factorization. Every partial
free factorization H has a cofactor which is a subgroup B < Γ such that Γ = (∗H) ∗ B
is a free factorization. We make no assumptions on the cofactor, but we do have the
following result. Let N (H) be the subgroup normally generated by the union of all the
subgroups of H.
Lemma 2.2. For any partial free factorization H of Γ and any realization Γ = (∗H) ∗ B
with cofactor B, there is a homomorphic retraction Γ → B with kernel N (H), and so B
is isomorphic to the quotient Γ/N (H).
Proof. The retraction on a reduced word w = γ1 · · · γl erases each letter γi in each
element of H, and multiplies out the surviving letters of B in order. Evidently N (H)
is in the kernel K. Conversely w can be rewritten by moving the letters of w lying in
B to the front of the word, preserving their order, at the expense of replacing the other
letters by conjugates; so if w ∈ K then after rewriting one sees that w ∈ N (H).
♦
Lemma 2.3. Consider a group Γ and two partial free factorizations H = {Hl }l∈L and
H′ = {Hl′ }l∈L of Γ with the same index set L. Consider also realizations Γ = (∗H) ∗ B =
(∗H′ ) ∗ B ′ with cofactors B, B ′ . If Hl is conjugate to Hl′ for all l ∈ L then the cofactors
B, B ′ are isomorphic. Furthermore there is an isomorphism Γ → Γ which restricts to a
conjugation from Hl to Hl′ and restricts to an isomorphism from B to B ′ .
Proof. Noting that N (H) = N (H′ ), apply Lemma 2.2 to conclude that each of B, B ′ is
isomorphic to Γ/N (H). After choosing conjugations Hl 7→ Hl′ and an isomorphism B 7→
B ′ , the lemma follows by applying the universality property for free factorizations. ♦
9
Remark. For an example which determines the extent to which cofactors can fail to be
well-defined up to conjugacy, see the discussion of Γ = A ∗ Z following Proposition 6.2
in which the non-well-definedness of the infinite cyclic cofactor Z is discussed in detail.
Definition 2.4 (Free factor systems.). A weak free factor system of Γ is a set of the form
A = {[Al ]}l∈L such that {Al }l∈L is a partial free factor system of Γ with free cofactor;
we make no assumption on the cardinality of the set A nor on the rank of the cofactor,
although by Lemma 2.3 the rank of the cofactor is well-defined. A free factor system
A is a weak free factor system which is finite and has a finite rank cofactor. Note that
the definition allows A = ∅ as a possible free factor system, but only if Γ is free of finite
rank. We usually write A = {[A1 ], . . . , [AL ]} so that A is realized as Γ = A1 ∗· · ·∗AL ∗B;
as usual, the cofactor B may be trivial. A free factor system A is proper if A =
6 {[Γ]}.
The individual elements [A1 ], . . . , [AL ] of A are called its components.
Remark. Recalling our blanket assumption that Γ is not infinite cyclic, nevertheless
an infinite cyclic group does have a unique proper free factor system, namely ∅.
Remark. In the case Γ = Fn every weak free factor system is a free factor system,
and the same is true for finitely generated groups, by Grushko’s Theorem. The reader
interested solely in Fn or other finitely generated Γ may therefore safely ignore the
adjective “weak”, which should cut down on the technical overload of this section. Also,
the Extension Lemma 2.11 will provide a relative setting in which we can also ignore
“weak”, which we shall do forever afterwards, once the Extension Lemma is proved.
2.2
The Kurosh Subgroup Theorem. Extension ⊏ and meet ∧.
The results obtained in this section by applying the Kurosh Subgroup Theorem are
standard in the case Γ = Fn ; see [BFH00].
The following foundational theorem can be proved using Bass-Serre theory; see for
example [Coh89]. The usual expression of this theorem is in the language of double
cosets. We provide a translation into the language of conjugacy of subgroups, as well as
a slightly more detailed conclusion, particularly in the case of a free factor A < Γ.
Kurosh Subgroup Theorem. For any group Γ, any free factorization Γ = ∗(Hl )l∈L ,
and any subgroup A < Γ, there exists for each l ∈ L a subset Ul ⊂ Γ consisting of
representatives u of distinct double cosets AuHl , and there exists a free subgroup C < A,
such that the following hold:
(1) A = ∗{A ∩ uHl u−1 l ∈ L, u ∈ Ul } ∗ C
(2) For each (l, v) ∈ L × Γ:
(a) The subgroup C ∩ vHl v −1 is trivial.
10
(b) The subgroup A ∩ vHl v −1 is nontrivial ⇐⇒ there exists u ∈ Ul such that the
subgroups A ∩ vHl v −1 and A ∩ uHl u−1 are conjugate in A ⇐⇒ there exists
u ∈ Ul such that AvHl = AuHl .
If furthermore A is itself a free factor then:
(3) For each (l, u), (m, v) ∈ L × Γ such that u ∈ Ul and v ∈ Um , if the subgroups
A ∩ uHl u−1 and A ∩ vHm v −1 are conjugate in Γ then l = m and u = v (and so in
particular those subgroups are equal).
Remarks. The statement of the Kurosh Subgroup Theorem found for example in
[Coh89] incorporates only item (1), but the others are easily proved. Item (2a) is easily
derived from the Bass-Serre theory proof found in [Coh89], as is the first equivalence
of item (2b). The second equivalence of (2b) is a calculation: if AvHl = AuHl then
u = avh for some a ∈ A, h ∈ Hl and so a(A ∩ vHl v −1 )a−1 = A ∩ uHl u−1 ; conversely if
a(A ∩ vHl v −1 )a−1 = A ∩ uHl u−1 for a ∈ A then A ∩ (av)Hl (av)−1 = A ∩ uHl u−1 and so,
by malnormality of Hl , we have u−1 av ∈ Hl implying that AvHl = AuHl . For proving
item (3), the conjugating element must be in A by malnormality of A, and l = m by
mutual malnormality of ∗{Hl }l∈L ; the rest follows from (2b).
One standard consequence of the Kurosh Subgroup Theorem is that for any partial
free factorization {Ai } of Γ and any subgroup B < Γ, if each Ai is a subgroup of B
then {Ai } is a partial free factorization of B. The following slight generalization, also
an immediate consequence of the Kurosh Subgroup Theorem, is needed for the proof of
the Extension Lemma 2.11.
Lemma 2.5. For any group Γ, any subgroup B < Γ, any partial free factorization
{Ai }i∈I of Γ, and any identically indexed set of subgroups {A′i }i∈I , if A′i is conjugate to
♦
Ai and if A′i < B for all i ∈ I, then {A′i }i∈I is a partial free factorization of B.
Extension ⊏ of free factor systems. Given two subgroups A, A′ ⊂ Γ with conjugacy
classes [A], [A′ ], let [A] ⊏ [A′ ] denote the well-defined relation that A is conjugate to a
subgroup of A′ . Define a partial ordering A ⊏ A′ on weak free factor systems systems by
requiring that for each [A] ∈ A there exists [A′ ] ∈ A′ such that [A] ⊏ [A′ ]. We express
the relation (this) ⊏ (that) in various ways: (this) is contained in (that); or (that) is an
extension of (this); or (this) ⊏ (that) is an extension; etc. An extension A ⊏ A′ such
that A 6= A′ is called a proper extension.
If A, B are free factor systems then we also express the relation A ⊏ B by saying
that B is a free factor system relative to A.
11
Meet of free factor systems.
systems defined by
A ∧ B = {[A ∩ uBu−1 ]
The meet ∧ is a binary operation on weak free factor
[A] ∈ A, [B] ∈ B, u ∈ Γ, A ∩ uBu−1 6= {Id}}
We shall prove the following using the Kurosh Subgroup Theorem:
Lemma 2.6 (Weak Meet Lemma). In any group Γ, the meet of any two weak free factor
systems is a weak free factor system.
We will need to strengthen the conclusion of this lemma by removing the word “weak”
in various situations. One such situation, for finitely generated groups, is described in
Corollary 2.8. Another “relativized” version is given in Proposition 2.12.
Before giving the proof of Lemma 2.6, here are two immediate corollaries.
Corollary 2.7. For any weak free factor systems A, B in any group Γ, their meet A ∧ B
can be characterized as the unique weak free factor system having the following properties:
(i) A ∧ B ⊏ A;
(ii) A ∧ B ⊏ B;
(iii) For every weak free factor system C, if C ⊏ A and C ⊏ B then C ⊏ A ∧ B.
♦
The next result, well known in the case of free groups from [BFH00], follows immediately
by combining Lemma 2.6, Grushko’s Theorem, and Corollary 2.7.
Corollary 2.8. In any finitely generated group Γ, for any two free factor systems A, B
of Γ, their meet A∧B is a free factor system. Furthermore if A, B are free factor systems
relative to a third free factor system C then A ∧ B is also a free factor system relative
to C.
♦
The second sentence of Corollary 2.8 is true in a general group; see Proposition 2.12.
Proof of the Weak Meet Lemma 2.6. Consider A = {[Ai ]}i∈I and B = {[Bj ]}j∈J with
respective realizations
(#)
Γ = ∗(Ai )i∈I ∗ A′
and
Γ = ∗(Bj )j∈J ∗ B ′
Applying the Kurosh Subgroup theorem to Ai using the given realization of B, we obtain
a free factorization
(##)
Ai = ∗(Aik )k∈Ki ∗ A′i
12
where A′i is a free group and the subgroups Aik are representatives of the Γ-conjugacy
classes of all nontrivial intersections of Ai with conjugates of the Bj ’s. It follows that
A ∧ B = {[Aik ] i ∈ I, k ∈ Ki }
Substituting (##) into (#) we obtain a free factorization
′
Γ = ∗ ∗{Aik }k∈Ki ∗ Ai
∗ A′
i∈I
′
′
= ∗{Aik }i∈I,k∈Ki ∗ ∗(Ai )i∈I ∗A
which, the factor in brackets [·] clearly being free, shows that A ∧ B is a weak free factor
system.
♦
2.3
Corank and the structure of extensions of free factor systems
In this section we prove the Extension Lemma 2.11 detailing the structure of any extension A ⊏ B of free factor systems. This will be applied in studying the depth of ⊏ in
Section 2.5, and when studying free splitting units in Sections 4.4 and 4.5.
Also, the Extension Lemma 2.11 guarantees that any weak free factor system that
is an extension of a free factor system is itself a free factor system, and Proposition 2.12
guarantees that for any free factor system A the meet of two free factor systems rel A is
also a free factor system rel A, which is how we generalize Corollary 2.8 to non finitely
generated groups. These results allow us henceforth to ignore the adjective “weak”.
Corank.
Define the corank of a free factor system A of a group Γ to be the integer
corank(A) = rank(Γ/N (A)) = rank(A′ ) ≥ 0
where A′ is the cofactor of any realization of A. When we wish to emphasize the ambient
group we also write corank(A; Γ). From Bass-Serre theory it follows that corank(A)
is equal to the topological rank of the underlying graph for any finite graph of groups
representation of Γ with trivial edge groups and with nontrivial vertex groups A1 , . . . , AK
so that A = {[A1 ], . . . , [AK ]}.
When Γ = Fn and A = {[A1 ], . . . , [AK ]}, the free factors A1 , . . . , AK are all free of
finite rank, and we have the following rank sum formula for the corank of A:
corank(A) = n −
K
X
k=1
13
rank(Ak )
This formula may be useful to the reader for deriving quick proofs of results to follow
in the special case Γ = Fn .
The following lemma defines what we shall call the containment function from one
free factor system to any of its extensions.
Lemma 2.9. Given an extension A ⊏ B of weak free factor systems of a group Γ, the
relation ⊏ between components of A and components of B defines a function A 7→ B,
called the containment function.
Proof. By definition, for any component [A] ∈ A there exists a component [B] ∈ B
such that [A] ⊏ [B]. By mutual malnormality of any realization of B, this [B] depends
uniquely on [A].
♦
The following result, in the special case Γ = Fn , is an evident consequence of the
rank sum formula for corank.
Proposition 2.10. For any nested pair of free factor systems A ⊏ A′ of Γ we have
corank(A) ≥ corank(A′ ). Equality holds if and only if the containment function A 7→ A′
is surjective and for each [A′j ] ∈ A′ there exists a free factorization with trivial cofactor
A′j = Aj1 ∗ · · · ∗ Ajkj so that the preimage of [A′j ] under the containment function is
{[Aj1 ], . . . , [Ajkj ]} ⊂ A.
The proof of Proposition 2.10 in the general case—where rank sum does not make
sense—will be given after the statement and proof of the following Extension Lemma.
For understanding the conclusions of the Extension Lemma we refer the reader to
Figure 1 which depicts those conclusions in tabular format. The proof of the Extension
Lemma is similar to the proof of the Weak Meet Lemma 2.6 but with more care taken
regarding cardinalities.
Lemma 2.11 (Extension Lemma). In any group Γ, if A is a free factor system, if A′
is a weak free factor system, and if A ⊏ A′ , then A′ is a free factor system. Moreover,
consider any realization Γ = A′1 ∗ · · · ∗ A′K ∗ B ′ of A′ = {[A′1 ], . . . , [A′K ]}, with indexing
chosen so that the image of the containment function A 7→ A′ equals {[A′1 ], . . . , [A′J ]},
where 0 ≤ J ≤ K. For 1 ≤ j ≤ J let Aj ⊂ A be the pre-image of [A′j ] under the
containment function, and let kj = |Aj |. Then there exists a realization of A of the
form
Γ = A11 ∗ · · · ∗ A1k1 ∗ · · · · · · ∗ AJ1 ∗ · · · ∗ AJkJ ∗ (B1 ∗ · · · ∗ BJ ∗ A′J+1 ∗ · · · ∗ A′K ∗ B ′ )
{z
}
|
B = cofactor of A
such that
Aj = {[Aj1 ], . . . , [Ajkj ]}
and
A′j = Aj1 ∗ · · · ∗ Ajkj ∗ Bj
14
(1 ≤ j ≤ J)
The subgroups B1 , . . . , BJ , A′J+1 , . . . , A′K , B ′ are all free of finite rank. By abuse of
notation (identifying conjugacy classes in Γ with conjugacy classes in A′j ) we may regard
Aj as a free factor system of the group A′j realized with cofactor Bj .
Proof. Since A is finite and A ⊏ A′ , any realization of the weak free factor system A′
can be listed as
(∗)
Γ = A′1 ∗ · · · ∗ A′J ∗ (∗{A′k }k∈K ) ∗ B ′ ,
J ≥0
so that B ′ is the cofactor, and so that the subset {[A′1 ], . . . , [A′J ]} ⊂ A′ is the image of
the containment map A 7→ A′ (we assume all free factors of (∗) are nontrivial, except
perhaps B ′ ). For 1 ≤ j ≤ J, let Aj ⊂ A be the preimage of [A′j ] under the containment
map A 7→ A′ , and let kj = |Aj | ≥ 1. We may choose pairwise nonconjugate subgroups
Aj1 , . . . , Ajkj < A′j so that Aj = {[Aj1 ], . . . , [Ajkj ]}. By Lemma 2.5 we have a free
factorization
(∗∗)j
A′j = Aj1 ∗ · · · ∗ Ajkj ∗ Bj
Substituting each (∗∗)j into (∗) and rearranging terms we obtain the following free
factorization of Γ, which is clearly a realization of A:
Γ = A11 ∗ · · · ∗ A1k1 ∗ · · · · · · ∗ AJ1 ∗ · · · ∗ AJkJ ∗ (B1 ∗ · · · ∗ BJ ∗ (∗{A′k }k∈K ) ∗ B ′ )
|
{z
}
B = cofactor of A
Since B is a finite rank free group, it follows that K is finite, and that the subgroups
A′k for k ∈ K and B1 , . . . , BJ , B ′ are all finite rank and free. It follows that A′ is a free
factor system of Γ with cofactor B ′ , and that Aj may be regarded as a free factor system
of A′j with cofactor Bj .
♦
Proof of Proposition 2.10. This is a quick application of Lemma 2.11. Following the
notation of that lemma we have corank(A) = rank(B) ≥ rank(B ′ ) = corank(A′ ), with
equality if and only if and only if none of B1 , . . . , BJ , A′J+1 , . . . , A′K exist: nonexistence of
A′J+1 , . . . , A′K is equivalent to J = K which is equivalent to surjectivity of A 7→ A′ ; and
nonexistence of the cofactor Bj is equivalent to existence of the desired free factorization
A′j = Aj1 ∗ · · · ∗ Ajkj without cofactor.
♦
Here is the promised relativization of Corollary 2.8.
Proposition 2.12. For any group Γ, any free factor system A, and any two free factor
systems B, C of Γ relative to A, their meet B ∧ C is a free factor system relative to A.
Proof. Applying Corollary 2.7, B ∧ C is a weak free factor system, and by item (iii) of
that corollary we have A ⊏ B ∧ C. By Lemma 2.11 it follows that B ∧ C is a free factor
system.
♦
15
j
free factorization of A′j
1
..
.
A11
···
..
.
A1k1
B1
..
.
J
J +1
..
.
A1J
· · · A1kJ
A′J+1
..
.
BJ
K
cofactor of A′
B′
A′K
Figure 1: The Extension Lemma 2.11 shows that for each extension A ⊏ A′ of free
factor systems, and for any realization of A′ , there exists a free factorization with terms
as depicted which simultaneously incorporates the following: the given realization of
A′ = {[A′1 ], . . . , [A′K ]} with cofactor B ′ ; for each j ≤ J a realization of a free factor
system of A′j , namely A′j = {[Aj1 ], . . . , [Ajkj ]}, with cofactor Bj ; and a realization of
A = {[A11 ], . . . , [A1k1 ], . . . . . . , [A1J ], . . . , [A1kJ ]} with cofactor B = B1 ∗ · · · ∗ BJ ∗ A′J+1 ∗
· · · ∗ A′K ∗ B ′ .
2.4
Grushko free factor systems
Recall Grushko’s theorem, which ways that every finitely generated group has a Grushko
decomposition. Grushko decompositions can also exist naturally outside of the realm of
finitely generated groups: any free product of a finite rank free group and finitely many
freely decomposable groups yields a Grushko decomposition. The following proposition
describes the behavior of general Grushko decompositions, expressed in terms of the ⊏
relation.
Proposition 2.13. For any group Γ and any free factor system A of Γ, the following
are equivalent:
(1) Some realization Γ = A1 ∗ · · · ∗ AK ∗ A′ of A is a Grushko decomposition.
(2) Any realization Γ = A1 ∗ · · · ∗ AK ∗ A′ of A is a Grushko decomposition.
(3) A is a minimum weak free factor system with respect to ⊏.
(4) For any weak free factor system B of Γ we have A ⊏ B. In particular A is the
unique minimum weak free factor system with respect to ⊏.
If these properties hold then we say that A is the Grushko free factor system of Γ.
16
Proof. Clearly (4) =⇒ (3) =⇒ (2) =⇒ (1). Assuming (1), in order to prove (4) it suffices
by Corollary 2.7 to prove that A = A ∧ B. Let C = A ∧ B ⊏ A. For each [C] ∈ C,
consider the unique component [Ak ] ∈ A such that [C] ⊏ [Ak ]. Applying the Kurosh
Subgroup Theorem, after conjugation it follows that C is a nontrivial free factor of Ak ,
but Ak is freely indecomposable, and so C = Ak . This proves that C is a subset of A. If
C=
6 A then there exists [Ak ] ∈ A such that [Ak ] 6∈ C, and by the Extension Lemma 2.11
it follows that [Ak ] is a free factor of some cofactor of C. But cofactors are free and Ak
is not free, a contradiction.
♦
2.5
Free factor system depth of a free factor system.
In general the depth of an element x of a partially ordered set is the cardinality L of
the longest ascending chain x = x0 ⊏ · · · ⊏ xL of order relations starting with the given
element. Given a group Γ we compute depth for the set of free factor systems of Γ with
respect to the partial ordering ⊏, and we derive some properties of this depth. These
could be immediately applied to define and compute depths of complexes of free factor
systems relative to a free factor system, but we shall delay that until Section 6.
Given a free factor system A = {[A1 ], . . . , [AK ]} of Γ define the free factor system
depth of A to be
DFF (A) = 2 corank(A) + |A| − 1 = 2 rank(B) + K − 1
where |·| denotes the cardinality, and B is any cofactor of any realization of A.
Assuming Γ = Fn , for any free factor system A = {[A1 ], . . . , [AK ]} we have
DFF (A) = 2 n −
K
X
1
K
X
rank(Ak ) + K − 1 = (2n − 1) −
2 rank(Ak ) − 1
1
Part of the content of Lemma 2.14 to follow is that DFF (A) is indeed the depth of A
with respect to the partial ordering ⊏. This is easily checked when Γ = Fn .
Here are some examples. The exceptional free factor systems A, defined to be those
for which DFF (A) ≤ 2, can be enumerated as follows:
• DFF (A) = 0 if and only if A is the improper free factor system A = {[Γ]}.
• DFF (A) = 1 if and only if corank(A) = 0 and |A| = 2, in which case A =
{[A1 ], [A2 ]} with Γ = A1 ∗ A2 . The possibility that corank(A) = 1 and |A| = 0 is
equivalent to Γ being infinite cyclic, which was ruled out.
• DFF (A) = 2 if and only if one of the following happens: either |A| = 1 and
corank(A) = 1, in which case A = {[A]} with realization Γ = A ∗ Z where the
cofactor Z is infinite cyclic; or |A| = 3 and corank(A) = 0 in which case A =
{[A1 ], [A2 ], [A3 ]} with realization Γ = A1 ∗ A2 ∗ A3 .
17
As we shall see in Proposition 6.2, the exceptional free factor systems A are those for
which the complex of free factor systems relative to A is exceptionally simple, either
empty or 0-dimensional.
We say that a proper extension A ⊏ A′ is elementary if one of the following holds:
(1) A′ = A ∪ {[Z]} where Z < Γ is infinite cyclic; or
(2) there is a realization Γ = A1 ∗ · · · ∗ AK ∗ B of A and two components [Ai ], [Aj ]
(i 6= j ∈ {1, . . . , K}) such that
A′ = A − {[Ai ], [Aj ]} ∪ {[Ai ∗ Aj ]}
Another part of Lemma 2.14 is that the statement “A ⊏ A′ is elementary” is equivalent
to DFF (A) = DFF (A′ ) + 1 which is equivalent to saying that no other free factor system
is properly contained between A and A′ . Again this is easily checked when Γ = Fn .
Lemma 2.14. The function DFF on free factor systems of Γ has the following properties:
(1) If A ⊏ A′ then DFF (A) ≥ DFF (A′ ) with equality if and only if A = A′ . As a
special case, DFF (A) ≥ 0 with equality if and only if A = {[Γ]}.
(2) If A ⊏ A′ is a proper extension then DFF (A) ≥ DFF (A′ ) + 1 with equality if and
only if A ⊏ A′ is an elementary extension.
(3) For any proper extension A ⊏ A′ there exists a free factor system C such that
A ⊏ C ⊏ A′ and such that A ⊏ C is elementary.
(4) For every chain of proper extensions of the form A = A0 ⊏ · · · ⊏ AK = {[Γ]}, its
length K satisfies K ≤ DFF (A). Equality holds if only if the chain is maximal, if
and only if every extension Ak−1 ⊏ Ak is an elementary extension.
Proof. Noting that item (4) is a consequence of the earlier items, it remains to prove (1),
(2) and (3). Assuming A ⊏ A′ , in items (1) and (2) we are interested in the difference
DFF (A) − DFF (A′ ) = 2(corank(A) − corank(A′ )) + |A| − A′
18
Applying Lemma 2.11 and adopting its notation, we have
corank(A) =
K
X
rank(A′j )
J
X
rank(Bj ) +
J
X
|Aj | − K
J
X
(|Aj | − 1) − (K − J)
j=J+1
j=1
|A| − A′ =
z }| {
rank(A′j ) + rank(B ′ )
rank(Bj ) +
j=1
corank(A) − corank(A′ ) =
corank(A′ )
K
X
J
X
j=J+1
j=1
=
j=1
DFF (A) − DFF (A′ ) =
J
X
j=1
2 rank(Bj ) +
| {z }
(a)j
J
K
X
X
(|Aj | − 1) +
(2 rank(A′j ) − 1)
| {z }
{z
}
|
j=1
j=J+1
(b)j
(c)j
From this it follows that DFF (A) ≥ DFF (A′ ) because each of the quantities (a)j , (b)j ,
(c)j is non-negative: for 1 ≤ j ≤ J the quantity (a)j is a non-negative even integer, and
the quantity (b)j is a non-negative integer because Aj 6= ∅; and for J + 1 ≤ j ≤ K the
quantity (c)j is an odd positive integer because A′j is free of rank ≥ 1. Furthermore:
• (a)j = 0 if and only if the free factorization A′j = Aj1 ∗ · · · ∗ Ajkj ∗ Bj has trivial
cofactor Bj (for 1 ≤ j ≤ J).
• (b)j = 0 if and only if |Aj | = kj = 1 if and only if Aj has exactly one component
(for 1 ≤ j ≤ J).
• (c)j > 0 (for J + 1 ≤ j ≤ K).
Thus DFF (A) = DFF (A′ ) if and only if no (c)j ’s exist, i.e. J = K, and Aj = {[Aj1 ]} for
each 1 ≤ j ≤ J, which happens if and only if A = A′ . This completes the proof of (1).
We next prove the “if” direction of item (2). Suppose that A ⊏ A′ is an elementary
extension. In one case we have A′ = A ∪ {[Z]} where Z is infinite cyclic, and it follows
that K = J + 1, that (a)j = (b)j = 0 for 1 ≤ j ≤ J, and that (c)J+1 = 1. In the
other case, there exists j0 ∈ {1, . . . , J} and two components [A], [A′ ] ∈ A such that up
to conjugacy we have A′j0 = A ∗ A′ , and A′ = (A − {[A], [A′ ]}) ∪ {[A′j0 ]}. It follows that
each (a)j = 0, that (b)j = 1 if j = j0 and (bj ) = 0 otherwise, and that there are no
(c)j ’s. In either case we have DFF (A) = DFF (A′ ) + 1.
Suppose now that A ⊏ A′ is a proper expansion, equivalently DFF (A)−DFF (A′ ) > 0,
equivalently at least one of the quantities (a)j , (b)j , (c)j is positive. In each case we
19
exhibit a free splitting C such that A ⊏ C ⊏ A′ , and A ⊏ C is elementary. Item (3) and
the remaining contentions of item (2) follow immediately.
Case 1: Some (a)j > 0 (1 ≤ j ≤ J) which means the free factorization A′j =
Aj1 ∗ · · · ∗ Ajkj ∗ Bj has nontrivial cofactor Bj . Let Z be rank 1 free factor of B and let
C = A ∪ {[Z]}.
Case 2: Some (b)j > 0 (1 ≤ j ≤ J) which means Aj = {[Aj1 ], [Aj2 ], . . . , [Ajkj ]} has
kj ≥ 2 components. Let C = (A − {[Aj1 ], [Aj2 ]}) ∪ {[Aj1 ∗ Aj2 ]}.
Case 3: Some (c)j > 0 exists, which means J < K. For J + 1 ≤ j ≤ K each of
the groups A′j is free of positive rank. Let Z < A′J+1 be a rank 1 free factor and let
C = A ∪ {[Z]}.
♦
3
The free splitting complex and its relativizations
In Sections 3.1–3.3, given an arbitrary freely decomposable group Γ we define free splittings of Γ and their partial ordering ≻ called the “collapse relation”. Also, using these
concepts we define free splitting complexes of Γ, both the “absolute” free splitting complex FS(Γ) and the free splitting complex FS(Γ; A) “relative to” a choice of free factor
system A. We also study a function which associates to each free splitting a free factor system called its “vertex stabilizer system”, and in Section 3.4 we study how this
function relates the partial orderings ⊏ and ≻. In Section 3.5 we study the depth of the
inverted partial ordering ≺. We apply that study to obtain a formula for the dimension
of FS(Γ; A), and to obtain a finer understanding of the partial ordering as it relates to
inclusion of simplices. Of particular importance is Proposition 3.6 that explains exactly
which free splittings are maximal and minimal with respect to the collapse relation ≻,
and which chains of the relation ≻ correspond to maximal simplices of FS(Γ; A).
The proofs in this section are primarily applications of Bass-Serre theory along with
basic topological manipulations of graphs and trees, and a few further applications of
the Kurosh Subgroup Theorem.
For the case of Γ = Fn , many of the results of this section, regarding basic concepts
of free splittings and the collapse partial ordering may be familiar to a reader of [HM13].
Nonetheless we examine these concepts from new points of view, in order to study relative
free splitting complexes. Throughout this section we try to view these points first from
the vantage of the special case Γ = Fn , before moving on the general formulation. This
is done so as to enable the reader interested mostly in Γ = Fn to get through this section
more quickly.
20
3.1
Basic terminology and notation regarding graphs.
A graph G is a 1-dimensional simplicial complex, a tree is a contractible graph, and a
subgraph of a graph G is a subcomplex of some simplicial decomposition of G. Given
p ∈ G, let Dp G denote the set of directions at p, meaning initial germs of locally
injective paths with initial point p. If p is a vertex then each element of Dp G is uniquely
represented by an oriented edge with initial vertex p.
Relatively natural cell structures. Subdivisions and edgelets. Suppose that
G is a connected graph, and P is a subset of the vertex set that includes all vertices
of valence 1 and which accumulates on all isolated ends of G (the only case of isolated
ends that matters at all to us is when G is homeomorphic to the line). In any setting
where P is fixed, there is a unique relatively natural cell structure on G which is the
CW structure on G whose 0-skeleton, the set of relatively natural vertices, is the union
of P with all points of valence ≥ 3; the 1-cells of this structure are called the relatively
natural edges. When P is understood we will often drop the adverb “relatively”.
Note that any CW structure on G whose vertex set contains P is a subdivision of
the relatively natural cell structure. In any context where one such CW structure is
specified we sometimes refer its edges as edgelets and we refer to that structure as an
edgelet subdivision of the relatively natural cell structure.
3.2
Free splittings and the partial order ≻.
A free splitting of Γ is a minimal simplicial action Γ y T of the group Γ on a simplicial
tree T such that T is not a point, the stabilizer of each edge is trivial, and there are
finitely many edge orbits. It follows that there are finitely many vertex orbits, and so
T /Γ is a finite graph of groups. It also follows, using minimality, that T has no valence 1
vertices.
Two free splittings S, T of Γ are equivalent, denoted S ≈ T , if there exists a Γequivariant homeomorphism f : S 7→ T . While this homeomorphism need not be simplicial, one can always make f be simplicial by first subdividing T along the image of
the vertex set of S, and then pulling the subdivided vertices of T back to obtain a subdivision of S. More generally a map f : S → T between free splittings is an equivariant
function which, with respect to some subdivision of the domain and range, is simplicial.
For any free splitting Γ y T we define the relatively natural cell structures on T
and on the quotient graph of groups T /Γ, so that the quotient map T 7→ T /Γ is a
cellular map taking relatively natural vertices to relatively natural vertices, and taking
relatively natural edges to relatively natural edges. On T the relatively natural vertices
are the points which either have valence ≥ 3 or have nontrivial stabilizer; and on T /Γ
the relatively natural vertices are the points which either have valence ≥ 3 or have a
nontrivial vertex group.
21
Remark. Note that a point p ∈ T has stabilizer isomorphic to Z/2 if and only p
has valence 2 and the stabilizer of p is nontrivial. Using this one can show that if Γ y T
is a free splitting and if T has an isolated end then the following conclusion holds: Γ
is the infinite dihedral group, T is a line, and Γ y T is the Bass-Serre tree of the free
decomposition Γ = Z/2 ∗ Z/2. Also, the same conclusions hold for any free splitting
of any group Γ that contains an infinite cyclic subgroup of finite index. Here we use
our convention, from the opening of Section 2, which rules out the possibility that Γ is
infinite cyclic (allowing Γ to be infinite cyclic, its unique free splitting up to equivalence
is its translation action on the line).
Vertex stabilizer systems. Associated to each free splitting Γ y T is a proper free
factor system of Γ denoted F(T ) and called the vertex stabilizer system of T , namely
the conjugacy classes of nontrivial vertex stabilizers. The fact that F(T ) is indeed a free
factor system follows from Bass-Serre theory, by using any isomorphism between Γ and
the fundamental group of the quotient graph of groups T /Γ. In the converse direction
we have the following fact, which will often be invoked silently:
Lemma 3.1. For every proper free factor system A of Γ there exists a free splitting
Γ y T such that A = F(T ).
Proof. Choose a realization Γ = A1 ∗ · · · ∗ AK ∗ B of A = {[A1 ], . . . , [AK ]}. Construct a
graph of groups with base point p, attaching to B a rose with rank equal to corank(A) =
rank(B), and attaching K additional edges to p with opposite vertices of valence 1 having
respective vertex groups A1 , . . . , AK . The fundamental group of this graph of groups
has an isomorphism to Γ = A1 ∗ · · · ∗ AK ∗ B. Letting T be the Bass-Serre tree of
this graph of groups with associated Γ action, we obtain a free splitting of Γ satisfying
F(T ) = A.
♦
Collapse maps and the partial ordering ≻. A partial ordering on the set of equivalence classes of free splittings of Γ is defined as follows. A collapse map f : T → S is a
map such that for each x ∈ S its inverse image f −1 (x) is connected. The union of those
inverse images f −1 (x) which are not single points is called the collapse forest σ ⊂ T ,
and so σ has no degenerate components, meaning no components that are single points.
Letting T 7→ T /σ denote the equivariant quotient map under which each component of
σ is collapsed to a single point, it follows that T /σ and S are equivalent free splittings.
[σ]
We sometimes incorporate σ into the notation by writing T −→ S.
[σ]
A collapse T −→ S is relatively natural if σ is a subcomplex of the relatively natural
cell structure, equivalently σ is a union of relatively natural edges. Note that if a collapse
[σ]
map T −→ S exists then a relatively natural collapse map exists, by replacing σ with
its unique maximal relatively natural cell subcomplex (which might be empty).
22
We define a relation denoted T ≻ S or S ≺ T to mean that there exists a collapse
map T 7→ S, equivalently there exists a relatively natural collapse map. This relation
is well-defined on equivalence classes. The relation T ≻ S is a partial order because
a composition of collapse maps is a collapse map. We express the relation T ≻ S in
various ways, such as T collapses to S, or S expands to T , or S ≺ T is an expansion,
or T ≻ S is a collapse. If furthermore T 6≈ S then the collapse or expansion is proper,
and this holds if and only if for some (all) collapse maps T 7→ S some point pre-image
contains more than one relatively natural vertex of T .
Note that for any map of free splittings f : S → T , each element of Γ that is elliptic
in S is also elliptic in T , and therefore F(S) ⊏ F(T ) (see Lemma 3.2 (3)). In particular,
if T ≺ S, equivalently if S ≻ T , then F(S) ⊏ F(T ).
Remark on abuses of notation. While a free splitting is formally denoted Γ y T ,
and we often use this notation to emphasize the action, also we often suppress the action
from the notation and simply write T . The action is always suppressed from the notation
for the equivalence class [T ], and sometimes we write just T for the equivalence class.
3.3
Free splitting complexes and their relativizations.
We define the (absolute) free splitting complex of Γ, denoted FS(Γ), to be the simplicial complex which is the geometric realization of the set of equivalence classes of free
splittings of Γ partially ordered by ≺. Thus FS(Γ) has a 0-simplex for each equivalence
class of free splittings Γ y T , denoted [T ]. In general FS(Γ) has a K-simplex for each
K + 1-tuple of distinct 0-simplices [T0 ], [T1 ], . . . , [TK ] such that T0 ≺ T1 ≺ · · · ≺ TK ;
this simplex is denoted [T0 ] ≺ [T1 ] ≺ · · · ≺ [TK ]. By our convention that Γ be freely
indecomposable, FS(Γ) is always nonempty.
Consider now a proper free factor system A of Γ. A free splitting of Γ rel A is a
free splitting Γ y T with the property that A ⊏ F(T ), equivalently A is elliptic with
respect to T meaning that each subgroup of Γ representing an element of A fixes some
point of T . The free splitting complex of Γ relative to A, denoted FS(Γ; A), is the flag
subcomplex of FS(Γ) consisting of all simplices [T0 ] ≺ · · · ≺ [TK ] such that A is elliptic
in each of the free splittings Γ y T0 , . . . , TK ; this is equivalent to requiring simply that
A is elliptic in TK , because F(TK ) ⊏ · · · ⊏ F(T0 ). The requirement that A be proper
implies that free splittings rel A exist (by Lemma 3.1) and so FS(Γ; A) is nonempty. In
Corollary 4.5 below we will see that FS(Fn ; A) is connected.
Note that if Γ has a proper Grushko decomposition, equivalently if there exists a
free factor system A which is minimal with respect to ⊏ (see Proposition 2.13), then
FS(Γ) = FS(Γ; A); this holds for example whenever Γ is finitely generated.
Remarks on terminology and notation. The notation [T ] is used both for the
equivalence class of a free splitting Γ y T and for the corresponding 0-simplex of FS(Γ).
23
Sometimes we abuse notation by writing things like “T ∈ FS(Γ; A)” which can be read
formally either as “T is a free splitting of Γ rel A” or as “[T ] is a 0-simplex of FS(Γ; A)”.
In [HM13] we used the notation FS(Fn ) a little differently, namely the complex
with one k-simplex for each equivalence class of free splittings T having k + 1-orbits
of natural edges, where the face inclusion is defined by the relation S ≺ T . Also,
we used the notation FS ′ (Fn ) for the first barycentric subdivision of FS(Fn ) which is
equivalent to the free splitting complex as defined in this section. But even in [HM13] we
worked primarily with this first barycentric subdivision, and since relative free splitting
complexes live naturally as subcomplexes of this first barycentric subdivision, in this
current work we switch the notation and we hope that this does not cause confusion.
3.4
Relations between the partial orders ⊏ and ≻.
In the following lemma we collect properties relating the partial order ⊏ on free factor
systems to the partial order ≻ on (equivalence classes of) free splittings. These properties
are all true as well when they are specialized by choosing a free factor system A and
putting in the qualifier “relative to A”.
Lemma 3.2. For any Γ the following hold:
(1) For any map of free splittings f : T → S we have an extension F(T ) ⊏ F(S) of
free factor systems. In particular if S ≺ T then F(T ) ⊏ F(S).
(2) For any free factor system A of Γ and any two free splittings Γ y S, T rel A there
exists a free splitting Γ y U rel A and a relatively natural collapse map f : U → T
such that F(U ) = F(S) ∧ F(T ) and such that for each x ∈ T , if the subgroup
StabT (x) is nontrivial then its action on f −1 (x) ⊂ U is equivalent to its action on
its minimal subtree in S.
(3) For any free splitting Γ y T and any free factor system B ⊏ F(T ) there exists a
free splitting U and a collapse map U 7→ T such that F(U ) = B.
(4) More generally, for each sequence of extensions A0 ⊏ A1 ⊏ · · · ⊏ AK of free factor
systems rel A, and each free splitting SK such that F(SK ) = AK there exists a
sequence of free splittings and collapses S0 ≻ S1 ≻ · · · ≻ SK such that F(Sk ) = Ak
for each k = 0, . . . , K.
Proof. Item (1) is evident since Stab(x) < Stab(f (x)). Clearly (2) =⇒ (3) by taking S
to be any free splitting such that F(S) = B and using that B ⊏ C implies B ∧ C = B.
Also clearly (3) =⇒ (4).
Item (2) says intuitively that one can always “blow up” T to get some U so that
for each x ∈ T the actions of Stab(x) on its blowup in U is a copy of its action on its
24
minimal subtree in S. The proof of item (2) is an elaboration of the Bass-Serre theory
proof of the Kurosh Subgroup Theorem (see e.g. [Coh89]); here are a few details. First
apply Proposition 2.12 to conclude that B = F(S) ∧ F(T ) is a free factor system rel A.
Consider x ∈ T such that StabT (x) is nontrivial, let S x ⊂ S be the minimal subtree for
the action StabT (x) y S, and suppose that S x is not a point. Blow up the vertex x
using S x : detach each of the directions of Dx T from x, then remove x, then reattach
the directions of Dx T to a copy of the tree S x in a StabT (x)-equivariant manner. Now
extend this “detachment–attachment” operation over the whole orbit of x, reattaching
the directions in a Γ-equivariant manner. Doing this for each orbit of such points x
results in the desired free splitting Γ y U .
♦
3.5
Free splitting depth of free factor systems and dimensions of relative free splitting complexes.
The absolute free splitting complex of a rank n free group FS(Fn ) has the following
easily proved properties. Define a free splitting Fn y T to be generic if every vertex
has valence 3. First, T has at most 3n − 3 natural edge orbits, the maximum being
attained if and only if T is generic. Also, the maximal number of natural vertex orbits
is the number attained for generic T which is 2n − 2. These are proved by simple Euler
characteristic calculations taking place in the quotient graph of groups T /Fn . Next,
given a D-simplex [T0 ] ≺ [T1 ] ≺ · · · ≺ [TD ] with corresponding sequence of relatively
natural collapse maps TD 7→ · · · 7→ T1 7→ T0 , the following are easily proved to be
equivalent:
(1) D = 3n − 4.
(2) TD is generic, each map Td 7→ Td−1 collapses exactly one orbit of natural edges,
and T0 has exactly one orbit of natural edges.
(3) The simplex [T0 ] ≺ [T1 ] ≺ · · · ≺ [TD ] is maximal, meaning it is not a proper face
of any other simplex.
As a consequence, the dimension of FS(Fn ) equals 3n − 4 and every simplex is a face
of some simplex of maximal dimension 3n − 4.
We now generalize, stating and proving analogous results for relative free splitting
complexes.
Definition 3.3. Let Γ be a group and A any free splitting of Γ.
(1) The free splitting depth of A is defined to be the number
DFS (A) = 3 corank(A) + 2 |A| − 4
25
(2) A free splitting Γ y T rel A is generic if F(T ) = A and for each vertex v the
following holds: if Stab(v) is trivial then v has valence ≤ 3; whereas if Stab(v) is
nontrivial then Stab(v) acts transitively on Dv T .
Note that for Γ y T to be generic, it is equivalent that in the quotient graph of groups
G = T /Fn the following hold: the nontrivial vertex groups are of the form A1 , . . . , AK
where A = {[A1 ], . . . , [AK ]}; and for every vertex V of G, if V has trivial vertex group
then V has valence 2 or 3, whereas if V has nontrivial vertex group then V has valence 1.
One can always choose the vertex groups to fit into a realization of A of the form
Γ = A1 ∗· · · ∗AK ∗B in such a way that B is identified with a lift to Γ of the fundamental
group of the underlying graph of G.
Proposition 3.4. For any free splitting Γ y T rel A the following hold:
(1) The number of relatively natural edge orbits of T satisfies E(T ) ≤ DFS (A) + 1.
(2) The following are equivalent:
(a) E(T ) = DFS (A) + 1.
(b) T is generic.
(c) [T ] is maximal with respect to the partial ordering ≺, that is, for every free
splitting Γ y U , if there exists a collapse map U 7→ T then [U ] = [T ].
(3) The number of relatively natural vertex orbits of T satisfies
V (T ) ≤ DFS (A) + 2 − corank(A) = 2 corank(A) + 2A − 2
with equality if and only if T is generic.
Proof. In this proof we assume that all vertices and all edges of free splittings are relatively natural, equivalently no valence 2 vertex has nontrivial stabilizer; if any such
vertices exist, just remove them from the 0-skeleton. Thus every vertex and every edge
of the quotient graph of groups is relatively natural, meaning that no valence 2 vertex
has trivial vertex group. Also, all collapse maps are relatively natural and are nontrivial
if and only if they are not homeomorphisms. Having done this, for any such free splitting
T with quotient G = T /Γ the numbers E = E(T ) and V = V (T ) are just the counts of
edge and vertex orbits of T , equivalent of edges and vertices of G. Let Vk = Vk (T ) be
the number of valence k vertices of G, equivalently the number of Γ-orbits of vertices
v ∈ T at which the set Dv Γ has exactly k orbits under the action of Stab(v).
We first prove (2b) =⇒ (2a). Assuming T is generic we have V = V1 + V3 and
V1 = |A|. We also have E = 21 (V1 + 3V3 ) and corank(A) = E − V + 1 (= rank(G)).
Eliminating V , V1 , and V3 gives E = DFS (A) + 1.
26
We next claim that for every free splitting Γ y T rel A there exists a generic free
[σ]
splitting Γ y S rel A and a relatively natural collapse map S −→ T . From this claim
we obtain the following consequences. First, item (1) holds because E(T ) ≤ E(S) =
DFS (A) + 1. Next, the implication (2a) =⇒ (2c) holds, because if (2c) does not hold
[σ]
then there exists U and a collapse U −→ T such that [U ] 6= [T ], and so σ is nontrivial,
implying by (1) that E(T ) < E(U ) ≤ DFS (A) + 1. Next, (2c) =⇒ (2b), because if T
is not generic then the collapse map S 7→ T is nontrivial and so [T ] is not maximal.
[σ]
Finally, item (3) follows because the collapse map S −→ T takes the relatively natural
vertices of S onto the relatively natural vertices of T and so V (S) ≥ V (T ), with equality
if and only if σ = ∅ if and only if [S] = [T ] if and only if T is generic, and
V (S) = 1 − rank(S/Γ) + E(S) = 1 − corank(A) + DFS (A) + 1
To prove the claim we do a sequence of expansions of T one at a time to build up
the properties of a generic free splitting rel A. First, by applying the expansion from
Lemma 3.2 (3) we may assume that Γ y T satisfies F(T ) = A.
Next, by expanding T we may assume that if v ∈ T is a vertex with nontrivial
stabilizer, and so [Stab(v)] ∈ A, then the number kv of Stab(v)-orbits in the set Dv Γ
satisfies kv = 1. Otherwise, if kv ≥ 2, choose orbit representatives d1 , . . . , dk ∈ Dv Γ, do
a simultaneous partial fold of these directions by identifying proper initial segments into
a single segment e, having one vertex with the same stabilizer as v and opposite vertex
of valence k + 1 and with trivial stabilizer. Extending these identifications equivariantly,
the resulting free splitting is an expansion of T because by collapsing the orbit of e we
recover T .
Finally, we may assume that if v is a vertex with trivial stabilizer and valence ≥ 3
then v has valence 3, for otherwise we may group Dv Γ into two sets of cardinality ≥ 2 and
expand T by pulling these two sets apart, inserting a new edge, and extending this expansion equivariantly over the orbit of v. This expansion decreases the lexicographically
ordered sequence (V3 (T ), V4 (T ), . . .).
♦
Definition 3.5. Let Γ be a group.
[σ]
(1) A relatively natural collapse map S −→ T of free splittings of Γ is elementary if σ
consists of a single orbit of relatively natural edges.
(2) A one edge free splitting is a free splitting Γ y T with exactly one relatively
natural edge orbit.
27
Proposition 3.6. For each D-simplex [T0 ] ≺ [T1 ] ≺ · · · ≺ [TD ] in FS(Γ; A) with
corresponding sequence of relatively natural collapse maps
[σD ]
[σD−1 ]
[σ2 ]
[σ1 ]
TD −−→ TD−1 −−−−→ · · · −−→ T1 −−→ T0
the following are equivalent:
(1) D = DFS (A).
(2) Each of the following holds: (a) TD is generic relative to A; (b) each collapse
map Td 7→ Td−1 is elementary, for d = 1, . . . , D; (c) T0 is a one-edge free splitting.
(3) The simplex [T0 ] ≺ [T1 ] ≺ · · · ≺ [TD ] is maximal, meaning it is not a face of any
other simplex.
As a consequence, the dimension of FS(Γ; A) equals DFS (A), and every simplex is a
face of a simplex of maximal dimension DFS (A).
Proof. As in the proof of Proposition (3.4), we assume that all edge and vertices are
relatively natural, and we continue to use the notation E(T ), V (T ) as in that proof.
The scheme of the proof is (1) ⇐⇒ (2) ⇐⇒ (3).
Assuming item (2) we shall prove (1). By applying Proposition 3.4 one concludes
E(TD ) = DFS (A) + 1, and then one notices that from (2) it follows that the edge orbits
of TD are collapsed one-at-a-time until only one remains, implying that the number D
of collapse maps equals DFS (A).
[σ]
Assuming (1) we shall prove (2). For any relatively natural collapse map S −→ T ,
letting E(σ) be the number of natural edge orbits of S contained in the Γ-equivariant
natural subforest σ, we have E(T ) + E(σ) = E(S); recall also that E(σ) = 0 ⇐⇒ σ =
∅ ⇐⇒ [S] = [T ]. Using that each of E(σD ), . . . , E(σ1 ), E(T0 ) is ≥ 1 we have
D + 1 ≤ E(σD ) + · · · + E(σ1 ) + E(T0 )
= E(TD )
≤ DFS (A) + 1
=D+1
(by Proposition 3.4 (1))
(by assumption of (1))
and so all inequalities are equations. Applying Proposition 3.4 (2), it follows TD is
generic. It also follows that E(σD ) = · · · = E(σ1 ) = E(T0 ) = 1, which proves (2).
Assuming (2) holds, we prove (3) as follows. Since T (D) is generic, there does
not exist any proper collapse map of the form S 7→ TD for that would imply E(S) >
DFS (A)+1, contradicting Proposition 3.4 (1). Since Td 7→ Td−1 is elementary, there exist
any factorization of Td 7→ Td−1 into proper collapse maps of the form Td 7→ S 7→ Td−1
28
because that would imply E(Td ) ≥ E(Td−1 )+2, contradicting that E(Td ) = E(Td−1 )+1.
Nor does there exist any proper collapse map of the form T0 7→ S, for that would imply
E(S) ≤ E(T0 ) − 1 = 1 − 1 = 0. It follows that the simplex [T0 ] ≺ · · · ≺ [TD ] is maximal.
Assuming (2) fails, we prove that (3) fails as follows. One of (a), (b), or (c) must
fail. If TD is not generic then by Proposition 3.4 (2c) there exists a free splitting S and
[σ]
a proper relatively natural collapse map S 7→ TD . If Td −→ Td−1 is not elementary then,
first collapsing a single edge orbit of σ, there a sequence of proper relatively natural
collapse maps Td 7→ S 7→ Td−1 . If T0 has more than one edge orbit then, collapsing just
one edge orbit, there exists a proper relatively natural collapse map T0 7→ S. In each case
we obtain a simplex of one dimension higher containing the simplex [T0 ] ≺ · · · ≺ [TD ]. ♦
3.6
The relative outer automorphism group Out(Γ; A).
Now that the sets of free factor systems and free splittings rel A have been defined
together with various relations and operations on them, we pause here to carefully
define the relative outer automorphism group Out(Γ; A) and its actions on those sets.
We also define the action of the group Out(Γ; A) on the relative free splitting complex
FS(Γ; A), although the definition of its action on the complex of free factor systems
rel A will await the definition of that complex to be given in Section 6.1.
The group Out(Γ) has a canonical left action on the set of free factor systems A,
namely: given φ ∈ Out(Γ), choosing a representative Φ ∈ Aut(Γ), and choosing a
realization Γ = A1 ∗ · · · ∗ AK ∗ B of A, one defines
φ(A) = {[Φ(A1 )], . . . , [Φ(AK )]}
This action is well-defined independent of choices, the left action equations φ(ψ(A)) =
(φψ)(A) and Id(A) = A hold, and the action preserves the extension partial order ⊏
and the meet operation ∧.
Relative outer automorphism groups. Given a free factor system A of Γ, the
subgroup of Out(Γ) that fixes A is denoted Out(Γ; A) and is called the outer automorphism group of Γ rel A. This is the group studied by Guirardel and Levitt in [GL07]
who derive information about the virtual cohomological dimension of Out(Γ; A) using
information about the virtual cohomological dimensions of the groups Ak , Aut(Ak ), and
Out(Ak ), k = 1, . . . , K.
Action on relative free splitting complexes. The group Out(Γ) has a canonical
right action on the set of equivalence classes of free splittings of Γ as follows. Consider
the equivalence class [T ] of a free splitting Γ y T with associated homomorphism
α : Γ → Aut(T ); incorporating α into the notation we write Γ yα T . Consider also φ ∈
Out(Γ) represented by Φ ∈ Aut(Γ). Precomposing α by Φ we obtain a homomorphism
α ◦ Φ : Γ → Aut(T ) which defines a free splitting Γ yα◦Φ T , the equivalence class of
29
which is defined to be [T ]·φ. This free splitting is well-defined, the right action equations
[T ] · (φψ) = ([T ] · φ) · ψ and [T ] · Id = [T ] hold, and the action preserves the collapse
partial order ≻. We obtain thereby an induced right action of Out(Γ) on linear chains
of free splittings as follows:
[T0 ] ≻ [T1 ] ≻ · · · ≻ [TK ] · φ = [T0 ] · φ ≻ [T1 ] · φ ≻ · · · ≻ [TK ] · φ
Finally, for any free factor system A of Γ, we obtain by restriction a right action of
Out(Γ; A) on linear chains of free splittings rel A. These chains define simplices of the
relative free splitting complex FS(Γ; A), and so we immediately obtain the right action
of Out(Γ; A) on FS(Γ; A) by simplicial isomorphisms.
Action on chains of relative free factor systems. The action of Out(Γ) on free
factor systems preserving ⊏ induces an action on linear chains of free factor systems:
φ A0 ⊏ A1 ⊏ · · · ⊏ AK = φ(A0 ) ⊏ φ(A1 ) ⊏ · · · ⊏ φ(AK )
For any given free factor system A we obtain by restriction a left action of Out(Γ; A) on
linear chains of free splittings rel A. Once the formal definitions are given in Section 6.1,
we will immediately obtain the left action of Out(Γ; A) on the relative complex of free
factor systems FF (Γ; A) by simplicial isomorphisms.
We record here one fact for later use, which is a simple consequence of the definitions:
Lemma 3.7. The function [T ] 7→ F[T ] satisfies the inverted equivariance condition
with respect to the actions of Out(Γ): given an equivalence class of free splittings [T ]
and φ ∈ Out(Γ) we have the following equation of free factor systems:
F [T ] · φ = φ−1 F[T ]
♦
4
Fold paths and free splitting units
In this section we fix a group Γ and a free factor system A in Γ, and we study fold
paths in the relative free splitting complex FS(Γ; A). Section 4.1 contains the basic
definitions, generalizing fold paths following [HM13] but also following [BF14b] to the
extent of dropping the “gate 3 condition” of [HM13]. In Section 4.2 we use fold paths
to give an explicit description of FS(Γ; A) in the simplest cases where the free factor
system A is very close to maximal in Γ. In Section 4.3 we generalize the concepts of
combing of fold paths following [HM13]. In Section 4.4 we consider a measurement of
the complexity of a Γ-invariant subforest of a free splitting Γ y T , and we study how
this complexity can change along a fold path. In Section 4.5 we use change of complexity
to define free splitting units along fold paths; in later sections these units are shown to
30
give efficient upper and lower bounds to distance along fold paths. We note that while
free splitting units as defined here are a fortiori comparable to free splitting units as
defined in [HM13], the definition here is somewhat simpler and easier to work with.
4.1
Fold sequences
Given two free splittings Γ y S, T and a map f : S → T which is injective on each
edgelet, for each p ∈ S there is an induced “derivative” dfp : Dp S → Df (p) T , which
maps the initial direction of each oriented edgelet E ⊂ S with initial vertex p to the
initial direction of the path f E. The point pre-images of the map dfp are called the
gates of f at p. We say that f : S → T is foldable if it is injective on each edgelet and
has at least 2 gates at each vertex. A foldable sequence is a sequence of maps
f1
f2
f
T0 −→ T1 −→ · · · −−K
→ TK
such that each map fji = fj ◦ · · · fi+1 : Ti → Tj is foldable. In discussing foldable
sequences we often restrict our attention to subsequences Ti 7→ · · · 7→ Tj parameterized
by an integer subinterval [i, j] = {k ∈ Z i ≤ k ≤ j}.
A foldable map f : S → T is a fold if there exist initial segments e, e′ ⊂ S of
oriented natural edges such that e ∩ e′ = v is their common initial point, and there
exists an orientation preserving homeomorphism h : e → e′ such that for all x 6= x′ ∈ S,
f (x) = f (x′ ) if and only if there exists γ ∈ Fn such that γ(x) ∈ e, γ(x′ ) ∈ e′ , and
h(γ(x)) = γ(x′ ).
We review the Bestvina-Feighn classification of folds given in [BF91] Section 2, with
simplifications as applied to free splittings. Consider free splittings Γ y S, T and a fold
map f : S → T which folds two edges e, e′ as above. Let w, w′ be the endpoints of e, e′
opposite their common initial endpoint v. Let π : S → S/Γ be the map to the quotient
graph of groups. The type I folds are as follows: f has type IA if π is one-to-one on
e ∪ e′ ; and f has type IB if (up to interchanging e, e′ ) the map π identifies v, w and
is otherwise one-to-one on e ∪ e′ . Type II folds do not occur in our setting, as they
involve nontrivial edge stabilizers. The type III folds are as follows: f has type IIIA if π
identifies w, w′ and is otherwise one-to-one on e ∪ e′ ; and f has type IIIB if π identifies
v, w, w′ and is otherwise one-to-one on e∪e′ . In all cases the extension F(S) ⊏ F(T ) can
be described explicitly. For types IA or IB: if at least one of Stab(w), Stab(w′ ) is trivial
then F(S) = F(T ) (and this is the only case of equality); otherwise [Stab(w)], [Stab(w′ )]
are two components of F(S), and F(T ) is obtained from F(S) by replacing those two
with the strictly larger component [hStab(w) ∪ Stab(w′ )i]. For a fold of type IIIA or
IIIB, letting g ∈ Γ be such that g(w) = w′ , F(T ) is obtained from F(S) by replacing the
component [Stab(w)] = [g −1 Stab(w′ )g] = [Stab(w′ )] with the strictly larger component
[hStab(w) ∪ {g}i].
31
A fold sequence is a foldable sequence denoted as above in which each of the maps
fi : Ti−1 → Ti is a fold map. Lemma 4.3 to follow is an instance of Stallings fold method,
and implies that every foldable sequence of K foldable maps (as denoted above) can be
interpolated by a fold sequence, meaning that for each k = 1, . . . , K the foldable map
Tk−1 7→ Tk may be factored as a fold sequence, and these K fold sequences may then be
concatenated to obtain a fold sequence from T0 to TK .
Remark on the “gate 3 condition”. In [HM13], in the setting of Γ = Fn , the
definition of a foldable map f : S → T had an additional requirement, the following
“gate 3 condition”: for any vertex p ∈ S of valence ≥ 3 the map f has at least three
gates at p. Here we follow Bestvina and Feighn [BF14b] to the extent of weakening
the definition of [HM13] by dropping the “gate 3 condition”. In what follows we will
occasionally explain how this change effects the proofs. For the most part these are
desirable changes, but see after the statement of Lemma 5.2 for a significant exception.
Two desirable effects of dropping the gate 3 condition are as follows. First, it allows for a
broader collection of fold sequences in the free splitting complex; this was an important
motivation for dropping that condition in [BF14b]. Second, the interpolation of the
previous paragraph does not generally work when foldable maps are required to satisfy
the gate 3 condition.
The following commonly used relativization tool is an immediate consequence of
Lemma 3.2 (3):
Lemma 4.1. If S ∈ FS(Γ; A) and T ∈ FS(Γ), and if there exists a map f : S → T ,
then T ∈ FS(Γ; A).
♦
Lemma 4.2 (cf. Lemma 2.4 of [HM13]). For any S, T ∈ FS(Γ; A) there exists S ′ , S ′′ ∈
FS(Γ; A) such that S ≺ S ′ ≻ S ′′ and such that a foldable map S ′′ 7→ T exists. If
F(S) ⊏ F(T ) then one can take S = S ′ .
Remark. The proof of the above lemma is considerably simpler than its [HM13]
analogue Lemma 2.4, due to the removal of the gate 3 condition.
Proof. Choose a free splitting Γ y S ′ such that S ≺ S ′ and F(S ′ ) ⊏ F(T ): if F(S) ⊏
F(T ) choose S ′ = S; otherwise, applying Lemma 3.2 (3), choose S ′ so that S ≺ S ′ and
F(S ′ ) = A ⊏ F(T ). In either case we have S ′ ∈ FS(Γ; A).
There exists a map S ′ 7→ T which on each edge of S is either constant or injective:
for each v ∈ S ′ choose f (v) ∈ T in a Γ-equivariant manner so that Stab(v) < Stab(f (v)),
and extend linearly over each edge; this is possible because F(S ′ ) = A ⊏ F(T ). For
each such map, let S ′ be subdivided so that each edgelet maps either to a vertex or an
edge of T . Amongst all such maps S ′ 7→ T , choose f : S ′ → T to minimize the number
32
f′
f ′′
of orbits of edgelets of S ′ on which f is nonconstant. Factor f as S ′ −→ S ′′ −→ T where
f ′ collapses to a point each component of the union of edgelets on which f is constant.
The map f ′′ is injective on each edgelet.
To prove that f ′′ is foldable it remains to show that at each vertex v ∈ S ′′ the
map f ′′ has at least two gates. Suppose to the contrary that f ′′ has only one gate
at v, let e1 , . . . , eI be the edgelets incident vertex v, and let w1 , . . . , wI be their opposite
endpoints. Let S ′′ 7→ S ′′′ be the quotient map obtained by collapsing to a point each
of e1 , . . . , eI and all edgelets in their orbits, so we get an induced action Γ y S ′′′ .
Noting that w1 , . . . , wI all map to the same point in T , there is an alternate description
of S ′′′ as follows: remove from S ′′ the point v and the interiors of e1 , . . . , eI , identify
w1 , . . . , wI to a single point, and extend equivariantly. From this description it follows
that the map f ′′ : S ′′ 7→ T induces a map S ′′′ 7→ T , and by construction the composition
f′
S ′ −→ S ′′ 7→ S ′′′ 7→ T is nonconstant on a smaller number of edgelet orbits than f is
nonconstant on. This contradicts minimality of the choice of f .
Applying Lemma 4.1 we have S ′′ ∈ FS(Γ; A).
♦
Lemma 2.7 of [HM13] shows in the case Γ = Fn that any foldable map of free
splittings S 7→ T factors into a fold sequence of free splittings, and the exact same proof
works for general Γ. Assuming in addition that S ∈ FS(Γ; A), by applying Lemma 4.1
inductively starting with S it follows that each term in the fold sequence is in FS(Γ; A).
This proves:
Lemma 4.3 (cf. Lemma 2.7 of [HM13]). For any S, T ∈ FS(Γ; A), any foldable map
S 7→ T factors as a fold sequence in FS(Γ; A).
♦
Lemma 4.4 (cf. Lemma 2.5 of [HM13]). Given S, T ∈ FS(Fn ; A), if there is a fold
S 7→ T then in FS(Fn ; A) we have d(S, T ) ≤ 2.
Proof. Let e, e′ ⊂ S be oriented segments with the same initial vertex p that are folded
by S 7→ T . In [HM13] Lemma 2.5 the following is proved in the case Γ = Fn , but the
proof applies in general: either there is an expansion S ≺ T ; or there is an expansion–
collapse S ≺ U ≻ T where U ≻ S collapses some edge e ⊂ U down to the point p, the
edge e has one vertex of valence 3, the opposite vertex of e has the same stabilizer as
p, and other vertex stabilizers outside the orbit of p are unaffected by this collapse. It
follows that F(U ) = F(S), so U ∈ FS(Γ; A) and d(S, T ) ≤ 2.
♦
A sequence of vertices (Ti )i∈I in FS(Γ; A), parameterized by some subinterval I ⊂ Z,
is called a fold path if for each i − 1, i ∈ I there exists a map fi : Ti−1 → Ti such that
fi−1
fi
fi+1
→ Ti −
−−
→ · · · is a fold sequence.
the sequence of maps · · · −−−→ Ti−1 −
By combining Lemmas 4.2, 4.3 and 4.4 (cf. remark following Theorem 3.1 of [HM13])
we have proved:
33
Corollary 4.5. FS(Fn ; A) is connected. Fold paths form an almost transitive sequence
of paths in FS(Fn ; A), meaning: for any S, T ∈ FS(Fn ; A) there is a fold path starting
at distance ≤ 2 from S, making jumps of distance ≤ 2, and ending at T .
♦
4.2
F S(Γ; A) in low complexity cases
Using the results of Section 4.1 we now give a complete description of free splitting
complexes FS(Γ; A) in two low complexity cases where FS(Γ; A) is a very specific finite
diameter tree. In all remaining cases we conjecture that FS(Γ; A) is of infinite diameter,
indeed that the action of Out(Γ; A) on FS(Γ; A) has loxodromic elements.
The first low complexity case is when DFF (A) = 0, which occurs if and only if
A = {[A1 ], [A2 ]} and Γ = A1 ∗ A2 . The second is when DFF (A) = 1 and |A| ≤ 1, which
occurs if and only if A = {[A]} and Γ = A ∗ Z where Z is infinite cyclic. We consider
these cases separately in Propositions 4.6 and 4.7 to follow.
Proposition 4.6. Suppose that DFF (A) = 0, equivalently A = {[A1 ], [A2 ]} has a realization of the form Γ = A1 ∗ A2 . In this case FS(Γ; A) is a single point, corresponding
to the Bass-Serre tree of the free factorization Γ = A1 ∗ A2 .
Remark. In the case that A1 , A2 are free of finite rank, this proposition is contained
in [BFH00] Corollary 3.2.2. The proof here is an extension of that proof.
Proof. We first note the fact that for any proper free factor system A′ of Γ, if A ⊏ A′
then A = A′ . It follows that for any free splitting Γ y T rel A, we have A = F(T ).
We next note the fact that since S has one edge orbit, if S ≻ S ′′ then S and S ′′ are
equivalent.
Given a vertex T ∈ FS(Γ; A) we must prove that S, T are equivariantly homeomorphic. Applying Lemma 4.2 and the facts noted above, it follows that there exists
a foldable map f : S → T . Applying Lemma 4.3, there exists a fold sequence from S
to T . However, at each vertex v ∈ S all of the directions at v are in the same orbit of
the subgroup Stab(v), because the quotient graph of groups S/Γ has two vertices each
of valence 1. A fold map cannot fold two directions in the same orbit. Thus the fold
sequence from S to T has length zero and S, T are equivalent.
♦
For describing the next case, we need a few definitions.
Consider a free product Γ = A ∗ Z where Z = hzi is infinite cyclic, and consider
the free factor system A = {[A]}. Define a monomorphism A ֒→ Out(Γ; A) denoted
a 7→ φa , where φa is represented by Φa ∈ Aut(Γ) which is characterized by Φa A = Id,
Φa (z) = za. Noting that the subgroups hzi and hzai are conjugate in Γ if and only if a
is trivial, it follows that the homomorphism a 7→ φa is injective.
34
In any 1-complex X, a star point is a 0-cell v such that the closure of each component
of X − v is an arc called a beam of X (we do not require a beam to consist of a single
edge). If a star point exists then X is a star graph.
Proposition 4.7. Suppose that DFF (A) = 1 and |A| = 1, equivalently A = {[A]} has a
realization of the form Γ = A ∗ Z where Z = hzi is infinite cyclic. In this case FS(Γ; A)
is a star graph with star point T such that each beam has the form T ≺ S ≻ R with
quotient graphs of groups as follows (assuming natural cell structures):
Loop type: T /Γ has one vertex labelled A and one edge forming a loop with both endpoints at the vertex.
Sewing needle type: S/Γ has two vertices, one labelled A and the other of valence 3
labelled with the trivial group, with one edge connecting the A vertex to the valence 3
vertex, and one edge forming a loop with both ends at the valence 3 vertex.
Edge type: There exists a realization Γ = A ∗ Z of A such that R/Γ has two vertices,
one labelled A and the other labelled by the infinite cyclic group Z, and one edge
connecting the two vertices.
Furthermore, under the monomorphism A ֒→ Out(Γ; A) given by a 7→ φa described
above, the induced action A y FS(Γ; A), is free and transitive on the set of beams,
allowing beams to be enumerated as follows:
• Every free factorization of the form Γ = A ∗ Z ′ satisfies Z ′ = hzai for a unique
a ∈ A.
• There are bijections: {beams of FS(Γ; A)} ↔ {edge-type free splittings rel A} ↔
{free factorizations Γ = A ∗ hzai, a ∈ A} ↔ A.
Since A is nontrivial, there are at least two beams and the diameter of FS(Γ; A) equals 4.
Remark. As was the case for Proposition 4.6, the proof of Proposition 4.7 is an
elaboration upon the proof of Corollary 3.2.2 of [BFH00] which is concerned with the
case that Γ is free of some finite rank n and A is free of rank n − 1.
Proof. The proof uses Bass-Serre theory [SW79] and the Bestvina–Feighn classification
of folds [BF91] that was reviewed earlier.
For any free splitting Γ y U representing a 0-simplex of FS(Γ; A), the free factor
system F(U ) satisfies either F(U ) = A, or F(U ) = A ∪ {[Z]} for some free factorization
Γ = A ∗ Z with Z infinite cyclic. It follows that U has a unique vertex v(U ) such that
Stab(v(U )) = A.
First we prove existence of a free splitting rel A of loop type. From the hypotheses
on A it follows that there exists a free factorization Γ = A ∗ Z with Z infinite cyclic, the
35
Bass-Serre tree of which is an edge type free splitting Γ y R. Expanding R by blowing
up the Z vertex of R/Γ into a loop one gets a free splitting Γ y S of sewing needle
type. Collapsing the non-loop edge of S/Γ one gets a free splitting of loop type.
Fix now a loop type free splitting Γ y T rel A. Consider any free splitting Γ y U
rel A. Since F(T ) ⊏ F(U ) and T has one edge orbit it follows, as in the proof of
Proposition 4.6, that there is a foldable map f : T → U . Note that f (v(T )) = v(U ).
The derivative dv(T ) : Dv(T ) T → Dv(U ) U is either one-to-one or two-to-one.
In the first case where dv(T ) is one-to-one, the map f is a homeomorphism and T ≡ U ,
just as in the proof of Proposition 4.6.
In the second case where dv(T ) is two-to-one, consider a fold sequence that factors
f1
f
the map f , given by T = T0 −→ T1 → · · · −−K
→ TK = U with K ≥ 1. Using the
Bestvina-Feighn classification of fold types described earlier, the folds in this sequence
are as follows. If K = 1 then f : T → U is either of type IA and U is of sewing needle
type, or f is of type IIIA and U is of edge type. If K ≥ 2 then each of f1 , . . . , fK−1 is
of type IA, and each of T1 , . . . , TK−1 is of sewing needle type; the final fold fK is either
of type IA and TK = U is also of sewing needle type, or fK is of type IIIA and TK = U
is of edge type.
Note in particular that if U is of loop type then dv(T ) is not two-to-one, and so any
foldable map T 7→ U is a homeomorphism and T ≡ U , so there is a unique loop-type
0-cell in FS(Fn ; A).
We have proved that each 0-cell in FS(Fn ; A) is represented by a free splitting of one
of the three types described. Each 0-cell of sewing needle type collapses to exactly two
other 0-cells, namely the unique one of loop type and one other of edge type. Each 0-cell
of edge type expands to exactly one other 0-cell, that being of sewing needle type. This
proves that the unique loop type 0-cell is a star point and each beam is as described.
To prove the “Furthermore” clause, by applying Proposition 4.6 to any free factor
system of the form {[A], [Z ′ ]} where Z ′ is a cofactor of A, it follows that there is an
Out(Γ; A)-equivariant bijection between the set of edge-type free splittings relative to
A and the set of conjugacy classes of cofactors of realizations of A. Each realization of
A is conjugate in Γ to one of the form Γ = A ∗ Z ′ . It therefore suffices to show that
each realization of the latter form is conjugate to a unique one of the form A ∗ hzai,
a ∈ A. Uniqueness follows from the observation that hzai is conjugate to hzbi if and
only if a = b. To prove existence, pick a generator Z ′ = hz ′ i. The two free factorizations
Γ = A ∗ hzi = A ∗ hz ′ i determine two loop type free splittings rel A, namely the BassSerre trees of the two HNN extensions of A over the trivial group, one with stable letter
z and the other with stable letter z ′ . But we proved above that any two loop type free
splittings rel A are equivalent, and it follows that z ′ = bz ±1 c for some b, c ∈ A. After
possibly replacing z ′ with its inverse we have z ′ = bzc, which is conjugate to zcb, and
taking a = cb we are done.
♦
36
4.3
Combing
Consider a foldable map f : S → T of free splittings of Γ rel A. Given a Γ-invariant
subgraph σT ⊂ T , a component of σT is is degenerate if it consists of a single point.
Assuming that σT has no degenerate component, the pullback of σT is the subgraph σS
obtained from f −1 (σT ) by removing degenerate components.
Following [HM13] Section 4.1 (but using the current definition of foldable sequences),
a combing rectangle in FS(Γ) is defined to be a commutative diagram of free splittings
of Γ of the form
S0
f1
/ ···
T0
/ Si−1
fi
[σi−1 ] πi−1
[σ0 ] π0
fi−1
g1
/ ···
gi−1
/ Ti−1
gi
/ Si
fi+1
/ ···
/ Ti
/ SK
[σK ] πK
[σi ] πi
fK
gi+1
/ ···
gK
/ TK
where the top and bottom rows are foldable sequences, each vertical arrow πk : Sk →
Tk is a collapse map with indicated collapse forest σk , and each σk is obtained from
k )−1 (σ ) by removing any components that degenerate to a point; we say that σ is
(fK
K
k
k . If A is a free factor system of Γ and each S , T
the pullback of σK under the map fK
k k
is in FS(Γ; A) then we also say this is a combing rectangle in FS(Γ; A).
Denoting a combing rectangle in shorthand as (Si ; Ti )0≤i≤K , two combing rectangles (Si ; Ti )0≤i≤K and (Si′ ; Ti′ )0≤i≤K ′ are said to be equivalent if K = K ′ and if there
are equivariant homeomorphisms Si ↔ Si′ and Ti ↔ Ti′ making all resulting squares
commute.
Lemma 4.8 (Relative combing by collapse, cf. [HM13] Proposition 4.3). For any combing rectangle, if its top row is in FS(Γ; A) then so is its bottom row. For any foldable
sequence S0 7→ · · · 7→ SK and any collapse πK : SK → TK in FS(Γ; A), there exists a
combing rectangle with that top row and right edge, and that combing rectangle is unique
up to equivalence.
Proof. The first sentence follows from Lemma 4.1. The existence statement in second
sentence is proved in the case Γ = Fn , A = ∅ in [HM13] Proposition 4.3, that proof
works without change to prove existence in our present setting, and the proof also gives
uniqueness. In outline: define σi ⊂ Si uniquely as required by the definition; use σi to
[σi ]
uniquely define the collapse map Si −−→ Ti ; check that there is a well-defined induced
map gi : Ti−1 → Ti which uniquely defines the bottom row; and then check that the
bottom row is a foldable sequence.
♦
Lemma 4.9 (Relative combing by expansion, cf. [HM13] Proposition 4.4). For any
foldable sequence T0 7→ · · · 7→ TK and any collapse map πK : SK → TK in FS(Γ; A)
37
there exists a combing rectangle in FS(Γ; A) with that bottom row and right edge, and
that combing rectangle is unique up to equivalence.
Proof. The existence proof in the case Γ = Fn , A = ∅ is found in [HM13], Proposition
4.4, “Step 1” and “Preparation for Step 2” (the further work in Step 2 of that proof
is entirely concerned with establishing the gate 3 condition for the S row, and so is
not relevant to us here). Following that proof, consider the fiber product of the two
free splittings Γ y Ti , Γ y SK with respect to the two Γ-equivariant maps Ti 7→ TK ,
SK 7→ TK . This fiber product is the subset of the Cartesian product Ti × SK consisting
of ordered pairs (x, y) such that the image of x in TK equals the image of y in TK . It
is a simplicial tree on which Γ acts with trivial edge stabilizers, and we define Sk to be
the minimal subtree for that action. The two projection maps of the Cartesian product
induce maps πi : Si → Ti and hiK : Si → SK . Exactly as in “Step 1”, the map πi is a
collapse map which collapses a subforest σi ⊂ Si , and σi is the set of nondegenerate
components of (hiK )−1 (σK ). And exactly as in “Preparation for Step 2”, the map hiK is
injective on edgelets and has ≥ 2 gates at each vertex, and so hiK is foldable according
to our current definition. We thus have a combing diagram in FS(Γ), and we need
to check that Si ∈ FS(Γ; A). For each subgroup A < Γ such that [A] ∈ A, since A
fixes unique points of Tk and of SK it follows that A fixes a unique point of the fiber
product tree; since A is nontrivial, that fixed point is in the minimal subtree Si , and so
Si ∈ FS(Fn ; A).
Uniqueness follows by noticing that for any combing rectangle, the maps πi : Si → Ti
i : S → S
and fK
i
K embed Si in the fiber product tree of the two maps Ti 7→ TK and
SK 7→ TK . Since the action of Γ on Si is minimal it follows that Si is identified with the
i
minimal subtree of the fiber product tree, and under this identification the maps πi , fK
are identified with the restrictions of the projection maps of the Cartesian product. The
desired uniqueness property is an immediate consequence.
♦
4.4
Invariant subgraphs of free splittings, and their complexity
This section is concerned with an important technical underpinning of the proof of hyperbolicity. The key idea is that as one moves along a fold path, one studies “pullback
subgraph sequences” along that path, meaning a sequence of Γ-equivariant subforests,
one in each free splitting along that fold path, such that the sequence is invariant under
pullback of the fold maps along that path. We focus on how the topology of the subforest varies along the sequence, and we use numerical measurements of “complexity” to
measure this change of topology. These subgraphs are just forests, of course: the only
aspects of their topology that concerns us are their component sets; and the maps on
component sets induced by foldable maps will be the only aspects of change of topology
that we consider.
38
The way the results of this section will be applied in what follows is to use upper and
lower bounds on the change of complexity along fold paths to obtain information about
upper and lower bounds on distance in FS(Γ; A) along folds paths; see the discussion
just below regarding the definition of complexity.
4.4.1
Definition of complexity.
Consider a free factor system A of Γ, a free splitting Γ y T relative to A, and a Γinvariant proper subgraph β ⊂ T with no degenerate components. We shall define a
positive integer valued complexity denoted C(β) which is a sum of several terms. This
complexity C(β) will be dominated by a single term C1 (β), equal to the number of
Γ-orbits of components of β: indeed the difference C(β) − C1 (β) is bounded above and
below by constants depending only on Γ and A, as we shall see in Lemma 4.10.
The definition of complexity is designed so that various upper and lower bounds on
C(β) can be used to obtain topological and metric conclusions. The most important of
these conclusions are as follows:
• From upper bounds on complexity we obtain upper bounds on diameters along
fold paths: see Lemma 4.13 (3a) and Lemma 4.14, and applications of those lemmas in later sections. Underlying these diameter bounds is the key technical result
Lemma 4.12. The terms forming the difference C(β) − C1 (β) are designed specifically to make Lemma 4.12 work.
• From lower bounds on C(β) we obtain lower bounds on C1 (β), from which we
deduce that some component of β is an arc in the interior of a natural edge of T :
see Lemma 4.11 and Fact 4.16 (5b). Ultimately this leads to lower bounds on
diameter along fold paths, as expressed in Theorem 5.4.
One may formally view the proof of hyperbolicity of FS(Γ; A) as a game in which upper
and lower bounds on complexity are played against each other, to obtain various upper
and lower bounds on distance as needed for proving hyperbolicity.
The complexity C(β) is defined by adding four non-negative integer summands:
C(β) = C1 (β) + C2 (β) + C3 (β) + C4 (β)
These summands are each tailored to cases in the proof of Lemma 4.12. For defining
them, recall the free splitting T /β obtained from T by collapsing to a point each component of β. The free factor system F(T /β) decomposes into two subsets F(T /β) =
F(β) ⊔ F(T − β) as follows: given [B] ∈ F(T /β), put [B] in F(β) if B stabilizes some
component of β, and put [B] in F(T − β) if B stabilizes some point of T − β.
• Define C1 (β) to be the number of Γ-orbits of components of β.
39
• Define C2 (β) = DFF (F(T /β)).
• Define C3 (β) to be the number of components [A] ∈ A satisfying the following:
the component of F(T /β) containing [A] is in the set F(T − β); equivalently [A]
stabilizes some vertex of T − β.
For defining C4 (β), first apply Lemma 2.11 using A and A′ = F(T /β), with the following
conclusions: the components of F(T /β) can that do not contain any component of A
can be listed as [B1 ], . . . , [BN ] where each of B1 , . . . , BN is free of finite rank and their
free product B1 , . . . , BN is a free factor of a cofactor of a realization of A (in the notation
of Lemma 2.11 these components are [A′J+1 ], . . . , [A′K ]). Up to re-indexing there exists
M ∈ {0, . . . , N } so that [B1 ], . . . , [BM ] ∈ F(T − β) and [BM +1 ], . . . , [BN ] ∈ F(β). Thus
[B1 ], . . . , [BM ] are precisely the components of F(T /β) that do not contain a component
of A and whose representative subgroups B1 , . . . , BM each fix some point of T −β. Since
each Bm is free of finite rank, it follows that the set F(T /β) − {[B1 ], . . . , [BM ]} is still
a free factor system, and it is still true that A ⊏ F(T /β) − {[B1 ], . . . , [BM ]}. Define
C4 (β) = corank F(T /β) − [B1 ], . . . , [BM ]
I removed a “−1”
from
C2 (β).
Compare Lemma
4.10, where the
second inequality
for C2 can be
equality only with
that “−1” being
removed.
Check
whether there are
any consequences
downstream,
for
various
bounds.
I
suspect
that
the bounds all
depend on Lemma
4.10 rather than
on direct use of
the definition of
C2 (β), and so we
will probably be
safe. — Lee
M
X
rank(Bm )
= corank F(T /β) +
m=1
We record for later use some estimates that are evident from the definitions:
Lemma 4.10. The three summands C2 (β), C3 (β), C4 (β) have the following bounds:
0 ≤ C2 (β) ≤ 2 corank(A) + |A| − 1
(by Lemma 2.14 (1))
0 ≤ C3 (β) ≤ |A|
0 ≤ C4 (β) ≤ corank(A)
4.4.2
(by Corollary 2.10)
♦
Consequence of a lower bound on complexity.
Lemma 4.11 to follow gives a very simple topological consequence for a specific lower
bound on the subgraph complexity. Further consequences of that lower bound are derived later in Proposition 4.16 (5b), and those consequences will play an important
role in the central arguments of Section 5, particularly in the statement and proof of
Proposition 5.3 where that constant is denoted b1 = 5 corank(A) + 4 |A| − 3.
Lemma 4.11. For any free splitting Γ y T rel A, and for any Γ-invariant subgraph
β ⊂ T , if C(β) > 5 corank(A)+4 |A|−3 then some component of β is an arc contained
in the interior of a relatively natural edge of T .
Proof. Combining the hypothesis with Lemma 4.10 it follows that
C1 (β) > 2 corank(A) + 2 |A| − 2
40
The definition of
C2 (β) not have
the “−1”, else the
second inequality
cannot possible be
an equation.
—
Lee
The right hand side is the maximal number of relatively natural vertex orbits amongst
all free splittings of Γ rel A, according to Proposition 3.4 (3). So β has more than that
number of component orbits, and hence one of those orbits must be disjoint from the
relatively natural vertices of T . Each component in that orbit is therefore contained in
the interior of some relatively natural edge.
♦
4.4.3
Complexity change under a foldable map.
We now turn to a study of subgraph complexity along a foldable sequence, starting with
its behavior under a single foldable map.
Lemma 4.12. (c.f. [HM13] Sublemma 5.3) Let S, T be free splittings of Γ rel A, let
f : S → T be a foldable map, let βT ⊂ T be a proper Γ-invariant subgraph, and let
βS ⊂ T be the pullback of βT under f . Then we have Ci (βS ) ≥ Ci (βT ) for i = 1, 2, 3, 4,
and hence C(βS ) ≥ C(βT ). Furthermore, if C(βS ) = C(βT ) then f induces a bijection
between the set of components of βS and the set of components of βT .
Proof. The Γ-equivariant surjection f : βS → βT induces a Γ-equivariant surjection of
component sets f∗ : π0 (βS ) → π0 (βT ) which induces in turn a surjection of component
orbit sets f∗∗ : π0 (βS )/Γ → π0 (βT )/Γ. It follows that C1 (βS ) ≥ C1 (βT ). By Lemma 4.8
the foldable map f : S 7→ T induces a foldable map f /β : S/βS → T /βS and hence
F(S/βS ) ⊏ F(T /βT ). Applying Lemma 2.14 (1) it follows that C2 (βS ) ≥ C2 (βT ).
To prove the inequality C3 (βS ) ≥ C3 (βT ), we use the fact that the composition of
containment functions A 7→ F(S/βS ) 7→ F(T /βT ) is the containment function A 7→
F(T /βT ). It follows that for each component [A] of A, if the component of F(S/βS )
containing [A] is in F(βS ) then the component of F(T /βT ) containing [A] is in F(βT ).
The inequality C3 (βS ) ≥ C3 (βT ) follows. Furthermore, for the equation C3 (βS ) =
C3 (βT ) to hold is equivalent to saying that for each [A] ∈ A, [A] is contained in F(S −βS )
if and only if [A] is contained in F(T − βT ).
We next prove the inequality C4 (βS ) ≥ C4 (βT ). Consider nested components [B] ⊏
′
[B ] of F(βS ) ⊏ F(βT ), respectively. Note that if B ′ stabilizes a point of T − βT then
B stabilizes a point of S − βS , and so if [B ′ ] ∈ F(T − βT ) then [B] ∈ F(S − βS ).
Let [B1 ], . . . , [BM ] be the components of F(S − βS ) not containing a component of A,
′ ] be the components of F(T − β ) not containing a component
and let [B1′ ], . . . , [BM
′
T
of A. It follows that we have an extension of free factor systems to which we apply
41
Lemma 2.14 (1):
′
F(S/βS ) − {[B1 ], . . . , [BM ]} ⊏ F(T /βT ) − {[B1′ ], . . . , [BM
(4.1)
′ ]}
′
corank F(S/βS ) − [B1 ], . . . , [BM ]
≥ corank F(T /βT ) − [B1′ ], . . . , [BM
′]
(4.2)
C4 (βS ) ≥ C4 (βT )
(4.3)
with equality holding in (3.2) if and only if it holds in (3.3).
Assuming that C(βS ) = C(βT ), and so Ci (βS ) = Ci (βT ) for i = 1, 2, 3, 4, it remains
to prove that f∗ is a bijection. From surjectivity of f : βS → βT it follows that f∗ is
also surjective, and what is left is to show that f∗ is injective. Consider a component
b′ of βT ; we must prove that there is exactly one nondegenerate component of f −1 (b′ ).
Since C1 (βS ) = C1 (βT ) it follows that f∗∗ is a bijection, and so all of the nondegenerate
components of f −1 (b′ ) are in the same Γ-orbit. If f −1 (b′ ) has more than one nondegenerate component then any element of γ taking one to the other is a nontrivial element
of Stab(b′ ); therefore if Stab(b′ ) is trivial then f −1 (b′ ) has only one nondegenerate component and we are done.
We have reduced to the case that Stab(b′ ) is nontrivial; by definition we have that
[Stab(b′ )] ∈ F(βT ). Since DFF (F(S/βS )) + 1 = C2 (βS ) = C2 (βT ) = DFF (F(T /βT )) + 1,
and since F(S/βS ) ⊏ F(T /βT ), by applying Lemma 2.14 (1) we have:
(∗)
F(S/βS ) = F(T /βT )
Consider the subcase that some nondegenerate component b of f −1 (b′ ) has nontrivial
stabilizer. Since Stab(b) < Stab(b′ ), it follows from (∗) that Stab(b) = Stab(b′ ). But
since all nondegenerate components of f −1 (b′ ) are in the same orbit, b must be the only
such component, because otherwise any γ ∈ Γ taking b to a different nondegenerate
component is an element of Stab(b′ ) but not of Stab(b). The proof is therefore complete
in this subcase.
We have further reduced to the subcase that all nondegenerate components of f −1 (b′ )
have trivial stabilizer, and in this subcase we shall derive a contradiction. It follows from
(∗) that there exists x ∈ S − βS such that Stab(x) = Stab(b′ ) ≡ H < Γ. By definition
we have [H] = [Stab(x)] ∈ F(S − βS ) whereas [H] = [Stab(b′ )] ∈ F(βT ). We now break
into two cases, depending on whether [H] contains some element of A.
Suppose first that [H] contains some [A] ∈ A, and so up to conjugacy we have
A < H. Since C3 (βS ) = C3 (βT ), it follows that A stabilizes a point of S − βS if and only
if A stabilizes a point of T − βT if and only if A does not stabilize any component of βT .
But A stabilizes the point x of S − βS and the component b′ of βT , a contradiction.
42
Suppose next that [H] contains no element of A. In the notation of (3.1), up to
conjugacy we have H = Stab(x) = Bm for some m = 1, . . . , M and so [H] is not an
element of the left hand side of (3.1), although [H] = [Stab(b′ )] is an element of the
right hand side. By Lemma 2.11 applied to the extension in (3.1), it follows that the
free factor system on the left hand side of (3.1) has a realization with a cofactor B
that freely factors into two or more nontrivial terms, one term up to conjugacy being
H = Bm (one of the terms denoted A′J+1 , . . . , A′K in Lemma 2.11), and another term
being a cofactor B ′ for a realization of the free factor system on the right hand side.
From this we obtain C4 (βS ) = rank(B) ≥ rank(B ′ ) + rank(Bm ) > rank(B ′ ) = C4 (βT ),
contradicting that C4 (βS ) = C4 (βT ).
♦
4.4.4
Complexity change along a foldable sequence.
f1
f2
fK
Given a foldable sequence T0 −→ T1 −→ · · · −−→ TK , a pullback subgraph sequence is a
sequence of Γ-invariant subgraphs βk ⊂ Tk , each with no degenerate components, such
k , equivalently for
that for each k ∈ {0, . . . , K} the graph βk is the pullback of βK via fK
i
0 ≤ i ≤ j ≤ K the graph βi is the pullback of βj via fj . For example, the sequence
of subgraphs occurring in a combing rectangle (see Section 4.3) is a pullback subgraph
sequence. Note that the complementary sequence ρk = cl(Tk −βk ) ⊂ Tk is also a pullback
subgraph sequence, and the subgraphs βk , ρk decompose the tree Tk in the sense that
every edgelet of Tk is in either βk or ρk ; such a sequence of decompositions Tk = βk ∪ ρk
is called a pullback blue–red decomposition.
The next lemma uses upper bounds on complexity to derive diameter bounds along
fold sequences.
Lemma 4.13 (cf. [HM13] Lemma 5.2). Given a pullback subgraph sequence of a foldable
sequence of free splittings of Γ rel A, as denoted above, the following holds:
(1) The quantities C1 (βk ), C2 (βk ), C3 (βk ), C4 (βk ), and C(βk ) are all nonincreasing
as functions of k.
(2) Equality C(βk−1 ) = C(βk ) implies that fk : Tk−1 → Tk restricts to a bijection from
components of βk−1 to components of βk .
(3) On any subinterval a ≤ k ≤ b along which C(βk ) is constant we have:
(a) The diameter of {Ta , . . . , Tb } in FS ′ (Γ; A) is at most 4.
(b) C1 (βa ) ≤ C1 (βb ) + (3 corank(A) + 2 |A| − 1)
Proof. Items (1) and (2) follow from Lemma 4.12. Item (3b) is an immediate consequence
of Lemma 4.10 combined with the equation C(βa ) = C(βb ) and the monotonicity of
C2 (βi ), C3 (βi ) and C4 (βi ).
43
The old version
had “if and only
if” instead of “implies that”. — Lee
To prove (3a), first apply item (2) to conclude that each fk induces a bijection from
the component set of βk−1 to the component set of βk . Now apply the proof of [HM13]
Lemma 5.2 (3) with little change; here is an outline. Given a ≤ i ≤ j ≤ b, there exists
0 ≤ P ≤ Q, a fold sequence Ti = U0 7→ · · · 7→ UP 7→ · · · 7→ UQ = Tj , free splittings X, Y ,
and a collapse expand sequence Ti = U0 ≻ X ≺ UP ≻ Y ≺ UQ = Tj . To see why, the
first part of the fold sequence U0 7→ · · · 7→ UP is done by folding only blue edgelet pairs
until the induced foldable map UP 7→ Tj is injective on blue edgelets. This is possible
because fji : Ti → Tj induces a bijection from components of βi to components of βj , and
so as one applies the construction of a fold factorization of fji , as long as the induced
foldable map to Tj is not yet injective there must exist a blue edgelet pair in some
component of βj which is ready to be folded, by virtue of having a common endpoint
and having the same image in Tj . The second part of the fold sequence from UP to
UQ may then be done by folding only red edgelet pairs. There is a single free splitting
Γ y X obtained by collapsing all blue edgelets of U0 or of UP , and a single free splitting
Γ y Y obtained by collapsing all red edgelets of UP or of UQ . Applying Lemma 4.1
and using that Ti ∈ FS(Γ; A), it follows, in order, that X, UP , Y ∈ FS(Γ; A), and hence
d(Ti , Tj ) ≤ 4.
♦
The construction in the following lemma is adapted from an argument of Bestvina
and Feighn, namely Lemma 4.1 of [BF14b], and translated into the language of complexity. In the context of FS(Fn ) this construction has the simplifying effect of enfolding
two upper bounds on distance from [HM13]—the “almost invariant edge bound” and the
“blue–red decomposition bound”—into a single distance bound. In the current context,
Lemma 4.14 will be used in the proof of Lemma 5.2 to get certain upper bounds to
distance along fold paths.
Lemma 4.14. For any foldable map f : S → T of free splittings of Γ rel A, and for
any point x ∈ T in the interior of some edgelet, there is a proper, Γ-invariant subgraph
βT ⊂ T with pullback subgraph βS ⊂ S such that
C(βS ) ≤ f −1 (x) + (3 corank(A) + 2 |A| − 1)
Proof. Let e be the edgelet whose interior contains x, so f −1 (e) is a union of f −1 (x)
edgelets, and by subdividing further we may assume that this is a disjoint union. Letting
βT = Γ · e, it follows that C1 (βS ) = f −1 (x) , and the conclusion follows immediately
from Lemma 4.10.
♦
4.5
Free splitting units
In [HM13] we defined free splitting units along fold paths of FS(Fn ), and showed that
they give a uniformly quasigeodesic parameterization of fold paths in FS(Fn ) (see Theorem 5.4). We shall do the same here in the relative setting of FS(Γ; A).
44
The manner in which free splitting units are applied in this paper is the same as in
[HM13]: any argument that bounds distance gives more information, by bounding free
splitting units. One can think of free splitting units as way of taking distance estimates
that are buried in the details of various proofs, bringing those estimates to the surface,
and using them to give explicit quasigeodesic parameterizations of fold paths, as will be
done in Theorem 5.4.
The manner in which free splitting units are defined in this paper is a little different
than in [HM13], with influences from [BF14b]. The definition of free splitting units along
fold paths in FS(Fn ), as given originally in [HM13] Section 5.2, involves two bounds: a
“blue–red” distance bound (c.f. Lemma 4.13(3a) in our present context); and an “almost
invariant edge” distance bound. As it turns out, the two bounds can be enfolded into a
single, simpler bound, a fact which we overlooked in [HM13]. A hint of this can be seen
in the simplifications of certain steps of the proof of hyperbolicity of FS(Fn ) that can
be found in [BF14b] and which are taken up in various places around the paper; see for
example Lemma 4.14 and the preceding discussion. Taking this hint, we present here a
simplified version of free splitting units, generalized to the setting of FS(Γ; A).
We expect this new definition of free splitting units seems to be more powerful and
to have more applications.
f1
f2
fK
Definition 4.15. Consider a fold sequence S0 −→ S1 −→ · · · −−→ SK in FS(Γ; A), and
consider 0 ≤ i ≤ j ≤ K.
(1) Define a collapse–expand diagram rel A over Si 7→ Sj to be a commutative diagram
of the form
/ Tj−1
/ ···
/ Tj
/ Ti+1
Ti
Si′
/ S′
fi+1
/ Si+1
/ S′
j−1
O
/ ···
i+1
O
O
Si
fi+2
/ ···
fj−1
/ Sj−1
/ S′
Oj
fj
/ Sj
where each horizontal row is a foldable sequence rel A and each of the two rectangles shown is a combing rectangle rel A. The diagram is trivial if all vertical
arrows are simplicial isomorphisms.
(2) We say that Si , Sj differ by < 1 free splitting unit rel A if there exists a collapse expand diagram over Si 7→ Sj , denoted as above, such that on the top row
Ti → · · · → Tj there exists a pullback subgraph sequence βk ⊂ Tk of constant
complexity C(βk ).
45
(3) More generally, the number of free splitting units rel A between Si and Sj is the
maximum length Υ = Υij of a subsequence i ≤ i(0) < · · · < i(Υ) ≤ j such that for
u = 1, . . . , Υ the number of free splitting units between Si(u−1) and Si(u) is not < 1.
Any such subsequence i(0) < · · · < i(Υ) of [i, j] is called a greedy sequence between
Si and Sj (while we do not require i = i(0) and i(Υ) = j, Proposition 4.16 (2)
guarantees that such a greedy sequence exists).
(4) For 0 ≤ i ≤ j ≤ K, the back greedy subsequence between i, j is the decreasing
sequence j = L0 > L1 > · · · > LU ≥ i defined inductively as follows: if Lu is
defined, and if there exists k with i ≤ k < u such that Lk , Lu differ by ≥ 1 free
splitting unit rel A, then Lu+1 is the largest such value of k. The front greedy
subsequence is the increasing sequence in [0, K] defined similarly.
(5) We extend free splitting units to a symmetric function by requiring Υij = Υji .
The following summarizes basic properties of free splitting units. Of particular importance is item (5b), which derives from Lemma 4.11, and which plays a central role in
the “big diagram argument”, the proof of Proposition 5.3.
f1
f2
f
Proposition 4.16. Consider a fold sequence S0 −→ S1 −→ · · · −−K
→ SK in FS(Γ; A),
and let Υij be the number of free splitting units rel A between Si , Sj . We have:
(1) For 0 ≤ i ≤ j ≤ K and i′ , j ′ ∈ [i, j], if Si , Sj differ by < 1 free splitting unit then
Si′ , Sj ′ differ by < 1 free splitting unit.
(2) (c.f. [HM13] after Definition 5.10) For any 0 ≤ i ≤ j ≤ K there exists a greedy
sequence between Si and Sj with first term i and with final term j. Also, the
front and back greedy sequences between Si and Sj are, indeed, greedy sequences,
in particular they have length equal to Υij .
(3) The“short triangle inequality” (c.f. [HM13] Lemma 5.12): For any i, j, k ∈ {0, . . . , K}
we have
Υik ≤ Υij + Υjk + 1
(4) The“long triangle inequality”: For any sequence 0 ≤ k0 < k1 < . . . < kL ≤ K we
have:
Υk0 , k1 + · · · + ΥkL−1 , kL ≤ Υk0 , kL ≤ Υk0 , k1 + · · · + ΥkL−1 , kL + L − 1
(5) For any collapse expand diagram as in Definition 4.15, and for any pullback sequence βk ⊂ Tk defined for i ≤ k ≤ j, we have:
(a) Υij ≤ C(βi ) − C(βj )
46
(b) If Υij ≥ 5 corank(A) + 4 |A| − 3 then some component of βi is arc in the
interior of a natural edge of Ti .
(6) (c.f. [HM13] Lemma 5.11) The diameter of {Si , . . . , Sj } is ≤ 10Υij + 8.
Proof. Item (1) is an immediate consequence of Definition 4.15. Items (2), (3) follow
exactly as in the references given above; their proofs are elementary.
To prove the first inequality of (4), for l = 1, . . . , L apply (2) to obtain a subsequence
of [kl−1 , kl ] which is a greedy sequence between Skl−1 and Skl , which starts with kl−1 ,
which ends with kl , and which has length Υk(l−1),k(l) . The union of these subsequences
is a sequence of length equal to the sum Υk0 ,k1 + · · · + ΥkL−1 ,kL , because the subsequence
of [kl−1 , kl ] ends with kl and the subsequence of [kl , kl+1 ] begins with kl . Between any
two terms of this sequence the number of free splitting units is ≥ 1, so a greedy sequence
between Sk0 and SkL has length no less than the sum.
The second inequality of (4) is a generalization of (3), and is proved as follows. If the
second inequality is false then there exists a greedy sequence between Sk0 and SkL whose
length is at least Υk0 , k1 + · · · + ΥkL−1 , kL + L. From the pigeonhole principle, for some
l = 1, . . . , L we obtain a greedy sequence between Skl−1 and Skl of length ≥ Υkl−1 ,kl + 1,
a contradiction.
Item (5b) follows from (5a) and Lemma 4.11 together with C(βj ) ≥ 1. To prove (5a),
apply (2) to obtain a greedy sequence
i = k(0) < k(1) < · · · < k(Υij ) = j
By Lemma 4.13 (1) we can uniquely decompose the interval [i, j] = [k(0), k(Υij )] as a
concatenation of M maximal subintervals on each of which C(βk ) is constant:
[ l(0) , l(1)], [l(1)+1, l(2)], . . . , [l(M −2)+1, l(M −1)], [l(M −1)+1, l(M ) ]
| {z }
|{z}
=k(Υij )
=k(0)
If Υij ≥ C(βi ) − C(βj ) then, since C(βi ) − C(βj ) ≥ M − 1, it follows Υij + 1 ≥ M , and
so there exists u ∈ [1, Υij ] and m ∈ [1, M ] such that k(u − 1), k(u) ∈ [l(m − 1) + 1, l(m)].
It follows further that C(βk ) is constant for k ∈ [k(u − 1), k(u)], contradicting that there
are ≥ 1 free splitting units between Sk(u−1) and Sk(u) .
Item (6) is proven just as in the reference given, except that one applies Lemma 4.4
and Lemma 4.13 (3a) in place of the analogous results of [HM13]: subdivide the interval
[i, j] into a concatenation of maximal subintervals on which C(βk ) is constant; apply
Definition 4.15(1,2) and Lemma 4.13(3a) to obtain diameter ≤ 8 over each subinterval;
and apply Lemma 4.4 to obtain distance ≤ 2 between incident endpoints of adjacent
subintervals.
♦
47
5
Hyperbolicity of relative free splitting complexes
In this section we prove hyperbolicity of the relative free splitting complex FS(Γ; A)
for any group Γ and any free factor system A. The proof uses the three Masur–Minsky
axioms for hyperbolicity of a connected simplicial complex, which are reviewed in Section 5.1, where one will also find specific details about how those axioms will be verified
for FS(Γ; A). Section 5.2 contains the proof of the first of those axioms, the Coarse
Retract Axiom; this where we pay the piper for dropping the “gate 3 condition” on
fold paths. Section 5.3 contains the statement of Proposition 5.3, which states that
certain properties of fold maps and free splitting units together imply the two remaining
Masur–Minsky axioms—the Coarse Lipschitz and the Strong Contraction Axioms. The
proof of Theorem 1.1 is thereby reduced to Proposition 5.3. Section 5.4 also applies
Proposition 5.3 to the proof of Theorem 5.4 which says that free splitting units give
a quasigeodesic parameterization along a fold path. Section 5.5 contains the proof of
Proposition 5.3, what we call the “Big Diagram” argument, an argument concerning the
large scale behavior of certain diagrams of combing rectangles in FS(Fn ; A).
The structure of this proof of Theorem 1.1, in particular the Big Diagram argument,
follows very closely the structure of the proof of hyperbolicity of FS(Fn ) given in [HM13].
But changing the definition of foldable maps by dropping the gate 3 condition has some
major effects on this structure: the proof of the Coarse Retract Axiom is quite a bit
more complex and so has needed to be rewritten from the beginning; and subtle changes
in the Big Diagram Argument make it necessary to re-present it from the beginning. In
both cases, we take up these changes from the version of the proof given by Bestvina
and Feighn in [BF14b].
5.1
The Masur–Minsky axioms.
Suppose one is given a connected, finite dimensional simplicial complex X with the
simplicial metric, a collection P of finite “paths” p : {0, . . . , Lp } → X (0) , and for each
p ∈ P a “projection map” πp : X (0) → {0, . . . , Lp }. Suppose that P is almost transitive
meaning that there is a constant A with two properties: for each p ∈ P and ℓ ∈
{1, . . . , Lp } we have d(p(ℓ − 1), p(ℓ)) ≤ A; and for each x, y ∈ X (0) there exists p ∈ P
such that d(x, p(0)) ≤ A and d(p(Lp ), y) ≤ A (as noted in [HM13], “almost transitivity”
yields equivalent axioms compared to “coarse transitivity” as used originally in [MM99]).
Suppose furthermore that there are constants a, b, c > 0 such that the following three
axioms hold for each p ∈ P :
Coarse Retract Axiom: For each ℓ ∈ {0, . . . , Lp } the diameter of p[ℓ, πp (p(ℓ))] is at
most c.
Coarse Lipschitz Axiom: For all x, y ∈ X (0) , if d(x, y) ≤ 1 then the diameter of
48
p[πp (x), πp (y)] is at most c.
Strong Contraction Axiom: For all x, y ∈ X (0) , if d(x, p[0, Lp ]) ≥ a, and if d(y, x) ≤
b · d(x, p[0, Lp ]), then the diameter of p[πp (x), πp (y)] is at most c.
The theorem proved by Masur and Minsky [MM99] is that if these axioms hold then X
is hyperbolic.
For verifying these axioms and hence proving hyperbolicity of FS(Γ; A), we let P be
the collection of all fold sequences rel A, for which we have already established Coarse
Transitivity in Corollary 4.5. In Section 5.2 we define the system of projection maps
πp , p ∈ P , and we prove the Coarse Retract Axiom (see Lemma 5.2). In Section 5.3
we shall reduce the Coarse Lipschitz and Strong Contraction Axioms to a single statement regarding fold sequences and free splitting units rel A (see Proposition 5.3). In
Section 5.4 we prove that fold paths are uniformly quasigeodesic when parameterized
by free splitting units (see Theorem 5.4, which also uses Proposition 5.3). Finally in
Section 5.5 we prove Proposition 5.3 using the “big diagram argument”.
5.2
Projection maps and the proof of the coarse retract axiom.
Given a fold path in FS(Γ; A) represented by a particular fold sequence, we now define
the projection map to that fold path, as required for the formulation of the Masur–
Minsky axioms. We then immediately turn to verification of the Coarse Retract Axiom,
which takes up the bulk of this section.
Definition 5.1. Given a fold sequence S0 7→ · · · 7→ SK and a free splitting T each in
FS(Γ; A), a projection diagram rel A from T to S0 7→ · · · 7→ SK is defined to be a
projection diagram as in [HM13] Section 4.1 in which all free splittings that occur are
restricted to lie in FS(Γ; A). This means a commutative diagram of free splittings and
maps of the form
/ TJ
/T
/ ···
T0
S0′
/ ···
/ S′
OJ
S0
/ ···
/ SJ
O
/ ···
/ SK
such that each free splitting is in FS(Γ; A), each row is a foldable sequence, and each of
the two rectangles shown is a combing rectangle. The integer J ∈ {0, . . . , K} is called
the depth of the projection diagram. The projection of T to S0 7→ · · · SK is an integer
π(T ) ∈ {0, . . . , K} defined as follows: if there exists a projection diagram rel A from T
to S0 7→ · · · SK then π(T ) is the maximal depth of such diagrams; otherwise π(T ) = 0.
49
Lemma 5.2 (The Coarse Retract Axiom). For any fold sequence S0 7→ · · · 7→ SK in
FS(Γ; A) and any 0 ≤ I ≤ K the number of free splitting units between SI and Sπ(SI ) ,
and the diameter of the fold sequence between SI and Sπ(SI ) , are both bounded above by
constants depending only on corank(A) and |A|.
By assuming the gate 3 condition, the proof of the Coarse Retract Axiom in [HM13]
was significantly simpler than the argument to be presented here. In lieu of that assumption, we instead adapt some concepts and arguments of Bestvina and Feighn from
[BF14b], namely the “hanging trees” of Proposition A.9; see “Claim (#)” below.
Proof. The proof starts as in [HM13]. Note that π(SI ) ≥ I, because there exists a
projection diagram rel A from SI to S0 , . . . , SK of depth I, namely the trivial diagram
defined by taking Ti = Si′ = Si for i = 0, . . . , I.
Choose a projection diagram rel A of maximal depth J = π(SI ) ≥ I from SI to
S0 7→ · · · 7→ SK , as follows:
/ ···
/ TI
/ TJ
/ ···
/ SI
T0
S0′
O
/ ···
/ S′
OI
/ ···
/ S′
OJ
/ SI
/ SJ
/ SK
/ ···
/ ···
/ ···
S0
Once we have bounded the number of free splitting units between SI and SJ = Sπ(SI ) ,
the diameter bound on the set {SI , . . . , Sπ(SI ) } follows from Proposition 4.16 (6).
The key observation is that in the foldable sequence TI 7→ · · · 7→ TJ 7→ SI , its
first and last terms TI , SI each collapse to the same free splitting, namely SI′ . This
observation will be combined with the following:
f1
f
L
Claim (#): Consider a fold sequence U0 −→ · · · −→
UL in FS(Γ; A). If there exists a
free splitting R ∈ FS(Γ; A) and collapse maps U0 7→ R, UL 7→ R, then there exist
integers 0 = ℓ0 ≤ ℓ1 ≤ ℓ2 = L, and for i = 1, 2 there exist xi ∈ Uℓi , such that
ℓ
the inverse image (fℓii−1 )−1 (xi ) ⊂ Uℓi−1 has cardinality bounded by a constant b#
depending only on corank(A) and |A|.
Before proving Claim (#), we apply it to finish the proof of the lemma, as follows.
By replacing each individual arrow in the foldable sequence TI 7→ · · · 7→ TJ 7→ SI by
a fold sequence that factors it, we obtain a fold sequence which contains TI , . . . , TJ , SI
as a subsequence. To that fold sequence we may then apply Claim (#), combined with
Lemma 4.14 followed by Lemma 4.13 (1), with the effect of subdividing the fold sequence between TI and TJ into at most two subintervals along each of which there is
50
pullback subgraph sequence of bounded complexity difference. Then applying Proposition 4.16 (5) we obtain a subdivision of the fold sequence between SI and SJ into at
most two subintervals along each of which the number of free splitting units rel A is
bounded. Applying Proposition 4.16 (3) we obtain an upper bound to the number of
free splitting units rel A between SI and SJ .
f =f 0
L
We turn to the proof of Claim (#). Denote V = U0 −−−→
UL = W . Choose oriented
relatively natural edges eV = [v− , v+ ] ⊂ V , eW = [w− , w+ ] ⊂ W which map onto the
same oriented relatively natural edge eR = [r− , r+ ] ⊂ R under collapse maps V, W 7→ R.
Decompose V \ eV = V− ∪ V+ and W \ eW = W− ∪ W+ so that v± is the frontier of V± ,
and w± is the frontier of W± , respectively. We have equations
(∗)
f (V+ ) = W+ ∪ [w+ , f (v+ )],
f (V− ) = W− ∪ [w− , f (v− )]
which are obtained by referring to [HM13] Lemma 5.5 and following the proof of the
implication (4) =⇒ (1), except that one may ignore the very last sentence which is the
only place in that proof where the gate 3 condition was used. We briefly outline the
proof of (∗) for f (V+ ). Decompose R \ eR = R− ∪ R+ so that r± is in the frontier of R± .
Let Γ+ be the set of elements of Γ acting loxodromically on R with axis contained
in R+ . First one shows the inclusion W+ ⊂ f (V+ ) by proving for each γ ∈ Γ+ that γ
acts loxodromically on V and W with axes contained in V+ and W+ respectively, and
that the union of such axes over γ ∈ Γ+ equals V+ , W+ respectively, and finally using
that for each γ ∈ Γ+ the f image of the axis of γ in V+ contains the axis of γ in W+ .
Next one shows, by bounded cancellation, that f (V+ ) is contained in a finite radius
neighborhood Nr (W+ ) of W+ . Finally, using that neighborhood, one shows that if (∗)
fails then f (V+ ) contains a valence 1 point distinct from f (v+ ), that point has the form
f (x) for some x ∈ V+ − v+ , and f has only one gate at x, contradicting foldability of f .
Consider the oriented segment f (eV ) = [f (v− ), f (v+ )] ⊂ W . If f (eV ) intersects
int(eW ) and preserves orientation, then f is one-to-one over some point x ∈ int(eW ), so
Claim (#) is proved with ℓ1 = L, x1 = x2 = x, and b# = 1. If f (eV ) intersects int(eW )
and reverses orientation, then f has only one gate at v− and at v+ , a contradiction.
Remark. Under the gate 3 hypothesis on the given fold sequence the proof of
Claim (#) ends here, because the segment f (eV ) must intersect int(eW ); see the last
lines of the proof of [HM13] Lemma 5.5. Without the gate 3 hypothesis our work
continues for rather a long while.
We may assume that f (eV ) is a subset of W− or of W+ ; up to reverse of orientation
we have
f (eV ) ⊂ W+ ,
f (V+ ) = W+ ,
and
eW = [w− , w+ ] ⊂ [w− , f (v− )]
{z
}
|
=α
51
We orient α with initial endpoint w− and terminal endpoint f (v− ), and we parameterize
α by simplicial distance from w− , inducing a linear order on α which lets us speak of
maxima and minima in α. Let Σ = V− ∩ f −1 α and ξ = Σ ∩ f −1 (w− ). Note that Σ − ξ
is connected: otherwise the closure of some component of Σ − ξ would not contain v− ,
its image in α would have a maximum value achieved at some x ∈ int(Σ), and x would
have one gate, a contradiction. It follows that Σ is connected, that each point of ξ
has valence 1 in Σ, and that ξ ∈ Fr(Σ). Furthermore Fr(Σ) = ξ ∪ {v− }, and the map
f : Σ → α takes Fr(Σ) to ∂α, mapping ξ to w− and v− to f (v− ).
Consider the initial edgelets of Σ meaning the edgelets incident to points of ξ, each
of which maps to the initial edgelet of eW . It follows that the initial edgelets of Σ are all
in different Γ-orbits, and so there are only finitely many of them, implying that ξ is finite
and so Σ is a finite tree. Since each vertex of Σ−Fr(Σ) has at least two gates with respect
to the map f Σ, and since f (Σ − Fr(Σ)) ⊂ α it follows that each vertex of Σ − Fr(Σ) has
exactly two gates and that f (Σ − Fr(Σ)) ⊂ int(α) = (w− , f (v− )). Assign an orientation
to each edgelet of Σ so as to point towards v− . By induction on distance to ξ it follows
that f maps each edgelet of Σ to an edgelet of α in an orientation preserving manner.
It follows that for each x ∈ ξ the map f takes [x, v− ] one-to-one onto α. Furthermore at
each y ∈ int(Σ) there is therefore a unique positive direction with respect to f , namely
the direction pointing towards v− , which is the unique direction at y whose image under
f is the direction at f (y) ∈ α pointing towards f (v− ). All other directions at y form
the negative gate, each mapping to the direction at f (y) ∈ α pointing back towards w− .
This gives Σ the structure of a “hanging tree” in the terminology of [BF14b]. It follows
that every edgelet of Σ that maps to the initial edgelet of eW is an initial edgelet of Σ.
We break into two cases depending on the behavior of the following subset of Γ:
Zb = {γ ∈ Γ int(Σ) ∩ int(γ · Σ) 6= ∅}
= {γ ∈ Γ Σ ∩ γ · Σ contains at least one edgelet}
b = {Id}. Consider the graph of groups V /Γ. It follows in Case 1 that
Case 1: Z
the orbit map V → V /Γ restricts to an injection on int(Σ). The images of the initial
edgelets of Σ are therefore all contained in distinct oriented relatively natural edges
of V /Γ. Applying Proposition 3.4 (1) it follows that the number of initial edgelets is
bounded by the number 2 DFS (A) + 2 = 6 corank(A) + 4 |A| − 6. Taking this number
to be b# , Claim (#) is proved with ℓ1 = L and with x1 = x2 = an interior point of the
initial edgelet of eW .
Case 2: Zb 6= {Id}. The action of each γ ∈ Zb on the tree W restricts as
τγ : (γ −1 · α) ∩ α → α ∩ (γ · α)
Furthermore, this map τγ is an isometry with respect to the parameterization of α
described earlier, so we may speak about whether γ preserves or reverses orientation,
52
and if γ preserves orientation we may also speak about the translation length of γ, all
by reference to what τγ does to the parameterization of α.
We next show:
(1) For each γ ∈ Zb the arc α ∩ γ · α has endpoints in the set ∂α ∪ γ · ∂α.
This follows from the earlier description of how f maps Σ to α and Fr(Σ) to ∂α, together
with the fact that Fr(Σ ∩ (γ · Σ)) ⊂ Fr(Σ) ∪ (γ · Fr(Σ)).
If γ reverses orientation it follows from (1) that γ 2 fixes the arc α ∩ γ · α and so,
b
since W is a free splitting, γ 2 is trivial, but that is a contradiction. Every element of Z
therefore preserves orientation.
We define γ ∈ Zb to be positive if γ has positive translation length with respect to
the parameterization of α; for γ to be negative is similarly defined by requiring negative
f
b
translation length. Thinking of the map Σ −
→ α as a “height function”, an element of Z
is positive if and only if it increases height in Σ, and negative if and only if it decreases
height.
b we shall show:
Letting Z < Γ be the group generated by Z,
b such that Z = hγi is infinite cyclic.
(2) There exists a positive γ ∈ Z
For any free splitting Γ y X in which the action of the cyclic group Z is not elliptic,
let Ax(X) denote the axis of that cyclic group. We show furthermore that:
(3) The set H = Z · Σ ⊂ V is a two-ended tree on which Z acts cocompactly, there is
a Z-equivariant deformation retraction H 7→ Ax(V ), and
f (H) = f (Z · Σ) = Z · f (Σ) = Z · α = Ax(W )
(4) The map f : H → Ax(W ) gives H the structure of a “bi-infinite hanging tree” as
follows: at each x ∈ H the map f H has a positive gate consisting of the unique
direction at x whose f -image points towards the positive end of Ax(W ), and if x
is not of valence 1 then all other directions at x are in a single negative gate whose
f -image points towards the negative end of Ax(W ).
(5) All translates of the tree H by elements of Γ − Z have disjoint interiors.
b whose translation distance on α is a
For the proofs of (2)–(5), pick a positive γ ∈ Z
b note that γ −1 δ is in Z
b and is non-negative. By
minimum. Given a positive δ ∈ Z,
−i
b
induction there exists i ≥ 0 such that γ δ ∈ Z and has translation number zero,
implying that it fixes an arc of α, and so δ = γ i . This proves (2), and (3) follow easily.
Item (4) follows from the analogous properties of the map f : Σ → [w− , f (v− )]. For (5),
suppose δ ∈ Γ has the property that the interiors of H and δ · H are not disjoint. Choose
53
integers i, j such that the interiors of γ i · Σ and δγ j · Σ are not disjoint, so the interiors of
Σ and γ −i δγ j Σ are not disjoint. By (2) we have γ −i δγ j ∈ Zb and so δ ∈ Z, proving (5).
From properties (2)–(5) it follows that the 1-complex H/Z deformation retracts to
the circle Ax(V )/Z. Furthermore, the induced map H/Z 7→ V /Γ is an embedding on the
complement of the valence 1 vertices. Define an initial edgelet of H to be an oriented
edgelet whose initial vertex has valence 1 in H, and so we have a bijection between initial
edgelets and valence 1 vertices. Define an initial edgelet of H/Z in a similar fashion. We
have a bijection between Z-orbits of initial edgelets of H and initial edgelets of H/Z.
Under the map H/Z 7→ V /Γ, the initial edgelets of H/Z all map into distinct oriented
natural edges of V /Γ. The number of Z-orbits of initial edgelets of H is therefore
bounded above by 2 DFS (A) + 2 = 6 corank(A) + 4 |A| − 6 (see Proposition 3.4 (1)), and
so the number of Z-orbits of valence 1 vertices of H has the same bound. A branch
of H is an oriented arc with initial endpoint at a valence 1 vertex, terminal endpoint
on Ax(V ), and interior disjoint from Ax(V ). Letting βv ⊂ H denote the branch with
initial vertex v, we have a Z-equivariant bijection v ↔ βv between valence 1 vertices
and branches, and so:
(6) The number of Z-orbits of branches of H is bounded by
2 DFS (A) + 2 = 6 corank(A) + 4 |A| − 6
We also have, as a consequence of property (4), the following:
(7) The map f : V → W is injective on each branch βv , mapping it homeomorphically
to an arc of Ax(W ). In particular, βv is legal with respect to f .
Consider now the whole fold sequence V = U0 7→ · · · 7→ UL = W . Choose x2 ∈ eW
to be in the interior of some edgelet. If fL0 is one-to-one over x2 then we are done with
x1 = x2 , ℓ1 = L, and b# = 1. Otherwise, let ℓ1 ∈ {0, . . . , L} be the largest integer such
that the map fLℓ1 : Uℓ1 → UL is not 1-to-1 over x2 . Since fLℓ1 +1 is 1-to-1 over x2 , and
since the fold map fℓ1 +1 is at worst 2-to-1 over the interior of each edgelet, it follows
that fLℓ1 is exactly 2-to-1 over x2 . Let y ∈ Uℓ1 +1 be the unique point of (fLℓ1 +1 )−1 (x2 ).
Note that y ∈ Ax(Uℓ1 +1 ), because
x2 ∈ α ⊂ Ax(W )
(see item (3))
= Ax(UL ) ⊂ fLℓ1 +1 (Ax(Uℓ1 +1 ))
Under the fold map fℓ1 +1 : Uℓ1 → Uℓ1 +1 the point y has exactly 2 pre-images, exactly
one of which denoted x1 is disjoint from Ax(Uℓ1 ). Let P = (fℓ01 )−1 (x1 ) ⊂ U0 , which is
disjoint from Ax(U0 ). It remains to show
Claim (∗) The cardinality of P is ≤ the number of Z-orbits of branches of H.
54
Applying this claim, it follows by (6) that the cardinality of P is bounded above by
2 DFS (A) + 2 = 6 corank(A) + 4 |A| − 4. Claim (#) is then proved by taking b# =
max{2, 2 DFS (A) + 2} = 2 DFS (A) + 2.
For proving Claim (∗), by applying item (5) we conclude that P ⊂ int(H), so P is
the inverse image of x1 under the restriction of fℓ01 to H. Consider Hℓ1 = fℓ01 (H) ⊂ Uℓ1 .
From items (3), (4), which describe the infinite hanging tree structure on H with respect
to the map f = fL0 : H → Ax(W ), it follows that Hℓ1 also has an infinite hanging tree
structure with respect to the map fLℓ1 : Hℓ1 → Ax(W ). For each branch βv ⊂ H, by (7)
the map f takes βv homeomorphically onto a subsegment of Ax(W ), from which it follows
that fℓ01 maps βv homeomorphically onto its image fℓ01 (βv ), a path which therefore takes
no illegal turns in Hℓ1 with respect to the map fLℓ1 : Hℓ1 → Ax(W ). Combining this
with the infinite hanging tree structure on Hℓ1 , it follows that if µ ⊂ βv is a subpath,
with homeomorphic image subpath fℓ01 (µ) ⊂ fℓ01 (βv ), and if the endpoints of fℓ01 (µ) are
disjoint from Ax(Uℓ1 ), then all of fℓ01 (µ) is disjoint from Ax(Uℓ1 ), because any path
between points in distinct components of an infinite hanging tree minus its axis must
contain an illegal turn.
If Claim (∗) fails then there exist b 6= b′ ∈ P , a branch βv ⊂ H, and γ i ∈ Z, such
that b ∈ βv and b′ ∈ γ i · βv . The path µ = [γ −i (b′ ), b] is contained in βv and so it
is mapped homeomorphically to the path fℓ01 (µ) = fℓ01 [γ −i (b′ ), b]. Also, the endpoints
γ −i (b′ ), b of µ are mapped by fℓ01 to the endpoints γ −i (x1 ), x1 of fℓ01 (µ), neither of which
are in Ax(Uℓ1 ). It follows that fℓ01 (µ) is disjoint from Ax(Uℓ1 ). In the tree U0 consider
the bi-infinite, γ i -invariant sequence of paths
· · · [γ −2i (b′ ), γ −i (b)], [γ −i (b′ ), γ 0 (b)], [γ 0 (b′ ), γ i (b)], [γ i (b′ ), γ 2i (b)], · · ·
{z
} |
{z
} |
{z
} |
{z
}
|
γ −i (µ)
µ
γ i (µ)
γ 2i (µ)
Since fℓ01 (γ −mi (b)) = γ −mi (x1 ) = fℓ01 (γ −mi (b′ )) for all m, it follows that the image of the
above sequence of paths under the map fℓ01 concatenates together to form a bi-infinite
γ i -invariant path in Uℓ1 which is disjoint from Ax(Uℓ1 ), a contradiction.
♦
5.3
Proof of Theorem 1.1: Reducing the coarse lipschitz and strong
contraction axioms to Proposition 5.3.
In [HM13] we proved hyperbolicity of FS(Fn ) by using fold sequences and free splitting
units to verify hyperbolicity axioms established by Masur and Minsky in [MM99]. We
follow the same method here to prove hyperbolicity of FS(Γ; A).
As alluded to earlier, Proposition 5.3 may be regarded as a translation of the Coarse
Lipschitz and Strong Contraction Axioms into a single statement regarding fold sequences and free splitting units. To state it we need one more definition.
55
/ ···
/ TJ
S0′
/ ···
/ S′
OJ
S0
/ ···
/ SJ
T0
O
/ TJ+1
/ ···
/ ···
/ SK
/ TL = T
Figure 2: An augmented projection diagram of depth J from T to S0 7→ · · · 7→ SK .
Given a fold sequence S0 7→ · · · 7→ SK and a free splitting T in FS(Γ; A), an augmented projection diagram over A of depth J from T to S0 7→ · · · 7→ SK is a commutative
diagram of free splittings and maps rel A of the form shown in Figure 2 such that each
horizontal row is a foldable sequence, the subsequence TJ 7→ · · · 7→ TL is a fold sequence,
and such that the two rectangles shown are combing rectangles. The diagram obtained
from Figure 2 by replacing the sequence TJ 7→ · · · 7→ TL with the composed foldable
map TJ 7→ TL is therefore an ordinary projection diagram as given in Definition 5.1.
Conversely, any projection diagram as given in Definition 5.1 can be converted into an
augmented projection diagram by simply factoring the map TJ 7→ TL as a fold sequence.
Proposition 5.3. [c.f. [HM13] Proposition 6.1] Let b1 = 5 corank(A) + 4 |A| − 3. Let
S0 7→ · · · 7→ SK be a fold sequence rel A, and let π : FS(Γ; A) → {0, . . . , K} be its
associated projection map. Let T be a free splitting rel A, and consider any augmented
projection diagram rel A of depth J = π(T ) (as denoted in Figure 2). Let Υ be the
number of free splitting units rel A between TJ and TL . For any free splitting R rel A,
if d(T, R) ≤ max{2⌊Υ/b1 ⌋, 1} and if the number of free splitting units rel A between S0
and SJ is at least b1 , then there exists ℓ ∈ [0, π(R)] such that the number of free splitting
units between Sℓ and SJ is at most b1 .
The conclusion says, in other words, that the projection of R to S0 → · · · → SK is no
further to the left of SJ than b1 free splitting units.
This proposition will be proved in the next section, using the Big Diagram argument.
For now we use it to prove our main results on hyperbolicity of FS(Γ; A) and on the
uniform quasigeodesic parameterization of fold paths using free splitting units.
Proof of Theorem 1.3. The argument follows closely the proof of hyperbolicity of FS(Fn )
given in [HM13] Section 6.1; here are a few details. We have already verified the Coarse
Retract Axiom in Lemma 5.2. Fixing free splittings T, R ∈ FS(Γ; A) we must verify the
Coarse Lipschitz Axiom and the Strong Contraction Axiom, which we do with constant
56
c = 10b1 + 8. After interchanging T, R we may assume π(R) ≤ π(T ) = J. We may also
assume that the number of free splitting units between S0 and SJ is at least b1 , for otherwise by applying Proposition 4.16 (6) the set {S0 , . . . , SJ } and its subset {Sπ(R) , . . . , SJ }
each have diameter ≤ 10b1 + 8 = c and the axioms follow.
For the Coarse Lipschitz Axiom, if d(T, R) ≤ 1 then Proposition 5.3 applies to
produce ℓ ≤ J such ℓ ≤ π(R) ≤ J and such that between Sℓ and SJ there are at most b1
free splitting units, so just as above the set {Sℓ , . . . , SJ } and its subset {Sπ(R) , . . . , SJ }
each have diameter ≤ 10b1 + 8 = c and the axiom follows.
For the Strong Contraction Axiom, one considers two cases. For the first case where
Υ < 2b1 , by Proposition 4.16 (6) we have d(T, TJ ) ≤ 20b1 +8 and so d(T, SJ ) ≤ 20b1 +10,
and taking a = 20b1 + 10 we may dispense with this case. For the second case where Υ ≥
2b1 ≥ 1, let b = 1/20b1 . Then follow exactly the final part of the proof of hyperbolicity of
FS(Fn ) given in [HM13] Section 6.1, with the conclusion that if d(T, R) ≤ b · d(T, S0 7→
· · · 7→ SK ) then d(T, R) ≤ Υ/b1 ≤ 2⌊Υ/b1 ⌋, and then Proposition 5.3 applies just as
above with the conclusion that {Sπ(R) , . . . , SJ } has diameter ≤ 10b1 + 8 = c, so the
axiom follows.
Having verified all of the Masur–Minsky axioms, hyperbolicity of FS(Γ; A) therefore
follows from [MM99].
♦
5.4
Theorem 5.4: Parameterizing fold paths using free splitting units
In the absolute case, [HM13] Proposition 6.2 shows that fold paths, when parameterized using free splitting units, are uniformly quasigeodesic in FS(Fn ). That argument
relativizes with very little change to the current setting using free splitting units rel A,
producing Theorem 5.4 below.
The free splitting unit parameterization of a fold path can be described with either a
discrete parameter or a continuous parameter. Consider a fold sequence S0 7→ · · · 7→ SM
rel A. Letting Υ be the number of free splitting units rel A from S0 to SM , one
chooses an integer sequence 0 = m0 < m1 < · · · < mΥ = M such that if 1 ≤ u ≤ Υ
then there is at least one free splitting unit between Smu−1 and Smu . The discrete
parameterization of this fold path by free splitting units is the map defined on the integer
interval [0, Υ] = {u ∈ Z 0 ≤ u ≤ Υ} 7→ X (1) define by u 7→ Smu . The continuous
parameterization is a function defined on the real interval {t ∈ R 0 ≤ u ≤ Υ}, whose
restriction to the subinterval u−1 ≤ t ≤ u parameterizes an interpolation of the fold path
Smu−1 7→ Smu−1 +1 7→ · · · 7→ Smu −1 7→ Smu , where each fold is replaced by an edge path in
X (1) of length at most 2 (c.f. Lemma 4.4). Since the set of vertices {Sm |mu−1 ≤ m ≤ mu }
has uniformly bounded diameter in FS(Γ; A) (by Proposition 4.16 (6)), uniform quasiisometry of the integer parameterization and of the real parameterization are equivalent
properties.
57
Theorem 5.4 (c.f. [HM13] Proposition 6.2). Parameterizations of fold paths by free
splitting units are uniform quasigeodesics in FS(Γ; A). That is, there exist constants
k ≥ 1, c ≥ 0 depending only on corank A and |A| such that for any fold path as denoted
above, the free splitting parameterization u 7→ Smu , defined for integers 0 ≤ u ≤ Υ, is a
(k, c) quasigeodesic in FS(Γ; A).
Proof. For this proof we switch from “subscript notation” to “function notation” along
the fold path, writing S(m) rather than Sm . By Proposition 4.16 (6), the map u 7→
S(mu ) is Lipschitz with constant 18. We must find constants k, c such that for each
u < v in [0, Υ], letting D = d(S(mu ), S(mv )), we have
|u − v| ≤ kD + c
Let π : FS(Fn ; A) → {0, . . . , M } denote the projection to the fold path S(0) 7→ · · · 7→
S(M ), and fix a projection diagram from S(mv ) to that fold path of maximal depth
π(S(mv )). Fix a geodesic edge path ρ in FS(Γ; A) between S(mu ) and S(mv ), having
length D. For any edge along this path, the number of free splitting units between their
π-images is uniformly bounded and so, by the “long triangle inequality” for free splitting
units, Proposition 4.16 (4), the number of free splitting units between S(π(S(mu ))) and
S(π(S(mv ))) is bounded above by kD for some uniform constant k. By the Coarse
Retract Axiom, Lemma 5.2, the number of free splitting units between S(π(S(mu )))
and S(mu ), and between S(π(S(mv ))) and S(mv ), is bounded above by some uniform
constant c′ . Again by the “long triangle inequality”, it follows that |u − v|, which is
the number of free splitting units between S(mu ) and S(mv ), is bounded above by
kD + c′ + 1.
♦
5.5
The proof of Proposition 5.3: Big Diagrams.
Throughout the proof we fix the constant b1 = 5 corank(A) + 4 |A| − 3, the geometric
significance of which was established in Lemma 4.11.
The proof of Proposition 5.3 is, in essence, a study of the large scale geometry
of certain diagrams of fold sequences and combing rectangles, diagrams that may be
regarded as living in the relative free splitting complex FS(Γ; A). We call these “big
diagrams”. We begin the proof by using the hypotheses of Proposition 5.3 to set up the
appropriate big diagram, and then we proceed to a study of its large scale geometry.
Constructing the Big Diagram, Step 0. The reader may refer to Figure 3 to follow
this construction.
Consider a fold sequence S0 7→ · · · 7→ SK in FS(Γ; A) with associated projection
map π : FS(Γ; A) → {0, . . . , K}. Consider also a free splitting T ∈ FS(Γ; A) with
augmented projection diagram over A of depth J = π(T ) as denoted in Figure 2 with
58
T = TL . Along the foldable sequence in the top horizontal line of that augmented
projection diagram, add a superscript 0, and so that sequence becomes
T00 7→ · · · 7→ TJ0 7→ · · · 7→ TL0 = T
Consider another free splitting R ∈ FS(Γ; A) and consider also any geodesic path
from TL0 to R in the 1-skeleton of FS(Γ; A). Since the concatenation of two collapse
maps is a single collapse map, a geodesic necessarily has the form of a zig-zag path
alternating between collapses and expansions. It is convenient for us to slightly alter the
geodesic path from TL0 to R so that it begins with a collapse and ends with an expansion:
in order to achieve this, prepend a trivial collapse and/or append a trivial expansion as
needed. The result is a path of even length D of the form
T = TL0 → TL1 ← TL2 → · · · ← TLD = R
where d(T, R) ≤ D ≤ d(T, R) + 2.
If D = 0 then T = R and we are done. Henceforth we assume D ≥ 2.
Construct a stack of D combing rectangles atop the foldable sequence T00 → · · · →
TL0 , by alternately applying relative combing by collapse, Lemma 4.8, and relative combing by expansion, Lemma 4.9, using the arrows in the path from TL0 to TLD , for a total
of D-applications. The result is the Big Diagram Step 0 depicted in Figure 3, in which
Tℓd denotes the entry in the “row d” and “column ℓ” of the stack of combing rectangles,
and in which we have highlighted certain columns and rows.
Here is the general idea of the proof of Proposition 5.3. Notice that each column of
the Big Diagram Step 0 is a zig-zag path, alternating between collapses and expansions.
The diagram has the shape of a piece of corrugated aluminum. The far right edge is, by
construction, a geodesic (except possibly for the first and last of its edges). The idea of
the proof is that as one sweeps leftward through the Big Diagram, one discovers shorter
vertical paths than the ones given in the diagram, allowing one to construct new Big
Diagrams with fewer corrugations between the top and bottom rows. Eventually enough
corrugations are removed to produce a projection diagram from R to S0 7→ · · · 7→ SK
from which one can estimate π(R).
For each even integer d with 2 ≤ d ≤ D − 2 we have a pair of collapse maps of the
[ρ]
[β]
form T d−1 ←− T d −→ T d+1 . If the subgraph ρ ∪ β ⊂ T d were proper in T d , then there
[β ′ ]
[ρ′ ]
would be a path T d−2 → T d−1 −−→ T h ←−− T d+1 ← T d+2 where T h is obtained by
[ρ∪β]
collapsing T d −−−→ T h , where β ′ is the image of β under T d 7→ T d−1 and ρ′ is the image
of ρ under T d 7→ T d+1 ; these images are proper, which is what allows this subpath to
exist. But by concatenating the two collapse maps from T d−2 to T h into a single collapse
map, and similarly for the two collapse maps from T d+2 to T h one obtains a shorter
path between TL0 and R, contradicting that the chosen path was geodesic. It follows
ρ ∪ β is not proper, that is T d = ρ ∪ β.
59
Letting Υ be the number of free splitting units rel A between TJ and TL , and letting
Ω = ⌊Υ/b1 ⌋, consider the sequence L = L0 > L1 > · · · > LΩ ≥ J which is obtained from
the right greedy sequence by taking only every bth
1 term. By induction it follows for each
1 ≤ ω ≤ Ω that Lω is the greatest integer ≤ Lω−1 such that between TLω and TLω−1
there are ≥ b1 free splitting units. Columns in big diagrams indexed by L0 , L1 , . . . , LΩ
will be emphasized as those diagrams evolve.
We have seen that we have a union TL20 = ρL0 ∪ βL0 . Knowing this, we may reduce
to the case that this union is a blue–red decomposition meaning that ρL0 ∩ βL0 contains
no edgelet: if this is not already so then we may alter the diagram to make it so, using
exactly the same normalization process described in [HM13] Section 6.2. In brief, one
replaces row T 2 by collapsing the intersection of red and blue along this row.
As in [HM13], the heart of the argument is an induction, starting with the Big
Diagram step 0 and producing Big Diagrams steps 1, 2, . . . , (D − 2)/2, each of which
consists of a stack of combing diagrams grouped into successive pairs forming collapse–
expand diagrams. At each step the number of combing rectangles decreases by 2, the
final diagram at step (D −2)/2 being just a stack of 2 combing diagrams forming a single
collapse–expand diagram. At all stages of the induction we highlight column TJ , the
number J being the projection of T onto S0 7→ · · · 7→ SK . Throughout the induction we
suppress the projection diagram atop which all big diagrams are formed. In particular
the foldable sequence T00 7→ · · · 7→ TJ0 is unaltered up until the case of the Big Diagram
step (D − 2)/2, at which point we again highlight the projection diagram, obtaining in
that case a stack of 4 combing rectangles. At that step we carry out one final alteration,
producing a stack of 2 combing rectangles forming a projection diagram from R to
S0 7→ · · · 7→ SK the depth of which is no more than b1 free splitting units to the left
of SJ .
For the induction step, assuming that D ≥ 4, we adopt variations introduced by
Bestvina and Feighn in [BF14b] for the method of successively producing the next Big
Diagram. We describe in detail the first step of the induction, going from step 0 in
Figure 3 to step 1 in Figure 8; further steps of the induction are then described very
briefly. The induction is complete at step (D − 2)/2, after which there will be one final
special alteration step, to be described in detail later.
The first induction step when D ≥ 4. Consider the collapse–expand diagram
defined by the subrectangle Tℓd for (ℓ, d) ∈ [L1 , . . . , L0 ] × [0, 1, 2], along the top row of
which we have an invariant blue–red decomposition Tℓ2 = βℓ ∪ ρℓ .
The key observation that gets the construction started is that βL1 , the collapse forest
[βL ]
for the map TL21 −−−1→ TL31 , has a component [x, y] which is a subarc of the interior of
some natural edge of the free splitting TL21 . This follows from the fact that there are ≥ b1
free splitting units between TL01 and TL00 , by applying Proposition 4.16 (5b). Let b ⊂ βL1
60
T0D
/ ···
/ TD
J
/ ···
/ TD
L1
/ ···
/ TD
L0
T04
/ ···
/ T4
J
/ ···
/ T4
L1
/ ···
/ T4
L0
/ ···
/ T3
OJ
/ ···
/ T3
L1
O
/ ···
/ T3
L0
O
T03
O
[β0 ]
T02
/ ···
/ T2
J
[ρ0 ]
[βL1 ]
[βJ ]
/ T2
L1
/ ···
[βL0 ]
/ ···
/ T2
L0
[ρL1 ]
[ρJ ]
[ρL0 ]
T01
/ ···
/ T1
OJ
/ ···
/ T1
L1
O
/ ···
/ T1
L0
O
T00
/ ···
/ T0
J
/ ···
/ T0
L1
/ ···
/ T0
L0
S0′
/ ···
/ S′
OJ
S0
/ ···
/ SJ
/ ···
/ SK
O
O
R
T
Figure 3: The Big Diagram, Step 0. Certain columns L = L0 , L1 , . . . are emphasized,
using free splitting units along the fold path TJ0 → · · · → TL0 . As the Big Diagram
evolves, and up until nearly the end of the evolution, the original projection diagram
atop which the diagram is built, which involves the S ′ and S rows, will not change. Those
rows will be suppressed in the meantime, returning only in the Penultimate Diagram of
Figure 9.
61
T03
O
/ ···
/ T3
OJ
/ ···
/ T3
L1
O
[Fn ·b]
T03b
O
/ ···
/ T 3b
JO
/ ···
/ T 3b
L1
O
′ ]
[βL
1
T02
/ ···
/ T2
J
/ ···
/ T2
L1
Figure 4: Factoring the combing rectangle between rows 2, 3 and columns 0, . . . , L1 .
be the blue edgelet in [x, y] with endpoint x. Let e be the red edgelet not in [x, y] with
[βL ]
1
endpoint x. Factor the collapse map TL21 −−−→
TL31 as a product of two collapse maps
as follows. The first factor collapses everything in βL1 except the orbit of b, collapsing
the subgraph βL′ 1 = βL1 \ Fn · b, and taking b to an edgelet b′ ⊂ TL3b1 . The second factor
collapses the orbit of b′ :
′ = β
[βL
L1 \Fn ·b]
[Fn ·b′ ]
TL21 −−−1−−−−−−−−→ TL3b1 −−−−→ TL31
Note that b′ is contained in the interior of some natural edge η ′ of TL3b1 . Also, letting
e′ ⊂ TL3b1 be the image of e, note that arc e′ ∪ b′ is also contained in η ′ .
Remark. The particular way in which the edgelets b and e are used in the above
paragraph is an innovation of Bestvina and Feighn in [BF14b], arising from dropping
the gate 3 condition on fold paths, and having the effect of simplifying the Big Diagram
argument.
[Fn ·b′ ]
The collapse map TL3b1 −−−−→ TL31 is equivariantly homotopic to a homeomorphism
h : TL3b1 7→ TL31 as follows. The homotopy is stationary off of the orbit of e′ ∪b′ . Restricted
to the arc e′ ∪ b′ , the collapse is a quotient map taking b′ to a point, and that quotient
map is homotopic, relative to the endpoints of the arc e′ ∪ b′ , to a homeomorphism;
extend that restricted homotopy over the orbit of e′ ∪ b′ .
Using the above concatenation of two collapse maps, the combing rectangle (ℓ, d) ∈
[0, L1 ]×[2, 3] factors it into a concatenation of two combing rectangles of the form shown
in Figure 4, whose right side is the above factorization of the collapse map TL21 7→ TL31
(here and later we silently apply the obvious generalizations to FS(Γ; A) of the results
of Section 4.3 of [HM13] which construct compositions and decompositions of combing
rectangles).
Now we proceed from step 0 to step 0.1, depicted in Figure 5. Starting from the step
0 diagram depicted in Figure 3, discard the portion of the diagram that lies strictly below
62
row 3 and right of column L1 , and the portion strictly above row 3 and left of column L1 .
Next, replace the combing rectangle (ℓ, d) ∈ [0, L1 ] × [2, 3] by inserting a certain portion
of the two concatenated combing rectangles from Figure 4, namely, the lower of the two
combing rectangles between row 2 and row 3b, plus the collapse map TL3b1 7→ TL31 ; do
not insert any part of row 3 to the left of TL31 , nor any of the vertical arrows to the left
of the collapse map TL3b1 7→ TL31 . And now replace the collapse map TL3b1 7→ TL31 by the
equivariant homeomorphism h : TL3b1 → TL31 , and using that homeomorphism identify the
free splittings TL3b1 ≈ TL31 . This ostensibly completes the construction of the Big Diagram
step 0.1 shown in Figure 5.
Unfortunately, the map h : TL3b1 ≈ TL31 is not simplicial, because h(x) is not a vertex
of TL31 . But h does become simplicial, after subdividing TL31 at the orbit of h(x). Unfortunately, after this subdivision the maps TL41 7→ TL31 7→ TL31 +1 are no longer simplicial.
To resolve this issue once and for all, we push the subdivision of TL31 up and to the right,
throughout the upper right rectangle of Figure 5 defined by (ℓ, d) ∈ [L1 , L0 ] × [3, D],
restoring that all maps in this rectangle are simplicial. Do this restoration by the following procedure: first push the subdivision forward along the row TL31 7→ · · · 7→ TL30 using
the fold maps of that row; then pull the subdivision back to the row TL41 7→ · · · 7→ TL40
under the collapse maps from row 4 to row 3; then push the subdivision forward to the
row TL51 7→ · · · 7→ TL50 under the collapse maps from row 4 to row 5; etc. Using the
simplicial homeomorphism h we may now identify TL3b1 and TL31 , truly completing the
construction of the Big Diagram step 0.1.
We must show that the following row in Figure 5 is a fold sequence:
T03b 7→ · · · 7→ TJ3b 7→ · · · 7→ TL3bΩ 7→ · · · 7→ TL3b1 ≈ TL31 7→ · · · 7→ TL30
where the homeomorphism h is used to identify TL3b1 ≈ TL31 . By construction it is a
fold sequence from T03b to TL3b1 and from TL31 to TL30 , and so it suffices to show that if
0 ≤ ℓ ≤ L1 then the map Tℓ3b 7→ TL30 has at least two gates at each vertex yℓ ∈ Tℓ3b . Let
various images of yℓ under the maps in Figure 4 be denoted yL1 ∈ TL3b1 , zℓ ∈ Tℓ3 , and
zL1 ∈ TL31 , and so we have zL1 = h(yL1 ). There are two cases depending on whether
yL1 ∈ Fn · b. If yL1 6∈ Fn · b then yℓ is not in the collapse graph of the map Tℓ3b 7→ Tℓ3 , and
so under this collapse map the directions at yℓ and at zℓ correspond bijectively as do the
directions at yL1 and at zL1 . The gates at yℓ and at zℓ for the maps to TL31 therefore also
correspond bijectively, and so the gates at yℓ and at zℓ for the maps to TL30 correspond
bijectively, but at zℓ there are at least two such gates, and so at yℓ there are also at least
two such gates. If yL1 ∈ Fn · b then yL1 has valence 2 as does zL1 , and at yℓ the map
to TL31 has exactly two gates, one for each direction at zL1 ; those two directions map to
two different directions in TL30 , and so there are two gates at yℓ for the map Tℓ3b → TL30 .
Next we proceed from step 0.1 to step 0.2, depicted in Figure 6. Starting from the
step 0.1 diagram depicted in Figure 5, apply relative combing by collapse, Lemma 4.8,
63
T03b
O
/ ···
/ T 3b
JO
/ ···
/ T2
J
[ρ0 ]
/ ···
/ TD
L0
TL41
/ ···
/ T4
L0
/ T 3b ≈ T 3
L1
O L1
/ ···
/ T3
L0
1
/ T2
L1
/ ···
[ρL1 ]
[ρJ ]
T01
/ ···
/ T1
J
O
/ ···
/ T1
L1
O
T00
/ ···
/ T0
J
/ ···
/ T0
L1
O
R
′ ]
[βL
[βJ′ ]
[β0′ ]
T02
/ ···
TLD1
Figure 5: The Big Diagram, step 0.1
and relative combing by expansion, Lemma 4.9. These are applied alternately to insert
D − 3 combing rectangles into the upper left corner of step 0.1, between row 3b and
row D and between column 0 and column L1 ; we also delete everything strictly below
row T 4 and right of TL1 ; the result is shown in Figure 6, with names Tid re-used in the
restored upper left corner. We note that for each 5 ≤ d ≤ D, the rectangle between
rows T d−1 and T d is a combing rectangle from column 0 to L1 , and from column L1 to
L0 , and these piece together to form a single combing rectangle from column 0 to L0 ,
as follows by applying the uniqueness clauses in the statements of relative combing by
collapse, Lemma 4.8, and relative combing by expansion, Lemma 4.9.
Next we proceed to the Big Diagram step 0.3, depicted in Figure 7. Notice that
in TL21 we have an edgelet disjoint union
TL21 = ρL1 ∪ βL′ 1 ∪(Fn · b)
| {z }
κL1
64
T0D
/ ···
/ TD
JO
/ ···
/ TD
L1
/ ···
/ TD
L0
T04
/ ···
/ T4
/ ···
/ T4
L1
/ ···
/ T4
L0
/ ···
/ T 3b
JO
/ ···
/ T 3b
L1
O
O
T03b
O
J
T02
/ ···
1
/ T2
/ T2
L1
/ ···
J
[ρ0 ]
′ ]
[βL
[βJ′ ]
[β0′ ]
[ρL1 ]
[ρJ ]
T01
/ ···
/ T1
J
O
/ ···
/ T1
L1
O
T00
/ ···
/ T0
/ ···
/ T0
L1
O
R
J
Figure 6: The Big Diagram, step 0.2
Define a commutative “baseball diagram” of collapse maps:
′ ]
[βL
1
TL3b1
TL21
⑥
⑥⑥
⑥
⑥
~⑥
⑥
[κL1 ]
❆❆
❆❆
❆❆
[ρL1 ] ❆❆
TLh1
❆❆
❆❆[ρL1 ]
❆❆
❆❆
TL11
⑥⑥
⑥⑥
⑥
⑥ ′
~⑥⑥ [ρL1 ]
Using Combing by Collapse on each of the five arrows in this diagram we obtain similar
baseball diagrams replacing L1 by any i ∈ [0, . . . , L1 ]. The combing diagrams that
correspond to the two arrows from 2nd base TL21 to 1st and 3rd bases TL11 and TL3b1
are the same as the two combing rectangles depicted in Figure 6 between rows T 2 and
rows T 1 and T 3b . The Big Diagram step 0.3 is now constructed by replacing those two
combing rectangles by the ones that correspond to the two arrows from 1st and 3rd
bases to home base TLh1 .
Finally, the Big Diagram step 1, depicted in Figure 8, is obtained from step 0.3 by
concatenating the two combing rectangles from row T 0 to T 1 and from row T 1 to T h
65
T0D
/ ···
/ TD
JO
/ ···
/ TD
L1
/ ···
/ TD
L0
T04
/ ···
/ T4
/ ···
/ T4
L1
/ ···
/ T4
L0
/ ···
/ T 3b
J
/ ···
/ T 3b
L1
T0h
/ ···
/ Th
J
O
/ ···
/ Th
L1
O
T01
/ ···
/ T1
J
O
/ ···
/ T1
L1
O
T00
/ ···
/ T0
/ ···
/ T0
L1
O
T03b
O
O
J
J
R
Figure 7: The Big Diagram, step 0.3
into a single coming rectangle from row T 0 to row T h , and by concatenating the two
combing rectangles from row T 4 to row T 3b and from row T 3b to row T h into a single
combing rectangle from row T 4 to row T h . This completes the first step of the induction,
constructing the Big Diagram step 1 from the Big Diagram step 0.
Further induction steps. Continuing to assume that D ≥ 4, each further induction step for 2 ≤ d ≤ (D − 2)/2 starts with the Big Diagram step d − 1, depicted as in
Figure 8 but with column subscript L1 replaced by Ld−1 and row superscript 4 replaced
by 2d. From there one constructs the Big Diagram step d, using a straightforward notational variation of the construction from step 0 to step 1. The key observation which
has a
gets the construction started is that the collapse forest for the map TL2dd 7→ TL2d+1
d
2d
component which is contained in the interior of a natural edge of TLd . This follows by
applying Proposition 4.16 (5b) together with the fact that the number of free splitting
units between TL0d and TL0d−1 is greater than or equal to b1 = 5 corank(A) + 4 |A| − 3.
The final step. When the induction is complete (which happens immediately if
D = 2), the Big Diagram step (D − 2)/2 consists of a single collapse–expand diagram.
From this diagram discard everything strictly right of column J and below the top row.
66
T0D
/ ···
/ TD
J
/ ···
/ TD
L1
/ ···
/ TD
L0
T04
/ ···
/ T4
J
/ ···
/ T4
L1
/ ···
/ T4
L0
T0h
/ ···
/ Th
J
O
/ ···
/ Th
L1
O
T00
/ ···
/ T0
J
/ ···
/ T0
L1
O
R
Figure 8: The Big Diagram, step 1
Also, from the projection diagram for T depicted in Figure 2 discard everything in the
T row strictly to the right of column J. Then glue these two diagrams together along
the two copies of the sequence T0 7→ TJ , resulting in the penultimate diagram shown in
Figure 9. In this diagram we emphasize also column I ∈ [0, . . . , J] which is defined so
that I is the largest integer for which there are ≥ b1 free splitting units between SI and
SJ , and hence there are exactly b1 free splitting units between SI and SJ ; the existence
of I follows from the hypothesis of Proposition 5.3 that there are ≥ b1 free splitting units
between S0 and SJ .
The final construction is triggered by the observation that the collapse forest for the
map from TI to TIh has a component that is contained in the interior of a natural edge
of TI , which follows by applying Proposition 4.16 (5b) together with the assumption
that between SI and SJ there are ≥ b1 free splitting units. Based on this observation,
we may now follow the same construction steps as above, the conclusion of which is a
diagram of the form shown in Figure 10 (where the names TiD , Tih for 0 ≤ i ≤ I have
been reused). This is a projection diagram from R to S0 7→ · · · 7→ SK of depth I, and so
the maximal depth of such a projection diagram, which by definition is the projection
π(R), satisfies π(R) ≥ I, finishing the proof of Proposition 5.3.
6
Hyperbolicity of relative free factor complexes
In this section, given a group Γ and a free factor system A of Γ, we define the complex
FF (Γ; A) of free factor systems of Γ relative to A (Section 6.1), we prove that FF (Γ; A)
is connected (Section 6.2), and we prove that it is hyperbolic (Section 6.3).
Our proof of hyperbolicity follows the method of Kapovich and Rafi developed in
[KR14] and used by them to derive hyperbolicity of FF (Fn ) from hyperbolicity of
67
T0D
/ ···
/ TD
/ ···
/ TD
J
T0h
/ ···
/ Th
IO
/ ···
/ Th
J
O
T0
/ ···
/ TI
/ ···
/ TJ
S0′
/ ···
/ S′
OI
/ ···
/ S′
OJ
S0
/ ···
/ SI
/ ···
/ SJ
O
O
I
/ ···
/ TD
L0
/ ···
/ SK
R
Figure 9: The Penultimate Diagram, aka the Big Diagram step (D−2)/2. The projection
diagram atop which all the Big Diagrams are constructed has been restored (except for
the portion of the T row to the right of column J). Column I is determined by requiring
that I is the largest integer ≤ J such that between SI and SJ there are exactly b1 free
splitting units.
/ ···
/ TD
T0h
/ ···
/ Th
IO
S0
/ ···
/ SI
T0D
O
I
/ ···
/ TD
J
/ ···
/ TD
L0
/ ···
/ SJ
/ ···
/ SK
R
Figure 10: The Ultimate Diagram. Notations TiD and Tih from Figure 9 have been
reused to denote new objects.
68
FS(Fn ). The way this method works is to derive hyperbolicity of a connected simplicial
complex Y from hyperbolicity of a given connected simplicial complex X, by exhibiting a
surjective Lipschitz map f : X 7→ Y satisfying a simple geometric condition. Intuitively
this condition says that if a geodesic in X has its endpoints mapped near each other in
Y by the map f , then the entire f -image of that geodesic is bounded. We construct the
required surjective Lipschitz map FS(Γ; A) → FF (Γ; A) in Section 6.2, and we prove
that it satisfies the needed condition on geodesics in Section 6.3.
6.1
The complex of free factor systems relative to a free factor system.
Fix a group Γ. Define the unreduced complex of free factor systems of Γ to be the simplicial realization of the set of free factor systems with respect to the partial ordering ⊏.
This complex has a 0-simplex for each free factor system A, and a K-simplex for each
chain of proper extensions of the form A0 ⊏ A1 ⊏ · · · ⊏ AK . The unreduced complex of
free factor system is a single point if and only if Γ is freely indecomposable. For present
purposes our interest in this unreduced complex is as a home for relative complexes of
free factor systems.
Given a free factor system A of Γ, the complex of free factor systems of Γ relative
to A, denoted FF (Γ; A), is the flag subcomplex of the unreduced complex of free factor
systems that contains a simplex A0 ⊏ · · · ⊏ AK if and only if there is a proper inclusion
A ⊏ A0 and a proper inclusion AK ⊏ {[Γ]}.
In the case that Γ has a Grushko free factor system A, for example when Γ is finitely
generated, it makes sense to define the (absolute) complex of free factor system FF (Γ)
to be FF (Γ; A). Note that this complex is obtained from the unreduced complex by
“reducing” it, by which we mean removing the minimal and maximal 0-simplices A
and {[Γ]} and the interiors of any incident simplices of positive dimension.
Recall the formula for the free factor system depth of a free factor system A:
DFF (A) = 2 corank(A) + |A| − 1
The following is an immediate corollary of Lemma 2.14 and the definition of relative free
factor systems:
Proposition 6.1. For any group Γ and any free factor system A such that DFF (A) ≥ 2,
the complex FF (Γ; A) has dimension DFF (A) − 2, and any simplex is contained in a
simplex of maximal dimension DFF (A) − 2.
♦
For the following result, which is a corollary of Proposition 6.1 together with a
straightforward case analysis, recall from Section 2.5 that a free factor system A of
Γ is exceptional if and only if DFF (A) ≤ 2. This result enumerates the exceptional
behavior of the topology of FF (Γ; A) when A is exceptional. We also enumerate the
1-dimensional complexes.
69
Proposition 6.2. For any group Γ, a free factor system A of Γ is exceptional if and
only if FF (Γ; A) is empty or 0-dimensional, and DFF (A) = 3 if and only if FF (Γ; A)
is 1-dimensional. More explicitly:
(1) FF (Γ; A) = ∅ ⇐⇒ DFF (A) ≤ 1 ⇐⇒ either A = {[Γ]}, or A = {[A1 ], [A2 ]} with
a realization Γ = A1 ∗ A2 .
(2) FF (Γ; A) is 0-dimensional ⇐⇒ DFF (A) = 2 ⇐⇒ one of the following occurs:
(a) A = {[A]} with realization Γ = A ∗ Z where Z is infinite cyclic; or
(b) or A = {[A1 ], [A2 ], [A3 ]} with realization Γ = A1 ∗ A2 ∗ A3 .
(3) FF (Γ; A) is 1-dimensional ⇐⇒ DFF (A) = 3 ⇐⇒ one of the following occurs:
(a) A = ∅ and with realization Γ = Z1 ∗ Z2 , each of Z1 , Z2 being infinite cyclic
(i.e. Γ is free of rank 2 and FF (Γ; A) is its absolute complex of free factor
systems); or
(b) A = {[A1 ], [A2 ]} with realization Γ = A1 ∗ A2 ∗ Z where Z has rank one; or
(c) A = {[A1 ], [A2 ], [A3 ], [A4 ]} with realization Γ = A1 ∗ A2 ∗ A3 ∗ A4 .
♦
If Γ has a Grushko free factor system A, for example when Γ is finitely generated,
then the unreduced complex of free factor systems is connected and has diameter ≤ 2,
because every other free factor system is an extension of A. If Γ has no Grushko free
factor system then the relation ⊏ has no minimum, in which case the depth of free factor
systems of Γ is unbounded, and the dimension of the unreduced complex of free factor
systems is infinite.
The complex FF (Fn ) (= FF (Fn ; ∅)) is related to the complex of free factors which
we denote F(Fn ), introduced by Hatcher and Vogtmann in [HV98] (and see [BF14a]).
Several other closely related complexes, known to be equivariantly quasi-isometric to
each other, are described for example in [KR14]. We recast the definition of F(Fn ) in
our present setting as follows. In ranks n ≥ 3, F(Fn ) is the subcomplex of FF (Fn )
consisting of all simplices A0 ⊏ · · · ⊏ AK each of whose 0-simplices A0 , . . . , AK has but
a single component. When n = 2 this definition would lead to a 0-dimensional complex
whose simplices have the form {[A]} where the free factor A < F2 has rank 1, but then
F(F2 ) itself is obtained by attaching a 1-simplex to each pair of 0-simplices [A], [B]
whenever there is a free factorization F2 = A ∗ B; clearly FF (F2 ) is the first barycentric
subdivision of F(F2 ).
Proposition 6.3. The inclusion F(Fn ) ֒→ FF (Fn ) is a quasi-isometry.
Proof. In this proof we shall abuse notation by identifying each 0-simplex A = {[A]} of
F(Fn ) with its single component [A].
70
We may assume n ≥ 3. It suffices to construct a Lipschitz retract r from the 0skeleton of FF(Fn ) to the 0-skeleton of F(Fn ). Given a 0-simplex A in FF(Fn ), choose
any component [A] ∈ A and define r(A) = [A]. If A has but a single component then
clearly r(A) = A (abusing notation).
Given a 1-simplex A ⊏ A′ in FF (Fn ), consider r(A) = [A] ∈ A and r(A′ ) = [A′ ] ∈
A′ . We must bound the distance between [A] and [A′ ] in F(Fn ). Letting [A′′ ] ∈ A′
be the unique element such that [A] ⊏ [A′′ ], since [A], [A′′ ] have distance at most 1 in
F(Fn ) it suffices to bound the distance between [A′ ] and [A′′ ]. If [A′ ] = [A′′ ] we are
done, so assume [A′ ] 6= [A′′ ]. If necessary, rechoose A′ , A′′ in their conjugacy classes so
that each is a term in a realization of A′ , and hence we have a free factorization of the
form Fn = A′ ∗ A′′ ∗ C. Picking rank 1 free factors B ′ < A′ , B ′′ < A′′ , we have a free
factorization Fn = B ′ ∗B ′′ ∗D, and since n ≥ 3 and B ′ , B ′′ each have rank 1 it follows that
the rank 2 subgroup B ′ ∗ B ′′ is a proper free factor. We therefore obtain a path in F(Fn )
of length at most 4 between [A′ ] and [A′′ ], namely [A′ ]—[B ′ ]—[B ′ ∗ B ′′ ]—[B ′′ ]—[A′′ ]. ♦
6.2
Connectivity of F F(Γ; A); a Lipschitz map F S(Γ; A) 7→ F F(Γ; A).
Consider a group Γ and a free factor system A such that DFF (A) ≥ 3, and so FF (Γ; A)
has dimension ≥ 1. We shall kill two birds (Proposition 6.5 (1) and (2)) with one
stone (Lemma 6.4): prove connectivity of FF(Γ; A); and describe a map FS(Γ; A) 7→
FF (Γ; A) which is Lipschitz with respect to simplicial metrics.
The free factor system A may be realized as Γ = A1 ∗ · · · ∗ AK ∗ B with A =
{[A1 ], . . . , [AK ]}, K = |A| ≥ 0, and while K is fixed we will vary such realizations as
needed.
Define the projection set map Π from the 0-skeleton of FS(Γ; A) to finite subsets of
the 0-skeleton of FF (Γ; A), as follows. Consider a 0-simplex [T ] ∈ FS(Γ; A) represented
by a free splitting Γ y T rel A. Define Π[T ] ⊂ FF (Γ; A) to be the set of all 0-simplices
of the form F(U ) such that Γ y U is a free splitting rel A, F(U ) 6= A, and there exists
a collapse map T 7→ U (which one may always choose to be relatively natural). Here are
a few properties of the set map Π that we will use without comment in what follows:
• Π[T ] is well-defined within the equivalence class of T .
• The sets Π[T ] cover the entire 0-skeleton of FF (Γ; A), as [T ] varies over the 0skeleton of FS(Γ; A).
• The inverted equivariance property: Π [T ]·φ = φ−1 Π[T ] for each φ ∈ Out(Γ; A).
• Π[T ] 6= ∅.
The first item is evident, the second follows from Lemma 3.1, and the third from
Lemma 3.7. The fourth follows from Proposition 3.6 (2)(c) which guarantees the exis71
tence of a collapse map T 7→ U such that U is a one-edge free splitting, together with
the fact that DFF (A) ≥ 2 which guarantees that F(U ) 6= A.
Using Bass-Serre theory, we next translate the definition of Π[T ] into the language
of graphs of groups. In the quotient graph of groups T /Γ, given a subgraph G ⊂ T /Γ,
consider the following two properties of G:
(1) G contains every vertex of T /Γ with nontrivial vertex group.
(2) Each vertex of valence 0 or 1 in G has nontrivial vertex group.
We say that G is a relative core graph if both of (1) and (2) hold. If only (1) holds then
G contains a unique maximal relative core graph denoted core(G): inductively remove
any vertex that violates (2) together with any incident edge. Every subgraph G ⊂ T /Γ
satisfying (1) represents a free factor system rel A that we denote [G]: to define [G], let
e ⊂ T be the total lift of G via the Bass-Serre universal covering map T 7→ T /Γ, and
G
e Note that
define [G] to be the set of conjugacy classes of stabilizers of components of G.
e Note also
[G] = F(U ) where the map T → U collapses to a point each component of G.
′
that [G] = [core(G)], and that if G ⊂ G ⊂ T /Γ both satisfy (1) then [G] ⊏ [G′ ] rel A,
with equality if and only if core(G) = core(G′ ). Note that [G] = {[Γ]} if and only if
G = T /Γ.
A relative core graph G ⊂ T /Γ is trivial if A has a realization Γ = A1 ∗ · · · ∗ AK ∗ B
such that G consists solely of K vertices v1 , . . . , vK with vertex groups A1 , . . . , AK ; a
trivial relative core graph of T /Γ exists if and only if F(T ) = A. The triviality property
is extended to arbitrary subgraphs G ⊂ T /Γ that satisfy (1) by requiring that Core(G)
be trivial. Note that [G] = A if and only if G and core(G) are trivial.
To complete the Bass-Serre translation, we note that Π[T ] is equal to the following
set of 0-simplices in FF (Γ; A):
Π[T ] = {[G] ∈ FF(Γ; A) G < T /Γ is a proper, nontrivial, relative core graph}
To prove this, in the discussion above we have already proved the inclusion ⊃. For the
[σ]
opposite inclusion ⊂, given F(U ) ∈ Π[T ] and a relatively natural collapse map T −→ U ,
let σ ′ be the union of σ with all vertices of T having nontrivial stabilizer, let G be
the image of σ ′ under the quotient map T 7→ T /Γ, and it follows that G is a proper,
nontrivial relative core graph and that [G] = F(U ).
Lemma 6.4. The set map Π has the following properties:
(1) For any collapse of free splittings S ≻ T we have Π[T ] ⊂ Π[S].
(2) For each [T ] ∈ FS(Γ; A) the set Π[T ] is contained in a connected subcomplex of
FF (Γ; A) of simplicial diameter ≤ 6.
72
Before proving Lemma 6.4 we apply it as follows. A function π from the 0-skeleton of
FS(Γ; A) to the 0-skeleton of FF(Γ; A) is called a projection map if π[T ] ∈ Π[T ] for each
[T ] ∈ FS(Γ; A). Projection maps always exist by simply choosing π[T ] ∈ Π[T ] 6= ∅. If it
is so desired, perhaps because of an aversion to wearing out the Axiom of Choice [Wei],
for a concretely given group such as Γ = Fn there are explicit constructions of projection
maps, based on explicit enumeration of the 0-skeleta of FS(Γ; A) and of FF (Γ; A) and
explicit computation of the set map Π.
Proposition 6.5. Assuming DFF (A) ≥ 3 the following hold:
(1) FF (Γ; A) is connected.
(2) For any projection map π from the 0-skeleton on FS(Γ; A) to the 0-skeleton of
FF (Γ; A) we have:
(a) π is Lipschitz, with constant depending only on corank(A) and |A|.
(b) π satisfies the “inverted coarse equivariance property”: d(π[T · φ], φ−1 (π[T ]))
has an upper bound depending only on corank(A) and |A|, for [T ] ∈ FS(Γ; A)
and φ ∈ Out(Γ; A).
Proof. To prove connectivity, for each 0-simplex [T ] ∈ FS(Γ; A) let Π1 [T ] be the union
of all edge paths having endpoints in Π[T ] and having length ≤ 6. Connectivity of Π1 [T ]
follows from Lemma 6.4 (2). This is the basis step of an inductive proof of the following
statement: for each edge path S0 —S1 —. . .—SL in FS(Γ; A) the set Π1 [S0 ]∪ · · · ∪ Π1 [SL ]
is connected. For the induction step one uses that either SL−1 ≻ SL or SL−1 ≺ SL and
therefore by Lemma 6.4 (1) the set Π[SL−1 ] ∪ Π[SL ] equals either Π[SL−1 ] or Π[SL ]
and so is connected. It follows that ∪{Π1 [T ] [T ] ∈ FS(Γ; A)} is connected, and this
includes the entire 0-skeleton of FF (Γ; A).
To prove π is Lipschitz it suffices to prove for any 1-simplex [S] ≺ [T ] in FS(Γ; A)
that d(π(S), π(T )) is bounded, but this follows from Lemma 6.4 which implies that
Π[S] ∪ Π[T ] = Π[T ] has diameter ≤ 6.
Inverted coarse equivariance for π follows from inverted equivariance for Π combined
with Lemma 6.4 (2).
♦
Proof of Lemma 6.4. To prove item (1) choose a collapse map S 7→ T . Each element
of Π[T ] has the form F(U ) for some collapse map T 7→ U , and since the composition
S 7→ T 7→ U is a collapse map it follows that F(U ) ∈ Π[S].
Having already proved (1), in order to prove (2) we may reduce to the case that the
free splitting T is generic (see Definition 3.3): by Proposition 3.6 there exists a generic
free splitting S such that S ≻ T , and applying (1) we see that property (2) for S implies
property (2) for T .
73
Henceforth we assume that T is generic. Again we may choose the realization Γ =
A1 ∗ · · · ∗ AK ∗ B of A and the vertex groups of T /Γ so that Vnt = {v1 , . . . , vK } with
vertex groups A1 , . . . , AK . The set of valence 1 vertices in T /Γ is precisely Vnt , and
every other vertex of T /Γ has valence 2 or 3. Also, there is at least one valence 3 vertex
in T /Γ because otherwise either K = 0 and Γ is a circle, or K = 2 and Γ = A1 ∗ A2 , and
each of these is ruled out by the hypothesis DFF (A) ≥ 3. Every relatively natural vertex
has valence 1 or 3 and hence is a natural vertex, so we shall drop the adverb “relatively”
from the phrase “relatively natural” for the rest of the proof.
For the proof, given two proper, nontrivial relative core graphs G1 6= G2 ⊂ T /Γ
we shall construct construct an edge path in FF(Γ; A) with endpoints [G1 ], [G2 ] and
length ≤ 6; the vertices along this edge path need not stay in Π[T ]. We start with some
easy cases:
The Nested Case: Suppose G1 ⊂ G2 . In this case we have a path [G1 ] ⊏ [G2 ] of
length 1 and we are done.
The Nontrivial Intersection Case: Suppose G1 ∩ G2 is nontrivial. In this case
we have a path [G1 ] ⊐ [G1 ∩ G2 ] ⊏ [G2 ] of length 2 and we are also done.
To complete the proof, after applying the Nested Case it suffices to connect [G1 ], [G2 ]
by a path in FF (Γ; A) of length ≤ 4 under the following assumption:
(a) Each of G1 , G2 ⊂ T /Γ is maximal with respect to inclusion.
Furthermore, after applying the Nontrivial Intersection Case we may also assume that:
(b) The subgraph G1 ∩ G2 is trivial.
From here the proof proceeds in two steps. Step 1 uses assumptions (a), (b) to show
that T /Γ with its natural cell structure is isomorphic one of three special graphs of low
complexity:
The clam: the rank 2 graph having two valence 3 vertices and three edges each with
its endpoints at distinct vertices.
The spindle: the rank 1 graph having two valence 3 vertices and two valence 1 vertices,
consisting of a circle with two edges attached each by identifying a single endpoint
of the edge to a distinct point on the circle.
The clam with an antenna: the rank 2 graph obtained from the clam by attaching
one endpoint of an edge to an interior point of one of the clam edges.
Step 2 constructs the needed path of length ≤ 4 in each of these three special cases.
Step 1. Note first that a proper relative core graph G ⊂ T /Γ is maximal if and
only if it is obtained from T /Γ by removing the interior of a single natural edge having
distinct endpoints. The “if” direction is clear. For the “only if” direction: if G is missing
74
the interiors of two natural edges e1 , e2 and if e1 has distinct endpoints then T − int(e1 )
is a proper relative core graph larger than G; and if G is missing the interior of a natural
edge e with both ends at some natural vertex p then, letting e′ be the edge having a
single endpoint at p, the graph G cannot contain e′ for otherwise p would have valence 1
in G and trivial vertex group, and so T /Γ − int(e′ ) is a proper relative core graph larger
than G.
Let Gi = T /Γ − int(ei ) for natural edges e1 6= e2 each with distinct endpoints. Each
set ∂e1 , ∂e2 has two points, so their union ∂e1 ∪ ∂e2 has four, three, or two points.
We handle those cases separately, and we also break into various subcases, in each of
which we find that T /Γ is a spindle, or a clam maybe with an antenna, or we find a
contradiction.
Case 1: ∂e1 ∪ ∂e2 is four points. In this case G1 ∩ G2 is a relative core graph,
because each of the four points ∂e1 ∪ ∂e2 has valence 2 in G1 ∩ G2 or valence 1 in T /Γ.
But G1 ∩ G2 is trivial, and so all four endpoints must have valence 1 in T /Γ. It follows
that T /Γ is the disjoint union of e1 and e2 and so is disconnected, a contradiction.
Case 2: ∂e1 ∪ ∂e2 is three points. Let ∂e1 ∩ ∂e2 = {p}, a single point. In this case
G1 ∩ G2 is not a core graph, because p has valence 1 in G1 ∩ G2 but valence 3 in T /Γ.
Letting e3 be the edge of G1 ∩G2 incident to p, and letting H = (G1 ∩G2 )−({p}∪int(e3 )),
it follows that core(H) = core(G1 ∩ G2 ), and so H is trivial but we cannot yet conclude
that H itself is a relative core graph. Let qi 6= p be the endpoint of ei opposite p. There
are two subcases, depending on whether q3 equals one of q1 , q2 .
If q3 is distinct from both q1 and q2 then each of q1 , q2 , q3 has valence 2 in H or
valence 1 in T /Γ and so H is a relative core graph. But H is trivial and so H = Vnt =
{q1 , q2 , q3 } implying that |A| = 3, and implying that T /Γ is a tree, more specifically a
triod, and so corank(A) = 0. But then DFF (A) = 2, a contradiction.
Suppose that q3 equals one of q1 or q2 , say q3 = q1 , a point of valence 3 in T /Γ and
of valence 1 in H, and so H is not a relative core graph. Let e′ be the edge of H incident
to q3 and let H ′ = H − ({q3 } ∪ int(e′ )), so core(H ′ ) = core(H) = core(G1 ∩ G2 ) and H ′
is trivial, but again H ′ need not be a relative core graph. Let q ′ be the endpoint of e′
opposite q3 . Depending on whether q2 = q ′ we will see that T /Γ is either a spindle or
a clam with an antenna. If q2 6= q ′ then each has valence 1 in T /Γ or valence 2 in H ′
and so H ′ is a relative core graph, but H ′ is trivial and so both q2 , q ′ have valence 1 in
T /Γ, and in this case T /Γ is a spindle. If q2 = q ′ then that point has valence 1 in H ′
and valence 3 in T /Γ, and so H ′ is not a relative core graph. Letting e′′ be the edge
of H ′ with endpoint q ′′ opposite q ′ it follows that H ′′ = H ′ − ({q ′ } ∪ int(e′′ )) satisfies
core(H ′′ ) = core(H ′ ) = core(G1 ∩ G2 ) and so H ′′ is trivial. Also, q ′′ has either valence 2
in H ′′ or valence 1 in T /Γ so H ′′ is, at last, a relative core graph. By triviality it follows
that q ′′ has valence 1 in T /Γ and that T /Γ is a clam with an antenna.
Case 3: ∂e1 ∪ ∂e2 = {p, q} is two points. These two points each have valence 1
in G1 ∩ G2 and valence 3 in T /Γ. Let ep , eq ⊂ T /Γ be the natural edges incident to p, q
75
respectively. If ep = eq then G1 ∩ G2 = ep , and so core(G1 ∩ G2 ) = ∅ and T /Γ is a clam.
We may therefore assume that ep 6= eq . Let H ′ = (G1 ∩G2 )−({p, q}∪int(ep )∪int(eq )},
so core(H ′ ) = core(G1 ∩ G2 ), implying that H ′ is trivial. Let p′ , q ′ be the endpoints of
ep , eq opposite p, q respectively. Depending on whether p′ = q ′ the graph T /Γ is either
a spindle or a claim with an antenna, which is proved exactly as in Case 2 but with the
notation changed to replace q2 in Case 2 with p′ in Case 3.
This completes Step 1.
Step 2. Knowing that T /Γ is the clam, spindle, or clam with an antenna, we now
consider these graphs one-at-a-time, and we consider the possibilities for the subgraphs
G1 , G2 . In each case we will obtain a path in FF(Γ; A) of length ≤ 4 connecting G1
and G2 .
Notational alert: The symbols e1 , e2 no longer assume their earlier meanings and are
freed up for new use.
T /Γ is a clam. We may denote its edges e1 , e2 , e′ so that G1 = e1 ∪e′ and G2 = e2 ∪e′ .
There is a collapse map T ≻ T ′ whose effect on T /Γ is to collapse e′ to a point, and
then there is an expansion T ′ ≺ T ′′ whose effect is to pull the two loops apart, so T ′′ /Γ
is a barbell graph with disjoint circles C1 , C2 connected by an edge, and [Gi ] = [Ci ]. We
obtain a length 2 path [G1 ] = [C1 ] ⊏ [C1 ∪ C2 ] ⊐ [C2 ] = [G2 ] in FF(Γ; A).
T /Γ is a spindle. Let the circle edges be denoted e1 , e2 with ∂e1 = ∂e2 = {p, q},
and let the edges ep , eq be attached to p, q with opposite endpoints P , Q respectively.
Consider the proper, nontrivial relative core graph C = e1 ∪ e2 ∪ {P, Q}, representing
the free factor system [C]. The graph G1 ∩ G2 , being trivial, cannot contain e1 ∪ e2 . By
symmetry we may therefore suppose that G1 = T /Γ − int(e1 ) = ep ∪ e2 ∪ eq .
Consider the case that G2 = T /Γ − int(e2 ) = ep ∪ e1 ∪ eq . For each choice of
i 6= j ∈ {1, 2} we may carry out the following operations. First collapse T ≻ T ′
with the effect on T /Γ of collapsing ej to a point and taking ei , ep , eq , P, Q ⊂ T /Γ to
e′i , e′p , e′q , P ′ , Q′ ⊂ T ′ /Γ respectively. The collapsed image of Gi is G′i = e′p ∪ e′q and the
collapsed image of C is C ′ = e′i ∪ {P ′ , Q′ }. Next, there is an expansion T ′ ≺ T ′′ whose
effect on T ′ /Γ is to pull apart the arc G′1 and the circle e′j , so that in the quotient graph
T ′′ /Γ the free factor systems [Gi ] and [C] are represented by relative core graphs G′′i , C ′′
having proper union G′′i ∪ C ′′ which is also a relative core graph. We obtain a length 2
path [Gi ] = [G′′i ] ⊏ [G′′i ∪ C ′′ ] ⊐ [C ′′ ] = [C]. Putting these together for i = 1, 2 we get a
length 4 path connecting [G1 ] and [G2 ].
By symmetry of notation it remains to consider the case that G2 = T /Γ − int(eq ).
From the argument of the previous paragraph we get a length 2 path connecting [G1 ] to
[C], and since C ⊂ G2 we get a length 1 path [C] ⊏ [G2 ], which together give a length 3
path connecting [G1 ] and [G2 ].
T /Γ is a clam with an antenna. Let the valence 1 vertex be R, let its incident edge
be eR with opposite vertex r, let other two incident edges to r be e1 , e2 with opposite
76
vertices p1 , p2 respectively, and let the two edges with endpoints p1 , p2 be e3 , e4 . Since
G1 ∩ G2 is trivial, it cannot contain a circle, and therefore it contains at most one of
e3 , e4 . We assume e4 6⊂ G1 ∩ G2 , and by symmetry we may assume e4 6⊂ G1 , and so
G1 = eR ∪ e1 ∪ e2 ∪ e3 . We have e4 ⊂ G2 . Exactly one of the edges eR , e1 , e2 , e3 is not
in G2 , and by symmetry we may assume e2 ⊂ G2 . If eR 6⊂ G2 then G1 ∩ G2 contains
the circle e1 ∪ e2 ∪ e3 , contradicting that G1 ∩ G2 is trivial. This shows that the edge
not in G2 is either e1 or e3 . It follows that G1 ∩ G2 = τ is a tree containing R: if
e1 6⊂ G2 then τ = eR ∪ e2 ∪ e3 ; whereas if e3 6⊂ G2 then τ = eR ∪ e1 ∪ e2 . Let T ≻ T ′
be the collapse map whose effect on T /Γ is to collapse the tree τ to a point. The graph
T ′ /Γ is a rank 2 rose whose rose point R′ is the unique vertex with nontrivial vertex
group, and whose petals C1′ , C2′ satisfy [G1 ] = [C1′ ] and [G2 ] = [C2′ ]. Let T ′ ≺ T ′′ be
the expansion which pulls the petals C1′ , C2′ apart into the disjoint circles C1′′ , C2′′ of the
barbell graph T ′′ /Γ, so that the arc connecting C1′′ to C2′′ is subdivided at its midpoint
R′′ , having nontrivial vertex group, into edges Ei′′ connecting Ci′′ to R′′ (i = 1, 2). We
have [Gi ] = [Ci′ ] = [Ci′′ ∪ Ei′′ ]. We then have a chain of proper, nontrivial relative core
graphs
C1′′ ∪ E1′′ ⊃ C1′′ ∪ R′′ ⊂ C1′′ ∪ R′′ ∪ C2′′ ⊃ R′′ ∪ C2′′ ⊂ E2′′ ∪ C2′′
producing a length 4 path
[G1 ] = [C1′′ ∪ E1′′ ] ⊐ [C1′′ ∪ R′′ ] ⊏ [C1′′ ∪ R′′ ∪ C2′′ ] ⊐ [R′′ ∪ C2′′ ] ⊏ [E2′′ ∪ C2′′ ] = [G2 ]
♦
6.3
Proof of hyperbolicity of F F(Γ; A)
We shall apply the following theorem of I. Kapovich and K. Rafi:
Theorem 6.6 ([KR14] Proposition 2.5). Let X be a connected simplicial complex which
is δ-hyperbolic with respect to the simplicial metric. Let Y be a connected simplicial
complex. Suppose that there exists a map of 0-skeleta π : X (0) → Y (0) with the following
properties:
(1) π is surjective
(2) π is K-Lipschitz
(3) There exists a constant D such that for all v, w ∈ X (0) , if dY (v, w) ≤ 1, and if
v = v0 , v1 , . . . , vL = w are the vertices along a geodesic in the 1-skeleton X (1)
between v and w, then diamY {π(v0 ), . . . , π(vL )} ≤ D.
It follows that Y is δ1 -hyperbolic with respect to the simplicial metric. Furthermore,
77
(4) If v0 , v1 , . . . , vL are the vertices along a geodesic in X (1) then {π(v0 ), π(v1 ), . . . , π(vL )}
is C-Hausdorff close to a geodesic in Y .
The constants δ1 and C depend only on δ, K, and D.
♦
By incorporating item (4) of the previous theorem into the proof of Theorem 1.4 we
get the following:
Theorem 6.7 (Enchanced version of Theorem 1.4). For any group Γ and any nonexceptional free factor system A of Γ, the complex FF (Γ; A) is nonempty, connected, and
hyperbolic. Furthermore, the image under π : FS(Γ; A) → FF (Γ; A) of any geodesic in
FS(Γ; A) is uniformly Hausdorff close to a geodesic in FF(Γ; A).
Proof. Choose a special projection map π : FS(Γ; A) → FF (Γ; A) having the property
that for each 0-simplex [T ] ∈ FS(Γ; A), if F(T ) 6= A then π[T ] = F(T ); such a map
exists since clearly F(T ) ∈ Π[T ]. Surjectivity of this special π follows from Lemma 3.1,
and π is Lipschitz by Proposition 6.5 (2a). These are hypotheses (1) and (2) of the
Kapovich–Rafi Theorem 6.6 above. It remains to verify hypothesis (3).
Consider 0-simplices [S], [T ] ∈ FS(Γ; A) such that d(π(S), π(T )) ≤ 1.
We first reduce to the case that F(S) = π(S) and that F(T ) = π(T ); by the special
choice of π this is equivalent to reducing to the case F(S) 6= A and F(T ) 6= A. From
the requirement that π[S] ∈ Π[S] and π[T ] ∈ Π[T ] it follows that there exist collapse
maps S ≻ S ′ and T ≻ T ′ such that π(S) = F(S ′ ) 6= A and π(T ) = F(T ′ ) 6= A, and
from the special choice of π it follows that F(S ′ ) = π(S ′ ) and F(T ′ ) = π(T ′ ). Note that
in FS(Γ; A) we have d(S, S ′ ) ≤ 1 and d(T, T ′ ) ≤ 1. By hyperbolicity of FS(Γ; A), any
geodesic connecting S to T stays uniformly Hausdorff close to any geodesic connecting
S ′ to T ′ , and since π is Lipschitz it follows that the π-images of these geodesics are
uniformly Hausdorff close in FF (Γ; A). Once we have verified that hypothesis (3) holds
for an S ′ , T ′ geodesic, it holds as well for an S, T geodesic, completing the reduction.
Henceforth we assume F(S) = π(S) and F(T ) = π(T ). Since d(F(S), F(T )) ≤ 1,
up to transposing notation we may assume F(S) ⊂ F(T ). Combining Lemma 4.2 and
Lemma 4.3, there exists a collapse map S ≻ S ′′ and a fold sequence from S ′′ to T . Since
d(S, S ′′ ) ≤ 1 it follows, just as in the previous paragraph, that once we have verified the
desired conclusions for an S ′′ , T geodesic, the conclusions for an S, T geodesic follow.
Henceforth we may assume that there exists a fold sequence from S to T , denoted
f1
f2
fL
S = S0 −→ S1 −→ · · · −→ SL = T
By Theorem 5.4 the sequence S0 , S1 , . . . , SL can be reparameterized as a uniform quasigeodesic. By hyperbolicity of FS(Γ; A) this quasigeodesic is uniformly Hausdorff close
in FS(Γ; A) to any S, T geodesic. And by the Lipschitz property for π the images of the
quasigeodesic and the geodesic are uniformly Hausdorff close in FF(Γ; A). It therefore
78
suffices to bound the diameter of the set {π(S0 ), π(S1 ), . . . , π(SL )}. By Lemma 3.2(3)
we have F(S0 ) ⊏ F(S1 ) ⊏ · · · ⊏ F(SL ) and so the set {F(S0 ), F(S1 ), . . . , F(SL )} has
diameter ≤ 1 in FF (Γ; A). Since A is properly contained in F(S) = F(S0 ), it follows
that A is properly contained in each of F(S0 ), F(S1 ), . . . , F(SL ), and so π(Si ) = F(Si )
for 0 ≤ i ≤ L. The set {π(S0 ), π(S1 ), . . . , π(SL )} therefore has diameter ≤ 1.
♦
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| 4 |
NEGATIVE CURVES OF SMALL GENUS ON SURFACES
arXiv:1105.1154v6 [math.GT] 5 Jun 2016
TED CHINBURG* AND MATTHEW STOVER**
Abstract. Let X be an irreducible smooth geometrically integral projective surface over
a field. In this paper we give an effective bound in terms of the Neron–Severi rank ρ(X)
of X for the number of irreducible curves C on X with negative self-intersection and
geometric genus less than b1 (X)/4, where b1 (X) is the first étale Betti number of X. The
proof involves a hyperbolic analog of the theory of spherical codes.
1. Introduction
Let X be an irreducible smooth geometrically integral projective surface over a field k.
1 (X, Q ) for any prime p different from
The first Betti number of X is b1 (X) = dimQp Hét
p
char(k). Let ρ(X) be the rank over Z of the Neron–Severi group NS(X) of X. The main
result of this paper is an effective proof of the following result.
Theorem 1.1. There are only finitely many irreducible curves C on X with negative selfintersection and geometric genus g(C) < b1 (X)/4. For sufficiently large ρ(X), the number
of such C is bounded above by 2 0.902 ρ(X) .
The condition g(C) < b1 (X)/4 is sharp in view of the following example.
Example 1.2. Let X be the direct product Y × Y , where Y is a smooth projective irreducible curve of genus g defined over Fp . Set k = Fp , and let Cn be the graph of σ n in
X(Fp ), where σ : Y −→ Y is the Frobenius automorphism. Then Cn is a reduced irreducible
curve on X of arithmetic genus g, and [4, Ex. V.1.10] implies that Cn2 −→ −∞ as n −→ ∞.
One has b1 (X) = 4g. In particular, there are infinitely many distinct reduced irreducible
curves on X of genus g = b1 (X)/4 with negative self-intersection.
The proof of Theorem 1.1 relies on a theory of codes in hyperbolic space, developed
in §3, which parallels the classical theory of spherical codes on Euclidean spheres. Recall
that NS(X) is a finitely generated abelian group, and the group Num(X) of divisors mod
numerical equivalence on X is the quotient of NS(X) by its torsion subgroup. Thus Num(X)
is a free Z-module of rank ρ(X). The curves C in Theorem 1.1 map bijectively to their
classes in Num(X)R = R ⊗Z Num(X). We show that these classes form a strict hyperbolic
code of angle at least π/2 in Num(X)R in the sense of Definition 3.4. The maximal number
of elements in such a code is the strict hyperbolic kissing number Rn (π/2). We bound
Rn (π/2) above by Kn−1 (φ0 ) + 2, where Kn−1 (φ0 ) is the classical kissing number for the
Euclidean unit sphere Sn−2 in Rn−1 associated with the angle φ0 = arccos(3/4). The bound
Date: January 19, 2018.
*Supported in part by NSF Grants DMS 1360767, DMS 1265290 and DMS 1100355, SaTC grant CNS1513671 and Simons fellowship 338379.
**Supported in part by NSF RTG grant DMS 0602191 and NSF grant DMS 1361000. The author acknowledges support from U.S. National Science Foundation grants DMS 1107452, 1107263, 1107367 “RNMS:
GEometric structures And Representation varieties” (the GEAR Network).
1
2
T. CHINBURG AND M. STOVER
in Theorem 1.1 then follows from an upper bound of Kabatiansky and Levenshtein on
Kn−1 (φ0 ).
We will also show that Rn (π/2) is bounded below by ⌊Kn−1 (2φ0 )/2⌋, the greatest integer
less than or equal to Kn−1 (2φ0 )/2. This lower bound grows exponentially with n by work
of Chabauty, Shannon and Wyner. However, we do not know if such large hyperbolic
codes can be realized by negative curves of small genus on surfaces. This lower bound on
Rn (π/2) shows that, to improve the upper bound 2 0.902 ρ(X) in Theorem 1.1 to one that
is subexponential in ρ(X), if such an improvement is possible, requires more information
about the distribution of negative curves inside Num(X) than is provided by the facts from
intersection theory recalled below. In view of the extensive literature on spherical codes
([2], [3]), it would be interesting to classify hyperbolic codes in small dimensions, as well as
the ones that can arise form negative curves of small genus on projective surfaces.
In the course of proving these results, we will show the following:
Theorem 1.3. Let F be a set of irreducible curves C on X for which C 2 < 0. If F contains
more than Rρ(X)−1 (π/2) elements, there are two elements C1 , C2 ∈ F together with positive
integers a, b such that aC1 + bC2 is an effective ample connected divisor on X.
It was shown in [6] that if the set F in Theorem 1.3 has more than ρ(X)2 + ρ(X) + 1
elements, then there is an ample effective divisor supported on the union of the elements of
F. However, using this replaces the genus bound b1 (X)/4 in Theorem 1.1 by the weaker
bound b1 (X)/(2ρ(X)2 + 2ρ(X) + 2). In particular, it is crucial for the proof of Theorem 1.1
that we reduce the number of curves in F involved in an effective divisor with positive selfintersection with each of its irreducible components down to two, which is clearly optimal.
We now outline the contents of the paper and the proofs of Theorems 1.3 and 1.1.
In §2 we recall the definition of spherical codes and some classical results concerning
them. We define hyperbolic codes in §3. In §4 we state our results concerning the relation
between negative curves of small genus and hyperbolic codes of angle at least π/2.
The proof of Theorem 1.3 involves the following steps. In §5 we study subsets D = {Di }i
of Num(X) for which there is a class h ∈ Num(X) with h2 > 0 such that Di2 < 0 ≤ Di · Dj
and h · Di > 0 ≥ (aDi + bDj )2 for all i 6= j and all integers a, b ≥ 0. We show D has
these properties if and only if it forms a strict hyperbolic code with angle at least π/2 in
the hyperbolic space of dimension n = ρ(X) − 1 associated with the intersection pairing
on Num(X). Now, suppose that F is a set of curves as in Theorem 1.3 with more than
Rn (π/2) elements. The map sending C ∈ F to its class [C] in Num(X) is injective, since
C1 · C2 ≥ 0 > C12 if C1 and C2 are distinct elements of F. Therefore D = {[C] : C ∈ F}
has more than Rn (π/2) elements. Taking h to be the class of an ample effective divisor, we
conclude that there are two curves C1 , C2 ∈ F and integers a, b ≥ 0 such that E = aC1 +bC2
has E 2 > 0. Since the Ci are irreducible, the Nakai–Moishezon criterion implies we can
adjust a and b so that E becomes an ample effective connected divisor. This will prove
Theorem 1.3.
To prove Theorem 1.1, we now let F be the set of irreducible curves C on X with C 2 < 0
and g(C) < b1 (X)/4. Suppose F has more than Rn (π/2) elements. We show in §8 that
this leads to a contradiction in the following way.
Theorem 1.3 implies there are C1 , C2 ∈ F and a, b ≥ 0 such that E = aC1 + bC2 is an
ample effective connected divisor. An étale Lefschetz theorem of Bost [1] implies that the
induced homomorphism of étale fundamental groups
π1ét (|E|, x) −→ π1ét (X, x)
NEGATIVE CURVES OF SMALL GENUS ON SURFACES
3
at some geometric point x ∈ E ∩ X has image of finite index in π1ét (X, x), where |E| is the
reduction of E. In Theorem 8.2 of §8 we use a motivic weight argument to show that the
natural morphism
Jac(C1♯ ) ⊕ Jac(C2♯ ) −→ Alb(X)
is surjective, where Jac(Ci♯ ) is the Jacobian of the normalization Ci♯ of Ci and Alb(X) is
the Albanese variety of X. Since g(Ci ) = g(Ci♯ ) = dim(Jac(Ci♯ )) this implies that
max(g(C1 ), g(C2 )) ≥ dim(Alb(X))/2 = b1 (X)/4.
This is impossible since the curves C in Theorem 1.1 are assumed to have geometric genus
strictly less than b1 (X)/4.
This reduces the proof of Theorem 1.1 to bounding Rn (π/2) from above. We do this in §5
and §6 using the upper half-space model of hyperbolic n-space to connect hyperbolic codes
to spherical codes. The connection comes about from the fact that geodesic half-spaces in
the upper half-space model are one side of either a vertical plane or a Euclidean sphere.
The latter spheres have centers on the Euclidean space Rn−1 of points at infinity different
from ∞. In §6 we prove an upper bound for the size of a strict hyperbolic code of angle
π/2 by showing that we can assume all the half-spaces associated with elements of the code
have boundaries that are Euclidean spheres, and we can place the center of the smallest
such sphere at the origin in Rn−1 . Then the rays outward from the origin to the centers
associated to other spheres must intersect a unit sphere Sn−2 in Rn−1 in a spherical code
with angle at least arccos(3/4) = φ0 . In §7 we prove a lower bound on the maximum size
of a hyperbolic code with angle at least π/2 by placing the above centers at a well-chosen
subset of a spherical code in Rn−1 withp
angle at at least 2φ0 and by taking the radii of all
the spheres around these centers to be 7/8.
Acknowledgements: The first author would like to thank the I.H.E.S., I.M.P.A. and the
University of Leiden for their support during the writing of this paper.
2. Spherical codes
In this section we recall some definitions and results concerning spherical codes. See [3]
for further details.
Definition 2.1. A spherical code is a subset S of the unit sphere Sn−1 in n-dimensional
Euclidean space Rn . If x, y ∈ S, the angle φ(x, y) between x and y is the unique number in
the range 0 ≤ φ(x, y) ≤ π such that cos(φ(x, y)) = x · y. Define the angle φ(S) of S to be
the infimum of φ(x, y) over all distinct x, y ∈ S. For 0 < φ ≤ τ ≤ π define
(2.1)
Kn (φ, τ ) = max{#S : φ ≤ φ(x, y) ≤ τ for all x, y ∈ S with x 6= y}.
We set Kn (φ) = Kn (φ, π).
Example 2.2. The kissing number Kn = Kn (π/3) is the maximum number of spheres of a
given positive radius that can touch a sphere of the same radius without having overlapping
interiors.
The following result is due to Kabatiansky and Levenshtein [5]:
Theorem 2.3. Suppose 0 < φ ≤ π/3. If n is sufficiently large, then Kn (φ) ≤ c(φ)n , where
1
p
.
(2.2)
c(φ) =
2 0.099 1 − cos(φ)
4
T. CHINBURG AND M. STOVER
In fact, [5] shows that the same conclusion holds for φ in the range from 0 to a number
slightly larger than π/3. The following lower bound is due Chabauty, Shannon and Wyner
[3, §1.6].
Theorem 2.4. Suppose 0 < φ < π/2 and 1 < c <
(2.3)
1
sin(φ) .
If n is sufficiently large, then
Kn (φ) ≥ cn .
3. Hyperbolic codes
The hyperbolic variant of spherical codes developed in this section is motivated by the
following observation. A point w in a spherical code W ⊂ Sn−1 determines and is determined
by the geodesic half-space
Z(w) = {w′ ∈ Sn−1 : hw, w′ i ≤ 0},
where h , i is the usual Euclidean inner product. One can thus reformulate spherical codes
as collections of geodesic half-spaces of Sn−1 whose outward normals form at least a certain
angle at their intersections.
This interpretation carries over directly to hyperbolic space. One complication is that
in hyperbolic space, half-spaces may not intersect and one half-space can properly contain
another. To formulate a precise definition, we first recall the definition of the hyperboloid
model Ln and the ball model B n of hyperbolic n-space. See [7, §3.2] for details.
Let h , i : Rn ⊗Rn −→ R be the usual Euclidean inner product with norm k k2 : Rn −→ R.
Define an inner product I : (Rn ⊥ R) ⊕ (Rn ⊥ R) −→ R by
I((v; u), (v ′ ; u′ )) = −hv, v ′ i + u · u′ .
For q = (v; u) write q 2 = I(q, q). They (upper) hyperboloid model of of hyperbolic space is
then
Ln = q = (v; u) ∈ Rn ⊥ R : q 2 = −kvk2 + u2 = 1 and u > 0 .
p
The line element for Ln is ds = (dv)2 − (du)2 .
Let 0 be the origin in Rn . Projection
π : (Rn ⊥ R) r {(0; −1)} −→ Rn
from the point (0; −1) identifies Ln with the ball model
B n = {v ∈ Rn : kvk2 < 1}
of hyperbolic space. The ideal boundary of B n is the closed unit sphere ∂B n = Sn−1 . Define
n
B = B n ∪ ∂B n , which is the closed unit ball in Rn .
Suppose w ∈ Rn ⊥ R is a negative vector, i.e., that I(w, w) = w2 < 0. The ray
determined by w is
r(w) = {tw : 0 < t ∈ R},
and the set of points
(3.4)
Y (w) = {π(q) ∈ B n : q = (v; u) ∈ Ln and I(w, q) ≤ 0}
is a closed geodesic half-space in B n . Let
W (w) = {π(q) ∈ B n : q = (v; u) ∈ Ln and I(w, q) = 0}
be the boundary of Y (w) in B n and define ∂Y (w) ⊂ Sn−1 (resp. ∂W (w) ⊂ Sn−1 ) to be
the ideal boundary of Y (w) (resp. W (w)). Then set Y (w) = Y (w) ∪ ∂Y (w) and W (w) =
W (w) ∪ ∂W (w). The following is well-known (e.g., see [8, §2.3]).
NEGATIVE CURVES OF SMALL GENUS ON SURFACES
r(w)
Rn ⊥ R
5
Y (w)
π
w
z
π(r)
0
Bn
0
Figure 1. Illustration of Lemma 3.1.
Lemma 3.1. The map identifying the ray r(w) with the closed half-space Y (w) defines a
bijection between the set of negative rays in Rn ⊥ R and the set of closed geodesic half-spaces
n
in B n . The set W (w) is the intersection of the closed unit ball B with a Euclidean sphere
or with a hyperplane of dimension n − 1. If n ≥ 2 then W (w) intersects Sn−1 at right angles
and W (w) ∩ Sn−1 = ∂W (w) is a Euclidean sphere of dimension n − 2 and positive radius.
Definition 3.2. Suppose that q ∈ W (w). If q ∈ W (w) ⊂ B n , let nw (q) be the outward
unit normal to Y (w) at q in the tangent space Tq B n of q in B n . If q ∈ ∂W (w) ⊂ Sn−1 and
n ≥ 2, let nw (q) be the outward unit normal to Y (w) ∩ Sn−1 = ∂Y (w) at q in the tangent
space Tq Sn−1 .
We now need to understand more about the properties of the subspaces associated with
a pair of negative vectors. Suppose that w1 , w2 ∈ Rn ⊥ R are negative vectors, and in
what follows set Wi = W (wi ) and Yi = Y (wi ), i = 1, 2. Suppose that there exists a point
q ∈ W 1 ∩ W 2 . It is well-known that the angle θ(w1 , w2 ) ∈ [0, π] between nw1 (q) and nw2 (q)
satisfies
−I(w1 , w2 )
.
(3.5)
cos(θ(w1 , w2 )) = p
I(w1 , w1 ) · I(w2 , w2 )
See [7, §3.2], and note that our I is the negative of the form used there. If W (w1 )∩W (w2 ) =
∅, define θ(w1 , w2 ) = −∞. We will need the following observations.
Lemma 3.3. Suppose w1 and w2 are two negative elements of Rn ⊥ R. The following
conditions are equivalent:
(i.) θ(w1 , w2 ) ≥ π/2;
(ii.) I(w1 , w2 ) ≥ 0 and there exists a point q ∈ W 1 ∩ W 2 ;
(iii.) I(w1 , w2 ) ≥ 0 and for all 0 ≤ a, b ∈ R one has
(3.6)
I(aw1 + bw2 , aw1 + bw2 ) ≤ 0.
Now suppose that any (and hence all) of these conditions hold and that there exists h ∈
Ln ⊂ Rn ⊥ R with I(wi , h) > 0 for i = 1, 2. Then π(h) ∈
/ Y1 ∪ Y2 . Let P ∈ Sn−1 = ∂B n be
n
the limit point of the geodesic ray in B starting at π(h) and perpendicular to W1 . Then P
is a point of Y 1 r W 1 that is not in Y 2 .
Proof. Since conditions (i), (ii) and (iii) are invariant under scaling, we can assume that
w12 = w22 = −1. The fact that (i) implies (ii) is clear from (3.5). If (ii) holds, then (iii)
6
T. CHINBURG AND M. STOVER
Y2
W2
w1
q
π(h)
θ
W1
Y1 ∩ Y2
w2
Y1
Figure 2. Geometric picture for Lemma 3.3.
follows from expanding I(aw1 + bw2 , aw1 + bw2 ) and using (3.5). Similarly, (iii) implies
|I(w1 , w2 )| ≤ 1, which means that W 1 and W 2 meet with angle given as in (3.5) and hence
(iii) implies (i).
We now suppose that (i), (ii) and (iii) hold and that there is an h ∈ Ln as in the last
part of the lemma. The fact that π(h) ∈
/ Y1 ∪ Y2 follows immediately from the definition
of the Yi . Intersecting with the appropriate totally geodesic B 2 inside B n , it suffices to
prove the claim for P in the hyperbolic plane. Then we have the geometric arrangement
shown in Figure 2. If the geodesic ray ℓ from π(h) intersecting W1 orthogonally were to
have endpoint on ∂Y 2 , then it would need to also meet W 2 .
Let z1 be the point at which ℓ meets W1 and z2 the point where ℓ meets W 2 . When
2
q, z1 , and z2 are distinct, they form a triangle in B , possibly with an ideal vertex at z2 ,
with interior angle θ at q and π/2 at z1 . Therefore the triangle has angle sum greater than
2
or equal to π, which is impossible for a triangle in B [7, §3.5]. In the degenerate case,
q = z1 = z2 , and the geodesic from π(h) to q visibly makes an angle
φ < π − θ ≤ π/2
with W1 at q, and hence cannot be orthogonal to W1 . Since π(h) is not in W1 and W1 is
totally geodesic, it is also clear that the endpoint of ℓ cannot be in W 1 . This completes the
proof of Lemma 3.3.
Definition 3.4. A hyperbolic code is a collection S of negative vectors w ∈ Rn ⊥ R. We
say that S is strict if the union over all w ∈ S of the half-spaces Y (w) is not all of B n .
Define θ(S) ∈ {−∞} ∪ [0, π] to be the greatest lower bound over all pairs w1 , w2 of distinct
elements of S of the angle θ(w1 , w2 ) defined above.
Definition 3.5. Let θ be an angle in the range 0 < θ ≤ π. The hyperbolic kissing number
(resp. strict hyperbolic kissing number ) Rn (θ) (resp. Rn (θ)) in Z ∪ {∞} is the supremum of
#S over all hyperbolic codes S (resp. strict hyperbolic codes S) for which θ(S) ≥ θ.
NEGATIVE CURVES OF SMALL GENUS ON SURFACES
7
4. Negative curves, hyperbolic codes, and Theorem 1.1.
As in the introduction, let X be an irreducible smooth projective surface over a field k.
The group Num(X) is torsion free and finitely generated. Let
Num(X)R = R ⊗Z Num(X).
The Hodge index theorem implies that the intersection pairing on Num(X) extends to a
pairing
I : Num(X)R × Num(X)R −→ R
with signature (1, n), where dim(Num(X)R ) = n + 1.
Definition 4.1. Let L(X) be the hyperboloid model of hyperbolic n-space associated with
the choice of isometry carrying I to the standard signature (1, n) pairing on Rn ⊥ R.
Definition 4.2. Let T (X) be the set of all irreducible curves C on X for which C 2 =
I(C, C) < 0 and g(C) < b1 (X)/4. For C ∈ T (X), let [C] be the class of C in Num(X)R
and define S(X) = {[C] : C ∈ T }.
The following is a well-known consequence of the fact that distinct curves have nonnegative intersection.
Lemma 4.3. The map T (X) −→ S(X) sending C to [C] is a bijection.
We prove the following theorem in §8.
Theorem 4.4. The set S(X) is a strict hyperbolic code in Num(X)R of angle at least π/2.
Recall that ρ(X) is the rank of Num(X), i.e., the real dimension of Num(X)R . We then
have the following conclusion.
Corollary 4.5. The number of elements of T (X) is bounded above by the strict hyperbolic
kissing number Rn (π/2) when n = ρ(X) − 1.
Recall that Kn−1 (θ) is the kissing number associated to the angle θ in the Euclidean
space Rn−1 . The first statement of following Theorem will be proved in §5. The second
statement will be proved in §6 and §7.
Theorem 4.6. For every n ≥ 2, one has
Rn (π/2) ≤ Rn (π/2) ≤ 2Rn (π/2).
√
Let 0 < φ < τ ≤ π be any choice of constants such that 2 sin(φ/2) = sin(τ /2). Then
Kn−1 (2φ0 )
(4.8)
max
, Kn−1 (φ, τ ) ≤ Rn (π/2) ≤ Kn−1 (φ0 ) + 2,
2
(4.7)
where φ0 = arccos(3/4).
From the results about spherical kissing numbers quoted in §2 we now have the following
conclusion.
Corollary 4.7. One has
2 0.011 n (1 + o(n)) ≤ Rn (π/2) ≤ 2 0.901 n (1 + o(n)),
where o(n) −→ 0 as n −→ ∞.
Proof of Theorem 1.1. Combine Corollary 4.5, Theorem 4.6, and Corollary 4.7.
8
T. CHINBURG AND M. STOVER
5. Hyperbolic codes and the upper half-space model
We now use the upper half-space model
H n = {(z1 , . . . , zn ) : zi ∈ R, zn > 0} ⊂ Rn .
of hyperbolic space to give another description of hyperbolic codes.
Recall from [7, §4.4] that there is an isometry f : Ln −→ H n defined in the following
way. For x ∈ Ln ⊂ Rn ⊥ R = Rn+1 , let y be the point (y1 , . . . , yn+1 ) on the unit (n + 1)sphere Sn in Rn+1 that is on the ray from the origin in Rn+1 to x. Then f (x) is the unique
point z = (z1 , . . . , zn ) ∈ H n such that (1, z1 , . . . , zn ) ∈ Rn+1 lies on the ray outward from
((−1, 0, 0, . . . , 0); 0) ∈ Rn+1 through y.
Consider the one-point compactification
∂H n = {∞} ∪ {(z1 , . . . , zn−1 , 0) : zi ∈ R}
of Rn−1 = {(z1 , . . . , zn−1 , 0) : zi ∈ R}. Then ∂H n is topological isomorphic to Sn−1 , and
the above construction identifies ∂H n with the boundary of H n . Geodesics in H n are the
intersection of H n with either circles or vertical lines in Rn that intersect ∂H n r{∞} = Rn−1
orthogonally. Geodesic hypersurfaces in H n are the intersection of H n with either
(i) vertical planes in Rn (i.e., planes intersecting ∂H n r {∞} = Rn−1 orthogonally), or
(ii) Euclidean spheres with center on ∂H n r {∞} (which then intersect ∂H n r {∞}
everywhere orthogonally).
Geodesic half-spaces are then formed by the set of all points of H n that lie either on one
n
chosen side of a geodesic hypersurface or on the hypersurface itself. Define H = H n ∪∂H n .
Definition 5.1. In (3.4) to each negative vector w ∈ Rn ⊥ R we defined a geodesic halfspace Y (w) in the open ball model B n of hyperbolic space with boundary W (w), a geodesic
hypersurface. Let Y ′ (w) be the corresponding geodesic half-space in H n with boundary
W ′ (w). Similarly, let ∂Y ′ (w) ⊂ ∂H n (resp. ∂W ′ (w) ⊂ ∂H n ) be the ideal boundary of
Y ′ (w) (resp. W ′ (w)). Finally, set Y ′ (w) = Y ′ (w) ∪ ∂Y ′ (w) and W ′ (w) = W ′ (w) ∪ ∂W ′ (w).
If W ′ (w) lies in a vertical plane we will say that the center z(w) of Y ′ (w) is the point ∞ of
∂H n and that the Euclidean radius of ∂Y ′ (w) is ∞. Otherwise, W ′ (w) is the intersection of
H n with a Euclidean sphere of some positive radius d(w) centered at a point z(w) ∈ Rn−1 =
∂H n r {∞}. If z(w) 6= ∞ and z(w) ∈ Y ′ (w), then Y ′ (w) is the intersection of H n with
the closed Euclidean ball of radius d(w) about z(w). Otherwise, Y ′ (w) is the intersection
of H n with the complement of the interior of this ball.
We now reformulate the condition that θ(w1 , w2 ) ≥ π/2 in Lemma 3.3 using the upper
half-space model. To simplify notation in what follows, given a negative vector wi ∈ Rn ⊥ R
we let Yi = Y (wi ) and similarly for the other notation from Definition 5.1.
Lemma 5.2. Suppose w1 and w2 are two negative elements of Rn ⊥ R such that neither
W1′ nor W2′ lie in a vertical plane. For z1 , z2 , d1 , d2 as in Definition 5.1, let |z1 − z2 | be the
′
Euclidean distance between z1 and z2 in Rn−1 . Define δi = 1 if zi ∈ Y i , and set δi = −1
otherwise. Then θ(w1 , w2 ) ≥ π/2 if and only if and only if
q
(5.9)
d21 + d22 ≤|z1 − z2 | ≤ d1 + d2
when δ1 δ2 = 1
q
|d1 − d2 | ≤|z1 − z2 | ≤ d21 + d22
(5.10)
when δ1 δ2 = −1.
NEGATIVE CURVES OF SMALL GENUS ON SURFACES
9
Finally, if θ(w1 , w2 ) ≥ π/2 and I(h, w1 ), I(h, w2 ) > 0 for some h ∈ Ln ⊂ Rn ⊥ R, then
z1 6= z2 .
′
n
Proof. For i = 1, 2, the half-space Y i is either H ∩B(zi , di ) (when δi = 1) or the complement
n
in H of the interior of B(zi , di ) (when δ(wi ) = −1).
Suppose first that θ(w1 , w2 ) ≥ π/2, so that there is a point q ∈ W 1 ∩ W 2 . If δ1 δ2 = 1,
the angle between theprays from q to z1 and from q to z2 is at least θ(w1 , w2 ) ≥ π/2.
′
′
Therefore |z1 − z2 | ≥ d21 + d22 . In this case, the existence of a point in W 1 ∩ W 2 implies
that |z1 − z2 | ≤ d1 + d2 . This proves (5.9).
If δ1 δ2 = −1, the angle between the rays from q to wp
1 and from q to w2 is at most π/2,
′
′
rather than being at least π/2. This leads to |z1 − z2 | ≤ d21 + d22 . Since W
p1 and W 2 must
intersect, we see that d1 + d2 ≥ |z1 − z2 | ≥ |d1 − d2 |. Note that |z1 − z2 | ≤ d21 + d22 already
implies d1 + d2 ≥ |z1 − z2 |. This gives (5.10).
For the converse, one reverses the above reasoning to show that (5.9) and (5.10) imply
that θ(w1 , w2 ) ≥ π/2. If z1 = z2 , then W1′ and W2′ are the the intersection of H n with
′
′
concentric spheres. Hence if θ(w1 , w2 ) ≥ π/2, we would have W 1 = W 2 and θ(w1 , w2 ) = π.
n
However, then Y 1 ∪ Y 2 = B , so there could be no h ∈ Ln with I(h, w1 ), I(h, w2 ) > 0.
We now have the following, which is one of the main technical results in this paper.
Theorem 5.3. For all integers m, n ≥ 1 , the following are are equivalent:
(1) There are elements w0 , · · · , wm ∈ Rn ⊥ R such that for some h ∈ Rn ⊥ R one has
(a)
(b)
(c)
(d)
I(h, h) > 0 > I(wi , wi ),
I(h, wi ) > 0,
I(wi , wj ) ≥ 0, and
I(awi +bwj , awi +bwj ) ≤ 0 for all distinct 0 ≤ i, j ≤ m and all positive a, b ∈ R.
(2) The subset {w0 , . . . , wm } ⊂ Rn ⊥ R is a strict hyperbolic code having m + 1 elements
and angle at least π/2.
(3) After replacing the m + 1 element subset {w0 , . . . , wm } ⊂ Rn ⊥ R by their image
′
′
under an isometry, the set {Y 0 , . . . , Y m } of half-spaces in H n has the following
description.
′
The ideal boundary of each Y i is a sphere centered at a point zi ∈ Rn−1 of some
′
radius di > 0. When i = 0, the point z0 is the origin 0 of Rn−1 , and Y 0 is the
n
exterior in H of the open ball of radius d0 .
′
n
If 1 ≤ i ≤ m, Y i is the intersection of H with the closed ball of radius di around
zi . Finally, the following inequalities hold:
(a) |zj |2 > max(0, d2j − d20 ) if 1 ≤ j ≤ m,
q
(b) |d0 − di | ≤ |zi | ≤ d20 + d2i , and
q
(c) d2i + d2j ≤ |zi − zj | ≤ di + dj if 1 ≤ i < j ≤ m
where |z − z ′ | is the Euclidean distance between points z, z ′ ∈ R.
Lastly, if {z1 , . . . , zm } is any set of m distinct points in Rn−1 for which there are positive constants d1 , . . . , dm > 0 such that condition (c) of part (3) holds, then there exist
h, w1 , . . . , wm ∈ Rn ⊥ R for which the statements in condition (1) hold for 1 ≤ i, j ≤ m.
10
T. CHINBURG AND M. STOVER
∞
Y0′
ℓ0
f (h)
Y0′ ∩ Y1′
Y1′
Y2′
z1
d1
Y1′ ∩ Y2′
0
d0
z2
Y0′ ∩ Y2′
d2
Figure 3. Arrangement of half-spaces in Theorem 5.3.
Proof. Lemma 3.3 shows the equivalence
of (1) and (2). Indeed, if h ∈ Rn ⊥ R and
p
I(h, h) > 0, we can replace h by h/ I(h, h) to make h an element of Ln .
To show that (1) implies (3), let h and w0 , . . . , wm be as in (1), where as above we can
assume h ∈ Ln . Let P̃ be the limit point on ∂H n of the geodesic ray in H n that starts
at f (h) and is perpendicular to the geodesic hypersurface W0′ , where f : Ln −→ H n is
the above isometry. This geodesic ray is part of a geodesic line with another limit point
P̃ ′ on ∂H n . Applying an isometry, we can assume that P̃ = ∞ and P̃ ′ is the origin 0 of
Rn−1 ⊂ ∂H n .
Translating the final statement of Lemma 3.3 to the upper half plane model, P̃ is a
′
′
′
point of Y 0 r W 0 that does not lie on Y j for any j > 0. Consider the points z0 , . . . , zm
associated with the wi by Definition 5.1. It is clear from our assumptions that each zj lies
in Rn−1 ⊂ ∂H n . Recall that Wj′ is the intersection of H n with a Euclidean sphere of radius
dj > 0 and center zj .
′
′
n
When j = 0, we know ∞ = P ′ ∈ Y 0 , so Y 0 must be the complement in H of the interior
B(z0 , d0 ) of the ball at z0 of radius d0 . Thus δ0 = −1 in the terminology of Lemma 5.2.
Furthermore, the sphere W0 is perpendicular to the geodesic with limit points ∞ and 0,
and this geodesic contains f (h), and we conclude that z0 = 0. Note also that now f (h)
must lie in the interior ℓ0 of the vertical line segment of Euclidean length d0 that has one
endpoint at 0. See Figure 3.
′
′
n
Suppose 1 ≤ j ≤ m. Then ∞ 6∈ Y j implies that Y j = H ∩ B(zj , dj ), so δj = 1 for
′
1 ≤ j ≤ m. Since f (h) is not contained in Y j , we find from the fact that f (h) is on ℓ0 that
d20 + |zj |2 > d2j .
We now apply the criterion in Lemma 5.2 to every pair wi , wj with 0 ≤ i 6= j ≤ m to
produce the inequalities in part (3) of Theorem 5.3.
Conversely, suppose all of the inequalities stated in part (3) of Theorem 5.3 are satisfied
n−1 and positive real numbers {d }m . Then z 6= z
with z0 = 0 and some of {zi }m
i i=0
0
i
i=1 ⊂ R
NEGATIVE CURVES OF SMALL GENUS ON SURFACES
11
for 1 ≤ i ≤ m because |zi | =
6 0 was assumed in part (3c) of Theorem 5.3. We can choose
′
n
negative vectors w0 , . . . , wm in Rn ⊥ R such that Y 0 is the complement in H of the open
′
n
unit ball of radius d0 about the origin z0 and Y i is H ∩ B(zi , di ) for 1 ≤ i ≤ m.
The assumption that d20 + |zi |2 > d2i for 1 ≤ i ≤ m in part (3) of Theorem 5.3 implies
that if we choose h ∈ Ln so that f (h) lies in B(0, 1) and is close enough to the point that
′
lies at distance 1 directly above the origin, then f (h) will not be in Y i for 1 ≤ i ≤ m or
′
in Y 0 . Now Lemma 5.2 shows that h, w0 , w1 , . . . , wm satisfy the conditions in part (1) of
Theorem 5.3.
The final statement we must prove is that if one has only points z1 , . . . , zm in Rn−1
and positive numbers d1 , . . . , dm for which part (c) of condition (3) holds, then there are
h, w1 , . . . , wm ∈ N (X)R for which condition (1) holds for 1 ≤ i, j ≤ m. In this case, we
′
n
choose wi so that Y i is H ∩ B(zi , di ) for 1 ≤ i ≤ m. Then the vertical heights of points
′
of each Y i are bounded, so we can find a point f (h) ∈ H n not in this union. Lemma
5.2 now shows that h, w1 , . . . , wm satisfy the conditions in part (1) of the theorem for
1 ≤ i, j ≤ m.
We now give a number of corollaries to Theorem 5.3.
Corollary 5.4. The strict hyperbolic kissing number Rn (π/2) is the supremum of m + 1
over all integers m for which there exist distinct points z1 , · · · , zm ∈ Rn−1 and positive real
constants d1 , · · · , dm for which
(a) max{0, d2i − 1} <q|zi |,
(b) |1 − di | ≤ |di | ≤ 1 + d2i , and
q
(c) d2i + d2j ≤ |zi − zj | ≤ di + dj .
Proof. The corollary follows from renormalizing the zi and di as in part (3) of Theorem 5.3
by dividing each by d0 , 1 ≤ i ≤ m.
Corollary 5.5. Suppose there is a is a possibly nonstrict hyperbolic code in Ln having m′
elements and angle at least π/2. If m = ⌈m′ /2⌉, then there is a strict hyperbolic code having
at least m elements and angle at least π/2.
Proof. Applying an isometry, suppose we have a nonstrict code S = {w1 , . . . , wm′ } with
angle at least π/2 such that, in the upper half-space model each wi gives a point zi ∈ Rn−1
together with a positive radius di . Removing at most half the wi , we can replace m′ by m
and assume that all of the constants δi are the same. In other words, all of the half-spaces
′
Y i come from either the interiors Ui of the open balls B(zi , di ), or all of them come from
the exterior of its closure of Ui . This means that we now satisfy the inequalities in (5.9) of
′
Lemma 5.2 for all 1 ≤ i, j ≤ m with i = j. Replacing each Y i by Ui now produces, via the
last statement of Theorem 5.3, a strict hyperbolic code with m elements, since the union of
all the Ui cannot be all of H n .
Corollary 5.6. Suppose that there is a largest positive integer m = m(n) for which the
equivalent conditions (1), (2), and (3) in Theorem 5.3 can be satisfied by some choice of
h, wi , zi and di as i ranges over 0 ≤ i ≤ m. Then m + 1 is the strict hyperbolic kissing
number Rn (π/2). The (nonstrict) hyperbolic kissing number Rn (π/2) satisfies
(5.11)
Rn (π/2) ≤ Rn (π/2) ≤ 2Rn (π/2).
12
T. CHINBURG AND M. STOVER
z3
z1
z2
Figure 4. Optimizing Lemma 6.1.
6. The upper bound on Rn (π/2).
We begin with the following technical estimate.
Lemma 6.1. Suppose n ≥ 2, z1 , z2 , z3 ∈ Rn−1 , 0 < d1 ≤ d2 ≤ d3 and that
q
(6.12)
d2i + d2j ≤ |zi − zj | ≤ di + dj for all i 6= j
as in condition (c) of Corollary 5.4 (cf. condition 2(c) of Theorem 5.3). Then n ≥ 3 and
z1 , z2 and z3 are not collinear. Let 0 < θ1 < π be the angle of the triangle (z1 , z2 , z3 ) at z1 .
Then θ1 ≥ φ0 = arccos(3/4).
Proof. Considering the subspace spanned by z1 , z2 , and z3 , we can reduce to the case where
n ≤ 3. If z1 , z2 , and z3 are collinear, we can assume n = 2 and z1 < z2 < z3 in Rn−1 = R.
Then (6.12) leads to a contradiction. Therefore after a translation and scaling, we can
assume n = 3, z1 = (0, 0) = 0 is the origin in R2 and 0 < d1 ≤ d2 ≤ d3 = 1. The input of
z2 , z3 , d1 and d2 is now specified by 6 real variables, and we want to maximize the function
cos(θ1 ) of these variables. It is a lengthy but elementary calculus exercise to show that
the maximum is obtained when cos(θ1 ) = 3/4. We list the steps involved here and include
complete details in an appendix.
Regarding d1 , d2 and d3 = 1 as fixed for the moment, let S(d1 , d2 ) be the set of
(z1 , z2 , z3 ) = (0, z2 , z3 ) that satisfy (6.12). One checks that cos(θ1 ) is a continuous function
on the compact set S(d1 , d2 ), so that it attains its maximum at some point (z1 , z2 , z3 ) =
(0, z2 , z3 ) in S(d1 , d2 ). To prove the lemma it suffices to show that θ1 ≥ φ0 .
The main fact we can now use is that since (z1 , z2 , z3 ) ∈ S(d1 , d2 ) maximizes cos(θ1 ), we
cannot move z1 , z2 and z3 in R2 and then translate z1 back to 0 in such a way that the
inequalities (6.12) still hold with the same d1 , d2 and d3 = 1 but with a smaller value for θ1 .
By considering such moves, we show in the appendix that the maximum value of cos(θ1 )
over all possible choices of 0 < d1 ≤ d2 ≤ d3 = 1 is attained by the example in Remark 6.2
below.
Remark 6.2. An angle of θ1 = φ0 can be achieved by setting d1 = d2 = d3√= 1, n = 3,
7/2). Then
z1 = (0, 0) ∈ Rn−1 = R2 , z3 = (2, 0) and z2 = (2cos(θ0 ), 2sin(φ
p 0 )) = (3/2,
√
|z1 − z3 | = d1 + d3 = 2 = d1 + d2 = |z1 − z2 | and |z2 − z3 | = d22 + d23 = 2. See Figure 4.
We now prove the following, which implies the upper bound in (4.8) of Theorem 4.6 as
well as all the bounds in (4.7) via Corollary 5.6.
NEGATIVE CURVES OF SMALL GENUS ON SURFACES
13
n−1 and d , . . . , d
Corollary 6.3. Suppose
1
m > 0 satisfy condition (c) of
q z1 , . . . , zm ∈ R
2
2
Corollary 5.4, so that di + dj ≤ |zi − zj | ≤ di + dj for all i 6= j. Then m ≤ Kn−1 (φ0 ) + 1.
In particular, the number m(n) from Corollary 5.6 satisfies m(n) ≤ Kn−1 (φ0 ) + 2.
Proof. Without loss of generality, we can order z1 , . . . , zm so that d1 ≤ di for all 1 ≤ i ≤ m.
By condition (c), the points zi are all distinct. Therefore, for 1 < i ≤ m the points
ξi = (zi − z1 )/|zi − z1 |
lie on the unit sphere Sn−2 in Rn−1 . Lemma 6.1 shows that for all 1 < i < j ≤ m, the angle
between the rays from the origin to ξi and to ξj must be at least φ0 . Therefore ξ2 , . . . , ξm
must form a spherical code with angular separation at least φ0 , so m − 1 ≤ Kn−1 (φ0 ). Then
the number m(n) from Corollary 5.6 is the number of points z0 , z1 , . . . , zm for which there
are d0 , . . . , dm as in Theorem 5.3, so we conclude m(n) ≤ m + 1 ≤ Kn−1 (φ0 ) + 2.
7. The lower bound on Rn (π/2).
√
Let 0 < φ < τ ≤ π be any choice of constants such that 2 sin(φ/2) = sin(τ /2) and
define m = Kn−1 (φ, τ ). We can therefore find a spherical code S = {z1 , . . . , zm } on the unit
sphere Sn−2 in Rn−1 such that the angular separation φ(zi , zj ) between the rays from the
origin to zi and to zj satisfies φ ≤ φ(zi , zj ) ≤ τ for all i 6= j. Therefore,
4sin2 (φ/2) = 2 − 2cos(φ) ≤ |zi − zj |2 = 2 − 2cos(φ(zi , zj )) ≤ 4sin2 (τ /2).
√
It follows that if we let dk = 2 sin(φ/2) = sin(τ /2) for all k = 1, . . . , m, then
q
d2i + d2j ≤ |zi − zj | ≤ di + dj
for all i 6= j, as in condition (c) of Corollary 5.4. Theorem 5.3 now says that there are
h, w1 , . . . , wm ∈ N (X)R for which the statements in condition (1) of Theorem 5.3 hold for
1 ≤ i, j ≤ m. Part (2) of Theorem 5.3 now says {w1 , . . . , wm } is a strict hyperbolic code
with angle at least π/2. Therefore Definition 3.5 gives that
m = Kn−1 (φ, τ ) ≤ Rn (π/2).
This is the first part of the lower bound (4.8) in Theorem 4.6.
To show the other lower bound in (4.8) of Theorem 4.6, it will suffice to show that when
φ0 = arccos(3/4), we have
Kn−1 (2φ0 )/2 ≤ Kn−1 (φ, τ )
for some φ and τ as above. Let φ = 2φ0 = 1.445... and τ = π − φ0 = 2.418..., so 0 < φ <
τ ≤ π. We then have
2sin2 (φ/2) = 2sin2 (φ0 ) = 2(1 − cos2 (φ0 )) = 2(1 − 9/16) = 7/8
sin2 (τ /2) = sin2 (π/2 − φ0 /2) = cos2 (φ0 /2) =
cos(φ0 ) + 1
3/4 + 1
=
= 7/8.
2
2
√
Thus 2 sin(φ/2) = sin(τ /2) since both of these numbers are positive.
Recall that if z and w are points on the unit sphere Sn−1 , φ(z, w) is the angle between the
rays z̃ and w̃ from the origin to z and to w, respectively. By the definition of ℓ = Kn−1 (2φ0 ),
we can find a spherical code S ′ = {r1 , . . . , rℓ } on Sn−2 such that
(7.13)
φ(ri , rj ) ≥ 2φ0
if
i 6= j.
14
T. CHINBURG AND M. STOVER
For each i, consider the open cone C(−ri ) of points z ∈ Sn−2 such that φ(−ri , z) < φ0 . If
there were two distinct points rj and rq in S ′ ∩ C(−ri ), then
φ(rj , rq ) ≤ φ(−ri , rj ) + φ(−ri , rq ) < 2φ0 ,
which contradicts (7.13). Therefore there is at most point point of the form rj in S ′ ∩C(−ri ),
and if such an rj exists, ri is the unique point in S ′ ∩ C(−rj ). Throwing away at most half
of the points in S ′ we then arrive at a spherical code S = {z1 , . . . , zℓ′ } with ℓ′ ≥ ℓ/2 =
Kn−1 (2φ0 )/2 such that S ∩ C(−zi ) = ∅ for all i. If j 6= i, then the angle φ(zi , zj ) can be
at most π − φ0 , since zj does not lie in C(−zi ). We therefore have φ = 2φ0 ≤ φ(zi , zj ) ≤
π − φ0 = τ , which shows that Kn−1 (2φ0 )/2 ≤ Kn−1 (φ, τ ). This finishes the proof of the
lower bound in (4.8) of Theorem 4.6.
8. The proofs of Theorems 1.3, 1.1, and 4.4.
Theorem 1.3 is equivalent to the following result.
Theorem 8.1. Suppose F is a set of irreducible curves C on X such that C 2 < 0 and
there is no ample connected effective divisor of the form pC1 + qC2 with 0 ≤ p, q ∈ Z and
C1 , C2 ∈ F. Then the set {[C] : C ∈ F} is a strict hyperbolic code of angle at least π/2.
Proof. Recall from Lemma 4.3 that the elements [C] are all distinct in Num(X). Let
pA be an
ample effective divisor on X. Then I([A], [C]) > 0 for C ∈ F. Therefore h = [A]/ I(A, A)
is an element of the hyperbolic space L(X) associated with the intersection pairing on
R ⊗Z Num(X), and it does not lie in any of the geodesic half-spaces
H([C]) = {q ∈ L(X) : I(q, [C]) ≤ 0},
hence
T =
[
H([C])
C∈F
is not all of L(X).
Suppose that T is not a strict hyperbolic code with angle at least π/2. Then θ([C1 ], [C2 ]) <
π/2 for some distinct elements C1 , C2 of F, and Lemma 3.3 shows that there are 0 ≤ a, b ∈ R
such that
I(a[C1 ] + b[C2 ], a[C1 ] + b[C2 ]) = αa2 + 2βab + γb2 > 0,
where
α = I([C1 ], [C1 ]) < 0
γ = I([C2 ], [C2 ]) < 0
β = I([C1 ], [C2 ]) ≥ 0.
Therefore β > 0 and β 2 > αγ. There will be positive integers p and q such that
0 < −γ/β < p/q < −β/α.
Then I([C1 ], p[C1 ] + q[C1 ]) = pα + βq > 0 and I([C2 ], p[C1 ] + q[C2 ]) = pβ + qγ > 0. The
Nakai–Moishezon criterion implies that pC1 + qC2 is an effective connected ample divisor,
contradicting the hypothesis of Theorem 8.1. This proves the Theorem.
NEGATIVE CURVES OF SMALL GENUS ON SURFACES
15
In view of the arguments in in §4 and the results we have already shown, the proof of
Theorem 1.1 is reduced to showing Theorem 4.4.
As in the statement of Theorem 4.4, let T (X) be the set of all irreducible curves C on
X for which C 2 = I(C, C) < 0 and g(C) < b1 (X)/4, and set
S(X) = {[C] : C ∈ T } ⊂ R ⊗Z Num(X).
We must show that S(X) is a strict hyperbolic code in L(X) with angle at least π/2. We
suppose throughout this section that this is not the case, and we will derive a contraction.
Theorem 8.1 implies there is an effective connected ample divisor on X of the form
pC1 + qC2 in which C1 and C2 are elements of T (X) and 0 < p, q ∈ Z. We will prove the
following result below:
Theorem 8.2. Suppose that E is an connected effective ample divisor on a smooth projective
geometrically integral surface X over a field k. Let E ♯ be the normalization of the reduction
|E| of E. Let J(E ♯ ) be the direct sum of the Jacobians of the irreducible components of E ♯ .
Then the natural morphism from J(E ♯ ) to the Albanese variety Alb(X) of X is surjective.
Before giving the proof, we note how it implies Theorem 4.4. If E = pC1 + qC2 as above,
we obtain a surjection
J(E ♯ ) = J(C1♯ ) ⊕ J(C2♯ ) −→ Alb(X).
Since Alb(X) has dimension b1 (X)/4 and J(Ci♯ ) has dimension the geometric genus g(Ci ),
we see that g(C1 ) + g(C2 ) ≥ b1 (X)/2. However, we supposed that every curve C ∈ T (X)
has g(C) < b1 (X)/4, and this contradiction proves Theorem 4.4. This also completes the
proof of Theorem 1.3.
Proof of Theorem 8.2. It suffices to prove the theorem for the base change of E and X to
an algebraic closure of k. We assume for the rest of the proof that k is algebraically closed.
Let f be the pullback morphism from the Picard variety Pic0,red (X) of X to the direct
sum Pic0,red (E ♯ ) of the Picard varieties of the irreducible components of E ♯ . By duality, it
will be enough to show that Ker(f ) is a finite group scheme. We suppose in what follows
that Ker(f ) is not finite and we will derive a contradiction.
Since Ker(f ) is a subgroup scheme of an abelian variety, it is an extension of an abelian
variety B of positive genus by a finite group scheme. Let ℓ be a prime different from the
characteristic of k. Then the ℓ-adic Tate module Tℓ (Ker(f )) is isomorphic to Tℓ (B), and it
is a positive rank submodule of Tℓ (Pic0,red (X)). The pullback morphism from Tℓ (Ker(f )) =
Tℓ (B) to Tℓ (Pic0,red (E ♯ )) is trivial.
We know from Bost [1] that the morphism
π1ét (|E|, x) −→ π1ét (X, x)
of étale fundamental groups at a geometric point x in the support of |E| is surjective, since
E is ample and effective. This means that
Hom(π1ét (X, x), Z/ℓn ) −→ Hom(π1ét (|E|, x), Z/ℓn )
is injective for all n. Since k is algebraically closed, Z/ℓn is isomorphic to the group scheme
µℓn of (ℓn )th roots of unity. Hence the Kummer sequence shows that
Hom(π1ét (X, x), Z/ℓn ) = Pic(X)[ℓn ] −→ Hom(π1ét (|E|, x), Z/ℓn ) = Pic(|E|)[ℓn ]
is injective for all n.
16
T. CHINBURG AND M. STOVER
Taking inverse limits over n we see that the pullback homorphism
Tℓ (Pic(X)) −→ Tℓ (Pic(|E|))
is injective. On the other hand, Tℓ (B) ⊆ Tℓ (Pic(X)) maps to 0 in Tℓ (Pic0,red (E ♯ )), so the
pullback of line bundles must induce an injection
(8.14)
ξ : Tℓ (B) −→ U = Ker Tℓ (Pic(|E|)) −→ Tℓ (Pic0,red (E ♯ ) .
We will derive our contradiction from this statement.
All of the above schemes are defined over finitely generated algebras over Z. By increasing
ℓ, if necessary, we can find a specialization of all of the above schemes over a finite field k′
of characteristic p not equal to ℓ so that it will suffice to show the map ξ in (8.14) is not
injective for k an algebraic closure of k′ .
We now analyze U using the map π : E ♯ −→ |E| coming from the fact that E ♯ is the
normalization of |E|. We have an exact sequence of sheaves of groups in the étale topology
of |E| given by
1 −→ Gm,|E| −→ π∗ Gm,E ♯ −→ V −→ 1
in which V has support of dimension 0. Since π is finite, when we take the étale cohomology
of this sequence, we find that U is a quotient of
M = lim
H 0 (k ⊗k′ |E|, V )[ℓn ],
←−
n
is the
torsion in the H 0 (k ⊗k′ |E|, V ).
where
⊗k′ |E|, V
Recall that ℓ is prime to the residue characteristic of the finite field k ′ over which we are
working. There is a filtration of H 0 (k ⊗k′ |E|, V ) by Gal(k/k′ )-stable submodules such that
each graded quotient is isomorphic to either k∗ or the additive group k+ . Therefore, if Φ is
the arithmetic Frobenius of Gal(k/k′ ), then the eigenvalues of Φ on M are all equal to the
order #k′ of k′ . This implies that the eigenvalues of Φ on U equal #k′ .
On the other hand Tℓ (B) is the Tate module of an abelian variety B over k′ , so the
eigenvalues of Φ on Tℓ (B) have absolute value the square root of #k′ by the Weil conjectures. It follows from this that ξ cannot be injective, since Tℓ (B) has positive rank. This
contradiction completes the proof.
H 0 (k
)[ℓn ]
ℓn
9. Appendix: A calculus exercise
In this appendix we complete the proof of Lemma 6.1, whose notation we now assume.
As in that proof, we begin by fixing 0 < d1 ≤ d2 ≤ d3 = 1. Let 0 = (0, 0) be the origin in
R2 = Rn−1 , and let S(d1 , d2 ) be the set of triples (z1 , z2 , z3 ) = (0, z2 , z3 ) with z2 , z3 ∈ R2
that satisfy (6.12). Then (6.12) implies that |z1 − z2 | = |z2 |, |z1 − z3 | = |z3 |, and |z2 − z3 |
are bounded above and below by positive constants. It follows that S(d1 , d2 ) is compact.
The law of cosines gives
(9.15)
cos(θ1 ) =
|z1 − z2 |2 + |z1 − z3 |2 − |z2 − z3 |2
,
2 · |z1 − z2 | · |z1 − z3 |
where the denominator on the right is bounded away from 0. Thus cos(θ1 ) is a continuous
function on S(d1 , d2 ), so it attains its maximum. We now assume this maximum occurs
at (z1 , z2 , z3 ) = (0, z2 , z3 ). As noted in §6, to prove Lemma 6.1 it will suffice to show that
θ1 ≥ φ0 .
Let θ2 and θ3 be the angles at z2 and at z3 between the sides of the triangle with vertices
at z1 , z2 , z3 , respectively. Since we observed in §6 that (6.12) implies that z1 , z2 , and z3 are
NEGATIVE CURVES OF SMALL GENUS ON SURFACES
17
not collinear, all of θ1 , θ2 and θ3 lie in the open interval (0, π). Suppose θ3 ≥ π/2. Then
(6.12) gives
(d1 + d2 )2 ≥ |z1 − z2 |2
(9.16)
≥ |z1 − z3 |2 + |z2 − z3 |2
≥
d21
+
d23
+
d22
+
d23 .
(since θ3 ≥ π/2)
This gives
2d1 d2 ≥ 2d23 = 2.
However, d1 ≤ d2 ≤ d3 = 1, so this is only possible if d1 = d2 = d3 = 1 and if all of the
inequalities in (9.16) are equalities. Hence (z1 , z2 , z3 ) = p
(0, z2 , z3 ) is a√right triangle with
side
lengths
|z
−
z
|
=
|z
|
=
d
+
d
=
2,
|z
−
z
|
=
d21 + d23 = 2, and |z2 − z3 | =
1
2
2
1
2
1
3
p
√
d22 + d23 = 2. This means that θ1 = π/4 ≥ φ0 in this case, as claimed.
As noted in §6, the main fact we can now apply is that since (z1 , z2 , z3 ) ∈ S(d1 , d2 )
minimizes θ1 , we cannot move z1 , z2 and z3 in R2 and then translate z1 back to 0 in such
a way that the inequalities (6.12) still hold with the same d1 , d2 and d3 = 1 but a smaller
value for θ1 .
We may assume that z3 is a point on the positive real axis by rotating both z2 and z3
around z1 = 0. Since 0 < θ1 < φ0 < π/2, the point z2 now lies in the upper right quadrant.
Let C1 be the circle of radius |z1 − z2 | = |z2 | around z2 in R2 . Suppose that
(9.17)
|z1 − z3 | < d1 + d3 .
Let z2′ = z2 and z3′ = z3 . We now move z1 = 0 to a point z1′ on C1 that lies in the upper
left quadrant and is very close to z1 . The only side length which changes is then |z1 − z3 |,
which becomes |z1′ − z3′ | > |z1 − z3 | since z3′ = z3 lies on the positive part of the real line.
Since |z1 − z3 | < d1 + d3 , all of the inequalities in (6.12) will hold with the same d1 , d2 , d3
when we replace (z1 , z2 , z3 ) by (z1′ , z2′ , z3′ ) = (z1′ , z2 , z3 ) if z1′ is a point on C1 that lies in
the upper left quadrant and is close enough to z1 . We now show that the angle θ1′ between
the sides meeting at z1′ of the triangle (z1′ , z2′ , z3′ ) satisfies θ1′ < θ1 . This will contradict the
minimality of θ1 and show that (9.17) cannot hold.
Let θ3′ be the angle between the sides of (z1′ , z2′ , z3′ ) meeting at z3′ = z3 . Since z1′ lies in
the upper left quadrant and on the same side of the line between z2′ = z2 and z3′ = z3 as
the origin z1 = (0, 0), we have 0 < θ3′ < θ3 < π/2. Thus 0 < sin(θ3′ ) < sin(θ3 ). The law of
sines now gives
sin(θ1′ )
sin(θ3′ )
=
=
(9.18)
=
|z2′ − z3′ |
|z1′ − z2′ |
|z2 − z3 |
|z1 − z2 |
sin(θ1 )
.
sin(θ3 )
Since sin(θ3′ ) < sin(θ3 ) we conclude that sin(θ1′ ) < sin(θ1 ). Since we took z1′ to be close to
z1 = (0, 0) on C1 , we can ensure that that θ1′ is close to θ1 . Since 0 < θ1 < φ0 < π/2, we
conclude that θ1′ < θ1 , contradicting the minimality of θ1 . Thus (9.17) is false, so
(9.19)
z1 = 0 = (0, 0)
and z3 = (d1 + d3 , 0)
after rotating z3 as above so that it lies on the positive real line.
18
T. CHINBURG AND M. STOVER
Now suppose that
d21 + d22 < |z1 − z2 |2 < (d1 + d2 )2 .
(9.20)
Recall that z2 is a point in the upper right quadrant, and that we have reduced to the case
in which (9.19) holds. We let z2′ = z2 and z3′ = z3 . Define C2 to be the circle with center
z3 = (d1 + d3 , 0) and radius d1 + d2 , so that C2 contains z1 = 0 by (9.19). Consider points
z1′ very close to z1 on C2 . The only edge distance that can change on replacing (z1 , z2 , z3 )
by (z1′ , z2′ , z3′ ) is |z1 − z2 |. Since |z1′ − z2′ | = |z1′ − z2 | will be close to |z1 − z2 | = |z2 | if z1′ is
close to z1 , we conclude from (9.20) that all the inequalities in (6.12) will hold if (z1 , z2 , z3 )
is replaced by (z1′ , z2′ , z3′ ) = (z1′ , z2 , z3 ) and z1′ is any point on C2 sufficiently close to z1 .
Recall that θ2 is the angle at z2 between the sides of the triangle (z1 , z2 , z3 ) adjoining z2 .
Let θ2′ be the corresponding angle for the triangle (z1′ , z2′ , z3′ ). If z1′ lies in the upper half
plane and is sufficiently close to z1 , it is on the other side of the line between z2 and z1 = 0
from z3 . It follows that θ2′ > θ2 in this case. We find similarly that θ2′ < θ2 in case z1′ is a
point of C2 that lies in the lower half plane and is sufficiently close to z1 . Thus we can in
either case choose a z1′ on C2 arbitrarily close to z1 for which
0 < sin(θ2′ ) < sin(θ2 ).
(9.21)
Since |z1 − z3 | = d1 + d2 = |z1′ − z3 | and |z2 − z3 | = |z2′ − z3′ |, the law of sines gives
sin(θ1′ )
sin(θ2′ )
=
=
=
(9.22)
|z2′ − z3′ |
|z1′ − z3′ |
|z2 − z3 |
|z1 − z3 |
sin(θ1 )
.
sin(θ2 )
Now (9.21) shows sin(θ1′ ) < sin(θ1 ). Since θ1′ will be close to θ1 < φ0 < π/2 for z1′ close to
z1 , we conclude that θ1′ < θ1 , which contradicts the minimality of θ1 . Thus the hypothesis
(9.20) must be false, and so
d21 + d22 = |z1 − z2 |2
(9.23)
or
|z1 − z2 |2 = (d1 + d2 )2 .
We now apply the law of cosines, together with (9.19) and d32 + d23 ≤ |z2 − z3 |2 from
(6.12). This gives
cos(θ1 ) =
≤
(9.24)
|z1 − z2 |2 + |z1 − z3 |2 − |z2 − z3 |2
2 · |z1 − z2 | · |z1 − z3 |
|z1 − z2 |2 + (d1 + d3 )2 − d22 − d23
2 · |z1 − z2 | · (d1 + d3 )
where d1 ≤ d2 ≤ d3 = 1.
Suppose first that d21 + d22 = |z1 − z2 | in (9.23). Then (9.24) becomes
(9.25)
cos(θ1 ) ≤
1
1
d1
d21 + d22 + (d1 + 1)2 − d22 − 1
p
≤√
=p
=p 2
2
2
2
2
2
1 + (d2 /d1 )
2 · d1 + d2 · (d1 + 1)
d1 + d2
since 0 < d1 ≤ d2 . This forces θ1 ≥ π/4, contradicting θ1 < φ0 < π/4.
NEGATIVE CURVES OF SMALL GENUS ON SURFACES
19
The remaining possibility in (9.23) is that |z1 − z2 | = d1 + d2 . Then (9.24) gives
cos(θ1 ) ≤
=
≤
(9.26)
≤
since 0 < d1 ≤ d2 ≤ d3 = 1. This
of Lemma 6.1.
(d1 + d2 )2 + (d1 + 1)2 − d22 − 12
2 · (d1 + d2 ) · (d1 + 1)
(d1 + d2 + 1)d1
(d1 + d2 ) · (d1 + 1)
1
1
(1 +
)·(
)
d1 + d2
1 + 1/d1
3
·
4
gives θ1 ≥ φ0 = arccos(3/4), which completes the proof
References
[1] J.-B. Bost. Potential theory and Lefschetz theorems for arithmetic surfaces. Ann. Sci. École Norm. Sup.
(4), 32(2):241–312, 1999.
[2] J. H. Conway and N. J. A. Sloane. Sphere packings, lattices and groups, volume 290 of Grundlehren der
Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag,
New York, third edition, 1999. With additional contributions by E. Bannai, R. E. Borcherds, J. Leech,
S. P. Norton, A. M. Odlyzko, R. A. Parker, L. Queen and B. B. Venkov.
[3] Thomas Ericson and Victor Zinoviev. Codes on Euclidean spheres, volume 63 of North-Holland Mathematical Library. North-Holland Publishing Co., Amsterdam, 2001.
[4] Robin Hartshorne. Algebraic geometry. Springer-Verlag, 1977. Graduate Texts in Mathematics, No. 52.
[5] G. A. Kabatiansky and V. I. Levenshtein. Bounds for packings on sphere and in space. Problemy Peredachi
Informatsii, 14(1):325, 1978.
[6] S. Müller-Stach, E. Viehweg, and K. Zuo. Relative proportionality for subvarieties of moduli spaces of
K3 and abelian surfaces. Pure Appl. Math. Q., 5(3, Part 2):1161–1199, 2009.
[7] John G. Ratcliffe. Foundations of hyperbolic manifolds, volume 149 of Graduate Texts in Mathematics.
Springer-Verlag, 1994.
[8] William P. Thurston. Three-dimensional geometry and topology. Vol. 1, volume 35 of Princeton Mathematical Series. Princeton University Press, Princeton, NJ, 1997. Edited by Silvio Levy.
Ted Chinburg, Department of Mathematics, University of Pennsylvania, Philadelphia, PA
19104, U.S.A.
E-mail address: ted@math.upenn.edu
Matthew Stover, Department of Mathematics, Temple University, Philadelphia, PA 19122,
U.S.A.
E-mail address: mstover@temple.edu
| 4 |
arXiv:1709.02435v1 [cs.AI] 7 Sep 2017
An Analysis of ISO 26262: Using Machine
Learning Safely in Automotive Software
Rick Salay
Rodrigo Queiroz
Krzysztof Czarnecki
University of Waterloo
ON, Canada
Email: rsalay@gsd.uwaterloo.ca
University of Waterloo
ON, Canada
Email: rqueiroz@gsd.uwaterloo.ca
University of Waterloo
ON, Canada
Email: kczarnec@gsd.uwaterloo.ca
Abstract—Machine learning (ML) plays an ever-increasing role
in advanced automotive functionality for driver assistance and
autonomous operation; however, its adequacy from the perspective of safety certification remains controversial. In this paper,
we analyze the impacts that the use of ML as an implementation
approach has on ISO 26262 safety lifecycle and ask what could be
done to address them. We then provide a set of recommendations
on how to adapt the standard to accommodate ML.
I. I NTRODUCTION
The use of machine learning (ML) is on the rise in many
sectors of software development, and automotive software
development is no different. In particular, Advanced Driver
Assistance Systems (ADAS) and Autonomous Vehicles (AV)
are two areas where ML plays a significant role [1], [2]. In
automotive development, safety is a critical objective, and the
emergence of standards such as ISO 26262 [3] has helped
focus industry practices to address safety in a systematic and
consistent way. Unfortunately, ISO 26262 was not designed to
accommodate technologies such as ML, and this has created a
tension between the need to innovate and the need to improve
safety.
In response to this issue, research has been active in several
areas. Recently, the safety of ML approaches in general
have been analyzed both from theoretical [4] and pragmatic
perspectives [5]. However, most research is specifically about
neural networks (NN). Work on supporting the verification
& validation (V&V) of NNs emerged in the 1990’s with a
focus on making their internal structure easier to assess by
extracting representations that are more understandable [6].
General V&V methodologies for NNs have also been proposed [7], [8]. More recently, with the popularity of deep
neural networks (DNN), verification research has included
more diverse topics such as generating explanations of DNN
predictions [9], improving the stability of classification [10]
and property checking of DNNs [11].
Despite their challenges, NNs are already used in high assurance systems (see [12] for a survey), and safety certification
of NNs has received some attention. Pullum et al. [13] give
detailed guidance on V&V as well as other aspects of safety
assessment such as hazard analysis with a focus on adaptive
systems in the the aerospace domain. Bedford et al. [14]
define general requirements for addressing NNs in any safety
standard. Kurd et al. [15] have established criteria for NNs to
use in a safety case.
The recent surge of interest in AV has also been driving
research in certification. Koopman and Wagner [2] identify
some of the key challenges to certification, including ML.
Martin et al. [16] analyze the adequacy of ISO 26262 for AV
but focus on the impact of the increased complexity it creates
rather than specifically the use of ML. Finally, Spanfelner
et al. [1] assess ISO 26262 from the perspective of driver
assistance systems.
The contribution of the current paper is complementary to
the above research. We analyze the impact that the use of MLbased software has on various parts of ISO 26262. Specifically,
we consider its impact in the areas of hazard analysis and in the
phases of the software development process. In all, we identify
five distinct problems that the use of ML creates and make
recommendations on steps toward addressing these problems
both through changes to the standard and through additional
research.
The remainder of the paper is structured as follows. In
Sec. II, we give the required background on ISO 26262 and
ML. Sec. III contains the analysis of the ISO 26262 safety
lifecycle with five subsections describing each impacted area
and the corresponding recommendations. Finally, in Sec. IV,
we summarize and give concluding remarks.
II. BACKGROUND
A. ISO 26262
ISO 26262 is a standard that regulates functional safety of
road vehicles. It recommends the use of a Hazard Analysis
and Risk Assessment (HARA) method to identify hazardous
events in the system and to specify safety goals that mitigate
the hazards. The standard has 10 parts, but we focus on Part
6: “product development at the software level”. The standard
follows the well-known V model for engineering shown in
Fig. 1.
An Automotive Safety Integrity Level (ASIL) refers to a risk
classification scheme defined in ISO 26262 for an item (e.g.,
subsystem) in an automotive system. The ASIL represents
the degree of rigor required (e.g., testing techniques, types of
documentation required, etc.) to reduce the risk of the item,
where ASIL D represents the highest and ASIL A the lowest
risk. If an element is assigned QM (Quality Management), it
does not require safety management. The ASIL assessed for a
given hazard is first assigned to the safety goal set to address
the hazard and is then inherited by the safety requirements
derived from that goal.
Specification of software
safety requirements
Software
testing
Verification of software
safety requirements
Design phase
verification
Software architectural
design
Software
testing
Software integration and
testing
Design phase
verification
Software unit design and
implementation
Software
testing
Software unit testing
Fig. 1. ISO 26262 part 6 - Product development at the software level.
Part 6 of the standard specifies the compliance requirements
for software development. For example, Fig. 2 shows the
error handling mechanisms recommended for use as part of
the architectural design. The degree of recommendation for a
method depends on the ASIL and is categorized as follows:
++ indicates that the method is highly recommended for the
ASIL; + indicates that the method is recommended for the
ASIL; and o indicates that the method has no recommendation
for or against its usage for the ASIL. For example, Graceful
Degradation (1b) is the only highly recommended mechanism
for an ASIL C item, while an ASIL D item would also require
Independent Parallel Redundancy (1c).
Methods
ASIL
A
B
C
D
1a Static recovery mechanism
+
+
+
+
1b Graceful degradation
+
+
++
++
1c
o
o
+
++
+
+
+
+
Independent parallel redundancy
1d Correcting codes for data
Fig. 2. ISO 26262 Part 6 - Mechanisms for error handling at the software
architectural level.
B. Machine learning
In this paper, we are concerned with software implementation using ML. We call a programmed component to be
one that is implemented using a programming language,
regardless of whether the programming was done manually or
automatically (e.g., via code generation). In contrast, an ML
component is one that is a trained model using a supervised,
unsupervised or reinforcement learning (RL) approach.
There are several characteristics of ML that can impact
safety or safety assessment.
Non-transparency. All types of ML models contain knowledge in an encoded form, but this encoding is harder to
interpret for humans in some types than others. For example, a
Bayesian Network for weather prediction is easier to interpret
since the nodes are random variables representing humandefined concepts such as “precipitation type”, “temperature”,
etc. In contrast, NN models are considered non-transparent,
and significant research effort has been devoted to making
them more transparent (e.g., [6], [9]). Increasing ML model
expressive power is typically at the expense of transparency
but some research efforts focus on mitigating this [17]. Nontransparency is an obstacle to safety assurance because it is
more difficult for an assessor to develop confidence that the
model is operating as intended.
Error rate. An ML model typically does not operate
perfectly and exhibits some error rate. Thus, “correctness” of
an ML component, even with respect to test data, is seldom
achieved and it must be assumed that it will periodically
fail. Furthermore, although an estimate of the true error rate
is an output of the ML development process, there is only
a statistical guarantee about the reliability of this estimate.
Finally, even if the estimate of the true error rate was accurate,
it may not reflect the error rate the system actually experiences
while in operation after a finite set of inputs because the
true error is based on an infinite set of samples [4]. These
characteristics must be considered when designing safe system
using ML components.
Training-based. Supervised and unsupervised learning
based ML models are trained using a subset of possible inputs
that could be encountered operationally. Thus, the training set
is necessarily incomplete and there is no guarantee that it is
even representative of the space of possible inputs. In addition,
learning may overfit a model by capturing details incidental
to the training set rather than general to all inputs. RL suffers
from similar limitations since it typically explores only a
subset of possible behaviours during training. The uncertainty
that this creates about how an ML component will behave is
a threat to safety. Another factor is that, even if the training
set is representative, it may under-represent the safety-critical
cases because these are often rarer in the input space [4].
Instability. More powerful ML models (e.g., DNN) are
typically trained using local optimization algorithms, and
there can be multiple optima. Thus, even when the training
set remains the same, the training process may produce a
different result. However, changing the training set also may
change the optima. In general, different optima may be far
apart structurally, even if they are similar behaviourally. This
characteristic makes it difficult to debug models or reuse parts
of previous safety assessments.
III. A NALYSIS OF ISO 26262
In this section, we detail our analysis of ML impacts on
ISO 26262. Since an ML component is a specialized type
of software component, we define an area of the standard
as impacted when it is relevant to software components and
the treatment of an ML component should differ from the
existing treatment of software components by the standard.
Applying this criterion to the ten parts of the standard resulted
in identifying five areas of impact in two parts: the hazard
analysis from the concept phase (Part 3) and the software
development phase (Part 6). We describe the five areas of
impact with corresponding recommendations in the following
subsections.
A. Identifying hazards
ISO 26262 defines a hazard as “a potential source of harm
caused by malfunctioning behaviour of the item where harm
is physical injury or damage to the health of persons” [3, Part
1]. The use of ML can create new types of hazards. One type
of such hazard is caused by the human operator becoming
complacent because they think the automated driver assistance
(often using ML) is smarter than it actually is [18]. For
example, the driver stops monitoring steering in an automated
steering function. On one level, this can be viewed as a
case of “reasonably forseeable misuse” by the operator, and
such misuse is identified in ISO 26262 as requiring mitigation [3, Part 3]. However, this approach may be too simplistic.
As ML creates opportunities for increasingly sophisticated
driver assistance, the role of the human operator becomes
increasingly critical to correct for malfunctions. But increasing
automation can create behavioural changes in the operator,
reducing their skill level and limiting their ability to respond
when needed [19]. Such behavioural impacts can negatively
impact safety even though there is no system malfunction or
misuse.
Other new types of hazards are due to the unique ways an
ML component can fail. For RL, faults in the reward function
can cause surprising failures. An RL-based component may
negatively affect the environment in order to achieve its
goal [5]. For example, an AV may break laws in order to
reach a destination faster. Another possibility is that the RL
component games the reward function [5]. For example, the
AV figures out that it can avoid getting penalized for driving
too close to other cars by exploiting certain sensor vulnerabilities so that it can’t “see” how close it is getting. Although
hazards such as these may be unique to ML components, they
can be traced to faults, and thus they fit within the existing
guidelines of ISO 26262.
Recommendations for ISO 26262: The definition of hazard
should be broadened to include harm potentially caused by
complex behavioural interactions between humans and the
vehicle that are not due to a system malfunction. The standard
itself takes note that the current definition is “restricted to the
scope of ISO 26262; a more general definition is potential
source of harm”[3, Part 1]. The definition and methods for
identifying such hazards should be informed by the research
specifically on behavioural impacts of ADAS [20] as well as
human-robot interaction (HRI)[21] more broadly. For example,
van den Brule et al. [22] study how a robot’s behavioural style
can affect the trust of humans interacting with it.
B. Faults and failure modes
ISO 26262 mandates the use of analyses such as Fault Mode
Effects Analysis (FMEA) to identify how faults lead to failures
that may cause harm (i.e., are hazards). We can ask whether
there are types of faults and failures that are unique to ML
and not found in programmed software. Specific fault types
and failure modes have been catalogued for NNs (e.g., [13],
[15]). Some of these are just “apparent” ML specific faults.
For example, a neuron that randomly changes its connection
in an operational NN is not really about neurons but rather a
conventional fault that can occur in the software on which the
NN runs. Others are distinctly ML-specific such as faults in
the network topology, learning algorithm or training set. This
creates the opportunity to develop focused tools and techniques
to help find faults independently of the domain for which the
ML model is being trained.
Although ML faults have some unique characteristics, this
cannot be said about failure modes. All faults can do is to
increase the error rate of the deployed component, and thus
cause one particular type of failure – an incorrect output for
some input. But since most software failures take the form
of incorrect output for a given input, we may conclude that
there is nothing different about the failure analysis of an ML
component as compared to a programmed component, and
existing ISO 26262 recommendations apply.
Recommendations for ISO 26262: Require the use of fault
detection tools and techniques that take into account the unique
features of ML. For example, Chakarov et al. [23] describe a
technique for debugging mis-classifications due to bad training
in data, while Nushi et al. [24] propose an approach for
troubleshooting faults due to complex interactions between
linked ML components.
C. The use of training sets
Spanfelner et al. [1] point out that there is an assumption in
ISO 26262, given by the left side of the V model (Fig. 1), that
component behaviour is fully specified and each refinement
can be verified with respect to its specification. Note that this
assumption is also made in other safety-critical domains such
as aerospace [25]. This is important to ensure that a safety
argument can trace the behaviour of the implementation to its
design, safety requirements and ultimately, to the hazards that
are mitigated.
This assumption is violated when a training set is used
in place of a specification since such a set is necessarily
incomplete, and it is not clear how to create assurance that
the corresponding hazards are always mitigated. Thus, an ML
component violates the assumption. Furthermore, the training
process is not a verification process since the trained model
will be “correct by construction” with respect to the training
set, up to the limits of the model and the learning algorithm.
A more careful analysis of the development lifecycle for
an ML component shows that there are multiple levels of
specification and implementation, some of which may satisfy
the assumption in the standard. High-level requirements for
the component, although abstract, can be expressed with
completeness and traced to up to hazards. For example, the
component may be required to “identify pedestrians” that the
AV should avoid harming. Detailed data requirements can
be specified carefully to ensure that an appropriate training,
validation and testing sets are obtained. Subsequently, the data
gathered can be verified with respect to this specification.
Completeness is still an issue but coverage can be used as
a surrogate, as it is with the design of test sets for software
testing.
A deeper issue, discussed by Spanfelner et al. [1], is that
many kinds of advanced functionality require perception of the
environment, and this functionality may be inherently unspecifiable. For example, what is the specification for recognizing
a pedestrian? We might observe that since a vehicle must
move around in a human world, advanced functionality must
involve perception of human categories (e.g., pedestrians).
There is evidence that such categories can only partially be
specified using rules (e.g., necessary and sufficient conditions)
and also need examples [26]. This suggests that ML-based
approaches may actually be required for implementing this
type of functionality.
Recommendations for ISO 26262: The approach to safe
implementation should be geared to the type of functionality
being implemented. If the functionality is fully specifiable,
then conventional programming can be required. In other
cases, such as advanced functionality requiring perception,
ML-based approaches should be used, and the complete specification requirement must be relaxed. Partial specifications
can be required, where possible. For example, if a pedestrian
must be less than 9 feet tall, then this property can be used to
filter out false positives. Such properties can be incorporated
into the training process or checked on models after training
(e.g., [11]).
Training set specifications and coverage metrics must be
required to improve training set quality. Ensemble methods
such as boosting and decision fusion can also be recommended
to improve the error rate. However, when the ASIL level is
high, it is unlikely that the error rate can ever be brought to
an acceptably low level only through increasing or improving the training set (due to the “curse of dimensionality”).
Therefore, fault tolerance strategies for software must be
required. For example, redundant pedestrian recognizers using
different ML models and training sets can be used to detect
potential recognition failures when there is disagreement.
Another possibility is to define a “safety envelope” of possible
known safe behaviours and limit the ML component to choose
among them [27]. Some of these recommendations may be
addressed in a forth-coming OMG standard relating to sensor
and perception issues [28].
D. Level of ML usage
Fig. 1 identifies an architectural level and a unit (i.e.,
component) level of implementation. ISO 26262 defines a
software architecture as consisting of components and their
interactions in a hierarchical structure [3, Part 6]. This component decomposition is important for safety because it allows
for easier comprehension of a complex system by human
assessors and it permits the use of compositional formal
analysis techniques.
ML could be used to implement an entire software system,
including its architecture, using an end-to-end approach. For
example, Bojarski et al. [29] train a DNN to make the
appropriate steering commands directly from raw sensor data,
side-stepping typical AV architectural components such as lane
detection, path planning, etc.
Here, we may assume that the unit level, in the conventional
sense of a distinct component that can be developed independently of the architecture, no longer exists. This is the case,
even if it is possible to extract and interpret the structure of
the trained model as consisting of units with distinct functions,
since this structure is emergent in the training process and
unstable. If the model is re-trained with a slightly different
training set, this structure can change arbitrarily. Note that
a DNN does have an architecture in a different sense – the
set of layers and their connections. However, since it is the
training that actually “implements” the required functionality,
this architecture is more of an generic execution layer. Thus,
an end-to-end approach deeply challenges the assumptions
underlying ISO 26262.
Another challenge with an end-to-end approach is that,
in some cases, the the size of the training set needs to be
exponentially larger than when a programmed architecture is
used [30]. This puts additional strain on the already challenging problem of obtaining an adequate training set for safetycritical contexts.
Finally, note that issues with an end-to-end approach can
also apply when ML is used at the component level, if
components are too complex. For example, at one extreme,
the architecture can consist of a single component. ISO
26262 specifically guards against this pitfall by mandating the
use of modularity principles such as restricting the size of
components and maximizing the cohesion within a component. However, the lack of transparency of ML components
can hamper the ability to assess component complexity and
therefore, to apply these principles. Fortunately, improving ML
transparency is an active research area (e.g., [9], [6]).
Recommendations for ISO 26262: Although using an
end-to-end approach has shown some recent successes with
autonomous driving (e.g., [29]), we recommend that an endto-end use of ML not be encouraged by ISO 26262, due to its
incompatibility with the assumptions about stable hierarchical
architectures of components.
E. Required software techniques
Part 6 of ISO 26262 deals with product development at
the software level and specifies 75 software development
techniques, such as shown in Fig. 2, that are used in various
phases of the development process in the V model (Fig. 1).
Of these, 34 apply at the unit level, and the remaining at
the architectural level. We performed an assessment of the
software techniques to determine their applicability to ML
components1 . Based on our recommendation in Sec III-D,
we assumed that ML was only used at the unit level and
programming is used at the architecture level to connect
components.
The charts in Fig. 3 show the results of the assessment
for the techniques dealing with the unit level. We classified
each technique into one of three categories based on the level
of applicability to ML. Category Ok means the technique
1 The
data is available at https://github.com/rsalay/safetyml
Percentage of techniques
(a) Averaged over ASILs
50%
40%
(8%) (2%)
(13%)
(1%)
30%
(6%)
20%
(2%)
10%
0%
Ok
Adapt
N/A
Degree of applicability
Highly recommended (++)
All recommended (+,++)
All (o,+,++)
(b) Highly recommended only
Percentage of techniques
is directly applicable without modification. Most of these
cases are due to the fact that they are black box techniques
(e.g., analysis of boundary values, error guessing, etc.) and
thus, the method of component implementation is irrelevant.
However, some white box techniques such as fault injection
also apply. For example, faults can be injected into an NN
by breaking links or randomly changing weights (e.g., [31]).
Category Adapt says that the technique can be used for an
ML component if it is adapted in some way. For example, the
technique walk-through can’t be used directly with an NN due
to the non-transparency characteristic. Finally, category N/A
indicates that the technique is fundamentally code-oriented and
does not apply to an ML component. For example, no multiple
use of variable names is meaningful for a program but has no
corresponding notion in an ML model.
The results in Chart (a) are grouped by the degree to
which the techniques are recommended. Recall from Sec. II
that each technique is marked as highly recommended (++),
recommended (+) or no recommendation (o) depending on the
ASIL level. The bars in each category show the percentage
of techniques that apply when considering all techniques
(0,+,++), only the recommended techniques (+, ++), and only
the highly recommended techniques (++). Since the degree
of recommendation varies by ASIL, each percentage is an
average value over all four ASILs with the standard deviation
in parentheses. Note that the standard deviation is 0 for
the “all” group since every technique is present for each
ASIL. Because of the high standard deviation for the highly
recommended group, we have included Chart (b) which gives
the actual data for each ASIL in this group.
Chart (a) shows that a significant part of the standard is
still directly applicable (category Ok) and there is an emphasis
on highly recommended techniques. However, the standard
deviation is high and Chart (b) shows that most of these highly
recommended techniques apply to the lower ASIL values – i.e.
they are less relevant from a safety critical perspective. Chart
(a) also shows that about 40% of the techniques do not apply at
all (category N/A) regardless of the degree of recommendation.
In general, techniques in the software part of the standard are
clearly biased toward imperative programming languages (e.g.,
C, Java, etc.) [25]. In addition to precluding ML components,
this bias makes it difficult to accept implementations in other
mature programming paradigms such as functional programming, logic programming, etc.
Recommendations for ISO 26262: One approach to addressing the gap in applicable techniques as well as the
imperative language bias without compromising safety may
be to specify the requirements for techniques based on their
intent and maturity rather than on their specific details. For
example, the intent of the no multiple use of variable names
technique is to reduce the possibility for confusion that may
prevent the detection of bugs. This helps humans understand
the implementation better and increase their confidence in
its correctness and safety. Thus, the standard can require the
use of “accepted clarity increasing” techniques instead of the
specific techniques.
70%
60%
50%
40%
30%
20%
10%
0%
ASIL A
ASIL B
Ok
ASIL C
Adapt
ASIL D
N/A
Fig. 3. Percentage of unit-level software techniques applicable to ML
components: (a) values averaged over the four ASIL levels with standard
deviation shown in parentheses; (b) values for each ASIL when only highly
recommended techniques are considered.
IV. S UMMARY AND C ONCLUSION
Machine learning is increasingly seen as an effective software implementation technique for delivering advanced functionality; however, how to assure safety when ML is used in
safety critical systems is still an open question. The ISO 26262
standard for functional safety of road vehicles provides a
comprehensive set of requirements for assuring safety but does
not address the unique characteristics of ML-based software.
In this paper, we make a step towards addressing this gap
by analyzing the places where ML can impact the standard
and providing recommendations on how to accommodate this
impact. Our results and recommendations are summarized as
follows.
Identifying hazards. The use of ML can create new
types of hazards that are not due to the malfunctioning of a
component. In particular, the complex behavioural interactions
possible between humans and advanced functionality implemented by ML can create hazardous situations that should be
mitigated within the system design. We recommend that ISO
26262 expands their definition of hazard to address these kinds
of situations.
Fault and failure modes. ML components have a development lifecycle that is different from other types of software.
Analyzing the stages in the lifecycle reveals distinct types
of faults they may have. We recommend that ISO 26262 be
extended to explicitly address the ML lifecycle and require the
use of fault detection tools and techniques that are customized
to this lifecycle.
The use of training sets. Because ML components are
trained from inherently incomplete data sets, they violate
the assumption in V model-based processes that component
functionality must be fully specified and that refinements are
verifiable. Furthermore, it is possible that certain types of ad-
vanced functionality (e.g., requiring perception) for which ML
is well suited are unspecifiable in principle. As a result, ML
components are designed with the knowledge that they have
an error rate and that they will periodically fail. Rather than
disqualifying this class of functionality, we recommend that
ISO 26262 provide different safety requirements depending
on whether the functionality is specifiable.
The level of ML usage. ML could be used broadly at
the architectural level with a system by using an end-to-end
approach or remain limited to use at the component level. The
end-to-end approach challenges the assumption that a complex
system is modeled as a stable hierarchical decomposition of
components each with their own function. This limits the
use of most techniques for system safety and we therefore
recommend that ISO 26262 only allow the use of ML at the
component level.
Required software techniques. ISO 26262 mandates the
use of many specific techniques for various stages of the
software development lifecycle. Our analysis shows that while
some of these remain applicable to ML components and others
could readily be adapted, many remain that are specifically
biased toward the assumption that code is implemented using
an imperative programming language. In order to remove this
bias, we recommend that the requirements be expressed in
terms of the intent and maturity of the techniques rather than
their specific details.
ACKNOWLEDGMENT
The authors would like to thank Atri Sarkar, Michael Smart,
Michal Antkiewicz, Marsha Chechik, Sahar Kokaly and Ramy
Shahin for their insightful comments.
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| 2 |
MultiParametric Statistical Method for Estimation of Accumulated
Fatigue by Sensors in Ordinary Gadgets
Nikita Gordienko,
PhysicalMathematical Lyceum 142, Kyiv, Ukraine
assasin.nik@gmail.com
Abstract
The new method is proposed to monitor the level of currently accumulated
fatigue and estimate it by the several statistical methods. The experimental
software application was developed and used to get data from sensors
(accelerometer, GPS, gyroscope, magnetometer, and camera), conducted
experiments, collected data, calculated parameters of their distributions (mean,
standard deviation, skewness, kurtosis), and analyzed them by statistical
methods (moment analysis, cluster analysis, bootstrapping, periodogram and
spectrogram analyses). The hypothesis 1 (physical activity can be estimated and
classified by moment and cluster analysis) and hypothesis 2 (fatigue can be
estimated by moment analysis, bootstrapping analysis, periodogram, and
spectrogram) were proposed and proved. Several “fatigue metrics” were
proposed: location, size, shape of clouds of points on bootstrapping plot. The
most promising fatigue metrics is the distance from the “rest” state point to the
“fatigue” state point (sum of 3 squared nonnormal distribution of
noncorrelated acceleration values) on the skewnesskurtosis plot. These
hypotheses were verified on several persons of various age, gender, fitness level
and improved standard statistical methods in similar researches. The method can
be used in practice for ordinary people in everyday situations (to estimate their
fatigue, give tips about it and advice on contextrelated information).
1.Background
The standard cardiology monitoring can show the instant state of cardiovascular
system, but unfortunately, cannot estimate the accumulated fatigue and physical
exhaustion. From 2009 FIFA persists that professional players should record
family history, heart rhythm, sounds, and ECG (electrocardiogram) results [1].
Despite this more than 60 professional football players died while playing a
game or training of a suspected heart attack or cardiac arrest during the last
decade only! On May 6, 2016, Patrick Ekeng (football team Dinamo București),
26, collapsed in a match 7 minutes after came on from the bench and died less
than two hours later of a suspected heart attack. Before the game he told his best
friend he was not able to play, he said he was very tired [2]. The common reason
for such sad statistics is the increased physical load and overtraining that lead to
dangerous fatigue and fatal outcome. In addition to sport, fatigue could be
crucial in many other areas (car driving, air and naval traffic control,
manufacture and service industry), where errors due to fatigue can lead to
decrease of working efficiency, manufacturing quality, and, especially,
workplace and customer safety. That is why people need some easily accessible
ways to estimate their fatigue and physical exhaustion. There are some
specialized commercial accelerometers, which are used to record the number of
steps, etc [34]. However, they are quite primitive in terms of data, and are not
able to assess the health state and measure fatigue [58].
2.Experimental Procedures and Methods of Statistical Analysis
The main aim of this research is to test theoretical possibility of monitoring the
health of individuals based on physical data gathered by sensors in usual
smartphones (and analysis of the data) for the assessment of human fatigue and
potential prediction of signs of some diseases, especially for the elderly.
To reach this aim the following tasks were performed: explore the possibility of
using sensors in conventional gadgets (smartphone, smart watches, smart
glasses) to monitor fatigue with their analysis of the physical activity; develop
and apply methods for measuring basic physical quantities that characterize
human mobility (acceleration, speed, etc.); create and use multiparametric
methods of statistical analysis, which use several parameters of distributions
(mean, standard deviation, skewness and kurtosis) of the physical data obtained
from sensors of conventional gadgets; offer practical ways for implementation
of these methods.
Experimental methodology is based on frequent measurements (every 1100
milliseconds) acceleration of the movements of certain parts of the body by the
axes X, Y, Z via Gsensor (acceleration sensor or accelerometer) in a
conventional mobile phone, or in a more convenient, but rare smart watches or
smart glasses. The amplitude acceleration values vary very widely for different
types of activity and operations. And it is very hard to find some patterns even
for very different activities. However, if you plot the distribution of physical
quantities, their difference is quite noticeable. For numerical characteristics of
this difference the standard parameters of distributions were selected: mean
value, standard deviation, skewness, and kurtosis. The statistical analysis of
experimental data was performed for distributions of acceleration values over a
short period of time (for 110 minutes), then the standard parameters of the
distributions were calculated and correlations with types of physical activity
were determined.
3.Results
3.1.Analysis of physical activity based on 2 and 3 parameters of distribution
Each point on the graphs below (Fig.12) corresponds to parameters of some
distribution of large number of measurements. Each point contains a label of the
body part, where sensors were located. For example, WALK_LEG_L indicates
measurement during walking (WALK), a sensor was located on the left leg
(LEG_L). It was noted that the parameters of the distribution are monotonically
dependent on the speed (Fig.12). This means that the motor activity (of species
listed in small text at the characters) can be classified according to the intensity
that is divided into groups (colored ovals) with close values of parameters
distributions: active behavior (sports, housework, walking highlighted in blue)
moderate behavior (letter, seat marked in green) and passive behavior (web
surfing, reading, sleeping highlighted in red) (Fig. 1315). Thus, it is possible
to characterize and classify the level of human mobility by means of the cluster
analysis.
Fig.1.Mean (MEAN) and standard
Fig.2.Skewness, kurtosis, and standard
deviation (STD) for various
deviation (STD) for various distributions,
distributions, corresponding to different
corresponding to different physical
physical activities and location of
activities and location of sensors.
sensors.
That is why the statistical analysis by 3 parameters of distributions can be more
reliable and accurate for determination of the level of physical activity. These
results are in good correspondence with the previous similar experiments on
measurements of physical activities by accelerometers [9].
Fig.3.Smart glasses EPSON Moverio BT200 (on the head) as a collector of
data; Conventional glasses with Smart Particle Photon controller with an
accelerometer (in the left red circle); Texas Instruments watches with
accelerometer (the medium blue circle); smart watches Samsung Gear 2 with
accelerometer (the right green circle).
3.2.Analysis of fatigue based on 2 and 3 parameters of distribution
To estimate fatigue, the interrelation of the muscle power to the mass of the
body part is very important. The lower muscle power and higher the mass of a
body part, the more pronounced effect of fatigue in the shape of tremor. For
example, tremor can be measured by subtle shakes of head or fingers of
outstretched hands. So the smart glasses and ordinary glasses with Particle
Photon controller with accelerometer were used to test the manifestations of
fatigue from shaking head (Fig.3).
The same multiparametric statistical analysis was applied for assessment of
human fatigue after physical exercises (squats for a minute) by smart glasses and
ordinary glasses (Fig.3). The measurements were made every 1 millisecond
(Fig.4) while sitting (letters and ellipse around them with index 1) and standing
for physical exercise (with index 2), and while standing (index 4) and seat
(index 5) after exercise. The parameters after exercise differ from values before
the exercise, especially at the initial moment when people have not had time to
recover.
Рис.4.Groups of parameters of
Fig.5.Parameters of partial samplings for
distributions enclosed by ellipses for
tremor of outstretched hands with a
individual stages of the experiment
smartphone in the hands in the rest states
(the explanations of symbols are
(compact clouds) and fatigue states
given in the text).
(elongated clouds)
3.3.Analysis of partial sampling (bootstrapping)
To analyze the stability of the obtained distribution parameters the method of
partial sampling (or bootstrapping method) from the original distribution of
acceleration values. In Fig.5 each point represents one of > 1000 partial samples
from the original distribution (hand tremor of the housewife, 45 years) with the
correspondent parameters: the standard deviation (axis is directed from the
depths of the page), the skewness (vertical axis) and kurtosis (horizontal axis).
Different colors mean different time of measurements (which is indicated in the
legend). In Fig.5 the qualitative difference is clearly seen: compact equiaxial
(nearly round) "clouds" of dots correspond to vigil state (morning and evening)
and elongated "cloud" tired state (after 10 km ski walk). The most promising
fatigue metrics is proposed as the distance from the “rest” state point, which is
actually position of Chisquare distribution (sum of 3 squared normal
distribution of noncorrelated acceleration values), to the “fatigue” state point
(sum of 3 squared nonnormal distribution of noncorrelated acceleration values)
on the skewnesskurtosis plot.
Fig.6.Spectrogram of tremor of outstretched hands with a smartphone.
3.4.Spectral analysis
The spectral analysis (i.e. amplitudes for the range of frequencies for a given
measurement interval 1215 seconds) is shown in Fig.6 for tremor of
outstretched hands with a smartphone during the day. The amplitude is denoted
by colors from blue (lowest) to red (highest). Slow increase of fatigue manifests
itself as an increase of the amplitude (red area at the bottom of the spectrogram)
of uncontrolled lowfrequency tremor: after the morning run 10 km (time 8:54),
fatigue accumulation up to the evening (17:27) and the greater fatigue at late
night (20:02). The slow relaxation (i.e. no explicit maximum of uncontrolled
lowfrequency tremor) is observed during the day except for the evening time.
For example, the largest fatigue levels are observed after the morning run of 10
km, before the delayed lunch and in the evening!
4.Conclusions
The experimental results obtained during measurements of body part
accelerations by ordinary and new gadgets allow to propose the new method to
monitor the level of currently accumulated fatigue and estimate it by the several
statistical methods. The experimental software application was developed and
used to get data from sensors (accelerometer, GPS, gyroscope, magnetometer,
and camera), conducted experiments, collected data, calculated parameters of
their distributions (mean, standard deviation, skewness, kurtosis), and analyzed
them by statistical methods (moment analysis, cluster analysis, bootstrapping,
periodogram and spectrogram analyses). The hypothesis 1 (physical activity can
be estimated and classified by moment and cluster analysis) and hypothesis 2
(fatigue can be estimated by moment analysis, bootstrapping analysis,
periodogram, and spectrogram) were proposed and proved. Several “fatigue
metrics” were proposed: location, size, shape of clouds of points on
bootstrapping plot. The most promising fatigue metrics is the distance from the
“rest” state point, which is actually position of Chisquare distribution (sum of 3
squared normal distribution of noncorrelated acceleration values), to the
“fatigue” state point (sum of 3 squared nonnormal distribution of
noncorrelated acceleration values) on the skewnesskurtosis plot. These
hypotheses were verified on several persons of various age (1649), gender
(M/F), fitness level (child, housewife, and … marathoner even) and improved
standard statistical methods in similar researches. The method can be used in
practice for ordinary people in everyday situations (to estimate their fatigue,
give tips about it and advice on contextrelated information).
References
[1] FIFA PreCompetition Medical Assessment (PCMA), 2009 (www.fifa.com)
[2] "Dinamo Bucharest midfielder Patrick Ekeng dies after collapsing on pitch".
The Guardian. 6 May 2016
[3] Yao Meng and HeeCheol Kim, A Review of AccelerometerBased Physical
Activity Measurement, в книге K. J. Kim and S. J. Ahn (eds.),
Proceedings of
the International Conference on IT Convergence and Security 2011
, Lecture
Notes in Electrical Engineering 120, Springer (2012).
[4] Lemoyne R et al., Implementation of an iPhone as a wireless accelerometer
for quantifying gait characteristics,
Proceedings 32nd annual International
Conference of IEEE EMBS
, 1: 3847–3851 (2010).
[5] Деменция. Информационный бюллетеньN°362 (2012).
[6] N.Zouba, F.Bremond, and M.Thonnat. A Computer System to Monitor Older
Adults at Home: Preliminary Results.
Gerontechnology
, 8(3):129–139 (2009).
[7] S.Chernbumroong, S.Cang, A.Atkins, H.Yu, Elderly Activities Recognition
and Classification for Applications in Assisted Living,
Expert Systems with
Applications
, doi:10.1016/j.eswa.2012.09.004 (2012).
[8] J.Liu et al, Local dynamic stability assessment of motion impaired elderly
using electronic textile pants.
IEEE Trans Autom. Sci. Eng.
5(4):696–702
(2008).
[9] Gordienko, N., Lodygensky, O., Fedak, G., & Gordienko, Y. (2015).
Synergy of volunteer measurements and volunteer computing for effective data
collecting, processing, simulating and analyzing on a worldwide scale. In
Proc.
38th International Convention on Information and Communication Technology,
Electronics and Microelectronics
(MIPRO), IEEE (pp. 193198), DOI:
10.1109/MIPRO.2015.7160263
.
| 5 |
PRICING VIRTUAL PATHS WITH QUALITY-OF-SERVICE
GUARANTEES AS BUNDLE DERIVATIVES
arXiv:cs/0106028v1 [cs.NI] 12 Jun 2001
LARS RASMUSSON
Abstra t. We des ribe a model of a ommuni ation network that allows us
to pri e omplex network servi es as nan ial derivative ontra ts based on
the spot pri e of the apa ity in individual routers. We prove a theorem of a
Girsanov transform that is useful for pri ing linear derivatives on underlying
assets, whi h an be used to pri e many omplex network servi es, and it is
used to pri e an option that gives a ess to one of several virtual hannels
between two network nodes, during a spe i ed future time interval. We give
the ontinuous time hedging strategy, for whi h the option pri e is independent
of the servi e providers attitude towards risk. The option pri e ontains the
density fun tion of a sum of lognormal variables, whi h has to be evaluated
numeri ally.
1. Introdu tion
1.1.
End-to-end quality of servi e.
Today, most tra
in
omputer networks
is handled by best eort routing; ea h network router passes on pa kets as long as
it
an, and when the buers are full, it drops in oming pa kets. When the network
load is low, all data streams get a high throughput, and when the load is high, all
streams experien e equal loss.
This works well for some data streams su h as le transfer, but less so for realtime data streams, i.e.
when data pa kets have hard deadlines.
audio/video streams [11℄, grid
gestion
omputing[16℄, and intera tive data streams. Con-
auses pa ket losses and retransmissions that result in jitter, suspended
omputation, and high laten y, respe tively.
network
Examples are
These problems arise be ause the
annot provide servi e quality guarantees and dierent servi e levels for
dierent kinds of tra . Ability to provide servi e guarantees requires that an individual user
an reserve some of the
apa ity in the
ongestion prone parts of the
network, be that routers, network links, or whatever.
The exibility of today's
logi
omputer network is due to the design hoi e to keep the
inside the network very simple, and to let all appli ation spe i
knowledge
be handled at the ends, by appli ations on top of the network layer.
In this
spirit, we advo ate that a reservation poli y should be implemented outside of
Date : 2001-06-12.
1
2
LARS RASMUSSON
A
B
C
D
Figure 1. An end-user wishes to reserve
apa ity in a path from
B to C, i.e. buy {B,A,C} or {B,D,C} or neither one, if the pri e is
too high. The total pri e depends on the pri es of the individual
resour es in a
omplex way.
the network, and that the network should only be a delivery vehi le for pa kets.
Current attempts to improve throughput rely on more
omplex internal statisti al
routing and network maintenan e. This intelligent network prin iple is dierent
to the end-to-end prin iple, and intelligent networks have not been as good as
end-to-end networks at supporting new appli ations and uses of the network.
1.2.
End-user bandwidth markets.
In today's parlan e, dis ussions of of band-
width markets often refer to the trading of spare trunk
apa ity among large tele om
ompanies, Internet servi e providers, et . See for instan e the bandwidth markets
at Band-X, RateX hange, Min-X, et . In these markets, the purpose of trading is
to maximize the pro t of the servi e providers, i.e. to let them ful ll their prior
lient obligations at a low
pri es only ae t the tra
ost. Sin e end-users are not ae ted by the
load on a
oarse s ale. Servi e providers
ost, these
an only guess
what the best buy would be, sin e they do not know the network requirements of
the appli ations running on the end-users'
omputers.
We propose a somewhat dierent approa h. To make an e ient market, the
bandwidth allo ation de isions should be a ne grained negotiation about a
the s ar e resour es, that takes pla e between the end-users, the a tual
This way, someone that really needs a parti ular resour e
ess to
onsumers.
an bid for it high enough
that someone else releases (sells) the resour e, and buys another resour e instead.
In most
ases, end-users require
omplex servi es, servi es that require
more than one router, and where these pri es intera t in a
apa ity in
omplex way to form
the total pri e of the servi e.
The resour e pri es are
orrelated in many ways.
In g.
1, the pri es of all
resour es that are on a potential path ae t the de ision on whether do buy any of
the other resour es. In g. 2, the demand of user 2 ae ts the pri e of resour e A,
not
on the user 2's potential paths.
There are some hurdles to over ome to make the resour e negotiation fast and
e ient:
PRICING VIRTUAL PATHS WITH QOS GUARANTEES AS BUNDLE DERIVATIVES
3
A
B
C
D
E
F
Figure 2. User 1 has reserved a path along {B,D,C}, and user 2
wants to reserve {E,D,F} but the pri e of D makes the path too
expensive. Then, if the pri es are appropriate, user 1 should sell
D and buy A, and user 2 buys D.
1. The negotiation between the end-users must be kept very simple. Bilateral
negotiations [11℄ is infeasible in real appli ations.
2. An end-user does not get a de nite pri e-quote for a
a path that involves
omplex servi e su h as
apa ity in several routers.
3. The end-users will generate an extreme amount of network tra
when they
buy and sell resour es, get pri e quotes, et .
These problems are addressed by the method presented here.
1.3.
Related work.
Related work on bandwidth markets to improve performan e
in networks are generally based on admission
Gibbens
et al.
ontrol at the edges, as done by
[6℄. A user is not admitted to the network if the network does not
provide su ient servi e quality. For instan e, Kelly [7℄ models inter onne tions as
reservations along a spe i ed path, and gives Lyapunov fun tions that show that the
system state stabilizes asymptoti ally, as end-users
to network load.
Cour oubetis [8℄ models a router as
derives the probability that a
[14℄ model admission
onsisting of
ertain fra tion of the tra
option that bounds the pri e a user has to pay for
et al.
hange their demand a
C
ording
hannels,
is lost, and pri es a
ontrol to a network over an exponential distributed
number of minutes as a number of n:th-pri e au tions on 1 minute time-slot a
and pri e an a
Lukose,
a
et al.
ess option as the sum of
[9℄ use a CAPM-like model to
ess with dierent servi e
all
apa ity in one router. Semret
ess,
all options on ea h of the time slots.
onstru t a mixed portfolio of network
lasses, in order to redu e the laten y varian e and
mean.
The above models either investigate the asymptoti
network behavior, or only
model the pri e for one network resour e. We are interested in handling the transient behavior of a network with several intera ting nodes.
4
LARS RASMUSSON
In neither of the models above do the underlying resour es
market, i.e. it is not possible to
onstitute a
omplete
reate a momentarily perfe tly balan ed (risk-less)
portfolio of options and underlying resour es. The pri e of the option is therefore
dependent on the risk-aversiveness of the network provider. In a
omplete market,
the pri e is independent of the risk-aversiveness sin e perfe t hedges
and the pri e
an be set more tightly, sin e anyone
an be
an trade and
reated,
ompete for the
bids.
We will present a model of a
omplete
ontinuous time market that allows us to
derive the pri e of an option that extends the
apabilities of the above options in
several ways
•
•
the pri e depends on more than one underlying resour es
•
the pri e is risk-neutral, using the Bla k-S holes' assumptions
the a tual path, or set of resour es assigned, does not have to be spe i ed in
advan e
Furthermore, we
hoose a market type in whi h the resour es are traded
ously, rather than in au tions with dis rete
Stru ture of the paper.
1.4.
learing times sin e that
ontinu-
auses laten y.
In se tion 2 we re apture relevant formulas from
the theory of derivative pri ing, in se . 3 we des ribe the model pri e pro ess for
whi h we pri e the derivatives, and models of the market and network resour es.
In se . 4 we state the main results, whi h are the de nitions and pri e formulas for
the network option (proofs are deferred to the appendix), and we
on lude with a
dis ussion in se . 5.
2. Preliminaries
Derivative pri ing.
2.1.
A standard way to pri e derivative ontra ts, a.k.a deriva-
tives, su h as options, futures, et . is to use arbitrage-free portfolio theory, whi h
says that an asset, known to be worth
e
−rT
ST
today. Here
r
is the
ontinuously
ST
at a future time
T,
is worth
S0 =
ompounded interest risk-free rate, the
loan/interest rate that you get from a bank or a government bond. The reasoning
is that if the asset was worth
X 6= S0 ,
whi h is less (more) than
ould make money by borrowing (lending)
resour e, wait to
X
rT
T,
X
to the rate
sell (buy ba k) the resour e for
) and keep the arbitrage pro t
ST − Xe
rT
ST ,
> 0 (Xe
r,
then anyone
buy (short sell) the
pay ba k
rT
S0 ,
XerT
− ST > 0).
(withdraw
This
annot
be possible in an arbitrage-free market. The argument requires that short selling
is allowed, and that transa tion fees are negligible.
Derivative pri ing often models the asset pri es as It pro esses
dS(t) = a(t, S(t))dt + b(t, S(t))dW (t)
S(t),
PRICING VIRTUAL PATHS WITH QOS GUARANTEES AS BUNDLE DERIVATIVES
where
W (t) is a Wiener pro
ess, and
a and b are su
5
iently bounded fun tions [15℄.
The Bla k-S holes method [1℄ pri es a derivative of an asset whi h pri e follows a
parti ular kind of It -pro ess. It is based on
onstru ting a portfolio that invests
some part of its money in the option, and some part in the asset. At ea h instant,
the portfolio is balan ed in su h a way that its value after
time goes and pri es
a derivative
f (t, s)
dt
is known exa tly. As
hange, the portfolio is rebalan ed. In short, it is shown that
that is a fun tion of a sto hasti
pro ess
dS(t) = a(t, S(t))dt + σS(t)dW (t)
follows the Bla k-S holes equation
∂f
σ 2 s2 ∂ 2 f
∂f
(t, s) − rf (t, s) = 0
(t, s) + rs (t, s) +
∂t
∂s
2 ∂t2
f (T, s) = g(s)
whi h from the Feynman-Ka
formula
an be seen to have the solution
f (t, s) = e−r(T −t) E Q [g(S(T ))|S(t) = s]
where
Q is the so
alled equivalent, or risk-free, martingale measure. The boundary
ondition, given by the fun tion
g(s),
spe i es the value of the option at the time
the derivative expires. For a traditional
K
all option,
is the strike pri e.
Under the
Q
S(T )
measure,
has the drift term
S(T ) = S(t)e
2
(r− σ2
W (T )
a(t, s) = rs,
and hen e, under
W (t)
t = 0.
Q,
is a Wiener pro ess, in other
is normal distributed with mean 0 and varian e
knowledge up until
where
)(T −t)+σ(W (T )−W (t))
Re all from the de nition of It pro esses that
words,
g(s) = max(s − K, 0),
This means that we
T
given
F0 ,
the
an pri e derivatives using Monte-
Carlo simulation to solve the dierential equations above. Another advantage with
Monte-Carlo is that it
onverges quite fast also for multidimensional problems,
something that is not the
ase for binomial-tree pri ing methods.
Rebalan ing a portfolio, or obtaining/selling assets to de rease its varian e is
alled 'hedging' the portfolio. The Bla k-S holes hedging method produ es a self nan ing hedge, i.e.
no additional
apital is needed to balan e the risk of the
portfolio. Another interesting ee t of Bla k-S holes hedging is that the formula
for the optimal
fun tion
a(t, s).
ontinuous-time hedge at
Hen e, the
t
given
S(t)
does not involve the drift
ontinuous-time hedging is the same for the mean-
reverting and the exponential drift pro esses.
2.2.
Applied pri ing.
The assumption that a portfolio
an be
ontinuously rebal-
an ed is violated in real markets. Market fri tions, su h as transa tion fees, often
6
LARS RASMUSSON
make it too
ostly to rebalan e a portfolio very often. For bandwidth markets, we
an eliminate market fri tion all together, sin e we are free to design the market to
our own liking. We
annot however guarantee that we
an rebalan e the portfolio at
every instant, sin e there are others that want to trade
and multiple other trade events may o
on urrently with ourselves,
ur between the rehedging events.
To understand the ee t of interval hedging
ompared to
ontinuous time hedg-
ing, we show the ee t on the portfolio value of hedging and rehedging a
on a single asset for three dierent pri e pro esses, hedged
all option
ontinuously and at
intervals.
Continuous time hedging.
2.2.1.
used to model sto k sto k pri e
S(t).
Its dynami s is
dS(t) = µS(t)dt + σS(t)dW
(1)
A derivative
µ = r under Q,
γ(t) derivatives and
ontra t, on an underlying asset obeying (1) and
an be hedged in
βi (t)
The lognormal Brownian motion pro ess is often
ontinuous time by
reating a portfolio of
assets. A perfe tly hedged, self nan ing, portfolio with a derivative
depending on
N
assets
S̄(t) = {S1 (t), ..., Sn (t)}
Π(t) = γ(t) f (t, S(t)) +
ontra t
has the value
N
X
βi (t) Si (t)
i=1
follows (for la k-of-arbitrage reasons) the value of a safe investment,
where
r
is the risk-free
ontinuously
Π(t) = Π(0)ert ,
ompounded interest rate. The hedging strat-
egy is
(2)
γ(t + dt)
(3)
βi (t + dt)
=
f (t) −
= −γ(t)
Π(t)
PN ∂f
i=1 ∂S (t) S(t)
∂f
(t)
∂Si
Consider a share whose pri e follows a lognormal Brownian motion, see g. 3.
The pri e is plotted in the left graph, together with the pri e of a
strike pri e 10 that expires at
t = 1.
The middle graph shows the
a perfe tly hedged portfolio. The topmost
part invested in shares, and under it is
urve is
β(t),
all option with
omposition of
the parts of the portfolio
γ(t), the part invested in options.
A negative
number means that the option should be sold short. To the right is a plot of the total
value of the portfolio that initially
onsisted of one option. Above and below are
plots of the total value of the portfolio holdings, in shares and options, respe tively.
Here
r = 0,
so the value of the portfolio should be
onstant even though the share
and option pri es u tuate. Sin e the portfolio value is
onstant in the rightmost
plot, the hedging works, and the portfolio yields the risk-free rate.
PRICING VIRTUAL PATHS WITH QOS GUARANTEES AS BUNDLE DERIVATIVES
18
0.4
5
share price
option price
16
7
beta
gamma
share value
option value
portfolio value
4
0.2
14
3
12
0
2
10
8
1
−0.2
6
0
4
−0.4
−1
2
0
0
0.2
0.4
0.6
0.8
−0.6
1
0
0.2
0.4
0.6
0.8
1
−2
0
0.2
Figure 3. To the left, the value of an asset and a
strike pri e
K = 10,
0.4
0.6
0.8
1
all option with
in the middle, the fra tion invested in ea h
resour e to get a balan ed portfolio, and to the right, the total
value of the portfolio ( onstant), and its parts.
mean
std deviation
2.05
0.8
2.04
0.7
2.03
0.6
2.02
0.5
2.01
0.4
2
0.3
1.99
0.2
1.98
0.1
1.97
0
pdf
1.4
1.2
1
0.8
0.6
0.4
0
0.5
1
0.2
0
0.5
1
0
−2
0
2
4
Figure 4. To the left and in the middle are plots of average value
and standard deviation for a portfolio at the rehedging times. To
the right is the portfolio value distribution when the option expires,
(t = 1).
Derived numeri ally (by Monte-Carlo simulation)
for a lognormal pri e pro ess (diamonds), and for a mean-reverting
pro ess (stars).
A well known result of the Bla k-S holes pri ing is the somewhat surprising
result that the derivative pri es is not dependent of the drift term. This is be ause
it is based on a rst order approximation, and the drift term is
diusion term is
√
O( dt).
O(dt)
while the
Sin e the drift term is the only dieren e between the
lognormal pro ess and the mean-reverting pro ess with multipli ative noise, the
hedging s heme works as equally well for both pro esses.
2.2.2.
Interval time hedging.
rather than in
For hedging at dis rete events with an interval
ontinuous time, the above formula does not give a
The ee t of hedging a
∆t > 0
omplete hedge.
all option with an unmodi ed strategy is shown, for two
dierent underlying pro esses, in g. 4. Above to the left, is a plot of the average
portfolio value from a Monte-Carlo simulation, of a lognormal pro ess, and a meanreverting pro ess with multipli ative noise. The middle plot shows the varian e of
8
LARS RASMUSSON
1.9
6
1.4
mean portfolio
std portfolio
s^ =0.945 s
1.88
pdf portfolio at T=1
1.2
5
1
1.86
4
1.84
3
1.82
2
1.8
1
0.8
0.6
0.4
1.78
0
0.5
1
0
0.2
0
0.5
0
−2
1
0
2
4
value
Figure 5. Plots of mean, standard deviation, and nal distribu-
tion of portfolio value (see g. 4) for a mean-reverting asset pri e
2
pro ess, rehedged 10 times with the adjusted varian e σ̂ .
the portfolio value at the 10 rebalan ing times. The varian e in reases with time,
and the varian e is higher for the mean-reverting pro ess than for the lognormal
pro ess. To the right is the density fun tion for the portfolio value at
rebalan ing of the portfolio is only done every 100 steps, i.e.
t = 1.
∆t = 0.1.
apparent that the portfolio value for the mean-reverting pro ess is not
The
It is
onstant. It
is hen e possible to make a statisti al arbitrage pro t on the derivatives. However,
the deviation in expe ted value is only a few per ent, so a small in rease in the
derivative pri e
an prote t the issuer from the arbitrage risks.
For some pro esses a modi ed hedging strategy
Bou haud
et al.
[3℄ have
onsidered time
an be derived.
orrelated sto hasti
Cornalba,
pro esses and shown
that for the Ornstein-Uhlenbe k pro ess
dS(t) = α(µ − S(t))dt + σdW (t)
the same hedging strategy
an be used, but with a modi ed varian e. A similar
−2α∆t
)
2
2 (1−e
derivation, inspired by Bou haud, gives that σ̂ = σ
. This is based on
2α∆t
a rst order approximation, hen e the modi ed volatility
small
∆t.
σ̂
is appropriate only for
For many pro esses, su h as pro esses with multipli ative noise, we do
not have modi ed strategies.
Fig. 5 shows plots, similar to g. 4, of the values for a mean-reverting pro ess
that is hedged with the modi ed measure of varian e
σ̂ 2 , and it
an be seen that the
adjustment is not perfe t, but still only a few per ent o. Its dynami s are more
omplex, and we do not know of a strategy with modi ed varian e that makes the
portfolio value pro ess repli ate that of a risk-free portfolio.
Sin e the portfolio
value is un ertain.
In
annot be made risk-free over all of
∆t,
the future portfolio
In terms of mathemati al nan e, the market is in omplete.
omplete markets, options
an be pri ed in a way that does not depend on the
risk-aversiveness of individual parti ipants, something that is not the
ase in an
PRICING VIRTUAL PATHS WITH QOS GUARANTEES AS BUNDLE DERIVATIVES
in omplete market. However in a te hni al system, designed to be
a market, we
9
ontrolled by
an require there to always be one or more trading programs that
behave risk-neutrally, or that
harge a spe i ed risk premium.
That guarantees
that derivatives are traded at pri es that make the market e ient, in the sense
that they maximize the expe ted utility of the end-users by providing low option
pri es.
3. Model
3.1.
Pri e pro esses.
In a omputer network, end-users share the limited
of the routers and links.
apa ity
To be able to provide QoS-dependent servi es, these
resour es must be managed by the end-users. To ea h end-user, the system load
appears to u tuate sto hasti ally, as the end-user does not have a
ess to the
omplete system state.
Our approa h is to view the state of the system as sto hasti
pro esses, and to
ontrol the use of s ar e resour es by trading the usage right at spot markets. On
these markets, pri es will appear to be sto hasti
pro esses. The pri es present an
an aggregated view of the system. With this view,
ontrolling a te hni al system
with intera ting subunits (not ne essarily a network), boils down to developing
suitable derivatives and hedging s hemes for the dierent servi es that the system
should provide. The implementation requires market-pla es for the individual resour es, and third party middle-men that sell derivatives to end-users and do the
a tual trading on the resour e level.
An average, sporadi
end-user is not willing to take the risk of paying an ex es-
sive amount for a network servi e, but rather get a de nite pri e quote. Trading
derivative
and the other part gets a xed
ontra ts
risk, where one part gets the risk and a premium,
ontra ts is a trade of
ash ow. With suitable market models, derivative
an be pri ed obje tively, at least when the pri e pro esses
an be de-
s ribed su iently well. So, instead of trading the a tual resour es, the end-users
buy derivative
ontra ts of a third party. The
ontra t guarantees the delivery of
the required set of resour es. The
ontra t may spe ify both future delivery of some
resour es, and deliveries that are
ontingent on future pri es, or fun tions thereof.
In a previous paper [13℄, we have simulated a simple bandwidth market without
derivatives, to determine the properties of the resulting sto hasti
was found to be very well des ribed by a
pri e pro ess. It
orrelated mean-reverting pro ess with
multipli ative noise,
dSi (t) = αi (µi − Si (t))dt + σi Si (t)dWi (t)
(4)
where the pri e
parameters
orrelation
α, µ, σ, ρ,
Corr[dWi (t), dWj (t)] = ρij .
We gave estimations of the
based on pri e history data from the simulation. Sin e this
10
LARS RASMUSSON
modeling was possible, it appears feasible to represent
terms of derivative
ontra ts on
ertain
omplex
omplex network servi es in
ombinations of resour es. Sin e
in general, introdu ing new assets in a market modi es the pri e dynami s, hen e
the parameters must be re-estimated when new derivatives are added.
The mean-reverting pro ess drifts ba k towards
µ
with a rate determined by
α.
As opposed to lognormal pro esses, mean-reverting pro esses are auto- orrelated
pro esses, and their the varian e per time unit,
in reasing
τ
1
τV
ar[S(t+τ )−S(t)] de
for the mean-reverting pro ess, while it is
reases with
onstant for the lognormal
pro ess. It is the fa t that the pri e pro ess has memory that
auses the deviation
in the expe ted value of the portfolio for interval hedging.
3.2.
Farmer markets.
The market pla es where resour e trading are of a spe ial
kind designed for very rapid markets that we
all Farmer markets, as the original
inspiration was found in [5℄. Ea h market handles one resour e, and is run by a
market maker that at ea h instan e guarantees to a
ept bids both to buy and to
sell.
There is no ba k-log of pending limit orders, only bids at market are a
This guarantees that the trading
epted.
an take pla e with very little overhead for the
market maker. Sin e only bids at-market are a
epted, the bidder does not know
at what pri e resour es will be traded, but pri es
an be estimated from the pri e-
quote history.
The
entral idea of this market design is that the resour es are ex hanged on
these markets, and that all more
omplex
ontra ts are expressed as derivative
ontra t on these resour es. For instan e, a limit order, i.e. an order to buy if the
pri e is less than a spe i ed amount, is a risky
ontra t sin e the bidder does not
know if the deal will go through or not.
3.3.
Resour e shares.
The
apa ity of ea h resour e is divided into equal well
de ned shares, that gives the owner the right to send a
on a short time
∆t
if he pays
ǫ S(t)∆t.
ertain amount of tra
From here on, we assume
apa ity of the shares must not surpass the total
ǫ = 1.
The total
apa ity of the resour e. Without
the payment from the resour e holder to the market maker, a holder of the resour e
would have no in entive to avoid
ongested resour es, whi h is shown in the pri ing
of the bundle options below.
For routers, statisti al multiplexing has shown to give a large throughput inrease, hen e it seems reasonable to mix two tra
lasses, 1) the guaranteed
lass, and 2) the best-eort
lass are guaranteed not to be dropped in
out valid
lasses. A router has two tra
ase of
lass. Pa kets in the rst
ongestion, while pa kets with-
redentials are handled in traditional best-eort manner.
Metro Pri ing S heme by Odlyzko [10℄, we only requires two tra
As with the
lasses, but in
PRICING VIRTUAL PATHS WITH QOS GUARANTEES AS BUNDLE DERIVATIVES
11
the Metro Pri ing s heme, pri es are determined outside of the system in su h a
way that there is no
ongestion in the rst
determined by demand, and rst
lass, while in our model, pri es are
lass pa kets get to go rst if there is
ongestion.
4. Results
In a
omputer network, end-users want to establish virtual paths with
ertain
guaranteed properties, su h as loss, laten y, et . The user wants to have the resour es at
T1
and to sell them again at
that delivers the resour es at
T1
T2 .
This
an be implemented as an option
together with a bundle of options that lets the user
sell the resour es at a guaranteed pri e. To nd the pri e of this option, we rst
establish the following
orollary. All proofs are deferred to the appendix.
Corollary. (1) The pri e of a future to buy shares of the resour es on the heapest
path between two network nodes at a T1 that are resold at T2 is zero.
However, to balan e the load, the pri e of the derivative must depend on the
resour e pri es, therefore the so- alled bundle future above is not suitable for loadbalan ing. Instead, we de ne a network option, in the following way.
De nition.
A network
all-option gives the holder the right to send pa kets with
a spe i ed intensity through nodes on a path between two network routers from
time
T1
The
to time
T2 ,
if the fee
K
is paid at
T1 .
all-option pri e depends on the pri e of the shares in all routers that are
on any of the possible paths. The following is a very useful theorem for deriving
pri es of options based on the
orrelated pri e pro esses.
Theorem. (3) Let
S(T ) = {S1 (T ), ..., (SN (T )} be an N -dimensional lognormal
pri e pro ess with orrelation ρij = Corr[dWi , dWj ] under probability measure Q.
Then
where
T
E Q [Sm (T )g(S(T ))|F0 ] = Sm,0 erT E Q g(ξm1
S1 (T ), ..., ξmN SN (T ))|F0
ξmi = eσi σm ρim = exp
This shows that linear derivatives
pro ess
1
Cov [log dSi (T ), log dSm (T )]
dt
an be pri ed as expe ted values of an adjusted
ξmi Si (T ).
With the help of this theorem, we
option, and its partial derivatives, and
a portfolio for the
an derive the value of the network
all-
al ulate the optimal rebalan e strategy for
ontinuous time hedge strategy, using Eq. (2) and (3).
12
LARS RASMUSSON
Corollary. (3) The value of a network all-option with strike pri e K is
= T C erT1
f (0, S̄)
N
X
Sm,0
m=1
M
X
i
h
vim Q i = argminj Ĉjm ∧ Ĉim < K
X
T1
Sk (T1 )
vik ξmi
i=1
−T C K Q [minj Cj > K]
where
Ĉim =
k
is the adjusted ost of path i, after the Girsanov transform to eliminate resour e
Sm (...),
TC =
e−rT1 − e−rT2
r
with limr→0 T C = T2 − T1 .
Corollary. (4) The partial derivative of the network option with strike-pri e K is
M
and
X
∂f
vin Q[Ĉin = minj Ĉjn ∧ Ĉin > K]
(0, S̄) = T C erT1
∂Sn,0
i=1
f (0, S̄) =
N
X
Sm,0
m=1
There is no
∂f
(0, S̄) − T C K Q [minj Cj > K]
∂Sn,0
losed form for the sum of lognormal variables [17℄, whi h makes it
di ult to redu e the
Q[...]-terms
the risk neutral measure
further, but sin e
Q, the option pri
e
S(T )
has a
losed form under
an be approximated with Monte-Carlo
simulation. Note that under the risk neutral measure one,
S(T )
an be simulated
without having to simulate the individual pri e traje tories for the mean-reverting
pro ess, something that is required for pri ing te hniques using the natural measure,
and whi h is very time
onsuming.
5. Dis ussion
In the proposed network model, a
ess to ea h node is traded in a dierent
market. This way, appli ations at the edge of the network an
in whi h way they
routing. By
hoose, i.e.
hoosing to trade
the fundamental
ombine the resour es
build broad ast trees, or failure safe multi-path
apa ity shares rather than time-slotted a
ess as
ommodity, we have only one market per router instead of one per
router and minute.
In the most popular alternative model, network a
edge of the network, hen e appli ations at the edges
ess is handled only at the
annot
reate new servi es.
PRICING VIRTUAL PATHS WITH QOS GUARANTEES AS BUNDLE DERIVATIVES
The
ost of a more ne grained
ontrol s heme is of
13
ourse more overhead, but with
the proposed s heme, the routers are relieved of mu h work, sin e the pa kets are
sour e routed.
The
entral idea is that simple resour es are ex hanged on very fast markets,
and that all more
omplex servi es are expressed as derivative
ontra t on these
resour es. Sin e we use Farmer markets, the trading generates very little overhead
but in urs a pri e risk for the trader, whi h must be hedged. Sin e end-users do
not hedge their risks themselves, but instead buy derivatives, the network will not
be ooded by bids and quotes between all end-users and all markets. When a user
requires a servi e, or a
ombination of servi es, the user simply tells a middle-man
that it is willing to pay a
and the user
marginal
ertain amount for a derivative that models the servi e,
an be informed dire tly whether or not it got the servi e, and of the
ost.
The apa ity pri es are assumed to be orrelated It pro esses with multipli ative
drift. We show how use a Martingale te hnique to pri e options on one-of-several
linear
ombinations of assets by proving a theorem on a Girsanov transform that
an be used to pri e several other options, and give a
egy that
ontinuous time hedge strat-
an be used by a trader to balan e out all risk.
We have not found a
omplete adjusted strategy for the mean-reverting pri e pro ess when the portfolio
is infrequently rebalan ed. A potential possibility to nd an adjusted strategy is to
look into pri ing of Bessel pro esses [2℄.
The proposed hedge s heme builds on the assumptions made in the Bla k-S holes
model, i.e. that the portfolio is self- nan ing, without arbitrage possibilities, and
that short selling is allowed. Other hedge s hemes use dierent assumptions, su h
as CAPM [18℄, whi h aims to minimize the varian e of the portfolio value while
maximizing its expe ted value. The proposed s heme has the advantage that the
pri e is invariant of risk-attitude, and
an be e iently evaluated using Monte-Carlo
te hniques.
Future work will
onsist of simulations of the
omplete bandwidth market in
order to determine the ee t of the network options on the pri e pro esses, to nd
better models for the nan ial risk of trading in Farmer markets, and to model
other network servi es as derivative servi es.
A knowledgement.
onstru tive
The author wishes to thank Erik Aurell for advi e and many
omments on the topi
of nan ial mathemati s, and Sverker Janson
for helpful dis ussions on agent based models. This work is funded in part by Vinnova/Nutek, The Swedish Agen y for Innovation Systems, program PROMODIS,
and in part by SITI, The Swedish Institute for Information Te hnology, program
Internet3.
14
LARS RASMUSSON
Referen es
[1℄ Fis her Bla k, and Myron S holes, The Pri ing of Options and Corporate Liabilities, Journal
of Politi al E onomy, (81:3), pp. 637-654, (1973).
[2℄ Hélyette Geman, and Mark Yor, Bessel Pro esses, Asian Options, and Perpetuities, Mathemati al Finan e, 3/4, (1993)
[3℄ Lorenzo Cornalba, Jean-Philippe Bou haud, and Mar Potters, Option Pri ing and Hedging
with Temporal Correlations, Nov (2000). http://xxx.lanl.gov/abs/ ond-mat/0011506
[4℄ James F. Kurose, and Rahul Simha, A Mi roe onomi Approa h to Optimal Resour e Allo ation in Distributed Computer Systems, IEEE Trans. on Computers, (38) no. 5, May
(1989).
[5℄ J. Doyne Farmer, Market for e, e ology, and Evolution, submitted to Journal of E onomi
Behavior and Organization, Feb. (2000). http://www.santafe.edu/~jdf/
[6℄ Ri hard J. Gibbens, and Frank P. Kelly, Resour e pri ing and the evolution of ongestion
ontrol, Automati a, 35, (1999) http://www.statslab. am.a .uk/~frank/evol.html
[7℄ Frank P. Kelly, Mathemati al modelling of the Internet. In "Mathemati s Unlimited - 2001
and Beyond" (Editors B. Engquist and W. S hmid). pp. 685-702, Springer-Verlag, Berlin,
(2001) . http://www.statslab. am.a .uk/~frank/mmi.html
[8℄ Costas Cour oubetis, Antonis Dimakis, and Marty Reiman, Providing Bandwidth Guarantees over a Best-eort Network: Call-admission and Pri ing, Pro . of Info om 2001.
http://info om.u sd.edu/papers/231.pdf
[9℄ Rajan M. Lukose, and Bernardo A. Huberman, A Methodology for Managing Risk in Ele troni Transa tions over the Internet, Pro . of the Third Int. Conf on Computational E onomi s, Palo Alto, June 1997.
http://www.par .xerox. om/istl/groups/iea/abstra ts/ECommer e/banking.html
[10℄ Andrew M. Odlyzko, Paris Metro Pri ing for the Internet, Pro eedings of the ACM Conferen e on Ele troni Commer e, pp. 140-147, (1999).
http://www.resear h.att. om/~amo/do /paris.metro.pri ing.ps
[11℄ Peyman Faratin, Ni holas Jennings, R. R. Bu kle, and Carlos Sierra, Automated negotiation
for provisioning virtual private networks using FIPA- ompliant agent s, Pro . of 5th Int.
Conf. of Pra tial Appli ations of Intelligent Agents and Multi-Agent Systems, Man hester,
UK, (2000).
[12℄ Pierre Collin Dufresne, William Keirstead, and Mi hael P. Ross, Pri ing Derivatives the
Martingale Way, May (1996). http://www.berkeley.edu/ nan e/WP/rpfabstra t.html
[13℄ Lars Rasmusson, and Erik Aurell, A Pri e Dynami s in Bandwidth Markets for Pointto-point Conne tions, unpublished, subm. to IEEE/Trans. on Networking, Jan 2001.
http://xxx.lanl.gov/abs/ s.NI/0102011
[14℄ Nemo Semret, and Aurel A. Lazar, Spot and derivative markets in admission ontrol, in
Pro . of ITC 16, P. Key and D. Smith, Eds. June 1999, pp. 925-941, Elsevier.
http:// iteseer.nj.ne . om/semret99spot.html
[15℄ Peter E. Kloeden, and E khard Platen, Numeri al Solution of Sto hasti Dierential Equations, Springer Verlag, Berlin, 1999.
[16℄ Klaus Krauter, and Muthu umaru Maheswaran, Ar hite ture for a Grid Operating System,
Pro . of 1st IEEE/ACM Int. Workshop on Grid Computing, Bangalore, India, LNCS 1971,
Springer Verlag, De . 2000.
http://www. sse.monash.edu.au/~rajkumar/Grid2000/grid2000/book/19710064.ps
PRICING VIRTUAL PATHS WITH QOS GUARANTEES AS BUNDLE DERIVATIVES
15
[17℄ Moshe Arye Milevsky, and Steven E. Posner, Asian Options, the Sum of Lognormals, and
the Re ipro al Gamma Distribution, Journal of Finan ial and Quantitative Analysis Vol. 33,
No. 3, September 1998
[18℄ Zvi Bode, AlexKane, and Alan J. Mar us, Investments, Irwin/M Graw-Hill, Boston, 1996.
16
LARS RASMUSSON
Appendix A.
Here we give the proofs of the theorems and lemmas needed for pri ing the
network option.
A.1.
Bundle futures.
De nition 1.
T1
and
T2 ,
A bundle future gives the holder a set of resour e shares between
given that an event
R
o
urs at
T1 .
Let
Q
be the equivalent martingale
measure.
Theorem 1. The pri e of the bundle future is zero.
Lemma 1. If W(T) is a Wiener pro ess, then
F0 is the natural ltration up to t = 0.
h
i
σ2
E e(− 2 )T +σW (T ) |F0 = 1, where
Lemma 2. The pri e of a future to buy a single resour e
Sj at T1 that is sold at
T2 is zero.
Proof:
pri e at
At
t=0
T2 ,
the future is worth
is
Aj = Sj (T2 ) − er(T2 −T1 ) Sj (T1 ).
Hen e the
e−rT2 E Q [Aj |F0 ]
i
h
σ2
σ2
e−rT2 s0 E Q e(r− 2 )T2 +σW (T2 ) − er(T2 −T1 ) e(r− 2 )T1 +σW (T1 ) |F0
i
h
σ2
σ2
s0 E Q e(− 2 )T2 +σW (T2 ) − e(− 2 )T1 +σW (T1 ) |F0
h
i
i
h
σ2
σ2
s0 E Q e(− 2 )T1 +σW (T1 ) |F0 E Q e(− 2 )(T2 −T1 )+σ(W (T2 )−W (T1 )) − 1|F0
|
{z
}
f (0, s0 ) =
=
=
=
=0
=
0
Lemma 3. The pri e of a derivative that delivers the future de ned in lemma 2,
given that event R o urs at T1 , is also zero.
Proof:
At
T2
the derivative is worth
Aj 1{R} .
Hen e the option pri e at
f (0, s0 ) = erT2 E Q [Aj 1{R} |F0 ]
= e
rT2
Q
Q
E E [Aj |F1 ] Q[R]|F0
| {z }
=0
= 0
where
Q[R]
is the probability of
R
under the probability measure
Q.
t=0
is
PRICING VIRTUAL PATHS WITH QOS GUARANTEES AS BUNDLE DERIVATIVES
Proof of Theorem 1:
Let
Ri
be the event that bundle
i
17
is bought. The pri e
of the bundle future is
N
X
vij Aj 1{Ri } |F0
e−rT2 E Q
f (0, S̄) =
j=1
=
0
Corollary 1. The pri e of a future to buy shares of the resour es on the heapest
path between two network nodes at a T1 that are resold at T2 is zero.
Proof:
Let
vij
event that path
i
j
be the amount of router
is the
required on path
T1 .
heapest path at
The
i,
and let
Ri
be the
orollary follows from Theorem
1.
A.2.
Network option (step one).
De nition 2.
A network option gives the holder the right to send pa kets with a
spe i ed intensity through nodes on a path between two network routers from time
T1
to time
T2 ,
if the fee
K
is paid at
T1 .
Lemma 4. The pri e of an arithmeti average (Asian) all option with strike pri e
zero and maturity at T is
f (0, s0 ) = s0
1 − e−rT
r
whi h be omes s0 T in the limit of r → 0+ .
Proof:
At
T,
fAsian (0, s0 )
the option is worth
= e−rT E Q
"Z
T
RT
0
S(t)dt|F0
0
=
lim∆t→0 e
−rT
S(t)dt, so
#
EQ
T /∆t−1
X
T /∆t−1
=
lim∆t→0 e
−rT
X
i=0
= e−rT
Z
T
s0 ert dt
0
1 − e−rT
= s0
r
s0 e
2
(r− σ2
)ti +σW (ti )
i=0
∆t|F0
i
h
σ2
s0 er ti ∆t E Q e(− 2 )ti +σW (ti ) |F0
{z
}
|
=1
18
LARS RASMUSSON
Theorem 2. Let Ci =
PN
m=1 vim Sj (T1 )
be the ost of the resour es for alternative
i at T1 . The pri e of a network option is
f (0, S̄) = T C E Q [max(mini (Ci ) − K, 0)|F0 ]
!
#
"M
X
Ci 1{Ci =mink Ck } 1{Ci >K} |F0 − K E Q 1{mini Ci >K} |F0
= T C EQ
i=1
where T C =
Proof:
T1 ,
e−rT1 −e−rT2
r
The
, for whi h limr→0 T C = T2 − T1 .
ost to send tra
pay the send fee from
T1
onsists of buying the required router shares at
T2
to
and sell ba k the shares at
to a bundle option and an Asian option from
sour e and the destination. The
path, i.e. where
i = argmink Ck
T1
to
T2
on some path between the
at
T1 .
At
T1
in-the-money, else it is worth 0. Hen e, on
t=0
f (0, S̄) = e
= e
−rT1
E
E
Q
Q
max
M X
N
X
i=1 m=1
"
max
K
if the option is
!
vim fAsian (T1 , S(T1 ))1{Ci =mink Ck } − K, 0 |F0
M
N
X
1 − e−r(T2 −T1 ) X
i=1
ost
the network option is worth the sum
heapest path minus
−rT1
This amounts
heapest option is the option over the least
of Asian options for the resour es on the
"
T2 .
r
m=1
#
!
vim S(T1 )1{Ci =mink Ck } − K, 0 |F0
#
"
M
X
e−rT1 − e−rT2 Q
Ci 1{Ci =mink Ck } − K, 0)|F0
=
E max(
r
i=1
!
#
"M
X
Q
Q
Ci 1{Ci =mink Ck } 1{Ci >K} |F0 − K E 1{mini Ci >K} |F0
= TC E
i=1
A.3.
The 1-dimensional Girsanov transform.
We start by showing the useful-
ness of the so- alled Girsanov transform for a one-dimensional sto hasti
in order to simplify the presentation of the proof of the
The one-dimensional pri ing of a
Lemma 5.
n-dimensional
all option was based on Dufresne
i
h
2
E Q [S(T )g(S(T ))|F0 ] = S0 erT E Q g(eσ T S(T ))|F0
pro ess,
transform.
et al.
[12℄.
#
PRICING VIRTUAL PATHS WITH QOS GUARANTEES AS BUNDLE DERIVATIVES
19
Proof:
Q
√
y = x − σ T.
∞
√
1
σ2
1 2
√ e− 2 x g S0 e(r− 2 )T +σ T x dx
2π
−∞
Z ∞
√ 2
√
1
σ2
1
√ e− 2 (x−σ T ) g S0 e(r− 2 )T +σ T x dx
= S0 erT
2π
−∞
Z ∞
√
√
σ2
1 2
1
√ e− 2 y g S0 e(r− 2 )T +σ T (y+σ T ) dx
= S0 erT
2π
−∞
i
h
2
= S0 erT E Q g(eσ T S(T ))|F0
E [S(T )g(S(T ))|F0 ] =
where
Z
S0 e(r−
σ2
2
√
)T +σ T x
Corollary 2. The value of a all option with strike pri e K , and maturity at T , is
√
e−rT E Q [max(S(T ) − K, 0)|F0 ] = S0 N (d1 + σ T ) − Ke−rT N (d1 )
σ2
where d1 = log Kσ√−T 2 T , N (x) is the umulative density fun tion for the standard
normal distribution, the urrent time t = 0, and S(0) = S0 .
S0
Proof:
From the de nition of the indi ator fun tion,
E Q [1{A} ] = Q[A].
The
orollary follows from seeing that
E Q [max(S(T ) − K, 0)|F0 ] = E Q [S(T ) 1{S(T )>K}|F0 ] − K E Q [1{S(T )>K} , 0)|F0 ]
and from the lemma,
Q
E [1{S(T )>K} |F0 ]
h
W (T )
<
= Q − √
T
= N (d1 )
S
log K0
|
2
−σ T i
√ 2
σ T
{z
}
=d1
and
E Q [S(T ) 1{S(T )>K} |F0 ]
i
h
= S0 erT E Q 1{eσ2 T S(T )>K} |F0
"
#
S
σ2
√
T
log 0 −
W (T )
rT
K
2
√
<
+σ T
= S0 e Q − √
T
σ T
√
= S0 erT N (d1 + σ T )
by doing a Girsanov transform, and using that
varian e
A.4.
W (T )
is normal distributed with
T.
The n-dimensional Girsanov transform of E[S g(S)].
Lemma 6. Let
2
σi
2
√
, i = i, ..., N , where xi are standard normal distributed variables, orrelated with {D}ij = Corr[xi , xj ] under the
{SQ (T )}i = Si,0 e(r−
)T +σi
T xi
20
LARS RASMUSSON
measure Q, and un orrelated under the measure R. Then
E Q [g(SQ (T ))|F0 ] = E R [g(SR (T ))|F0 ]
where {SR (T )}i = Si,0 e(r−
tion of D, i.e. P T P = D.
Proof:
tion
PN
2
σi
2
)T +σi
√
PN
T
j=1
Pij xj
, and P is the Cholesky fa toriza-
P T P = D, then D−1 P T P = I = P T D−1 P . The substitu√
P y = x makes xT D−1 x = yT y, dx = (det P )dy = det Ddy, and xi =
j=1
Sin e
Pij yj .
Z
E Q [g(S(T ))|F0 ] =
∞
dx g(SQ (T ))
1
√
(2π)N/2
det D
1
1 T
dy g(SR (T ))
e− 2 y y
N/2
(2π)
−∞
−∞
∞
1
e− 2 x
T
D−1 x
Z
=
E R [g(SR (T ))|F0 ]
=
Lemma 7. The multidimensional variant of the elimination of Sm for an un orrelated N-dimensional random variable S is
E R [{SR (T )}m g(SR (T ))|F0 ] = Sm,0 erT E R [g(SZ (T ))|F0 ]
where {SZ (T )}i = eσi σm
Proof:
=
Z
PN
j=1
Pij Pmj
{SR (T )}i .
E R [{SR (T )}m g(SR (T ))|F0 ]
∞
dxSm,0 e(r−
−∞
= Sm,0 e
(r−
2
σm
2
)T
Z
2
σm
2
∞
−∞
= Sm,0 e
)T +σm
dx
√
T
PN
j=1
= Sm,0 erT
Pn
2
1
− 12
j=1 xj g(SR (T ))
e
(2π)N/2
√
Pn
2
1
− 12
j=1 (xj −2σm T Pmj xj ) g(SR (T ))
e
(2π)N/2
=1
z }| {
N
X
σ2
2
T
Pmj Z
(r− 2m )T + 21 σm
j=1
Z
Pmj xj
∞
∞
−∞
dx
√
Pn
2
1
− 21
j=1 (xj −σm T Pmj ) g(SR (T ))
e
N/2
(2π)
1
1 T
dy
e− 2 y y g(SZ (T ))
N/2
q(2π)
−∞
= Sm,0 erT E R [g(SZ (T ))|F0 ]
PRICING VIRTUAL PATHS WITH QOS GUARANTEES AS BUNDLE DERIVATIVES
where
√
xi = yi + σm T Pmi ,
{SZ (T )}i
Lemma 8.
Proof:
therefore
2
σi
2
)T +σi
√ PN
√
T
j=1 Pij (yj +σm T Pmj )
=
Si,0 e(r−
=
eσi σm T
=
eσi σm T
=
eσi σm T {D}im {SR (T )}i
σi σm {D}im =
1
dt
PN
j=1
PN
j=1
2
σi
2
Pij Pmj
Si,0 e(r−
Pij Pmj
{SR (T )}i
)T +σi
√
T
PN
j=1
Pij yj )
CovQ [log dSi (T ), log dSm (T )].
By It 's Formula (see for instan e, theorem 3.3.2 in [15℄)
a dt + σi dWi
for some bounded fun tion
Q
Cov [log
21
log dSi =
a.
dSi (T ), log dSm (T )] =
=
Q
Cov [σi dWi , σj dWj ]
+ O(dt3/2 )
σi σj {D}ij dt + O(dt3/2 )
Theorem 3. Let S(T ) = {S1 (T ), ..., (SN (T )} be an N -dimensional lognormal pri e
pro ess with
Then
{D}ij = Corr[dWi , dWj ] under probability measure Q.
orrelation
E Q [Sm (T )g(S(T ))|F0 ] =
T
Sm,0 erT E Q [g(ξm1
S1 (T ), ..., ξmN SN (T ))|F0 ]
1
where ξmi = eσi σm {D}im = exp( dt
CovQ [log dSi (T ), log dSm (T )]).
Proof:
Follows from using lemma 6, lemma 7, and lemma 6 again, in the other
dire tion.
A.5.
Network option (step two).
Corollary 3. The value of a network all-option with strike pri e K is
f (0, S̄)
= T C erT1
N
X
Sm,0
m=1
M
X
i=1
vim Q[i = argminj Ĉjm ∧ Ĉim > K]
−T C K Q[minj Cj > K]
P
T1
Sk (T1 ) is the adjusted ost of path i, after the Girsanovwhere Ĉim = k vik ξmi
transform to eliminate resour e Sm (...).
Proof:
f (0, S̄) = T C E
|
Q
#
"M
X
Ci 1{Ci =mink Ck } 1{Ci >K} |F0 − K E Q [1{mini Ci >K} |F0 ]
|
{z
}
i=1
{z
}
=V2
=V1
22
LARS RASMUSSON
where
V2 = K Q[mini Ci > K]
V1
= EQ[
M
X
i=1
= EQ[
and
Ci 1{Ci =mink Ck } 1{Ci >K} |F0 ]
M X
N
X
i=1 m=1
=
M X
N
X
i=1 m=1
= erT1
vim Sm (T1 )1{Ci =mink Ck ∧Ci >K} |F0 ]
vim S0.m erT1 E Q [1{Ĉim =mink Ĉkm ∧Ĉim >K} |F0 ]
N
X
S0.m
m=1
M
X
i=1
vim Q[Ĉim = mink Ĉkm ∧ Ĉim > K]
by using theorem 3 in step three.
Corollary 4. The partial derivative of the network option with strike-pri e K is
M
X
∂f
vin Q[Ĉin = minj Ĉjn ∧ Ĉin > K]
(0, S̄) = T C erT1
∂Sn,0
i=1
and
f (0, S̄) =
N
X
m=1
Proof:
m
all
∗
Sm,0
∂f
(0, S̄) − T C K Q[minj Cj > K]
∂Sn,0
∂Q∗
= 0 nearly everywhere, for
The rst statement follows from that ∂S
in
∗
and n, for both ases when Q = Q[Ĉin = minj Ĉjn ∧ Ĉin > K], and when
Q = Q[minj Cj > K].
The se ond follows trivially from the value of the network
all-option and the rst statement.
Swedish Inst. of Computer S ien e, Box 1263, S-16429 Kista, Sweden
E-mail address :
Lars.RasmussonSICS.se
| 5 |
arXiv:1703.07970v1 [math.AC] 23 Mar 2017
The strong Lefschetz property of monomial
complete intersections in two variables
Lisa Nicklasson
Abstract
In this paper we classify the monomial complete intersection algebras,
in two variables, and of positive characteristic, which has the strong Lefschetz property. Together with known results, this gives a complete classification of the monomial complete intersections with the strong Lefschetz
property.
1
Background
L
A graded algebra A =
i≥0 Ai is said to have the strong Lefschetz property
(SLP) if there is a linear form such that multiplication by any power of this
linear form has maximal rank in every degree. Let A be a monomial complete
intersection, that is A = K[x1 , . . . , xn ]/(xd11 , . . . , xdnn ), where K is a field and
d1 , . . . , dn some positive integers. In characteristic zero, A always has the SLP,
which was first proved by Stanley in [12]. When the characteristic is positive,
the algebraPdoes not always have the SLP. A first result is that A has the SLP
when p > (di − 1), where p is the characteristic. This was proved in the case
n = 2 by Lindsey in [8], and later in the general case by Cook II in [5].
A classification of all monomial complete intersections in three or more variables with the SLP is provided in [9]. Notice that the problem is trivial when
n = 1, so the remaining case is n = 2, which will be treated in this paper. The
sufficient conditions in [9] hold also in two variables, but it turns out that there
is an additional class of algebras K[x, y]/(xa , y b ) with the SLP. This is indicated
by Cook II in [5], where the two special cases, when a = b, and when the characteristic is two, is studied. Cook II solves these cases, under the assumption
that the residue field K is infinite.
The main result of this paper is Theorem 3.2, which is a classification of the
algebras K[x, y]/(xa , y b ) with the SLP, where K is a field of characteristic p ≥ 3.
The classification is given in terms of the base p digits of the integers a and b.
Together with the mentioned earlier results, this gives a complete classification
of the monomial complete intersections with the SLP, see Theorem 3.4.
The technique used both in [5] and in this paper, is the theory of the syzygy
gap function, introduced by Monsky in [10]. The syzygy gap function deals with
the degrees of the relations on xa , y b and (x + y)c . This can then be connected
to the SLP using results of Brenner and Kaid in [1] and [2]. In [1], [2], and [10]
the residue field is required to be algebraically closed. We will see in Section
4 that this assumption can be dropped. We will also give a new proof of the
connection to the SLP, when working with monomial complete intersections.
1
2
The strong Lefschetz property
L
Let A = i≥0 Ai be a graded algebra. A linear map Ai → Aj is said to have
maximal rank if it is injective or surjective. Each homogeneous element f ∈ Ad
induces a family of linear maps Ai → Ai+d by a 7→ f · a. Let such maps be
denoted by ·f : Ai → Ai+d . For short, we say that multiplication by f has
maximal rank in every degree, if all the maps induced by f have maximal rank.
Definition 2.1. A graded algebra A is said to have the strong Lefschetz property (SLP) if there exists an ℓ ∈ A1 such that the maps ·ℓm : Ai → Ai+m have
maximal rank for all i ≥ 0 and all m ≥ 1. In this case, ℓ is said to be a strong
Lefschetz element.
We say that A has the weak Lefschetz property (WLP) if there exists an
ℓ ∈ A1 such that the maps ·ℓ : Ai → Ai+1 , have maximal rank for all i ≥ 0. In
this case, ℓ is said to be a weak Lefschetz element.
Let now K be a field, and A = K[x1 , . . . , xn ]/I, where I is a monomial
ideal. In [9, Proposition 4.3] it is proved that A has the WLP if and only if
x1 + · · · + xn is a weak Lefschetz element. The corresponding is also true for
the strong Lefschetz property.
Theorem 2.2. Let R = K[x1 , . . . , xn ], where K is a field, and let I ⊂ R be a
monomial ideal. Then R/I has the SLP (WLP) if and only if x1 + . . . + xn is
a strong (weak) Lefschetz element.
P
Proof. Suppose that i∈Λ ci xi , for some Λ ⊆ {1, . . . , n} and 0 6= ci ∈ K, is a
strong Lefschetz element of A = R/I. The monomial ideal I is left
Punchanged
under a change of variables ci xi 7→ xi . This shows that ℓ =
i∈Λ xi also
is a strong Lefschetz element. If Λ = {1, . . . , n} we are done. Assume that
Λ ⊂ {1, . . . , n}, and j ∈
/ Λ. The next step is to prove that xj + ℓ, is also a
strong Lefschetz element. For this purpose we introduce a new element a in an
extension field of the type K ′ = K(a) ⊃ K. We will prove that axj + ℓ is a
strong Lefschetz element in A′ = A ⊗K K ′ . Let, for each i, Bi be the vector
space basis for Ai that consists of monic monomials. This is also a basis for A′i ,
as a vector space over K ′ . Let M be the matrix of the multiplication map
·(axj + ℓ)m : A′i → A′i+m ,
w. r. t. the bases Bi and Bi+m . The entries of M are polynomials in a. Let M0
be the matrix we obtain by substituting a = 0 in M . If M does not have maximal
rank, neither does M0 . But M0 is the matrix of the map ·ℓm : Ai → Ai+m ,
which has maximal rank. This shows that M has maximal rank, and axj + ℓ
is a strong Lefshetz element of A′ . But then, since a is a non-zero element of
the field K ′ , so is xj + ℓ. The coefficients of xj + ℓ are in K, so it is also a
strong Lefschetz element of A. It follows that x1 + · · · + xn is a strong Lefschetz
element of A.
L
The Hilbert function of a graded algebra A =
i≥0 Ai with residue field
K is a function HFA : Z≥0 → Z≥0 definied by HFA (i) = vdimK Ai , i. e. the
vector space dimension of Ai over K. The Hilbert series of A, denoted
HSA , is
P
the generating function of the sequence HF(i), that is HSA (t) = i≥0 HF(i)ti .
Let now A be a monomial complete intersection,
A
=
K[x1 , . . . , xn ]/(xd11 , . . . , xdnn ), for some positive integers d1 , . . . , dn .
Let
2
P
t = ni=1 (di − 1). This is the highest possible degree of a monomial in A, and
hence HFA (i) = 0 when i > t. It can also be seen that the Hilbert function is
symmetric about t/2, and that HFA (i) ≤ HFA (i + d) when i ≤ (t − d)/2. For a
multiplication map to have maximal rank in every degree in A, it shall then be
injective up to some degree i, and surjective for larger i. It can be proved that
the injectiveness in this case implies the surjectiveness.
Pn
Proposition 2.3. Let A = K[x1 , . . . , xn ]/(xd11 , . . . , xdnn ) and t = i=1 (di − 1),
and let f ∈ A be a form of degree d. The maps ·f : Ai → Ai+d all have maximal
rank if and only if the maps with i ≤ (t − d)/2 are injective.
Proof. See e. g. [9, Proposition 2.6].
In other words, multiplication by a form f has maximal rank in every degree
if all homogeneous zero divisors of f are of degree greater than (t−d)/2. Another
interesting fact is that if we consider forms of the type ℓd , and t − d is even,
then multiplicaton by ℓd+1 has maximal rank in every degree if multiplication
by ℓd does. This result will be important for the classification of algebras with
the SLP when n = 2.
Pn
Proposition 2.4. Let A = K[x1 , . . . , xn ]/(xd11 , . . . , xdnn ) and t = i=1 (di − 1).
Let ℓ ∈ A be a linear form, and d a positive integer such that t − d is even. If
the maps ·ℓd : Ai → Ai+d have maximal rank for all i ≥ 0, so does the maps
·ℓd+1 : Ai → Ai+d+1 .
Proof. Assume that ·ℓd : Ai → Ai+d have maximal rank for all i ≥ 0. By
Proposition 2.3 all zero divisors of ℓd are of degree at least (t − d)/2. Suppose
that there is a homogeneous element f such that ℓd+1 f = 0. By Proposition
2.3, we are done if we can prove that deg(f ) > (t − (d + 1))/2 = (t − d)/2 − 1/2.
Since t − d is even, the right hand side is not an integer, and it is enough to
prove deg(f ) > (t − d)/2 − 1. Consider first the case when ℓd f = 0. That is, f
is a zero divisor of ℓd , and it follows that deg(f ) > (t − d)/2. Consider instead
the case when ℓd f 6= 0. We know that ℓd+1 f = 0, that is ℓf is a homogeneous
zero divisor of ℓd . Then deg(ℓf ) > (t − d)/2, and deg(f ) > (t − d)/2 − 1, which
finishes the proof.
Proposition 2.5. The algebra A = K[x, y]/(xa , y b ) has the SLP if and only if
the maps
·(x + y)a+b−2c : Ai → Ai+a+b−2c
have maximal rank for all i ≥ 0 and 1 ≤ c < min(a, b).
Proof. The ”only if”-part follows from Theorem 2.2.
The numbers t and d in Proposition 2.4 are here t = a+b−2, and d = a+b−
2c. We see that t − d = 2c − 2 is even, so if multiplication by (x + y)a+b−2c has
maximal rank in every degree, so does multiplication by (x+y)a+b−2c+1 . If c ≤ 0
then Ai+a+b−2c = {0}, and obviously any map Ai → Ai+a+b−2c is surjective.
This is why we only need to consider c ≥ 1. Without loss of generality, we
can assume that a = min(a, b). To complete the proof we need to show that
multiplication by (x + y)a+b−2c has maximal rank in every degree when c ≥ a.
Suppose there is a non-zero homogenous f ∈ A such that (x + y)a+b−2c f = 0.
3
By Proposition 2.3 multiplication by (x + y)a+b−2c has maximal rank in every
degree if we can prove that
deg(f ) >
a + b − 2 − (a + b − 2c)
= c − 1.
2
Let F be a homogeneous element in K[x, y] whose image in A is f . Then
(x + y)a+b−2c F = gxa + hy b , for some g, h ∈ K[x, y].
We can not have h = 0, because that would imply that F is divisible by xa ,
and f = 0 in A. Hence h 6= 0 and deg((x + y)a+b−2c F ) ≥ b, which is equivalent
to deg(F ) ≥ 2c − a. If c ≥ a this implies deg(f ) = deg(F ) ≥ c, and we are
done.
3
Classifying the monomial complete intersections with the strong Lefschetz property
A classification of the monomial complete intersections with the SLP, in three or
more variables, is given in [9, Theorem 3.8]. Here we give a slightly reformulated
version of the theorem, to make the notation similar to that used later in the
case of two variables. We will prove that the formulation here is equivalent to
that in [9].
Theorem 3.1. Let A = K[x1 , . . . , xn ]/(xd11 , . . . , xdnn )
all i, and K is a field of characteristic p > 0. Let t
d1 = max(d1 , . . . , dn ). Write d1 = N1 p + r1 with 0 ≤
SLP if and only if one of the following two conditions
where
Pn n ≥ 3, di ≥ 2 for
= i=1 (di − 1) and let
r1 < p. Then A has the
hold
1. t < p,
2. d1 ≥ p, di < p for i = 2, . . . , n and
Pn
i=2 (di
− 1) ≤ min(r1 , p − r1 ).
Proof. The difference, compared to [9, Theorem 3.8], is that in [9] the bound
for r1 is 0 < r1 ≤ p, and the second condition is
d1 > p, di ≤ p for i = 2, . . . , n and
n
X
(di − 1) ≤ min(r1 , p − r1 ).
i=2
It is easy to see that both definitions of r1 gives the same value min(r1 , p − r1 ).
When d1 = p condition 2 of [9, Theorem 3.8] is not satisfied.
Neither is condition
P
2 in Theorem 3.1, because min(r1 , p − r1 ) = 0, and ni=2 (di − 1) ≥ n − 1 ≥ 2.
When di = p, for some i > 1, conditionP
2 in Theorem 3.1 is not satisfied. Neither
n
is 2 in [9, Theorem 3.8], because then i=2 (di − 1) ≥ p, and min(r1 , p − r1 ) < p
in general. This shows that both formulations agree.
The two conditions in Theorem 3.1 above can be genaralized to the case
n = 2. Next we will prove that in two variables, and characteristic p > 2, the
algebra A has the SLP in these two cases, but also in an additional one.
4
Theorem 3.2. Let A = K[x, y]/(xa , y b ), where a, b ≥ 2 and K is a field of
characteristic p > 2. Write a and b in base p, that is a = ak pk + · · · + a1 p + a0
and b = bℓ pℓ + · · · + b1 p + b0 , where 0 ≤ ai , bi < p, and ak , bℓ 6= 0. We may
assume that ℓ ≥ k. The classification of the algebras with the SLP is divided
into three cases.
1. When a, b < p, A has the SLP if and only if a + b ≤ p + 1.
2. When a < p and b ≥ p, A has the SLP if and only if a ≤ min(b0 , p−b0 )+1.
3. When a, b ≥ p, A has the SLP if and only if the following three conditions
are satisfied.
(a) a0 =
p±1
2 ,
(b) ai = bi =
and b0 =
p−1
2
p±1
2 ,
for i = 1, 2, . . . , k − 1,
(c) ak + bk ≤ p − 1, and bk ≥ ak when ℓ > k.
Notice that there are no restrictions on bi for i > k, in the case ℓ > k. The
theorem will be proved later in this section.
In [5, Theorem 4.9] Cook II proves the special case a = b of Theorem 3.2.
Cook II also proves the characteristic two case.
Theorem 3.3 ([5, Corollary 4.8]). Let A = K[x, y]/(xa , y b ), where 2 ≤ a ≤ b
and K is a field of characteristic two. A has the SLP if and only if one of the
two following conditions hold.
1. a = 2 and b is odd,
2. a = 3 and b ≡ 2 mod 4.
Theorem 3.1, Theorem 3.2, and Theorem 3.3 can now be combined into a
complete classification of the monomial complete intersections with the SLP.
Theorem 3.4. Let A = K[x1 , . . . , xn ]/(xd11 , . . . , xdnn ), where all di ≥ 2 and K
is a field of characteristic p > 0. Write each di in base p as di = ciki pki + · · · +
ci1 p + ci0 , with ciki 6= 0. The algebra A has the SLP if and only if one of the
following conditions hold.
1. n = 1,
2. n = 2, p = 2, and one of the following holds, for d1 ≤ d2
• d1 = 2 and c20 = 1,
• d1 = 3, c21 = 1, and c20 = 0,
3. n = 2, p > 2 and all the following conditions are satisfied, for k1 ≤ k2
• c10 =
• c1j =
p±1
p±1
2 , c20 = 2 ,
c2j = p−1
2 , for j
= 1, . . . , k1 − 1,
• c1k1 + c2k1 < p, and c2k1 ≥ c1k1 if k1 < k2 ,
Pn
4. n ≥ 2, and i=1 (di − 1) < p,
5
5. n
P ≥ 2, and there is a j such that dj ≥ p, di < p for all i 6= j, and
i6=j (di − 1) ≤ min(cj0 , p − cj0 ).
Proof. The case n = 1 is trivial. Condition 3 is condition 3 of Theorem 3.2, and
Condition 4 is Theorem 3.3 with b = d2 written in base 2. The conditions 4
and 5 are the conditions 1 and 2 from Theorem 3.1 and Theorem 3.2 combined.
Notice that 4 and 5 are not satisfied when p = 2.
Both proofs of [5, Corollary 4.8] and [5, Theorem 4.9] uses Theorem 3.5
below. This will also be the key to the proof of Theorem 3.2.
Theorem 3.5. Let K be a field of characteristic p > 0.
K[x, y]/(xa , y b ) has the SLP if and only if
The algebra
|a − upi | + |b − vpi | + |a + b − 2c − wpi | ≥ pi
for all integers i ≥ 0, 1 ≤ c < min(d1 , d2 ), and u, v, w such that u + v + w is
odd.
Theorem 3.5 is proved in Section 4.
We will now prove that Theorem 3.5 can be reformulated as the following
proposition.
Proposition 3.6. Let A = K[x, y]/(xa , y b ), where K is a field of characteristic
p > 0. For each integer i ≥ 1 we can write a = mi pi + ri , and b = ni pi + si ,
where 0 ≤ ri , si < pi . The algebra A has the SLP if and only if the following
conditions hold for all i.
1. If mi > 0, then ri ≥ si − 1,
2. If ni > 0, then si ≥ ri − 1,
3. If mi > 0 and ni > 0, then ri + si ≥ pi − 1,
4. ri + si ≤ pi + 1.
Proof. We shall prove that the conditions above is equivalent to that in Theorem
3.5. Let us investigate for which a and b it can happen that
|a − upi | + |b − vpi | + |a + b − 2c − wpi | < pi .
Write a = mi pi + ri and b = ni pi + si , as in the proposition. Notice that
ri
when u = mi
i
|a − up | =
pi − ri when u = mi + 1.
For all other values of u we get |a − upi | ≥ pi , and then of course |a − upi | +
|b − vpi | + |a + b − 2c − wpi | ≥ pi . Therefore we only need to consider u = mi
and u = mi + 1. The corresponding is also true for |b − vpi |. This gives us four
cases to examine.
I. u = mi and v = ni
Here
|a − upi | + |b − vpi | = ri + si .
6
To obtain |a − upi | + |b − vpi | + |a + b − 2c − wpi | ≤ pi − 1 it is necessary
that ri + si ≤ pi − 1.
Suppose first that ri + si = pi − 1. Since u + v + w = mi + ni + w is
supposed to be odd, we must have w = mi + ni − 2d + 1, for some integer
d. Then
a + b − 2c − wpi = ni pi + ri + mi pi + si − 2c − (mi + ni − 2d + 1)pi
= ri + si − 2c + (2d − 1)pi = 2dpi − 2c − 1,
which is an odd number, and thus |a + b − 2c − wpi | ≥ 1. We get
|a − upi | + |b − vpi | + |a + b − 2c − wpi | ≥ pi − 1 + 1 = pi ,
and we can conclude that |a − upi | + |b − vpi | + |a + b − 2c − wpi | ≥ pi for
all w and c, when ri + si = pi − 1.
Now suppose that ri + si ≤ pi − 2. We want to find out what the smallest
possible value of |a+b−2c−wpi | is. For this purpose we choose the largest
w such that u + v + w is odd, and a + b − wpi > 0. After that we choose
the value for c that makes |a + b − 2c − wpi | as small as possible. Since
ri +si ≤ pi −2, the largest w with the required properties is w = mi +ni −1.
Then
a + b − wpi = pi + ri + si .
If mi = 0, then min(a, b) = min(ri , b) ≤ ri and c ≤ ri − 1. Then
a + b − 2c − wpi ≥ pi + ri + si − 2(ri − 1) = pi − ri + si + 2, and
|a − upi | + |b − vpi | + |a + b − 2c − wpi | ≥ ri + si + pi − ri + si + 2 > pi .
In a similar way we see that |a − upi | + |b − vpi | + |a + b − 2c − wpi | > pi
if ni = 0. Suppose now that mi > 0 och ni > 0. Then we choose
c = [(pi + ri + si )/2], where [...] denotes the integer part. This gives
a + b − 2c − wpi = 0 or 1, and
|a − upi | + |b − vpi | + |a + b − 2c − wpi | ≤ ri + si + 1 ≤ pi − 1.
The conclusion, in this case, is that |a−upi |+|b−vpi |+|a+b−2c−wpi | < pi ,
exactly when mi , ni > 0 and ri +si ≤ pi −2. This corresponds to condition
3 in the proposition.
II. u = mi and v = ni + 1
Here
|a − upi | + |b − vpi | = ri + pi − si .
To obtain |a − upi | + |b − vpi | + |a + b − 2c − wpi | ≤ pi − 1 it is necessary
that ri + pi − si ≤ pi − 1, that is ri ≤ si − 1. Let us first consider the case
when ri = si − 1. Since u + v + w is supposed to be odd we must have
w = ni + mi − 2d, for some integer d. This gives
a + b − wpi = ri + si + 2dpi = 2ri + 1 + 2dpi ,
which is odd. Then |a + b − 2c − wpi | ≥ 1, and
|a − upi | + |b − vpi | + |a + b − 2c − wpi | ≥ ri + pi − si + 1 = pi .
7
Suppose instead that ri ≤ si − 2. We use that same idea as in case 1, and
choose first w, and then c, such that |a + b − 2c − wpi | has the smallest
possible value. The best option for w is w = ni + mi . This gives
a + b − wpi = ri + si .
If mi = 0, then min(a, b) = min(ri , b) = ri , thus c = ri − 1 is the largest
allowed value of c. Then
a + b − 2c − wpi = ri + si − 2(ri − 1) = si − ri + 2, and
|a − upi | + |b − vpi | + |a + b − 2c − wpi | = ri + pi − si + si − ri + 2 = pi + 2.
If mi > 0 on the other hand, we are allowed tho choose c = si − 1. Then
we get
a + b − 2c − wpi = ri − si + 2
instead. Note that this is a non-positive number. This gives
|a − upi | + |b − vpi | + |a + b − 2c − wpi | = ri + pi − si + si − ri − 2 = pi − 2.
The conclusion, in this case, is that |a−upi |+|b−vpi |+|a+b−2c−wpi | < pi ,
exactly when mi > 0 and ri ≤ si − 2. This corresponds to condition 1 in
the proposition.
III. u = mi + 1 and v = ni
In the same way as above, we see that this corresponds to condition 2.
IV. u = mi + 1 och v = ni + 1
Here
|a − upi | + |b − vpi | = 2pi − ri − si ,
so for this to be smaller than pi we must have 2pi − ri − si ≤ pi − 1, which
is ri + si ≥ pi + 1. Consider first the case when ri + si = pi + 1. Then we
must choose w = mi + ni − 2d + 1, for some integer d. Then
a + b − wpi = ri + si + (2d − 1)pi = 2dpi + 1,
and |a + b − 2c − wpi | ≥ 1. Then we get
|a − upi | + |b − vpi | + |a + b − 2c − wpi | ≥ 2pi − ri − si + 1 = pi .
Suppose now that ri + si ≥ pi + 2. We choose w = mi + ni + 1 and
c = [(ri + si − pi )/2], because this gives
a + b − 2c − wpi = ri + si − pi − 2c = 0 or 1, and
|a − upi | + |b − vpi | + |a + b − 2c − wpi | ≤ 2pi − ri − si + 1 ≤ pi − 1.
This shows that |a − upi | + |b − vpi | + |a + b − 2c − wpi | < pi when
ri + si ≥ pi + 2, which is condition 4.
8
Proposition 3.6 will be used later in this section to prove Proposition 3.7,
which says something about the structure of an algebra that does not have the
SLP. Now we shall use Proposition 3.6, with p > 2, to prove Theorem 3.2.
Proof of Theorem 3.2. Let A = K[x, y]/(xa , y b ), and suppose throughout this
proof that the characteristic of K is greater than 2. Write a and b in base p as
a = ak pk + · · · + a1 p + a0 and b = bℓ pℓ + · · · + b1 p + b0 , where 0 ≤ ai , bi < p. We
assume that ℓ ≤ k. With the notation a = mi pi + ri from Proposition 3.6 we
have ri = ai−1 pi−1 + · · · + a1 p + a0 , and mi = ak pk−i + ak−1 pk−i−1 + · · · + ai ,
and similar for b.
If a, b < p then ni = mi = 0 in Proposition 3.6, for all i, and the conditions
1, 2 and 3 are trivially satisfied. Since a + b < 2p condition 4 is satisfied for
i > 1. The only restriction we get comes from condition 4 when i = 1, and
states that A has the SLP if and only if a + b ≤ p + 1.
If a < p and b ≥ p we get b0 ≥ a0 − 1 and a0 + b0 ≤ p + 1 from the
conditions 2 and 4 with i = 1. These two inequalities can be written as a0 ≤
min(b0 , p − b0 ) + 1. In condition 1 and 3 there is nothing to check, and for
i > 1 all conditions are satisfied. We get that A has the SLP if and only if
a0 ≤ min(b0 , p − b0 ) + 1.
Assume now that a, b ≥ p. The idea now is to translate the four conditions
of Proposition 3.6 into the base p digits of a and b.
Let us first look at i = 1 in Proposition 3.6. We know that m1 , n1 > 0, so 1
and 2 gives a0 − 1 ≤ b0 ≤ a0 + 1. The conditions 3 and 4 gives p − 1 ≤ a0 + b0 ≤
p±1
p+ 1. Both these intequalites are satisfied exactly when a0 = p±1
2 and b0 = 2 .
This is condition (a) in Theorem 3.2. Suppose that this is the case, and move
on to i = 2. If k ≥ 2 then m2 and n2 are positive. The conditions 1 and 2 gives
a1 p + a0 − 1 ≤ b1 p + b0 ≤ a1 p + a0 + 1,
which implies a1 = b1 . For 3 and 4 to be satisfied
p2 − 1 ≤ (a1 + b1 )p + (a0 + b0 ) ≤ p2 + 1
is required. This is true if and only if a1 +b1 = p−1. Hence we get a1 = b1 = p−1
2 .
We suppose that this is true and continue with i = 3, . . . k. In the same way as
above we get a2 = · · · = ak−1 = b2 = · · · = bk−1 = p−1
2 . This is condition (b)
in Theorem 3.2.
Suppose that the conditions for i = 1, 2 . . . , k are satisfied, and move on to
i = k + 1. Now mk+1 = 0, so in condition 1 and 3 there is nothing to check. If
ℓ > k then nk+1 > 0. In this case condition 2 says
bk pk + · · · + b1 p + b0 ≥ ak pk + · · · + a1 p + a0 − 1,
which holds if and only if bk ≥ ak . Condition 4 says
(ak + bk )pk + · · · + (a1 + b1 )p + (a0 + b0 ) ≤ pk+1 + 1,
which holds if and only if ak + bk ≤ p − 1. This proves (c).
We must also show that there are no futher restrictions on bj for j > k, when
such bj exist. Suppose that the four conditions of Proposition 3.6 are satisfied
for i = 1, 2, . . . , k + 1. We continue by looking at i = k + 2. The conditions
1 and 3 are satisfied, since mi = 0. Notice also that rk+2 = rk+1 = a, and
9
sk+2 ≥ sk+1 . This means that if condition 2 is satisfied for i = k + 1, so it is for
i = k + 2. Condition 4 requires
bk+1 pk+1 + (ak + bk )pk + · · · + (a1 + b1 )p + (a0 + b0 ) ≤ pk+2 + 1.
But this is no restriction on bk+1 , other than bk+1 < p. The same reasoning
works for larger i.
The proof in [9] of when an algebra in three or more variables does not have
the SLP, is carried out by finding a monomial zero-divisor of (x1 + · · · + xn )m ,
for some m. We will now see that this can also be done in two variables. This
gives an alternative proof of the ”only if”-part of Theorem 3.2.
L
Proposition 3.7. Let A = K[x1 , . . . , xn ]/(xd11 , . . . , xdnn ) = i≥0 Ai be an algebra of characteristic p > 0 which does not possess the SLP. Let ℓ be a linear
form in A. Then there are integers d and m such that HFA (d) ≤ HFA (d + m),
and the kernel of the multiplication map ·ℓm : Ad → Ad+m contains a non-zero
monomial.
Proof. For the case n ≥ 3, see [9].
Assume n = 2, and let ℓ = c1 x1 + c2 x2 for some c1 , c2 ∈ K. Recall that
HFA (d) ≤ HFA (d + m) when d ≤ (d1 + d2 − 2 − m)/2. We shall prove that when
one of the conditions in Proposition 3.6 fails, we can find a monomial of degree
low enough, which is a zero divisor of some power of ℓ. Write d1 = mi pi + ri
and d2 = ni pi + si , for some i, as in Proposition 3.6, and suppose that condition
1 fails for this i. This means that mi > 0 and ri ≤ si − 2. Then ri < d1 , and
therefore xr1i 6= 0. Recall that
i
i
i
i
i
i
ℓp = (c1 x1 + c2 x2 )p = cp1 xp1 + cp2 xp2 ,
since we are in a ring of characteristic p. Also,
i
i
i
i
i
i
(cp1 xp1 + cp2 xp2 )mi +ni = ex1mi p xn2 i p
in A, for some e ∈ K
β
since all the other terms in the expansion will be of the form cxα
1 x2 where either
α ≥ d1 or β ≥ d2 . We have
i
i
i
i
i
i
i
i
ip
xn2 i p xr1i = ex1mi p
ℓ(mi +ni )p xr1i = (c1p xp1 +cp2 xp2 )mi +ni xr1i = exm
1
+ri ni pi
x2
= 0.
In other words, xr1i is a monomial in the kernel of the multiplication map
i
·ℓ(mi +ni )p : Ari → Ari +(mi +ni )pi , and since
ri ≤ si − 2
⇐⇒
ri ≤
ri + si − 2
2
⇐⇒
ri ≤
d1 + d2 − 2 − (mi + ni )pi
2
we have HFA (ri ) ≤ HFA (ri + (mi + ni )pi ).
If instead conditions 2 of Proposition 3.6 fails, the proof is carried out in the
same way, but with xr1i replaced by xs2i . Suppose now that condition 3 fails for
some i. That is mi , ni > 0, and ri + si ≤ pi − 2. Then xr1i xs2i 6= 0. We have
i
i
i
i
(mi −1)pi ni pi
x2
i
ℓ(mi +ni −1)p = (cp1 xp1 + cp2 xp2 )mi +ni −1 = e1 x1
10
i
(ni −1)pi
ip
x2
+ e2 xm
2
i
for some e1 , e2 ∈ K, and we see that ℓ(mi +ni −1)p xr1i xs2i = 0. Also,
ri +si ≤ pi −2 ⇐⇒ ri +si ≤
ri + si − 2 + pi
d1 + d2 − 2 − (mi + ni − 1)pi
=
,
2
2
which implies that HFA (ri + si ) ≤ HFA (ri + si + (mi + ni − 1)pi ).
At last, suppose that condition 4 of Proposition 3.6 fails. Then ri +si ≥ pi +2.
This implies that d1 + d2 − 2 = mi pi + ri + ni pi + si ≥ (mi + ni + 1)pi , and
HF((mi + ni + 1)pi ) ≥ 1. But
i
i
i
i
i
ℓ(mi +ni +1)p = (cp1 xp1 + cp2 xp2 )mi +ni +1 = 0,
β
since all terms in the expansion will be of the form cxα
1 x2 where either α ≥
d1 or β ≥ d2 . This shows that 1 is in the kernel of the multiplication map
i
·ℓ(mi +ni +1)p : A0 → A(mi +ni +1)pi . Since HF((mi + ni + 1)pi ) ≥ 1 = HF(0),
this completes the proof.
4
The Syzygy gap
The main purpose of this section is to prove Theorem 3.5. If we require the
residue field to be algebraically closed, the theorem follows from combining a
theorem by Han [6] and results by Brenner and Kaid in [1] and [2]. Han’s result is
also proved in a different way by Monsky in [10]. Monsky deals with the syzygy
module of three pairwise relatively prime polynomials in two variables, and the
so called ”syzygy gap”, while Brenner and Kaid connects this to the Lefschetz
properties. We will go through the results from [10], and give a new proof of
the connection to the SLP in the case of monomial complete intersections. The
reason to go though the results of [10] is to prove that the residue field does
not need to be algebraically closed, but also to give a deeper understanding of
Theorem 3.5 and the theory behind it.
4.1
Mason-Stothers’ Theorem
First we need a review of Mason-Stothers’ Theorem. Suppose f is a polynomial
in K[x1Q
, . . . , xn ], where K is some field. The polynomial f can be factorized
s
as f Q
= i=1 pei i , where the pi ’s are distinct irreducible factors. Define r(f ) =
s
deg( i=1 pi ). Note that r(f g) ≤ r(f ) + r(g), with equality when f and g are
relatively prime. Let fx′ j denote the formal derivative of f w. r. t. the variable
xj . When in a polynomial ring with just one variable, we write f ′ for the
derivative. Mason-Stothers’ theorem is usually formulated over one variable, as
follows.
Theorem 4.1 (Mason-Stothers). Let K be a field, and let f, g and h be polynomials in K[x] such that
• f, g and h are pairwise relatively prime,
• f ′ , g ′ and h′ are not all zero,
• f + g + h = 0.
Then max(deg(f ), deg(g), deg(h)) ≤ r(f gh) − 1.
11
An elementary proof can be found in [11]. There is also a version of this
theorem for homogeneous polynomials in two variables. For clairity we will
prove how it can be deduced from Theorem 4.1.
Theorem 4.2. Let K be a field, and let f, g and h be homogeneous polynomials
of degree d in K[x, y] such that
• f, g and h are pairwise relatively prime,
• fx′ , fy′ , gx′ , gy′ , h′x and h′y are not all zero,
• f + g + h = 0.
Then d ≤ r(f gh) − 2.
Proof. Let K ′ be the splitting field of f . Over this field f can be factorized as
follows
f (x, y) =
d
X
αi xi y d−i = y d
i=0
d
X
i=0
αi
x i
y
= yd
d
d
Y
Y
x
(ui x − vj y),
uj − vj =
y
j=1
j=1
where the αi , uj and vj ’s are elements in K ′ . After a possible linear change of
Q
variables, we can assume that f (x, y) = y m d−m
j=1 (rj x − sj y), where m ≥ 1.
Qd−m
ˆ
Let f (x) = f (x, 1) = j=1 (rj x − sj ), ĝ(x) = g(x, 1) and ĥ(x) = h(x, 1). Then
r(fˆ) = r(f ) − 1, while r(g) = r(ĝ) and r(h) = r(ĥ). Note also that deg(ĝ) = d.
By Theorem 4.1 it now follows that
d = deg(ĝ) ≤ r(fˆĝ ĥ)−1 = r(fˆ)+r(ĝ)+r(ĥ)−1 = r(f )+r(g)+r(h)−2 = r(f gh)−2,
which we wanted to prove.
4.2
The syzygy gap
Let now R = K[x, y], where K is any field. Let f1 , f2 and f3 be non-zero
homogeneous, pairwise relatively prime, polynomials in R, with di = deg(fi ),
and let I = (f1 , f2 , f3 ). The R-module R/I has a free resolution of length 2,
by Hilbert’s syzygy theorem. If {f1 , f2 , f3 } is a minimal set of generators of I,
then
φ
0 → ker φ → R3 → R → R/I → 0,
(1)
where φ is given by the matrix f1 f2 f3 , is an exact sequence of free modules. We have rank ker φ = 3 − 1 = 2. That is, ker φ = Syz(f1 , f2 , f3 ) is
generated by two homogeneous elements. If {f1 , f2 , f3 } is not a minimal set
of generators of I, we have e. g. f3 = g1 f1 + g2 f2 , for some homogeneous
polynomials g1 and g2 . Then every relation Af1 + Bf2 + Cf3 = 0 can be written as (A + Cg1 )f1 + (B + Cg2 )f2 = 0. Since f1 and f2 are relatively prime
A + Cg1 = hf2 , and B + Cg2 = −hf1 , for some homogeneous h. It follows that
ker φ is generated by (f2 , −f1 , 0) and (g1 , g2 , −1). This shows that (1) is always
a free resolution (but not necessarily minimal), and ker φ is generated by two
homogeneous elements of degrees, say α and β. We have a graded resolution
0 → R(−α) ⊕ R(−β) → R(−d1 ) ⊕ R(−d2 ) ⊕ R(−d3 ) → R → R/I → 0,
12
of R/I. Define ∆(f1 , f2 , f3 ) = |α−β|. This is the syzygy gap function introduced
in [10]. From the graded resolution we see that the Hilbert series of R/I is
HSR/I (t) =
1 − td 1 − td 2 − td 3 + tα + tβ
.
(1 − t)2
We also know that R/I has dimension 0, thus the Hilbert series is a polynomial,
say HSR/I (t) = p(t). Then
(1 − t)2 p(t) = 1 − td1 − td2 − td3 + tα + tβ .
By taking the derivative of both sides, and substituting t = 1 we get 0 =
−d1 − d2 − d3 + α + β, that is α + β = d1 + d2 + d3 . This is one of the so called
Herzog-Kühl equations, see e. g. [4]. From this follows also the below lemma.
Lemma 4.3. Let f1 , f2 and f3 be non-zero, pairwise relatively prime homogeneous polynomials in K[x, y], with di = deg(fi ). Then ∆(f1 , f2 , f3 ) ≡ d1 +d2 +d3
mod 2.
We shall also see some other properties of the function ∆.
Lemma 4.4. Let K be a field of characteristic p > 0, and let f1 , f2 and f3 be
non-zero, pairwise relatively prime homogeneous polynomials in K[x, y]. Then
s
s
s
∆(f1p , f2p , f3p ) = ps ∆(f1 , f2 , f3 ),
for all non-negative integers s.
We will give two proofs of this lemma. The first proof uses the Frobenius
functor. The second proof is new, and uses elementary methods.
Proof 1. Let R = K[x, y], and I = (f1 , f2 , f3 ). For a fixed s, let q = ps . Let ϕ
denote the s:th power of the Frobenius endomorphism on R, that is ϕ(a) = aq .
Consider now R as an R-bimodule, denoted Rϕ , where left multiplication by an
element r ∈ R is usual multiplication, while right multiplication is multiplication
by ϕ(r). The Frobenius functor F is a functor on the category of left R-modules
defined by F (M ) = Rϕ ⊗R M . For a more extensive review of the Frobenius
functor, see e. g. [3]. Two well knows properties of this functor is that F (Rm ) ∼
=
Rm and F (R/I) ∼
= R/I (q) where I (q) is the ideal generated by the q:th powers of
the elements in I. One can also prove that if ψ : Rm → Rn is a homomorphism
of free modules represented by a matrix (aij ), then the induced map F (ψ) is
represented by the matrix (aqij ). It follows from [7, Corollary 2.7] that F is an
exact functor. Now, suppose Syz(f1 , f2 , f3 ) is generated by (A1 , A2 , A3 ) and
(B1 , B2 , B3 ), of degrees α and β. When we apply F to the resolution
A1
A
2
B1
B2
f1 f2 f3
A3 B3
0 → R2 −−−−−−−−−−→ R3 −−−−−−−−−−−−−→ R → R/I → 0
we get an exact sequence
Aq1
q
A
2
Aq3
B1q
B2q
q
f1 f2q f3q
B3q
0 → R2 −−−−−−−−−−→ R3 −−−−−−−−−−−−−→ R → R/I (p) → 0.
13
This proves that Syz(f1q , f2q , f3q ) is generated by (Aq1 , Aq2 , Aq3 ) and (B1q , B2q , B3q ).
Then
∆(f1q , f2q , f3q ) = |aq − bq| = q|a − b| = q∆(f1 , f2 , f3 ),
which we wanted to prove.
Proof 2. Suppose Syz(f1 , f2 , f3 ) is generated by (A1 , A2 , A3 ) and (B1 , B2 , B3 ),
of degrees α and β. Notice first that
A1 f1 + A2 f2 + A3 f3 = 0
s
s
s
s
s
s
Ap1 f1p + Ap2 f2p + Ap3 f3p = 0,
⇐⇒
s
s
s
and the corresponding for (B1 , B2 , B3 ), since (A1 f1 +A2 f2 +A3 f3 )p = Ap1 f1p +
s
s
s
s
s
s
s
Ap2 f2p + Ap3 f3p . However, this does not prove that Syz(f1p , f2p , f3p ) is geners
s
s
s
s
s
ated by (Ap1 , Ap2 , Ap3 ) and (B1p , B2p , B3p ).
s
s
s
Claim: The two syzygies generating Syz(f1p , f2p , f3p ) consist of polynomials
s
s
in xp and y p .
These two generators give two syzygies in Syz(f1 , f2 , f3 ) by the change of
s
s
variables
xps 7→s x and y p 7→ y. It follows that the two relations that generate
s
Syz(f1p , f2p , f3p ) has degrees αps and βps . Then
s
s
s
∆(f1p , f2p , f3p ) = |aps − bps | = ps |a − b| = ps ∆(f1 , f2 , f3 ),
which we wanted to prove.
It remains to prove the claim. We will first prove that the two syzygies
consist of polynomials in xp and y p . Assume that
s
s
s
s
s
s
C1 f1p + C2 f2p + C3 f3p = 0 and D1 f1p + D2 f2p + D3 f3p = 0
s
s
(2)
s
are the two relations that generate Syz(f1p , f2p , f3p ). We may assume that
(C1 , C2 , C3 ) is the one of lowest degree. Let f ′ denote the derivative of f w. r.
t. x. Notice that
s
s
s
(Cf p )′ = C ′ f p + ps Cf p
and hence
s
s
s
f = C ′f p ,
−1 ′
s
C1′ f1p + C2′ f2p + C3′ f3p = 0.
But there can not be a relation of lower degree than (C1 , C2 , C3 ), thus Ci′ = 0
for i = 1, 2, 3. The same holds when we take the derivative w. r. t. y. This
means that C1 , C2 , C3 are polynomials in xp and y p .
If we derivate the other relation w. r. t. x we get
s
s
s
D1′ f1p + D2′ f2p + D3′ f3p = 0.
Here we can not conclude that all Di′ = 0, because there are elements of lower
degree, namely those generated by (C1 , C2 , C3 ). Suppose there is a g such that
b i , where G
Di′ = gCi , for i = 1, 2, 3. Recall that Ci′ = 0. This gives Di = GCi + D
′
′
b
is some homogeneous polynomial such that G = g, and Di = 0. We can assume
b 1, D
b 2, D
b 3 ) is not a multiple of (C1 , C2 , C3 ), because that would imply
that (D
that (D1 , D2 , D3 ) is also a multiple of (C1 , C2 , C3 ). Therefore we can replace
b 1, D
b 2, D
b 3 ). Since D
b ′ = 0, D
b i is a homogeneous polynomial
(D1 , D2 , D3 ) by (D
i
p
b
in x and y. Suppose Di has degree mi p + ri , where 0 ≤ ri < p. Then the
14
b i is divisible by
b i looks like cxαi p y (mi −αi )p+ri , which means that D
terms in D
s
b i f p ) = mi p + di ps + ri . Since
y ri . Also deg(D
i
s
s
s
b 3f p )
b 2 f p ) = deg(D
b 1 f p ) = deg(D
deg(D
3
2
1
b1, D
b 2 and D
b 3 by y r1 and get a syzygy of
r1 = r2 = r3 . If r1 > 0 we can divide D
lower degree, which is not a multiple of (C1 , C2 , C3 ). But this is a contradiction,
b 1, D
b 2 and D
b 3 also are polynomials in
so r1 = r2 = r3 = 0. This means that D
xp and y p . A change of variables xp 7→ x and y p 7→ y in (2) gives two elements
s−1
s−1
s−1
in Syz(f1p , f2p , f3p ). The claim now follows by induction.
Let us now investigate what happens with ∆(f1 , f2 , f3 ) when, for example,
f1 is replaced by ℓf1 , for some linear form ℓ. By Lemma 4.3, ∆(f1 , f2 , f3 ) and
∆(ℓf1 , f2 , f3 ) has different parity, so they can not be equal. If we have a relation
A1 f1 + A2 f2 + A3 f3 = 0, we also get a relation on ℓf1 , f2 , f3 by multiplying the
expression by ℓ. This means that the two elements that generates Syz(ℓf1 , f2 , f3 )
can have degrees at most α + 1 and β + 1. On the other hand, a relation A1 ℓf1 +
A2 f2 + A3 f3 = 0 on ℓf1 , f2 , f3 can also be considered a syzygy (A1 ℓ, A2 , A3 )
on f1 , f2 , f3 . Hence, the two generators of Syz(ℓf1 , f2 , f3 ) have degrees at least
α and β. This shows that ∆ must either increase of decrease by 1 when f1 is
replaced by ℓf1 . We summarize this in a lemma.
Lemma 4.5. Let f1 , f2 and f3 be non-zero, pairwise relatively prime homogeneous polynomials in K[x, y]. Let ℓ be a linear form, relatively prime to f2 and
f3 . Then
∆(ℓf1 , f2 , f3 ) = ∆(f1 , f2 , f3 ) ± 1.
We shall look more carefully into two special cases where Lemma 4.5 applies.
Let (A1 , A2 , A3 ) be the element in Syz(f1 , f2 , f3 ) of the lowest degree α. If ℓ|A1
then (ℓ−1 A1 , A2 , A3 ) is a syzygy of ℓf1 , f2 , f3 of degree α. The other generating
syzygy can have degree β or β + 1, as we saw above. But since ∆(ℓf1 , f2 , f3 ) 6=
∆(f1 , f2 , f3 ) it must have degree β + 1. Hence, ∆(ℓf1 , f2 , f3 ) = ∆(f1 , f2 , f3 ) + 1
in this case.
It follows also from Lemma 4.5 that ∆(ℓ−1 f1 , f2 , f3 ) = ∆(f1 , f2 , f3 ) ± 1, if
ℓ|f1 . If, in addition, ℓ|A2 , it follows from the equality A1 f1 + A2 f2 + A3 f3 = 0
that ℓ also divides A3 . Then we can divide the whole expression by ℓ, and get
a syzygy (A1 , ℓ−1 A2 , ℓ−1 A3 ) on ℓ−1 f1 , f2 , f3 , of degree α − 1. We see that we
must have ∆(ℓ−1 f1 , f2 , f3 ) = ∆(f1 , f2 , f3 ) + 1, in this case.
This, together with Theorem 4.2, can now be used to prove the following
proposition.
Proposition 4.6 ([10, Theorem 8]). Let K be a field of characteristic p > 0. Let
f1 , f2 , and f3 be homogeneous relatively prime polynomials in K[x, y]. Assume
there is a linear form ℓ such that f1 = ℓm h, where l ∤ h and p ∤ m. Assume
also that ∆(f1 , f2 , f3 ) decreases when f1 is replaced by ℓf1 or ℓ−1 f1 . Then
∆(f1 , f2 , f3 ) ≤ r(f1 f2 f3 ) − 2.
Proof. Let (A1 , A2 , A3 ) be one of the two generators of Syz(f1 , f2 , f3 ) of minimal
degree α. We saw above that if ℓ|A1 then ∆(ℓf1 , f2 , f3 ) = ∆(f1 , f2 , f3 ) + 1. We
also saw that if ℓ|A2 then ∆(ℓ−1 f1 , f2 , f3 ) = ∆(f1 , f2 , f3 ) + 1. The same holds
15
if ℓ|A3 . By assumption, none of this is the case, and hence A1 , A2 and A3 are
not divisible ℓ. Let M = gcd(A1 f1 , A2 f2 , A3 f3 ). Then
A2 f2
A3 f3
A1 f1
+
+
=0
M
M
M
and the three terms Ai fi /M are relatively prime. Notice that every irreducible
factor of M must divide one of f1 , f2 or f3 . Also ℓ does not divide M , since ℓ
does not divide A2 , A3 , f2 or f3 . We shall now see that the formal derivative of
A1 f1 /M w. r. t. x or y is non-zero, so that we can use Theorem 4.2. One of ℓ′x
and ℓ′y must be non-zero, otherwise ℓ = 0. Say that ℓ′x = c 6= 0. Then
A f ′
A h ′
A h ′
A1 h
1 1
1
1
+ ℓm
= ℓm
= mcℓm−1
.
M x
M x
M
M x
The two terms can not cancel each other, and the first one is non-zero, since
m 6= 0 in K. Hence (A1 f1 /M )′x 6= 0. By Theorem 4.2
deg
A f A f A f
A f
1 1 2 2 3 3
1 1
≤r
− 2.
M
M3
(3)
We know that deg(A1 f1 /M ) = α − deg(M ). Let di = deg(fi ), for i = 1, 2, 3,
and recall that d1 + d2 + d3 = α + β, where β is the degree of the other generator
of Syz(f1 , f2 , f3 ). We have
A A A
A f A f A f
f f f
1 2 3
1 1 2 2 3 3
1 2 3
+
deg
r
≤
r
M3
M
M2
≤ r(f1 f2 f3 ) + deg(A1 ) + deg(A2 ) + deg(A3 ) − 2 deg(M )
= r(f1 f2 f3 ) + (α − d1 ) + (α − d2 ) + (α − d3 ) − 2 deg(M )
= r(f1 f2 f3 ) + 3α − (α + β) − 2 deg(M )
= r(f1 f2 f3 ) + 2α − β − 2 deg(M ).
Inserted in (3), this gives
α − deg(M ) ≤ r(f1 f2 f3 ) + 2α − β − 2 deg(M ) − 2,
which is rewritten as
β − α ≤ r(f1 f2 f3 ) − deg(M ) − 2.
We can now conclude that ∆(f1 , f2 , f3 ) = β − α ≤ r(f1 f2 f3 ) − 2.
4.3
Application of the syzygy gap function to monomial
complete intersection algebras
We will now specialize to the case f1 = xd1 , f2 = y d2 , and f3 = (x + y)d3 . This
is allowed, since these polynomials are pairwise relatively prime. For an easier
notation we introduce a new function δ : Z3+ → Z≥0 defined by δ(d1 , d2 , d3 ) =
∆(xd1 , y d2 , (x+y)d3 ). We will now see how the theory of the syzygy gap connects
to the SLP.
Proposition 4.7. Let S = K[x, y]/(xd1 , y d2 ). The maps ·(x+y)d3 : Si → Si+d3 ,
with d3 < d1 + d2 , have maximal rank for all i if and only if δ(d1 , d2 , d3 ) ≤ 1.
16
This result can be proved for general f1 , f2 and f3 using [1, Theorem 2.2]
and [2, Corollary 3.2]. Below follows an easier proof for this special case.
Proof. We know that the syzygy module Syz(xd1 , y d2 , (x + y)d3 ) is generated
by two homogenous elements (A1 , A2 , A3 ) and (B1 , B2 , B3 ) of degrees α and β.
We may assume that α ≤ β. Provided that A3 6= 0, this can be formulated as
(x + y)d3 A3 = 0 in S, and A3 is a homogenous element of lowest degree with
this property. The degree of A3 is α − d3 . By Proposition 2.3 multiplication by
(x + y)d3 has maximal rank in every degree if and only if
α − d3 >
d1 + d2 − 2 − d3
d1 + d2 + d3 − 2
or equivalently α >
.
2
2
Recall that α+β = d1 +d2 +d3 . This inserted in the above inequality gives, after
simplification, α > β −2. Since α ≤ β this is exactly the property δ(d1 , d2 , d3 ) =
β − α ≤ 1.
It remains to prove that A3 6= 0. If A3 = 0 we would have a relation
A1 f1 + A2 f2 = 0. Since f1 and f2 are relatively prime, this gives A1 = cf2 and
A2 = −cf1 , for some c ∈ K. Then α = d1 + d2 , and since α + β = d1 + d2 + d3 ,
we get β = d3 . But β ≥ α and d3 < d1 + d2 yields a contradiction.
This result combined with Proposition 2.5 now gives the following.
Theorem 4.8. The algebra K[x, y]/(xd1 , y d2 ) has the SLP if and only if
δ(d1 , d2 , d1 + d2 − 2c) = 0 for all 1 ≤ c < min(d1 , d2 ).
Proof. It follows directly from Proposition 4.7 and Proposition 2.5 that
K[x, y]/(xd1 , y d2 ) has the SLP if and only if δ(d1 , d2 , d1 + d2 − 2c) ≤ 1. By
Lemma 4.3 δ(d1 , d2 , d1 + d2 − 2c) is even, so it must be 0 in this case.
The problem now is to determine for which d1 , d2 , d3 we have δ(d1 , d2 , d3 ) =
0. Let us define
L = {(u, v, w) ∈ Z3+ | 2 max(u, v, w) ≤ u + v + w}.
Also, let L= be the subset of L where equality holds, and L< = L \ L= .
Lemma 4.9. Let (d1 , d2 , d3 ) ∈ L= . Then δ(d1 , d2 , d3 ) = 0.
Proof. Suppose d1 ≤ d2 < d3 = d1 + d2 . We are in the situation when xd1 ,
y d2 , (x + y)d3 is not a minimal generating set; there are polynomials g and
h such that (x + y)d1 +d2 = gxd1 + hy d2 As we saw in the beginning of Section 4.2, the module Syz(xd1 , y d2 , (x + y)d1 +d2 ) is, in this case, generated by
(g, h, −1) and (y d2 , −xd1 , 0). Both these relations have degree d1 + d2 , which
gives δ(d1 , d2 , d3 ) = 0.
The case when d1 or d2 is the largest among d1 , d2 , d3 follows from the above
after a linear change of the variables x and y.
Lemma 4.10. For any two points (c1 , c2 , c3 ) and (d1 , d2 , d3 ) in Z3+ it holds that
|δ(c1 , c2 , c3 ) − δ(d1 , d2 , d3 )| ≤ |c1 − d1 | + |c2 − d2 | + |c3 − d3 |.
17
(4)
Moreover, for (d1 , d2 , d3 ) ∈ L< we can find a point (c1 , c2 , c3 ) such that
δ(c1 , c2 , c3 ) = δ(d1 , d2 , d3 ) + |c1 − d1 | + |c2 − d2 | + |c3 − d3 |,
and δ(c1 , c2 , c3 ) decreases when any ci is replaced by ci ± 1.
Proof. Recall from Lemma 4.5 that δ(d1 , d2 , d3 ) increases or decreases by 1 when
we ”take a step” in Z3+ , that is when one di is replaced by di ± 1. This proves
(4).
Imagine now that we start in the point (d1 , d2 , d3 ), and take a step in some
direction, if it makes the value of δ increase. We continue in this way, as long
as we can make the value of δ increase in each step. What we want to prove is
that such a path can not be infinitely long. Let us fix a point (d′1 , d′2 , d′3 ) on our
path. Any other path between (d1 , d2 , d3 ) and (d′1 , d′2 , d′3 ) must give the same
value of δ at (d′1 , d′2 , d′3 ). It follows that a path where the value of δ increases
in each step must be of minimal length, among all paths between these two
points. Any other path of minimal length must also have the property that δ
increases in each step. Hence we can replace our path by the path that first
increases/decreases d1 , then d2 and last d3 . But when d2 and d3 are fixed, we
can only increase of decrease d1 a finite number of times, before we hit L= . The
corresponding holds for d2 and d3 . At L= the value of δ is zero, as we saw in
Lemma 4.9, so δ must have decreased. This shows that there is a bound for the
length of a path that starts in a given point (d1 , d2 , d3 ) ∈ L< , and increases δ
in each step. Eventually we will reach a point (c1 , c2 , c3 ) such that
δ(c1 , c2 , c3 ) = δ(d1 , d2 , d3 ) + |c1 − d1 | + |c2 − d2 | + |c3 − d3 |,
and δ(c1 , c2 , c3 ) decreases when any ci is replaced by ci ± 1.
d2 = d1 + d3
δ=0
L
+1
+1
+1 +1
d3
+1
+1
+1
+1 +1
+1
+1
+1
+1
+1
+1
+1 +1
+1 +1
d3
The set L, and a path with increasing δ, for a fixed d3 .
18
d1 = d2 + d3
δ=0
Theorem 4.11. Let the function δ be defined over K[x, y] where K is a field
of characteristic p > 0. Let d1 , d2 , d3 be positive integers, such that d1 ≤ d2 ≤
d3 < d1 + d2 . Then δ(d1 , d2 , d3 ) = 0 if and only if
|d1 − ups | + |d2 − vps | + |d3 − wps | ≥ ps
for all integers s, u, v, w such that s ≥ 0 and u + v + w is odd.
Proof. Assume first that δ(d1 , d2 , d3 ) = 0. Let s be a non-negative integer, and
u, v, w integers with odd sum. From Lemma 4.3 we know that δ(u, v, w) is odd,
in particular δ(u, v, w) ≥ 1. Lemma 4.4 and Lemma 4.10 now gives
|d1 − ups | + |d2 − vps | + |d3 − wps | ≥ |δ(ups , vps , wps ) − δ(d1 , d2 , d3 )|
= δ(ups , vps , wps ) = ps δ(u, v, w) ≥ ps .
For the other implication, assume that
|d1 − ups | + |d2 − vps | + |d3 − wps | ≥ ps
for all s ≥ 0 and integers u, v, w with odd sum. By Lemma 4.10 there is a point
(c1 , c2 , c3 ) such that
δ(c1 , c2 , c3 ) = δ(d1 , d2 , d3 ) + |d1 − c1 | + |d2 − c2 | + |d3 − c3 |,
and δ(c1 , c2 , c3 ) decreases by one if we replace any ci by ci ± 1. Write c1 = ps u,
c2 = ps v and c3 = ps w, such that (at least) one of u, v, and w is not divisible
by p. Notice that δ(u, v, w) also must decrease when u, v or w is increased or
degreased by one. Otherwise we would have e. g. δ(u, v, w + 1) = δ(u, v, w) + 1,
which implies δ(c1 , c2 , c3 + ps ) = δ(c1 , c2 , c3 ) + ps . This can only hold if δ
increases in each step from (c1 , c2 , c3 ) to (c1 , c2 , c3 + ps ), which is not the case.
Now we can use Proposition 4.6 on δ(u, v, w) with ℓ = x, y, or x + y, depending
on which of u, v and w are not divisible by p. Since r(xu y v (x + y)w ) = 3
we get δ(u, v, w) ≤ 1. Since δ(u, v, w) − 1 = δ(u, v, w + 1) ≥ 0, we must have
δ(u, v, w) = 1. By Lemmma 4.3 u + v + w is odd, and we can use our assumption
to get
δ(d1 , d2 , d3 ) = δ(c1 , c2 , c3 ) − (|d1 − c1 | + |d2 − c2 | + |d3 − c3 |)
= ps δ(u, v, w) − (|d1 − ups | + |d2 − vps | + |d3 − wps |)
= ps − (|d1 − ups | + |d2 − vps | + |d3 − wps |) ≤ 0.
By definition δ(d1 , d2 , d3 ) ≥ 0, so we can conclude δ(d1 , d2 , d3 ) = 0.
Proof of Theorem 3.5. By Theorem 4.8, K[x, y]/(xd1 , y d2 ) has the SLP if and
only if
δ(d1 , d2 , d1 + d2 − 2c) = 0 for all 1 ≤ c < min(d1 , d2 ).
With d3 = d1 + d2 − 2c, clearly d1 ≤ d3 , d2 ≤ d3 and d3 < d1 + d2 , so we can use
Theorem 4.11. Substituting d3 = d1 + d2 − 2c into the inequality in Theorem
4.11, gives Theorem 3.5.
19
References
[1] H. Brenner and A. Kaid, Syzygy bundles on P2 and the weak Lefschetz
property, Illinois Journal of Mathematics 51 (2007) 1299-1308.
[2] H. Brenner, Looking out for stable syzygy bundles, Advances in Mathematics 219(2) (2008) 401-427.
[3] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge studies in
advanced mathematics 39 (1998).
[4] Clark, T., Cooper, S., Fløystad, G., et al., Progress in Commutative Algebra 1. Combinatorics and Homology. Berlin, Boston: De Gruyter (2012).
[5] D. Cook II, The Lefschetz properties of monomial complete intersections
in positive characteristic, Journal of Algebra 369 (2012) 42-58.
[6] C. Han, The Hilbert-Kunz function of a diagonal hypersurface, Ph.D. thesis, Brandeis University (1991).
[7] E. Kunz, Characterizations of regular local rings of characteristic p, American Journal of Mathematics 91 (1969) 772-784.
[8] M. Lindsey, A class of Hilbert Series and the strong Lefschetz property,
Proceedings of the American Mathematical Society 139 (1) (2011) 79-92.
[9] S. Lundqvist and L. Nicklasson, On the structure of monomial complete
intersections in positive characteristic, arXiv:1604.06820.
[10] P. Monsky, Mason’s theorem and syzygy gaps, Journal of Algebra 303
(2006) 373-381.
[11] N. Snyder, An Alternate proof of Mason’s Theorem, Elemente der Mathematik 55 (2000) 93-94.
[12] R. Stanley, Weyl groups, the hard Lefschetz theorem and the Sperner property, SIAM Journal on Algebraic Discrete Methods 1 (1980) 168-184.
20
| 0 |
arXiv:1509.01763v5 [cs.CR] 30 Jun 2017
Implementing Support for Pointers to Private Data in a
General-Purpose Secure Multi-Party Compiler
Yihua Zhang
Department of Computer Science and Engineering
University of Notre Dame
yzhang16@nd.edu
Marina Blanton
Department of Computer Science and Engineering
State University of New York at Buffalo
mblanton@buffalo.edu
Ghada Almashaqbeh∗
Department of Computer Science
Columbia University
ghada@cs.columbia.edu
Abstract
Recent compilers allow a general-purpose program (written in a conventional programming
language) that handles private data to be translated into secure distributed implementation of
the corresponding functionality. The resulting program is then guaranteed to provably protect
private data using secure multi-party computation techniques. The goals of such compilers are
generality, usability, and efficiency, but the complete set of features of a modern programming
language has not been supported to date by the existing compilers. In particular, recent compilers PICCO and the two-party ANSI C compiler strive to translate any C program into its
secure multi-party implementation, but currently lack support for pointers and dynamic memory allocation, which are important components of many C programs. In this work, we mitigate
the limitation and add support for pointers to private data and consequently dynamic memory
allocation to the PICCO compiler, enabling it to handle a more diverse set of programs over
private data. Because doing so opens up a new design space, we investigate the use of pointers
to private data (with known as well as private locations stored in them) in programs and report
our findings. Besides dynamic memory allocation, we examine other important topics associated
with common pointer use such as reference by pointer/address, casting, and building various
data structures in the context of secure multi-party computation. This results in enabling the
compiler to automatically translate a user program that uses pointers to private data into its distributed implementation that provably protects private data throughout the computation. We
empirically evaluate the constructions and report on performance of representative programs.
1
Introduction
Recent advances in secure multi-party computation make it feasible to securely compute with
private data belonging to different organizations even for complex functionalities. Furthermore,
together with ubiquitous proliferation of cloud computing services, these techniques give rise to
secure computation outsourcing. For these reasons, the research community has recently developed
a number of compilers for transforming a general-purpose program into the corresponding secure
distributed implementation (see, e.g., [9, 23]). These tools aim at generality and are designed to
∗
Work done while at the University of Notre Dame.
1
translate a program written in a conventional programming language into an equivalent program
that uses secure computation techniques to protect private data. They also aid usability and make it
easier for a programmer without extensive knowledge of secure computation techniques to produce
a protocol that can be securely executed in a distributed environment.
It has been long known that any computable function can be securely evaluated by multiple
participants if it is represented as an arithmetic or Boolean circuit. This representation, however,
is not always obvious or known or may otherwise significantly increase the program size. Existing
compilers remove the need for the programmer to perform this translation manually and assemble
secure implementations from efficient building blocks for elementary operations. Thus, efficiency of
the resulting secure computation is also one of the goals that compilers target. Furthermore, the
ability to support both private (i.e., protected) and public (i.e., not protected) data or variables in
a single program adds a level of complexity to the implementation because of the need to support
interaction between public and private variables and secure data flow enforcement.
While the design goal of several compilers was to support any feature of a general-purpose programming language (such as C in [9] and [23]), all such compilers we are aware of have limitations.
In particular, the original version of the PICCO compiler [23] provided no direct support for C
pointers (i.e., pointers were supported only in the form of static arrays) and, as a result, no support for dynamic memory allocation other than static arrays. Similarly, the original version of the
two-party compiler for ANSI C [9] supported pointers only in the form of statically allocated arrays restricted to a constant size and had additional limitations (such as support for floating point
arithmetic was not available in the open source CBMC that the compiler builds upon). Thus,
support for C-like pointers – or, in other programming languages, support for the features that
pointers enable such as dynamic memory allocation, reference by pointer or address, and building
data structures – is the most crucial part of a general-purpose program that is currently unavailable
in existing compilers. Adding this support is thus the focus of our work.
In this paper, we extend the PICCO compiler [23] with pointer support. PICCO1 is a sourceto-source translator that takes as an input a program written in the C programming language
with variables to be protected marked as private and produces a C program that implements the
computation using secure multi-party computation techniques based on linear secret sharing. We
view PICCO as an attractive compiler choice because of the flexibility of the setting it uses. In
particular, the setting assumes three groups of participants: (i) input parties who hold private
inputs into the computation, (ii) computational parties who perform secure computation on secretshared data, and (iii) output parties who are entitled to learning the result of the computation. The
composition of these three groups can be arbitrary (in particular, including the same, overlapping,
or non-overlapping groups), which makes the setting suitable for secure multi-party computation
(SMC), delegation of the computation by multiple data owners to a subset of them or other suitable
entities or secure computation outsourcing by one or more parties. This flexibility follows from the
use of secret sharing techniques and may or may not be present in tools that build on alternative
secure computation techniques (such as, e.g., garbled circuit evaluation).
With linear secret sharing, before secure computation can commence, each input party splits
her private inputs into n > 2 secret shares, where n is the number of computational parties, and
communicates each share to a respective computational party. The computational parties then proceed with evaluating the function on secret-shared data and communicate their shares of the result
to the output parties who reconstruct the output using their shares. Any linear combination of
secret-shared integers is performed locally, but multiplication of secret-shared integers constitutes
the elementary interactive operation. Performance is then measured in the total number of inter1
Available from GitHub at https://github.com/picco-team/picco.
2
active operations as well as the number of sequential interactions or rounds, and recent solutions
based on secret sharing aim at minimizing overhead using both metrics.
When PICCO is used to perform source-to-source translation, the input program is a conventional C program where each variable is marked to be either private or public. All computation
with private variables is transformed into secure arithmetic on shared data, while operations with
public variables that do not interact with private data are left unchanged. In addition to specifying
private/public qualifies for each variable, for performance reasons PICCO also allows the programmer to mark the places where computation can proceed concurrently (i.e., to decrease the number
of computation rounds), which also extends the conventional C syntax.
Adding pointer to support to a program that manipulates private data not only extends the
compiler to handle the full range of C programs (that do not violate secrecy of private data),
but also permits important features of programming languages, treatment of which, to the best
of our knowledge, has not been done before. As part of this work, we thus explore how pointers
to private data (including pointers with private locations) can be implemented and discuss our
design decisions. Having added support of pointers to private data, we further study common uses
of pointers in programs and the impact our implementation has on those language features. For
example, we evaluate passing arguments by references, dynamic memory allocation, and pointer
casting. Based on our analysis as well as empirical evaluation, several of these features introduce
only marginal costs. Also, one of the important topics studied in this is work is the use of pointerbased data structures written for private data. Our results indicate that the use of pointers (to
private data) is very attractive and maintains high efficiency for several popular data structures.
In some other cases, in particular when working with sorted data, privately manipulating pointers
increases complexity of data structure operations and it might be desirable to pursue alternative
implementations.
We would like to emphasize that it is not the goal of this paper to try to develop most efficient
implementations for different data structures. Instead, the goal is to determine how pointers to
private data can be supported at a low possible cost and to what performance of typical programs
that might lead. We note that, depending on the program structure, asymptotic complexity of a
translated program might be higher than that of the original. For example, consider an if-then-else
statement with a private condition (e.g., conditional statements used in traversing a binary tree).
When data privacy is not required, only one of the two branches will be executed, but with any
compiler that produces a secure implementation both branches will have to be evaluated to hide
the result of the private condition. Then with a sequence of n nested if-then-else statements, in
the worst case the secure program might have to execute O(2n ) instructions where the original
program would execute only O(n). This means that the general translation approach can lead to
an exponential increase in the runtime for programs of practical relevance. As part of this work
we show that data structures that utilize pointers to private data cover the entire spectrum of
possibilities: in one extreme, they result in no asymptotic increase over conventional non-secure
counterparts, and in another extreme, the increase is exponential. This provides insights on when
natural pointer use is very attractive and when other, alternative implementations might be desired.
While for many data structures alternative, non-pointer-based implementations may be possible,
we note that our extension of PICCO with pointers enables support for an important aspect of
modern programs otherwise not available in any secure multi-party compiler we are aware of, which
is dynamic memory allocation. Dynamic memory allocation is essential for a general-purpose
programming language, but has not been systematically studied in the context of secure multiparty computation (e.g., even publications and compilers that run secure computation on data sets
of a large size assume that the size of the data is fixed and known at compilation time). As an
example, consider an application in which data items arrive over time. A pointer-based linked
3
list will allow for a natural and graceful mechanism of memory management, that unlike statically
allocated arrays does not require complex provisions for allocating a larger buffer when the current
one becomes full, moving the data, or merging previously allocated buffers.
The rest of this paper is organized as follows: We first give a brief overview of related work in
Section 2. After presenting background information in Section 3, we proceed with presenting our
solution for supporting pointers to private data in Section 4. In Section 5, we discuss common uses
of pointers in programming, such as passing arguments by reference, dynamic memory allocation,
array manipulation, and pointer casting and their underlying implementation in our framework.
Section 6 next summarizes operations with pointers to private data and formally shows security of
the design. Section 7 analyzes various data structures built using pointers to private data. Lastly,
Section 8 presents the results of performance evaluation of representative programs that utilize
pointers (to private data) and Section 9 concludes this work.
2
Related Work
In this section we review the most closely related work on SMC compilers and secure/oblivious
data structures. Regarding the compilers, Fairplay [15] was a pioneer work that enables compilation of secure two-party protocols based on garbled circuits. Its extension to multiple parties,
FairplayMP [3], implements secure computation using Boolean circuits and secret sharing techniques. TASTY [8] is another two-party SMC compiler that combines garbled circuit techniques
with those based on homomorphic encryption. Sharemind [5] and VIFF [7] are multi-party compilers based on custom additive 3-party secret sharing and standard threshold linear secret sharing,
respectively. All of the above compilers use custom domain-specific languages to represent user
programs. The two-party compiler for ANSI C [9] and PCF [12] both use two-party garbled circuit
techniques, where the former’s goal is to support general purpose C programs, while the latter
uses a new circuit format and employs optimizations to reduce the compilation time and storage.
Lastly, TinyGarble [20] uses hardware synthesis to optimize garbled circuits for two-party computation. All of these compilers require linear in the size of memory work to access memory at a
private location. SCVM [13], on the other hand, is an automated compiler that utilizes oblivious
RAM (ORAM) and targets two-party computation. ObliVM [14] is another ORAM-based secure
two-party computation compiler that transforms programs written in high level abstractions to
optimized garbled circuit implementations. Finally, a recent compiler Frigate [17] was designed to
guarantee correctness of programs compiled into circuits for secure two-party computation.
To support data structures in the SMC framework, several solutions [21, 10, 16, 22] have been
proposed. The main motivation of this line of work is the need to store and manipulate private
data in an efficient and flexible manner. Toft [21] proposed a private priority queue that has
a deterministic access pattern as opposed to randomized ones in ORAM-based data structures.
On the other hand, Keller and Scholl [10] introduced implementations of arrays, dictionaries, and
priority queues based on various flavors of ORAM implementations. Mitchell and Zimmerman
[16] also provide implementations of stacks, queues, and priority queues based on oblivious data
compaction and an offline variant of ORAM. Wang et al. [22] proposed implementations of maps,
sets, priority queues, stacks, and deques based on ORAM techniques modified for specific data access
patterns. Different from all of these publications, our work includes extending the PICCO compiler
to support dynamic data structures in a generic way as found in general purpose programming
languages. That is, the programmer has the basic tools and primitives that enable her to build any
desired data structure.
In our implementation, a pointer to private data may store one or more locations where the
4
data might reside, which in the worst case is linear in the program’s memory size. ORAM-based
techniques, on the other hand, guarantee that when an item is accessed at a private location,
the number of accessed memory locations is polylogarithmic in the total memory size. Thus, our
general solution may or may not be faster than using ORAM, depending on both the program and
the data size. As we discuss later in this work, employing ORAM techniques can be beneficial for
certain data structures (and sufficiently large data sets). Building custom data structures, however,
is beyond the scope of this work.
One of the applications that the compiler can naturally be used for once support for pointers
to private data is in place is evaluation of a context-free grammar on private data (implemented
as a shift-reduce parser using a stack). The grammar can be either public or private, and in the
latter case execution will correspond to evaluation of private expressions/programs on private data.
Techniques for evaluation of private programs (on private data) are a separate area of research,
discussion of which is beyond the scope of this work, but the reader may refer to recent results in
this areas such as those in [11, 19].
3
Background Information
PICCO uses Shamir secret sharing [18] for implementing secure arithmetic and other operations
on private data. It is an (n, t)-threshold linear secret sharing scheme, in which a private value is
represented using n > 2 secret shares, one held by each computational party. Then any t + 1 or
more shares can be used to reconstruct the private value, while t or fewer parties cannot learn any
information about the shared value (which is perfectly protected in the information-theoretic sense).
In a linear secret sharing scheme, a linear combination of secret-shared values can be performed
by each computational party locally, without any interaction, while multiplication of secret-shared
values requires communication between all of them. With Shamir secret sharing, computation takes
place over a field of a desired size (larger than any value that needs to be represented). A secret
s is represented using a random polynomial of degree t with the free coefficient set to s, and each
share corresponds to the evaluation of the polynomial on a distinct non-zero point. Given t + 1
or more shares, the secret can be reconstructed using Lagrange interpolation. Then addition or
subtraction of secret-shared values, or multiplication of a secret-shared value by a known integer can
be performed by each party locally using its shares. Multiplication involves multiplying two shares,
which raises the corresponding polynomial degree to 2t, and resharing and interpolating the result
to bring the degree of the corresponding polynomial from 2t to t. This imposes the requirement
that t < n/2. With the way multiplication is performed, it is also possible to evaluate any multivariate polynomial of degree 2 over secret-shared integers with a single interaction. That is, we
first evaluate the polynomial and re-share the overall result instead of doing so for the intermediate
products. This serves as a powerful optimization tool when, for instance, the dot product of two
secret shared vectors/arrays needs to be computed, which results only in a single interaction.
While the regular field operations achieve perfect secrecy, implementation of some of the basic
operations used in PICCO (such as comparisons, division, etc.) is statistically secure, which requires
the bitlength of the field elements to be increased by the statistical security parameter. This
slightly increases the cost of field operations, as well as the amount of communication associated
with transmitting field elements. The optimal size of field elements is automatically determined by
PICCO for each program it compiles.
Performance of these techniques is measured in terms of the total number of elementary interactive operations (field multiplications or reconstructions of a value from its shares) as well as
the number of sequential interactions or rounds. For that reason, PICCO supports a number of
5
optimizations to reduce the round complexity of programs that it outputs through concurrent or
batch execution.
Support for pointers to private data (and the corresponding functionalities such as dynamic
memory management) was the only missing functionality in PICCO. Thus, we modify the compiler
to enable it to compile user programs that contain pointers to private data, which was not previously
available. Our changes to the compiler affect only pointers to private data and the introduction of
two built-in functions for memory allocation and deallocation (called pmalloc and pfree) associated
with pointers to private data.
4
Adding Pointer Support
Recall that in C a pointer is a variable of a special type that stores a location in memory at which
data of a particular type can be located. Because a pointer stores an address, it can be treated
very generally with the possibility of directly manipulating pointers, changing the addresses they
store, dereferencing a pointer to access the data to which it points, casting a pointer of a particular
type to a pointer of another type, and using pointers to functions.
4.1
Working Toward a Solution
When working with pointers in the presence of private data, besides traditional C pointers to
public variables, we can distinguish between pointers to private data that (i) point to a single
known location where the private data is stored and (ii) point to a memory pool or a number
of locations where private data is stored and the location of the private data is not known. In
determining how this can be implemented in a C-like programming language, we considered preallocating memory pools for pointers with private locations. Such memory pools would be required
for each data type to ensure that we can store and extract private data correctly. This approach,
however, has severe disadvantages, which are:
1. Using memory pools unnecessarily increases the program’s memory footprint, where one
pool will be needed for each used data type including complex types defined via the struct
construct. Furthermore, it is not clear to what size each pool should be set to optimize
performance.
2. It would also often incur unnecessarily large computation costs due to the need to touch all
locations within a memory pool per single access (or touch several locations when the pool is
implemented using more complex ORAM techniques). As will be evident later in the paper,
there are large classes of programs, applications, and data structures, where a pointer to
private data always corresponds to a single location, which removes the need to use secure
multi-party computation techniques for pointer manipulation. Allowing a pointer to store a
single known location drastically improves program performance compared to using pointer
pools.
3. Memory pools would also not work in the presence of pointer casting.
Then if we do not want a pointer to initially point to a pre-allocated memory pool, would the
decision to properly declare a pointer as pointing to a single (known) location or a set of locations
be left to the programmer? This is going to introduce an additional burden for a programmer
who would need to know at a variable declaration time whether the variable of a pointer type will
require protecting its value. This happens if the pointer is used inside a conditional statement with
6
private condition, which then requires protecting the location assigned to the pointer to protect
the result of the condition evaluation.
To ease programming burden and at the same time avoid consuming unnecessary (memory and
computation) resources, our solution is to use the same programming interface for all pointers that
are to point to private data. When the pointer is being declared or initialized, it has one known
location associated with it (if the pointer is not initialized, that location is set to the default value
corresponding to uninitialized pointers). Throughout the computation, the pointer, however, may
be pointing to multiple locations, one of which is its true location. This happens when the pointer’s
value is modified inside conditional statements with private conditions as illustrated next. Suppose
we declare variables a and b to be private integers followed by the code below:
1. private int *p;
2. p = &a;
3. if (priv-cond) then p = &b;
We see that variable p was declared as a pointer to a private integer, but the type of the pointer
with respect to whether the location itself is private is implicit. After executing lines 1–2, p has a
single known location, but after executing line 3, p is associated with a list of two locations (the
address of a and the address of b) and the value of the true location is protected. That is, a pointer
always starts with a single publicly known location and the location to which it is pointing may
become private, but the user does not declare the pointer itself as public or private. In the rest of
this work, we use the term “public location” in reference to a pointer to private data to mean that
the pointer has a single known location (either initialized or uninitialized) and we use the term
“private location” to mean that the pointer has a list of public locations, but which location is in
use remains private.
When we consider interaction of public and private values in connection to the use of pointers,
a number of questions arise, which we address next.
1. Can a pointer that was declared to point at private data be assigned address of public data?
Note that without the use of pointers, the equivalent actions are generally allowed. That is, a
variable declared to hold private data can be assigned a known value, which is consequently
converted into protected form. The same does not hold for pointers and we disallow assigning
locations of public variables to pointers which were declared to point to private data. To
see why, suppose that a user program contains the code below where a was declared to be a
private integer, while b is a public integer:
1. p = &a;
2. if (priv-cond) then p = &b;
3. *p += 1;
After executing lines 1–2, p stores two addresses and the true location of where it is pointing
out of these two addresses is protected. On line 3, however, the pointer is dereferenced and
the result of private condition priv-cond evaluation is revealed by examining the value of b
before and after line 3. Thus, to eliminate information leakage, pointers to private data can
be assigned only locations that store private values.
2. Can a pointer declared to point to public data be modified inside conditional statements with
private conditions and as a result become pointing to multiple locations? The answer to this
question is No. If a pointer to public data is updated in the body of a conditional statement
7
with private condition, it must be treated as a pointer to private data (otherwise, using its
dereferenced value reveals unauthorized information). Allowing such uses and performing the
conversion implicitly by the compiler will be confusing to the programmer (who no longer
can use the pointer to store addresses of public data). For that reason, we disallow updates
to pointers to public data within the body of conditional statements with private conditions.
4.2
Pointer Implementation
We next proceed with describing how pointers to private data are implemented to realize the ideas
outlined above. We note that all program transformations that we describe preserve semantics of
the original program and, given that a program can be compiled into the corresponding secure
implementation, the transformed program will always produce the same output as the original
program. There are some restrictions that user programs must meet in order to be compiled into
secure implementations with no information leakage. Such restrictions include the two cases at the
interaction of public and private data described above and additional restrictions inherited from
PICCO (e.g., the fact that the body of a conditional statement with a private condition cannot have
public side effects, a loop termination condition should be public or made public, etc.). This is to
ensure that no information leakage in the compiled program can take place, and the programs that
do not meet the requirements are aborted at the compilation time. Once these constraints are met,
our extension of PICCO will allow any user program to be compiled into its secure counterpart.
Pointer representation. As we incorporate support for pointers, we first note that pointers to
public data will not need to be modified and their implementation remains the same as in the
C programming language. The most significant change in implementing pointers to private data
comes from the need to maintain multiple locations. For that reason, the data structure that we
maintain for pointers to private data consists of (i) an integer field that stores the number α (≥ 1)
of locations associated with the pointer; (ii) a list of α addresses where the data is stored; and
(iii) a list of α private (i.e., secret-shared) tags, one of which is set to 1 (true data location) and
all others are set to 0. For the important special case of α = 1, the pointer has known (public)
location and the tags are not used.
We formalize the above pointer representation using the following invariant, which is maintained
throughout various pointer operations: among all locations stored with a pointer to a private object,
there is exactly one true location of the object and the tag corresponding to that location is set to
1, while the tags corresponding to all other locations are set to 0. This invariant is true of all
well-formed programs and may be violated only in the case of dangling pointers as detailed later.
Because we would like to employ a uniform data structure for pointers to private data of any
data type such as integer, floating point values, etc. and even pointers to a pointer, the data
structure we maintain needs to include two additional fields: (iv) an integer flag that determines
the type of data associated with the pointer (i.e., integer = 1, float = 2, struct = 3, etc.) and (v)
an integer field that indicates the indirection level of the pointer. For instance, if a pointer refers
to a private value of a non-pointer type, its indirection level is set to 1; and if it refers to a pointer
whose indirection level is k (for k ≥ 1), its level will be set to k + 1. A pointer to a struct also has
indirection level 1 regardless of the types of the struct’s fields (which can be pointers themselves).
Pointer updates. Initially, at the pointer declaration time, the number of locations α associated
with the pointer is set to 1 and the address is set to to a special constant used for uninitialized
pointers. Then every time the pointer is modified (including simultaneously with pointer declaration), its data structure is updated. When the pointer is assigned a new location using a public
constant, a variable’s address, or a memory allocation mechanism (e.g., as in p = 0, p = &a, or p
8
= malloc(size)), α in the pointer’s data structure is set to 1 and the associated address is stored
in the pointer’s address list. When a pointer is updated using another pointer (as in p = p1), the
latter’s data structure is copied and stored with the former.
Such simple manipulations are used only when the assignment does not take place inside the
body of a conditional statement with a private condition. Pointer assignments inside conditional
statements with a private condition present the most interesting case when the list of pointer
locations gets modified. Updating values modified in the body of a conditional statement with
a private condition already requires special handling in PICCO, and all we need is to support
a specific procedure when a variable of pointer type is being modified. We need to distinguish
between if-then and if-then-else statements, which we consequently discuss.
Consider the following code with an if-then statement:
1. p = p1;
2. if (priv-cond) then p = p2;
where p, p1, and p2 are pointers (to private data) of the same type. This is the most general
case, where on line 2 both p and p2 can have any number of locations associated with each of
them (recall that all other assignment types use a single location). When this code is written for
ordinary (private) variables of the same type a, a1 , and a2 , a generic way to implement this update
in PICCO and similar compilers is to first set [a] = [a1 ] and then compute
[a] = [c] · [a2 ] + (1 − [c]) · [a] = [c] · ([a2 ] − [a]) + [a],
where c is a bit equal to the result of evaluating priv-cond. We use notation [x] to indicate that
the value of x is protected via secret sharing and computation takes place on its shares. In the
case of pointers, such a simple update does not work because this procedure would turn addresses
into secret shared values preventing the pointer from being dereferenced (without touching all
possible memory locations). Thus, after executing the assignment p = p1, we combine the (public)
locations of p and p2 and set the tags in p based on the current tags of p and p2 and the result
c of evaluating priv-cond. Let pointer p after executing the first assignment contain α1 locations
stored as L1 = {ℓ1 , . . ., ℓα1 } with corresponding tags T1 = {[t1 ], . . ., [tα1 ]} (i.e., this information was
copied from p1). Similarly, let pointer p2 store α2 , L2 = {ℓ′1 , . . ., ℓ′α2 }, and T2 = {[t′1 ], . . ., [t′α2 ]}.
Note that the ordering of addresses in each L is arbitrary, but the tag ti in T must correspond
to the address ℓi at the same position i in L. Then as a result of the conditional assignment, we
compute p’s new content as given in Algorithm 1.
In the algorithm, L3 is composed of all locations appearing in L1 or L2 (repeated locations are
stored only once). We use notation L.find to retrieve the position of the element of L provided as
the argument or special symbol ⊥ is the element is not found. The tags in the output T3 are set
based on three different cases: (i) a location in L3 is found in both L1 and L2 ; (ii) it is found in
L1 , but not in L2 ; and (iii) it is found in L2 , but not L1 . Because only tags in T1 and T2 and c are
private, only lines 7, 9, and 11 correspond to private computation.
If the conditional statement is of the form if-then-else, but p is not updated in the body of the
else clause, then the computation in Algorithm 1 is applied unchanged. If the pointer is instead
updated only in the body of the else clause, then the computation is performed similarly, but
Algorithm 1 is called with the value of 1 − c instead of c.
Lastly, if the pointer is updated in both clauses of the if-then-else statement, the pointer content
prior to that statement needs to be disregarded. The pointer values used in the two assignments
are then merged as in Algorithm 1 using the result c of private condition evaluation. To better
illustrate this, consider the following code segment:
9
Algorithm 1 hα3 , L3 , T3 i ← CondAssign(hα1 , L1 , T1 i, hα2 , L2 , T2 i, [c])
1: L3 = L1 ∪ L2 ;
2: α3 = |L3 |;
3: for every ℓ′′
i ∈ L3 do
4:
pos1 = L1 .find(ℓ′′i );
5:
pos2 = L2 .find(ℓ′′i );
6:
if (pos1 6=⊥ and pos2 6=⊥) then
7:
[t′′i ] = [c] · [t′pos2 ] + (1 − [c]) · [tpos1 ];
8:
else if (pos2 =⊥) then
9:
[t′′i ] = (1 − [c]) · [tpos1 ];
10:
else
11:
[t′′i ] = [c] · [t′pos2 ];
12:
end if
13: end for
′′
′′
14: set T3 = {[t′′
1 ], [t2 ], . . ., [tα3 ]};
15: return hα3 , L3 , T3 i;
1. p = p1;
2. if (priv-cond) then p = p2;
3. else p = p3;
After we assign p1 to p on the first line, p’s content is be overwritten with the content of either
p2 or p3 depending on the result c of evaluating priv-cond. We can see that before entering
the if-clause, the current content of p (i.e., that copied from p1) can be safely disregarded without affecting its correctness. In other words, to update p inside the conditional statement, we
call CondAssign(hα2 , L2 , T2 i, hα3 , L3 , T3 i, c) in Algorithm 1, where hα2 , L2 , T2 i and hα3 , L3 , T3 i are
contents of pointers p2 and p3, respectively.
These constructions compose in presence of nested conditional statements with private conditions. For instance, after executing the code:
1. if (priv-cond1) then p = p1;
2. else
3.
p = p2;
4.
if (priv-cond2) then p = p3;
5.
else p = p4;
p will contain the combined content of pointers p1, p3, and p4. That is, Algorithm 1 is first called
with the content of pointers p3 and p4 and the result c2 of evaluating priv-cond2, after which
Algorithm 1 is called on the result of its previous execution, the content of p1, and the result c1 of
evaluating priv-cond1.
As evident from the description above, all modifications to variables of all types (including pointers as well as data) inside conditional statements with private conditions require special handling
inside the compiler. For each such conditional statement, PICCO examines the list of variables
modified inside the body of the statement and updates them differently from when the modification
is not surrounded by a private condition. Thus, in the case of pointers we specify how pointers
need to be updated inside such statements using Algorithm 1 and compiler will process all variables
inside the body of conditional statements with private conditions.
Note that each pointer starts with a single location (i.e., α is set to 1) at the time of its
declaration, and the list and the number of locations α are updated during pointer assignments
10
as described above. This information is maintained only during program execution and thus the
locations that a pointer might store or their number is not necessarily known at compile time.
Pointer dereferencing. When pointer p with a private location is being dereferenced, its dereferenced value is privately computed from α, L = {ℓ1 , . . ., ℓα }, and T = {[t1 ], . . ., [tα ]} stored at p.
Let [aiP
] denote the value stored at location ℓi ∈ L. Then we compute the dereferenced value as
[v] = αi=1 [ai ] · [ti ]. Note that with linear secret sharing this operation can be implemented as
an inner product that costs only a single interactive operation resulting in a profound impact on
performance.
When the dereferenced value is being updated, all locations in L need to be touched, but the
content of only one of them is being changed. If we, as before, use [ai ] to denote the value stored
at ℓi ∈ L and let [anew ] denote the value with which the dereferenced value is being updated, then
we update the content of each location ℓi as [ai ] = [ti ] · [anew ] + (1 − [ti ]) · [ai ]. That is, the true
location (ti = 1) will be set to anew , while all others (ti = 0) will be kept unchanged.
In the current form, the above procedures are applicable only to pointers with the indirection
level equal to 1. That is, if pointer p is associated with a list of private locations of pointers, the
above computation will result in producing secret shared locations and the information looses its
semantic meaning. Thus, for pointers with indirection level > 1 different computation is used.
That is, now each ℓi ∈ L stores an address of a pointer pi and let each pi be associated with αi ,
(i)
(i)
(i)
(i)
Li = {ℓ1 , . . ., ℓαi }, and Ti = {[t1 ], . . ., [tαi ]}. To retrieve the dereferenced value of p, we first
(i)
compute [ti ] · [tj ] for 1 ≤ i ≤ α and 1 ≤ j ≤ αi and merge all lists Li for 1 ≤ i ≤ α. The resulting
list is thus set to L′ = L1 ∪ L2 ∪ · · · ∪ Lα and let α′ = |L′ |. For any location in L′ , we compute its
(i)
corresponding tag as the sum of all [ti ] · [tj ] values matching that location in the individual lists
Li . (We can simply use the sum because only one tag can be set to 1.) The result is α′ , L′ and the
corresponding tags T ′ .
To illustrate this on an example, let α = 3, T = ([0], [0], [1]), and L store the addresses of
pointers p1 , p2 , p3 with α1 = 1, L1 = (123), T1 = (1); α2 = 2, L2 = (189, 245), T2 = ([0], [1]);
and α3 = 3, L3 = (123, 176, 207), T3 = ([0], [1], [0]). The result of this operation is a pointer with
L′ = L1 ∪ L2 ∪ L3 = (123, 176, 189, 207, 245), α′ = |L′ | = 5, and T ′ = ([0], [1], [0], [0], [0]).
To update the dereferenced value of p through an assignment as in *p = p’, each pointer pi
stored at address ℓi ∈ L needs to be updated with p’’s information. In particular, for each pi each
(i)
(i)
(i)
tag [tj ] (for location ℓj ) is updated to (1 − [ti ]) · [tj ]. We also compute tag [ti ] · [t′j ] for each
location ℓ′j in p’’s list of locations. We then merge the location list of each pi with that of p’ to
form pi ’s new list. For any new location inserted into Li , its tag is set to the computed [ti ] · [t′j ]
for the appropriate choice of j, and any location that appears on both pi and p’ lists, the value
[ti ] · [t′j ] is added to pi ’s updated tag for that location. In other words, if ti is true, we take p’’s
value and otherwise keep pi ’s value.
If pointer p with a private location is being dereferenced m > 1 times, the above dereference
algorithms are naturally applied multiple times with the first m − 1 instances being the version that
produces a pointer and the last instance producing either a pointer or a private value depending on
p’s indirection level. p can then be treated as the root of a tree with its child nodes being locations
of pointers stored in its list and the leaves of the tree eventually pointing to private data (of a
non-pointer type). To perform an m-level dereferencing operation, we traverse the top m + 1 levels
of the tree and consolidate the values stored at those levels (and update the values at the (m + 1)st
level if the dereferenced value is to be updated).
Secrecy of pointers to private data. As previously discussed, the value of a pointer to private
data is treated as public when it stores a single location (α = 1), and it is private otherwise (α > 1).
11
More generally, if pointers to private data are used in predicates or similar expressions, the result
of a predicate evaluation is public if its outcome can be determined using only public data. For
example, the outcome of an expression that compares two pointers to private data for equality is
public if (i) both pointers store a single location in their lists L1 and L2 or (ii) at least one of the
pointers stores multiple locations, but L1 ∩ L2 = ∅. In other circumstances, the outcome depends
on private tags and is treated as private.
Note that when the result of a predicate evaluation on pointers is private, it can be naturally
computed by privately determining the true location of each pointer and applying the predicate
to them. This computation, however, can be optimized for certain types of predicates to result
in faster performance. For example, in the case of pointer equality, the general solution is to
P
P
(1) (1)
(2) (2)
compute [ℓ1 ] = ℓ(1) ∈L ℓi [ti ] and [ℓ2 ] = ℓ(2) ∈L ℓi [ti ] and then compare [ℓ1 ] and [ℓ2 ] for
1
2
i
i
P
(1) (2)
(1)
(2)
equality, while an optimized implementation computes ℓ(1) ∈L ∩L [ti ][tj ], where ℓi = ℓj for
i
(1)
ℓi
1
2
each
∈ L1 ∩ L2 . The latter can be implemented as the inner product that costs a single
interactive operation and is significantly lower than the cost of comparing two private integers.
Because it is not always possible to determine at compile time whether a predicate evaluated
on one or more pointers (to private data) will have a public or private status (which, for example,
may depend on program’s public input), some checking will need to be deferred to run time. In
particular, if pointers to private data are used in a predicate to form a conditional statement and the
result of its evaluation is private, the usual constraints for the body of conditional statements with
private conditions apply. We address this by evaluating the body of such conditional statements for
public side effects at compile time (as in the original PICCO design). If the body contains public
side effects, the transformed program will include checks for the status of the conditional statement
at run time. If the result of evaluating the conditional statement is determined to be private at run
time (and public side effects are present in the body of the statement), execution will be aborted
with an error due to a possible information leak.
Note that the fact whether the execution is aborted or not never leaks private information,
as this decision is solely based on public data. That is, an abort takes place when (i) the result
of evaluating a predicate on two or more pointers is treated as private and (ii) the body of the
conditional statement that uses the predicate contains public side effects. Whether the result
of evaluating the predicate is public or not is public knowledge because it is determined by the
public locations stored in the pointers and the predicate itself. Similarly, whether the body of the
conditional statement has public side effects or not is based on the public code that forms the body
of the conditional statement.
Optimizations. Because the computation involved in computing a pointer’s dereferenced value is
interactive (and thus is relatively expensive) when the pointer stores multiple locations, we considered caching and reusing its result. That is, when a pointer’s dereferenced value is computed, we
can store it in the pointer structure and reuse the value on consecutive dereference operations when
there are no changes to the values stored at all pointer’s locations L between such operations. Once
any value stored at one of the pointer’s locations is modified, the cached dereferenced value needs
to be marked as out of date. A similar caching technique can also be applied to the computation on
private tags that takes place during pointer update and dereference operations. Note that private
tags can be viewed as aggregation of conjunctions of 1-bit private variables that denote the evaluation results of private conditions (or their negations) in user programs. Because, once computed,
the variables will have fixed values, the conjunction results of those variables can be cached in our
framework in a lookup table and allow for their retrieval when the same conjunctions need to be
repeatedly computed. This optimization will result in considerable savings when multiple private
12
pointers are updated or dereferenced within the body of a conditional statement with a private
condition. The savings due to either of the above caching optimizations are, however, applicationdependent and require additional program analysis at program parse and translation time. Thus,
these optimizations are not presently a part of our implementation.
Another possible optimization can lower a program’s memory footprint by reusing data structures created for pointers to private data. In particular, when a pointer is assigned another pointer’s
value (as in p = p1), we could have both pointers pointing to the same data structure instead of
creating its copy. When, however, one of the pointers with the shared data structure is being
modified, it should be unlinked from the shared data structure and its data structure modified
accordingly. Implementing this would require that each pointer data structure is associated with
a list of pointer variables which are using it. Furthermore, a data structure can be reused only
when all information stored in it is identical for multiple pointers (i.e., not only the locations in
L, but also the private tags in T ). Because different pointers often have distinct roles in user
programs, the expected savings are not very large. This optimization is presently not a part of our
implementation.
4.3
Pointers to Struct
We next discuss design and implementation of pointers to structs, including their representation
and the associated algorithms. Pointers to complex data types declared using struct constructs are
common for building data structures such as linked lists, stacks, and trees, and thus pointers to
structs deserve special attention.
As before, if a complex data type contains no private fields, no transformations are needed.
However, when dealing with pointers to struct with private fields, we need to address the following
questions:
1. A struct groups together a number of different variables that can be either private or public,
but the complex data type itself declared using struct is not associated with any particular
type of secrecy. When declaring a pointer to a complex data type, we thus need to determine
if a pointer to it can be treated as a pointer to private data or if it has to be treated as a
conventional pointer to a public variable.
2. When designing representation of a private pointer that points to struct, we need to take into
account the fact that fields of a complex data type can be accessed and modified independently
of each other or the struct itself. Thus, it remains as a question whether we should maintain
a separate list of addresses for each struct field or maintain only a single list of addresses for
all possible struct variables associated with the pointer.
3. The last question is whether we can reuse the previously described algorithms for working
with private pointers for updating or dereferencing pointers to structs on the individual fields
of a struct or if modifications are needed.
In what follows, we thus focus on answering these questions.
Secrecy of pointers to struct. Secrecy of a pointer to struct is implicitly determined by the
protection modes of the struct’s fields. We determined that a pointer to a complex data type can
be treated as a pointer to private data only if all fields in its declaration are private. It means that
if at least a single field of a struct is public, pointers to this data type can be of public type only.
This treatment is necessary to eliminate information leakage when pointers to structs are modified
inside conditional statements with private conditions. Consider, for example, a data type containing
13
one private and one public field. If we treat a pointer to this data type as a pointer to private
data, it can be modified inside an if-statement with a private condition and have multiple locations
associated with the pointer. However, by dereferencing and observing the value of the public field,
one can determine the true location of the pointer and thus learn unauthorized information about
the result of the private condition.
Because a complex data type may contain other struct variables as its fields, the variables in
the data type will need to be checked recursively to determine whether at least one public field is
present (with provisions to skip cycles in the declarations). If none are found, pointers to this data
type are treated as pointers to private data.
Pointer to struct representation. To implement private pointers to structs, we needed to
determine whether a single list of locations is sufficient for all fields of the complex data type
(recall that all fields are private) or separate lists must be maintained. In working to answer this
question, we determine that there is no need to maintain multiple lists of locations, because the
list of locations associated with each field in the struct must be the same (adjusted for the offset of
the field within the struct). That is, values of a struct’s fields can be modified individually (e.g., as
in p->x = y), but the only way to access or modify the location of a field is through the location
of the entire struct. In other words, the list of addresses associated with a pointer to struct p (and
thus the addresses corresponding to all of its fields) can be modified only by directly updating p,
as operations of the type p->x = y do not affect the list of addresses associated with the field x.
Storing a single list has the added benefit that we can employ the same representation of pointers
to private data as for simple data types. This treatment also implies that a pointer to a struct
object will have indirection level 1 even if all fields of the struct are pointers themselves.
Operations on private pointers to struct. We represent pointers to a struct record in the same
way as other pointers. This means that operations for using pointers and updating their values
remain unchanged. To dereference a specific field of a pointer as in p->x and retrieve the value
of the variable x, also only minor changes to the previously described algorithms are needed. In
particular, all we need is to determine the offset f of the variable’s address within the record and
perform the dereferencing procedure in the same way as for pointer p itself, but instead of using
locations ℓi from L, we use locations ℓi + f . The same modification applies to the case when the
dereferenced value is modified through assignment.
If we would like to dereference p and retrieve the entire record as in rec = *p, we need to
iterate through each field of the struct and retrieve the dereference value of each field as described
above for p->x. Similarly, to update a dereferenced pointer p as in *p = rec, we need to perform
the equivalent of p->x = rec.x for each field x of the struct.
5
Pointer Uses in Programming
In this section we discuss many common uses of pointers in programming and how they are translated to our environment of computing with private data. The topics we cover are passing arguments by reference, dynamic allocation of memory, array manipulation, and pointer casting. Data
structures also constitute a common use of pointers, but we discuss them separately in Section 7.
5.1
Passing Arguments by Reference
Function calls contribute to the basic software engineering principles of modular program design,
but could be expensive in terms of stack memory usage for the passed arguments. This has led to
differentiating between function calls where the arguments are passed by value and by reference.
14
In the latter case, the function typically takes a pointer to the argument and all updates to the
dereferenced pointer will be visible after completing the function call (thus, arguments passed by
reference can be used for either input or output).
Passing private variables to functions by reference inherits the same benefits as for conventional
(public) variables in the programming language. The good news is that no special provisions are
needed for passing private variables by reference, resulting in efficient implementations. Furthermore, because often to pass an argument by reference, its address is supplied to a function call (as
opposed to supplying an existing pointer), the resulting pointer will have a single known location.
This allows us to enjoy the benefits of avoiding using extra resources without the slowdown of
working with pointers with private locations.
5.2
Dynamic Memory Allocation
Pointers are often used in programming to dynamically allocate memory on the free store and
deallocate it when it is no longer in use. Here we focus on C-style malloc() and free() used with
pointers to public variables and show what modifications are needed to support dynamic memory
allocation with pointers to private variables.
malloc() in C allocates the requested number of bytes on the heap which are passed as an
argument to the function malloc(). The result of this function is the address of the allocated
variable or the first array element in case of dynamic array allocation, which is stored in a pointer.
To support dynamic memory allocation for private variables, we start with the following code in C:
1. int* p = (int*) malloc(sizeof(int));
2. int* p1 = (int*) malloc(10*sizeof(int));
Here p points to single variable, while p1 points to a dynamic array of size 10. The assignment
operator directly saves the malloc result into the pointer because they are of compatible types.
However, this is not the case for pointers to private variables because a private pointer is represented
using multiple fields. Consequently, we cannot assign the malloc result directly to a private pointer
and use a modified interface for pointers to private variables. In particular, we use a function
pmalloc2 to implement private malloc, which is invoked as:
1. private int* p = pmalloc(10, private int);
As shown, pmalloc takes two arguments, which are the requested number of dynamic variables
and the data type. The function returns the data structure used for private pointers in our implementation with α = 1 and the only location in L set to the address of the first variable in
the allocated array (when the first argument to the function is > 1). Specifying the private data
type is necessary to properly allocate and initialize the memory. For example, in PICCO a private
integer is represented using one variable of type mpz t from the GMP library [1] and a private float
is represented using four mpz t variables. Once memory for the necessary number of variables is
allocated, each of them also needs to be initialized before it can be used in computation.
Calling free with a pointer in C allows to deallocate the memory (for either a variable or
dynamic array) to which the pointer is pointing. To support similar functionality for private
variables, we implement a function pfree that similarly takes a pointer (to a private variable or
dynamic array) as its only argument. With pfree we distinguish between two different cases: the
2
Note that the choice of the function is not crucial and it can be called malloc instead to simplify programmers’
effort for transforming an existing program to an equivalent program that computes with private data. We, however,
prefer to use pmalloc to make it explicit that the computation refers to private data.
15
pointer provided as an argument to the function has a single known location (i.e., α = 1) or it has
a private location out of a public list (α > 1).
Handling the first case is simple and efficient: we can simply call the free command to deallocate
memory associated with the address stored in the pointer. Pointers to private data with public
locations are very common in programs that use pointers to private data or build data structures
from private data (e.g., linked lists, stacks). Freeing memory used by pointers to private data in
such cases is thus going to be extremely efficient and does not introduce additional overhead.
Handling the second case well, however, is very challenging. This is because deallocating physical
memory results in publicly observable outcomes, and we must be extremely careful not to reveal
the true location stored in a pointer with a private location while at the same time reducing the
program’s memory usage. For example, a simple strategy of deallocating memory associated with
all locations on a pointer’s list of addresses will not be acceptable for some programs. To illustrate
this, consider a dummy example with two pointers p1 and p2, for each of which we allocate memory
using pmalloc. Then the locations to which the pointers are pointing are swapped based on the
result of a private condition evaluation. We obtain that both p1 and p2 now contain two identical
locations in their lists of addresses, but their true addresses are distinct. Suppose we process the
data to which p1 points and want to deallocate the corresponding memory. If we deallocate both
addresses on p1’s list, p2 becomes a dangling pointer and the data to which it was pointing is no
longer accessible. Thus, such an implementation of pfree would be too restrictive to permit its
general use.
Thus, calling pfree(p) should result in deallocating memory associated with only one address
on p’s list of addresses. Furthermore, the address being deallocated cannot depend on any private
data (but can be any function of public data). This means that we are not necessarily deallocating
memory associated with the true location of the pointer and other pointers that store the same
location on their lists must be adjusted to preserve correctness of the computation (which involves
additional resources). For example, we can choose to deallocate the fist location ℓ1 on a pointer’s
list, but if this was not the pointer’s true location (which we can privately check), the data stored at
ℓ1 needs to be relocated and other pointers storing ℓ1 on their lists need to be updated accordingly.
We next describe in more detail how we can realize this idea.
First, if the pointer p on which pfree was called contains the default location corresponding
to uninitialized pointers on its list of addresses L (which is public knowledge), we choose not to
perform memory deallocation. This is to ensure that no memory is being deactivated (which may be
in use by other pointers) if p happens to be uninitialized. Otherwise, we free the first location ℓ1 on
p’s list.3 (Alternatively, the location used by the smallest number of pointers can be freed.) Before
we can actually free the memory, we need to privately update the values stored at the remaining
locations in L using the value stored at ℓ1 to maintain correctness. We will need to ensure that
(i) if ℓ1 happens to be the true location, the values stored in the remaining locations will remain
unchanged and (ii) if ℓ1 is not the true location, the value stored at ℓ1 can be found at p’s true
location, while the values stored at all other locations remain unchanged. The rationale for doing
this as follows: if ℓ1 is indeed p’s true location, no additional work would be required if this fact
was public (i.e., it is the programmer’s job to ensure that freeing p does not affect other variables
still in use). If ℓ1 , however, was not p’s true location, it may be in use by other pointers and the
value stored at ℓ1 needs to be relocated to p’s true location prior to memory deallocation (and the
pointers that contain ℓ1 in their lists need to be updated accordingly).
Let p at the time of calling pfree store α, L = {ℓ1 , . . ., ℓα }, T = {[t1 ], . . ., [tα ]} and A =
3
In the event that data stored at the locations contained in the pointer have different sizes, the location with the
data of the smallest size should be chosen instead.
16
{[a1 ], . . ., [aα ]} denote values stored at locations in L.4 To obliviously update [ai ]’s for 2 ≤ i ≤ α,
we compute
[ai ] = [a1 ] · [ti ] + [ai ] · (1 − [ti ]).
This satisfies the above two requirements as follows: if t1 is true (ℓ1 is the true location) and thus
ti is false, the result will be ai for any i; if ti is true and thus t1 is false, the result will be a1 ; if
both t1 and ti are false, the result will be ai . Surprisingly the formula does not depend on t1 .
Second, we need to update private pointers that store the freed location ℓ1 in their lists (and
are still in use), but no computation needs to be performed for pointers that store any of ℓ2 , . . ., ℓα
from L, but not ℓ1 itself. The rationale for doing this as follows: if ℓ1 is indeed p’s true location, no
additional work would be required if this fact was public (i.e., it is programmer’s job to ensure that
freeing p does not affect other variables still in use). If ℓ1 , however, was not p’s true location, it may
be in use by other pointers and the value stored at ℓ1 is moved to p’s true location prior to memory
deallocation. We thus need to replace ℓ1 in other pointers’ lists with locations that are guaranteed
to include the value originally stored at ℓ1 and update the locations’ tags accordingly. Thus, for
each pointer p’ that stores ℓ1 in its list L′ , we retrieve ℓ1 ’s position pos in L′ and its corresponding
tag t′pos . We then replace ℓ1 in L′ with {ℓ2 , . . ., ℓα } and t′pos in T ′ with {[t′pos ] · [t2 ], . . ., [t′pos ] · [tα ]}.
If any of ℓi for i = 2, . . ., α already appears in L′ , that location is not included the second time and
its tag is set to the sum of the tag already present in T ′ for location ℓi and [t′pos ] · [ti ].
Returning to our example with p1 and p2, we have that prior to calling pfree(p1), p1 stores
α1 = 2, L1 = (ℓ1 , ℓ2 ), T1 = (t1 , t2 ), and p2 stores α2 = 2, L2 = (ℓ2 , ℓ1 ), T2 = (t′1 , t′2 ). Then either
t1 = t′1 = 1 and t2 = t′2 = 0 or t1 = t′1 = 0 and t2 = t′2 = 1. Once pfree(p1) is called, ℓ1 is
scheduled for deallocation. If t1 = 1, no changes take place; otherwise (t2 = 1), the data from
location ℓ1 is copied into location ℓ2 . We obtain that location ℓ1 is being removed from L′ (and
the corresponding tag t′2 from T ′ ) and location ℓ2 is being added to L′ with the corresponding tag
t′2 · t2 . Because ℓ2 is already present in L′ , it is stored once and the tag becomes t′1 + t′2 · t2 . Thus,
we have that L′ now stores a single location and the tag is 1 for any possible set of original tags.
Note that the second step of updating pointers that store location ℓ1 in their lists is more complex
when the pointer p being freed points to a struct or an array. In those cases, multiple addresses
are processed in this step (ℓ1 and other valid addresses that store data at fixed offsets from ℓ1 )
depending on the type of data to which p points. In our implementation, we gather all interactive
operations associated with the execution of a call to pfree and perform them simultaneously in a
single round.
If the user program is written correctly (i.e., does not leave dangling pointers after a call to
free), our implementation of pfree will maintain that for each pointer exactly one location’s tag
is set to 1 and all other locations’ tags are set to 0. When, however, a call to deallocate memory
corresponding to a pointer results in dangling pointers, all tags in such pointers can be 0. For
that reason, if a call to pfree causes the number of addresses for some pointer to reduce to 1, we
do not treat the corresponding tag as public. That is, when a program is not correctly written,
opening the value of the tag may reveal private information, while assuming that the tag is 1 may
modify the program’s behavior. Thus, our implementation maintains privacy even in the presence
of programming errors that result in dangling pointers.
We also note that the use of pmalloc or pfree will not be allowed inside conditional statements
with private conditions because these functions have public side effects.
4
Although in the current discussion we assume p is a private pointer that points to a non-pointer data type, the
same idea will apply when p points to a pointer. In particular, if p points to a pointer, the procedure will include
merging the lists of pointers stored at locations ℓ1 and ℓi and updating the tags similar to the formula for simple
data types. Furthermore, when p is a pointer to a struct, each field is updated separately according to its type.
17
5.3
Accessing Array Elements
The next common use of pointers in programming is manipulating arrays using pointers. Even
for statically allocated arrays, the array name is treated as a constant pointer that points to the
first element of the array. Hence, arrays and pointers are tightly coupled and pointers are used
extensively to work with arrays.
Array indexing. Because arrays are based on pointers, array indexing also applies to pointers.
Thus, we can see constructions such as p = a and p[i], where p is a pointer and a is an array, and
need to support them for pointers to private data. Pointer indexing p[i] with a pointer p to private
data and a public index i is implemented naturally, where we iterate through all locations in the
address list L of p, advance each of them by i multiplied by the size of the data type, retrieve the
data at the determined positions, and combine all of them using private tags for each location to
obtain the result. In other words, the computation is very similar to that of pointer dereferencing,
where instead of retrieving data at the positions specified in L, we advance each position by i data
items. (As C permits the use of negative indices, when i in p[i] is negative each location in L is
decremented by the necessary amount during this operation.)
Pointers as arrays with known bounds. In PICCO, statically allocated arrays of private
variables have the array size stored with them (which is known at the array creation time). Knowing
the size of the arrays allows the compiler to support of a number of important operations on arrays.
Most significantly, this permits the use of private indexing with arrays, when an element at a private
position i is retrieved from an array a using syntax a[i]. (The size of the array must be known to
support private indexing, regardless of what technique is used to implement it.) This also permits
the use of other operations such as inner (or dot) products on two arrays, which were introduced
to optimize runtime of compiled programs.
We treat private indexing as an essential part of secure computation with private data and
would like to see it supported for arrays dynamically allocated on the heap. This means that we
would like to offer pointer indexing p[i] with private i and private pointer p. The main challenge
that we need to overcome is the fact that the size of the memory pointed by p is not available in
C. Furthermore, a location stored in p may be arbitrary and do not correspond to a valid memory
address (i.e., be unaccessible by the program, correspond to memory marked as not being in use
or any location from the program’s stack, etc.). This means that a pointer can take on many
addresses which were not allocated for variable use and for which the corresponding size cannot
be meaningfully determined (i.e., accessing such addresses would trigger invalid memory access
exceptions in safe programming languages). The size of properly allocated memory, however, can
be determined and utilized to implement private indexing (and other operations that require array
size) with pointers to private data. In particular, all memory that malloc allocates on the heap is
marked with the size of each allocated block. Thus, we can use the information that malloc/free
maintain to determine whether a pointer content falls within a properly allocated memory block,
and if it is the case, access the block’s size and use it to implement private indexing.
In more detail, in addition to using private indexing with statically allocated arrays (as already
implemented in PICCO), we permit private indexing to be used with pointers to private data.
The latter is only successful if the location stored in the pointer5 was allocated via a prior call to
pmalloc (and it was not deallocated during a call to pfree). Because the secure implementation
that PICCO produces makes more calls to malloc than once per call to pmalloc, the program
5
The current discussion refers to a single location stored in a pointer, which we view as the most common use of
private indexing. When the pointer contains multiple locations, the operation is performed on each location separately
and the results are combined in the same way as during pointer dereferencing.
18
internally maintains a list of addresses returned by malloc that correspond to memory requested
by the user program (and an address is taken off the list if it is being freed). Then when private
indexing p[i] is called in the user program and the pointer stores address ℓ, we iterate through
the list of maintained addresses. For each such address l, we retrieve the corresponding block size
s from the information stored by malloc and check whether l ≤ ℓ < l + s and the offset of ℓ from l
is a multiple of the data type size. If these checks succeed for at least one location on the allocated
address list, s is adjusted for the data type size and is used as the size of the array to which p
points. Note that with this implementation ℓ does not have to correspond to the beginning of the
memory block. Then when ℓ is not the address of the beginning of the array, i can legitimately
take negative values.
Under the circumstances when the address ℓ does not fall within any memory block dynamically
allocated by the user, private indexing operation is not performed and the returned result is set to
be secret-shares of 0 (note that, regardless to what value the result is set, it is not guaranteed to be
interpreted as an error). We thus proceed with the computation despite the error, but send signal
SIGBUS6 and store an error message in a fixed location, so that the program can catch the signal and
act on it. We note that the address that each call to pmalloc returns is always public information
and the programmer can avoid using invalid addresses. Ideally, the fact that the private indexing
operation cannot be carried out on the given address is determined before the program is run, at
compile time. Unfortunately, this will not always be possible and for some incorrectly written user
programs the error will not be triggered until the program is executed (i.e., even programming
languages that perform static analysis of user programs do array-bounds checking dynamically).
The best we can do is to perform static program analysis at compile time and warn the user about
places where such an error might be possible.
Pointer arithmetic. Pointers can be modified by setting the address to which they point to
the result of an arithmetic expression evaluation. While in C pointers can be used in arbitrary
expressions similar to the way integer variables are used, only a limited set of operations on pointer
variables is meaningful when they are used to store and manipulate addresses within the program.
For example, pointer arithmetic can be relied upon to increment or decrement a pointer value
by an integer amount to move to a different position within an array or between struct fields.
Many other arithmetic operations on pointer variables are not meaningful, and moving between
different variables using pointer arithmetic is unreliable and error-prone. Thus, in PICCO’s default
configuration we chose to disable pointer arithmetic involving pointers to private data in user
programs that the compiler processes. We introduce this as a mechanism for eliminating a large
class of programming errors without constraining expressiveness of user programs. That is, if we
want to change the pointer’s position within an array, instead of using p = p-i or p = p1+4*k+1,
the program will be written as p = &p[-i] and p = &p1[4*k+1], respectively. We note that
disabling pointer arithmetic for pointers to private data in the default configuration should not
be treated as a limitation of the compiler or our approach, but was a deliberate choice to reduce
programming errors without constraining expressiveness of user programs.
Nevertheless, turning off pointer arithmetic in PICCO makes it deviate from standard C. Furthermore, there is a small class of functionalities that become disabled without pointer arithmetic.
One example of such functionalities is the use of embedded linked lists as, for example, implemented
in the Linux kernel. Embedded linked lists might rely on pointer arithmetic to move between different fields of a struct. Thus, if the need for this or similar functionality when working with private
data arises in applications that use PICCO, the compiler can be configured with a command-line
flag to enable support for pointer arithmetic involving pointers to private data. We implement such
6
Alternatively, custom SIGUSR1 or SIGUSR2 can be triggered if the user program is known not to use it.
19
functionality as described below.
As mentioned before, in regular C, pointers can store any integer values and using pointer
variables in arithmetic expressions will result in evaluating the expressions on the integer values
stored in such variables. In the context of pointers to private data, we however distinguish between
pointer objects and (integer) addresses that pointer objects store. Thus, pointer content is no
longer equivalent to integer values, and we support pointer arithmetic with pointer objects only for
the purposes of using pointers to store and manipulate addresses. Such arithmetic operations can
be categorized into two groups:
1. Using pointers to private data in expressions of the type p + exp and p - exp, where exp
is an expression that evaluates to a (public) integer value. This operation produces the same
output as executing &p[exp], i.e., all locations stored in p are advanced by the amount of
space occupied by exp elements of the array.
2. Using pointers to private data in offset computation as in p1 - p2. This operation is straightforward to implement when both pointers store a single location. When, however, at least
one of them has multiple locations, our implementation computes the private difference between the true locations of the pointers. This option, in our opinion, implements the right
semantic value as opposed to other variants (such as computing pair-wise differences between
all addresses stored in the pointers) which bear little meaning.
Note that expressions of the type p + exp, and equivalently &p[exp], where exp evaluated to a
private integer, are not meaningful and not supported.
5.4
Pointer Casting
Variable casting refers to the ability to treat a variable of one type as a variable of another type.
Casting a constant or variable of one type to a constant or variable of another type typically results
in the value being preserved after the conversion (if possible) even if the two types use different
data representations. This means that conversion is likely to involve computation. In PICCO,
conversion between floating point and integer values is based on the algorithms given in [2], while
conversion between integer types of different sizes and floating point types of different sizes requires
minimal to no work (assuming no overflow or underflow detection is required when casting a value
to a shorter representation).
Pointer casting is handled differently and C is unique in the sense of allowing pointer-based
in-memory casting from one data type to another. Pointer casting involves no data conversion: the
memory is read as is and is interpreted as a sequence of elements of another type. Thus, pointer
casting is meaningful between a limited number of data types. In order to support pointer casting
in PICCO, we need to resolve the main question: because data representation of private data types
differs from data representation of the corresponding public data types, we need to determine how
to mimic sizes of public data types when working with blocks of private data without modifying the
data itself. That is, all secret shared values in PICCO are represented as elements of the same field,
which means that, for example, shares of a 16-bit integer and shares of a 64-bit integer have the
same bitlength. A programmer who casts memory storing an array of 64-bit integers to a pointer to
an array of 16-bit integers, however, expects to extract four 16-bit integers from each 64-bit integer.
This means that to meet the programmer’s expectations, private data will need to be processed
and assembled in a different form. We, however, cannot modify the original data because only the
pointer was cast, not the data itself.
20
Instead of duplicating the memory and performing conversion at the time of casting, our solution
is to do the necessary computation at the time of pointer dereferencing. This means that we need
to record information about the data type from which casting was performed (to the data type of
the pointer) at the time of casting, but delay conversion until the pointer is dereferenced. We store
casting data type information with the pointer and use it to extract the relevant portion of the
memory at pointer dereferencing time. Note that in presence of a sequence of casts, only a single
data type needs to be maintained because the memory layout does not change.
Because in PICCO simple data types can be defined to have any bitlength, casting, for example,
a pointer of one integer type to a pointer of another integer type does not guarantee that one data
type will have a bitlength multiple of another. In that case we still calculate what the relevant
portion of the memory is based on the position of the memory being dereferenced, but the last,
partially filled, element might not be reliably extracted. For example, suppose some memory was
filled as a 3-element 30-bit integer array. When it is cast to an array of 20-bit integers, the fourth
elements will be extracted as bits 61–80 of the original data, while retrieving the fifth element might
result in memory violation because there is not enough data in the original array to fully form that
element.
5.5
Pointers to Functions
Similar to pointers to ordinary data types, we need to distinguish between pointers to functions that
will be treated as pointers to private data and pointers to functions that will be treated as pointers
to public data. The former can be used inside conditional statements with private conditions (as
a result of which they acquire multiple locations and the true location becomes private) and are
restricted to functions with no public side effects. The latter can contain pointers to functions of any
type, but cannot be modified or dereferenced inside conditional statements with private conditions.
The distinction is made at the time of pointer declaration using private/public qualifiers with void
data type. That is, by using private void *p, p will be treated similar to other pointers to private
data, while all pointers declared syntax public void *p will be treated as conventional C pointers.
Private pointers to functions are supported naturally in our implementation. When a pointer
stores a single location, the function is invoked as in conventional program execution. If, however,
a pointer acquires multiple locations as a result of its modification inside conditional statements
with private conditions, at the time of pointer dereferencing all functions stored in the pointer will
be executed, but only the effects of one of them will be applied. Conceptually this is the same as
executing branching statements with private conditions: all branches are executed, but only the
effects of one of them are applied depending on the result of private condition evaluation. That is,
when a pointer p storing α locations L = (ℓ1 , . . . , ℓα ) with the corresponding tags T = (t1 , . . . , tα )
is being dereferenced, each function fi stored at address ℓi is being invoked. Then each (private)
variable a that fi modifies is set to a = ai · ti + aorig (1 − ti ), where aorig and ai are its original and
newly computed by
Pfi values. If a is modified
P by multiple fi ’s with indices i1 through ik , its value
is updated as a = kj=1 aij · tij + aorig (1 − kj=1 tij ) (recall that only one ti can be set to 1).
6
Analysis
After discussing multiple aspects of private pointer design and its uses in programming, in this
section we summarize the notion of pointer to private data and operations on it and formally show
that program execution that involves pointers to private data complies with a standard definition
of security used in secure multi-party computation.
21
A pointer to private data is defined as a C-style pointer to a private object storing location
information of the object, where a private object can be one of the following types:
1. a primitive private data type (private int, float, etc.);
2. a composite data type (C-style struct), each element of which is a private object;
3. a function with no public side effects;
4. a pointer to a private object.
The second and forth categories define a pointer to private data recursively, which means that
a pointer can have any indirection level, nested struct types, and any combination of primitive,
composite, and pointer data types. The previously defined operations on pointers to private objects
are:
1. Pointer read and update with value v, denoted as read(p) and update(p, v), respectively.
2. Dereferenced pointer read and update, denoted as read(*p) and update(*p, v), respectively.
3. Pointer update inside a conditional statement with a private condition. Following prior work,
we use multiplexor notation to denote this operation as mux(p, v1 , v2 , cond), where p is set
to v1 if private cond evaluates to 1 and to v2 otherwise. Pointer read inside a conditional
statement with a private condition is processed identically to a conventional read read(p)
and thus is not listed as a separate operation.
4. Dereferenced pointer update inside a conditional statement with a private condition, which
we denote as mux(*p, v1 , v2 , cond). Similar to the previous case, processing of dereferenced
pointer reads is not affected by the presence of conditional statements.
5. Dynamic memory allocation in the form of malloc; for an assignment p = pmalloc(n, type)
we use notation alloc(p, n, type).
6. Dynamic memory deallocation as in pfree(p), denoted as dealloc(p).
7. Array indexing p[i] with a public index i. This is treated as a generalization of pointer
dereferencing, and we use notation read(p, i), update(p, i, v), mux(p, i, v1 , v2 , cond) to
denote read, update, and update inside a conditional statement with a private condition,
respectively.
8. Array indexing p[i] with a private index i is also divided into three operations readp(p, i),
updatep(p, i, v), and muxp(p, i, v1 , v2 , cond). These operations can only be performed
on locations stored in a pointer that correspond to arrays with known bounds (i.e., allocated
using the pmalloc interface or static array declaration).
9. Evaluation of predicate f on one or more pointers p1 , . . ., denoted as pred(f , p1 , . . . ).
10. Pointer casting, denoted as cast(p, type), where type is the data type to which p is cast.
In the above, v, v1 , and v2 correspond to either values associated with private objects or data
structures corresponding to pointers to private objects, depending on the context. The value of
cond is always a private bit, while variables n, i, and type are public. Any variable can be read
inside a conditional statement with a private condition, but updates can be performed only as
specified using the mux operations.
22
This list implicitly defines operations that cannot be performed on pointers to private objects
and will be rejected by the compiler. That is, addresses of public objects cannot be used to update
a pointer to private data (either via update or mux); a pointer to public data or a mix of private
and private fields defined with a struct construct cannot be modified inside a conditional statement
with a private condition (i.e., there is no corresponding mux operation); pmalloc and pfree cannot
be called inside conditional statements with private conditions (i.e., there is no mux operations for
them); similarly, casting cannot be called inside conditional statements with private conditions as
it modifies publicly stored data.
Recall that our implementation of pointers maintains the invariant that in a well-formed program there is exactly one true location associated with a private object. The invariant may be
violated only when memory associated with a pointer is being deallocated when the pointer is still
in use (in such a case, no tag is set to 1 and dereferencing the pointer will return no data).
In showing security of pointer-related operations that reference private data, we use a traditional
simulation-based definition of security. Because PICCO is built on top of (n, t)-threshold secret
sharing techniques with n ≥ 3 computational parties, we utilize the same setup in our security
analysis. Similarly, because the underlying techniques offer information theoretic security, we utilize
statistical (as opposed to computational) indistinguishability in the security definition.
Definition 1. Let parties p1 , . . ., pn engage in a protocol Π that evaluates program P on a mix of
public and private data. Let VIEWΠ (pi ) denote the view of pi during the execution of Π, which is
formed by its input, internal random coin tosses ri , and messages m1 , . . ., mk passed between the
parties during protocol execution: VIEWΠ (pi ) = (ini , ri , m1 , . . ., mk ). Let I denote a subset of the
participants of size t and VIEWΠ (I) denote the combined view of the participants in I during the
execution of Π. Protocol Π is said to be t-private in the presence of semi-honest adversaries if for
each coalition of size at most t < n/2 there exists a probabilistic polynomial time simulator SI that
given the input of the parties in I, P , and P ’s output, produces a view statistically indistinguishable
from VIEWΠ (I) together with the output of the parties in I.
Theorem 1. Any program P augmented with pointers to public and private data and no out-ofboundary access compiled by PICCO is translated into a t-private protocol for any t < n/2 when
the computation is carried out by n parties.
Proof. Our proof proceeds by evaluating each operation involving a pointer to a private object as
summarized above. After building a simulator for each pointer-related operation, we apply the
composition theorem of Canetti [6] to the result, which would guarantee that any combination of
these operations (and other secure operations in PICCO) results in security of the overall program
P that the computational parties execute. Building our simulator requires only the use of t-private
implementations of addition/subtraction and multiplication operations (which is met in PICCO by
the underlying linear secret sharing scheme with t < n/2).
We describe a simulator for each operation involving a pointer to a private object in turn. Prior
to each operation, each party in the adversarial coalition I holds a share of each relevant private
data item (including private input data, private fields of pointer data structures, etc.) and at the
time of operation termination each party in I holds a share of the output and/or updated private
items. Many functions also modify data publicly available to each party (including the simulator).
• read(p): This operation simply retrieves the data structure stored in p and is local to each
computational party. The simulator does not interact with the parties in I.
• update(p, v): The data structure contained in v is simply copied into p by each party. The
simulator does not interact with the parties in I.
23
• read(*p): When p stores only a single location, the value stored at that location is retrieved
and the simulator does not interact with the parties in I. When p stores multiple locations,
each party is instructed to iterate through the locations extracting values stored in them and
then combine the values using the private tags associated with each location as described in
Sections 4.2 and 4.3. Most of the procedure operates on public values (such as locations) and
the only private computation consists of multiplying tags with private data (or other tags in
case of pointers to pointers). The simulator thus participates in each multiplication operation
on behalf of honest users by invoking a simulator corresponding to the multiplication operation
the necessary number of times.
When p is a pointer to an object of a complex data type declared using struct and a single
field of p is being dereferenced (as in p->x), the way the simulator interacts with the parties
in I is not affected. (Only the locations from which values are retrieved are locally modified
by a known offset by each party.)
• update(*p, v): Similar to reading a referenced pointer, when p is associated with only one
location, value v is stored in that location without the parties interacting with the simulator.
When p stores multiple locations, each party in I iterates through all possible locations and
updates all locations using secure multiplications as described in Sections 4.2 and 4.3. Because
the locations (and their number) are always public, the simulator only needs to engage in a
pre-determined number of secure multiplications simulating all honest users, which is does by
invoking a simulator of the secure multiplication protocol.
When p points to a struct and only one field p->x is to be updated, the simulator’s interaction
with the parties in I is identical (but the valued are retrieved and the computed values are
placed not at the locations stored in p but the locations adjusted by the offset of x in the
struct).
• mux(p, v1 , v2 , cond): To implement conditional assignment, each party (and the simulator)
first locally merges the lists stored in v1 and v2 , after which the (private) tags needs to be
updated based on the private condition cond as described in Algorithm 1. The simulator only
needs to participate in a certain number of secure multiplications determined by the public
contents of v1 and v2 , which it performs as before by invoking a multiplication operation
simulator.
• mux(*p, v1 , v2 , cond): Conditional update of a dereferenced pointer involves modifying the
value stored at each location in p with a fixed function of v1 , v2 and cond (see Section 4.2).
Thus, the simulator participates in a pre-determined number of secure multiplications simulating all honest users using a simulator for secure multiplication.
• alloc(p, n, type): Allocating memory for a number n of private objects of type type does not
involve secure computation and thus the simulator does not interact with the parties in I for
the purpose of this operation.
• dealloc(p): First, if one of the locations stored in p corresponds to the special “uninitialized”
value, no party (including the simulator) performs any operation. Otherwise, the first location
ℓ1 is removed from the data structure that p stores and the values at all other locations
are updated using two secure multiplications (as described in Section 5.2). The simulator
produces communication on behalf of all honest users to simulate invocations of the secure
multiplication protocol as before. The parties consequently locate other pointers that store
ℓ1 and update their locations and tags using a procedure that depends only on public data.
24
The simulator is invoked to simulate a necessary number of secure multiplications on behalf
of honest users using the simulator of secure multiplication.
• read(p, i): Similar to a dereferenced pointer read, the simulator will need to simulate 0 or
more invocations of secure multiplications on behalf of honest users. The only difference from
the simulation of read(*p) is that the pointer locations are (locally) adjusted by the space
occupied by i items by each party (including the simulator) during the computation.
• update(p, i): This procedure is also very similar to update(*p), where the difference is only
in the local (publicly available) computation.
• mux(p, i, v1 , v2 , cond): Similar to mux(*p v1 , v2 , cond), the simulator will need to participate
in a pre-determined number of secure multiplications simulating all honest parties using its
simulator after some computation on public data.
• readp(p, i): To retrieve an element of an array represented by a pointer using a private
index, for each location stored in p that corresponds to a properly allocated memory block,
the parties together with the simulator perform a private table lookup. This operation is
implemented in PICCO by reading each element of the array and can be easily simulated
once the array size is determined using information publicly stored by each computational
party. Thus, for each eligible address stored in p the simulator simulates private table lookup
operation on behalf of honest users by invoking a private table lookup simulator, after which
the results from multiple locations (if present) are combined together using private tags,
during which the simulator engages in secure multiplication simulation.
• updatep(p, i, v): This operation proceeds similar to readp(p, i), but with some operations
performed in a different order. Each party in I and the simulator first locally determine
eligible addresses in p using public information, after which the parties and the simulator
need to modify the data being stored for each location using the location’s tag through secure
multiplication and then engages in the interaction for table update at a private location.
Thus, the simulator needs to engage in a pre-determined number of secure multiplication
simulations and then in simulating interaction corresponding to private table updates. As
before, the simulator can simply invoke the corresponding simulators for secure multiplication
and table updates at a private location.
• muxp(p, i, v1 , v2 , cond): Conditional update of an array element at a private index is
performed similar to the regular update. Here, the parties and the simulator can first combine
data of v1 and v2 using cond into a single v, where the simulator will need to engage in
simulating a fixed number of secure multiplications. After this they can follow the steps of
updatep(p, i, v) with the simulator producing messages as described for the simulation of
that operation.
• cast(p, type): This operation only updates public information associated with pointer p and
the simulator does not interact with the parties in I.
• pred(f , p1 , . . . ): The simulator proceeds by evaluating the operations in f in order. For
any operation that uses a single pointer, the simulation proceeds according to one of the
above cases. For any operation that uses two pointers (most significantly, pointer equality
or inequality testing), the simulator first computes the public/private status of the result.
If the status is public, the result is determined locally. Otherwise, the simulator engages in
the computation associated with determining the (private) result by invoking the simulators
25
corresponding to the operations used in the computation. Once any intermediate result in
evaluating f is computed to be private, any computation that uses the result is carried out
on private data by invoking simulators corresponding to the operations used in f .
This covers all possible operations with pointers to private objects and concludes the proof.
We note that our security result above is stated for programs that PICCO successfully compiles
and that do not contain out-of-boundary array accesses. As previously discussed, the compiler will
reject almost all types of improperly written programs during static analysis at compilation time.
For example, if a pointer to public data is modified inside a conditional statement with a private
condition, the program will not compile. This means that the compiler will filter out the majority
of programs with errors, while any well-formed program enjoys simulatable security.
The main type of errors that the compiler may not be able to detect during static analysis deal
with memory access to invalid locations (e.g., out of boundary array access, access to a hard-coded
invalid location, etc.). When such accesses are triggered, one or more computational parties might
not be able to continue with the execution and quit on error, but no privacy violations take place.
This is because while incorrect shares of private data might be read, the data will still remain in
the protected form and cannot be reconstructed without the programmer’s original intent. Note
that this discussion applies only to dereferencing a pointer with no offset (e.g., as in *p) or with
a publicly known offset (e.g., as in p[i] with public i) because in our implementation accessing
array at a private location never results in out of boundary access (i.e., calling this operation on
an improperly allocated array will be skipped at runtime).
Because the memory layout may differ on different platforms, illegal memory accesses might
result in different behavior of the computational parties (e.g., execution at one party might result
in a segmentation fault faster than at another party). As our simulator is not guaranteed to run in
an identical environment to those of the honest parties, the simulation might be distinguishable for
programs with illegal memory accesses based on the point of execution when any given party aborts
the computation. Thus, we exclude such programs from the security statement of Theorem 1, but
still guarantee that any program that compiles in our framework will not result in privacy violations.
To summarize, our solution guarantees that any program that can be compiled in our framework
never reveals any unauthorized information about private data. Simulation of a poorly written
program with illegal memory accesses may not be indistinguishable from its real execution, but any
properly written program enjoy simulation-based security.
7
Pointer-Based Data Structures
There are several popular data structures typically built using pointers. In this section we discuss
how they would be implemented using pointers to private data and in what complexities their
performance results. In particular, we explore linked lists, trees, stacks, and queues.
7.1
Linked Lists
A linked list consists of a sequentially linked group of nodes. For a singly linked list, each node is
composed of data and a reference in the form of a pointer to the next node in the sequence, while
for more complex variants such as doubly and circular linked lists the reference field incorporates
additional links. A linked list allows for efficient node insertion and removal, which makes it an
ideal candidate for implementation of stacks and queues as well as representation of graphs that
uses an adjacency list. In what follows, we discuss implementations of linked lists that store private
26
data. We start by analyzing various operations in standard linked lists and then elaborate on the
special case when a linked list stores sorted data. The latter does not represent a typical use of
linked lists in programming (and does not necessarily have attractive features), but is provided as
a relatively simple way to demonstrate what form working with sorted data can take in a secure
computation framework.
Standard linked lists. Because of ubiquitous use of linked lists in programming, we analyze
different possible uses of linked lists and the corresponding operations. When a linked list stores
public data, node insertion has cost O(1) as a node is inserted in a fixed place (e.g., beginning or
end of the list). Performing a search requires O(n) time, where n is the number of nodes in the list,
because the nodes are traversed sequentially. Deleting a node from a fixed place (i.e., beginning
or end of the list as done in the case of stacks and queues) involves O(1) time, but when deletion
is preceded by a search (and the found node is deleted), the search together with deletion require
O(n) time.
When a linked list stores private data, the reference field holds a pointer to private data (i.e.,
a record of the same type) and at the time of node creation, the pointer stores a single location.
Without loss of generality, let nodes be inserted in the beginning of the list. Node insertion then
places a new node in the beginning of the list manipulating pointers as before, which still takes
O(1) time and is very efficient. Searching a list involves n private comparisons and all nodes need
to be processed as not to reveal the result of individual comparisons on private data and the total
work is O(n). Similarly, when a node is deleted from the beginning (or end) of the list, the time
complexity of the operation is O(1) and each node’s pointer still stores a single location. It is only
when nodes need to be removed from varying positions in the list and the position itself needs to
be protected, pointers can start acquiring multiple locations, which causes the time complexity of
list traversal and deletion after a search to go up. However, when the fact whether the searched
data was found in the list or not must remain private, we cannot remove any node, but instead
need to erase the content of a found node (if present) with a value that indicates “no data”. In this
case, all pointers still contain a single location and the cost of list search and other operations do
not change, but the list will never reduce in its size. We defer the discussion of the case when the
node is guaranteed to be found in a search and needs to be removed from a private location until
the end of this subsection.
Throughout this section, we express complexities of the operations as a function of n, where n
is the “visible” list size. This value will correspond to the actual list size when delete operations
remove an element from the list so that its size is reduced, while it will correspond to the number of
insertions (i.e., the maximum list size) when delete operations only mark data as deleted (to hide
whether the element was found or not), but not reduce the list size. The same notation applied to
other data structures discussed in this section when insertions/deletions are based on the result of
private conditions.
Sorted linked lists. As mentioned before, we discuss sorted linked lists only as a means of
demonstrating how sorted data might be processed using a general-purpose secure computation
compiler and it should be understood that this is not a typical use of linked lists or even not the
best way of working with sorted data. We use the results of this discussion in our consecutive
description.
Now when a node is being inserted in a linked list, the insertion position must be determined
based on the data stored in the list, which involves O(n) time with public data (and the complexities
of other operations are the same as before). When we work with private data, the location where
the node is being inserted must remain private (since it depends on private data) and the execution
needs to simulate node insertion at every possible position. Consider the following two ways of
27
inserting a node and the performance in which they result:
1. Pointer updates: The first is a traditional implementation of node insertion in a linked list,
where if the correct insertion point is found, we update the pointer of the found node in the
list to point to the new node and the pointer of the new node to point to the next node in the
list. Because this conditional statement is based on private data, this will result in adding
one location to the pointer in the found node and one location to the pointer in the node
being inserted. After executing this operation for every node of the list, the pointer of each
node in the original list stores 2 locations and the pointer in the newly inserted node stores n
locations. When this operation is performed repeatedly, each node in the list acquires more
and more locations (to the maximum of the current list size). This means that if the list is
built by inserting one node at a time, the cost of node insertion and list traversal becomes
O(n2 ). Node deletion after a search also takes O(n2 ) time, while node deletion from a fixed
location is bounded by O(n). When, however, only a constant number of nodes are inserted
into an existing list (which, e.g., can be provided as sorted input into the program), the
complexity of all operations are unchanged from the public data case.
2. Data updates: Another possible implementation of sorted linked lists is to always insert a
new node at the beginning of the list and keep swapping its content with the next node
on the list until the correct insertion point is found. When this algorithm is implemented
obliviously on private data using an SMC compiler, the computation processes each node on
the list starting with the newly inserted node and based on the result of private comparison
of current and new data either performs the swap or keeps the data unchanged. After each
node insertion the reference field of each node still points to a single node in the list and
therefore the complexity of all operations are unchanged from their public versions.
Thus, it is clear that we want to avoid acquiring a large number of locations in each reference
field of a pointer-based data structure and privately moving data (as opposed to privately moving
pointers) is preferred when working with sorted data.
Node deletion in standard linked lists. We can now return to the question of deleting a node
from a private location in a standard (unsorted) linked list when it is known that the searched node
is present and needs to be removed from the list. The above two approaches of inserting a node in a
private position also apply to deleting a node from a private position. The first, standard, approach
of manipulating pointers will result in acquiring multiple locations at each pointer, which degrades
performance of all operations. Using the second approach of data updates, we can obliviously place
the data to be deleted into the first node on the list (after scanning the nodes and swapping values
based on private data comparisons) and then simply remove it from the list. This will maintain
optimal complexities of all operations. The above tells us that traditional implementations of
data structures can exhibit performance substantially worse than alternative implementations in
a secure computation framework and our analysis can be viewed as a step in making informed
decisions about implementation needs.
7.2
Trees
Trees implement hierarchical data structures commonly used to store sorted data and make searching it easy. A tree node is typically comprised of data and a list of references to its child nodes. In
an n-node balanced search tree, all of searching, node insertion, and node deletion take O(log n)
time. Unfortunately, these complexities greatly change when we write a program to implement a
28
search tree on private data. In what follows, we distinguish between trees that are pre-built using
the information available prior to the start of the computation and trees built gradually using
information that becomes available as the computation proceeds:
1. Pre-built trees. Consider a balanced binary search tree and suppose that we want to perform
a search on the tree. A traditional implementation involves O(log n) conditional statements
to traverse the tree from the root to a leaf choosing either the left or right child of the current
node. When the data is private, such statements use private conditions and thus both branches
of the computation must be executed. The result is that the sequence of O(log n) nested
private conditions results in executing all possible O(n) branches of the computation and
touches all nodes in the tree. This is an exponential increase in the complexity compared to
working with public data, even if we do not consider node insertions and deletions that result
in node rotations to balance the tree (which are discussed next together with gradually-built
trees).
2. Gradually-built trees. By analogy with inserting nodes into a sorted linked list, we can either
manipulate pointers to insert a new node at the appropriate place in the tree or insert the
node in a fixed location and move the data in place. The complexity of the latter option is
O(n) for insertions, deletions, and search and we take a closer look at the former. As we
traverse the tree looking for the place to insert the new node, similar to searching, all nodes
will be touched (as a result of nested private conditions). Furthermore, because the execution
cannot reveal the place into which the new node is inserted, pointers in all nodes will acquire
new locations. If we add computation associated with node rotations when the tree becomes
unbalanced, pointers will be acquiring new locations even faster (to the maximum of n − 1 per
pointer). After repeatedly calling insert to gradually build the tree, eventually each node will
point to all other nodes resulting in O(n2 ) complexity for insertions, deletions, and searching.
Such complexity is clearly avoidable and alternative implementations should be pursued.
Search trees represent the worst possible scenario where implementing an algorithm on private
data using a general-purpose compiler incurs an exponential increase in its runtime compared to
the public data counterpart. As is evident from our discussion of linked lists and trees, searching
an n-element store for a single element cannot be performed in less than linear time using generic
techniques, regardless of whether the data is stored sorted or not. It means that without custom,
internally built implementations of specific data structures it is conceptually simpler and more
efficient to maintain data in unsorted form, use append for insertion (O(1) time), and shift data to
implement deletion.
7.3
Stacks
A stack is characterized by the last-in, first-out (LIFO) behavior, which is achieved using push and
pop operations. It has several fundamental applications such as parsing expressions (e.g., parsing
programs in compilers), backtracking, and implementing function calls within an executable program. To the best of our knowledge, despite its popularity, this data structure has not been studied
in the context of secure multi-party computation before and our analysis and consecutive implementation of stack that works with private data demonstrate its appeal for secure computation.
A pointer-based implementation of a stack is built using a linked list, where a node is always
inserted at the head of the list and is always removed from the head as well, either of which
takes O(1) time. As was discussed in section 7.1, implementing these operations on private data
maintains constant time complexities.
29
When using a stack with private data, we also consider the possibility that push and pop
operations might be performed inside conditional statements with private conditions, in which case
it is not publicly known whether the operation takes place and what record might be on top of the
stack. Then if we implement a conditional private push operation by manipulating pointers, the
top of the stack will store m + 1 locations when the last m push operations were based on private
conditions. Implementing a push operation is then equivalent to executing the code:
1. node p = new node();
2. if (priv-cond)
3.
p->next = top;
4.
top = p;
Because both p and p->next store only a single location at the time of conditional push, merging
the lists of p->next and top takes O(m) time. Similarly, merging the lists of top and p takes O(m)
time.
Implementing a pop operation within a private condition involves executing code:
1. if (priv-cond)
2.
temp = top;
3.
top = top->next;
4.
// use temp
The complexity of this operation is dominated by the second assignment. Because top points to
O(m) locations, and the next field of each of its locations can store O(m) locations as well, the
overall complexity of that assignment is O(m2 ). This means that the worst time complexity of a
conditional push becomes O(n) for a stack containing n records and it is O(n2 ) for a conditional
pop.
If we instead implement push and pop operations that depend on private conditions by maintaining a single chain of records (with pointers containing a single location) and data update, push
and pop operations result in O(1) and O(n) work, respectively. That is, we can always insert a
new node (with data or no data depending on the private condition) into the stack and take O(n)
time during pop to privately locate the first node with data (and erase the data as necessary).
7.4
Queues
Queue is another important data structure used to maintain a set of entities or events in a specified
order which are waiting to be served. We can distinguish between first-in, first-out (FIFO), last-in,
first-out (LIFO), and priority queues. Implementing a queue involves maintaining two pointers:
the head and the tail. The head points to the beginning of the queue, i.e., the element that will be
removed by a dequeue operation, and the tail points to the last element added to the queue using
an enqueue operation.
Similar to the stack, when enqueue and dequeue operations in a FIFO queue are implemented
on public data or private data outside of private conditional statements, their complexities are O(1).
Their complexities for enqueue and dequeue operations are also O(n) and O(n2 ), respectively, when
implemented through private pointer manipulation (the implementation needs to maintain two
pointers for the head and tail of the queue, but updating the second pointer does not asymptotically
increase the amount of work) and O(1) and O(n), respectively, when private data update is used.
In a priority queue, each node additionally stores priority (which we assume is private) and
dequeue removes a node with the highest priority. The complexity of priority queue operations
30
Data structure
Linked list
Linked list (delete at private location)
Search tree
Stack or queue
Stack (conditional private push & pop) or
queue (conditional private enqueue & dequeue)
Priority queue
Priority queue (conditional private enqueue & dequeue)
Insert
O(1)
O(1)
O(n)
O(1)
Delete
O(1)
O(n)
O(n)
O(1)
Search
O(n)
O(n)
O(n)
—
O(1)
O(n)
—
O(1)
O(n)
O(1)
O(n)
O(1)
O(n)
—
—
—
Table 1: Performance of various data structures using pointers to private data.
depends on the underlying data structure used to implement it. The best known complexities for
public data are O(log n) for enqueue (O(1) average case) and O(log n) for dequeue using a heap.
Suppose for now that all operations are outside conditional statements with private conditions.
If we use a linked list to store queue nodes, the best performance can be achieved using O(1) for
enqueue and O(n) for dequeue (i.e., always store a newly inserted node in the beginning and remove
the highest priority node from a private location as a result of the search for the highest priority
element) or O(n) for enqueue and O(1) for dequeue (i.e., store the list sorted and always remove the
first node during dequeue). Then if the operations depend on private conditions, we can maintain
O(1) for enqueue and O(n) for dequeue if the operations depend on private conditions using a very
similar approach to that of regular queues and stacks. That is, we always insert an element into the
beginning of the queue as a result of a conditional enqueue (if the condition is false, the element is
empty), and during dequeue we scan the queue for the highest priority element and erase it from
the queue if the condition is true.
If the underlying implementation is a heap, we insert a new node in a fixed leaf location and use
O(log n) compare-and-exchange operations to maintain the invariant of a max-heap to implement
enqueue. Realizing dequeue, however, requires O(n) work because it cannot be revealed what path
was traversed from the root to a leaf (since the path choice depends on private priorities). Similar
to other implementations, we can maintain these complexities even when enqueue and dequeue are
performed as a result of private condition evaluation.
7.5
Summary
Before we conclude this section, we would like to summarize performance of different data structures
that can be implemented on private data using newly introduced pointers to private data or records.
Table 1 lists the best performance we could achieve using a pointer-based implementation of the data
structures discussed in this section. Recall that in all data structures with conditional operations
performance depends on the number of insertions as opposed to the actual data size. We note that
the complexities in Table 1 can be directly linked to the amount of memory consumed by those
data structures, with small fixed constants hidden behind the asymptotic notation.
These data structures can also be evaluated using alternative mechanisms. For example, our
analysis suggests that implementing these data structures using arrays of private data instead of
pointers to private data would result in the same complexities (which is often the case for public
data as well). Also, utilizing ORAM-based implementation can improve asymptotic complexity
of some (but not all) data structures and can lead to faster runtime in practice at least for large
enough data sets. The most pronounced benefit of using ORAM will be observed for implementing
search trees, where all operations can be performed in polylogarithmic (in n) time (e.g., using the
31
solution in [22]). On the other hand, using ORAM for linked lists can only increase the complexity
of its operations (even the complexity of a delete at a private location following a search cannot be
reduced below O(n)). Other data structures that can benefit from ORAM-based implementations
are stacks and queues where the operations that update the data structures are performed inside
private conditional statements. ORAM techniques, however, involve larger constants behind the
big-O notation than simple operations and their initial setup cost is also significant. We thus
leave a thorough comparison of ORAM vs. pointer or array based implementations of various data
structures in this framework as a direction of future work.
8
Performance Evaluation
In this section, we report on the results of our implementation and evaluation of a number of
representative programs that utilize pointers to private data. Because such programs have not been
previously evaluated in the context of secure multi-party computation, we cannot draw comparisons
with prior work. In some cases, however, we are able to measure the cost of using pointers, or
the cost of a pointer-based data structure, in a program by implementing the same or reduced
functionality that makes no use of pointers. Note that because PICCO can be used for both secure
multi-party computation and outsourcing, the inputs in these programs can come from one or more
input parties/clients.
The programs that we implemented and evaluated as part of this work are:
1. The first program constructs a linked list from private data read from the input and then
traverses the list to count the number of times a particular data value appears in the list.
This is a traditional implementation of a linked list, where each record with private data is
prepended to the beginning of the list when building it. The program is given in Figure 1.
We next notice that this program is sub-optimal in terms of its run time because it does
not utilize concurrent execution capabilities provided in PICCO. For that reason, we also
implement an optimized version of this program. The difference is that all private comparisons
during the list traversal are executed in a single round using PICCO’s batch constructs.
2. To evaluate pointer-based implementations that work with private data maintained in a sorted
form, and more generally privately manipulating pointer locations vs. obliviously moving
data, we build a program for a sorted linked list. The functionality of this program is similar
to that of the first program (i.e., create a linked list and then traverse it to count the number
of occurrences of a given data item in it) and the difference is in the way the list is build. We
evaluate two variants of the program corresponding to pointer update (PU) and data update
(DU) as described in section 7.1. The program for the DU variant is given in Figure 2, and
the program for the PU variant is given in Figure 3.
3. The third program implements mergesort that takes an array of unsorted integers as its input.
The program makes an extensive use of pointers to private data to pass data by reference to
a function that conditionally swaps two data items based on their values (i.e., performs the
so-called compare-and-exchange operations). Mergesort was chosen not necessarily because it
provides the best performance for an oblivious sort, the objective instead was to demonstrate
how performance of a program that utilizes pointers to private data (and exercises modular
design of a program) compares to a similar program that does not use pointers. We thus
also evaluate another version of mergesort that performs compare-and-exchange operations
in place (without calling any function) and makes no use of pointers. The pointer-based
mergesort is given in Figure 4 and its non-pointer-based implementation is given in Figure 5.
32
struct node {
private int data;
struct node *next;
};
public int count = 128;
public int main() {
public int i;
private int array[count], output;
struct node *ptr, *head = 0;
smcinput(array, 1, count);
//construct the list
for (i = 0; i < count; i++) {
ptr = pmalloc(1, struct node);
ptr->data = array[i];
ptr->next = head;
head = ptr;
}
//traverse the list
privaate int val = 10;
ptr = head;
for (i = 0; i < count; i++) {
if (ptr->data == val)
output = output+1;
ptr = ptr->next;
}
smcoutput(output, 1);
return 0;
}
Figure 1: Construction and traversal of a linked list.
4. Our last program implements a shift-reduce parser for a context-free grammar (CFG) on
private data. This is one of fundamental applications that can now be naturally implemented
using the compiler by building and maintaining a stack, once support for pointers to private
data is in place. We choose a CFG that corresponds to algebraic expressions consisting of
additions, multiplications, and parentheses on private integer variables, which is specified as
follows:
statement → statement | statement + term
term → term | term * factor
factor → var | (statement)
Here, all variables are shown in italics, while terminals are set in bold font. The grammar can
obviously be generalized to more complex expressions and programs that work with private as
well as public variables of different types. We view this application as enabling one to evaluate
a custom function on private data without writing and compiling a separate program for each
function. That is, both the function to be evaluated and its input (consisting of private data)
are provided as input to the parser. We note that it is possible for the function or the grammar
rules to be private as well, but this would result in an increase in the program performance.
33
struct node {
int data;
struct node *next;
};
public int count = 128;
public int main() {
public int i, j;
private int array[count], output, tmp;
struct node *head, *ptr1, *ptr2;
smcinput(array, 1, count);
//construct the list
head = pmalloc(1, struct node);
head->data = array[0];
for (i = 1; i < count; i++) {
ptr1 = pmalloc(1, struct node);
ptr1->data = array[i];
ptr1->next = head;
head = ptr1;
ptr2 = head;
for (j = 0; j < i; j++) {
if (ptr2->data > ptr2->next->data) {
tmp= ptr2->data;
ptr2->data = ptr2->next->data;
ptr2->next->data = tmp;
}
ptr2 = ptr2->next;
}
}
//traverse the list
private int val = 10;
ptr1 = head;
for (i = 0; i < count; i++) {
if (ptr1->data == val)
output = output+1;
ptr1 = ptr1->next;
}
smcoutput(output, 1);
return 0;
}
Figure 2: Construction and traversal of a sorted linked list (using data update).
Our parser uses one lookahead character, and due to the complexity of the implementation,
the program itself is not included in the paper.
To approximate performance overhead associated with using a pointer-based stack, we create
a program that performs only arithmetic operations on private data which are given to the
parser and which the parser executes. Note that unlike evaluation of mergesort, these are
34
Program
Field
size (bits)
Linked list
(list building, traversal,
and optimized traversal)
Shift-reduce parser
Arithmetic operations
81
33
33
25
0.0004 ± 5%
0.086 ± 1%
0.026 ± 7%
0.005 ± 9%
0.005 ± 9%
28
0.003 ±
0.661 ±
0.140 ±
0.039 ±
0.038 ±
4%
1%
3%
2%
3%
Data
211
0.014 ± 3%
5.30 ± 3%
1.019 ± 1%
0.307 ± 1%
0.294 ± 1%
size
214
0.097 ±
42.27 ±
8.051 ±
2.439 ±
2.336 ±
2%
1%
2%
1%
1%
217
0.760 ±
337.8 ±
63.75 ±
19.73 ±
18.85 ±
1%
1%
1%
2%
1%
220
5.40 ± 1%
2,692 ± 2%
513.8 ± 1%
157.2 ± 1%
150.8 ± 1%
Table 2: Performance of representative programs with unsorted data structures measured in seconds.
Program
Sorted linked list (DU)
(list building and traversal)
Sorted linked list (PU)
(list building, traversal,
and head node removal)
Mergesort without pointers
Mergesort with pointers
Field
size (bits)
81
81
81
81
Data size
24
0.466 ±
0.036 ±
1.464 ±
0.051 ±
0.005 ±
0.053 ±
0.053 ±
1%
1%
1%
1%
1%
5%
4%
25
1.908 ±
0.071 ±
9.956 ±
0.149 ±
0.015 ±
0.121 ±
0.122 ±
1%
1%
1%
2%
1%
5%
5%
26
7.750 ±
0.142 ±
85.51 ±
0.613 ±
0.044 ±
0.271 ±
0.272 ±
1%
1%
2%
2%
2%
5%
5%
27
31.24 ±
0.284 ±
918.6 ±
5.285 ±
0.174 ±
0.625 ±
0.638 ±
1%
1%
2%
2%
2%
4%
6%
28
125.5 ±
0.567 ±
9,900 ±
45.93 ±
0.720 ±
1.453 ±
1.503 ±
1%
1%
3%
2%
3%
4%
5%
29
565.5 ± 1%
1.311 ± 1%
N/A
N/A
N/A
3.124 ± 4%
3.201 ± 5%
Table 3: Performance of representative programs with sorted data structures measured in seconds.
not equivalent functionalities. That is, one program is much more complex, parses its input
according to the CF grammar, maintains a stack, etc., while the other only performs additions
and multiplications.
Note that most of these programs already exercise dynamic memory allocation (i.e., all linked list
programs and the shift-reduce parser). However, to provide a more complete evaluation of dynamic
memory management, we also include experiments that measure the overhead of dynamic memory
deallocation. Thus, we incorporate calls to pfree to two programs: (i) we call pfree as part of
the shift-reduce parser at the end of each pop operation and (ii) we evaluate the cost of removing
the head node in a sorted linked list built using pointer update (Figure 3). These were chosen as
natural applications of memory deallocation, where pointers to private objects contain a single and
multiple locations, respectively. In the second case, the head stores locations of all nodes on the
list and the overhead of pfree includes updating the structures of other pointers on the list upon
memory deallocation.
Each program was compiled using PICCO, extended with pointer support as described in this
work, and run in a distributed setting with three computational parties. All compiled programs
utilize the GMP library for large number arithmetic and OpenSSL to implement secure channels
between each pair of computational parties. We ran all of our experiments using three 2.4 GHz
6-core machines running Red Hat Linux and connected through 1Gb/s Ethernet.
Each experiment was run 10 times, and we report the mean time over all runs and the corresponding deviation from the mean observed in the experiments. The results of the experiments for
working with unsorted and sorted data are given in Tables 2 and 3, respectively.
As can be seen from the tables, each program was run on data of different sizes. For all linked
lists programs as well as mergesort, the data size corresponds to the number of elements in the
input set, while for the shift-reduce parser and arithmetic operations the size corresponds to the
number of arithmetic operations in the formula, which were a mix of 90% multiplications and 10%
additions. All linked list experiments contain two different times, which correspond to the times to
build and traverse the linked list, respectively. The tables also report the size of field elements in
bits used to represent secret shared values. While all programs were written to work with 32-bit
35
integers, most programs in the table use statistically secure comparisons, which requires the length
of the field elements to be increased by the statistical security parameter (which we set to 48).
(The size of the field elements needs to be the size of the data plus one bit to ensure that all data
values can be represented.)
The results tell us that working with linked lists (the first program in Table 2) in the secure
computation framework is very efficient. That is, building a linked list that consists of thousands
of elements takes a fraction of a second. Traversing a linked list is also rather quick, where going
through a linked list of size 211 about 1 second in our optimized program.
Performance of the sorted linked lists (the first two programs in Table 3) characterizes performance expected from different data structures where it is necessary to hide the place where a
new node or data item is being inserted. As previously mentioned, there is no good reason to
implement the PU variant of different data structures and it is provided here for sorted linked lists
for illustration purposes only. The DU version of sorted linked list has the same list traversal time
as the regular (unsorted) linked lists, and the reported time for sorted linked lists can be further
optimized in the same way as it was done for regular linked lists. When we are building a sorted
linked list via DU, each operation takes O(n) time and thus the time to perform this operation
for all n elements of the input is O(n2 ). This quadratic performance is also observed empirically
where increasing the size of the data set by a factor of 2 results in four-time increase in the list
building time (all insertion operations are performed sequentially). As far as the head node removal
operation in a sorted linked list with PU goes, it consists of two pointer dereferences (i.e., using
data and next fields) and one call to pfree, where the overhead of pfree was between 76.4% to
81.3% of that operation’s time. In this particular experiment, each pointer stores O(n) locations,
which contributes to the complexity of both memory deallocation and pointer dereferencing, but
the latter operation can be performed more efficiently.
If we next look at mergesort (the last two programs in Table 3), we see that the variant that
uses pointers to private data and makes a function call to a compare-and-exchange operation for
each comparison and the variant with no pointers and corresponding function calls differ in their
performance by a very small amount. The non-pointer version that performs less work is faster by
0.4–2.4%.
Lastly, the performance of our shift-reduce parser (the second program in Table 2) is extremely
fast and is almost entirely consists of the time it takes to evaluate the provided formula on private
data (the second program in Table 2). That is, despite having a more complex functionality and
employing pointer-based stack, the time to perform arithmetic operations only is almost the same as
the time the parser takes. Also, adding pfree to the program does not effect the runtime (because
the pointer stores a single address) and the times with memory deallocation are omitted from the
table.
As far as memory consumption goes, the introduction of pointers to private data only marginally
affects the amount of allocated memory for programs with pointers storing a single memory location
(linked list, shift-reduce parser, and mergesort). The amount of memory needed to store and process
sorted linked lists is quadratic in the data size and matches in its complexity list traversal. Removing
a node from the list and calling pfree reduces the memory consumed by the data. While in general
calls to pfree can increase memory consumption, in this case all pointers store the same lists of
O(n) locations and removing a node and merging the lists in pfree decreases the size of each list.
In general, we can say that memory consumption is at most quadratic in the amount of data and
user-declared variables in any program.
We note that all functionalities used for our experiments have alternative implementations using
arrays. For linked lists, mergesort, and a shift-reduce parser, we expect array-based implementation
to exhibit very similar performance to that based on pointers because all pointers store a single
36
location. For sorted linked lists, we expect array-based programs to have performance similar to
our data update implementation (with the same asymptotic complexities). To confirm this finding,
we evaluated performance of array-based sorted linked lists, the source code of which can be found
in Figure 6. Building the sorted list took about 20% less time using arrays for most data sizes, while
list traversal was about 9% slower using arrays for most data sizes. Thus, both implementations
exhibit comparable performance. Memory consumption is also similar in most programs, with the
exception of array-based sorted list implementation that uses memory linear in the data size.
Performance of our pointer-based programs can also be compared to that of array-based implementations using another system or compiler. Sharemind [5] is a powerful system that supports a
wide range of programs and, similar to PICCO, builds on (a different type of) information-theoretic
secret sharing, which is hand-optimized to work with three computational parties. Despite similarities of the setup, Sharemind programs exhibit significantly different performance characteristics. In
particular, the implementation is optimized for performing a large number of identical operations
in a batch, while the cost of performing only a single operation is high (e.g., on the order of 100ms
for a single integer equality test [4]). As such, Sharemind programs will perform significantly worse
(i.e., orders of magnitude slower) on our programs that perform sequential execution, such as unoptimized linked list traversal, the shift-reduce parser, and building a sorted linked list (mergesort
is also slower as reported in [23]). In the case of optimized linked list traversal, on the other hand,
Sharemind implementations will still be slower for small data sets (such as 25 ), but significantly
faster for large data sets (up to two orders of magnitude faster for 220 elements).
All of these experiments demonstrate that pointers have a great potential for their use in
general-purpose programs evaluated over private data. Some pointer-based data structures can
exhibit substantially higher performance in this framework than their public-data counterparts,
and custom, internally built implementations for such data structures are recommended.
9
Conclusions
In this work, we introduce the first solution that incorporates support for pointers to private
data into a general-purpose secure multi-party computation compiler. To maintain efficiency of
pointer-based implementations, we distinguish between pointers with public addresses and pointers
with private addresses and introduce the latter only when necessary. We provide an extensive
evaluation of the impact of our design on various features of the programming language as well as
evaluate performance of commonly used pointer-based data structures. Our analysis and empirical
experiments indicate that the cost of using pointers to private data is minimal in many cases.
Several pointer-based data structures retain their best known complexities when they are used
to store private data. Complexity of others (most notably balanced search trees) increases due
to the use of private data flow, and custom, internally built implementations of oblivious data
structures that work with sorted data are recommended. We hope that this work provides valuable
insights into the use of various programming language features when developing programs for
secure computation using a general-purpose compiler, as well as highlight benefits and limitations
of pointer-based designs for SMC compiler developers.
Acknowledgments
We would like to thank Ethan Blanton for discussions at early stages of this work and anonymous
reviewers for the valuable feedback. This work was supported in part by grants CNS-1319090
and CNS-1223699 from the National Science Foundation and FA9550-13-1-0066 from the Air Force
37
Office of Scientific Research. Any opinions, findings, and conclusions or recommendations expressed
in this publication are those of the authors and do not necessarily reflect the views of the funding
agencies.
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39
//global declarations are the same as in Fig. 2
public int main() {
public int i, j;
private int array[count], output;
struct node *head, *ptr1, *ptr2;
smcinput(array, 1, count);
//construct the list
ptr1 = pmalloc(1, struct node);
ptr2 = pmalloc(1, struct node);
ptr1->data = array[0];
ptr2->data = array[1];
if (array[0] < array[1]) {
head = ptr1;
head->next = ptr2;
} else {
head = ptr2;
head->next = ptr1;
}
for (i = 2; i < count; i++) {
ptr1 = pmalloc(1, struct node);
ptr1->data = array[i];
ptr1->next = 0;
ptr2 = head;
if (ptr1->data < ptr2->data){
ptr1->next = ptr2;
head = ptr1;
}
for (j = 0; j < i; j++) {
if ((ptr2->data < array[i]) &&
(ptr2->next->data > array[i])) {
ptr1->next = ptr2->next;
ptr2->next = ptr1;
}
ptr2 = ptr2->next;
}
if (ptr2->data < ptr1->data)
ptr2->next = ptr1;
}
//traversal code is the same as in Fig. 2
// remove the head node
val = head->data;
ptr1 = head;
head = head->next;
pfree(ptr1);
smcoutput(val, 1);
return 0;
}
Figure 3: Construction and traversal of a sorted linked list (using pointer update).
40
public int K = 128;
public void swap(private int* A, private int* B) {
private int tmp;
if (*A > *B) {
tmp = *A;
*A = *B;
*B = tmp;
}
}
void mergesort(private int *A, public int l, public int r) {
public int i, j, k, m, size;
size = r - l + 1;
if (r > l) {
m = (r + l)/2;
[ mergesort(A, l, m); ]
[ mergesort(A, m + 1, r); ]
for (i = size >> 1; i > 0; i = i >> 1)
for (j = 0; j < size; j += 2*i) [
for (k = j; k < j + i; k++) [
swap(&A[k+l], &A[k+i+l]);
]
]
}
}
public int main() {
public int median = K/2;
private int A[K];
smcinput(A, 1, K);
mergesort(A, 0, K-1);
smcoutput(A[median], 1);
return 0;
}
Figure 4: Mergesort median program with pointers.
41
public int K = 128;
private int A[K];
void mergesort(public int l, public int r) {
public int i, j, k, m, size;
size = r - l + 1;
int tmp[size];
if (r > l) {
m = (r + l)/2;
[ mergesort(l, m); ]
[ mergesort(m + 1, r); ]
for (i = size >> 1; i > 0; i = i >> 1)
for (j = 0; j < size; j += 2*i) [
for (k = j; k < j + i; k++) [
tmp[k] = A[k+l];
if (A[k+l] > A[k+i+l]) {
A[k+l] = A[k+i+l];
A[k+i+l] = tmp[k];
}
]
]
}
}
public int main() {
public int median = K/2;
smcinput(A, 1, K);
mergesort(0, K-1);
smcoutput(A[median], 1);
return 0;
}
Figure 5: Mergesort median program without pointers.
42
public int count = 128;
public int main() {
public int i, j;
private int input[count], data[count], a, tmp, output;
smcinput(input, 1, count);
// build the sorted array
data[0] = input[0];
for (i = 1; i < count; i++) {
// move the data if necessary
a = input[i];
for (j = 0; j < i-1; j++) {
if (a < data[j]) {
tmp = data[j];
data[j] = a;
a = tmp;
}
}
data[i-1] = a;
}
// traverse the array searching for all instances of the value stored in a
a = 5;
output = 0;
for (i = 0; i < count; i++) {
if (data[i] == a)
output = output+1;
}
smcoutput(output, 1);
return 0;
}
Figure 6: Construction and traversal of a sorted list (using a static array).
43
| 6 |
Vehicle Routing with Subtours
Stephan Held1, Jochen Könemann2 and Jens Vygen1
1
arXiv:1801.04991v1 [cs.DS] 15 Jan 2018
2
Research Institute for Discrete Mathematics, University of Bonn
Department of Combinatorics & Optimization, University of Waterloo
held@dm.uni-bonn.de, jochen@uwaterloo.ca, vygen@dm.uni-bonn.de
January 17, 2018
When delivering items to a set of destinations, one can save time and cost
by passing a subset to a sub-contractor at any point en route. We consider a
model where a set of items are initially loaded in one vehicle and should be
distributed before a given deadline ∆. In addition to travel time and time
for deliveries, we assume that there is a fixed delay for handing over an item
from one vehicle to another.
We will show that it is easy to decide whether an instance is feasible, i.e.,
whether it is possible to deliver all items before the deadline ∆. We then
consider computing a feasible tour of minimum cost, where we incur a cost
per unit distance traveled by the vehicles, and a setup cost for every used
vehicle. Our problem arises in practical applications and generalizes classical
problems such as shallow-light trees and the bounded-latency problem.
Our main result is a polynomial-time algorithm that, for any given > 0
and any feasible instance, computes a solution that delivers all items before
time (1 + )∆ and has cost O(1 + 1 )OPT, where OPT is the minimum cost
of any feasible solution.
Known algorithms for special cases begin with a cheap solution and decompose it where the deadline is violated. This alone is insufficient for our
problem. Instead, we also need a fast solution to start with, and a key feature of our algorithm is a careful combination of cheap and fast solutions.
We show that our result is best possible in the sense that any improvement
would lead to progress on 25-year-old questions on shallow-light trees.
1. Introduction
Logistics companies are exploring innovative ways of delivering items to customers. In
particular with an increasing number of crowdsourced delivery startups [19] comes new
1
flexibility in designing delivery routes; e.g., [12] studies delivery applications where
crowdsourcing is used to facilitate last-leg delivery of items. In the classical setting,
a set of vehicles are initially located at a central depot that in turn serves as the starting
point for all tours serving customers. Where deadlines are tight, this model naturally
leads to a large number of tours, and implementing these solutions necessitates the
maintenance of large vehicle fleets.
Our work is motivated by instances where the depot is relatively far from many customers. In designing delivery schedules, one would therefore want to use bigger vehicles
to transport a large set of items closer to clusters of customers at which point one would
then utilize smaller vehicles for the final leg of the delivery process. To model this situation, we introduce a new kind of vehicle routing problem and study its approximability.
Let us assume that a set of items is initially loaded on one vehicle and that they need
to be delivered to their destinations by a given deadline ∆. The vehicle can deliver items
itself, but it can also – at any place en route – hand over a subset of its items to another
vehicle. This vehicle can then deliver these items or can again hand over subsets to new
vehicles en route.
Besides the normal travel cost per unit distance, we assume a fixed setup cost for every
vehicle that we use. This assumption is of course a simplification, but some companies
have vehicles at many different locations (almost everywhere) or hire sub-contractors
that are not paid for their way to the meeting point where items are handed over to
him. For the same reason, the way back home is not paid; therefore our tours are paths
and not circuits. We remark that essentially the same problem arises when collecting
items (or students) and transporting them to a central location (or school). However,
to avoid confusion, we will speak of the delivery problem only.
We do not consider vehicle capacities here, although implicitly the number of items
that a vehicle can handle is bounded by the deadline. We not only account for the travel
time and the time to deliver items, but also consider a hand-over time proportional to
the number of items exchanged.
Assuming that items can only be handed over from a vehicle to one other vehicle
simultaneously, then the resulting route structure can best be described as an arborescence with maximum out-degree two, where the root corresponds to the starting point of
the initial vehicle, vertices with out-degree two correspond to a hand-over, and all other
vertices correspond to a delivery. We note that the imposed bound on the out-degree of
our vertices is not restrictive, as we allow placing multiple vertices of the arborescence
at the same geographical location.
1.1. Problem definition
Let us now describe the problem formally and introduce our notation. An instance
consists of a finite set P of item destinations, a root r ∈
/ P , and a metric space (M, c).
We are also given a map µ : {r} ∪ P → M that assigns root and items in P to locations
in the metric space. Finally, we are given non-negative parameters δ, σ and ∆, capturing
delivery time, setup cost, and deadline, respectively.
We represent a schedule by an arborescence (W, A) rooted at r with P ⊂ W and
2
p7
p1
1
1 4
2
4
p2
2
2
r
8
1
p4
4 1 4
p6
4
1
3
4
p3
3
1
p5
2
4
Figure 1: Example of a schedule. Edges are labeled by their distance (green). Vertices
are labeled by hand-over delay (blue) or delivery delay (red, δ = 4). The initial
vehicle delivers p1 , p2 and p3 . It hands over parcels p4 , p5 and p6 to a second
vehicle and later p7 to a third vehicle. The second vehicle hands over p6 to a
fourth vehicle en route. The delay of this schedule is 30 (attained at p3 ). Its
cost is 23 + 4σ.
µ : W \ (P ∪ {r}) → M . An arc (x, y) ∈ A means that a vehicle travels from x to y,
causing delay c(µ(x), µ(y)). At a vertex p ∈ P we deliver the item p and incur delay δ.
At a vertex w ∈ W \ P with out-degree 2 we split off a subtour and incur a hand-over
delay equal to the number of items handed over. Note that it is always better to hand
over at most half of the items that are currently in the vehicle. We disallow out-degree
greater than 2 because we need to specify the order of hand-overs. Assuming that a
vehicle cannot simultaneously be involved in a delivery and a hand-over, we also forbid
out-degree 2 for vertices in P . However, we allow that multiple vertices are mapped to
the same point in the metric space, so multiple subtours can start at the same point in
the metric space. A schedule allows that items are handed over multiple times on their
path from the root to their destination.
The delay of a schedule is the maximum
P delay to an item. The cost of a schedule is c(A, µ) + s(W, A), where c(A, µ) := (x,y)∈A c(µ(x), µ(y)) is the travel cost and
s(W, A) := σ · |W0 | is the setup cost for vehicles; here W0 denotes the set of leaves in
(W, A). The goal is to find a minimum-cost schedule with delay at most ∆.
Figure 1 shows an example with seven items and a schedule with four vehicles.
1.2. Overview of results and techniques
As we will see in Section 1.4, our problem is a common generalization of several problems
studied in the literature, including shallow-light trees and the bounded-latency problem.
However, the possibility of starting subtours far away from the root and the handover
delays make the problem more difficult.
Many previous approaches for special cases and similar problems, including the bestknown algorithms for shallow-light trees and the bounded-latency problem (cf. Section
1.4), begin with a cheap solution (minimum spanning tree or short TSP tour), traverse
it, and split the tree or tour whenever necessary to make the solution fast enough,
connecting the next vertex directly to the root. However, this approach fails for our
3
problem because connecting to the root can be too expensive and the order in which we
split off subtours matters due to the handover delays.
We still use this tour-splitting technique to group the items, but in addition we need
to begin with a fast solution. Therefore we first show that there always exists a fastest
schedule with caterpillar structure, i.e., where all deliveries occur at leaves and the
subgraph induced by the vertices with out-degree 2 is a path. As a corollary, one can
easily compute such a solution and check whether a given instance is feasible (cf. Section
2).
In Section 3 we describe our algorithm. It first uses the tour-splitting technique to
partition the items into groups each of which is spanned by a short path and does not
contain too many items. Next, our algorithm constructs a two-level caterpillar so that
items in the same group are consecutive, without violating the deadline much. Although
this helps saving length, this schedule still uses a separate vehicle for every item and
thus is usually very expensive. In a final step, we cluster subsets of groups to reduce the
number of vehicles.
We compare our solution to a lower bound, composed of the length of a minimum
cost tree spanning {r} ∪ P (in the following denoted by MST) and a lower bound on the
number of required vehicles. This will yield our main result:
Theorem 1. Given a feasible instance and > 0, we can compute a solution with delay
at most (1 + )∆ and cost O(1 + 1 )OPT in polynomial time, where OPT is the minimum
cost of a feasible schedule.
In Section 4 we will give an example that the tradeoff in Theorem 1 is unavoidable
unless we use a significantly stronger lower bound. In fact, the example also shows that
the 25-year old result of Cong et al. [4] on shallow-light trees is best possible. Since this
is a special case of our problem, improving the dependency on by more than a constant
factor would require new lower bounds for shallow-light trees and immediately lead to
progress on this very well-studied problem.
1.3. Comments on our model
Before we move on, let us comment on two subtle assumptions that we made in our
model, and argue why they are reasonable and necessary to obtain such a result.
First, we assume δ ≥ 1, that is, delivering an item to its final destination takes at
least as long as handing it over to another vehicle (we assumed the hand-over time to be
1 per item, which is of course no loss of generality by scaling). This is certainly realistic
in practice. Some assumption on δ is also necessary for our main result: with δ = 0,
there is no polynomial-time algorithm delivering all items of a feasible instance before
time (1 + )∆ for arbitrary small > 0 (unless P = NP). To see this, scale down the
distances so that the shortest path beginning in µ(r) and containing µ(p) for all p ∈ P
has length less than 1. Then the earliest deadline that we can meet is the length of a
shortest such path. However, determining this length is APX-hard (by straightforward
reduction from the classical TSP [18]; cf. Appendix A).
4
Another subtle point in our model is that we pay σ for every vehicle, including the
initial one. Again, this looks reasonable from the practical point of view, although
one might also think of the setup cost of the initial vehicle as already paid. But this
assumption too is necessary for our main result. If we had assumed the initial vehicle
to be free, then a very large value of σ would force any reasonably cheap solution to
be a path, and finding a path almost meeting a deadline is then equivalent to finding
an almost shortest tour starting at r and visiting all item destinations. As above, this
problem is APX-hard.
Finally, our tours and subtours are paths rather than circuits. This is not very important though, because if every vehicle had to return to its starting point, the cost can
at most double.
1.4. Related work
The problem discussed in this paper generalizes several well-studied, classical network
design problems. For example, our problem contains the Steiner tree problem (set σ = 0
and ∆ = ∞) and is thus APX-hard [3].
An interesting special case of our problem arises when σ = 0 and c is large compared
to δ. In other words, the traveled distance dominates the delay and the cost of any
schedule. In this case, the goal would be to compute a Steiner tree (or if M = µ({r}∪P )
a spanning tree) that balances cost and the maximum distance from the root.
Awerbuch et al. [2] first showed that every finite metric space contains a spanning tree
whose diameter is at most a constant times that of the underlying metric space, and
whose weight is at most a constant times that of a minimum-cost spanning tree. Such
trees are called shallow-light trees. Cong et al. [4] improved these results; they showed
how to find, for any > 0, a spanning tree of length at most (1 + 2 )MST in which the
path from r to any other vertex is no longer than 1+ times the maximum distance from
r. Khuller, Raghavachari and Young [15] generalized this work to obtain, for any > 0,
a tree T with total cost at most (1 + 2 )MST such that, for every v ∈ P , the distance in
T from r to v is at most 1 + times the distance between r and v in the metric space.
Khuller et al. [15] also gave an example showing that the obtained tradeoff is best
possible. In Section 4 we generalize their example to prove that this is true even for
instances where all vertices have the same distance from r; implying that the result of
[4] on shallow-light trees is also best possible. This will also show that Theorem 1 is
best possible unless we use a stronger lower bound than MST.
A further generalization was given by Held and Rotter [9], considering Steiner trees
and having an additional distance penalty per bifurcation. However, in all of the above
algorithmic variants the result may have many leaves and the delay models differ substantially from hand-over delays in our model.
There has been a tremendous amount of work on solving optimization problems arising
in the general context of vehicle routing, and we cannot provide an adequate survey here.
We focus on the most closely related work that we are aware of, and refer the reader to
Toth and Vigo’s book [20] for a more comprehensive introduction.
5
Another interesting special case of our problem arises when delays are dominated by
travel times and cost is dominated by the setup cost. Then it does not harm to start all
subtours at the position of the root, and the problem reduces to covering the items by
as few paths as possible, each starting at the root and having length at most ∆. Jothi
and Raghavachari [11] called this problem the bounded-latency problem. They observed
that the tour-splitting technique yields a solution violating the deadline by at most a
factor of (1 + ) and using at most 2 times the optimum number of paths.
A similar problem is the distance-constrained vehicle routing problem (DVRP), where
the goal is to cover the items by a minimum number of closed tours (returning to the
root), each having length at most ∆. Khuller, Malekian, and Mestre [16] and independently Nagarajan and Ravi [17] gave an (1 + , O(log 1 ))-bicriteria approximation algorithm. This also works for the bounded-latency problem: partition the items according
to their distance from r: items at distance more than (1−2)∆ are in group 1, and items
at distance between (1 − 2j )∆ and (1 − 2j−1 )∆ are in group j (j = 2, . . . , dlog 1 e).
Then each group j is covered by (unrooted) paths, each of which has length at most
2j−1 ∆ and can be completed by an edge to r, exceeding length ∆ by at most ∆ (only
in group 1). The number of these paths can be minimized up to a factor 3 using an
algorithm of [1]. If OPT is the number of tours in an optimal solution, we can cover
each group by 2OPT paths (shortcutting those that contain at least one item this group
and splitting it into two if necessary). Thus we end up with at most 6OPT paths in
each group, and 6dlog 1 eOPT paths in total.
Connecting all paths to the root can, however, be much more expensive than splitting
off subtours elsewhere: for instance, if all items are to be delivered at the same position
far away from the root, and the (relaxed) deadline prevents any tour from delivering
more than one item, then the total length increases by a factor |P | if we insist that all
paths start at the root. See also Appendix B for a similar example.
Friggstad and Swamy [6] studied regret-bounded variants of vehicle routing problems
and provided an O(log ∆/ log log ∆) approximation for DVRP under the assumption that
the minimum distance in the underlying metric is at least one. Gørtz et al. [8] considered
various vehicle routing problems in the setting where vehicles have non-uniform speeds
and capacities. Among other things, the authors study the variant of DVRP where the
vehicles have finite capacity and non-uniform speeds, and where the goal is to minimize
the deadline. Gørtz et al. provide a constant-factor approximation algorithm for this
problem.
Closely related to DVRP are vehicle routing problems with min-max objective. A typical such problem is the min-max X-cover problem, where X ∈ {path, tree, tour, . . .}.
Here, one is given a metric space on n points, and a parameter k, and the goal is now
to find a collection of k subgraphs of type X to cover all points so that the maximum
length of any of these subgraphs is smallest. The problem is APX-hard in the case of
trees [21] and constant-factor approximation algorithms are known [1, 5, 14]. Xu, Xu and
Li [22] study the min-max path cover problem and obtain constant-factor approximation
algorithms for several variants, also including delivery times.
Another notable variant is the preemptive multi-vehicle dial-a-ride problem, where
n items have to be transported by a fixed number of vehicles, which are located at
6
given depots. Item i ∈ {1, . . . , n} has to be picked up at si and delivered to ti . Items
may be passed from one vehicle to another on their journey. Gørtz, Nagarajan, and
Ravi [7] present an O(log3 n)-approximation algorithm for minimizing the makespan
under capacity constraints, and an O(log t)-approximation algorithm without vehicle
capacities, where t is the number of distinct depots.
2. Deciding Feasibility
2.1. Notation
Let us call an arborescence proper if its root is r and has out-degree 1, all elements
of P are vertices with out-degree 0 or 1, and all other vertices have out-degree 2. We
may restrict ourselves to schedules with proper arborescences because if the root has
out-degree 2 we can introduce an extra vertex at the same location without changing
delay or cost.
For a proper arborescence (W, A) and x ∈ W we use the following notation: (Wx∗ , Ax∗ )
is the maximal subarborescence of (W, A) rooted at x. For y ∈ Wx∗ we denote by
(Wxy , Axy ) the path from x to y in (W, A). We have W = W0 ∪ W1 ∪ W2 , where Wi
contains the vertices of out-degree i (i = 0, 1, 2). The elements of W0 are called leaves,
and the elements of W2 are the bifurcation nodes. Note that |W0 | = |W2 | + 1.
With this notation, we extend the definition of delay of an item in a schedule (W, A, µ)
to any vertex y ∈ W :
X
delay(W,A,µ) (r, y) := c(Ary , µ) + δ|P ∩ Wry | +
min+ |P ∩ Wx∗ |.
w∈Wry ∩W2
(w,x)∈δ (w)
The first term is the delay of traversing edges, the second term is the time for delivering
items on the r-y-path, and the third term is the time to hand over a subset of items to
a subtour. Now, the delay of a schedule (W, A, µ) can be written as
delay(W, A, µ) := max delay(W,A,µ) (r, p).
p∈P
We denote by n := |P | the number of items.
2.2. Caterpillar structure
If we disregard cost, we can afford a separate vehicle for every item, going straight from
the root to the item’s destination. This certainly minimizes the first two components
(travel time and delivery time) of the delay to each item. Then the only question is how
the tree structure should look like, because it will determine the handover delays. It
turns out that there is always a fastest solution with a caterpillar structure. This allows
us to determine efficiently whether a feasible solution exists.
We first introduce the leafication step that takes a vertex with fanout one and branches
it off as a single leaf.
7
x
x
y0
y
y
leafication
z
z
Figure 2: Leafication of y.
Definition 2 (Leafication). Consider a schedule (W, A, µ). Let y ∈ W1 \ {r}, and let
(x, y), (y, z) ∈ A be the two arcs incident to y in (W, A). The leafication of y is the
˙ 0 }, where y 0 is a new vertex,
new schedule (W 0 , A0 , µ0 ) with W 0 = W ∪{y
A0 = A \ {(x, y), (y, z)} ∪ {(x, y 0 ), (y 0 , y), (y 0 , z)} ,
µ0 (u) = µ(u) for all u ∈ W , and µ0 (y 0 ) = µ(y).
See Figure 2. The leafication does not increase the delay of the schedule:
Lemma 3. If (W 0 , A0 , µ0 ) is obtained from (W, A, µ) through a leafication, then
delay(W 0 , A0 , µ0 ) ≤ delay(W, A, µ).
Proof: Let y be the leafication vertex and (x, y), (y, z) ∈ A its incident arcs. First note
0
that the leafication changes delays only in Wy? = Wz?
∪ {y}. Furthermore, c(Arp , µ) =
0
0
c(Arp , µ ) for all p ∈ P . Now for y and any w ∈ Wz? ∩ P we have
delay(W 0 ,A0 ,µ0 ) (r, y) ≤
=
≤
≤
delay(W 0 ,A0 ,µ0 ) (r, w)
delay(W,A,µ) (r, w) − δ + 1
delay(W,A,µ) (r, w)
max delay(W,A,µ) (r, p),
p∈P
where the first inequality follows from the fact that w collects at least the handover
delay as y and at least its own delivery time. The equality follows from replacing the
delivery time by the handover time for y on the path to z. The second last inequality
follows from δ ≥ 1. We conclude delay(W 0 , A0 , µ0 ) ≤ delay(W, A, µ).
2
By leafication we can get rid of out-degree 1 vertices (except for the root). In order to
obtain the caterpillar structure we need to move all out-degree 2 vertices onto a single
“heavy” path:
Definition 4 (Heavy path). Given a proper arborescence (W, A), a vertex x ∈ W is
called heavy if x = r or |Wx∗ ∩ P | ≥ |Wy∗ ∩ P | for every child y of the predecessor of x.
A heavy path in (W, A) is maximal set of heavy vertices that induce a path.
8
Note that any heavy path begins at the root and ends at a leaf. We will move all
bifurcation nodes onto a heavy path by the following operation.
Definition 5 (Flip). Consider a schedule (W, A, µ), and H a heavy path in (W, A). Let
w ∈ W2 ∩ H, and let h be the child of w that belongs to H. Suppose that the other child
x of w is a bifurcation node with µ(h) = µ(x). Let yh and yl be the two children of x,
where yh is heavy. The flip at x is the new schedule (W, A0 , µ) with
A0 := A \ {(w, h), (x, yl )} ∪ {(w, yl ), (x, h)}.
See Figure 3. Note that H ∪ {x} is a heavy
path in (W, A0 ). We now show that a flip does
not make a schedule slower.
Lemma 6. If (W, A0 , µ) is obtained from
(W, A, µ) through a flip, then
delay(W 0 , A0 , µ0 ) ≤ delay(W, A, µ).
w
w
h
x
yl
yh
h
flip
x
yl
yh
Proof: First note that c(A0rp , µ) ≤ c(Arp , µ0 )
for all p ∈ P due to µ(h) = µ(x) and the triFigure 3: Flip operation.
angle inequality. We show that the total handover delay to any item does not increase. By
construction the delays of items outside Wx∗ coincide for (W, A, µ) and (W, A0 , µ). The
handover delays of the flipped schedule are reduced by |Wyl ∗ ∩ P | for all items in Wyh ∗
and by |Wx∗ ∩ P | for all items in Wyl ∗ .
2
In graph theory, caterpillars are trees for which deleting all leaves results in a path.
We use the term for arborescences in a slightly different way:
Definition 7 (Caterpillar). A proper arborescence (W, A) is a caterpillar if W1 = {r}
and the subgraph induced by {r} ∪ W2 is a path. For P = {p1 , . . . , pn } we denote by
C(pn , . . . , p1 ) the caterpillar in which the r-pi -path in (W, A) has min{n, n + 2 − i} edges
for all i = 1, . . . , n.
See Figure 4. Now we can show the main result of this section.
Theorem 8. There exists a schedule (W, A, µ) with minimum delay such that (W, A) is
a caterpillar and µ(w) = µ(r) for all w ∈ W2 .
Proof: If n = 1, the statement is clearly true, so let n > 1. Let (A0 , W 0 , µ0 ) be a
schedule with minimum delay. Recall that W00 ∪ W10 = {r} ∪ P .
Step 1: By iteratively applying a leafication step to all y ∈ W10 \ {r}, we can transform
it into a schedule (A1 , W 1 , µ1 ) with W11 = {r} that still has minimum delay. Then all
items are delivered at leaves, i.e., P = W01 .
Step 2: We transform (A1 , W 1 , µ1 ) into (A2 , W 2 , µ2 ), by setting A2 = A1 , W 2 = W 1 and
µ2 (y) = µ1 (r) for all y ∈ W22 . As (A1 , W 1 ) = (A2 , W 2 ), only the delays for traversing
9
arcs change in (A2 , W 2 , µ2 ). But as c(A2rp , µ2 ) = c(r, p) for all p ∈ P , this part of the
delay is minimum possible and so (A2 , W 2 , µ2 ) still has minimum delay.
Step 3: Finally, we use the flip operation to obtain the caterpillar structure. Let H be
a heavy path. As long as there is a bifurcation node outside H, there is such a vertex x
such that its predecessor w belongs to H. Then applying the flip operation at x increases
the cardinality of H by one, so after finitely many steps, the heavy path contains all
bifurcation nodes.
2
2.3. Consequences
Corollary 9. For any feasible instance we have
(a) ∆ ≥ min c(r, q) + δ + min{|Q|, n − 1} : q ∈ Q for every nonempty subset Q ⊆ P ;
(b) ∆ ≥ n;
(c) ∆ ≥ max{c(r, p) : p ∈ P }.
Proof: For (a), let Q ⊆ P , and consider a feasible caterpillar (which exists by Theorem 8). At
least one of the |Q| items, say q, will have at least
min{|Q|, n − 1} bifurcation nodes on its path. The
path to q has length c(r, q) and it also pays δ for
delivering q.
(b) follows from (a) by setting Q = P and using
δ ≥ 1. (c) is obvious.
2
Theorem 8 allows us to determine efficiently
whether a feasible schedule exists.
r
p1
pn
pn
1
pn
2
p2
Figure 4: Caterpillar C(pn , . . . , p1 ).
Corollary 10. We can find a schedule meeting the deadline or decide that none exists
in time O(n log n + θn), where θ is the time to evaluate distances in (M, c).
Proof: If n = 1, verifying the feasibility of the deadline is easy, so let n > 1. By
Theorem 8 there is a fastest solution (A, W, µ) such that (A, W ) is a caterpillar. This
caterpillar is unique up to the order in which the items are attached as leaves. Furthermore, the distribution of total handover delays for the leaves is predetermined and each
item suffers a single delivery delay of δ for its own delivery. So the leaves have delivery
plus handoff delay 1 + δ, 2 + δ, . . . , n − 2 + δ, n − 1 + δ, n − 1 + δ.
Thus it suffices to find an assignment of the items to the leaves that meets the deadline.
The best we can do is to iteratively assign an item with the maximum distance from
r to the closest available leaf in the arborescence. To this end we sort the items by
their distance from r. Let p1 , p2 , . . . , pn be the ordering of the items by non-decreasing
distance from r, i.e. c(r, pi ) ≤ c(r, pi+1 ) for all i ∈ {1, . . . , n − 1}.
We assign the items p1 , p2 , . . . , pn one by one to a not yet occupied leaf vertex that
has a maximum number of arcs on its path from r (cf. Figure 4). The deadline ∆ can
be met if and only if the generated schedule meets it.
10
The running time is dominated by sorting the items, which can be done in O(n log n)
time, and by computing all root to item distances, which takes θ · n time.
2
The schedule from Theorem 8 is fast but also expensive. It has the maximum possible
setup cost of σ · n and also high travel costs, as each item is transported individually
from the root location to its destination.
3. Algorithm
Our algorithm first groups the items by splitting a short tour into paths similar to [1].
The items in each group will have similar distance from r. Therefore, rearranging the
fastest solution with the caterpillar structure (Theorem 8) so that the items in each
group are consecutive does not make the schedule much slower. Next, we design a twolevel caterpillar, where the items in each group are served by a caterpillar and the groups
are served by a top-level caterpillar. In order to avoid that all subtours begin at the
position of the root, we make the main tour of each sub-caterpillar drive to all locations
of items in that group. Finally, we avoid too many subtours by merging tours in each
subcaterpillar.
3.1. Grouping items
Lemma 11. Let MST denote the length of a minimum cost tree with vertex set {r} ∪ P ,
and let s ∈ P be an item at maximum distance from r. Let > 0 and 0 < ∆ ≤ MST.
Then there is a forest (P, F ) whose components are vertex-disjoint paths such that
(a) the number of paths is at most 1 +
n+2 MST−c(r,s)
;
∆
(b) no path is longer than ∆;
(c) no path contains more than 1 + ∆ items;
(d) c(F ) ≤ 2 MST − c(r, s).
Such a forest can be found in polynomial time.
Proof: Take any approximately cheapest path with vertex set {r} ∪ P from r to s.
We can find it by taking a minimum cost spanning tree for {r} ∪ P in (M, c), doubling
all edges except those of the r-s-path, finding an Eulerian r-s-walk, and shortcutting.
The resulting r-s-path ({r} ∪ P, F0 ) has total cost at most 2 MST − c(r, s). From now
on, we will only delete edges, yielding (d).
To satisfy (b) and (c), we start with F = ∅ and traverse the path r-s-path, ignoring
r and the first edge. In each step, we add the next edge to F unless this would violate
(b) or (c).
The conditions (b), (c), and (d) are then satisfied by construction. Whenever we drop
an edge e, one of the conditions (b) and (c) would be violated for P ∪ {e}, where P is a
connected component of F . If (b), the length of P ∪ {e} exceeds ∆, so this can happen
11
pkq
r
1
pkq
group q
pkq
1 +1
1
2
pkq
1
group 1
pk1 pk1 +1 pk2 2
pk 2
1
rq
rq
r2
1
pkq+1
pkq pkq +1
group q
2
pkq+1
pk2 pk2 +1
pk 3
group 2
1
2
pk 3
1
Figure 5: Schedule S1 with a sub-caterpillar for each group.
at most
c(F0 )
∆
times. If (c), the number of items in P exceeds ∆, so this can happen at
times. So we drop at most 2 MST−c(r,s) + n edges (in addition to the initial
n
most ∆
one). This yields (a).
∆
∆
2
Note that (b) immediately implies that items in the same path have similar distances
from r: if p and p0 are in the same path, the triangle inequality yields c(r, p) ≤ c(r, p0 ) +
∆.
3.2. Towards a cheaper schedule
Let us call the vertex sets of the connected components of (P, F ) from Lemma 11 groups;
they form a partition of P . For p ∈ P , let
d(p) := max{c(r, p0 ) : p and p0 are in the same group}
be the distance from r to the most remote item in the same group as p (note that p0 = p
is not excluded). Order the items P = {p1 , . . . , pn } such that
d(p1 ) ≤ · · · ≤ d(pn )
and
F ⊆ {{pi , pi+1 } : i = 1, . . . , n − 1},
so items of the same group are consecutive, every edge in F connects two consecutive
items, and groups containing more remote items come later.
Let P = {p1 , . . . , pn } be the items ordered as above, and let 1 = k1 < k2 < · · · <
kq+1 = n + 1 such that {pki , . . . , pki+1 −1 } (i = 1, . . . , q) are the groups. Let S1 be the
schedule resulting from a path with vertices r, rq , rq−1 , . . . , r2 in this order by identifying
12
the root of the caterpillar C(pki , . . . , pki+1 −1 ) with rmax{2,i} (i = 1, . . . , q; see Figure 5).
The bifurcation nodes rq , . . . , r2 are placed at µ(r), but the bifurcation nodes of the
subcaterpillars are placed at the position of the item that splits off there: the j-th
bifurcation node of the subcaterpillar for group i is placed at µ(pki +j−1 ).
Lemma 12. If the instance is feasible, then delay(S1 ) ≤ (1 + 3)∆.
Proof: Consider group i with items pki , . . . , pki+1 −1 . The maximum delay of an item
in this group is the one to pki+1 −1 , which is at most
ki+1 −2
c(r, pki ) +
X
l=ki
c(pl , pl+1 ) + n − kmax{2,i} + 1 + ki+1 − ki − 1 + δ,
which by Lemma 11(b) and (c) is at most
d(pki ) + ∆ + n − kmax{2,i} + 1 + ∆ + δ.
(1)
By Corollary 9 (a), there exists a j ∈ {ki , . . . , n} with ∆ ≥ c(r, pj )+n−max{2, ki }+1+δ.
Since d(pki ) ≤ d(pj ) ≤ c(r, pj ) + ∆, we have
∆ ≥ c(r, pj ) + n − max{2, ki } + 1 + δ
≥ c(r, pj ) + n − kmax{2,i} + 1 + δ
≥ d(pki ) − ∆ + n − kmax{2,i} + 1 + δ.
Hence our bound (1) on the maximum delay of an item in group i yields that this delay
is at most (1 + 3)∆.
2
We can bound the length of S1 as follows:
Lemma 13. S1 has length at most (2 + 2 )MST.
P
Proof: The length of the schedule is qi=1 c(r, pki ) (which is the initial edges of the
sub-caterpillars) plus c(F ) (remaining length of the subcaterpillars). By Lemma 11,
−c(r,s)
c(F ) ≤ 2 MST − c(r, s) and the number q of groups is at most 1 + n+2 MST
.
∆
Moreover c(r, pki ) ≤ c(r, s) for all i by the choice of s. Hence the length of the schedule
is at most
q c(r, s) + 2 MST − c(r, s) ≤
n + 2 MST − c(r, s)
c(r, s) + 2 MST.
∆
Since n ≤ ∆ and c(r, s) ≤ ∆ by Corollary 9 (b) and (c), this is at most
∆
2 MST − c(r, s)
2
MST.
c(r, s) +
∆ + 2 MST = 2 +
∆
∆
2
13
3.3. Saving vehicles
The schedule is already short, but it still contains a separate vehicle for each item.
However, we can deliver up to m := 1 + b ∆
c items by the same vehicle by replacing the
δ
edge entering an item pj in group i by the edge (pj−1 , pj ) unless j − ki is a multiple of
m. The resulting schedule S2 is the output of our algorithm, except that we remove the
non-item nodes that now have out-degree 1, by shortcutting.
Lemma 14.
at most (1 + 4)∆ and length at most (4 + 2 )MST. It has at
S2 has delay
+nδ
vehicles.
most 1 + 2 MST
∆
Proof: Going from S1 to S2 increases the maximum delay of an item in a group by
at most (m − 1)δ ≤ ∆, because the maximum travel time and the maximum handover
delay in each group cannot increase.
The length of S2 is at most c(F ) longer than S1 . Since c(F ) ≤ 2MST, the length
bound follows.
The total number of vehicles is at most
δn
n + 2 MST − c(r, s) δn
2 MST + nδ
n
≤q+
≤1+
+
≤1+
,
q+
m
∆
∆
∆
∆
2
where we used δ ≥ 1 in the last inequality.
We compare this to the following lower bound.
1
Lemma 15. Every feasible schedule has length at least 12 MST and uses at least 2
vehicles, where MST is the length of a minimum spanning tree for µ({r} ∪ P ).
MST+nδ
∆
Proof: Any schedule connects {r}∪P , so the lower bound 21 MST for the length follows
from the Steiner ratio. For the bound on the number of vehicles, fix any feasible schedule
(W ∗ , A∗ , µ∗ ), say with l∗ vehicles numbered 1, . . . , l∗ . Let Di be the delay of the last
item that vehicle i delivers; this is at most ∆. As each edge needs to be traversed and
each item delivered by one of the vehicles,
∗
l
X
1
∗
∗
MST + nδ ≤ c(A , µ ) + nδ ≤
Di ≤ l∗ ∆.
2
i=1
2
Lemma 14 and 15 imply that the cost of our schedule is at most 8 + 4 times the cost
of an optimum schedule. This proves Theorem 1.
Our algorithm is very fast – the running time is dominated by computing a minimum
spanning tree and sorting the groups.
We remark that the constants can be improved, for example by beginning with a
Steiner tree that is a better approximation than MST. However, any improvement by
more than a constant factor would imply an improvement over the 25-year old bicriteria
algorithm for shallow-light trees by Cong et al. [4]. We will demonstrate this in the
following.
14
4. An almost tight example
We now show that our bicriteria result is the best we can hope for up to constant factors.
To this end, we modify an example of Khuller, Raghavachari, and Young [15] to make
it work for a uniform deadline.
Theorem 16. Let > 0 and 1 ≤ α < 1 + 1 . There is an undirected graph G = (V, E)
with weights c : E → R>0 and r ∈ V such that dist(G,c) (r, v) ≤ 1 for all v ∈ V , and for
each spanning tree F in which every path from r has length at most 1 + , the total length
of F is more than α · MST, where MST is the length of a minimum spanning tree.
Proof: For sufficiently large k ∈ N (in particular k > 1 + ), consider the graph
G = (V, E) with vertex set V = {r, s, p11 , . . . , p1k , p21 , . . . , p2k , . . . , pk1 , . . . , pkk } shown
in Figure 6. The edge set E contains a red edge from r to every vertex other than r,
and blue edges {s, pi1 } and {pij , pi(j+1) } for all i = 1, . . . , k and j = 1, . . . , k − 1. Red
. Thus, dist(G,c) (r, v) = 1 for all
edges have weight 1, and blue edges have weight k−1
v ∈ V \ {r}.
For each 1 ≤ i ≤ k, unless the tree uses one of the red edges {r, pij } (j = 1, . . . k),
k > 1 + . Therefore, for any tree F in
the tree distance from r to pik is at least 1 + k−1
which every path from r has length at most 1 + , F has total length at least
k + k(k − 1)
+
> k(1 + ).
k−1 k−1
On the other hand, a minimum spanning tree consists of one red and all blue edges;
. Thus, the ratio between the total length of a tree whose
it has length MST = 1 + k 2 k−1
paths from r have length at most 1 + and a minimum spanning tree is at least
k(1 + ) k→∞ 1 +
1
−−→
= 1 + > α.
−
2
1 + k k−1
So for sufficiently large k, the ratio is greater than α.
2
This shows that the result of Cong et al. [4] mentioned in Section 1.4 is best possible
up to a constant factor.
This example applies not only to shallow-light trees, but also to a special case of our
problem, namely when M = µ({r} ∪ P ), σ = 0, and c is large compared to δ so that
delivery times and handover delays can be neglected. We see that unless we use a much
stronger lower bound than MST on the length of a feasible schedule, the tradeoff in
Theorem 1 is unavoidable.
References
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vehicle routing problems. Journal of Algorithms, 59(1):1–18, 2006.
15
r
1
...
p1k
...
k−1
p12 p11
p21
s
pk1 pk2
p(k−1)1
.
..
p2k
.
...
..
p22 p(k−1)2
pkk
p(k−1)k
Figure 6: An almost tight example
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and shortest-path trees. Algorithmica, 14(4):305–321, 1995.
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17
r
s
n2
n2 +2n
2−
p1
p2
pn
Figure 7: An example where using a Steiner node reduces the cost very much.
Appendices
A. APX-hardness of TSP with one fixed endpoint
It is well-known that the metric TSP is APX-hard [18]. More precisely, it is NP-hard
1
to approximate it with any ratio better than 1 + , where = 122
[13]. We deduce from
this inapproximability thresholds for the variants studied by Hoogeveen [10], where we
look for a path with 0, 1, or 2 endpoints fixed.
First, the same threshold holds for the path TSP if both endpoints are fixed. Given a
TSP instance, just guess an edge {r, s} of an optimal tour and approximate the r-s-path
TSP instance.
Second, we get the threshold 1 + 3 if only one endpoint r is fixed. This works as
follows. Given an instance of the path TSP with both endpoints r and s fixed, let U
be an upper bound on the optimum, say at most 53 OPT. Add a vertex t with distances
3+
U . Then consider the path TSP with
c(v, t) := c(v, s)+M for all cities v, where M = 3−
only one endpoint r fixed. The optimum has length at most OPT + M . In fact equal
to this, because any tour not ending in t will have length more than 2M = (1 + 3 )M +
3+
U ≥ (1 + 3 )(M + OPT). Therefore, any algorithm with approximation ratio
(1 − 3 ) 3−
less than 1 + 3 will find a tour ending in t and have length less than (1 + 3 )(OPT + M ).
Without loss of generality it visits s just before t (it is on the way anyway). After deleting
the edge {s, t} we get a tour for the original r-s-path TSP instance, and it has length less
than (1+ 3 )(OPT+M )−M = (1+ 3 )OPT+ 3 3+
U < (1+ 3 )OPT+ 3 · 65 U ≤ (1+)OPT,
3−
1
in the strict inequality.
where we used < 11
Third, if no endpoint is fixed, we can apply the same trick twice, appending q at r
and (as before) t at s, and forcing any cheap path to be a path from q to t.
B. Steiner nodes
Our model allows to place bifurcation nodes at positions in M \ (µ({r} ∪ P ), but our
algorithm does not do this. We remark that using such Steiner nodes would also be
necessary to improve on Theorem 1 by more than a constant factor. Let n ∈ N, 0 < <
1
, and M := {r, s, p1 , . . . , pn } with µ(v) = v for v ∈ {r, p1 , . . . , pn } and c(r, s) = n2 and
2
2 +2n
1
c(pi , pj ) = c(s, pi ) = n2−
= c(r, pi ) − n2 for all 1 ≤ i, j ≤ n (cf. Figure 7). Let σ = 0,
2
18
δ = 1 and ∆ = n2 + n +
2
n2 +2n
= 2n +4n−n
. Then traveling first to pi and then to pj
2−
2−
2 +n)+2n
2 +4n−n)
2(n2 +2n)
+ 2− = ∆ + (2n 2−
> ∆ + (2n 2−
= (1 + )∆.
(j 6= i) takes time ∆ − n
Unless we allow violating the deadline by more than a factor 1 + , no tour can deliver
more than one item, and no subtour can start at any pi . Hence either n tours start at
r, or subtours start at s, which decreases the cost by a factor
n2 +2n
)
2−
2
+2n
)
n( n2−
n(n2 +
n2 +
2n3 + 2n2
=
n3 + (4 − )n2
19
−−−→
n→∞
2
.
| 8 |
arXiv:1301.3545v2 [cs.LG] 16 Mar 2013
Metric-Free Natural Gradient for Joint-Training of
Boltzmann Machines
Guillaume Desjardins, Razvan Pascanu, Aaron Courville and Yoshua Bengio
Département d’informatique et de recherche opérationnelle
Université de Montréal
Abstract
This paper introduces the Metric-Free Natural Gradient (MFNG) algorithm for
training Boltzmann Machines. Similar in spirit to the Hessian-Free method of
Martens [8], our algorithm belongs to the family of truncated Newton methods and
exploits an efficient matrix-vector product to avoid explicitly storing the natural
gradient metric L. This metric is shown to be the expected second derivative
of the log-partition function (under the model distribution), or equivalently, the
covariance of the vector of partial derivatives of the energy function. We evaluate
our method on the task of joint-training a 3-layer Deep Boltzmann Machine and
show that MFNG does indeed have faster per-epoch convergence compared to
Stochastic Maximum Likelihood with centering, though wall-clock performance
is currently not competitive.
1
Introduction
Boltzmann Machines (BM) have become a popular method in Deep Learning for performing feature extraction and probability modeling. The emergence of these models as practical learning
algorithms stems from the development of efficient training algorithms, which estimate the negative
log-likelihood gradient by either contrastive [4] or stochastic [18, 19] approximations. However, the
success of these models has for the most part been limited to the Restricted Boltzmann Machine
(RBM) [6], whose architecture allows for efficient exact inference. Unfortunately, this comes at the
cost of the model’s representational capacity, which is limited to a single layer of latent variables.
The Deep Boltzmann Machine (DBM) [15] addresses this by defining a joint energy function over
multiple disjoint layers of latent variables, where interactions within a layer are prohibited. While
this affords the model a rich inference scheme incorporating top-down feedback, it also makes training much more difficult, requiring until recently an initial greedy layer-wise pretraining scheme.
Since, Montavon and Muller [9] have shown that this difficulty stems from an ill-conditioning of
the Hessian matrix, which can be addressed by a simple reparameterization of the DBM energy
function, a trick called centering (an analogue to centering and skip-connections found in the deterministic neural network literature [17, 14]). As the barrier to joint-training 1 is overcoming a
challenging optimization problem, it is apparent that second-order gradient methods might prove to
be more effective than simple stochastic gradient methods. This should prove especially important
as we consider models with increasingly complex posteriors or higher-order interactions between
latent variables.
To this end, we explore the use of the Natural Gradient [2], which seems ideally suited to the stochastic nature of Boltzmann Machines. Our paper is structured as follows. Section 2 provides a detailed
derivation of the natural gradient, including its specific form for BMs. While most of these equations
1
Joint-training refers to the act of jointly optimizing θ (the concatenation of all model parameters, across all
layers of the DBM) through maximum likelihood. This is in contrast to [15], where joint-training is preceded
by a greedy layer-wise pretraining strategy.
1
have previously appeared in [3], our derivation aims to be more accessible as it attempts to derive the
natural gradient from basic principles, while minimizing references to Information Geometry. Section 3 represents the true contribution of the paper: a practical natural gradient algorithm for BMs
which exploits the persistent Markov chains of Stochastic Maximum Likelihood (SML) [18], with
a Hessian-Free (HF) like algorithm [8]. The method, named Metric-Free Natural Gradient (MFNG)
(in recognition of the similarities of our method to HF), avoids explicitly storing the natural gradient
metric L and uses a linear solver to perform the required matrix-vector product L−1 Eq [∇ log pθ ].
Preliminary experimental results on DBMs are presented in Section 4, with the discussion appearing
in Section 5.
2
The Natural Gradient
2.1
Motivation and Derivation
The main insight behind the natural gradient is that the space of all probability distributions P =
{pθ (x); θ ∈ Θ, x ∈ χ} forms a Riemannian manifold. Learning, which typically proceeds by
iteratively adapting the parameters θ to fit an empirical distribution q, thus traces out a path along
this manifold. An immediate consequence is that following the direction of steepest descent in the
original Euclidean parameter space does not correspond to the direction of steepest descent along P.
To do so, one needs to account for the metric describing the local geometry of the manifold, which
is given by the Fisher Information matrix [1], shown in Equation 4. While this metric is typically
derived from Information Geometry, a derivation more accessible to a machine learning audience
can be obtained as follows.
The natural gradient aims to find the search direction ∆θ which minimizes a given objective function, such that the Kullback–Leibler divergence KL(pθ k pθ+∆θ ) remains constant throughout
optimization. This constraint ensures that we make constant progress regardless of the curvature
of the manifold P and enforces an invariance to the parameterization of the model. The natural
gradient for maximum likelihood can thus be formalized as:
∇N := ∆θ∗
←
s.t.
argmin∆θ Eq [− log pθ+∆θ (x)]
KL(pθ k pθ+∆θ ) = const.
(1)
In order to derive a useful parameter update rule, we will consider the KL divergence under the
assumption ∆θ → 0. We also assume we have a discrete and bounded domain χ over which we
define the probability mass function2 pθ . Taking the Taylor series expansion of log pθ+∆θ around θ,
∂f
and denoting ∇f as the column vector of partial derivatives with ∂θ
as the i-th entry, and ∇2 f the
i
Hessian matrix with
∂2f
∂θi ∂θj
KL(pθ k pθ+∆θ ) ≈
in position (i, j), we have:
X
X
1
T
pθ log pθ −
pθ log pθ + (∇ log pθ ) ∆θ + ∆θT ∇2 log pθ ∆θ
2
χ
χ
1 T
∆θ Epθ −∇2 log pθ ∆θ
(2)
2
P
P
pθ
∂
with the transition stemming from the fact that χ pθ ∂ log
= ∂θ
x∈χ pθ (x) = 0. Replacing
∂θi
i
the objective function of Equation 1 by its first-order Taylor expansion and rewriting the constraint
as a Lagrangian, we arrive at the following formulation for L(θ, ∆θ), the loss function which the
natural gradient seeks to minimize.
λ
T
L(θ, ∆θ) = Eq [− log pθ ] + Eq [−∇ log pθ ] ∆θ + ∆θT Epθ −∇2 log pθ ∆θ.
2
=
Setting
∂L
∂∆θ
to zero yields the natural gradient direction ∇N :
∇N = L−1 Eq [∇ log pθ ] with L =
or equivalently L =
2
Epθ −∇2 log pθ
Epθ ∇ log pθ ∇T log pθ
When clear from context, we will drop the argument of pθ to save space.
2
(3)
(4)
While its form is reminiscent of the Newton direction, the natural gradient multiplies the estimated
gradient by the inverse of the expected Hessian of log pθ (Equation 3) or equivalently by the Fisher
Information matrix (FIM, Equation 4). The equivalence between both expressions can be shown
trivially, with the details appearing in the Appendix. We stress that both of these expectations
are computed with respect to the model distribution, and thus computing the metric L does not
involve the empirical distribution in any way. The FIM for Boltzmann Machines is thus not equal
to the uncentered covariance of the maximum likelihood gradients. In the following, we pursue our
derivation from the form given in Equation 4.
2.2
Natural Gradient for Boltzmann Machines
Derivation. Boltzmann machines define a joint distribution over
P a vector of binary
Prandom variables x ∈ {0, 1}N by way of an energy function E(x) = − k<l Wkl xk xl − k bk xk , with
weight matrix W ∈ RN ×N and bias vector b ∈ RN . Energy and probability are related by the
Boltzmann
distribution, such that p(x) = Z1 exp (−E(x)), with Z the partition function defined by
P
Z = x exp (−E(x)).
Starting from the expression of L found in Equation 3, we can derive the natural gradient metric for
Boltzmann Machines.
L(BM ) = Epθ ∇2 E(x) + ∇2 log Z = Epθ ∇2 log Z
The natural gradient metric for first-order BMs takes on a surprisingly simple form: it is the expected
Hessian of the log-partition function. With a few lines of algebra (whose details are presented in the
Appendix), we can rewrite it as follows:
h
i
T
L(BM ) = Epθ (∇E(x) − Epθ [∇E(x)]) (∇E(x) − Epθ [∇E(x)])
(5)
L(BM ) is thus given by the covariance of ∇E, measured under the model distribution pθ . Concretely,
if we denote Wkl and Wmn as the i and j-th parameters of the model respectively, the entry Lij will
take on the value −E [xk xl xm xn ] + E [xk xl ] E [xm xn ].
Discussion. When computing the Taylor expansion of the KL divergence in Equation 2, we
glossed over an important detail. Namely, how to handle latent variables in pθ (x), a topic first discussed in [3]. If x =P
[v, h], we could
Pjust as easily have derived the natural gradient by considering
the constraint KL ( h pθ (v, h) k h pθ+∆θ (v, h)) = const. Alternatively, since the distinction
between visible and hidden units is entirely artificial (since the KL divergence does not involve the
empirical distribution), we may simply wish to consider the distribution obtained by analytically
integrating out a maximal number of random variables. In a DBM, this would entail marginalizing
over all odd or even layers, a strategy employed with great success in the context of AIS [15]. In this
work however, we only consider the metric obtained by considering the KL divergence between the
full joint distributions pθ and pθ+∆θ .
3
Metric-Free Natural Gradient Implementation
We can compute the natural gradient ∇N by first replacing the expectations of Equation 5 by a finite
sample approximation. We can do this efficiently by reusing the model samples generated by the
persistent Markov chains of SML. Given the size of the matrix being estimated however, we expect
this method to require a larger number of chains than is typically used. The rest of the method is
similar to the Hessian-Free (HF) algorithm of Martens [8]: we exploit an efficient matrix-vector
implementation combined with a linear-solver, such as Conjugate Gradient or MinRes[12], to solve
the system Ly = Eq [∇ log pθ ] for y ∈ RN . Additionally, we replace the expectation on the rhs.
of this previous equation by an average computed over a mini-batch of training examples (sampled
from the empirical distribution q), as is typically done in the stochastic learning setting.
For Boltzmann Machines, the matrix-vector product Ly can be computed in a straightforward manner, without recourse to Pearlmutter’s R-operator [13]. Starting from a sampling approximation to
Equation 5, we simply push the dot product inside of the expectation as follows:
3
−
Algorithm 1 MFNG iteration(θ, X + , Zold
)
θ: parameters of the model. N :=| θ |.
X + : mini-batch of training examples, with X + = {xm ; m ∈ [1, M ]}.
(1)
(2)
−
Zold
: previous state of Markov chains, with Z = {zm := (vm , hm , hm ); m ∈ [1, M ]}
(1)+
(2)+
+ :=
• Generate positive phase samples Z + = {zm
(xm , hm , hm ); m ∈ [1, M ]}
−
−
• Initializing M Markov chains from state Zold
, generate negative phase samples Znew
.
−
∂E(zm
)
, ∀m.
∂θ
P
1
+
−
Compute negative log-likelihood gradient as g = M
m (sm − sm ).
P −
1
M ×N
−
Denote S ∈ R
as the matrix with rows sm and S̄ = M m sm .
• Compute the vectors s+
m =
•
•
+
∂E(zm
)
∂θ
and s−
m =
# Solve the system “Ly = g” for y, given L = (S − S̄)T (S − S̄) and an initial zero vector.
# computeLy is a function which performs equation 6, without instantiating L.
• ∇N θ ← CGSolve(computeLy, S, g, zeros(N ))
L(BM ) y
with
and
and
≈
S − S̄
T
S − S̄ y
(6)
∂E(xm )
∂θj
X
1
smj
S̄ ∈ RN , the vector with entries sj =
M m
S ∈ RM ×N , the matrix with entries smj =
xm ∼ pθ (x), m ∈ [1, M ].
By first computing the matrix-vector product (S − S̄)y, we can easily avoid computing the full
N × N matrix L. Indeed, the result of this operation is a vector of length M , which is then leftmultiplied by a matrix of dimension N × M , yielding the matrix-vector product Ly ∈ RN . A
single iteration of the MFNG is presented in Algorithm 1. A full open-source implementation is
also available online.3 .
4
Experiments
We performed a proof-of-concept experiment to determine whether our Metric-Free Natural Gradient (MFNG) algorithm is suitable for joint-training of complex Boltzmann Machines. To this
end, we compared our method to Stochastic Maximum Likelihood and a diagonal approximation of
MFNG on a 3-layer Deep Boltzmann Machine trained on MNIST [7]. All algorithms were run in
conjunction with the centering strategy of Montavon and Muller [9], which proved crucial to successfully joint-train all layers of the DBM (even when using MFNG) 4 . We chose a small 3-layer
DBM with 784-400-100 units at the first, second and third layers respectively, to be comparable to
[10]. Hyper-parameters were varied as follows. For inference, we ran 5 iterations of either meanfield as implemented in [15] or Gibbs sampling. The learning rate was kept fixed during training and
chosen from the set {5 · 10−3 , 10−3 , 10−4 }. For MinRes, we set the damping coefficient to 0.1 and
used a fixed tolerance of 10−5 (used to determine convergence). Finally, we tested all algorithms
on minibatch sizes of either 25, 128 or 256 elements 5 . Finally, since we are comparing optimization algorithms, hyper-parameters were chosen based on the training set likelihood (though we still
report the associated test errors). All experiments used the MinRes linear solver, both for its speed
and its ability to return pseudo-inverses when faced with ill-conditioning.
3
https://github.com/gdesjardins/MFNG
The centering coefficients were initialized as in [9], but were otherwise held fixed during training.
5
We expect larger minibatch sizes to be preferable, however simulating this number of Markov chains in
parallel (on top of all other memory requirements) was sufficient to hit the memory bottlenecks of GPUs.
4
4
Figure 1: Estimated model likelihood as a function of (left) epochs and (right) CPU-time for MFNG,
its diagonal approximation (MFNG-diag) and SML. All methods were run in conjunction with the
DBM centering trick [9], with centering coefficients held fixed during training. Our grid search
yielded the following hyper-parameters: batch size of 256/128 for MFNG(-diag)/SGD; 5 steps of
mean-field / sampling-based inference for MFNG(-diag)/SGD and a learning rate of 5 · 10−3 .
Figure 1 (left) shows the likelihood as estimated by Annealed Importance Sampling [15, 11] as a
function of the number of epochs 6 . Under this metric, MFNG achieves the fastest convergence,
obtaining a training/test set likelihood of −71.26/−72.84 nats after 94 epochs. In comparison,
MFNG-diag obtains −73.22/−74.05 nats and SML −80.12/−79.71 nats in 100 epochs. The picture
changes however when plotting likelihood as a function of CPU-time, as shown in Figure 1 (right).
Given a wall-time of 8000s for MFNG and SML, and 5000s for MFNG-diag7 , SML is able to
perform upwards of 1550 epochs, resulting in an impressive likelihood score of −64.94 / −67.73.
Note that these results were obtained on the binary-version of MNIST (thresholded at 0.5) in order
to compare to [10]. These results are therefore not directly comparable to [15], which binarizes
the dataset through sampling (by treating each pixel activation as the probability p of a Bernouilli
distribution).
Figure 2 shows a breakdown of the algorithm runtime, for various components of the algorithm.
These statistics were collected in the early stages of training, but are generally representative of the
bigger picture. While the linear solver clearly dominates the runtime, there are a few interesting
observations to make. For small models and batch sizes greater than 256, a single evaluation of Ly
appears to be of the same order of magnitude as a gradient evaluation. In all cases, this cost is smaller
than that of sampling, which represents a non-negligible part of the total computational budget.
This suggests that MFNG could become especially attractive for models which are expensive to
sample from. Overall however, restricting the number of CG/MinRes iterations appears key to
computational performance, which can be achieved by increasing the damping factor α. How this
affects convergence in terms of likelihood is left for future work.
5
Discussion and Future Work
While the wall-clock performance of MFNG is not currently competitive with SML, we believe there
are still many avenues to explore to improve computational efficiency. Firstly, we performed almost
no optimization of the various MinRes hyper-parameters. In particular, we ran the algorithm to
convergence with a fixed tolerance of 10−5 . While this typically resulted in relatively few iterations
(around 15), this level of precision might not be required (especially given the stochastic nature of
6
While we do not report error margins for AIS likelihood estimates, the numbers proved robust to changes in
the number of particles and temperatures being simulated. To obtain such robust estimates, we implemented all
the tricks described in Salakhutdinov and Hinton [15] and [16]: pA a zero-weight base-rate model whose biases
(1−β ) (β )
are set by maximum likelihood; interpolating distributions pi ∝ pA i pB i , with pB the target distribution;
and finally analytical integration of all odd-layers.
7
This discrepancy will be resolved in the next revision.
5
Figure 2: Breakdown of algorithm runtime, when we vary (left) the batch size (with fixed model
architecture 784 − 400 − 100) and (right) the model size (with fixed batch size of 256). Runtime
is additive in the order given by the labels (top to bottom). Dotted lines denote intermediate steps,
while continuous lines denote full steps. Data was collected on a Nvidia GTX 480 card.
the algorithm). Additionally, it could be worth exploiting the same strategy as HF where the linear
solver is initialized by the solution found in the previous iteration. This may prove much more
efficient than the current approach of initializing the solver with a zero vector. Pre-conditioning
is also a well-known method for accelerating the convergence speed of linear solvers [5]. Our
implementation used a simple diagonal regularization of L. The Jacobi preconditioner could be
implemented easily however by computing the diagonal of L in a first-pass.
Finally, while our single experiment offers little evidence in support of either conclusion, it may very
well be possible that MFNG is simply not computationally efficient for DBMs, compared to SML
with centering. In this case, it would be worth applying the method to either (i) models with known
ill-conditioning, such as factored 3-rd order Boltzmann Machines or (ii) models and distributions
exhibiting complex posterior distributions. In such scenarios, we may wish to maximize the use
of the positive phase statistics (which were obtained at a high computational cost) by performing
larger jumps in parameter space. It remains to be seen how this would interact with SML, where the
burn-in period of the persistent chains is directly tied to the magnitude of ∆θ.
Appendix
We include the following derivations for completeness.
5.1
Expected Hessian of log Z and Fisher Information.
2
∂ log p(x)
Epθ −
∂θi ∂θj
1 ∂p(x) ∂p(x)
1 ∂ 2 p(x)
Epθ
−
p(x)2 ∂θj ∂θi
p(x) ∂θi ∂θj
1 ∂p(x)
1 ∂ 2 p(x)
1 ∂p(x)
−
Epθ
p(x) ∂θi
p(x) ∂θj
p(x) ∂θi ∂θj
X
∂ log p(x) ∂ log p(x)
1 ∂ 2 p(x)
Epθ
−
p(x)
∂θi
∂θj
p(x) ∂θi ∂θj
x
X 2
∂ p(x)
∂ log p(x) ∂ log p(x)
Epθ
−
∂θi
∂θj
∂θi ∂θj
x
P
2
∂
∂ log p(x) ∂ log p(x)
x p(x)
Epθ
−
∂θi
∂θj
∂θi ∂θj
∂ log p(x) ∂ log p(x)
Epθ
∂θi
∂θj
=
=
=
=
=
=
6
5.2
Derivation of Equation 5
log p(x) =
∂ log p(x)
=
∂θi
=
2
∂ log p(x)
Epθ −
=
∂θi ∂θj
=
−E(x) − log Z
∂E(x)
1 X ∂
−
−
[exp (−E(x))]
∂θi
Z x ∂θi
∂E(x)
∂E(x)
+ Epθ
−
∂θi
∂θi
∂E(x)
∂E(x)
∂E(x)
∂E(x)
Epθ
− Epθ
− Epθ
∂θi
∂θi
∂θj
∂θj
∂E(x) ∂E(x)
∂E(x)
∂E(x)
− Epθ
Epθ
Epθ
∂θi
∂θj
∂θi
∂θj
References
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pages 33–40.
[5] Chapelle, O. and Erhan, D. (2011). Improved preconditioner for hessian free optimization. NIPS Workshop
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recognition. Proc. IEEE.
[8] Martens, J. (2010). Deep learning via Hessian-free optimization. pages 735–742.
[9] Montavon, G. and Muller, K.-R. (2012). Deep Boltzmann machines and the centering trick. In G. Montavon, G. Orr, and K.-R. Muller, editors, Neural Networks: Tricks of the Trade, volume 7700 of Lecture
Notes in Computer Science, pages 621–637.
[10] Montavon, G. and Müller, K.-R. (2012). Learning feature hierarchies with centered deep Boltzmann
machines. CoRR, abs/1203.4416.
[11] Neal, R. M. (2001). Annealed importance sampling. Statistics and Computing, 11(2), 125–139.
[12] Paige, C. C. and Saunders, M. A. (1975). Solution of Sparse Indefinite Systems of Linear Equations.
SIAM Journal on Numerical Analysis, 12(4), 617–629.
[13] Pearlmutter, B. (1994). Fast exact multiplication by the Hessian. Neural Computation, 6(1), 147–160.
[14] Raiko, T., Valpola, H., and LeCun, Y. (2012). Deep learning made easier by linear transformations in
perceptrons. In AISTATS’2012.
[15] Salakhutdinov, R. and Hinton, G. (2009). Deep Boltzmann machines. In Proceedings of the Twelfth
International Conference on Artificial Intelligence and Statistics (AISTATS 2009), volume 8.
[16] Salakhutdinov, R. and Murray, I. (2008). On the quantitative analysis of deep belief networks. volume 25,
pages 872–879.
[17] Schraudolph, N. N. (1998). Centering neural network gradient factors. In G. B. Orr and K.-R. Muller,
editors, Neural Networks: Tricks of he Trade, pages 548–548. Springer.
[18] Tieleman, T. (2008). Training restricted Boltzmann machines using approximations to the likelihood
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[19] Younes, L. (1999). On the convergence of Markovian stochastic algorithms with rapidly decreasing
ergodicity rates. Stochastics and Stochastic Reports, 65(3), 177–228.
7
| 9 |
1
Coding Method for Parallel Iterative Linear Solver
arXiv:1706.00163v3 [cs.IT] 5 Jun 2017
Yaoqing Yang, Student Member, IEEE, Pulkit Grover, Senior Member, IEEE, and Soummya Kar
Abstract—Computationally intensive distributed and parallel
computing is often bottlenecked by a small set of slow workers
known as stragglers. In this paper, we utilize the emerging
idea of “coded computation” to design a novel error-correctingcode inspired technique for solving linear inverse problems
under specific iterative methods in a parallelized implementation
affected by stragglers. Example applications include inverse
problems in machine learning on graphs, such as personalized
PageRank and sampling on graphs. We provably show that our
coded-computation technique can reduce the mean-squared error
under a computational deadline constraint. In fact, the ratio of
mean-squared error of replication-based and coded techniques
diverges to infinity as the deadline increases. Our experiments
for personalized PageRank performed on real systems and real
social networks show that this ratio can be as large as 104 .
Further, unlike coded-computation techniques proposed thus far,
our strategy combines outputs of all workers, including the
stragglers, to produce more accurate estimates at the computational deadline. This also ensures that the accuracy degrades
“gracefully” in the event that the number of stragglers is large.
Index Terms—Distributed computing, error-correcting codes,
straggler effect, iterative linear inverse, personalized PageRank
I. I NTRODUCTION
Although modern distributed computing systems have developed techniques for maintaining fault tolerance, the performance of such computing systems is often bottlenecked
by a small number of slow workers. This “straggling” effect
of the slow workers [1]–[4] is often addressed by replicating
tasks across workers and using this redundancy to ignore some
of the stragglers. Recently, concepts from error-correcting
codes have been used for speeding up distributed computing
[5]–[18], which build on classical works on algorithm-based
fault-tolerance [19]. The key idea is to treat slow workers as
“erasures” and use error-correcting codes to retrieve the result
after a subset of fast workers have finished. In some cases,
(e.g. [6], [8] for matrix multiplications), coding techniques
(i.e., techniques that utilize error-correcting codes) achieve
scaling-sense speedups in average computation time compared
to replication. In this work, we propose a novel coding-inspired
technique to deal with the straggler effect in distributed computing of linear inverse problems using iterative solvers [20],
such as personalized PageRank [21], [22] and signal recovery
on large graphs [23]–[25]. We focus on iterative methods for
solving these linear systems. For the personalized PageRank
problem, we study the power-iteration method which is the
most classical PageRank algorithm [21]. For the problem of
signal recovery on graphs, we study linear iterative algorithms
such as the projected gradient descent algorithm [23]–[25].
Existing algorithms that use coding to deal with stragglers
treat straggling workers as erasures, that is, they ignore comY. Yang, P. Grover and S. Kar are with the Department of Electrical and
Computer Engineering, Carnegie Mellon University, Pittsburgh, PA, 15213.
Classical coded computation
r1
r2
E
n
c
o
d
e
r1
r2
r1+r2
Processor1
Processor2
(slow)
Processor3
(slow)
waitEforEtwoE
fastEworkers
*1
D
e
c
o
ignore!
d
x1*+x2*
e
x
x*2
fail
Proposed coded method
r1
r2
E
n
c
o
d
e
r1
r2
r1+r2
DeadlineETdl
Processor1
x1* +e1
Processor2
(slow)
x2* +e2
Processor3
(slow)
x1* +x2* +e3
D
e weightedE
c combination
o
d
e
Fig. 1. A toy example of the comparison between the existing scheme in [6]
and the proposed algorithm.
putation results of the stragglers. Interestingly, in the iterative
solvers for linear inverse problems, even if the computation
result at a straggler has not converged, the proposed algorithm
does not ignore the result, but instead combines it (with
appropriate weights) with results from other workers. This is
in part because the result of the iterative method converges
gradually to the true solution. We use a small example shown
in Fig. 1 to illustrate this idea. Suppose we want to solve
two linear inverse problems with solutions x∗1 and x∗2 . We
“encode the computation” by adding an extra linear inverse
problem, the solution of which is x∗1 + x∗2 (see Section III-A),
and distribute these three problems to three workers. Using
this method, the solutions x∗1 and x∗2 can be obtained from
the results of any combination of two fast workers that are
first to come close to their solutions.
But what if we have a computational deadline, Tdl , by which
only one worker converges to its solution? In that case, the natural extension of existing coded-computation strategies (e.g.,
[6]) will declare a computation failure because it needs at least
two workers to respond. However, our strategy does not require
convergence: even intermediate results from workers, as long
as they are received, can be utilized to estimate solutions. In
other words, our strategy degrades gracefully as the number
of stragglers increases, or as the deadline is pulled earlier.
Indeed, we will show that it is suboptimal to ignore stragglers
as erasures, and will design strategies that treat the difference
from the optimal solution as “soft” additive noise (see Fig. 3,
and Section III-A). We use an algorithm that is similar to
weighted least squares algorithm for decoding, giving each
worker a weight based on its proximity to convergence. In
this way, we can expect to fully utilize the computation results
from all workers and obtain better speedup.
Theoretically, we show that for a specified deadline time
Tdl , under certain conditions on worker speed, the coded linear
inverse solver using structured codes has smaller mean squared
2
error than the replication-based linear solver (Theorem IV.4).
In fact, when the computation time Tdl increases, the ratio of
the mean-squared error (MSE) of replication-based and coded
linear solvers can get arbitrarily large (Theorem IV.5)!
For validation of our theory, we performed experiments to
compare the performance of coded and replication-based personalized PageRank (respectively using coded and replicationbased power-iteration method) on the Twitter and Google Plus
social networks under a deadline on computation time using
a given number of workers on a real computation cluster
(Section VI-A). We observe that the MSE of coded PageRank
is smaller than that of replication by a factor of 104 at
Tdl = 2 seconds. From an intuitive perspective, the advantage
of coding over replication is that coding utilizes the diversity
of all heterogeneous workers. To compare with existing coded
technique in [6], we adapt its technique to inverse problems
by inverting only the partial results from the fast workers.
However, from our experiments, if only the results from the
fast workers are used, the error amplifies due to inverting an illconditioned submatrix during decoding (Section VI-A). This
ill-conditioning issue of real-number erasure codes has also
been recognized in a recent communication problem [26]. In
contrast, our novel way of combining all the partial results
including those from the stragglers helps bypass the difficulty
of inverting an ill-conditioned matrix.
The focus of this work is on utilizing computations to
deliver the minimal MSE in solving linear inverse problems.
Our algorithm does not reduce the communication cost. However, because each worker performs sophisticated iterative
computations, such as the power-iteration computations in our
problem, the time required for computation dominates that
of communication (see Section V-C). This is unlike some
recent works (e.g. [7], [11], [27]–[30]) where communication
costs are observed to dominate. However, in these works, the
computational tasks at each worker are simpler, and sometimes
the communication cost is large because of data-shuffling (e.g.
in MapReduce [31]).
Finally, we summarize our main contributions in this paper:
A. Preliminaries on Solving Parallel Linear Systems using
Iterative Methods
Consider the problem of solving k inverse problems with
the same linear transform matrix and different inputs ri :
Mxi = ri , i = 1, 2, . . . k.
When M is a square matrix, the closed-form solution is
xi = M−1 ri .
•
•
•
We propose a coded algorithm for distributed computation for multiple instances of a linear inverse problem;
We theoretically analyze the mean-squared error of
coded, uncoded and replication-based iterative linear
solvers under a deadline constraint;
We compare mean-squared error of coded linear solver
with uncoded and replication-based linear solver and
show scaling sense advantage in theory and orders of
magnitude smaller error in data experiments;
This is the first work that treats stragglers as soft errors
instead of erasures, which leads to graceful degradation
in the event that the number of stragglers is large.
II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
In this section, we present the model of parallel linear
systems that we will apply the idea of error correcting codes.
Then, we provide two applications that can be directly formulated in the form of parallel linear systems.
(2)
When M is a non-square matrix, the solution to (1) is
interpreted as
2
2
xi = arg min kMx − ri k + λ kxk , i = 1, 2, . . . k,
(3)
with an appropriate regularization parameter λ. The closedform solution of (3) is
xi = (M> M + λI)−1 M> ri .
(4)
Computing matrix inverse in (2) or (4) directly is often hard
and commonly used methods are often iterative. In this paper,
we study two ordinary iterative methods, namely the Jacobian
method for solving (1) and the gradient descent method for
solving (3).
1) The Jacobian Method for Square System: For a square
matrix M, one can decompose M = D + L, where D is
diagonal. Then, the Jacobian iteration is written as
(l+1)
xi
(l)
= D−1 (ri − Lxi ).
(5)
Under certain conditions of D and L (see [20, p.115]), the
computation result converges to the true solution.
2) Gradient Descent for Non-square System: For the `-2
minimization problem (3), the gradient descent solution has
the form
(l+1)
xi
(l)
= ((1 − λ)I − M> M)xi + M> ri ,
(6)
where is an appropriate step-size.
From these two problem formulations, we can see that the
iterative methods have the same form
(l+1)
•
(1)
xi
(l)
= Bxi + Kri , i = 1, 2, . . . k,
(7)
for two appropriate matrices B and K. Denote by x∗i the
solution to (1) or (3). Then,
x∗i = Bx∗i + Kri , i = 1, 2, . . . k.
(8)
(l)
(l)
Then, from (8) and (7), the computation error ei = xi − x∗i
satisfies
(l+1)
(l)
ei
= Bei .
(9)
B. Distributed Computing and the Straggler Effect
Consider solving k linear inverse problems in k parallel
workers using the iterative method (7), such that each processor solves one problem. Due to the straggler effect, the
computation of the linear inverse problem on different workers
can have different computation speeds. Suppose after the
deadline time Tdl , the i-th worker has completed li iterations
in (7). Then, the residual error at the i-th worker is
(l )
(0)
ei i = Bli ei .
(10)
3
For our theoretical results, we sometimes need the following
assumption.
Assumption 1. We assume that the optimal solutions x∗i , i =
1, 2, . . . k are i.i.d.
Denote by µE and CE respectively the mean and the
covariance of each x∗i . Assume we start with the initial
(0)
estimate xi = µE , which can be estimated from data. Then,
(0)
(0)
ei = xi − x∗i has mean 0N and covariance CE .
Note that Assumption 1 is equivalent to the assumption that
the inputs ri , i = 1, 2, . . . k are i.i.d., because the input and the
true solution for a linear inverse problem are related by a linear
transform. We provide an extension of this i.i.d. assumption
in Section III-C.
C. Two Motivating Examples
Now we study two examples of the general linear inverse
problems. Both of them related to data processing of graphs.
1) PageRank as a Square System: For a directed graph
G = (V, E) with node set V and edge set E, the PageRank
algorithm aims to measure the importance of the nodes in
V by computing the stationary distribution of a discretetime “random walk with restart” on the graph that mimics
the behavior of a web surfer on the Internet. At each step,
with probability 1 − d, the random walk chooses a random
neighbor on the graph with uniform probability to proceed
to (e.g. d = 0.15 in [21]). With probability d, it jumps to
an arbitrary node in the graph. The probability d is often
called the “teleport probability”. From [21], the problem of
computing the stationary distribution is equivalent to solving
the following linear problem
d
1N + (1 − d)Ax,
(11)
N
where N is the number of nodes and A is the columnnormalized adjacency matrix, i.e., for a directed edge vi → vj ,
1
, where deg(vj ) is the in-degree of vj .
Aij = deg(v
j)
The personalized PageRank problem [22] considers a more
general linear equation
x=
x = dr + (1 − d)Ax,
(12)
for any possible vector r ∈ RN that satisfies 1> r = 1. Compared to the original PageRank problem [21], personalized
PageRank [22] utilizes both the structure of the graph and
the personal preferences of different users. The solution x
is also the limiting distribution of a random walk, but with
different restarting distribution. That is, with probability d,
instead of jumping to each node with uniform probability, the
random walk jumps to different nodes according to distribution
r. Intuitively, difference in the vector r represents different
preferences of web surfers.
A classical method to solve PageRank is power-iteration,
which iterates the following computation until convergence
(0)
(usually with initial condition xi = N1 1N ):
x(l+1) = dr + (1 − d)Ax(l) .
(13)
One can see that the power-iteration method is exactly the
same as the Jacobian iteration (5).
2) “Graph Signal Reconstruction” as a Non-Square System: The emerging field of signal processing on graphs [32],
[33] is based on a neat idea of treating “values” associated
with the nodes of a graph as “a signal supported on a graph”
and apply techniques from signal processing to solve problems
such as prediction and detection. For example, the number
of cars at different road intersections on the Manhattan road
network can be viewed as a graph signal on the road graph
G = (V, E) where V is the set of road intersections and E is
the set of road segments. For a directed or undirected graph
G = (V, E), the graph signal has the same dimension as the
number of nodes |V| in the graph, i.e., there is only one value
associated with each node.
One important problem of graph signal processing is that
of recovering the values on the remaining nodes of a graph
given the values sampled on a particular node subset S ⊂ V
under the assumption of “bandlimited” graph signal [23]–[25].
One application of graph signal reconstruction can be that of
reconstructing the entire traffic flow using the observations
from a few cameras at some road intersections [34]. The graph
signal reconstruction problem can be formulated as a leastsquare solution of the following linear system (see equation
(5) in [24])
α1
f (S)
u1 (S) u2 (S) . . . uw (S) α2
=
,
f (S c )
u1 (S c ) u2 (S c ) . . . uw (S c ) ...
αw .
(14)
where f (S) is the given part of the graph signal f on the
set S, f (S c ) is the unknown part of the graph signal f to
be reconstructed, [α1 , α2 , . . . αw ] are unknown coefficients of
f (S)
the graph signal f =
represented in the subspace
f (S c )
spanned by u1 , u2 , . . . uw which are the first w eigenvectors
of the graph Laplacian matrix. It is shown in [24] that this
least-square reconstruction problem can be solved using an
iterative linear projection method (see equation (11) in [24])
f (S)
fk+1 = UU> fk + J> J(
− fk ) ,
(15)
0
where U = [u1 , u2 , . . . uw ]. This iteration can be shown to
converge to the least square solution given by
−1
f (S c ) = (U)S c ((U)>
(U)tS c f (S).
S c (U)S c )
(16)
Note that we may want to reconstruct multiple instances of
graph signal, such as road traffic at different time, which brings
in the formulation of distributed computing.
D. Preliminaries on Error Correcting Codes
We borrow concepts and terminology from error-correcting
codes [35]. E.g., we will use “encode” and “decode” to denote
preprocessing and post-processing before and after parallel
computation. Although classically encoding and (specially)
decoding can be nonlinear operations, in this paper, the
encoder multiplies the inputs to the parallel workers with a
“generator matrix” G and the decoder multiplies the outputs
4
of the workers with a “decoding matrix” L. We call a code
an (n, k) code if the generator matrix has size k × n.
In this paper, we often use generator matrices G with
orthonormal rows, which means
Gk×n G>
n×k = Ik .
(17)
An example of such a matrix is the submatrix formed by any
k rows of an n×n orthonormal matrix (e.g., a Fourier matrix).
Under this assumption, Gk×n can be augmented to form an
n×n orthonormal matrix using
another matrix H(n−k)×n , i.e.
Gk×n
the square matrix Fn×n =
satisfies F> F = In .
H(n−k)×n
This structure assumption is not necessary when we compare
the error exponents of different linear inverse algorithms when
the computation time Tdl goes to infinity (e.g., coded and
uncoded). However, it is useful when we present theorems
on finite Tdl .
III. C ODED D ISTRIBUTED C OMPUTING OF L INEAR
I NVERSE P ROBLEMS
A. The Coded Linear Inverse Algorithm
The proposed coded linear inverse algorithm is shown in
Algorithm 1. The algorithm has three stages: preprocessing
(encoding) at the central controller, parallel computing at n
parallel workers, and post-processing (decoding) at the central
controller. As we show later in the analysis of computing
error, the entries in the diagonal matrix Λ are the expected
mean-squared error at each worker prior to decoding. The
decoding matrix Lk×n in the decoding step (21) is chosen to
be (GΛ−1 G> )−1 GΛ−1 to reduce the mean-squared error of
the estimates of linear inverse solutions by assigning different
weights to different workers based on the estimated accuracy
of their computation (which is what Λ provides).
B. Bounds on Performance of the Coded Linear Inverse Algorithm
Define l = [l1 , l2 , . . . ln ] as the vector of the number
of iterations at all workers. We use the notation E[·|l] to
denote the conditional expectation taken with respect to the
randomness of the optimal solution x∗i (see Assumption 1)
but conditioned on fixed iteration number li at each worker,
i.e., for a random variable X,
E[X|l] = E[X|l1 , l2 , . . . ln ].
(24)
Theorem III.1. Define E = X̂ − X∗ , i.e., the error of the
decoding result (21). Assuming that the solutions for each
linear inverse problem are chosen i.i.d. (across all problems)
according to a distribution with covariance CE . Then, the error
covariance of E satisfies
2
E[kEk |l] ≤ σmax (G> G) trace (GΛ−1 G> )−1 , (25)
where the norm k·k is the Frobenius norm, σmax (G> G) is the
maximum eigenvalue of G> G and the matrix Λ is defined in
(22). Further, when G has orthonormal rows,
2
E[kEk |l] ≤ trace (GΛ−1 G> )−1 ,
(26)
Algorithm 1 Coded Distributed Linear Inverse
Input: Input vectors [r1 , r2 , . . . , rk ], generator matrix
Gk×n , the linear system matrices B and K defined in (7).
Initialize (Encoding): Encode the input vectors
[r1 , r2 , . . . , rk ] and the initial estimates by multiplying
with the generator matrix G:
(0)
[s1 , s2 , . . . , sn ] = [r1 , r2 , . . . , rk ] · G.
(18)
(0)
(19)
(0)
(0)
(0)
[y1 , y2 , . . . , yn(0) ] = [x1 , x2 , . . . , xk ] · G.
Parallel Computing:
for i = 1 to n do
(0)
Send si and yi to the i-th worker. Compute the solution
of (1) or (3) using the specified iterative method (7) with
(0)
initial estimate yi at each worker in parallel until a
deadline Tdl .
end for
(l )
After Tdl , collect all linear inverse results yi i from these
(l )
n workers. The superscript li in yi i represents that the ith worker finished li iterations within time Tdl . Denote by
Y(Tdl ) the collection of all results
(T )
(l )
(l )
dl
YN ×n
= [y1 1 , y2 2 , . . . , yn(ln ) ].
(20)
Post Processing (Decoding):
Compute an estimate of the linear inverse solutions using
the following matrix multiplication:
X̂> = L · (Y(Tdl ) )> := (GΛ−1 G> )−1 GΛ−1 (Y(Tdl ) )> ,
(21)
where the estimate X̂N ×k = [x̂1 , x̂2 , . . . , x̂k ], the matrix Λ
is
Λ = diag [trace(C(l1 )), . . . , trace(C(ln ))] ,
(22)
where the matrices C(li ), i = 1, . . . , n are defined as
C(li ) = Bli CE (B> )li .
(23)
In computation of Λ, if trace(C(li )) are not available, one
can use estimates of this trace as discussed in Section V-B.
Proof. See appendix Section VIII-A for a detailed proof. Here
we provide the main intuition by analyzing a “scalar version”
of the linear inverse problem, in which case the matrix B is
equal to a scalar a. In appendix Section VIII-A, we make the
generalization to all B matrices using matrix vectorization.
For B = a, the inputs and the initial estimates in (18) and
(19) are vectors instead of matrices. As we show in appendix
Section VIII-A, if we encode both the inputs and the initial
estimates using (18) and (19), we also “encode” the error
(0)
(0)
(0)
(0)
(0)
[1 , 2 , . . . , (0)
n ] = [e1 , e2 , . . . , ek ] · G =: E0 G, (27)
(0)
(0)
where i = yi − yi∗ is the initial error at the i-th worker,
(0)
(0)
ei = xi − x∗i is the initial error of the i-th linear inverse
(0) (0)
(0)
(0)
problem, and E0 := [e1 , e2 , . . . ek ]. Suppose var[ei ] =
ce , which is a scalar version of CE after Assumption 1. From
5
(10), the error satisfies:
(l )
i i
=
C. Bounds on the Mean-squared Error beyond the i.i.d. Case
(0)
ali i , i
= 1, 2, . . . n.
(28)
Denote by D = diag{al1 , al2 , . . . aln }. Therefore, from (27)
and (28), the error before the decoding step (21) can be written
as
(l ) (l )
(0) (0)
(0)
n)
[1 1 , 2 2 , . . . (l
n ] =[1 , 2 , . . . n ] · D = E0 GD. (29)
We can show (see appendix Section VIII-A for details) that
after the decoding step (21), the error vector is also multiplied
by the decoding matrix L = (GΛ−1 G> )−1 GΛ−1 :
h
i>
(l ) (l )
n)
(30)
E> = L 1 1 , 2 2 , . . . (l
= LD> G> E>
n
0.
Thus,
2
E[kEk |l] =E[trace[E> E]|l]
>
= trace[LD> G> E[E>
0 E0 |l]GDL ]
(a)
= trace[LD> G> ce Ik GDL> ]
=ce trace[LD> G> GDL> ]
(31)
(b)
≤ce σmax (G> G) trace[LD> DL> ]
= σmax (G> G) trace[LΛL> ]
(d)
= σmax (G> G) trace[(GΛ−1 G> )−1 ],
(0)
where (a) holds because E0 := [e1 , e2 , . . . ek ] and
(0)
var[ei ] = ce , (b) holds because G> G σmax (G> G)In ,
(c) holds because ce D> D = Λ, which is from the fact that
for a scalar linear system matrix B = a, the entries in the Λ
matrix in (22) satisfy
li
> li
2li
trace(C(li )) = a ce (a ) = ce a
,
(32)
which is the same as the entries in the diagonal matrix
ce D> D. Finally, (d) is obtained by directly plugging in H :=
(GΛ−1 G> )−1 GΛ−1 . Finally, inequality 26 holds because
when G has orthonormal rows, σ(G> G) = σ(G> G) =
1.
Corollary III.1. Suppose the i.i.d. Assumption 1 holds and the
matrix Gk×n is a submatrix of an n × n
orthonormal
matrix,
Gk×n
i.e. there exists a matrix Fn×n =
satisfies
H(n−k)×n
F> F = In . Assume that the symmetric matrix FΛF> has
the block form
J1 J2
FΛF> = >
,
(33)
J2 J4 n×n
that is, (J1 )k×k is GΛG> , (J2 )k×(n−k) is GΛH> , and
(J4 )(n−k)×(n−k) is HΛH> . Then, we have
2
>
E[kEk |l] ≤ trace(J1 ) − trace(J2 J−1
4 J2 ).
(37)
Under this assumption, we have to change the coded linear
inverse algorithm slightly. The details are shown in Algorithm 2.
(c)
(0)
for some random vector x̄ and an i.i.d. vector zi across
different queries (different i). The common part x̄ is random
because the common fashion topic itself can be random. This
model can be generalized to the following “stationary” model.
Assumption 2. Assume the solutions x∗i ’s of the linear inverse
problems have the same mean µE and stationary covariances,
i.e.,
E[x∗i (x∗i )> ] = CE + CCor , ∀1 ≤ i ≤ k,
(36)
E[x∗i (x∗j )> ] = CCor , ∀1 ≤ i, j ≤ k.
=σmax (G> G) trace[L(ce D> D)L> ]
(0)
Until now, we based our analysis on the i.i.d. assumption 1.
For the PageRank problem discussed in Section II-C, this
assumption means that the PageRank queries are independent
across different users. Although the case when the PageRank
queries are arbitrarily correlated is hard to analyze, we may
still provide concrete analysis for some specific cases. For
example, a reasonable case when the PageRank queries are
correlated with each other is when these queries are all affected
by some “common fashion topic” that the users wish to
search for. In mathematics, we can model this phenomenon by
assuming that the solutions to the i-th linear inverse problem
satisfies
x∗i = x̄ + zi ,
(35)
(34)
Proof. The proof essentially relies on the Schur complement.
See appendix Section VIII-B for details.
Algorithm 2 Coded Distributed Linear Inverse (Stationary
Inputs)
Call Algorithm 1 but replace the Λ matrix with
Λ̃ = σmax (G> G)Λ+diag{G> 1k }·Ψ·diag{G> 1k }> , (38)
where σmax (G> G) is the maximum eigenvalue of G> G, and
Ψn×n = [Ψi,j ] satisfies
Ψi,j = trace[Bli Ccor (B> )lj ].
(39)
For the stationary version, we can have the counterpart
of Theorem III.1 as follows. Trivial generalizations include
arbitrary linear scaling x∗i = αi x̄ + βi zi for scaling constants
αi and βi .
Theorem III.2. Define E = X̂ − X∗ , i.e., the error of the
decoding result (21) by replacing Λ defined in (22) with Λ̃
in (38). Assuming that the solutions for all linear inverse
problems satisfy Assumption 2. Then, the error covariance of
E satisfies
h
i
2
E[kEk |l] ≤ trace (GΛ̃−1 G> )−1 .
(40)
where the norm k·k is the Frobenius norm.
Proof. See appendix Section VIII-C.
In Section IV, we compare coded, uncoded and replicationbased linear inverse schemes under the i.i.d. assumption.
However, we include one experiment in Section VI-A to show
that Algorithm 2 also works in the stationary case.
6
IV. C OMPARISON WITH U NCODED S CHEMES AND
R EPLICATION - BASED S CHEMES
trace(J1 ) =
j=1 i=1
Here, we often assume (we will state explicitly in the
theorem) that the number of iterations li at different workers
are i.i.d.. Ef [·] denotes expectation on randomness of both the
linear inverse solutions x∗i and the number of iterations li .
Assumption 3. Within time Tdl , the number of iterations of
linear inverse computations at each worker follows an i.i.d.
distribution li ∼ f (l).
A. Comparison between the coded and uncoded linear inverse
before a deadline
First, we compare the coded linear inverse scheme with an
uncoded scheme, in which case we use the first k workers
to solve k linear inverse problems in (8) without coding. The
following theorem quantifies the overall mean-squared error
of the uncoded scheme given l1 , l2 , . . . , lk . The proof is in
appendix Section VIII-D.
Theorem IV.1. In the uncoded scheme, the overall error is
h
i
2
(l1 ) (l2 )
(lk )
2
l
E kEuncoded k |l = E [e1 , e2 . . . , ek ]
=
k
X
(41)
trace (C(li )) .
i=1
Further, when the i.i.d. Assumption 3 holds,
h
i
2
Ef kEuncoded k = kEf [trace(C(l1 ))].
(42)
Then, we compare the overall mean-squared error of coded
and uncoded linear inverse algorithms. Note that this comparison is not fair because the coded algorithm uses more
workers than uncoded. However, we still include Theorem IV.2
because we need it for the fair comparison between coded and
replication-based linear inverse.
Theorem IV.2. (Coded linear inverse beats uncoded) Suppose
the i.i.d. Assumption 1 and 3 hold and suppose G is a k ×
n submatrix of an n × n Fourier transform matrix F. Then,
expected error of the coded linear inverse is strictly less than
that of uncoded:
h
i
h
i
2
2
>
Ef kEuncoded k − Ef kEcoded k ≥ Ef [trace(J2 J−1
4 J2 )],
(43)
where J2 and J4 are defined in (33).
Proof. From Corollary III.1, for fixed li , 1 ≤ i ≤ n,
2
>
E[kEcoded k |l] ≤ trace(J1 ) − trace(J2 J−1
4 J2 ).
(44)
We will show that
h
i
2
Ef [trace(J1 )] = Ef kEuncoded k ,
k X
n
X
(45)
n
|gji |2 trace(C(li )) =
kX
trace(C(li )).
n i=1
Therefore,
Ef [trace(J1 )] = kEf [trace(C(l1 ))].
(47)
which, along with (46), completes the proof of (45), and hence
also the proof of Theorem IV.2.
B. Comparison between the replication-based and coded linear inverse before a deadline
Consider an alternative way of doing linear inverse using
n > k workers. In this paper, we only consider the case
when n − k < k, i.e., the number of extra workers is only
slightly bigger than the number of problems (both in theory
and in experiments). Since we have n − k extra workers, a
natural way is to pick any (n − k) linear inverse problems
and replicate them using these extra (n − k) workers. After
we obtain two computation results for the same equation, we
use two natural “decoding” strategies for this replication-based
linear inverse: (i) choose the worker with higher number of
iterations; (ii) compute the weighted average
using weights
p
w2
w1
and
,
where
w
=
1/
trace(C(l
1
1 )) and
w1 +w2
p w1 +w2
w2 = 1/ trace(C(l2 )), and l1 and l2 are the number of
iterations completed at the two workers.
Theorem IV.3. The replication-based schemes satisfies the
following lower bound on the mean-squared error:
h
i
h
i
2
2
Ef kErep k >Ef kEuncoded k
(48)
− (n − k)Ef [trace(C(l1 ))].
Proof overview. Here the goal is to obtain a lower bound on
the MSE of replication-based linear inverse and compare it
with an upper bound on the MSE of coded linear inverse.
Note that if an extra worker is used to replicate the computation at the i-th worker, i.e., the linear inverse problem
with input ri is solved on two workers, the expected error
of the result of the i-th problem could at best reduced from
Ef [trace(C(l1 ))] to zero1 . Therefore, (n − k) extra workers
make the error decrease by at most (and strictly smaller than)
(n − k)Ef [trace(C(l1 ))].
Using this lower bound, we can provably show that coded
linear inverse beats replication-based linear inverse when
certain conditions are satisfied. One crucial condition is that
the distribution of the random variable trace(C(l)) satisfies a
“variance heavy-tail” property defined as follows.
Definition 1. The random variable trace(C(l)) is said to have
a “ρ-variance heavy-tail” property if
varf [trace(C(l))] > ρE2f [trace(C(l))],
(49)
which completes the proof. To show (45), first note that from
(42),
h
i
2
Ef kEuncoded k = kEf [trace(C(l1 ))].
(46)
for some constant ρ > 1. For the coded linear inverse, we
will use a Fourier code the generator matrix G of which is a
submatrix of a Fourier matrix. This particular choice of code
is only for ease of analysis in comparing coded linear inverse
Since G := [gj,i ] is a submatrix of a Fourier matrix, we have
|gji |2 = 1/n. Thus, J1 = GΛG> satisfies
1 Although this is clearly a loose bound, it makes for convenient comparison
with coded linear inverse
7
and replication-based linear inverse. In practice, the code that
minimizes mean-squared error should be chosen.
Theorem IV.1, the computation error of uncoded and coded
linear inverse are respectively
Theorem IV.4. (Coded linear inverse beats replication) Suppose the i.i.d. Assumption 1 and 3 hold and G is a k × n
submatrix of√an n × n Fourier matrix F. Further, suppose
(n − k) = o( n). Then, the expected error of the coded linear
inverse satisfies
i
h
ii
1 h h
2
2
lim
Ef kEuncoded k − Ef kEcoded k
n→∞ n − k
(50)
varf [trace(C(l1 ))]
≥
.
Ef [trace(C(l1 ))]
k
h
i X
2
E kEuncoded k |l =
trace (C(li )) ,
Moreover, if the random variable trace(C(l)) satisfies the ρvariance heavy-tail property for ρ > 1, coded linear inverse
outperforms replication-based linear inverse in the following
sense,
h h
i
h
ii
1
2
2
Ef kEuncoded k − Ef kErep k
lim
n→∞ (n − k)
h h
i
h
ii
1
1
2
2
<
lim
Ef kEuncoded k − Ef kEcoded k
.
ρ n→∞ (n − k)
(51)
Proof overview. See appendix Section VIII-E for a complete
and rigorous proof. Here we only provide the main intuition
behind the proof. From Theorem IV.2, we have
h
i
h
i
2
2
>
Ef kEuncoded k − Ef kEcoded k ≥ Ef [trace(J2 J−1
4 J2 )].
(52)
To prove (50), the main technical difficulty is to simplify
>
the term trace(J2 J−1
F, we
4 J2 ). For a Fourier matrix
are
J
J
1
2
able to show that the matrix FΛF> =
(see
J>
J4
2
Corollary III.1) is a Toeplitz matrix, which provides a good
structure for us to study its behavior. Then, we use the
Gershgorin circle theorem [36] (with some algebraic derivation) to show that the maximum eigenvalue of J4 satisfies
σmax (J4 ) ≈ Ef [trace(C(l1 ))], and separately using some
algebraic manipulations, we show
trace(J2 J>
2 ) ≈ (n − k)varf [trace(C(l1 ))],
(53)
for large matrix size n. Since
>
−1 >
trace(J2 J−1
J2 )
4 J2 ) ≥ trace(J2 (σmax (J4 ))
1
trace(J2 J>
=
2 ),
σmax (J4 )
(54)
we obtain
>
trace(J2 J−1
4 J2 ) ≥
(n − k)varf [trace(C(l1 ))]
,
Ef [trace(C(l1 ))]
(55)
for large n. Then, (50) can be proved by plugging (55) into
(52). After that, we can combine (50), (48) and the variance
heavy-tail property to prove (51).
C. Asymptotic Comparison between Coded, Uncoded and
Replication-based linear inverse as the Deadline Tdl → ∞
Consider the coded and uncoded linear inverse when the
overall computation time Tdl → ∞. From Theorem III.1 and
(56)
i=1
2
E[kEcoded k |l] ≤ σmax (G> G) trace (GΛ−1 G> )−1 , (57)
where the matrix Λ is
Λ = diag [trace(C(l1 )), . . . , trace(C(ln ))] ,
(58)
and the matrices C(li ), i = 1, . . . , n are defined as
C(li ) = Bli CE (B> )li .
(59)
Assumption 4. We assume the computation time of one power
iteration is fixed at each worker for each linear inverse
computation, i.e., there exist n random variables v1 , v2 , . . . vn
such that li = d Tvdli e, i = 1, 2, . . . n.
The above assumption is validated in experiments in appendix Section VI-C.
The k-th order statistic of a statistic sample is equal to its kth smallest value. Suppose the order statistics of the sequence
v1 , v2 , . . . vn are vi1 < vi2 < . . . vin , where {i1 , i2 , . . . in } is a
permutation of {1, 2, . . . n}. Denote by [k] the set {1, 2, . . . k}
and [n] the set {1, 2, . . . n}.
Theorem IV.5. (Error exponent comparison when Tdl → ∞)
Suppose the i.i.d. Assumption 1 and Assumption 4 hold.
Suppose n − k < k. Then, the error exponents of the coded
and uncoded computation schemes satisfy
1
2
1
2
,
log E[kEcoded k |l] ≥
log
T
v
1
−
d
dl
i
e
k
(60)
1
2
lim
log E[kEuncoded k |l]
−
T
Tdl
Tdl →∞,li =d vdl e
−
lim
T
Tdl →∞,li =d vdl
i
i
=
−
lim
T
Tdl →∞,li =d vdl
i
e
1
2
1
2
,
log E[kErep k |l] =
log
Tdl
maxi∈[k] vi
1−d
(61)
The error exponents of uncoded, replication and coded linear
inverse satisfy coded>replication=uncoded.
Here the expectation E[·|l] is only taken with respect to the
randomness of the linear inverse sequence xi , i = 1, 2, . . . k,
and conditioned on the number of iterations l. The limit
limTdl →∞ is taken under the Assumption 4, i.e., li = d Tvdli e.
Proof overview. See appendix Section VIII-F for a detailed
proof. The main intuition behind this result is the following:
when Tdl approaches infinity, the error of uncoded computation is dominated by the slowest worker among the first
k workers, which has per-iteration time maxi∈[k] vi . For the
replication-based scheme, since the number of extra workers
n − k < k, there is a non-zero probability (which does not
change with Tdl ) that the n − k extra workers do not replicate
the computation in the slowest one among the first worker.
Therefore, replication when n − k < k does not improve
the error exponent, because the error is dominated by this
slowest worker. For coded computation, we show in appendix
Section VIII-F that the slowest n−k workers among the overall
8
V. A NALYZING THE C OMPUTATIONAL C OMPLEXITY
We first show that the encoding and decoding complexity
of Algorithm 1 are in scaling-sense smaller than that of the
computation at each worker. This is important to ensure that
straggling comes from the parallel workers, not the encoder or
decoder. The proof of the following Theorem is in appendix
Section VIII-H.
Theorem V.1. The computational complexity for the encoding
and decoding is Θ(nkN ), where N is the number of rows
in the matrix B and k, n depend on the number of available
workers assuming that each worker performs a single linear
inverse computation. For a general dense matrix B, the computational complexity of computing linear inverse at each worker
is Θ(N 2 l), where l is the number of iterations in the specified
iterative algorithm. The complexity of encoding and decoding
is smaller than that of the computation at each user for large
B matrices (large N ).
The complexity of encoding and decoding can be further
reduced if the Coppersmith-Winograd algorithm for matrixmatrix multiplication is used [37]. In our experiment on the
Google Plus graph for computing PageRank, the computation
time at each worker is 30 seconds and the encoding and
decoding time at the central controller is about 1 second.
B. Computing the Matrix Λ
One difficulty in our coded linear inverse algorithm is
computing the entries trace(C(l)) = trace Bl CE (B> )l
in the weight matrix Λ in (22), which involves a number
of matrix-matrix multiplications. One way to side-step this
problem is to estimate trace(C(l)) using Monte Carlo simulations. Concretely, choose m i.i.d. N -variate random vectors
a1 , a2 , . . . am that are distributed the same as the initial error
e(0) after Assumption 1. Then, compute the statistic
m
1 X
Bl aj
m j=1
Twitter graph
100
Uncoded
Uncoded
Replication-2
10-2
Replication-1
10-5
Replication-2
Replication-1
10-4
10-10
10-6
Coded
0
10
20
30
10-8
Computation deadline Tdl (sec)
A. Encoding and decoding complexity
γ̂m,l =
Google Plus graph
100
Average mean-squared error
n workers do not affect the error exponent, which means that
the error is dominated by the k-th fastest worker, which has
per-iteration time vik . Since the k-th fastest worker among all
n workers can not be slower than the slowest one among the
first (unordered) k workers, the error exponent of coded linear
inverse is larger than that of the uncoded and the replicationbased linear inverse.
Coded
0
0.5
1
1.5
2
Computation deadline Tdl (sec)
Fig. 2. Experimentally computed overall mean squared error of uncoded,
replication-based and coded personalized PageRank on the Twitter graph and
Google Plus graph on a cluseter with 120 workers. The ratio of MSE for
repetition-based schemes and coded PageRank increase as Tdl increases.
change is that we have to compute Ψi,j in (39) for all possible
li , lj such that 1 ≤ li , lj ≤ Tu . We also choose m i.i.d.
N -variate random vectors b1 , b2 , . . . bm that are distributed
with mean 0N and covariance Ccor , which is the same as the
correlation part according to Assumption 2. Then, compute
the statistic
m
1 X
γ̂m,(li ,lj ) =
bu Blj Bli bu , 1 ≤ li , lj ≤ Tu .
(63)
m u=1
Then, the lemma in appendix Section VIII-G shows that
γ̂m,(li ,lj ) is also an unbiased and asymptotically consistent
estimator of Ψi,j .
C. Analysis on the cost of communication versus computation
In this work, we focus on optimizing the computation cost.
However, what if the computation cost is small compared to
the overall cost, including the communication cost? If this is
true, optimizing the computation cost is not very useful. In
what follows, we show that the computation cost is larger
than the communication cost in the scaling-sense.
Theorem V.2. The ratio between computation and communication at the i-th worker is COSTcomputation /COSTcommunication =
¯ operations per integer, where li is the number of
Θ(li d)
iterations at the i-th worker, and d¯ is the average number
of non-zeros in each row of the B matrix. See appendix
Section VIII-I for a complete proof.
VI. E XPERIMENTS AND S IMULATIONS
2
, l = 1, 2, . . . Tu ,
(62)
where Tu is an upper bound of the number of iterations
in a practical iterative computing algorithm. The lemma in
appendix Section VIII-G shows that γ̂m,l is an unbiased and
asymptotically consistent estimator of trace(C(l)) for all l.
In our experiments on PageRank, for each graph we choose
m = 10 and estimate trace(C(l)) before implementing the
coded linear inverse algorithm (in this case it is the coded
power-iteration algorithm), which has the same complexity as
solving m = 10 extra linear inverse problems.
For the correlated case, we have to compute a slightly
modified weighting matrix denoted by Λ̃ in (38). The only
A. Experiments on Real Systems
We test the performance of the coded linear inverse algorithm for the PageRank problem on the Twitter graph and
the Google Plus graph from the SNAP datasets [38]. The
Twitter graph has 81,306 nodes and 1,768,149 edges, and the
Google Plus graph has 107,614 nodes and 13,673,453 edges.
We use the HT-condor framework in a cluster to conduct
the experiments. The task is to solve k = 100 personalized
PageRank problems in parallel using n = 120 workers. The
uncoded algorithm picks the first k workers and uses one
worker for each PageRank problem. The two replication-based
schemes replicate the computation of the first n − k PageRank
problems in the extra n − k workers (see Section IV-B). The
9
Average mean-squared error
Average mean-squared error
10
Comparison between Algorithm 1
and [6] on Gplus graph
5
Original coded
method in [6]
100
10-5
Algorithm 1
Extension of coded
method in [6]
0
5
10
15
20
25
30
Computation deadline Tdl (sec)
Fig. 3. Experimental comparison between an extended version of the
algorithm in [6] and Algorithm 1 on the Google Plus graph. The figure shows
that naively extending the general coded computing in [6] using matrix inverse
increases the computation error.
Comparison of Different
Codes on Twitter graph
Average mean-squared error
Uncoded
10-5
10-10
Coded
Replication 1
Replication 2
10-15
0
10
20
30
Computation deadline Tdl (sec)
10-10
10-15
Correlated queries on Google Plus graph
100
100
DFT
Random binary
Random sparse
Gaussian
10-5
0
0.5
1
1.5
2
Computation deadline Tdl (sec)
Fig. 4. Experimental comparison of four different codes on the Twitter graph.
In this experiment the DFT-code out-performs the other candidates in mean
squared error.
coded PageRank uses n workers to solve these k = 100
equations using Algorithm 1. We use a (120, 100) code where
the generator matrix is the submatrix composed of the first
100 rows in a 120 × 120 DFT matrix. The computation results
are shown in Fig. 2. Note that the two graphs of different
sizes so the computation in the two experiments takes different
time. From Fig. 2, we can see that the mean-squared error of
uncoded and replication-based schemes is larger than that of
coded computation by a factor of 104 .
We also compare Algorithm 1 with the coded computing
algorithm proposed in [6]. The original algorithm proposed
in [6] is not designed for iterative algorithms, but it has a
natural extension to the case of computing before a deadline.
Fig. 3 shows the comparison between the performance of
Algorithm 1 and this extension of the algorithm from [6].
This extension uses the results from the k fastest workers
to retrieve the required PageRank solutions. More concretely,
suppose S ⊂ [n] is the index set of the k fastest workers.
Then, this extension retrieves the solutions to the original k
Fig. 5. Experimentally computed overall mean squared error of uncoded,
replication-based and coded personalized PageRank on the Twitter graph on
a cluster with 120 workers. The queries are generated using the model from
the stationary model in Assumption 2.
PageRank problems by solving the following equation:
YS = [x∗1 , x∗2 , . . . , x∗k ] · GS ,
(64)
where YS is the computation results obtained from the fastest
k workers and GS is the k × k submatrix composed of
the columns in the generator matrix G with indexes in S.
However, since there is some remaining error at each worker
(i.e., the computation results YS have not converged yet),
when conducting the matrix-inverse-based decoding from [6],
the error is magnified due to the large condition number of GS .
This is why the algorithm in [6] cannot be naively applied in
the coded PageRank problem.
Finally, we test Algorithm 2 for correlated PageRank queries
that are distributed with the stationary covariance matrix in the
form of (36) and (37). Note that the only change to be made
in this case is on the Λ matrix (see equation (38)). The other
settings are exactly the same as the experiments that are shown
in Figure 2. The results on the Twitter social graph are shown
in Figure 4. In this case, we also have to compute
One question remains: what is the best code design for the
coded linear inverse algorithm? Although we do not have a
concrete answer to this question, we have tested different codes
(with different generator matrices G) in the Twitter graph
experiment, all using Algorithm 1. The results are shown in
Fig. 3 (right). The generator matrix used for the “binary” curve
has i.i.d. binary entries in {−1, 1}. The generator matrix used
for the “sparse” curve has random binary sparse entries. The
generator matrix for the “Gaussian” curve has i.i.d. standard
Gaussian entries. In this experiment, the DFT-code performs
the best. However, finding the best code in general is a
meaningful future work.
B. Simulations
We also test the coded PageRank algorithm in a simulated
setup with randomly generated graphs and worker response
times. These simulations help us understand looseness in
our theoretical bounding techniques. They can also test the
performance of the coded Algorithm for different distributions.
We simulate Algorithm 1 on a randomly generated ErdösRényi graph with N = 500 nodes and connection probability
0.1. The number of workers n is set to be 240 and the number
of PageRank vectors k is set to be 200. We use the first
0.02
mean square error
Simulation
Theoretical error
0.015
0.01
0.005
Number of iterations at each worker
10
Computation speed at different workers
150
100
50
0
0
5
10
15
20
25
30
Computation time/s
0
0
50
100
150
200
200 different personalized PageRank problems
Fig. 9. This figure shows the number of PageRank power iterations completed
at different workers in 30 seconds in the Google Plus experiment.
Fig. 6. This simulation result shows the mean squared error of the computation results for k = 200 different problems in the uncoded scheme.
7
×10-4
Simulation
Theoretical error
mean square error
6
5
4
3
2
1
0
0
50
100
150
200
200 different personalized PageRank problems
Fig. 7. This simulation result shows the mean squared error of the computation results for k = 200 different problems in the coded scheme.
k = 200 rows of a 240 × 240 DFT-matrix as the G matrix
in the coded PageRank algorithm in Section III-A. In Fig.
6 and Fig. 7, we show the simulation result on the mean
squared error of all k = 200 PageRank vectors in both uncoded
and coded PageRank, which are respectively shown in Fig. 6
and Fig. 7. The x-axis represents the computation results for
different PageRank problems and the y-axis represents the
corresponding mean-squared error. It can be seen that in the
uncoded PageRank, some of the PageRank vectors have much
higher error than the remaining ones (the blue spikes in Fig.
6), because these are the PageRank vectors returned by the
slow workers in the simulation. However, in coded PageRank,
the peak-to-average ratio of mean squared error is much lower
than in the uncoded PageRank. This means that using coding,
we are able to mitigate the straggler effect and achieve more
uniform performance across different PageRank computations.
From a practical perspective, this means that we can provide
fairness to different PageRank queries.
We compare the average mean-squared error of uncoded,
replication-based and coded PageRank algorithms in Fig. 8.
The first simulation compares these three algorithms when the
processing time of one iteration of PageRank computation is
exponentially distributed, and the second and third when the
number of iterations is uniformly distributed in the range from
1 to 20 and Bernoulli distributed at two points 5 and 20 (which
we call “delta” distribution). It can be seen that in all three
different types of distributions, coded PageRank beats the other
two algorithms.
C. Validating Assumption 4 using Experiments
Here we provide an experiment that validates Assumption 4
in Section IV-C, i.e., the computation time of one poweriteration at the same worker is a constant over time. In
Fig. 9, we plot the number of power-iterations completed at
different workers versus computation time. We can see that
the computation speed is indeed constant, which means that
Assumption 4 is valid2 . Note that there is a non-zero time cost
for loading the graph at each worker. This amount of time does
not exist if the network graph is already loaded in the cache
of distributed workers for online queries.
VII. C ONCLUSIONS
average mean square error
1
×10-3
uncoded
repetition
coded
0.8
0.6
0.4
0.2
0
exponential
uniform
delta
different worker speed distribution
Fig. 8. This figure shows the mean squared error of uncoded, replicationbased and coded PageRank algorithms.
Coded computing for distributed machine learning and data
processing systems affected by the straggler effect has become
an active research area in recent years [5]–[8]. By studying
coding for iterative algorithms designed for distributed inverse
problems, we aim to introduce new applications and analytical
tools to the problem of coded computing with stragglers.
Since these iterative algorithms designed for inverse problems
2 In this work, we assume that the statistics of speed distributions are
unknown. However, from Fig. 9, it may seem that the speeds at different
workers are quite predictable. In fact, each time when scheduling tasks to
the pool of parallel workers, the central controller assign the tasks through
virtual machines instead of actual physical addresses. Therefore, the machines
assigned to the same task can be different, and this assignment is transparent
to the end-users. Thus, the statistics of speed distributions are generally
unobtainable.
11
commonly have decreasing error with time, the partial computation results at stragglers can provide useful information
for the distributed computing of the final outputs. By incorporating these partial results, we expect to have improved
error reduction compared to treating the results as erasures.
Note that this is unlike recent works on coding for multistage computing problems [39]–[41], where the computation
error can accumulate with time and coding has to be applied
repeatedly to suppress this error accumulation. Our next goal
is to study the problem of designing and applying coding
techniques to distributed computing under different situations
of error accumulation/decay, especially in iterative and multistage computations. The exponential decay of error considered
in this paper is a special case of the general problem mentioned
above.
VIII. A PPENDICES
A. Proof of Theorem III.1
We first introduce some notation and preliminary properties
that we will use in this proof. Denote by vec(A) the vector that
is composed of the concatenation of all columns
in a matrix
1 2 3
A. For example, the vectorization of A =
is the
4 5 6
column vector vec(A) = [1, 4, 2, 5, 3, 6]> . We will also use
the Kronecker product defined as
a11 B a12 B . . . a1n B
..
..
..
Am×n ⊗ B = ...
(65)
.
.
.
am1 B am2 B . . .
where each Aij is a square matrix of size N × N . Then, for
an arbitrary matrix L of size k × n,
trace (L ⊗ IN ) · A · (L ⊗ IN )>
trace[A11 ] . . . trace[A1n ]
> (72)
..
..
..
= trace L ·
· L .
.
.
.
trace[An1 ] . . .
trace[Ann ]
Proof. See appendix Section VIII-J.
1) Computing the explicit form of the error matrix E:
From (18), we have encoded the input ri to the linear inverse
problem in the following way:
[s1 , s2 , . . . , sn ] = [r1 , r2 , . . . , rk ] · G.
(73)
Since x∗i is the solution to the linear inverse problem, we have
x∗i = Cinv ri ,
(74)
where Cinv is either the direct inverse in (2) for square linear
inverse problems or the least-square solution in (4) for nonsquare inverse problems. Define yi∗ as the solution of the
inverse problem with the encoded input si . Then, we also have
yi∗ = Cinv si .
(75)
Left-multiplying Cinv on both LHS and RHS of (73) and
plugging in (74) and (75), we have
amn B.
We now state some properties of the vectorization and Kronecker product.
Lemma VIII.1. Property 1: if A = BC, then
vec(A) = (C ⊗ IN )vec(B).
(66)
Property 2: vectorization does not change the Frobenius norm,
i.e.,
kAk = kvec(A)k .
(67)
Property 3: The following mixed-product property holds
(A ⊗ B)(C ⊗ D) = (A · C) ⊗ (B · D),
(68)
if one can form the matrices AC and BD.
Property 4: If A and B are both positive semi-definite, A ⊗ B
is also positive semi-definite.
Property 5: Suppose C is positive semi-definite and A B.
Then,
A ⊗ C B ⊗ C.
(69)
Property 6: (commutative property) Suppose Am×n and Bp×q
are two matrices. Then,
(Am×n ⊗Ip )·(In ⊗Bp×q ) = (Im ⊗Bp×q )·(Am×n ⊗Iq ). (70)
Property 7: Suppose A is an nN × nN matrix that can be
written as
A11 A12 . . . A1n
A21 A22 . . . A2n
AnN ×nN = .
(71)
..
.. ,
..
..
.
.
.
An1 An2 . . . Ann
[y1∗ , y2∗ , . . . , yn∗ ] = [x∗1 , x∗2 , . . . , x∗k ] · G = X∗ · G.
(l)
(76)
(l )
Define i = yi i − yi∗ , which is the remaining error at
the i-th worker after li iterations. From the explicit form (10)
of the remaining error of the executed iterative algorithm, we
have
(l )
(l )
(0)
yi i = yi∗ + i i = yi∗ + Bli i .
(77)
Therefore,from the definition of Y(Tdl ) (see (20)) and equation (76) and (77),
(l )
(l )
Y(Tdl ) =[y1 1 , y2 2 , . . . , yn(ln ) ]
(l )
(l )
n)
=[y1∗ , y2∗ , . . . , yn∗ ] + [1 1 , 2 2 , . . . , (l
n ]
∗
=X · G +
(78)
(0)
[Bl1 1 , . . . , Bln (0)
n ].
Plugging in (21), we get the explicit form of E = X̂> − X∗ :
X̂> =
= (GΛ−1 G> )−1 GΛ−1 (Y(Tdl ) )>
h
i
(0)
>
= (GΛ−1 G> )−1 GΛ−1 G> (X∗ )> + [Bl1 1 , . . . , Bln (0)
n ]
h
i>
(0)
= (X∗ )> + (GΛ−1 G> )−1 GΛ−1 Bl1 1 , . . . , Bln (0)
.
n
(79)
(0)
From (19), (76) and the definition i
(0)
xi − x∗i , we have
(0)
(0)
(0)
(0)
(0)
(l)
= yi − yi∗ and ei =
(0)
[1 , 2 , . . . , (0)
n ] = [e1 , e2 , . . . , ek ] · G.
(80)
12
2) Vectorization of the error matrix E: From property 2 of
Lemma VIII.1, vectorization does not change the Frobenius
norm, so we have
Therefore, from (92), we have
(0)
(0)
2
(0)
(0)
(0)
vec([e1 , e2 , . . . , ek ])> |l] · (G> ⊗ IN )>
2
E[kEk |l] = E[kvec(E)k |l]
= E trace vec(E)vec(E)> |l .
(0)
>
E[E0 E>
0 |l] =(G ⊗ IN ) · E[vec([e1 , e2 , . . . , ek ])·
(81)
Therefore, to prove the conclusion
of this theorem,
i.e.,
2
E[kEk |l] ≤ σmax (G> G) trace (GΛ−1 G> )−1 , we only
need to show
E trace vec(E)vec(E)> |l
(82)
≤σmax (G> G) trace (GΛ−1 G> )−1 .
3) Express the mean-squared error using the vectorization
form: Now we prove (82). From (79), we have
(0)
>
E> = (GΛ−1 G> )−1 GΛ−1 [Bl1 1 , . . . , Bln (0)
n ] , (83)
which is the same as
(0)
−1 > −1
E = [Bl1 1 , . . . , Bln (0)
G ) GΛ−1 ]> . (84)
n ] · [(GΛ
From property 1 of Lemma VIII.1, (84) means
vec(E)
(0)
= (GΛ−1 G> )−1 GΛ−1 ⊗ IN · vec([Bl1 1 , . . . , Bln (0)
n ])
−1 > −1
−1
= (GΛ G ) GΛ ⊗ IN
(0)
· diag[Bl1 , . . . , Bln ] · vec([1 , . . . , (0)
n ]).
(85)
(a)
= (G> ⊗ IN ) · (Ik ⊗ CE ) · (G> ⊗ IN )>
=(G> ⊗ IN ) · (Ik ⊗ CE ) · (G ⊗ IN )
(b)
=(G> · Ik · G) ⊗ (IN · CE · IN )
=G> G ⊗ CE
(c)
σmax (G> G)In ⊗ CE ,
(94)
where (a) is from (93), (b) and (c) follow respectively from
property 3 and property 5 of Lemma VIII.1.
If G has orthonormal rows, the eigenvalues of G> G (which
is an n×n matrix) are all in (0, 1]. This is why we can remove
the term σmax (G> G) in (26) when G has orthonormal rows.
In what follows, we assume G has orthonormal rows, and
the result when G does not have orthonormal rows follows
naturally.
Assuming G has orthonormal rows, we have
E[E0 E>
0 |l] In ⊗ CE .
Plugging (95) into (90), we have
E trace vec(E)vec(E)> |l
≤ trace (L ⊗ IN ) · D(In ⊗ CE )D> (L ⊗ IN )> ,
(95)
(96)
where D = diag[Bl1 , . . . , Bln ]. Therefore,
Define
L = (GΛ−1 G> )−1 GΛ−1 ,
l1
(86)
ln
D = diag[B , . . . , B ],
(87)
D(In ⊗ CE )D>
=diag[Bl1 CE (B> )l1 , . . . , Bln CE (B> )ln ].
(97)
From the definition of C(li ) in (23),
and
(0)
E0 = vec([1 , . . . , (0)
n ]).
(88)
Then,
vec(E) = (L ⊗ IN ) · D · E0 .
(89)
Therefore,
E trace vec(E)vec(E)> |l
>
= trace (L ⊗ IN · D)E[E0 E>
.
0 |l](L ⊗ IN · D)
(90)
4) Bounding the term E[E0 E>
0 |l] using the maximum eigen(0)
(0)
value σmax (G> G): Note that E0 = vec([1 , . . . , n ]).
From (80), we have
(0)
(0)
(0)
(0)
(0)
[1 , 2 , . . . , (0)
n ] = [e1 , e2 , . . . , ek ] · G.
(91)
Therefore, using property 1 of Lemma VIII.1, we have
(0)
(0)
(0)
E0 = (G> ⊗ IN ) · vec([e1 , e2 , . . . , ek ]).
(0)
From Assumption 1, the covariance of ei
(0)
(0)
(92)
is
E[ei (ei )> |l] = CE , i = 1, . . . , k.
D(In ⊗ CE )D> = diag[C(l1 ), . . . , C(ln )].
(98)
5) Reducing the dimensionality of D(In ⊗ CE )D> in the
trace expression using property 7 in Lemma VIII.1: From
Property 7 in Lemma VIII.1, we can simplify (99):
E trace vec(E)vec(E)> |l
≤ trace (L ⊗ IN ) · D(In ⊗ CE )D> (L ⊗ IN )>
(a)
= trace (L ⊗ IN ) · diag[C(l1 ), . . . , C(ln )](L ⊗ IN )>
(b)
= trace L · diag[trace(C(l1 )), . . . , trace(C(l1 ))]L>
(c)
= trace LΛL> ,
(99)
where (a) is from (98), (b) is from Property 7 and (c) is
from the definition of Λ in (22). Equation (99) can be further
simplified to
E trace vec(E)vec(E)> |l ≤ trace LΛL>
(a)
= trace (GΛ−1 G> )−1 GΛ−1 Λ((GΛ−1 G> )−1 GΛ−1 )>
= trace((GΛ−1 G> )−1 ),
(93)
(100)
13
where (a) is from the definition of the decoding matrix
L = (GΛ−1 G> )−1 GΛ−1 . Thus, we have completed the
proof of Theorem III.1 for the case when G has orthonormal
rows. As we argued earlier, the proof when G does not have
orthonormal rows follows immediately (see the text after (94)).
which is obtained by adding the second term (1k 1>
k ) ⊗ Ccor
in (106) into the step (a) in (94). From Property 6 of
Lemma VIII.1, (108) can be simplified to
Σ = (G> 1k 1>
k G) ⊗ Ccor .
(109)
Therefore, from the definition of (87)
B. Proof of Corollary III.1
DΣD>
First, note that
G = [Ik , 0k,n−k ] F.
(101)
=diag[Bl1 , . . . , Bln ] · (G> 1k 1>
k G) ⊗ Ccor
> l1
(110)
> ln
· diag[(B ) , . . . , (B ) ].
Therefore,
>
GΛ−1 G> = [Ik , 0k,n−k ] FΛ−1 F> [Ik , 0k,n−k ]
(a)
>
= [Ik , 0k,n−k ] (FΛF> )−1 [Ik , 0k,n−k ] ,
(102)
where (a) is from F> F = In . Now take the inverse of both
sides of (33), we have
(J1 − J2 J−1
J>
)−1 ∗
> −1
2
4
(FΛF ) =
,
(103)
∗
∗ n×n
where ∗ is used as a substitute for matrices that are unimportant for our argument. Thus, comparing (102) and (103),
> −1
GΛ−1 G> = (J1 − J2 J−1
,
4 J2 )
(104)
>
(GΛ−1 G> )−1 = J1 − J2 J−1
4 J2 .
(105)
which means
From (26) and (105), the theorem follows.
C. Proof of Theorem III.2
The proof follows the same procedure as the proof of
Theorem III.1. Basically, we can obtain exactly the same
results from (73) to (92) except that all Λ are replaced with
Λ̃. However, now that we assume the solutions of the linear
inverse problems satisfy 2, we have
(0)
(0)
(0)
(0)
(0)
(0)
E[vec([e1 , e2 , . . . , ek ])vec([e1 , e2 , . . . , ek ])> |l]
= Ik ⊗ CE + (1k 1>
k ) ⊗ Ccor .
(106)
Note that the first part Ik ⊗ CE is exactly the same as in the
proof of Theorem III.1, so all conclusions until (99) can still
be obtained (note that σmax (G> G) should be added in the
general case) for this part. More specifically, this means (99)
can be modified to
E trace vec(E)vec(E)> |l
(107)
≤ trace (L ⊗ IN ) · DΣD> (L ⊗ IN )>
>
>
+ σmax (G G) trace LΛL ,
where the second term σmax (G> G) trace LΛL> is the
same as in (99) because of the first part Ik ⊗ CE in (106).
However, the first term trace (L ⊗ IN ) · DΣD> (L ⊗ IN )>
is from the correlation between different inputs, and the matrix
Σ is
>
>
Σ = (G> ⊗ IN ) · ((1k 1>
k ) ⊗ Ccor ) · (G ⊗ IN ) ,
(108)
Define the column vector h = G> 1k := [h1 , h2 , . . . hn ]> .
Then, (G> 1k 1>
k G) ⊗ Ccor can be written as a block matrix where the block on the i-th row and the j-th column is hi h∗j Ccor . Therefore, After left-multiplying the block
diagonal matrix diag[Bl1 , . . . , Bln ] and right-multiplying
diag[(B> )l1 , . . . , (B> )ln ], we obtain
DΣD> = Ψ̃ = [Ψ̃i,j ],
(111)
where the block Ψ̃i,j on the i-th row and the j-th column is
hi h∗j Bli Ccor (B> )lj . From Property 7 of Lemma VIII.1, we
have
trace[(L ⊗ IN ) · DΣD> (L ⊗ IN )> ]
h h
i
i
(a)
= trace L trace[Ψ̃i,j ] L>
(b)
= trace L hi h∗j trace[Bli Ccor (B> )lj ] L>
(c)
= trace Ldiag(h) trace[Bli Ccor (B> )lj ] diag(h> )L>
(d)
= trace Ldiag{G> 1k } · Ψ · diag{G> 1k }> L> ,
(112)
where step
h (a) is from
i Property 7 of Lemma VIII.1 and the
notation trace[Ψ̃i,j ] means the n × n matrix with entries
trace[Ψ̃i,j ], (b) is from the definition of Ψ̃i,j below (112), (c)
is from the definition h = G> 1k := [h1 , h2 , . . . hn ]> , and (d)
is from the definition of the matrix Ψ in (39). Plugging (112)
into (107), we obtain
E trace vec(E)vec(E)> |l
≤ trace Ldiag{G> 1k } · Ψ · diag{G> 1k }> L>
(113)
+ σmax (G> G) trace LΛL>
= trace[LΛ̃L> ],
where Λ̃ = σmax (G> G)Λ+diag{G> 1k }·Ψ·diag{G> 1k }> ,
which is the same as in (38). Therefore
E trace vec(E)vec(E)> |l ≤ trace LΛ̃L>
(a)
= trace (GΛ̃−1 G> )−1 GΛ̃−1 Λ̃((GΛ̃−1 G> )−1 GΛ̃−1 )>
= trace((GΛ̃−1 G> )−1 ),
(114)
where (a) is from the definition of the decoding matrix L =
(GΛ̃−1 G> )−1 GΛ̃−1 .
14
D. Proof of Theorem IV.1
In this section, we compute the residual error of the uncoded
linear inverse algorithm. From (9), we have
(l+1)
ei
(l)
= Bei .
(115)
Therefore, in the uncoded scheme, the overall error is
h
i
2
E kEuncoded k |l
2
(l1 ) (l2 )
(lk )
=E [e1 , e2 . . . , ek ] |l
=
=
k
X
(l )
E [ei i ]
i=1
k
X
2
where (a) follows from (118) and (b) follows from (119).
Also note that after we prove (50), then using (48), we have
h
i
h
i
2
2
Ef kEuncoded k − Ef kErep k
(121)
≤(n − k)Ef [trace(C(l1 ))],
so we have
h h
i
h
ii
1
2
2
Ef kEuncoded k − Ef kErep k
n→∞ (n − k)
≤Ef [trace(C(l1 ))]
lim
(a) 1
varf [trace(C(l1 ))]
ρ Ef [trace(C(l1 ))]
h h
i
h
ii
1
1
2
2
Ef kEuncoded k − Ef kEcoded k
,
≤ lim
ρ n→∞ (n − k)
(122)
|l
≤
h
i
(l ) (l )
trace E ei i (ei i )> |l
i=1
k
h
i
(a) X
(0)
(0)
=
trace E Bli ei (Bli ei )> |l
=
i=1
k
X
(116)
h
i
(0) (0)
trace Bli E ei (ei )> |l (Bli )>
i=1
=
=
k
X
i=1
k
X
trace Bli · CE · (Bli )>
trace Bli CE (Bli )>
i=1
k
(b) X
trace (C(li )) ,
=
i=1
where (a) is from (115) and (b) is from the definition of C(li )
in (23). Thus, we have proved (41). To prove (42), we note
that from the i.i.d. assumption of li ,
" k
#
h
i
X
2
Ef kEuncoded k =Ef
trace (C(li ))
(117)
i=1
=kEf [trace(C(l1 ))].
which means coded computation beats uncoded computation.
Note that step (a) holds because of the variance heavy-tail
property.
Therefore, we only need to prove (119). The proof of (119)
is divided into two steps, and intuition behind each step is
provided along the proof. The main intuition is that the Fourier
structure of the matrix F makes the matrix J4 concentrates
around its mean value, which makes the most tricky term
>
Ef [trace(J2 J−1
4 J2 )] analyzable.
1) Exploiting the Fourier structure to obtain a Toeplitz
covariance matrix: First, we claim that when Fn×n is the
Fourier transform matrix, the matrix FΛF> in (33)
J1 J2
>
FΛF = >
,
(123)
J2 J4 n×n
is a Toeplitz matrix composed of the Fourier coefficients of
the sequence (vector) s = [trace(C(l1 )), . . . , trace(C(ln ))].
In what follows, we use the simplified notation
sj := trace(C(lj+1 )), j = 0, 1, . . . , n − 1.
Lemma VIII.2. If
F=
E. Proof of Theorem IV.4
From Theorem IV.2,
h
i
h
i
2
2
>
Ef kEuncoded k − Ef kEcoded k ≥ Ef [trace(J2 J−1
4 J2 )].
(118)
We now argue that to show (50), we only need to show
varf [trace(C(l1 ))]
1
Ef [trace(J2 J−1
J>
,
lim
2 )] ≥
4
n→∞ n − k
Ef [trace(C(l1 ))]
(119)
because then, we have
h h
i
h
ii
1
2
2
lim
Ef kEuncoded k − Ef kEcoded k
n→∞ (n − k)
(a)
1
>
≥ lim
Ef [trace(J2 J−1
4 J2 )]
n→∞ (n − k)
(b) var [trace(C(l ))]
f
1
≥
,
Ef [trace(C(l1 ))]
(120)
(124)
wpq
√
n
,
(125)
p,q=0,1,...,n−1
where w = exp(−2πi/n), then
FΛF> = Toeplitz[s̃p ]p=0,1,...,n−1 ,
where
s̃p =
n−1
1 X −pj
w sj
n j=0
(126)
(127)
Proof. The entry on the l-th row and the m-th column of
FΛF> is
[FΛF> ]l,m =
n−1
X
j=0
wlj w−mj
√ √ sj
n
n
n−1
1 X (l−m)j
=
w
sj .
n j=0
Thus, Lemma VIII.2 holds.
(128)
15
Therefore, the variance of all entries of FΛF> is the same
because
n−1
X
1
varf [s̃p ] =varf
w−pj sj
n j=0
(129)
1
1
= varf [s0 ] =: v.
n
n
Lemma VIII.4. When n − k = o(n), with high probability (in
1 − O( n−k
n ))
1
k
trace(J2 J>
v − .
(137)
2)≥
n−k
n
Proof. Since (J2 )k×(n−k) := [Ji,j ] (Ji,j represents the entry
on the i-th row and the j-th column) is the upper-right
submatrix of FΛF> = Toeplitz[s̃p ]p=0,1,...,n−1 ,
Further, the means of all diagonal entries of FΛF> are
Ef [s̃0 ] = Ef [s0 ] =: µ,
(130)
trace(J2 J>
2)=
=
(131)
2) Using the concentration of J4 to obtain the error when
n → ∞: From an intuitive perspective, when n → ∞, the
submatrix J4 concentrates at µIn−k (see the above computation on the mean and variance of all entries). In this case
1
>
>
Ef [trace(J2 J−1
4 J2 )] ≈ Ef [trace(J2 J2 )]
µ
1
= k(n − k)var[s̃p ]
µ
k
n−k
v· .
=
µ
n
(132)
lim
1
v
varf [s0 ]
>
Ef [trace(J2 J−1
=
.
4 J2 )] =
n−k
µ
Ef [s0 ]
(133)
for any > 0.
After we prove Lemma VIII.3, we obtain a bound on
expectation using the fact that
lim
n→∞
(138)
2
|s̃m−l | .
l=1 m=k+1
Since all entries in J2 have zero mean (because l 6= m ever in
(138) and from (131) all off-diagonal entries have zero mean)
and have the same variance nv (see (129)),
1
1
trace(J2 J>
)
=
· k(n − k)Ef [|s̃1 |2 ]
Ef
2
n−k
n−k
1
k
(a)
=
· k(n − k)varf [s̃1 ] = v,
n−k
n
(139)
where (a) holds because Ef [s̃1 ] = 0. To prove (137), we
compute the variance of trace(J2 J>
2 ) and use Chebyshev’s
inequality to bound the tail probability. Define
k(n − k)
v,
n
where (a) follows from (139). From (138), we have
(a)
Now, we formalize the above intuitive statement. In fact,
we will show a even stronger bound than the bound on the
expected error.
√
Lemma VIII.3. When n − k = o( n), with high probability
2
(in 1 − O( (n−k)
)),
n
1
1
k
−1 >
trace(J2 J4 J2 ) ≥
v− ,
(134)
n−k
µ+ n
1
>
Ef [trace(J2 J−1
4 J2 )]
n−k
(n − k)2
1
k
≥ (1 − O(
))
v− .
n
µ+ n
√
Thus, when n → ∞ and n − k = o( n),
k
n
X
X
µB := Ef [trace(J2 J>
2 )] =
Therefore, we have
n→∞
|Ji,j |2
i=1 j=1
while the means of all off-diagonal entries are
n−1
1 X −pj
Ef [s̃p ] =
w Ef [sj ] = 0, ∀p 6= 0.
n j=0
k n−k
X
X
(135)
1
v−
varf [s0 ] −
>
Ef [trace(J2 J−1
=
,
4 J2 )] ≥
n−k
µ+
Ef [s0 ] +
(136)
for all > 0, which completes the proof of Theorem IV.4.
The proof of Lemma VIII.3 relies on the concentration of
trace(J2 J>
2 ) and the concentration of J4 . In particular, when
we prove the concentration of J4 , we use the Gershgorin circle
theorem [36]. First, we show the following Lemma.
trace(J2 J>
2 ) ≤(n − k)
n−1
X
(140)
|s̃p |2
p=1
n−1
X
1
= (n − k)
s2 − |s̃0 |2 ,
n j=0 j
(141)
(a)
where the last equality (a) holds due to Parseval’s
equality
Pn−1
for the Fourier transform, which states that n1 j=0 s2j =
Pn−1
2
p=0 |s̃p | . Then,
1
>
trace(J2 J2 )
varf
n−k
"
2 #
1
1
>
2
>
=Ef
trace(J2 J2 )
− Ef
trace(J2 J2 )
n−k
n−k
2
n−1
X
(a)
k2
1
≤ Ef
s2j − |s̃0 |2 − 2 v 2
n j=0
n
2
n−1
n−1
1 X 2 k2 2
(b)
1X 2
=Ef
sj − (
sj )
− 2v ,
n j=0
n j=0
n
(142)
where (a) follows from (139) and (141) and (b) follows from
(127). Note that
n−1
n−1
1X 2
1X 2
n−1 2
sj − (
sj ) =
s ,
n j=0
n j=0
n
(143)
16
where
n−1
1 X
s :=
(sj − s̄)2 ,
n − 1 j=0
2
(144)
is the famous statistic called “unbiased sample variance”, and
its variance is (see Page 229, Theorem 2 in [42])
1
n−3 2
var[s2 ] =
µ4 −
µ2 ,
(145)
n
n−1
where
µ4 = E[(s0 − µ)4 ],
(146)
µ2 = E[(s0 − µ)2 ] = var[s0 ] = v.
(147)
Next, we show that with high probability the largest
eigenvalue of (J4 )(n−k)×(n−k) is smaller than (1 + )µ.
Note that the matrix J4 is a principle submatrix of the
Toeplitz matrix FΛF> = Toeplitz[s̃p ]p=0,1,...,n−1 , so J4 =
Toeplitz[s̃p ]p=0,1,...,n−k−1 is also Toeplitz. Using the Gershgorin circle theorem, all eigenvalues of J4 := [J̃ij ] must
lie in the union of (n − k) circles, in which the i-th circle
is
the diagonal entry J̃ii = s̃0 and has radius
P centered atP
j6=i |J̃ij | =
j6=i |s̃j−i |. These (n−k) circles are all within
Pn−k−1
the circle centered at s̃0 with radius 2 p=1 |s̃p |. Therefore,
the maximum eigenvalue of J4 satisfies
and
σmax < s̃0 + 2
Also note that the sample variance is unbiased, which means
Ef [s2 ] = v.
(148)
1
n
n−3 2
µ4 −
v + v2 ,
n−1
(149)
so we have
1
>
varf
trace(J2 J2 )
n−k
2
n−1
n−1
X
X
(a)
1
1
k2
≤ Ef
s2j − (
sj )2 − 2 v 2
n j=0
n j=0
n
k2
n−1 2 2
s ) − 2 v2
=Ef (
(150)
n
n
(n − 1)2
k2
=
Ef [(s2 )2 ] − 2 v 2
2
n
n
2
n−3 2
(n − 1)2 − k 2 2
(c) (n − 1) 1
µ
−
v
+
v
=
4
n2
n
n−1
n2
1
(n − 1)2 − k 2 2
=O
+
v ,
n
n2
|s̃p |.
(152)
p=1
Thus,
Therefore, we have
Ef [(s2 )2 ] = var[s2 ] + (Ef [s2 ])2 =
n−k−1
X
n−k−1
X
Pr(σmax > µ + ) < Pr s̃0 + 2
!
|s̃p | > µ +
p=1
= Pr s̃0 − µ + 2
n−k−1
X
!2
|s̃p |
> 2
p=1
!2
n−k−1
X
1
s̃0 − µ + 2
|s̃p |
≤ 2E
p=1
(b) 1
(n − k)2
v
1
≤ 2 (2n − 2k − 1)2 = 2 O
,
n
n
(a)
(153)
(b)
where (a) follows from (142), (b) follows from (143) and (c)
follow from (149).
Note that we have computed the expectation of
k
1
>
n−k trace(J2 J2 ), which is n v (see (139)). Using the
Chebyshev’s inequality
1
k
>
Pr
trace(J2 J2 ) − v ≥
n−k
n
1
1
≤ 2 var
trace(J2 J>
)
2
n−k
(a) 1
1
1 (n − 1)2 − k 2 2
≤ 2O
+ 2
v
n
n2
(151)
1
1
1 (n − k − 1)(n + k − 1) 2
= 2O
+ 2
v
n
n2
(b) 1
1
2 n−k−1 2
< 2O
+ 2
v
n
n
1
n−k
= 2O
.
n
where (a) is from (150) and (b) is because n + k − 1 < 2n.
Therefore, the proof of (137) is over.
where (a) is from the Markov inequality and (b) is due to the
fact that var[s̃p ] = nv for all p and E[s̃0 ] = µ and E[s̃p ] = 0
for all p 6= 0.
From Lemma VIII.4 and (153), when n → ∞ and
(n − k)2 = o(n), with high probability (which is 1 −
(n−k)2
1
),
2 O
n
1
k
trace(J2 J>
v − ,
2)≥
n−k
n
(154)
and at the same time
J−1
4
1
In−k .
µ+
(155)
From concentration of trace(J2 J>
2 ) and the lower bound of
J−1
4 , we have, with high probability,
1
1
k
−1 >
trace(J2 J4 J2 ) ≥
v− ,
(156)
n−k
µ+ n
for all . This concludes the proof of Lemma VIII.3 and hence
completes the proof of Theorem IV.4 (see the details from after
Lemma VIII.3 to equation (136)). This lemma is a formal
statement of equality (133).
F. Proof of Theorem IV.5
1) Uncoded linear inverse problem:
Consider the eigenvalue decomposition
B = PΘP−1 ,
(157)
17
where
Θ = diag{γ1 , γ2 , . . . γN },
(158)
and without the loss of generality, assume γ1 is the maximum
eigenvalue. Then, from the definition C(li ) = Bli CE (B> )li
in (23),
C(li ) = PΘli P−1 CE (P> )−1 Θli P> .
(159)
Since P−1 CE (P> )−1 is a positive definite matrix, all of its
eigenvalues are positive real numbers. Suppose the maximum
eigenvalue and the minimum eigenvalue of P−1 CE (P> )−1
are respectively emax and emin . Then, (159) gives the upper
and lower bounds
trace(C(li )) ≤ emax trace(PΘ2li P> ),
(160)
trace(C(li )) ≥ emin trace(PΘ2li P> ).
(161)
and
Suppose the maximum and minimum eigenvalues of P> P
are respectively cmax and cmin . Then, (160) and (161) can be
further simplified to
≤ cmax emax trace(Θ2li )
= cmax emax
2
1
v
2
log E[kEuncoded k |l] = max log γ1 i
Tdl →∞ Tdl
i∈[k]
2
1
=−
log .
maxi∈[k] vi
γ1
(165)
lim
Therefore, the error exponent is determined by the worker with
the slowest speed (maximum vi ).
2) replication-based linear inverse:
Now we look at the replication-based linear inverse scheme.
At first, we do not know the order of the random sequence
v1 , v2 , . . . vn . Therefore, when we assign the extra n − k < k
workers to replicate the computations of the last n − k linear
inverse problems, there is a non-zero probability that the
slowest worker of the first k workers does not have any other
copy. More precisely, denote by E the above event. Then, if
we uniformly choose n−k workers to replicate, the probability
of E is
Pr(E) =
(162)
2
E[kErep k |l] ≥ cmin emin Pr(E)
γj2li ,
and
trace(C(li )) ≥ emin trace(Θli P> PΘli )
≥ cmin emin trace(Θ2li )
= cmin emin
(163)
γj2li ,
j=1
where the last equality in the above two inequalities are from
the definition of Θ in (158). Therefore,
1
2
log E[kEuncoded k |l]
Tdl
!
k
X
1
(a)
= lim
log
trace (C(li ))
Tdl →∞ Tdl
i=1
N X
k
X
1
(b)
log
γj2li
= lim
Tdl →∞ Tdl
j=1 i=1
N X
k
Tdl
X
2d v e
1
= lim
log
γj i
Tdl →∞ Tdl
j=1 i=1
lim
N
X
2d max
γj
Tdl
i∈[k] vi
e
,
(167)
Tdl
where the exponent 2d maxi∈[k]
vi e is because we are lowerbounding the error of replication-based scheme using only the
error of the slowest worker in the first k workers, and Pr(E)
is the probability that this particular worker is not replicated
using any of the n − k extra workers.
Using the fact that maxj γj = γ1 and the fact that
cmin emin Pr(E) is a constant that does not change with Tdl ,
we have
1
2
1
2
log E[kErep k |l] ≥
log .
Tdl →∞ Tdl
maxi∈[k] vi
γ1
lim
2
(168)
2
Note that E[kErep k |l] ≤ E[kEuncoded k |l], so we also have
Tdl →∞
(c)
(166)
j=1
j=1
N
X
k−1
n−k
.
k
n−k
This is also a constant that does not depend on the time Tdl .
Therefore,
trace(C(li )) ≤ emax trace(Θli P> PΘli )
N
X
exponent in a log-sum form. Since the maximum eigenvalue
of the matrix B is γ1 , we have
1
2
log E[kErep k |l]
Tdl
1
2
≤ lim
log E[kEuncoded k |l]
Tdl →∞ Tdl
2
1
=
log .
maxi∈[k] vi
γ1
lim
Tdl →∞
(164)
2
vi
= max log(γj ) ,
(169)
Therefore,
1
2
1
2
log E[kErep k |l] =
log .
Tdl →∞ Tdl
maxi∈[k] vi
γ1
lim
(170)
i∈[k],j
where (a) is from (41), (b) is obtained by plugging in (162)
and (163) and the fact that the constants emin , cmin , emax and
cmin do not change the error exponent when Tdl increases, and
(c) is from the fact that the maximum term dominates the error
3) Coded linear inverse algorithm:
For the coded linear inverse algorithm,
2
E[kEcoded k |l] ≤ σmax (G> G) trace (GΛ−1 G> )−1 .
(171)
18
From (162), we have
trace(C(li )) ≤cmax emax
N
X
γj2li
(172)
j=1
≤cmax emax N γ12li .
Plugging into (171), we have
2
> −1
E[kEcoded k |l] ≤ σmax (G> G) trace (GΛ−1
,
2 G )
(173)
where
Λ2 :=diag{cmax emax N γ12l1 , . . . cmax emax N γ12ln }
=N cmax emax diag{γ12l1 , . . . γ12ln }.
(174)
1
2
log E[kEcoded k |l]
Tdl
1
> −1
log trace (GΛ−1
,
≤ lim
3 G )
Tdl →∞ Tdl
(175)
where
T
2d vdl e
=diag{γ1
1
(176)
T
dl e
2d vn
, . . . γ1
}.
1
γ1
min
i∈S
2d Tvdl e
i
=
1
γ1
2d vTdl e
ik
.
(177)
For i ∈ [n] \ S = {ik+1 , . . . in },
1
γ1
2d Tvdl e
i
≥ 0.
(178)
Λ−1
3
1
γ1
2d vTdl e
ik
diag{c1 , c2 , . . . cn },
(179)
where ci is the indicator
ci = δ(i ∈ S).
(180)
Define GT as the submatrix of G composed of the columns
in G with indexes in T ⊂ [n]. Use σmin (X) to denote the
minimum eigenvalue of a matrix X. Define
smin =
G. Computing the Matrix Λ
Recall that the statistic γ̂m,l is defined as
Therefore, from the definition of the diagonal matrix Λ3 in
(176), the entries of Λ−1
3 can be lower-bounded by (177) for
i ∈ S, and can be lower-bounded by (178) for i ∈ [n] \ S.
Thus,
−2d vTdl e
ik
1
1
log
= lim
Tdl →∞ Tdl
γ1
2
1
=−
log ,
vi k
γ1
h
i
1
k
where (a) is because trace smin
Ik = smin
is a constant and
does not change the error exponent. Thus, we have completed
the proof of Theorem IV.5.
(a)
Define S = {i1 , . . . ik }, i.e., the index set of the fastest k
workers. Then,
1
2
log E[kEcoded k |l]
Tdl
1
> −1
≤ lim
log trace (GΛ−1
3 G )
Tdl →∞ Tdl
−2d vTdl e
ik
1
1
1
≤ lim
log trace
Ik
Tdl →∞ Tdl
γ1
smin
−2d vTdl e
(183)
i
1
1
1
k
trace
log
Ik
= lim
Tdl →∞ Tdl
γ1
smin
Tdl →∞
lim
Λ3 :=diag{γ12l1 , . . . γ12ln }
where (a) is from (179), (b) is from (180), and (c) is from
(181). Thus, plugging (182) into (175) (note that there is an
inverse inside the trace of (175))
lim
Since N , cmax and emax are all constant numbers,
Tdl →∞
only on the generator matrix G and does not change with the
overall time Tdl . Therefore,
2d vTdl e
(a)
ik
1
−1 >
Gdiag{c1 , c2 , . . . cn }G>
GΛ3 G
γ1
2d vTdl e
ik
1
(b)
(182)
=
GS G>
S
γ1
2d vTdl e
(c)
ik
1
smin Ik .
γ1
min
T ⊂[n],|T |=k
σmin (GT G>
T ).
(181)
Since G is a matrix with orthonormal rows, any arbitrary GT
that satisfies |T | = k must have full rank. This means that
smin > 0. Note that smin > 0 is a constant that depends
m
γ̂m,l =
1 X
Bl aj
m j=1
2
, l = 1, 2, . . . Tu .
(184)
The computational complexity of computing γ̂m,l , l =
1, 2, . . . Tu is the same as the computation of m linear inverse
problems for Tu iterations. The computation has low complexity and can be carried out distributedly in m workers before
the main algorithm starts. Additionally, the computation results
can be used repeatedly when we implement the coded linear
inverse algorithm multiple times. In the data experiments, we
use m = 10, which has the same complexity as solving
m = 10 extra linear inverse problems.
The following Lemma shows that γ̂m,l , l = 1, 2, . . . Tu
is an unbiased and asymptotically consistent estimate of
trace(C(l)) for all l.
Lemma VIII.5. The statistic γ̂m,l is an unbiased and asymptotically consistent estimator of trace(C(l)). More specifically,
the mean and variance of the estimator γ̂m,l satisfies
E[γ̂m,l |l] = trace(C(l)),
(185)
19
h
i
1
4
4
Bl F E kaj k .
m
Proof. The expectation of γ̂m,l satisfies
m
i
1 X h l
2
E[γ̂m,l ] =
E B aj
m j=1
h
i
2
=E Bl a1
l >
=E trace(Bl a1 a>
1 (B ) )
vart [γ̂m,l ] ≤
(186)
(187)
(a)
l >
= trace(Bl E[a1 a>
1 ](B ) )
= trace(Bl CE (Bl )> )
= trace(C(l)),
where (a) is from the fact that a1 has covariance CE . To bound
the variance of γ̂m,l , note that for all j,
l
B aj
2
≤
2
Bl F
2
kaj k .
(188)
Therefore,
m
1 X
2
Bl aj ]
m j=1
h
i
(a) 1
2
= var Bl aj
m
h
i
(b) 1
4
≤ E Bl aj
m
h
i
(c) 1
4
4
≤
Bl F E kaj k ,
m
var[γ̂m,l ] =var[
extra linear inverse problems. Additionally, it is a one-time
cost in the pre-processing step. Thus, we do not take into
account the complexity of computing Λ for the analysis of
encoding and decoding.
I. Proof of Theorem V.2
We assume that the matrix B and K have already been
stored in each worker before the computation of the linear
inverse problems. For the PageRank problem, this means that
we store the column-normalized adjacency matrix A in each
worker.
In Algorithm 1, the i-th worker requires the central controller to communicate a vector ri with length N to compute
the linear inverse problem. Thus
COSTcommunication = N
INTEGERS.
The computation cost at each worker is equal to the number
of operations in one iteration multiplied by the number of
iterations in the specified iterative algorithm. In each iteration,
the number of operations also roughly equals to the number
of non-zeros in B. Thus
COSTcomputation ≈ 2 · |E| · li OPERATIONS,
(189)
where (a) holds because all kaj k are independent of each
other, and (b) holds because var[X] ≤ E[X 2 ], and (c) is from
the Cauchy-Schwartz inequality.
H. Proof of Theorem V.1
The computational complexity at each worker is equal to
the number of operations in one iteration multiplied by the
number of iterations. The number of iterations is l. In each
iteration, the number of operations is equal to the number of
non-zeros in B because each iteration x(l+1) = Kr + Bx(l)
requires at least scanning through the non-zeros in B once to
compute Bx(l) . For general matrices, the number of entries
is in the order of N 2 , where N is the number of rows in B.
Therefore, the overall number of operations at each worker is
in the order of Θ(N 2 l).
The encoding and decoding steps in Algorithm 1 are all
based on matrix-matrix multiplications. More specifically,
for encoding, we multiply the generator matrix Gk×n with
the input matrix and the initial estimates, which both have
size N × k. Thus, the complexity scales as O(nkN ). For
decoding, the computation of the decoding matrix L =
(G> Λ−1 G)−1 GΛ−1 is has complexity Θ(k 3 ) (matrix inverse) plus Θ(k 2 n) (matrix multiplications). Multiplying the
decoding matrix Lk×n with linear inverse results that have
size N × n has complexity Θ(nkN ). Therefore, for large N ,
the computational complexity is in the order of Θ(nkN ).
The computation of the matrix Λ, as we have explained in
Section V, has the same complexity as computing m ≈ 10
(190)
(191)
where li is the number of iterations completed at the i-th
worker, |E| is the number of non-zeros, and 2· is because
we count both addition and multiplication. From Fig. 9, the
typical number of li is about 50.
Thus, the ratio between computation and communication is
COSTcomputation /COSTcommunication
≈ li d¯ OPERATIONS/INTEGERS,
(192)
where d¯ is the average number of non-zeros in each row of the
B matrix. Since li is about 50, we expect that the computation
cost is much larger than communication.
J. Proof of Lemma VIII.1
Property 1 and property 2 can be directly examined from
the definition. Property 3 is Theorem 3 in [43]. To prove
property 4, we note that the eigenvalues of A ⊗ B equals
to the pairwise products of the eigenvalues of A and the
eigenvalues of B (from Theorem 6 in [43]). Therefore, since
the eigenvalues of A and the eigenvalues of B are all nonnegative, the eigenvalues of A ⊗ B are also non-negative.
Property 5 follows directly from property 4 because when
B − A 0 and C 0, (B − A) ⊗ C 0.
To prove property 6, we can repeatedly use property 3:
(Am×n ⊗ Ip ) · (In ⊗ Bp×q )
=(Am×n · In ) ⊗ (Ip · Bp×q )
=(Im · Am×n ) ⊗ (Bp×q · Iq )
(193)
=(Im ⊗ Bp×q ) · (Am×n ⊗ Iq ).
To prove property 7, we first assume that
L11 L12 . . . L1n
L21 L22 . . . L2n
Lk×n = .
..
.. .
..
..
.
.
.
Ln1 Ln2 . . . Lkn
(194)
20
Then,
trace (L ⊗ IN ) · A · (L ⊗ IN )>
k
n X
n
X
X
(a)
Lki Aij Lkj
=
trace
i=1 j=1
l=1
k
n X
n
X
X
=
Lki trace[Aij ]Lkj
l=1
i=1 j=1
...
..
.
trace[An1 ] . . .
trace[A1n ]
>
..
· L ,
.
trace[A11 ]
(b)
..
= trace L ·
.
trace[Ann ]
(195)
where (a) and (b) hold both because the trace can be computed
by examining the trace on the diagonal (or the diagonal
blocks).
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| 7 |
Frequency Distribution of Error Messages
David Pritchard
arXiv:1509.07238v1 [cs.SE] 24 Sep 2015
Center for Education in Math and Computing, University of Waterloo, Canada ∗
dagpritchard@uwaterloo.ca
Abstract
Which programming error messages are the most common?
We investigate this question, motivated by writing error explanations for novices. We consider large data sets in Python
and Java that include both syntax and run-time errors. In
both data sets, after grouping essentially identical messages,
the error message frequencies empirically resemble ZipfMandelbrot distributions. We use a maximum-likelihood approach to fit the distribution parameters. This gives one possible way to contrast languages or compilers quantitatively.
Categories and Subject Descriptors D.3.4. [Programming
Language Processors]: Compilers, Run-time environments
Keywords Error messages, empirical analysis, usability,
education.
1.
Introduction
This work started as an offshoot of Computer Science Circles (CS Circles) [33, 34], a website with 30 lessons and 100
exercises teaching introductory programming in Python. It
contains a system where students can ask for help if they
are stuck on a programming exercise. Often, students reported being stuck because they could not comprehend an
error message, asking for a better explanation of what the
compiler/runtime was trying to say. E.g., the message
SyntaxError: can’t assign to function call
might not be understood by a novice who wrote sqrt(y)=x.
Motivated by this, we decided to systematically improve
the error messages that students received. There is copious
literature on writing good error messages [14, 25, 26, 29,
∗ Work
started while located at Princeton University and completed at
U. Southern California. Currently located at Google Los Angeles.
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classroom use is granted without fee provided that copies are not made or distributed
for profit or commercial advantage and that copies bear this notice and the full citation
on the first page. Copyrights for components of this work owned by others than ACM
must be honored. Abstracting with credit is permitted. To copy otherwise, or republish,
to post on servers or to redistribute to lists, requires prior specific permission and/or a
fee. Request permissions from permissions@acm.org.
PLATEAU ’15, Month d–d, 20yy, City, ST, Country.
Copyright c 2015 ACM 978-1-nnnn-nnnn-n/yy/mm. . . $15.00.
http://dx.doi.org/10.1145/nnnnnnn.nnnnnnn
39, 40], but how can this advice be incorporated into the
programming ecosystem? One approach would be making
upstream improvements to the compiler/runtime, but this
can take a long time, and not all audiences would appreciate the changes that would most benefit novices. A second approach would be to write a tool that analyzes code
from scratch, looking for common syntactic bugs or likely
semantic mistakes. The literature includes many such tools:
see checkstyle, findbugs and [10, 15, 20, 22, 23, 36].
We chose a more lightweight approach: augmenting the
normal error messages with additional explanations. To wit,
we compile and execute the code as usual, and then add a
beginner-appropriate elaboration of the resulting error message, implemented by rendering the normal error with a
clickable pop-up link to the explanation. This augmentingexplanation approach has been previously used on a small
scale with Java compiler errors [6, §5.2], Python runtime errors [13, §5.2.1], and C++ STL compiler errors [41].
It has long been observed that “a few types of errors account for most occurrences” [35], see also [7]. In order to
make sure that a small number of explanations would be
useful as often as possible, we had to answer the following
question: what error messages are the most common? Counting error message frequencies has a long history, starting
from assembler [4, 29] and SP/k [14], with renewed interest
more recently, using much larger data sets [1, 7, 17, 18, 38].
Using with the history of all previous submissions, we
determined the essentially distinct error messages and their
frequencies (see Section 2), available online at http://
daveagp.github.io/errors. We wrote explanations for
the 36 most common messages. Regular expressions were
used to aid the implementation. At the most basic level,
some errors were made more readable by elaborating them
into a full paragraph of text rather than a one-line message.
Some explanations include concrete examples of code that
causes the same error message, and a description of how to
fix it. See [14, 25, 26, 29, 39, 40] for advice on writing error
messages. Given a large data set, the work involved in this
group-and-explain approach is modest and not technically
challenging, so we would recommend it in any beginnerfacing system. Moreover, in internationalized settings, one
can then add explanations in other languages (this has been
implemented in CS Circles’ Lithuanian translation).
This paper compares and contrasts the most common error messages in CS Circles with those in another programming language. The Blackbox project [1, 21] is a large-scale
data collection-and-sharing project using BlueJ, a Java programming environment oriented at beginners. We obtained
the error messages from all recorded compilation and execution events, grouping and counting the essentially different
messages like we did for the Python data set. Comparing the
two data sets, we found that both error message frequency
distributions resembled the same family of distributions, the
Zipf-Mandelbrot distribution [24]. For these data sets, this
means that for any integer k, the frequency of the kth most
common error is approximately proportional to 1/(k + t)γ
where t and γ are parameters of the data set. In order to
determine the best values for these parameters, we propose
using a simple maximum-likelihood approach.
1.1
Discussion and Other Related Work
Orthogonal to purely quantitative analysis, a large body of
work focuses on manual categorization of errors. This allows
researchers to get more accurate results, and to precisely
understand the psychological state of the user, rather than
focus on the compiler-generated error messages themselves.
Good reasons for doing this include that “A single error may,
in different context, produce different diagnostic messages”
and that “The same diagnostic message may be produced by
entirely different and distinct errors,” see [27]. This analysis
also helps measure whether a compiler’s error messages are
appropriate (e.g., see [19, 35]). This analysis is important
for compiler designers, language designers and educational
research, but it is not our focus.
The comparison of error message frequencies between
different languages raises many interesting open-ended
questions. Even within the same language, some compilers are significantly better or worse than others; see Brown’s
amusing crowdsourcing of Pascal error messages [2, 3] as
well as [31, 40]. One way to view different error message
distributions is to imagine the extremes: the worst possible language would only ever say “?” without elaborating
(this has been formally evaluated, see [39]), while the best
possible language would, like a human tutor, always give a
perfectly adapted explanation. The exponent γ in our work
is one way to measure where a language sits between these
extremes. However, simpler measures such as entropy could
also be used. Also, a single quantitative measure should not
be treated as paramount without context. When comparing
languages/compilers (e.g., [28]), statistical fitness is less important than overall usability, including measures like time
between errors and time to achieve user goals.
A notable alternative approach to improving student feedback based on large-scale data, rather than focusing on error
messages, is the HelpMeOut system [12], which uses a detailed repository of past student work sessions to find old
errors similar to new ones and make suggestions of how to
fix them.
To our knowledge, this paper is the first one to examine any link between programming error messages and statistical distributions. The special case t = 0 of the ZipfMandelbrot distribution is known as the power law distribution. It arises empirically in data sets such as the frequency
of distinct words in books, of links in webpages, and of citations in literature. Caveats apply here [5, 11, 30], including: that generative explanations of how these distributions
could arise are tenuous; that near-power-law data sets may
be even closer still to other distributions; and that analyzing
such data sets has common pitfalls like using linear regression. Another caveat for our work is that the distributions of
error messages will depend on the nature of the users, and
the kind of setting in which the work is collected. In our
case, both data sets come from a very large, open project intended for beginners. We anticipate that a data set where students only work on a fixed set of exercises could be skewed
in some way, but both BlueJ and the CS Circles “console”
allow students to do any sort of open-ended programming.
See [32] for discussion of power laws in runtime objectreference graphs of industry-scale computer programs.
2.
Data Sets
Our first data set is the Python corpus from CS Circles.
Amongst the first 1.6 million code submissions, about
640000 resulted in an error. Our second data set is the Java
corpus from BlueJ Blackbox. We specifically considered
the “compile” events, of which there were about 8 million,
half of which produced an error, and the “invoke” events, of
which there were about 5 million, about 260000 of which
produced a syntax error and 180000 of which produced a
run-time error. We did not include the codepad or unit test
events, both of which are an order of magnitude smaller.
In both cases, following [31], we only counted the first error message. This tends to be the most accurate error (since
a syntax error can cause new valid parts of the program below to be reported as errors) and it is also the error that the
programmer is most likely to pay attention to and fix first.
Moreover, CS Circles only shows the first error message in
its user interface; and even for a UI like BlueJ that shows
multiple errors, beginner students often (by habit or by instruction) fix only one at a time and then recompile/re-run.1
After obtaining these raw data sets of hundreds of thousands of error messages, we had to count how many time
each distinct message occurred. It is necessary to “sanitize”
the data by removing parts that pertained to specifics of user
code rather than the kind of error. For instance, NameError:
name ’x’ is not defined should be understood by our
system to be essentially the same error as NameError:
1 It would not be invalid to investigate data sets where all errors are reported
and counted, but a worry is that it might say more about the statistics of
chain effects in syntax errors and less about the actual underlying bugs. Two
other strategies, “count-all” and “count-distinct,” are used in [38], though
their study participants were professionals and not novices.
tion will appear in either case. The sanitization was an iterative process. Simple heuristics handled most cases correctly,
and in total we needed about 20 sanitization rules for Python
and 50 for Java, implemented using regular expressions.
There is a question of how far one should sanitize. Should
these two error messages be considered the same?
1000000
Rank vs Frequency, CS Circles Errors
100000
10000
Frequency
name ’sum’ is not defined so that the same explana-
1000
100
10
Overall we tended to use fewer sanitization rules rather
than more (considering the above to be different); a similar
approach was used in [38]. Conceptually, to fix a single objective goal for sanitization, we imagined that each category
should uniquely correspond to a single line of source code
of the compiler/runtime where the error is first detected.
Another step in sanitization was to remove any nonEnglish error messages, to avoid inadvertently seeing the
same patterns repeated in multiple languages, which might
affect the results. This was done by removing all messages
with non-ASCII characters, and manual filtering.
2.1
Overview of Data Sets
The Python data set yielded 309710 syntax errors and
333538 compile-time errors. The Java data set yielded
4002822 compile-time errors and 129650 run-time errors.
Note that the Java data set has a much smaller proportion of
run-time errors than Python (only about 3% rather than almost half). But to a degree, this difference is inherent in the
language, since many errors that would occur at compiletype in Java’s strict typing-and-scoping system are not encountered until run-time in Python.
After sanitization and grouping, the Python data set
yielded 283 distinct error messages. Of these, 17 occurred
exactly twice and 42 occurred only once (for example,
ValueError: Format specifier missing precision
and SyntaxError: can’t assign to Ellipsis). The
Java data set yielded 572 error messages in total; 65 occurred
exactly twice and 127 occurred only once (for example,
com.vmware.vim25.InvalidArgument and cannot
create array with type arguments).
Errors are not completely parallel for both languages. For
example, Java allows function overloading, i.e. two functions with distinct signatures but the same name. In Python,
this must instead be implemented by a single function that
takes different actions depending on the runtime number and
type of its argument(s). It is the function’s responsibility to
generate the error message. It turns out that not all such functions generate identical messages and so the single Java error
message no suitable method found corresponds to
more than one distinct Python error message:
f argument must be a string or number, not T
and f arg 1 must be a type or tuple of types.
1
1
10
100
1000
Rank of Error
1000000
Rank vs Frequency,
BlueJ Errors
100000
10000
1000
Frequency
RuntimeError: maximum recursion depth exceeded
while getting the repr of a list
RuntimeError: maximum recursion depth exceeded
while getting the repr of a tuple
100
10
1
1
10
100
1000
Rank of Error
Figure 1. The two data sets for our study. The CS Circles
data set is Python, while BlueJ is Java. The plots are log-log.
The 5 most common Python errors were:
179624
97186
76026
26097
20758
SyntaxError: invalid syntax
NameError: name ’NAME’ is not defined
EOFError: EOF when reading a line
SyntaxError: unexpected EOF while parsing
IndentationError: unindent does not match
any outer indentation level
The 5 most common Java errors were:
702102
407776
280874
197213
183908
cannot find symbol - variable NAME
’;’ expected
cannot find symbol - method NAME
cannot find symbol - class NAME
incompatible types
We plot both data sets in Figure 1. The x-axis measures
the rank of each error message (with 1 being the most frequent) and the y-axis measures the number of times each error occurred. Using a logarithmic scale is necessary for the
changes in the y-axis to be visible, and we also use a logarithmic scale for the x-axis. Notice that both data sets give
rise to similar distributions; in the rest of the paper we will
try to describe them in a common framework.
2.2
Notation
For any given data set, we will use N to denote the total
number of errors logged, and M for the number of distinct
error types. For example, the Python data set has N =
643248 and M = 283. Let Fk denote the number of times
that the kth-most common error occurred, e.g. F1 = 179624
for Python. We will also write Fmax := F1 as an alternate
symbol for the same value, when we wish to emphasize that
3
4 5
19 13 6
31 17 18
6
4
15
Table 1. Number of f -legomena in each data set.
it is the maximum frequency. The smallest frequency FM
is 1 for both of our data sets. In lexicography, the items
occurring just once are known as the hapax legomena of the
corpus. An f -legomenon is any error message that appears
exactly f times. We will use the symbol
#F −1 (f )
to denote the number of f -legomena. The first few counts of
f -legomena in our data sets is listed in Table 1.
3.
Power Law Distributions
When studying frequency counts of different objects, a discrete power law distribution is one in which the frequency
Fk of the kth most common item is proportional to 1/k γ .
As mentioned in the introduction, power law distributions
provide good fits to many unrelated empirical distributions.
A common example is that in the novel Moby Dick, the frequency of the kth-commonest word is approximately proportional to 1/k 1.05 . “Zipf’s law” is sometimes used as a synonym for the discrete power law, but sometimes also refers
to the special case Fk ∝ 1/k where γ = 1. There is also
a large body of work on continuous power laws, where one
sorts items by some magnitude that takes on continuous values, and examines the relationship between rank and magnitude. See many examples of both types in [5].
Note that sanitization of error messages is particularly important because of the fact that many natural languages follow power-law curves. If we did absolutely no sanitization,
then our error message distributions would have significant
aspects determined by the frequency distribution of variable
names chosen by users, and finding a power law describing
the latter would be less surprising, given that natural language is already known to exhibit power law behaviour.
We now turn to analyzing our data sets from the power
law perspective. Do they approximately satisfy a power law?
This did not appear to be the case: a power law, when plotted
on a log-log scale, should give a straight line, but it is clear
from Figure 1 that this is not an accurate description of our
data set.
3.1
Zipf-Mandelbrot Distributions
There is a generalization of power law distributions called
the Zipf-Mandelbrot family of distributions. These distributions are defined, using two parameters t and γ, by
Fk ∝
1
.
(k + t)γ
Such a sequence should appear linear on a log-log plot provided that along the x-axis, we plot the logarithmic positions
of (k + t) rather than of k. When we tested plotting these
distributions in this modified way, for an appropriate value
of t, we obtained a much more persuasive fit: in Figure 2,
which has the shift t = 60, the points very nearly fall on a
line. This shift was obtained by trial-and-error, and the line
drawn in has slope −γ = −6.3. In the rest of the paper we
aim to give a more principled way of estimating t and γ.
Is a Zipf-Mandelbrot distribution plausible? Here is one
argument that, if we accept that power laws can arise in
natural settings, that there is reason to suspect that ZipfMandelbrot laws can too. It is not meant to give an exhaustive explanation, just an argument for plausibility. Suppose we start with a power law, and then coalesce several
items together. I.e., replace several distinct error messages
with a single unified message having the sum of their frequencies. (In a list of English words, the analogy would
be that a single word has multiple meanings.) The effects
of this message-merging would be twofold: the resulting
new message would be an outlier to the original power law
curve; and the remaining data points, when plotted on a rankfrequency scale, would be shifted several positions to the
left, i.e. they would follow a Zipf-Mandelbrot distribution instead of a power law distribution. This is indeed a plausible
scenario for the Python data set! The most common error,
SyntaxError: invalid syntax, is very generic. It
can be obtained by writing two tokens in a row (such as
forgetting a comma or quote marks), using an assignment
statement in place of a conditional expression (such as using
if a=b: instead of using ==), by mismatching parentheses, etc.
3.2
Consequences of a Zipf-Mandelbrot model
What behaviour does a Zipf-Mandelbrot model predict? It
postulates that there is some innate ordering of error messages, from most frequent to least frequent, so that the inherent probability F`∗ of the `th most frequent error message
is proportional to (` + t)−γ . The reason that we use the sub-
Shifted rank-frequency plot
6
5
Log(frequency)
#F −1 (f ) where f is: 1
2
Python
42 17
Java
127 65
4
3
2
1
10
Rank
100
1000
0
-1
1.75
1.95
2.15
2.35
2.55
Log(rank+t)
Figure 2. The Python data set with the (pre-logarithmic)
x-axis shifted by t = 60, and a straight line with slope
−γ = −6.3 that approximately fits most of the data.
script ` here is that our concrete data set arrives via sampling
from the inherent distribution. So like a sampling error, the
observed ordering of messages from most to least frequent
is not necessarily the exact same as the innate ordering.
An interesting aspect of this model is that it assumes
F`∗ ∝ (` + t)−γ continues to hold for arbitrarily large `.
Can this be plausible: is the total number of possible errors infinite? We will accept this as a reasonable hypothesis,
which if not literally true, could continue long enough to
be consistent with the size of any measured data set, for the
following reason, using Python as an example. The most
common errors we see are the ones in the Python core code
(the syntax errors, and runtime errors from the “builtins”).
Less frequently we start to see errors from Python modules, such ValueError: math domain error within the
sqrt function of the math module. While this is the only
module taught on the site, users have occasionally submitted code using other common modules like time, random
and functools, each of which comes with its own specific errors. Moving on, there would be errors from rarelyused modules, then even after this, modules that users may
with more or less frequency import (or copy in) themselves.
For example, we observed a mainfile: error: must
provide name of pdb file error caused by someone
who copied in a Python program for use with the X-PLOR
biomolecular structure determination software [37]. The
same phenomenon happens in the Java data set. So errors
with arbitrarily small inherent frequency are not unreasonable despite the finite size of the languages.
4.
Analysis
To fit our data to a Zipf-Mandelbrot distribution, several approaches are possible. For power laws, the naive approach,
using a least-squares fit to a linear log-log plot (c.f. Figure 2)
is known to introduce errors [11]. Rather, we will follow
Newman [30], who considered maximum likelihood estimation methods for power laws. Some work will be needed to
extend this to Zipf-Mandelbrot distributions.
The method in [30] involves a particular way of processing the data; let us mention the motivation. The direct approach to maximum likelihood estimation
would be to deQ
γ Fk
termine the γ and t that maximize k (C/(k
P∞ + t) ) −γwhere
C is the normalizing constant with C · k=1 (k + t) = 1.
Izsák [16] suggests this. But trying this approach gives unsatisfactory results with any of our data sets — the curves
produced fit the data very poorly except in the regime of
F1 . The calculation goes wrong because it is too heavily biased by the highest-frequency errors. (For Zipf-Mandelbrot
in particular, if the argument in Section 3.1 were to be true,
then it should be no surprise that fitting to F1 would be
problematic, since F1 would be an outlier from the norm.)
Also, the most likely fit entails that the innate order exactly matches the observed frequency-ordering of error messages [16], which is itself unlikely.
4.1
Probabilities of Frequencies
This motivates the maximum likelihood method on frequencies [5, 30]. It starts by taking a different view of the data set.
Using the Python data set as a concrete example, we imagine
the frequency vector F = (179624, 97186, . . . , 1, 1) itself as
being an unordered set of M data points from a parameterized distribution — given a new error message, how frequent
is it? This distribution-on-frequencies is a transformed version of the inherent distribution F ∗ , and also depends on the
data set size. The goal, then, is to choose the parameters so
as to maximize the likelihood of observing F.
The analysis in [5, 30] primarily achieves rigor for continuous distributions. For discrete distributions, it turns out
that the distribution-on-frequencies is actually given by another distribution which seems to have been first studied by
Evert [8]. To describe it we recall the Γ function, which is
the (shifted) analytic continuation of the factorial function,
satisfying Γ(n) = (n − 1)! at positive integer values and
Γ(n) = (n−1)Γ(n−1) on its whole domain. The beta function, another standard function, is a continuous analogue of
the binomial coefficient, defined by
B(x, y) := Γ(x)Γ(y)/Γ(x + y).
Then, finally, the Evert distribution is the frequency distribution, parameterized by one parameter α, defined by
frequency of f ∝ B(f + 1 − α, α).
It is involved in our analysis for the following reason:
Proposition 1. Suppose that we draw samples from a discrete Zipf-Mandelbrot distribution with parameters γ and t.
If the number of samples is large, then for all small f , the expected number of f -legomena is proportional to B(f + 1 −
α, α) where α = 1 + 1/γ.
Paraphrasing, this says that the distribution-on-frequencies
for a discrete Zipf-Mandelbrot distribution is the Evert distribution. This result was obtained by Evert [8] though he
expressed it in terms of “type density functions.” We reprove it in Appendix A.
We remark that the proof of Proposition 1 remains valid
even if the discrete Zipf-Mandelbrot distribution is perturbed
by altering some of the highest probabilities, which ensures
that it is still valid even if outliers à la Section 3.1 occur.
4.1.1
Remarks
In [5, 30], the focus of the analysis is on continuous power
law distributions, and for that, the analogue of Proposition 1
is to use a simple power law with exponent α instead of an
Evert distribution. Though the Evert distribution is not mentioned in [30], it is remarked that the another distribution, the
Yule distribution, is an “an alternative and often more convenient form” of the discrete power law. These conveniences
are mathematical in nature: the normalizing constant, expectation, variance, and moments of the Yule distribution have
nicer closed forms than a pure power law. (And it is reasonable to use in power law analysis because up to a scaling factor, the Yule distribution becomes a discrete power law in the
limit.) These conveniences holds for the Evert distribution
too, since Evert and Yule differ onlyPby a shift. For instance,
Fmax
B(f +1−α, α) =
our fitting code utilizes the identity f =1
B(2 − α, α − 1) − B(Fmax + 2 − α, α − 1).
4.2
Maximum Likelihood
The Evert distribution allows us to compute the most likely
value of α for the collection of frequencies F. Writing Efα
for B(f + 1 − α, α), and C α for the normalizing constant
PFmax α
Ef = 1, we seek the α that maximizes
with C α · f =1
M
Y
Java
Python
Pythonw/o
Max. likelihood
α = 1.216, t = 33.1
α = 1.165, t = 44.7
α = 1.131, t = 99.8
χ2 min.
α = 1.225, t = 25.7
α = 1.143, t = 65.7
α = 1.133, t = 92.9
Table 2. Results of fitting our data sets to Zipf-Mandelbrot
distributions with both methods. Pythonw/o indicates the
Python data set with the 3 commonest messages removed.
C α · EFαk .
k=1
This can be determined numerically using binary search,
using logarithms since the numbers involved are very small.
Then we determine the parameter γ using γ = 1/(α − 1).
The only remaining issue is how to determine the value
of the shift parameter t that has maximum likelihood. Proposition 1 does not help since t plays no role in its conclusion.
(The reason for this apparent paradox is that the approximation guarantee of Proposition 1 is only valid for small frequencies.) Nonetheless, we can determine a value for t using
some ideas from the analysis of the continuous case [5, 30].2
Proposition 2. Let α = 1 + 1/γ. Suppose we draw a
sample from the bounded continuous power-law distribution
with exponent −α and domain (1, ( t+Mt +1 )−γ ). Then the
(M + 1)-quantiles of this random variable are proportional
to (t + 1)−γ , (t + 2)−γ , . . . , (t + M )−γ . Furthermore, the
choice of t that maximizes the likelihood of observing F is
1/γ
t = (M + 1)/(Fmax − 1).
Figure 3. Plot of the Python data set on shifted log-log
axes, with shift t from maximum likelihood estimation.
Red: observed frequencies; blue: Chi-squared fitted ZipfMandelbrot distribution; green: maximum likelihood fitted
Zipf-Mandelbrot distribution.
The first conclusion says that a “typical” draw of M
items from this continuous distribution-on-“frequencies” is
a model for the Zipf-Mandelbrot distribution. The second
conclusion gives us the rule that we use to compute t in our
statistical fitting. We prove the proposition in Appendix B.
5.
Fitting the Data
In Evert’s paper [8], rather than using maximum-likelihood,
he proposes estimating α using a Chi-squared test on the first
few #F −1 (1), #F −1 (2), . . . values. This approach is implemented by the R library zipfR of Evert and Baroni [9].
We fit our data sets to the Zipf-Mandelbrot family of
distributions, using both the Chi-squared approach, and the
maximum-likelihood method of Propositions 1 and 2 (implemented in Maple). The results of the fitting are shown in
2 The
authors of [5, 30] note that the continuous model reasonably resembles the discrete model when thinking about larger frequencies; the smallest
continuous variables are the ones that would have to be distorted the most
in order to become quantized. Thus, the inaccuracies of Proposition 2 are
complementary to those of Proposition 1.
Figure 4. Plot of the Java data set, analogous to Figure 3.
Table 2. The fit for the Python data set improved greatly by
treating the three most common errors as outliers (c.f. Figure 1). In Figures 3 and 4 we show shifted log-log plots of
the observed fits (the Python plot omits the outliers). For the
Python-without-outliers data set, both methods give a good
fit. For the Java data set, the maximum-likelihood method
gives a significantly better fit than the Chi-squared method.
6.
Future Work
A few very short questions for future work are: (1) can the
good fit be replicated in other Java/Python systems? (2) if so,
what properties of the user base or programming ecosystem
affect the α and t parameters? (3) do error messages in other
languages also follow a Zipf-Mandelbrot distibution?
In the context of the hypothetical extreme languages of
Section 1.1, Python’s slightly smaller value of α suggests
that it tends to give more distinctive error messages. Is it actually giving more information in its errors? Could it alternatively be explained due to artefacts like the non-parallelism
mentioned in Section 2.1?
It would be interesting to re-analyze the discrete data
sets in [30] using the Evert maximum likelihood method.
Specifically, this could be done for the data sets for word
frequency, web hits, telephone calls, and citations, which are
discrete distributions coming from a population that is large
enough to be effectively infinite. Additionally, it would be
interesting to apply the Kolmogorov-Smirnov test suggested
in [30] to the Evert maximum likelihood method, to be more
rigorous in our approach.
From a more practical perspective, it would be not hard,
and of a great potential benefit, to release a systematic data
set of good beginner-friendly explanations of the top errors
in different programming languages. Further work could try
to quantify if this improves the ability of beginner students
to program independently.
Acknowledgment
We thank the SIGCSE Special Projects committee, whose
Summer 2013 grant for CS Circles provided funding when
this work was initiated [34], and the Blackbox project for
their work on providing accessible huge data sets. We thank
undergraduate assistants Ayomikun (George) Okeowo and
Pallavi Koppol for their work on sanitizing and writing
explanations. Thanks to Jurgis Pralgauskis for translating
the Python explanations to Lithuanian. We also thank the
PLATEAU referees for their helpful suggestions.
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A.
Proof of Proposition 1
Fix a constant f and consider N as growing to infinity. By
linearity of expectation, the expected number E[#F −1 (f )]
of f -legomena is equal to the sum, over all `, of the probability that word ` occurs exactly f times in our sample.
For large N , any word with frequency bigger than a constant has vanishingly small probability of occurring only f
times. So for a word that may become an f -legomenon, its
(N C(` + t)−γ )f
.
f ! exp(N C(` + t)−γ )
We approximate this infinite sum with the infinite integral
Z ∞
(N C(x + t)−γ )f
dx.
E[#F −1 (f )] =
f ! exp(N C(x + t)−γ )
1
To evaluate it, we substitute y = N C(x + t)−γ , i.e. x =
( NyC )−1/γ − t and so dx = − γ1 (CN )1/γ y −1−1/γ dy, giving
E[#F −1 (f )] =
(CN )1/γ
f !γ
Z
0
N C −1 t−γ
1
y f − γ −1
dy.
ey
Again assuming N large, the above integral is well-approximated
by replacing the upper bound by +∞. Therefore, taking the
terms that do not depend on f into the constant of proportionality, we find that
E[#F
[38] H. Seo, C. Sadowski, S. Elbaum, E. Aftandilian, and R. Bowdidge. Programmers’ build errors: A case study (at Google).
In Proc. 36th ICSE, pages 724–734, 2014.
[39] B. Shneiderman. Designing computer system messages. Communications of the ACM, 25(9):610–611, 1982.
∞
X
`=1
[34] D. Pritchard, S. Graham, and T. Vasiga. The state of CS Circles: Open source and outreach with an introductory Python
website (Poster). In Proc. 46th SIGCSE, page 688, 2015.
[35] G. D. Ripley and F. C. Druseikis. A statistical analysis of
syntax errors. Computer Languages, 3(4):227–240, 1978.
(N C(` + t)−γ )f
.
f ! exp(N C(` + t)−γ )
B.
−1
1
Z
1 +∞ y f − γ −1
dy
(f )] ∝
f! 0
ey
Γ(f − 1/γ)
=
Γ(f + 1)
∝ B(f − 1/γ, 1 + 1/γ) = B(f + 1 − α, α).
Proof of Proposition 2
Let U (a, b) denote a random variable from the uniform
distribution on (a, b). Our starting observation is that the
continuous power-law distribution with exponent −α and
unbounded domain (1, +∞) is identical in distribution to
U (0, 1)−γ . See, for instance, [5, App. D].
Therefore, adding the bound to get the continuous powerlaw in the hypothesis of the theorem, said distribution is
identical in distribution to U ( t+Mt +1 , 1)−γ .
The (M + 1)-quantiles of U ( t+Mt +1 , 1)−γ are ((t +
k)/(t+M +1))−γ for k = 1, . . . , M , so the first conclusion
follows.
Finally, the smaller the domain (1, ( t+Mt +1 )−γ ), the
larger the probability density function at the observed F
values, except that we need ( t+Mt +1 )−γ ≥ Fmax for Fmax
to be observable at all. This proves the second conclusion.
| 6 |
1
User Association and Resource Allocation in
Unified Non-Orthogonal Multiple Access
Enabled Heterogeneous Ultra Dense Networks
arXiv:1801.08198v1 [cs.IT] 24 Jan 2018
Zhijin Qin, Member, IEEE, Xinwei Yue, Student Member, IEEE, Yuanwei Liu, Member, IEEE,
Zhiguo Ding, Senior Member, IEEE, and Arumugam Nallanathan, Fellow, IEEE
Abstract
Heterogeneous ultra dense networks (HUDNs) and non-orthogonal multiple access (NOMA) have
been identified as two proposing techniques for the fifth generation (5G) mobile communication systems
due to their great capabilities to enhance spectrum efficiency. This article investigates the application
of NOMA techniques in HUDNs to support massive connectivity in 5G systems. Particularly, a unified
NOMA framework is proposed, including power-domain NOMA and code-domain NOMA, which can
be configured flexibly to serve different applications scenarios. As a further advance, the unified NOMA
framework enabled HUDNs is further investigated, with particular focuses on the user association and
resource allocation. Two case studies are provided for demonstrating the effectiveness of the unified
NOMA enabled HUDNs. Finally, some main challenges and promising research directions in NOMA
enabled HUDNs are identified.
Index Terms
Heterogeneous ultra dense networks, non-orthogonal multiple access, massive connectivity, user
association, and resource allocation.
I. I NTRODUCTION
The last decade has witnessed the densification of wireless networks due to the various types of wireless
communication services. In order to support explosive data traffic, the concept of heterogeneous network
Z. Qin and Z. Ding are with Lancaster University, Lancaster, UK, LA1 4YW, email: {zhijin.qin, z.ding}@lancaster.ac.uk.
X. Yue is with Beihang University, Beijing 100191, China, email: xinwei yue@buaa.edu.cn.
Y. Liu and A. Nallanathan are with Queen Mary University of London, London, UK, E1 4NS, email: {yuanwei.liu,
a.nallanathan}@qmul.ac.uk.
2
(HetNet) has been proposed by overlaying small cells with low transmit power to macro cells. Through
the dense deployment of small cells, throughput and spectrum efficiency of cellular networks can be
enhanced significantly [1–3]. Moreover, it is predicated that Internet of Things (IoT) will bring critical
challenges for the fifth generation (5G) communication systems as billions of devices are to be connected.
In order to support massive connectivity with heterogeneous quality of service (QoS), non-orthogonal
multiple access (NOMA) has attracted extensive attentions due to its potential capability to enhance
spectrum efficiency [4–7]. The key idea of NOMA is to enable multi-user transmission within the same
resource block (RB), i.e., frequency/time, by using various power levels and/or different codes. Driven by
the key characters of heterogeneous ultra dense networks (HUDNs) and NOMA, it is natural to invoke
NOMA technique in HUNDs to support heavy data traffic as well as provide massive connectivity.
Existing NOMA can be mainly categorized into power-domain NOMA (PD-NOMA) and code-domain
NOMA (CD-NOMA) including low-density spreading CDMA (LDS-CDMA), low-density spreading
OFDM (LDS-OFDM), and sparse code multiple access (SCMA), which distinguish users by different
power levels and codes, respectively. Despite of the growing attempts and extensive efforts on NOMA,
most of the studies have focused on the performance analysis of various NOMA techniques individually,
such as PD-NOMA and CD-NOMA. However, different scenarios have different preferred NOMA techniques. For example, if users experience very bad channel conditions due to the near-far effect or in a
moving network, PD-NOMA can be a better candidate. If users experience poor channel conditions but
requiring high reliability, SCMA is preferred due to its shaping gain and near-optimal message passing
algorithm (MPA) detection. Therefore, it is desired to design a unified NOMA framework for 5G systems
to support various scenarios. The core idea of the proposed unified NOMA is to provide a multiple access
(MA) framework, which is capable of supporting massive connectivity with heterogeneous QoS by using
the same hardware infrastructure.
The goal of this article is to provide an unified NOMA framework and investigate its application in
HUNDs to support massive connectivity for 5G and IoT. As both the dense deployment of small cells
and the non-orthogonality in resource sharing bring severe interference, user association and resource
allocation are very challengeable to support massive connectivity. Particularly, the following two issues
should be addressed when designing the unified NOMA enabled HUNDs:
1) User association: user association process should consider both intra interference from the same
cell and inter interference introduced by the same cell as well as neighboring cells. Therefore,
controlling the number of users assigned to each cell can be an efficient approach to control
interference.
2) Resource allocation: once users are allocated into different cells, how to assign them with the most
3
suitable cell and proper transmit power for NOMA users within the same cell becomes critical.
Thus, efficient resource allocation and interference control schemes are more than desired in NOMA
enabled HUNDs.
The rest of this article is organized as follows. We first introduce the concepts of HUNDs and NOMA
techniques. Then we explore how NOMA techniques can be applied in HUNDs to enhance spectrum
efficiency and support massive connectivity. Subsequently, we propose a unified NOMA framework and
investigate its application in HUNDs in section III. Specifically, we provide both the uplink and downlink
cases for the unified NOMA enabled HUNDs. We discuss user association and resource allocation in
NOMA enabled HUNDs. In section IV, we provide related case studies for the proof of concept. We
also identify some potential research challenges in unified NOMA enabled HUNDs in Section V before
giving the conclusion remarks in Section VI.
II. OVERVIEW
OF
NOMA-E NABLED HUDN S
In this section, we first introduce the basic principles of HUDNs and NOMA techniques, respectively.
Then we discuss the general architecture of NOMA enabled HUDNs to support massive connectivity.
A. HUDNs
The density of wireless networks is invoked by the large number of devices, such as tablets, smart
phones, and IoT devices. To provide higher throughput and spectrum efficiency, HUDN technique has
attracted extensive research interest [1]. Particularly, HUDNs refer to networks that involve many different
types of small cells to make the access points getting as close as possible to end users [2]. Besides macro
cells, HUDNs contain cells with various sizes, such as pico cells, femto cells, and relays, which normally
transmit at lower power than macro cells and can offload data traffic from the macro cells. With the
increasing density of small cells, the backhaul network capacity and spectrum efficiency can be enhanced
significantly. However, it is unrealistic to deploy small cells with infinite density in practical scenarios.
Therefore, extensive research efforts are required to capture the reality of densification networks.
B. NOMA
In order to satisfy the requirements of massive connectivity and higher spectrum efficiency, NOMA has
been identified as a proposing technique in 5G systems [4–7]. Compared with the orthogonal multiple
access (OMA) techniques, such as FDMA and TDMA, NOMA breaks the orthogonality by allowing
multiple users sharing the same physical resource. More particularly, the virtue of superposition coding
is adopted to generate signals for multiple users at transmitters. At receivers, according to the adopted MA
4
approaches, different multi-user detection (MUD) algorithms, such as successive interference cancellation
(SIC) and MPAs, can be implemented to remove co-channel interference. In general, MA techniques can
be classified into PD-NOMA and CD-NOMA. Extensive research work has been carried out on PDNOMA [4, 6, 8–11] and CD-NOMA [12–14], respectively.
C. Understanding NOMA in HUDNs
Driven by the above overview, both HUDNs and NOMA are considered as promising techniques in 5G
systems to enhance spectrum efficiency and to support massive connectivity. Therefore, NOMA enabled
HUDNs are capable of further improving the spectral efficiency by offering more access opportunities.
Fig. 1 illustrates the architecture of NOMA enabled HUDNs. It can be observed that small cells,
including femto cells, pico cells, and relays, are densely deployed through the whole network, which can
enhance the system capacity significantly. In NOMA enabled HUDNs, NOMA technique is adopted in
each small cell to enable multiple users sharing the same RB, while the massive multiple-input multipleoutput (MIMO) technique is employed by macro cells. More particularly, macro cells can be connected
to core networks by optical fiber or wireless backhaul networks, and it is assumed that the number of
antennas equipped at each macro BS is much larger than the number of users. For small cells, each user
is equipped with single antenna and NOMA techniques are adopted to support multi-user transmission
over the same RB.
More specifically, the pico cell in Fig. 1 gives an example of PD-NOMA, which has low-complexity
receivers and is preferred by applications with less restriction on reliability. It can be observed that pico
BS allocates different power levels for PD-NOMA users according to their channel conditions. At User
1, who has best channel condition and lowest transmit power, by applying SIC, signals for the other users
with higher transmit power levels will be detected first and abstracted from the received signal. Then the
desired signal for User 1 can be obtained. While for User n assigned with highest transmit power, signals
for other users will be treated as noise when detecting its own signal. Moreover, femto cell shown in
Fig. 1 gives an illustration of multi-user transmission by invoking CD-NOMA, which is more suitable
for cases requiring higher reliability. We observe that users in femto cell carry out MUD-based MPA
individually to alleviate error propagation effects. After MPA detection, the soft information of users
is output to Turbo decoder, and the iterative process between MPA detector and Turbo decoder further
enhances the detection performance.
In HUDNs, users may experience very different channel conditions when connecting to BSs from
different tiers. It has been identified that different NOMA techniques have different performance in
terms of supporting massive connectivity under various scenarios, i.e., applications requiring different
5
Core networks
User 1 signal
detection
Subtract the signal
of user 1
SIC
User n signal
detection
MPA
User n signal
detection
Pico BS
User 1
Detector
Macro BS
Turbo
Decoder
User’s
signal
Macro BS
ĂĂ
Femto BS
ĂĂ
User m
User n
Detector
Relay
User 1
Turbo
Decoder
User 1
User’s
signal
MPA
ĂĂ
User k
Fig. 1. Illustration of the NOMA enabled HUDNs.
reliabilities and/or transmission rates. The same user may need different NOMA techniques when it
experiences various channel conditions and has different transmission requirements. Therefore, different
hardware architectures are required to support such various scenarios, which brings bottleneck for the
real implementation of NOMA techniques. It is necessary to provide a unified NOMA framework for
HUNDs, which can be implemented on the same hardware architecture but with flexible capability to
support various scenarios.
III. A U NIFIED NOMA F RAMEWORK
FOR
HUDN S
In this section, we first propose a unified NOMA framework for HUDNs, in which both the uplink
and the downlink are investigated, respectively. Then we address user association and resource allocation
issues in the considered HUDNs with the proposed unified NOMA framework.
A. Overview of HUDNs with Unified NOMA
In this part, we propose a unified NOMA framework, which contains both PD-NOMA and CD-NOMA
techniques. As shown in Fig. 2, we first map the superposed signals of multiple users to single RB or
multi-RB over a sparse matrix, in which most of the elements are zero. Note that single RB and multi-RB
correspond to single carrier and multi-carrier, respectively. In other words, single carrier NOMA (PDNOMA) is the special case of multi-carrier NOMA (CD-NOMA). The rows and columns of the sparse
6
matrix represent different RB and different users, respectively. Here, “1” represents the user occupies
the corresponding RB and “0” otherwise. The use of such a sparse matrix is essential to capture the
features of SCMA [12] and pattern division multiple access (PDMA) [13], where the optimal design
of sparse matrix for CD-NOMA is capable of reducing detection complexity at receivers. In particular,
SCMA and PDMA are belong to CD-NOMA’s different types of forms, in which the equal and unequal
column weight sparse matrixes are employed, respectively. It is worth noting that in HUDNs, each cell
can choose PD-NOMA or CD-NOMA scheme based on the pre-configuration of the BS. We will present
the details of PD-NOMA and CD-NOMA, i.e., SCMA and PDMA, and use a two user case to illustrate
how the unified NOMA framework works in the following.
Regarding PD-NOMA, a transmitter is capable of multiplexing multiple users via different power levels
within the single subcarrier, i.e., the first row of sparse matrices illustrated in Fig. 2. At receivers, PDNOMA exploits SIC to remove the multi-user interference. Additionally, PD-NOMA can be also realized
in multi-carriers, with the aid of appropriate user scheduling and power allocation approaches [8]. It is
worth noting that downlink multi-user superposition transmission (MUST), which is essentially a special
case of PD-NOMA, has been standardized.
CD-NOMA can be regarded as a special extension that directly maps data streams of multiple users
into multiple carriers by using the sparse matrix or low density spreading code at transmitters, as shown
in the multi-carrier case in Fig. 2. At receivers, multiple users are distinguished by MPA to obtain
the multiplexing or coding gains. For example, SCMA utilizes a sparse matrix, in which each column
can be selected from the predefined codebooks. With the multidimensional constellations to optimize
codebooks, SCMA is able to achieve enhanced shaping and coding gains. While for PDMA, the core
concept is to jointly optimize transmitters with sparse pattern design and receivers with MPA-based
detection. The design of sparse pattern can provide disparate diversity for multiple users and further
reduce the complexity of detection. Additionally, phase shifting is an effective way to obtain constellation
shaping gain. It is worth noting that the pivotal difference between these two schemes is that the number
of RBs occupied by each user has to be the same in SCMA, while PDMA allows a variable number of
RBs to be occupied by the same user. For another CD-NOMA scheme, muti-user shared access (MUSA)
[14], each user’s data symbols are spread by a special spread sequence to facilitate SIC implementation.
This kind of spread sequence can be selected from the sparse matrix as illustrated in Fig. 2, which requires
special design to achieve low cross-correlation. Note that multiple spreading sequences constitute a pool,
where each user can select one sequence from it.
7
Multiple carrier
…
User 1 User 2
SIC
RB 1
MUD
based on
MPA
…
RB 2
RB K
User N
Multiple carrier
User 1
0
1 L 1 L 0 L 0
M
M
0
0
0
1
0
1K×N
1
0
1
RB 1
RB 2
…
User detection
1
1
M
1
Single carrier
RB K
The signal
of user
Pico/Femto BS
User 1
1
1
M
1
User 2
1
1
M
0
…
User N
Single carrier
User 2
64
4744
8 644744
8
0
1
0
1 1
1 1 L 1 L 0 L 0
M M
0
0
M
1
0
1 K × N
1 0
Pico/Femto BS
0
0
0
1
User 1
User 2
SIC
Signal of
superposition
(a) Uplink NOMA
Detector
Decoder
The signal
of user
MPA
(b) Downlink NOMA
Fig. 2. Uplink and downlink NOMA systems
B. Uplink and Downlink Design for NOMA Enabled HUDNs
1) Uplink Design: To facilitate understanding the uplink design in the unified NOMA framework, we
present specific examples in the following. For uplink PD-NOMA, multiple users transmit messages to the
BS by the same RB. As shown in Fig. 2(a), when using the sparse matrix for PD-NOMA, multiple users’
signals are mapped into RBs in the first row of sparse matrix, while the rest rows of the sparse matrix
are set to zeros. Notice that power control strategy is a vital issue in the uplink NOMA transmission,
especially when powers received at users are significantly distinct. We employ SIC at BSs to decode and
subtract the information of the nearby user first, then decode the message of the distant user. By doing
so, the data rate of the distant user can be guaranteed. For CD-NOMA, each user selects one or several
columns from the sparse matrix and then maps its information to multiple RBs by using the spreading
code. For example, considering the case that each user selects only one column and spreads its message
to “1”, then the data streams of N users are superposed through K RBs to construct the sparse matrix
with dimensions of K × N at the BS. The sparse properties of matrix is conducive for implementing the
MPA detection. To guarantee the fairness among users, the nearby user selects one column with smaller
column weight, while the distant user selects column with larger column weight. Such operation can
enhance reliability of received signals for multiple users.
8
2) Downlink Design: The key feature of the downlink design in the unified NOMA framework is that
multiple data streams of different users are superimposed at BSs. Then BSs transmit the superimposed
signals to multiple users simultaneously. More particularly, a BS maps multiple users’ signals into single
or multiple carriers over the sparse matrix. In the following, we use a two-user case to illustrate how the
downlink of the unified NOMA framework works.
For PD-NOMA shown in Fig. 2(b), the BS maps signals of multiple users into a single RB by utilizing
one row of the sparse matrix with dimension of 1 ×
N
2
and then transmits superposed signals to two
users. It is observed that one row including multiple “1”s denotes that the data streams from one user can
be spread into multiple layers, which can provide more flexibility for resource mapping. While for CDNOMA, a user’s data streams are directly mapped into multi-RB by sharing multiple spread sequences. As
a further advance, the superposed signals of two users are formulated at the BS, which will be transmitted
to the destination. Similar to uplink design, optimal design of the sparse matrix can further enhance the
detection performance. At the receiver, MPA is implemented to recover the desirable user’s signals. From
a practical perspective, computational complexity at the reciter will grow exponentially with the number
of users increasing. Hence, how to reduce the detection complexity for CD-NOMA should be taken into
account in the 5G standardization process.
C. User Association in NOMA Enabled HUDNs
The distinct characteristics of the HUDNs with unified NOMA inevitably necessitate the redesign of
user association algorithms. In contrast to the conventional user association approaches, on the one hand,
the dense deployment of small cells introduce severe inter-interference as the neighbouring cells share
the same RB. On the other hand, NOMA brings extra intra-interference from the same BS, hence making
the user association design more challengeable.
In order to address these two issues, we propose a flexible user association design for the unified NOMA
enabled HUDNs, in which a NOMA user is allowed to access the BS of any tier in order to achieve
the best coverage. As shown in Fig. 3, we take the PD-NOMA as a specific example. For simplicity, we
consider that all BSs of HUDNs operate over the same orthogonal RB. Assuming that each user connects
with one BS at most, while one BS can serve two users by adopting NOMA techniques. Particularly, we
propose to associate users to BSs based on the maximum average power received at each NOMA user.
In other words, each user not always access the nearest BS. It is allowed to access any tier BS. Such
a user association scheme is fundamentally different from the conventional approach, which associates
users with the nearest BS and may lead to the association of most users with small cells as their BSs
are much closer to end users. As illustrated in Fig. 3, each BS has been associated with some users.
9
When a new user joints the network, its association should be determined by considering the effects of
both transmit power disparity of HUDNs and power sharing coefficients of NOMA users associated to
the same BS. Based on this flexible user association approach, network performance of NOMA enabled
HUDNs has been investigated in [9], which has analytically demonstrated that NOMA enabled HUNDs
outperform the conventional OMA enabled one.
D. Resource Allocation in NOMA Enabled HUDNs
Resource allocation is another significant aspect for designing NOMA enabled HUDNs. Note that
the implementation of NOMA brings more sophisticated co-channel interference to existing HUDNs,
such distinct characteristics lead the resource allocation problems more challengeable. Fig. 4 provides
an illustration of resource allocation for our proposed unified NOMA enabled HUDNs framework. More
particularly, date streams of different users can be spread over multiple RBs, where “1” and “0” denote
whether there exists a resource mapping between the corresponding user and RB. More specifically,
the shaded blocks refer to the RB occupied by users’ data, which indicates a mapping. Technically,
each user can select one column from the sparse matrix randomly. However, to improve the detection
performance, distant user prefers to select one column with larger column weight for resource allocation.
While nearby user tends to select one column with smaller column weight. Furthermore, by multiplying
a power sharing coefficient with each column, network performance can be further enhanced. Finally, we
employ MUD-based SIC/MPA to detect and output the information for the desired user.
For resource allocation in NOMA enabled HUDNs, several problems should be jointly considered for
intelligently tackling the intra-BS and inter-BS interferences: i) the number of users to be allocated in
the same RB; ii) which users should be allocated into which RB; and iii) the power sharing coefficient
for each RB as well as for users sharing the same RB. For example, for single carrier system, the
superposed signal of multiple users can be mapped into a single carrier over different power levels. The
power allocation between NOMA users should be considered carefully. For multi-carrier system, the
superposed signal can be mapped into multiple sub-carriers, where the NOMA users can select which
sub-carrier to employ based on their requirements and then consider power allocation. All these problems
can be addressed by properly designing the sparse matrix and its corresponding power sharing coefficients.
Actually, due to the unique character of intra-BS interference brought by NOMA, resource allocation
in NOMA enabled HUNDs becomes mix integer non-convex optimization problems, which usually tend
to be NP-hard. Hence, efficient resource allocation algorithms are more than desired. Matching theory
can be invoked as an effective approach for achieving good tradeoff between system performance and
computational complexity. With invoking matching theory, the authors in [11] have proposed an effective
10
User
associations
User 4
User m
Pico BS
User 1
User 3
User 2
Femto BS
User n
Fig. 3. User association for NOMA enabled HUDNs.
!1
#1
#
#!
#
%1
1
0
1
1
1
0
!
!
0
0
1
0
0"
0$
$
0$
$
1& K
Sparse matrix
N
Resource
management
ĂĂ
SIC/MPA
Resource
mapping
Fig. 4. Resource allocation for NOMA enabled HUDNs.
resource allocation approach to show that NOMA enabled HUNDs scheme is capable of achieving a
higher sum rate compared to the OMA-enabled one.
IV. C ASE S TUDY
FOR
NOMA E NABLED HUDN S
In this section, we evaluate the unified NOMA enabled HUDNs by simulations. For simplicity, we consider the uplink transmission of PD-NOMA. In the following, we provide two case studies to demonstrate
user association and resource allocation in the unified NOMA enabled HUDNs, respectively.
A. User Association in NOMA Enabled HUDNs
In this study, we illustrate how the density of small cells influences the user association in NOMA
enabled HUDNs. Here, we consider user association in the case where the proposed unified NOMA
framework is applied in HUDNs based on a stochastic geometry model. More particularly, the locations
of BSs and users follow homogeneous Poisson point processes. In the consider network, macro cells
employ massive MIMO and small cells adopt NOMA to support massive connectivity, and the maximum
11
average received power approach is adopted to determine user association as aforementioned in section
III. More details of the considered network configurations can be found in [9].
Fig. 5 plots the user association probability of the unified NOMA enabled HUDNs with three layers,
i.e., K = 3. In this case, the density of macro BSs is fixed, while the densities of pico BSs and femto
BSs vary correspondingly. It can be observed that NOMA users prefer macro cells when the density of
small cells is relatively low. With the densification of small cells, which is the case of HUDNs, NOMA
users show a higher intention to be associated with BSs in pico and femto cells. It is also worth noting
that NOMA users have a higher probability to be associated with BSs in femto cells even though BSs
in pico cells transmit at a higher power level. This is caused by the dense deployment of femto cells,
which also revels the effectiveness and benefits of the NOMA enabled HUDNs.
B. Resource Allocation in NOMA Enabled HUDNs
In this study, we compare the performance of our unified NOMA enabled HUDNs with the conventional
OMA enabled scheme in terms of both fairness and sum rates. We consider a HetNet with two tiers,
in which the macro cell and small cells reuse the same set of RBs. In other words, we can refer to the
small cells as the underlay tier. We allow each small cell BS to serve two users via NOMA by using the
same RB. Our goal is to maximize the sum rate of NOMA enabled small cell BSs via proper RB and
power sharing coefficients schemes. More particularly, we adopt the matching theory for user allocation
and sequential convex programming for power control [11].
Fig. 6 plots the fairness and sum rate of resource allocation versus the total number of small cell BSs,
respectively. Here, τ is the maximum number of small cell BSs occupying the same RB for restricting the
co-channel interference. Jains fairness [15] index is adopted to evaluate the performance of the considered
networks. We can observe that the fairness performance decreases with the number of small cell BSs
in Fig. 6(a). This is because that large number of small cell BSs leads to more severe competition for
limited spectrum resources. Consequently, more small cell BSs with poor channel conditions can not be
accessed. We can also note that as τ increases, a higher fairness rate can be achieved. This is attributed to
the fact that more small cell BSs can be multiplexed on each RB, which increases the multi-user diversity
gain. It is also worth noting that the unfied NOMA enabled HUDNs have superior performance than the
conventional OMA scheme both in terms of fairness and sum rate, which demonstrates the effectiveness
the proposed structure.
12
User association probability
1
Macro cells
Pico cells
Femto cells
0.8
0.6
0.4
0.2
0 −7
10
−6
10
−5
−4
10
10
Density of BSs in pico cells
−3
−2
10
10
Fig. 5. User association probability versus density of BSs in pico cells, K = 3 tiers, number of antennas equipped in macro
cell BS M = 200, data streams N = 15, macro BS transmit power is P1 = 40 dBm, pico BS transmit power is P2 = 30
dBm, and femto BS transmit power is P3 = 20 dBm, power sharing coefficients for NOMA users are am = 0.6 and an = 0.4,
respectively, density of macro BS
0.5
density of small cell BS λf emto = 5 × λpico .
24
NOMA, τ=2
NOMA, τ=1
OMA, τ=2
OMA, τ=1
22
Sum rate (bits/(s*Hz))
Jain‘s fairness index
0.6
1
,
2×500 2 π
0.4
0.3
0.2
NOMA
OMA
20
18
16
14
12
0.1
20
25
30
35
Number of base statioin in each small cell
10
10
40
12
14
16
18
Number of base station in each small cell
20
(a) Fairness comparison, transmit power at macro BSs is 43 (b) Sum rate comparison, maximum number of small cells
dBm, the transmit power at small cell BSs is 23 dBm.
occupying the same RB τ = 2.
Fig. 6. Resource allocation comparison between the unified NOMA enabled HUDNs and the OMA enabled scheme.
V. R ESEARCH C HALLENGES
IN
NOMA E NABLED HUDN S
To support massive connectivity and enhance spectrum efficiency in 5G systems, the following research
problems should be addressed in the NOMA enabled HUNDs:
•
Energy efficiency in NOMA enabled HUNDs: one of the potential applications of NOMA enabled
HUNDs is IoT for smart cities, in which massive number of devices need to be connected. The unified
NOMA enabled HUNDs provide a practical infrastructure to offer massive access opportunities for
13
such large number of devices, especially for the cases that each device only needs to send a small
amount of data periodically. However, these devices are normally restricted to power consumption
as they are powered by battery. In order to extend the battery lifetime of these IoT devices,
i.e., devices are expected to keep live for ten years, the energy efficiency is under investigated.
Particularly, higher data transmission rate results in shorter airtime, however, the power consumption
for data transmission becomes higher. With lower power consumption, the achieved transmission
rate becomes lower, which extends the airtime. Therefore, the tradeoff between data rate and airtime
should be considered to maximize battery lifetime of devices.
•
Big Data aided Adaptive NOMA in HUNDs: IoT devices normally have limited processing
capability, while some devices, i.e., mobile phones, are capable of performing more complex tasks.
Meanwhile, it is noted that different NOMA schemes requires different complexity levels at the user
side. For instance, SIC receiver is relatively simple, which makes PD-NOMA more suitable for IoT
devices. Therefore, a software-defined NOMA network architecture is desired to achieve adaptive
NOMA with awareness on complexity to support different user scenarios. Machine learning can
be invoked to predict the data traffic for different user scenarios. Adaptive MA technique can be
categorized into two types, including pre-settings and real-time settings. Pre-settings refer to assign
the MA technique according to historical social media information, such as the number of users
within a given area in several months or one year. The real-time settings adjust the MA technique
based on the real-time feedback from social media.
•
Testbed for NOMA enabled HUNDs: even though extensive research has been carried out on the
performance analyses and algorithm designs for NOMA enabled HUNDs, there is still a large gap
to the real implementation of NOMA in HUNDs. For the proof of concept, a testbed is more than
desired to demonstrate the effectiveness of the proposed unified NOMA enabled HUNDs. More
particularly, the idea of software defined radio (SDR) can be adopted to enable BSs to select the
proper NOMA technique smartly according to different application scenarios.
VI. C ONCLUSIONS
This article has envisioned NOMA enabled HUDNs as a promising solution to support massive
connectivity in 5G systems. Instead of focusing on specific NOMA techniques individually, we have
proposed a unified NOMA framework. Moreover, we have investigated the application of the proposed
unified NOMA framework in HUDNs. We have further explored the critical challenges on user association
and resource allocation in NOMA enabled HUDNs, as both the dense deployment of small cells and the
non-orthogonality in resource sharing bring severe interference. Additionally, we have carried out related
14
case studies, which have provided important insights for the future design of NOMA enabled HUDNs
to support massive connectivity in 5G systems.
R EFERENCES
[1] H. Zhang, Y. Dong, J. Cheng, M. J. Hossain, and V. C. M. Leung, “Fronthauling for 5G LTE-U ultra dense cloud small
cell networks,” IEEE Wireless Commun., vol. 23, no. 6, pp. 48-53, Dec. 2016.
[2] X. Ge, S. Tu, G. Mao, and C. X. Wang, “5G ultra-dense cellular networks,” IEEE Wireless Commun., vol. 23, no. 1, pp.
72-79, Feb. 2016.
[3] H. Zhang, S. Huang, C. Jiang, K. Long, V. C. M. Leung and H. V. Poor, “Energy efficient user association and power
allocation in millimeter-wave-based ultra dense networks with energy harvesting base stations,” IEEE J. Sel. Areas Commun.,
vol. 35, no. 9, pp. 1936-1947, Sep. 2017.
[4] Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li and K. Higuchi, “Non-orthogonal multiple access (NOMA) for
cellular future radio access,” IEEE Veh. Technol. Conf. (VTC Spring’13), Dresden, Germany, Jun. 2013, pp. 1-5.
[5] L. Dai, B. Wang, Y. Yuan, S. Han, C. l. I and Z. Wang, “Non-orthogonal multiple access for 5G: solutions, challenges,
opportunities, and future research trends,” IEEE Commun. Mag., vol. 53, no. 9, pp. 74-81, Sep. 2015.
[6] Z. Ding, Y. Liu, J. Choi, Q. Sun, M. Elkashlan, I. Chih-Lin, and H. V. Poor, “Application of non-orthogonal multiple access
in LTE and 5G networks,” IEEE Commun. Mag., vol. 55, no. 2, pp. 185–191, Feb. 2017.
[7] Y. Cai, Z. Qin, F. Cui, G. Y. Li, J. A. McCann, “Modulation and multiple access for 5G networks,” IEEE Commun. Surveys
Tut., vol. PP, no. 99, pp. 1-1. Oct. 2017.
[8] Y. Sun, D. W. K. Ng, Z. Ding, and R. Schober, “Optimal joint power and subcarrier allocation for full-duplex multicarrier
non-orthogonal multiple access systems,” IEEE Trans. Commun., vol. 65, no. 3, pp. 1077-1091, Mar. 2017.
[9] Y. Liu, Z. Qin, M. Elkashlan, A. Nallanathan, and J. A. McCann, “Non-orthogonal multiple access in large-scale
heterogeneous networks,” IEEE J. Sel. Areas Commun., vol. 35, no. 12, pp. 2667-2680, Dec. 2017.
[10] F. Fang, H. Zhang, J. Cheng and V. C. M. Leung, “Energy-efficient resource allocation for downlink non-orthogonal
multiple access network”, IEEE Trans. Commun., vol. 64, no. 9, pp. 3722-3732, Sep. 2016.
[11] J. Zhao, Y. Liu, K. Chai, A. Nallanathan, Y. Chen, and Z. Han, “Resource allocation for non-orthogonal multiple access
in heterogeneous networks”, in Proc. IEEE Int. Conf. Commun. (ICC’17), Paris, France, May 2017.
[12] H. Nikopour and H. Baligh, “Sparse code multiple access,” in Proc. IEEE Annual Int. Symp. Personal Indoor Mobile Radio
Commun. (PIMRC’13), London, UK, Sep. 2013, pp. 332–336.
[13] S. Chen, B. Ren, Q. Gao, S. Kang, S. Sun, and K. Niu, “Pattern division multiple access PDMA - A novel non-orthogonal
multiple access for 5G radio networks,” IEEE Trans. Veh. Technol., vol. 66, no. 4, pp. 3185-3196, Apr. 2017.
[14] Z. Yuan, G. Yu, W. Li, Y. Yuan, X. Wang, and J. Xu, “Multi-user shared access for Internet of Things,” in Proc. IEEE
Veh. Technol. Conf. (VTC Spring’16), Nanjing, China, May 2016, pp. 1–5.
[15] R. Jain, D.-M. Chiu, and W. R. Hawe, “A quantitative measure of fairness and discrimination for resource allocation in
shared computer system,” Eastern Research Laboratory, Digital Equipment Corporation Hudson, MA, USA, Sep. 1984, vol.
38.
| 7 |
arXiv:1711.06961v1 [math.AC] 19 Nov 2017
SYSTEMS OF SETS OF LENGTHS OF PUISEUX MONOIDS
FELIX GOTTI
Abstract. In this paper we study the system of sets of lengths of non-finitely generated atomic Puiseux monoids (a Puiseux monoid is an additive submonoid of Q≥0 ).
We begin by presenting a BF-monoid M with full system of sets of lengths, which
means that for each subset S of Z≥2 there exists an element x ∈ M whose set of
lengths L(x) is S. It is well known that systems of sets of lengths do not characterize
numerical monoids. Here, we prove that systems of sets of lengths do not characterize
non-finitely generated atomic Puiseux monoids. In a recent paper, Geroldinger and
Schmid found the intersection of systems of sets of lengths of numerical monoids.
Motivated by this, we extend their result to the setting of atomic Puiseux monoids.
Finally, we relate the sets of lengths of the Puiseux monoid P = h1/p | p is primei
with the Goldbach’s conjecture; in particular, we show that L(2) is precisely the set
of Goldbach’s numbers.
1. Introduction
Factorization theory originated from algebraic number theory, in particular, from the
fact that the ring of integers OK of an algebraic number field K fails in general to be
factorial. An integral domain is called half-factorial if any two factorizations of the same
element involve the same number of irreducibles. In the 1950’s, L. Carlitz [3] proved
that a ring of integers OK is half-factorial if and only if its class group C(OK ) has order
at most 2. He also proposed to study further connections between the phenomenon
of non-unique factorizations of OK and the structure of C(OK ). In general, many
algebraic invariants of Noetherian domains can also be used to understand to which
extent such integral domains fail to be factorial.
If the set Z(x) of factorizations into irreducibles of a given element x in an integral
domain is not a singleton, many interesting questions naturally emerge to understand
how bizarre Z(x) might be. For instance, what is the size of Z(x) or what is the
subset L(x) ⊂ N consisting of the lengths of elements of Z(x)? Or, perhaps, how
similar or close are two given factorizations in Z(x)? Many statistics and algebraic
invariants have been introduced to answer these and other similar questions induced
by the phenomenon of non-unique factorizations. The study of such algebraic invariants
in integral domains (see [2]) and, more recently, in the abstract setting of commutative
cancellative monoids (see [11] and references therein) have given shape to the modern
factorization theory.
Date: November 21, 2017.
Key words and phrases. Puiseux monoids, system of sets of lengths, sets of lengths, realization
theorem, characterization problem, atomic monoids.
1
2
F. GOTTI
Perhaps the most investigated factorization invariant is the system of sets of lengths.
If M is a commutative cancellative monoid and x ∈ M can be written as a product
of n irreducibles, then n is called a length of x, and the set L(x) comprising all the
lengths of x is called the set of lengths of x. In addition, the set {L(x) | x ∈ M} is
called the system of sets of lengths of M. The reader might have noticed that if M is
factorial, then |L(x)| = 1 for all x ∈ M. Clearly, |L(x)| = 1 for all x ∈ M means that
M is half-factorial.
An extensive family of commutative cancellative monoids with a wild atomic structure and a rich factorization theory is hidden inside the set of nonnegative rational
numbers. The members of this family, additive submonoids of Q≥0 , are called Puiseux
monoids. The atomic structure of Puiseux monoids has only been studied recently (see
[16], [17], [18]). In the present manuscript, we study the system of sets of lengths of
non-finitely generated atomic Puiseux monoids.
We begin the next section by establishing the notation of commutative semigroup
theory we shall be using later. Then we recall the concepts of factorization theory that
are required to follow this paper. In particular, we formally define what a factorization
is, and we introduce the factorization invariants that will play some role in the following sections. Finally, a brief insight into the atomic structure of Puiseux monoids is
provided.
In Section 3, we begin our journey into the system of sets of lengths of Puiseux
monoids. It was recently proved in [13] that for each S ⊆ Z≥2 we can find a numerical
monoid N and x ∈ N with L(x) = S. We study the same question, but in the setting
of non-finitely generated Puiseux monoids. As a result, we construct a BF-monoid M
whose system of sets of lengths is as large as we can expect, i.e., for each S ⊆ Z≥2
there exists x ∈ M such that L(x) = S. Note that in our setting the monoid does not
depend on the choice of the set S.
Can we characterize the members of certain given family of atomic monoids by their
systems of sets of lengths? This is an important question in factorization theory, which
is known as the Characterization Problem. For example, it is conjectured that Krull
monoids over finite abelian groups with prime divisors in all classes whose Davenport
constant is at least 4 can be characterized by their systems of sets of lengths. On
the other hand, it was proved in [1] that systems of sets of lengths do not characterize
numerical monoids. In Section 4, we show that non-finitely generated Puiseux monoids
cannot be characterized by their systems of sets of lengths.
In Section 5, we construct an atomic Puiseux monoid M that is completely non-halffactorial, meaning that each x ∈ M \ {0} that is not irreducible satisfies |L(x)| ≥ 2.
Then, motivated by [13, Section 4], we study the intersection of systems of sets of
lengths of atomic Puiseux monoids. The construction of a completely non-half-factorial
Puiseux monoid will allow us to give a version of [13, Theorem 4.1] in the setting of
atomic Puiseux monoids.
SYSTEMS OF SETS OF LENGTHS OF PUISEUX MONOIDS
3
As most of the results in this paper are about cardinality restrictions and partial
descriptions of sets of lengths rather than their explicit determination, we feel the need
to argue how complex are explicit computations of sets of lengths, even for the simplest
atomic Puiseux monoids. Thus, in Section 6 we show that in the elementary Puiseux
monoid h1/p | p is primei the set of lengths L(2) is precisely the set of Goldbach’s
numbers. This, in particular, implies that an explicit description of L(2) is as hard as
the Goldbach’s conjecture.
2. Background
To begin with let us introduce the fundamental concepts related to our exposition
as an excuse to establish the notation we need. The reader can consult Grillet [19]
for information on commutative semigroups and Geroldinger and Halter-Koch [10] for
extensive background in non-unique factorization theory of commutative domains and
monoids. These two references also provide definitions for most of the undefined terms
we mention here.
Throughout this sequel, we let N denote the set of positive integers, and we set
N0 := N ∪ {0}. For X ⊆ R and r ∈ R, we set X≤r := {x ∈ X | x ≤ r}; with a similar
spirit we use the symbols X≥r , X<r , and X>r . If q ∈ Q>0 , then we call the unique
a, b ∈ N such that q = a/b and gcd(a, b) = 1 the numerator and denominator of q and
denote them by n(q) and d(q), respectively.
The unadorned term monoid always means commutative cancellative monoid. As
most monoids here is assumed to be commutative, unless otherwise specified we will
use additive notation. We let M • denote the set M \{0}. For a, c ∈ M, we say that a
divides c in M and write a |M c provided that c = a + b for some b ∈ M. We write
M = hSi when M is generated by a set S. We say that M is finitely generated if it
can be generated by a finite set; otherwise, M is said to be non-finitely generated.
A non-invertible element a ∈ M is an atom (or irreducible) if for each pair of elements
u, v ∈ M such that a = u + v either u or v is invertible. Let A(M) denote the set of
atoms of M. Every monoid M in this paper will be reduced, which means that 0 is the
only invertible element of M. This clearly implies that A(M) will be contained in each
generating set of M. If A(M) generates M, then M is said to be atomic. Monoids
addressed in this article are all atomic. We say that a multiplicative monoid F is free
abelian on P ⊂ F if every element a ∈ F can be written uniquely in the form
Y
a=
pvp (a) ,
p∈P
where vp (a) ∈ N0 and vp (a) > 0 only for finitely many elements p ∈ P . It is well known
that for each set P , there exists a unique (up to canonical isomorphism) monoid F such
that F is free abelian on P . When we want to emphasize the relation between P and
F , we denote F by F (P ). It follows by the fundamental theorem of arithmetic that
4
F. GOTTI
the multiplicative monoid N is free abelian on the set of prime numbers. In this case,
we can extend vp to Q≥0 as follows. For r ∈ Q>0 let vp (r) := vp (n(r)) − vp (d(r)) and
set vp (0) = ∞. The map vp : Q≥0 → Z, called the p-adic valuation on Q≥0 , satisfies
the following two properties:
(2.1)
(2.2)
vp (rs) = vp (r) + vp (s) for all r, s ∈ Q≥0 ;
vp (r + s) ≥ min{vp (r), vp (s)} for all r, s ∈ Q≥0 .
The free abelian monoid on A(M), denoted by Z(M), is called the factorization
monoid of M, and the elements of Z(M) are called factorizations. If z = a1 . . . an is
a factorization in Z(M) for some n ∈ N0 and a1 , . . . , an ∈ A(M), then n is called the
length of z and is denoted by |z|. The unique homomorphism
φ : Z(M) → M satisfying φ(a) = a for all a ∈ A(M)
is called the factorization homomorphism of M, and for each x ∈ M the set
Z(x) := φ−1 (x) ⊆ Z(M)
is called the set of factorizations of x. By definition, we set Z(0) = {0}. Note that the
monoid M is atomic if and only if Z(x) is nonempty for all x ∈ M. For each x ∈ M,
the set of lengths of x is defined by
L(x) := {|z| | z ∈ Z(x)}.
A monoid M is half-factorial if |L(x)| = 1 for all x ∈ M. On the other hand, we say
that the monoid M is completely non-half-factorial if |L(x)| = ∞ for all x ∈ M • \A(M).
If L(x) is a finite set for all x ∈ M, then we say that M is a BF-monoid. The system
of sets of lengths of M is defined by
L(M) := {L(x) | x ∈ M}.
In [11] the interested reader can find a friendly introduction to sets of lengths and the
role they play in factorization theory. In general, sets of lengths and systems of sets
of lengths are factorization invariants of atomic monoids that have received significant
attention in recent years (see, for instance, [1, 5, 14]).
A very special family of atomic monoids is that one comprising all numerical monoids,
cofinite submonoids of N0 . Each numerical monoid N has a unique minimal set of generators, which is finite; such a unique minimal generating set is precisely A(N). As a
result, every numerical monoid is atomic and contains only finitely many atoms. An
introduction to the realm of numerical monoids can be found in [8].
Recall that an additive submonoid of Q≥0 is called a Puiseux monoid. Puiseux
monoids are a natural generalization of numerical monoids. However, in general, the
atomic structure of Puiseux monoids differs significantly from that one of numerical
monoids. Puiseux monoids are not always atomic; for instance, consider h1/2n | n ∈ Ni.
On the other hand, if an atomic Puiseux monoid M is not isomorphic to a numerical
monoid, then A(M) is infinite. It is also useful to know that if a Puiseux monoid
SYSTEMS OF SETS OF LENGTHS OF PUISEUX MONOIDS
5
does not contain 0 as a limit point, then it is atomic. Indeed, the following stronger
statement, which is a direct consequence of [15, Theorem 4.5], holds.
Theorem 2.1. Let M be a Puiseux monoid. If 0 is not a limit point of M, then M is
a BF-monoid.
The atomic structure and factorization theory of Puiseux monoids has only been
studied recently (see [17] and references therein).
3. A BF-Puiseux monoid with full system of sets of lengths
Given a factorization invariant f of atomic monoids (resp., of elements of atomic
monoids) and certain class of atomic monoids C, the question of whether there exists
M ∈ C (resp., M ∈ C and x ∈ M) such that f(M) (resp., fM (x)) equals some prescribed
value is called a realization problem. Besides the sets of lengths, there are many factorization invariants, including the set of distances and the catenary degree (definitions
can be found in [10]), for which the realization problem restricted to several classes of
atomic monoids have been studied lately. Indeed, theorems in this direction have been
established in [4, 12, 22, 23].
In this section, we study a realization problem when the factorization invariant f is
the system of sets of lengths and the class C is that one comprising all Puiseux monoids
which are also BF-monoids. We take our prescribed value to be the collection of sets
S = {0}, {1}, S | S ⊆ Z≥2 and |S| < ∞ .
As the only nonzero elements of an atomic monoid M having factorizations of length 1
are the atoms, it follows that L(M) ⊆ {{0}, {1}, S | S ⊆ Z≥2 }. Therefore, when M is
a BF-monoid we obtain that L(M) ⊆ S.
Definition 3.1. We say that a BF-monoid M has full system of sets of lengths provided
that L(M) = S.
We positively answer our realization question by constructing (in the proof of Theorem 3.6) a Puiseux monoid with full system of sets of lengths. Note that families
of monoids and domains having full systems of sets of lengths have been found and
studied before. It was proved by Kainrath [21] that Krull monoids having infinite class
groups with primes in each class have full systems of sets of lengths. On the other
hand, Frish [6] proved that the subdomain Int(Z) of Z[x] also has full system of sets of
lenghts; this result has been recently generalized [7], as we show in Example 3.3.
Example 3.2. Let M be Krull monoid, and let G be the class group of M. Suppose
that G is infinite and that every class contains at least a prime. Therefore for each
nonempty finite subset L of Z≥2 and every function f : L → N, there exists x ∈ M
satisfying the following two conditions:
6
F. GOTTI
(1) L(x) = L, and
(2) |{z ∈ Z(x) | |z| = k}| ≥ f (k) for each k ∈ L.
In particular, M has full system of sets of lengths. In this example we illustrate a
simplified version of [21, Theorem 1]. We refer the reader to [11, Section 3] not only
for the definition of Krull monoids but also for a variety of examples showing up in
diverse areas of mathematics.
Example 3.3. [7, Theorem 4.1] Let OK be the ring of integers of a given number field
K. In addition, take m1 , . . . , mn ∈ N such that m1 < · · · < mn . Let Int(OK ) denote
the subring of integer-valued polynomials of K[x] (i.e., the subring of polynomials of
K[x] stabilizing OK ). Then there exists p(x) ∈ Int(OK ) and z1 , . . . , zn ∈ Z(p(x))
satisfying that |zi | = mi + 1 for each i = 1, . . . , n. As a result, the domain Int(OK ) has
full system of sets of lengths.
The following result, which is a crucial tool in our construction, is a simplified version
of a recent realization theorem by Geroldinger and Schmid.
Theorem 3.4. [13, Theorem 3.3] For every nonempty finite subset S of Z≥2 , there
exists a numerical monoid N and x ∈ N such that L(x) = S.
Theorem 3.4 implies, in particular, that every nonempty finite subset S of Z≥2 can be
realized as the set of lengths of an element inside certain Puiseux monoid. In principle,
the choices of both the Puiseux monoid and the element depend on the set S. However,
the existence of a Puiseux monoid with full system of sets of lengths will eliminate the
former of these two dependences. We will create such a Puiseux monoid by ”gluing”
together a countable family of numerical monoids, each of them containing an element
whose set of lengths is a specified finite subset of Z≥2 . Clearly, we should glue the
numerical monoids carefully enough so that none of the specified sets of lengths is lost
in the process. First, let us state the next lemma.
Lemma 3.5. If N is a submonoid of (N0 , +) and q ∈ Q>0 , then qN is a finitely
generated (Puiseux) monoid satisfying that LN (x) = LqN (qx) for every x ∈ N.
Proof. It suffices to notice that, for every q ∈ Q>0 , multiplication by q yields an
isomorphism from N to qN.
Theorem 3.6. There is an atomic Puiseux monoid with full system of sets of lengths.
Proof. Because the collection of all finite subsets of Z≥2 is countable, we can list them
in a sequence, say {Sn }. Now we recursively construct a sequence of finitely generated
Puiseux monoids {Mn }, a sequence of rational numbers {xn }, and a sequence of odd
prime numbers {pn } satisfying the following conditions:
(1) xn ∈ Mn and LMn (xn ) = Sn ;
(2) the set An minimally generating Mn satisfies that d(a) = pn for every a ∈ An ;
SYSTEMS OF SETS OF LENGTHS OF PUISEUX MONOIDS
7
(3) pn > max 2xn , 2a | a ∈ An ;
(4) max An < min An+1 for every n ∈ N.
To do this, use Theorem 3.4 to find a numerical monoid M1′ minimally generated by A′1
such that L(x′1 ) = S1 for some x′1 ∈ M1′ . Then take p1 to be a prime number satisfying
that p1 > max{2x′1 , 2a′ | a′ ∈ A′1 }. Now define
p1 − 1 ′
p1 − 1 ′
M1 and x1 =
x1 .
M1 =
p1
p1
By Lemma 3.5, the element x1 satisfies condition (1). Conditions (2), (3), and (4)
follow immediately. Suppose now that we have already constructed a set of Puiseux
monoids {M1 , . . . , Mn }, a set of rational numbers {x1 , . . . , xn }, and a set of prime
numbers {p1 , . . . , pn } such that the above conditions are satisfied. By Theorem 3.4,
′
there exists a submonoid Mn+1
of (N0 , +) minimally generated by A′n+1 which contains
′
(x′n+1 ) = Sn+1 . By Lemma 3.5, we can assume that the
an element x′n+1 with LMn+1
′
elements of An+1 are large enough that max An < min A′n+1 . Now choose a prime
number pn+1 sufficiently large such that pn+1 ∤ a for any a′ ∈ A′n+1 ,
pn+1 − 1 ′
pn+1 − 1 ′
′
′
pn+1 > max 2
(3.1)
xn+1 , 2
a a ∈ An+1 ,
pn+1
pn+1
and
pn+1 − 1
(3.2)
min A′n+1 .
max An <
pn+1
Finally, set
pn+1 − 1 ′
pn+1 − 1 ′
Mn+1 =
Mn+1 and xn+1 =
xn+1 .
pn+1
pn+1
′
By Lemma 3.5, it follows that LMn+1 (xn+1 ) = LMn+1
(x′n+1 ) = Sn+1 , which is condition
(1). The fact that pn+1 ∤ a for any a ∈ A′n+1 yields condition (2). Finally, conditions
(3) and (4) follows from (3.1) and (3.2), respectively. Hence our sequence of finitely
generated Puiseux monoids {Mn } satisfies the desired conditions.
Now consider the Puiseux monoid M = hAi, where A := ∪n∈N An . In addition, let
{kn } be a sequence of positive integers such that
pn − 1
An =:
ani 1 ≤ i ≤ kn ,
pn
where ani ∈ N for every n and i = 1, . . . , kn . We verify now that M is atomic and
A(M) = A. To do so, suppose that for some m ∈ N and j ∈ {1, . . . , km } we can write
s
(3.3)
k
n
XX
pn − 1
pm − 1
cni
amj =
ani ,
pm
pn
n=1 i=1
where s ∈ N and cni ∈ N0 for every n = 1, . . . , s and i = 1, . . . , kn . If m > s, then the
pm -adic valuation of the right-hand side of (3.3) would be nonnegative, contradicting
8
F. GOTTI
that pm ∤ (pm − 1)amj . Therefore assume that m ≤ s. Now set qs = p1 . . . ps . After
multiplying (3.3) by qs and taking modulo pm , we find that
k
m
pm − 1
pm − 1 X
cmi ami .
qs amj −
qs
pm
pm
i=1
pm
(3.4)
Since gcd pm , pmpm−1 qs = 1, there exists N ∈ N0 such that
amj = Npm +
km
X
cmi ami .
i=1
The fact that pm > max 2Am ≥ am (by condition (3)) now implies that N = 0 and,
therefore, one obtains that amj = cm1 am1 + · · · + cmkm amkm . As Am generates Mm
minimally, it follows that cmj = 1 and cmi = 0 for each i 6= j. This, along with (3.3),
implies that cni = 0 for every (n, i) 6= (m, j). As a result, A(M) = A.
Now we show that every nonempty finite subset of Z≥2 is a set of lengths of M. This
amounts to verifying that LM (xℓ ) = LMℓ (xℓ ) for every ℓ ∈ N. Recall that
xℓ =
pℓ − 1 ′
xℓ , where LMℓ′ (x′ℓ ) = Sℓ .
pℓ
Suppose that for some ℓ, t ∈ N with ℓ ≤ t we have
t
(3.5)
k
n
pn − 1
pℓ − 1 ′ X X
cni
xℓ =
ani =
pℓ
p
n
n=1 i=1
k
X
n∈[t]\{ℓ}
k
n
ℓ
pn − 1 X
pℓ − 1 X
cni ani +
cℓi aℓi ,
pn i=1
pℓ i=1
where t ∈ N and cni ∈ N0 for every n = 1, . . . , t and i = 1, . . . , kn . Multiplying the
equality (3.5) by qt = p1 . . . pt and taking modulo pℓ , one can see that
(3.6)
pℓ
kℓ
pℓ − 1
pℓ − 1 X
′
qt xℓ −
qt
cℓi aℓi .
pℓ
pℓ
i=1
) implies the existence of N ′ ∈ N0 such that
Once again, the fact that gcd(pℓ , pℓp−1
ℓ
x′ℓ
′
= N pℓ +
kℓ
X
cℓi aℓi .
i=1
x′ℓ
However, as pℓ > 2xℓ ≥
(by condition (3)), we have that N ′ = 0 and, as a consequence, cℓ1 + · · · + cℓkℓ ∈ LMℓ′ (x′ℓ ). The fact that x′ℓ = cℓ1 aℓ1 + · · · + cℓkℓ aℓkℓ , along with
equality (3.5), immediately implies that cni = 0 for every n 6= ℓ and i = 1, . . . , kn . As
a result,
kℓ
t X
kn
X
X
cni =
cℓi ∈ LMℓ′ (x′ℓ ) = LMℓ (xℓ ).
n=1 i=1
i=1
SYSTEMS OF SETS OF LENGTHS OF PUISEUX MONOIDS
9
Then the inclusion LM (xℓ ) ⊆ LMℓ (xℓ ) holds. As the reverse inclusion clearly holds,
we conclude that LM (xℓ ) = LMℓ (xℓ ) for every ℓ ∈ N. Thus, each subset of Z≥2 can
be realized as the set of lengths of some element in M. Because {0} and {1} can be
obviously realized, we get that
(3.7)
L(M) ⊇ {0}, {1}, S ⊂ Z≥2 | |S| < ∞ .
Finally, it is easily seen that condition (4) ensures that M does not contain 0 as a limit
point. So it follows by Theorem 2.1 that M is a BF-monoid. Hence every set of lengths
of M is finite, which yields the reverse inclusion of (3.7).
Remark 3.7. Theorem 3.6 does not follow from Example 3.2 via transfer homomorphisms; indeed, it was proved in [17] that the only nontrivial Puiseux monoids that are
transfer Krull are those isomorphic to (N0 , +). We refer the reader to [11, Section 4]
for the definition and applications of transfer homomorphisms.
4. A Few Words on the Characterization Problem
The question of whether the arithmetical information encoded in the phenomenon
of nun-unique factorization of the ring of integers OK of a given algebraic number field
K suffices to characterize the class group C(OK ) dated back to the mid-nineteenth
century. In the 1970’s, Narkiewicz proposed the more general question of whether the
arithmetic describing the non-uniqueness of factorizations in a Krull domain could be
used to characterize its class group. For affirmative answer to this, the reader might
want to consult [10, Sections 7.1 and 7.2]. The next conjecture, also known as the
Characterization Problem, is still open. However, for an overview of results where the
statement of the conjecture holds under certain extra conditions, we refer the reader
to [11, Theorem 23].
Conjecture 4.1. Let M and M ′ be Krull monoids with respective finite abelian class
groups G and G′ each of their classes contains at least one prime divisor. Assume also
that D(G) ≥ 4. If L(M) = L(M ′ ), then M ∼
= M ′.
Because the system of sets of lengths encodes significant information about the arithmetic of factorizations of an atomic monoid, further questions in the same spirit of the
above conjecture naturally arise. For instance, we might wonder whether, in a specified family F of atomic monoids, a member is determined up to isomorphisms by its
system of sets of lengths. It was proved in [1] that the answer is negative when F is
the family of all numerical monoids. In this section, we use Theorem 3.6 to answer the
same question when F is taken to be the family of all non-finitely generated atomic
Puiseux monoids.
Lemma 4.2. [17, Proposition 3.1] The homomorphisms of Puiseux monoids are precisely those given by rational multiplication.
10
F. GOTTI
Lemma 4.3. Let P and Q be disjoint infinite sets of primes, and let MP = hap | p ∈ P i
and MQ = hbq | q ∈ Qi be Puiseux monoids such that for all p ∈ P and q ∈ Q, the
denominators d(ap ) and d(bq ) are nontrivial powers of p and q, respectively. Then
MP ≇ MQ .
Proof. Suppose, by way of contradiction, that MP ∼
= MQ . By Lemma 4.2, there exists
r ∈ Q>0 such that MP = rMQ . If q is a prime number in Q such that q ∤ n(r), then
rbq would be an element of MP such that d(rbq ) is divisible by a nontrivial power of q
and, therefore, q ∈ P . But this contradicts the fact that P ∩ Q is empty.
A Puiseux monoid M is bounded if it can be generated by a bounded set of rational
numbers; otherwise, M is said to be unbounded.
Lemma 4.4. [18, Lemma 3.4] Let M be a nontrivial Puiseux monoid. Then d(M • ) is
bounded if and only if M is finitely generated.
Theorem 4.5. There exist two non-isomorphic non-finitely generated atomic Puiseux
monoids with the same system of sets of lengths.
Proof. Consider two infinite sets P and Q consisting of prime numbers such that P ∩ Q
is empty. Now let us construct, as in the proof of Theorem 3.6, a Puiseux monoids
MP using only prime numbers in P such that MP has full system of sets of lengths.
The way we constructed the Puiseux monoid MP ensures that d(MP• ) is unbounded.
So Lemma 4.4 implies that MP is non-finitely generated. Similarly, we can construct a
non-finitely generated Puiseux monoid MQ with full system of sets of lengths by using
only prime numbers in Q. As P and Q are disjoint, Lemma 4.3 guarantees that MP
and MQ are non-isomorphic. The fact that MP and MQ both have full systems of sets
of lengths completes the proof.
5. Intersections of Systems of Sets of Lengths
In their recent paper [13], Geroldinger and Schmid studied the intersections of systems of sets of lengths of numerical monoids. In particular, they proved the following
result.
Theorem 5.1. [13, Theorem 4.1] We have
\
L(H) = {0}, {1}, {2} ,
where the intersection is taken over all numerical monoids H ⊂ N0 . More precisely,
for every s ∈ Z≥6 , we have
\
L(H) = {0}, {1}, {2} ,
|A(H)|=s
SYSTEMS OF SETS OF LENGTHS OF PUISEUX MONOIDS
11
and, for every s ∈ {2, 3, 4, 5}, we have
\
L(H) = {0}, {1}, {2}, {3} ,
|A(H)|=s
where the intersections are taken over all numerical monoids H with the given properties.
In this section, we study the intersection of the systems of sets of lengths of atomic
Puiseux monoids. We offer two versions of Theorem 5.1, namely Corollary 5.3 and
Corollary 5.7. We will also construct a completely non-half-factorial Puiseux monoid.
Proposition 5.2. Let M be a Puiseux monoid such that 0 is not a limit point of M • .
Then {2} ∈ L(M).
Proof. Let q = inf M • . As 0 is not a limit point of M, it follows that M is atomic (by
[16, Theorem 3.10]) and q 6= 0. If q ∈ M, then q must be an atom. In this case, the
minimality of q ensures that {2} = L(2q) ∈ L(M). Suppose, otherwise, that q ∈
/ M.
In this case, there must be an atom a such that q < a < 3q/2. As the sum of any
three atoms is greater than 3q, the element 2a only contains factorizations of lengths 2.
Hence {2} = L(2a) ∈ L(M).
Because the family of Puiseux monoids strictly contains the family of numerical
monoids, Theorem 5.1 guarantees that the intersection of all systems of sets of lengths
of nontrivial atomic Puiseux monoids is contained in {{0}, {1}, {2}}, the following
corollary is an immediate consequence of Proposition 5.2.
Corollary 5.3. We have
\
L(M) = {0}, {1}, {2} ,
where the intersection is taking over all nontrivial atomic Puiseux monoids M not
having 0 as a limit point.
In contrast to Proposition 5.2, we will construct an atomic Puiseux monoid that
does not contain the singleton {2} as a set of lengths.
Lemma 5.4. Let {Mn } be a sequence of atomic Puiseux monoids satisfying that
A(Mn ) ⊂ A(Mn+1 ) for each n ∈ N. Then M = ∪n∈N Mn is an atomic Puiseux monoid
and
[
A(Mn ).
A(M) =
n∈N
Proof. Because Mn is atomic for each n ∈ N, the inclusion A(Mn ) ⊂ A(Mn+1 ) implies
that Mn ⊂ Mn+1 . As a consequence, M is a Puiseux monoid. Let A = ∪n∈N A(Mn ). It
is clear that A generates M. Therefore A(M) ⊂ A. On the other hand, take a ∈ A(Mn )
for some n ∈ N. Since A(M) ⊂ A, it follows that ZM (a) ⊆ ZMℓ (a) for some ℓ > n.
Now the fact that a ∈ A(Mn ) ⊂ A(Mℓ ) guarantees that a has only one factorization
12
F. GOTTI
in M, which has length 1. Thus, a ∈ A(M). Because A(M) = A, we finally conclude
that M is atomic.
For S ⊆ Q>0 , we define
dp (S) = {prime p | p divides d(s) for some s ∈ S}.
Theorem 5.5. There exists an atomic Puiseux monoid M such that {2} ∈
/ L(M).
Proof. We will construct inductively a sequence of positive rational numbers {an } and
an increasing sequence of natural numbers {kn } so that each set An = {a1 , a2 , . . . , akn }
minimally generates the Puiseux monoid Mn = hAn i. Take (a1 , k1 ) = (1, 1) and
(a2 , k2 ) = (2/3, 2). Clearly, A(M1 ) = {1} = A1 and A(M2 ) = {1, 2/3} = A2 . Now
suppose that we have already found k1 , . . . , kn and a1 , . . . , akn satisfying the desired
conditions. Let
kn + 1
(si1 , si2 ) 1 ≤ i ≤
2
be an enumeration of the set {(a, b) ∈ [1, kn ] × [1, kn ] | a ≤ b}. Now let us choose kn2+1
/ dp (Mn ) and pi ∤ n(asi1 + asi2 ) for any
odd prime numbers p1 , . . . , p(kn +1) such that pi ∈
2
i ∈ {1, . . . , kn2+1 }. Finally, define
akn +i =
asi1 + asi2
pi
for every i ∈ 1, . . . , kn2+1 , and take kn+1 = kn + kn2+1 .
We verify now that An+1 := {a1 , a2 , . . . , akn+1 } minimally generates the Puiseux
monoid Mn+1 := hAn+1 i. Suppose, by contradiction, that for some j ∈ {1, . . . , kn+1},
kn+1
(5.1)
aj =
X
ci ai ,
i=1
where c1 , . . . , ckn+1 ∈ N0 and cj = 0. Notice that if j > kn , then we would obtain that
the pt -valuation (for t = j − kn ) of the right-hand side of (5.1) is nonnegative while the
fact that p ∤ n(ast1 + ast2 ) implies that the pt -valuation of the left-hand side is negative,
a contradiction. Thus, assume that j ≤ kn . Because the set
kn + 1
dp (Mn ) ∩ pi 1 ≤ i ≤
2
is empty, for i ∈ 1, . . . , kn2+1
the prime number pi divides the denominator of
only one of the am ’s on the right-hand side of (5.1), namely akn +i . Therefore, after
applying the pi -adic valuation map to both sides
of (5.1), we find that pi | ckn +i . After
simplifying all the possible pi ’s (1 ≤ i ≤ kn2+1 ) in the denominators of the right-hand
side of (5.1), it becomes a sum of elements of An containing at least two summands,
which contradicts the fact that An generates Mn minimally.
SYSTEMS OF SETS OF LENGTHS OF PUISEUX MONOIDS
13
Now define
M=
[
Mn .
n∈N
Because A(Mn ) = An and An ⊂ An+1 for each n ∈ N, Lemma 5.4 implies that M
is an atomic Puiseux monoid with A(M) = {an | n ∈ N}. Finally, we verify that
{2} ∈
/ L(M). It suffices to show that |L(x)| > 1 for all x ∈ M such that 2 ∈ L(x).
Take an element x ∈ M such that 2 ∈ L(x), and choose two indices i, j ∈ N such that
x = ai + aj . Let m be the minimum natural number such that ai , aj ∈ Am . By the way
we constructed the sequence {An }, there exist a ∈ Am+1 and an odd prime number p
such that
ai + aj
a=
.
p
Because a ∈ A(M), the element x contains a factorization of length p, namely pa.
Since p > 2, it follows that |L(x)| ≥ |{2, p}| = {2}, as desired.
Recall that an atomic monoid M is said to be completely non-half-factorial if for
each x ∈ M • \ A(M) one has |L(x)| > 1. It is not hard to argue that the monoid
provided by Theorem 5.5 is completely non-half-factorial. Indeed, it satisfies condition
(5.2), which is stronger than completely non-half-factoriality.
Corollary 5.6. There exists an atomic Puiseux monoid M such that
(5.2)
{|L(x)| | x ∈ M} = {1, ∞}.
Proof. Let M be the Puiseux monoid constructed in Theorem 5.5. Take x ∈ M such
that |L(x)| > 1. This means that x is neither zero nor an atom. Let a1 and a2 be two
atoms of M such that y = a1 + a2 divides x in M. It follows by the construction of M
in the proof of Theorem 5.5 that |L(y)| = ∞. The fact that y divides x in M implies
now that |L(x)| = ∞, which concludes the proof.
Corollary 5.7. We have
\
L(M) = {0}, {1} ,
where the intersection is taking over all nontrivial atomic Puiseux monoids.
Proof. This is an immediate consequence of Theorem 5.1 and Theorem 5.5.
6. Relation with the Goldbach’s Conjecture
We conclude this paper providing evidence that explicit computations of sets of
lengths is an extremely hard problem even in particular cases of atomic Puiseux
monoids. Specifically, we shall prove that finding L(M) for M = hp | p is prime i
is as hard as the famous longstanding Goldbach’s conjecture.
14
F. GOTTI
Conjecture 6.1 (Goldbach’s conjecture). Every even n ≥ 4 can be expressed as the
sum of two prime numbers.
The following weaker version of the Goldbach’s conjecture, called the Goldbach’s
weak conjecture, was proved in 2013 by Helfgott [20].
Theorem 6.2. Every odd n ≥ 7 can be written as the sum of three prime numbers.
We call the Puiseux monoid M = h1/p | p is prime i the elementary Puiseux monoid.
It was proved in [18] that M is hereditarily atomic (i.e., every submonoid of M is
atomic). On the other hand, it follows immediately that M is not a BF-monoid. For
every n ∈ N and k = 1, . . . , n, set
Sn,k := {(a1 , . . . , ak ) ∈ Nk | a1 + · · · + ak = n}.
In the following proposition we characterize the sets of lengths of the elements of M ∩N
in terms of the Sn,k ’s.
Proposition 6.3. Let M be the elementary Puiseux monoid. Then for each n ∈ M ∩N,
we have that
k
n X
[
(6.1)
ai pi (a1 , . . . , ak ) ∈ Sn,k and p1 , . . . , pk are primes .
L(n) =
k=1
i=1
Proof. Take n ∈ M ∩ N, and denote the right-hand side of (6.1) by R. Note that
assuming the extra condition p1 < · · · < pk does not change R. Suppose that ℓ ∈ L(n).
Then there exist k ∈ N and distinct prime numbers p1 , . . . , pk such that
1
1
n = c1 + · · · + ck ,
(6.2)
p1
pk
where the right-hand side of (6.2) has length ℓ = c1 +· · ·+ck when seen as a factorization
of n. Applying the pi -valuation map in both sides of (6.2), we find that pi | ci for
i = 1, . . . , k. Now setting ai := ci /pi for i = 1, . . . , k, we get that a1 +· · ·+ak ∈ Sn,k and,
therefore, ℓ = a1 p1 + · · · + ak pk ∈ R. Thus, L(n) ⊆ R. Conversely, if (a1 , . . . , ak ) ∈ Sn,k
and p1 , . . . , pk are distinct prime numbers, then it immediately follows that
z :
k
X
i=1
(ai pi )
1
pi
is a factorization of n = a1 + · · · + ak ∈ M ∩ N with |z| = a1 p1 + · · · + ak pk . Therefore
R ⊆ L(n), which completes the proof.
A natural number is said to be a Goldbach’s number if it can be expressed as the sum
of two prime numbers (not necessarily distinct). Let G denote the set of Goldbach’s
numbers. If M is the elementary primary Puiseux monoid, then an explicit description
of L(2) is as hard as the Goldbach’s conjecture.
SYSTEMS OF SETS OF LENGTHS OF PUISEUX MONOIDS
15
Theorem 6.4. Let M be the elementary primary Puiseux monoid. Then
(1) L(2) = G.
(2) L(3) = Z≥7
Proof. First, we verify part (1). To see that L(2) ⊆ G, take z ∈ Z(2). If z consists of
copies of only one atom, say 1/pi , then |z| = 2pi ∈ G. Otherwise, by Proposition 6.3, z
is the formal sum of copies of two atoms, that is, 2 = a1 (1/p1 ) + a2 (1/p2) for distinct
prime numbers p1 and p2 and positive coefficients a1 and a2 . In this case, p1 | a1 and
p2 | a2 , which force a1 = p1 and a2 = p2 . As a result, |z| = a1 + a2 = p1 + p2 ∈ G. On
the other hand, for each Goldbach number p1 + p2 , the expression p1 (1/p1 ) + p2 (1/p2 )
yields an element of Z(2) of length p1 + p2 , which implies that G ⊆ L(2). Thus, (1)
follows. The proof of part (2) uses Theorem 6.2; however it follows similarly to the
proof of part (1) and so it is left to the reader.
7. Acknowledgements
While working on this paper, the author was supported by the NSF-AGEP fellowship. The author would like to thank Alfred Geroldinger for helpful suggestions.
References
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Theory and Additive Group Theory (A. Geroldinger and I. Ruzsa, eds.), Advanced Courses in
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Analytic Theory, Pure and Applied Mathematics Vol. 278, Chapman & Hall/CRC, Boca Raton,
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(2017) 188–198.
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Department of Mathematics, UC Berkeley, Berkeley, CA 94720
E-mail address: felixgotti@berkeley.edu
| 0 |
Tight Approximation Bounds for the Seminar
Assignment Problem ⋆
Amotz Bar-Noy and George Rabanca
arXiv:1610.04785v1 [cs.DS] 15 Oct 2016
Department of Computer Science, Graduate Center, CUNY, New York, USA
Abstract. The seminar assignment problem is a variant of the generalized assignment problem in which items have unit size and the amount of
space allowed in each bin is restricted to an arbitrary set of values. The
problem has been shown to be NP-complete and to not admit a PTAS.
However, the only constant factor approximation algorithm known to
date is randomized and it is not guaranteed to always produce a feasible
solution.
In this paper we show that a natural greedy algorithm outputs a solution with value within a factor of (1 − e−1 ) of the optimal, and that
unless N P ⊆ DT IM E(nlog log n ), this is the best approximation guarantee achievable by any polynomial time algorithm.
Keywords: general assignment · budgeted maximum coverage · seminar
assignment problem
1
Introduction
In the Seminar Assignment problem (SAP) introduced in [8] one is
given a set of seminars (or bins) B, a set of students (or items) I, and for
each seminar b a set of integers Kb specifying the allowable number of
students that can be assigned to the seminar. Unless otherwise specified,
we assume that 0 ∈ Kb for any b ∈ B. For each student i and seminar
b ∈ B let p(i, b) ∈ R represent the profit generated from assigning student
i to seminar b. A seminar assignment is a function A : J → B where
J ⊆ I and we say that the assignment is feasible if |A−1 (b)|∈ Kb for all
b ∈ B, where A−1 is the pre-image of A. The goal is to find a feasible
seminar assignment A that maximizes the total profit:
X
p(i, A(i)).
p(A) =
i∈J
The problem has been introduced in [8] in a slightly less general version.
In the original version, for each b ∈ B the set Kb equals to {0}∪{lb , ..., ub }
for some lower and upper bounds lb , ub ∈ N. The more general setting
⋆
This work is supported in part by grants from NSF CNS 1302563, by Navy N0001416-1-2151, by NSF CNS 1035736, by NSF CNS 1219064. Any opinions, findings, and
conclusions or recommendations expressed here are those of the authors and do not
necessarily reflect the views of sponsors.
considered in this paper can be useful for example when a seminar doesn’t
just require a minimum number of students and has a fixed capacity,
but in addition requires students to work in pairs and therefore would
allow only an even number of students to be registered. In addition, this
generalization also simplifies notation.
SAP is a variant of the classic General Assignment problem (GAP)
in which one is given m bins with capacity B1 , ..., Bm and n items. Each
item i has size s(i, b) in bin b and yields profit p(i, b). The goal is to find
a packing of the items into the bins that maximizes total profit, subject
to the constraint that no bin is overfilled. A GAP instance with a single
bin is equivalent to the knapsack problem, and a GAP instance with
unit profit can be interpreted as a decision version of the bin packing
problem: can all items be packed in the m bins?
SAP is also related to the Maximum Coverage problem (MC). In the
classic version of the MC problem one is given a collection of sets S =
{S1 , ..., Sm } and a budget B. The goal is to select a subcollection S ′ ⊆ S
with cardinality less than or equal to B such that |∪S∈S ′ S| is maximized.
The algorithms with the best approximation ratio for both MC and GAP
are greedy algorithms and the approximation bounds have been proved
with similar techniques. In this paper we show how to extend these analysis techniques to SAP.
Related Work. In [8] the authors show that SAP is NP-complete
even when Kb = {0, 3} for all b ∈ B and p(i, b) ∈ {0, 1} for any i ∈ I.
Moreover, they show that SAP does not admit a PTAS by providing
a gap-preserving reduction from the 3-bounded 3-dimensional matching
problem. In [1] the authors investigate the approximability of the problem and provide a randomized algorithm which they claim outputs a
solution that in expectation has value at least 1/3.93 of the optimal.
In [2] this result is revised and the authors show that for any c ≥ 2,
their randomized algorithm outputs a feasible solution with probability
c−1
e−1
at least 1 − min{ 1c , e cc } and has an approximation ratio of (2c−1)·e
.
The GAP is well studied in the literature, with [3] and [9] surveying the
existing algorithms and heuristics for multiple variations of the problem.
In [11] the authors provide a 2-approximation algorithm for the problem and in [4] it is shown that any α-approximation algorithm to the
knapsack problem can be transformed into a (1 + α)-approximation algorithm for GAP. In [6] tight bounds for the GAP are given showing
that no polynomial time algorithm can guarantee a solution within a
factor better than (1 − e−1 ), unless P = N P , and providing an LP-based
approximation which for any ǫ > 0 outputs a solution with profit within
a (1 − e−1 − ǫ) factor of the optimal solution value.
The GAP with minimum quantities, in which a bin cannot be used if
it is not packed at least above a certain threshold, is introduced in [8].
Because items have arbitrary size, it is easy to see that when a single bin
is given and the lower bound threshold equals the bin capacity, finding
a feasible solution with profit greater than zero is equivalent to solving
Subset Sum. Therefore, in its most general case the problem cannot be
approximated in polynomial time, unless P = N P .
In [10] and [5] the authors study the problem of maximizing a nondecreasing submodular function f satisfying f (∅) = 0 under a cardinality constraint. They show that a simple greedy algorithm achieves an
approximation factor of (1 − e−1 ) which is the best possible under standard assumptions. Vohra and Hall note that the classic version of the
maximum coverage problem belongs to this class of problems [13]. When
each set Si in the MC problem is associated with a cost c(Si ) the Budgeted Maximum Coverage problem asks to find a collection of sets S ′
covering theP
maximum number of elements under the (knapsack) constraint that Si ∈S ′ c(Si ) ≤ B for some budget B ∈ R. In [7] the authors
show that the greedy algorithm combined with a partial enumeration of
all solutions with small cardinality also achieves a (1 − e−1 ) approximation guarantee, and provide matching lower bounds which hold even in
the setting of the classic MC problem (when all sets have unit cost). In
[12] Sviridenko generalizes the algorithm and proof technique to show
that maximizing any monotone submodular function under a knapsack
constraint can be approximated within (1 − e−1 ) as well.
Contributions. In Section 2, by a reduction from the Maximum Coverage problem, we show that there exists no polynomial time algorithm
that guarantees an approximation factor larger than (1 − e−1 ), unless
N P ⊆ DT IM E(nlog log n ). In Section 4 we present a greedy algorithm
that outputs a solution that has profit at least 21 · (1 − e−1 ) of the optimal solution. The algorithm is based on the observation that when the
required number of students in each seminar is fixed, the problem is solvable in polynomial time. Finally, in Section 5 we show how this algorithm
can be improved to guarantee an approximation bound of (1 − e−1 ).
2
Hardness of Approximation
In this section we show that the problem is hard to approximate within
a factor of (1−e−1 +ǫ), ∀ǫ > 0, even for the case when for each b ∈ B the
set Kb equals {0, n} for some integer n, and the profit for assigning any
student to any seminar is either 0 or 1. We prove this result by showing
that such restricted instances of SAP are as hard to approximate as the
Maximum Coverage problem defined below.
Definition 1. Given a collection of sets S = {S1 , ..., Sm } and an integer
k, the Maximum Coverage (MC) problem is to find a collection of sets
S ′ ⊆ S such that |S ′ |≤ k and the union of the sets in S ′ is maximized.
In [7] it is shown that the MC problem is hard to approximate within
a factor of (1 − e−1 + ǫ), unless N P ⊆ DT IM E(nlog log n ). We use this
result to prove the following:
Theorem 1. For any ǫ > 0 the SAP is hard to approximate within a
factor of (1 − e−1 + ǫ) unless N P ⊆ DT IM E(nlog log n ).
Proof. To prove the theorem we create a SAP instance for any given MC
instance and show that from any solution of the SAP instance we can
create a solution for the MC instance with at least equal value, and that
the optimal solution of the SAP instance has value at least equal to the
optimal solution of the MC instance. Therefore, an α-approximation algorithm for SAP can be transformed into an α-approximation algorithm
for MC.
Given a MC instance, let U = ∪S∈S S and n = |U |. For each set S ∈ S let
bS be a seminar with the allowable number of students Kb = {0, n}, and
for each element e ∈ U let ie be a student in I. The profit of a student ie
assigned to a seminar bS is 1 if the element e belongs to the set S and 0
otherwise. In addition, let d1 , ..., dn∗(k−1) be dummy students that have
profit 0 for any seminar.
We first show that any feasible assignment A corresponds to a valid
solution to the given MC instance. Since every seminar requires exactly
n students and there are exactly k · n students available, clearly at most
k seminars can be assigned students in any feasible assignment. Let S ′ =
{S ∈ S : A(bS ) > 0}. It is easy to see that the number of elements in
∪S∈S ′ S is at least equal to the profit p(A) since a student ie has profit
1 for a seminar bS only if the set S covers element e.
It remains to show that for any solution to the MC instance there exists
a solution to the corresponding SAP instance with the same value. Fix
a collection of sets S ′ ⊆ S with |S ′ |≤ k. For every e ∈ ∪S∈S ′ S let Se be
a set in S ′ that contains e and let A(ie ) = bSe . Then, assign additional
dummy students to any seminar with at least one student to reach the
required n students per seminar. Clearly, the profit of the assignment A
is equal to the number of elements covered by the collection S ′ , which
proves the theorem.
3
Seminars of Fixed Size
In this section we show that when the allowable number of students that
can be assigned to any seminar b is a set K = {0, kb } for some integer
kb , SAP can be approximated within a factor of (1 − e−1 ) in polynomial
time. This introduces some of the techniques used in the general case in
a simpler setting.
For an instance of the SAP, a seminar selection is a function S : B → N
with
P the property that S(b) ∈ Kb for any b ∈ B. We say that S is feasible
if
b∈B S(b) ≤ |I|. In other words, a seminar selection is a function
that maps each seminar to the number of students to be assigned to
it. A seminar selection S corresponds to an assignment A if for any
seminar b the number of students assigned by A to b is S(b). We slightly
abuse notations and denote by p(S) the maximum profit over all seminar
assignments corresponding to the seminar selection S; we call p(S) the
profit of S. In the remainder of this paper for a graph G = (V, E) we
denote the subgraph induced by the vertices of X ⊆ V by G[X].
Definition 2. Given a SAP instance let Vb = {vb,1 , ..., vb,kb } for every
b ∈ B and let V = ∪b∈B Vb . The bipartite representation of the instance is the complete bipartite graph G = (V ∪ I, E) with edge weights
ω(vb , i) = p(i, b) for every vb ∈ Vb . The bipartite representation of a
seminar selection S is the graph G[VS ∪ I] where VS = ∪b∈B VS,b and
VS,b = {vb,1 , ..., vb,S(b) } for every b ∈ B.
Lemma 1. For any SAP instance and any feasible seminar selection
S, p(S) is equal to the value of the maximum weight matching in the
bipartite representation of S.
Proof. Let GS = (VS ∪ I, E) be the bipartite representation of S. First
observe that any matching M of GS that matches all the vertices of
VS can be interpreted as an assignment AM of equal value by setting
AM (i) = b whenever vertex i ∈ I is matched by M to a vertex in VS,b .
Since GS is complete and has non-negative edge weights, there exists a
maximum weight matching that matches all the vertices of VS .
Similarly, any feasible assignment for the SAP instance can be interpreted as a matching MA of equal value, which proves the lemma.
Definition 3. For a given finite set A, a set function f : 2A → R is
submodular if for any X, Y ⊆ A it holds that:
f (X) + f (Y ) ≥ f (X ∪ Y ) + f (X ∩ Y ).
Sviridenko shows that certain submodular functions can be maximized
under knapsack constraints, which will be useful in proving Theorem 3:
Theorem 2 ([12]). Given a finite set A, a submodular, non-decreasing,
non-negative, polynomially computable function f : 2A → R, a budget
L ≥ 0, and costs ca ≥ 0, ∀a ∈ A, the following optimization problem is
approximable within a factor of (1 − e−1 ) in polynomial time:
)
(
X
cx ≤ L
max f (X) :
X⊆A
x∈X
We relate now the value of a maximum weight matching in a bipartite
graph to the notion of submodularity.
Definition 4. For an edge weighted bipartite graph G = (A ∪ B, E), the
partial maximum weight matching function f : 2A → R maps any set
S ⊆ A to the value of the maximum weight matching in G[S ∪ B].
Lemma 2. Let f be the partial maximum weight matching function for
a bipartite graph G = (A ∪ B, E) with non negative edge weights. Then
f is submodular.
Proof. Fix two sets X, Y ⊆ A and let M∩ and M∪ be two matchings for
the graphs G[(X ∩ Y ) ∪ B] and G[(X ∪ Y ) ∪ B] respectively. To prove
the lemma it is enough to show that it is possible to partition the edges
in M∩ and M∪ into two disjoint matchings MX and MY for the graphs
G[X ∪ B] and G[Y ∪ B] respectively.
The edges of M∩ and M∪ form a collection of alternating paths and
cycles. Let C denote this collection and observe that no cycle of C contains
vertices from X \ Y or Y \ X. This holds because M∩ does not match
those vertices.
Let PX be the set of paths in C with at least one vertex in X \ Y and
let PY be the set of paths in C with at least one vertex in Y \ X. Two
such paths are depicted in Fig. 1.
Claim 1. PX ∩ PY = ∅.
Proof of claim: Assume by contradiction that there exists a path P ∈
PX ∩ PY . Let x be a vertex in X \ Y on path P and similarly let y be
a vertex in Y \ X on path P . Observe that since neither x nor y belong
to X ∩ Y they do not belong to the matching M∩ by definition, and
therefore they are the endpoints of the path P . Moreover, since both x
and y are in A, the path P has even length and since it is an alternating
path, either the first or last edge belongs to M∩ . Therefore M∩ matches
either x or y contradicting its definition.
PX
PY
X
Y
Fig. 1: MX∪Y matches each vertex in X ∪ Y to the vertex directly
above it. MX∩Y is depicted with contiguous segments, MX with
dotted segments and MY with dashed segments. Two alternating
paths of P are shown in light gray.
For a set of paths P we let E(P) = {e ∈ P : P ∈ P}. Moreover, let
MX = (E(PX ) ∩ M∪ ) ∪ (E(C \ PX ) ∩ M∩ )
and
MY = (E(PX ) ∩ M∩ ) ∪ (E(C \ PX ) ∩ M∪ ).
It is clear that MX ∪ MY = M∩ ∪ M∪ and MX ∩ MY = M∩ ∩ M∪ .
To prove the theorem it remains to show that MX and MY are valid
matchings for G[X ∪ B] and G[Y ∪ B] respectively. To see that MX is a
valid matchings for G[X ∪ B] observe first that that no vertex of Y \ X is
matched by MX since PX does not intersect Y \ X by Claim 1, and M∩
does not intersect Y \ X by definition. Therefore, MX only uses vertices
of X ∪ B. Second observe that every vertex x ∈ X is matched by at most
one edge of MX since otherwise x belongs to either two edges of M∪ or
two edges of M∩ , contradicting the definition. This proves that MX is a
valid matching for G[X ∪ B]; showing that MY is a valid matchings for
G[Y ∪ B] is similar.
Theorem 3. Any instance of SAP in which |Kb |≤ 2 for all b ∈ B can
be approximated in polynomial time to a factor of (1 − e−1 ).
Proof. Fix a SAP instance and for any X ⊆ B let SX be the seminar
selection which allocates kb students to any seminar in S and 0 students
to any seminar in B \ S. Moreover, let G be the bipartite representation
of the SAP instance and f be the partial maximum weight matching
function for graph G. Denote by G[VX ∪ I] the bipartite representation
of SX and let g(X) = f (VX ). Since f is submodular by Lemma 2, it is
easy to see that g is submodular as well. Assume by contradiction that
there exist sets X, Y ⊆ B such that the submodularity condition for g
doesn’t hold:
g(X) + g(Y ) < g(X ∪ Y ) + g(X ∩ Y ).
(1)
Therefore, by definition of g we have
f (VX ) + f (VY ) < f (VX ∪ VY ) + g(VX ∩ VY ),
contradicting the submodularity of f proven in Lemma 2.
Clearly g is also monotone, non-negative and polynomially computable.
Let cb = kbP
, ∀b ∈ B, let L = |I|, and observe that SX is feasible if
and only if
x∈X cx ≤ L. Moreover, by Lemma 1 and the definition
of g, g(X) = p(SX ) whenever the seminar selection SX is feasible and
therefore the proof follows from Theorem 2.
4
A Constant Factor Greedy Algorithm
The algorithm presented in this section sequentially increments the number of students allocated to each seminar in a greedy fashion. It is similar
in nature to the greedy algorithm of [7] and [12] but the details of the
approximation guarantee proof are different. In the rest of this section
we denote by AS an optimal assignment for the seminar selection S. Remember that Lemma 1 shows that given feasible seminar selection S, an
optimal seminar assignment AS can be found in polynomial time.
We say that a seminar selection T is greater than a selection S (denoted
by T ≻ S) if T (b) ≥ S(b), ∀b ∈ B, and there exists b ∈ B s.t. T (b) >
S(b). The cost of a seminar selection S is denoted by c(S) and equals
P
b∈B S(b). When T ≻ S we define the marginal cost of T relative to S
as the difference between the cost of T and the cost of S:
cS (T ) = c(T ) − c(S)
Similarly, we define pS (T ) = p(T )−p(S), the marginal profit of T relative
to S. We say that T is an incrementing selection for a seminar selection
S if T ≻ S and there exists a single seminar for which the selection T
allocates more students than selection S; more precisely, the cardinality
of the set {b ∈ B : T (b) > S(b)} is 1. For a selection S we denote the set
of incrementing seminar selections that are feasible by inc(S).
We are now ready to present our algorithm:
Greedy
1. S0 = initial seminar selection;
2. i = 0;
3. While inc(Si ) 6= ∅:
(a) Si+1 ← arg maxS ′ ∈inc(Si ) (p(S ′ ) − p(Si ))/(c(S ′ ) − c(Si ));
(b) i ← i + 1
4. A1 ← ASi ;
5. A2 ← maximum assignment to any single seminar b for which
S0 (b) = 0;
6. Return max A1 , A2 ;
In this section we analyze the algorithm starting from an empty initial
seminar selection. In the following section we show that by running the
algorithm repeatedly with different initial seminar selections, the approximation guarantee can be improved.
Observe that the cardinality of inc(S) is never greater than |B|·|I| and is
therefore polynomial in the size of the input. Thus, using the maximum
weight matching reduction from the proof of Lemma 1, step 3(a) of the
algorithm can be performed efficiently.
Definition 5. For a seminar selection S and a tuple (b, kb ) with b ∈ B
and kb ∈ N, let S ⊕ (b, kb ) denote the seminar selection S ′ with S ′ (b) =
max{kb , S(b)} and S ′ (b′ ) = S(b′ ) for any b′ ∈ B, b′ 6= b.
Lemma 3. For any feasible seminar selections S and T , if for every
seminar b ∈ B the seminar selection S ⊕ (b, T (b)) is feasible, then it
holds that:
X
[p(S ⊕ (b, T (b))) − p(S)] ≥ p(T ) − p(S).
b∈B
Proof. For a fixed SAP instance let G be its bipartite representation and
let G[VS ∪ I] and G[VT ∪ I] be the bipartite representations of S and T
respectively. Moreover, let MS and MT be two maximum weight matchings in G[VS ∪ I] and G[VT ∪ I] respectively. Remember that according
to Lemma 1 it holds that p(S) = ω(MS ) and p(T ) = ω(MT ). To prove
the lemma we create matchings M = {Mb }b∈B for the bipartite representations of assignments p(S ⊕ (b, T (b)), such that each edge of MT is
used in exactly one of the matchings in M and each edge of MS is used
in exactly |B|−1 of the matchings in M.
Let C be the collection of isolated components formed by the union of
the edges of MS and MT . Since both MS and MT are matchings in G,
each element of C is a path or cycle in G. For every b ∈ B let Pb = {P ∈
C : V (P ) ∩ Vb ∩ (V (MT ) \ V (MS )) 6= ∅}, where V (P ) denotes the vertices
of component P (Fig. 2).
Claim 2. For any a 6= b ∈ B, Pa ∩ Pb = ∅.
Proof of claim: To prove the claim, assume that there exist P ∈ Pa ∩ Pb
for some a 6= b ∈ B. Then by definition there exist va ∈ Va and vb ∈ Vb
such that va , vb ∈ V (P ) and va , vb ∈
/ V (MS ) and therefore va and vb are
the endpoints of the alternating path P . Since neither of the endpoints of
the path belong to MS , P must have an odd number of edges. However,
because both endpoints of P belong to the same partition of the bipartite
graph G, the path P must have an even number of edges, hence the claim
holds by contradiction.
P1
b1
b1
P2
b2
b2
b3
b3
P3
(a)
(b)
(c)
(d)
Fig. 2: An example with 3 seminars, b1 , b2 , b3 . (a) Two assignments
MS (dashed edges) and MT (dotted edges); the three alternating
paths formed by MS ∪ MT (light gray). q(P1 ) = b1 because it
only intersects vertices from Vb1 ; q(P2 ) = b1 because P2 contains
a vertex V (MT ) \ V (MS ) that is in Vb1 ; r(P3 ) = b2 . (b), (c) and
(d) assignments for seminar selections S ⊕ (b1 , 3), S ⊕ (b2 , 2) and
S ⊕ (b3 , 2) combining edges of MS and MT .
Let q : C → B be a map of the isolated components to the seminars with
the following properties:
1. q(P ) ∈ {b ∈ B : V (P ) ∩ Vb 6= ∅};
2. if P ∈ Pb for any b ∈ B, q(P ) = b.
Since Pb are disjoint by the previous claim and since for any seminar b
it holds by definition that V (P ) ∩ Vb 6= ∅ whenever P ∈ Pb , it is clear
that such a mapping q exists.
For every b ∈ B let Mb be the matching of G that uses all the edges of
MT from the alternating paths P ∈ C mapped by q to the seminar b, and
all the edges of MS from the paths P ∈ C mapped by q to some other
seminar:
Mb = [MT ∩ E(q −1 (b))] ∪ [MS ∩ (E(C) \ E(q −1 (b)))].
Observe that any edge of MT belongs to at least one matching Mb for
some b ∈ B and that any edge of MS belongs to all but one of the
matchings Mb . Therefore,
X
ω(Mb ) ≥ ω(MT ) + (|B|−1) · ω(MS ).
b∈B
Moreover, observe that for each b ∈ B, Mb is a matching in the bipartite
representation of the seminar selection S ⊕ (b, T (b)). Therefore p(S ⊕
(b, T (b))) = ω(Mb ) and the lemma follows.
Lemma 4. Let S and T be two seminar selections such that S ⊕(b, T (b))
is feasible for every b ∈ B. Let S ∗ = arg maxS ′ ∈inc(S) (p(S ′ )−p(S))/(c(S ′ )−
c(S)). Then it holds that:
p(S ∗ ) − p(S)
p(T ) − p(S)
≥
.
c(S ∗ ) − c(S)
c(T )
Proof. By Lemma 3 we have that
X
[p(S ⊕ (b, T (b))) − p(S)] ≥ p(T ) − p(S).
(2)
b∈B
P
P
Since b∈B [c(S ⊕ (b, T (b))) − c(S)] ≤ b∈B T (b) = c(T ), inequality (2)
implies that
P
[p(S ⊕ (b, T (b))) − p(S)]
p(T ) − p(S)
Pb∈B
≥
.
[c(S
⊕
(b,
T
(b)))
−
c(S)]
c(T )
b∈B
(3)
Then, there exists at least one seminar b∗ ∈ B such that
p(T ) − p(S)
p(S ⊕ (b∗ , T (b∗ ))) − p(S)
≥
.
c(S ⊕ (b∗ , T (b∗ ))) − c(S)
c(T )
(4)
Since S ⊕ (b∗ , T (b∗ ))) is clearly in inc(S) the lemma follows directly from
Eq. (4) and the definition of S ∗ .
Lemma 5. Let T be a feasible seminar selection and let r ∈ N be such
that Si ⊕ (b, T (b)) is feasible for every i < r and b ∈ B. Then for each
i ≤ r the following holds:
"
#
c(Sk+1 ) − c(Sk )
1−
p(Si ) − p(S0 ) ≥ 1 −
· p(T ) − p(S0 ) .
c(T )
k=0
i−1
Y
Proof. We prove the lemma by induction on the iterations i. By the
definition of the algorithm, S1 is the seminar selection with maximum
marginal density in inc(S0 ), and thus Lemma 4 shows that the inequality
holds for i = 1. Suppose that the lemma holds for iterations 1, ..., i. We
show that it also holds for iteration i + 1. For ease of exposition, for the
c(Si+1 )−c(Si )
.
remainder of this proof let αi =
c(T )
p(Si+1 ) − p(S0 ) = p(Si ) − p(S0 ) + p(Si+1 ) − p(Si )
≥ p(Si ) − p(S0 ) + αi · (p(T ) − p(Si ))
= (1 − αi )p(Si ) + αi · p(T ) − p(S0 )
!
i−1
Y
(1 − αk ) (p(T ) − p(S0 ))
≥ (1 − αi ) · 1 −
k=0
+ (1 − αi ) · p(S0 ) + αi · p(T ) − p(S0 )
!
i
Y
(1 − αk ) (p(T ) − p(S0 ))
1 − αi −
=
k=0
+ αi · (p(T ) − p(S0 ))
!
i
Y
(1 − αk ) (p(T ) − p(S0 )).
1−
=
k=0
Where the first inequality follows from Lemma 4 and the second inequality follows from the induction hypothesis.
Theorem 4. When
S0 is the empty assignment the
is a 21 · 1 − e−1 approximation for SAP.
Greedy algorithm
Proof. Let OP T be the seminar selection of a fixed optimal assignment
solution for the given SAP instance. Let b∗ ∈ B be the seminar that
is allocated the most students in OP T and let OP T ′ be the seminar
selection for which OP T ′ (b∗ ) = 0 and OP T ′ (b) = OP T (b) for any b 6=
b∗ ∈ B. Let r be the first iteration of the algorithm for which c(Sr ) >
c(OP T ′ ). Clearly, Si ⊕ (b, OP T (b)) is feasible for every i < r and b ∈ B.
Since p(S0 ) = 0, by applying Lemma 5 to iteration r we obtain:
"
#
r−1
Y
c(Sk+1 ) − c(Sk )
1−
p(Sr ) ≥ 1 −
· p(OP T ′ )
c(OP T ′ )
k=0
"
#
r−1
Y
c(Sk+1 ) − c(Sk )
· p(OP T ′ ).
(5)
1−
≥ 1−
c(Sr )
k=0
P
Observe that c(Sr ) = P r−1
k=0 c(Sk+1 ) − c(Sk ) and that for any real numbers a0 , ..., ar−1 with r−1
k=0 ak = A it holds that:
r−1
Y
k=0
ak
1−
≤
A
1
1−
r
r
< e−1 .
(6)
Therefore Eq. (5) implies p(Sr ) > (1 − e−1 ) · p(OP T ′ ). Since the profit
of A2 is at least p(b∗ , OP T (b∗ )) it holds that
A1 + A2 > (1 − e−1 ) · p(OP T ′ ) + p(b∗ , OP T (b∗ ))
≥ (1 − e−1 ) · p(OP T )
and therefore either A1 or A2 has profit at least
1
2
· (1 − e−1 )p(OP T ).
5
Improving the Approximation
In this section we show that the algorithm can be improved by starting
the greedy algorithm not from an empty seminar selection, but from
a seminar selection that is part of the optimal solution. The improved
algorithm is less efficient but achieves the optimal approximation ratio
of (1 − e−1 ). Let Aopt be an optimal seminar assignment and for any
b ∈ B let popt (b) be the profit obtained in this assignment from seminar
b:
X
p(i, b).
popt (b) =
−1
i∈Aopt (b)
P
Clearly, the profit of the optimal solution is
b∈B popt (b). W.l.o.g, let
b1 , b2 , b3 be the three seminars of the optimal solution with highest profit
and let S ∗ be a seminar selection such that S ∗ (b) = OP T (b) if b ∈
{b1 , b2 , b3 }, and S ∗ (b) = 0 otherwise.
Theorem 5. When S0 = S ∗ the
algorithm is a 1 − e−1 approximation for SAP.
Greedy
Proof. Let OP T be the seminar selection corresponding to Aopt . Let b∗
be the seminar that is allocated the most students in OP T and is not
allocated students in S ∗ . Moreover, let OP T ′ be the seminar selection
for which OP T ′ (b∗ ) = 0 and OP T ′ (b) = OP T (b) for any b 6= b∗ ∈ B.
Let r be the first iteration of the algorithm for which c(Sr ) > c(OP T ′ ).
Clearly, the seminar selection Si ⊕ (b, OP T (b)) is feasible for every i < r
and b ∈ B. By applying Lemma 5 to iteration r we obtain:
"
#
r−1
Y
c(Sk+1 ) − c(Sk )
1−
· p(OP T ′ ) − p(S ∗ )
p(Sr ) − p(S ∗ ) ≥ 1 −
′
c(OP T )
k=0
"
#
r−1
Y
c(Sk+1 ) − c(Sk )
1−
≥ 1−
· p(OP T ′ ) − p(S ∗ ) .
c(Sr )
k=0
By applying Eq. (6) we obtain that
p(Sr ) − p(S ∗ ) ≥ (1 − 1/e) · p(OP T ′ ) − p(S ∗ ) ,
and therefore
p(Sr ) ≥ (1 − 1/e) · p(OP T ′ ) + p(S ∗ )/e
≥ (1 − 1/e) · p(OP T ) − popt (b∗ ) + p(S ∗ )/e.
(7)
∗
By hypothesis S selects the three seminars with maximum profit in the
optimal assignment and allocates exactly as many students to each as
OP T does. Then, since popt (b∗ ) ≤ popt (bi ) for i = 1, ..., 3 it holds that
p(S ∗ ) ≥ 3 · popt (b∗ ) > e · popt (b∗ ) and the theorem follows.
Observe that the number of feasible seminar selections assigning students to at most three seminars is polynomial in the size of the input.
Therefore, by repeatedly calling the greedy algorithm with all possible
such selections our main result follows:
Corollary 1. There exists a polynomial time (1 − e−1 )-approximation
algorithm for SAP.
References
1. M. Bender, C. Thielen, and S. Westphal. A constant factor approximation for the generalized assignment problem with minimum quantities and unit size items. Mathematical Foundations of Computer
Science, pages 135–145, 2013.
2. M. Bender, C. Thielen, and S. Westphal. Erratum: A constant factor approximation for the generalized assignment problem with minimum quantities and unit size items. Mathematical Foundations of
Computer Science, pages E1–E3, 2013.
3. D. Cattrysse and L. Van Wassenhove. A survey of algorithms for the
generalized assignment problem. European Journal of Operational
Research, 60(3):260 – 272, 1992.
4. R. Cohen, L. Katzir, and D. Raz. An efficient approximation for the
generalized assignment problem. Inf. Process. Lett., 100(4), November 2006.
5. M. Conforti and G. Cornuéjols. Submodular set functions, matroids
and the greedy algorithm: Tight worst-case bounds and some generalizations of the Rado-Edmonds theorem. Discrete Applied Mathematics, 7(3):251 – 274, 1984.
6. L. Fleischer, M. Goemans, V. Mirrokni, and M. Sviridenko. Tight approximation algorithms for maximum general assignment problems.
In Proceedings of the Seventeenth Annual ACM-SIAM Symposium
on Discrete Algorithm, SODA ’06, 2006.
7. S. Khuller, A. Moss, and J. Naor. The budgeted maximum coverage
problem. Inf. Process. Lett., 70(1):39–45, April 1999.
8. S. Krumke and C. Thielen. The generalized assignment problem with
minimum quantities. European Journal of Operational Research,
228(1):46–55, 2013.
9. O. Kundakcioglu and S. Alizamir. Generalized assignment problem.
In C. Floudas and P. Pardalos, editors, Encyclopedia of Optimization,
pages 1153–1162. Springer US, 2009.
10. G. Nemhauser and L. Wolsey. Maximizing submodular set functions: Formulations and analysis of algorithms. In P. Hansen, editor, Studies on Graphs and Discrete Programming, pages 279 – 301.
North-Holland, 1981.
11. D. Shmoys and É. Tardos. An approximation algorithm for the
generalized assignment problem. Math. Program., 62(3):461–474,
December 1993.
12. M. Sviridenko. A note on maximizing a submodular set function
subject to a knapsack constraint. Oper. Res. Lett., 32(1), 2004.
13. R. Vohra and N. Hall. A probabilistic analysis of the maximal covering location problem. Discrete Appl. Math., 43(2), May 1993.
| 8 |
arXiv:1709.00734v2 [math.GR] 6 Nov 2017
Worst-case approximability of functions on finite
groups by endomorphisms and affine maps
Alexander Bors∗
November 7, 2017
Abstract
For a finite set M and functions f, g : M → M , say that g approximates f to
the degree N if and only if f (x) = g(x) for at least N elements x ∈ X. In this
paper, we study the minimum degree to which any function on a given finite group
G can be approximated by a suitable endomorphism of G, and also the analogous
minimum approximability degree by affine functions on G, a certain generalization
of endomorphisms. We give general bounds on these two approximability degrees
and prove results concerning their asymptotic behavior as |G| → ∞.
1
1.1
Introduction
Motivation and main results
In this paper, whenever we speak of a function on some set M , we mean a function
M → M . As a motivation for the main results of this paper, consider the following
general concept:
Definition 1.1.1. Let M1 , M2 be finite sets, F a set of functions M1 → M2 . For a
function g : M1 → M2 , we denote by appF (g) := maxf ∈F |{x ∈ M1 | f (x) = g(x)}|
the F-approximability of f and set appF (M1 , M2 ) := ming:M1 →M2 appF (g), the minimum (or worst-case) F-approximability between M1 and M2 . In case M1 = M2 =
M , we also write appF (M ) instead of appF (M1 , M2 ) and call it the minimum (or
worst-case) F-approximability on M .
∗
School of Mathematics and Statistics, University of Western Australia, Crawley 6009, WA, Australia.
E-mail: alexander.bors@uwa.edu.au
The author is supported by the Austrian Science Fund (FWF), project J4072-N32 “Affine maps on finite
groups”. The work on this paper was started while the author still enjoyed the support of FWF Project
F5504-N26, a part of the Special Research Program “Quasi-Monte Carlo Methods: Theory and Applications”.
2010 Mathematics Subject Classification: Primary: 20D60, 20F69. Secondary: 20D25, 20D30, 20E22,
20P05.
Key words and phrases: Finite groups, Affine Maps, Affine Functions, Hamming metric, Nonlinearity.
1
Alexander Bors
Worst-case approximability
Various authors have studied appF (f ) for particular choices of F and f when M1
and M2 are (general or particular) finite groups. For example, there is a particularly
rich literature on the Aut(G)-approximability of power functions on a finite group
G, particularly the inversion, squaring and cubing function, see [3, Subsection 1.1]
for an overview. Furthermore, the main result of [8] can be viewed as providing
nontrivial upper bounds on the approximability of word maps on nonabelian finite
simple groups S by constant functions (here, M1 = S d with d the number of variables
in the word, and M2 = S).
In this paper, we will mainly be concerned with the case when M1 = M2 = G
for a finite group G (we will prove one general combinatorial result for functions
M1 → M2 though, Lemma 3.1). Our goal is to study the minimum approximability
of a function on a finite group by endomorphisms and by functions of a slightly more
general type, which we call affine maps:
Definition 1.1.2. Let G be a group, g ∈ G, ϕ an endomorphism of G. Then the
function Ag,ϕ : G → G, x 7→ gϕ(x), is called the (left-)affine map of G with respect
to g and ϕ. We denote the set of affine maps on G by Aff(G).
We note that the notion of an affine map on a group and the notation Aff(G)
already appeared in the author’s paper [2], where Aff(G) denoted something different,
namely {Ag,α | g ∈ G, α ∈ Aut(G)}, the set of bijective affine maps on G, which forms
a subgroup of the symmetric group on G. We also note the earlier paper [7], which
used the terminology “affine transformation” instead of “bijective affine map”.
In order to motivate the study of the approximability of functions on finite groups
by affine maps and not just endomorphisms, observe the following:
1. For many, though not all, finite groups G, determining the precise value of
appEnd(G) (G) is a relatively easy problem (see, for instance, Proposition 2.7 and
the remarks thereafter), whereas appAff(G) (G) is more delicate in general (due to
the greater “freedom in mapping behavior” which affine maps enjoy compared
to endomorphisms), thus leading to interesting problems and questions even in
cases where appEnd(G) (G) is trivial.
2. For special choices of the finite groups G1 , G2 (mostly elementary abelian pgroups, i.e., vector spaces over the finite prime field Fp ), the problem of finding
functions f : G1 → G2 which are “as far away from being affine as possible”
(with different precise definitions of this) is heavily studied in cryptography,
inter alia due to the need of encryption procedures which resist so-called linear
attacks, see [4, Introduction]. One of these measures is simply the Hamming
distance of f to the set Aff(G1 , G2 ) := {Ag,ϕ : x 7→ gϕ(x) | g ∈ G2 , ϕ ∈
Hom(G1 , G2 )} of affine functions G1 → G2 ,
distAff(G1 ,G2 ) (f ) :=
min
g∈Aff(G1 ,G2 )
|{x ∈ G1 | f (x) 6= g(x)}|,
which is complementary to our notion of approximability in the sense that
distAff(G1 ,G2 ) (f ) + appAff(G1 ,G2 ) (f ) = |G1 |. It is of intrinsic interest to study
generalizations of this cryptographic problem to larger classes of algebraic structures such as finite groups in general (see also, for instance, the paper [9]).
2
Alexander Bors
Worst-case approximability
In the following, we will present our main results in the form of Theorem 1.1.4
below, but before, we introduce some more notation for a more concise formulation:
Notation 1.1.3. Let G be a finite group, f a function on G.
1. We denote by endapp(f ) := appEnd(G) (f ) the endomorphic approximability of
f.
2. We denote by affapp(f ) := appAff(G) (f ) the affine approximability of f .
3. We denote by endapp(G) := appEnd(G) (G) = minf :G→G endapp(f ) the minimum (or worst-case) endomorphic approximability on G.
4. We denote by affapp(G) := appAff(G) (G) = minf :G→G affapp(f ) the minimum
(or worst-case) affine approximability on G.
Note that endapp(G) ≤ affapp(G), as all endomorphisms are affine maps, and that
since Aff(G) contains all constant functions on G, we always have affapp(G) ≥ 1,
whereas there are various examples of finite groups G such that endapp(G) = 0 (see
Proposition 2.7 and the remarks thereafter).
In this paper, we will show the following bounds and asymptotic results on endapp
and affapp:
Theorem 1.1.4. The following hold for all finite groups G:
1. If G is nontrivial, then 0 ≤ endapp(G) ≤ ( log1 2 +
( log1 2
2
2
log |G| ) log |G|.
affapp(G) ≤
+
affapp(G) = o(|G|) as |G| → ∞.
2
1
log |G| ) log |G|
and 1 ≤
In particular, we have endapp(G) ≤
2. There is an infinite class of finite groups H with endapp(H) ≥ log2 |H| and
affapp(H) ≥ log2 |H| + 1. In particular, we have lim sup|G|→∞ endapp(G) =
lim sup|G|→∞ affapp(G) = ∞.
3. There is an infinite class of finite groups with both minimum endomorphic approximability 0 and minimum affine approximability 1. In particular, we have
lim inf |G|→∞ endapp(G) = 0 and lim inf |G|→∞ affapp(G) = 1.
Theorem 1.1.4(3) is interesting in view of the aforementioned connections to cryptography, since we will see later (in Corollary 2.5) that in the abelian setting in which
cryptographers usually work, one cannot bring the endomorphic (resp. affine) approximability of a function below 1 (resp. 2), whereas Theorem 1.1.4(2) asserts that it is
possible to do so on suitably chosen (nonabelian) finite groups. Therefore, at least
with respect to that particular “measure of non-affineness”, one can do a little bit
better on some nonabelian groups than in the well-studied abelian setting.
Moreover, we note that the infinite class of finite groups which we will give as
an example to prove Theorem 1.1.4(3) also has another interesting property, which
was already noted by the author in his unpublished preprint [1]: These groups are
nonabelian with commutative endomorphism monoid. Nonabelian groups with the
weaker property of having an abelian automorphism group have been studied by
several authors before (see [5, Introduction] for an overview), and “our” groups with
commutative endomorphism monoid were first introduced and studied in [6], where
it was shown that their automorphism groups are abelian.
3
Alexander Bors
1.2
Worst-case approximability
Overview of this paper
Sections 2–4 of this paper serve to prove the three parts of Theorem 1.1.4.
In Section 2, we discuss some methods to prove lower bounds on endapp(G) and
affapp(G), which will allow us to prove Theorem 1.1.4(2). As a further application,
we determine the precise values of endapp(G) and affapp(G) for |G| ≤ 7, which will
be used in the proof of Theorem 1.1.4(1) in Section 3.
Section 3 consists mainly of the proof of Lemma 3.1, which provides some general
bounds on the worst-case F-approximability of functions M1 → M2 where M1 and
M2 are sets and F is a “small” family of functions M1 → M2 . Theorem 1.1.4(1) will
follow swiftly from this and the case study of groups up to order 7 from the previous
section.
Section 4 is dedicated to the proof of Theorem 1.1.4(3) by a careful examination
of the groups already studied by Jonah and Konvisser in [6].
Finally, Section 5 provides some open problems and questions for further research.
1.3
Notation and terminology
The notation and terminology defined in this subsection will be used throughout the
paper without further explanation; more notation and terminology will be explicitly
introduced throughout the text where appropriate.
We denote by N the set of natural numbers, including 0, and by N+ the set of
positive integers. For a function f , the image of f is denoted by im(f ), and the
restriction of f to a set M by f|M .
The exponent of a finite group G is denoted by exp(G), its center by ζG and
its derived (or commutator) subgroup by G′ . D2n and Dic4n respectively denote
the dihedral group of order 2n and the dicyclic group of order 4n respectively. The
symmetric and alternating groups of degree n are denoted by Sn and An respectively.
The kernel of a group homomorphism ϕ is denoted by ker(ϕ). For a prime p, the
finite field with p elements is denoted by Fp .
As usual, Euler’s constant is denoted by e, and for a positive real number c 6= 1,
logc denotes the base c logarithm, with log := loge .
2 On Theorem 1.1.4(2): Lower bounds on endapp(G)
and affapp(G)
The following simple lemma, whose morale is that just as in the abelian case, all
affine maps on finite groups are “difference-preserving”, can be used in arguments
for both upper and lower bounds on affapp(G):
Lemma 2.1. Let G be a group, X ⊆ G, f a function on G. The following are
equivalent:
1. There exists an affine map A on G such that f|X = A|X .
2. There exists an endomorphism ϕ of G such that for all x, y ∈ X, we have
ϕ(y −1 x) = f (y)−1 f (x).
4
Alexander Bors
Worst-case approximability
3. There exists an endomorphism ϕ of G and x ∈ X such that for all y ∈ X,
ϕ(y −1 x) = f (y)−1 f (x).
Proof. For “(1) ⇒ (2)”: Write A = Ag,ϕ . Then for all x, y ∈ X, it follows that
f (y)−1 f (x) = A(y)−1 A(x) = (gϕ(y))−1 gϕ(x) = ϕ(y)−1 ϕ(x) = ϕ(y −1 x), as required.
For “(2) ⇒ (3)”: Trivial.
For “(3) ⇒ (1)”: Set g := f (x)ϕ(x)−1 . Then for all y ∈ X, it follows that
f (y) = f (x)ϕ(y −1 x)−1 = f (x)ϕ(x)−1 ϕ(y) = gϕ(y), so A := Ag,ϕ does the job.
The following is also useful for reduction arguments:
Lemma 2.2. Let G be a finite group, f a function on G. If A is any bijective affine
map on G, then affapp(f ) = affapp(f ◦ A) = affapp(A ◦ f ).
Proof. Noting that A−1 is a bijective affine map on G as well, one sees that it suffices
to show affapp(f ) ≤ affapp(A ◦ f ) and affapp(f ) ≤ affapp(f ◦ A). To this end, let
X ⊆ G with |X| = affapp(f ) and B ∈ Aff(G) with f|X = B|X . Then (A ◦ f )|X =
(A◦B)|X , which proves the first inequality, and (f ◦A)|A−1 [X] = (B ◦A)|A−1 [X] , which
proves the second inequality.
From Lemma 2.1, one can immediately derive a sufficient criterion for the simulataneous validity of endapp(G) ≥ l and affapp(G) ≥ l + 1 for some fixed l ∈ N+
based on the following concepts:
Definition 2.3. Let G be a group.
1. A universal element in G is an element u ∈ G such that for all g ∈ G, there
exists ϕ = ϕg ∈ End(G) such that ϕ(u) = g.
2. More generally, for l ∈ N+ , a universal l-tuple in G is an l-tuple (u1 , . . . , ul ) ∈
Gl such that for all (g1 , . . . , gl ) ∈ Gl , there exists ϕ = ϕ(g1 ,...,gl ) ∈ End(G) such
that ϕ(ui ) = gi for i = 1, . . . , l.
Note that a universal element in a finite group G is necessarily of order exp(G).
Proposition 2.4. Let G be a nontrivial finite group.
1. If G has a universal element, then endapp(G) ≥ 1 and affapp(G) ≥ 2.
2. More generally, if, for some l ∈ N+ , G has a universal l-tuple, then endapp(G) ≥
l and affapp(G) ≥ l + 1.
Proof. It suffices to show point (2). Let (u1 , . . . , ul ) be a universal l-tuple in G. Then
endapp(G) ≥ l holds because by the definition of “universal l-tuple”, every function
on G agrees with a suitable endomorphism of G on the set {u1 , . . . , ul } (and the ui
must be pairwise distinct). For the bound on affapp(G), it suffices to show that any
−1
function on G agrees with some affine map on G on the subset {1, u−1
1 , . . . , ul }. But
for this, it is, by Lemma 2.1, sufficient to check that there is some endomorphism ϕ
−1 · 1) = f (u−1 )−1 f (1), which is clear
of G such that for i = 1, . . . , l, we have ϕ((u−1
i )
i
by universality.
5
Alexander Bors
Worst-case approximability
We note two interesting consequences of Proposition 2.4, the latter of which also
directly implies Theorem 1.1.4(2):
Corollary 2.5. Let G be a nontrivial finite abelian group. Then endapp(G) ≥ 1
and affapp(G) ≥ 2.
Proof. This follows from Proposition 2.4(1), since by the structure theorem for finite
abelian groups, it is clear that every such group G has a universal element (actually, by [10, 4.2.7, p. 102], any cyclic subgroup of G generated by an element of
order exp(G) always admits a direct complement in G, so that any such element is
universal).
Corollary 2.6. For m, r ∈ N+ , m ≥ 2, we have endapp((Z/mZ)r ) ≥ r and
affapp((Z/mZ)r ) ≥ r + 1.
Proof. This follows from Proposition 2.4(2) by observing that the “standard” generators of (Z/mZ)r form a universal r-tuple in that group.
Proof of Theorem 1.1.4(2). This follows from Corollary 2.6 with m := 2.
The rest of this section serves either as direct preparation for determining the
precise values of endapp(G) and affapp(G) for small G at the end of the section
(see Proposition 2.14) or to raise some other interesting points. First, let us note
the following simple fact, which shows that in many finite groups, the problem of
determining the minimum endomorphic approximability is trivial:
Proposition 2.7. Let G be a finite group. The following are equivalent:
1. endapp(G) ≥ 1.
2. G has a universal element.
Proof. For “(1) ⇒ (2)”: We show the contraposition. So assume that G has no
universal element. Then we can choose, for every element g ∈ G, an element f (g) ∈ G
which is not an image of g under any endomorphism of G. The resulting function f
on G clearly has endomorphic approximability 0.
For “(2) ⇒ (1)”: This implication is part of Proposition 2.4(1).
Hence endapp(G) = 0 whenever G has no universal element, which holds true for
example when G is any nonabelian finite simple group. The problem of determining
the minimum affine approximability of a function on a finite group seems less trivial.
For example, below, we will give another criterion on nontrivial finite groups G which
is sufficient for affapp(G) ≥ 2 (Proposition 2.9) and which holds, for example, for
G = A4 , which also has no universal element (see also Example 2.11(1) below). The
new criterion is based on the following concepts:
Definition 2.8. Let G be a nontrivial finite group.
1. We call the orbits of the natural action of Aut(G) on G the automorphism
orbits of G, and {1G } the trivial automorphism orbit of G.
6
Alexander Bors
Worst-case approximability
2. A dominating automorphism orbit of G is a nontrivial automorphism orbit O
of G such that |O| > 12 (|G| − 1).
3. A function f on G with f (1) = 1 is called automorphism orbit avoiding (henceforth abbreviated by a.o.a.) if and only if for all g ∈ G \ {1}, g and f (g) lie in
different automorphism orbits of G.
Proposition 2.9. Consider the following conditions on nontrivial finite groups G:
1. G has a dominating automorphism orbit.
2. G has no a.o.a. functions which are bijective (i.e., permutations on G).
3. affapp(G) ≥ 2.
Conditions (1) and (2) are equivalent, and either of them implies condition (3).
The following combinatorial lemma will be used in the proof of Proposition 2.9:
Lemma 2.10. For a partition P on a nonempty finite set M , call a function f on M
P-avoiding if and only if for all x ∈ M , x and f (x) lie in different partition classes
from P. Then the following are equivalent:
1. One of the elements of P is of size larger than 12 |M |.
2. There is no P-avoiding permutation on M .
Proof. For “(1) ⇒ (2)”: Let P denote the unique element of P of size larger than
1
2 |M |. Assume, by contradiction, that there is a P-avoiding permutation f on M .
Then f would have to map P injectively into the smaller set M \ P , which is impossible.
For “(2) ⇒ (1)”: We show the contraposition of this implication, i.e., that there is
a P-avoiding permutation on M if all partition classes from P have size at most 12 |M |,
by induction on |M |. To this end, one first verifies directly that this holds for |M | ≤ 5.
Now assume that |M | ≥ 6. Note that P must consist of at least two nonempty
partition classes, and choose distinct P1 , P2 ∈ P such that |P1 | ≥ |P2 | ≥ |P | for
all P ∈ P \ {P1 , P2 }. Fix p1 ∈ P1 and p2 ∈ P2 , and set M ′ := M \ {p1 , p2 } and
P′ := {P \ {p1 , p2 } | P ∈ P}. Then P′ is a partition of M ′ , and it still has the
property that none of its members has size larger than 12 |M ′ |. Indeed, an element
of P′ is either obtained from P1 or P2 by deleting an element and thus has size at
most |P1 | − 1 ≤ 21 |M | − 1 = 21 (|M | − 2) = 21 |M ′ |, or it is equal to an element of
P distinct from P1 and P2 , whence it can only have size at most 13 |M |, which is
less than or equal to 21 |M | − 1 by the assumption |M | ≥ 6. Hence by the induction
hypothesis, there exists a P′ -avoiding permutation g on M ′ , and it is clear that
f := g ∪ {(p1 , p2 ), (p2 , p1 )} (less formally: the permutation on M obtained by adding
the transposition of p1 and p2 to g) is a P-avoiding permutation on M .
Proof of Proposition 2.9. The equivalence of (1) and (2) follows immediately from
Lemma 2.10 with M := G \ {1} and P the collection of nontrivial automorphism
orbits of G.
For “(2) ⇒ (3)”: Let f be any function on G. We need to show that f agrees
with a suitable affine map on G on some subset of G of size at least 2. Since all
7
Alexander Bors
Worst-case approximability
constant functions on G are affine, this is clear if f is not a permutation on G, so
assume that f : G → G is bijective. Composing f with a suitable translation on
G, we may, by Lemma 2.2, also assume w.l.o.g. that f (1) = 1. But since G has no
a.o.a. permutations by assumption, it follows that for some g ∈ G \ {1} and some
automorphism α of G, f (g) = α(g). Hence f agrees with α on {1, g}, and we are
done.
Example 2.11. We now give some applications of Proposition 2.9, some of which will
also be used later, and in point (4), we give an example which shows that the condition
affapp(G) ≥ 2 is not equivalent to either of conditions (1) or (2) in Proposition 2.9.
1. The alternating group A4 has no universal element, whence endapp(A4 ) = 0
by Proposition 2.7. On the other hand, A4 has a dominating automorphism
orbit, namely the one consisting of elements of order 3, which has size 8. Hence
affapp(A4 ) ≥ 2 by Proposition 2.9.
2. In any finite dihedral group D2n = hr, s | r n = s2 = 1, srs−1 = r −1 i with n ≥ 3,
the reflections, i.e., the elements of the form sr k with k ∈ {0, . . . , n − 1}, form
a dominating automorphism orbit, so affapp(D2n ) ≥ 2 for all n ≥ 3. Similarly,
one shows that affapp(Dic4n ) ≥ 2 for n ≥ 2.
(1)
3. Let p be an odd prime, and denote by Gp := (Z/pZ)2 ⋊ Z/pZ = hx, y, t |
xp = y p = [x, y] = tp = [x, t] = 1, tyt−1 = xyi the unique nonabelian group
of order p3 and exponent p. We claim that the elements of the form xk1 y k2 tl
with k1 , k2 ∈ {0, . . . , p − 1} and l ∈ {1, . . . , p − 1} form an automorphism
(1)
orbit of Gp , which clearly is dominating. It is easy to verify that for any
fixed k1 , k2 , l as above, the map x 7→ x, y 7→ y, tl 7→ xk1 y k2 tl extends to an
automorphism of G, so it suffices to argue why for each l ∈ {1, . . . , p − 1}, t
and tl lie in the same automorphism orbit. But for this, it is sufficient to show
that the automorphisms of (Z/pZ)2 which t and tl induce by conjugation are
conjugate.
Now
these
automorphisms are represented by the (2 × 2)-matrices
1 1
1 l
and
over Fp , and these are conjugate because they must have
0 1
0 1
the same rational canonical form (namely the Frobenius companion matrix of
(1)
the polynomial (X − 1)2 ∈ Fp [X]). Hence we conclude that affapp(Gp ) ≥ 2
for all odd primes p.
(2)
4. Again, let p be an odd prime and consider now Gp := Z/p2 Z ⋊ Z/pZ = hx, t |
2
xp = tp = 1, txt−1 = x1+p i, the unique nonabelian group of order p3 and
(2)
exponent p2 . As all elements of Gp outside hxi ∼
= Z/p2 Z have order p, hxi
(2)
(2)
is characteristic in Gp , and thus, since elements in Gp from different cosets
of hxi act on hxi via conjugation by distinct (and hence, by commutativity
of Aut(hxi), by non-conjugate) automorphisms, such elements cannot be in the
(2)
(2)
same automorphism orbit of Gp , so that Gp has no dominating automorphism
(2)
orbit. However, it is not difficult to check that x is a universal element in Gp ,
(2)
so that affapp(Gp ) ≥ 2 by Proposition 2.4(1) nonetheless. This shows that for
8
Alexander Bors
Worst-case approximability
nontrivial finite groups G, having a dominating automorphism orbit is indeed
only sufficient (not necessary) for affapp(G) ≥ 2.
Points (2)–(4) of Example 2.11 together with Corollary 2.5 also imply the following:
Proposition 2.12. Let p be a prime and G a finite group of order pk , k ∈ {1, 2, 3}.
Then affapp(G) ≥ 2.
Note, however, that not all nontrivial finite p-groups have affapp-value at least 2,
as the examples for Theorem 1.1.4(3) are of order p8 .
We can also say something about endapp- and affapp-values of finite cyclic groups:
Lemma 2.13. Let n ∈ N+ . Then the following hold:
1. endapp(Z/nZ) = 1.
2. If n is a prime, then affapp(Z/nZ) = 2.
3. If n = pk , p a prime and k ∈ N+ , then affapp(Z/nZ) = affapp(Z/pk Z) ≤ p.
Proof. For (1): Note that by Corollary 2.5, endapp(Z/nZ) ≥ 1, so it suffices to give
an example of a function f on Z/nZ such that endapp(f ) ≤ 1. To this end, fix
a generator g of Z/nZ, and consider the following function f on Z/nZ: It maps
all non-generators of Z/nZ (i.e., elements that generate a proper subgroup) to g,
and it maps each generator x of Z/nZ to x2 (squaring modulo n). Then any set
X ⊆ Z/nZ on which f agrees with some endomorphism ϕ of Z/nZ, say ϕ(t) = a · t
for a suitable fixed a ∈ Z/nZ and all t ∈ Z/nZ, cannot contain any non-generators,
and it also cannot contain two distinct generators x1 , x2 , since that would imply
x21 = f (x1 ) = ϕ(x1 ) = ax1 , and thus a = x1 , although one can analogously show
a = x2 as well.
For (2): Again by Corollary 2.5, we have affapp(Z/nZ) ≥ 2, so it suffices to give
an example of a function f on Z/nZ with affapp(f ) ≤ 2. Let f be the square function
of the ring Z/nZ. Note that any fixed affine map on Z/nZ is of the form x 7→ ax + b
with a, b ∈ Z/nZ fixed, so that the elements of Z/nZ on which f and that affine map
agree are the solutions to the quadratic equation x2 = ax + b in the ring Z/nZ. Since
n is a prime, that ring is a field, whence each such equation has at most 2 solutions
in Z/nZ, as required.
For (3): We give an example of a function f on Z/pk Z such that affapp(f ) ≤ p. To
define f , take as the underlying set of the group Z/pk Z the standard representatives
0, 1, . . . , pk − 1 of the integer residue classes modulo pk , and as the group operation
addition modulo pk . By integer division, every element x of Z/pk Z can be uniquely
written as r(x) + q(x) · p with r(x) ∈ {0, 1, . . . , p − 1} and q(x) ∈ {0, 1 . . . , pk−1 − 1}.
Then we define f through f (x) := r(x) + q(x). Let us argue why f cannot agree with
an affine map of Z/pk Z on any subset X with |X| ≥ p + 1. Indeed, such a subset X
contains two distinct elements x and y such that r(x) = r(y), say w.l.o.g. x ≥ y in
Z. Then by Lemma 2.1, it would follow that some endomorphism ϕ of Z/pk Z maps
x − y = p · (q(x) − q(y)) to f (x) − f (y) = q(x) − q(y), which is impossible, because the
order in Z/pk Z of q(x) − q(y) is strictly larger than the order of p · (q(x) − q(y)).
9
Alexander Bors
Worst-case approximability
We are now ready for determining the precise endapp- and affapp-values of groups
of order up to 7:
Proposition 2.14. The precise values of endapp(G) and affapp(G) for finite groups
G with |G| ≤ 7 are as in the following table:
G
endapp(G)
affapp(G)
{1}
1
1
Z/2Z
1
2
Z/3Z
1
2
Z/4Z
1
2
(Z/2Z)2
2
3
Z/5Z
1
2
Z/6Z
1
2
S3
0
2
Z/7Z
1
2
Proof of Proposition 2.14. By Lemma 2.13, all the assertions on endapp(G) and
affapp(G) in the cases where G is cyclic are clear except for the one assertion
affapp(Z/6Z) = 2. To see that this holds, it suffices to give an example of a function
f on G = Z/6Z such that affapp(f ) ≤ 2. Using that Z/6Z ∼
= Z/2Z × Z/3Z, we write
the elements of G as (x, y) with x ∈ {0, 1} and y ∈ {0, 1, 2}. Consider the following
function f on G, which is a kind of “component swap”: f (x, y) := (y (mod 2), x) for
all x ∈ {0, 1} and y ∈ {0, 1, 2}. We argue why f cannot agree with any affine function
of G on any subset of size 3. Note that since the component subgroups Z/2Z and
Z/3Z of G are fully invariant, any affine map of G can be written as a product (in the
sense of component-wise application) of an affine map on Z/2Z and an affine map on
Z/3Z. Consequently, any affine map of G maps pairs with the same first (resp. second) coordinate to pairs whose first (resp. second) coordinates agree as well. But by
its definition, f never maps distinct pairs with the same second coordinate (which are
necessarily of the form (0, x) and (1, x) for some x ∈ {0, 1, 2}) to such pairs. Hence if
there are any three pairwise distinct elements (x1 , y1 ), (x2 , y2 ), (x3 , y3 ) of G on which
f agrees with some affine map, then their second coordinates must be pairwise distinct, so that we can assume w.l.o.g. that y1 = 0, y2 = 1 and y3 = 2. Hence it remains
to show that no affine map A = Ag,ϕ on G can show the following mapping behavior
for some x1 , x2 , x3 ∈ {0, 1}: (x1 , 0) 7→ (0, x1 ), (x2 , 1) 7→ (1, x2 ) and (x3 , 2) 7→ (0, x3 ).
If {x1 , x2 , x3 } = {0, 1}, then choosing p1 , p2 ∈ {(x1 , 0), (x2 , 1), (x3 , 2)} with the same
first coordinate and using that by the proof of Lemma 2.1, ϕ(p1 −p2 ) = f (p1 )−f (p2 ),
we see that ϕ|Z/3Z is trivial, and thus A|Z/3Z is constant, contradicting the fact that
in the images of the three pairs (x1 , 0), (x2 , 1) and (x3 , 2), there appear two distinct
values in the second coordinates. Hence x1 = x2 = x3 so that all three pairs must
be mapped to pairs with the same first coordinate, which is not the case, the final
contradiction.
We now turn to the two non-cyclic groups in the list: (Z/2Z)2 and S3 ∼
= D6 .
2
For G = (Z/2Z) , write the four elements of G as (0, 0), (1, 0), (0, 1), (1, 1), and
note that by Corollary 2.6, endapp(G) ≥ 2 and affapp(G) ≥ 3. It is easy to check
that endapp(f ) ≤ 2 for the following function f on (Z/2Z)2 :
x
f (x)
(0, 0)
(1, 0)
(1, 0)
(1, 0)
(0, 1)
(0, 1)
(1, 1)
(0, 1)
For affapp(G) = 3, we only need to argue that there is some non-affine function
2
on G, which is clear, since G has precisely 22 = 16 endomorphisms, thus precisely
4 · 16 = 64 affine maps, but there are 44 = 256 functions on G altogether.
10
Alexander Bors
Worst-case approximability
∼ D6 , we note that endapp(G) = 0 holds by Proposition
Finally, for G = S3 =
2.7, since G has no elements of order exp(G) = 6 and thus no universal elements.
Moreover, affapp(G) ≥ 2 by Example 2.11(2), so it suffices to give an example of a
function f on G such that affapp(f ) ≤ 2. Since G = D6 = hr, s | r 3 = s2 = 1, srs−1 =
r −1 i, we can write the elements of G in normal form as 1, r, r 2 , s, sr, sr 2 . We define
f via the following table:
x
f (x)
1
1
r
r
r2
r
s
1
sr
r2
sr 2
r2
Let X ⊆ G with |X| = 3. We show by contradiction that f cannot agree with
any affine map A = Ag,ϕ of G on X. First, consider the case when X consists only
of rotations, i.e., X = {1, r, r 2 }. Then if f|X = A|X , we would in particular have
A(1) = 1 and thus that A = ϕ is an endomorphism of G, but then the mapping
behavior of A on {r, r 2 } is contradictory. Next, consider the case when X contains
both a rotation r k and a reflection sr l for suitable k, l ∈ {0, 1, 2}. Then by the
proof of Lemma 2.1, ϕ(sr k+l ) = ϕ(r −k sr l ) = f (r k )−1 f (sr l ) ∈ hri, which is only
possible if ϕ is the trivial endomorphism of G so that A is constant. But by the
definition of f , A certainly takes at least two distinct values on the three elements of
X, another contradiction. The only case left is when X only consists of reflections,
i.e., X = {s, sr, sr 2 }. Denoting by µs the left translation by s on G, we get in
this case that A ◦ µs is an affine map of G showing the following mapping behavior:
1 7→ 1, r 7→ r 2 , r 2 7→ r 2 . From this, we can derive a contradiction like we did in the
case X = {1, r, r 2 }.
3 On Theorem 1.1.4(1): Upper bounds on endapp(G)
and affapp(G)
Recall the general approximability notion appF (M1 , M2 ) from Definition 1.1.1. We
will now show the following general combinatorial lemma:
Lemma 3.1. Let M1 and M2 be finite sets of cardinality at least 2, f : N → (0, ∞)
and F a set of functions M1 → M2 .
1|
1. If F contains all constant functions M1 → M2 , then appF (M1 , M2 ) ≥ max{1, |M
|M2 | }.
2. If |F| ≤ |M2 |f (|M1 |) , then
appF (M1 , M2 ) ≤ max{e2 ·
|M1 |
, f (|M1 |) log |M2 | + log |M1 |}.
|M2 |
Proof. For (1): Each function g : M1 → M2 can be approximated at each single
argument x ∈ M1 by the constant g(x) function M1 → M2 , so appF (M1 , M2 ) ≥ 1.
|M1 |
Similarly, one sees that appF (M1 , M2 ) ≥ |M
as at least one of the fibers of g must
2|
be of size at least
|M1 |
|M2 | .
11
Alexander Bors
Worst-case approximability
For (2): Consider the Hamming metric dist on the set M2M1 of all functions
M1 → M2 . For k ∈ N and G ⊆ M2M1 , denote by Nk (G) := {h ∈ M2M1 | ∃g ∈ G :
dist(g, h) ≤ k} the k-neighborhood of G in M2M1 with respect to dist. For g ∈ M2M1 ,
we also write Nk (g) instead of Nk ({g}) for the k-ball around g, and we denote by
Ck (g) := {h ∈ M2M1 | dist(g, h) = k} the k-circle around g. | Nk (g)| and | Ck (g)| do
P
not depend on g; indeed, | Ck (g)| = |Mk1 | ·(|M2 |− 1)k , and | Nk (g)| = ki=0 | Ci (g)| =
Pk
|M1 |
· (|M2 | − 1)i . Henceforth, we will denote by νk resp. γk the size of the
i=0
i
k-ball resp. k-circle around any function M1 → M2 .
The proof is based on the following observation: For each k ∈ {0, . . . , |M1 | − 1},
appF (M1 , M2 ) ≤ |M1 | − (k + 1) is equivalent to the inclusion Nk (F) ⊆ M2M1 being
proper. We thus want to show | Nk (F)| < |M2M1 | = |M2 ||M1 | for as large k as possible.
Now
| Nk (F)| = |
[
f (|M1 |)
Nk (g)| ≤ |F| · νk ≤ |M2 |
·
k
X
γi ≤ |M2 |f (|M1 |) · |M1 | max γi ,
i=0
g∈F
i=0,...,k
and so, setting L := log|M2 | (|M1 |), for | Nk (F)| < |M2 ||M1 | to hold, it is sufficient
to have γi = |Mi 1 | · (|M2 | − 1)i < |M2 ||M1 |−f (|M1 |)−L for i = 0, . . . , k. We make the
ansatz i = |M1 | − l and transform the substituted inequality
|M1 |
· (|M2 | − 1)|M1 |−l < |M2 ||M1 |−f (|M1 |)−L
(1)
|M1 | − l
to obtain a (preferably small) lower bound on l, which is also an upper bound on
appF (M1 , M2 ). Using that
|M2 |L·l
|M1 |
|M1 |
|M1 |l
=
,
=
≤
l!
l!
|M1 | − l
l
we see that for Formula (1) to hold, it is sufficient to have
|M2 |f (|M1 |)+l(L−1)+L < l!.
Now l! >
(l/e)l
(2)
(see, for instance, [11]), so Formula (2) is implied by
l
|M2 |f (|M1 |)+l(L−1)+L ≤ ( )l ,
e
which by taking logarithms on both sides and bringing all summands involving l
as a factor on one side is equivalent to
(f (|M1 |) + L) · log |M2 | ≤ l · (log l − 1 − (L − 1) log |M2 |).
(3)
Finally, we note that for Formula (3) to hold, it is sufficient to have l ≥ (f (|M1 |)+
L) · log |M2 | and log l − 1 − (L − 1) log |M2 | ≥ 1, i.e.,
l ≥ max{(f (|M1 |)+L)·log |M2 |, e2 ·|M2 |L−1 } = max{(f (|M1 |)+L)·log |M2 |, e2 ·
This concludes the proof.
12
|M1 |
}.
|M2 |
Alexander Bors
Worst-case approximability
Proof of Theorem 1.1.4(1). For |G| = 2, . . . , 7, the validity of the asserted inequalities can be checked case by case using Proposition 2.14, so we may assume that
|G| ≥ 8. Note that since endomorphisms of G are determined by their values on any
generating subset of G and the size of a minimal (with respect to inclusion) generating subset of G is always bounded from above by log2 |G| due to Lagrange’s theorem,
we have | End(G)| ≤ |G|log2 |G| and | Aff(G)| = |G| · | End(G)| ≤ |G|1+log2 |G| .
We can therefore apply Lemma 3.1(2) with M1 := M2 := G and f : x 7→ log2 x
(resp. f : x 7→ 1 + log2 x) to conclude that
endapp(G) ≤ max{e2 , log2 |G| log |G| + log |G|} = max{e2 , (
1
1
+
) log2 |G|}
log 2 log |G|
resp.
affapp(G) ≤ max{e2 , (1+log2 |G|) log |G| + log |G|} = max{e2 , (
2
1
+
) log2 |G|}.
log 2 log |G|
As it is easily checked that for all real numbers x ≥ 8, ( log1 2 + x1 ) log2 x ≥ e2 ,
the asserted upper bounds on endapp(G) and affapp(G) follow from the just derived
inequalities.
4 On Theorem 1.1.4(3): Finite groups G minimizing endapp(G) and affapp(G)
The following finite groups are the ones studied by Jonah and Konvisser in [6], as
mentioned in the Introduction:
Definition 4.1. Let p be a prime, λ = (λ1 , λ2 ) an element of the set {(1, 0), (0, 1),
(1, 1), . . . , (p − 1, 1)} of representatives of 1-dimensional subspaces of F2p . The JKgroup JKp,λ is defined as the p-group of nilpotency class 2 generated by 4 elements
a1 , a2 , b1 , b2 subject to the following additional relations:
ap1 = [a1 , b1 ], ap2 = [a1 , bλ1 1 bλ2 2 ], bp1 = [a2 , b1 b2 ], bp2 = [a2 , b2 ], [a1 , a2 ] = [b1 , b2 ] = 1.
We now collect some basic facts on the JKp,λ which Jonah and Konvisser already
used in their proof that Aut(JKp,λ ) is abelian. It is elementary to check that every
element of JKp,λ has a unique normal form representation as
ak11 ak22 bl11 bl22 [a1 , b1 ]r1 [a1 , b2 ]r2 [a2 , b1 ]r3 [a2 , b2 ]r4
with k1 , k2 , l1 , l2 , r1 , . . . , r4 ∈ {0, . . . , p − 1}, so that JKp,λ is a special p-group
of order p8 with ζ JKp,λ = JK′p,λ elementary abelian of order p4 with Fp -basis
[a1 , b1 ], [a1 , b2 ], [a2 , b1 ], [a2 , b2 ]. The central quotient of JKp,λ is elementary abelian
of order p4 too, with Fp -basis the images of a1 , a2 , b1 , b2 under the canonical projection.
13
Alexander Bors
Worst-case approximability
Jonah and Konvisser showed that all automorphisms of JKp,λ are central, i.e., of
the form g 7→ (id +ϕ)(g) := gϕ(g) for some fixed homomorphism ϕ : JKp,λ → ζ JKp,λ
(and conversely, all such maps on JKp,λ actually are automorphisms, so that the
automorphisms of JKp,λ are completely understood), which immediately implies that
any two automorphisms of JKp,λ commute, since im(ϕ) ≤ ζ JKp,λ = JK′p,λ ≤ ker(ϕ)
for every homomorphism ϕ : JKp,λ → ζ JKp,λ , whence the composition of any two
such homomorphisms is the trivial endomorphism of JKp,λ .
The author’s approach in [1] to show that for “most” of the JKp,λ , even any two
endomorphisms of JKp,λ commute, is analogous to the one of Jonah and Konvisser,
i.e., it essentially consists of gaining a complete understanding of all endomorphisms
of those JKp,λ in the form of the following key lemma:
Lemma 4.2. Let p be an odd prime, λ1 ∈ {0, . . . , p − 1}, λ2 := 1 and λ := (λ1 , λ2 ).
Then every endomorphism of JKp,λ which is not an automorphism is a homomorphism JKp,λ → ζ JKp,λ .
In other words, every endomorphism of such a JKp,λ is of one of the two forms ϕ
or id +ϕ for a homomorphism ϕ : JKp,λ → ζ JKp,λ , which implies the commutativity
of End(JKp,λ ) in a simple case distinction. It is also this complete understanding of the endomorphisms of “most” JKp,λ which will allow us to prove that both
endapp(JKp,λ ) = 0 and affapp(JKp,λ ) = 1 (the latter via Lemma 2.1).
For the reader’s convenience, we now recall the proof of Lemma 4.2 as in [1].
Proof of Lemma 4.2. Since JKp,λ is nilpotent of class 2, p is odd and JK′p,λ has
exponent p, it follows that JKp,λ satisfies the identity (xy)p = xp y p (see also [10,
5.3.5, p. 141]). Using this, the defining relations and that λ2 = 1 by assumption, it
follows that
(ak11 ak22 bl11 bl22 [a1 , b1 ]r1 [a1 , b2 ]r2 [a2 , b1 ]r3 [a2 , b2 ]r4 )p =
[a1 , b1 ]k1 +λ1 ·k2 [a1 , b2 ]k2 [a2 , b1 ]l1 [a2 , b2 ]l1 +l2 .
(4)
But the (linear) map
(Z/pZ)4 → (Z/pZ)4 , (k1 , k2 , l1 , l2 ) 7→ (k1 + λ1 k2 , k2 , l1 , l1 + l2 ),
is a bijection. Hence by Equation (4), it follows that the elements of order a
divisor of p in JKp,λ are just those from JK′p,λ = ζ JKp,λ and that each such element
has precisely one p-th root in JKp,λ of the form ak11 ak22 bl11 bl22 .
Now let ϕ be an endomorphism of JKp,λ with nontrivial kernel. Fix x ∈ JKp,λ of
order p in ker(ϕ), and let y = as11 as22 bt11 bt22 , (s1 , s2 , t1 , t2 ) ∈ {0, . . . , p−1}4 \{(0, 0, 0, 0)},
be a p-th root of x in JKp,λ . Then ϕ maps y to an element of JKp,λ of order a divisor
of p, i.e., to an element of ζ JKp,λ .
Note that for showing im(ϕ) ⊆ ζ JKp,λ , by the defining relations of JKp,λ , it
suffices to show that at least one of the three elements a1 , a2 , b1 gets mapped into
ζ JKp,λ by ϕ. Moreover, for any element g ∈ JKλ,p , if ϕ(g) ∈ ζ JKλ,p , then all
commutators of the form [h, g] with h ∈ JKλ,p are in ker(ϕ).
14
Alexander Bors
Worst-case approximability
We now agree on the following notational conventions: “(k1 , k2 , l1 , l2 ) 7→ ζ” is
an abbreviation for “ϕ(ak11 ak22 bl11 bl22 ) ∈ ζ JKλ,p ”, and “(K1 , K2 , L1 , L2 ) 7→ 1” abbreviates “[a1 , b1 ]K1 [a1 , b2 ]K2 [a2 , b1 ]L1 [a2 , b2 ]L2 ∈ ker(ϕ)”. Then the following implications
hold: Firstly, by “taking brackets” with the generators a1 , a2 , b1 , b2 ,
(k1 , k2 , l1 , l2 ) 7→ ζ ⇒(l1 , l2 , 0, 0) 7→ 1, (0, 0, l1 , l2 ) 7→ 1, (k1 , 0, k2 , 0) 7→ 1 and
(0, k1 , 0, k2 ) 7→ 1.
(5)
Secondly, using Formula (4), the observation that an element is mapped into the
center if and only if its p-th power is in the kernel of ϕ translates as
(k1 , k2 , l1 , l2 ) 7→ ζ ⇔ (k1 + λ1 k2 , k2 , l1 , l1 + l2 ) 7→ 1,
which is equivalent to
(K1 , K2 , L1 , L2 ) 7→ 1 ⇔ (K1 − λ1 K2 , K2 , L1 , L2 − L1 ) 7→ ζ.
(6)
Our assumption that ϕ(y) ∈ ζ JKp,λ translates to (s1 , s2 , t1 , t2 ) 7→ ζ. We now
make a case distinction according to the values of s1 , s2 , t1 , t2 . First, assume that
t1 6= 0. By Formula (5), we have (0, 0, t1 , t2 ) 7→ 1, and by Formula (6), this implies
(0, 0, t1 , t2 − t1 ) 7→ ζ. Applying Formula (5), we deduce from this that (0, 0, t1 , t2 −
t1 ) 7→ 1. Iteration of this argumentation yields (0, 0, t1 , t2 − n · t1 ) 7→ 1 for all n ∈ N,
i.e., (0, 0, t1 , t) 7→ 1 for all t ∈ {0, . . . , p − 1}. In particular, (0, 0, l1 , 0) 7→ 1, which
by l1 6= 0 implies (0, 0, 1, 0) 7→ 1, and thus (0, 0, 1, −1) 7→ ζ by Formula (6). Spelled
out, this means ϕ(b1 b−1
2 ) ∈ ζ JKλ,p . But also, by an appropriate subtraction among
the (0, 0, t1 , t1 − n · t2 ) 7→ 1, we derive (0, 0, 0, 1) 7→ 1, which by Formula (6) yields
(0, 0, 0, 1) 7→ ζ, or explicitly, ϕ(b2 ) ∈ ζ JKλ,p . In combination with the already derived
ϕ(b1 b−1
2 ) ∈ ζ JKλ,p , this yields ϕ(b1 ) ∈ ζ JKλ,p , so that we are done in this case.
Now assume t1 = 0. The subcase s2 6= 0 reduces to the first case, since Formula (5)
yields (s1 , 0, s2 , 0) 7→ 1, which in turn gives (s1 , 0, s2 , −s2 ) 7→ ζ by Formula (6). Hence
we can assume that t1 = s2 = 0. But then another successive application of Formulas
(5) and (6) yields (s1 , 0, 0, 0) 7→ ζ, which in case s1 6= 0 implies (1, 0, 0, 0) 7→ ζ, i.e.,
the sufficient ϕ(a1 ) ∈ ζ JKλ,p . Hence we can assume s1 = s2 = t1 = 0 and t2 6= 0. In
this case, (t1 , t2 , 0, 0) 7→ 1, valid by Formula (5), simplifies to (0, t2 , 0, 0) 7→ 1, which
by Formula (6) yields (−λ1 l2 , l2 , 0, 0) 7→ ζ and hence reduces the situation to the case
t1 = 0, s2 6= 0 dealt with before.
By Proposition 2.7, it is now clear that the JKp,λ with p > 2 and λ 6= (1, 0) satisfy
endapp(JKp,λ ) = 0. Indeed, a hypothetical universal element u in JKp,λ would need
to be of order exp(JKp,λ ) = p2 , so that u ∈ JKp,λ \ζ JKp,λ , and thus can only be
mapped to elements from ζ JKp,λ ∪uζ JKp,λ ( JKp,λ by Lemma 4.2 and Jonah and
Konvisser’s results. It remains to show affapp(JKp,λ ) = 1, which is done by the proof
of the following proposition:
Proposition 4.3. Let p be an odd prime, λ1 ∈ {0, . . . , p − 1}, λ2 := 1, λ := (λ1 , λ2 ),
and σ any fixed-point free automorphism of (Z/pZ)4 = F4p (for example, σ could be
chosen as a Singer cycle). Denote, for i = 1, 2, 3, 4, by πi : (Z/pZ)4 → Z/pZ the
15
Alexander Bors
Worst-case approximability
projection onto the i-th coordinate. Then the following function f on JKp,λ , defined
on elements in normal form, satsfies affapp(f ) = 1:
f (ak11 ak22 bl11 bl22 [a1 , b1 ]r1 [a1 , b2 ]r2 [a2 , b1 ]r3 [a2 , b2 ]r4 ) :=
π (σ(k1 ,k2 ,l1 ,l1 )) π2 (σ(k1 ,k2 ,l1 ,l2 )) π3 (σ(k1 ,k2 ,l1 ,l2 )) π4 (σ(k1 ,k2 ,l1 ,l2 ))
a2
b1
b2
·
[a1 , b1 ]π1 (σ(r1 ,r2 ,r3 ,r4 )) [a1 , b2 ]π2 (σ(r1 ,r2 ,r3 ,r4 )) [a2 , b1 ]π3 (σ(r1 ,r2 ,r3 ,r4 )) [a2 , b2 ]π4 (σ(r1 ,r2 ,r3 ,r4 )) .
a1 1
In other words, if we identify the elements of JKp,λ via their normal forms
with octuples (k1 , k2 , l1 , l2 , r1 , r2 , r3 , r4 ) ∈ F8p , then f consists in applying σ to both
(k1 , k2 , l1 , l2 ) and (r1 , r2 , r3 , r4 ) and concatenating the resulting images.
Proof of Proposition 4.3. By Lemma 2.1, we need to show that there do not exist
two distinct elements x, y ∈ JKp,λ such that some endomorphism of JKp,λ maps y −1 x
to f (y)−1 f (x). We do so in a case distinction:
1. Case: x, y lie in a common coset of ζ JKp,λ . Then by definition of f and choice
of σ, y −1 x and f (y)−1 f (x) are distinct nontrivial elements of ζ JKp,λ . By Jonah
and Konvisser’s results, any automorphism of JKp,λ is of the form id +ϕ with
ζ JKp,λ ≤ ker(ϕ), in particular fixes ζ JKp,λ element-wise, and by Lemma 4.2,
any endomorphism of JKp,λ which is not an automorphism maps all elements
of ζ JKp,λ to 1. Hence no endomorphism of JKp,λ can map y −1 x to f (y)−1 f (x)
in this case, as required.
2. Case: x, y lie in different cosets of ζ JKp,λ . Then by definition of f and choice
of σ, y −1 x and f (y)−1 f (x) lie in different cosets of ζ JKp,λ , both distinct from
ζ JKp,λ itself. Jonah and Konvisser’s result that all automorphisms of JKp,λ
are central just means that they leave all cosets of ζ JKp,λ invariant, and by
Lemma 4.2, all other endomorphisms have their image contained in ζ JKp,λ , so
no endomorphism mapping y −1 x to f (y)−1 f (x) can exist in this case either.
5
Concluding remarks
We conclude this paper with some open questions and problems for further research.
Note that while we gained a deeper understanding of the asymptotic behavior of
endapp(G) and affapp(G) as |G| → ∞ in this paper, determining the precise values of
the two functions on a given finite group remains a challenging problem (just consider
the effort we had to put into it for small groups such as S3 or Z/6Z in the proof of
Proposition 2.14). Nonetheless, it would be nice to know these precise values at least
on a few “basic” classes of groups. For example, consider the following question,
which is, to the author’s knowledge, open in this generality:
Question 5.1. Is affapp(Z/nZ) = 2 for all n ∈ N, n ≥ 2?
16
Alexander Bors
Worst-case approximability
By Lemma 2.13(2,3), we do know that affapp(Z/nZ) = 2 when n is prime or a
power of 2, and it also holds for n = 6 by Proposition 2.14.
In this context, it would also be nice to extend our list of endapp(G) and affapp(G)
for “small” G from Proposition 2.14 further, which might also lead to more interesting
conjectures about their behavior on certain classes of finite groups:
Problem 5.2. Determine the precise values of endapp(G) and affapp(G) for all
finite groups G of order up to N , for N ∈ N as large as possible.
Finally, it would be interesting to determine provably asymptotically best possible
upper bounds on endapp(G) and affapp(G) in general. In this context, we note the
following: If a finite group G has a universal k-tuple, then |G|log2 |G| ≥ | End(G)| ≥
|G|k , so that the lower bounds on endapp(G) and affapp(G) which we can prove with
our current methods from Section 2 are at best logarithmic in |G|. This leads to the
question whether we hit this boundary for a good reason:
Question 5.3. Is endapp(G) ≤ log2 |G| and affapp(G) ≤ 1 + log2 |G| for all nontrivial finite groups G? If not, is it at least the case that affapp(G) = O(log |G|) as
|G| → ∞ for finite groups G?
At least, we do know by Theorem 1.1.4(1) that affapp(G) = O(log2 |G|) as |G| →
∞.
References
[1] A. Bors, On the endomorphism monoids of some groups with abelian automorphism group, preprint (2014), arXiv:1411.4190 [math.GR].
[2] A. Bors, Classification of Finite Group Automorphisms with a Large Cycle,
Comm. Algebra 44(11):4823–4843 (2016).
[3] A. Bors, Finite groups with an automorphism inverting, squaring
or cubing a non-negligible fraction of elements, submitted (2016),
an online version of the submitted manuscript is available on
http://www.sfb-qmc.jku.at/fileadmin/publications/Bors_InvSquCub.pdf.
[4] C. Carlet and Y. Tarannikov, Covering Sequences of Boolean Functions and Their
Cryptographic Significance, Des. Codes Cryptogr. 25(3):263–279 (2002).
[5] V.K. Jain, P.K. Rai and M.K. Yadav, On finite p-groups with abelian automorphism group, Internat. J. Algebra Comput. 23(5):1063–1077 (2013).
[6] D. Jonah and M. Konvisser, Some non-abelian p-groups with abelian automorphism groups, Arch. Math. (Basel) 26(1):131–133 (1975).
[7] D. Jonah and B.M. Schreiber, Transitive affine transformations on groups, Pacific
J. Math. 58(2):483–509 (1975).
17
Alexander Bors
Worst-case approximability
[8] M. Larsen and A. Shalev, Fibers of word maps and some applications, J. Algebra
354 (2012), 36–48.
[9] L. Poinsot, Non Abelian bent functions, Cryptogr. Commun. 4:1–23 (2012).
[10] D.J.S. Robinson, A Course in the Theory of Groups, Graduate Texts in Mathematics 80, Springer (2nd. ed. 1996).
[11] T. Tao,
254A, Notes 0a:
Stirling’s formula,
online notes,
https://terrytao.wordpress.com/2010/01/02/254a-notes-0a-stirlings-formula/.
18
| 4 |
High-Accuracy Real-Time
Whole-Body Human Motion Tracking
Based on Constrained Nonlinear Kalman Filtering
Jannik Steinbringa , Christian Manderyb , Nikolaus Vahrenkampb , Tamim Asfourb ,
Uwe D. Hanebecka
a
Intelligent Sensor-Actuator-Systems Laboratory (ISAS)
Institute for Anthropomatics and Robotics
Karlsruhe Institute of Technology (KIT), Germany
arXiv:1511.04278v1 [cs.SY] 13 Nov 2015
b
High Performance Humanoid Technologies Lab (H2 T)
Institute for Anthropomatics and Robotics
Karlsruhe Institute of Technology (KIT), Germany
Abstract
We present a new online approach to track human whole-body motion from motion capture
data, i.e., positions of labeled markers attached to the human body. Tracking in noisy data can be
effectively performed with the aid of well-established recursive state estimation techniques. This allows
us to systematically take noise of the marker measurements into account. However, as joint limits
imposed by the human body have to be satisfied during estimation, first we transform this constrained
estimation problem into an unconstrained one by using periodic functions. Then, we apply the Smart
Sampling Kalman Filter to solve this unconstrained estimation problem. The proposed recursive state
estimation approach makes the human motion tracking very robust to partial occlusion of markers
and avoids any special treatment or reconstruction of the missed markers. A concrete implementation
built on the kinematic human reference model of the Master Motor Map framework and a Vicon
motion capture system is evaluated. Different captured motions show that our implementation can
accurately estimate whole-body human motion in real-time and outperforms existing gradient-based
approaches. In addition, we demonstrate its ability to smoothly handle incomplete marker data.
1. Introduction
Understanding human whole-body motion has been a fundamental research interest with numerous
applications in the robotics community. Great efforts have been done to establish procedures for
capturing, representation, processing, and transfer of human motion in robotics, such as the Master
Motor Map (MMM) framework [1] providing a unifying reference model of the human body, which
this work builds upon.
Today, several commercial systems offer an easy way to capture human motion and provide
accurate Cartesian measurements of labeled markers attached to the human body. Based on those
measurements, the whole-body human motion can be tracked by estimating certain parameters of a
given kinematic model, e.g., root pose and joint angles (see Fig. 1). However, like all measurements,
the captured marker positions suffer from noise. Hence, stochastic approaches are needed to obtain a
precise estimation of all the kinematic parameters of interest. Additionally, joint limits imposed by
the human body have to be satisfied during estimation.
Email addresses: jannik.steinbring@kit.edu (Jannik Steinbring), mandery@kit.edu (Christian Mandery),
vahrenkamp@kit.edu (Nikolaus Vahrenkamp), asfour@kit.edu (Tamim Asfour), uwe.hanebeck@ieee.org
(Uwe D. Hanebeck)
Figure 1: Single frame of a tracked motion from labeled markers. Measured markers on the left. Pose
estimated with the proposed approach on the right.
Tracking the human motion is equivalent to estimating the state of a discrete-time stochastic
nonlinear dynamic system, where the system state is the human pose. This allows us to employ a
recursive nonlinear state estimator to solve the human motion tracking problem. The advantage of
such an estimator is that it maintains a probability distribution of the state estimate rather than
only a simple set of values. The estimator uses this distribution to optimally fuse the current state
estimate with newly available noisy measurements to obtain an updated distribution. Popular state
estimators are (nonlinear) Kalman filters [2, 3, 4, 5] or particle filters [6, 7]. Due to the many degrees
of freedom (DoF) required for a detailed human pose, particle filters are not suitable for real-time
tracking as they would need a huge amount of particles to get meaningful state estimates. Hence,
we choose to use the Smart Sampling Kalman Filter (S2 KF) [8] to estimate all parameters of the
kinematic model in real-time. Furthermore, we transform the constrained estimation problem into
an unconstrained problem with the aid of periodic functions. This is necessary to satisfy all the
imposed joint limits as Kalman filters can only estimate unconstrained quantities. Furthermore, the
problem of missing marker positions, e.g., due to occlusions, is addressed. Thanks to the recursive
state estimation approach, it is possible to ignore missing marker measurements while still obtain
good tracking results based on the remaining measurements only.
In [9], the authors solve the more general human motion tracking problem consisting of unlabeled
markers and an a priori unknown skeleton that has to be fitted to that marker data. Here, however,
we assume that the marker associations are already known and that we can make use of knowledge
about the kinematic model to get highly accurate estimates. Therefore, their special initialization
phase, i.e., a T-pose performed by the human, is not required by our approach. Moreover, the authors
are not clear about the real-time capabilities of their approach.
The problem of missing marker positions is addressed by the authors of [10]. Their solution is to
predict missing marker positions, use previously known marker positions, and get information based
on rigid body assumptions. Nonetheless, our proposed recursive state estimation approach implicitly
takes information of previous frames into account to be able to handle missing markers more easily.
2
Moreover, their tracking approach does not work with a fixed kinematic model. That is, they do
online joint localization in the marker point cloud, which results in a time-varying kinematic model.
A force-based approach in the fields of computer graphics is taken in [11]. The authors solve the
tracking problem with a physical simulation. Unfortunately, their approach does not consider the
noise of the measurements.
The remainder of this paper is structured as follows. In the next Section, we briefly present the
aforementioned MMM framework. After that, in Sec. 3, we present a general way to track whole-body
human motion over time from noisy marker measurements using a constrained nonlinear Kalman
filter. Based on the MMM framework, we formulate a concrete implementation of this general tracking
approach in Sec. 4. We evaluate the implementation with a complex motion in Sec. 5. Finally, the
conclusions are given in Sec. 6.
2. The Master Motor Map Framework
The Master Motor Map (MMM) framework [12, 1] provides an open-source framework for the
capturing, representation and analysis of human motion, and its reproduction on humanoid robots. It
has been used in a number of different robotics research applications that leverage human motion, e.g.
[13, 14, 15]. At its core, the MMM framework provides the MMM reference model, a whole-body model
containing both kinematic and dynamic specifications for the human body based on well-established
biomechanical literature by Winter [16] and others. This reference model allows the representation of
human motion using 6 DoF for the root pose, 52 DoF for torso, extremities, head, and eyes, and 2 ×
23 DoF for both hands. Fig. 2 shows the kinematics of the MMM reference model.
A central proposition of the MMM approach is to use the MMM reference model as a unique
intermediate model that allows the unifying representation of human motion provided by different
motion input sources, e.g., marker-less or marker-based motion capture, visual approaches based
on 2D or depth images, or data captured from inertial measurement units. Therefore, procedures
are needed that allow to transfer raw input data to the kinematic embodiment of the MMM model,
reconstructing position, orientation, and joint angle values of the model. In the terminology of the
MMM, these modules are called converters.
For the use of motion capture data as the input source, the MMM framework provides procedures
for the recording of motion that include a marker set for human whole-body motion capture comprising
a total of 56 marker locations at characteristic anatomical landmarks of the human body1 . Virtual
markers that match the markers placed on the subject are then added to the MMM reference model
at the corresponding locations. One converter is already provided by the MMM framework for the
reconstruction of human motion from motion capture [1]. This converter uses a framewise gradientbased optimization approach based on the Jacobian matrix of the MMM kinematics to fit the model
pose and is compared to the new approach proposed in this paper in Sec. 5.
3. Whole-Body Human Motion Tracking
In this Section, we present a new way to track the motion of a human based on noisy marker
measurements using a constrained nonlinear Kalman filter. The proposed approach is not limited to a
certain kinematic model of the human motion or how the measurements of the markers are obtained.
3.1. Problem Formulation
Given a kinematic model of the human body parameterized by J joint angles
(1)
(J)
θk = [θk , . . . , θk ]>
1
In-depth specifications of the marker set are available online: https://motion-database.humanoids.kit.edu/
marker_set/
3
Figure 2: Whole-body kinematics of the MMM reference model, including joints abbreviations
(from [1]).
and root pose consisting of position
rk = [rkx , rky , rkz ]>
in Cartesian space and orientation
ok = [ork , opk , oyk ]>
in roll, pitch, and yaw angles2 , our goal is to estimate the pose of a human at discrete time steps k.
The estimation relies on several unique markers attached to the human body at known positions, e.g.,
on a shoulder or hand that are observed and measured by a tracking system. At each time step k, the
tracking system provides us with a set
(1)
(M )
Mk = {m̃k , . . . , m̃k
}
(i)
of M labeled noisy marker positions m̃k in Cartesian space. In addition, due to human joint
limitations, the estimated pose has to satisfy the bound constraint
(j)
lj ≤ θk ≤ uj
∀j ∈ {1, . . . , J}
(1)
(j)
for all joint angles θk with lower bound lj and upper bound uj .
3.2. From Model Parameters to Marker Positions
In order to infer the kinematic model parameters θk , rk , and ok from the received marker positions
Mk based on a nonlinear Kalman filter, we need a mapping from these parameters to each individual
marker position. This mapping consists of two parts. First, given a concrete pose (in form of the
parameters θk , rk , and ok ), for each marker the forward kinematics of the respective kinematic chain
2
Vectors are underlined and matrices are printed bold face.
4
has to be computed. As a result, it is known where to expect all markers if the human has that
concrete pose. Second, like all measurements, the measured marker positions suffer from noise. Hence,
that noise has to be taken into account in order to obtain good estimation results, especially in case
of strong noise. Both together leads to the desired mapping
(i)
(i)
mk = h(i) (rk , ok , θk ) + v k
,
(2)
(i)
where mk denotes the i-th marker position in Cartesian space, h(i) (·, ·, ·) the forward kinematics for
(i)
(i)
the i-th marker, and v k additive, zero-mean, and white Gaussian noise with covariance matrix Rk .
(i)
The choice of Rk depends on the utilized tracking system. Moreover, it is assumed that the noise
(i)
(j)
(i)
vectors v k and v k with i 6= j are mutually independent. Note also the difference between mk and
(i)
m̃k : the first one is a random vector, whereas the latter one is a realization of that random vector.
3.3. Satisfying the Joint Angle Bound Constraints
The considered estimation task poses an additional challenge for Kalman filters: a (nonlinear)
Kalman filter, by design, only estimates unconstrained quantities. That is, directly estimating θk
with a Kalman filter will violate the constraints (1). Recall that the estimate of a Kalman filter is
represented by a mean vector and a covariance matrix. In order to take the bound constraints properly
into account, it is necessary that (i) the mean vector must always lie inside the bounded region of the
state space, and (ii) the covariance matrix also has to reflect that the state space is bounded, that
is, the covariance matrix has to be smaller compared to an unconstrained state space. In literature,
there exist various approaches to incorporate constraints into Kalman filters.
• Perfect measurements [4] are designed for equality constraints and are not suitable for inequality
constraints. Hence, they cannot be applied to the considered bound constraint problem.
• Projection techniques [4] correct the posterior state mean after a Kalman filter prediction/update
step. Unfortunately, they cannot correct the posterior state covariance matrix as well.
• Pdf truncation [4] is an elegant way to respect linear inequality constraints and corrects both
posterior state mean and covariance matrix. However, it is computationally expensive for large
state dimensions as it requires several Gram-Schmidt orthogonalizations and eigendecompositions
of the state covariance matrix, which is not guaranteed to converge, and hence, makes this
approach unreliable.
• The sampling-based approach proposed in [17] can be seen as a numerical approximation of the
pdf truncation approach. The problem here is that situations may occur where all samples lie
outside of the constrained region and no constrained estimate can be obtained. This is analogous
to the known sample degeneracy problem of particle filters.
As we seek a real-time capable and accurate human motion tracking, we choose another way to
satisfy (1) for all joint angles. We perform a parameter transformation using a periodic function
that is defined on (−∞, ∞) but its range is limited to the interval [−1, 1]. We introduce a new joint
(j)
(j)
parameter Θk for each joint angle θk according to the mapping
(j)
(j)
θk = gj (Θk ) =
uj − lj
lj + uj
(j)
sin(Θk ) +
.
2
2
(j)
As a result, Θk can take any value, i.e., it is unconstrained, while (1) is always satisfied. It should be
(j)
noted that this periodic approach, however, is sensitive to large uncertainties in the parameters Θk ,
that is, its uncertainty should not be larger than the period of the periodic function to get meaningful
estimation results.
5
Alternatively, sigmoid functions like the hyperbolic tangent could also be used for such a transformation. However, a nonlinear Kalman filter has problems to properly update a joint angle estimate in
situations where it is near to a bound constraint as the gradient of a sigmoid function becomes very
small for large parameters.
Analogously to the vector θk , we define the joint parameter vector
(1)
(J)
Θk = [Θk , . . . , Θk ]> .
We also introduce the vector-valued function
(1)
g1 (Θk )
..
θk = g(Θk ) =
.
(3)
(J)
gJ (Θk )
that transforms all joint parameters back to their corresponding joint angles.
3.4. State Estimation with the Smart Sampling Kalman Filter
At this point, we can introduce the system state
>
> >
xk = [r>
k , ok , Θk ]
(4)
that fully describes the constrained whole-body human motion at time step k. This state vector can
now be estimated with a usual nonlinear Kalman filter. A Kalman filter is a recursive state estimator
consisting of two alternating parts: (i) the prediction step that propagates the state estimate, that is,
mean and covariance matrix, from the last time step k − 1 to the current time step k and (ii) the
measurement update that corrects the predicted state estimate given a set of measurements.
First, we concentrate on the measurement update. Here, we have to define what the actual
measurement is that will be processed by the Kalman filter, and the measurement equation that
maps xk to that measurement. To obtain the measurement, we stack the received single marker
(i)
measurements m̃k to a 3M -dimensional measurement vector
(1)
(M ) > >
m̃k = [(m̃k )> , . . . , (m̃k
) ]
(5)
.
For the measurement equation, we combine the M marker mappings (2) with (3), and stack them in
the same manner of (5) which yields
(1)
(1)
(1)
h (rk , ok , g(Θk ))
mk
vk
.
..
..
+ . ,
. =
.
.
(M )
(M )
(M
)
mk
vk
h (rk , ok , g(Θk ))
| {z
} |
{z
} | {z }
mk
(6)
vk
h(xk )
where the zero-mean noise vector v k has the block diagonal covariance matrix
(1)
(M )
Rk = diag(Rk , . . . , Rk
) .
Of course, the order of stacking has to be the same for both (5) and (6). Otherwise, marker positions
would not be related to their corresponding measurements. Now, given a predicted state estimate by
state mean x̂pk and state covariance matrix Cpk , the following moments are computed based on the
6
measurement model (6)
Z
h(xk ) · N (xk ; x̂pk , Cpk ) dxk
Z
(h(xk ) − m̂k ) · (h(xk ) − m̂k )> ·
m̂k =
Cm
k
=
(7)
N (xk ; x̂pk , Cpk ) dxk + Rk
Cx,m
=
k
Z
(xk − x̂pk ) · (h(xk ) − m̂k )> ·
N (xk ; x̂pk , Cpk ) dxk ,
where N (xk ; x̂pk , Cpk ) denotes the Gaussian probability density function.
Together with the measurement (5), we get the posterior, i.e., the corrected, state estimate by
computing the posterior state mean according to
−1
(Cm
(m̃k − m̂k ) ,
x̂ek = x̂pk + Cx,m
k )
k
and the posterior state covariance matrix according to
−1
>
Cek = Cpk − Cx,m
(Cm
(Cx,m
.
k )
k
k )
Second, the prediction step requires a system equation that models the temporal evolution of the
system state between two measurement updates. Here, this means (slight) changes in the human
motion from one time step to the next one, e.g., a movement of the root position or moving an
extremity. A general system equation is given by
xk = ak (xk−1 ) + wk ,
(8)
where wk denotes zero-mean white Gaussian noise with covariance matrix Qk . The function ak (·)
models the actual changes in the kinematic model parameters over time and should take any prior
knowledge about the human motion scenario into account. For example, it can rely on velocities to
predict where the human or its extremities will be in the next time step. Note that such velocities can
be estimated together with the actual kinematic model parameters by augmenting the state system (4)
with such velocities. The noise, and therefore Qk , incorporates modelling errors into the prediction
and depends on the used tracking system and the elapsed time between measurement updates. Given
the system equation (8) and the state estimate from the last time step by state mean x̂ek−1 and state
covariance matrix Cek−1 , the Kalman filter computes the predicted state mean according to
Z
p
x̂k = ak (xk−1 ) · N (xk−1 ; x̂ek−1 , Cek−1 ) dxk−1 ,
(9)
and the predicted state covariance matrix according to
Z
p
Ck = (ak (xk−1 ) − x̂pk ) · (ak (xk−1 ) − x̂pk )> ·
N (xk−1 ; x̂ek−1 , Cek−1 ) dxk−1
(10)
+ Qk .
Up to this point, no concrete nonlinear Kalman filter has been chosen to compute the multidimensional integrals in (7), (9), and (10). Basically, every nonlinear Kalman filter such as the extended
Kalman filter (EKF) [4] or the unscented Kalman filter (UKF) [3] could be used. However, on the
one hand, the EKF relies on explicit linearization of the measurement model around the predicted
state mean, which requires the Jacobian matrix of (6) and (8). Moreover, it does not incorporate the
uncertainty of the predicted state estimate into this linearization, making this approach sensitive to the
predicted state mean. On the other hand, the UKF relies on statistical linearization, which incorporates
7
the prior uncertainty and does not need any Jacobian matrix. Instead, the UKF propagates a set of
samples through the measurement/system model to compute the integrals. Unfortunately, the number
of samples is fixed and cannot be increased to obtain more reliable and more accurate state estimates.
To get rid of the limitations imposed by the EKF and UKF, we use the Smart Sampling Kalman
Filter (S2 KF) [18, 8]. The S2 KF also relies on samples to compute the integrals, but it can use an
arbitrary number of optimally placed samples3 . Hence, we can use more samples than the UKF to
improve the estimation quality, but only as many as possible to guarantee real-time capability.
Finally, to start with the recursive state estimation, an initial state estimate with initial mean
x̂e0 and initial covariance matrix Ce0 is required. These initial values depend on the quality of the
measurements and other prior knowledge of the human motion scenario, e.g., if it is known where the
human starts or what its initial pose is.
3.5. Working with Incomplete Measurement Sets
(i)
If the position m̃k for the i-th marker cannot be obtained by the tracking system at time step k,
e.g., due to occlusions, we omit it from (5) and the corresponding measurement function h(i) (·) from
(6), and use only the remaining measured marker positions for the measurement update. That is, an
estimation of root pose and joint angles is still possible for that time step. However, due to the lack
of certain marker position measurements, the filter has less information about the current human
pose. Consequently, the estimation quality can be less accurate. Nonetheless, an estimation is still
possible thanks to prior knowledge of the pose, i.e., the predicted state estimate.
4. A New Converter for the MMM Framework
After describing the general whole-body human motion tracking approach based on constrained
nonlinear Kalman filtering in Sec. 3, we implement this approach for the kinematic model presented
in Sec. 2. Here, we select a subset of the joints from the MMM reference model based on the parts
of motion that can be estimated using our motion capture setup. That is, hands, eyes, and some
joints on shoulders and feet have been excluded. In total, J = 40 joints are used for the kinematic
model resulting in a system state dimension of 46. Moreover, root position and marker positions
are measured in millimeters and root orientation and joint angles are measured in radians (this is
(i)
important as it also defines the units of the noise covariance matrices Rk and Qk ).
In addition, the MMM marker set describes the placement on markers on the human body and
the MMM reference model provides the corresponding forward kinematics h(i) (·, ·, ·) required for the
measurement model (6). We use M = 50 of those markers. Their positions are measured with a
Vicon MX10 system using ten T10 cameras. It is an optical motion capture system based on passive
(reflective) markers. The system records at 100 Hz, that is, every 10 ms we get a new set of markers
Mk . For the measurement update, the measurement noise properties of the Vicon system, i.e., the
(i)
covariance matrices Rk , have to be known. Experimentally, we have found that the marker positions
provided by the Vicon system are disturbed approximately with
(i)
Rk = 10−4 I3 ,
where I3 denotes the identity matrix of dimension three. To perform the measurement update, the
S2 KF is configured to use 301 samples. This is more than six times the number of samples that would
be used by the UKF.
The highly accurate marker position measurements provided by the Vicon system allows us to
model the temporal evolution of the human motion with a simple random walk system model according
to
xk = xk−1 + wk .
3
An open-source MATLAB implementation of the S2 KF is available online:
nonlinearestimation/toolbox/
8
https://bitbucket.org/
That is, it is assumed that the system state does not evolve significantly in the 10 ms between two
measurement updates (only the uncertainty of the estimate will increase) and that the measurement
update corrects the state estimate adequately. As a consequence, the Kalman filter prediction can be
computed easily in closed-form with
x̂pk = x̂ek−1
Cpk = Cek−1 + Qk ,
that is, no sampling (like for the measurement update) is required at all. Furthermore, the system
noise covariance matrix is set to the time-invariant diagonal matrix
Qk = diag(25, 25, 25, 10−10 , . . . , 10−10 ) .
For the initial system state estimate, we assume no prior knowledge about the human motion
scenario. This makes this implementation very versatile to track any human motion without a special
treatment. More precisely, we use the first set of available measurements M0 to initialize the S2 KF.
The root position and its covariance is obtained according to
M
1 X (i)
r̂0 =
m̃0 ,
M
Cr0 =
1
M
i=1
M
X
(i)
(i)
(m̃0 − r̂0 ) · (m̃0 − r̂0 )> ,
i=1
the orientation mean and covariance is set to
ô0 = [0, 0, 0]> ,
Co0 = 10−6 I3 ,
and the joint parameters and their uncertainties are initialized with
Θ̂0 = [0, . . . , 0]> ,
−10
CΘ
I40 ,
0 = 10
that is, each initial joint angle is set to the average of its bound constraints. Then, the initial system
state estimate is given by
> >
>
x̂e0 = [r̂>
,
0 , ô0 , Θ̂0 ]
Ce0 = diag(Cr0 , Co0 , CΘ
0) .
5. Evaluation
In this Section, we evaluate the implemented human motion tracking approach from Sec. 4 with
a performed handstand. Its Vicon motion recording was taken from the KIT Whole-Body Human
Motion Database [19], which provides a rich collection of raw human motion capture recordings for
numerous kinds of motion tasks in the industry standard C3D file format. The recording contains 675
frames (time steps). In all frames, the entire set of marker positions is available. That means that we
can work with a complete measurement set in all time steps.
We compare the new approach with the Jacobian-based MMM converter [1] mentioned in Sec. 2.
In order to assess the human motion tracking performance, for each estimated pose (one for each time
step), we compute the expected marker positions using their respective forward kinematics. Then, we
compute the distances between the expected and the measured marker positions and subsequently
build the average over all these distances.
9
Average marker distance in cm
60
50
40
30
20
Jacobian-based approach all markers
New approach all markers
New approach hidden markers
10
0
100
200
300
400
500
600
Frame
Figure 3: Averaged distances between measured marker positions and marker positions expected by
the estimated pose. The red lines indicate the frames shown in Figures 5 and 6.
Runtime in ms
8
7
6
5
100
200
300
400
500
600
Frame
Figure 4: Estimation runtime of the new approach. The evaluation is performed on an Intel Core
i7-3770 CPU.
The marker distances for the new approach (orange) and the Jacobian-based approach (green) are
depicted in Fig. 3. On the one hand, it can be seen that the new approach requires approximately
20 frames (only 200 ms) to converge. This can be explained with the nonlinear Kalman filter that
gradually improves its initial state estimate over time. However, after convergence, the marker
distances of the new approach do not change much over time, even when the pose changes drastically
at beginning and end of the handstand. It offers a total average marker distance of only 4 to 5 cm.
It is important to note that a further reduction in the marker distances is not straightforward. The
problem is that markers attached to the human body will never exactly coincide with the corresponding
marker positions defined in the reference model. On the other hand, the Jacobian-based approach
has problems to track the handstand motion. Although it can handle ordinary motions, like human
bipedal locomotion tasks well, it clearly has problems with such a complex motion as the handstand.
At beginning and end of the handstand, its average marker distances are over 60 cm. Also the general
marker distance level is much higher compared to the new approach.
In Fig. 4, the runtime of the new approach is shown. Its runtime varies over time but stays always
below 8 ms. As the Vicon systems records at 100 Hz, the proposed approach can be used to track a
whole-body human motion in real-time. Moreover, Fig. 5 illustrates the recorded marker positions
and the corresponding poses estimated by the new approach for selected frames. Despite the short
runtimes, the handstand is tracked very accurately.
10
Figure 5: Selected frames of a performed handstand. In the top row, measured markers. In the
bottom row, corresponding poses estimated by the proposed approach.
Figure 6: Same frames as in Fig. 5, but with hidden markers. In the top row, measured markers,
where hidden markers are not shown. In the bottom row, corresponding poses estimated by the
proposed approach.
11
Number of hidden markers
30
25
20
15
10
Run A
Run B
Run C
Average
5
0
100
200
300
400
500
600
Frame
Figure 7: Number of simulated hidden markers.
In contrast to the Jacobian-based approach, the new approach can handle incomplete measurement
sets as described in Sec. 3.5. To simulate markers that cannot be observed by the Vicon system for
some period of time, we randomly remove markers from the previously used handstand recording. More
precisely, in every frame each marker will become invisible for the next frames with a probability of
0.5 %. The number of frames is determined by drawing a random number from a Poisson distribution
with parameter λ = 100, i.e., on average a marker will be hidden for 100 frames (one second). Hence,
over time the total number of hidden markers will vary from frame to frame. In order to get meaningful
results, we simulate this in 100 Monte Carlo runs and try to track the human motion in each run with
the randomly modified marker sets. Fig. 7 depicts the number of hidden markers over time for three
different simulation runs as well as the average over all runs. As can be seen, there are frames where
more than 25 of the 50 markers are not available for the motion estimation. Averaged over all runs
and frames, 16 markers cannot be observed.
Although markers are assumed to be hidden for the estimation, we still know the originally
recorded position. Hence, we can compute the same marker distances as described above (also for the
markers that were not available for the state estimation). The results (blue area) are also given in
Fig. 3. The upper bound of the blue area denotes the largest averaged marker distances occurred
for a run, whereas the lower bound denotes the smallest averaged marker distance. We see that the
missed markers can slightly increase the distance up to 9 cm. Nonetheless, the results are still very
good. Looking at the motion estimated from one simulation run in Fig. 6, including the remaining
markers available for processing, we see that the motion is very similar to the one in Fig. 5.
6. Conclusions
In this paper, we proposed a new way to track whole-body human motion using measurements from
labeled markers attached to the human body. Tracking a human motion is equivalent to estimating
the state of a stochastic dynamic system. Hence, we chose to rely on the Smart Sampling Kalman
Filter (S2 KF) to perform the human motion tracking. This recursive state estimation approach makes
it possible to systematically take the uncertainty of the marker measurements into account while
being at the same time very robust to the partial collusion of markers. However, before we could
apply the filter, we had to incorporate the joint limits imposed by the human body into the estimation
procedure. This was done by transforming the constraint estimation problem into an unconstrained
problem using periodic functions. An implementation of the proposed approach was built around
the kinematic reference model of the Master Motor Map and a Vicon motion capture system. The
evaluations showed that the proposed approach offers highly accurate estimates of complex whole-body
human motions, even if half of the markers could not be observed.
12
Acknowledgment
The research leading to these results has received funding from the European Union Seventh
Framework Programme under grant agreement no 611909 (KoroiBot) and the European Union H2020
Programme under grant agreement no 643666 (I-SUPPORT).
References
[1] O. Terlemez, S. Ulbrich, C. Mandery, M. Do, N. Vahrenkamp, and T. Asfour, “Master Motor Map (MMM)
- framework and toolkit for capturing, representing, and reproducing human motion on humanoid robots,” in
IEEE-RAS International Conference on Humanoid Robots (Humanoids), 2014, pp. 894–901.
[2] K. Ito and K. Xiong, “Gaussian Filters for Nonlinear Filtering Problems,” IEEE Transactions on Automatic Control,
vol. 45, no. 5, pp. 910–927, May 2000.
[3] S. J. Julier and J. K. Uhlmann, “Unscented Filtering and Nonlinear Estimation,” in Proceedings of the IEEE,
vol. 92, Mar. 2004, pp. 401–422.
[4] D. Simon, Optimal State Estimation, 1st ed. Wiley & Sons, 2006.
[5] P. Stano, Z. Lendek, J. Braaksma, R. Babuska, C, de Keizer, and A. J. den Dekker, “Parametric Bayesian Filters
for Nonlinear Stochastic Dynamical Systems: A Survey,” IEEE Transactions on Cybernetics, vol. 43, no. 6, pp.
1607–1624, Dec. 2013.
[6] B. Ristic, S. Arulampalam, and N. Gordon, Beyond the Kalman Filter: Particle Filters for Tracking Applications.
Artech House Publishers, 2004.
[7] A. Doucet and A. M. Johansen, “A Tutorial on Particle Filtering and Smoothing: Fifteen Years Later,” in Oxford
Handbook of Nonlinear Filtering, 2011, pp. 656–704.
[8] J. Steinbring and U. D. Hanebeck, “LRKF Revisited: The Smart Sampling Kalman Filter (S2 KF),” Journal of
Advances in Information Fusion, vol. 9, no. 2, pp. 106–123, Dec. 2014.
[9] J. Meyer, M. Kuderer, J. Müller, and W. Burgard, “Online Marker Labeling for Fully Automatic Skeleton Tracking
in Optical Motion Capture,” in Proceedings of the IEEE International Conference on Robotics & Automation
(ICRA), Hong Kong, China, May 2014, pp. 5652–5657.
[10] A. Aristidou and J. Lasenby, “Real-Time Marker Prediction and CoR Estimation in Optical Motion Capture,” The
Visual Computer, vol. 29, no. 1, pp. 7–26, 2013.
[11] V. B. Zordan and N. C. Van Der Horst, “Mapping Optical Motion Capture Data to Skeletal Motion Using a
Physical Model,” in Proceedings of the 2003 ACM SIGGRAPH/Eurographics Symposium on Computer Animation,
2003, pp. 245–250.
[12] P. Azad, T. Asfour, and R. Dillmann, “Toward an unified representation for imitation of human motion on
humanoids,” in IEEE International Conference on Robotics and Automation (ICRA), April 2007, pp. 2558–2563.
[13] M. Do, T. Asfour, and R. Dillmann, “Towards a unifying grasp representation for imitation learning on humanoid
robots,” in IEEE International Conference on Robotics and Automation (ICRA), May 2011, pp. 482–488.
[14] T. Asfour, M. Do, K. Welke, A. Bierbaum, P. Azad, N. Vahrenkamp, S. Gärtner, A. Ude, and R. Dillmann,
“From sensorimotor primitives to manipulation and imitation strategies in humanoid robots,” in Robotics Research,
ser. Springer Tracts in Advanced Robotics, C. Pradalier, R. Siegwart, and G. Hirzinger, Eds. Springer Berlin
Heidelberg, 2011, vol. 70, pp. 363–378.
[15] C. Mandery, J. Borràs, M. Jöchner, and T. Asfour, “Analyzing whole-body pose transitions in multi-contact
motions,” in IEEE-RAS International Conference on Humanoid Robots (Humanoids), 2015.
[16] D. A. Winter, Biomechanics and motor control of human movement, 4th ed. Hoboken, NJ: Wiley, 2009.
[17] O. Straka, J. Dunı́k, and M. Šimandl, “Truncation Nonlinear Filters for State Estimation with Nonlinear Inequality
Constraints,” Automatica, vol. 48, no. 2, pp. 273–286, Feb. 2012.
[18] J. Steinbring and U. D. Hanebeck, “S2 KF: The Smart Sampling Kalman Filter,” in Proceedings of the 16th
International Conference on Information Fusion (Fusion 2013), Istanbul, Turkey, Jul. 2013, pp. 2089–2096.
[19] C. Mandery, O. Terlemez, M. Do, N. Vahrenkamp, and T. Asfour, “The KIT Whole-Body Human Motion Database,”
in International Conference on Advanced Robotics (ICAR), July 2015, pp. 329–336.
13
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arXiv:1705.03980v2 [math.AC] 25 Jan 2018
AUSLANDER MODULES
PEYMAN NASEHPOUR
Dedicated to my father, Maestro Nasrollah Nasehpour
Abstract. In this paper, we introduce the notion of Auslander modules, inspired from Auslander’s zero-divisor conjecture (theorem) and give some interesting results for these modules. We also investigate torsion-free modules.
0. Introduction
Auslander’s zero-divisor conjecture in commutative algebra states that if R is a
Noetherian local ring, M is a nonzero R-module of finite type, and finite projective
dimension, and r ∈ R is not a zero-divisor on M , then r is not a zero-divisor on
R [6, p. 8] and [5, p. 496]. This “conjecture” is in fact a theorem, after Peskine
and Szpiro in [15] showed that Auslander’s zero-divisor theorem is a corollary of
their new intersection theorem and thereby proved it for a large class of local rings.
Also see [16, p. 417]. Note that its validity without any restrictions followed when
Roberts [17] proved the new intersection theorem in full generality. Also see Remark
9.4.8 in [1].
Let M be an arbitrary unital nonzero module over a commutative ring R with
a nonzero identity. Inspired from Auslander’s zero-divisor theorem, one may ask
when the inclusion ZR (R) ⊆ ZR (M ) holds, where by ZR (M ), we mean the set of
all zero-divisors of the R-module M . In Definition 1.1, we define an R-module M
to be Auslander if ZR (R) ⊆ ZR (M ) and in Proposition 1.2, we give a couple of
examples for the families of Auslander modules. The main theme of §1 is to see
under what conditions if M is an Auslander R-module, then the S-module M ⊗R S
is Auslander, where S is an R-algebra (see Theorem 1.4, Theorem 1.6, and Theorem
1.10). For example, in Corollary 1.11, we show that if M is an Auslander R-module,
B a content R-algebra, and M has property (A), then M ⊗R B is an Auslander
B-module. For the definition of content algebras refer to [14, Section 6].
On the other hand, let us recall that an R-module M is torsion-free if the natural
map M → M ⊗ Q is injective, where Q is the total quotient ring of the ring R
[1, p. 19]. It is easy to see that M is a torsion-free R-module if and only if
ZR (M ) ⊆ ZR (R). In §2, we investigate torsion-free property under polynomial and
power series extensions (see Theorem 2.1 and Theorem 2.2). We also investigate
torsion-free Auslander modules (check Proposition 2.5, Theorem 2.7, and Theorem
2.9).
In this paper, all rings are commutative with non-zero identities and all modules
are unital.
2010 Mathematics Subject Classification. 13A15, 13B25, 13F25.
Key words and phrases. Auslander modules, Auslander’s zero-divisor conjecture, content algebras, torsion-free modules.
1
2
PEYMAN NASEHPOUR
1. Auslander Modules
We start the first section by defining Auslander modules:
Definition 1.1. We define an R-module M to be an Auslander module, if r ∈ R
is not a zero-divisor on M , then r is not a zero-divisor on R, or equivalently, if the
following property holds:
ZR (R) ⊆ ZR (M ).
Let us recall that if M is an R-module, the content of m ∈ M , denoted by c(m),
is defined to be the following ideal:
\
c(m) = {I ∈ Id(R) : m ∈ IM },
where by Id(R), we mean the set of all ideals of R. The R-module M is said to be
a content R-module, if m ∈ c(m)M , for all m ∈ M [14]. In the following, we give
some families of Auslander modules:
Proposition 1.2 (Some Families of Auslander Modules). Let M be an R-module.
Then the following statements hold:
(1) If R is a domain, then M is an Auslander R-module.
(2) If M is a flat and content R-module such that for any s ∈ R, there is an
x ∈ M such that c(x) = (s). Then M is an Auslander R-module.
(3) If M is an R-module such that Ann(M ) = (0), then M is an Auslander
R-module.
(4) If for any nonzero s ∈ R, there is an x ∈ M such that s · x 6= 0, i.e.
Ann(M ) = (0), then HomR (M, M ) is an Auslander R-module.
(5) If N is an R-submodule of an R-module M and N is Auslander, then M
is also Auslander.
(6) If M is an Auslander R-module, then M ⊕ M ′ is an Auslander R-module
for any R-module M ′ . In particular, if {Mi }i∈Λ is a family of R-modules
and there
L is an i ∈QΛ, say i0 , such that Mi0 is an Auslander R-module,
then i∈Λ Mi and i∈Λ Mi are Auslander R-modules.
Proof. The statement (1) is obvious. We prove the other statements:
(2): Let r ∈ ZR (R). By definition, there is a nonzero s ∈ R such that r · s = 0.
Since in content modules c(x) = (0) if and only if x = 0 [14, Statement 1.2] and
by assumption, there is a nonzero x ∈ M such that c(x) = (s), we obtain that
r · c(x) = (0). Also, since M is a flat and content R-module, by [14, Theorem 1.5],
r · c(x) = c(r · x). This implies that r ∈ ZR (M ).
(3): Suppose that r · s = 0 for some nonzero s in R. By assumpstion, there exists
an x in M such that s · x 6= 0, but r · (s · x) = 0, and so r is a zero-divisor on M .
(4): Let r ∈ ZR (R). So, there is a nonzero s ∈ R such that r · s = 0. Define
fs : M −→ M by fs (x) = s · x. By assumption, fs is a nonzero element of
HomR (M, M ). But rfs = 0. This means that r ∈ ZR (Hom(M, M )).
(5): The proof is straightforward, if we consider that Z(N ) ⊆ Z(M ).
The statement (6) is just a corollary of the statement (5).
Proposition 1.3. Let M be an Auslander R-module and S a multiplicatively closed
subset of R contained in R − ZR (M ). Then, MS is an Auslander RS -module.
Proof. Let ZR (R) ⊆ ZR (M ) and S be a multiplicatively closed subset of R such
that S ⊆ R − ZR (M ). Take r1 /s1 ∈ ZRS (RS ). So there exists an r2 /s2 6= 0/1 such
AUSLANDER MODULES
3
that (r1 · r2 )/(s1 · s2 ) = 0/1. Since S ⊆ R − ZR (R), we have r1 · r2 = 0, where
r2 6= 0. But ZR (R) ⊆ ZR (M ), so r1 ∈ ZR (M ). Consequently, there is a nonzero
m ∈ M such that r1 · m = 0. Since S ⊆ R − ZR (M ), m/1 is a nonzero element of
MS . This point that r1 /s1 · m/1 = 0/1, causes r1 /s1 to be an element of ZRS (MS )
and the proof is complete.
Let us recall that an R-module M has property (A), if each finitely generated
ideal I ⊆ ZR (M ) has a nonzero annihilator in M [11, Definition 10]. Examples
of modules having property (A) include modules having very few zero-divisors [11,
Definition 6]. Especially, finitely generated modules over Noetherian rings have
property (A) [7, p. 55]. Homological aspects of modules having very few zerodivisors have been investigated in [12]. Finally, we recall that if R is a ring, G a
monoid, and f = r1 X g1 + · · · + rn X gn is an element of the monoid ring R[G], then
the content of f , denoted by c(f ), is the finitely generated ideal (r1 , . . . , rn ) of R.
Theorem 1.4. Let the R-module M have property (A) and G be a commutative,
cancellative, and torsion-free monoid. Then, M [G] is an Auslander R[G]-module if
and only if M is an Auslander R-module.
Proof. (⇒): Let r ∈ ZR (R). So, r ∈ ZR[G] (R[G]) and by assumption, r ∈
ZR[G] (M [G]). Clearly, this means that there is a nonzero g in ZR[G] (M [G]) such
that rg = 0. Therefore, there is a nonzero m in M such that rm = 0.
(⇐): Let f ∈ ZR[G] (R[G]). By [11, Theorem 2], there is a nonzero element
r ∈ R such that f · r = 0. This implies that c(f ) ⊆ ZR (R). But M is an Auslander
R-module, so ZR (R) ⊆ ZR (M ), which implies that c(f ) ⊆ ZR (M ). On the other
hand, M has property (A). Therefore, c(f ) has a nonzero annihilator in M . Hence,
f ∈ ZR[G] (M [G]) and the proof is complete.
Note that a semimodule version of Theorem 1.4 has been given in [10].
It is good to mention that if R is a ring and f = a0 + a1 X + · · · + an X n + · · ·
is an element of R[[X]], then Af is defined to be the ideal of R generated by the
coefficients of f , i.e.
Af = (a0 , a1 , . . . , an , . . .).
One can easily check that if R is Noetherian, then Af = c(f ). The following lemma
is a generalization of Theorem 5 in [4]:
Lemma 1.5. Let R be a Noetherian ring, M a finitely generated R-module, f ∈
R[[X]], g ∈ M [[X]] − {0}, and f g = 0. Then, there is a nonzero constant m ∈ M
such that f · m = 0.
Proof. Define c(g), the content of g, to be the R-submodule of M generated by its
coefficients. If c(f )c(g) = (0), then choose a nonzero m ∈ c(g). Clearly, f · m = 0.
Otherwise, by Theorem 3.1 in [2], one can choose a positive integer k, such that
c(f )c(f )k−1 c(g) = 0, while c(f )k−1 c(g) 6= 0. Now for each nonzero element m in
c(f )k−1 c(g), we have f · m = 0 and the proof is complete.
Theorem 1.6. Let R be a Noetherian ring and the R-module M have property (A).
Then, M [[X]] is an Auslander R[[X]]-module if and only if M is an Auslander Rmodule.
Proof. By Lemma 1.5, the proof is just a mimicking of the proof of Theorem 1.4.
4
PEYMAN NASEHPOUR
Since finitely generated modules over Noetherian rings have property (A) [7, p.
55], we have the following corollary:
Corollary 1.7. Let R be a Noetherian ring and M be a finitely generated R-module.
Then, M [[X]] is an Auslander R[[X]]-module if and only if M is an Auslander Rmodule.
Remark 1.8 (Ohm-Rush Algebras). Let us recall that if B is an R-algebra, then
B is said to be an Ohm-Rush R-algebra, if f ∈ c(f )B, for all f ∈ B [3, Definition
2.1]. It is easy to see that if P is a projective R-algebra, then P is an OhmRush R-algebra [14, Corollary 1.4]. Note that if R is a Noetherian ring and f =
a0 + a1 X + · · · + an X n + · · · is an element of R[[X]], then Af = c(f ), where Af is
the ideal of R generated by the coefficients of f . This simply implies that R[[X]] is
an Ohm-Rush R-algebra.
Now we go further to define McCoy algebras, though we don’t go through them
deeply in this paper. McCoy semialgebras (and algebras) and their properties
have been discussed in more details in author’s recent paper on zero-divisors of
semimodules and semialgebras [10].
Definition 1.9. We say that B is a McCoy R-algebra, if B is an Ohm-Rush Ralgebra and f · g = 0 with g 6= 0 implies that there is a nonzero r ∈ R such that
c(f ) · r = (0), for all f, g ∈ B.
Since any content algebra is a McCoy algebra [14, Statement 6.1], we have plenty
of examples for McCoy algebras. For instance, if G is a torsion-free abelian group
and R is a ring, then R[G] is a content - and therefore, a McCoy - R-algebra [13].
For other examples of McCoy algebras, one can refer to content algebras given in
Examples 6.3 in [14]. Now we proceed to give the following general theorem on
Auslander modules:
Theorem 1.10. Let M be an Auslander R-module and B a faithfully flat McCoy
R-algebra. If M has property (A), then M ⊗R B is an Auslander B-module.
Proof. Let f ∈ ZB (B). So by definition, there is a nonzero r ∈ R such that
c(f ) · r = (0). This implies that c(f ) ⊆ ZR (R). But M is an Auslander R-module.
Therefore, c(f ) ⊆ ZR (M ). Since c(f ) is a finitely generated ideal of R [14, p. 3]
and M has property (A), there is a nonzero m ∈ M such that c(f ) · m = (0). This
means that c(f ) ⊆ AnnR (m). Therefore, c(f )B ⊆ AnnR (m)B. Since any McCoy
R-algebra is by definition an Ohm-Rush R-algebra, we have that f ∈ c(f )B. Our
claim is that AnnR (m)B = AnnB (1 ⊗ m) and here is the proof: Since
0 −→ R/ AnnR (m) −→ M
is an R-exact sequence and B is a faithfully flat R-module, we have the following
B-exact sequence:
0 −→ B/ AnnR (m)B −→ M ⊗R B,
with AnnR (m)B = Ann(m ⊗R 1B ). This means that f ∈ ZB (M ⊗R B) and the
proof is complete.
Corollary 1.11. Let M be an Auslander R-module and B a content R-algebra. If
M has property (A), then M ⊗R B is an Auslander B-module.
AUSLANDER MODULES
5
Proof. By definition of content algebras [14, Section 6], any content R-algebra is
faithfully flat. Also, by [14, Statement 6.1], any content R-algebra is a McCoy
R-algebra.
Question 1.12. Is there any faithfully flat McCoy algebra that is not a content
algebra?
2. Torsion-Free Modules
Let us recall that if R is a ring, M an R-module, and Q the total ring of fractions
of R, then M is torsion-free if the natural map M → M ⊗ Q is injective [1, p.
19]. It is starightforward to see that M is a torsion-free R-module if and only if
ZR (M ) ⊆ ZR (R). Therefore, the notion of Auslander modules defined in Definition
1.1 is a kind of dual to the notion of torsion-free modules.
The proof of the following theorem is quite similar to the proof of Proposition
1.4. Therefore, we just mention the proof briefly.
Theorem 2.1. Let the ring R have property (A) and G be a commutative, cancellative, and torsion-free monoid. Then, the R[G]-module M [G] is torsion-free if
and only if the R-module M is torsion-free.
Proof. (⇒): Let r ∈ ZR (M ). Clearly, this implies that r ∈ ZR[G] (M [G]). But
the R[G]-module M [G] is torsion-free. Therefore, ZR[G] (M [G]) ⊆ ZR[G] (R[G]). So,
r ∈ ZR (R).
(⇐): Let f ∈ ZR[G] (M [G]). By [11, Theorem 2], there is a nonzero m ∈ M
such that c(f ) · m = 0, which means that c(f ) ⊆ ZR (M ). Since M is torsion-free,
c(f ) ⊆ ZR (R), and since R has property (A), f ∈ ZR[G] (R[G]) and the proof is
complete.
Theorem 2.2. Let R be a Noetherian ring and M be a finitely generated R-module.
Then, the R[[X]]-module M [[X]] is torsion-free if and only if the R-module M is
torsion-free.
Proof. (⇒): Its proof is similar to the proof of Theorem 2.1 and therefore, we don’t
bring it here.
(⇐): Let f ∈ ZR[[X]] (M [[X]]). By Lemma 1.5, there is a nonzero element
m ∈ M such that f · m = 0. By Remark 1.8, this implies that c(f ) ⊆ ZR (M ). But
M is torsion-free, so ZR (M ) ⊆ ZR (R), which implies that c(f ) ⊆ ZR (R). On the
other hand, since every Noetherian ring has property (A) (check [7, Theorem 82,
p. 56]), c(f ) has a nonzero annihilator in R. This means that f ∈ ZR[[X]] (R[[X]]),
Q.E.D.
We continue this section by investigating torsion-free Auslander modules.
Remark 2.3. In the following, we show that there are examples of modules that are
Auslander but not torsion-free and also there are some modules that are torsion-free
but not Auslander.
(1) Let R be a ring and S ⊆ R − ZR (R) a multiplicatively closed subset of
R. Then, it is easy to see that ZR (R) = ZR (RS ), i.e. RS is a torsion-free
Auslander R-module.
6
PEYMAN NASEHPOUR
(2) Let D be a domain and M a D-module such that ZD (M ) 6= {0}. Clearly,
ZD (D) = {0} and therefore, M is Auslander, while M is not torsion-free.
For example, if D is a domain that is not a field, then D has an ideal I
such that I 6= (0) and I 6= D. It is clear that ZD (D/I) ⊇ I.
(3) Let k be a field and consider the ideal I = (0)⊕k of the ring R = k ⊕k. It is
easy to see that ZR (R) = ((0)⊕k)∪(k⊕(0)), while ZR (R/I) = (0)⊕k. This
means that the R-module R/I is torsion-free, while it is not Auslander.
Proposition 2.4 (Some Families of Torsion-free Auslander Modules). Let M be
an R-module. Then, the following statements hold:
(1) If R is a domain and M is a flat R-module, then M is torsion-free Auslander R-module.
(2) If M is a flat and content R-module such that for any s ∈ R, there is an
x ∈ M such that c(x) = (s). Then M is a torsion-free Auslander R-module.
(3) If R is a Noetherian ring and M is a finitely generated flat R-module and
for any nonzero s ∈ R, there is an x ∈ M such that s · x 6= 0. Then
HomR (M, M ) is a torsion-free Auslander R-module.
(4) If M is an Auslander R-module, and M and M ′ are both flat modules, then
M ⊕ M ′ is a torsion-free Auslander R-module. In particular, if {Mi }i∈Λ is
a family of flat R-modules and
L there is an i ∈ Λ, say i0 , such that Mi0 is an
Auslander R-module, then i∈Λ Mi is a torsion-free Auslander R-module.
(5) If R is a coherent ring and {Mi }i∈Λ is a family of flat R-modules and
Q there
is an i ∈ Λ, say i0 , such that Mi0 is an Auslander R-module, then i∈Λ Mi
is a torsion-free Auslander R-module.
Proof. It is trivial that every flat module is torsion-free. By considering Proposition
1.2, the proof of statements (1) and (2) is straightforward.
The proof of statement (3) is based on Theorem 7.10 in [9] that says that each
finitely generated flat module over a local ring is free. Now if R is a Noetherian
ring and M is a flat and finitely generated R-module, then M is a locally free Rmodule. This causes HomR (M, M ) to be also a locally free R-module and therefore,
HomR (M, M ) is R-flat and by Proposition 1.2, a torsion-free Auslander R-module.
The proof of the statements (4) and (5) is also easy, if we note that the direct
sum of flat modules is flat [8, Proposition 4.2] and, if R is a coherent ring, then the
direct product of flat modules is flat [8, Theorem 4.47].
Proposition 2.5. Let both the ring R and the R-module M have property (A)
and G be a commutative, cancellative, and torsion-free monoid. Then, M [G] is a
torsion-free Auslander R[G]-module if and only if M is a torsion-free Auslander
R-module.
Proof. By Theorem 1.4 and Theorem 2.1, the statement holds.
Corollary 2.6. Let R be a Noetherian ring and M a finitely generated R-module,
and G a commutative, cancellative, and torsion-free monoid. Then, M [G] is a
torsion-free Auslander R[G]-module if and only if M is a torsion-free Auslander
R-module.
Theorem 2.7. Let M be a flat Auslander R-module and B a faithfully flat McCoy
R-algebra. If M has property (A), then M ⊗R B is a torsion-free Auslander Bmodule.
AUSLANDER MODULES
7
Proof. By Theorem 1.10, ZB (B) ⊆ ZB (M ⊗R B). On the other hand, since M is
a flat R-module, by [8, Proposition 4.1], M ⊗R B is a flat B-module. This implies
that ZB (M ⊗R B) ⊆ ZB (B) and the proof is complete.
Corollary 2.8. Let M be a flat Auslander R-module and B a content R-algebra.
If M has property (A), then M ⊗R B is a torsion-free Auslander B-module.
Theorem 2.9. Let R be a Noetherian ring and M a finitely generated R-module.
Then, M [[X]] is a torsion-free Auslander R[[X]]-module if and only if M is a
torsion-free Auslander R-module.
Proof. Since M is finite and R is Noetherian, M is also a Noetherian R-module.
This means that both the ring R and the module M have property (A). Now by
Theorem 1.6 and Theorem 2.2, the proof is complete.
Acknowledgements
The author was partly supported by the Department of Engineering Science at
Golpayegan University of Technology and wishes to thank Professor Winfried Bruns
for his invaluable advice. The author is also grateful for the useful comments by
the anonymous referee.
References
[1] W. Bruns and J. Herzog, Cohen-Macaulay Rings, revised edn., Cambridge University Press,
Cambridge, 1998.
[2] N. Epstein and J. Shapiro, A Dedekind-Mertens theorem for power series rings, Proc. Amer.
Math. Soc. 144 (2016), 917–924.
[3] N. Epstein and J. Shapiro, The Ohm-Rush content function, J. Algebra Appl., 15, No. 1
(2016), 1650009 (14 pages).
[4] D. E. Fields, Zero divisors and nilpotent elements in power series rings, Proc. Amer. Math.
Soc. 27 (1971), no. 3, 427–433.
[5] M. Hochster, Intersection Problems and Cohen-Macaulay Modules, in Algebraic Geometry:
Bowdoin 1985 (part 2) (1987), 491–501.
[6] M. Hochster, Topics in the homological theory of modules over commutative rings, CBMS
Regional Conf. Ser. in Math., 24, Amer. Math. Soc, Providence, RI, 1975.
[7] I. Kaplansky, Commutative Rings, Allyn and Bacon, Boston, 1970.
[8] T. Y. Lam, Lectures on Modules and Rings, Springer-Verlag, Berlin, 1999.
[9] H. Matsumura, Commutative Ring Theory, Vol. 8., Cambridge University Press, Cambridge,
1989.
[10] P. Nasehpour, On zero-divisors of semimodules and semialgebras, arXiv preprint (2017),
arXiv:1702.00810.
[11] P. Nasehpour, Zero-divisors of semigroup modules, Kyungpook Math. J., 51 (1) (2011),
37–42.
[12] P. Nasehpour and Sh. Payrovi, Modules having few zero-divisors, Comm. Algebra 38 (2010),
3154–3162.
[13] D. G. Northcott, A generalization of a theorem on the content of polynomials, Proc. Cambridge Phil. Soc. 55 (1959), 282–288.
[14] J. Ohm and D. E. Rush, Content modules and algebras, Math. Scand. 31 (1972), 49–68.
[15] C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Publ. Math. IHES
42, (1973), 47-119.
[16] P. Roberts, Intersection theorems, in: Commutative Algebra, in: Math. Sci. Res. Inst. Publ.,
Vol. 15, Springer-Verlag, Berlin, 1989, 417–436.
[17] P. Roberts, Le théoreme d’intersection, CR Acad. Sci. Paris Ser. I Math 304 (1987), 177–180.
8
PEYMAN NASEHPOUR
Department of Engineering Science, Golpayegan University of Technology, Golpayegan, Iran
E-mail address: nasehpour@gut.ac.ir, nasehpour@gmail.com
| 0 |
1
Capturing the Future by Replaying the Past
Functional Pearl
arXiv:1710.10385v1 [cs.PL] 28 Oct 2017
JAMES KOPPEL, MIT
ARMANDO SOLAR-LEZAMA, MIT
Delimited continuations are the mother of all monads! So goes the slogan inspired by Filinski’s 1994 paper, which showed
that delimited continuations can implement any monadic effect, letting the programmer use an effect as easily as if it was
built into the language. It’s a shame that not many languages have delimited continuations.
Luckily, exceptions and state are also the mother of all monads! In this Pearl, we show how to implement delimited
continuations in terms of exceptions and state, a construction we call thermometer continuations. While traditional implementations of delimited continuations require some way of "capturing" an intermediate state of the computation, the insight of
thermometer continuations is to reach this intermediate state by replaying the entire computation from the start, guiding it
using a "replay stack" it so that the same thing happens until the captured point.
Along the way, we explain delimited continuations and monadic reflection, show how the Filinski construction lets
thermometer continuations express any monadic effect, share an elegant special-case for nondeterminism, and discuss why
our construction is not prevented by theoretical results that exceptions and state cannot macro-express continuations.
CCS Concepts: •Theory of computation → Control primitives; Functional constructs; •Software and its engineering → Functional languages; Control structures; General programming languages;
Additional Key Words and Phrases: monads, delimited continuations
ACM Reference format:
James Koppel and Armando Solar-Lezama. 2016. Capturing the Future by Replaying the Past. 1, 1, Article 1 (January 2016),
27 pages.
DOI: 10.1145/nnnnnnn.nnnnnnn
1
INTRODUCTION
In the days when mainstream languages have been adopting higher-order functions, advanced monadic effects
like continuations and nondeterminism have held out as the province of the bourgeois programmer of obscure
languages. Until now, that is.
Of course, there’s a difference between effects which are built into a language and those that must be encoded.
Mutable state is built-in to C, and so one can write int x = 1; x += 1; int y = x + 1;. Curry 1 is nondeterministic,
and so one can write (3 ? 4) * (5 ? 6) , which evaluates to all of {15, 18, 20, 24}. This is called the direct style.
When an effect is not built into a language, the monadic, or indirect, style is needed. In the orthodox indirect
style, after every use of an effect, the remainder of the program is wrapped in a lambda. For instance, the nondeterminism example would be rendered in Scala as List(3,4).flatMap(x ⇒List(5,6).flatMap(y ⇒List(x * y))).
Effects implemented in this way are called monadic. "Do-notation," as seen in Haskell, makes this easier, but still
inconvenient.
1 Hanus
et al. (1995)
Permission to make digital or hard copies of part or all of this work for personal or classroom use is granted without fee provided that
copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page.
Copyrights for third-party components of this work must be honored. For all other uses, contact the owner/author(s).
© 2016 Copyright held by the owner/author(s). XXXX-XXXX/2016/1-ART1 $15.00
DOI: 10.1145/nnnnnnn.nnnnnnn
, Vol. 1, No. 1, Article 1. Publication date: January 2016.
We show that, in any language with exceptions and state, you can implement any monadic effect in direct
style. With our construction, you could implement a ? operator in Scala such that the example (3 ? 4) * (5 ? 6)
will run and return List(15,18,20,24). Filinski showed how to do this in any language that has an effect called
delimited continuations or delimited control 2 . We first show how to implement delimited continuations in
terms of exceptions and state, a construction we call thermometer continuations. Filinski’s result does the rest.
Continuations are rare, but exceptions are common. With thermometer continuations, you can get any effect in
direct style in 9 of the TIOBE top 10 languages 3 .
Here’s what delimited continuations look like in cooking: Imagine a recipe for making chocolate nut bars. Soak
the almonds in cold water. Rinse, and grind them with a mortar and pestle. Delimited continuations are like a step
that references a sub-recipe. Repeat the last two steps, but this time with pistachios. Delimited continuations can
perform arbitrary logic with these subprograms ("Do the next three steps once for each pan you’ll be using"), and
they can abort the present computation ("If using store-bought chocolate, ignore the previous four steps"). They
are "delimited" in that they capture only a part of the program, unlike traditional continuations, where you could
not capture the next three steps as a procedure without also capturing everything after them, including the part
where you serve the treats to friends and then watch Game of Thrones. Implementing delimited continuations
requires capturing the current state of the program, along with the rest of the computation up to a "delimited"
point. It’s like being able to rip out sections of the recipe and copy them, along with clones of whatever ingredients
have been prepared prior to that section. This is a form of "time travel" that typically requires runtime support —
if the nuts had not yet been crushed at step N, and you captured a continuation at step N, when it’s invoked, the
nuts will suddenly be uncrushed again.
The insight of thermometer continuations is that every subcomputation is contained within the entire computation, and so there is an alternative to time travel: just repeat the entire recipe from the start! But this time, use
the large pan for step 7. Because the computation contains delimited control (which can simulate any effect), it’s
not guaranteed to do the same thing when replayed. Thermometer continuations hence need state for a replay
stack, which makes each replayed effectful function call do the same thing as a previous invocation. Additionally,
like a recipe step that overrides previous steps, or that asks you to let it bake for an hour, delimited continuations
can abort or suspend the rest of the computation. Implementing this uses exceptions.
This approach poses an obvious limitation: the replayed computation can’t have any side effects, except for
thermometer continuations. And replays are inefficient. Luckily, thermometer continuations can implement all
other effects, and there are optimization techniques that make it less inefficient. Also, memoization — a "benign
effect" — is an exception to the no-side-effects rule, and makes replays cheaper.
Here’s what’s in the rest of this Pearl: Our construction has an elegant special case for nondeterminism,
presented in Section 2, which also serves as a warm-up to full delimited control. In the following section, we
give an intuition for how to generalize the nondeterminism construction with continuations. Section 4 explains
thermometer continuations. Section 5 explains Filinski’s construction and how it combines with thermometer
continuations to get arbitrary monadic effects. Section 6 discusses how to optimize the fusion of thermometer
continuations with Filinski’s construction, and provides a few benchmarks showing that thermometer continuations are not entirely impractical. Finally, Section 7 discusses why our construction does not contradict a
theoretical result that exceptions and state cannot simulate continuations.
2 Filinski
3 TIOBE
(1994)
Software BV (2017)
2
SML code
Closest Haskell equivalent
int option
type
α
f =
α
*
Maybe Int
α
type F a = (a, a)
!r
val x
let
:
readIORef r
int = 1
<decls> in <expr> end
1 :: 2 :: []
x
let
<decl1> in
let
:: Int; x = 1
<decl2> in <expr>
1 : 2 : []
f [1,2] @ g [3,4]
f [1,2] ++ g [3,4]
"abc" ^ "def"
"abc" ++ "def"
f o g
f . g
#1 (a,b)
fst (a,b)
Fig. 1. The SML/Haskell Cheat Sheet
2
WARM-UP: REPLAY-BASED NONDETERMINISM
Nondeterminism is perhaps the first effect students learn which is not readily available in a traditional imperative language. This section presents replay-based nondeterminism, a useful specialization of thermometer
continuations, and an introduction to its underlying ideas.
When writing the examples in this paper, we sought an impure language with built-in support for exceptions
and state, and which has a simple syntax with good support for closures. We hence chose to present in SML.
The cheat sheet in Figure 1 explains SML’s less obvious syntax by giving their closest Haskell equivalents. Also
note that ML functors are module-valued functions, and are substantially different from Haskell-style functors.
Throughout this paper, we will assume there is no concurrency.
Nondeterminism provides a choice operator choose such that choose [x1, x2, . . .] may return any of the x i . Its
counterpart is a withNondeterminism operator which executes a block that uses choose, and returns the list of values
resulting from all executions of the block.
withNondeterminism (fn ()
⇒
(choose [2,3,4]) * (choose [5,6]))
(* val it = [10,12,15,18,20,24]
:
int list *)
In this example, there are six resulting possible values, yet the body returns one value. It hence must run six
times. The replay-based implementation of nondeterminism does exactly this: in the first run, the two calls to
choose return 2 and 5, then 2 and 6 in the second, etc. In doing so, the program behaves as if the first call to choose
was run once but returned thrice. We’ll soon show exactly how this is done. But first, let us connect our approach
to the one most familiar to Haskell programmers: achieving nondeterminism through monads.
In SML, a monad is any module which implements the following signature (and satisfies the monad laws):
signature MONAD = sig
type
α
m;
val return
val bind
:
: α → α m;
α m → (α → β
m)
→β
m;
end;
Here is the implementation of the list monad in SML:
3
structure ListMonad
type
α
m =
α
:
MONAD = struct
list
fun return x = [x]
fun bind []
|
f = []
bind (x :: xs) f = f x @ bind xs f
end;
The ListMonad lets us rewrite the above example in monadic style. In the direct style, choose [2,3,4] would
return thrice, causing the rest of the code to run thrice. Comparatively, in the monadic style, the rest of the
computation is passed as a function argument to ListMonad.bind, which invokes it thrice.
open ListMonad;
bind [2,3,4] (fn x
bind [5,6]
(fn y
⇒
⇒
return (x * y)))
(* val it = [10,12,15,18,20,24]
:
int list *)
Let’s look at how the monadic version is constructed from the direct one. From the perspective of the invocation
the rest of the expression is a function awaiting its result, which it must invoke thrice:
choose [2, 3, 4],
C = (fn
⇒
* choose [5, 6])
This remaining computation is the continuation of choose [2,3,4]. Each time choose [2,3,4] returns, it invokes
this continuation. The monadic transformation captured this continuation, explicitly turning it into a function.
This transformation captures the continuation at compile time, but it can also be captured at runtime with the
callcc "call with current continuation" operator: if this first call to choose were replaced with callcc (fn k ⇒. . .),
then k would be equivalent to C. So, the functions being passed to bind are exactly what would be obtained if the
program were instead written in direct style and used callcc.
This insight makes it possible to implement a direct-style choose operator. The big idea is that, once callcc
has captured that continuation C in k, it must invoke k thrice, with values 2, 3, 4. This implementation is a little
verbose in terms of callcc, but we’ll later see how delimited continuations make this example simpler than with
callcc-style continuations.
Like the monadic and callcc-based implementations of nondeterminism, replay-based nondeterminism invokes
the continuation (fn x ⇒x * choose [5,6]) three times. Since the program is left in direct style (choose [2,3,4] * choose
and it cannot rely on a built-in language mechanism to capture the continuation, it does this by running the entire
block multiple times, with some bookkeeping to coordinate the runs. We begin our explanation of replay-based
nondeterminism with the simplest case: a variant of choose which takes only two arguments, and may only be
used once.
2.1
The Simple Case: Two-choice nondeterminism, used once
We begin by developing the simplified choose2 operator. Calling choose2
branches, returning x in the first branch and y in the second. For example:
(withNondeterminism2 (fn ()
=⇒
=⇒
⇒
3 * choose2 (5, 6)))
[3 * 5, 3 * 6]
[15, 18]
4
(x,y)
splits the execution into two
[5,6]) ,
br_idx
0
1
br_idx
1
1
pos=2
21 0
1 0
0
1
21 0
1 0
br_idx
1
0
1
0
0
pos=0
1 0
21 0
pos=0
(a)
(b)
(c)
Fig. 2. Several points in the execution of the replay-based nondeterminism algorithm.
This execution trace hints at an implementation in which withNondetermism2 calls the block twice, and where
returns the first value in the first run, and the second value in the second run. withNondeterminism2 uses a
single bit of state to communicate to choose2 whether it is being called in the first or second execution. As long as
the block passed to withNondeterminism2 is pure, with no effects other than the single use of choose (and hence no
nested calls to withNondeterminism2), each execution will have the same state at the time choose is called.
choose2
val firstTime = ref false
(* choose2
: α
*
α → α
*)
fun choose2 (x1,x2) = if !firstTime then x1 else x2
(* withNondeterminism2
→ α ) → α list *)
: = true ; f
(firstTime : = false; f
(unit
:
fun withNondeterminism2 f = [(firstTime
withNondeterminism2 (fn ()
(* val it = [15,18]
:
⇒
()),
())]
3 * choose2 (5,6))
int list *)
While simple and restrictive, this implementation contains the insights that allow arbitrary nondeterminism. View a nondeterministic computation as a tree, where each call to choose is a node, and the sequence of
values returned by calls to choose identify a branch. This section gives the special case where the tree has but
two branches. Here, withNondeterminism2 picks the branch, and choose2 follows it. Because there are only two
branches of execution, withNondeterminism2 and choose2 share only a single bit of state. What’s needed for arbitrary
nondeterminism? More state.
2.2
Towards Arbitrary Nondeterminism
Replay-based nondeterminism executes every branch in the computation tree, replaying the program once for
each. It’s like a depth-first tree traversal, except that reaching each leaf requires traversing the entire path from
the root. For instance, consider this program:
withNondeterminism (fn ()
if choose [3,4]
=
⇒
3 then
choose [5,6]
else
choose [7,8,9])
5
There are five branches in the execution tree of this program. In the first run, the two calls to choose return
(3, 5). In the second, they return (3, 6), followed by (4, 7), (4, 8) and (4, 9). Each branch is identified by a branch
index. Figure 2 depicts this execution tree with branch indices used at different points in the algorithm. The gist
of our algorithm is: withNondeterminism will run the block once for every branch index; within a run, each call to
choose uses the current branch index to pick which alternative to return.
Or, in code: br_idx stores the current branch index, while pos tells the next call to choose which element of the
branch index to look at.
val br_idx
int list ref = ref [];
:
val pos = ref 0;
A branch index is a list of numbers, each number indicating which alternative to return at a call to choose.
These choices are numbered right-to-left, so that, when this number reaches 0, the algorithm knows that all
choices at that call to choose have been exhausted. Like the stacks used to traverse trees, the first element of a
branch index corresponds to the last call to choose in that branch, which makes updating to the next branch index
simple. For instance, the first branch, in which the calls to choose return 3 and 5, has branch index [1, 1], and the
next branch, returning 3 and 6, has branch index [0, 1]. These are shown in Figures 2a and 2b. The decr inputs a
branch index, and returns the index of the next branch.
fun decr (0 :: ns) = decr ns
|
|
decr (n :: ns) = (n-1) :: ns
decr []
= []
After executing the branch with index [0, 1], withNondeterminism updates the current branch index to [0]. This
is merely a prefix of the actual branch index: it instructs the first call to choose to return 4, but withNondeterminism
does not yet know that doing so causes execution to encounter the call choose [7,8,9]. Instead, that call to choose
will extend the branch index to [2, 0]. Figure 2c depicts this.
Do note that nested calls to withNondeterminism will not work because they both use global state and will
interfere with each other. The thermometer continuations in Section 4.4 lack this problem.
We can now implement choose. If the branch index records a choice for the current invocation of choose, it
selects that alternative. Else, it extends the branch index to pick the first alternative of the current choice.
open List;
(* choose
: α
list
→ α
*)
fun choose xs = if not (0 = !pos) then
let val idxFromEnd = nth (!br_idx, !pos - 1) in
(pos
:=
(!pos) - 1;
nth (xs, (length xs - 1) - idxFromEnd))
end
else
(br_idx
:=
(length xs - 1)
::
!br_idx;
hd xs)
The withNondeterminism function repeatedly executes the given block. It picks a different branch each time until
the branches have been exhausted, and concatenates the results together. Note that no initialization code is
needed, because the state is always returned to its initial value upon completing a call of withNondeterminism.
6
(* withNondeterminism
(unit
:
→ α) → α
list *)
fun withNondeterminism f =
let val v = [f()] in
br_idx : = decr (!br_idx);
length (!br_idx);
_
case !br idx of
pos
|
:=
[]
_
⇒
⇒
v
v @ withNondeterminism f
end
With this implementation in place, we can now finally run the examples.
withNondeterminism (fn ()
⇒
choose [2,3,4] * choose [5, 6])
(* val it = [10,12,15,18,20,24]
withNondeterminism (fn ()
if choose [3,4]
=
:
int list *)
⇒
3 then
choose [5,6]
else
choose [7,8,9])
(* val it = [5,6,7,8,9]
2.3
:
int list *)
But What About the Empty Case?
The above implementation can handle nondeterminism with 1 or more alternatives. But having 0 alternatives is
fundamentally different: the previous implementation assumes one value per branch, but choose allows branches
with no values.
What to do when a program calls choose []? Returning a dummy value is not an option: choose has type
α list →α , and there is no value of type α to return. We look to the monadic version as a guide. A call to
choose [] is translated into:
bind [] (fn x
⇒ . . .)
Here, the function argument of bind represents the continuation of choose. bind never invokes it, which is
equivalent to aborting the continuation: it’s like choose never returns. The replay-based implementation of
choose [] achieves this by raising an exception.
Supporting empty choices requires minimal modification to our implementation. When a program calls
choose [], choose raises an exception to pass control to withNondeterminism, which moves on to the next branch.
7
exception Empty;
fun choose [] = raise Empty
|
choose xs =
...
fun withNondeterminism f =
let val v = [f()] handle Empty
⇒
[] in
...
end
fun fail () = choose []
withNondeterminism (fn ()
⇒
let val x = chooose [2,3,4] * choose [5,7] in
if x
≥
20 then x
else fail () end);
(* val it = [21,20,28]
3
:
int list *)
CONTINUATIONS IN DISGUISE
The previous section showed a trick for implementing direct-style nondeterminism in deterministic languages.
Now, we delve to the deeper idea behind it, and surface the ability to generalize from nondeterminism to
any monadic effect. We now examine how replay-based nondeterminism stealthily manipulates continuations.
Consider evaluating this expression:
val e = withNondeterminism (fn ()
⇒
choose [2,3,4] * choose [5, 6])
Every subexpression of e has a continuation, and when it returns a value, it invokes that continuation. After
the algorithm takes e down the first branch and reaches the point T = 2 * choose [5, 6], this second call to choose
has continuation C = fn ⇒2 * .
choose must invoke this continuation twice, with two different values. But C is not actually a function that can
be repeatedly invoked: it’s a description of what the program does with a value after it’s returned, and returning
a value causes the program to keep executing, consuming the continuation. choose invokes this continuation
the first time normally, returning 5. To copy this ephemeral continuation, it re-runs the computation until it’s
reached a point identical to T, evaluating that same call to choose with a second copy of C as its continuation —
and this time, it invokes the continuation with 6.
So, the first action of the choose operator is capturing the continuation. And what happens next? The
continuation is invoked once for each value, and the results are later appended together. We’ve already seen
another operation that invokes a function once on each value of a list and appends the results: the ListMonad.bind
operator. Figure 3 depicts how applying ListMonad.bind to the continuation produces direct-style nondeterminism.
So, replay-based nondeterminism is actually a fusion of two separate ideas:
(1) Capturing the continuation using replay
(2) Using the captured continuation with operators from the nondeterminism monad
In Section 4, we extract the first half to create thermometer continuations, our replay-based implementation of
delimited control. The second half — using continuations and monads to implement any effect in direct style
— is Filinski’s construction, which we explain in Section 5. These produce something more inefficient than the
8
Fig. 3. To implement
multiple times.
=
[
1
C
, , ]
2
C
3
C
[1,2,3]
>>=
choose:
first capture the continuation, and then use the list monad’s
C
bind
operator to evaluate it
replay-based nondeterminsm of Section 2, but we’ll show in Section 6 how to fuse them together into something
equivalent.
4
THERMOMETER CONTINUATIONS: REPLAY-BASED DELIMITED CONTROL
In the previous section, we explained how the replay-based nondeterminism algorithm actually hides a mechanism
for capturing continuations. Over the the remainder of this section, we extract out that mechanism, developing
the more general idea of thermometer continuations. But first, let us explain the variant of continuations that our
mechanism uses: delimited continuations.
4.1
What is delimited control?
When we speak of "the rest of the computation," a natural question is "until where?" For traditional continuations,
the answer is: until the program halts. This crosses all abstraction boundaries, making these "undelimited
continuations" difficult to work with. Delimited continuations on the other hand, introduced by Felleisen 4 ,
only represent a prefix of the remaining computation. Just as callcc makes continuations first class, allowing a
program to modify its continuation to implement many different global control-flow operators, the shift and
reset constructs make delimited continuations first-class, and can be used to implement many local control-flow
operators.
In the remainder of this section, we’ll denote a one-holed context C with the notation C[x], i.e.: C[x+1] is some
expression containing x+1. This notation makes it easy to give the semantics for shift and reset.
Consider an expression which is about to evaluate a shift. In the special case where there is only one shift in
the code, it evaluates as follows:
C 1 [reset (fn () ⇒C 2 [shift (fn k ⇒E)])]
=⇒C 1 [(fn k ⇒E)(fn x ⇒C 2 [x])]
C 1 [reset (fn () ⇒E)] =⇒E if E does not contain
a shift
Figure 4 depicts this evaluation. The name "shift" is illustrative. Suppose shift and reset were normal functions
rather than control operators. Then the delimited continuation of the shift up until the reset is C 2 , which in
turn has a delimited continuation of C 3 [x] = x. What shift does is, well, shift these. Continuation C 3 replaces
Continuation C 2 . This means that, after the shift returns, control jumps straight to the reset. In that regard, it
acts like a C-style return operator. C 2 , however, gets turned into a function and saved in the variable k.
E is free to use k in interesting ways. It can invoke k with several values. This essentially makes the shift
"return" multiple times, and can implement nondeterminism a la Figure 3. It can stuff k in a data-structure to be
"resumed" later, similar to a Python-style yield. It can even "chain" calls to k, like in this example:
1 + reset (fn ()
4 Felleisen
⇒
2 * (shift (fn k
⇒
k (k 5))))
(1988)
9
shift (λk.E)
k = λx
E
C
reset boundary
x
C
reset boundary
Fig. 4. Graphical depiction of the action of the shift operator.
Let’s rewrite this with C 1 [x]=
now evaluate it:
1+x
and C 2 [x]
= 2 * x
so that it matches the semantics we gave earlier. We can
⇒ 2 * (shift (fn k ⇒ k (k 5))))
C 1 [reset (fn () ⇒C 2 [shift (fn k ⇒k (k 5))])]
=⇒C 1 [(fn k ⇒k (k 5))(fn x ⇒C 2 [x])]
=⇒ 1 + (fn k ⇒ k (k 5))(fn x ⇒ 2 * x) end
=⇒ 21
1 + reset (fn ()
=
The definition above only works when there is only one shift. We now give the full semantics, which can
handle code with multiple shift’s. In the case where there is only one shift, this is equivalent to the previous
semantics.
C 1 [reset (fn () ⇒C 2 [shift (fn k ⇒E)])]
=⇒C 1 [reset (fn () ⇒(fn k ⇒E)(fn x ⇒ reset (fn
C 1 [reset (fn () ⇒E)] =⇒E if E does not contain a
()
⇒ C 2 [x])))]
shift
When the captured delimited continuation is invoked, it gets delimited by the inner reset. So, any shift in C_2
will return to the body of the first shift E, instead of discarding the computation in E. Meanwhile, if E contains a
shift, the inner shift will only capture the continuation up to the outer reset rather than the entire remainder of
the program, because the outer reset gets left in place.
Here’s an example of multiple shift’s in a row.
reset (fn ()
⇒
shift (fn k
⇒
=⇒ reset (fn
=⇒ 1 + reset
=⇒ 1 + reset
=⇒ 1 + 1 + 2
=⇒ 8
()
⇒
1 + k 2) (fn x
(fn k
(fn ()
(fn ()
⇒
⇒
⇒
1 + k 2) * shift (fn k’
2 * shift (fn k’
(fn k’
⇒
⇒
⇒
⇒
1 + k’ 3))
reset (fn ()
⇒
x * shift (fn k’
⇒
1 + k’ 3))))
1 + k’ 3))
1 + k’ 3)(fn x
⇒
reset (fn ()
⇒
2 * x)))
* 3
The outer reset gets left in place. This means that if there are nested shift’s (i.e.: a shift in E),
And here’s an example of nested shift’s. Note how the two delimited continuations fn x ⇒2 + x and fn
get applied in reverse-order.
1 + reset (fn ()
=⇒
=⇒
=⇒
=⇒
⇒
2 + (shift (fn k
1 + reset (fn ()
1 + reset (fn ()
1 + reset (fn ()
⇒
⇒
⇒
(fn k
⇒
⇒
3 * shift (fn l
(fn l
⇒
3 * shift (fn l
3 * shift (fn l
⇒
⇒
⇒
3 * x))
37
10
⇒3
l (k 10)))))
l (k 10)))(fn x
l (2 + 10)))
l 12)(fn x
⇒
x
⇒
reset (fn ()
⇒
2 + x)))
* x
As the previous examples show, shift and reset are quite versatile. Indeed, they can implement any other
control operator.
shift and reset are termed delimited control operators, and are encoded in the following SML signature. There
are also other equivalent formulations of delimited control using different operators5 .
signature CONTROL = sig
type ans
val reset
val shift
:
:
→ ans) → ans
→ ans) → ans) → α
(unit
(( α
end;
4.2
Baby Thermometer Continuations
Programming with continuations requires being able to capture and copy an intermediate state of the program. We
showed in Section 3 that replay-based nondeterminism implicitly does this by replaying the whole computation,
and hence needs no support from the runtime. We now see how to do this more explicitly.
This section presents a simplified pseudocode version of thermometer continuations. This version assumes the
reset block only contains one shift, and also ignores type-safety, but can still handle the example of delimited control from Section 4.1. Consider an expression containing a single shift, C 1 [reset (fn () ⇒C 2 [shift (fn k ⇒E)])]
The tough part of delimited continuations is to capture the continuation of the shift, namely C[t] = 2 * t, as
a function fn t ⇒C[t]. But suppose shift used mutable state so that change_state x; shift f evaluates to x. If C
doesn’t use any mutable state, then change_state x commutes with everything in C, so that change_state x;C [shift f]
is equivalent to C [change_state x; shift f]. Then, since block () is equal to C [shift (fn k ⇒E)], the function
fn x ⇒(change_state x; block () ) is equivalent to fn x ⇒C [x], the continuation to capture!
The other part of shift is to return a value directly to the enclosing reset (to "abort the current continuation").
It can do this by raising an exception. So, in totality, here are the semantics of shift and reset implemented in
this fashion:
C 1 [reset (fn () ⇒C 2 [shift (fn k ⇒E)])]
=⇒C 1 [C 2 [raise (Done (fn k ⇒E)(fn x ⇒(change_state x; C 2 [shift
handle (Done x ⇒ x)]
≡ C 1 [(fn k ⇒E)(fn x ⇒C 2 [change_state x; shift (fn k ⇒E)])]
≡ C 1 [(fn k ⇒E)(fn x ⇒C 2 [x])]
(fn k
⇒E)])))]
This is exactly the semantics of shift and reset given in Section 4.1 for the case when there is only one shift.
We will later add support for multiple shift’s.
For the remainder of this section, define block = (fn () ⇒2 * shift (fn k ⇒1 + k 5)). We consider the example
reset block. It evaluates as follows:
reset block
=⇒ let
=⇒ 11
val k = (fn t
⇒
2 * t) in 1 + k 5 end
⇒1 + k 5)] .
We now show a pseudocode implementation of shift and reset. shift checks a piece of mutable state to see
if it should ignore its argument and return some other value. For ease of extension, we use a stack, called the
replay_stack. If the replay_stack is nonempty, it does so; else, it does a "normal shift."
C[shift (fn k
5 Dyvbig
et al. (2007)
11
fun shift f = if
<replay_stack
is not empty>
pop replay_stack
else
normal_shift f
normal_shift calls its argument with "the captured continuation." In this example, (push replay_stack x; block ())
is equivalent to 2 * x. So, the captured continuation is equivalent to (fn x ⇒(push replay_stack x; block ())) .
Then normal_shift f must invoke f with this "proto thermometer continuation." shift then transfers control, along
with its result, to the enclosing reset — this is raising an exception.
fun normal_shift f = raise (Done (f (fn x
⇒
(push replay_stack x;
block ()))))
And reset must catch this exception so as to receive control.
reset f = (f ()) handle (Done x
⇒
x)
This is not a real definition of reset. With our definition of shift, reset f only works when f
ignoring how to handle multiple nested calls to reset until the real implementation.
The example is now translated as follows:
= block.
We’re
reset block
=⇒ (2
* (raise (Done (1 +
(push replay_stack 5; block ())))))
handle (Done x
=⇒
⇒
x)
11
This implementation also handles the example from Section 4.1 nicely. The example
1 + reset (fn ()
⇒
2 * (shift (fn k
⇒
k (k 5))))
becomes
⇒ 2 * (shift (fn k ⇒ k (k 5)))) in
1 + (2 * (raise (Done (push replay_stack
(push replay_stack 5; block ());
let val block = (fn ()
block ())))
handle (Done x
⇒
x))
end
==⇒ 21
4.3
Multiple (non-nested) shift’s
Replaying the block becomes more interesting when there are multiple shift’s. We’ll now show how to extend
our pseudocode implementation to handle multiple non-nested shift’s, introducing the record_stack. We’ll use
the following, which we explained in Section 4.1:
val block = (fn ()
⇒
shift (fn k
* shift (fn k’
⇒
⇒
1 + k
2)
1 + k’ 3))
reset block
12
With our previous implementation of shift, calling block () when the replay_stack contains [x, y] will return
x ∗ y. What happens when replay_stack only contains one value, [x]? It will run (pop replay_stack) * (shift . . .) .
x gets popped off the replay_stack before the second shift runs and pushes on y, so the block is never invoked
with more than one value on the replay_stack. This motivates the second record_stack. When shift pops a value
off the replay_stack, it saves it on the record_stack. This way, it can "remember" that value of x when it invokes
block the second time, moving values back from the record_stack to the replay_stack.
fun shift f = if
<replay_stack
is not empty>
let val x = pop replay_stack in
(push record_stack x;
x)
end
else
normal_shift f
fun normal_shift f = raise (Done (f (fn x
⇒
(replay_stack
record_stack
:=
:=
reverse (x
::
(!record_stack));
[];
reset block)))
This is enough to run our multi-shift example:
reset block
=⇒
(raise (Done (1 + (replay_stack : = reverse (2
record_stack : = [];
::
[]);
reset block)))
=⇒
=⇒
* shift (fn k’ ⇒ 1 + k’ 3)) handle (Done x ⇒ x)
1 + (push replay_stack 2; reset block)
1 + (push replay_stack 2; (((push record_stack 2; pop replay_stack)
* raise (Done (1 + (replay_stack : = reverse (3
record_stack : = [];
::
(!record_stack));
block ()))))
handle (Done x
=⇒
⇒
x)))
8
In Section 4.2, we could represent a continuation fn t ⇒C [t] as fn t ⇒(push replay_stack t; block ()). In this
section, if the continuation of a shift is fn t ⇒C [t], this continuation is represented by fn t ⇒(replay_stack : = reverse
where s is some stack. Hence, the continuation C can be represented by the pair (s, block). This motivates our
definition of a thermometer continuation.
Definition 4.1. A thermometer continuation for a continuation C is a pair of a stack and a computation,
(s, block), so that for appropriate functions change_state and run, change_state x s; run block evaluates to C [x].
So, the definitions of shift and reset in this section already implement thermometer continuations. Figure
5 depicts a thermometer continuation. For instance, the continuation of the second shift in this example,
(fn y ⇒2 * y) , is equivalent to invoking block with a replay stack of [2,y], and hence is represented by the
thermometer continuation ([2], block) .
Figure 5 shows a thermometer continuation. Figure 6 animates the execution of a thermometer continuation:
as block executes, each element of the replay stack pairs with a call to shift. When it’s finished, the remaining
13
(x
::st);
record
a
b
c
d
e
Function
Replay Record
Stack Stack
Fig. 5. A thermometer continuation before being invoked
e
d
c
b
stmt1 a
stmt2
stmt3
e
d
c
stmt1 b
stmt3
e
d
c
b
a
...
a
Thermometer
Continuation
Context
Fig. 6. Graphical depiction of running a thermometer continuation
execution of block is equivalent to that continuation C. The left side of Figure 6 resembles a thermometer sticking
out of the function, which inspired the name "thermometer continuations."
The version in this section, of course, still has limitations. We’ve been assuming the variable block was magically
set to the body of the reset, which ignores nested reset’s. It also ignores nested shift’s. It doesn’t work if the
captured continuation escapes the reset, which is needed for implementing the state monad (Section 5.4) and for
implementing iterators (e.g.: Python’s yield). And we’ve been assuming that replay_stack and record_stack only
contain integers; they need to store any value.
The general algorithm presented in the next section solves all these problems. The replay and record stacks
have a universal type. It’s careful about saving old values in closures. For nested shift’s, the replay stack can
store markers, indicating that the replay should re-enter a shift. And, to evaluate a nested reset, it must be able
to stow away the current block, record_stack, and replay_stack, evaluate the nested reset, and then restore them
afterwards. It does this with a third stack, called the reset_stack.
4.4
Real Thermometer Continuations
The previous two sections gave a quick sketch of thermometer continuations. Now we polish it, producing real
code that can handle anything that can be done with shift and reset. We start by defining the stack operations
we used in the previous sections. A stack is a mutable list:
(* push
: α
(* pop
: α
→ α → ()
: = x :: !st)
list ref
fun push st x = (st
list ref
→ α
*)
option *)
fun pop st = case !st of
(x :: xs)
⇒
(st
:=
xs;
14
SOME x)
|
⇒
[]
NONE
One problem with the record and replay stacks is that the values recorded may be of different types. So that
we may store many types of values on a single stack, following Filinski 6 , we define a universal type, which all
other types may be converted to and from:
signature UNIVERSAL = sig
type u;
val to_u
: α → u;
: u → α;
val from_u
end;
structure Universal
:
UNIVERSAL = struct
datatype u = U;
val to_u = Unsafe.cast;
val from_u = Unsafe.cast;
end;
Regrettably, this implementation uses Unsafe.cast, but shift and reset are carefully designed so that these casts
never fail. While Filinski was able to upgrade these definitions to a type-safe implementation of the universal
type in his follow-up paper 7 , the heavy use of replay in our construction unfortunately prevents that solution
from working. We chose not to search for another type-safe version of the universal type in order to keep this
paper focused.
A thermometer continuation is a (function, replay stack) pair. It can’t be called like a function directly. We will
provide a function called invoke_cont that invokes a thermometer continuation, running the function using the
replay stack as long as it lasts. We will present the definitions out-of-order, saving the invoke_cont function for
last, but the overall setup is:
functor Control (type ans)
:
CONTROL = struct
type ans = ans
(*
*
...
type, exception, and state definitions
*)
(* invoke_cont : (unit
fun invoke_cont f st =
fun reset f =
...
fun shift f =
...
→ ans) →
...
stack
→
ans *)
end;
6 Filinski
7 Filinski
(1994)
(1999)
15
The key state is the replay_stack and the record_stack. The entire executing computation is also stored in
mutable state.
To handle nested shift’s, the stacks store a frame type instead of raw values. Previously, when replaying an
expression that contains shift (fn k ⇒shift (fn l ⇒E)), there would be no way to enter the first shift, but make
the second shift return a value x. Now, this can be done with the replay stack [ENTER, RET x].
exception MissingFun
datatype frame = RET of Universal.u
|
ENTER
type stack = frame list
val record_stack
val replay_stack
val cur_fun
:
:
:
stack ref = ref []
stack ref = ref []
(unit
→
ans) ref = ref (fn ()
⇒
raise MissingFun)
The reset function is implemented as a small wrapper around invoke_cont: it invokes a computation with an
empty replay stack, causing the computation to execute from the start.
fun reset f = invoke_cont f []
The shift of Section 4.3 is almost complete. shift f just needs a couple casts to deal with the universally-typed
stacks, and it needs to dereference and it needs to dereference and capture the values of the record-stack and
cur_fun before invoking f. Then the captured continuation becomes observationally pure, so it can escape the
current reset.
exception Done of ans
fun shift f = case pop replay_stack of
(SOME (RET v)) ⇒ (push record_stack (RET v);
Universal.from_u v)
|
_
⇒
let val st = !record_stack
val g = !cur_fun in
(push record_stack ENTER;
raise (Done (f (fn v
⇒
invoke_cont g (RET (Universal.to_u v )
::
st)))))
end
One funny thing about this implementation is how it behaves the same when it encounters an ENTER frame
versus when the replay_stack is exhausted. Whether it was told to do so by the replay_stack or if it’s entering the shift for the first time, the correct thing to do is push an ENTER frame to the record stack. Note
how it captures the current state of the record_stack before pushing on the ENTER frame. So, in the expression
reset (fn () ⇒shift (fn k ⇒shift (fn l ⇒E))), invoking k x will replay the computation with a replay_stack
of [RET x], and will evaluate to reset (fn () ⇒x). Meanwhile, invoking l x will replay it with a replay_stack of
[ENTER, RET x], evaluating to reset (fn () ⇒shift (fn k ⇒x)).
We need one additional piece of state in order to implement invoke_cont. If a thermometer continuation is
invoked while executing another computation, the program must be able to save and restore the current values
in the replay and record stacks. Nested reset calls will also change the function currently being invoked, so this
must be saved as well. To accomplish this, invoke_cont uses a third stack, the reset stack.
type reset_stack = ((unit
→
ans) * stack * stack) list
16
val reset_stack
:
reset_stack ref = ref []
We are now ready to present invoke_cont f st. It is similar to the definition of reset given in 4.2, except that it
also saves and restores state to the reset_stack.
(* invoke_cont : (unit → ans) → stack → ans *)
fun invoke_cont f st =
(push reset_stack (!cur_fun, !record_stack, !replay_stack);
record_stack : = [];
replay_stack : = rev st;
cur_fun : = f;
let val res = (f () handle (Done x) ⇒ x)
val (SOME (f’, rec_stack’, rep_stack’)) = pop reset_stack in
cur_fun : = f’;
record_stack
replay_stack
:=
:=
rec_stack’;
rep_stack’;
res
end)
With this implementation finished, we can now run our earlier examples:
structure C = Control(type ans = int);
1 + C.reset (fn ()
(* val it = 21
:
C.reset (fn ()
⇒
⇒
2 * (C.shift (fn k
C.shift (fn k
* C.shift (fn k’
(* val it = 8
1 + C.reset (fn ()
5
k (k 5))))
⇒
⇒
1 + k
2)
1 + k’ 3))
int *)
:
(* val it = 37
⇒
int *)
:
⇒
2 + C.shift (fn k
⇒
3*(C.shift (fn l
⇒
l (k 10)))));
int *)
ARBITRARY MONADS
In 1994, Filinski showed how to use delimited continuations to express any monadic effect in direct style8 . The
explanation is quite obtuse, heavy on notation and light on examples. Dan Piponi has a blog post which is more
readable9 , but also missing big concepts like monadic reflection. In this section, we hope to convey a better intuition
for Filinski’s construction, and also discuss what it looks like when combined with thermometer continuations.
The code in this section comes almost verbatim from Filinski. This section is helpful for understanding the
optimizations of Section 6, in which we explain how to fuse thermometer continuations with the code in this
section.
8 Filinski
9 Dan
(1994)
Piponi (2008)
17
5.1
Monadic Reflection
In SML and Java, there are two ways to program with mutable state. The first is to use the language’s built-in
variables and assignment. The second is to use the monadic encoding, programming similar to how a pure
language like Haskell handles mutable state. A stateful computation is a monadic value, a pure value of type
s → (a, s).
These two approaches are interconvertible. The program can take a value of type s → (a, s) and run it,
yielding a stateful computation of return type a. This operation is called reflect. Conversely, it can take a stateful
computation of type a, and reify it into a pure value of type s → (a, s). Together, the reflect and reify operations
give a correspondence between monadic values and effectful computations. This correspondence is termed
monadic reflection.
reflect and reify generalize to arbitrary monads. Consider nondeterminism, where a nondeterministic computation is either an effectful computation of type a, or a monadic value of type [a]. Then the reflect operator
would take the input [1, 2, 3] and nondeterministically return 1, 2, or 3 — this is the choose operator from Section
2). reify would take a computation that nondeterministically returns 1, 2, or 3, and return the pure value [1, 2, 3]
— this is withNondeterminism.
So, for languages which natively support an effect, reflect and reify convert between effects implemented by
the semantics of the language, and effects implemented within the language. Curry is a language with built-in
nondeterminism, and it has these operators, calling them anyOf and getAllValues. SML does not have built-in
nondeterminism, but, for our previous example, one can think of the code within a withNondeterminism block as
running in a language extended with nondeterminism. So, one can think of the construction in the next section
as being able to extend a language with any monadic effect.
In SML, monadic reflection is given by the following signature:
signature RMONAD = sig
structure M
: MONAD
: α M.m → α
: (unit → α ) → α
val reflect
val reify
M.m
end;
5.2
Monadic Reflection through Delimited Control
Filinski’s insight was that the monadic style is similar to an older concept called continuation-passing style. We
can see this by revisiting an example from Section 2.
Consider this expression:
withNondeterminism (fun ()
⇒
(choose [2,3,4]) * (choose [5,6]))
It is transformed into the monadic style as follows:
bind [2,3,4] (fn x
bind [5,6]
(fn y
⇒
⇒
return (x * y)))
The first call to choose has continuation fn ⇒ * (choose [5,6]). If x is the value returned by the first call to
the second has continuation fn ⇒x * . These continuations correspond exactly to the functions bound
in the monadic style. The monadic bind is the "glue" between a value and its continuation. Nondeterministically
choosing from [2,3,4] wants to return thrice, which is the same as invoking the continuation thrice, which is the
same as binding to the continuation.
choose,
18
So, converting a program to monadic style is quite similar to converting a program to this "continuation-passing
style." Does this mean a language that has continuations can program with monads in direct style? Filinski
answers yes.
The definition of monadic reflection in terms of delimited control is short. The overall setup is as follows:
functor Represent (M
:
MONAD)
RMONAD = struct
:
structure C = Control(type ans = Universal.u M.m)
structure M = M
...
...
fun reflect m =
fun reify t =
end;
Figure 3 showed how nondeterminism can be implemented by binding a value to the (delimited) continuation.
The definition of reflect is a straightforward generalization of this.
fun reflect m = C.shift (fn k
⇒
M.bind m k)
If reflect uses shift, then reify uses reset to delimit the effects implemented through shift. This implementation
requires use of type casts, because reset is monomorphized to return a value of type Universal.u m. Without these
type casts, reify would read
fun reify t = C.reset (fn ()
⇒
M.return (t ()))
Because of the casts, the actual definition of reify is slightly more complicated:
⇒ M.return (Universal.to_u (t ()))))
(M.return o Universal.from_u)
fun reify t = M.bind (C.reset (fn ()
5.3
Example: Nondeterminism
Using this general construction, we immediately obtain an implementation of nondeterminism equivalent to the
one in Section 2.3 from Section 2’s definition of ListMonad.
structure N = Represent(ListMonad)
fun choose xs = N.reflect xs
fun fail () = choose []
N.reify (fn ()
⇒
let val x = choose [2,3,4] * choose [5,7] in
if x
≥
20 then x
else fail () end);
(* val it = [21,20,28]
:
int list *)
It’s worth thinking about how this generic implementation executes on the example, and contrasting it with
the direct implementation of Section 2.3. The direct implementation executes the function body 6 times, once
for each branch of the computation. The generic one executes the function body 10 times (once with a replay
stack of length 0, 3 times with length 1, and 6 times with length 2). In the direct implementation, choose will
return a value if it can. In the generic one, choose never returns. Instead, it invokes the thermometer continuation,
causes the desired value to be returned at the equivalent point in the computation, and then raises an exception
19
containing the final result. So, 4 of those times, it could just return a value rather than replaying the computation.
This is the idea of one of the optimizations we discuss in Section 6. This, plus one other optimization, let us
derive the direct implementation from the generic one.
5.4
Example: State monad
State implemented through delimited control works differently from SML’s native support for state.
functor StateMonad (type state)
type
α
m = state
fun return x = fn
fun bind m f = fn
: MONAD = struct
→ α * state
s ⇒ (x, s)
s ⇒ let val (x, s’) = m s
in f x s’ end
end;
structure S = Represent (StateMonad (type state = int) )
fun tick () = S.reflect (fn s
⇒
((), s+1))
= S.reflect (fn s ⇒ (s, s))
fun put n = S.reflect (fn _ ⇒ ((), n))
fun get ()
#1 (S.reify (fn ()
⇒
(put 5; tick ();
2 * get ()))
0)
(* val it = 12
:
int *)
Let’s take a look at how this works, starting with the example reify
(reify (fn ()
=⇒
=⇒
=⇒
=⇒
=⇒
=⇒
⇒
(reset (fn ()
⇒
⇒
⇒
3 * (reflect (fn s
⇒
⇒
* get ()).
(s, s))))) 2
return (3 * (shift (fn k
⇒ (s,s))
⇒ fn s ⇒
bind (fn s
(let val k = (fn x
(fn s
⇒3
3 * get ())) 2
(reify (fn ()
(fn k
(fn ()
k end)(fn x
⇒
⇒
bind (fn s
⇒
(s, s)) k))))) 2
return (3*x)) 2
(3*x, s)) in (fn s
⇒
k s s)) 2
(3*s, s)) 2
(6, 2)
The get in reify (fn () ⇒3 * get ()) suspends the current computation, causing the reify to return a function
which awaits the initial state. Once invoked with an initial state, it resumes the computation (multiplying by 3).
What does reify (fn () ⇒(tick (); get ())) do? The call to tick () expands into shift (fn k ⇒fn s ⇒k () (s+1)).
It again suspends the computation, awaiting the state s. Once it receives s, it resumes it, returning () from tick.
The call to get suspends the computation again, returning a function that awaits a new state; tick supplies s+1.
Think for a second about how this works when shift and reset are implemented as thermometer continuations.
The get, put, and tick operators do not communicate by mutating state. They communicate by suspending the
computation, i.e.: by raising exceptions containing functions. So, although the implementation of state in terms
of thermometer continuations uses SML’s native support for state under the hood, it only does so tangentially, to
capture the continuation.
20
6
OPTIMIZATIONS
Section 5.3 compared the two implementations of nondeterminism, and found that the generic one using
thermometer continuations replayed the computation gratuitously. Thermometer continuations also replay the
program in nested fashion, consuming stack space. In this section, we sketch a few optimizations that make
monadic reflection via thermometer continuations less impractical, and illustrate the connections between the
two implementations of nondeterminism.
6.1
CPS-bind: Invoking the Continuation at the Top of the Stack
The basic implementation of thermometer continuations wastes stack space. Look at the last example of Section
4.3, and notice how it calls block () three nested times. And yet, the outer two calls to block () will be discarded
by a raised exception as soon as the inner one completes. So, the implementation could save a lot of stack space
by raising an exception before replaying the computation. Indeed, we did this when symbolically evaluating that
example in 4.3 to make it easier to read.
So, when a program invokes a thermometer continuation, it will need to raise an exception to transfer control
to the enclosing reset, and thereby signal reset to replay the computation. While the existing Done exception
signals that a computation is complete, it can do this with a second kind of exception, which we call Invoke.
However, the shift and reset functions do not invoke a thermometer continuation: the body of the shift does.
In the case of monadic reflection, this is the monad’s bind operator. Raising an Invoke exception will discard the
remainder of bind, so it must somehow also capture the continuation of bind. We can do this by writing bind itself
in the continuation-passing style, i.e.: with the following signature:
val bind
: α
m
→ ( α → (β
m) cont)
→ (β
m) cont;
where (β m) cont = forall δ . (β m →δ ) →δ )
The above is not valid SML because SML lacks the rank-2 polymorphism (i.e.: the nested forall) required by
the continuation-passing style. Nonetheless, we have implemented this in both SML, using additional unsafe
casts, and in OCaml, which does support rank-2 polymorphism.
The supplementary material contains code with this optimization, and uses it to implement nondeterminism
in a way that executes more similarly to the direct implementation. We give here some key points. Here’s what
the CPS’d bind operator for the list monad would look like if SML hypothetically had rank-2 polymorphism:
fun bind []
|
f d = d []
bind (x :: xs) f d = f x (fn a
⇒
bind xs f
(fn b
⇒
d (a @ b)))
When used by reflect, f becomes a function that raises the Invoke exception, transferring control to the
enclosing reset, which then replays the entire computation, but at the top level. The continuations of the bind d
get nested in a manner which is harder to describe, but ultimately get evaluated at the very end, also at the
top level. So the list appends in d (a @ b) actually run at the top level of the reset, similar to how, in direct
nondeterminism, it is the outer call to withNondeterminism that aggregates the results of each branch.
While this CPS-monad optimization as described here can be used to implement many monadic effects, it
cannot be used to implement all of them, nor general delimited continuations. Consider the state monad from
Section 5.4: bind actually returns a function which escapes the outer reify. Then, when the program invokes that
function and it tries to invoke its captured thermometer continuation, it will try to raise an Invoke exception to
transfer control to its enclosing reify, but there is none. This CPS-monad optimization as described does not work
if the captured continuation can escape the enclosing reset. With more work, it could use mutable state to track
whether it is still inside a reset block, and then choose to raise an Invoke exception or invoke the thermometer
continuation directly.
21
6.2
Direct Returns
In our implementation, a reset body C[reflect (return 1)] expands into C[raise (Done (C[1] handle (Done x ⇒x)))].
So, the entire computation up until that reflect runs twice. Instead, of replaying the entire computation, that
reflect could just return a value. C[reflect (return 1)] could expand into C[1].
reflect (return 1) expands into shift (fn k ⇒bind (return 1) k). By the monad laws, this is equivalent to
shift (fn k ⇒k 1). Tail-calling a continuation is the same as returning a value, so this is equivalent to 1. So, it’s
the tail-call that allows this instance of reflect to return a value instead of replaying the computation.
Implementing the direct-return optimization is a small tweak to the CPS-bind optimization. The signature for
bind is further modified to:
val bind
: α
m
→ ( α → (β
m) cont)
→ ( α → (β
m) cont)
→ (β
m) cont;
where (β m) cont = forall δ . (β m →δ ) →δ ) . This variant of bind takes two arguments of type α →(β m) cont.
One raises an Invoke exception, as described in Section 6.1. The other returns a value directly, after updating the
internal state of the thermometer continuation implementation. So, the first time bind invokes the continuation,
it may do so by directly returning a value, and thereafter instead raises an Invoke exception.
The bind operator for the list monad never performs a tail-call (it must always wrap the result in a list), but,
after converting it to CPS, it always performs a tail-call. So this direct-return optimization combines well with
the previous CPS-monad optimization. Indeed, applying them both transforms the generic nondeterminsm of
Section 5.3 into the direct nondeterminism of Section 2. In Section 6.4, we show benchmarks showing that this
actually gives a faster implementation of nondeterminism than the code in Section 2.
In the supplementary material, we demonstrate this optimization, providing optimized implementations of
nondeterminism (list monad) and failure (maybe monad).
6.3
Memoization
While the frequent replays of thermometer continuations can interfere with any other effects in a computation,
it cannot interfere with observationally-pure memoization. Memoizing nested calls to reset can save a lot of
computation, and any expensive function can memoize itself without integrating into the implementation of
thermometer continuations.
6.4
Benchmarks
To get a better understanding of the performance cost of thermometer continuations and the effect of our
benchmarks, we implemented several benchmarks with different monadic effects.
There are four benchmarks: NQUEENS, INTPARSE-GLOB, INTPARSE-LOCAL, and MONADIC-ARITHPARSE. These four benchmarks use three different monads. Depending on the monad, we gave three to six
implementations of each benchmark. Each Direct implementation implements the program pure-functionally,
without monadic effects. The ThermoCont and Filinski implementations use monadic reflection, implemented
via thermometer continuations and Filinski’s construction, respectively. For the nondeterminism and failure
monads, our optimizations apply, given in Opt. ThermoCont. For the nondeterminism monad, we can also
use our Replay-based Nondet construction from Section 2. These were all implemented in SML. Finally,
for nondeterminism, we also compared to an implementation in Curry, which provides native support for
nondeterminism.
The SML solutions were all run using the SML/NJ interpreter v110.8010 . Although MLTON11 , the whole-program
optimizing compiler for SML, is far more efficient than SML/NJ, we could not easily port our implementation of
10 Appel
11 Weeks
and MacQueen (1991)
(2006)
22
thermometer continuations to MLTON because it lacks Unsafe.cast. We also tried using our OCaml implementation
of thermometer continuations; surprisingly, we got a stack overflow error even for relatively small inputs, even
though the implementation uses only shallow recursion, and the SML version ran fine. We ran the Curry
implementation in KiCS212 v0.5.1. Our informal experiments show that a different Curry compiler, PAKCS13 , was
much slower.All experiments were conducted on a 2015 MacBook Pro with a 2.8 GHz Intel Core i7 processor. All
times shown are the average of 5 trials, except for MONADIC-ARITH-PARSE, as discussed below.
The first benchmark NQUEENS is the problem of enumerating all solutions to the n-queens problem. Table 1
reports the times for each implementation for different n. While the direct implementation was unsurprisingly the
fastest, thermometer continuations beat Filinski’s construction for small n, and the optimized version remained
within a factor of 2 until n = 10. Optimized thermometer continuations beat replay-based nondeterminism, likely
because the optimized thermometer continuation solution replaces list allocations and operations with closures.
Surprisingly, Curry performed by far the worst; for n = 11, we killed the process after running it for 20 minutes.
The twin benchmarks INTPARSE-GLOB and INTPARSE-LOCAL both take a list of numbers as strings, parse
each one, and return their sum. They both use a monadic failure effect (like Haskell’s Maybe monad), and differ
only in their treatment of strings which are not valid numbers: INTPARSE-GLOB returns failure for the entire
computation, while INTPARSE-LOCAL will recover from failure and ignore any malformed entry. Table 2
gives the results for INTPARSE-GLOB. For each input size n, we constructed both a list of n valid integers, as
well as one which contains an invalid string halfway through. Table 3 gives the results for INTPARSE-LOCAL.
For each n, we constructed lists of n strings where every 1/100th, 1/10th, or 1/2nd string was not a valid int.
For INTPARSE-GLOB, unoptimized thermometer continuations wins, as it avoids Filinski’s cost of callcc, the
optimized version’s reliance on closures, as well as the direct approach’s cost of wrapping and unwrapping
results in an option type. For INTPARSE-LOCAL, unoptimized thermometer continuations lost out to the direct
implementation. Note that thermometer continuations here devolves into raising an exception for bad input, but
with a clean relation to the pure monadic monadic version.
Finally, benchmark MONADIC-ARITH-PARSE is a monadic parser in the style of Hutton and Meijer14 . These
programs input an arithmetic expression, and return the list of results of evaluating any prefix of the string which
is itself a valid expression. The Filinski and ThermoCont implementations closely follow Hutton and Meijer,
executing in a "parser" monad with both mutable state and nondeterminism (equivalent to Haskell’s StateT List
monad). The direct implementation inlines the monad definition, passing around a list of (remaining input, parse
result) pairs. We did not provide an implementation with optimized thermometer continations, as we have not
yet found how to make our optimizations work with the state monad. Note that all three implementations use
the same algorithm, while producing beautiful code (for the monadic versions), is exponentially slower than the
standard LL/LR parsing algorithms.
Table 4 reports the average running time of each implementation on 30 random arithmetic expressions with a
fixed number of leaves. There was very high variance in the running time of different inputs. For inputs with
40 leaves, the fastest input took under 25ms for all implementations, while the slowest took approximately 25
minutes on the direct implementation, 5 hours and 43 minutes for Filinski’s construction, and 9 hours 50 minutes
for thermometer continuations.
Overall, these benchmarks show that the optimizations of this section can provide a substantial speedup, and
there are many computations for which thermometer continuations do not pose a prohibitive cost. Thermometer
continuations are surprisingly competitive with Filinski’s construction, even though SML/NJ is known for its
efficient callcc, and yet thermometer continuations can be used in far more programming languages.
12 Braßel
et al. (2011)
et al. (2003)
14 Hutton and Meijer (1998)
13 Hanus
23
Direct
Replay-based Nondet
Filinski
ThermoCont
Opt. ThermoCont
Curry
4
0.006s
0.006s
0.010s
0.006s
0.006s
0.002s
5
0.005s
0.006s
0.009s
0.006s
0.006s
0.004s
6
0.005s
0.007s
0.010s
0.007s
0.006s
0.014s
7
0.006s
0.007s
0.011s
0.009s
0.008s
0.119s
8
9
10
0.006s 0.008s
0.015s
0.017s 0.089s
0.506s
0.014s 0.022s
0.046s
0.017s 0.052s
0.248s
0.013s 0.041s
0.198s
1.270s 13.593s 154.987s
11
12
0.064s
0.310s
2.606s 8m54.364s
0.199s
1.063s
1.497s
9.462s
1.210s
7.809s
>20m
>20m
Table 1. Benchmark NQUEENS
Bad input?
Y
Direct
N
Y
Filinski
N
Y
ThermoCont
N
Y
Opt. ThermoCont
N
1000
0.000s
0.000s
0.000s
0.000s
0.000s
0.000s
0.000s
0.000s
10,000
0.001s
0.002s
0.004s
0.004s
0.003s
0.003s
0.000s
0.002s
50,000
0.004s
0.014s
0.018s
0.020s
0.008s
0.011s
0.013s
0.015s
100,000
0.026s
0.051s
0.012s
0.014s
0.024s
0.029s
0.015s
0.024s
500,000
0.159s
0.213s
0.172s
0.221s
0.167s
0.223s
0.197s
0.247s
1,000,000
0.260s
0.371s
0.258s
0.360s
0.260s
0.367s
0.293s
0.364s
5,000,000
36.260s
1m54.072s
28.719s
1m26.603s
27.461s
1m20.841s
27.298s
1m23.433s
1000 10,000 50,000 100,000 500,000 1,000,000
0.000s 0.002s 0.011s
0.060s
0.232s
0.404s
0.000s 0.002s 0.002s
0.053s
0.221s
0.375s
0.000s 0.000s 0.001s
0.014s
0.176s
0.257s
0.000s 0.001s 0.048s
0.053s
0.264s
0.456s
0.000s 0.004s 0.043s
0.048s
0.234s
0.420s
0.000s 0.004s 0.008s
0.050s
0.182s
0.301s
0.000s 0.002s 0.045s
0.064s
0.344s
0.470s
0.000s 0.003s 0.028s
0.064s
0.266s
0.445s
0.000s 0.001s 0.023s
0.067s
0.214s
0.353s
0.000s 0.002s 0.030s
0.059s
0.227s
0.403s
0.000s 0.002s 0.030s
0.058s
0.218s
0.386s
0.000s 0.002s 0.023s
0.055s
0.183s
0.289s
5,000,000
2m04.316s
1m40.251s
15.515s
3m19.265s
3m22.245s
23.632s
4m28.700s
3m26.809s
22.750s
3m01.981s
2m37.321s
27.546s
Table 2. Benchmark INTPARSE-GLOB
% bad input
0.01
Direct
0.10
0.50
0.01
Filinski
0.10
0.50
0.01
ThermoCont
0.10
0.50
0.01
Opt. ThermoCont
0.10
0.50
Table 3. Benchmark INTPARSE-LOCAL
Direct
Filinski
ThermoCont
10
0.011s
0.116s
0.184s
20
0.163s
1.638s
2.540s
30
1.908s
19.035s
30.229s
40
2m39.860s
25m18.794s
39m58.183s
Table 4. Benchmark MONADIC-ARITH-PARSE
24
7
BUT ISN’T THIS IMPOSSIBLE?
In a 1990 paper, Matthias Felleisen presented formal notions of expressibility and macro-expressibility of one
language feature in terms of others, along with proof techniques to show a feature cannot be expressed 15 . Hayo
Thielecke used these to show that exceptions and state together cannot macro-express continuations 16 . This is
concerning, because, at first glance, this is exactly what we did.
First, a quick review of Felleisen’s concepts: Expressibility and macro-expressibility help define what should be
considered core to a language, and what is mere "syntactic sugar." An expression is a translation from a language
L containing a feature F to a language L 0 without it which preserves program semantics. A key restriction is
that an expression may only rewrite AST nodes that implement F and the descendants of these nodes. So, an
expression of state may only rewrite assignments, dereferences, and expressions that allocate reference cells. The
whole-program transformation that transforms a stateful program into a pure one that constantly passes around
an ever-updating "state" variables is not an expression. A macro-expression is an expression which may rewrite
nodes from F , but may only move or copy the children of such nodes (technically speaking, it must be a term
homomorphism). A classic example of a macro-expression is implementing the += operator in terms of normal
assignment and addition. A classic example of an expression which is not a macro-expression is desugaring
for-loops into while-loops (it must dig into the loop body and modify every continue statement). Another one is
implementing Java finally blocks (which need to execute an action after every return statement).
There are a couple reasons why Thielecke’s proof does not immediately apply. First, it only concerns macroexpressibility. Second, it concerns callcc-style continuations rather than delimited continuations. So, there are
two ways it could be extended to forbid our construction. First, one could extend Thielecke’s results to general
expressibility and delimited continuations. Second, we could limit the discussion to programs wrapped in an
all-encompassing reset, so that callcc will itself be macro-expressible using shift. We need not worry about their
combination: If we limit ourselves to programs that are entirely enclosed by a reset, then an "expression" of
shift/reset may rewrite the entire program.
It turns out that neither extension applies either. First, implementing effects with monadic reflection is not
an expression. An expression for mutable state may rewrite assignments and dereferences in terms of other
operations, but our construction must also enclose that entire program fragment in a reify. Filinski’s construction
is not an expression either for the same reason.
Second, even without the issue of wrapping with reify aside, our construction is still not a macro-expression.
Let’s take a look at Thielecke’s proof and see where it fails. Thielecke’s proof is based on showing that, in a
language with exceptions and state but not continuations, all expressions of the following form with different j
are operationally equivalent:
R j = λ f .((λx .λy.(f 0; x := !y; y := j; !x))(ref 0)(ref 0))
The intuition behind this equivalence is that the two reference cells are allocated locally and then discarded,
and so the value stored in them can never be observed. However, with continuations, on the other hand, f could
cause the two assignments to run twice on the same reference cells.
This example breaks down because it cannot be expressed in our monadic reflection framework as is. The
monadic reflection framework assumes there are no other effects within the program other than the ones
implemented via monadic reflection. To write the R j using thermometer continuations and monadic reflection,
the uses of ref must be changed from the native SML version to one implemented using the state monad. Then,
when the computation is replayed, repeated calls to ref may return the same reference cell, allowing the state to
escape, thereby allowing different R j to be distinguished.
15 Felleisen
(1990)
(2001)
16 Thielecke
25
8
RELATED WORK
Our work is most heavily based on Filinski’s work expressing monads using delimited control 17 . We have also
discussed theoretical results regarding the inter-expressability of exceptions and continuations in Section 7. Other
work on implementing continuations using exceptions relate the two from a runtime-engineering perspective
and from a typing perspective.
Continuations from stack inspection. Oleg Kiselyov’s delimcc library 18 provides an implementation of delimited
control for OCaml, based on the insight that the stack-unwinding facilities used to implement exceptions are
also useful in implementing delimited control. Unlike our approach, delimcc works by tightly integrating with
the OCaml runtime, exposing low-level details of its virtual machine to user code. Its implementation relies on
copying portions of the stack into a data structure, repurposing functionality used for recovering from stack
overflows. It hence would not work for e.g.: many implementations of Java, which recover from stack overflows
by simply deleting activation records. On the other hand, its low-level implementation makes it efficient and lets it
persist delimited continuations to disk. A similar insight is used by Pettyjohn et al 19 to implement continuations
using a global program transformation.
Typing power of exceptions vs. continuations. Lillibridge 20 shows that exceptions introduce a typing loophole
that can be used to implement unbounded loops in otherwise strongly-normalizing languages, while continuations
cannot do this, giving the slogan "Exceptions are strictly more powerful than call/cc." As noted by other authors 21 ,
this argument only concerns the typing of exceptions rather than their execution semantics, and is inapplicable
in languages that already have recursion.
9
CONCLUSION
Filinski’s original construction of monadic reflection from delimited continuations, and delimited continuations
from normal continuations plus state, provided a new way to program for the small fraction of languages which
support first-class continuations. With our demonstration that exceptions and state are sufficient, this capability
is extended to a large number of popular languages, including 9 of the TIOBE 1022 . While languages like Haskell
with syntactic support for monads may not benefit from this construction, bringing advanced monadic effects
to more languages paves the way for ideas spawned in the functional programming community to influence a
broader population.
In fact, the roots of this paper came from an attempt to make one of the benefits of monads more accessible.
We built a framework for Java where a user could write something that looks like a normal interpreter for a
language, but, executed differently, it would become a flow-graph generator, a static analyzer, a compiler, etc.
Darais 23 showed that this could be done by writing an interpreter in the monadic style (concrete interpreters
run programs directly; abstract interpreters run them nondeterministically). We discovered this concurrently
with Darais, and then discovered replay-based nondeterminism so that Java programmers could write normal,
non-monadic programs.
Despite the apparent inefficiency of thermometer continuations, the optimizations discussed in Section 6,
combined with the oft-unused speed of modern machines, provide hope that the ideas of this paper can find their
17 Filinski
(1994)
(2010)
19 Pettyjohn et al. (2005)
20 Lillibridge (1999)
21 Thielecke (2001)
22 TIOBE Software BV (2017)
23 Darais et al. (2017)
18 Kiselyov
26
way into practical applications. Indeed, Filinski’s construction is actually known as a way to make programs
faster 24
Overall, we view finding a way to bring delimited control into mainstream languages as a significant achievement. We hope to see a flourishing of work with advanced effects now that they can be used by more programmers.
Working code for all examples and benchmarks, as well as the CPS-bind and direct-return optimizations, is
available from https://github.com/jkoppel/thermometer-continuations .
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24 Hinze
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27
| 6 |
Linear State Estimation via 5G C-RAN Cellular
Networks using Gaussian Belief Propagation
arXiv:1710.08671v2 [cs.IT] 2 Feb 2018
Mirsad Cosovic, Dejan Vukobratovic, Vladimir Stankovic
Abstract—Machine-type communications and large-scale
information processing architectures are among key
(r)evolutionary enhancements of emerging fifth-generation (5G)
mobile cellular networks. Massive data acquisition and
processing will make 5G network an ideal platform for
large-scale system monitoring and control with applications in
future smart transportation, connected industry, power grids,
etc. In this work, we investigate a capability of such a 5G
network architecture to provide the state estimate of an
underlying linear system from the input obtained via
large-scale deployment of measurement devices. Assuming that
the measurements are communicated via densely deployed
cloud radio access network (C-RAN), we formulate and solve
the problem of estimating the system state from the set of
signals collected at C-RAN base stations. Our solution, based
on the Gaussian Belief-Propagation (GBP) framework, allows
for large-scale and distributed deployment within the emerging
5G information processing architectures. The presented
numerical study demonstrates the accuracy, convergence
behavior and scalability of the proposed GBP-based solution to
the large-scale state estimation problem.
I. I NTRODUCTION
With transition towards fifth generation (5G), mobile
cellular networks are evolving into ubiquitous systems for
data acquisition and information processing suitable for
monitoring and control of large-scale systems. At the
forefront of this evolution is the transformation of radio
access network (RAN) to support massive-scale
machine-type communications (MTC) [1] and transformation
of core network (CN) to support large-scale centralized or
distributed information processing through Cloud-RAN
(C-RAN) and Fog-RAN (F-RAN) architecture [2], [3]. MTC
services in 5G will offer both massive-scale data acquisition
from various machine-type devices through massive MTC
(mMTC) service, but also, provide ultra reliable and
low-latency
communication
(URLLC)
service
for
mission-critical applications [4]. Complemented with
ultra-dense RAN deployment and flexible and virtualized
signal processing architecture, novel 5G network services
that are particularly suitable for large-scale system
monitoring and control of various smart infrastructures are
emerging [5].
In this work, we focus on a generic state estimation
problem placed in the context of a future 5G-inspired
M. Cosovic is with Schneider Electric DMS NS, Novi Sad, Serbia
(e-mail: mirsad.cosovic@schneider-electric-dms.com). D. Vukobratovic
is with Department of Power, Electronic and Communications
Engineering, University of Novi Sad, Novi Sad, Serbia (e-mail:
dejanv@uns.ac.rs). V. Stankovic is with Department of Electronic and
Electrical Engineering, University of Strathclyde, Glasgow, UK (email:vladimir.stankovic@eee.strath.ac.uk).
C-RAN-based cellular network. We consider an underlying
large-scale physical system characterized by the state vector
s that contains values of N system state variables. The state
variables are observed through the set of M measurements x
of physical quantities collected at the measurement devices
spread across the system. This paper considers linear system
model in which measured quantities are linear functions of
the (sub)set of state variables. Further, we assume
measurements are wirelessly communicated across
C-RAN-based cellular network. In C-RAN, large number of
spatially distributed remote radio heads (RRH) constitutes an
ultra-dense RAN infrastructure that receives signals from
densely populated MTC devices (e.g., the measurement
devices under consideration) [2]. The signal vector y
collected at RRHs is forwarded via backhaul links to a
central C-RAN location where it is fed into a collection of
base-band units (BBU) for signal detection and recovery. In
the standard C-RAN signal detection problem, the goal is to
recover the signal x transmitted by the set of MTC devices
from the signal y received at RRHs and gathered centrally at
BBUs [6] [7]. However, in this paper, focusing on widely
applicable linear system state estimation, we extend this goal
and investigate the problem of recovering the system state s
directly from the signal y collected across the C-RAN.
The problem we observe represents a concatenation of the
two well-studied problems: the linear system state estimation
problem (see, e.g., [8], for the case of power system state
estimation) and the problem of uplink signal detection in
C-RAN [6]. For the joint problem, it is straightforward to
derive (and implement at a central location) the standard
minimum mean-square error (MMSE) estimator, however,
such a solution comes with prohibitive O(N 3 )-complexity
that hinders its application for large-scale systems. By
exploiting inherent sparsity within both of the component
problems, an approximate MMSE solution for each problem
can be obtained using the tools from probabilistic graphical
models, as recently investigated for both (power system)
state estimation [9] and uplink signal detection in C-RANs
[7]. In particular, an instance of the Belief-Propagation (BP)
algorithm, called Gaussian BP (GBP) [10], can be applied to
produce an exact MMSE estimate with O(N )-complexity,
thus scaling the MMSE solution to large-scale system
scenarios.
In this paper, we motivate, formulate and solve the linear
system state estimation problem considered jointly with the
signal detection problem in C-RAN-based cellular networks.
We cast the problem of estimating the system state s from
the received vector y into an equivalent maximum
a-posteriori (MAP) problem, and place it into the framework
of a popular class of probabilistic graphical model called
factor graphs. The state estimate ŝ is then derived as a
solution of the GBP algorithm applied over a specific
bi-layer structure of the factor graph. Throughout the paper,
we use state estimation in power systems with the
measurements collected via 5G-inspired C-RAN network as
a running example. Our initial numerical results demonstrate
the viability of the proposed approach, both in terms of
accuracy and convergence.
The paper is organized as follows. In Section II, we present
the joint state estimation and C-RAN uplink communication
system model. In Section III, this model is mapped into a
corresponding factor graph, and the state estimate is obtained
via GBP. Section IV provides numerical results of the proposed
GBP state estimator. The paper is concluded in Section V.
II. S YSTEM M ODEL
We consider a generic state estimation problem where a
set of state variables s is estimated from a set of observed
noisy linear measurements x. However, unlike the traditional
setup where the measurements in x are assumed available at
a central node, here we assume they are transmitted via radio
access network (RAN) of a mobile cellular system based on
a cloud-RAN (C-RAN) architecture. The received signal y is
collected at a large-number of densely deployed remote radio
heads (RRHs) and jointly processed at the C-RAN base-band
units (BBUs). The problem we consider is that of estimating
the system state s from the received signal y.
Linear system measurements model: We consider a
system described via the set of N state variables
s = (s1 , s2 , . . . , sN )T ∈ CN ×1 . The system is observed via a
set of M measurements x = (x1 , x2 , . . . , xM )T ∈ CM×1 .
Each measurement is a linear function of the state variables
additionally corrupted by the additive noise, i.e.,
xi
= ai · s + ni , 1 ≤ i ≤ M , where
ai = (ai,1 , ai,2 , . . . , ai,N ) ∈ C1×N is a vector of coefficients,
while ni ∈ C is a complex random variable. Overall, the
system is represented via noisy linear observation model
x = A · s + n, where the matrix A ∈ CM×N contains
vectors ai , 1
≤
i
≤
M, as rows, while
n = (n1 , n2 , . . . , nM )T ∈ CM×1 is a vector of additive noise
samples. We assume noise samples ni are independent
identically distributed (i.i.d.) Gaussian random variables with
variance σn2 i . For simplicity, we assume the measurement
noise variances are equal, i.e., σn2 i = σn2 , 1 ≤ i ≤ M . In
other words, n represents a complex Gaussian random vector
with zero means µn = 0 and the covariance matrix
Σn = σn2 I ∈ CM×M (I is an identity matrix).
C-RAN uplink communication model: In the standard
state estimation models, measurements are either assumed
available, or they are communicated to the central node,
where the state estimation problem is solved.
Communication models typically involve point-to-point
communication links between the measurement devices and
the central node, affected by communication impairments
such as delays, packet losses, limited bit rates, etc. In the
cellular networks context, this assumes reservation of uplink
resources and subsequent non-orthogonal transmission,
which typically incurs significant communication delays.
Inspired by the recent evolution of massive MTC and
ultra-dense C-RAN architectures in upcoming 5G mobile
cellular networks, in this work, we consider different
grant-less and non-orthogonal communication model, as we
detail next. Note that the following C-RAN cellular network
model could provide ultra-low latency for the state
estimation application under consideration. In other words,
such an architecture could produce the system state estimate
at the central network node with very low delay after the
measurements are acquired, which is crucial for emerging
mission-critical 5G MTC use cases [11].
In the mobile cellular system under consideration,
measurements are collected by the measurement devices that
we refer to as MTC user equipment (MTC-UE). We consider
uplink transmission of M single-antenna MTC UEs towards
L single-antenna RRHs. We assume both MTC UEs and
RRHs are randomly and uniformly distributed across a given
geographic area (note that this placement model is somewhat
refined in the numerical results section). The signal
x ∈ CM×1 , representing the set of collected noisy
measurements, is transmitted1 by M MTC UEs, while the
signal y = (y1 , y2 , . . . , yL ) ∈ CL×1 is received at L RRHs,
where y = H · x + m. The matrix H ∈ CL×M represents
the channel matrix, where hi,j represents a complex channel
coefficient between the j-th MTC UE and the i-th RRH,
while m = (m1 , m2 , . . . , mL ) ∈ CL is a vector of additive
noise samples. As for the measurement process, for the
communication process we also assume noise samples mi
are i.i.d. zero-mean Gaussian random variables with variance
2
σm
, i.e., the mean value and the covariance matrix of m is
2
I ∈ CL×L , respectively.
given as µm = 0 and Σm = σm
Linear system state estimation problem: From the
received signal y collected across RRHs, we are interested in
finding an estimate ŝ of the state vector s. In this paper, we
focus on the centralized C-RAN architecture, where all the
BBUs are collocated at the central C-RAN node. Thus, due
to availability of y at the central location, we consider
centralized algorithms for the state estimation problem.
However, the solution we propose in this paper is based on
GBP framework, thus it is easily adaptable to a distributed
scenario that we refer to as fog-RAN (F-RAN), where BBUs
are distributed across different geographic locations closer to
the MTC UEs. We refer the interested reader to our recent
overview of distributed algorithms for solving the state
estimation problem in the context of upcoming 5G cellular
networks [5].
A common centralized approach to solve the state
estimation problem is to provide the minimum mean-square
error (MMSE) estimate. One can easily obtain the MMSE
estimate ŝ for the underlying linear model in the form:
−1
H −1
ŝ = Σ−1
· (HA) (HA)H Σ−1 y, (1)
s + (HA) Σ
1 Note that, at this point, one can insert specific linear modulation scheme
xm = fm (x). For simplicity, we assume fm (x) = x.
C-RAN BBUs
C-RAN RRHs
Power System
Fig. 1. System model example: State estimation in Smart Grid via C-RAN-based mobile cellular network.
where (·)H is the conjugate-transpose matrix operation,
Σ = HΣn HH + Σm , and where we assume prior
distribution of s is i.i.d. Gaussian with mean µs = 0 and
Σs = σs2 I ∈ CN ×N . However, solving (1) scales as O(N 3 )
which makes linear MMSE state estimation inapplicable in
large-scale systems which are of interest in this paper.
In the following, we cast the MMSE estimation into an
equivalent MAP state estimator as follows:
ŝ = arg max P (s|y),
s∈CN
(2)
where P (s|y) is the posterior probability of the state s after
the signal y is observed at the RRHs. As we will demonstrate
in the sequel, if certain sparsity arguments are applicable in the
system model under consideration, the solution of the MAP
problem can be efficiently calculated using the framework of
factor graphs and belief-propagation (BP) algorithms.
System model example (Smart Grid): Before
continuing, it is useful to consider an example of the above
state estimation setup. We consider the state estimation
problem in an electric power system, where the goal is to
estimate the state of the power system s, containing complex
voltages of N system buses, via the set of measurements x
obtained using measurement devices. Measurement devices
are geographically distributed across the power system and
we assume they are equipped with wireless cellular
interfaces, i.e., they represent MTC UEs connected to the
C-RAN based cellular network. The signal y received from
the set of RRHs densely deployed across the cellular
network coverage area is processed centrally within the
C-RAN system architecture. Fig. 1 illustrates the smart grid
example that we will further refine in Section IV and use as
a running example throughout this paper.
III. S TATE E STIMATION VIA G AUSSIAN B ELIEF
P ROPAGATION
In this section, we provide a solution to the combined state
estimation and uplink signal detection problem defined in (2),
by applying factor graphs and GBP framework. In fact, for
both constituent scenarios: the conventional state estimation
(in case of power systems) and the uplink signal detection in
C-RAN, the GBP has already been proposed and analyzed (see
details in [9] and [7]). Thus in this work, we propose using
GBP in a joint and combined setup of extracting the system
state directly from the observed C-RAN signals.
We note that the various properties of the proposed GBP
approach (e.g., complexity, convergence, etc.) will strongly
depend on the structure of the underlying factor graph. For
example, in terms of complexity (we will come back to
convergence in the next section), for GBP to scale well to
large-scale systems, it is fundamental that both matrices A
and H defining the two linear problems are sparse, i.e., that
for both A and H, the number of non-zero entries scales as
O(N ). In many real-world scenarios, the sparsity typically
arises from geographic constraints and reflect locality that is
typically present in both the measurement and the
communication part of the system model. More detailed
=
arg
(4)
s∈CN
max
P (s, x, y)
s∈CN ,x∈CM
Note that the distribution P (s, x, y) is jointly Gaussian, and
in addition, due to the problem structure where s and y are
conditionally independent given x, we obtain:
ŝ =
arg
=
arg
max
P (y|s, x)P (s, x) =
(5)
max
P (y|x)P (x|s)P (s).
(6)
s∈CN ,x∈CM
s∈CN ,x∈CM
As noted before, in many real-world systems of interest, a
measurement xj is a linear function of a small subset of local
state variables sN (xj ) , where N (xj ) is the index set of the
state variables that affect xj , and sN (xj ) = {si |i ∈ N (xj )}.
In other words, the row-vector aj has non-zero components
only on a small number of positions indexed by the set N (xj ),
thus making the matrix A sparse2 . Using this fact and the fact
that the measurements xj are mutually independent, we obtain:
M
Y
P (x|s) =
P (xj |sN (xj ) ).
(7)
j=1
P (y|x) =
P (yi |xN (yi ) ).
FH
X
FA
S
.
.
.
L
Y
Y
Fx
.
.
.
On the other hand, in the C-RAN communication part,
although in theory the received signal yi depends on all the
transmitted symbols in x, the channel coefficients between a
RRH and a geographically distant MTC UE can be
considered negligible, thus leading to matrix H sparsification
[7]. Upon distance-based sparsification proposed in [7], the
received symbol yi depends only on a small number of
symbols xN (yi ) , where N (yi ) is the index set of symbols
transmitted by the set of MTC UEs in geographic proximity
of the i-th RRH. Taking the channel sparsification into
account, we obtain:
Fy
.
.
.
(3)
s∈CN
.
.
.
arg max P (s|y) ∝ arg max P (s, y) =
.
.
.
=
.
.
.
ŝ
The factor graph representation of the MAP problem
follows the factorization presented in (9) and is illustrated in
Fig. 2. Factor graph G = G(V ∪ F , E) is a bipartite graph
consisting of the set of variable nodes V, the set of factor
nodes F , and the set of edges E. In our setup, the set V can
be further divided as V = S ∪ X ∪ Y, where
S
= {s1 , s2 , . . . , sN } is the set of state nodes,
X = {x1 , x2 , . . . , sM } is the set of measurement nodes,
while Y = {y1 , y2 , . . . , yL } is the set of received symbol
nodes. The set of factor nodes can be divided as
F
=
FH ∪ FA ∪ Fy ∪ Fx ∪ Fs , where
FH = {fh1 , fh2 , . . . , fhL } and FA = {fa1 , fa2 , . . . , faM }
represent factor nodes that capture linear relationships
between variable nodes described by the rows of matrices H
and A, respectively. In addition, Fy = {fy1 , fy2 , . . . , fyL }
and Fs = {fs1 , fs2 , . . . , fsN } represent the factor nodes that
provide inputs due to observations of y and the prior
knowledge
about
x,
respectively,
while
Fx = {fx1 , fx2 , . . . , fxM } serve as virtual inputs needed for
initialization of measurement nodes. Similarly, the set of
edges E can be divided as E = EH ∪ EA ∪ Ey ∪ Ex ∪ Es ,
where EH ∪ Ey ∪ Ex and EA ∪ Es ∪ Ex can be considered as
the set of edges of two bipartite subgraphs
GH = (Y ∪ X ∪ FH ∪ Fy ∪ Fx , EH ∪ Ey ∪ Ex ) and
GA = (X ∪ S ∪ FA ∪ Fs ∪ Fx , EA ∪ Es ∪ Ex ), obtained as
the subgraphs of G induced from the set of factor nodes
FH ∪ Fy ∪ Fx and FA ∪ Fs ∪ Fx , respectively3.
.
.
.
account on the sparsity of matrices A and H clearly depends
on the specific scenario under consideration, and we relegate
these details to Section IV where we will explicitly deal with
the smart grid example introduced earlier.
Factor Graph System Representation: Assuming that
the system state s has a Gaussian prior, and given that the
measurement and communication noise is assumed Gaussian,
the MAP problem can be rewritten as follows:
(8)
i=1
Fig. 2. Factor graph representation of the system model.
Finally, assuming that the state vector is apriori given as
a set of i.i.d Gaussian random variables, we obtain the final
factorized form of the initial MAP problem:
P (sk ). (9)
As noted earlier, the state estimation problem using GBP
over factor graph GA , and the uplink C-RAN signal detection
problem using GBP over factor graph GH , have been recently
investigated in detail in [9] and [7], respectively.
Gaussian Belief-Propagation and GBP Messages: To
estimate the state variables s, we apply message-passing
GBP algorithm [10]. GBP operates on the factor graph G by
exchanging messages between factor nodes and variable
precisely, the number of state variables that affect certain
measurement is limited by a constant, independently of the size N of the
system.
3 As noted in footnote 1, if the signal x is modulated prior to transmission,
one can easily add an additional “layer” to the factor graph in Fig. 2 containing
a set of M modulated signal variable nodes Xm connected via modulation
factor nodes Fm with the corresponding measurement nodes X .
ŝ = arg
max
s∈CN ,x∈CM
L
Y
P (yi |xN (yi ) )·
i=1
M
Y
j=1
2 More
P (xj |sN (xj ) ) ·
N
Y
k=1
nodes in both directions. As a general rule, at any variable or
factor node, an outgoing message on any edge is obtained as
a function of incoming messages from all other edges, using
the message calculation rules presented below. In general, the
underlying factor graph describing joint state estimation and
signal detection problem (Fig. 2) will contain cycles, thus the
resulting GBP will be iterative, which means that all nodes
will iteratively repeat message updates on all of the outgoing
edges according to a given message-passing schedule. We
provide details on message-passing schedule, correctness and
convergence of GBP on loopy graphs later in this section.
Let us consider a variable node vi ∈ V incident to a factor
node fj ∈ F . Let N (vi ) denote the index set of factor nodes
incident to vi , and N (fj ) denote the index set of variable
nodes incident to fj . We denote messages from vi to fj and
from fj to vi as µvi →fj (vi ) = (mvi →fj , σv2i →fj ) and
µfj →vi (vi ) = (mfj →vi , σf2j →vi ), respectively. Note that, in
the GBP scenario, all messages exchanged across the factor
graph represent Gaussian distributions defined by the
corresponding mean-variance pairs (m, σ 2 ). Thus to describe
processing rules in a variable and a factor node, it is
sufficient to provide equations that map input (m, σ 2 )-pairs
into the output(m, σ 2 )-pair, as detailed below.
Message from a variable node to a factor node: the
equations
below
are
used
to
calculate
µvi →fj (vi ) = (mvi →fj , σv2i →fj ):
!
X mfk →vi
(10a)
σv2i →fj
mvi →fj =
σf2k →vi
k∈N (vi )\j
1
σv2i →fj
=
X
k∈N (vi )\j
1
σf2k →vi
.
(10b)
Message from a factor node to a variable node: In the
setup under consideration, factor nodes represent linear
relations between variable nodes. Thus, e.g., for a factor
node fj , we can write the corresponding linear relationship
as:
X
Ck vk .
fj (vN (fj ) ) = Ci vi +
(11)
k∈N (fj )\i
With this general notation, the equations below provide
µfj →vi (vi ) = (mfj →vi , σf2j →vi ):
!
X
1
mfj →vi =
(12a)
Ck mvk →fj
Ci
k∈N (fj )\i
!
X
1
2 2
2
(12b)
Ck σvk →fj .
σfj →vi = 2
Ci
k∈N (fj )\i
Calculation of marginals: Applying the above rules in
variable and factor nodes of the factor graph results in the
sequence of updates of messages exchanged across the edges
of the graph. To complete description of loopy GBP, we
need to define message initialization at the start, and
message scheduling during the course of each iteration,
which is done next. After sufficient number of GBP
iterations, the final marginal distributions of the random
variables corresponding to variable nodes is obtained as:
!
X mfk →vi
(13a)
σv2i
m̂vi =
σf2k →vi
k∈N (vi )
1
=
σ̂v2i
X
k∈N (vi )
1
.
σf2k →vi
(13b)
GBP Message-Passing Schedule, Correctness and
Convergence: We adopt standard synchronous GBP
schedule in which variable node processing is done in the
first half-iteration, followed by the factor node processing in
the second half-iteration. The iterations are initialized by
input messages from Fy generated from the received signal
y, and initial messages from Fx and Fs that follow certain
prior knowledge (as detailed in the next section).
GBP performance on linear models defined by loopy
factor graphs is fairly well understood. For example, if the
GBP converges, it is known that the GBP solution will
match the solution of the MMSE estimator. The convergence
criteria can also be derived in a straightforward manner, by
deriving recursive fixed point linear transformations that
govern mean value and variance updates through the
iterations and investigating spectral radius of such
transformations. Due to space restrictions, we leave the
details of the convergence analysis in our scenario for the
future work.
IV. N UMERICAL C ASE S TUDY: S MART G RID S TATE
E STIMATION IN 5G C-RAN
In this section, we specialize our state estimation setup for a
case study in which we perform power system state estimation
by collecting measurements via 5G-inspired C-RAN.
Power system state estimation - DC model: For the sake
of simplicity, in the following, we consider the linear DC
model of a power system. The DC model is an approximate
model obtained as a linear approximation of the non-linear
AC model that precisely follows the electrical physical laws
of the power system. In the DC model, the power system
containing N buses is described by N state variables
s = (s1 , s2 , . . . , sN )T , where each state variable si = θi
represents the voltage angle θi (in the DC model, the
magnitudes of all voltage phasors are assumed to have unit
values). In the DC model, the measurements include only
active power flow Prk at the branch (r, k) between the bus r
and the bus k, active power injection Pr into the bus r, and
the voltage angle θr . Collecting M of such arbitrary
measurements across the power system, we obtain the
measurement vector x = (x1 , x2 , . . . , xM )T , where each
measurement xi ∈ {Prk , Pr , θr } is a linear function4 of the
(sub)set of state variables s, additionally corrupted by
additive Gaussian noise of fixed (normalized) noise standard
deviation of σn per unit (p.u.). The noisy measurements x
are then transmitted via C-RAN network as described below.
4 More precisely, we have that P
rk = −brk (θr − θk ) and Pr =
P
− k∈Nr brk (θr − θk ), where Nr is the set of adjacent buses of the bus
r and brk is susceptance of the branch (r, k).
28
23
29
30
26
25
14
15
18
19
27
16
12
13
17
20
24
22
11
10
9
1
3
21
4
6
8
2
5
7
Fig. 3. The IEEE 30 bus test case divided into disjoint sub-rectangles.
C-RAN cellular network model: The set of M
MTC-UEs simultaneously transmit their measurements to the
set of L RRHs during a given allocated time-frequency slot
shared by all MTC-UEs. We assume MTC-UEs and RRHs
are placed uniformly at random following independent
Poisson Point Process (PPP) in a unit-square area, however,
with slight refinement of the PPP placement strategy.
Namely, to account for neighboring relations within logical
topology of IEEE 30 bus test case, we first divide a
unit-square into w × q disjoint sub-rectangles as shown in
Fig. 3, and then we assign M MTC-UEs to one of w · q
sub-rectangles. We also balance the number of RRHs per
sub-rectangle, thus allocating ∼ L/(w · q) RRHs per
sub-rectangle. Finally, all RRHs and MTC-UEs allocated to
a given sub-rectangle are placed using the PPP within a
given sub-rectangle.
After the placement, we assume M MTC-UEs transmit
their signals x, where each measured signal is normalized to
its expected normalization value5 . For the channel
coefficients between the MTC UEs and RRHs, we assume
the following model:
use channel sparsification approach
proposed in [7], with
p
threshold distance set to d0 = w2 + q 2 (i.e., equal to the
diagonal length of each sub-rectangle). The received signal
y = (y1 , y2 , . . . , yL ) collected at L RRHs is additionally
corrupted by additive Gaussian noise, whose standard
deviation is selected so as to provide fixed and pre-defined
signal-to-noise ratio (SNR) value. Finally, noisy received
signal y is forwarded via high-throughput backhaul links to
C-RAN BBUs.
GBP-based State Estimation: Using the approach
presented in Section III, we apply GBP across the factor
graph illustrated in Fig. 2 to recover the state estimate x
from the received signal y. More precisely, for each random
measurement configuration, we generate the part of the
factor graph GA and, similarly, from known MTC-UE and
RRH random positions, we derive6 the part of the factor
graph GH . Upon reception of y, the GBP runs until it
converges. We adopt a synchronous scheduling of GBP
messages where messages are synchronously flooded from
the factor nodes to variable nodes and back within a single
GBP iteration. For a linear model, it is well known that if
the GBP converges, it will converge to the minimum
mean-square error (MMSE) estimate of the state x.
Simulation Results: In the first set of experiments, we fix
the relative RRH density L/M = 1, SNR = 10, and
redundancy M/N = 3. We investigate the accuracy of the
GBP solution of state estimate as a function of different
values of measurements noise σn = {10−1 , 10−2 , 10−3 ,
10−4 }.
8.00
6.00
RMSE
We illustrate the methodology using the IEEE test bus case
with 30 buses shown in Fig. 3 (N = 29, since one of the bus
voltage angles is set to the reference value zero) that we use
in the simulations. The example set of M measurements is
selected in such a way that the system is observable with the
redundancy M/N . For each simulation scenario, we generate
1000 random (observable) measurement configurations.
4.00
2.00
0.00
10−1
10−2
10−3
10−4
Measurement noise σn (p.u.)
Fig. 4. The root mean square error of estimate vector of power system state
variables obtained by C-RAN and without C-RAN model.
where γi,j is the i.i.d. Rayleigh fading coefficient with zero
mean and unit variance, di,j is distance between i-th
MTC-UE and j-th RRH and α is the path loss exponent. We
Fig. 4 shows the root mean square error
RMSE = (1/N )||ŝc − ŝc̄ ||2 , where ŝc and ŝc̄ are estimate
vectors of power system state variables obtained with and
without C-RAN model discussed in this paper, respectively,
for different values of measurement noise σn . For the case
without C-RAN model, we assume measurements x are
available at BBUs as they are, i.e., without additional noise
or errors. In practice, this could be obtained via standard
grant-based uplink procedures where each MTC UE is
5 We assume normalization constants are known in advance at MTC-UEs
and C-RAN nodes, either as a prior knowledge or by long-term averaging.
6 We note that, in case the small-scale fading is included in the model, one
can assume that the channel state information is available at the C-RAN.
−α
hi,j = γi,j di,j
,
(14)
Unobservable Topologies (%)
allocated separate orthogonal resources. However, such a
strategy incurs significant delay as the underlying system
scales, due to message exchange delay, resource allocation
delay, as well as ARQ-based error-correction strategies. Note
that the C-RAN model described in this paper admits very
low latency as all MTC UEs transmit their signals
immediately and concurrently. According to the box plot in
Fig. 4, the C-RAN approach is able to reach nearly identical
solution as the approach without C-RAN (e.g., RMSE → 0),
if the value of measurements noise is sufficiently low. Note
that the typical value (standard deviation) of the
measurement noise for devices located across a power
system are in the range between 10−2 p.u. and 10−3 p.u., for
legacy measurement devices, and between 10−4 p.u. and
10−5 p.u., for phasor measurement units. Consequently, the
presented approach is suitable for the state estimation in
power systems.
In the next simulation experiment, we investigated the
system observability as a function of the number of RRHs L
deployed in the system, for different values of redundancy
M/N . We start with L/M = 0.2 and increase the RRH
density in order to evaluate its effect on the system
observability.
100
M/N
1.5
2.0
2.5
3.0
3.4
50
V. C ONCLUSIONS
Motivated by the development of 5G massive MTC and
large-scale distributed 5G C-RAN architecture, in this paper,
we proposed a scalable and efficient linear state estimation
framework. The proposed framework is based on the GBP
algorithm and jointly combines linear state estimation with
signal detection in 5G C-RANs. The advantage of GBP
solution is accuracy that matches the MMSE estimation, low
complexity due to lack of scheduling MTC-UE
transmissions, low latency due to simultaneous data transfer,
scalability to large-scale systems (due to the fact that the
underlying factor graph is usually sparse), and ease of
parallelization and distributed implementation in future
distributed F-RAN architectures. For the future work, we aim
to provide rigorous convergence analysis of GBP in the
presented framework, motivated by similar analysis in [7]
and [9], and provide extensive numerical simulation study.
R EFERENCES
0
0.2
RRHs L. In contrast, for the scenario where a number of
MTC UEs is small, the number of RRHs must be increased
for successful reconstruction. In addition, simulation results
point to capability of the proposed scheme to provide
successful reconstruction if the underlying system is
observable (i.e., of full rank), while the accuracy of
reconstruction (i.e., the accuracy of the state estimator) will
depend on the parameters such as channel sparsification,
SNR, measurement standard deviations and number of MTC
UEs and RRHs. We leave detailed study of these
inter-dependencies for our future work.
0.4
0.6
0.8
1
1.2
Relative RRH density L/M
Fig. 5. The fraction of unobservable system topologies for different values
of measurement redundancies M/N versus relative RRH density L/M .
Fig. 5 shows the fraction of instances GBP was not able
to converge due to insufficient rank of the underlying system
as a function of the number of base stations L. Note that the
fundamental condition for the system to have full rank is
that L ≥ N . By slightly expanding this condition, we get
(L/M ) · (M/N ) ≥ 1. For all the points in Fig. 5 for which
this condition is not satisfied, the system is unobservable. If
the condition is satisfied, then in each simulation run, a
random measurement configuration is verified to provide an
observable system, thus the rank insufficiency may only
appear as a consequence of the C-RAN topology and the
channel matrix sparsification. According to Fig. 5, for the
parameters used in our simulations, we can see that GBP
generally performs well, however, in the region where
(L/M ) · (M/N ) is slightly above 1, rank insufficiency may
deteriorate the performance.
Overall, for systems with large number of MTC UEs M ,
the state can be estimated with relatively small number of
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J. Bergman, T. Tirronen, and E. Yavuz, “An overview of 3GPP
enhancements on machine to machine communications,” IEEE
Communications Magazine, vol. 54, no. 6, pp. 14-21, June 2016.
[2] A. Checko, H. L. Christiansen, Y. Yan, L. Scolari, G. Kardaras, M. S.
Berger, and L. Dittmann, “Cloud RAN for mobile networks a technology
overview,” IEEE Communications Surveys Tutorials, vol. 17, no. 1, pp.
405-426, 2015.
[3] F. Bonomi, R. Milito, J. Zhu, and S. Addepalli, “Fog computing and its
role in the internet of things,” in Proc. ACM Workshop on Mobile cloud
computing (MCC), pp. 13-16, 2012.
[4] H. Shariatmadari, R. Ratasuk, S. Iraji, A. Laya, T. Taleb, R. Juntti,
and A. Ghosh, “Machine-type communications: current status and future
perspectives toward 5G systems,” IEEE Communications Magazine, vol.
53, no. 9, pp. 10-17, September 2015.
[5] M. Cosovic, A. Tsitsimelis, D. Vukobratovic, J. Matamoros and C.
Anton-Haro, ”5G Mobile Cellular Networks: Enabling Distributed State
Estimation for Smart Grids,” in IEEE Communications Magazine, vol.
55, no. 10, pp. 62-69, October 2017.
[6] C. Fan, Y. Zhang, X. Yuan,“Advances and challenges towards scalable
cloud radio access network,” IEEE Communications Magazine, vol. 54,
no. 6, pp. 29–35, June 2016.
[7] C. Fan, Y. Zhang, X. Yuan,“Scalable uplink signal detection in CRANs via Randomized Gaussian Message Passing,” IEEE Trans. Wireless
Communications, vol. 16, no. 8, pp. 5187–5200, August 2017.
[8] A. Monticelli, Electric power system state estimation, Proceedings of the
IEEE, vol. 88, no. 2, pp. 262-282, 2000.
[9] M. Cosovic, D. Vukobratovic, “Distributed Gauss-Newton Method for
AC State Estimation Using Belief Propagation,” submitted, arXiv:
https://arxiv.org/abs/1702.05781v2.
[10] H. A. Loeliger, J. Dauwels, J. Hu, S. Korl, L. Ping, and F.
R. Kschischang, “The factor graph approach to model-based signal
processing,” Proc. of the IEEE, vol. 95, no. 6, pp. 12951322, 2007.
[11] Lien S.-Y., Hung S.-C., Chen K.-C., Liang Y.-C., “Ultra-low latency
ubiquitous connections in heterogeneous cloud radio networks,” IEEE
Wireless Communications Magazine, vol. 22, no. 3, pp. 2231, Jun. 2015.
| 7 |
L2 –INVISIBILITY OF SYMMETRIC OPERAD GROUPS
arXiv:1407.7450v3 [math.AT] 2 Apr 2016
WERNER THUMANN
Abstract. We show a homological result for the class of planar or symmetric
operad groups: We show that under certain conditions, group (co)homology
of such groups with certain coefficients vanishes in all dimensions, provided it
vanishes in dimension 0. This can be applied for example to l2 –homology or
cohomology with coefficients in the group ring. As a corollary, we obtain explicit vanishing results for Thompson-like groups such as the Brin–Thompson
groups nV .
1. Introduction
In [9] it is shown that a certain class of groups acting on compact ultrametric
spaces, the so-called dually contracting local similarity groups, are l2 –invisible. The
latter means that group homology with group von Neumann algebra coefficients
vanishes in every dimension, i.e.
Hk G, N (G) = 0
for all k ≥ 0 where N (G) denotes the group von Neumann algebra of G. If G is
of type F∞ , i.e. there is a classifying space for G with finitely many cells in each
dimension, then this is equivalent to
Hk G, l2 (G) = 0
for all k ≥ 0 by [8, Lemmas 6.98 on p. 286 and 12.3 on p. 438].
In [10] the author proposed to study fundamental groups of categories naturally
associated to operads. This class of groups, called operad groups, contains a lot
of Thompson-like groups already existent in the literature. Among these are the
above mentioned local similarity groups (see [10, Subsection 3.5]).
This article is mainly concerned with generalizing the results of [9] to the setting
of symmetric operad groups which form a much larger class of groups. The proof
in [9] consisted of constructing a suitable simplicial complex on which the group in
question acts and then applying a spectral sequence associated to this action which
computes the homology of the group in terms of the homology of the stabilizer
subgroups. The proof in the case of operad groups goes exactly the same way.
However, it is a priori unclear how to construct the simplicial complex. The reason
is the following: A local similarity group is defined as a representation, i.e. as a
group of homeomorphisms of a compact ultrametric space. This space is used to
construct the simplicial complex as a poset of partitions of this space. The case of
operad groups is more abstract. A priori, there is no canonical space comparable
to these ultrametric spaces on which an operad group acts. However, these spaces,
called limit spaces, are conjectured to exist if the operad satisfies the calculus of
fractions (see [10, Subsection 3.3] for the latter notion). We don’t use these limit
spaces here. Instead, we will take the conjectured correspondence between calculus
of fractions operads and their limit spaces as a motivation to mimic the necessary
2010 Mathematics Subject Classification. Primary 20J05; Secondary 22D10, 18D50.
Key words and phrases. Operad groups, Thompson groups, group homology, l2 –homology.
1
2
WERNER THUMANN
notions for the construction of the desired simplicial complex in terms of the operad
itself.
As in [9], we briefly want to discuss the relationship between these results and
Gromov’s Zero-in-the-spectrum conjecture (see [7]). The algebraic version of this
conjecture states that if Γ = π1 (M ) is the fundamental group of a closed aspherical Riemannian manifold, then there always exists a dimension p ≥ 0 such that
Hp (Γ, N Γ) 6= 0 or equivalently Hp (Γ, l2 Γ) 6= 0. Conjecturally, the fundamental
groups of closed aspherical manifolds are precisely the Poincaré duality groups G of
type F , i.e. there is a compact classifying space for G and a natural number n ≥ 0
such that
(
0 if i 6= n
i
H (G, ZG) =
Z if i = n
(see [5]). Dropping Poincaré duality and relaxing type F to type F∞ , we arrive at a
more general question which has been posed by Lück in [8, Remark 12.4 on p. 440]:
If G is a group of type F∞ , does there always exist a p with Hp (G, N G) 6= 0? In [10]
we discuss conditions for operads which imply that the associated operad groups
are of type F∞ . Combining this with the results in the present article, we obtain a
large class of groups of type F∞ which are also l2 –invisible. This class contains the
well-known symmetric Thompson group V and consequently, Lück’s question has
to be answered in the negative. Unfortunately, all these groups G are neither of
type F nor satisfy Poincaré duality since, as another corollary of our main theorem
(Theorem 2.5), we can show H k (G, ZG) = 0 for all k ≥ 0.
1.1. Prerequisites. The present article is based on Sections 2 and 3 of [10].
1.2. Notation and Conventions. When f : A → B and g : B → C are two
composable arrows, we write f ∗ g for the composition A → C instead of the usual
notation g ◦ f . Consequently, it is often better to plug in arguments from the left.
When we do this, we use the notation x⊲f for the evaluation of f at x. However,
we won’t entirely drop the usual notation f (x) and use both notations side by side.
Objects of type Aut(X) will be made into a group by the definition f g = f ·g := f ∗g.
Conversely, a group G is considered as a groupoid with one object and arrows the
elements in G together with the composition f ∗ g := f · g.
1.3. Acknowledgments. I want to thank my PhD adviser Roman Sauer for the
opportunity to pursue mathematics, for his guidance, encouragement and support
over the last few years. I also gratefully acknowledge financial support by the DFG
grants 1661/3-1 and 1661/3-2.
2. Statement of the main theorem
Definition 2.1. Let MG be a ZG–module for every group G. We call M
• Künneth if for every two groups G1 , G2 and n1 , n2 ∈ Z with ni ≥ −1 the
following is satisfied:
∀k≤n1 Hk (G1 , MG1 ) = 0
=⇒ ∀k≤n Hk (G, MG) = 0
∀k≤n2 Hk (G2 , MG2 ) = 0
where G := G1 × G2 and n := n1 + n2 + 1.
• inductive if whenever H and G are groups with H a subgroup of G and
k ≥ 0, then we have that
Hk (H, MH) = 0 implies Hk (H, MG) = 0
L2 –INVISIBILITY OF SYMMETRIC OPERAD GROUPS
3
Let P be a property of groups. Then we say that M is P–Künneth if the property
Künneth has to be satisfied only for P–groups G1 , G2 . We say that M is P–
inductive if the property inductive has to be satisfied only for P–subgroups H of
the arbitrary group G. Furthermore, one can formulate these two properties also
in the cohomological case.
Definition 2.2. Let O be a planar or symmetric or braided operad and X an
object in S(O). We say that X is
• split if there are objects A1 , A2 , A3 and an arrow A1 ⊗X ⊗A2 ⊗X ⊗A3 → X
in S(O).
• progressive if for every arrow Y → X there are objects A1 , A2 and an arrow
A1 ⊗ X ⊗ A2 → Y such that the coordinates of X are connected to only
one operation in this arrow (see Figure 1).
A1
Y
X
A2
Figure 1. An arrow A1 ⊗ X ⊗ A2 → Y such that X is only
connected to one operation.
Remark 2.3. If X is just a single color, then X is split if and only if there is an
operation with output color X and and at least two inputs of color X. If O is
monochromatic and X 6= I is an object of S(O), then X is split if and only if there
is at least one operation in O with at least two inputs. So in the monochromatic
case, the condition split is in fact a property of O.
Remark 2.4. If X is just a single color, then X is progressive if and only if for
every operation θ with output color X there is another operation φ with at least
one input of color X and at least one input of θ has the same color as the output of
φ. Now assume that O is monochromatic. Then an object X 6= I in S(O) (which
is just a natural number X > 0, e.g. X = 3) is progressive if and only if there is an
operation in O with at least X inputs (e.g. 3 inputs). Note that X = 1 is always
progressive in the monochromatic case.
Theorem 2.5. Let O be a planar or symmetric operad which satisfies the calculus
of fractions. Let M be a coefficient system which is Künneth and inductive. Let X
be a split progressive object of S(O). Set Γ := π1 (O, X). Then
H0 (Γ, MΓ) = 0
=⇒
∀k≥0 Hk (Γ, MΓ) = 0
The same is true for cohomology.
More generally, let P be a property of groups which is closed under taking products. Then the statement is true also for coefficient systems M which are only
P–Künneth and P–inductive, provided that Γ satisfies P.
Remark 2.6. Let X, Y be objects in S(O). Generalizing the notion of progressiveness, we say that X is Y –progressive if for every arrow Z → X there is an arrow
4
WERNER THUMANN
A1 ⊗ Y ⊗ A2 → Z and the coordinates of Y are connected to only one operation in this arrow (call this the link condition). In particular, there is an arrow
A1 ⊗ Y ⊗ A2 → X.
With this notion, we can formulate a slightly more general version of Theorem
2.5: Let O, P, M be as in the theorem. Let X be an object of S(O) and set
Γ = π1 (O, X). Assume there is a split object Y such that X is Y –progressive,
Υ := π1 (O, Y ) satisfies P and H0 (Υ, MΥ) = 0. Then Hk (Γ, MΓ) = 0 for each
k ≥ 0. The same is true for cohomology.
3. Proof of the main theorem
We start with two general lemmas concerning the calculus of fractions.
Lemma 3.1. Let C be a category satisfying the calculus of fractions. Then two
square fillings of a given span can be combined to a common square filling. This
means: Let x, y be two arrows with the same codomain and assume having two
square fillings as in the diagram
x
• o_❅
❅❅ j
❅❅
❅
•o
i
•
h
•❅
❅❅ g
❅❅
❅
•
y
then we can complete this diagram to the commutative diagram
α ❢ ❜ ❴ •
♣
♠ ✐
t⑦ ⑦ ✣
q
②
ǫ
|✈
⑦ ⑦ δ ✛
•
• o_❅
☎
❅❅
✘
✠
❅❅
✔β
❅ ✌
✏
•❅
❅❅
☞
❅❅
❅ ✞
•o
•
Proof. Let c, d be a square filling of a := ix = hy, b := jx = gy, i.e. ca = db. Then
ch and dg are two parallel arrows which are coequalized by y, i.e. (ch)y = (dg)y.
By the equalization property we find an equalizing arrow k with k(ch) = k(dg).
By the same reasoning we find an arrow l with l(ci) = l(dj). Let m, n be a square
filling of l, k, i.e. ml = nk =: p. Then one can easily calculate that the arrows
δ = pc
α = pci
β = pch
ǫ = pd
fill the diagram as required.
Lemma 3.2. Let C be a category satisfying the calculus of fractions. Let x̄ and ȳ
be two arrows A → C. Assume that there are arrows x, y : A → B and a : B → C
such that xa = x̄ and ya = ȳ.
C
x̄
ȳ
vo
a
x̄
ȳ
Bo
/B
x
A
y
a
/( C
x
y
Then the span C ←
−A−
→ C is null-homotopic if and only if the span B ←
−A−
→B
is null-homotopic.
L2 –INVISIBILITY OF SYMMETRIC OPERAD GROUPS
x
5
y
Proof. First note that a span like B ←
−A−
→ B is null-homotopic if and only if the
parallel arrows x and y are homotopic. Since C satisfies the calculus of fractions, this
is the case if and only if there is an equalizing arrow, i.e. an arrow d : D → A with
dx = dy. Now if x and y are homotopic then clearly also x̄ and ȳ are homotopic.
On the other hand, assume that x̄ and ȳ are homotopic and d : D → A equalizes x̄
and ȳ. Then we have
(dx)a = d(xa) = dx̄ = dȳ = d(ya) = (dy)a
Then by the equalization property we find an arrow e : E → D with e(dx) = e(dy).
Consequently, the arrow ed equalizes x and y and thus, x and y are homotopic.
We now turn to the proof of Theorem 2.5. In the following, let O be a planar or
symmetric operad satisfying the calculus of fractions with set of colors C and let
S := S(O).
3.1. Marked objects. Let c = (c1 , ..., cn ) be an object of S, i.e. c1 , ..., cn are
colors in C. First we define a marking on c in the symmetric case. It assigns to
each coordinate of c a symbol. A symbol can be assigned several times and not
every coordinate has to be marked by a symbol. More precisely, a marking of c is
a set S of symbols together with a subset I ⊂ {1, ..., n} and a surjective function
f : I → S. In the planar case, we additionally require the marking to be ordered.
This means that whenever i⊲f = j⊲f for i < j then also i⊲f = k⊲f = j⊲f for
i < k < j.
Let m1 , m2 be two markings of c with symbol sets S1 , S2 . We say m1 ⊂ m2 if
there is a function i : S1 → S2 and every coordinate which is marked with s1 ∈ S1
is also marked with s1 ⊲i ∈ S2 . We say m1 and m2 are equivalent if m1 ⊂ m2 and
m2 ⊂ m1 . This means that there is a bijection i : S1 → S2 and a coordinate is
marked with s1 ∈ S1 if and only if it is marked with s1 ⊲i ∈ S2 . By slight abuse of
notation, we identify equivalent markings and write m1 = m2 if they are equivalent.
Then ⊂ becomes a partial order on the set of markings on c (see the first paragraph
of Subsection 3.3).
3.2. Marked arrows. Let α : c → d be an arrow in S with objects c = (c1 , ..., cn )
and d = (d1 , ..., dm ). A marking on α is a marking on the domain c. A comarking
on α is a marking on the codomain d. A comarking on α induces a marking on α:
Let (σ, X) be a representative of α where σ is either an identity or a colored permutation, depending on whether O is planar or symmetric. Write X = (X1 , ..., Xm ).
The comarking yields a marking on the operations Xi : Mark each input of Xi with
the same symbol. Now push the markings through σ to obtain a marking on the
domain c. Figure 2 illustrates this procedure. If m is the comarking, then we denote this pull-backed marking by α∗ (m). Observe that this pull-back is functorial,
i.e. we have
(αβ)∗ (m) = α∗ (β ∗ (m))
Furthermore, we have
m1 ⊂ m2 ⇐⇒ α∗ (m1 ) ⊂ α∗ (m2 )
Now fix an object x in S.
Let (α1 , m1 ) and (α2 , m2 ) be two marked arrows with codomain x, i.e. αi : ci → x
is an arrow and mi is a marking on ci . We write
(α1 , m1 ) ⊂ (α2 , m2 )
6
WERNER THUMANN
⋆
⋆
♦
⋆
♦
♦
♦
Figure 2. A comarking (left) and the pull-backed marking (right).
if there is a square filling
d
β2
/ c2
α2
β1
c1
β1∗ (m1 )
α1
/ x
β2∗ (m2 ).
with
⊂
Observe that then every square filling satisfies this: Let
(γ1 , γ2 ) be another square filling of (α1 , α2 ). Then choose a common square filling
(δ1 , δ2 ) as in Lemma 3.1. It is not hard to see that the property δ1∗ (m1 ) ⊂ δ2∗ (m2 )
is inherited from the square filling (β1 , β2 ). On the other hand, this forces the
property onto the square filling (γ1 , γ2 ), i.e. we have γ1∗ (m1 ) ⊂ γ2∗ (m2 ).
Remark 3.3. This observation implies also the following: Let (α1 , m1 ) ⊂ (α2 , m2 )
and assume that α1 = α2 . Then we necessarily have m1 ⊂ m2 . Indeed, we can
choose β1 = id = β2 in the above square filling.
Proposition 3.4. The relation ⊂ on the set of marked arrows is reflexive and
transitive.
Proof. Reflexivity is clear. For transitivity assume (α1 , m1 ) ⊂ (δ, m) and (δ, m) ⊂
(α2 , m2 ). Choose two square fillings
a1
δ1
β1
/do
δ2
a2
β2
δ
c1
α1
/ x o
α2
c2
with β1∗ (m1 ) ⊂ δ1∗ (m) and δ2∗ (m) ⊂ β2∗ (m2 ). Choose a square filling of (δ1 , δ2 )
e❅
⑦⑦ ❅❅❅ γ2
⑦
❅❅
⑦
η
❅❅
⑦⑦
~⑦⑦
/do
a1
a2
γ1
δ1
β1
δ2
β2
δ
c1
α1
/ x o
α2
c2
Now we have
(γ1 β1 )∗ (m1 ) =
⊂
γ1∗ (β1∗ (m1 ))
γ1∗ (δ1∗ (m))
=
(γ1 δ1 )∗ (m)
L2 –INVISIBILITY OF SYMMETRIC OPERAD GROUPS
=
=
η ∗ (m)
(γ2 δ2 )∗ (m)
=
⊂
γ2∗ (δ2∗ (m))
γ2∗ (β2∗ (m2 ))
=
(γ2 β2 )∗ (m2 )
This proves (α1 , m1 ) ⊂ (α2 , m2 ).
7
3.3. Balls and partitions. A transitive and reflexive relation 4 on a set Z is not
a poset in general since a 4 b together with b 4 a does not imply a = b in general.
We can repair this in the following way: Define a, b ∈ Z to be equivalent if a 4 b
and b 4 a. This is indeed an equivalence relation because 4 is assumed to be
reflexive and transitive. Now if a and b are two equivalence classes, we write a ≤ b
if there are representatives a and b respectively with a 4 b. One can easily show
that then any two representatives satisfy this. Using this, it is not hard to see that
≤ is indeed a partial order on the set of equivalence classes. In particular, we have
a = b if and only if a ≤ b and b ≤ a.
We want to apply this observation to the reflexive and transitive relation ⊂ on
the set of marked arrows. We say that two marked arrows (α1 , m1 ) and (α2 , m2 )
with common codomain x are equivalent if both (α1 , m1 ) ⊂ (α2 , m2 ) and (α2 , m2 ) ⊂
(α1 , m1 ) hold. We remark that this is equivalent to the existence of a square filling
d
β2
/ c2
α2
β1
c1
with
β1∗ (m1 )
α1
/ x
β2∗ (m2 )
=
and moreover that every square filling satisfies this.
• A semi-partition is an equivalence class of marked arrows.
• A partition is a semi-partition with fully marked domain for some (and
therefore for every) representative of the semi-partition. Here, an object in
S is fully marked if every coordinate is marked.
• A multiball is a semi-partition with an uni-marked domain for some (and
therefore for every) representative of the semi-partition. Here, an object in
S is uni-marked if there is only one symbol in the marking.
• A ball is a semi-partition such that there is a single-marked representative.
Here, an object in S is single-marked if only one coordinate is marked.
Note that these definitions depend on the base point x. Following the remarks in
the first paragraph, we write P ⊂ Q for two semi-partitions P and Q if there are
representatives p of P and q of Q satisfying p ⊂ q. Then for all such representatives
p, q we have p ⊂ q. It follows that ⊂ is a partial order on the set of semi-partitions.
In particular, we have P = Q if and only if P ⊂ Q and Q ⊂ P.
We now investigate the relationship between semi-partitions and multiballs. Let
P be a semi-partition with representative (α, m). Picking out a symbol s of m
and removing all markings except those with the chosen symbol s gives a unimarked arrow (α, ms ). The corresponding equivalence class is a multiball and is
independent of the chosen representative (α, m) in the following sense: If we choose
another representative (β, n), then (α, m) ∼ (β, n) and to the chosen symbol s of
m corresponds a unique symbol r of n. Deleting all markings in n except those
with the symbol r gives a uni-marked arrow (β, nr ) which is equivalent to (α, ms ).
Multiballs arising in this way are called submultiballs of P and we write P ∈ P for
submultiballs. Note that Remark 3.3 implies that two submultiballs P1 , P2 coming
8
WERNER THUMANN
from a representative of P by choosing two different symbols satisfy P1 6⊂ P2 and
P2 6⊂ P1 , in particular P1 6= P2 . It follows that there is a canonical bijection
between the set {P ∈ P} of submultiballs of P and the set of symbols of P (which
is, by definition, the set of symbols of the marking of any representative for P).
Moreover, any two submultiballs P1 , P2 ∈ P with P1 6= P2 satisfy the stronger
property (P1 6⊂ P2 ) ∧ (P2 6⊂ P1 ). Equivalently, whenever P1 ⊂ P2 or P2 ⊂ P1 , then
already P1 = P2 .
Proposition 3.5. Let P, Q be semi-partitions, then
Q⊂P
⇐⇒ ∀Q∈Q ∃P ∈P Q ⊂ P
In particular, P = Q if and only if {Q ∈ Q} = {P ∈ P}.
Proof. We first prove the last statement since it is a formal consequence of the
previous statement and the remarks preceding the proposition. First recall that
P = Q is equivalent to P ⊂ Q and Q ⊂ P. The first statement of the proposition
says that there is a function i : {Q ∈ Q} → {P ∈ P} with the property that
Q ⊂ Q⊲i for each Q ∈ Q. Since we also have P ⊂ Q, there is another function
j : {P ∈ P} → {Q ∈ Q} with the property that P ⊂ P ⊲j for each P ∈ P. We have
Q ⊂ Q⊲i ⊂ (Q⊲i)⊲j = Q⊲(ij)
for all Q ∈ Q. Since both the left and right side are submultiballs of Q, the remarks
preceding the proposition imply Q = Q⊲(ij) for all Q ∈ Q. We then have
Q ⊂ Q⊲i ⊂ Q
and therefore also Q = Q⊲i for all Q ∈ Q. This shows {Q ∈ Q} ⊂ {P ∈ P}. With
a similar argument applied to ji, we also obtain {Q ∈ Q} ⊃ {P ∈ P}. So we have
indeed {Q ∈ Q} = {P ∈ P}. The converse implication also follows easily from the
first statement of this proposition.
Now let’s turn to the first statement: Assume Q ⊂ P. By the square filling
technique, we know that we can choose a common arrow α : c → x with markings
mQ ⊂ mP such that [α, mQ ] = Q and [α, mP ] = P. If Q ∈ Q, then we find a
symbol sQ of the marking mQ which corresponds to Q. But since mQ ⊂ mP , there
is a unique symbol sP of the marking mP such that the coordinates of c marked by
sQ are also marked by sP . The submultiball obtained from (α, mP ) corresponding
to the symbol sP is the one we are looking for.
Conversely, assume that there is a function i : {Q ∈ Q} → {P ∈ P} such that
Q ⊂ Q⊲i for every Q ∈ Q. Using the square filling technique, we find a common
arrow α : c → x with markings mQ , mP such that [α, mQ ] = Q and [α, mP ] =
P. We want to show mQ ⊂ mP . Let s be any symbol of mQ . To this symbol
corresponds exactly one submultiball Q ∈ Q such that Q = [α, msQ ] where msQ is
the submarking of mQ with all markings removed except those with the symbol s.
To the submultiball Q⊲i ∈ P corresponds exactly one symbol r of mP such that
Q⊲i = [α, mrP ]. Since Q ⊂ Q⊲i we have (α, msQ ) ⊂ (α, mrP ) and therefore msQ ⊂ mrP
by Remark 3.3. It follows mQ ⊂ mP and thus Q ⊂ P.
3.4. The action on the set of semi-partitions. Here we will define an action
of Γ = π1 (S, x) on the set of semi-partitions over x. So let γ ∈ Γ and P be a
semi-partition over x. We will define another semi-partition γ · P over x. Recall
γd
γn
that γ is represented by a span x ←− a −→ x (the d refers to denominator and the
n refers to nominator ) and that P is represented by a marked arrow (α : c → x, m).
L2 –INVISIBILITY OF SYMMETRIC OPERAD GROUPS
9
First choose a square filling of (γn , α)
γd
γn
o
/xo
x g❖
a_
❖❖
❖❖
❖❖
β1
❖❖
δ
❖❖
❖❖
❖
b
α
?c
β2
and then define δ := β1 γd : b → x. Endow this arrow with the marking µ := β2∗ (m).
Finally, define γ · P := [δ, µ]. We have to show that this is well-defined, i.e. we have
to show that the resulting class is independent of
1. the square filling (β1 , β2 )
2. the marked arrow (α, m) as a representative of P
3. the span (γd , γn ) as a representative of γ
We now prove these points one by one.
1. Assume we have two square fillings of (γn , α) as in the following diagram:
b′
β2′
β1′
xo
γd
af
x
γn
/xo
β1
α
&
8c
β2
b
Choose a common square filling as in Lemma 3.1:
xo
γd
e ❩
⑥ ✯ ❄ ❄ η′ ❯ ❖
✯ ❄
☎
❍δ2
✡
✯
❄
❅
δ1 ✎
′
✯
b
✼
✔
✯
β1′
β2′
✘
✶
✯η
✯
✜ w
γn
α
&
ag
8c
✯/xo
✯
✯
β1
β2
b
Now the marked arrow (β1 γd , β2∗ (m)) is equivalent to the marked arrow (δ1 γd , δ2∗ (m))
∗
via η. Analogously, the marked arrow (β1′ γd , β2′ (m)) is equivalent to (δ1 γd , δ2∗ (m))
′
∗
via η and therefore equivalent to (β1 γd , β2 (m)), q.e.d.
2. Let (α′ , m′ ) be another marked arrow equivalent to (α, m) and choose a
∗
square filling (β, β ′ ) such that β ∗ (m) = β ′ (m′ ) =: µ as in the following diagram:
♣cg
♣♣♣
♣
♣
♣♣♣
♣♣♣
w
δ
♣
/ x f◆
o
◆◆◆
◆◆◆
◆◆
α′ ◆◆◆◆
x
c′
α
xo
γd
a
γn
β
e
β′
First choose a square filling (η1 , η2 ) of (γn , α) and then a square filling (ν1 , ν2 ) of
(η2 , β). Analogously, choose a square filling (η1′ , η2′ ) of (γn , α′ ) and then a square
10
WERNER THUMANN
filling (ν1′ , ν2′ ) of (η2′ , β ′ )
z ▼
♣♣
■
♣
❈
♣
♣
❂ν2
♣
η
2
w♣
❴ ❴ ❴ ❴ ❴ ❴/ c f
✼
y
♣
♣♣
♣
♣
✸
♣
η1 ♣
β
α ♣♣
♣
♣
♣
✴
♣
♣
♣
w♣♣♣
w♣ ♣ γn
δ
/
o
a f▼
x f◆◆
eI
▼▼
◆◆◆
✏
◆◆◆
▼▼
◆◆◆
✌
▼▼
β′
η1′
α′
◆◆ x
✟
❴ ❴ ❴ ❴ ❴ ❴ / c′
y ′ f▼
▼ ▼ η2′
✄ ′
⑥ ν2
▼▼
✇
▼▼
ν1′
s
z′
ν1
xo
γd
The marked arrow (η1 γd , η2∗ (m)) is equivalent to Λ := (ν1 η1 γd , ν2∗ (µ)) via ν1 . On the
∗
∗
other side, the marked arrow (η1′ γd , η2′ (m′ )) is equivalent to Λ′ := (ν1′ η1′ γd , ν2′ (µ))
′
′
via ν1 . The marked arrows Λ and Λ are both constructed from the same marked ar∗
row (δ, µ) and so are equivalent by 1. Consequently, (η1 γd , η2∗ (m)) and (η1′ γd , η2′ (m′ ))
are equivalent, q.e.d.
3. Let (γd′ , γn′ ) be another representing span of γ homotopic to the span (γd , γn ).
Then recall that the two spans can be filled by a diagram as follows:
♦♦ aO ❖❖❖❖❖
❖❖γ❖n
η
❖❖❖
♦♦♦
♦
❖❖'
♦
δn
wo♦♦ δd
/xo
x f◆◆
e
◆◆◆
♣♣8
♣
♣
◆◆◆
♣
η′
◆◆◆
♣♣♣
γd′
◆◆ ♣♣♣♣♣ γn′
a′
γd ♦♦♦♦
α
c
Now choose a square filling (ν1 , ν2 ) of (δn , α) and note that (ǫ, ν2 ), where ǫ := ν1 η,
gives a square filling of (γn , α).
♦z❂
♦♦♦
❂
♦
♦
❂
♦♦♦
♦
♦
❂
w♦♦
ν
ν
2
1
❂
♦ aO ❖❖❖❖
♦
♦
❂
❖❖❖γn
γd ♦♦♦
η
❖❖❖
❂
♦♦
♦
❖
♦
❂
❖❖'
δn
wo♦♦♦ δd
α
/xo
x f◆◆
e
c
8
◆◆◆
♣♣♣
♣
◆◆◆
♣
η′
◆◆◆
♣♣♣
γd′
◆◆ ♣♣♣♣♣ γn′
a′
ǫ
The marked arrow (ǫγd , ν2∗ (m)) is equivalent to (ν1 δd , ν2∗ (m)). Similarly, define
ǫ′ = ν1 η ′ and note that (ǫ′ , ν2 ) gives a square filling of (γn′ , α). Again, the marked
arrow (ǫ′ γd′ , ν2∗ (m)) is equivalent to (ν1 δd , ν2∗ (m)). Therefore, (ǫγd , ν2∗ (m)) and
(ǫ′ γd′ , ν2∗ (m)) are equivalent, q.e.d.
Now we want to show that this is indeed an action, i.e. 1·P = P and γ 1 ·(γ 2 ·P) =
(γ γ ) · P. The first property is easy to see. The second property is not entirely
trivial but straightforward. We will be explicit for completeness. Choose two
representing spans (γd1 , γn1 ) and (γd2 , γn2 ) for γ 1 and γ 2 respectively. Let (α, m)
represent P. To get a representing span for the composition γ1 γ2 , choose a square
1 2
L2 –INVISIBILITY OF SYMMETRIC OPERAD GROUPS
11
filling (β1 , β2 ) of (γn1 , γd2 ) and take the span (β1 γd1 , β2 γn2 ). This span acts on (α, m)
as before and is sketched diagrammatically as follows:
❣❣ z ❁
❁❁
❣❣❣❣❣
❣
❣
❣
❣
❁❁
❣❣
❣
❣
❣
❣
❁❁
❣❣
❣
❣
❣
❣
❁❁ν
❣❣
s❣
❱
y
❱
❤
❁❁
▼
❱
❤
▼
❱
❤
q
❱
❤
▼
q
▼▼▼❱❱❱❱❱❱ δ
❁❁
❤❤❤ q
δ1 ❤❤❤❤❤ qqq
2
❱
▼▼▼
❱❱❱❱
❁❁
q
❤❤
❤
q
❱
❤
▼
❱
❤
q
❁❁
❱❱❱❱
▼▼▼
q β1
β2
❤❤❤
q
❤
❱
❤
q
❱
❤
& 2
xq
❱❱/+ o
o ❤❤
/xo
x s❤
a1
a
x
c
η
γd1
γd2
1
γn
α
2
γn
So a representative of (γ 1 γ 2 ) · P is given by (ηδ1 , ν ∗ (m)). Now a representative for
γ 2 ·P is given by (ηβ2 γd2 , ν ∗ (m)) because (ηβ2 , ν) is a square filling for (γn2 , α). Since
(ηβ1 , idz ) is a square filling for (γn1 , ηβ2 γd2 ), we obtain that (ηβ1 γd1 , id∗z (ν ∗ (m))) is a
representative of γ 1 · (γ 2 · P). But this last marked arrow is equal to (ηδ1 , ν ∗ (m)),
q.e.d.
Remark 3.6. It is not hard to see that P ⊂ Q implies γ · P ⊂ γ · Q.
Remark 3.7. The submultiballs of γ · P are the multiballs γ · P with P ∈ P.
3.5. Pointwise stabilizers of partitions. Let P be a partition over x. By the
pointwise stabilizer of P we mean the subgroup
Λ := {γ ∈ π1 (S, x) | γ · P = P for all P ∈ P}
Fix some representative (α, m) of P. We can assume without loss of generality that
the marking m on the domain c of α is ordered. That means that if f : I → S is the
marking function of m and whenever i⊲f = j⊲f for i < j, then also i⊲f = k⊲f = j⊲f
for every k with i < k < j. This is true in the planar case by definition. In the
symmteric case, we can choose a colored permutation σ ∈ Sym(C) with σ ∗ (m)
ordered and replace (α, m) by the equivalent marked arrow (σα, σ ∗ (m)).
Proposition 3.8. Each symbol of the marking m determines a subword of the
word c = dom(α). Denote these subwords by c1 , ..., ck and order them such that
c = c1 ⊗ ... ⊗ ck . Then we have a well-defined isomorphism of groups
Ξ : π1 (S, c1 ) × ... × π1 (S, ck ) → Λ
which is given by applying the tensor product of paths and then conjugating with
the arrow α. More explicitly, it is given by sending representing spans p1 , ..., pk to
the homotopy class represented by the path
α
x ←−−−−−−
p≺
p≻
c1
⊗
..
.
1
←−
−
⊗
..
.
a1
⊗
..
.
1
−−→
⊗
..
.
c1
⊗
..
.
⊗
ck
⊗
⊗
←−
−
a
k
≺
⊗
−−→
≻
⊗
ck
pk
pk
α
−−−−−−→ x
is the arrow pointing to the left and p≻
where
i the arrow pointing to the right in
the span pi .
p≺
i
Proof. It is not hard to see that the map is independent of the chosen representing
spans pi and that it is a group homomorphism. Injectivity follows from Lemma
3.2 and Lemma 3.9 below. Before we prove surjectivity, we want to see that the
image really lies in the subgroup Λ. We can use the representative (α, m) to extract representatives of submultiballs P ∈ P. The subwords ci are in one to one
correspondence with the submultiballs P ∈ P. A representative (α, mi ) of P ∈ P
12
WERNER THUMANN
corresponding to ci is obtained from (α, m) by removing all markings except the
markings on the subword ci . The representing span of Ξ(p1 , ..., pk ) pictured above
≺
≻
≻
can be written as (p≺ α, p≻ α) where p≺ = p≺
= p≻
1 ⊗ ... ⊗ pk and p
1 ⊗ ... ⊗ pk .
≻
Letting this span act on (α, mi ), we can choose (id, p ) as a square filling and the
∗
resulting representative is (p≺ α, p≻ (mi )). But this is equivalent to (α, mi ) because
≺∗
≻∗
p (mi ) = p (mi ).
Now we prove surjectivity. Let γ ∈ Λ which can be represented by a path of the
form
z≺
α
z≻
α
x ←−−−−−− c ←−−−−−−− a −−−−−−−→ c −−−−−−→ x
Observe the representatives (α, mi ) of the submultiballs P ∈ P from above. A
∗
representative of γ · [α, mi ] is given by (z ≺ α, z ≻ (mi )). So we have (α, mi ) ∼
≺
≻∗
≺
(z α, z (mi )). Of course, (z , id) is a square filling of (α, z ≺ α) and thus
∗
∗
z ≺ (mi ) = z ≻ (mi )
Now assume for the moment that the operad O is planar. Then it follows easily
from these equalities that the span (z ≺ , z ≻ ) splits as a product according to the
decomposition c = c1 ⊗ ... ⊗ ck , i.e. there are zi≺ : ai → ci and zi≻ : ai → ci with
z ≺ = z1≺ ⊗...⊗zk≺ and z ≻ = z1≻ ⊗...⊗zk≻. By construction, the spans (zi≺ , zi≻ ) give a
preimage of γ under Ξ. If, on the other hand, O is symmetric, then there is colored
permutation σ ∈ Sym(C) such that the span (σz ≺ , σz ≻ ), which is homotopic to
(z ≺ , z ≻ ), splits as above and we can finish the proof also in this case.
q
p
Lemma 3.9. Let a ←
−b−
→ a be a span in S which is a tensor product of k spans
pi
qi
ai ←− bi −→ ai for i = 1, ..., k, i.e. q = q1 ⊗ .... ⊗ qk and p = p1 ⊗ ... ⊗ pk . Then the
span (q, p) is null-homotopic if and only if each (qi , pi ) is null-homotopic.
Proof. It is clear that if each (qi , pi ) is null-homotopic, then (q, p) is null-homotopic.
So we prove the converse. We can assume without loss of generality that qi 6=
idI 6= pi where I is the monoidal unit in S, i.e. the empty word. First observe
that p, q are parallel arrows and since S satisfies the calculus of fractions, they
are homotopic if and only if there is an arrow r : c → b with rq = rp. Now, by
precomposing with an arrow in Sym(C) if necessary, we can assume that r is an
arrow in S(Opl ), i.e. a tensor product of operations in O. Observe that in S we
have α1 ⊗ ... ⊗ αl = β1 ⊗ ... ⊗ βm for arrows αi 6= idI 6= βi if and only if l = m
and αi = βi for each i = 1, ..., l. Now it follows easily that r gives arrows r1 , ..., rk
such that ri qi = ri pi for each i = 1, ..., k. Thus, qi is homotopic to pi for each
i = 1, ..., k.
3.6. The poset of partitions. From now on, fix some base object x which is split
and progressive. More generally, in view of Remark 2.6:
Let y be a split object such that x is y–progressive.
Furthermore, let n ∈ N.
Two objects a, b in S are called equivalent if they are isomorphic in π1 (S),
i.e. there is a path (equivalently, a span) between them in S. Of course, π1 (S, a) ∼
=
π1 (S, b) in this case.
Let c = (c1 , ..., ck ) with ci ∈ C an object in S and m be a uni-marking on
c, i.e. there is only one symbol in m. Then m determines another object c(m)
by deleting all ci ’s which are not marked by m. If α : a → c is an arrow, then
c(α∗ (m)) ∼ c(m) in the above sense.
Let B be a multiball. If (α, m) and (α′ , m′ ) are representatives, then c(m) ∼
c(m′ ). Thus, each multiball B gives an equivalence class cc(B) of objects.
L2 –INVISIBILITY OF SYMMETRIC OPERAD GROUPS
13
We say that a partition P (over x) satisfies the n–condition with respect to y
if at least n submultiballs P ∈ P satisfy y ∈ cc(P ). The n–condition with respect
to y is preserved by the action of Γ = π1 (S, x) on the partitions: If P satisfies the
n–condition with respect to y, then also γ · P satisfies it.
We define a poset (P, ≤): The objects of P are partitions over x and P ≤ Q if
and only if P ⊃ Q. The group Γ = π1 (S, x) acts on this poset via the action on
partitions. Because of Remark 3.6, the action indeed respects the relation ≤.
Since the n–condition with respect to y is invariant under the action of Γ, we can
define the invariant subposet (Pn , ≤) to be the full subpost consisting of partitions
satisfying the n–condition with respect to y. Next, we want to show that
1. Pn 6= ∅ and
2. (Pn , ≤) is filtered.
This implies that the poset Pn is contractible.
1. Since x is y–progressive, there is an arrow a1 ⊗ y ⊗ a2 → x. Apply y’s
split condition (n − 1) times to find an arrow z → x where z has a tensor product
decomposition with at least n factors equal to y. Mark each of these factors with
a different symbol and the rest with yet another symbol. This yields a partition
P ∈ Pn .
2. Let P, Q ∈ Pn . We have to find R ∈ Pn with P, Q ≤ R. First we find
one in P. Let (αP , mP ) and (αQ , mQ ) be representatives of P and Q respectively.
Choose a square filling (βP , βQ ) of (αP , αQ ) and set δ = βP αP = βQ αQ . Now
∗
∗
find a full marking µ ⊂ βP
(mP ), βQ
(mQ ), for example by marking each coordinate
of dom(δ) with a different symbol. Then R = [δ, µ] is a common refinement of P
and Q. Now use that x is y–progressive to find an arrow η : z → dom(δ) where
z has a tensor product decomposition with at least one factor equal to y. Then
apply y’s split condition (n − 1) times to obtain an arrow ν : w → z where w has
a tensor product decomposition with at least n factors equal to y. Observe the
marked arrow (νηδ, (νη)∗ (µ)). The so-called link condition in Remark 2.6 ensures
that the n factors of w equal to y are marked with the same symbol in the marking
(νη)∗ (µ). Refine this marking such that these factors are marked with new different
symbols. This gives a representative of a partition satisfying the n–condition with
respect to y, refining R and thus refining both P and Q.
A simplex σ in the poset Pn is a finite ascending sequence of objects, written
[P0 < P1 < ... < Pp ]. We now observe the stabilizer subgroup Γσ of such a
simplex. By definition, an element γ is in this stabilizer subgroup if and only if
{P0 , ..., Pp } = {γ · P0 , ..., γ · Pp }. But since the action of γ respects ≤, this is
equivalent to γ · Pi = Pi for each i = 0, ..., p. So each γ ∈ Γσ fixes σ vertex-wise.
Observe the subgroup
Λσ := {γ ∈ Γ | γ · P = P for all P ∈ Pp } < Γ
By Proposition 3.8, we know that Λσ ∼
= π1 (S, c1 ) × ... × π1 (S, ck ) for appropriate
objects ci . Since Pp satisfies the n–condition with respect to y, at least n of these
objects are equivalent to y and thus at least n of the factors in the product decomposition of Λσ are isomorphic to Υ := π1 (S, y). So we find a normal subgroup
Λ′σ ⊳ Λσ with Λ′σ ∼
= Υn . Below, we will show that Λσ is a normal subgroup of Γσ .
So we arrive at the following situation
Υn ∼
= Λ′σ ⊳ Λσ ⊳ Γσ
Lemma 3.10. Let R1 , R2 be semi-partitions and P be a partition with P ⊂ R1 .
Assume
∀R1 ∈R1 ∃R2 ∈R2 ∀P ∈P P ⊂ R1 =⇒ P ⊂ R2
14
WERNER THUMANN
Then we have R1 ⊂ R2 .
Proof. By applying the square filling technique twice, we find an arrow δ with three
markings mP , mR1 , mR2 on its domain such that (δ, mP ) represents P and (δ, mRi )
represents Ri . Since P ⊂ R1 we have (δ, mP ) ⊂ (δ, mR1 ) and therefore mP ⊂ mR1 .
Note that P is a partition and therefore mP is a full marking. Now the assumption
of the statement implies mR1 ⊂ mR2 and thus R1 ⊂ R2 .
We first show that Λσ is contained in Γσ . So let γ ∈ Λσ , i.e. γ · P = P for all
P ∈ Pp . In particular, we have γ · Pp = Pp (Proposition 3.5 and Remark 3.7). We
have to show γ · Pi = Pi also for the other i’s. Write P := Pp and R := Pi for some
other i. Then we have P ⊂ R. We want to apply the above lemma to R1 = R and
R2 = γ · R and deduce R ⊂ γ · R. So let R ∈ R and observe γ · R ∈ γ · R. Let
P ∈ P with P ⊂ R. Then P = γ · P ⊂ γ · R and the assumption of the lemma is
satisfied. Similarly, we get R ⊂ γ −1 · R and thus γ · R ⊂ R. This yields γ · R = R,
q.e.d.
Now we show that Λσ is normal in Γσ . Let γ ∈ Γσ and α ∈ Λσ . We have to show
γ −1 αγ ∈ Λσ , i.e. γ −1 αγ ·P = P for all P ∈ Pp =: P or equivalently α·(γ ·P ) = γ ·P
for all P ∈ P. Since γ · P = P, we have a bijection f : {P ∈ P} → {P ∈ P} such
that γ · P = P ⊲f for all P ∈ P (Proposition 3.5). Consequently, if P ∈ P,
α · (γ · P ) = α · (P ⊲f ) = P ⊲f = γ · P , q.e.d.
3.7. End of the proof. Let P be a property of groups which is closed under taking
products and let M be a coefficient system which is P–Künneth and P–inductive.
We will only give the proof for homology. Using analogous devices for cohomology,
we obtain a proof of the cohomological version of the statement.
Our main tool will be a spectral sequence explained in Brown’s book [3, Chapter
VII.7] (see also [9, Subsection 4.1]). If we plug in our Γ–complex (Pn , ≤) and the
k
with
ZΓ–module MΓ, we obtain a spectral sequence Epq
M
1
Hq (Γσ , MΓ) ⇒ Hp+q (Γ, MΓ)
Epq
=
σ∈Σp
where Σp is set of p–cells representing the Γ–orbits of (Pn , ≤). This uses that the
poset Pn is contractible and that the cell stabilizers fix the cells pointwise.
We assumed that Υ satisfies P and that H0 (Υ, MΥ) = 0. Applying the P–
Künneth property (n − 1) times, we obtain Hk (Υn , MΥn ) = 0 for k ≤ n − 1.
So we have Hk (Λ′σ , MΛ′σ ) = 0 for k ≤ n − 1. The property P–inductive yields
Hk (Λ′σ , MΓ) = 0 for k ≤ n− 1. Since Λ′σ ⊳ Λσ ⊳ Γσ , we can apply the Hochschild–
Serre spectral sequence twice to obtain Hk (Γσ , MΓ) = 0 for k ≤ n − 1. The above
spectral sequence now yields Hk (Γ, MΓ) = 0 for k ≤ n − 1. Since n was arbitrary,
the result follows.
4. Non-amenability and infiniteness
In this section we use the techniques from Section 3 to prove non-amenability
and infiniteness of some operad groups. Note that semi-partitions and the action
on the set of semi-partitions can also be defined in the braided case.
Lemma 4.1. If O satisfies the calculus of fractions, then the action of the colored
permutations in AutSym(C) (X) or the colored braids in AutBraid(C) (X) on the set
of arrows HomS(O) (X, Y ) is free. In particular, in the operad O, the action of the
symmetric groups or the braid groups on the sets of operations is free.
Proof. Let [α, Θ] be an element in HomS(O) (X, Y ) and σ ∈ AutSym(C) (X) or σ ∈
AutBraid(C) (X). We have to show that [σ, id] ∗ [α, Θ] = [α, Θ] implies that σ is
trivial. From this equality and the equalization property of S(O), we obtain an
L2 –INVISIBILITY OF SYMMETRIC OPERAD GROUPS
15
arrow z := [δ, Ψ] with z ∗ [σ, id] = z. We can assume without loss of generality
that δ = id. We then have z ∗ [σ, id] = [σ̄, Ψ̄] with σ̄ = Ψyσ and Ψ̄ = Ψxσ.
Consequently, the pairs (σ̄, Ψ̄) and (id, Ψ) are equivalent in Sym(C) × S(Opl ) or
Braid(C) × S(Opl ). This is only possible if σ is trivial.
Let O be a symmetric or braided operad. Let α be an arrow in S(O). For
any colored permutation σ ∈ Sym(C) or colored braid σ ∈ Braid(C) with suitable
domain and codomain, we can form the group element γ represented by the span
(α, [σ, id]∗α). Recall that the first arrow always denotes the denominator, i.e. points
to the left.
Lemma 4.2. Assume O satisfies the calculus of fractions. Then
σ 6= 1 =⇒ γ 6= 1
Proof. First consider the symmetric case. Observe the semi-partition R represented
by the marked arrow (α, m) where m is a marking on the domain of α with only one
marked coordinate and this coordinate is non-trivially permuted by σ −1 . It is easy
to see that γ · R is represented by (α, [σ, id]∗ (m)). The marking m′ := [σ, id]∗ (m) is
different from m because σ −1 maps the only marked coordinate of m to a different
coordinate by assumption. From Remark 3.3 it follows that the marked arrow
(α, m) cannot be equivalent to (α, m′ ) and thus γ · R 6= R. Consequently γ 6= 1.
Now if O is braided we can apply the above argument verbatim if we require that
the braid σ has a non-trivial permutation part. But there are of course non-trivial
braids which are trivial as permutations (so-called pure braids). Assume that σ is
such a pure braid and that γ = 1. Note that the latter means that the parallel
arrows σα := [σ, id] ∗ α and α are homotopic. Since S(O) satisfies the calculus of
fractions, this means that there is an arrow δ with δ ∗ σα = δ ∗ α. We can assume
without loss of generality that δ ∈ S(Opl ), i.e. that δ = [id, Θ]. Composing in S(O),
we get δ ∗ [σ, id] = [σ̄, Θ̄] where σ̄ = Θyσ and Θ̄ = Θxσ. Using that σ is pure we
immediately see Θ̄ = Θ. So we have
[σ̄, id] ∗ [id, Θ] ∗ α = δ ∗ [σ, id] ∗ α = [id, Θ] ∗ α
Lemma 4.1 now gives σ̄ = 1 and thus σ = 1.
α
α
Denoting the element γ suggestively by ←
− σ −
→, the lemma implies that two
α
α
α
α
such group elements ←
−σ−
→ and ←
− σ′ −
→ are equal if and only if σ = σ ′ . We will
use this now to give a proof of the following proposition.
Proposition 4.3. Let O be a symmetric operad satisfying the calculus of fractions
and let X be a split object of S(O). Then Γ = π1 (O, X) contains a non-abelian free
subgroup and is therefore non-amenable.
Proof. Using the split condition on X, we will explicitly construct two non-trivial
elements γ1 , γ2 ∈ Γ of order 2 and 3 respectively. Then we will define two disjoint
subsets A1 , A2 of the set of semi-partitions over X such that γ1 · A2 ⊂ A1 and
γ2n · A1 ⊂ A2 for n = 1, 2. The Ping–Pong Lemma then shows that the subgroup
hγ1 , γ2 i generated by the two elements γ1 and γ2 is isomorphic to the free product
hγ1 i ∗ hγ2 i. So we have found a subgroup which is isomorphic to Z2 ∗ Z3 . Since
Z2 ∗ Z3 contains a free non-abelian subgroup, the proof of the proposition is then
complete.
We now give the constructions. Because X is split, there is an arrow
̟ : A1 ⊗ X ⊗ A2 ⊗ X ⊗ A3 → X
16
WERNER THUMANN
For better readability, we assume that X is a single color and A1 = A2 = A3 = I.
The construction goes the same way in the general case (with obvious modifications). So we assume that ̟ is just an operation with two inputs of color X and
an output of color X. Define γ1 to be the following element
̟
̟
and γ2 to be the following element
̟
̟
̟
̟
Lemma 4.2 implies that γ1 is of order 2 and γ2 is of order 3. Let B1 be the ball
represented by the marked arrow
̟
⋆
and let B2 be the ball represented by the marked arrow
⋆
̟
Composing the operation ̟ several times, one gets operations that look like binary
trees. Call them ̟–tree operations. Now define A1 to be the set of all balls B ⊂ B1
which are represented by ̟–tree operations. Similarly, define A2 to be the set of
all balls B ⊂ B2 which are represented by ̟–tree operations. For example, the
following marked arrows represent balls in A1
̟
̟
̟
⋆
⋆
̟
̟
⋆
and the following marked arrows represent balls in A2
⋆
̟
̟
̟
⋆
̟
̟
⋆
It is straightforward to check γ1 · A2 ⊂ A1 and γ2 · A1 ⊂ A2 and γ22 · A1 ⊂ A2 , so
the proof is completed.
Next we give sufficient conditions for infiniteness of operad groups.
L2 –INVISIBILITY OF SYMMETRIC OPERAD GROUPS
17
Proposition 4.4. Let O be a planar, symmetric or braided operad satisfying the
calculus of fractions and let X be a split object of S(O). Then Γ := π1 (O, X)
contains an infinite cyclic subgroup and is therefore infinite.
Proof. Because X is split, there is an arrow
̟ : A1 ⊗ X ⊗ A2 ⊗ X ⊗ A3 → X
For better readability, we assume that X is a single color and A1 = A2 = A3 = I.
The construction goes the same way in the general case (with obvious modifications). So we assume that ̟ is just an operation with two inputs of color X and
an output of color X. Define γ to be the following element
̟
̟
̟
̟
Formally, γ is represented by the span (̟, id) ∗ ̟, (id, ̟) ∗ ̟ . We claim that γ
has infinite order. The element γ n is represented by the span (for better readability,
we use the same symbol id for different identities)
(̟, id) ∗ ... ∗ (̟, id) ∗ ̟, (id, ̟) ∗ ... ∗ (id, ̟) ∗ ̟
By Lemma 3.2, this span is null-homotopic if and only of the span (remove the last
̟ in both arrows)
(̟, id) ∗ ... ∗ (̟, id), (id, ̟) ∗ ... ∗ (id, ̟) =: (̟1 , ̟2 )
is null-homotopic. This is true if and only if there is an arrow r with r̟1 = r̟2 .
The arrow r can be chosen to lie in S(Opl ). But note that ̟1 splits as
̟1 = (̟, id) ∗ ... ∗ (̟, id) ∗ ̟ ⊗ idX
and ̟2 splits as
̟2 = idX ⊗ (id, ̟) ∗ ... ∗ (id, ̟) ∗ ̟
It can easily be seen that such an arrow r cannot exist because otherwise operations
with a different number of inputs must be equal. Consequently, each γ n is nontrivial and therefore γ has infinite order.
5. Applications
Observe the 1–dimensional planar cube cutting operads and the d–dimensional
symmetric cube cutting operads from [10, Subsection 3.5]. They all satisfy the
(cancellative) calculus of fractions. Furthermore, they are monochromatic and possess operations of arbitrarily high degree. From Remarks 2.3 and 2.4 it follows
that all objects (except the uninteresting unit object) are split and progressive. So
Theorem 2.5 is applicable to these operads. Furthermore, the corresponding operad
groups are all infinite by Proposition 4.4 and non-amenable in the symmetric case
by Proposition 4.3.
Observe now the local similarity operads. Let SimX be a finite similarity structure on the compact ultrametric space X. When choosing a ball in each SimX –
equivalence of balls, we obtain a symmetric operad with transformations O. The
colors of O are the chosen balls. We choose X for the SimX –equivalence class
[X]. We already know that O satisfies the (cancellative) calculus of fractions. In
[9, Definition 3.1] we called SimX dually contracting if there are two disjoint proper
subballs B1 , B2 of X together with similarities X → Bi in SimX . This easily implies
that X is split.
18
WERNER THUMANN
Lemma 5.1. The color X is progressive provided SimX is dually contracting.
Proof. Let θ = (f1 , ..., fl ) be an operation with output X. This means that the
fi : Bi → X are SimX –embeddings (i.e. fi yields a similarity in SimX when the
codomain is restricted to the image) such that the images of the fi are pairwise
disjoint and their union is X. So the images im(fi ) form a partition P of X into
balls. If we apply [9, Lemma 3.7] to this partition, we find a j and a small ball
B which is SimX –equivalent to X and such that B ⊂ im(fj ). Using this, we can
construct an operation ψ = (g1 , ..., gk ) with codomain Bj such that g1 : X → Bj .
From Remark 2.4 it now follows that X is progressive.
Consequently, Theorem 2.5 is applicable to dually contracting local similarity
operads. Furthermore, the corresponding operad groups based at X are all infinite
by Proposition 4.4 and non-amenable by Proposition 4.3.
5.1. L2 –homology. For a group G, let l2 G beP
the Hilbert space with Hilbert base
G. Thus, elements in l2 G are formal sums
g∈G λg g with λg ∈ C such that
P
2
|λ
|
<
∞.
Left
multiplication
with
elements
in G induces an isometric G–
g
g∈G
action on l2 G. Denote the set of G–equivariant linear bounded operators l2 G → l2 G
by B G (l2 G), a subalgebra of the algebra of all bounded linear operators B(l2 G).
Right multiplication with an element γ ∈ G induces a G–equivariant linear bounded
operator γ⊲ρ : l2 G → l2 G. This induces a homomorphism ρ : CG → B(l2 G) from
the group ring into the algebra of bounded linear operators, i.e. 1⊲ρ = id and
(γ1 γ2 )⊲ρ = (γ1 ⊲ρ) ∗ (γ1 ⊲ρ). The closure of the image of this map with respect
to the weak or strong operator topology is called the von Neumann algebra N G
associated to G. It is equal to the subalgebra of all G–equivariant bounded linear
operators B G(l2 G) ⊂ B(l2 G) [8, Example 9.7].
We will cite some known results about this von Neumann algebra in order to
deduce a corollary for l2 –homology.
• (N is inductive) Let H be a subgroup of G and A ∈ B H (l2 H). Then
CG ⊗CH l2 H ⊂ l2 G is a dense G–invariant subspace and
idCG ⊗CH A : CG ⊗CH l2 H → CG ⊗CH l2 H
is a G–equivariant linear map which is bounded with respect to the norm
coming from l2 G. Consequently, it can be extended to an element in
B G (l2 G). We obtain a map N H → N G which is an injective ring homomorphism. So if H < G, then N H is a subring of N G. Even more is
true: It is a faithfully flat ring extension [8, Theorem 6.29]. From this, it
follows easily that the coefficient system N is inductive.
• (N is Künneth) If H1 , H2 are two subgroups of G which commute in G,
i.e. h1 h2 = h2 h1 for all h1 ∈ H1 and h2 ∈ H2 , then N H1 and N H2 commute
in N G. In particular, N H1 ⊗C N H2 is a subring of N G. This implies, using
a standard homological algebraic argument [8, Lemma 12.11(3)], that N is
Künneth.
• (H0 and amenability) Going back to a result of Kesten, the 0’th group
homology of a group G with coefficients in the von Neumann algebra N G
vanishes if and only if G is non-amenable [8, Lemma 6.36]. So we have
H0 (G, N G) = 0
⇐⇒
G non-amenable
• (Relationship with l2 –homology) From [8, Lemma 6.97] or [8, Theorem
6.24(3)] we get for groups G of type F∞ and every k ≥ 0
Hk (G, N G) = 0
⇐⇒
Hk (G, l2 G) = 0
Applying Theorem 2.5 to these observations, we get the following corollary.
L2 –INVISIBILITY OF SYMMETRIC OPERAD GROUPS
19
Corollary 5.2. Let O be a planar or symmetric operad which satisfies the calculus
of fractions. Let X be a split progressive object of S(O). Set Γ := π1 (O, X) and
assume that Γ is non-amenable. Then
Hk (Γ, N Γ) = 0
for all k ≥ 0. If Γ is also of type F∞ (e.g. if the conditions in [10, Theorem 4.3]
are satisfied), we also have
Hk (Γ, l2 Γ) = 0
for all k ≥ 0.
From Proposition 4.3, we get the following corollary.
Corollary 5.3. Let O be a symmetric operad which satisfies the calculus of fractions. Let X be a split progressive object of S(O). Set Γ := π1 (O, X). Then
Hk (Γ, N Γ) = 0
for all k ≥ 0. If Γ is also of type F∞ , we have
Hk (Γ, l2 Γ) = 0
for all k ≥ 0.
From the remarks at the beginning of this section and from [10, Subsection 4.6],
we get the following corollary.
Corollary 5.4. Let O be a symmetric cube cutting operad or a local similarity
operad coming from a dually contracting finite similarity structure SimX . In the
first case, let A be any object in S(O) different from the monoidal unit I. In the
second case, let A be the object X. Set Γ = π1 (O, A). Then
Hk (Γ, N Γ) = 0
for all k ≥ 0. Assume furthermore that SimX is rich in ball contractions [6, Definition 5.12], in other words, the associated operad O is color-tame in the sense of
[10, Definition 4.2]. Then we also have
Hk (Γ, l2 Γ) = 0
for all k ≥ 0.
In particular, we obtain that the Higman–Thompson groups Vn,r and the higherdimensional Thompson groups nV (see [1]) are l2 -invisible. This answers a question
posed by Lück [8, Remark 12.4]: The Zero-in-the-spectrum conjecture by Gromov
says that whenever M is an aspherical closed Riemannian manifold, then there is
always a dimension p such that zero is contained in the spectrum of the minimal
closure of the Laplacian acting on smooth p–forms on the universal covering of M :
f) → L2 Ωp (M
f)
∃p≥0 0 ∈ spec cl(∆p ) : D ⊂ L2 Ωp (M
By [8, Lemma 12.3], this is equivalent to
∃p≥0 Hp (G, N G) 6= 0
for G = π1 (M ). Dropping Poincaré duality from the assumptions, we arrive at the
following question: If G is a group of type F (i.e. there exists a compact classifying
space for G), then is there a p with Hp (G, N G) 6= 0? Relaxing the assumption
on the finiteness property, we arrive at the following question: If G is a group
of type F∞ , then is there a p with Hp (G, N G) 6= 0? Corollary 5.4 gives explicit
counterexamples to this question.
20
WERNER THUMANN
5.2. Cohomology with coefficients in the group ring. We want to apply the
cohomological version of Theorem 2.5 to MG := ZG. To this end, we want to
show that ZG is F P∞ –Künneth and F P∞ –inductive (in cohomology). The first
property follows from [9, Proposition 4.3]. The second property follows from the
observation that ZG is a free ZH–module if H < G and that group cohomology
of groups of type F P∞ commutes with direct limits in the coefficients [2, Theorem
VIII.4.8]. From Theorem 2.5, Proposition 4.4 and H 0 (G, ZG) = (ZG)G = 0 for
infinite G, we obtain:
Corollary 5.5. Let O be a planar or symmetric operad which satisfies the calculus
of fractions. Let X be a split progressive object of S(O). Set Γ := π1 (O, X) and
assume that Γ is of type F P∞ (e.g. if the conditions in Theorem [10, Theorem 4.3]
are satisfied). Then
H k (Γ, ZΓ) = 0
for all k ≥ 0.
Recall that type F∞ implies type F P∞ and note that, in this case, H k (Γ, ZΓ) = 0
for all k ≥ 0 implies that Γ has infinite cohomological dimension [2, Propositions
VIII.6.1 and VIII.6.7]. Unfortunately, this tells us that none of these groups can be
of type F and consequently, we cannot find such a group which is also l2 -invisible.
From the remarks at the beginning of this section and from [10, Subsection 4.6],
we get the following corollary.
Corollary 5.6. Let O be a planar or symmetric cube cutting operad or a local
similarity operad coming from a dually contracting finite similarity structure SimX
which is also rich in ball contractions. In the first two cases, let A be any object in
S(O) different from the monoidal unit I. In the last case, let A be the object X.
Set Γ = π1 (O, A). Then
H k (Γ, ZΓ) = 0
for all k ≥ 0.
In particular, we obtain H k (F, ZF ) = 0 and H k (V, ZV ) = 0 for all k ≥ 0. This
has been shown before in [4, Theorem 7.2] and in [3, Theorem 4.21].
References
[1] M. G. Brin, Higher dimensional Thompson groups, Geom. Dedicata 108 (2004), 163–192.
[2] K. S. Brown, Cohomology of groups, Graduate Texts in Mathematics, vol. 87, Springer-Verlag,
New York, 1994.
[3]
, Finiteness properties of groups, Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 45–75.
[4] K. S. Brown and R. Geoghegan, An infinite-dimensional torsion-free FP∞ group, Invent.
Math. 77 (1984), no. 2, 367–381.
[5] M. W. Davis, Poincaré duality groups, Surveys on surgery theory, Vol. 1, Ann. of Math.
Stud., vol. 145, Princeton Univ. Press, Princeton, NJ, 2000, pp. 167–193.
[6] D. S. Farley and B. Hughes, Finiteness properties of some groups of local similarities, Proc.
Edinb. Math. Soc. (2) 58 (2015), no. 2, 379–402.
[7] M. Gromov, Large Riemannian manifolds, Curvature and topology of Riemannian manifolds
(Katata, 1985), Lecture Notes in Math., vol. 1201, Springer, Berlin, 1986, pp. 108–121.
[8] W. Lück, L2 -invariants: theory and applications to geometry and K-theory, Ergebnisse der
Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics,
vol. 44, Springer-Verlag, Berlin, 2002.
[9] R. Sauer and W. Thumann, l2 -invisibility and a class of local similarity groups, Compos.
Math. 150 (2014), no. 10, 1742–1754.
[10] W. Thumann, Operad groups and their finiteness properties, arXiv:1409.1085v3 (2016).
Karlsruhe Institute of Technology, Karlsruhe, Germany
| 4 |
arXiv:1603.04212v1 [math.GR] 14 Mar 2016
AMENABILITY AND PARADOXICAL DECOMPOSITIONS FOR
PSEUDOGROUPS AND FOR DISCRETE METRIC SPACES
TULLIO CECCHERINI-SILBERSTEIN, ROSTISLAV I. GRIGORCHUK,
AND PIERRE DE LA HARPE
Abstract. This is an expostion of various aspects of amenability and paradoxical decompositions for groups, group actions and metric spaces. First, we review the formalism
of pseudogroups, which is well adapted to stating the alternative of Tarski, according to
which a pseudogroup without invariant mean gives rise to paradoxical decompositions, and
to defining a Følner condition. Using a Hall-Rado Theorem on matchings in graphs, we
show then for pseudogroups that existence of an invariant mean is equivalent to the Følner
condition; in the case of the pseudogroup of bounded perturbations of the identity on a
locally finite metric space, these conditions are moreover equivalent to the negation of the
Gromov’s so-called doubling condition, to isoperimetric conditions, to Kesten’s spectral
condition for related simple random walks, and to various other conditions. We define also
the minimal Tarski number of paradoxical decompositions associated to a non-amenable
group action (an integer ≥ 4), and we indicate numerical estimates (Sections II.4 and
IV.2). The final chapter explores for metric spaces the notion of supramenability, due for
groups to Rosenblatt.
I. Introduction
The present exposition shows various aspects of amenability and non-amenability. Our
initial motivation comes from a note on the Banach-Tarski paradox where Deuber, Simonovitz and Sós indicate one kind of paradoxical decomposition for metric spaces, in
relation with what they call an “exponential growth” property [DeSS]. Our first purpose is to revisit their work which, in our view, relates paradoxical decompositions with
amenability rather than with growth (see in particular Observation 33 below).
For this, we recall in Chapter II the formalism of set-theoretical pseudogroups which is
well adapted to showing the many aspects of amenability: existence of invariant finitely
additive measures, absence of paradoxical decompositions, existence of Følner sets and
isoperimetric estimates. We also state one version of the basic Tarski alternative: a pseudogroup is either amenable or paradoxical.
In Chapter III, we specialize the discussion to metric spaces and pseudogroups of bounded
perturbations of the identity; metric spaces, there, are locally finite (except at the very end
of the chapter). On one hand, this is an interesting class, with many examples given by
Date: March 15, 2016.
2000 Mathematics Subject Classification. 43A07, 22F05, 54E40, 05C63.
Key words and phrases. amenability, paradoxical decomposition, pseudogroup, metric space.
The authors acknowledge support from the Fonds National Suisse de la Recherche Scientifique.
1
2
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
finitely generated groups. On the other hand, it provides a convenient setting for proving
Følner characterization as stated in Chapter II. We discuss also the Kesten characterization
in terms of simple random walks.
For a group G which is not amenable, we estimate in Chapter IV the Tarski number
T (G) ∈ {4, 5, . . . , ∞}, which indicates the minimal number of pieces involved in a paradoxical decomposition of G. It is known that T (G) = 4 if and only if G has a subgroup which
is free non abelian. We show that one has 5 ≤ T (G) ≤ 34 [respectively 6 ≤ T (G) ≤ 34] for
some torsion-free groups
[resp. for some torsion groups] constructed by Ol’shanskii [Ol1],
and 6 ≤ T B(m, n) ≤ 14 for B(m, n) a Burnside group on m ≥ 2 generators of odd
exponent n ≥ 665 [Ady2].
Building upon the seminal 1929 paper by von Neumann [NeuJ], Rosenblatt has defined
for groups a notion of supramenability. He has shown that supramenable groups include
those of subexponential growth, and it is not known whether there are others. In Chapter
V, we investigate supramenability for pseudogroups and for locally finite metric spaces; in
particular, we describe a simple example of a graph which is both supramenable and of
superexponential growth.
We are grateful to Joseph Dodziuk, Vadim Kaimanovich, Alain Valette and Wolfgang Woess for useful discussions and bibliographical informations, as well as to Laurent
Bartholdi for Presentation 12, Example 74, and his critical reading of a preliminary version
of this work.1
II. Amenable pseudogroups
II.1. Pseudogroups
1. Definition. In the present set-theoretical context, a pseudogroup G of transformations
of a set X is a set of bijections γ : S → T between subsets S, T of X which satisfies the
following conditions (as listed, e.g., in [HS1]):
(i) the identity X → X is in G,
(ii) if γ : S → T is in G, so is the inverse γ −1 : T → S,
(iii) if γ : S → T and δ : T → U are in G, so is their composition δγ : S → U,
(iv) if γ : S → T is in G and if S ′ is a subset of S, the restriction γ|S ′ : S ′ → γ(S ′ ) is in G,
(v) if γ : S → T is a bijection between two subsets S, T of X and if there exists a finite
partition S = ⊔1≤j≤n Sj with γ|Sj in G for j ∈ {1, . . . , n}, then γ is in G (where ⊔
denotes a disjoint union).
Property (v) expresses the fact that G is closed with respect to finite gluing up; together
with (iv), they express the fact that, for a bijection γ, being in G is in some sense a local
condition.
1This
post on arXiv is the published version (Proc. Steklov Inst. Math. 224 (1999), 57–97) with
the following changes: (i) the caution following Definition 28, (ii) the addition of a missing hypothesis
in Proposition 38 (we are grateful to Volker Diekert for having pointed out this omission to us), (iii) the
updating of some references, and (iv) the correction of a few minor typos.
Moreover, we have collected comments on several items in a new Chapter VI, after the first list of
references.
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
3
For γ : S → T in G, we write also α(γ) for the domain S of γ and ω(γ) for its range T .
For “a pseudogroup G of transformations of a set X” , we write shortly “a pseudogroup
(G, X)”, or even “a pseudogroup G”.
2. Examples. (i) Any action of a group G on a set X generates a pseudogroup GG,X . More
precisely, a bijection γ : S → T is in GG,X if there exists a finite partition S = ⊔1≤j≤n Sj
and elements g1 , . . . , gn ∈ G such that γ(x) = gj (x) for all x ∈ Sj , j ∈ {1, . . . , n}. If there
exists such a γ, the subsets S, T of X are sometimes said to be G-equidecomposable (or
“endlich zerlegungsgleich” in [NeuJ]).
In case G = X acts on itself by left multiplications, we write GG instead of GG,G .
(ii) Piecewise isometries of a metric space X constitute a pseudogroup PiIs(X), generated (in the obvious way) by the partial isometries between subsets of X. Observe that it
may be much larger than the pseudogroup associated as in the previous example with the
group of isometries of X; see for example the metric space obtained from the real line by
gluing two hairs of different length at two distinct points of the line.
(iii) For a metric space X, the pseudogroup W(X) of bounded perturbations of the identity
consists of bijections γ : S → T such that supx∈S d(γ(x), x) < ∞. In agreement with the
main example of [DeSS], we like to call W(X) the pseudogroup of wobbling bijections; the
notion seems to come from the important work by Laczkovich [Lacz]. See also Item 0.5.C1′′
in [Gro3].
(iv) Given a pseudogroup G of transformations of a set X and a subset A of X, the set of
bijections γ ∈ G with α(γ) ⊂ A and ω(γ) ⊂ A constitute a pseudogroup of transformations
of A, denoted below by G(A) .
(v) From a pseudogroup (G, X) and an integer k ≥ 1, one obtains a pseudogroup Gk of
transformations of the direct product Xk of X and {1, . . . , k}, generated by the bijections
of the form
(
S × {j} −→ T × {j ′ }
(x, j)
where γ : S → T is in G and 1 ≤ j, j ′ ≤ k.
7−→
(γ(x), j ′ )
3. Remarks. The above notion of pseudogroup of transformations is strongly motivated
by the study of Banach-Tarski paradoxes, as shown by the first three observations below.
(i) The very definition of a paradoxical decomposition with respect to a group action
involves the associated pseudogroup as in Example 2.(i).
(ii) Pseudogroups are easily restricted on subsets as in Example 2.(iv). This is important
for the study of supramenability (see Chapter V below).
(iii) Pseudogroups are easily induced on oversets, as in Example 2.(v). This is useful in
the setting of a pseudogroup constituted by bijections with domains and range required to
be in a given algebra (or σ-algebra) of subsets of X (for example the measurable sets of a
measure space), and in corresponding variations on the Tarski alternative [HS1].
4
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
(iv) For a pseudogroup (G, X), the set
R = (x, y) ∈ X × X there exists γ ∈ G such that x ∈ α(γ) and y = γ(x)
is an equivalence relation. A natural problem is to study the existence of measures µ
on X such that, for each measurable subset A of X of measure zero, the saturated set
{x ∈ A | there exists a ∈ A with (x, a) ∈ R} has also measure zero, see [CoFW], [Kai2],
[Kai3].
(v) In a topological context, Conditions (iv) and (v) in Definition 1 are usually replaced
by a condition involving restrictions to open subsets; see [Sac] and page 1 of [KoNo].
(vi) Consider a metric space X, the pseudogroup W(X) of Example 2.(iii), and a subspace A of X. It is then remarkable (though straightforward to check) that the pseudogroup
W(A) coincides with the restriction of W(X) to A in the sense of Example 2.(iv).
II.2. Amenability and paradoxical decompositions - the Tarski alternative
Let (G, X) be a pseudogroup. We denote by P(X) the set of all subsets of X.
4. Definitions. A G-invariant mean on X is a mapping µ : P(X) → [0, 1] which is
(f a)
(in)
(no)
finitely additive: µ(S1 ∪ S2 ) =
µ(S1 ) + µ(S2 ) for S1 , S2 ⊂ X with S1 ∩ S2 = ∅,
invariant: µ ω(γ) = µ α(γ) for all γ ∈ G,
normalized: µ(X) = 1.
More generally, for A ⊂ X, a G-invariant mean on X normalized on A is a mapping
µ : P(X) → [0, ∞] which satisfies Conditions (f a) and (in) above, as well as
(no′ )
µ(A) = 1.
The pseudogroup G is amenable if there exists a G-invariant mean on X, and the triple
(G, X, A) is amenable if there exists a G-invariant mean on X normalized on A. These
notions are essentially due to von Neumann [NeuJ].
5. Definition. A paradoxical G-decomposition of X is a partition X = X1 ⊔ X2 such that
there exist γj ∈ G with α(γj ) = Xj and ω(γj ) = X (j = 1, 2).
A pseudogroup (G, X) is paradoxical if it has a paradoxical G-decomposition, or equivalently (because of Theorem 7 below) if it is not amenable.
6. Remarks. (i) There cannot exist such paradoxical G-decomposition if G is amenable.
This is obvious, because (with the notation of Definitions 4 and 5) one cannot have
1 = µ(X) = µ(X1 ) + µ(X2 ) = 2 !
It is remarkable that there is no further obstruction, as Theorem 7 shows.
(ii) Let G and H be two pseudogroups of transformations of the same set X, with G ⊂ H.
If H is amenable, then so is G; if G is paradoxical, then so is H. This will be used for
example in the proof of Theorem 25 (Item 36).
!!
(iii) In short-hand, Definition 5 reads 2[X] = [X]. It has variations in the literature; for
!!
example, one may ask (n + 1)[X] ≤ n[X], or more precisely:
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
5
there exists an integer n ≥ 1 and elements γ1 , . . . , γN ∈ G such
Pthat
|{j ∈ {1, . . . , N} | x ∈ α(γj )}| ≥ n + 1 for all x ∈ X, namely kj=1 [α(γj )] ≥ (n + 1)[X],
and
P
{j ∈ {1, . . . , N} | x ∈ ω(γj )} ≤ n for all x ∈ X, namely kj=1 [ω(γj )] ≤ n[X].
Then Remark (i) still holds for the same obvious kind of reason. Indeed, the variation is
equivalent to Definition 5, as can be seen either with manipulations à la Cantor-Bernstein
(see for example [HS1]) or as a consequence of the following theorem.
7. Theorem (Tarski alternative). Let G be a pseudogroup of transformations of a set
X. Exactly one of the following holds:
- either G is amenable,
- or there exists a paradoxical G-decomposition of X.
Let moreover A be a non-empty subset of X and let G(A) be the pseudogroup obtained by
restriction of G, as in Example 2.(iv). Exactly one of the following holds:
- either there exists a G-invariant mean on X normalized on A,
- or there exists a paradoxical G(A) -decomposition of A.
The theorem originates in Tarski’s work: see [Tar3], as well as earlier papers by Tarski
([Tar1], [Tar2]).
One proof for pseudogroups has been written up in [HS1]. Its starting point is an
application of the Hahn-Banach theorem, to the Banach space ℓ∞ (X) of bounded realvalued functions onX, to the subspace d∞ (X) of finite linear combinations of functions
of the form χ ω(γ) − χ α(γ) for some γ ∈ G (where χ(A) denotes the characteristic
function of A), and to the open cone C of functions F ∈ ℓ∞ (X) such that inf x∈X F (x) > 0;
one has to observe that G has an invariant mean if and only if d∞ (X) ∩ C = ∅. This
proof uses also ideas of Banach, Cantor-Bernstein, Hausdorff, König, Kuratowski and von
Neumann.
We give here another proof, based on what we call the Hall-Rado theorem (Theorem
35), which is essentially the “König theorem” of [Wag]. More precisely, the first statement
of Theorem 7 is a straightforward consequence of Theorem 25 and Theorem 32, and the
second statement follows (see the sketch below).
Much more complete information on all this can be found in Wagon’s book (see [Wag],
in particular Corollary 9.2 on page 128). Important more recent work in this area include
[DouF].
Let us sketch the proof of the second statement of the theorem. Assume that the
pseudogroup G(A) is not paradoxical, so that, by the first statement, there exists a G(A) invariant mean µA : P(A) → [0, 1]. Define then a mapping µ : P(X) → [0, ∞] as follows;
for a subset Y of X, if there exists a partition Y = ⊔1≤j≤n Yj P
and elements γ1 : Y1 →
B1 , . . . , γn : Yn → Bn in G with B1 , . . . , Bn ⊂ A, then set µ(Y ) = nj=1 µA (Bj ); otherwise,
set µ(Y ) = ∞. Then one checks that µ is well defined and that it is a G-invariant mean
on X normalized on A.
6
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
8. Remark. A famous theorem of E. Hopf can be expressed very much like Tarski’s
alternative.
Let T : X −→ X be an ergodic non-singular transformation of a finite probability space
(X, B, m), with m non-atomic. Let [[T ]] denote the set of all 1-1 non-singular transformations φ : U → V such that φ(x) belongs to the T -orbit of x for all x ∈ U (with U, V ∈ B);
this [[T ]] is the full groupoid of T of Katznelson and Weiss [KaWe, page 324]. For two
measurable subsets A, B of X, say that A is dominated by B, and write A ≺ B, if there
exists a measurable subset B ′ of B with m(B \ B ′ ) > 0 and a bijective transformation
φ : A −→ B ′ in [[T ]].
Hopf alternative. (i) In the situation above, exactly one of the following holds:
- there exists a T -invariant probability measure on (X, B) equivalent to m,
- one has X ≺ X.
(ii) Also, exactly one of the following holds:
- there exists a T -invariant infinite measure on (X, B) equivalent to m,
- one has X ≺ X, and there exists A ∈ B with m(A) > 0 such that A is not dominated
by A.
In other words, (i) says that there is a finite invariant measure in the measure class m if
and only if X itself is not “Hopf-compressible”, and (ii) that there is an infinite invariant
measure in the measure class m if and only if X is Hopf-compressible and some measurable
subset of X of positive measure is not Hopf-compressible [Weis].
If there exists a T -invariant probability measure [respectively infinite measure] on (X, B)
equivalent to m, then T is said to be of type II1 [resp. of Type II∞ ].
II.3. The case of groups
For any group G, we consider first the pseudogroup GG which is associated with the action
of G on itself on the left, as in Example 2.(i).
Let now G be a group generated by a finite set S. Let ℓS : G → N denote the corresponding word length function; thus ℓS associates with g ∈ G the smallest integer n ≥ 0 for
which there exist s1 , . . . , sn ∈ S ∪ S −1 with g = s1 . . . sn . Let dL and dR denote respectively
the left and right invariant metrics on G defined by
dL (x, y) = ℓS x−1 y
dR (x, y) = ℓS xy −1
for all x, y ∈ G.
Besides GG , we consider also the pseudogroup PiIs(G) of piecewise isometries of the
metric space (G, dL ), as in Example 2.(ii), as well as the pseudogroup W(G) of bounded
perturbations of the identity of the metric space (G, dR ), as in Example 2.(iii). It is easy
to check that the pseudogroup W(G) does not depend on the choice of S.
9. Observation With the notation above, one has GG = W(G) for any finitely generated
group G.
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
7
Proof. It is obvious that GG ⊂ W(G). Conversely, let γ : U → V be in W(G). Set
k = sup dR (γ(x), x)
x∈U
B = { g ∈ G | ℓS (g) ≤ k }
and observe that B is a finite subset of G. For each g ∈ B, set
Ug = { x ∈ U | γ(x) = gx }.
One has U = ⊔g∈B Ug and γ(x) = gx for all x ∈ Ug . Hence γ ∈ GG .
It is clear that GG ⊂ PiIs(G). It is also clear that GG 6= PiIs(G) in general (example:
for G = Z generated by {1}, the isometry n 7→ −n is not in GZ ).
10. Definition. A group G is amenable if the pseudogroup GG is amenable.
If G is finitely generated, the previous observation shows that one may equivalently
define G to be amenable if the pseudogroup W(G) is amenable.
11. On the class of amenable groups. Amenability may be viewed as a finiteness
condition. One of the main problems is to understand various classes of amenable groups,
for example those which are finitely generated or finitely presented. (Recall that a group
is amenable if and only if all its finitely generated subgroups are amenable; see Theorem
1.2.7 in [Gre1] and Observation 19 below.)
The following question, implicit in [NeuJ], was formulated explicitely by Day, at the
end of Section 4 in [Day1]: does every non-amenable group contain a free group on 2
generators? As much as we know and despite several misleading allusions in the literature
to some “von Neumann conjecture”, von Neumann himself has never conjectured that a
non-amenable group should contain a non-abelian free subgroup!
Day’s question was answered negatively by A. Yu. Ol’shanskii [Ol1], Adyan [Ady2] and
Gromov [Gro2, Corollary 5.6.D]; the first two use cogrowth criteria (see Item 52 below)
and Gromov uses Property (T). For infinite groups, this Property (T) of Kazhdan [Kaz] is
(among other things) a strong form of non-amenability: see [Sch] and [CoWe]. However,
when restricted to the class of linear groups (i.e. of groups which have faithful finitedimensional linear representations), Day’s question can be answered positively: this follows
from an important result due to Tits [Tit].
M. Day has defined the class EG of “elementary amenable groups”, which is the smallest class of groups which contains finite groups and abelian groups, and which is closed
under the four operations of (i) taking subgroups, (ii) forming factor groups, (iii) group
extensions and (iv) upwards directed unions. He has asked (again in [Day1]) whether the
class EG coincides with the class AG of all amenable groups (see also [Cho]).
Today, we know that there are finitely generated groups in AG which are not in EG;
this has first been shown using growth estimates ([Gri2], [Gri3]), and more recently by an
elegant argument of Stepin (see [Ste], based on [Gri2]).
8
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
One knows also finitely presented groups in AG which are not in EG; more precisely, the
finite presentation
+
*
a2 = b2 = c2 = d2 = bcd = (ad)4 = (adacac)4 = 1
G =
a, b, c, d, t
t−1 at = aca t−1 bt = d t−1 ct = b t−1 dt = c
defines an amenable group which is not elementary amenable ([Gri6], [Gri7]).
12. Bartholdi’s presentation. It has later been shown that the group G of [Gri6] has a
presentation with two generators only (namely a and t) and four relations of total length
109 = 2 + 19 + 32 + 56. Here are Bartholdi’s computations, where T stands for t−1 .
The relations c = aT ata, d = tcT and b = T ct show first that the relations c2 = d2 =
2
b = 1 may be deleted in the presentation above, and second that the generators b, c, d
may also be deleted. Thus
+
*
a2 = T ctctcT = (atcT )4 = (atcT acac)4 = 1
G =
a, t
T 2 ct2 = tcT
where c holds for aT ata. The relation T ctctcT = 1 implies T 2 ctctc = 1 = tcT 2 ctc (by
conjugation), hence also (using c−1 = c)
−1
1 = T 2 ctctc tcT 2 ctc
= T 2 ct2 (tcT )−1
using free simplifications, so that the relation T 2 ct2 = tcT may also be deleted. Finally,
one observes that atcT is conjugate to T atc = (T ata)2 so that (atcT )4 = 1 may be written
(T ata)8 = 1, and one observes also that atcT acac is equal to ataT ataT aaT ataaaT ata, so
is conjugate to T 2 ataT at2 aT ata. One obtains finally Bartholdi’s presentation
2
8
2
2
4
G =
a, t a = T aT atataT atataT ataT = (T ata) = (T ataT at aT ata) = 1 .
13. Categorical considerations. For a given integer k, let Fk be the free group on k
generators {s1 , . . . , sk } and let Xk denote the space of all marked groups on k generators,
namely of all data Fk ։ Γ, where ։ indicates a homomorphism onto. There is an
appropriate topology on Xk , for which two quotients π : Fk ։ Γ and π ′ : Fk ։ Γ′ are
“near” each other if the corresponding Cayley graphs have balls of “large” radius around
the unit element which are isomorphic. This makes Xk a compact space; one shows for
example that the closure of the subset of Xk corresponding to finite groups contains the
subset of Xk corresponding to residually finite finitely generated groups. For details, see
[Gri2], [Cha] and [Ste].
It would be interesting to find pairs (Y, Z) where
• Y is a compact subspace of Xk ,
• Z is a “small” (e.g. countable) subset of Y , consisting of amenable groups,
• Y \ Z consists of non-elementary amenable groups, or more generally
the set of elementary amenable groups in Y \ Z is of first category.
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
9
The point is that the space Y contains a dense Gδ consisting of amenable groups which are
not elementary amenable. (As usual a Gδ in Y is a countable intersection of open subsets
of Y .)
One such pair has been constructed in [Gri2] and analyzed in [Ste], with Z a countable
set of virtually 2-step solvable groups and with Y \ Z consisting of infinite torsion groups.
Understanding other such pairs would probably help us understanding the closures of AGk
and of EGk in Xk , where AGk [respectively EGk ] denotes the subspace of Xk containing
marked groups π : Fk ։ Γ with Γ amenable [resp. elementary amenable].
14. Variation on one question of Day. Let us denote by BG the smallest class of
groups containing finitely generated groups of subexponential growth (see Definition 64)
and closed with respect to the four operations of Day listed in 11 above, namely with
respect to (i) taking subgroups, (ii) forming factor groups, (iii) group extensions and (iv)
upwards directed unions.
Question: does one have BG=AG ?
15. Other definitions of amenability for groups; topological groups. The natural
setting for amenability of groups is that of topological groups, mainly locally compact
groups. A substantial part of the theory consists in showing the equivalence of a large
number of definitions.
Let G be a Hausdorff topological group. Denote by C b (G) the Banach space of bounded
continuous functions from G to C, with the supremum norm. For ξ ∈ C b (G) and g ∈ G,
let g ξ ∈ C b (G) be the function x → f (g −1x). Denote by UC b (G) the closed subspace of
C b (G) of functions ξ for which the mapping g 7→ g ξ from G to C b (G) is continuous. The
following are known to be equivalent (see Theorem 3 in [Day2] and Theorem 4.2 in [Ric2]):
• there exists a left-invariant mean on UC b (G),
• any continuous action G×Q → Q of G by affine transformations of a non-empty compact
convex subset Q of a Hausdorff locally convex topological vector space has a fixed point.
The group G is amenable if these properties hold. In case G is assumed to be locally
compact, here is a short list of other equivalent properties:
• there exists a left-invariant mean on C b (G),
• there exists a left-invariant mean on L∞ (G),
• the unit representation of G is weakly contained in the left regular representation of G
on L2 (G),
• for any continuous action G×X → X of G by homeomorphisms of a non-empty compact
space X, there exists a G-invariant probability measure on X.
The last point, on G-invariant measures, goes back to a paper by Bogolyubov, see [Bogl],
quoted by Anosov [Ano]. This paper, published in Ukrainian in 1939, has remained unnoticed; the paper does not quote von Neumann [NeuJ], and it is conceivable that Bogolyubov has introduced independently the notion of amenability. About relations between
amenability, growth and existence of invariant measures, we would also like to quote [Bekl].
10
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
The list above is very far from being complete! (See 16; other items could be: several
formulations of the Følner property for locally compact groups, the Reiter-Glicksberg property, the existence of approximate units in the Fourier algebra, . . .) See, e.g., the books
[Gre1], [Pat] and [Wag], as well as [Rei, Chapter 8], [Eym2], [Zim, Chapter 4], [Wag, in
particular Theorem 10.11] and [Lub, Chapter 2]. In case of a countable group (with the
discrete topology), here is the most recent characterization of amenability with which one
of the authors has been involved: a countable group G is amenable if and only if, for any
action of G by homeomorphisms on the Cantor discontinuum K, there exists a probability
measure on K which is invariant by G [GiH2].
We would like to point out that some attention has been given to topological groups
which are not locally compact (in [Ric2, Section 4] among other places). For example, let
U(H)st be the group of unitary operators on a separable infinite dimensional Hilbert space
H, with the strong topology; then U(H)st is amenable, namely there exists a left invariant
mean on UC b (U(H)st ), but there does not exist any left invariant mean on C b (U(H)st )
[Har1, Har2]. Moreover, this group does have closed subgroups which are not amenable;
indeed, if H = ℓ2 (Fn ) for a free group Fn of rank n ≥ 2, then U(H)st has clearly a discrete
subgroup isomorphic to Fn , as observed in [Har3]. Here is another example involving non
locally compact topologies; let G be the group of real points of an R-algebraic group and
let Γ be a subgroup of G which is dense for the Zariski topology; if Γ is amenable, so is G
(see [Moo], and Theorem 4.1.15 in [Zim]).
Let us mention the following: for a locally compact group G which is almost connected
(this means that the quotient of G by the connected component of 1 is compact), the three
properties
G is amenable,
G does not contain a discrete subgroup which is free on 2 generators,
G/r(G) is compact,
are equivalent. This is due to Rickert: Theorem 5.5 in [Ric2], building on [Ric1]; see also
Theorem 3.8 in [Pat]. Recall that the solvable radical r(G) of a locally compact group
G is the largest connected closed normal solvable subgroup of G [Iwa]. (One may define
similarly the amenable radical of G as the largest amenable closed normal subgroup of G;
see Lemma 1 of Section 4 in [Day1] and Proposition 4.1.12 in [Zim].)
This result of Rickert reduces in some sense the problem of understanding the class
of amenable locally compact groups to totally disconnected groups; we believe moreover
that the most important (and difficult) part of the problem is that which concerns finitely
generated groups.
16. Cohomological definitions of amenability. There are various (co)homological
characterizations of amenability.
One is that of Johnson: a group G is amenable if and only if H 1 (ℓ1 (G), M ∗ ) is reduced
to {0} whenever M ∗ is a G-module dual to some Banach G-module M [Joh]. It follows
that the bounded cohomology of an amenable group is always reduced to {0}; this is given
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
11
by Gromov (Section 3.0 in [Gro1]) together with a reference to an unpublished explanation
of Philip Trauber - hence the name “Trauber theorem”.
Another one is in terms of “uniformly finite homology”; it applies to finitely generated
groups, and indeed to metric spaces in a much broader class. Such a space X is not
amenable if and only if the group H0uf (X) is reduced to {0} (in this statement, one may
take R as coefficients, or equivalently Z); this is one way to express that the Følner
condition does not hold in X [BlW1].
It seems also appropriate to quote here a theorem of Brooks: let G be the covering group
of a normal covering M of a compact manifold X; then G is amenable if and only if 0 is
in the spectrum of the Laplace-Beltrami operator acting on the space of square-integrable
functions on M (see [Bro], or the exposition in [Lot]).
There are other conditions in terms of other “coarse” (co)homology theories of the
groups, or in terms of K-theory of appropriate algebras associated with the group (see
various papers by G. Elek, including [Ele2]).
Let us mention that there are interesting cohomological consequences of amenability. For
example, let G be a group which has an Eilenberg-MacLane space K(G, 1) which is a finite
complex; if G is amenable, then G has Euler characteristic χ(G) = 0 (a particular case of
Corollary 0.6 of Cheeger and Gromov [ChGr], who use ℓ2 -cohomology methods, and also
a result of B. Eckmann, who uses other methods [Eck]). Also, let G be the fundamental
group of some closed 4-manifold M; if G is infinite and amenable, then χ(M) ≥ 0 [Eck].
17. Variations on amenability of groups. There are standard variations on the
pseudogroup GG and the notion of amenability.
One is to consider the pseudogroup GG×G associated as in Example 2.(i) with the action
of G × G on G defined by (x, y) ◦ g = xgy −1. It is classical that GG×G is amenable if and
only if GG is amenable. In other words: G has a left invariant mean if and only if G has a
two-sided invariant mean (Lemma 1.1.1 and Lemma 1.1.3 in [Gre1]).
Another variation is to consider the action of G on G \ {1} defined by x ◦ g = xgx−1
and the notion of inner amenability for a group. It is obvious that an amenable group
is inner amenable. Straightforward examples (such as non-trivial direct products of free
groups and amenable groups) show that there are non-amenable groups which are inner
amenable. More on this in [BeHa], [Eff], [GiH1] and [HS2].
A third variation is to consider a subgroup H of G and the pseudogroup GG/H associated
with the natural action of G on G/H. The subgroup H is said to be co-amenable in G
if GG/H is amenable. There is a comprehensive analysis of this notion in [Eym1]; see also
[Bekk], in particular Theorem 2.3. In case G = Fm is a free group of finite rank, a criterion
for co-amenability of a subgroup in terms of cogrowth is given in [Gri1] (see Item 52 below).
One may generalize actions of G on G/H to actions of G on locally compact spaces; coamenability of H is then a particular case of a notion of amenability for actions known as
amenability in the sense of Greenleaf [Gre2].
The notion of amenability for a group and that of co-amenability for a subgroup may
both be viewed as particular cases of a notion for G-mappings, for which we refer to [AnaR].
In case of a group G with the discrete topology, it can be defined as follows. Let X, Y be
12
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
two Borel spaces given with measure classes µ, ν and with actions of G by non-singular
invertible Borel mappings, and let φ : X → Y be a surjective Borel mapping such that
φ∗ (µ) = ν; thus there is a canonical linear isometric mapping by which we identify the
Banach space L∞ (Y, ν) with a closed G-invariant subspace of L∞ (X, µ). Say the mapping
φ is amenable if there exists a G-equivariant linear mapping E : L∞ (X, µ) → L∞ (Y, ν)
which is a conditional expectation, namely which is positive and which restricts to the
identity on L∞ (Y, ν). Example 1: X = G and Y is reduced to one point; then X → Y
is amenable if and only if G is amenable. Example 2: X = G/H for a subgroup H of G
and Y is reduced to a point; then X → Y is amenable if and only if H is co-amenable in
G. Example 3: X = G × Z for a G-space Z (with G acting from the left on itself and
diagonally on the product G × Z); then the projection G × Z → Z is amenable if and only
if the action of G on Z is amenable in the sense of Zimmer [Zim, Section 4.3].
There are other notions, including the three following ones: K-amenability [Cun], weak
amenability à la Cowling-Haagerup [CowH], and a-T-menability à la Gromov. (See 7.A
and 7.E in [Gro3], and [BekCV]; in fact Gromov rediscovered the class of groups having
“Property 3B” of Akemann and Walter in [AkWa].)
II.4. Tarski number of paradoxical group actions
Consider more generally the pseudogroup GG,X associated with a group action G × X → X
(see again Example 2.(i)).
18. Definition. For γ : S → T in GG,X , define the Tarski number of γ as the smallest “number of pieces” n such that there exists a partition S = ⊔1≤j≤n Sj and elements
g1 , . . . , gn in G with γ(x) = gj (x) for all x ∈ Sj , j ∈ {1, . . . , n}.
The Tarski number of a paradoxical GG,X -decomposition
G
X = X1 X2 , γ 1 : X1 → X , γ 2 : X2 → X
as above is the sum of the Tarski number of γ1 and of that of γ2 . It is clear that such a
sum is an integer ≥ 4.
When GG,X is not amenable, we define the Tarski number T (G, X) of the action G×X →
X as the minimum of the Tarski numbers of the paradoxical GG,X -decompositions of X;
when GG,X is amenable, we set T (G, X) = ∞. For a group G acting on itself by left
multiplication, we write T (G) rather than T (G, G).
19. Observation. Let G be a group given together with a subgroup G′ and a quotient
group G′′ . It is straightforward that one has
T (G) ≤ T (G′ )
T (G) ≤ T (G′′ ).
For example, for the first of these inequalities, view G as a disjoint union of cosets of G′ .
Each group G has a finitely generated subgroup G′ such that T (G′ ) = T (G). Indeed,
assuming G to be non-amenable, consider a paradoxical decomposition
G = X1 ⊔ . . . ⊔ Xm ⊔ Y1 . . . ⊔ Yn = g1 X1 ⊔ . . . ⊔ gm Xm = h1 Y1 ⊔ . . . ⊔ hn Yn
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
13
containing m + n = T (G) pieces (where X1 , . . . , Xm , Y1 , . . . , Yn are subsets of G and
g1 , . . . , gm , h1 , . . . , hn are elements of G). Let G′ be the subgroup of G generated by
{g1 , . . . , gm , h1 , . . . , hn }. Set Xi′ = Xi ∩ G′ for all i ∈ {1, . . . , m} and Yj′ = Yj ∩ G′ for
all j ∈ {1, . . . , n}. Then
′
′
G′ = X1′ ⊔ . . . ⊔ Xm
⊔ Y1′ . . . ⊔ Yn′ = g1 X1′ ⊔ . . . ⊔ gm Xm
= h1 Y1′ ⊔ . . . ⊔ hn Yn′
so that T (G′ ) ≤ T (G). With the first inequality of the present observation, this shows that
T (G′ ) = T (G). (One may observe a fortiori that X1′ , . . . , Yn′ are non-empty.) It follows
that one has
T (G) = inf T (G′ )
where the infimum is taken over all finitely generated subgroups G′ of G.
It should be interesting to study how the Tarski number behaves with respect to other
group theoretical constructions such as extensions and HNN-constructions. In particular,
for the latter, we have in mind some presentations of the Richard Thompson’s F group
[CaFP]; recall that F is a group which does not have non-abelian free subgroups, which is
an HNN-extension of itself [BrGe], that F is inner-amenable [Jol], that F has non-abelian
free subsemigroups so that it is not supramenable (see Chapter V below), and that one
does not know whether F is amenable or not.
20. Proposition (Jonsson, Dekker). For a group G, one has T (G) = 4 if and only
if G contains a non-abelian free subgroup.
Proof. For the free group F2 on 2 generators g and h, it is classical that T (F2 ) = 4; see,
e.g., Figure 4.1 in [Wag]. We recall this as follows. Set
A1 = W g
A2 = W g −1
B1 = W h ∪ 1, h−1 , h−2 , . . .
B2 = W h−1 r h−1 , h−2 , . . .
where W (x) denotes the subset of F2 consisting of reduced words on {g, h} with x as first
letter on the left, for x ∈ {g, g −1, h, h−1 }. Then
G G G
G
G
F2 = A1 A2 B1 B2 = A1 gA2 = B1 hB2 .
It follows that T (F2 ) = 4.
Observation 19 shows that T (G) = 4 for any group G containing a subgroup isomorphic
to F2 .
Conversely, let G be a group with T (G) = 4, so that there exist subsets X1 , X2 , Y1, Y2
and elements g1 , g2 , h1 , h2 in G such that
G
G
G G
G
G = X1 X2 Y1 Y2 = g1 X1 g2 X2 = h1 Y1 h2 Y2 .
14
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
Set g = g1−1g2 and h = h−1
1 h2 . Then, one has successively
G
G
X1 = G r gX2 = gX1 gY1 gY2
X1 ⊃ gX1 ⊃ . . . ⊃ g k−1 X1 ⊃ g k Yj
(k ≥ 1 and j = 1, 2)
G
G
X2 = G r g −1 X1 = g −1X2 g −1Y1 g −1 Y2
X2 ⊃ g −1 X2 ⊃ . . . ⊃ g −k+1X2 ⊃ g −k Yj
(k ≥ 1 and j = 1, 2)
so that
g k Y j ⊂ X1 ∪ X2
for all
k ∈ Z , k 6= 0 and j = 1, 2.
for all
k ∈ Z , k 6= 0 and j = 1, 2.
One has similarly
hk Xj ⊂ Y1 ∪ Y2
Hence g and h generate in G a free subgroup of rank 2, by a classical lemma going back essentially to F. Klein, and sometimes known as the “table-tennis lemma” (see, e.g., [Har4]).
The argument above is our rephrasing of the proof of Theorem 4.8 in [Wag].
Proposition 20 is an unpublished work from the 40’s by B. Jonsson (a student of Tarski)
and is a particular case of results of Dekker published in the 50’s. For precise references,
see the Notes of Chapter 4 in [Wag].
Let us also mention that, for a group G containing a non abelian free group and for an
action G × X → X with stabilizers {g ∈ G | gx = x} which are abelian for all x ∈ X, the
corresponding Tarski number is also given by T (G, X) = 4 (Theorem 4.5 in [Wag]).
21. Proposition. For a non-amenable torsion group G, one has T (G) ≥ 6.
Proof. By Proposition 20 we know that T (G) ≥ 5. We assume that T (G) = 5, and we
will reach a contradiction.
The hypothesis implies that there exist subsets X1 , X2 , Y1, Y2 , Y3 and elements g1 , g2 , h1 , h2 , h3
in G such that
G
G G G
G
G
G
G = X1 X2 Y1 Y2 Y3 = g1 X1 g2 X2 = h1 Y1 h2 Y2 h3 Y3 .
Let n denote the order of g + g1−1g2 . As in the proof of Proposition 11, one has
G G
X1 ⊃ gX1 ⊃ . . . ⊃ g n−1 X1 ⊃ g n Y1 Y2 Y3 .
But now g n = 1 and this is absurd. Hence T (G) > 5.
22. Question. Does there exist a group G with Tarski number T (G) equal to 5 ? to 6 ?
More generally, what are the possible values of T (G) ?
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
15
II.5. Følner condition for pseudogroups
Let (G, X) be a pseudogroup of transformations. For a subset R of G and a subset A of
X, we define the R-boundary of A as
)
(
there exists ρ ∈ R ∪ R−1 such that
.
∂R A = x ∈ X r A
x ∈ α(ρ) and ρ(x) ∈ A
23. Definition. The pseudogroup (G, X) satisfies the Følner condition if
for any finite subset R of G and for any real number ǫ > 0
there exists a finite non-empty subset F = F (R, ǫ) of X
such that |∂R F | < ǫ|F |
where |F | denotes the cardinality of the set F .
24. Ahlfors and Følner. Ideas underlying the Følner condition go back at least to
Ahlfors. (Følner does not refer to this work.) Ahlfors defines an open Riemann surface S
to be regularly exhaustible if, for some appropriate metric g in the conformal class defined
by the complex structure of S, there
S exists a nested sequence Ω1 ⊂ Ω2 ⊂ . . . of domains
with smooth boundaries such that n≥1 Ωn is the whole surface and such that
|∂Ωn |g
= 0
n→∞ |Ωn |g
lim
where |Ω|g denotes the area of a domain Ω and where |∂Ω|g denotes the length of its
boundary, both with respect to g. (A lemma of Ahlfors shows that this does not depend
on the choice of g.) These sequences may be used to define averaging processes, as Ahlfors
did first and as Følner did later.
Using this notion, Ahlfors has developped a geometric approach to the Nevanlinna theory
of distribution of values of meromorphic functions, known as Ahlfors theory of covering
surfaces. In particular, he gave a generalization of the second main theorem of Nevanlinna
on defect. (See Section 25 in Chapter III of [Ahl]; see also Chapter XIII in [Nev], Chapter
5 in [Hay], Theorem 6.5 on page 1223 of [Oss], [Sto] and [ZoK].)
Amenability of coverings of Riemann surfaces can also be expressed in terms of Teichmüller spaces [McM2].
25. Theorem. A pseudogroup of transformations is amenable if and only if it satisfes the
Følner condition.
Følner’s original proof (for a group acting on itself by left multiplications) goes back to
1955 [Fol]. The proof has been simplified by Namioka [Nam] (who generalized Følner’s
result to one-sided cancellative semigroups), and extended to group actions by Rosenblatt
[Ros1]; the best place to read it is probably Section 2.1 of [Co1]. In case of a group G
acting by conjugation on G r {1}, the proof can also be found in [BeHa], and it applies
verbatim to an action of G on any set X. All these references use essentially techniques
16
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
of functional analysis. (See also Wagon’s comment about the implication (6) =⇒ (1) in
Theorem 10.11 of [Wag].)
The proof below, in Items 26 and 36, uses completely different techniques.
26. Beginning of the proof of Theorem 25. We prove here the implication “Følner
condition ⇒ existence of an invariant mean”.
Let M(X) denote the set of all means on X, namely of all finitely additive probability
measures on X (see Conditions (fa) and (no) in Definition 4). Let ℓ∞ (X) denote the
Banach space of all bounded functions on X, with the norm of uniform convergence; it is
standard2 that M(X) can be identified with a subset of the unit ball in the dual space of
ℓ∞ (X). It is also standard that the weak∗ -topology makes M(X) into a compact space.
For each finite non-empty subset F of X, we consider the mean
−→
[0, 1]
P(X)
µF :
|A ∩ F |
A
7−→
|F |
in M(X). Consider also the set
ordered by
N = { (R, ǫ) |R ⊂ G is finite, and ǫ ∈ R, ǫ > 0}
(R, ǫ) ≤ (R′ , ǫ′ )
if
R ⊂ R′ and ǫ ≥ ǫ′ .
Notation being as in Definition 23 of the Følner condition (which is now assumed to hold),
(∗)
µF (R,ǫ) (R,ǫ)∈N
becomes a net. By compacity of M(X), this net has a cluster point, say µ (we use the
terminology of [Kel, Chapter 2]). The proof consists in showing that µ is G-invariant; in
other words, given a subset A of X and a transformation γ in G with A ⊂ α(γ), one has
to show that µ γ(A) = µ(A).
We choose a number δ > 0. As µ is a cluster point of the family (∗), there exists
(R, ǫ) ∈ N such that
(i)
(R, ǫ) ≥ ({γ}, δ), i.e., R ∋ γ and ǫ ≤ δ,
(ii)
(iii)
|µF (R,ǫ)(A) − µ(A)| ≤ δ,
µF (R,ǫ) γ(A) − µ γ(A) ≤ δ.
From now on, we write F instead of F (R, ǫ). Define
Ai,i = { a ∈ A | a ∈ F
Ai,o = { a ∈ A | a ∈ F
and γ(a) ∈ F } = A ∩ F ∩ γ −1 (F ),
and γ(a) ∈ ∂R F } = A ∩ F ∩ γ −1 (X r F ),
Ao,i = { a ∈ A | a ∈ ∂R F
Ao,o = { a ∈ A | a ∈
/F
2See
(1905).
and γ(a) ∈ F } = A ∩ (X r F ) ∩ γ −1 (F ),
and γ(a) ∈
/ F } = A ∩ (X r F ) ∩ γ −1 (X r F )
footnote 37 in [NeuJ], where von Neumann refers in turn to Lebesgue’s “Leçons sur l’intégration”
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
17
(think of “inside” for “i” and of “outside” for “o”). Observe that A = Ai,i ⊔Ai,o ⊔Ao,i ⊔Ao,o ,
with the first three sets being finite. Observe also that
(iv)
A ∩ F = Ai,i ⊔ Ai,o
so that
|A ∩ F | = |Ai,i | + |Ai,o|
(v)
γ
induces a bijection Ai,i ⊔ Ao,i → γ(A) ∩ F
so that
(vi)
|γ(A) ∩ F | = |Ai,i | + |Ao,i |
∂R F ⊃ ∂{γ,γ −1 } F ⊃ γ(Ai,o ) ∪ Ao,i
so that
|Ai,o | + |Ao,i | ≤ 2|∂R F | ≤ 2ǫ|F |.
It follows from (iv) to (vi) that
(vii)
|γ(A) ∩ F | − |A ∩ F | ≤ 2ǫ|F |.
Using the definition of the mean µF and the conclusion of the Følner condition, one may
rewrite (vii) as
µF γ(A) − µF (A) ≤ 2ǫ
(viii)
so that one obtains finally
µ γ(A) − µ(A) ≤ 2δ + 2ǫ ≤ 4δ
using (ii), (iii) and (viii). As the choice of δ is arbitrary, this ends the proof of one
implication of Theorem 25.
27. Remark. In case of a locally finite graph X with finitely many orbits of vertices under
the full automorphism group (for example in case of a Cayley graph), Følner condition is
equivalent to the existence of a nested sequence F1 ⊂ F2 ⊂ . . . of finite subsets of the
vertex set X 0 such that ∪n≥1 Fn = X 0 and limn→∞ |∂Fn |/|Fn | = 0; see our Section III.2 for
amenable graphs and for the notation ∂Fn , and Theorem 4.39 in [Soa] for the equivalence.
In the case of a group G acting on a set X, the Følner condition is most often expressed
in a way involving the symmetric difference between a finite subset F of X and its image
gF by some g ∈ G; for the equivalence of this with the analogue of our Definition 23, see
Proposition 4.3 in [Ros1].
For groups, Følner condition implies the existence of Følner sets with extra tiling properties, and this is useful for showing extensions to amenable groups of the Rohlin theorem
from ergodic theory [OrWe].
III. Amenability and paradoxical decompositions for metric spaces
III.1. Gromov condition and doubling condition
Let X be a metric space and let d denote the distance on X.
For S, T ⊂ X, a mapping φ : S → T (not necessarily a bijection) is a bounded perturbation
of the identity if supx∈S d(φ(x), x) < ∞. We will denote by
B(X)
the collection of all these maps. (This would be an example of a “pseudo-semigroup”, but
we will not use this term again below.)
18
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
As in Example 2.(iii), we denote by W(X) the pseudogroup of all bijections, between
subsets of X, which are bounded perturbations of the identity.
For a subset A of X and a real number k > 0, we denote by
Nk (A) = { x ∈ X | d(x, A) ≤ k }
the k-neighbourhood of A in X.
Recall that a metric space is locally finite 3 if its subsets of finite diameter are finite.
28. Definitions. A locally finite metric space X is said to be amenable [respectively
paradoxical] if the pseudogroup W(X) is amenable [resp. paradoxical].
Caution. This definition is not convenient for non-locally finite metric spaces, because
the pseudogroup W(R) is paradoxical. Indeed, the bijections
[
[
γeven :
[2n, 2n + 1[ −→ R
and
γodd :
[2n + 1, 2n + 2[ −→ R
n∈Z
n∈Z
defined by γeven |[2n,2n+1[ (t) = 2t − (2n + 1/2) and γodd |[2n+1,2n+2[ (t) = 2t − (2n + 3/2), are
in W(R) and define a paradoxical decomposition of R.
A notion of amenability for some non-locally finite metric spaces is suggested in Remark
42.
29. Definition. A locally finite metric space X is said to satisfy the Gromov condition if
there exists a mapping φ : X → X in B(X) such that
for all x ∈ X.
φ−1 (x) ≥ 2
This terminology refers in particular to the “lemme 6.17” in [GrLP], introduced there
as “le meilleur moyen de montrer qu’un groupe est non-moyennable”; see also Item 0.5.C1′′
in [Gro3].
30. Definition. The locally finite metric space X satisfies the doubling condition if there
exists a constant K > 0 such that
NK (F ) ≥ 2|F |
for any non-empty finite subset F of X.
It is of course equivalent to ask that there exists a constant k > 0 and a number ǫ > 0
such that
Nk (F ) ≥ (1 + ǫ)|F |
for any non-empty finite subset F of X; indeed, this implies |NK (F )| ≥ 2|F | for any
non-empty finite subset F of X, with K = nk and n an integer such that (1 + ǫ)n ≥ 2.
3The
VI.
terminology “discrete” of the 1999 published version is not appropriate. More on this in Chapter
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
19
31. Bipartite graphs and matchings. Let B = Bip(Y, Z; E) be a bipartite graph
with two classes Y, Z of vertices and with edge set E; by definition of “bipartite”, any edge
e ∈ E is incident with one vertex in Y and one vertex in Z; we consider here simple graphs,
namely graphs without loops and without multiple edges. Recall that, for integers k, l ≥ 1,
a perfect (k, l)-matching of B is a subset M of E such that any y ∈ Y [respectively any
z ∈ Z] is incident to exactly k edges in M [resp. l edges in M].
For a set F of vertices of B, we denote by ∂E F the set of vertices in B which are not in
F , and are connected to some vertex of F by some e ∈ E.
Let again X be a metric space, as earlier in the present section. With two subsets
S, T ⊂ X and a real number K ≥ 0, one associates the bipartite graph BK (S, T ) with vertex
classes S and T , and with an edge connecting x ∈ S and y ∈ T whenever d(x, y) ≤ K;
note that, by definition, S and T are disjoint in the vertex set of BK (S, T ), even if they
need not be as subsets of X. Observe that X is locally finite if and only if BK (X, X) is
locally finite for all K ≥ 0.
32. Theorem. For a locally finite metric space X, the following conditions are equivalent (with B(X) as before Definition 28).
(i) The space X is paradoxical.
(ii) There exists a mapping φ : X → X in B(X) such that φ−1 (x) = 2 for all x ∈ X.
(iii) There exists a mapping φ : X → X in B(X) such that φ−1 (x) ≥ 2 for all x ∈ X
(namely X satisfies the Gromov condition).
(iv) The space X satisfies the doubling condition.
(v) There exists a real number K > 0 for which the bipartite graph BK (X, X) has a
perfect (2, 1)-matching.
(vi) The pseudogroup W(X) does not satisfy the Følner condition.
33. Observations. As there are amenable groups of exponential growth, for example
finitely generated solvable groups which are not virtually nilpotent, Conditions (ii) and
(iii) are not connected to growth, as suggested in [DeSS], but indeed to amenability, as
already observed in our Introduction.
For a recent survey on growth and related matters, see [GriH].
Some of the implications of Theorem 32 may be made more precise. See for example
Proposition 54 below.
34. Proof of Theorem 32.
(i) ⇐⇒ (ii). If X is paradoxical, there exists a partition X = X1 ⊔ X2 and two
bijections γj : Xj → X in W(X). The mapping φ : X → X defined by φ(x) = γj (x)
for x ∈ Xj (j = 1, 2) satisfies (ii).
Conversely, given a mapping φ : X → X as in (ii), one uses the axiom of choice to order
the two points of φ−1 (x) for each x ∈ X, say as φ−1 (x) = γ1−1 (x), γ2−1 (x) . This provides
a paradoxical decomposition involving the mappings γ1 and γ2 .
The implications (ii) =⇒ (iii) =⇒ (iv) are straightforward. Condition (v) is nothing
but a rephrasing of Condition (ii).
20
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
(vi) =⇒ (iv). If W(X) does not satisfy the Følner condition, there exists ǫ > 0 and a
non-empty finite subset R of W(X) such that, for any non-empty finite subset F of X,
one has |F ∪ ∂R F | ≥ (1 + ǫ)|F |. Setting
C =
max
sup d(ρ(x), x)
ρ∈R∪R−1 x∈α(ρ)
(see Definition 1 for the notation α(ρ)), one has a fortiori
NC (F ) ≥ (1 + ǫ)|F |
for any non-empty finite subset F of X.
(i) =⇒ (vi). The contraposition not(vi) =⇒ not(i) may be checked as follows: if the
pseudogroup W(X) satisfies the Følner condition, it is amenable by Proof 26, so that
W(X) is not paradoxical by the straightforward part of the Tarski alternative (Remark
6.(i)).
We have now shown all but the right lowest ⇒ in the following diagram:
(v)
(vi)
m
⇓
(vi) ⇐ (i) ⇔ (ii) ⇒ (iii) ⇒ (iv) ⇒ (v)
For the last implication (iv) =⇒ (v), we follow [DeSS] and call upon a form of the HallRado Theorem. More precisely, with the notation of Theorem 35 below and with k = K,
(iv) implies that ∂E F ≥ 2|F | for any subset F of Y or of Z, so that (v) follows.
All what we will need about the Hall-Rado theorem can be found in [Mir] but, as a
first background, we recommend also the discussion in Section III.3 of [Bol]. (Recall that
“Hall” refers to Philip Hall.)
35 Theorem (Hall-Rado). Let B = Bip(Y, Z; E) be a locally finite bipartite graph and
let k ≥ 1 be an integer. Assume that one has
∂E F ≥ k|F |
∂E F ≥ |F |
for all finite subsets F of Y
for all finite subsets F of Z.
Then there exists a perfect (k, 1)-matching of B.
On the proof. Consider the bipartite graph Bk = B (⊔1≤j≤k Yj , Z; Ek ) where ⊔1≤j≤k Yj
denotes a disjoint union of k copies of Y , and where, for each edge e ∈ E with ends y ∈ Y
and z ∈ Z, there is one edge ej ∈ Ek with ends the vertex yj ∈ Yj corresponding to y and
the vertex z, this for each j ∈ {1 . . . k}.
One the one hand, the hypothesis implies that
∂Ek F ≥ |F |
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
21
for all finite subset F of ⊔1≤j≤k Yj or of Z. On the other hand, there exists a perfect
(k, 1)-matching of B if and only if there exists a perfect (1, 1)-matching of Bk . It follows
that one may assume k = 1 without loss of generality.
By the most usual form of the Hall-Rado theorem, there are subsets MY , MZ of E such
that the edges in MY [respectively in MZ ] are pairwise disjoint, and such that each y ∈ Y
[resp. each z ∈ Z] is incident with exactly one edge in MY [resp. in MZ ]; see, e.g.,
Theorem 4.2.1 in [Mir]. Thus MY ∪ MZ define a spanning subgraph of B whose connected
components are either edges, or simple polygons with a number of edges which is even and
at least 4, or infinite lines. (This argument is standard: see e.g. the middle of page 317 in
[Nas].)
One may color the edges of the latter subgraph in black and white such that each vertex
of B is incident to exactly one black edge. The set of black edges thus obtained is a perfect
(1, 1)-matching of B.
If k = 1, observe that the condition of the Theorem is also necessary for the existence of
a perfect (1, 1)-matching. If k ≥ 2, it is not so (consider a complete bipartite graph with
|Y | = 1 and |Z| = k), despite the statement following Definition 6 of [DeSS].
36. End of proof of Theorem 25. We show here the implication “existence of an
invariant mean ⇒ Følner condition”, or rather its contraposition: we assume that (G, X)
does not satisfy the Følner condition, and we have to prove that X has no G-invariant
mean.
First case: X is a metric space and G is the pseudogroup W(X). Implication (vi) =⇒ (i)
of Theorem 32 shows that X is paradoxical, hence that X is not amenable. The proof of
Theorem 25 is complete in this case.
General case. If (G, X) does not satisfy the Følner condition, there exists a number
ǫ > 0 and a non-empty finite subset R of G such that
|∂R F | > ǫ|F |
for any non-empty finite subset F of X. Define a metric dR on X by
there exists ρ1 , . . . , ρn ∈ R ∪ R−1 such that
dR (x, y) = min n ∈ N
ρn ρn−1 . . . ρ1 (x) . . . is defined and is equal to y
with the understanding that dR (x, y) = ∞ if there exists no such n. One has a posteriori
|N1 (F )| ≥ (1 + ǫ)|F |
for any non-empty finite subset F of X, where the neighborhood N1 (F ) refers to the metric
dR (for the definition of N1 , see before Definition 28). Hence the pseudogroup W(X, dR )
is not amenable by the previous case. As W(X, dR ) ⊂ G, the pseudogroup G itself is not
amenable either.
22
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
37. Definition. Recall that two metric spaces X, Y are quasi-isometric if there exist
constants λ ≥ 1 , C ≥ 0 and a mapping φ : X → Y such that
1
d(x1 , x2 ) − C ≤ d φ(x1 ), φ(x2 ) ≤ λd(x1 , x2 ) + C
λ
for all x1 , x2 ∈ X and
d y, φ(X) ≤ C
for all y ∈ Y .
Recall also that X and Y are Lipschitz equivalent if there exists a constant λ ≥ 1 and a
bijection ψ : X → Y such that
1
d(x1 , x2 ) ≤ d ψ(x1 ), ψ(x2 ) ≤ λd(x1 , x2 )
λ
for all x1 , x2 ∈ X. (See also Item 0.2.C in [Gro3].)
38. Proposition. Let X and Y be two uniformly locally finite metric spaces which are
quasi-isometric. Then X is amenable [respectively paradoxical] if and only if Y is so.
Proof. For uniformly locally finite4 metric spaces, the Gromov condition of Definition 29
is clearly invariant by quasi-isometry.
39. Examples. For each prime p, there are uncountably many 2-generated p-groups
which are amenable and pairwise not quasi-isometric; see [Gri2] for p = 2 and [Gri3] for
p ≥ 2.
40. Examples. There are uncountably many 2-generated torsion-free groups which are
paradoxical and pairwise not quasi-isometric [Bow].
41. Remark. It is a result due independently to Volodymyr Nekrashevych [Nek1] and
Kevin Whyte [Why] that two uniformly discrete non-amenable metric spaces X and Y are
quasi-isometric if and only if they are Lipschitz equivalent. This answers a question of
Gromov (Item 1.A′ in [Gro3]); see also [Pap] and [Bogp] for partial answers.
42. Remark. Let (Ω, dΩ ) be a metric space. A subset X of Ω is a separated net if there
exists a constant r > 0 for which the two following properties hold: (i) dΩ (x, y) ≥ r for
all x, y ∈ X, x 6= y, and (ii) X is a maximal subset of Ω for this property (this implies
dΩ (ω, X) ≤ 2r for all ω ∈ Ω). Such nets exist by Zorn’s Lemma.
If the metric space Ω is “slim and well-behaved” in the sense of [MaMT], for example if
Ω is a Riemannian manifold with Ricci curvature bounded from below and the injectivity
radius of the exponential map positive, then two nets in Ω are quasi-isometric to each other.
(See Theorems 3.3 and 3.4 in [MaMT], as well as [Kan1], [Kan2] and [Nek1], [Nek2].) For
such slim and well-behaved spaces, there are natural notions of amenability and paradoxes,
4We
are grateful to Volker Diekert for having pointed out to us the omission of uniform local finiteness
in the hypotheses in our previous version.
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
23
defined via their nets; this has appeared in several places, including [BlW1]. Proposition
38 carries over to these spaces, by definition.
43. Examples. There are uncountably many Riemann surfaces of constant curvature
−1 which are amenable as metric spaces, and which are pairwise not quasi-isometric [Gri5].
III.2. Graphs as metric spaces, isoperimetric constants
Let X = (X 0 , X 1 ) be a graph with vertex set X 0 and with edge set X 1 (say X has no
loops and no multiple edges, for simplicity). If X is connected, X 0 is naturally a uniformly
discrete metric space, the distance d(x, y) between two vertices x, y ∈ X 0 being the minimal
number of edges in a path between them.
For a disconnected graph X, there are also notions of combinatorial distances. For
example, if X is a subgraph of a connected graph Y which is clear from the context, one
may restrict to X 0 the distance defined on Y 0 as above. One may also set d(x, y) = ∞ for
x, y in different connected components of X.
In this section, we assume that X = (X 0 , X 1 ) is a graph given together with a metric
d : X 0 × X 0 −→ R+ such that d(x, y) is the combinatorial distance whenever x, y are
vertices in the same connected component of X.
44. Definition. A locally finite graph X is said to be amenable or paradoxical if the
metric space X 0 is so in the sense of Definition 28.
For a subset F of X 0 , the boundary ∂E F defined in graph theoretical terms in Item 31
(here E = X 1 ) coincides with N1 (F ) \ F , where N1 (F ) is the neighborhood defined in
metrical terms before Definition 28. We will write
below.
∂F = N1 (F ) \ F
45. Definition. The isoperimetric constant of the graph X is
|∂F |
0
ι(X) = inf
F ⊂ X is finite and non-empty .
|F |
For example, ι(X) = 0 as soon as X is finite.
46. Variations. There are several variations on the definition of the isoperimetric constant
in the literature, because a boundary ∂F could be defined using
either vertices outside F as here (before Definition 45) or in [BeSc] and [McM1],
or vertices inside F as in [Dod] or [CoSa],
or vertices both inside and outside F as in [OrWe, page 24],
or edges connecting vertices inside F to those outside F as in [BiMS] or [Kai1].
24
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
For example, denoting by ∂∗ F the set of edges connecting a vertex of F to a vertex
outside F , there is another isoperimetric constant
|∂∗ F |
0
F ⊂ X is finite and non-empty .
ι∗ (X) = inf
|F |
for the graph X. One has ι∗ (X) ≥ ι(X); if X has maximal degree k, one has also ι∗ (X) ≤
kι(X).
47. Example Let d be an integer, d ≥ 3. For a tree T in which every vertex is of degree
at least d, the isoperimetric constant satisfies the inequality
ι(T ) ≥ d − 2.
If T is regular of degree d, then ι(T ) = d − 2.
Proof. As we have not found a convenient published reference for this very standard fact,
we indicate now a proof. We denote by T (d) the regular tree of degree d.
Let F be a finite subset of the vertex set of T , let X denote the subgraph of T induced
by F , let X1 , . . . , XN denote its connected components, and let Fi denote the vertex set of
Xi , for i ∈ {1, . . . , N}. We claim that
|∂F | ≥ (d − 2)|F | + 2.
Assume first that X is connected. We proceed by induction on |F |. If |F | = 1, then
|∂F | ≥ d = (d − 2)|F | + 2 and the claim is obvious. Assume now that |F | = k ≥ 2; let
y ∈ F be a vertex of X-degree 1, and let Y be the subgraph of X induced by F \ {y}. One
has
∗
|∂F | ≥ |∂(F \ {y})| + d − 2 ≥ (d − 2) |F | − 1 + 2 + d − 2 = (d − 2)|F | + 2
∗
where ≥ holds because of the induction hypothesis. (It is easy to check that |∂F | =
(d − 2)|F | + 2 in case T = T (d).)
Assume now that X has N ≥ 2 connected components, and proceed by induction on N.
As T is a tree, one may assume
of the Fi ’s such that ∂F1 has at most
S the enumeration
one vertex in common with ∂
2≤i≤N Fi . Then
∂F ≥ |∂F1 | + ∂
∗∗
[
2≤i≤N
Fi
∗∗
− 1 ≥ (d−2)|F1 |+2+(d−2)
N
X
i=2
|Fi |+2−1 > (d−2)|F |+2
where ≥ holds because of the induction hypothesis.
It follows that ι(T ) ≥ d − 2, with equality for a d-regular tree.
Recall that a hanging chain of length k in a graph X is a path of length k (with k + 1
vertices, k − 1 so-called inner ones and the two end-vertices) with all inner vertices of
degree 2 in X. It is obvious that, if X has hanging chains of arbitrarily large lengths, then
ι(X) = 0. The following is a kind of converse, for trees.
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
25
48. Example Let T be a connected infinite locally finite tree without end-vertices and let
k be an integer, k ≥ 2.
If T has no hanging chain of length > k, then
1
.
2k
Also ι(T ) = 0 if and only if T has arbitrarily long hanging chains.
ι(T ) ≥
Proof: see the proof of Corollary 4.2 in [DeSS].
Other interesting estimates of isoperimetric constants appear, for example, in Section 4
of [McM1].
49. Definitions. On a locally finite graph X, there is a natural simple random walk
with corresponding Markov operator T . Suppose for simplicity that X is connected and of
bounded
the Hilbert space ℓ2 (X 0 , deg) of functions h from X 0 to C such
P degree. Consider
2
that x∈X 0 deg(x)|h(x)| < ∞, and the bounded self-adjoint operator T defined on this
Hilbert space by
1 X
(T h)(x) =
h(y)
deg(x) y∼x
for h ∈ ℓ2 (X 0 , deg), x ∈ X 0 , where y ∼ x indicates a summation over the neighbours y of
the vertex x. The spectral radius of X is
ρ(X) = sup hh|T hi h ∈ ℓ2 (X 0 , deg) , khk2 ≤ 1
= sup |λ|
λ is in the spectrum of T .
Observe that 1 − T is a natural analogue on X of a Laplacian, so that 1 − ρ(X) is often
referred to as the first eigenvalue of the Laplacian or (more appropriately) as the bottom
of its spectrum.
It is also known that, for a real number λ, the following are equivalent:
1 X
(i)
there exists F : X 0 →]0, ∞[ such that
F (y) = λF (x),
deg(x) y∼x
1 X
(ii)
there exists F : X 0 →]0, ∞[ such that
F (y) ≤ λF (x),
deg(x) y∼x
(iii)
one has λ ≥ ρ(X),
so that (i) and (ii) indicate alternative definitions of the spectral radius. In terms of
the Laplace operator, (i) and (ii) are respectively conditions about (1 − λ)-harmonic and
(1 − λ)-superharmonic functions. (For a proof in terms of graphs, see Proposition 1.5 in
[DoKa]. But there are earlier proofs in the literature on irreducible stationary discrete
Markov chains. The equivalence of (ii) and (iii) is standard; the equivalence with (i) is
more delicate: [Harr] and [Pru].)
26
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
For x, y ∈ X 0 and for an integer n ≥ 0, denote by p(n) (x, y) the probability that a simple
random walk starting at x is at y after n steps. Then one has also
q
ρ(X) = lim sup n p(n) (x, y);
n→∞
in particular, the value of this lim sup is independent on x and y. From this probabilistic
interpretation of ρ(X), one deduces
√ easily that, for a connected graph X which is regular
of degree d ≥ 2, one has ρ(X) ≥ 2 d − 1/d; equality holds if and only if X is a tree.
0
0
(More generally, for any
P transition kernel p : X × X → [0, ∞[ with reversible measure
0
µ : X →]0, ∞[, so that z∈X 0 p(x, z) = 1 and µ(x)p(x, y) = p(y, x)µ(y) for all x, y ∈ X 0 ,
one introduces the Hilbert space ℓ2 (X 0 , µ), and the self-adjoint operator p
T defined by the
2
0
kernel p on ℓ (X , µ). Then the norm of T is again equal to lim supn→∞ n p(n) (x, y).)
50. Lemma (an isoperimetric inequality). For a graph X which is regular of degree
d ≥ 2, one has
1 − ρ(X)
.
ι(X) ≥ 4
ρ(X)
Proof. Let X1 denote the set of oriented edges of X. (If X is finite, the cardinal of X1 is
twice the number of geometric edges of X.) Each e ∈ X1 has a head e+ ∈ X 0 and a tail
e− ∈ X 0 . For a function h ∈ ℓ2 (X 0 , deg) with real values, one has
2
X
X
X
1 X
h(e+ )h(e− ) = khk2 −
h(x)
h(y) =
hh|T hi =
h(e+ ) − h(e− ) .
2
1
0
1
y∼x
e∈X
x∈X
Let now F be a finite non-empty subset of X 0 ,
h ∈ ℓ2 (X 0 , deg) defined by
1
√
d
1
h(x) =
√
2 d
0
e∈X
with boundary ∂F . Consider the function
if x ∈ F
if x ∈ ∂F
otherwise
One has clearly
(∗)
1
ι(X)
khk = |F | + |∂F | ≥ |F | 1 +
.
4
4
2
One has also
2
2
X X
1
1 X
h(e+ ) − h(e− ) =
h(y) − h(x) ≤ |∂F | d .
2
4d
1
y∈∂F x∼y
e∈X
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
27
Together with (∗ ), this implies that
ρ(X) ≥
Taking the infimum over
|∂F |
|F |
|∂F |
hh|T hi
.
≥
1
−
ι(X)
khk2
4|F | 1 + 4
one obtains
ρ(X) ≥ 1 −
1
ι(X)
4
+ ι(X)
4
and the lemma follows.
The previous lemma appears in several places (see No 51 below). It is related to Theorem
3.1 of [BiMS], which is stated in terms of the constant ι∗ (X) of our Item 46, and which
shows that ι∗ (X) ≥ 4(1−ρ(X)). Recently, T. Smirnova-Nagnibeda has improved the latter
to
d2
ι∗ (X) ≥
(1 − ρ(X))
d−1
(the improvement comes from choosing a test-function, playing the role of the function h
in the proof above, which is more efficient than the one chosen in [BiMS]).
For a majoration of ι(X) in terms of 1 − ρ(X) and d (namely for an analogue of the
“Cheeger’s inequality”), see Theorem 2.3 in [Dod] or Theorem 3.2 in [BiMS] (in each case
with normalizations different from ours).
51. Theorem. Let X be a connected graph which is of bounded degree. The following are
equivalent:
(i)
X
is paradoxical
(see Definition 44),
(ii)
ι(X) > 0
(see Definition 45),
(iii)
ρ(X) < 1
(see Definition 49),
(iv)
p(n) (x, y) = o(σ n )
for some σ ∈]0, 1[ and for all x, y ∈ X 0
and they imply that
(v)
the simple random walk on X is transient.
On the proof. The equivalence (i) ⇐⇒ (ii) is a reformulation of Theorem 25 on the Følner
condition.
The equivalence (ii) ⇐⇒ (iii) may be viewed as a discrete analogue of the CheegerBuser inequalities for Riemannian manifolds [Che], [Bus]. For graphs as in the present
theorem, it can be found in [Dod], [Var], [DoKe], [DoKa], [Ger], [Anc], [Kai1]; there are
also similar arguments showing appropriate estimates for finite graphs in several papers by
Alon et alii, quoted in [Lub] (in particular near Propositions 4.2.4 and 4.2.5).
28
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
For (iii) ⇐⇒ (iv) and for equivalence with other conditions, see Theorem 4.27 in Soardi’s
notes on Networks [Soa].
The implication (iii) =⇒ (v) is obvious.
For groups, the equivalence
amenability
⇐⇒
ρ(X) = 1
goes back to the pioneering papers of Kesten [Kes1], [Kes2]. See also [Day3] and the review
in [Woe].
There are other conditions equivalent to (i) to (iv) above, for example in terms of norms
of Markov operators on ℓp -spaces; see [Kai1].
For locally finite graphs which are not necessarily of bounded degree, one has to modify
some of the definitions above. Thus, for a finite set F of vertices of a graph X, one considers
the sum kF k of the degrees of the vertices in F , the number k∂F k of edges with one end in
F and the other end outside F , and the infimum ι̃(X) of the quotients k∂F k/kF k (compare
with Definition 45). For graphs of bounded degree, one has ι̃(X) = 0 ⇐⇒ ι(X) = 0, but
in general5 on may have ι̃(X) = 0 and ι(X) > 0. By a particular case of a result of
Kaimanovich (Theorem 5.1 in [Kai1]), one has ι̃(X) > 0 ⇐⇒ ρ(X) < 1.
Graphs of unbounded degree are also covered by the arguments in [DoKa] and [DoKe].
Graphs give rise to several kinds of algebras, and it is a natural question in each case to
ask how the properties of Theorem 51 translate. For Gromov’s translation algebras (see the
end of 8.C2 in [Gro3]), there is a hint in [Ele1]. For other algebras associated with graphs
(and more generally with oriented graphs), see [KPRR] and [KPR]. Amenable properties
of certain kind of graphs (more precisely of bipartite graphs with appropriate weights) are
also important in the study of subfactors; see various works by S. Popa, including [Pop1]
and [Pop2].
Amenability has of course been one of the most important notions in the theory of
operator algebras since the works of von Neumann. We will not discuss more of this here,
but only refer to [Co2] and [Hel].
5Here
is an example shown to us by Vadim Kaimanovich. Let (hj )j≥1 be a sequence of integers, all at
least 2, and consider first a rooted tree Y in which a vertex at distance n of the root is of degree
k
X
k + 2 if n =
hj for some k ≥ 1,
j=1
3
otherwise.
P
Consider then the graph X obtained from Y by adding, for each vertex x of Y at distance n = kj=1 hj
from the root (for some k), the 12 (k + 1)(k + 2) edges between the successors of x in Y . Then one has
ι(X) > 0 (because Y is a spanning tree for X) and ι̃(X) = 0 (because X contains induced subgraphs
which are complete graphs on k + 2 vertices for k arbitrarily large). One has also ρ(X) = 1.
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
29
IV. Estimates of Tarski numbers
IV.1. From relative growth to Tarski number of paradoxical
decompositions
Let G be a finitely generated group, given as a quotient
π : Fm −→ G
of the free group Fm on m generators s1 , . . . , sm , for some m ≥ 1. The purpose of the
present section is to review notions which will be used in IV.2.
52. Recall: relative growth, spectral radius and isoperimetric constant. Let
ℓ : Fm → N denote the word length on Fm with respect to s1 , . . . , sm . For each integer
k ≥ 0, let σ(ker(π), k) denote the cardinality of the set { w ∈ ker(π) | ℓ(w) = k }. The
relative growth of ker(π) (some authors say “the cogrowth of G”!) is, by definition,
p
αker(π) = lim sup k σ(ker(π), k).
k→∞
√
If ker(π) 6= {1}, it√is easy to check that 2m − 1 ≤ αker(π) ≤ 2m − 1, and one shows
more precisely that 2m − 1 < αker(π) , see [Gri1].
The corresponding Cayley graph (with vertex set G and with an edge between two
vertices x, y if and only if ℓ(xy −1 ) = 1) has a spectral radius given by the formula
√
√
2m − 1
if 1 ≤ α ≤ 2m − 1
m
√
√
ρ =
√
2m − 1
2m − 1
α
+ √
2m − 1 < α ≤ 2m − 1
if
2m
α
2m − 1
[Gri1]. It follows that the three conditions
α = 2m − 1
ρ = 1
G is amenable
are equivalent; the equivalence of the last two is due to Kesten, as already recalled in the
proof of Theorem√51. (In the present setting for the formula giving ρ as a function of α,
one has 1 ≤ α ≤ 2m − 1 if and only if α = 1, if and only if ker(π) = {1}; but the formula
makes sense and
√ is correct for subgroups of Fm which need not be normal, and then the
range 1 ≤ α ≤ 2m − 1 is meaningful.)
53. Isoperimetric constant and doubling characteristic distance. Let X be a
graph, with its set X 0 of vertices viewed as a metric space for the combinatorial distance
d as in Section III.2. A doubling characteristic distance for X is (if it exists) an integer K
for which the doubling condition of Definition 30 holds, namely an integer K such that
|NK (F )| ≥ 2|F |
30
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
for any non-empty finite subset F of X 0 . If the isoperimetric constant ι(X) of Definition
45 is strictly positive, the integer
log 2
KX =
log(1 + ι(X))
is clearly a doubling characteristic distance, where ⌈t⌉ indicates the least integer larger
than or equal to t.
54. Proposition. Let X be a graph with isoperimetric constant ι(X) > 0; define KX
as in the previous paragraph. Then there exists a paradoxical decomposition involving a
partition X 0 = X10 ⊔ X20 and two bounded perturbations of the identity φi : Xj0 → X 0 in
W(X 0 ) such that
(j = 1, 2).
sup d(φj (x), x) ≤ KX
x∈Xj0
Proof: this is a quantitative phrasing of the implication (iv) =⇒ (i) of Theorem 32, and
follows from our Proof 34.
55. Four functions. Let m be an integer, √m ≥ 2. √
√
2m−1
For α ∈] 2m − 1, 2m − 1], set ρm (α) = 2m−1
+
2m
α
For ρ ∈]0, 1], set ι(ρ) =
4 1−ρ
ρ
l
For ι ∈ [0, ∞[, set K(ι) =
√ α
2m−1
∈
i√
2m−1
,1
m
∈ [0, ∞[.
m
log 2
∈ {1, 2, 3, . . . , ∞}, with ⌈. . .⌉ as in 53.
log(1+ι)
i
.
K
−1
.
For K ∈ {1, 2, 3, . . . , ∞}, set bm (K) = m(2m−1)
m−1
Observe that α 7→ ρm (α) and K 7→ bm (K) are increasing, while ρ 7→ ι(ρ) and ι 7→ K(ι) are
decreasing. Observe also that, in the Cayley graph of a group G with respect to a set of m
generators, a ball of radius K has at most bm (K) elements, and precisely bm (K) elements
in case G is free on m generators.
56. Theorem. Let G = Fm /N be a group given as a quotient of the free group on m
generators by a normal subgroup N 6= {1}. Let αG denote the corresponding relative growth
and let ι(X) denote the isoperimetric constant of the corresponding Cayley graph X (see
Definition 45 and Item 52). Using the notation of the previous number, one has:
(i) if αG ≤ α for some α ≤ 2m − 1, the Tarski number of G satisfies
T (G) ≤ 2bm K ι (ρm (α)) ,
(ii) if ι(X) ≥ ι for some ι ≥ 0, then
T (G) ≤ 2bm K(ι) .
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
31
Proof. For (i), one has ι(X) ≥ ι (ρm (α)) by the formula of Item 52 and by the isoperimetric
inequality of Lemma 50, and this implies KX ≤ K (ι (ρm (α))). If φj : Xj0 → X 0 are as in
Proposition 54, one may write Xj0 as a finite disjoint union of the sets
Aj,g = x ∈ Xj0 | φj (x) = gx
for g in the ball B G (KX ) = {g ∈ G | ℓ(g) ≤ KX } (compare with Observation 9), this for
j = 1 and j = 2. As |B G (KX )| ≤ bm (KX ), this ends the proof of (i). The end of the
argument shows also (ii).
57. Comments and examples. Observe that we have argued with the Cayley graph
of G related to the right-invariant distance d(x, y) = ℓ (xy −1 ) on G, so that the leftmultiplications x 7→ gx are bounded perturbations of the identity.
Let us now test the inequalities of Theorem 56.
(i) Let F2 denote the free group of rank 2 and let X denote the Cayley graph of F2 with
respect to some free basis (X is of course a regular tree of degree 4). Kesten [Kes1]√ has
3
computed the spectral value of the corresponding simple random
walk
m as ρ(X) = 2 ≈
l
log 2
0.86603 so that ι(X) ≥ 4 1−ρ(X)
≈ 0.6188. Hence KX = log(1.6188)
ρ(X)
characteristic distance. The resulting estimate
T (F2 ) ≤ 2|B F2 (2)| = 2 2. 32 − 1 = 34
= 2 is a doubling
compares rather badly with the correct value T (F2 ) = 4.
A similar
computation with the Cayley graph Y of F3 with respect to a free basis gives
√
5
ρ(Y ) = 3 ≈ 0.7454, so that ι(Y ) ≥ 1.366. Hence K = 1 is a doubling characteristic
distance. Consequently T (F3 ) ≤ 2|B F3 (1)| = 14. As F3 is a subgroup of F2 one may
improve the previous estimate to
T (F2 ) ≤ 14
by Observation 19.
(ii) Consider again the Cayley graph X of F2 . Its
constant is precisely
m
l isoperimetric
log 2
ι(X) = deg(X)−2 = 2 by Example 47. Hence KX = log 3 = 1 is a doubling characteristic
distance; thus
T (F2 ) ≤ 2|B F2 (1)| = 10,
which compares better than the previous estimate with T (F2 ) = 4.
These computations indicate that some effort should be given to sharpen the isoperimetric inequality of Lemma 50 used above (see Question 62.(a)).
IV.2. Tarski number for Ol’shanskii groups and for Burnside groups
58. On Ol’shanskii groups. We consider first a family of groups investigated in [Ol1].
(See also [Ol2] both for this family and for other ones, discovered by the same author, and
32
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
relevant for the subject discussed here.) For any ǫ > 0, there exists one of these groups
given as a quotient π : F2 ։ G for which the relative growth αG satisfies
√
√
3 < αG ≤ 3 + ǫ
and which is consequently non-amenable. Moreover Ol’shanskii has shown that these
groups do not have any non-abelian free subgroups; thus their Tarski number satisfy
T (G) ≥ 5, and T (G) ≥ 6 in case of torsion groups (Proposition 21). From the relative growth extimate above and from Theorem 56 (see also the first computation of Item
57), one obtains the following.
59. Proposition. There exists a two-generator non-amenable torsion-free group G without non-abelian free subgroup, for which the Tarski number T (G) satisfies
5 ≤ T (G) ≤ 34.
There exist a two-generator non-amenable torsion group G, with all proper subgroups cyclic,
for which
6 ≤ T (G) ≤ 34.
(The constructions of these groups are due to Ol’shanskii.)
60. On Burnside groups. We consider next the Burnside group B(m, n), given as
the quotient of the free group Fm of rank m ≥ 2 by the normal subgroup generated by
{xn }x∈Fm , for an odd integer n ≥ 665. It is obvious that B(m, n) does not contain any
free group not reduced to {1}. It is known that B(m, n) is infinite, indeed of exponential
growth (see VI.2.16 in [Ady1]), and indeed not amenable [Ady2].
From Theorem 3 and the last but one line in6 [Ady2], one has the relative growth estimate
1
where
1
2
+
1
15
+
5.69
57
1
α ≤ (2m − 1) 2 + 15 +
5.69
57
is strictly smaller than, but near, 23 .
√
3
√
3
4
√
3
√
3
9
For m = 2, Theorem 56 shows that one has successively α < 9, hence ρ <
l
m
log 2
0.881, hence ι(X) ≥ 4 1−ρ(X)
≈
0.540,
hence
K
=
= 2, hence finally
ρ(X)
log(1.540)
T (B(2, n)) ≤ 2|B F2 (2)| = 2 2. 32 − 1 = 34.
√ √
√
5
For m = 3, the corresponding computations give α < 3 25, hence ρ < 65 √
+
3
25
0.772, hence ι ≥ 1.181, hence K = 1, hence finally
+
√
3
√9
3
√
3
√25
5
≈
T (B(3, n)) ≤ 2|B F3 (1)| = 14.
6There
are printing mistakes in the English version of [Ady2]. In Theorem 3 of this paper, first the C
should read G, and second the exponent of (2m − 1) should read
"
!#
δR
4
β
1
log2m−1 e 1 +
+
+
2 γR
δR
4γR
(with the largest parenthesis () as above). Also, in the last but one line of the paper,
1
1
replaced by 15
+ 5.69
57 , which is indeed a number strictly smaller than 6 !
1
15
+
6
57
should be
≈
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
33
Let m1 , m2 be such that 2 ≤ m1 ≤ m2 ≤ ∞ and let n be as above. It follows from general
principles on relatively free groups in varieties of groups that B(m2 , n) has a subgroup
isomorphic to B(m1 , n); see [NeuH], Statements 12.62 and 13.41. It is also known that
B(m1 , n) has a subgroup isomorphic to B(m2 , n); see [Sir],
and also § 35.2 in [Ol2]. Thus,
it follows from Observation 10 that one has T B(m, n) = T B(3, n) for any m ≥ 2.
This and Proposition 21 show the following.
61. Theorem. For m ≥ 2 and for n odd and at least 665, the Tarski number of the
Burnside group B(m, n) satisfies
6 ≤ T B(m, n) ≤ 14.
Let us mention that it is unknown whether, for n large, B(m, n) has infinite amenable
quotients. (A question of Stepin, which is Problem 9.7 of [Kou].) Similarly, one could ask
what are the Tarski numbers of non-amenable quotients of these groups.
62. Questions of continuity.
Question (a): given ǫ > 0, does there exist δ > 0 such that, for any quotient group G
of a free group F with spectral radius satisfying ρ(G) < ρ(F ) + δ, one has necessarily an
estimate ι(G) > ι(F ) − ǫ for the isoperimetric constants ? More generally, can one sharpen
the inequality ι(X) ≥ 4 1−ρ(X)
of Lemma 50?
ρ(X)
Question (b): given δ > 0, does there exist η > 0 such that, for any quotient group G of
a free group F with minimal growth rate satisfying ω(G) > ω(F ) − η, one has necessarily
an estimate ρ(G) < ρ(F ) + δ?
(For ω(G), see [GriH]. If the free group F above is of rank
m and is considered together
√
2m−1
with a free basis, recall that ω(F ) = 2m − 1, ρ(F ) = m , and ι(F ) = 2m − 2. The
coefficients ρ(G) and ι(G) are of course taken with respect to the images in G of free
generators in F .)
Assume the two questions above have affirmative answers; then: (i) for a convenient
group G of Ol’shanskii, ι(G) ≥ 2 − ǫ, and K = 1, and consequently T (G) ≤ 10; (ii) for the
Burnside groups B(2, n) of Theorem 61 with n large enough, one would
have ω(G) ≥ 3 − ǫ
(VI.2.16 in [Ady1]), and K = 1, and consequently also T B(m, n) = T B(2, n) ≤ 10
for any m ≥ 2 and n odd large enough.
V. Supramenability
V.1. Supramenability and subexponential growth
63. Definition. A pseudogroup (G, X) is supramenable if the pseudogroup (G(A) , A) defined in Example 2.(iv) is amenable for any nonempty subset A of X.
In case of a pseudogroup W(X), Remark 3.(vi) shows that one may read this definition
in two ways. More precisely, a locally finite metric space X is supramenable if, for any
subspace A of X, one has
34
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
(i) the metric space A is amenable, i.e. the pseudogroup W(A) is amenable,
or equivalently
(ii) the restriction W(X)(A) of the pseudogroup W(X) to A is amenable.
Observe that supramenability of uniformly locally finite metric spaces is invariant by quasiisometry, because of Proposition 38.
A finitely generated group is supramenable if it so as a metric space, for the combinatorial
distance on its Cayley graph with respect to a finite generating set (this definition of
supramenability does not depend on the choice of the generating set).
This notion, due to Rosenblatt [Ros2], carries over to not necessarily finitely generated
groups, and indeed to topological groups, but we will not use this below.
64. Definition. Let X be a locally finite metric space; for a point x ∈ X and a number
r ≥ 0, we denote by βxX (r) the cardinality of the closed ball of radius r around x in X. The
space X is of
p
subexponential growth if
lim sup r βxX (r) = 1
r→∞
p
exponential growth if
1 < lim sup r βxX (r) < ∞
r→∞
p
superexponential growth if
lim sup r βxX (r) = ∞.
r→∞
Observe7 that any of these holds for some x ∈ X if and only if it holds for all x ∈ X, and
also if and only if it holds for any pair (X ′ , x′ ) with X ′ quasi-isometric to X. In particular,
subexponential growth and exponential growth make sense for finitely generated groups,
without any mention of a generating set.
65. Lemma. Inside a locally finite metric space of subexponential growth, any subspace
is also of subexponential growth.
Proof. For a subspace Y of a space X, one may choose in the previous definition the point
x inside Y . Then the lemma follows from the obvious inequality βxY (r) ≤ βxX (r), for all
r ≥ 0.
For historical perspective, let us recall that a simple argument going back to [AdVS]
shows that a finitely generated group which is of subexponential growth is amenable, and
indeed supramenable (Theorem 4.6 in [Ros2]).
As a consequence, one has ι(X) = 0 for any Cayley graph X of a finitely generated group
of subexponential growth. There are further connections between growth and isoperimetry,
due to Varopoulos and others. More precisely, consider for example a finitely generated
group G generated by a finite set S, the corresponding growth function βSG defined by
βSG (n) = | { g ∈ G | the S-word length of g is at most n } |
7Unlike
in some other places of this paper (such as Proof 36), we insist here that the distance between
two points of X is always finite.
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
for all n ≥ 0, and the isoperimetric profile ISG defined by
ISG (n) = max
min
m≤n
F ⊂X 0 , |F |=m
35
|∂F |
for all n ≥ 1, where X 0 denotes the vertex set of the Cayley graph of G with respect to
S (namely X 0 = G!); then, for various classes of groups, there are quite precise estimates
relating the growth function βSG and the isoperimetric profile ISG ; see in particular [CoSa]
and [PiSa].
In our context, the argument of [AdVS] provides the following result.
66. Theorem. A locally finite metric space of subexponential growth is supramenable.
Proof. Let X be a locally finite metric space of subexponential growth. By the previous
lemma, it is enough to show that X is amenable; we will show that X satisfies the Følner
condition.
Consider a finite subset R in the pseudogroup W(X), a point x0 ∈ X and a number
ǫ > 0. Set
'
&
C =
max
sup d(x, ρ(x)) .
ρ∈R∪R−1 x∈α(ρ)
As
lim sup
r→∞
q
r
βxX0 (r) = 1
there exists a strictly increasing sequence of integers (rk )k≥1 such that
βxX0 (rk + C)
= 1.
lim
k→∞
βxX0 (rk )
Set
Fk = ball of radius rk
centered at x0
in X
for all k ≥ 1.
As ∂R Fk ⊂ NC Fk r Fk for all k ≥ 1, one has
|∂R Fk |
= 0
k→∞ |Fk |
lim
so that (Fk )k≥1 is a “Følner sequence” (see Definition 23), and this ends the proof.
The following criterium for graphs will be used in Section V.2. Recall that a metric
space X is long-range connected if there is a constant C > 0 such that every two points x
and y in X can be joined by a finite chain of points
x0 = x , x1 , . . . , xn = y
such that
d(xi−1 , xi ) ≤ C
for all i ∈ {1, . . . , n} (see Item 0.2-A2 in [Gro3]).
36
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
67. Proposition. A connected locally finite graph is supramenable if and only if all its
long-range connected subgraphs are amenable.
Proof of the non-trivial implication. Given a graph X which is not supramenable, we have
to show that there exists a long-range connected subset Z of its vertex set X 0 which is not
amenable (as a metric space, for the combinatorial distance of X).
By hypothesis, there exists a subset Y of X 0 and a mapping φ : Y → Y such that
supy∈Y d(φ(y), y) ≤ C for some constant C ≥ 0, and such that |φ−1(y)| ≥ 2 for all y ∈ Y .
Set Z = NC (Y ), and let (Zi )i∈I be an enumeration of the connected components of Z.
For all i ∈ I, set Yi = Y ∩ Zi . As φ is a C-bounded perturbation of the identity, one has
φ−1 (Yi ) ⊂ Zi , and it follows that φ−1 (Yi ) ⊂ Yi , for all i ∈ I. Hence Yi is paradoxical for
each i ∈ I.
V.2. Examples with trees
Let S2 denote the free semigroup on two generators. From the natural word length, one
defines on S2 a metric making it a uniformly locally finite metric space which is of exponential growth, and indeed paradoxical. Thus, any finitely generated group containing a
subsemigroup isomorphic to S2 has a paradoxical subspace (the group being viewed as a
metric space), and consequently is not supramenable.
68. Question. Does there exist a finitely generated group which is amenable, not supramenable, and without subsemigroup isomorphic to S2 ?
This question is due to Rosenblatt, who conjectured the answer to be negative (see [Ros2],
just after Theorem 4.6 and after Corollary 4.20); he also observed the following alternative
for a finitely generated solvable group: either the group has a nilpotent subgroup of finite
index, and then the group is supramenable, or the group contains S2 as a subsemigroup,
and then the group is not supramenable (Theorems 4.7 and 4.12 in [Ros2]).
However, Question 68 has been answered positively by the second author as follows.
69. Examples [Gri4]. For each prime p, there exist uncountably many finitely generated
p-groups which are
• of exponential growth,
• without any subsemigroup isomorphic to S2 ,
• amenable,
• not supramenable.
On the proof. This involves wreath products8 G = Cp ≀ H, where Cp denotes a cyclic
group of order p and where H is one of the p-groups of intermediate growth constructed
in [Gri2, Gri3].
To show that G is not supramenable, the idea is to construct a paradoxical binary subtree
in an appropriate Cayley graph of G. As a torsion group, G does not contain S2 . The two
other claims are straightforward.
8In
the English translation of [Gri4], the Russian word for “wreath product” has been incorrectly
translated as “amalgamated product”!
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
37
70. Question. Does there exist a finitely generated group which is supramenable and of
exponential growth ?
This question, formulated as Item 12.9.a and Problem C.12 of [Wag], is still open.
One way to make the question more precise is recorded as Problem 16.11 in the Kourovka
Notebook [Kou]: does there exist a finitely generated semigroup S with cancellation having
subexponential growth and such that the group of left quotients G = S −1 S has exponential
growth? (The group of quotients would exist, because the so-called “Ore condition” holds;
see for example Sections 1.10 and 12.4 in [ClPr].) The point is that such a semigroup of
subexponential growth is supramenable and that a group of quotients of a supramenable
semigroup is a supramenable group.
Here is however a straightforward construction.
71. Example. There exists a locally finite metric space which is of superexponential
growth and which is supramenable.
Proof. Consider a sequence (dk )k≥0 of integers ≥ 2 and a sequence (hk )k≥1 of integers ≥ 1.
Let X be a rooted tree in which a vertex at distance n of the root is of degree
k
X
d
if
n
=
hj
k
j=1
2
otherwise
(given the two sequences, this completely defines the tree up to isomorphism).
If lim inf k→∞ hk = ∞, a long-range connected subspace Y of the vertex set of X cannot
satisfy the Gromov condition (compare with Proposition 35 above, i.e. with Corollary 4.2
of [DeSS]). It follows from Proposition 67 that X is supramenable.
Now the growth sequence of X with respect to the root, say x0 , satisfies
βxX0 (n + 1) ≥
k
Y
j=0
dj
for
n =
k
X
hj ,
j=1
so that, if the sequence (dk )k≥0 is increasing rapidly enough, one has
q
lim sup m βxX0 (m) = ∞
m→∞
P
j
!, then
h
and X is of superexponential growth. For example, if dj =
i
i=1
βxX0 (n + 1) ≥ dk = n!
p
P
whenever n = kj=1 hj , and this implies lim supm→∞ m βxX0 (m) = ∞ by Stirling’s formula.
72. Variation on the previous example. There exists a graph of bounded degree which
is of exponential growth and which is supramenable.
38
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
Proof. Consider a rooted tree X in which a vertex at distance n of the root is of degree
2 if (k − 1)k ≤ n < k 2 for some k ≥ 1,
3 if k 2 ≤ n < k(k + 1) for some k ≥ 1.
The growth function of X with respect to the root satisfies
k(k+1)
k(k+1)
2 2 ≤ β X k(k + 1) ≤ 3 2
for all k ≥ 1, so that X is clearly of exponential growth. Example 48 implies that X is
supramenable.
73. Question. Let G and H be two finitely generated groups which are supramenable; is
the product G × H supramenable ?
This question appears in [Ros2] (just before Proposition 4.21), and the answer is still
unknown. Here is however an example, for which we are grateful to Laurent Bartholdi.
74. Example. There exist two supramenable locally finite metric spaces X, Y such that
the direct product X × Y is not supramenable, for the metric defined by
dX×Y (x1 , y1), (x2 , y2 ) = dX (x1 , x2 ) + dY (y1 , y2).
Proof. Let (hk )k≥1 be a strictly increasing sequence of integers ≥ 1. Let X be a rooted
tree in which a vertex at distance n of the root is of degree
2k+1
2k
X
X
3
hj ≤ n <
hj
for some k ≥ 0,
if
j=0
j=1
2
otherwise
P2k
(with j=0 hj = 0 for k = 0). And let Y be a rooted tree in which a vertex at distance n
of the root is of degree
2k+1
2k+2
X
X
3
if
hj ≤ n <
hj
for some k ≥ 0,
j=1
j=1
2
otherwise.
Observe that both X and Y are supramenable, because each of their infinite connected
subgraphs has arbitrarily large hanging chains. Observe also that, for each integer n, there
is either in X or in Y a vertex of degree 3 at distance n of the relevant root. It follows that
the product of the two metric spaces defined by X and Y , for the distance dX×Y defined
above, contains a paradoxical tree. Consequently, X × Y is not supramenable.
75. Paradoxical subtrees in paradoxical graphs. It is known that a paradoxical
graph contains a paradoxical tree [BeSc]. It is unknown whether a connected paradoxical
graph necessarily contains a paradoxical tree which is spanning, i.e. which contains all
vertices of the original graph (this is Problem 2 in Section 4 of [DeSS]).
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
39
However, Benjamini and Schramm have shown that, if X is a paradoxical graph with
ι(X) ≥ n for some integer n ≥ 2, then X has a spanning forest of which every connected
component is a tree with one vertex of degree n − 1 and all other vertices of degree n + 1.
This implies that X has a paradoxical spanning tree.
76. A question of V. Trofimov. This appears as Problem 12.87 in the Kourovka
Notebook [Kou]. Let X be a connected undirected graph without loops and multiple edges
and suppose that its automorphism group Aut(X) acts transitively on the vertices. Is it
true that one of the following holds?
(i) the stabilizer of a vertex of X is finite,
(ii) the action of Aut(X) on the vertices of X admits a non-trivial imprimitivity system σ
with finite blocks for which the stabilizer of a vertex of the factor-graph X/σ in Aut(X/σ)
is finite,
(iii) there exists a natural number n such that the graph, obtained from X by adding
edges connecting distinct vertices the distance between which in X is at most n, contains
a tree all of whose vertices have valence 3.
If the answer to this question was positive, this would imply that a graph of subexponential growth having a transitive group of automorphisms is essentially a Cayley graph
of a group.
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VI. Comments and corrections (March 2016)
On the article by Deuber, Simonovits and Sós, and their terminology of exponential growth. There is an annotated version of the 1995 version, dated 2004 [DeSS–04],
and an exposition of related material [ElSo–05]. In the annotated version, the authors observe that their terminology of exponential growth is not standard in the group theory
literature.
On No. 11 and elementary amenable groups. Let B0 denote the class consisting of
all finite groups and the infinite cyclic group. The following fact was shown by Chou and
refined by Osin [Osin–02, Theorem 2.1]: the class EG of elementary amenable groups is
the smallest class of groups which contains the trivial group {1}, which is closed under
taking direct limits, and which is such that a group G is in EG whenever there exists an
extension {1} → N → G → Q → {1} with N ∈ EG and Q ∈ B0 .
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
45
Nekrashevych [Nekr] has recently discovered examples of finitely generated infinite groups
that are simple, periodic, and of intermediate growth. In particular they are amenable (because of intermediate growth) and not elementary amenable (either because infinite finitely
generated simple groups cannot be elementary amenable, or because infinite finitely generated periodic groups cannot be elementary amenable – both observations go back to
Chou).
On No. 13 and the space of marked groups. The space of marked groups has
received a considerable amount of attention. Besides the articles cited in No. 13, we
indicate [ChGu–05], [CoGP–07], and [BCGS–14].
See also [WeWi, Corollary 6.25] for an original proof of the existence of finitely generated
groups that are amenable and are not elementary amenable: in the appropriate space of
marked groups, the set of amenable groups is Borel and the set of elementary amenable
groups is not.
On No. 14 and subexponentially amenable groups. There are amenable groups
(i.e. groups in AG) that are not subexponentially amenable (i.e. not in BG). Indeed, the
so-called Basilica group was first shown to be not in BG [GrZu–02], and later shown to be
amenable [BaVi–05]. The method of Bartholdi and Virag was streamlined and generalized
in [Kaim–05]. Further examples can be found in [Ersc–06], [Brie–09], [BaKN–10]. The
finitely generated amenable simple groups that appear in [JuMo–13] are also amenable
and not subexponentially amenable.
On Nos. 15 and 24, Ahlfors’ notion of regular exhaustion, and Bogolyubov’s
ideas on amenability for topological groups. In [Roe–88], there is a discussion of
regular exhaustion, introduced by Ahlfors in 1935 [Ahl]. In [GrHa], there is a discussion of
Bogolyubov’s ideas on amenability, in his 1939 article [BogL] which went almost unnoticed.
There is an exposition of basic material on amenability of topological groups (and the
important case of locally compact groups) in Chapter II.G of [BeHV–08].
On No. 15, amenability of groups and cellular automata. Let G be a group and
A a finite set. Equip AG = {u : G → A} with its prodiscrete topology (i.e. the topology
of pointwise convergence) and with the shift action of G defined by gu(h) := u(g −1 h) for
all g, h ∈ G and u ∈ AG . A cellular automaton over G is a continuous map τ : AG → AG
that is G-equivariant, i.e., satisfies τ (gu) = gτ (u) for all g ∈ G and u ∈ AG . For two maps
u, v ∈ AG write u ≈ v if they coincide outside of a finite subset of G. It is clear that ≈
is an equivalence relation. A map τ : AG → AG is said to be pre-injective if its restriction
to each ≈-equivalence class is injective. Then the Garden of Eden theorem [CeMS–99]
(see also [Gro–99b]), originally established by Moore [Moo–63] and Myhill [Myh–63] for
G = Z, states that a cellular automaton over an amenable group is surjective if and only
if it is pre-injective. It follows from[Bart–10] that if a group G is non-amenable then there
exist cellular automata over G that are surjective but not pre-injective. Thus, the Garden
of Eden theorem yields a characterization of amenability for groups in terms of cellular
automata. For more on the Garden of Eden theorem (consequences and variations) we
refer to [CeCo–10].
46
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
On No. 17, inner amenability, and coamenability. An old question on inner amenability from [Eff] has been solved in [Vaes–12].
Several claims of Theorem 5 in [BeHa] have to be corrected, as in [Stal–06, Section 3].
For the notion of coamenability of a subgroup of a group, see also [MoPo–03] and
[Pest–03].
On Definition 18, Question 22, and Tarski numbers. There are other definitions
of Tarski numbers, see [ErSP–15, Appendix A]. Since 1999, there has been some progress
on understanding of Tarski numbers. For example, there are 2-generated non-amenable
groups with arbitrarily large Tarski numbers, there are groups which we know have Tarski
number exactly 5, or 6, and every number τ ≥ 4 is the Tarski number of some faithful
transitive action of a finitely generated free group. See [OzSa–13], [ErSP–15], [Gola–a],
and [Gola–b].
On Definition 29 and the reference [GrLP]. In its second edition, this book has been
considerably expanded [Gro–99a].
On Definition 30 and the terminology “doubling condition”. It is unfortunate
that this terminology is used in several incompatible meanings. Some authors use them in
our sense, see e.g. [Kapo–02]. But many more authors use them in a completely different
meaning, most often for metric spaces with measures, and occasionally for metric spaces
as such; see e.g. [Gro–99a], [Hein–01] and [LoVi–07].
More precisely, a metric space X is called doubling if there exists a constant C > 0 such
that, for all d > 0, any subset of X of diameter at most d can be covered by C subsets of
X of diametrer at most d/2 [Hein–01, Definition 10.13]. The doubling metric spaces are
precisely the spaces of finite Assouad dimension; compare [Hein–01, Definition 10.15].
In retrospect, our terminology for the notion of Defintion 30 was unfortunate.
A change of terminology there should have some effect on the terminology “doubling
characteristic distance” of No. 53.
On Section III.1, discrete and locally finite metric spaces, and uniform notions.
In the published version, just before Defintion 28, we have unfortunately used the word
“discrete” for what should be “locally finite”.
For a metric space (X, d), the four following properties should not be confused:
• (X, d) is discrete if, for every x ∈ X, there exists δx > 0 such that d(x, y) ≥ δx
for all y ∈ X r {x}; note that (X, d) is discrete if and only if the topology on
X defined by d is discrete;
• (X, d) is uniformly discrete if there exists δ > 0 such that d(x, y) ≥ δ for all
x, y ∈ X such that x 6= y;
• (X, d) is locally finite if every subset of X of finite diameter is finite;
• (X, d) is uniformly locally finite if, for every D ≥ 0, there exists a constant C
such that every subset of X of diameter at most D has at most C elements.
Note that a discrete metric space is locally finite if and only if it is proper, i.e. if and only
if its closed balls are compact. Note also that a uniformly locally finite metric space need
not be uniformly discrete (example: the subspace {n ∈ Z | n ≥ 1} ∪ {n + 2−n | n ≥ 1} of
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
47
the real line). Let X be a connected graph, and (X 0 , d) its vertex set together with the
combinatorial distance function; then X 0 is always uniformly discrete, and X 0 is locally
finite [respectively uniformly locally finite] if and only if X is locally finite [respectively of
bounded degree].
In the present version (unlike in the published version), we have used “locally finite”
instead of “discrete” in Nos. 28 to 36. Proposition 38, on invariance of amenability by
quasi-isometries, holds for uniformly locally finite metric spaces.
An example in [DiMW]. Here is the last example of [DiMW], showing that the hypothesis
of uniform local finiteness cannot be deleted in Proposition 38.
Consider the graph X defined as follows:
· it has vertices (n, 1) for all n ∈ Z with n ≥ −1,
and (n, k) for all n ≥ 1 and k ∈ N such that 2 ≤ k ≤ 2n ;
· it has edges connecting (n, 1) to (n + 1, 1) for all n ∈ Z with n ≥ −1,
and (n, 1) to (n, k) for all n ≥ 1 and k ∈ N such that 2 ≤ k ≤ 2n .
Observe that X is locally finite and not uniformly locally finite. Denote by X 0 the vertex
set of this graph, considered as a metric space for the combinatorial metric, say d; observe
that X 0 is a uniformly discrete metric space. Let Φ : X 0 7−→ X 0 be the mapping defined
as follows:
· Φ(−1, 1) = Φ(0, 1) = (−1, 1);
· Φ(n, k) = Φ(n, k + 2n−1 ) = (n − 1, k) for all n ≥ 1 and k with 1 ≤ k ≤ 2n−1 .
Then Φ is a bounded perturbation of the identity and all its fibers have two elements, in
other terms
d(Φ(x), x) ≤ 3 and |Φ−1 (x)| = 2 for all x ∈ X 0 .
Hence X 0 satisfies the Gromov condition of Definition 29, and X 0 is paradoxical by Theorem 32. Consider also the subgraph of this graph with vertex set Y 0 = {(n, 1) | n ∈ Z, n ≥
−1} and edges connecting (n, 1) to (n + 1, 1) for all n ∈ Z with n ≥ −1; observe that this
graph is a half line, and that the corresponding discrete metric space Y is amenable.
There is an obvious quasi-isometry from X 0 to Y 0 , that maps (n, k) to (n, 1) for all
(n, k) ∈ X 0 . Yet X 0 is paradoxical and Y 0 amenable.
On metric spaces for which amenability could make sense. Logically, Definitions
28, 29, 30 would make sense for every metric space. In Theorem 32 implications
(v)
(vi)
m
⇓
(vi) ⇐ (i) ⇔ (ii) ⇒ (iii) ⇒ (iv)
would still be correct. But local finiteness is important for our proof of (iv) ⇒ (v). Indeed,
Hall-Rado Theorem, No. 35, does not carry over to arbitrary bipartite graphs, as the
following example shows.
48
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
Consider the bipartite graph B = B(Y, Z; E) of which the vertex set is the disjoint union
of two sets given with bijections with the integers, say α : Y −→ N and β : Z −→ N
(recall that N contains 0), and the edge set is
E = {(y, z) ∈ Y × Z | α(y) = β(z) + 1} ⊔ {(y, z) ∈ Y × Z | α(y) = 0}.
in other terms, α−1 (n + 1) ∈ Y has a unique neighbour β −1 (n) for all n ≥ 0, and the set
of neighbours of α−1 (0) is the whole of Z. Then |∂E F | ≥ |F | for all every finite subset
F of either Y or Z, but there does not exist any (1, 1)-matching of B, i.e. B satisfies the
hypothesis of Hall-Rado Theorem, but not the conclusion.
We wish to stress that local finiteness is important for our Theorem 32, and than an
even stronger condition, uniform local finiteness, is important for Proposition 38.
On Remark 42 and metric spaces for which amenability does make sense. A
subspace Y of a metric space X is cobounded if supx∈X d(x, Y ) < ∞. A subspace Y of
X which is both uniformly discrete and cobounded is a net, as defined in Remark 42,
also called a metric lattice [CoHa, Section 3.C]. An application of Zorn Lemma shows
that every metric space contains metric lattices. A metric space X is uniformly coarsely
proper if there exists R0 ≥ 0 such that, for every R ≥ 0, there exists an integer N such
that every ball of radius R in X can be covered by N balls of radius R0 , equivalently if
X contains a uniformly locally finite metric lattice (for this equivalence, and others, see
[CoHa, Proposition 3.D.16]).
For uniformly coarsely metric spaces, amenability makes good sense, and is invariant by
coarse equivalence, in particular is invariant by quasi-isometry.
On Lemma 50 and an isoperimetric inequality. A better inequality than that of
the end of No. 50 appears in [Moha–88, Theorem 3.1(b)]. Particularized to our situation
(regular graph) and with our notation, it reads
ι∗ (X) ≥
d2
(1 − ρ(X))
d − 1 − (1 − ρ(X))
(note that ρ(X) ≤ 1). We are grateful to T. Nagnibeda for this reference.
On No. 52 and the formula expressing ρ in terms of α. For an elaboration of this
formula, see [Bart–99]. There, Bartholdi establishes an equality between two generating
functions, one related to numbers of circuits of length n in some appropriate graph, and the
other related to numbers of circuits of length n with no backtracking in the same graph;
the formula of No. 52 is then obtained as the equality between the radii of convergence of
these two formal power series.
On No. 61 and infinite amenable quotients of Burnside groups. The existence of
such quotients still appears as an open question in a more recent edition of the Kourovka
Notebook [Kour–15].
On Question 62.b and amenable quotients of Fm having large growth rate. Let
G be a finitely generated group and S a finite generating set. For every integer k ≥ 0,
denote by βSG (k) the number of elements g ∈ G that can be written as products of at
AMENABILITY AND PARADOXICAL DECOMPOSITIONS
49
most k elements inpS ∪ S −1 . The exponential growth rate of the pair (G, S) is the limit
ω(G, S) = limk→∞ k βSG (k); the existence of the limit follows from the submultiplicativity
of the sequence (βSG (k))k≥0 .
The answer to the analogue for exponential growth rates of Question (b) in No. 62 is
negative; indeed the following is shown in [ArGG–05]. Consider an integer m ≥ 2 and the
free group Fm of rank m; there exists a sequence (Nn )n≥1 of normal subgroups of Fm such
that the quotient group Gn := Fm /Nn is amenable for all n ≥ 1 and limn→∞ ω(Gn , Sn ) =
2m − 1, where Sn stands for the image of a free generating set of Fm by the canonical
projection of Fm onto Gn . Moreover, the sequence (Nn )n≥1 can be chosen such that Gn is
abelian-by-nilpotent for all n ≥ 1, or metabelian-by-finite for all n ≥ 1.
The minimal growth rate of a finitely generated group G is the number ω(G) = inf S ω(G, S),
where the infimum is taken over all finite generating sets S of G. For a group which can
be generated by m elements, it is standard that 1 ≤ ω(G) ≤ 2m − 1, with equality on the
right if and only if G is free of rank m. For this and more on minimal growth rates, see
[GriH].
As much as we know, Question 62.b itself, on ω(G), is still open: For m ≥ 2, does
there exist a sequence (Nn )n≥1 of normal subgroups of Fm such that the quotient group
Gn = Fm /Nn is amenable for all n ≥ 1 and limn→∞ ω(Gn ) = 2m − 1?
A misprint in No. 62. Watch out: with the normalization
chosen in √
our paper (the
√
same as in the original paper [Kest–59]), ρ(Fm ) = 2m − 1/m; the value 2m − 1 of the
published version of our paper is a misprint.
On No. 63, and equivalent definitions of supramenability for groups. The following is established (among other things) in [KeMR–13]. For a group G (which need not
be finitely generated), the following conditions are equivalent:
• G is supramenable,
• every cocompact action of G on a locally compact Hausdorff space admits a nonzero invariant Radon measure,
• there is no injective Lipschitz map from the free group of rank two to G.
A map f from a group G to a group H is Lipschitz if, for every finite subset S of G, there
exists a finite subset T of H such that f (x)f (y)−1 ∈ T for every x, y ∈ G with xy −1 ∈ S.
On No. 64, and types of growth for locally finite metric spaces. The notions of this
definition, i.e. subexponential growth, exponential growth, and superexponential growth,
should be restricted to uniformly locally finite metric spaces (rather than to discrete metric
spaces as in the 1999 publication).
Note that they are meaningfull for locally finite metric spaces, but in this context they
are not invariant by quasi-isometry. See the example given above in the comment on
Proposition 38, or [CoHa, Example 3.D.7].
On isoperimetric profiles, as in the remark that follows Lemma 65. For precise
estimates of various isoperimetric profiles – or, equivalently, of the corresponding Følner
functions – see [Ersc–03].
50
T. CECCHERINI-SILBERSTEIN, R.I. GRIGORCHUK, AND P. DE LA HARPE
On the terminology used for Proposition 67: coarsely connected metric spaces.
Rather than “long-range connected”, here is the terminology used in various places, including [CoHa]: a metric space X is coarsely connected if there exists a constant C > 0
such that for every pair of points (x, x′ ) in X, there exists a finite sequence of points
(x0 = x, x1 , . . . , xn = x′ ) in X such that d(xi−1 , xi ) ≤ C for all i ∈ {1, . . . , n}. The point
is that coarse connectedness is invariant by coarse equivalence.
On Questions 70 and 73 on supramenable groups. At the best of our knowledge,
these two questions are still open:
• does there exist a supramenable group of exponential growth? (Rosenblatt’s
question);
• is it true that the direct product of two supramenable groups is always supramenable?
On Question 76, of Trofimov. This appears still as an open question in a more recent
edition of the Kourovka Notebook [Kour–15].
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Dipartimento di Ingegneria, Università del Sannio, C.so Garibaldi 107, 82100 Benevento, Italy
E-mail address: tceccher@mat.uniroma3.it
Mathematics Department, Texas A & M University, College Station, TX 77843-3368,
USA
E-mail address: grigorch@math.tamu.edu
Section de Mathématiques, Section de mathématiques, 2–4 rue du Lièvre C.P. 64. CH
1211 Genève 4. Switzerland
E-mail address: Pierre.delaHarpe@unige.ch
| 4 |
Reachability problems for PAMs
⋆
Oleksiy Kurganskyy1 and Igor Potapov2
1
Institute of Applied Mathematics and Mechanics, NAS of Ukraine
Department of Computer Science, University of Liverpool, UK
arXiv:1510.04121v1 [cs.NA] 14 Oct 2015
2
Abstract. Piecewise affine maps (PAMs) are frequently used as a reference model to show the openness of the reachability questions in other
systems. The reachability problem for one-dimentional PAM is still open
even if we define it with only two intervals. As the main contribution of
this paper we introduce new techniques for solving reachability problems
based on p-adic norms and weights as well as showing decidability for
two classes of maps. Then we show the connections between topological
properties for PAM’s orbits, reachability problems and representation of
numbers in a rational base system. Finally we show a particular instance
where the uniform distribution of the original orbit may not remain uniform or even dense after making regular shifts and taking a fractional
part in that sequence.
Keywords: Reachability problems, piecewise affine maps (PAMs), βexpansion, p-adic analysis
1
Introduction
The simplification of real programs shows that there is a number of quite basic
models/fragments for which we have fundamental difficulties in the design of
verification tools. One of them is the model of iterative map that appears in
many different contexts, including discrete-event/discrete-time/hybrid systems,
qualitative biological models, chaos-based cryptography, etc [1, 8, 16, 23].
The one-dimensional affine piecewise iterative map is a very rich mathematical object and at the same time one of the simplest dynamical system producing
very complex and sensitive effects. A function f : Q → Q is a one-dimensional
piecewise-affine map (PAM) if f is of the form f (x) = ai x + bi for x ∈ Xi where
all coefficients ai ,bi and the extremities of a finite number of bounded intervals
Xi are rational numbers. Let us consider the sequence of iterations starting from
a rational point x: x, f (x), f 2 (x) = f (f (x)), and so on. The reachability in PAM
is a problem to decide for a given f and two rational points x and y whether y
is reachable from x. In other words, is there an n ∈ N such that f n (x) = y?
The decidability of the reachability problem for one dimensional piecewiseaffine map is still an open problem, which is related to other challenging questions
in the theory of computation, number theory and linear algebra [7, 14, 15]. This
⋆
This research is supported by EPSRC grant Reachability problems for words, matrices and maps (EP/M00077X/1)
model plays a crucial role in the recent research about verification of hybrid
systems [2, 3], timed automata [2] control systems [9, 10], representation of numbers in a rational base (β-expansions) [19, 21], discounted sum automata [11]. In
particular PAM is often used as a reference model to show the openness of the
reachability questions in other systems. It also has a very natural geometrical interpretation as pseudo-billiard system [17] and Hierarchical Piecewise Constant
Derivative (HPCD) system [3]. The reachability problem for one-dimensional
PAM is still open even if we define it with only two intervals [2, 3, 5, 12].
The primary goal of this paper is to demonstrate new approaches for solving
reachability problem in PAMs, connecting reachability questions with topological
properties of maps and widening connections with other important theoretical
computer science problems. First, we show new techniques for decidability of the
reachability problem in PAMs based on p-adic norms and weights. We illustrate
these techniques showing decidability of two classes of PAMs. The algorithm in
Theorem 1 solves point to point reachability problem for two-interval injective
PAM under the assumption that a PAM has bounded invariant densities. While
our numerical experiments shows that the sequence of invariant densities converge to smooth functions it is not yet clear whether it holds for all PAMs or if
not whether this property can be algorithmically checked.
Following the proposed approach based on p-adic weights in Theorem 2 we
define another fragment of PAMs for which the reachability problem is decidable. In particularly we remove the condition on bounded invariant densities and
injectivity of piecewise-affine map and consider a PAM f with a constraint on
linear coefficients in affine maps. This class of PAMs is also related to encoding
of rational numbers in the rational base (β-expansions). The decidability of the
point-to-point problem for this class is shown in Theorem 2 and decidability of
point-to-set problem for the same class can give someone an answer to the open
problem related to β-expansions.
Then we establish the connections of topological properties for PAM’s orbits
with reachability problems and representation of numbers in a rational base system. We show that the reachability problems for above objects tightly connected
to questions about distribution of the fractional parts in the generated sequences
and moreover about distribution of the fractional part after regular shifts.
2
Preliminaries and Notations
In what follows we use traditional denotations N, Z, Z+ = {0, 1, 2, . . .}, P, Q
and R for sets of natural, integers, positive integers, primes, rational and real
numbers, respectively. Let us denote by S 1 = Q/Z the unit circle which consists
only rational numbers. By {x}, ⌊x⌋ and ⌈x⌉ we denote the fractional part 3 of a
number, floor and ceiling functions.
Let Y be a set of numbers and x is a single number, then we define their
addition and multiplication as follows: Y + x = x + Y = {x + y|y ∈ Y } and
3
It will be clear from the context if brackets are used in other conventional ways, for
example, to indicate a set of numbers.
xY = Y x = {xy|y ∈ Y }. The application of a function f : X → Y to a set
X ′ ⊆ X is defined as f (X ′ ) = {f (x)|x ∈ X ′ }. If f ⊆ X × Y is a nondeterministic
map, i.e. S
f : X → 2Y and x ∈ X, X ′ ⊆ X, we define f (x) = {y|(x, y) ∈ f } and
′
f (X ) = x∈X ′ f (x).
p-adic norms and weights: Let us consider an arbitrary finite set of prime
numbers F = {p1 , p2 , . . . , pk } ⊂ P in ascending order and define the product of
prime numbers from F by m = p1 p2 . . . pk . Let x be a positive rational number
that Q
can be represented by primes from a set F. Then its prime factorization is
x = p∈F pαp , where αp ∈ Z, p ∈ F.
Any nonzero rational number x can be represented by x = (pαp r)/s, where
p is a prime number, r and s are integers not divisible by p, and αp is a unique
integer. The p-adic norm of x is then defined by |x|p = p(−αp ) . The p-adic weight
of x is defined as kxkp = logp (|x|p ), i.e. kxkp = −αp . The following properties
of p-adic weights are directly follows from the properties of p-adic norm:
kxkp = kykp ⇒ kx + ykp ≤ kxkp .
(1)
kxkp < kykp ⇒ kx + ykp = kykp ,
(2)
kx · ykp = kxkp + kykp ,
(3)
kxr kp = rkxkp ,
(4)
If there is a prime p ∈
/ F such that kxkp > 0, then we define kxkm = +∞,
otherwise kxkm = max kxkp .
p∈F
By m-weight and m-vector-weight of x in respect to a set F we denote kxkm
and (kxk)m =(kxkp1 , kxkp2 , . . . , kxkpk )T respectively. Informally speaking the
m-weight of a number x (if kxkm > 0) is the number of digits after the decimal
point in the representation of x in base m, i.e. x can be written as x = y·m−kxkm ,
where y is an integer which the last digit in the m-ary representation is non zero.
If kxkm ≤ 0, then x is an integer number.
Without loss of generality let us consider from now on only such x ∈ X for
which kxkm < +∞. Alternatively if kxkm = +∞, it is enough to change the set
F = {p1 , . . . , pk } in order to fulfill the requirements of kxkm < +∞.
Lemma 1. For all rational x ∈ [0, 1] with an upper bound a ∈ Z+ on kxkm , i.e.
kxkm < a, there is a lower bound b ∈ Z based on a and F such that kxkp ≥ b for
all p ∈ P.
P
Proof. Let us denote two sums of weights: α = − p∈P,kxkp ≤0 kxkp and β =
P
p∈F,kxkp ≥1 kxkp . Assuming that 2 is the smallest possible prime number and
pk is the largest number in F, we have the following inequality:
2α
pka
k
≤
2α
pβ
k
Then −α ≥ −kalog2 pk and if b = −α we have kxkp ≥ b for all p ∈ P.
≤ x ≤ 1.
⊓
⊔
Corollary 1. For any a ∈ Z there is only a finite number of rational x ∈ [0, 1]
for which kxkm < a.
Reachability problem for PAMs: We say that f : S 1 → S 1 is a onedimensional piecewise affine map (PAM) whenever f is of the form f (x) =
ai x + bi , where ai , bi ∈ Q ⇔ x ∈ Xi , S 1 = X1 ∪ X2 ∪ . . . ∪ Xl and where
{X1 , . . . , Xl } is a finite family of disjoint (rational) intervals. If the intervals are
not disjoint we call it non-deterministic piecewise affine map and by default a
piecewise-affine mapping is understood to be deterministic. The derivative f ′ of
a PAM f we define as f ′ (x) = ai for x ∈ Xi , 1 ≤ i ≤ l.
If f is deterministic PAM, an orbit (trajectory) of a point x is denoted by
Of (x) and will be understood either as a set Of (x) = {f i (x)|i ∈ Z+ } or as a
sequence, i.e. Of (x) : Z+ → S 1 , Of (x)(i) = f i (x). We also define that a point y
is reachable from x if y ∈ Of (x).
In general the reachability problem for PAM can be defined as follows. Given
a PAM f , x ∈ S 1 and Y ⊆ S 1 , decide whether the intersection Y ∩ Of (x) is
empty. If Y is a finite union of intervals, we name the reachability problem as
point-to-set (interval) problem. If Y is a one element set (i.e. a single point), the
reachability problem is known as point-to-point reachability. 4
In this paper we only consider one-dimensional PAMs and by the reachability
problem for PAM we understand point-to-point reachability and explicitly state
the type of the problem when we need to refer to other reachability questions.
Note that the point-to-interval reachability can be reduced to point-to-point
reachability problem by extending a map with a few intervals in which the current value is just sequentially deleted. It works for all PAMs but may not preserve
the properties and the form of the original map.
Generally speaking the piecewise-affine mapping does not need to be defined
as f : X → X, where X = S 1 = [0, 1). However if the set X is a union of any
finite number of bounded intervals we always can scale it to S 1 . If f : X → X is
such that X 6= [0, 1), and X ⊆ [a, b), then by applying conjugation h(x) = x−a
b−a
the original reachability problem for f is reduced to the reachability problem for
the mapping g = h ◦ f ◦ h−1 from [0, 1) to [0, 1). Moreover the interest to PAMs
as f : S 1 → S 1 is also motivated by their use in the research of chaotic systems.
3
Decidability using p-adic norms
It is well know in dynamical systems research that due to complexity of orbits in
iterative maps it is less useful, and perhaps misleading, to compute the orbit of
a single point and it is more reasonable to approximate the statistics of the underlying dynamics [13, 20]. This information is encoded in the so-called invariant
measures, which specify the probability to observe a typical trajectory within a
certain region of state space and their corresponding invariant densities.
Let us consider a density as an ensemble of initial starting points (i.e. initial
conditions). The action of the dynamical system on this ensemble is described by
the Perron-Frobenius operator. The ensembles which are fixed under the linear
4
Also in a similar way it is possible to define set-to-point and set-to-set reachability
problems.
Perron-Frobenius operator is known as invariant densities or in other words, they
are eigenfunctions with eigenvalue 1 [13].
Formally under an ensemble A we understand an enumerated set (sequence)
of points in phase space. With ensemble we can associate the distribution function and the density function. Let I be a set of points. We denote by FIA (n) =
|{i ∈ Z+ |i ≤ n, A(i) ∈ I}| the number of elements in the sequence A which belong to the set I and which indexes are less or equal n. The distribution function
FA
(n)
of the ensemble A is defined as ΦA (x) = limn→∞ (−∞,x)
, if the limits exist.
n
The density function φA of the ensemble A is defined as φA (x) = Φ′A (x).
Suppose given an ensemble A0 with density φ0 . If we apply PAM f to each
point of the ensemble, we get a new ensemble A1 with some density distribution
φ1 . We say that the function φ1 is obtained from φ0 using the Frobenius-Perron
or transfer operator, which we denote by Lf . It is known that
φ1 (x) = Lf (φ0 )(x) =
X
y∈f −1 (x)
φ0 (y)
.
|f ′ (y)|
If φ1 = φ0 we say that φ0 is a f -invariant density function or an eigenfunction
of the transfer operator Lf .
We prove that if for an injective PAM f there exists an invariant bounded
density function then the reachability problem for f is decidable.
Lemma 2. Let f be an injective PAM, and φ be a f -invariant density function.
If there are Kmin > 0 and Kmax < +∞ such that for any x from the domain
of f the following inequality holds: Kmin < φ(x) < Kmax , then for an arbitrary
Kmin
≤ |c1 ·
segment of the orbit x1 , x2 , . . . xn+1 , where xi+1 = f (xi ), we have K
max
Kmax
c2 · . . . · cn | ≤ Kmin , where ci = aj if xi ∈ Xj .
Proof. Let φ be an eigenfunction of the Perron-Frobenius operator for an injective PAM f . Then injectivity of f and the fact that y = f (x) implies that φ(y) =
φ(x)
φ(x1 )
′
|f ′ (x)| . We denote f (xi ) by ci . Then φ(xn+1 ) = |c1 ·c2 ·...·cn | and |c1 · c2 · . . . · cn | =
φ(x1 )
φ(xn+1 ) .
Now we can bound |c1 · c2 · . . . · cn | by
Kmin
Kmax
and
Kmax
Kmin .
Theorem 1. Given an injective PAM f with two intervals and the existence of
a f -invariant density function φ such that there are Kmin > 0 and Kmax < +∞
and the following inequality holds Kmin < φ(x) < Kmax for all x from the
domain of f . Then the reachability problem for f is decidable.
Proof. A piecewise-affine map f with two intervals X1 and X2 is defined as
follows: f (x) = ai x + bi , if x ∈ Xi . Note that we only consider PAMs over
rational numbers where a starting point as well as all coefficients and borders
of intervals are rational numbers. Let us consider an arbitrary pair of points
x, y ∈ X such that y ∈ Of (x) and the sequence of points x1 , x2 , . . . , xn+1 from
the orbit Of (x) such that x1 = x, xn+1 = y, xi+1 = f (xi ), 1 ≤ i ≤ n.
If it would be possible to find a computable upper bound M on the length n
of such reachability sequence from x to y (based on Kmin , Kmax a1 , a2 , b1 , b2 ,
x, y) we could solve the reachability problem by considering only initial segment
of the reachability path of length less or equal to M .
In general the existence of such bound is not obvious, however if we can show
that in the path x1 , x2 , . . . , xn+1 where x1 = x and xn+1 = y the p-adic weights
kxi km for all i ∈ {1, . . . , n + 1} are bounded by some value M1 then we can
restrict M as follows: n < M < mM1 as kxi km is the number of decimal places
in the representation of xi in base m. In particular if there is a orbit’s segment
of length that is greater than mM1 and which consists of numbers with p-adic
weights less than M1 then it always contains two identical numbers and the orbit
loops.
So in order to prove a computable upper bound on kxi km it is sufficient
to prove that for any p ∈ F p-adic weights (i.e. logarithmic norms) kxi kp are
bounded from above by a computable number M2 for all i ∈ {1, . . . , n + 1}. Note
that by the definition of the logarithmic m-norm, M2 = M1 .
Let us define a number h = max{kb1 km + 1, kb2km + 1, kx1 km , kxn+1 km }.
Let us take an arbitrary p ∈ F and select a subsequence xj , xj+1 , . . . , xr in the
sequence x1 , x2 , . . ., xn+1 such that its elements x ∈ {xj+1 , xj+2 , . . . , xr−1 } have
the following property kxkp > h, but at the same time kxj kp ≤ h and kxr kp ≤ h.
For simplicity, but without loss of generality, we assume that j = 1, r = n + 1
and kx1 kp = kxn+1 kp = h.
Later we either will find the computable upper bound on n or will show a
computable bound M2 on p-adic weights based on Kmin , Kmax , a1 , a2 and h. As
stated above, it is enough for proving the theorem.
Let us define xi+1 = f (xi ) = ci xi + di , where ci ∈ {a1 , a2 }, di ∈ {b1 , b2 }.
From the properties (1) and (2) of the p-adic weights and the fact that kxi kp >
kbj kp , i ∈ {1, 2, . . . , n}, j ∈ {1, 2} follows that kxi+1 kp = kxi kp + kci kp . This
Pn
P2
implies kxn+1 kp = kx1 kp + i=1 kci kp = kx1 kp + i=1 αi kai kp for some nonnegative numbers α1 and P
α2 , where α1 + α2 = n. Taking into account that
2
kx1 kp = kxn+1 kp we have i=1 αi kai kp = 0.
Let r be the greatest common divisor of α1 and α2 . We define α = α1 /r and
β = α2 /r, i.e. α and β are smallest non-negative integers such that αka1 kp +
P
βka2 kp = 0. Thus, we obtain 2i=1 αi kai kp = r(αka1 kp + βka2 kp ). By Lemma 2
Kmin
Kmin
Kmax
α β r
max
we have: K
≤ |c1 · c2 · . . . · cn | ≤ K
Kmin , i.e. Kmax ≤ |a1 a2 | ≤ Kmin ,
max
β
α β
Now we consider two cases |aα
1 a2 | 6= 1 and |a1 a2 | = 1. The first case corresponds to linearly independent columns of the matrix Af , rank(A) = 2, and the
second case to linear dependence, rank(Af ) < 2.
β
α β
α β
α β
Let |aα
1 a2 | 6= 1, then either |a1 a2 | > 1 or |a1 a2 | < 1. If |a1 a2 | > 1 then from
β r
|aα
1 a2 | ≤
Kmax
Kmin
β r
from |aα
1 a2 | ≥
follows that r ≤
Kmin
Kmax
follows r ≤
Kmax
Kmin
β
ln |aα
1 a2 |
Kmin
ln Kmax
β
ln |aα
1 a2 |
ln
β
. On the other hand if |aα
1 a2 | < 1, then
.
β
Now suppose that |aα
1 a2 | = 1. The only interesting case is when a1 6= 1 and
a2 6= 1 as the cases where |a1 | = 1, |a2 | = 1 or both correspond to the trivial
case of piecewise-affine mapping (i.e. with no more than one linear factor).
Without loss of generality, we assume that a1 > 1 and a2 < 1. Note also that
ka2 kp = − α
β ka1 kp . Let us now consider an arbitrary subsequence of consecutive
points x1 x2 . . . xj+1 , j ≤ n of the original reachability path. Assuming that α1 ,
α2 such that |c1 c2 . . . cj | = |a1 |α1 |a2 |α2 we have that
α
kxj+1 kp = kx1 kp + α1 ka1 kp + α2 ka2 kp = (α1 − α2 )ka1 kp .
β
On the other hand from Lemma 2 we know that:
β
Since |aα
1 a2 | = 1,then |a2 | = |a1 |
−α
β
Kmin
Kmax
≤ |a1 |α1 |a2 |α2 ≤
Kmax
Kmin .
and
α
Kmax
Kmin
≤ |a1 |α1 −α2 β ≤
.
Kmax
Kmin
ln
Kmax
Kmin
Taking into account assumption that |a1 | > 1, we get α1 − α2 α
β ≤ ln |a1 | .
Now it follows that:
!
max
ln K
α
Kmin
kxj+1 kp = kx1 kp + (α1 − α2 )ka1 kp ≤ h +
ka1 km .
β
ln |a1 |
Thus, we have shown a computable upper bound for p-adic weights of the orbital
elements that can reach a point y. Finally, in view of provided reasoning, we
shown that the reachability problem for this type of PAMs is decidable.
⊓
⊔
The theorem can be applied for a larger class of PAMs if more information
would be known about the convergence of density functions under the action of
the Perron-Frobenius operator. Let us call an ensemble A to be statistically fixed
with respect to f , if φA = Lf (φA ). E.g. if someone can show that in injective
PAM all statistically fixed ensembles have identical distribution functions then
Theorem 1 can be applied to show decidability in injective PAMs.
Following the proposed approach based on p-adic weights we define another
fragment of PAMs for which the reachability problem is decidable. In particularly
we remove the condition on eigenfunction of the transfer operator and injectivity
of piecewise-affine map and consider a PAM f with only constraints on linear
coefficients in affine maps. More specifically we require that the powers of prime
numbers from prime factorizations of linear coefficients should have the same
signs (i.e. two sets of prime numbers used in nominator and denominator are
disjoint). Let us denote for a PAM f a matrix Af with values (aji ), where
aji = kai kpj , 1 ≤ i ≤ l, 1 ≤ j ≤ k. The rank of Af is denoted by rank(Af ).
Theorem 2. The reachability problem for a PAM f is decidable if every row of
a matrix Af contains values of the same sign, (i.e. aji · aj ′ i ≥ 0, for all i, j such
that 1 ≤ i ≤ l, 1 ≤ j, j ′ ≤ k).
Proof. Let us consider a PAM f of the form f (x) = ai x + bi for x ∈ Xi where
all coefficients ai ,bi and the extremities of a finite number of bounded intervals
Xi are rational numbers. Let us define h = max{kb1 km , kb2 km , . . . , kbl km }. The
condition of the theorem means that for any prime p ∈ F all linear coefficients
of the map f have non-zero p-adic weights of the same sign.
In this case, if p-adic weights of linear coefficients of f are non-negative,
then for any x ∈ X from kxkp > h follows that kf (x)kp ≥ kxkp and therefore
kf (x)km ≥ kxkm (i.e. m-adic weight does not decrease). If p-adic weight of
linear coefficients of the mapping are negative, then for any x ∈ X we have
kf (x)kp ≤ max{kxkp , h}.
Thus, in the sequence of reachable points for an orbit of a map f either all
points of the orbit have m-adic weights bounded from above by h, then we have
a cyclic orbit, or from some moment when m-adic weight of a reachable point
exceeds h it does not decrease and again, either orbit loops or m-adic weight
increases indefinitely.
Thus, in order to decide whether y is reachable, i.e. y ∈ Of (x), it is sufficient
to start generating a sequence of reachable points in the orbit Of (x) and wait
for one of the events, where either 1) a point in the orbit is equal to y ( y is
reachable ), or 2) the orbit will loop and y ∈
/ Of (x) ( y is not reachable ), or 3)
a point x′ is reachable, such that kx′ km > max{h, kykm}, and then y ∈
/ Of (x) (
y is not reachable ).
⊓
⊔
Definition 1. A piecewise affine mapping f : S 1 → S 1 is complete if for a set
of disjoint intervals S 1 = X1 ∪ X2 ∪ . . . ∪ Xn , f (Xi ) = S 1 for any i = 1..n.
Definition 2. Let be F : R → R is the lifting of a continuous map f : S 1 → S 1
on R, i.e. f ({x}) = {F (x)}. Then by the degree deg(f ) of a map f we denote
the number F (x + 1) − F (x), which is independent from the choice of the point
x and the lifting F .
Corollary 2. The reachability problem for complete piecewise affine mappings
with two intervals 5 is decidable.
Proof. The condition of a piecewise affine map with two intervals f : S 1 → S 1
1
1
to be complete
m means that Sm = X1 ∪ X2 and f (X1 ) = f (X2 ) = S . Thus,
n
if X1 = 0, n and X2 = n , 1 , then f (x) = a1 x + b1 , where a1 = ± m
,
n
when x ∈ X1 , and f (x) = a2 x + b2 , where a2 = ± n−m at x ∈ X2 , m, n ∈ N,
gcd(m, n) = 1. It is clear that n, m, n− m are relatively prime. So the conditions
of Theorem 2 are satisfied.
⊓
⊔
4
PAM representation of β-expansions
Given a rational non-integer β > 1 and the number x ∈ [0, 1]. The target
discounted-sum 0-1 problem [22, 11] is defined P
as follows: Is there a sequence
∞
w : N → {0, 1} of zeros and ones such that x = i=1 w(i) β1i .
For any
x ∈ S 1 , there exists β-expansion w : N → {0, 1, . . . , ⌈β⌉ − 1} such
P∞
that x = i=1 w(i) β1i . If w(i) ∈ {0, 1} we call it (0, 1) − β-expansion. Therefore, when β ≤ 2 the answer to the target discounted-sum problem is always
5
In particularly the continuous piecewise affine mapping of degree two
positive. Therefore, the only interesting case is when β > 2. We denote D =
{0, 1, . . . , ⌈β⌉ − 1}. Then the minimal and maximal numbers, which
P∞are representable in the basis β with digits from the alphabet D, are min = i=1 0 β1i = 0
P∞
and max = i=1 (⌈β⌉ − 1) β1i = ⌈β⌉−1
β−1 . When β > 2 then max is always less
then two. Let us denote by Xd the interval [ min+d
, max+d
) for each d ∈ D. If β
β
β
is not an integer number then two intervals Xd and Xd+1 intersect. Also taking
into account that max < 2, then the intervals Xd and Xd+2 have no common
points. Finally from the above construction we get the next lemma:
Lemma 3. If β > 2 and β is rational/non-integer number: Xd ∩ Xd+1 6= ∅,
d < ⌈β⌉ − 1; Xd ∩ Xd+2 = ∅, d < ⌈β⌉ − 2; [min, max) = ∪d∈A Xd .
Proposition 1. For any β-expansion there is a non-deterministic PAM where a
symbolic dynamic of visited intervals (i.e. a sequence of symbols associated with
intervals) from an initial point x0 corresponds to its representation in base β.
Proof. Let us define the piecewise affine mapping f ⊆ [min, max) × [min, max)
as follows f = {(x, βx − d)|x ∈ Xd , d ∈ D}. It directly follows from this
definition that f (Xd ) = [min, max). Let us consider an orbit f i (x) = x(i),
Fig. 1. A non-deterministic PAM for
5
-expansion
2
i ∈ Z+ . We say that di ∈ D such that x(i) ∈ Xdi . Then for any n ∈ N
min < β n x − d1 β n−1 − d2 β n−2 − . . . − dn β 0 < max, and in other form min
βn <
P
Pn
n
1
x − i=1 (di β1i ) < max
i=1 (di β i ) → 0, n → ∞, and therefore
β n . So x −
P∞
P
1
x = i=1 (di β1i ). Let us consider it in other direction. Let x = ∞
i=1 (di β i ), then
the sequence x(i), where x(0) = x, x(i + 1) = βx(i) − di , is the orbit of x in
PAM f . Let us name the constructed map as the β-expansion PAM.
⊓
⊔
The nondeterministic β-expansion can be translated into deterministic maps
corresponding to greedy and lazy expansions as follows:
Definition 3. A function f : [min, max) → [min, max) is the greedy β-expansion
PAM if the domain [min, max) is divided on intervals Xd′ , d ∈ {0, 1, . . . , ⌈β⌉−1}
′
′
such that X⌈β⌉−1
= X⌈β⌉−1 , Xd−1
= Xd−1 − Xd , d ∈ {1, 2, . . . , ⌈β⌉ − 1} and
′
f (x) = βx − d iff x ∈ Xd .
Since Xd = [ min+d
, max+d
) then Xd′ = [ min+d
, min+d+1
) = [ βd , d+1
β
β
β
β
β ), d ∈
1
′
{0, 1, . . . , ⌈β⌉ − 2} and the length of the interval Xd is equal to β , d < ⌈β⌉ − 1.
Fig. 2. Deterministic greedy (on the left) and lazy (on the right)
5
-expansion
2
PAM
Definition 4. A function f : [min, max) → [min, max) is the lazy β-expansion
PAM, if the domain [min, max) is divided into intervals Xd′′ , d ∈ {0, 1, . . . , ⌈β⌉−
1}, such that X0′′ = X0 , Xd′′ = Xd − Xd−1 , d ∈ {1, 2, . . . , ⌈β⌉ − 1}, and f (x) =
βx − d iff x ∈ Xd′′ .
Proposition 2. Let f and g are greedy and lazy β-expansion PAM’s respectively. f and g are (topologically) conjugate by the homeomorphism h : h(x) =
h−1 (x) = max − x, i.e. f = h ◦ g ◦ h.
′′
Proof. The statement holds since Xd′ = max − X⌈β⌉−1−d
, d ∈ {0, 1, . . . , ⌈β⌉ − 1}
We would like to highlight that the questions about reachability as well as
representation of numbers in rational bases are tightly connected with questions
about the density of orbits in PAMs. Moreover if the density of orbits are the
same for all non-periodic points then it may be possible to have a wider application of p-adic techniques provided in the beginning of the paper. Let us formulate
a hypothesis that goes along with our experimental simulations in PAMs:
Hypothesis 1 The orbit of any rational point in any expanding deterministic
PAM is either finite or dense on the whole domain.
Lemma 4. Any (0, 1) − β-expansion is greedy.
Proof. Let f be a β-expansion PAM. Assume that there is a point x and the orbit
x(i), where x(0) = x, x(i + 1) = βx(i) − di in the map f such that di ∈ {0, 1}
for all i ∈ N and the orbit does not correspond to the β greedy expansion of x.
The intersection of intervals X0 and X1 is an interval X01 = [ minβ+1 , max
β ).
Applying a map y = βx to X01 we see that X01 is scaled into [min +1, max) =
⌈β⌉−1
[1, ⌈β⌉−1
β−1 ). The interval [1, β−1 ) does not have any common points with X0
as the point 1 lies on the right side of the left border of the interval X2 =
[ minβ+2 , maxβ +2 ) and by Lemma 3 Xi ∩ Xi+2 = ∅.
1
we have βx − 1 > x. Let us assume that for some i
Note that when x > β−1
x(i) ∈ X01 and x(i + 1) = βx(i) − 0, i.e. we did not followed a greedy expansion
and therefore x(i + 1) ∈ [1, ⌈β⌉−1
β−1 ). Then x(i + 2) = βx(i + 1) − 1 > x(i + 1)
and x(i + 2) ∈
/ X0 , etc. In this case starting from x(i + 1) there is monotonically
increasing sequence of orbital points in the interval X1 . So points in such orbit
should eventually leave the interval X1 and reach Xd , where d > 1. This gives
us a contradiction with the original assumption.
⊓
⊔
Corollary 3. Since the greedy expansion can be expressed by a deterministic
map then (0, 1) − β-expansion is unique and greedy.
Theorem 3. If Hypothesis 1 holds then a non-periodic (0, 1)− β-expansion does
not exist.
Proof. Any (0, 1)−β-expansion can be constructed by expanding 6 deterministic
greedy β-expansion PAM. If the orbit of a rational point in greedy β-expansion
PAM is non-periodic, then by Hypothesis 1 it should be dense and therefore
should intersect all intervals and cannot provide (0, 1) − β-expansion.
⊓
⊔
Theorem 4. If Hypothesis 1 holds then for any rational number its deterministic β-expansion is either eventually periodic or it contains all possible patterns
(finite subsequences of digits) from {0, 1, . . . , ⌈β⌉ − 1}.
Proof. The statement is obvious as Hypothesis 1 implies that the orbit is either
periodic or it is dense and the dense orbit is visiting all intervals.
⊓
⊔
It looks that the point-to-interval problem is harder than the point-to-point
reachability problem for the expanding PAMs, as for example Theorem 2 gives an
algorithm for the point-to-point reachability problem in the β-expansion PAMs,
but not for the point-to-interval reachability that is equivalent to the β-expansion
problem.
Note that in the β-expansion PAMs all linear coefficients are the same, so
the density of the orbit correspond to the density of the following sequence
x(n) = f n (x0 ), where f (x) = {βx}. For example when β = 25 and x0 = 1 we get
the sequence:
{ 25 }, { 25 { 25 }}, { 25 { 25 { 52 }}}, . . .
2
3
The question about the distribution of a similar sequence { 23 }, { 232 }, { 323 }, ... ,
where the integer part is removed once after taking a power of a fraction (for
example 3/2) is known as Mahler’s 3/2 problem, that is a long standing open
problem in analytic number theory.
5
Density of orbits and its geometric interpretation
It is well known that x(n) = {αn}, where α is an irrational number, has an
uniform distribution. Let us give some geometric interpretation of the orbit density. Consider the Cartesian plane with the y-axis x and and the x-axis y (just
6
I.e. with linear coefficients that are greater than 1
swapping their places). Now let us divide the set of lines x = n, n ∈ N, by
integer points on the segments of the unit length. The set of points (y, x), where
m ≤ y < m + 1, x = n, i.e. the interval [m, m + 1) × n on the line x = n, will be
denoted by Sm,n , m ∈ Z+ , n ∈ N. In other words, Sm,n = (m + [0, 1)) × n. Let I
be an interval such that I ⊆ [0, 1) and by Im,n let us denote the set (m + I) × n.
Fig. 3. Left: An example for two sets Sm,n and Im,n ; Right: A dynamic interval I(n).
Two points of the plane are defined to be equivalent if they belong to a
same line passing through the origin. We call α as homogeneous coordinate of a
point (y, x) if y = αx. SBy H(I) we denote the set of homogeneous coordinates
Im,n . The sequence x(n) = {αn} is dense in [0, 1)
of all points from
m∈Z+ ,n∈N
if and only if for any interval I ⊆ [0, 1) there are m and n, such that the line
y = αx intersects the set Im,n . It is known that [0, 1) − H(I) ⊆ Q for any
interval
S I ⊆ [0, 1), i.e. for any irrational α > 0 the line y = αx intersect the set
Im,n . Moreover in the case of irrational factors it is known that the
m∈Z+ ,n∈N
frequency of occurrence of x(n) = {αn} in the interval I is equal to its length.
In some sense the interval I, in the above example, can be named as static
because it does not change in time n. However in order to study and describe previously mentioned problems such as the target discounted-sum problem, PAMs
reachability problems, the Mahler’s 3/2 problem we require the notion of “dynamic intervals“.
Let x be a sequence
of numbers from [0, 1). What is the distribution of a
sequence x′ (n) = pk(n) x(n) , where k : Z+ → Z+ is a non-decreasing sequence?
For example, if k(n) = n− 1 and the number x(n) has in the base p the following
form x(n) = 0.an1 an2 . . . ann an,n+1 . . ., then x′ (n) = 0.ann an,n+1 . . ..
Let us assume that I ⊆ S 1 and k : Z+ → Z+ is a non-decreasing sequence,
p ∈ N. Now we define “dynamical intervals” as an evolving infinite sequence
I(1), I(2), I(3), . . . :
I(1) = I, I(n) =
pk(n)
[−1
j=0
I +j
.
pk(n)
By FIx (n) = |{i ∈ Z+ |i ≤ n, x(i) ∈ I(i)}| we denote a function representing
a frequency of hitting dynamical interval I by the sequence x. In contrast to
FIx (n) which only counts the number of hittings to a fixed interval I, our new
function FIx counts the number of hittings when both points and intervals are
changing in time.
′
Proposition 3. The following equation holds: FIx (n) = FIx (n).
The phenomenon that significant digit distribution in real data are not accruing randomly known as Benford’s Law. For example the sequence p1 , p2 , p3 ,..
satisfies Benford’s Law, under the condition that log10 p is an irrational number, which is a consequence of the Equidistribution theorem (proved separetly
by Weyl, Sierpinski and Bohl). The Equidistribution theorem states that the
sequence {α}, {2α}, {3α}, . . . is uniformly distributed on the circle R/Z , when
a is an irrational number. It gives us the fact that each significant digit of numbers in (pn ) sequence will correspond to the interval R/Z and the length of the
interval related to the frequency for each appearing digit.
However the question about the distribution of the sequence {(3/2)n } is
different in the way that it is not about the distribution of the first digits of 3n
n
in base 2, i.e. not about the distribution of the sequence 2⌈n 3log2 3⌉ , but related
n
to the sequence of digits after some shift of the number 2⌈n 3log2 3⌉ corresponding
to the multiplication by a power of 2.
So in the above notations the distribution of numbers in the sequence x′ (n) =
′
{(3/2)n } corresponds to the nFIx (n) for the logarithmic (Benford’s law) distributed sequence x(n) = 2⌈n 3log2 3⌉ , p = 2 and k(n) = ⌈n log2 3⌉ − n.
Now we will show that even if the sequence {α}, {2α}, {3α}, . . . is uniformly
distributed on the circle R/Z, the irrationality of α is not enough to guarantee
uniform distribution or even density of the sequence x′ (n) on the circle corresponding to the linear shifts k(n) = n.
P∞ 1
where ∆1 = 1, ∆i+1 = 2∆i + ∆i ,
Theorem 5. Let us define α = i=1 2∆
i
i ≥ 1 (http://oeis.org/A034797). Then for all n ∈ N ∪ {0} the sequence {2n nα}
is not dense in the interval [0, 1] and {2n nα} < 21 .
Proof. Let us consider a sequence ∆, which initial elements are ∆1 = 1, ∆2 = 3,
∆3 = 11, ∆4 = 2059 etc.
x(0) = 0,
x(1) = 0,
x(2) = 0,
x(3) = 0,
x(4) = 0,
x(5) = 0,
x(6) = 0,
x(7) = 0,
x(8) = 0,
x(9) = 0,
x(10) = 0,
x(11) = 0,
∆1
0
1
0
0
1
1
1
1
0
0
0
0
0
0
1
1
0
0
1
1
0
0
0
0
∆2
0
1
0
1
0
1
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1
0
0
0
0
0
0
1
1
0
0
1
1
0
0
1
1
∆3
0
1
0
1
0
1
0
1
0
1
0
1
0 ...
0 ...
0 ...
0 ...
0 ...
0 ...
0 ...
0 ...
0 ...
0 ...
0 ...
0 ...
x′ (0) = 0, 0 0 0 0 0 0 0 0 0 0 0 0 . . .
x′ (1) = 0, 0 1 0 0 0 0 0 0 0 1 0 . . .
x′ (2) =
0, 0 0 0 0 0 0 0 1 0 0 . . .
x′ (3) =
0, 0 0 0 0 0 0 1 1 0 . . .
x′ (4) =
0, 0 0 0 0 1 0 0 0 . . .
x′ (5) =
0, 0 0 0 1 0 1 0 . . .
x′ (6) =
0, 0 0 1 1 0 0 . . .
x′ (7) =
0, 0 1 1 1 0 . . .
x′ (8) =
0, 0 0 0 0 . . .
x′ (9) =
0, 0 1 0 . . .
x′ (10) =
0, 0 0 . . .
x′ (11) =
0, 0 . . .
Let us prove that when 0 ≤ n ≤ ∆i the inequality {2n nα} < 21 if and
Pi
only if {2n n j=1 2∆1 j } < 12 . The implication from left to right follows from
P
{2n n ij=1 2∆1 j } < {2n nα}. Let us show that it also holds in other direction.
P
P
Assume that {2n n ij=1 2∆1 j } < 12 then {2n n ij=1 2∆1 j +x} < 21 for any 0 ≤ x ≤
Pi
1
1
. In this case it is enough to show that 2n nα − 2n n j=1 2∆1 j < 2∆i +1−n
2∆i +1−n
holds when n ≤ ∆i . In fact we have that
2n nα − 2n n
=
i
∞
X
X
1
1
1
n
=
2
n
< 2n n ∆i+1 −1 =
∆
∆
j
j
2
2
2
j=1
j=i+1
1
2∆i+1 −n−log2 (n)−1
≤
1
2∆i+1 −∆i −log2 (∆i )−1
==
1
∆
22 i −log2 (∆i )−1
.
1
Finally we have 2∆i −log1 (∆ )−1 < 2∆i +1−n
when n ≤ ∆i and i > 1.
2
i
2
Now for proving
the
theorem
it
is
enough
to show that for 0 ≤ n ≤ ∆i the
P
inequality {2n n ij=1 2∆1 j } < 21 holds. When i = 1 the statement is trivial. Let
us assume that it holds for i = k − 1. We show now
Pkthat it holds for i = k, i.e. we
show that for ∆k−1 ≤ n ≤ ∆k inequality {2n n j=1 2∆1 j } < 21 holds or taking
into account ∆k−1 ≤ n also we have inequality {2n n 2∆1 k } < 21 .
For ∆k−1 ≤ n ≤ ∆k − ∆k−1 − 1 we get 2n n 2∆1 k ≤ 2∆k −∆k−1 −1 (∆k − ∆k−1 −
1) 2∆1 k . Since ∆k = 2∆k−1 + ∆k−1 , then 2∆k −∆k−1 −1 (∆k − ∆k−1 − 1) 2∆1 k =
1
1
1
1
(2∆k−1 − 1) 2∆k−1
+1 = 2 − ∆k−1 +1 < 2 .
2
Consider the case when ∆k −∆k−1 ≤ n ≤ ∆k , i.e. 2∆k−1 ≤ n ≤ 2∆k−1 +∆k−1
and define m = n − 2∆k−1 . We have that 0 ≤ m ≤ ∆k−1 and 2n n 2∆1 k =
∆k−1
2m+2
(m + 2∆k−1 )
1
∆k−1
22
+∆k−1
= 2m (m + 2∆k−1 ) 2∆1k−1 = 2m m 2∆1k−1 + 2m .
1
}, where 0 ≤ m ≤ ∆k−1 . Now by the inducTherefore {2n n 2∆1 k } = {2m m 2∆k−1
1
1
m
tion we have that {2 m 2∆k−1 } < 2 .
While the question about the distributions for PAM orbits remains open we
have unexpectedly shown that in a very similar system, operating with irrational
numbers, the uniform distribution of original orbits in maps may not remain
uniform or even dense when taking the fractional part after regular shifts. This
makes the questions about PAMs even more “mysterious” as it is not clear
whether such property may hold for a sequence of points generated by PAMs,
β-expansion and Mahler’s problem.
References
1. A. Aswani and C. Tomlin (2007), Reachability algorithm for biological piecewiseaffine hybrid systems, HSCC,2007,633-636.
2. E. Asarin, V. Mysore, A. Pnueli, G. Schneider. Low dimensional hybrid systems
– decidable, undecidable, don’t know. Inf. Comput. 211: 138-159 (2012)
3. E.Asarin and G.Schneider, Widening the boundary between decidable and undecidable hybrid systems. CONCUR, 2002,193-208 .
4. E.Asarin, O. Maler, A. Pnueli. Reachability analysis of dynamical systems having
piecewise-constant derivatives. Theoretical Computer Science, 138, 1995, 35 - 66.
5. Paul C. Bell , Shang Chen, Lisa Jackson: Reachability and Mortality Problems
for Restricted Hierarchical Piecewise Constant Derivatives. RP 2014: 32-44
6. P. Bell, I. Potapov. On undecidability bounds for matrix decision problems Theoretical Computer Science 391 (1), 3-13, 2008.
7. Amir M. Ben-Amram: Mortality of iterated piecewise affine functions over the
integers: Decidability and complexity. Computability 4(1): 19-56 (2015)
8. M. Blank and L. Bunimovich. Switched flow systems: pseudo billiard dynamics.
Dynamical Systems. V.19, No.4 (2004) 359-370
9. V. Blondel, O. Bournez, P. Koiran, C. Papadimitriou and J. Tsitsiklis. Deciding
stability and mortality of piecewise affine dynamical systems. Theoretical Computer Sci. 255: (1-2), 687-696, 2001.
10. V. Blondel, O. Bournez, P. Koiran, J. Tsitsiklis: The Stability of Saturated Linear
Dynamical Systems Is Undecidable. J. Comput. Syst. Sci. 62(3): 442-462 (2001)
11. Udi Boker and Thomas A. Henzinger and Jan Otop. The Target Discounted-Sum
Problem. LICS 2015: 750-761 (2015)
12. Hugo Bazille, Olivier Bournez, Walid Gomaa, Amaury Pouly: On The Complexity
of Bounded Time Reachability for Piecewise Affine Systems. RP 2014: 20-31
13. M. Dellnitz, G. Froyland, and S. Sertl, On the isolated spectrum of the PerronFrobenius operator. Nonlinearity, vol. 13, no. 4, pp. 1171-1188, July 2000.
14. P. Koiran, M. Cosnard, and M. Garzon. Computability with low-dimensional
dynamical systems. Theoretical Computer Science, 132:113-128, 1994
15. P. Koiran: The topological entropy of iterated piecewise affine maps is uncomputable. Discrete Mathematics and Theor. Computer Science 4(2): 351-356 (2001)
16. V. Blondel, J. Tsitsiklis. A survey of computational complexity results in systems
and control. Automatica, 36, 2000, 1249-1274.
17. O. Kurganskyy, I. Potapov, F. Sancho-Caparrini: Reachability Problems in LowDimensional Iterative Maps. Int. J. Found. Comput. Sci. 19(4): 935-951 (2008)
18. O. Kurganskyy, I. Potapov. Computation in One-Dimensional Piecewise Maps
and Planar Pseudo-Billiard Systems. LNCS, v.3699, 2005, 169-175.
19. A. Renyi. Representations for real numbers and their ergodic properties. Acta
Mathematica Academiae Scientiarum Hungarica, 8(3-4):477-493, 1957.
20. G. Setti, G. Mazzini, R. Rovatti, and S. Callegari, Statistical modeling of discretetime chaotic processes: basic finite dimensional tools and applications, Proccedings of the IEEE, vol. 90, no. 5, May 2002, pp. 662-690.
21. N. Sidorov. Almost Every Number Has a Continuum of β-Expansions. The American Mathematical Monthly 110(9): 838-842 (2003)
22. Mickael Randour, Jean-Francois Raskin, Ocan Sankur. Percentile Queries in
Multi-Dimensional Markov Decision Processes. Dagstuhl seminar “Non-zero-sum
games and control”, 5.02.2015.
23. J. Ouaknine, J. Sousa Pinto, J. Worrell: On Termination of Integer Linear Loops.
SODA 2015: 957-969
| 8 |
Improved Hardness for Cut, Interdiction,
and Firefighter Problems
arXiv:1607.05133v1 [cs.CC] 18 Jul 2016
Euiwoong Lee∗
Computer Science Department
Carnegie Mellon University
Pittsburgh, PA 15213.
Abstract
We study variants of the classic s-t cut problem and prove the following improved hardness
results assuming the Unique Games Conjecture (UGC).
• For any constant k ≥ 2 and ǫ > 0, we show that Directed Multicut with k source-sink pairs
is hard to approximate within a factor k − ǫ. This matches the trivial k-approximation
algorithm. By a simple reduction, our result for k = 2 implies that Directed Multiway Cut
with two terminals (also known as s-t Bicut) is hard to approximate within a factor 2 − ǫ,
matching the trivial 2-approximation algorithm. Previously, the best hardness factor for
these problems (for constant k) was 1.5 − ǫ [EVW13, CM16] under the UGC.
• For Length-Bounded Cut and Shortest Path Interdiction, we show that both problems are
hard to approximate within any constant factor, even if we allow bicriteria approximation. If we want to cut vertices or the graph is directed, our hardness factor for LengthBounded Cut matches the best approximation ratio up to a constant. Previously, the best
hardness factor was 1.1377 for Length-Bounded Cut [BEH+ 10] and 2 for Shortest Path
Interdiction [KBB+ 07].
• Assuming a variant of the UGC (implied by another variant of Bansal and Khot [BK09]),
we prove that it is hard to approximate Resource Minimization Fire Containment within any
constant factor. Previously, the best hardness factor was 2 [KM10].
Our results are based on a general method of converting an integrality gap instance to a lengthcontrol dictatorship test for variants of the s-t cut problem, which may be useful for other problems.
∗
Supported by the Samsung Scholarship, the Simons Award for Graduate Students in TCS, and Venkat Guruswami’s
NSF CCF-1115525. euiwoonl@cs.cmu.edu
1 Introduction
One of the most important implications of the Unique Games Conjecture (UGC, [Kho02]) is the
results of Khot et al. [KKMO07] and Raghavendra [Rag08], which say that for any maximum
constraint satisfaction problem (Max-CSP), an integrality gap instance of the standard semidefinite
programming (SDP) relaxation can be converted to the NP-hardness result with the same gap.
These results initiated the study of beautiful connections between power of convex relaxations and
hardness of approximation, from which surprising results for both subjects have been discovered.
While their results hold for problems in Max-CSPs, the framework of converting an integrality
gap instance to hardness has been successfully applied to covering and graph cut problems. For
graph cut problems, Manokaran et al. [MNRS08] showed that for Undirected Multiway Cut and its
generalizations, an integrality gap of the standard linear programming (LP) relaxation implies the
hardness result assuming the UGC. Their result is further generalized by Ene et al. [EVW13] by
formulating them as Min-CSPs. On the other hand, Kumar et al. [KMTV11] studied Strict CSPs
and showed the same phenomenon for the standard LP relaxation.
One of the limitations of the previous CSP-based transformations from LP gap instances to
hard instances is based on the fact that they do not usually preserve the desired structure of the
constraint hypergraph.1 For example, consider the Length-Bounded Edge Cut problem where the
input consists of a graph G = (V, E), two vertices s, t ∈ V , and a constant l ∈ N, and the goal is to
remove the fewest edges to ensure there is no path from s to t of length less than l. This problem
can be viewed as a special case of Hypergraph Vertex Cover (HVC) by viewing each edge as a vertex
of a hypergraph and creating a hyperedge for every s-t path of length less than l. While HVC is in
turn a Strict CSP, but its integrality gap instance cannot be converted to hardness using Kumar et
al. [KMTV11] as a black-box, since the set of hyperedges created in the resulting hard instance is
not guaranteed to correspond to the set of short s-t paths of some graph.
For Undirected Multiway Cut, Manokaran et al. [MNRS08] bypassed this difficulty by using
2-ary constraints so that the resulting constraint hypergraph becomes a graph again. For Undirected Node-weighted Multiway Cut, Ene et al. [EVW13] used the equivalence to Hypergraph Multiway Cut [OFN12] so that the resulting hypergraph does not need to satisfy additional structure.
These problems are then formulated as a Min-CSP by using many labels which are supposed to
represent different connected components. For Directed Multiway Cut, to the best of our knowledge, the existence of an analogous formulation as a Min-CSP is unknown.
We study variants of the classical s-t cut problem in both directed and undirected graphs that
have been actively studied, including the aforementioned Length-Bounded Cut and Directed Multiway Cut. We prove the optimal hardness or the first super-constant hardness for them. See
Section 1.1 for the definitions of the problems and our results. All our results are based on the
general framework of converting an integrality gap instance to a length-control dictatorship test.
The structure of our length-control dictatorship tests allows us to naturally convert an integrality
gap instance for the basic LP for various cut problems to hardness based on the UGC. Section 1.2
provides more detailed intuition of this framework. While these problems have slightly different
characteristics that make it hard to present the single result for a wide class of problems like CSPs,
we hope that our techniques may be useful to prove hardness of other cut problems.
1
One of notable exceptions we are aware is the result of Guruswami et al. [GSS15], using Kumar et al. [KMTV11] to
show that k-Uniform k-Partite Hypergraph Vertex Cover is hard to approximate within a factor k2 − ǫ for any ǫ > 0.
1
1.1 Problems and Results
Directed Multicut and Directed Multiway Cut. Given a directed graph and two vertices s and
t, one of the most natural variants of s-t cut is to remove the fewest edges to ensure that there is
no directed path from s to t and no directed path from t to s. This problem is known as s-t Bicut
and admits the trivial 2-approximation algorithm by computing the minimum s-t cut and t-s cut.
Directed Multiway Cut is a generalization of s-t Bicut that has been actively studied. Given
a directed graph with k terminals s1 , . . . , sk , the goal is to remove the fewest number of edges
such that there is no path from si to sj for any i 6= j. Directed Multiway Cut also admits 2approximation [NZ01, CM16]. If k is allowed to increase polynomially with n, there is a simple
reduction from Vertex Cover that shows (2 − ǫ)-approximation is hard under the UGC [GVY94,
KR08].
Directed Multiway Cut can be further generalized to Directed Multicut. Given a directed graph
with k source-sink pairs (s1 , t1 ), . . . , (sk , tk ), the goal is to remove the fewest number of edges such
that there is no path from si to ti for any i. Computing the minimum si -ti cut for all i separately
gives the trivial k-approximation algorithm. Chuzhoy and Khanna [CK09] showed Directed Mul1−ǫ
1−ǫ
ticut is hard to approximate within a factor 2Ω(log n) = 2Ω(log k) when k is polynomially grow11
ing with n. Agarwal et al. [AAC07] showed Õ(n 23 )-approximation algorithm, which improves
the trivial k-approximation when k is large.
Very recently, Chekuri and Madan [CM16] showed simple approximation-preserving reductions from Directed Multicut with k = 2 to s-t Bicut (the other direction is trivially true), and
(Undirected) Node-weighted Multiway Cut with k = 4 to s-t Bicut. Since Node-weighted Multiway
Cut with k = 4 is hard to approximate within a factor 1.5 − ǫ under the UGC [EVW13] (matching
the algorithm of Garg et al. [GVY94]), the same hardness holds for s-t Bicut, Directed Multiway
Cut, and Directed Multicut for constant k. To the best of our knowledge, 1.5 − ǫ is the best hardness factor for constant k even assuming the UGC. In the same paper, Chekuri and Madan [CM16]
asked whether a factor 2 − ǫ hardness holds for s-t Bicut under the UGC.
We prove that for any constant k ≥ 2, the trivial k-approximation for Directed Multicut might
be optimal. Our result for k = 2 gives the optimal hardness result for s-t Bicut, answering the
question of Chekuri and Madan.
Theorem 1.1. Assuming the Unique Games Conjecture, for every k ≥ 2 and ǫ > 0, Directed Multicut
with k source-sink pairs is NP-hard to approximate within a factor k − ǫ.
Corollary 1.2. Assuming the Unique Games Conjecture, for any ǫ > 0, s-t Bicut is hard to approximate
within a factor 2 − ǫ.
Length-Bounded Cut and Shortest Path Interdiction. Another natural variant of s-t cut is the
Length-Bounded Cut problem, where given an integer l, we only want to cut s-t paths of length
strictly less than l.2 Its practical motivation is based on the fact that in most communication /
transportation networks, short paths are preferred to be used to long paths [MM10].
Lovász et al. [LNLP78] gave an exact algorithm for Length-Bounded Vertex Cut (l ≤ 5) in undirected graphs. Mahjoub and McCormick [MM10] proved that Length-Bounded Edge Cut admits
an exact polynomial time algorithm for l ≤ 4 in undirected graphs. Baier et al. [BEH+ 10] showed
that both Length-Bounded Vertex Cut (l > 5) and Length-Bounded Edge Cut (l > 4) are NP-hard
2
It is more conventional to cut s-t paths of length at most l. We use this slightly nonconventional way to be more
consistent with Shortest Path Interdiction.
2
√
to approximate within a factor 1.1377. They presented O(min(l, nl )) = O( n)-approximation algo2 √
rithm for Length-Bounded Vertex Cut and O(min(l, nl2 , m)) = O(n2/3 )-approximation algorithm
for Length-Bounded Edge Cut, with matching LP gaps. Length-Bounded Cut problems have been
also actively studied in terms of their fixed parameter tractability [GT11, DK15, BNN15, FHNN15].
If we exchange the roles of the objective k and the length bound l, the problem becomes Shortest
Path Interdiction, where we want to maximize the length of the shortest s-t path after removing at
most k vertices or edges. It is also one of the central problems in a broader class of interdiction problems, where an attacker tries to remove some edges or vertices to destroy a desirable property (e.g.,
short s-t distance, large s-t flow, cheap MST) of a network (see the survey of [SPG13]). The study
of Shortest Path Interdiction started in 1980’s when the problem was called as the k-most-vital-arcs
problem [CD82, MMG89, BGV89] and proved to be NP-hard [BGV89]. Khachiyan et al. [KBB+ 07]
proved that it is NP-hard to approximate within a factor less than 2. While many heuristic algorithms were proposed [IW02, BB08, Mor11] and hardness in planar graphs [PS13] was shown,
whether the general version admits a constant factor approximation was still unknown.
Given a graph G = (V, E) and s, t ∈ V , let dist(G) be the length of the shortest s-t path. For
′
V ⊆ V , let G \ V ′ be the subgraph induced by V \ V ′ . For E ′ ⊆ E, we use the same notation
G \ E ′ to denote the subgraph (V, E \ E ′ ). We primarily study undirected graphs. We first present
our results for the vertex version of both problems (collectively called as Short Path Vertex Cut
onwards).
Theorem 1.3. Assume the Unique Games Conjecture. For infinitely many values of l ∈ N, given an
undirected graph G = (V, E) and s, t ∈ V where there exists C ∗ ⊆ V \ {s, t} such that dist(G \ C ∗ ) ≥ l,
it is NP-hard to perform any of the following tasks.
1. Find C ⊆ V \ {s, t} such that |C| ≤ Ω(l) · |C ∗ | and dist(G \ C) ≥ l.
√
2. Find C ⊆ V \ {s, t} such that |C| ≤ |C ∗ | and dist(G \ C) ≥ O( l).
ǫ
3. Find C ⊆ V \ {s, t} such that |C| ≤ Ω(l 2 ) · |C ∗ | and dist(G \ C) ≥ O(l
1+ǫ
2
) for some 0 < ǫ < 1.
The first result shows that Length Bounded Vertex Cut is hard to approximate within a factor
Ω(l). This matches the best 2l -approximation up to a constant. [BEH+ 10]. The second result shows
√
that Shortest Path Vertex Interdiction is hard to approximate with in a factor Ω( OPT), and the
third result rules out bicriteria approximation — for any constant c, it is hard to approximate both l
and |C ∗ | within a factor of c.
The above results hold for directed graphs by definition. Our hard instances will have a natural layered structure, so it can be easily checked that the same results (up to a constant) hold for
directed acyclic graphs. Since one vertex can be split as one directed edge, the same results hold
for the edge version in directed acyclic graphs.
For Length-Bounded Edge Cut and Shortest Path Edge Interdiction in undirected graphs (collectively called Short Path Edge Cut onwards), we prove the following theorems.
Theorem 1.4. Assume the Unique Games Conjecture. For infinitely many values of the constant l ∈ N,
given an undirected graph G = (V, E) and s, t ∈ V where there exists C ∗ ⊆ E such that dist(V \ C ∗ ) ≥ l,
it is NP-hard to perform any of the following tasks.
√
1. Find C ⊆ E such that |C| ≤ Ω( l) · |C ∗ | and dist(G \ C) ≥ l.
2
2. Find C ⊆ E such that |C| ≤ |C ∗ | and dist(G \ C) ≥ l 3 .
3
2ǫ
3. Find C ⊆ E such that |C| ≤ Ω(l 3 ) · |C ∗ | and dist(G \ C) ≥ O(l
2+2ǫ
3
) for some 0 < ǫ < 21 .
√
√
Our hardness factors for the edge versions, Ω( l) for Length-Bounded Edge Cut and Ω( 3 OPT)
for Shortest Path Edge Interdiction, are slightly weaker than those for their vertex counterparts,
but we are not aware of any approximation algorithm specialized for the edge versions. It is an
interesting open problem whether there exist better approximation algorithms for the edge versions.
RMFC. Resource Minimization for Fire Containment (RMFC) is a problem closely related to LengthBounded Cut with the additional notion of time. Given a graph G, a vertex s, and a subset T
of vertices, consider the situation where fire starts at s on Day 0. For each Day i (i ≥ 1), we
can save at most k vertices, and the fire spreads from currently burning vertices to its unsaved
neighbors. Once a vertex is burning or saved, it remains so from then onwards. The process is
terminated when the fire cannot spread anymore. RMFC asks to find a strategy to save k vertices
each day with the minimum k so that no vertex in T is burnt. These problems model the spread
of epidemics or ideas through a social network, and have been actively studied recently [CC10,
ACHS12, ABZ16, CV16].
RMFC, along with other variants, is first introduced by Hartnell [Har95]. Another well-studied
variant is called the Firefighter problem, where we are only given s ∈ V and want to maximize the
number of vertices that are not burnt at the end. It is known to be NP-hard to approximate within
a factor n1−ǫ for any ǫ > 0 [ACHS12]. King and MacGillivray [KM10] proved that RMFC is hard
√
to approximate within a factor less than 2. Anshelevich et al. [ACHS12] presented an O( n)approximation algorithm for general graphs, and Chalermsook and Chuzhoy [CC10] showed that
RMFC admits O(log∗ n)-approximation in trees. Very recently, the approximation ratio in trees
has been improved to O(1) [ABZ16]. Both Anshelevich et al. [ACHS12] and Chalermsook and
Chuzhoy [CC10] independently studied directed layer graphs with b layers, showing O(log b)approximation.
Our final result on RMFC assumes Conjecture 7.5, a variant of the Unique Games Conjecture
which is not known to be equivalent to the original UGC. Given a bipartite graph as an instance
of Unique Games, it states that in the completeness case, all constraints incident on (1 − ǫ) fraction
of vertices in one side are satisfied, and in the soundness case, in addition to having a low value,
9
1
fraction of vertices on one side have at least a 10
fraction of vertices on the other side as
every 10
neighbors. Our conjecture is implied by the conjecture of Bansal and Khot [BK09] that is used to
prove the hardness of Minimizing Weighted Completion Time with Precedence Constraints and requires
a more strict expansion condition. See Section 7 for the exact statement.
Theorem 1.5. Assuming Conjecture 7.5, it is NP-hard to approximate RMFC in undirected graphs within
any constant factor.
Again, our reduction has a natural layered structure and the result holds for directed layered
graphs. With b layers, we prove that it is hard to approximate with in a factor Ω(log b), matching
the best approximation algorithms [CC10, ACHS12].
1.2 Techniques
All our results are based on a general method of converting an integrality gap instance to a dictatorship test. This method has been successfully applied by Raghavendra [Rag08] for Max-CSPs,
Manokaran et al. [MNRS08] and Ene et al. [EVW13] for Multiway Cut and Min CSPs, and Kumar
4
et al. [KMTV11] for strict CSPs, and by Guruswami et al. [GSS15] for k-uniform k-partite Hypergraph Vertex Cover. As mentioned in the introduction, the previous CSP-based results do not
generally preserve the structure of constraint hypergraphs or use ingenious and specialized tricks
to reduce the problem to a CSP, so they are not applicable as a black-box to the graph cut problems
we consider.
We bypass this difficulty by constructing a special class of dictatorship tests that we call lengthcontrol dictatorship tests. Consider a meta-problem where given a directed graph G = (V, E), some
terminal vertices, and a set P of desired paths between terminals, we want to remove the fewest
number of non-terminal vertices to cut every path in P. The integrality gap instances we use in
this work [SSZ04, BEH+10, MM10, CC10] share the common feature that every p ∈ P is of length
at least r, and the fractional solution cuts 1r fraction of each non-terminal vertex so that each path
p ∈ P is cut. This gives a good LP value, and additional arguments are required to ensure that
there is no efficient integral cut.
Given such an integrality gap instance, we construct our dictatorship test instance as follows.
We replace every non-terminal vertex by a hypercube ZR
r and put edges such that for two vertices (v, x) and (w, y) where v, w ∈ V and x, y ∈ ZR
,
r there is an edge from (v, x) to (w, y) if
(1) (v, w) ∈ E and (2) yj = xj + 1 for all j ∈ [R]. The set of desired paths P ′ is defined to be
{(s, (v1 , x1 ), . . . , (vl , xl ), t) : (s, v1 , . . . , vl , t) ∈ P} (s, t denote some terminals). Note that each path
in P ′ is also of length at least r. We want to ensure that in the completeness case (i.e., every hypercube reveals the same influential coordinate), there is a very efficient cut, while in the soundness case
(i.e., no hypercube reveals an influential coordinate), there is no such efficient cut.
In the completeness case, let q ∈ [R] be an influential coordinate. For each vertex (v, x) where
v ∈ V, x ∈ ZR
r , remove (v, x) if xq = 0. Consider a desired path p = (s, (v1 , x1 ), . . . , (vl , xl ), t) ∈
P ′ for some terminals s, t and some vj ∈ V, xj ∈ ZR
r (1 ≤ j ≤ l), and let yj = (xj )q . By our
construction, yj+1 = yj + 1 for 0 ≤ j < l. Since p is desirable, l ≥ r, so there exists j such
that yj = (xj )q = 0, but (vj , xj ) is already removed by our previous definition. Therefore, every
desired path is cut by this vertex cut. Note that this cut is integral and cuts exactly 1r fraction of
non-terminal vertices. This corresponds to the fractional solution to the gap instance that cuts 1r
fraction of every vertex.
For the soundness analysis, our final dictatorship test has additional noise vertices and edges
to the test defined above. If no hypercube reveals an influential coordinate, the standard application of the invariance principle [Mos10] proves that we can always take an edge between two
hypercubes unless we almost completely cut one hypercube. We can then invoke the proof for the
integrality gap instance to show that there is no efficient cut.
This idea is implicitly introduced by the work of Svensson [Sve13] for Feedback Vertex Set
(FVS) and DAG Vertex Deletion (DVD) by applying the It ain’t over till it’s over theorem to ingeniously constructed dictatorship tests with auxiliary vertices. Guruswami and Lee [GL16] gave
a simpler construction and a new proof using the invariance principle instead of the It ain’t over
till it’s over theorem. Our results are based on the observation that length-control dictatorship
tests and LP gap instances fool algorithms in a similar way for various cut problems as mentioned
above, so that the previous LP gap instances can be plugged into our framework to prove matching hardness results.
This method for the above meta-problem can be almost directly applied to Directed Multicut.
For Length-Bounded Cut and RMFC in undirected graphs, we use the fact that the known integrality gap instances have a natural layered structure with s in the first layer and t in the last
layer. Every edge is given a natural orientation, and the similar analysis can be applied. For
Length-Bounded Cut, another set of edges called long edges are added to the dictatorship test.
More technical work is required for edge cut versions in undirected graphs (Short Path Edge Cut),
5
and the notion of time (RMFC).
Our framework seems general enough so that they can be applied to integrality gap instances
to give strong hardness results. Since each problem has slightly different characteristics as mentioned above, each application needs some specialized ideas and sometimes leads to sub-optimal
results. It would be interesting to further abstract this method of converting integrality gap instances to length-bounded dictatorship tests, as well as to apply it to other problems whose approximability is not well-understood.
2 Preliminaries
Graph Terminologies. Depending on whether we cut vertices or edges, we introduce weight
wt(v) for each vertex v, or weight wt(e) for each edge e. Some weights can be ∞, which means
that some vertices or edges cannot be cut. For vertex-weighted graphs, we naturally have wt(s) =
wt(t) = ∞. To reduce the vertex-weighted version to the unweighted version, we duplicate each
vertex according to its weight and replace each edge by a complete bipartite graph between corresponding copies. To reduce the edge-weighted version to the unweighted version, we replace a
single edge with parallel edges according to its weight. To reduce to simple graphs, we split each
parallel into two edges by introducing a new vertex.
For the Length-Bounded Cut problems, we also introduce length len(e) for each edge e. It
can be also dealt with serially splitting an edge according to its weight. We allow weights to be
rational numbers, but as our hardness results are stated in terms of the length, all lengths in this
work will be a positive integer.
For a path p, depending on the context, we abuse notation and interpret it as a set of edges or
a set of vertices. The length of p is always defined to be the number of edges.
Gaussian Bounds for Correlated Spaces. We introduce the standard tools on correlated spaces
from Mossel [Mos10]. Given a probability space (Ω, µ) (we always consider finite probability
spaces), let L(Ω) be the set of functions {f : Ω → R} and for an interval I ⊆ R,P
LI (Ω) be the
set of functions {f : Ω → I}. For a subset S ⊆ Ω, define measure of S to be µ(S) := ω∈S µ(ω). A
collection of probability spaces are said to be correlated if there is a joint probability distribution on
them. We will denote k correlated spaces Ω1 , . . . , Ωk with a joint distribution µ as (Ω1 ×· · ·×Ωk , µ).
Given two correlated spaces (Ω1 × Ω2 , µ), we define the correlation between Ω1 and Ω2 by
ρ(Ω1 , Ω2 ; µ) := sup {Cov[f, g] : f ∈ L(Ω1 ), g ∈ L(Ω2 ), Var[f ] = Var[g] = 1} .
Given a probability space (Ω, µ) and a function f ∈ L(Ω) and p ∈ R+ , let kf kp := Ex∼µ [|f (x)|p ]1/p .
Consider a product space (ΩR , µ⊗R ) and f ∈ L(ΩR ). The Efron-Stein decomposition of f is given
by
X
f (x1 , . . . , xR ) =
fS (xS )
S⊆[R]
where (1) fS depends only on xS and (2) for all S 6⊆ S ′ and all xS ′ , Ex′ ∼µ⊗R [fS (x′ )|x′S ′ = xS ′ ] = 0.
The influence of the ith coordinate on f is defined by
Inf i [f ] :=
[Var[f (x1 , . . . , xR )].
E
x1 ,...,xi−1 ,xi+1 ,...,xR xi
The influence has a convenient expression in terms of the Efron-Stein decomposition.
X
X
Inf i [f ] = k
fS k22 =
kfS k22 .
S:i∈S
S:i∈S
6
We also define the low-degree influence of the ith coordinate.
X
Inf ≤d
[f
]
:=
kfS k22 .
i
S:i∈S,|S|≤d
For a, b ∈ [0, 1] and ρ ∈ (0, 1), let
Γρ (a, b) := Pr[X ≤ Φ−1 (a), Y ≥ Φ−1 (1 − b)],
where X and Y are ρ-correlated standard Gaussian variables and Φ denotes the cumulative distribution function of a standard Gaussian. The following theorem bounds the product of two
functions that do not share an influential coordinate in terms of their Gaussian counterparts.
Theorem 2.1 (Theorem 6.3 and Lemma 6.6 of [Mos10]). Let (Ω1 × Ω2 , µ) be correlated spaces such
that the minimum nonzero probability of any atom in Ω1 × Ω2 is at least α and such that ρ(Ω1 , Ω2 ; µ) ≤ ρ.
R
Then for every ǫ > 0 there exist τ, d depending on ǫ and α such that if f : ΩR
1 → [0, 1], g : Ω2 → [0, 1]
≤d
≤d
satisfy min(Inf i [f ], Inf i [g]) ≤ τ for all i, then E(x,y)∈µ⊗R [f (x)g(y)] ≥ Γρ (Ex [f ], Ey [g]) − ǫ.
Organization. Section 3 shows the dictatorship tests for Directed Multicut. We present our dictatorship tests for Short Path Edge Cut and Short Path Vertex Cut in Section 4 and Section 5 respectively. The dictatorship tests for RMFC are presented in Section 6. These tests will be used in
Section 7 to prove hardness results based on the UGC.
3 Directed Multicut
We propose our dictatorship test for Directed Vertex Multicut that will be used for proving Unique
Games hardness. Note that hardness of Directed Edge Multicut easily follows from that of the
vertex version by splitting each vertex. Our dictatorship test is inspired by the integrality gap for
the standard LP constructed by Saks et al. [SSZ04], and parameterized by positive integers r, k, R
and small ǫ > 0, where k in this section denotes the number of (si , ti ) pairs for Directed Multicut.
All graphs in this section are directed.
M
= (V, E) be the graph defined as follows.
For positive integers r, k, R, and ǫ > 0, define Dr,k,R,ǫ
Consider the probability space (Ω, µ) where Ω := {0, . . . , r − 1, ∗}, and µ : Ω 7→ [0, 1] with µ(∗) = ǫ
and µ(x) = 1−ǫ
r for x 6= ∗.
• V = {si , ti }1≤i≤k ∪ {vxα }α∈[r]k ,x∈ΩR . Let v α denote the set of vertices {vxα }x∈ΩR .
• For α ∈ [r]k and x ∈ ΩR , wt(vxi ) = µ⊗R (x). Note that the sum of weights is r k .
• For any i ∈ [k], there are edges from si to {vxα : α ∈ [r]k , αi = 1, x ∈ ΩR }, and edges from
{vxα : α ∈ [r]k , αi = r, x ∈ ΩR } to ti .
• For α, β ∈ [r]k and x, y ∈ ΩR , we have an edge from vxα to vyβ if α 6= β and
– For any 1 ≤ i ≤ r: αi − βi ∈ {−1, 0, +1}.
– For any 1 ≤ j ≤ R: [yj = (xj + 1) mod r] or [yj = ∗] or [xj = ∗].
7
Completeness. We first prove that vertex cuts that correspond to dictators behave the same as
the fractional solution that gives 1r to every vertex. For any q ∈ [R], let Vq := {vxα : α ∈ [r]k , xq =
k−1 (1 + ǫr).
∗ or 0}. Note that the total weight of Vq is r k (ǫ + 1−ǫ
r )≤r
Lemma 3.1. After removing vertices in Vq , there is no path from si to ti for any i.
Proof. Fix i and let p = (si , vxα11 , . . . , vxαzz , ti ) be a path from si to ti where αj ∈ [r]k and xj ∈ ΩR for
each 1 ≤ j ≤ z. Let yj := (xj )q for each 1 ≤ j ≤ z. The construction ensures that yj+1 = (yj + 1)
mod r, so after removing vertices in Vq , z must be strictly less than r. Since any path from si to ti
must contain at least r non-terminal vertices, there must be no path from si to ti .
Soundness. To analyze soundness, we define a correlated probability space (Ω1 × Ω2 , ν) where
both Ω1 , Ω2 are copies of Ω = {0, . . . , r − 1, ∗}. It is defined by the following process to sample
(x, y) ∈ Ω2 .
• Sample x ∈ {0, . . . , r − 1}. Let y = (x + 1) mod r.
• Change x to ∗ with probability ǫ. Do the same for y independently.
1
Note that the marginal distribution of both x and y is equal to µ. Assuming ǫ < 2r
, the minimum
2
probability of any atom in Ω1 × Ω2 is ǫ . We use the following lemma to bound the correlation
ρ(Ω1 , Ω2 ; ν).
Lemma 3.2 (Lemma 2.9 of [Mos10]). Let (Ω1 × Ω2 , µ) be two correlated spaces such that the probability
of the smallest atom in Ω1 × Ω2 is at least α > 0. Define a bipartite graph G = (Ω1 ∪ Ω2 , E) where
2
(a, b) ∈ Ω1 × Ω2 satisfies (a, b) ∈ E if µ(a, b) > 0. If G is connected, then ρ(Ω1 , Ω2 ; µ) ≤ 1 − α2 .
In our correlated space, the bipartite graph on Ω1 ∪ Ω2 is connected since every x ∈ Ω1 is
4
connected to ∗ ∈ Ω2 and vice versa. Therefore, we can conclude that ρ(Ω1 , Ω2 ; ν) ≤ ρ := 1 − ǫ2 .
Γρ ( ǫ , ǫ )
3 3
) to get τ and d. We will later apply this
Apply Theorem 2.1 (ρ ← ρ, α ← ǫ2 , ǫ ←
2
theorem with the parameters obtained here. Fix an arbitrary subset C ⊆ V , and let Cα := C ∩ v α .
For α ∈ [r]k , call v α blocked if µ⊗R (Cα ) ≥ 1 − ǫ. The number of blocked v α ’s is at most wt(C)
1−ǫ .
Consider the following graph D = (VD , ED ), which is the original integrality gap instance
constructed by Saks et al. [SSZ04].
• VD = {si , ti }i∈[k] ∪ {v α }α∈[r]k .
• For any i ∈ [k], there are edges from si to {v α : α ∈ [r]k , αi = 1}, and edges from {v α : α ∈
[r]k , αi = r} to ti .
• For α, β ∈ [r]k , we have an edge from v α to v β if α 6= β and 1 ≤ i ≤ r: αi − βi ∈ {−1, 0, +1}.
Saks et al. [SSZ04] proved the following theorem in their analysis of their integrality gap.
Theorem 3.3. Let C ′ be a set of less than k(r − 1)k−1 vertices. There exists a path (si , v α1 , . . . , v αz , ti ) for
some i that does not intersect C ′ .
Setting C ′ := {v α ∈ VD : v α is not blocked.}, and applying Theorem 3.3 concludes that unless
wt(C) ≥ (1 − ǫ) · k · (r − 1)k−1 , there exists a path (si , v α1 , . . . , v αz , ti ) where each v αi is unblocked
for some i ∈ [k].
αj−1 αj
For 1 ≤ j ≤ z, let Sj ⊆ v αj be such that x ∈ Sj if there exists a path (s, vxα11 , . . . , vxj−1
, vx ) for
1
j−1
R
some x , . . . , x . For 1 ≤ j ≤ z, let fj : Ω 7→ {0, 1} be the indicator function of Sj . We prove
that if none of fj reveals any influential coordinate, µ⊗R (Sz ) > 0, which shows that there exists a
si -ti path even after removing vertices in C.
8
⊗R (S ) > 0.
Lemma 3.4. Suppose that for any 1 ≤ j ≤ z and 1 ≤ i ≤ R, Inf ≤d
z
i [fj ] ≤ τ . Then µ
Proof. We prove by induction that µ⊗R (Sj ) ≥ 3ǫ . It holds when j = 1 since v α1 is unblocked.
Assuming µ⊗R (Sj ) ≥ 3ǫ , since Sj does not reveal any influential coordinate, Theorem 2.1 shows
that for any subset Tj+1 ⊆ v αj+1 with µ⊗R (Tj+1 ) ≥ 3ǫ , there exists an edge from Sj and Tj+1 . If
′
′
Sj+1
⊆ v αj+1 is the set of out-neighbors of Sj , we have µ⊗R (Sj+1
) ≥ 1− 3ǫ . Since v αj+1 is unblocked,
′
µ⊗R (Sj+1
\ C) ≥ 2ǫ
3 , completing the induction.
In summary, in the completeness case, if we cut vertices of total weight r k−1 (1 + ǫr), we cut
every si -ti pair. In the soundness case, unless we cut vertices of total weight at least (1 − ǫ) · k ·
k−1
(r − 1)k−1 , we cannot cut every si -ti pair. The gap is k(1−ǫ)(r−1)
. For a fixed k, increasing r and
(1+ǫr)r k−1
decreasing ǫ faster makes the gap arbitrarily close to k.
4 Short Path Edge Cut
We propose our dictatorship test for Short Path Edge Cut that will be used for proving Unique
Games hardness. It is parameterized by positive integers a, b, r, R. It is inspired by the integrality
gap instances by Baier et al. [BEH+ 10] Mahjoub and and McCormick [MM10], and made such
that the edge cuts that correspond to dictators behave the same as the fractional solution that cuts
1
r fraction of every edge. All graphs in this section are undirected.
E
For positive integers a, b, r, R, we construct Da,b,r,R
= (V, E). Let Ω = {0, . . . , r − 1}, and
1
µ : Ω 7→ [0, 1] with µ(x) = r for each x ∈ Ω. We also define a correlated probability space
(Ω1 × Ω2 , ν) where both Ω1 , Ω2 are copies of Ω. It is defined by the following process to sample
(x, y) ∈ Ω2 .
• Sample x ∈ {0, . . . , r − 1}. Let y = (x + 1) mod r.
• With probability 1− 1r , output (x, y). Otherwise, resample x, y ∈ Ω independently and output
(x, y).
Note that the marginal distribution of both x and y is equal to µ. Given x = (x1 , . . . , xR ) ∈ ΩR and
Q
E
y = (y1 , . . . , yR ) ∈ ΩR , let ν ⊗R (x, y) = R
i=1 ν(xi , yi ). We define Da,b,r,R = (V, E) as follows.
• V = {s, t} ∪ {vxi }0≤i≤b,x∈ΩR . Let v i denote the set of vertices {vxi }x∈ΩR .
• For any x ∈ ΩR , there is an edge from s to vx0 and an edge from vxb to t, both with weight ∞
and length 1.
• For 0 ≤ i < b, x ∈ ΩR , there is an edge (vxi , vxi+1 ) of length a and weight ∞. Call it a long edge.
• For any 0 ≤ i < b x, y ∈ ΩR , there is an edge (vxi , vyi+1 ) of length 1 and weight ν ⊗R (x, y).
Note that ν ⊗R (x, y) > 0 for any x, y ∈ ΩR . Call it a short edge. The sum of finite weights is b.
Completeness. We first prove that edge cuts that correspond to dictators behave the same as the
fractional solution that gives 1r to every edge. Fix q ∈ [R] and let Eq be the set of short edges
defined by
Eq := {(vxi , vyi+1 ) : 0 ≤ i < b, yq 6= xq + 1 mod R or (xq , yq ) = (0, 1)}.
When (x, y) ∈ Ω1 ×Ω2 is sampled according to ν, the probability that yq 6= xq +1 mod R or (xq , yq ) =
(0, 1) is at most 2r . The total weight of Eq is 2b
r .
9
Lemma 4.1. After removing edges in Eq , the length of the shortest path is at least a(b − r + 1).
Proof. Let p = (s, vxi11 , . . . , vxizz , t) be a path from s to t where ij ∈ {0, . . . , b} and xj ∈ ΩR for each
1 ≤ j ≤ z. Let yj := (xj )q ∈ {0, . . . , r − 1} for each 1 ≤ j ≤ z.
For each 1 ≤ j < z, the edge (pj , pj+1 ) is either a long edge or a short edge, and either taken
forward (i.e., ij < ij+1 ) or backward (i.e., ij > ij+1 ). Let zLF , zSF , zLB , zSB be the number of long
edges taken forward, short edges taken forward, long edges taken backward, and shot edges taken
backward, respectively (zLF + zSF + zLB + zSB = z − 1). By considering how ij changes,
zLF + zSF − zLB − zSB = b.
(1)
Consider how yj changes. Taking a long edge does not change yj . Taking a short edge forward
increases yj by 1 mod r, taking a short edge backward decreases yj by 1 mod r. Since Eq is cut, yj
can never change from 0 to 1. This implies
zSF − zSB ≤ r − 1.
(2)
(1) − (2) yields zLF − zLB ≥ b − r + 1. The total length of p is at least a · zLF ≥ a(b − r + 1).
Soundness. We first bound the correlation ρ(Ω1 , Ω2 ; ν). The following lemma of Wenner [Wen13]
gives a convenient way to bound the correlation.
Lemma 4.2 (Corollary 2.18 of [Wen13]). Let (Ω1 × Ω2 , δµ + (1 − δ)µ′ ) be two correlated spaces such
that the marginal distribution of at least one of Ω1 and Ω2 is identical on µ and µ′ . Then,
p
ρ(Ω1 , Ω2 ; δµ + (1 − δ)µ′ ) ≤ δ · ρ(Ω1 , Ω2 ; µ)2 + (1 − δ) · ρ(Ω1 , Ω2 ; µ′ )2 .
When (x, y) is sampled from ν, they are completely independent with probability 1r . Therefore,
q
we have ρ := ρ(Ω1 , Ω2 ; ν) ≤ 1 − 1r . By Sheppard’s Formula,
∞
X
(2n)!
1
1
1
1
1
1
1
1 1
arcsin(−ρ) ≥ −
arccos( √ ) =
( √ )2n+1 ≥ √ .
Γρ ( , ) = +
2 2
4 2π
4 2π
r
4n (n!)2 (2n + 1) r
r
n=0
Γρ ( 1 , 1 )
2 2
Apply Theorem 2.1 (ρ ← ρ, α ← r13 , ǫ ←
) to get τ and d. We will later apply this
3
theorem with the parameters obtained here.
Fix an arbitrary subset C ⊆ E of short edges. For 0 ≤ i < b, let Ci = C ∩ (v i × v i+1 ). Call a pair
Γρ ( 1 , 1 )
2 2
. Let b′ be the number of
(i, i + 1) as the ith layer, and say it is blocked when ν ⊗R (Ci ) ≥
2
blocked layers. For 0 ≤ i ≤ b, let Si ⊆ v i be such that x ∈ Si if there exists a path (s, p0 , . . . , pi = vxi )
such that
′
• For 0 ≤ i′ ≤ i, pi′ ∈ v i .
• For 0 ≤ i′ < i, (pi′ , pi′ +1 ) is short if and only if the i′ th layer is unblocked.
Let fi : ΩR 7→ [0, 1] be the indicator function of Si . We prove that if none of fi reveals any
influential coordinate, Sb is nonempty, implying that there exists a path using b′ long edges and
b − b′ short edges. . Therefore, even after removing edges in C, the length of the shortest path is at
most 2 + ab′ + (b − b′ ).
Lemma 4.3. Suppose that for any 0 ≤ i ≤ b and 1 ≤ j ≤ R, Inf ≤d
j [fi ] ≤ τ . Then Sb 6= ∅.
10
Proof. Assume towards contradiction that Sb = ∅. Since S0 = ΩR and Si = Si+1 if the ith layer
is blocked (and we use long edges), there must exist i such that the ith layer is unblocked and
µ⊗R (Si ) ≥ 21 , µ⊗R (Si+1 ) < 12 . All short edges between Si and v i+1 \ Si+1 are in Ci . Theorem 2.1
implies that ν ⊗R (Ci ) > 32 Γρ ( 12 , 12 ). This contradicts the fact that the ith layer is unblocked.
In summary, in the completeness case, if we cut edges of total weight k := k(a, b, r) = 2b
r , the
length of the shortest path is at least l := l(a, b, r) = a(b − r + 1). In the soundness case, even
′
′ √r layers are blocked, the length of the
after cutting edges of total weight k′ , at most Γ 2k
1 1 ≤ 2k
(
,
)
ρ 2 2
√
√
shortest path is at most l′ = 2 + (b − 2k′ r) + 2ak ′ r.
√
• Let a = 4, b = 2r −√
1 so that k ≤ 4, l = 4r. Requiring l′ ≥ l results in k′ = Ω( r), giving a
√
gap of Ω( r) = Ω( l) between the completeness case and the soundness case for LengthBounded Edge Cut.
√
• Let a = r, b = 2r − 1 so that k ≤ 4, l = r 1.5 . Requiring k′ ≤ 4 results in l′ = O(r), giving
√
a gap of Ω( r) = Ω(l1/3 ) for Shortest Path Interdiction. Generally, k′ ≤ O(r ǫ ) results in
l′ ≤ O(r 1+ǫ ), giving an (O(r ǫ ), O(r 1/2−ǫ ))-bicriteria gap for any ǫ ∈ (0, 21 ).
5 Short Path Vertex Cut
We propose our dictatorship test for Short Path Vertex Cut that will be used for proving Unique
Games hardness. It is parameterized by positive integers a, b, r, R and small ǫ > 0. It is inspired
by the integrality gap instances by Baier et al. [BEH+ 10] Mahjoub and and McCormick [MM10],
and made such that the vertex cuts that correspond to dictators behave the same as the fractional
solution that cuts 1r fraction of every vertex. All graphs in this section are undirected.
V
= (V, E) be the graph defined as
For positive integers a, b, r, R, and ǫ > 0, define Da,b,r,R,ǫ
follows. Consider the probability space (Ω, µ) where Ω := {0, . . . , r − 1, ∗}, and µ : Ω 7→ [0, 1] with
µ(∗) = ǫ and µ(x) = 1−ǫ
r for x 6= ∗.
• V = {s, t} ∪ {vxi }0≤i≤b,x∈ΩR . Let v i denote the set of vertices {vxi }x .
• For 0 ≤ i ≤ b and x ∈ ΩR , wt(vxi ) = µ⊗R (x). Note that the sum of weights is b + 1.
• For any 0 ≤ i ≤ b, there are edges from s to each vertex in vi with length ai + 1 and edges
from each vertex in vi to t with length (b − i)a + 1.
• For x, y ∈ ΩR , we call that x and y are compatible if
– For any 1 ≤ j ≤ R: [yj = (xj + 1) mod r] or [yj = ∗] or [xj = ∗].
• For any 0 ≤ i < b and compatible x, y ∈ ΩR , we have an edge (vxi , vyi+1 ) of length 1 (called a
short edge).
• For any i, j such that 0 ≤ i < j − 1 < b and compatible x, y ∈ ΩR , we have an edge (vxi , vyj )
of length (j − i)a (called a long edge).
11
Completeness. We first prove that vertex cuts that correspond to dictators behave the same as
the fractional solution that gives 1r to every vertex. For any q ∈ [R], let Vq := {vxi : 0 ≤ i ≤ b, xq =
∗ or 0}. Note that the total weight of Vq is (b + 1)(ǫ + 1−ǫ
r ).
Lemma 5.1. After removing vertices in Vq , the length of the shortest path is at least a(b − r + 2).
Proof. Let p = (s, vxi11 , . . . , vxizz , t) be a path from s to t where ij ∈ {0, . . . , b} and xj ∈ ΩR for each
1 ≤ j ≤ z. Let yj := (xj )q ∈ {0, . . . , r − 1} for each 1 ≤ j ≤ z.
i
i
For each 1 ≤ j < z, the edge (vxjj , vxj+1
j+1 ) is either a long edge or a short edge, and either
taken forward (i.e., ij < ij+1 ) or backward (i.e., ij > ij+1 ). Let zLF , zSF , zLB , zSB be the number of
long edges taken forward, short edges taken forward, long edges taken backward, and shot edges
taken backward, respectively (zLF + zSF + zLB + zSB = z − 1). For 1 ≤ j ≤ zLF (resp. zLB ), consider
i
′
i
′
the jth long edge taken forward (resp. backward) — it is (vxjj′ , vxjj′+1
) for some j ′ . Let sFj (resp. sB
+1
j)
be |ij ′ − ij ′ +1 |. The following equality holds by observing how ij changes.
i1 +
zLF
X
j=1
sFj
+ zSF −
zLB
X
sB
j
j=1
− zSB = iz
⇒
i1 +
zLF
X
j=1
sFj + zSF − zLB − zSB − iz ≥ 0.
(3)
Consider how yj changes. Taking any edge forward increases yj , and taking any edge backward
decreases yj . Since yj can never be 0 or ∗, we can conclude that
zLF + zSF − zLB − zSB ≤ r − 2.
(4)
(3) − (4) yields
i1 − iz +
zLF
X
j=1
(sFj
− 1) ≥ 2 − r
⇒
i1 − iz +
sFj
zLB
X
zLF
X
j=1
sFj ≥ 2 − r.
(5)
The total length of p is
2 + a i1 + b − iz +
≥
a i1 + b − iz +
≥
a(b − r + 2).
zLF
X
zLF
X
j=1
+
sB
j ) + zSF + zSB
j=1
sFj )
j=1
Soundness. To analyze soundness, we define a correlated probability space (Ω1 × Ω2 , ν) where
both Ω1 , Ω2 are copies of Ω = {0, . . . , r − 1, ∗}. It is defined by the following process to sample
(x, y) ∈ Ω2 .
• Sample x ∈ {0, . . . , r − 1}. Let y = (x + 1) mod r.
• Change x to ∗ with probability ǫ. Do the same for y independently.
12
1
, the minimum
Note that the marginal distribution of both x and y is equal to µ. Assuming ǫ < 2r
2
probability of any atom in Ω1 × Ω2 is ǫ . Furthermore, in our correlated space, ν(x, ∗) > 0 for
all x ∈ Ω1 and ν(∗, x) > 0 for all x ∈ Ω2 . Therefore, we can apply Lemma 3.2 to conclude that
4
ρ(Ω1 , Ω2 ; ν) ≤ ρ := 1 − ǫ2 .
Γρ ( ǫ , ǫ )
3 3
Apply Theorem 2.1 (ρ ← ρ, α ← ǫ2 , ǫ ←
) to get τ and d. We will later apply this theorem
2
with the parameters obtained here. Fix an arbitrary subset C ⊆ V , and Ci := C ∩ v i . For 0 ≤ i ≤ b,
i
′
call v i blocked if µ⊗R [Ci (x)] ≥ 1 − ǫ. At most ⌊ wt(C)
1−ǫ ⌋ v ’s can be blocked. Let k be the number of
blocked v i ’s, and z = b + 1 − k′ be the number of unblocked v i ’s. Let {v i1 , . . . , v iz } be the set of
unblocked v i ’s with i1 < i2 < · · · < iz .
i
For 1 ≤ j ≤ z, let Sj ⊆ v ij be such that x ∈ Sj if there exists a path (p0 = s, p1 , . . . , pj−1 , vxj )
such that each pj ′ ∈ v ij′ \ C (1 ≤ j ′ < j). For 1 ≤ j ≤ z, let fj : ΩR 7→ [0, 1] be the indicator function
of Sj .
We prove that if none of fj reveals any influential coordinate, µ⊗R (Sz ) > 0. Since any path
passing v i1 , . . . , v iz (bypassing only blocked v i ’s) uses short edges at least b − 2k′ times, so the
length of the shortest path after removing C is at most 2 + (b − 2k′ ) + 2ak ′ .
⊗R (S ) > 0.
Lemma 5.2. Suppose that for any 1 ≤ j ≤ z and 1 ≤ i ≤ R, Inf ≤d
z
i [fj ] ≤ τ . Then µ
Proof. We prove by induction that µ⊗R (Sj ) ≥ 3ǫ . It holds when j = 1 since v i1 is unblocked.
Assuming µ⊗R (Sj ) ≥ 3ǫ , since Sj does not reveal any influential coordinate, Theorem 2.1 shows
that for any subset Tj+1 ⊆ v ij+1 with µ⊗R (Tj+1 ) ≥ 3ǫ , there exists an edge between Sj and Tj+1 . If
′
′
Sj+1
⊆ v ij+1 is the set of neighbors of Sj , we have µ⊗R (Sj+1
) ≥ 1 − 3ǫ . Since v ij+1 is unblocked,
2ǫ
′
µ⊗R (Sj+1
\ C) ≥ 3 , completing the induction.
In summary, in the completeness case, if we cut vertices of total weight k := k(a, b, r, ǫ) =
(b + 1)(ǫ + 1−ǫ
r ), the length of the shortest path is at least l := l(a, b, r, ǫ) = a(b − r + 2). In the
soundness case, even after cutting vertices of total weight k′ , the length of the shortest path is at
k′
k′
) + 2a( 1−ǫ
).
most 2 + (b − 1−ǫ
• Let a = 4, b = 2r − 2 and ǫ small enough so that k ≤ 2, l = 4r. Requiring l′ ≥ l results in
k′ = Ω(r), giving a gap of Ω(r) = Ω(l) for Length Bounded Cut.
• Let a = r, b = 2r − 2 and ǫ small enough
so that k ≤ 2, l = r 2 . Requiring k′ ≤ 2 results in
√
′
l = O(r), giving a gap of Ω(r) = Ω( l) for Shortest Path Interdiction. Generally, k′ ≤ O(r ǫ )
results in l′ ≤ O(r 1+ǫ ), giving an (O(r ǫ ), O(r 1−ǫ ))-bicriteria gap for any ǫ ∈ (0, 1).
6 RMFC
We present our dictatorship test for the RMFC problem. Our test is inspired by the integrality gap
example in Chalermsook and Chuzhoy [CC10], which is suggested by Khanna and Olver. This
test will be used in Section 7 to prove the hardness result based on Conjecture 7.5. All graphs in
this section are undirected. We will prove hardness of RMFC where T = {t} for a single vertex t.
P
Given positive integers b and R, let B = (b!) · ( bi=1 b!i ), Ω = {∗, 1, . . . , B}R . Consider the
probability space (Ω, µ) where µ : Ω 7→ [0, 1] with µ(∗) = ǫ and µ(x) = 1−ǫ
B for x 6= ∗. We define
F
Db,R,ǫ = (V, E) as follows.
• V = {s, t} ∪ ({vxi }1≤i≤b,x∈ΩR ). Let v i := {vxi }x∈ΩR . The weight a vertex vxi is i · µ⊗R (x).
• There is an edge from s to each vertex in L1 , from each vertex in Lb to t.
13
• For x, y ∈ ΩR , we call that x and y are compatible if
– For any 1 ≤ j ≤ R: [yj = xj ] or [yj = ∗] or [xj = ∗].
• For any 0 ≤ i < b and compatible x, y ∈ ΩR , we have an edge (vxi , vyi+1 ).
Completeness.
We first prove that vertex cuts that correspond to dictators are efficient. Let Hi =
Pi
i!
j=1 j
Hi
B and B0 = 0.
be the ith harmonic number. For 1 ≤ i ≤ b, let Bi = H
b
Pb b!
Pb
Each Bi is an integer since B = (b!) · ( i=1 i ), and i=1 (Bi − Bi−1 ) = B.
For any q ∈ [R], we consider the solution where on Day i (1 ≤ i ≤ b), we save
1+
1
2
+ ··· +
1
i
=
i!
Vqi := {vxi : xq = ∗ or Bi−1 + 1 ≤ xq ≤ Bi }.
Note each day the total weight that the total weight of Vq is i(ǫ +
Lemma 6.1. In above solution, t is never burnt.
1
i·Hb )
≤ bǫ +
1
Hb .
Proof. Fix an arbitrary p = (s, vxi11 , . . . , vxizz , t) from s to t, and let yj = (xj )q (1 ≤ j ≤ z). Since ij ≤ j
i
for any j, Vqi is saved before we arrive vxjj . Therefore y1 = y2 = · · · = yz . There exists r ∈ {1, . . . , b}
such that y ∈ {Br−1 + 1, . . . , Br }. p intersects Vqr .
Soundness. To analyze soundness, we define a correlated probability space (Ω1 × Ω2 , ν) where
both Ω1 , Ω2 are copies of Ω = {∗, 1, . . . , B}. It is defined by the following process to sample
(x, y) ∈ Ω2 .
• Sample x ∈ {1, . . . , B}. Let y = x.
• Change x to ∗ with probability ǫ. Do the same for y independently.
1
, the minimum
Note that the marginal distribution of both x and y is equal to µ. Assuming ǫ < 2B
probability of any atom in Ω1 × Ω2 is ǫ2 . Furthermore, in our correlated space, ν(x, ∗) > 0 for
all x ∈ Ω1 and ν(∗, x) > 0 for all x ∈ Ω2 . Therefore, we can apply Lemma 3.2 to conclude that
Γρ ( 1 , 1 )
4
3 3
) to get τ and d. We will
ρ(Ω1 , Ω2 ; ν) ≤ ρ := 1 − ǫ2 . Apply Theorem 2.1 (ρ ← ρ, α ← ǫ2 , ǫ ←
2
later apply this theorem with the parameters obtained here.
Fix an arbitrary solution where we save Ci ⊆ V on Day i with wt(Ci ) ≤ k′ . Let Si ⊆ v i be the
set of vertices of v i burnt at the end of Day i. Let fi : ΩR 7→ [0, 1] be the indicator function of Si
(1 ≤ i ≤ b). We prove that if none of fi reveals any influential coordinate, unless k′ is large, µ⊗R (Si )
is large for all i, so t will be burnt on Day b + 1.
1
′
⊗R (S ) ≥
Lemma 6.2. Suppose that for any 1 ≤ i ≤ b and 1 ≤ j ≤ R, Inf ≤d
i
j [fi ] ≤ τ . If k ≤ 3 , µ
all 1 ≤ i ≤ b.
1
3
for
Proof. We prove by induction on i. It is easy to see µ⊗R (S1 ) ≥ 13 since the wt(v 1 ) = 1 but k′ ≤ 13 .
Suppose that the claim holds for i. For any T ⊆ v i+1 with µ⊗R (T ) ≤ 13 , since Si does not reveal any
influential coordinate, Theorem 2.1 shows that there exists an edge between Si and T . It implies
that µ⊗R (N (Si )) ≥ 32 , where N (Si ) ⊆ v i+1 denotes the set of neighbors of Si in v i+1 . The total
weight of saved vertices up to Day i is at most ik′ ≤ 3i . Since wt(v i ) = i, even if all saved vertices
are in v i , µ⊗R (v i ∩ (C1 ∪ · · · ∪ Ci )) ≤ 31 . Since Si+1 = N (Si ) \ (C1 ∪ · · · ∪ Ci ), µ⊗R (Si+1 ) ≥ 31 , the
induction is complete.
In summary, in the completeness case, we save vertices of total weight at most bǫ + H1b and
save t. In the soundness case, we fail to save t unless we spend total weight at least 31 each day. By
taking ǫ small enough, the gap becomes Ω(log b).
14
7 Unique Games Hardness
7.1 UGC and Variant
We introduce the Unique Games Conjecture and its equivalent variant.
Definition 7.1. An instance L(B(UB ∪ WB , EB ), [R], {π(u, w)}(u,w)∈EB ) of Unique Games consists of
a biregular bipartite graph B(UB ∪ WB , EB ) and a set [R] of labels. For each edge (u, w) ∈ EB there is a
constraint specified by a permutation π(u, w) : [R] → [R]. The goal is to find a labeling l : UB ∪ WB → [R]
of the vertices such that as many edges as possible are satisfied, where an edge e = (u, w) is said to be satisfied
if l(u) = π(u, w)(l(w)).
Definition 7.2. Given a Unique Games instance L(B(UB ∪WB , EB ), [R], {π(u, w)}(u,w)∈EB ), let Opt(L)
denote the maximum fraction of simultaneously-satisfied edges of L by any labeling, i.e.
Opt(L) :=
1
|EB |
max
l:UB ∪WB →[R]
| {e ∈ E : l satisfies e} |.
Conjecture 7.3 (The Unique Games Conjecture [Kho02]). For any constants η > 0, there is R = R(η)
such that, for a Unique Games instance L with label set [R], it is NP-hard to distinguish between
• opt(L) ≥ 1 − η.
• opt(L) ≤ η.
To show the optimal hardness result for Vertex Cover, Khot and Regev [KR08] introduced the
following seemingly stronger conjecture, and proved that it is in fact equivalent to the original
Unique Games Conjecture.
Conjecture 7.4 (Khot and Regev [KR08]). For any constants η > 0, there is R = R(η) such that, for a
Unique Games instance L with label set [R], it is NP-hard to distinguish between
• There is a set W ′ ⊆ WB such that |W ′ | ≥ (1 − η)|WB | and a labeling l : UB ∪ WB → [R] that
satisfies every edge (u, w) for v ∈ UB and w ∈ W ′ .
• opt(L) ≤ η.
For RMFC, we use the following variant of Unique Games, which is not known to be equivalent to the original Conjecture.
Conjecture 7.5. For any constants η > 0, there is R = R(η) such that, for a Unique Games instance L
with label set [R], it is NP-hard to distinguish between
• There is a set W ′ ⊆ WB such that |W ′ | ≥ (1 − η)|WB | and a labeling l : UB ∪ WB → [R] that
satisfies every edge (u, w) for v ∈ UB and w ∈ W ′ .
• opt(L) ≤ η. Moreover, the instance satisfies the following expansion property: For every set S ⊆
9
|UB |, where N (S) := {v ∈ UB : ∃w ∈ S, (v, w) ∈ EB }.
WB , |S| = |W10B | , we have |N (S)| ≥ 10
Conjecture 7.5 is similar to that of Bansal and Khot [BK09], under which the optimal hardness
of Minimizing Weighted Completion Time with Precedence Constraints is proved. Their conjecture
requires that in the soundness case, ∀S ⊆ WB with |S| = δ|WB |, we must have |N (S)| ≥ (1−δ)|UB |
for arbitrarily small δ. Our conjecture is a weaker (so more likely to hold) since we require this
1
.
condition for only one value δ = 10
15
7.2 General Reduction
We now introduce our reduction from Unique Games to our problems Short Path Edge Cut, Short
E
,
Path Vertex Cut, Directed Multicut, and RMFC. We constructed four dictatorship tests for Da,b,r,R
F
M
V
Da,b,r,R,ǫ , Dr,k,R,ǫ , and Db,R,ǫ . Fix a problem, and let D = (VD , ED ) be the dictatorship test for
the problem with the chosen parameters. D E is edge-weighted and D V , D M and D F are vertexweighted, and our reduction will take care of this difference whenever relevant.
Given an instance L of Unique Games, we describe how to reduce it to a graph G = (VG , EG ).
We assign to each vertex w ∈ WB a copy of VD — formally, VG := {s, t} ∪ (WB × VD ) for Short Path
Edge Cut, Short Path Vertex Cut, RMFC, and VG := {si , ti }i∈[k] ∪ (WB × VD ) for Directed Multicut.
wt(v)
|WB | , so that the
and r k for Directed
For any w ∈ WB , v ∈ VD , the vertex weight of (w, v) is
sum of vertex weights is
b + 1 for Short Path Vertex Cut and b(b+1)
for RMFC,
Multicut.
2
For a permutation σ : [R] → [R], let x ◦ σ := (xσ(1) , . . . , xσ(R) ). To describe the set of
edges, consider the random process where u ∈ UB is sampled uniformly at random, and its
two neighbors w1 , w2 are independently sampled. For each edge (vxi11 , vxi22 ) ∈ ED , we create an
edge ((w1 , vxi11 ◦π(u,w1 ) ), (w2 , vxi22 ◦π(u,w2 ) )). Call this edge is created by u. For Short Path Edge Cut ,
the weight of each edge is the weight in D E times the probability that (u, w1 , w2 ) are sampled.
The sum of weights is b. For each edge incident on a terminal (i.e., (X, vxi ) or (vxi , X) where
X ∈ {s, t} ∪ {si , ti }i ), we add the corresponding edge (X, (w, vxi )) or ((w, vxi ), X) for each w ∈ WB .
For Short Path Edge Cut , their wegiths are ∞ as in D E .
7.3 Completeness
Suppose there exists a labeling l and a subset W ′ ⊆ WB with |W ′ | ≥ (1 − η)|WB | such that l satisfy
every edge incident on W ′ .
Short Path Edge Cut . For every triple (u, w1 , w2 ) such that u ∈ UB and (u, w1 ), (u, w2 ) ∈ EB , we
cut the following edges.
{((w1 , vxi ), (w2 , vyi+1 ) : 0 ≤ i < b, yl(w2 ) 6= xl(w1 ) + 1 mod R or (xl(w1 ) , yl(w2 ) ) = (0, 1)}.
For w ∈
/ W ′ , we additionally cut every edge incident on {w} × D. The total cost is at most 2b
r + 2ηb.
The completeness analysis for the dictatorship test ensures that the length of the shortest path is
at least a(b − r + 1). The proof of Lemma 4.1 works if we have yj = xjl(wj ) .
Short Path Vertex Cut .
For every w ∈ W ′ , we cut the following vertices.
{(w, vxi ) : 0 ≤ i ≤ b, xl(w) = ∗ or 0}.
For w ∈
/ W ′ , we cut every vertex in {w} × D. The total cost is (b + 1)(ǫ + 1−ǫ
r ) + η(b + 1). The
completeness analysis for the dictatorship test ensures that the length of the shortest path is at
least a(b − r + 2). The proof of Lemma 5.1 works if we have yj = xjl(wj ) .
Directed Multicut.
For every w ∈ W ′ , we cut the following vertices.
{(w, vxα ) : α ∈ [r]k , xl(w) = ∗ or 0}.
k
k
For w ∈
/ W ′ , we cut every vertex in {w} × D. The total cost is at most (ǫ + 1−ǫ
r )r + ηr ≤
k−1
r (1 + rǫ + rη). The completeness analysis for the dictatorship test, Lemma 5.1, ensures that
there is no path from si to ti for any i.
16
RMFC. For w ∈ W ′ , on Day i(1 ≤ i ≤ b), we save every vertex in
{(w, vxi ) : xl(w) = ∗ or Bi−1 ≤ xl(w) ≤ Bi },
Hi
B. For w ∈
/ W ′ , on Day i (1 ≤ i ≤ b), we save every vertex in (w, v i ). This ensures
where Bi = H
b
that fire never spreads to vertices associated with w ∈
/ W ′ . Each day, the total cost of saved vertices
1
is at most bǫ + Hb + bη. The completeness analysis for the dictatorship test ensures that t is saved
in this case. The proof of Lemma 6.1 works if we have yj = xjl(wj ) .
7.4 Soundness for Cut / Interdiction Problems
We present the soundness analysis for Short Path Edge Cut, Short Path Vertex Cut, and Directed
Multicut. The soundness analysis of RMFC is in Section 7.5. We first discuss how to extract an
influential coordinate for each u ∈ UB .
Short Path Edge Cut . Fix an arbitrary C ⊆ EG with the total cost k′ , and consider the graph
after cutting edges in C. We will show that if the length of the shortest path is greater than l′ =
√
√
2 + b − 4k′ r + 4ak′ r, we can decode influential coordinates for many vertices of the Unique
Games instance.
For each w ∈ WB , 0 ≤ j ≤ b, and a sequence c = (c1 , . . . , cj ) ∈ {L, S}j , let gw,j,c : ΩR 7→
{0, 1} such that gw,j,c(x) = 1 if and only if there exists a path p = (s, p0 = (w0 , vx00 ), . . . , pj−1 =
j
0
j−1 ∈ ΩR such that (p ′ , p ′ )
(wj−1 , vxj−1
j −1 j
j−1 ), pj = (w, vx )) for some w0 , . . . , wj−1 ∈ WB and x , . . . , x
′
is long if and only if cj ′ = L for 1 ≤ j ≤ j.
For u ∈ UB , 0 ≤ j ≤ b, and c ∈ {L, S}j , let fu,j,c : ΩR 7→ [0, 1] be such that
fu,j,c(x) =
E
w∈N (u)
[gw,j,c(x ◦ π −1 (u, w))],
where N (u) is the set of neighbors of u in the Unique Games instance.
Let γ(u) be the sum of weights of the edges created by u in C. Eu [γ(u)] = k′ , so at least 12
fraction of u’s have Eu [γ(u)] ≤ 2k′ . For such u, since the length of the shortest path is greater than
√
√
l′ = 2 + b − 4k ′ r + 4ak ′ r, the soundness analysis for the dictatorship test shows that there exist
j ∈ {0, . . . , b}, q ∈ [R], c such that Inf ≤d
q [fu,j,c ] ≥ τ (d and τ do not depend on u).
Short Path Vertex Cut . Fix an arbitrary C ⊆ VG with the total cost k′ , and consider the graph
after cutting vertices in C. We will show that if the length of the shortest path is greater than
l′ = 2 + (b − 4k′ ) + 8ak ′ , we can decode influential coordinates for many vertices of the Unique
Games instance.
For each w ∈ WB , 1 ≤ j ≤ b, and a sequence i = (i1 < · · · < ij ) ∈ {0, . . . , b}j , let gw,j,i : ΩR 7→
i
i
j
{0, 1} such that gw,j,i (x) = 1 if and only if there exists a path p = (s, (w1 , vxi11 ), . . . , (wj−1 , vxj−1
j−1 ), (w, vx ))
for some w1 , . . . , wj−1 ∈ WB and x1 , . . . , xj−1 ∈ ΩR .
For u ∈ UB , 0 ≤ j ≤ b, and i ∈ {0, . . . , b}i , let fu,j,i : ΩR 7→ [0, 1] be such that
fu,j,i (x) =
E
w∈N (u)
[gw,j,i (x ◦ π −1 (u, w))],
where N (u) is the set of neighbors of u in the Unique Games instance.
Let γ(u) be the expected weight of C∩({w}×D), where w is a random neighbor of u. Eu [γ(u)] =
′
k , so at least 12 fraction of u’s have Eu [γ(u)] ≤ 2k′ . For such u, Since the length of the shortest path
is greater than l′ = 2 + (b − 4k′ ) + 8ak ′ , the soundness analysis for the dictatorship test shows that
there exists q ∈ [R], 1 ≤ j ≤ b, i such that Inf ≤d
q [fu,j,i ] ≥ τ (d and τ do not depend on u).
17
Directed Multicut. Fix an arbitrary C ⊆ VG with the total cost k′ , and consider the graph after
cutting vertices in C. Let β > 0 be another small parameter to be determined later. If k′ ≤
k(1 − ǫ)(1 − β)(r − 1)k−1 , we prove that we can decode influential coordinates for many vertices
of the Unique Games Instance.
For each w ∈ WB , i ∈ [k], 1 ≤ j ≤ r k , and a sequence α = (α1 , . . . , αj ) ∈ ([r]k )j , let gw,j,α :
αj−1
α
R
), (w, vx j ))
Ω 7→ {0, 1} such that gw,j,α(x) = 1 if and only if there exists a path p = (s, (w1 , vxα11 ), . . . , (wj−1 , vxj−1
for some w1 , . . . , wj−1 ∈ WB and x1 , . . . , xj−1 ∈ ΩR .
For u ∈ UB , 0 ≤ j ≤ b, and α ∈ ([r]k )j , let fu,j,α : ΩR 7→ [0, 1] be such that
fu,j,α(x) =
E
w∈N (u)
[gw,j,α(x ◦ π −1 (u, w))],
where N (u) is the set of neighbors of u in the Unique Games instance.
Let γ(u) be the expected weight of C∩({w}×D), where w is a random neighbor of u. Eu [γ(u)] =
′
k ≤ k(1 − ǫ)(1 − β)(r − 1)k−1 , so at least β fraction of u’s have Eu [γ(u)] ≤ k(1 − ǫ)(r − 1)k−1 . For
such u, since any si -ti pair is disconnected, the soundness analysis for the dictatorship test shows
that there exists q ∈ [R], 1 ≤ j ≤ r k , α such that Inf ≤d
j ′ [fu,j,α] ≥ τ (d and τ do not depend on u).
Finishing Up. The above analyses for Short Path Edge Cut, Short Path Vertex Cut, and Directed
Multicut can be abstracted as follows. Each vertex u ∈ UB is associated with {fu,h : ΩR 7→
[0, 1]}h∈T for some index set H (|H| is upper bounded by some function of b for Short Path Edge
Cut and Short Path Vertex Cut, and some function of r and k for Multicut). For at least β fraction
of u ∈ UB (β = 21 for Short Path Edge Cut and Short Path Vertex Cut ), there exist h ∈ H and
q ∈ [R] such that Inf ≤d
q [fu,h ] ≥ τ . Set l(u) = q for those vertices. Since
X
≤
X
αq 6=0,|α|≤d
2
fd
u,h (α) =
αq 6=0,|α|≤d
≤d
−1
2
E[fd
w,h (π(u, w) (α)) ] = E[Inf π(u,w)−1 (q) (fw,h )],
Inf ≤d
q (fu,h ) =
X
−1
2
(E[fd
w,h (π(u, w) (α))] )
αq 6=0,|α|≤d
w
w
w
at least τ /2 fraction of u’s neighbors satisfy Inf ≤d
(f ) ≥ τ /2. There are at most 2d/τ
π(u,w)−1 (q) w,h
coordinates with degree-d influence at least τ /2 for a fixed h, so their union over h ∈ H yields
coordinates. Choose l(w) uniformly at random among those coordinates (if there is
at most 2d·|H|
τ
τ
) fraction of all
none, set it arbitrarily). The above probabilistic strategy satisfies at least β( τ2 )( 2d·|H|
edges. Taking η smaller than this quantity proves the soundness of the reductions.
7.5 Soundness for RMFC
Fix an arbitrary solution C1 , . . . , Cb ⊆ V such that Ci is saved on Day i and the weight of each Ci
1
is at most k′ = 10
. Suppose that t is saved. We will prove that the Unique Games instance admits
a good labeling.
For each w ∈ WB , 1 ≤ i ≤ b, let gw,i : ΩR 7→ {0, 1} such that gw,i (x) = 1 if and only if (w, vxi )
is burning on Day i. Let Day i∗ be the first day where Ew,x [gw,i∗ (x)] ≥ 12 and Ew,x [gw,i∗ +1 (x)] ≤ 12 .
Such i∗ must exist since Ew,x [gw,1 ] ≥ 1 − k′ ≥ 12 but Ew,x [gw,b ] = 0. For each w ∈ WB , let gw := gw,i∗
and let fw : ΩR 7→ {0, 1} be such that fw (x) = 1 if and only if there exists (w′ , x′ ) such that the
∗
∗
∗
vertex (w′ , vxi ′ ) is burning on Day i and there exists an edge ((w′ , vxi ′ ), (w, vxi +1 )). We must have
1
1
= 53 , since we can save at most k′ = 10
fraction of {gw,i∗ +1 }w before Day
Ew,x [fw (x)] ≤ 21 + 10
∗
i + 1.
18
By an averaging argument, at least 14 fraction of w ∈ WB satisfies Ex [gw,i∗ ] ≥ 41 . Call them heavy
9
vertices. By the expansion of the Unique Games instance, at least 10
fraction of u ∈ UB has a heavy
9
′
neighbor, and at least 10 fraction of w ∈ WB has a heavy w ∈ WB such that (u, w), (u, w′ ) ∈ EB
for some u ∈ UB (say w is reachable from w′ ).
By Theorem 2.1, there exist τ and d such that for each heavy w′ , if Inf ≤d
j [gw ′ ] ≤ τ for all j ∈ [R],
9
′
′
all w reachable from w should satisfy Ex [fw (x)] ≥ 10 (say w reveals an influential coordinate if such
1
= 0.15 fraction of w′ are heavy and reveal an influential coordinate, since
j exists). At least 41 − 10
9 2
otherwise by the expansion Ew,x [fw (x)] ≥ ( 10
) > 53 .
9
fraction of u ∈ UB is a neighbor of heavy
Another expansion argument ensures that at least 10
w with an influential coordinate. Call such u good and let hu : ΩR 7→ {0, 1} such that hu (x) =
9
. Since Ew,x [fw (x)] ≤ 53 , the fraction
gw (x ◦ π −1 (u, w)). Finally, call w ∈ WB good if Ex [fw (x)] ≤ 10
1
of good w is at least 3 . Theorem 2.1 ensures that if there is (u, w) ∈ EB where both u and w are
≤d
good, there exists j ∈ [R] such that min(Inf ≤d
j [hu ], Inf π(u,w)−1 (j) [fw ]) ≥ τ .
Our labeling strategy for Unique Games is as follows. Each good u will get a random label
≤d
from {j : Inf ≤d
j [hu ] ≥ τ }, and each good w will get a random label from {j : Inf j [fw ] ≥ τ }. Other
9
fraction of u ∈ UB are good, 31 fraction of w ∈ WB
vertices get an arbitrary label. Since at least 10
9
− 23 ≥ 51 fraction of edges are
are good, and the Unique Games instance is biregular, at least 10
between good vertices. For each fw or hu , the number of coordinates j with degree-d influence at
least τ is at most τd . Therefore, this strategy satisfies at least 51 ·( τd )2 fraction of edges in expectation.
Taking η smaller than this quantity proves the soundness of the reduction.
7.6 Final Results
Combining our completeness and soundness analyses and taking ǫ and η small enough, we prove
our main results.
Short Path Edge Cut . It is hard to distinguish the following cases.
• Completeness: There is a cut of weight at most k :=
shortest path after the cut is at least l := a(b − r + 1).
2b
r
+ 2ηb such that the length of the
• Soundness: For every cut of weight k′ , the length of the shortest path is at most l′ := 2 + b −
√
√
4k′ r + 4ak ′ r.
√
Setting a = 4, b = 2r − 1 yields k ≤ 4 and l = 4r. Since l′ ≥ 4r implies k′ = Ω( r), we prove the
√
first case of Theorem 1.4. Setting a = r and b = 2r − 1 yields k ≤ 4 and l = r 1.5 . Since l′ = O(k′ r),
we prove the last two cases of Theorem 1.4.
Short Path Vertex Cut .
It is hard to distinguish the following cases.
• Completeness: There is a cut of weight at most k := (b + 1)(ǫ + 1−ǫ
r ) + η(b + 1) such that the
length of the shortest path after the cut is at least l := a(b − r + 2).
• Soundness: For every cut of weight k′ , the length of the shortest path is at most l′ := 2 + (b −
4k′ ) + 8ak ′ .
Setting a = 4, b = 2r − 2 yields k ≤ 2 and l = 4r. Since l′ ≥ 4r implies k′ = Ω(r), we prove the first
case of Theorem 1.3. Setting a = r and b = 2r − 2 yields k ≤ 2 and l = r 2 . Since l′ = O(k′ r), we
prove the last two cases of Theorem 1.3.
19
Directed Multicut.
It is hard to distinguish the following cases.
• Completeness: There is a cut of weight at most r k−1 (1 + rǫ + rη) that separates every si and
ti .
• Soundness: Every multicut must have weight at least k(1 − ǫ)(1 − β)(r − 1)k−1 .
This immediately implies Theorem 1.1 by taking large r and small ǫ, β, η.
RMFC. It is hard to distinguish the following cases.
• Completeness: There is a solution where we save vertices of cost bǫ + H1b + bη = O( log1 b ) each
day to eventually save t.
• Soundness: Saving vertices of
1
10
each day cannot save t.
This immediately implies Theorem 1.5 by taking small ǫ and η.
Acknowledgments The author thanks Konstantin Makarychev for useful discussions on Directed Multicut, and Marek Elias for introducing Shortest Path Interdiction.
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10
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| 8 |
Submitted to the Bernoulli
arXiv: arXiv:1407.0335
arXiv:1407.0335v3 [math.ST] 23 Jan 2017
A general approach to posterior contraction
in nonparametric inverse problems
BARTEK KNAPIK1,* and JEAN-BERNARD SALOMOND2,**
1
Department of Mathematics, Vrije Universiteit Amsterdam, De Boelelaan 1081, 1081 HV Amsterdam,
The Netherlands. E-mail: * b.t.knapik@vu.nl
2
Université Paris-Est, Laboratoire d’Analyse et de Mathématiques Appliquées (UMR 8050), UPEM,
UPEC, CNRS, F-94010, Créteil, France. E-mail: ** jean-bernard.salomond@u-pec.fr
In this paper we propose a general method to derive an upper bound for the contraction rate of the posterior
distribution for nonparametric inverse problems. We present a general theorem that allows us to derive contraction rates for the parameter of interest from contraction rates of the related direct problem of estimating
transformed parameter of interest. An interesting aspect of this approach is that it allows us to derive contraction rates for priors that are not related to the singular value decomposition of the operator. We apply
our result to several examples of linear inverse problems, both in the white noise sequence model and the
nonparametric regression model, using priors based on the singular value decomposition of the operator,
location-mixture priors and splines prior, and recover minimax adaptive contraction rates.
Keywords: Bayesian nonparametrics, nonparametric inverse problems, posterior distribution, rate of contraction, modulus of continuity.
1. Introduction
Statistical approaches to inverse problems have been initiated in the 1960’s and since then many
estimation methods have been developed. Inverse problems arise naturally when one observes the
object of interest only indirectly. Mathematically speaking, this phenomenon is easily modeled
by the introduction of an operator K modifying the object of interest f , such that the observation
at hand comes from the model
n
Y n ∼ PKf
,
(1.1)
where f is assumed to belong to a parameter space F, and n → ∞ reflects the increasing
amount of information in the observation. In many applications the operator K is assumed to be
injective. However, in the most interesting cases its inverse is not continuous, thus the parameter
of interest f cannot be reconstructed by a simple inversion of the operator. Such problems are said
to be ill-posed. Several methods dealing with the discontinuity of the inverse operator have been
proposed in the literature. The most famous one is to conduct the inference while imposing some
regularity constraints on the parameter of interest f . These so-called regularization methods have
been widely studied in the literature both from a theoretical and applied perspective, see [12, 18]
for reviews.
A Bayesian approach to inverse problems is therefore particularly interesting, as it is well
known that putting a prior distribution on the functional parameter yields a natural regularization.
1
imsart-bj ver. 2011/11/15 file: inverse_general_BJ_rev3.tex date: January 24, 2017
2
Bartek Knapik and Jean-Bernard Salomond
This property of the Bayesian approach is particularly interesting for model choice, but it has
proved also useful in many estimation procedures, as shown in [35] in the case of overfitted
mixtures models, in [9] in the case of nonparametric models where regularization is necessary,
in [37] in the semiparametric problem of estimating a monotone density at the boundaries of its
support, or in [28] in the white noise setting.
In this paper we study the behaviour of the posterior distribution when the amount of information goes to infinity (e.g. when the number of data points n goes to infinity or when the level of
the noise goes to 0) under the frequentist assumption that the data Y n are generated from model
(1.1) for some true unknown parameter f0 . Asymptotic properties of the posterior distribution in
nonparametric models have been studied for many years. Some first results about consistency of
Bayes procedures date back to Schwartz [38]. Her ideas were further refined and extended in an
unpublished work of Barron [5], and can be also found in other works, e.g., [4], [22]. The next
natural step is to consider the rate at which the neighborhoods of the truth can shrink, yet still
capture most of the posterior mass. In other words, the interest lies in finding an upper bound for
the rate at which the posterior concentrates around f0 . This is also the main focus of this paper.
The aforementioned consistency results served as a starting point for two seminal papers on rates
of convergence of posterior distributions by Ghosal et al. [23] and Shen and Wasserman [42].
Understanding of the whole posterior distribution is necessary for uncertainty quantification, see
a recent paper by Szabó et al. [43] for an overview, but is also directly related to asymptotic properties of Bayes point estimators, see, e.g., Theorem 2.5 in [23]. In Bayesian nonparametrics it is
important to understand the impact of the prior distribution on the posterior. In particular, some
aspects of the prior may be inherited by the posterior when the amount of information grows to
infinity and may thus be highly influential for the quality and speed of recovery.
Asymptotic properties of the Bayesian approach to nonparametric linear inverse problems
have recently received a growing interest. Knapik et al. [28], Agapiou et al. [1], and Florens and
Simoni [21] were the first to study posterior contraction rates under conjugate prior in the socalled mildly ill-posed setting (in the terminology of [12]). These were followed by two papers by
Knapik et al. [29] and Agapiou et al. [2], studying Bayesian recovery of the initial condition for
heat equation and related extremely and severely ill-posed inverse problems. One type of priors
studied in [29] leads to a rate-adaptive Bayesian procedure. The paper by Ray [33] was the first
study of the posterior contraction rates in the non-conjugate sequence setting. Considering nonconjugate prior is particularly interesting as it allows some additional flexibility of the model.
However, the approach presented in [33] is only valid for priors that are closely linked to the
singular value decomposition (SVD) of the operator. Moreover, in [33] several rate-adaptive
priors were considered, both in the mildly and severely ill-posed setting. It should be noted,
however, that some of the bounds on contraction rates in the severely ill-posed setting obtained
in that paper are not optimal and do not agree with the bounds found in [29] or [2], probably
due to proof techniques. Similar adaptive results, in the conjugate mildly ill-posed setting, using
empirical and hierarchical Bayes approach were obtained in [27].
There is a rich literature on the problem of deriving posterior contraction rate in the direct
problem setting, i.e. estimating Kf in (1.1). Since the seminal papers of Ghosal et al. [23]
and Shen and Wasserman [42], general conditions on the prior distribution for which the posterior contracts at a certain rate have been derived in various cases. In particular, Ghosal and
van der Vaart in [24] give a number of conditions for non independent and identically distributed
imsart-bj ver. 2011/11/15 file: inverse_general_BJ_rev3.tex date: January 24, 2017
Posterior contraction in inverse problems
3
data. However, such results cannot be applied directly to ill-posed inverse problems, and to the
authors’ best knowledge, no analogous results exist in the inverse problem literature. In this
paper we propose a unified general approach to posterior contraction in nonparametric inverse
problems, and illustrate it for specific linear inverse problems.
To understand why the existing general posterior contraction results are not suited for nonparametric inverse problems consider an abstract setting in which the parameter space F is
an arbitrary metrizable topological vector space and let K be a continuous injective mapping
K : F 3 f 7→ Kf ∈ KF. Let d and dK denote some metrics or semi-metrics on F and KF,
respectively. Any prior Π on f imposes a prior on Kf through the continuous mapping K. Recall
that the true parameter of interest f0 belongs to F. General posterior contraction results (e.g., in
[23] or [24]) rely on several natural metrics related to the model (1.1) and therefore control the
distance between Kf0 and Kf in the dK metric. On the other hand, our interest lies in the recovery of f0 , and therefore the control of the distance between f0 and f in the d metric is desirable.
Since the operator K does not have a continuous inverse and the problem is ill-posed, even if
dK (Kf, Kf0 ) is small, the distance d(f, f0 ) between f and the true f0 can be arbitrarily large.
In other words, there is no equivalence between the metrics d and dK and therefore the existing
theory of posterior contraction does not allow obtaining bounds on posterior contraction rates for
the recovery of f0 .
Even if the problem is ill-posed, there exist subsets Sn of F such that the inverse of the operator K restricted to KSn is continuous. We can thus easily derive posterior contraction rate for
f ∈ Sn from posterior contraction rate for Kf by inverting the operator K. For suitably chosen
priors, the sets Sn will capture most of the posterior mass, and we can thus extend the contraction result to the whole parameter space F. The sets Sn , thought of as sieves approximating the
parameter space F, have already been considered in [23] allowing some additional flexibility
and are often incorporated in results on posterior contraction for various models. However, their
principal role was not to enable the change of metrics, but rather alleviate the usual entropy condition. In our approach we first assume the existence of a contraction result for the so-called
direct problem (that is the recovery of Kf ) that can be derived using general posterior contraction literature. Next, we choose a sequence of subsets Sn in such a way that the inversion of the
operator K on KSn , so also the change of metrics, can be controlled and at the same this sets
are big enough (in terms of the posterior mass). The latter condition can be verified by imposing
additional sufficient conditions on the prior (on f ). We are then able to show that the posterior
distribution for the parameter of interest f contracts at a given rate.
The rest of the paper is organized as follows: we present the main result in Section 2 and
discuss how it relates to other results using the concept of sieves to control contraction in other
metrics. We then apply our result in various settings. We first consider the white noise sequence
model in Section 3, where we present a general construction of the sets Sn , and recover many
of the existing results with much less effort. We also observe an interesting interplay between
optimality of Bayesian procedures for estimating f and Kf . In Section 4 we apply our method
in the nonparametric inverse regression setting, considering new families of priors that need not
be related to the SVD, and leading to optimal Bayesian procedures. Proofs of Sections 2–4 are
placed in Section 5. We conclude the paper with a discussion in Section 6.
For two sequences (an ) and (bn ) of numbers, an bn means that |an /bn | is bounded away
from zero and infinity as n → ∞, an . bn means that an /bn is bounded, an ∼ bn means that
imsart-bj ver. 2011/11/15 file: inverse_general_BJ_rev3.tex date: January 24, 2017
4
Bartek Knapik and Jean-Bernard Salomond
an /bn → 1 as n → ∞, and an bn means that an /bn → 0 as n → ∞. For two real numbers a
and b, we denote by a ∨ b their maximum, and by a ∧ b their minimum. For a sequence of random
variables X n = (X1 , . . . , Xn ) ∼ Pfn and any measurable function ψ with respect to Pfn , we
denote by Ef ψ the expectation of ψ(X n ) with respect to Pfn and when f = f0 we will write E0
instead of Ef0 .
2. General theorem
n
Assume that the observations Y n come from model (1.1) and that PKf
admit densities pnKf
n
relative to a σ-finite measure µ . To avoid complicated notations, we drop the superscript n in
the rest of the paper. Let F and KF be metric spaces, and let d and dK denote metrics on both
spaces, respectively.
In this section we present the main result of this paper which gives an upper bound on the
posterior contraction rate under some general conditions on the prior. We call the estimation
of Kf given the observations Y the direct problem, and the estimation f given Y the inverse
problem. The main idea is to control the change of metrics dK and d. If the posterior distribution
concentrates around Kf0 for the metric dK at a certain rate in the direct problem, applying the
change of metrics will give us an upper bound on the posterior contraction rate for the metric
d in the inverse problem. However, since the operator K does not posses a continuous inverse,
the change of metrics cannot be controlled over the whole space KF. A way to circumvent this
issue is to only focus on a sequence of sets of high posterior mass for which the change of metric
is feasible. More precisely, for a set S ⊂ F, f0 ∈ F and a fixed δ > 0 we call the quantity
ω(S, f0 , d, dK , δ) := sup d(f, f0 ) : f ∈ S, dK (Kf, Kf0 ) ≤ δ
(2.1)
the modulus of continuity. We note that in this definition we do not assume f0 ∈ S. This is thus
a local version of the modulus of continuity considered in [17] or [26]. On the one hand, the sets
Sn need to be big enough to capture most of the posterior mass. On the other hand, one has to be
able to control the distance between the elements of Sn and f0 , given the distance between Kf
and Kf0 is small. Since the operator K is unbounded, this suggests that the sets Sn cannot be
too big.
Theorem 2.1. Let n → 0 and let Π the prior distribution on f be such that
E0 Π Snc | Y n → 0,
(2.2)
for some sequence of sets (Sn ), Sn ⊂ F, and for any positive sequence Mn
E0 Π f : dK (Kf, Kf0 ) ≥ Mn n | Y n → 0.
(2.3)
Then
E0 Π f : d(f, f0 ) ≥ ω(Sn , f0 , d, dK , Mn n ) | Y n → 0.
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Posterior contraction in inverse problems
5
The proof is elementary and can be found in Section 5.1.
The idea behind Theorem 2.1 is simple and was used to change metrics also in direct problems.
For instance Castillo and van der Vaart [11] considered the multivariate normal mean model in the
situation that the mean vector is sparse. They use the fact that the posterior concentrates along
certain subspaces on which it is easy to control an `q -like metric with the standard Euclidean
metric for q < 2. Hoffmann et al. [26] also use concentration of the posterior on specific sets to
control the L∞ metric with the L2 metric in the white noise model.
Castillo et al. [10] extended the ideas of [11] to the sparse linear regression model, in which
the recovery of the parameter of the model is an inverse problem. Similar reasoning was also
used in [44] to study posterior contraction in the special case of Gaussian elliptic inverse problem, and in [16] to investigate asymptotic properties of empirical Bayes procedures for density
deconvolution. However, these papers consider specific inverse problems only, whereas Theorem 2.1 allows deriving contraction rates for a wide variety of inverse problem models for which
the prior is not necessarily related to the spectral decomposition of the operator K, e.g., when
the operator does not admit singular value decomposition, as in Section 4.1.
The interpretation of the theorem is the following: given a properly chosen sequence of sets
Sn , the rate of posterior contraction Mn n in the direct problem restricted to the given sequence
can be translated to the rate of posterior contraction in the inverse setting. Note that the sequence
Mn is often chosen to grow to infinity as slowly as needed (see, e.g., in [23] or [24]), making
n the effective rate of posterior contraction. Also, in both contraction results of Section 4, the
sequence Mn need not be diverging and is chosen to be constant. Since the operator K is injective
and continuous, any prior Π on f induces a prior on Kf , and the general posterior contraction
results can be applied to obtain the rate of contraction in the direct problem of estimating Kf .
Next, the choice of Sn is crucial as it is the principal component in the control of the change
of metric. In particular, the contraction rate Mn n for the direct problem may not be optimal, and
still lead to an optimal contraction rate ω(Sn , f0 , d, dK , Mn n ) for the inverse problem with a
well-suited choice of Sn . As shown in Section 3.3, it is possible in some cases to obtain optimal
recovery of f without having optimal recovery of Kf . In this example, we can choose Sn small
enough so that the change of metrics can be control very precisely. This widens the possible
choice of priors leading to optimal contraction rates and shows that the change of metric is the
crucial part here. However, in most cases, the priors considered in this paper lead to optimal
recovery for both f and for Kf .
To control the posterior mass of the sets Sn we can usually alter the proofs of contraction
results for the direct problems. Here we present a standard argument leading to (2.2). Define the
usual Kullback–Leibler neighborhoods by
Z
n
pKf
dµ ≤ n2 ,
Bn (Kf0 , ) = f ∈ F : − pKf0 log
pKf0
Z
(2.4)
o
pKf 2
2
pKf0 log
dµ ≤ n , ,
pKf0
The following lemma adapted from [24] gives general conditions on the prior such that (2.2) is
satisfied.
Lemma 2.1 (Lemma 1 in [24]). Let n → 0 and let (Sn ) be a sequence of sets Sn ⊂ F. If Π is
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6
Bartek Knapik and Jean-Bernard Salomond
the prior distribution on f satisfying
Π(Snc )
. exp(−2n2n ),
Π(Bn (Kf0 , n ))
then
E0 Π Snc | Y n → 0.
For clarity of presentation the results in this section are stated for a fixed f0 , but we note
that they are easily extended to uniform results over certain sets, i.e., balls of fixed radius and
regularity, or union of balls of fixed radii over compact range of regularity parameter (see results
of Section 3).
3. Sequence white noise model
Our first examples are based on the well-studied infinite-dimensional normal mean model. In the
Bayesian context the problem of direct estimation of infinitely many means has been studied,
among others, in [7, 24, 42, 45].
We consider the white noise setting, where we observe an infinite sequence Y n = (Y1 , Y2 , . . .)
satisfying
1
(3.1)
Yi = κi fi + √ Zi ,
n
where Z1 , Z2 , . . . are independent standard normal random variables, f = (f1 , f2 , . . .) ∈ `2 is
the infinite-dimensional parameter of interest and (κi ) is a known sequence that may converge
to 0 as i → ∞. If this is the case (so when the operator K does not possess a continuous inverse)
the modulus of continuity defined in (2.1) is infinite when S = F.
Even though this model is rather abstract, it is mathematically tractable and it enables rigorous
results and proofs. Moreover, it can be seen as an idealized version of other statistical models
through equivalence results see, e.g., [8, 31, 32]. Both white noise examples of inverse problems
presented in this section have already been studied in the Bayesian literature. We present them
here for several reasons. First, the direct version of the normal mean model attracted a lot of
attention in the Bayesian literature, e.g. providing contraction results for estimation of Kf in the
mildly ill-posed setting. Therefore, we choose this example to illustrate how Theorem 2.1 works
in practice. In particular, it allows us to make it clear how one could construct a sequence of sets
Sn . In the severely ill-posed case we study truncated (or sieve) priors leading to optimal recovery
of the parameter of interest. Our results improve the findings of [29] and [2]. In addition, we can
show that optimal contraction for f does not necessarily require optimal recovery of Kf .
3.1. Computation of a modulus
In this section we first present an example of the sequence of sets Sn , and later present how
the modulus of continuity for this sequence can be computed in a standard inverse problem
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7
Posterior contraction in inverse problems
setting. We now suppose that F and KF are separable Hilbert spaces, denoted (H1 , k · kH1 ) and
(H2 , k · kH2 ) respectively. We note that the sets Sn resemble the sets Pn considered in [33].
As already noted, the operator K restricted to certain subsets of the domain H1 might have a
finite modulus of continuity defined in (2.1). Clearly, one wants to construct a sequence of sets
Sn that in a certain sense approaches the full domain H1 . This is understood in terms of the
remaining prior mass condition in Theorem 2.1. Moreover, since we do not require f0 to be in
Sn , we need to be able to control the distance between f0 and Sn .
A natural guess is to consider finite-dimensional projections of H1 . In this section we go
beyond this concept. To get some intuition, consider the Fourier basis of H1 . The ill-posedness
can be then viewed as too big an amplification of the high frequencies through the inverse of the
operator K. Therefore, one wants to control the higher frequencies in the signal, and thus in the
parameter f .
Since H1 is a separable Hilbert space, there exist an orthonormal basis (ei ) and each element
f ∈ H1 can be viewed as an element of `2 and
kf kH1 =
∞
X
fi2 .
i=1
For given sequences of positive numbers kn → ∞ and ρn → 0, and a constant c ≥ 0 we define
n
o
X
Sn := f ∈ `2 :
fi2 ≤ cρ2n .
(3.2)
i>kn
If the operator K is compact, then the spectral decomposition of the self-adjoint operator
K T K : H1 → H1 provides a convenient orthonormal basis. In the compact case the operator
K T K possesses countably many positive eigenvalues κ2i and there is a corresponding orthonormal basis (ei ) of H1 of eigenfunctions, and the sequence (ẽi ) defined by Kei = κi ẽi forms an
orthonormal conjugate basis of the range of K in H2 . Therefore, both f and Kf can be associated with sequences in `2 . Since the problem is ill-posed when κi → 0, we can assume without
loss of generality that the sequence κi is decreasing.
Let kn , ρn , and c in the definition of Sn be fixed. Then for any g ∈ Sn
kgk2H1 =
∞
X
i=1
≤
X
gi2 =
X
gi2 +
i≤kn
gi2
+
cρ2n
≤
gi2
i>kn
=
i≤kn
κ−2
kn
X
X
2 2
2
κ−2
i κi gi + cρn
i≤kn
X
κ2i gi2
2
2
+ cρ2n ≤ κ−2
kn kKgkH2 + cρn .
i≤kn
Let fn be the projection of f0 on the first kn coordinates, i.e., fn,i = f0,i for i ≤ kn and
0 otherwise. Moreover, we assume that f0 belongs to some smoothness class described by a
decreasing sequence (si ):
∞
X
2
kf0 k2s =
s−2
i f0,i < ∞.
i=1
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8
Bartek Knapik and Jean-Bernard Salomond
For instance, the usual Sobolev space of regularity β is defined in that way with si = i−β .
Therefore, we have
kfn − f0 kH1 ≤ skn kf0 ks ,
kKfn − Kf0 kH2 ≤ skn κkn kf0 ks .
Using the triangle inequality twice and keeping in mind that f − fn ∈ Sn we obtain
kf − f0 kH1 ≤ kf − fn kH1 + kfn − f0 kH1
√
cρn + skn kf0 ks
≤ κ−1
kn kKf − Kfn kH2 +
√
−1
≤ κkn kKf − Kf0 kH2 + κkn skn kf0 ks + cρn + skn kf0 ks
√
cρn + 2kf0 ks skn .
= κ−1
kn kKf − Kf0 kH2 +
(3.3)
We then find an upper bound for the modulus of continuity with this specific choice of Sn is
ω(Sn , f0 , k · kH1 , k · kH2 , δ) . κ−1
kn δ + ρn + skn .
(3.4)
3.2. Mildly ill-posed problems
In this section we consider the model (3.1), where C −1 i−p ≤ κi ≤ Ci−p for some p ≥ 0 and
C ≥ 1. Since the κi ’s decay polynomially, the operator is mildly ill-posed. Such problems are
well studied in the frequentist literature, and we refer the reader to [12] for a comprehensive
overview. There are also several papers on properties of Bayes procedures for such problems.
The first studies of posterior contraction in mildly ill-posed operators were obtained in [28]
and [1]. Later, adaptive priors leading to the optimal minimax rate of contraction (up to slowly
varying factors) were studied in [33] and [27]. Similar problem, with a different noise structure,
has been studied in [21]. The main purpose of this section is to show how Theorem 2.1 can be
applied to such problems and how existing results on contraction rates for Kf in the sequence
setting can be used to obtain posterior contraction rates for f without explicit computations as in
aforementioned papers.
We put a product prior on f of the form
Π=
∞
O
N (0, λi ),
i=1
where λi = i−1−2α , for some α > 0. Furthermore, the true parameter f0 is assumed to belong
to S β for some β > 0:
n
o
X
S β = f ∈ `2 : kf k2β :=
fi2 i2β < ∞ .
(3.5)
Therefore, kKf0 k2β+p is finite, the prior on f induces the prior on Kf such that (Kf )i ∼
N (0, λi κ2i ), and one can deduce from the results of [45] and [7] that
(α∧β)+p
sup
E0 Π f : kKf − Kf0 k ≥ Mn n− 1+2α+2p
Y n → 0.
kKf0 kβ+p ≤R
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Posterior contraction in inverse problems
In order to apply Theorem 2.1 we need to construct the sequence of sets Sn and verify condition (2.2). We use the construction as in (3.2), and we verify the remaining posterior mass
condition along the lines of Lemma 2.1.
Theorem 3.1.
Mn → ∞
Suppose the true f0 belongs to S β for β > 0. Then for every R > 0 and
(α∧β)
sup
E0 Π f : kf − f0 k ≥ Mn n− 1+2α+2p
Y n → 0.
kf0 kβ ≤R
The proof of this theorem is postponed to Section 5.2.1.
The upper bound on the posterior contraction rate obtained in this theorem agrees with the
ones already obtained in the existing literature (see, for instance, [27, 28, 33]). We note that the
prior used above requires the knowledge of the true regularity parameter β in order to achieve
minimax optimal rate of recovery. Moreover, we note that the prior with α = β leads to optimal
recovery of both f and Kf .
The prior used in this section is rather simple and is not hierarchical, i.e., is not aimed at adaptive recovery. We have already pointed out that [33] and [27] studied adaptive Bayesian approach
to mildly ill-posed inverse problems and obtained optimal rates (up to logarithmic factors). We
would also like to point out that recent studies of adaptive approaches to the sequence white
noise model [e.g. 3, 27] already consider its inverse version (i.e., allowing κi 6= 1). In a recent
work Belitser [6] even obtained adaptive posterior contraction rate in a setting equivalent to the
one considered here that could be used both for the estimation of Kf and the estimation of f .
Therefore, even though one could consider the existing approaches studied in the literature to
achieve adaptation (by first showing optimal contraction for Kf and then applying Theorem 2.1
to prove contraction for f ), this will not be treated here for the sake of simplicity (in the latter
cases also to avoid rather artificial application of Theorem 2.1).
3.3. Severely and extremely ill-posed problems
We again consider the sequence white noise setting, where we observe an infinite sequence Y n =
(Y1 , Y2 , . . .) as in (3.1) where κi exp(−γip ) for some p ≥ 1 and γ > 0. We first consider
estimation of Kf0 that will be later used to obtain the rate of contraction of the posterior around
f0 . We put a product prior on f of the form
Π=
kn
O
N (0, λi ),
i=1
where λi = i−α exp(−ξip ), for α ≥ 0, ξ > 0, and some kn → ∞. We choose kn solving
1 = nλi exp(−2γip ) = ni−α exp(−(ξ + 2γ)ip ). Using the Lambert function W one can show
that
p p(ξ + 2γ) 1/p log n
1/p
α
kn =
W nα
=
+ O(log log n)
,
(3.6)
p(ξ + 2γ)
α
ξ + 2γ
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10
Bartek Knapik and Jean-Bernard Salomond
see also Lemma A.4. in [29]. Note that in this case we have exp(knp ) = (nkn−α )1/(ξ+2γ) , so we
can avoid exponentiating kn . Therefore, we do not have to specify the constant in front of the
log log n term in the definition of kn , and we may assume that kn is of the order (log n)1/p .
Note that the hyperparameters of the prior do not depend on f0 , but only on K, which is
known. For Sn as in (3.2) with kn as above and c = 0, the prior is supported on Sn and the
first condition of Theorem 2.1 is trivially satisfied. Regardless of the choice of ξ and α (as long
as α ≥ 0 and ξ > 0) the following theorem shows that the posterior contracts at the optimal
minimax rate (log n)−β/p for the inverse problem of estimating f0 (cf. [29] or [2] and references
therein), so the prior is rate-adaptive.
In this section we consider deterministically truncated Gaussian priors. Similar priors in the
extremely ill-posed setting are considered in [33], but in this paper the truncation level is endowed with a hyper-prior and the bound on the posterior contraction is suboptimal. Other papers
on Bayesian approach to severely and extremely ill-posed inverse problems do not consider truncated priors. In [29] the optimal rate is achieved for the priors with exponentially decaying or
polynomially decaying variances (in the latter case the speed of decay leading to optimal rate
is closely related to the regularity of the truth). Similar results for the priors with polynomially decaying variances are presented in [33] and [2]. However, in the former case the rate for
undersmoothing priors is worse than the rate obtained in the other papers.
Theorem 3.2. Suppose the true f0 belongs to S β for β > 0. Then for every R > 0 and
Mn → ∞
β
sup E0 Π f : kf − f0 k ≥ Mn (log n)− p Y n → 0.
kf0 kβ ≤R
The proof of this Theorem is postponed to Section 5.2.2. The prior considered in this theorem
might seem unnatural, since λi ’s do not coincide with the type of regularity of the truth and the
prior puts mass only on analytic functions of growing complexity. However, similar approaches
are quite common in the Bayesian literature, for instance when finite mixtures models are considered. Moreover, this prior has also some computational advantages, since the corresponding
posterior can be handled numerically.
Inspection of the proof shows that the deterministic truncation is suboptimal for the estimation of Kf0 , since the resulting upper bound is polynomially slower than the minimax
rate n−1/2 (log n)1/2p . It sheds light on an interesting, although counterintuitive property of the
Bayesian approach to inverse problems: one may not need optimal contraction for the estimation of Kf0 to get optimal contraction for the estimation of f0 . This phenomenon should be
interpreted in the following way: since the operator K regularizes the parameter f0 , one could
compensate the suboptimal contraction of the posterior for the direct problem, by a sharper control of the deviation between f and f0 in (3.3) when f is in Sn . Indeed, when ξ increases (which
slows down the upper bound on the posterior contraction for Kf0 ), the truncation level kn decreases. As a result, the sets Sn become smaller, so the sharper control of d(f, f0 ) is indeed
possible. In the specific setting of sequence white noise model it might seem artificial. However,
this observation could prove useful in more complex settings, especially because it widens the
class of possible prior distributions giving optimal contraction rates.
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Posterior contraction in inverse problems
Remark 3.1. If an upper bound βe on the regularity of the true f0 is known, one can also take
ξ = 0 and α ≥ 1 + 2βe and the assertion of Theorem 3.2 stays valid. In this case the upper bound
on the posterior contraction rate for Kf0 is logarithmically slower than the minimax rate.
4. Regression
We now consider the inverse regression model with Gaussian residuals
Yi = Kf (xi ) + σi ,
iid
i ∼ N (0, 1),
i = 1, . . . , n,
(4.1)
where the covariates xi are fixed in a covariate space X . In the sequel, we either choose X =
[0, 1] or X = R. In the following we consider the noise level σ > 0 to be known although
one could also think of putting a prior on it and estimate it in the direct model. Nonparametric
regression models have been studied in the literature for direct problems, and frequentist properties of the posterior distribution are well known for a wide variety of priors. In [24], Ghosal and
van der Vaart give general conditions on the prior such that the posterior contracts at a given rate.
Nonparametric inverse regression models are also used in practice, for instance in econometrics
where one considers instrumental variable as in [20]. However, to the authors’ best knowledge,
contraction rates for these models have only been considered in [44].
In this setting, a common choice for the metrics d and dK are the usual l2 norms
d(f, g)2 = n−1
n
X
(f (xi ) − g(xi ))2 = kf − gk2n ,
dK (f, g) = d(Kf, Kg).
i=1
For a ∈ Rk , k ∈ N∗ , and f ∈ L2 , we denote the standard Euclidean and L2 norms by
kakk =
k
X
a2i
1/2
,
kf k =
Z
f2
1/2
,
i=1
respectively.
We now consider two examples of inverse regression problems, namely numerical differentiation and deconvolution on R. For these sampling models, we study the frequentist properties of
the posterior distribution for standard prior that have not been considered for inverse regression
problems so far.
4.1. Numerical differentiation using spline prior
In this section, we consider the inverse regression problem (4.1) with the operator K between
L1 [0, 1] and the space of functions differentiable almost everywhere on the interval [0, 1] (see
also Chapter 7 of [36]) defined by
Z x
Kf (x) =
f (t)dt,
for x ∈ [0, 1].
(4.2)
0
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Bartek Knapik and Jean-Bernard Salomond
We note that the operator K is not defined between two Hilbert spaces, hence goes beyond
the concept of singular value decomposition. This model is particularly useful for numerical
differentiation, for instance, and has been well studied in the literature. In particular, in [12] a
related problem of estimating a derivative of a square integrable function is presented and it is
shown that the SVD basis is the Fourier basis. Moreover, the operator is mildly ill-posed of degree
1 (cf. Section 3). We consider a prior on f that is well-suited if the true regression function f0
belongs to the Hölder space H(β, L) for some β > 0, that is f0 is β0 = bβc times differentiable
and
|f (β0 ) (x) − f (β0 ) (y)|
≤ L.
kf0 kβ = sup
|x − y|β−β0
x6=y
Since Kf0 is (β0 + 1) times differentiable, it also holds that f0 ∈ H(β, L) implies then Kf0 ∈
H(β + 1, L).
We construct a prior on f by considering its decomposition in a B-spline basis. A definition
of the B-spline basis can be found in [13]. For a fixed positive integer q > 1 called the degree
of the basis, and a given partition of [0, 1] in m subintervals of the form ((i − 1)/m, i/m], the
space of splines is a collection of function f (0, 1] → R that are q − 2 times differentiable and if
restricted to one of the sets ((i − 1)/m, i/m], are polynomial of degree at most q. An interesting
feature of the space of splines is that it forms a J-dimensional linear space with the so called
B-spline basis denoted (B1,q , . . . , BJ,q ), for J = m + q − 1. Priors based on the decomposition
of the function f in the B-spline basis of order q have been considered in the regression setting
in, e.g., [24] and [40], and are commonly used in practice. Here we construct a different version
of the prior that will prove to be useful to derive contraction rate for the direct problem and the
inverse problem.
Let the prior distribution on f be defined as follows:
J ∼ ΠJ
Π := a1 , . . . aJ iid
(4.3)
∼ Πa,J
PJ−1
f (x) = J j=1 (aj+1 − aj )Bj,q−1 (x).
Given the definition of Bj,q in [13], standard computations give
0
Bj,q
(x) = J Bj,q−1 (x) − Bj+1,q−1 (x)
which in turn gives
Kf (x) =
J
X
aj Bj,q (x).
(4.4)
j=1
This explains why we choose a prior as in (4.3) since it leads to the usual spline prior on Kf .
Note that the condition that Kf (0) = 0 can be imposed by a specific choice of nodes for the Bspline basis (see [13]). To compute the modulus of continuity for this model, we need to impose
some conditions on the design. Let Σqn be a matrix defined by its elements
n
(Σqn )i,j =
1X
Bi,q (xl )Bj,q (xl ),
n
i, j = 1, . . . , J.
l=1
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Posterior contraction in inverse problems
Similarly to [24], we ask that the design points satisfy the following conditions:
D1 for all v1 ∈ RJ
J −1 kv1 k2J v10 Σqn v1
D2 for all v2 ∈ RJ−1
(J − 1)−1 kv2 k2J−1 v20 Σn(q−1) v2 .
Condition D1 is natural when considering B-splines priors in a regression setting, and both conditions are satisfied for a wide variety of designs. Consider for instance the uniform design
xi = i/n for i = 1, . . . , n. Then given Lemma 4.2 in [23], we get that for v1 ∈ RJ , v2 ∈ RJ−1
kv1 k2J J −1 .
J
X
2
v1,j Bj,q
. kv1 k2J J −1 ,
j=1
kv2 k2J−1 (J − 1)−1 .
J−1
X
2
v2,j Bj,q−1
. kv2 k2J−1 (J − 1)−1 ,
j=1
and the constants depend only on q. Furthermore, we have that
J
X
2
v1,j Bj,q
= v10 Σqn v1 + O n−1 ,
j=1
where the remainder depends only on q. Similarly,
J−1
X
2
v2,j Bj,q−1
= v20 Σnq−1 v2 + O n−1 .
j=1
Thus D1 and D2 are satisfied for the uniform design for all J = o(n).
We now go on and derive conditions on the prior such that the posterior contracts at the
minimax adaptive rate (up to a log n factor). The prior we consider is not conjugate, and does not
depend on the singular value decomposition of the operator K for obvious reasons.
Theorem 4.1. Let Y n = (Y1 , . . . , Yn ) be a sample from (4.1) with X = [0, 1] and Π be a prior
for f as in (4.3). Suppose that ΠJ is such that for some constants cd , cu > 0 and t ≥ 0,
exp(−cd j(log j)t ) ≤ ΠJ (j ≤ J ≤ 2j),
ΠJ (J > j) . exp(−cu j(log j)t ),
(4.5)
J
for all J > 1, and suppose that Πa,J is such that for all a0 ∈ R , ka0 k∞ ≤ H, there exists a
constant c2 depending only on H such that
Πa,J (ka − a0 kJ ≤ ) ≥ exp(−c2 J log(1/))
(4.6)
Let Θ(β, L, H) = {f ∈ H(β, L), kf k∞ ≤ H}. If the design (x1 , . . . , xn ) satisfies conditions
D1 and D2, then for all L and for all β ≤ q there exits a constant C > 0 that depends only on q,
L, H and Π such that if f0 ∈ H(β, L), then
β
sup
sup
E0 Π kf − f0 kn ≥ Cn− 2β+3 (log n)3r Y n → 0,
(4.7)
β≤q−1 f0 ∈Θ(β,L,H)
with r = (1 ∨ t)(β + 1)/(2β + 3).
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Bartek Knapik and Jean-Bernard Salomond
Condition (4.5) is for instance satisfied by the Poisson or geometric distribution. A similar
condition is considered in [40]. Condition (4.6) is satisfied for usual choices of priors, such as
the product of J independent copies of a distribution that admits a continuous density. Similar
results hold for functions that are not uniformly bounded, with additional conditions on the tails
of Πa,J . This will only require additional computations similar to those in [40], and will thus not
be treated here.
This theorem gives theoretical validation for a family of priors that are widely used in practice
for regression problems and are easy to implement. A key feature here is that we can control the
transformation of a spline basis function by the operator K through (4.4), which in turn allows
us to control the change of norms. This point is highly interesting as it gives guidelines for the
construction of priors for inverse problems. Namely, it suggests that a prior whose geometry does
not change too much through the application of the operator K could lead to optimal contraction
for the inverse problem.
4.2. Deconvolution using mixture priors
In this section, we consider the model (4.1), where K is the convolution operator in R. This
model is widely used in practice, especially when considering auxiliary variables in a regression
setting or for image deblurring. For a convolution kernel λ ∈ L2 (R) symmetric around 0, and
for all f ∈ L2 (R), we define K as
Z
Kf (x) = λ ? f (x) =
f (u)λ(x − u)du,
for x ∈ R.
(4.8)
R
To the authors’ best knowledge, theoretical properties of Bayesian nonparametric approach to
this nonparametric regression model have not been studied in the literature. In this setting we
consider a mixture type prior on f , and derive an upper bound for the posterior contraction rate.
Mixture priors are common in the Bayesian literature: [25], [24] and [41] consider mixtures
of Gaussian kernels, [30] consider location scale mixture and [34] studies mixtures of betas.
Nonetheless, since they do not fit well into the usual setting based on the SVD of the operator,
mixture priors have not be considered in the literature for ill-posed inverse problems. In our case,
they proved particularly well suited for the deconvolution problem.
Let Y n = (Y1 , . . . , Yn ) be sampled from model (4.1) for a true regression function f0 ∈
L2 (R) with X = R, and assume that for cx > 0, for all i = 1, . . . , n, xi ∈ [−cx log n, cx log n].
It is equivalent to imposing tail conditions on the design distribution in the random design setting.
We choose a prior that is well suited for f0 in the Sobolev ball W β (L), for some β > 0. To
avoid technicalities, we will also assume that f0 has finite support, that we may choose to be
[0, 1] without loss of generality. Similar results should hold for function with support on R with
additional assumptions on the tails of f0 but are not treated here.
For a collection of kernels Ψv that depend on the parameter v, a positive integer J and a sequence of nodes (z1 , . . . , zJ ) we consider the following decomposition of the regression function
f from the model (4.1)
J
X
f (·) =
wj Ψv (· − zj ),
j=1
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15
Posterior contraction in inverse problems
where (w1 , . . . , wJ ) ∈ RJ is a sequence of weights. We choose Ψj proportional to a Gaussian
kernel of variance v 2 and the uniform sequence of nodes zj = j/J for j such that j/J ∈
[−2cx log n, 2cx log n]
Ψj,v (x) = Ψv (x − zj ) = √
1
2πv 2
e−
(x−j/J)2
2v 2
,
The choice of a Gaussian kernel is fairly natural in the nonparametric literature. In our specific
case it will prove to be particularly well suited. The main advantage of Gaussian kernels in this
case is that we can easily compute the Fourier transform of f and thus use a similar approach as
in Section 3.1 to control the modulus of continuity. We consider the following prior distribution
on f
J ∼ ΠJ
Π := v ∼ Πv
(4.9)
NJ
w1 , . . . , wJ |J ∼ j=1 N (0, 1)
We use a specific Gaussian prior for the weights (w1 , . . . , wJ ) in order to use the results on Reproducing Kernel Hilbert Spaces following [14] to derive contraction rate for the direct problem.
However, we believe that the following result should hold for more general classes of priors, but
the computations would be more involved.
Following [19], we define the degree of ill-posedness of the problem through the Fourier
transform of the convolution kernel. For p > 0, we say that the problem is mildly ill-posed of
degree p if there exist some constants c, C > 0 such that for λ̂, the Fourier transform of λ,
Z
λ̂(t) = λ(u)eitu du,
we have for |t| sufficiently large
c|t|−p ≤ |λ̂(t)| ≤ C|t|−p ,
p ∈ N∗ ,
(4.10)
For all f0 ∈ W β (L), we have that Kf0 ∈ W β+p (L0 ) for L0 = LC. Under these conditions, the
following Theorem gives an upper bound on the posterior contraction rate.
Theorem 4.2. Let Y n = (Y1 , . . . , Yn ) be sampled from (4.1) with X = R and assume that
the design satisfies (x1 , . . . , xn ) ∈ [−cx log n, cx log n]n . Let f0 be such that for β ∈ N∗ and
M > 0, f0 ∈ W β (L) with support on [0, 1] and kf0 k∞ ≤ M . Consider K as in (4.8) with λ
satisfying (4.10). Let Π be a prior distribution as in (4.9) with
ΠJ (J = j) j −s ,
c
c
u
d
v −q exp − log(1/v)u . Πv (v) . v −q exp − log(1/v)u ,
v
v
for some positive constants s, cu , cd , q, and u. Then there exist constants C and r depending
only on Π, L, K and M such that
β
E0 Π kf − f0 k ≥ Cn− 1+2β+2p (log n)r Y n ) → 0.
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16
Bartek Knapik and Jean-Bernard Salomond
Note that the prior does not depend on the regularity β of the true f0 and the posterior contracts
at the minimax rate. Our approach is thus adaptive. Moreover, the prior does not depend on the
degree of ill-posedness either. It is thus well suited for a wide variety of convolution kernels. In
particular, this can be useful when the operator is only partially known, as in this case when the
regularity of the kernel may not be accessible. However, this is beyond the scope of this article.
We prove Theorem 4.2 by applying Theorem 2.1 together with Lemma 2.1. A first difficulty
is to define the sets Sn on which we can control the modulus of continuity. A second problem
is to derive the posterior contraction rate for the direct problem, given that in our setting Kf
is supported on the real line: [14] derived the posterior contraction rate only for Hölder smooth
functions with bounded support. However, their results directly extend to the case of convolution
of Sobolev functions with bounded support given the results of [39]. The complete proof of this
Theorem is postponed to Section 5.3.2.
5. Proofs
5.1. Proof of the main theorem
Proof of Theorem 2.1. By the definition of the modulus of continuity
E0 Π f : d(f, f0 ) ≥ ω(Sn , f0 , d, dK , Mn n ) | Y n
≤ E0 Π f ∈ Sn : d(f, f0 ) ≥ ω(Sn , f0 , d, dK , Mn n ) | Y n + E0 Π(Snc | Y n )
≤ E0 Π f ∈ Sn : dK (Kf, Kf0 ) ≥ Mn n | Y n + E0 Π(Snc | Y n ).
Together with (2.2) and (2.3) it completes the proof.
5.2. Proofs of Section 3
5.2.1. Mildly ill-posed problems
Proof of Theorem 3.1. We first note that if kf kβ ≤ R, then kKf kβ+p ≤ CR. Next we verify
the condition of Lemma 2.1. Let
1
(α∧β)
kn = n 1+2α+2p ,
(α∧β)+p
ρn = n− 1+2α+2p ,
n = n− 1+2α+2p .
Note that
n2n = n · n−
2(α∧β)+2p
1+2α+2p
=n
1+2α−2(α∧β)
1+2α+2p
−
= n
1+2α−2(α∧β)
(α∧β)+p
,
hence Π(Bn (Kf0 , n )) & exp(−C2 n2n ) uniformly over a Sobolev ball of radius R (see Lemma 5.1
at the end of this subsection).
Note also that
2(α∧β)
1+2α
ρ2n kn1+2α = n− 1+2α+2p · n 1+2α+2p = n
1+2α−2(α∧β)
1+2α+2p
= n2n ,
and given c ≥ 2(1 + 2α)/α we have Π(Snc ) ≤ exp(−(c/8)n2n ) by Lemma 5.2.
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17
Posterior contraction in inverse problems
Hence
c
Π(Snc )
. exp −
− C2 n2n ,
Π(Bn (Kf0 , n ))
8
uniformly over a ball of radius R. The condition of Lemma 2.1 is verified upon choosing c =
8(2 + C2 ) ∨ 2(1 + 2α)/α.
Finally, we note that (cf. (3.4))
ω(Sn , f0 , k · k,k · k, Mn n )
(α∧β)+p
p
(α∧β)
β
. Mn n 1+2α+2p · n− 1+2α+2p + n− 1+2α+2p + n− 1+2α+2p
(α∧β)
. Mn n− 1+2α+2p ,
which ends the proof.
Lemma 5.1. Suppose f0 ∈ S β . Then for every R > 0 there exist positive constants C1 , C2
such that for all ∈ (0, 1),
1+2α−2(α∧β)
inf Π(Bn (Kf0 , )) ≥ C1 exp −C2 − (α∧β)+p
.
kf0 kβ ≤R
Proof. This proof is adapted from [7]. Recall that in the white noise model the `2 balls and
Kullback–Leibler neighborhoods are equivalent. By independence, for any N ,
∞
X
(κi fi − κi f0,i )2 ≤ 2
Π
i=1
∞
N
X
X
(κi fi − κi f0,i )2 ≤ 2 /2 .
(κi fi − κi f0,i )2 ≤ 2 /2 Π
≥Π
i=1
Also
∞
X
(5.1)
i=N +1
2
(κi fi − κi f0,i ) ≤ 2
i=N +1
∞
X
κ2i fi2
+2
i=N +1
∞
X
2
κ2i f0,i
.
(5.2)
i=N +1
The second sum in the display above is less than or equal to
2N −2β−2p
∞
X
i=N +1
2
i2β f0,i
≤ 2N −2β−2p kf0 k2β <
2
,
4
whenever N > N1 = (8kf0 k2β )1/(2β+2p) −1/(β+p) .
By Chebyshev’s inequality, the first sum on the right-hand side of (5.2) is less than 2 /4 with
probability at least
1−
∞
∞
8 X
8 X −1−2α−2p
4
2 2
E
(κ
f
)
=
1
−
i
≥1−
> 1/2
Π
i
i
2
2
(α + p)N 2(α+p) 2
i=N +1
i=N +1
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18
Bartek Knapik and Jean-Bernard Salomond
if N > N2 = (8/(α + p))1/(2α+2p) −1/(α+p) .
To bound the first term in (5.1) we apply Lemma 6.2 in [7] with ξi = κi f0,i and δ 2 = 2 /2.
Note that
N
X
i1+2α+2p ξi2
=
i=1
N
X
2
i1+2α+2p · i−2p f0,i
i=1
=
N
X
2 2β
i1+2α−2β f0,i
i ≤ N (1+2α−2β)∨0 kf0 k2β .
i=1
Therefore,
Π
N
X
(κi fi −κi f0,i )2 ≤ 2 /2
i=1
log 2
N exp −N (1+2α−2β)∨0 kf0 k2β
≥ exp − 1 + 2α + 2p +
2
N
X
× Pr
Vi2 ≤ 2δ 2 N 1+2α+2p .
i=1
The last term, by the central limit theorem, is at least 1/4 if 2δ 2 N 1+2α+2p > N and N is
large, that is, N > N3 = −1/(α+p) and N > N4 , where N4 does not depend on f0 . Choosing
N = max{N1 , N2 , N3 , N4 } we obtain
Π(f : kKf −Kf0 k ≤ )
log 2
1
≥ exp − 1 + 2α + 2p +
N exp −N (1+2α−2β)∨0 kf0 k2β .
8
2
Consider α ≥ β. Then exp(−N ) ≥ exp(−N (1+2α−2β) ) so
Π(f : kKf − Kf0 k ≤ ) ≥
1
exp −C3 N (1+2α−2β) ,
8
for some constant C3 that depends only on α, β, p and kf0 k2β . Moreover, since < 1 and α ≥ β,
N is dominated by −1/(β+p) and we can write
1+2α−2β
1
,
Π(f : kKf − Kf0 k ≤ ) ≥ exp −C4 − β+p
8
where C4 depends on f0 again through kf0 k2β only.
Now consider α < β. Similar arguments lead to
Π(f : kKf − Kf0 k ≤ ) ≥
1
1
exp −C5 − α+p ,
8
for some constant C5 that depends only on α, β, p and kf0 k2β .
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19
Posterior contraction in inverse problems
Lemma 5.2. Let ρn be an arbitrary sequence tending to 0, c be an arbitrary constant, and let
the sequence kn → ∞ satisfy kn2α ≥ 2(1 + 2α)/(αcρ2n ). Then
c
Π(Snc ) ≤ exp − ρ2n kn1+2α .
8
Proof. For W1 , W2 , . . . independent standard normal random variables
X
Π(Snc ) = Pr
λi Wi2 > cρ2n .
i>kn
For some t > 0
X
Pr
λi Wi2 > cρ2n
i>kn
X
X
= Pr exp t
λi Wi2 > exp(tcρ2n ) ≤ exp(−tcρ2n )E exp t
λi Wi2
i>kn
=
exp(−tcρ2n )
i>kn
Y
E exp(tλi Wi2 )
i>kn
=
exp(−tcρ2n )
Y
(1 − 2tλi )−1/2 .
i>kn
We first applied Markov’s inequality, and later used properties of the moment generating function. Here we additionally assume that 2tλi < 1 for i > kn .
We take the logarithm of the right-hand side of the previous display. Since log(1 − y) ≥
−y/(1 − y), we have
X
−tcρ2n +
log(1 − 2tλi )−1/2
i>kn
= −tcρ2n −
1 X 2tλi
1 X
log(1 − 2tλi ) ≤ −tcρ2n +
.
2
2
1 − 2tλi
i>kn
i>kn
2tkn−1−2α
We continue with the latter term, noticing that 1 − 2tλi > 1 −
for i > kn
X
X
2tλi
1
t
≤
i−1−2α .
−1−2α
2
1 − 2tλi
1
−
2tk
n
i>k
i>k
n
Since x
n
−1−2α
X
i>kn
is decreasing, we have that
Z ∞
k −2α
1 + 2α
i−1−2α ≤
,
x−1−2α dx + kn−1−2α = n + kn−1−2α ≤ kn−2α
2α
2α
kn
noting that kn > 1 for n large enough. Finally
X
1 + 2α
t
−tcρ2n +
log(1 − 2tλi )−1/2 ≤ −tcρ2n +
k −2α .
2α 1 − 2tkn−1−2α n
i>kn
Thus for t =
since kn2α
kn1+2α /4
c
c
1 + 2α
Π(Snc ) ≤ exp − ρ2n kn1+2α +
kn ≤ exp − ρ2n kn1+2α ,
4
4α
8
≥ 2(1 + 2α)/(αcρ2n ).
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20
Bartek Knapik and Jean-Bernard Salomond
5.2.2. Severely and extremely ill-posed problems
Proof of Theorem 3.2. Assume for brevity that we have the exact equality κi = exp(−γip ).
Dealing with the general case is straightforward, but makes the proofs somewhat lengthier.
Since Yi |fi ∼ N (κi fi , n−1 ) and fi ∼ N (0, λi ) for i ≤ kn , the posterior distribution (for
√
Kf ) can be written as (Kf )i |Y n ∼ N ( nti,n Yi , vi,n ) for i ≤ kn , where
vi,n =
λi κ2i
,
1 + nλi κ2i
ti,n =
nλ2i κ4i
.
(1 + nλi κ2i )2
Since the posterior is Gaussian, we have
Z
X
d − Kf0 k2 +
kKf − Kf0 k2 dΠ(Kf |Y n ) = kKf
vi,n ,
(5.3)
i≤kn
d denotes the posterior mean and can be rewritten as:
where Kf
√
nλ κ2
nλ κ3 f
k n
nλi κ2i Zi kn
i i
i i 0,i
Yi
=
+
2
2
1 + nλi κi
1 + nλi κi
1 + nλi κ2i i=1
i=1
p
d + ( ti,n Zi )kn .
=: EKf
d=
Kf
i=1
By Markov’s inequality the left side of (5.3) is an upper bound to Mn2 ε2n times the desired
posterior probability. Therefore, in order to show that Π(f : kKf − Kf0 k ≥ Mn εn |Y n ) goes
to zero in probability, it suffices to show that the expectation (under the true f0 ) of the right hand
side of (5.3) is bounded by a multiple of ε2n . The last term is deterministic. As for the first term
we have
X
d − Kf0 k2 = kEKf
d − Kf0 k2 +
EkKf
ti,n .
i≤kn
We also observe
d − Kf0 k2 =
kEKf
X
i≤kn
−1
2
X
κ2i f0,i
+
κ2i f02 .
2
2
(1 + nλi κi )
i>kn
−1
Note that ti,n ≤ n and si,n ≤ n , hence
X
1
vi,n . n−1 kn n−1 (log n) p ,
i≤kn
X
1
ti,n . n−1 kn n−1 (log n) p .
i≤kn
By Lemma 5.3
X
i≤kn
2
X
2β
2γα
κ2i f0,i
2γ
2
+
κ2i f0,i
. kf0 k2β n− ξ+2γ (log n)− p + p(ξ+2γ) .
2
2
(1 + nλi κi )
i>kn
Therefore, the posterior contraction rate for the direct problem is given by
β
γα
γ
(log n)− p + p(ξ+2γ) n− ξ+2γ ,
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21
Posterior contraction in inverse problems
and is uniform over Sobolev balls of fixed radius. [This bound is also valid if ξ = 0 and α ≥
1 + 2β.]
By (3.4) an upper bound for the modulus of continuity is given by
ω(Sn , f0 , k · k, k · k, Mn n ) . Mn exp(γknp )n + kn−β
γα
γ
β
. Mn n ξ+2γ (log n)− p(ξ+2γ) n + (log n)− p
β
. Mn (log n)− p ,
which ends the proof.
Lemma 5.3.
It holds that
2
X
κ2i f0,i
i≤kn
(1 +
nλi κ2i )2
+
X
2γ
2
κ2i f0,i
. kf0 k2β n− ξ+2γ (log n)−
2β
2γα
p + p(ξ+2γ)
.
i>kn
Proof. As for the first sum we have
X
i≤kn
2
X
κ2i f0,i
−2 −2β 2β 2
−2
≤
n
λ−2
i f0,i
i κi i
2
(1 + nλi κi )2
i≤kn
X
2
= n−2
i2(α−β) exp(2(ξ + γ)ip )i2β f0,i
,
i≤kn
2(α−β)
and for kn large enough all terms i2(α−β) exp(2(ξ + γ)ip ) are dominated by kn
γ)knp ), so
2
X
κ2i f0,i
≤ n−2 kn2(α−β) exp(2(ξ + γ)knp )kf0 k2β .
(1 + nλi κ2i )2
exp(2(ξ +
(5.4)
i≤kn
As for the second sum we note that
X
X
2
2
κ2i f0,i
=
exp(−2γip )i−2β i2β f0,i
,
i>kn
i>kn
and since exp(−2γip )i−2β is monotone decreasing
X
2
κ2i f0,i
≤ exp(−2γknp )kn−2β kf0 k2β .
(5.5)
i>kn
Recall that exp(knp ) = (nkn−α )1/(ξ+2γ) and therefore we can rewrite the bounds in (5.4) and
(5.5) as
2γα
2(ξ+γ)
2γ
−2β+ ξ+2γ
n−2 kn2(α−β) nkn−α ξ+2γ = n− ξ+2γ kn
,
and
kn−2β nkn−α
2γ
− ξ+2γ
2γ
2γα
−2β+ ξ+2γ
= n− ξ+2γ kn
Finally, since kn in this case can be taken of the order (log n)
bound.
1/p
.
, we obtain the desired upper
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22
Bartek Knapik and Jean-Bernard Salomond
5.3. Proofs of Section 4
5.3.1. Numerical differentiation using spline prior
We first compute an upper bound for the modulus of continuity. For a ∈ RJ we define ∆(a) ∈
RJ−1 such that ∆(a)i = ai+1 − ai , for i = 1, . . . , (J − 1). Given conditions D1 and D2 we get,
kf k2n = J 2 ∆(a)0 Σnq−1 ∆(a) . J 2
. J2
1
k∆(a)k2J−1
J −1
1
kak2J . J 2 kKf k2n .
J −1
To apply Theorem 2.1, we first need to derive a contraction rate for Kf . Note that in this case
we simply have a standard non parametric regression model with a spline prior. This model has
been extensively studied in the literature (see, e.g., [24] or [15]) and we can easily adapt their
results to derive minimax adaptive contraction rates.
Lemma 5.4. Let Π be as in Theorem 4.1. Let Yn be sampled form model 4.1 with f = f0 and
assume that f0 ∈ Θ(β, L, H) with β ≤ q − 1. Then there exists a constant C depending only on
H, L, Π, and q such that
β+1
E0 Π kKf − Kf0 kn ≥ Cn− 2β+3 (log n)r Yn → 0
with r = (1 ∨ t)β/(2β + 1).
Similar results have been proved in [40], however the authors do not give a direct proof of their
result. Here this lemma gives us directly the posterior contraction rate for the direct problem. The
proof of this lemma is postponed to the end of this subsection.
Proof of Theorem 4.1. We now derive the posterior contraction rate of the posterior distribution
for the inverse problem. We first get an upper bound for the modulus of continuity, for f ∈ Sn .
Using standard approximation results on splines (e.g. [13]), we have that for all J there exists
a0 ∈ RJ such that
f0 −
J−1
X
(a0j+1 − a0j )(Bj,q−1 )
j=1
and
Kf0 −
J
X
j=1
a0j Bj,q
∞
∞
≤ (J − 1)−β kf0 k∞ ,
≤ J −β−1 kKf0 k∞ .
We thus deduce that for J ≥ 2,
kf − f0 kn ≤ kf − fa0 kn + kfa0 − f0 kn
≤ CJ −1 kKf − Kfn kn + kfa0 − f0 kn
≤ CJ −1 kKf − Kf0 kn + kKfa0 − Kf0 kn + kfa0 − f0 kn .
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23
Posterior contraction in inverse problems
We can thus deduce an upper bound for the modulus of continuity
ω(Sn , f0 , k · kn , k · kn , δ) ≤ Jn δ.
Applying Theorem 2.1 gives
β
E0 Π kf − f0 kn ≥ Cn− 2β+3 (log n)r Y n → 0,
for a constant C > 0 depending only on kf0 k∞ , r, and Π.
Proof of Lemma 5.4. We prove the lemma using Theorem 4 in [24]. Let β ≤ q and f0 be
in H(β, L) and set n = Cn−(β+1)/(2β+3) (log n)r with r = (1 ∨ t)β/(2β + 1). Set Jn :=
J0 n2n log(n)−t for a fixed constant J0 > 0 and consider the sets Sn defined by
Sn := J ≤ Jn , a ∈ RJ
We first control the local entropy function N (, {J, a ∈ Sn : kKf − Kf0 kn ≤ n }, k · kn ). By
using the same reasoning as in the proof of Theorem 12 in [24], for all J ∈ Sn we get
log(N (, {J, a ∈ Sn : kKf − Kf0 kn ≤ n }, k · kn )) ≤ n2n .
The prior mass of the set Sn is easily controlled using condition (4.5):
Π(Snc ) = ΠJ (J > Jn ) ≤ exp(−cu Jn (log Jn )t ).
We now need to control the prior mass of Kullback–Leibler neighborhoods of Kf0 . Note that
this condition will also be useful to apply Lemma 2.1 and thus derive the posterior contraction
rate for the direct problem. Let Bn (Kf0 , ) be defined as in (2.4).
Using the results of Section 7.3 in [24], setting J˜n = Jn (log n)−r/β we deduce that for some
constant c depending only on σ
Bn (Kf0 , n ) ⊃ {J˜n ≤ J ≤ 2J˜n , kKf − Kf0 k2n ≤ c2n }.
Standard approximation results on splines give that for all J there exists a sequence a0 =
(a0,1 , . . . , a0,J ) such that
Kf0 −
J
X
j=1
a0,j Bj,q
n
≤ J −β−1 kKf0 kβ ≤ J −β−1 L.
Given condition D1 on the design, we thus have that for a constant c0 > 0 depending only on σ
and L
Bn (Kf0 , n ) ⊃ J˜n ≤ J ≤ 2J˜n , ka − a0 kJ˜n ≤ c0 J˜n1/2 n .
Therefore, we obtain a lower bound on the prior mass of a Kullback–Leibler neighbourhood of
Kf0 :
Π(Bn (Kf0 , n )) ≥ Π J˜n ≤ J ≤ 2J˜n , ka − a0 kn ≤ c0 J˜n1/2 n
≥ exp −J˜n (cd (log J˜n )t + c2 log(J˜n−1/2 −1
n )) .
imsart-bj ver. 2011/11/15 file: inverse_general_BJ_rev3.tex date: January 24, 2017
24
Bartek Knapik and Jean-Bernard Salomond
We thus have for C2 > 0,
Π(Snc )
≤ exp(−C2 Jn (log Jn )t ),
Π(Bn (Kf0 , n ))
(5.6)
which together with Theorem 4 in [24] ends the proof.
5.3.2. Deconvolution using mixture priors
Proof of Theorem 4.2. We first specify the sets Sn for which we can control the modulus of
continuity. Denoting fˆ the Fourier transform of f , for any sequence an going to infinity and
In = [−an , an ] we define for a > 0
Z
n Z
o
Sn = f :
|fˆ(t)|2 dt ≥ a
(5.7)
|fˆ(t)|2 dt .
c
In
In
We control the modulus of continuity ω(Sn , f0 , k · k, k · k, δ) in a similar way as in Section 3.1.
First consider f ∈ Sn , and denote fˆn (·) = fˆ(·)IIn (·). We then have
Z
2
kf k2 = kfˆk2 ≤ (1 + a)kfˆn k2 . a2p
|fˆ|2 |λ̂|2 . a2p
n
n kKf k .
In
Note that for f0 ∈ W β (L) we have for f0,n (x) =
kf0 − f0,n k ≤ 2a−β
n L,
R
fˆ0,n (t)e−itx dt
kKf0 − Kf0,n k ≤ 2an−(β+p) L0 ,
which gives
ω(Sn , f0 , k · k, k · k, δ) . apn δ + a−β
n .
(5.8)
We now control the prior mass of Snc in order to apply Lemma 2.1. Denote by ln = ban /(2ΠJ)c,
Ln = dan /(2ΠJ)e. We have
Z
|fˆ(t)|2 dt ≥ 2πJ
Z
ln
e−4π
2 2 2
t v
−Ln
In
= 2πJ
ln
X
= 2πJ
0
≥ 2πJ
1
e−4π
t v
wj e2πjt
e−4π
wj e2πjt dt
j=1
j=1
l=−Ln
J
X
2 2 2
l
J
X
ln
X
wj e2πjt dt
j=1
l+1
Z
l=−Ln
Z
J
X
ln
X
e−4π
2
(t+l)2 v 2
dt
l=−Ln
2
(1+|l|)2 v 2
Z
0
1
J
X
wj e2πjt dt,
j=1
imsart-bj ver. 2011/11/15 file: inverse_general_BJ_rev3.tex date: January 24, 2017
25
Posterior contraction in inverse problems
and similarly we get
Z
|fˆ(t)|2 dt ≤ 2πJ
c
In
1
Z
0
≤ 2πJ
J
X
wj e2πjt
−L
Xn
j=1
−L
Xn
e
e−4π
2
(t+l)2 v 2
+
l=−∞
−4π 2 l2 v 2
+
l=−∞
∞
X
e
∞
X
e−4π
2
(t+l)2 v 2
dt
l=ln
−4π 2 l2 v 2
1
Z
J
X
0
l=ln
wj e2πjt dt.
j=1
0
We thus deduce that for absolute constants C > 0
Π(Snc ) ≤ Π(v ≤ J/an ) . e−C
0
an log an
.
We end the proof by combining this result (choosing an = n2n ) with Lemma 2.1, Lemma 5.5,
and Theorem 2.1.
Lemma 5.5. Let Y n be sample from (4.1) with K defined by (4.8). Let Π be as in Theorem 4.2.
For all β ∈ N∗ if f0 ∈ W β (L) with support on [0, 1] and ||f0 ||∞ ≤ M , we have for C > 0 large
enough if n = n−(β+p)/(1+2β+2p) (log n)r , where r is some constant,
E0 Π(kKf − Kf0 k ≥ Cn |Y n ) → 0,
and
2
Π(kKf − Kf0 k ≤ n ) ≥ e−nn .
Proof. This proof is based on the results of [14] and [39]. We adapt the results of [14] to our
setting in order to control the posterior mass of the Kullback–Leibler neighbourhoods of Kf0
and the posterior contraction rate for the direct problem. Following their notation we have that
KΨv ∈ P∞ , and thus the small ball probability Π(kf k∞ ≤ ) can be controlled by their Lemma
3.3. We then extend their Lemma 3.5 to our setting. Note that with Lemma 9 of [39], Lemma 3.4
of [14] holds for the same Tα,v with α = β + p. Choosing h to be as in the proof of Lemma 3.5
of [14] and denoting ω0 = f0 ? λ, we have
X
1 x − j/J
h(x) =
Tα,v (ω0 ) Ψ
,
Jv
v
j/J∈[−2cx log n,2cx log n]
and thus deduce
khk2H J,v ≤ 2cx kTα,v (ω0 )k2 log n.
Using their decomposition (3.8), we control |h(x) − Ψv ? Tα,v (ω0 )(x)| along the same lines as
in their computations on page 3312. We have
|h(x) − Ψv ? Tα,v (ω0 )(x)|
Z 2cx log n
≤ h(x) −
Tα,v (ω0 )(y)Ψv (x − y)dy
Z
−2cx log n
−2cx log n
Z
∞
Tα,v (ω0 )(y)Ψv (x − y)dy +
+
−∞
Tα,v (ω0 )(y)Ψv (x − y)dy
2cx log n
imsart-bj ver. 2011/11/15 file: inverse_general_BJ_rev3.tex date: January 24, 2017
26
Bartek Knapik and Jean-Bernard Salomond
The first term on the right hand side of the above display can be controlled as in the proof of
Lemma 3.5 of [14]. For the last two terms, we have
Z ∞
Z −2cx log n
Tα,v (ω0 )(y)Ψv (x − y)dy
Tα,v (ω0 )(y)Ψv (x − y)dy +
−∞
2cx log(n)
. kTα,v (ω0 )k∞ e
c2 (log n)2
− x 2v2
v −1 .
Following the same proof of Theorem 2.2 of [14], we get
β+p
E0 Π(kKf − Kf0 k ≥ Cn− 1+2β+2p (log n)r |Y n ) → 0,
and similarly to their equation (2.5) we get, with n = n−(β+p)/(1+2β+2p) (log n)r , where r is
some constant,
2
Π(kKf − Kf0 k ≤ n ) ≥ e−nn .
6. Discussion
In this paper we propose a new approach to the problem of deriving posterior contraction rates
for linear ill-posed inverse problems. More precisely, we put a prior on the parameter of interest
f that naturally imposes the prior on Kf , leading to a certain rate of contraction in the direct
problem. Next, we consider a sequence of sets on which the operator K possesses a continuous
inverse. Then, we impose additional conditions on the prior (or the posterior itself) under which
the posterior contracts at a certain rate in the inverse problem setting.
This is a great advantage of the Bayesian approach in this setting as when the posterior distribution is known to contract at a given rate in the direct problem, one only has to consider subset
of high prior mass for which the norm of the inverse of the operator may be handled. Our result
seems to show that the main difficulty when considering linear inverse problems is to control the
change of metrics form dK to d, which is dealt here by considering the modulus of continuity as
introduced in [17] and [26]. It is also to be noted that contrariwise to existing methods, we do
not require a Hilbertian structure for the parameter space, see for instance the example treated
in Section 4.1. This could be particularly useful when considering nonlinear operators, and is of
potential interest when considering the case of partially known operators.
We recovered (a subset of) the existing results from [28], [29], [1], [2], and [33]. Our approach
should be viewed as a generalization of the ideas presented in the last paper and the existing sieve
method used in the literature on posterior contraction. Furthermore, we were able to go beyond
the sequence setting as well as derive posterior contraction rates for prior distributions that were
not covered by the existing theory. We also treated an operator that does not admit singular value
decomposition. In this sense, the approach proposed in this paper is more general, and we believe
more natural, than the existing ones.
imsart-bj ver. 2011/11/15 file: inverse_general_BJ_rev3.tex date: January 24, 2017
Posterior contraction in inverse problems
27
Acknowledgements
The authors would like to thank the editor, the associate editor, and the referees for their comments which helped to improve this paper. The authors are also grateful to Judith Rousseau and
Eduard Belitser for helpful discussions and comments. This work is partially funded by the STAR
cluster, ANR Bandhits, VICI Safe Statistics and Labex ECODEC. This work is part of the second
author’s PhD.
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| 10 |
The Robust Reading Competition Annotation and
Evaluation Platform
Dimosthenis Karatzas, Lluis Gómez Marçal Rusiñol
arXiv:1710.06617v1 [cs.CV] 18 Oct 2017
Computer Vision Centre, Universitat Autonoma de Barcelona, Barcelona, Spain;
{dimos, lgomez, marcal}@cvc.uab.es
Abstract—The ICDAR Robust Reading Competition (RRC),
initiated in 2003 and re-established in 2011, has become the defacto evaluation standard for the international community.
Concurrent with its second incarnation in 2011, a continuous
effort started to develop an online framework to facilitate the
hosting and management of competitions.
This short paper briefly outlines the Robust Reading Competition Annotation and Evaluation Platform, the backbone of the
Robust Reading Competition, comprising a collection of tools and
processes that aim to simplify the management and annotation
of data, and to provide online and offline performance evaluation
and analysis services.
I. I NTRODUCTION
The Robust Reading Competition (RRC) series1 addresses
the need to quantify and track progress in the domain of
text extraction from a variety of text containers like borndigital images, real scenes, and videos. The competition
was initiated in 2003 by Simon Lucas et al. [1] initially
focusing only on scene text detection and recognition, and
extended to include challenges on born-digital images [2],
video sequences [3], and incidental scene text [4]. The 2017
edition of the Competition, under way at the time of writing,
introduces five new challenges: a challenge on scene text
detection and recognition based on the COCO-Text dataset, the
largest scene text dataset currently available [5]; a challenge on
text extraction from biomedical literature figures based on the
DeText dataset [6]; a challenge on video scene text localization
and recognition on the Downtown Osaka Scene Text (DOST)
dataset [7]; a challenge on constrained real world end-to-end
scene-text understanding based on the > 1M images French
Street Name Signs (FSNS) dataset [8]; and a challenge on
Multi-lingual scene text detection and script identification [9].
Over the past six years, the competition has grown steadily,
reaching more than 3,000 registered users from more than 80
countries by mid-2017, who have submitted more than 10,000
results that have been automatically evaluated online. Out of
these, 424 have been made public by their authors. A summary
of the submissions made is given in Table I. The portal receives
and evaluates on average 10-20 new submissions per day. In
terms of visibility, the RRC Web portal has received 360K
page views from 21K users over the past four years.
To manage all the above Challenges and respond to the
increasing demand, the Computer Vision Centre has invested
1 http://rrc.cvc.uab.es/
significant resources to the development of the RRC Annotation and Evaluation Platform, which is the backbone of the
competition and is briefly introduced next.
II. T HE RRC A NNOTATION AND E VALUATION P LATFORM
The RRC Annotation and Evaluation Platform, is a collection of tools and processes that aim to facilitate the generation
of data, the definition of performance evaluation metrics for
different research tasks and the visualisation and analysis
of results. The interface has evolved over time to support
image annotation at different levels, provide version control
and coordination mechanisms between ground-truthers and
facilitate the verification of the final annotations. All online
software tools are implemented as HTML5 interfaces, while
specialised processing (e.g. the calculation of performance
evaluation metrics) takes place on the server side and is
principally coded in python. Key features of the platform
include:
• A comprehensive range of ground truthing tools
• Centralised management of the annotation process
• Quality control and versioning
• Streamlined definition of evaluation scenarios
• Calculation of performance evaluation metrics
• In-depth results visualisation including intermediate evaluation steps
An earlier version of the annotation platform was made
public in 2013, and is described in detail in [10]. In 2015,
key updates were made to support the definition of nonaxis oriented, quadrilateral boxes for words and text lines, as
required for the Incidental Scene Text dataset. In the following
section we briefly highlight some of the key aspects of the
RRC Annotation and Evaluation Platform.
A. Dataset Management
Datasets of images can be created and managed through online interfaces, supporting the direct uploading of images, but
also offering tools to harvest images online. As an example, a
Google Street View crawler is integrated in the interface and
can be used to automatically harvest images from Street View
as seen in Figure 1.
Figure 2 shows a screenshot of the ground truthing management tool. The platform presents a searchable list to the ground
truth manager that allows one to keep track of the overall
progress, respond to specific comments that ground truthers
TABLE I
N UMBER OF SUBMISSIONS TO THE DIFFERENT DATASETS OF THE RRC.
Born Digital
Focused Scene Text
Text in Video
Incidental Scene Text
New 2017 Challengesb
Public
Submissions
Private
Submissions
63
142
18
93
108
1,403
6,445
435
1,734
44
a Activity shown is for period from 2013 to August 2017.
b Preliminary figures, as the competition is still under way
Years
Active
2011
2003
2013
2015
2017
-
2017
2017
2017
2017
a
at the time of writing.
defined text parts is displayed on the left-hand side of the
interface. In the example shown, text atoms are defined at the
pixel level in terms of their area and skeleton, and grouped
together to form words and text lines. Alternatively, annotations directly at the bounding box level (axis oriented or 4point quadrilaterals), and at different granularities (characters,
words, text blocks) are supported.
Fig. 1. The integrated Street View crawler.
make and assign a quality rating to each image. The Quality
Manager can make use of this information to provide feedback
and ensure consistency of the ground truthing process. The
same interface allows assigning images to the different subsets
(training, validation, public and sequestered test) that are
subsequently used for defining evaluation scenarios.
Fig. 2. A reduced screenshot of the ground truth management tool.
B. Image Annotation
Using the RRC Annotation and Evaluation platform it is
possible to generate ground truth that represents everything
from the pixel level to text lines in an image. behind the
scenes, the RRC Annotation and Evaluation Platform stores
such ground truth in a hierarchical tree using a combination
of XML files for all metadata and transcription information
and image files for pixel level annotations.
A screenshot of one of the Web-based annotation tools can
be seen in Figure 3. The hierarchy of textual content and the
A number of tools are provided to ensure consistency and
quality. For example, in the particular case of 4-point quadrilateral bounding boxes, when text with perspective distortion
is annotated, it is often difficult for annotators to agree on
what is a good annotation. To ensure consistency in the ground
truth definition, a real time preview of a rectified view of the
region is provided, and annotators are required to adjust the
quadrilateral so that the rectified word appears correct (see
Figure 4). This process improves substantially the consistency
between different annotators.
All annotated elements, apart from their transcription, can
have any number of custom defined associated metadata like
script information, quality metrics etc. A special element
type is text that should be excluded from the competitive
process, and is thus marked as do not care. Depending on
the challenge, such cases can include text which is partially
cut, low resolution text, text in scripts other than the ones
the challenge focuses on, or indeed any other text that the
annotator deems as unreadable text.
Judging whether a text should be marked as do not care is
challenging and in some cases similar text might be treated
differently by individual annotators. At the same time, there
are many cases where text can be read because the context
is clear (e.g. if the words on the left and right are readable
the middle word can be easily guessed), and annotators have
trouble deciding whether such text should be actually marked
as do not care or not. To reduce such subjective judgements
different verification processes are available through the RRC
platform, including interfaces for verifying words on their
own, out the context, which has been shown to eliminate the
inherent bias of annotators to use textual context to guess the
transcription (see Figure 5).
The Web-based annotation functionality is available to use
for research purposes by contacting with the RRC organisers.
Fig. 3. A screenshot of one of the Web-based annotation tools.
Fig. 4. Annotators are shown a real time preview of a rectified version of
the word region being defined.
C. Evaluation and Visualisation of Results
The online portal permits users to upload results of their
methods against a public validation or test dataset and obtain
evaluation results online. Apart from ranked tables of quantitative results of submitted methods, users can see per sample
visualisations of their results along with insights about the
intermediate evaluation steps, as seen in Figure 6. Through the
same interface users can hot-swap between different methods
to easily compare behaviours.
The python evaluation scripts used by the RRC platform
are publicly available for each of the tasks. In addition, a
standalone pack integrating the evaluation and visualisation
interface is available to download. The pack can run offline
on the user’s machine and provides a Web-based graphical
interface similar to the RRC portal’s.
III. C ONCLUSION
The RRC Annotation and Evaluation Platform is the backbone of the Robust Reading Competition’s online portal. Many
Fig. 5. Do not care regions appear in red, normal regions appear in green. Do
not care regions do not have to respect the granularity of the rest of the ground
truth. In the example, words have gone through a second-stage verifications
where their readability was judged individually to eliminate any annotation
bias introduced by contextual information (e.g. words that can be guessed to
say ”roast chicken” due to the visual context were judged as unreadable when
seen individually).
of the functionalities are exposed to the public (e.g. evaluation and visualisation of results), while others are accessible
through contacting with the authors (e.g. annotation tools and
dataset management). We strive to provide code when possible,
although this is not always feasible due to the tight integration
of certain functionalities with the Web portal. Nevertheless,
a full version of the RRC Web portal was made public
in the past [10], while more recently standalone interfaces
for evaluation and visualisation were also made available to
download.
[7] M. Iwamura, N. Morimoto, K. Tainaka, D. Bazazian, L. Gomez, and
D. Karatzas, “ICDAR2017 robust reading challenge on omnidirectional
video,” in Document Analysis and Recognition (ICDAR), 2017 14th
International Conference on. IEEE, 2017.
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and S. Lin, “End-to-end interpretation of the french street name signs
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(ECCV). Springer, 2016, pp. 411–426.
[9] N. Nayef, F. Yin, I. Bizid, H. Choi, Y. Feng, D. Karatzas, Z. Luo, U. Pal,
C. Rigaud, J. Chazalon, W. Khlif, M. L. Muzzamil, J.-C. Burie, C.-l.
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[10] D. Karatzas, S. Robles, and L. Gomez, “An on-line platform for ground
truthing and performance evaluation of text extraction systems,” in
International Workshop on Document Analysis Systems (DAS), 2014,
pp. 242–246.
Fig. 6. Per-image results interface for text localisation.
ACKNOWLEDGEMENTS
This work is supported by Spanish project
TIN2014-52072-P and the CERCA Programme / Generalitat
de Catalunya.
R EFERENCES
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| 1 |
arXiv:1703.10959v3 [cs.DC] 22 Dec 2017
Parallelism, Concurrency and Distribution in Constraint Handling
Rules: A Survey
THOM FRÜHWIRTH
University of Ulm, Germany
December 25, 2017
Abstract
Constraint Handling Rules (CHR) is both an effective concurrent declarative programming language and a versatile computational logic formalism. In CHR, guarded reactive rules rewrite a multiset of constraints. Concurrency is
inherent, since rules can be applied to constraints in parallel.
In this comprehensive survey, we give an overview of concurrent, parallel as well as distributed CHR semantics,
standard and more exotic, that have been proposed over the years at various levels of refinement. These semantics
range from the abstract to the concrete. They are related by formal soundness results. Their correctness is proven as
a correspondence between parallel and sequential computations.
On the more practical side, we present common concise example CHR programs that have been widely used
in experiments and benchmarks. We review parallel and distributed CHR implementations in software as well as
hardware. The experimental results obtained show a parallel speed-up for unmodified sequential CHR programs. The
software implementations are available online for free download and we give the web links.
Due to its high level of abstraction, the CHR formalism can also be used to implement and analyse models for
concurrency. To this end, the Software Transaction Model, the Actor Model, Colored Petri Nets and the Join-Calculus
have been faithfully encoded in CHR. Finally, we identify and discuss commonalities of the approaches surveyed and
indicate what problems are left open for future research.
KEYWORDS: Parallelism, Concurrency, Distribution,
Declarative Programming, Concurrent Constraint Programming, Constraint Handling Rules,
Semantics, Rewriting, Concurrency Models.
Contents
1
Introduction
2
Parallel Abstract Operational Semantics of CHR
2.1 Semantics of CHR and their Properties . . . . . . . . .
2.2 Abstract Syntax of CHR . . . . . . . . . . . . . . . .
2.3 Sequential Abstract Operational Semantics of CHR . .
2.4 Extension to Parallel Abstract Semantics . . . . . . . .
2.5 Properties: Monotonicity, Soundness and Serializability
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3.1 Algorithms of Erastothenes, Euclid, von Neumann, Floyd and Warshall
3.2 Classical Models and Classical Algorithms with Statefulness . . . . . .
3.3 Parallel Preflow-Push Algorithm . . . . . . . . . . . . . . . . . . . . .
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Parallel CHR with Transactions
4.1 Transactions in Parallel CHR . . . . . . . . . . . . . .
4.2 Abstract Syntax and Semantics of CHRt . . . . . . . .
4.3 Properties: Monotonicity, Soundness and Serializability
4.4 Encoding Transactions in Standard CHR . . . . . . . .
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Refined Parallel CHR Semantics
5.1 Syntax for Refined Parallel CHR . . . . . . . . . . . .
5.2 Sequential Refined CHR Semantics . . . . . . . . . .
5.3 Extension to Parallel Refined CHR Semantics . . . . .
5.4 Properties: Monotonicity, Soundness and Serializability
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Parallel CHR Implementation in Haskell
6.1 Implementation Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Massively-Parallel Set-Based CHR Semantics
7.1 Massively-Parallel Set-Based Semantics CHRmp . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.2 Example Programs under Exhaustive Parallelism . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7.3 Properties: Soundness under Deletion-Acyclicity . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Parallel Hardware Implementations of CHR
8.1 Basic Compilation of CHR to Procedural Languages . . . . . . . . . . . . . . . . . . . . . . . . . .
8.2 Compiling CHR to Parallel Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Distribution in CHR
9.1 Distributed Set-Based Goal Stores in CHRd . . . . . .
9.2 Distributed Parallel CHRe and its Syntax . . . . . . . .
9.3 Refined Semantics of CHRe . . . . . . . . . . . . . .
9.4 Properties: Monotonicity, Soundness and Serializability
9.5 Encoding 1- and n-Neighbor Rules in Local Rules . . .
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10 Models of Concurrency in CHR
10.1 Software Transactional Memory STM
10.2 Colored Petri Nets CPN . . . . . . . .
10.3 Actor Model . . . . . . . . . . . . . .
10.4 Join-Calculus and Join-Patterns . . . .
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11 Discussion and Future Work
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12 Conclusions
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1 Introduction
Parallelism has become an eminent topic in computer science again with the widespread arrival of multi-core processors. With the proliferation of mobile devices and the promises of the internet-of-things, distribution is another
major topic, intertwined with parallelism. Parallel and distributed programming is known to be difficult. Declarative
programming languages promise to ease the pain. This survey shows how parallelism and distribution are addressed
in the declarative language Constraint Handling Rules.
Basic Notions. Before we start with our survey, we shortly clarify the essential concepts at stake and introduce
Constraint Handling Rules. The technical terms of concurrency, parallelism and distribution have an overlapping
2
meaning, and processes are another central notion in this context. Due to their generality they are hard to define
precisely:
Concurrency allows for logically more or less independent computations, be they sequential or parallel. This abstract
concept thus supports the modular design of independent program components that can be composed together.
Parallelism allows for computations that happen simultaneously, at the same time, thus hopefully improving performance. On the downside, sequential programs usually have to be rewritten to be able to run in parallel. With the
arrival of multi-core processors, it has become a dominant computation model. The processors may have access
to a shared memory to exchange information.
Distribution allows for program components that are located on physically distributed decentralized networked processors. Each processor has its own local memory (distributed memory). Personal computers, the internet and
mobile devices have enforced this computational paradigm. Distribution introduces modularity and potential
parallelism, but also the need for communication between the components.
Processes are programs that are executed independently but can interact with each other. Processes can either execute local actions or communicate, coordinate and synchronize by passing (sending and receiving) messages.
Depending on context and level of abstraction, processes are also called threads, workers, tasks, activities or
even agents.
Concurrency and distribution are easier with declarative programming languages, since they are compositional: different computations can be composed into one without unintended interference. Moreover, declarative languages offer a
wealth of program analysis and reasoning techniques.
Constraint Handling Rules (CHR). CHR is both an effective concurrent declarative constraint-based programming language and a versatile computational logic formalism [Frü09, SVWSDK10, FR11, Frü15, Frü16]. CHR has its
roots in constraint logic programming and concurrent constraint programming, but also integrates ideas from multiset
transformation and rewriting systems. While conceptually simple, CHR is distinguished by a remarkable combination
of desirable features:
• a semantic foundation in classical logic as well as in linear logic [Bet14],
• an effective and efficient sequential and parallel execution model [FR11],
• a proof that every algorithm can be expressed with best known time and space complexity [SSD09],
• up to a million rule applications per second due to CHRs novel rule execution strategy based on lazy matching
without conflict resolution [VW10],
• guaranteed properties like the anytime algorithm and online algorithm properties [AFM99],
• program analysis methods for deciding essential properties like confluence and program equivalence [AF99].
The given references are meant to serve as starting points into the respective themes. One could continue with their
references but also the papers that reference them.
Information on CHR can be found online at http://www.constraint-handling-rules.org, including news,
tutorials, papers, bibliography, online demos and free downloads of the language.
Minimum Example. Assume we would like to compute the minimum of some numbers, given as multiset
min(n1), min(n2),..., min(nk ). We interpret the constraint (predicate) min(ni) to mean that the number ni
is a candidate for the minimum value. We make use of the following CHR rule that filters the candidates.
min(N) \ min(M) <=> N=<M | true.
The rule consists of a left-hand side, on which a pair of constraints has to be matched, a guard check N=<M that has to
be satisfied, and an empty right-hand side denoted by true. In effect, the rule takes two min candidates and removes
the one with the larger value (constraints after the \ symbol are to be removed). Starting with a given initial state, CHR
rules are applied exhaustively, resulting in a final state. Note that CHR is a committed-choice language, i.e. there is
3
no backtracking in the rule applications. Here the rule keeps on going until only one, thus the smallest value, remains
as single min constraint. Note that the min constraints behave both as operations (removing other constraints) and as
data (being removed). This abstraction is characteristic of the notion of constraint.
A state is a multiset of constraints. In a sequential computation, we apply one rule at a time to a given state. A
possible computation sequence is (where we underline constraints involved in a rule application):
min(1), min(0), min(2), min(1) 7→
min(0), min(2), min(1) 7→
min(0), min(1) 7→
min(0)
The final state is called answer. The remaining constraint contains the minimum value, in this case zero.
By the way, CHR insists on multisets so one can directly model resources as constraints, for example:
buy : cup \ euro, euro <=> coffee.
This rule expresses that we get a coffee for two euros if we have a cup. As we will see, there are also some semantics
and implementations of CHR that are set-based.
Concurrency and Parallelism in CHR. One of the main features of CHR is its inherent concurrency. Intuitively,
in a parallel execution of a CHR program, rules can be applied to separate parts of a state in parallel. As we will see,
CHR rules can even be applied in parallel to overlapping parts of a state, in principle without the need to change the
program. This is referred to as logical parallelism or declarative concurrency.
The rule of min can be applied in parallel to different parts of the state:
min(1), min(0),
min(0),
min(2), min(1) 7→
min(1) 7→
min(0)
We arrive at the answer in less computation steps than with the sequential execution.
The rule can also be applied in parallel to overlapping parts of the state, provided the overlap is not removed by any
rule. For example, let the overlap be the constraint min(0). Then the three pairs min(0), min(1), min(0), min(1)
and min(0), min(2) can be matched to different rule instances. (Note that we always match the same min(0), but
that we have two copies of min(1).) These rules can be applied at the same time, since the common (overlapping)
constraint min(0) is not removed.
min(0),
min(1), min(2), min(1) 7→
min(0)
So this is another, even shorter way to arrive at the same answer.
In CHR, concurrently executing processes are CHR constraints that communicate via a shared built-in constraint
store. The built-in constraints take the role of (partial) messages and variables take the role of communication channels.
Guaranteed Properties of CHR. First of all, the essential monotonicity property of CHR means that adding
constraints to a state cannot inhibit the applicability of a rule. (Rule matching and guards check for presence of certain
constraints, never absence.) Among other things, this monotonicity enables decidable program analyses and helps
declarative concurrency. Most, but not all semantics that we introduce enjoy the monotonicity property.
Now assume that while the program runs, we add another constraint. It will eventually participate in the computation in that a rule will be applied to it. The answer will be as if the newly added constraint had been there from the
beginning but ignored for some time. This property of a CHR program is called incrementality or online algorithm
property and directly follows from monotonicity.
Furthermore, in CHR, we can stop the computation at any time and observe the current state as intermediate
answer. We can then continue by applying rules to this state without the need to recompute from scratch. If we stop
again, we will observe a next intermediate answer that is closer to the final answer. This property of a CHR program
is called the anytime algorithm property. Note that by this description, an anytime algorithm is also an approximation
algorithm, since intermediate answers more and more approximate the final answer.
4
Desirable Property of Confluence. This property of a program guarantees that any computation starting from a
given initial state results in the same answer no matter which of the applicable rules are applied. There is a decidable,
sufficient and necessary syntactic condition to analyse confluence of terminating programs and to detect rule pairs that
lead to non-confluence when applied. Among other things, confluence implies that rules can be applied in parallel,
with the same result as any sequential computation, without the need for any modification of the given program. If
on the other hand a program is not confluent, it may have to be rewritten to ensure proper parallel execution. This
rewriting is aided by the method of completion, which automatically adds rules to a program to make it confluent (but
may not terminate).
An introduction into all these properties can be found in [Frü09]. In the next section we will discuss desirable
properties that characterize the correspondence between different semantics of CHR.
Overview of the Survey and its Structure. The richness of topics in this survey, from formal semantics to
hardware implementation and more, poses a challenge for the structure of this text. We decided to go from abstract to
concrete while making sure concepts are introduced in sections before they are referred to in later sections.
Section 2-4: Abstract Parallel CHR Semantics, Example Programs, Extension by Transactions. In the next section
we define abstract syntax and abstract operational semantics for CHR. One sequential transition describes rule
applications, another one parallel transitions, a trivial third one connects the two. The essential correctness
properties of monotonicity, soundness and serializability are introduced. In Section 3, we present common
classic CHR example programs based on well-known algorithms. Often one rule suffices. All but one of the
programs can be run in parallel without change. In Section 4, we extend abstract parallel CHR with transactions,
a popular and essential concept in concurrency.
Section 5-6: Refining the Parallel Semantics and its Implementation. In Section 5, we refine our abstract semantics
by differentiating between a goal and a constraint store. The goal holds active constraints to execute them as
processes in operation, the constraint store holds inactive constraints as data. This implies that we now have to
account for the in-activation (suspension) and re-activation (wake-up) of user-defined constraints. In Section 6,
we describe an implementation of the refined semantics in Haskell using software transactions and the result of
benchmark experiments showing parallel speed-ups.
Section 7-8: Excursion: Set-Based Massive Parallelism and Hardware Implementations. Section 7 introduces a
more exotic abstract semantics that is massively parallel. It is also set-based. This theoretical model in the
extreme case allows to find primes in constant time and to solve SAT problems in linear time. This comes
with a cost: soundness only holds under a certain condition. We then move on to more mundane fast hardware
implementations of the parallel CHR semantics introduced in Section 8 and again present some experimental
evidence. It is typically one order of magnitude faster than the fastest software implementations. The translation
scheme of the hardware implementations also applies to procedural languages like C and Java.
Section 9: Distribution in CHR. In Section 9 we discuss two distributed semantics for CHR, where the constraint
store and computations are decentralized by introducing the notion of locations. Distribution requires a syntactic
restriction on CHRs rule heads to ensure shared variables as communication channels among locations. The first
semantics is informal and set-based, the second one full-fledged. Both semantics allow for propagation rules.
Both semantics have been implemented.
Section 10: Concurrency Models in CHR. Last but not least, in Section 10 we shortly show the high-level encoding
common formal models of concurrency in CHR on four concrete models: the Software Transaction Model,
the Actor Model, Colored Petri Nets and the Join-Calculus have been faithfully embedded in CHR to enable
comparison and further investigation by the program analyses available in CHR. The embeddings have been
proven correct. Some embeddings are available online.
Section 11-12: Discussion and Conclusions. We end the paper with a discussion, directions for future work and in
Section 11 with conclusions.
Within the sections, we also try to follow a standard structuring where applicable: We define the parallel or
distributed semantics at hand and discuss its correspondence to the standard sequential CHR semantics. This usually
5
done by proving the properties of soundness and serializability, which are notions of correctness. Another property
of interest is monotonicity, which is also enjoyed by standard CHR. For software and hardware implementations, we
give free download links and we summarize experimental results found in the literature. We illustrate the approaches
to semantics and implementation with additional examples.
For a better reading experience, we use the editorial we throughout. Of course it refers to different authors in
different sections of this paper.
2 Parallel Abstract Operational Semantics of CHR
We will present the sequential equivalence-based abstract CHR semantics and extend it with parallelism. We just need
a sequential transition describes rule applications, another one parallel transitions, a trivial third one that connects the
two. We also introduce the three properties that prove the correctness of a given semantics with regard to a more
abstract or a sequential semantics: monotonicity, soundness and serializability. We assume basic familiarity with firstorder predicate logic and state transition systems. Readers familiar with CHR can skip most of this section. We start
with some preliminaries.
2.1 Semantics of CHR and their Properties
Structural Operational Semantics (SOS) is a common inductive approach to describe the behavior of programming
languages, in particular concurrent ones. In SOS, a state transition system specifies the computations. Transitions
rewrite states and take the form of inference rules. All semantics of CHR, sequential or parallel, employ this approach.
Semantics for sequential CHR. They exist in various formulations and at various levels of refinement, going from
the abstract to the concrete (refined) [Frü09, BRF10]:
• The very abstract semantics [Frü09] is close to modus ponens of predicate logic.
• The abstract semantics [AFM99] is the classical basis for CHR program analysis and its properties.
• The more recent state-equivalence-based abstract semantics [RBF09] will be the starting point of our survey.
We will extend it with parallelism.
• The refined semantics [DSGH04] describes more concretely the actual behavior of CHR implementations. All
more concrete parallel semantics of CHR are based on it.
In addition, several alternative operational semantics for sequential CHR have been proposed.
Soundness and Serializability. The correctness of a more refined semantics is shown by its soundness with regard
to a more abstract semantics. This means that for each computation in the refined semantics, there is a corresponding
computation in the abstract semantics. The converse (completeness) typically does not hold, because refined semantics
are more concrete and thus rule out certain computations. When we introduce a parallel semantics for CHR, it will be
related by soundness to a more abstract semantics and/or the sequential part of the semantics.
Actually, the interleaving semantics approach to concurrency is defined by the fact that for each possible parallel
computation, there exists a corresponding sequential computation with the same result. The sequential computation
uses interleaving of the different parallel computations. This means that a parallel computation step can be simulated
by a sequence of sequential computation steps. This correspondence property is called serializability (sequential
consistency). Most semantics we discuss are correct in this way.
2.2 Abstract Syntax of CHR
Constraints are relations, distinguished predicates of first-order predicate logic. We differentiate between two kinds of
constraints: built-in (pre-defined) constraints and user-defined (CHR) constraints which are defined by the rules in a
CHR program. Built-in constraints can be used as tests in the guard as well as for auxiliary computations in the body
of a rule. In this survey, besides the trivial constraint true, we will have syntactical equality = between logical terms
and equations between arithmetic expressions.
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Definition 2.1. A goal is a conjunction of built-in and user-defined constraints. A state is also a goal. Conjunctions
are understood as multisets of their conjuncts. We will use letters such as A, B,C, D, E, . . . for goals and S and T for
states.
A CHR program is a finite set of rules. A (generalized) simpagation rule is of the form
r : H1 \H2 ⇔ C|B
where r : is an optional name (a unique identifier) of a rule. In the rule head (left-hand side), H1 and H2 are conjunctions
of user-defined constraints, the optional guard C| is a conjunction of built-in constraints, and the body (right-hand side)
B is a goal.
In the rule, H1 are called the kept constraints, while H2 are called the removed constraints. At least one of H1 and
H2 must be non-empty. If H1 is empty, the rule corresponds to a simplification rule, also written
s : H2 ⇔ C|B.
If H2 is empty, the rule corresponds to a propagation rule, also written
p : H1 ⇒ C|B.
Interestingly, most parallel semantics do not allow for propagation rules, while distributed semantics do. This will
be discussed in Section 11.
Ground CHR. Most implementations and some semantics assume that variables are substituted by ground
(variable-free) terms at run-time. This requirement can be captured by a common syntactic fragment of CHR: In
Ground CHR, every variable in a rule (also) occurs in the head of the rule. We also say that the rule is range-restricted.
This condition can be relaxed by allowing for local variables in the body of rule, provided they first occur in built-in
constraints that always bound them to ground values at run-time (e.g. arithmetic functions). So given a ground initial states, all states in a computation will stay ground. As we will see, this greatly simplifies refined semantics and
implementations, since then it is not necessary to account for the suspension and wake-up of user-defined constraints
during computations. It is worth noting that Ground CHR without propagation rules is still Turing-complete: it can
implement a Turing machine with just one rule as we will see in Section 3.2.
2.3 Sequential Abstract Operational Semantics of CHR
The semantics follows [RBF09, Bet14]. It relies on a structural equivalence between states that abstracts away from
technical details in a transition.
State Equivalence. The equivalence relation treats built-in constraints semantically and user-defined constraints
syntactically. Basically, two states are equivalent if they are logically equivalent (imply each other) while taking into
account that user-defined constraints form a multiset, i.e. multiplicities matter. For a state S, the notation Sbi denotes
the built-in constraints of S and Sud denotes the user-defined constraints of S.
Definition 2.2 (State Equivalence). Two states S1 = (S1bi ∧S1ud ) and S2 = (S2bi ∧S2ud ) are equivalent, written S1 ≡ S2 ,
if and only if
|= ∀(S1bi → ∃ȳ((S1ud = S2ud ) ∧ S2bi)) ∧ ∀(S2bi → ∃x̄((S1ud = S2ud ) ∧ S1bi))
with x̄ those variables that only occur in S1 and ȳ those variables that only occur in S2 .
The CHR state equivalence is defined by two symmetric implications and moreover syntactically equates the
conjunctions of user-defined constraints as multisets. For example,
X=<Y ∧Y =<X ∧ c(X,Y ) ≡ X=Y ∧ c(X, X) 6≡ X=Y ∧ c(X, X) ∧ c(X, X).
Transition. Using this state equivalence, the abstract CHR semantics is defined by a single transition that is the
workhorse of CHR program execution. It defines the application of a rule. Let the rule (r : H1 \H2 ⇔ C|B) be a variant
of a rule from a given program P. A variant (renaming) of an expression is obtained by uniformly replacing its
variables by fresh variables.
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(Apply) S ≡ (H1 ∧ H2 ∧C ∧ G) (r : H1 \H2 ⇔ C|B) ∈ P
S 7→r T
(H1 ∧C ∧ B ∧ G) ≡ T
Upper-case letters stand for (possibly empty) conjunctions of constraints in this section. The goal G is called context
of the rule application. It is left unchanged.
In a transition (computation step) S 7→r T , S is called source state and T is called target state. We may drop the
reference to the program P and rule r to simplify the presentation.
If the source state can be made equivalent to a state that contains the head constraints and the guard built-in
constraints of a variant of a rule, then we delete the removed head constraints from the state and add the rule body
constraints to it. Any state that is equivalent to this target state is in the transition relation.
A computation (derivation) of a goal S in a program P is a connected sequence Si 7→ Si+1 beginning with the initial
state (query) S0 that is S and ending in a final state (answer, result) or the sequence is non-terminating (diverging).
The notation 7→∗ denotes the reflexive and transitive closure of 7→.
Note that the abstract semantics does not account for termination of propagation rules: If a state can fire a propagation rule once, it can do so again and again, ad infinitum. This is called trivial non-termination of propagation
rules. Most parallel semantics rule out propagation rules. Propagation rules and their termination will be discussed for
distributed CHR in Section 9, though.
For the minimum example, here is a possible (Apply) transition from a state S = (min(0) ∧ min(2) ∧ min(1)) to a
state T = (min(0) ∧ min(1)):
S ≡ (min(X) ∧ min(Y ) ∧ X ≤ Y ∧ (X = 0 ∧Y = 2 ∧ min(1)))
(min(X)\min(Y ) ⇔ X ≤ Y |true)
(min(X) ∧ X ≤ Y ∧ true ∧ (X = 0 ∧Y = 2 ∧ min(1))) ≡ T
S 7→ T
2.4 Extension to Parallel Abstract Semantics
We extend the abstract semantics by parallelism. We interpret conjunction as parallel operator. As we have seen for
the minimum example, CHR rules can also be applied simultaneously to overlapping parts of a state, as long as the
overlap (shared, common part) is not removed by any rule. Following [Frü05a], CHR parallelism with overlaps is
called strong. It can be defined as follows, see also Chapter 4 in [Frü09].
(Strong) Parallelism (with Overlap). We denote parallel transitions by the relation Z⇒. The transition (IntroPar) says that any sequential transition is also a parallel transition. The transition (Parallel) combines two parallel
transitions using conjunction into a single parallel transition where the overlap E is kept.
(Intro-Par) A 7→ C
A Z⇒ C
(Parallel) A ∧ E Z⇒ C ∧ E
B ∧ E Z⇒ D ∧ E
A ∧ B ∧ E Z⇒ C ∧ D ∧ E
Again, back to the minimum example:
(Parallel) min(1) ∧ min(0) Z⇒ true ∧ min(0) min(2) ∧ min(0) Z⇒ true ∧ min(0)
min(1) ∧ min(2) ∧ min(0) Z⇒ true ∧ true ∧ min(0)
Here the overlap is the goal min(0).
2.5 Properties: Monotonicity, Soundness and Serializability
The monotonicity property of CHR states that adding constraints to a state cannot inhibit the applicability of a rule
[AFM99]. It is easy to see from the context of the sequential (Apply) transition and from the overlap of the (Parallel)
transition that a rule can be applied in any state that contains its head and guard.
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Theorem 2.1 (Monotonicity of CHR). If A 7→ B then A ∧ G 7→ B ∧ G. If A Z⇒ B then A ∧ E Z⇒ B ∧ E.
The correctness of the abstract parallel semantics can be established by proving the following theorem.
Theorem 2.2 (Soundness and Serializability). If A Z⇒ B, then there exists a sequential computation A 7→∗ B.
The essential aspect of the truth is that the (Parallel) transition can be simulated sequentially: If A ∧ E 7→ B ∧ E
and C ∧ E 7→ D ∧ E, then A ∧C ∧ E 7→ S 7→ B ∧ D ∧ E, where S is either A ∧ D ∧ E or B ∧C ∧ E, i.e. the two transitions
commute.
3 Parallel CHR Example Programs
These exemplary CHR programs are mostly folklore in the CHR community, see e.g. Chapters 2 and 7 in [Frü09].
These are concise and effective implementations of classical algorithms and problems starting with finding primes,
sorting, including Turing machines and ending with Preflow-Push and Union-Find. Often one type of constraint and
one rule will suffice, and we will not need more then six rules. Due to the guaranteed properties of CHR, these
programs are also incremental anytime online approximation algorithms. Typically they run in parallel without any
need for modifying the program. An exception is Union-Find, which is known to be hard to parallelize. We do it with
the help of confluence analysis.
These sequential programs are in the subset of Ground CHR without propagation rules and can therefore be understood in all parallel semantics and executed in all parallel implementations surveyed without modification. On the other
hand, most example programs may require some modification for distributed semantics and their implementations. As
we will see, the experimental results report parallel speed-ups.
3.1 Algorithms of Erastothenes, Euclid, von Neumann, Floyd and Warshall
Here we introduce some classical algorithms over numbers and graphs. They are implemented as simple multiset
transformations reminiscent of the Chemical Abstract Machine (CHAM). Typically, they can be implemented with
one kind of constraint and a single rule in CHR that can be applied in parallel to pairs of constraints. Our running
example of minimum falls into this category. These programs are confluent when run as intended, with ground goals.
Correctness of each implementation can be shown by contradiction: given the specified initial goal, if the resulting
answer were not of the desired form, the rule would still be applicable.
Prime Numbers. The following rule is like the rule for minimum, but the guard is different, more strict. In effect,
it filters out multiples of numbers, similar to the Sieve of Erastothenes.
sift : prime(I) \ prime(J) <=> J mod I =:= 0 | true.
If all natural numbers from 2 to n are given, only the prime numbers within this range remain, since non-prime
numbers are multiples of other numbers greater equal to 2. Obviously, the rules can be applied to pairs of prime
number candidates in parallel. In a parallel step, we can try to remove each prime by associating it with another prime
such that the sift rule is applicable. This gives a maximum, linear parallel speed-up without the need to modify the
program. This was confirmed experimentally for both a software and a hardware implementation [Lam11a, TORF12].
Greatest Common Divisor (GCD). The following rule computes the greatest common divisor of natural numbers
written each as gcd(N).
gcd(N) \ gcd(M) <=> 0<N,N=<M | gcd(M-N).
The rule replaces M by the smaller number M − N as in Euclid’s algorithm. The rule maintains the invariant that the
numbers have the same greatest common divisor. Eventually, if N = M, a zero is produced. The remaining nonzero
gcd constraint contains the value of the gcd. The rules can be applied to pairs of gcd numbers in parallel. Note that
to any pair of gcd constraints, the rule will always be applicable. A parallel speed-up was observed in a hardware
implementation [TORF12], and even a super-linear speed-up in a software implementation [Lam11a].
Merge Sort. The initial goal state contains arcs of the form a->V for each value V, where a is a given smallest
(dummy) value.
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msort : A->B \ A->C <=> A<B, B<C | B->C.
The rule only updates the first argument of the arc constraint, never the second. The first argument is replaced by a
larger value and the two resulting arcs form a small chain A->B, B->C. The rule maintains the invariant that A=<B. So
eventually, in each arc, a number will be followed by its immediate successor, and thus the resulting chain of arcs is
sorted.
For sorting with optimal run-time complexity, we prefer merging arc chains of the same length. To this end, we
precede each chain with its length, written as special arc N=>FirstNode. We also have to add a rule to initiate merging
of chains of the same length:
N=>A, N=>B <=> A<B | N+N=>A, A->B.
In the initial goal we now introduce constraints of the form 1=>V for each value V. The rules can be applied to pairs of
arcs in parallel similar to the previous examples.
Floyd-Warshall All-Pair Shortest Paths. Our implementation finds the shortest distance between all connected
pairs of nodes in the transitive closure of a directed graph whose edges are annotated with non-negative distances.
shorten : arc(I,K,D1), arc(K,J,D2) \ arc(I,J,D3) <=>
D3>D1+D2 | arc(I,J,D1+D2).
Clearly we can shorten arc distances in parallel by considering triples of arc constraints that match the head of the rule.
In each parallel step, we can try to remove each arc by associating it with a corresponding pair of arc constraints and
by checking if the rule is applicable then.
3.2 Classical Models and Classical Algorithms with Statefulness
These algorithms about abstract problems are characterized by their statefulness, i.e. their essence is a state change,
an update. While other declarative languages may not have an efficient way to update, CHR has a proven one by
constant-time updating (i.e. removing and adding) user-defined constraints [SSD09].
Turing Machine. The Turing machine is the classical model of computability used in theoretical computer science.
One rule suffices to implement it efficiently in CHR.
st(QI,SI,SJ,D,QJ) \ state(I,QI), cell(I,SI) <=> state(I+D,QJ), cell(I,SJ).
The state transition steps of the Turing machine are given as constraints st(QI,SI,SJ,D,QJ): in the current state QI
reading tape symbol SI, write symbol SJ and move in direction D to be in state QJ. The direction is either left or right,
we move along the cells of a tape. We represent cells as an array, so positions are numbers and the direction is either
+1 or −1. A Turing machine with one tape is inherently sequential, since we can only be in one state at a time. Still
parallelism can be employed to find the matching state transition constraint.
The implementation of the Turing machine shows Turing-completeness of the Ground CHR fragment with constants only and without propagation rules, actually with a single rule [Sne08].
Dijkstras Dining Philosophers. In this classical problem in concurrency, several philosophers sit at a round table.
Between each of them a fork is placed. A philosopher either thinks or eats. In order to eat, a philosopher needs two
forks, the one from his left and the one from his right. After a while, an eating philosopher will start to think again,
releasing the forks and thus making them available to his neighbors again.
think_eat : think(X), fork(X), fork(Y) <=> Y =:= (X+1) mod n | eat(X).
eat_think : eat(X) <=> Y =:= (X+1) mod n | think(X), fork(X), fork(Y).
In the implementation, we assume a given number n of philosophers (and forks). They are identified by a number from
zero to n-1. The rules are inverses of each other, the constraints simply switch sides.
The problem is to design a concurrent algorithm that is fair, i.e. that no philosopher will starve. Here we are
mainly interested in the inherent parallelism of the problem. Disjoint pairs of neighboring forks can be used for eating
in one parallel computation step. (For the experiments, time counters for eating and thinking were introduced into the
program to introduce termination.)
Blocks World. Blocks World is a classical planning problem in Artificial Intelligence. It simulates robot arms
re-arranging stacks of blocks.
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grab : grab(R,X), empty(R), clear(X), on(X,Y) <=> hold(R,X), clear(Y).
putOn : putOn(R,Y), hold(R,X), clear(Y) <=> empty(R), clear(X), on(X,Y).
The operation constraints grab and putOn specify the action that is taken. The other constraints are data constraints
holding information about the scenario. Operation constraints update data constraints. The rule grab specifies that
robot arm R grabs block X if R is empty and block X is clear on top and on block Y. As a result, robot arm R holds block
X and block Y is clear. The rule putOn specifies the inverse action. The data constraints in the rule switch sides. At any
time, only one of the actions is thus possible for a given robot arm. Parallelism is induced by introducing several robot
arms and multiple actions for them. Different robot arms can grab different clear blocks in parallel or put different
blocks on different clear blocks in parallel.
3.3 Parallel Preflow-Push Algorithm
Next we present two non-trivial algorithms, Preflow-Push and Union-Find. Both algorithms are acknowledged in the
literature to be hard to parallelize. To maintain the focus of the survey, we cannot explain these algorithms in detail.
The Preflow-Push algorithm [GT88] solves the maximum-flow problem. Intuitively the problem can be understood
as a system of connected water-pipes, where each pipe has a restricted given capacity. The system is closed except for
one source and one sink valve. The problem now is to find the maximum capacity the system can handle from source
to sink and to find the routes the water actually takes.
A flow network is a directed graph, where each edge is assigned a non-negative capacity. We want to find a
maximum flow through the network from a source to a sink node under the capacity restrictions. The Preflow-Push
algorithm moves flow locally between neighboring nodes until a maximum flow is reached.
In [Mei07], we present and analyse a concise declarative parallel implementation of the preflow-push algorithm
by just four rules. In the code listing below, comment lines start with the symbol %.
% increase node height by one, remove minimum
lift : n(U,N), e(U,E) \ h(U,_), m(U,M,C)
<=> U \= source, U \= sink, 0 < E, C =:= N+E | h(U,M+1).
% replace K by HU in unchecked egde, insert minimum
up
: h(U,HU), h(V,HV) \ r(U,V,K)
<=> HU =< HV, K < HU | m(U,HV,1), r(U,V,HU).
% push flow downwards by one unit, insert minimum, reverse edge
push : h(U,HU), h(V,HV) \ e(U,EU), e(V,EV), r(U,V,_)
<=> 0 < EU, HV < HU | e(U,EU-1), e(V,EV+1), m(V,HU,1), r(V,U,HV).
% compute minimum for node, count for completeness
min : m(U,M1,C1), m(U,M2,C2) <=> m(U,min(M1,M2),C1+C2).
The variable U stands for a node, N is its number of outward capacity edges, E is its current excess flow, HU is its current
height. The constraint m(U,M,C) encodes a minimum candidate with value M for node U, where the counter C allows
to detect if the minimum of all outward edges has been computed. The constraint r(U,V,K) encodes a residual edge
from nodes U to V with remaining capacity K.
The implementation described in [Mei07] simulates parallel computations sequentially using an interleaving semantics approach and time stamps for user-defined constraints. The active elements (nodes with excess flow) can be
processed in parallel as long as their neighborhoods (set of nodes connected to them through an edge) do not overlap.
In the simulation, we greedily, randomly and exhaustively apply as many rules as possible at a given time point t before
progressing to time t + 1. A speed-up in experiments with random graphs was consistently observed. The speed-up
depends on the total amount of flow units, its distribution on disjoint nodes, and the density of the flow network. A
parallel speed-up was also confirmed in the experiments of [TORF12].
3.4 Parallel Union-Find Algorithm
This classical union-find (also: disjoint set union) (UF) algorithm [TL84] efficiently maintains disjoint sets under the
operation of union. Each set is represented by a rooted tree, whose nodes are the elements of the set. Union-Find is
acknowledged in the literature to be hard to parallelize.
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In [Frü05a], we implement the UF algorithm in CHR with optimal time and space complexity and with the anytime
online algorithm properties. This effectiveness is believed impossible in other pure declarative programming languages
due to their inability to express destructive assignment in constant time. When the UF algorithm is extended by rules
that deal with function terms (rational trees), it can be used for optimal complexity unification [MF07]. Last but not
least, a generalization of Union-Find yields novel incremental algorithms for simple Boolean and linear equations
[Frü06]. See chapter 10 in [Frü09] for an overview of Union-Find in CHR.
Parallelizing Basic Union-Find. We only discuss the basic Union-Find (UF) algorithm here, not the optimized
one, since the former has been used in experiments [SL08]. In CHR, the data constraints root and arc -> represent
the tree data structure. With the UF algorithm come several operation constraints: find returns the root of the tree in
which a node is contained, union joins the trees of two nodes, link performs the actual join.
union
: union(A,B) <=> find(A,X), find(B,Y), link(X,Y).
findNode : A->B \ find(A,X) <=> find(B,X).
findRoot : root(A) \ find(A,X) <=> found(A,X).
linkEq
: link(X,Y), found(A,X), found(A,Y) <=> true.
linkRoot : link(X,Y), found(A,X), found(B,Y), root(A) \ root(B) <=> B->A.
The second argument of the find operation find holds a fresh variable as identifier. When the root is found, it is
recorded in the constraint found.
CHR confluence analysis [AF04, AF98] produces abstract states that reveal a deadlock: when we are about to
apply the linkRoot rule, another link operation may remove one of the roots that we need for linking. From the
non-confluent states we can derive an additional rule for found that mimics the rule findNode: the found constraint
now keeps track of the updates of the tree so that its result argument is always a root.
foundUpdate : A->B \ found(A,X) <=> found(B,X).
Linking for disjoint node pairs can now run in parallel. While this seems an obvious result, this semi-automatic
confluence-based approach yields a non-trivial parallel variant of the optimized UF algorithm with path compression.
Correctness of the parallelisation is proven in both cases in [Frü05a]. A parallel speed-up is reported in [Lam11a].
4 Parallel CHR with Transactions
We now extend parallel CHR by transactions. Transactions will also be used for the implementation of parallel CHR
in Section 6 and for encoding of a transaction-based concurrency model in CHR in Section 10.1.
Transactions. They alleviate the complexity of writing concurrent programs by offering entire computations to
run atomically and in isolation. Atomicity means that a transaction either proceeds un-interrupted and successfully
commits or has to rollback (undo its side-effects). In optimistic concurrency control, updates are logged and only
committed at the end of a transaction when there are no update conflicts with other transactions. Isolation means that
no intermediate update is observable by another transaction. The highest level of isolation is serializability, the major
correctness criterion for concurrent transactions: for each parallel execution there is a sequential execution with the
same result.
4.1 Transactions in Parallel CHR
The paper [SS08b] proposes CHRt as a conservative extension of CHR with atomic transactions. An atomic transaction
is denoted as a meta-constraint atomic(C) where C is a conjunction of CHR constraints. Atomic transactions may
appear in goals.
Example 4.1. Consider these CHR rules for updating a bank account:
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balance(Acc,Bal), deposit(Acc,Amt) <=> balance(Acc,Bal+Amt).
balance(Acc,Bal), withdraw(Acc,Amt) <=> Bal>Amt | balance(Acc,Bal-Amt).
transfer(Acc1,Acc2,Amt) <=> withdraw(Acc1,Amt), deposit(Acc2,Amt).
The balance constraint is a data constraint, and the deposit and withdraw constraints are operation constraints.
The guard ensures that withdrawal is only possible if the amount in the account is sufficient. The transfer constraint
rule combines deposit and withdrawal among two accounts.
Now assume a transfer between two accounts:
balance(acc1,500), balance(acc2,0), transfer(acc1,acc2,1000)
We can execute the deposit, but we cannot execute the withdrawal due to insufficient funds. The transaction gets stuck.
It has a deadlock and cannot proceed till the end. This is clearly not the desired behavior of a transfer.
In CHRt, we can introduce a transaction to avoid this problem. The transfer constraint in the goal is wrapped
by the meta-constraint atomic.
balance(acc1,500), balance(acc2,0), atomic(transfer(acc1,acc2,1000))
Now the incomplete transaction will be rolled back, no money will be transferred.
4.2 Abstract Syntax and Semantics of CHRt
We assume Ground CHR. We classify CHR constraints into operation constraints and data constraints. The distinction
appeals to the intuitive understanding that operation constraints update data constraints. Thus the head of a CHRt rule
must contain exactly one operation constraint. It requires one more transition for transactions. The (Atomic) transition
executes any number of atomic transactions in parallel in a common context T of data constraints.
(Atomic)
(T ∧ S1 ∧C1 7→∗ T ∧ S1′ ), . . . (T ∧ Sn ∧Cn 7→∗ T ∧ Sn′ )
T ∧ S1 ∧ . . . Sn ∧ atomic(C1 ) ∧ . . . atomic(Cn ) Z⇒ T ∧ S1′ ∧ . . . Sn′
In the transition, T, Si , and Si′ must be data constraints. The parallel step considers the separate evaluation of each
Ci in isolation. The transactions only share the common data constraints T , which serves as a context. Note that
each transaction may perform arbitrary many computation steps. Each transaction is fully executed until there are no
operation constraints. It does not get stuck. So there are only data constraints in the target state.
4.3 Properties: Monotonicity, Soundness and Serializability
For CHRt programs, the following properties are proven to hold in [SS08b].
Serializability For each (Atomic) transition with n concurrent transactions, there is a corresponding computation of
n consecutive sequential (Atomic) transitions each with only one transaction.
Soundness For any computation in CHRt, there is a corresponding computation in CHR where the atomic wrappers
are dropped.
Monotonicity Although not proven in the paper, it follows from Soundness and the context T of the (Atomic) transition.
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4.4 Encoding Transactions in Standard CHR
We want to execute CHRt in standard parallel CHR, i.e without the (Atomic) transition. The straightforward way
is to execute atomic transactions only sequentially. Thus, we trivially guarantee the atomic and isolated execution
of transactions. We identify two special cases where we can erase the atomic wrappers and still allow for parallel
execution: bounded and for confluent transactions.
Bounded Transactions. A bounded transaction is one that performs a finite, statically known number of transitions. We eliminate a bounded transaction atomic(G) from a program by adding a rule to the program of the form
atomic(G) <=> G. Then we unfold the rule [Frü05b, GMTW13, FH03] until no more operation constraints appear
in its body. Since the transaction is bounded, unfolding will eventually stop.
In the running example, we can replace the atomic transfer rule (since it is bounded) by the following rule.
balance(Acc1,Amt1),balance(Acc2,Amt2),atomic(transfer(Acc1,Acc2,Amt)) <=>
Amt1>Amt | balance(Acc1,Amt1-Amt), balance(Acc2,Amt2+Amt).
The rule head expresses the fact that an atomic transfer requires exclusive access to both accounts involved.
Confluent Transactions. The paper proves that if a CHRt program is confluent when we ignore atomic wrappers,
then it can be executed in standard parallel CHR provided the initial goal never gets stuck (deadlocks). Confluence
then guarantees that isolation is not violated.
Consider the example of the stuck transaction that attempts to overdraw an account. The withdraw rule can be fixed
if we drop its guard (and hence allow negative balances): Any two consecutive transfers commute now. Regardless of
the order they are performed in, they yield the same final result (even if the intermediate results differ). Hence, we can
safely erase the atomic wrappers.
5 Refined Parallel CHR Semantics
A refined semantics for parallel CHR is developed and implemented in [SL08, LS09, Lam11a]. This semantics can
be seen as a refinement of the parallel abstract semantics given before. In states, we now differentiate between the
goal that holds active constraints to be processed, and the constraint store that holds inactive suspended constraints as
data. This means that we have to account for the in-activation (suspension) and re-activation (wake-up) of user-defined
constraints due to built-in constraints on shared variables. As before, the semantics is given in two parts, the sequential
transitions and the parallel transitions and the properties of monotonicity, soundness and serializability are shown.
5.1 Syntax for Refined Parallel CHR
Built-In Constraint
CHR Constraint
Goal Constraint
Store Constraint
State
e
c ::= p(t)
g ::= c | e | nc
sc ::= e | nc
σ ::= hG, Sni
Identified (CHR) Constraint
Goal (Store)
(Constraint) Store
Matched Constraints
nc ::=U
c#i
G ::= g
S
Sn ::= sc
δ ::= Sn \ Sn
Figure 1: Refined Parallel CHR Syntax
Figure 1 describes the syntax for the refined semantics. The notation a denotes a sequence of a’s. We only
consider built-in constraints that are syntactic equalities or arithmetic equations. To distinguish multiple occurrences
(copies, duplicates) of CHR constraints, they are extended by a unique identifier. We call c#i an identified constraint.
Conjunctions are modeled as (multi)sets. Unlike in the abstract semantics, a state is now a pair: we distinguish between
a goal (store) (a multiset of constraints) and the (constraint) store (a set of built-in and identified CHR constraints).
Correspondingly, there are goal and store constraints. We also introduce matched constraints that are pairs of store
constraints which we will need as an annotation to transitions.
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5.2 Sequential Refined CHR Semantics
The sequential part of the semantics in Figure 2 is a generalization of the refined CHR semantics of [DSGH04]. The
semantics assumes generalized simpagation rules that are not propagation rules.
Constraints from the goal are executed one by one. A constraint currently under execution is called active constraint. It tries to apply rules to itself. To try a rule, the active constraint is matched against a head constraint of the
rule. The remaining head constraints are matched with partner constraints from the constraint store. If there is such
a complete matching and if the guard is satisfied under this matching, then the rule applies (fires). The constraints
matching the removed constraints of the head are deleted atomically and the body of the rule is added to the state.
Because of the role of the active constraint, we call the semantics goal-based semantics.
W = WakeU p(e, Sn)
(Solve+Wake)
W \{}
h{e} ⊎ G | Sni hW ⊎ G | {e} ∪ Sni
i is a fresh identifier
(Activate)
{}\{}
h{c} ⊎ G | Sni h{c#i} ⊎ G | {c#i} ∪ Sni
Variant of (r : HP′ \HS′ <=> t | B′ ) ∈ P such that
∃φ Eqs(Sn) |= φ (t) φ (HP′ ) = DropIds(HP )
φ (HS′ ) = {c} ⊎ DropIds(HS) δ = HP \{c#i} ∪ HS
h{c#i} ⊎ G | {c#i} ∪ HP ∪ HS ∪ Sni
(Apply-Remove)
δ
hφ (B′ ) ⊎ G | HP ∪ Sni
Variant of (r : HP′ \HS′ <=> t | B′ ) ∈ P such that
∃φ Eqs(Sn) |= φ (t) φ (HS′ ) = DropIds(HS )
φ (HP′ ) = {c} ⊎ DropIds(HP) δ = {c#i} ∪ HP\HS
h{c#i} ⊎ G | {c#i} ∪ HP ∪ HS ∪ Sni
(Apply-Keep)
δ
hφ (B′ ) ⊎ {c#i} ⊎ G | {c#i} ∪ HP ∪ Sni
(Apply-Remove) and (Apply-Keep) do not apply to c#i in Sn
(Suspend)
{}\{}
h{c#i} ⊎ G | Sni hG | Sni
where Eqs(S)
= {e | e ∈ Sn, e is a built-in costraint}
DropIds(Sn)
= {c | c#i ∈ Sn} ⊎ {e | e ∈ Sn}
WakeU p(e, Sn) = {c#i | c#i ∈ Sn ∧ φ m.g.u. of Eqs(Sn)∧
θ m.g.u. of Eqs(Sn ∪ {e}) ∧ φ (c) 6= θ (c)}
δ
Figure 2: Parallel CHR Semantics (Sequential Part )
δ
Transitions. A transition σ σ ′ maps the CHR state σ to σ ′ involving the CHR constraint goals in δ . The
transition annotation δ holds the constraints that where matched with the rule head. It will be needed in the parallel
part of the semantics.
The first transition (Solve+Wake) moves a built-in constraint, an equation or equality e, into the store and wakes up
(reactivates) identified constraints in the store which could now participate in a rule application. This is the case when
15
the built-in constraint effects variables in a user-defined constraint, because then the re-activated (woken) constraint
may now be able to match a rule head and satisfy the guard of the rule. The function WakeU p(e, Sn) computes a
conservative approximation of the reactivated constraints, where m.g.u. denotes the most general unifier induced by a
set of syntactic equations.
In transition (Activate), a CHR constraint goal becomes active by annotating it with a fresh unique identifier and
adding it to the store.
Rules are applied in transitions (Apply-Remove) and (Apply-Keep). They are analogous, but distinguish if the active constraint c#i is kept or removed. In both cases, we seek for the missing partner constraints in the store, producing
a matching substitution φ in case of success. The guard t must be logically entailed by the built-in constraints in the
store under the substitution φ . Then we apply the rule instance of r by atomically removing the matching constraints
HS and adding the rule body instance φ (B) to the goal. We also record the matched identified constraints HS and
HP in the transition annotation. In transition (Apply-Remove), the matching constraints HS include c#i. Since c#i is
removed, we drop it from both the goal and the store. In transition (Apply-Keep), c#i remains and so can possibly fire
further rules.
Finally, in transition (Suspend), we put an active constraint to sleep. We remove the active identified constraint
from the goal if no (more) rules apply to the constraint. Note that the constraint is kept suspended in the store and may
be woken later on.
5.3 Extension to Parallel Refined CHR Semantics
Figure 3 presents the parallel part of the refined operational semantics. It is a refinement of the parallel transition for
the abstract semantics. We allow for multiple goal stores to be combined while the constraint store is shared among
the parallel computations.
δ
hG | Sni hG′ | Sn′ i
(Intro-Par)
δ
hG | Sni || hG′ | Sn′ i
δ1
hG1 | HS1 ∪ HS2 ∪ Sni || hG′1 | HS2 ∪ Sni
δ2
(Parallel-Goal)
hG2 | HS1 ∪ HS2 ∪ Sni || hG′2 | HS1 ∪ Sni
δ1 = HP1 \HS1 δ2 = HP2 \HS2
HP1 ⊆ Sn HP2 ⊆ Sn δ = HP1 ∪ HP2 \HS1 ∪ HS2
HS1 ∩ (HP2 ∪ HS2 ) = {} HS2 ∩ (HP1 ∪ HS1 ) = {}
δ
hG1 ⊎ G2 ⊎ G | HS1 ∪ HS2 ∪ Sni || hG′1 ⊎ G′2 ⊎ G | Sni
δ
Figure 3: Parallel CHR Semantics (Parallel Part || )
In the (Intro-Par) transition, we turn a sequential computation into a parallel computation. Transition
(Parallel-Goal) parallelizes two parallel computations operating on the same shared store, if their matched constraints
δ1 and δ2 do not have an overlap that involves removed constraints. They may overlap in the kept constraints. This
makes sure that parallel computations remove distinct constraints in the store. The identifiers of constraints make
sure that we can remove multiple but different copies of the same constraint. The matched constraints δ1 and δ2 are
composed by the union of the kept and removed components, respectively, forming δ . Note that a context G is added
to the goals in the resulting parallel transition, implying monotonicity.
16
5.4 Properties: Monotonicity, Soundness and Serializability
The following results are proven in the appendix of [LS09].
Monotonicity holds for the goal store, but not for the constraint store. In an enlarged constraint store, the (Suspend)
transition may not be possible anymore, because a new rule becomes applicable to the active constraint. The monotonicity is still sufficient though, because in the semantics, the constraint store is only populated via the goal store.
Serializability holds: Any parallel computation can be simulated by a sequence of sequential computations in the
refined semantics.
Furthermore, soundness holds: any parallel computation has a correspondence in a suitable variant of the sequential
abstract semantics. For the upcoming theorem, let us note that an initial state is of the form hG, {}i, a final state is of
the form h{}, Sni. Given a computation hG | {}i ∗|| hG′ | Sni, the state hG′ | Sni is called a reachable state.
Theorem 5.1 (Soundness). For any reachable state hG | Sni,
if
then
hG | Sni ∗|| hG′ | Sn′i
(NoIds(G) ⊎ DropIds(Sn)) 7→∗ (NoIds(G′ ) ⊎ DropIds(Sn′))
where 7→∗ denotes transitions in the sequential abstract semantics and where NoIds = {c | c ∈
G, c is a CHR constraint} ⊎ {e | e ∈ G, e is a built-in constraint}.
6 Parallel CHR Implementation in Haskell
The parallel refined semantics from the previous Section 5 has been implemented in the lazy functional programming
language Haskell [SL07, LS07, SL08, LS09, Lam11a]. Concretely, we use the Glasgow Haskell Compiler for implementing parallel Ground CHR because of its good support for shared memory and multi-core architectures. The
implementation is available online for free download at https://code.google.com/archive/p/parallel-chr/.
In principle, the system can be reimplemented in mainstream procedural languages such as C and Java.
6.1 Implementation Principles
Our implementation follows the principles of standard sequential implementations of CHR where possible [HGSD05,
VW10]. The goal store is realized as a stack, the constraint store as a hash table. We implement common CHR
optimizations, such as constraint indexing (hashing) and optimal join ordering for finding partner constraints with
early guard scheduling.
Parallel goal execution must not remove constraints in overlaps that participate in several rule head matchings.
We discuss two approaches of concurrency control to implement this kind of parallel rule-head matching, locking and
transactions, before we settle for a hybrid approach.
Fine-grained Lock-based Parallel Matching. Pessimistic concurrency control uses locking as the basic serialization mechanism. We restrict the access to each constraint in the shared store with a lock. When an active constraints
finds an applicable rule, it will first try to lock its matching removed partner constraints. Kept constraints can be used
by several rules simultaneously, so they need not be locked. Locking fails if any constraint in the complete rule head
matching is already locked by another active constraint. If locking fails, the active constraint releases all its locks and
tries to redo the rule application. If locking succeeds, the rule is applied. No unlocking is necessary since locked
constraints are removed. This locking mechanism can avoid deadlocks and cyclic behavior using standard techniques
for these problems such as timestamps or priorities.
Software Transactional Memory (STM). Optimistic concurrency control is based on transactions that can either
commit or rollback and restart. We use the STM transactions provided in Haskell. The principles of transaction have
been introduced in Section 4. The idea of STM is that atomic program regions are executed optimistically. That is, any
read/write operations performed by the region are recorded locally and will only be made visible when the transaction
is completed. Before making the changes visible, the underlying STM protocol will check for read/write conflicts with
other atomically executed regions. If there are update conflicts among transactions, the STM protocol will randomly
commit one of the atomic transactions and rollback the others [ST97]. Committing means that the programs updates
17
become globally visible. Rollback means that we restart the program. The disadvantage of STM is that unnecessary
rollbacks can happen. We will meet STM again in Section 10.1, when it is specified in CHR.
Hybrid STM-based Locking Scheme. In the implementation, we use both Software Transactional Memory
(STM) and traditional shared memory access locking techniques. The search for matching partner constraints is
performed outside STM to avoid unnecessary rollbacks. When a complete rule head matching is found, we perform
an STM procedure that we call atomic rule-head verification (ARV). It checks that all the constraints are still available
and marks the constraints to be removed as deleted. These deleted constraints will be physically delinked from the
constraint store, either immediately or later. Both behaviors can be implemented with standard concurrency primitives
(such as compare-and-swap and locks).
Thread Pool. The naive way to implement a parallel CHR system is to spawn an active thread for each goal
constraint in a state. Each thread tries to find its partner constraints. However, the thread and its later partner constraints
would then compete for the same rule application. Moreover, the number of threads would be unbounded, as the
number of constraints in a state is unbounded. Our implementation uses a bounded number of active threads. A thread
pool maintains threads waiting for tasks to be allocated for parallel execution.
6.2 Experimental Results
Experiments were performed on an Intel Core quad core processor with hyper-threading technology (that effectively
allows it to run 8 parallel threads) . We measure the relative performance of executing with 1, 2, 4, 8 and an unbounded
number of threads against our sequential CHR implementation in Haskell. The table in Figure 4 gives some exemplary
results with these two optimizations: each goal thread searches store constraints in a unique order to avoid matching
conflicts and a special goal ordering for Merge Sort and Gcd is used (explained below).
Number of Threads
Merge Sort
Gcd
Parallel Union-Find
Blocks World
Dining Philosophers
Prime
Fibonacci
Turing Machine
1
121%
109%
125%
123%
119%
115%
125%
111%
2
94%
37%
82%
77%
74%
73%
85%
63%
4
70%
18%
52%
54%
49%
46%
59%
78%
8
52%
12%
32%
39%
41%
30%
39%
70%
Unbounded
>200%
123%
>200%
>200%
>200%
155%
>200%
>200%
Figure 4: Experimental results, with optimal configuration (on 8 threaded Intel processor)
There are several general observations to be made with regard to the number of threads. Executing with one
goal thread is clearly inferior to the sequential implementation because of the wasted overhead of parallel execution.
Executions with 2, 4 and 8 goal threads show a consistent parallel speed-up, with exception of the Turing Machine.
It is inherently single-threaded. Interestingly, we still obtain improvements from parallel execution of administrative
procedures (for example dropping of goals due to failed matching). Unbounded thread pooling is always slower than
the sequential implementation. Furthermore we observed a super-linear speed-up for the Gcd example with a queuebased goal ordering instead of the usual stack-based ordering in the goal store. In merge sort, we stack -> constraints
and queue just => for optimal performance. Last but not least, experiments also confirmed that there is a speed-up
when a multi-core processor instead of a single-core processor is used.
7 Massively-Parallel Set-Based CHR Semantics
A CHR semantics is set-based if conjunctions of constraints are considered as set instead of multiset. In [RF10], we
present a parallel execution strategy for set-based CHR. The use of sets instead of multisets has a dramatic impact: it
allows for multiple removals of constraints. This means that overlaps can be removed several times. We show that
18
the resulting refined semantics is not sound in general anymore, but sound if the program is deletion-acyclic (i.e.,
when its simpagation rules do not allow for mutual removal of constraints). CHRmp programs for the computation of
minimum, prime numbers, and sorting can run in constant time, given enough processors. We describe a program for
SAT solving in linear time.
7.1 Massively-Parallel Set-Based Semantics CHRmp
As in the parallel abstract semantics, there are no restrictions on the syntax of CHR. Reconsider the essential (Parallel)
transition of the abstract CHR semantics. Keep in mind that conjunctions of constraints are now interpreted as sets of
constraints.
(Parallel) A ∧ E Z⇒ C ∧ E
B ∧ E Z⇒ D ∧ E
A ∧ B ∧ E Z⇒ C ∧ D ∧ E
Consider the program
a <=> b,c.
a <=> b,d.
Then the following transition for the goal a ∧ e is possible in the set-based interpretation:
a ∧ e Z⇒ b ∧ c ∧ e a ∧ e Z⇒ b ∧ d ∧ e
a ∧ e Z⇒ b ∧ c ∧ d ∧ e
This means that a is removed twice and b is only produced once.
When we generalize this observation, we see that overlaps between rule matchings can be removed arbitrary many
times, leading to a kind of massive parallelism.
Refined CHRmp Semantics. We refine this set-based semantics now. We assume CHR without propagation
rules. In the body of a rule, we distinguish between CHR constraints Bc and built-in constraints Bb , and write Bc , Bb .
A CHRmp state S (or T ) is of the form hG; Bi, where the goal (store) G is a set (not multiset) of constraints and the
(built-in) constraint store B is a conjunction of built-in constraints. c and d are atomic constraints. We adapt the state
equivalence ≡ in the obvious way to CHRmp states.
Definition 7.1 (Massively Parallel Transition). Given a CHRmp state S = hG; Bi. Let R be the smallest set such that
for each rule variant r : H1 \H2 ⇔ G | Bc , Bb where S ≡ hH1 ∪ H2 ∪ G′ ; G ∧ B′ i it holds that (H1 , H2 , Bc , Bb , B′ ) ∈ R.
We then define for any non-empty subset R ⊆ R:
- the set of removed constraints D = {c | ∃( , H2 , , , B′ ) ∈ R, c ∈ G : H2 ∧ B′ → c}
- the set of added constraints A = {c | ∃( , , Bc , , ) ∈ R : c ∈ Bc }
- the conjunction of added built-in constraints B =
V
( , , ,Bb ,B′ )∈R
B′ ∧ Bb
A massively parallel transition (step) of S = hG; Bi using R is then defined as:
(Massive-Apply) hG; Bi ։R h(G \ D) ∪ A; B ∧ Bi
If the specific set R is not of importance we write ։ instead of ։R .
The idea is that in the set R we collect all possible rule applications and then we apply any subset of them at once in
one parallel computation step. In this way, multiple removals of the same constraint are possible. In the extreme case,
R = R, so all possible rule applications are performed simultaneously. We call this exhaustive parallelism. With such
an execution strategy, any CHRmp program is trivially confluent, because there are no conflicting rule applications.
On the other hand, if R is a singleton set, only one rule is applied and we are back to sequential CHR.
19
Example 7.1. Reconsider the CHR program for computing prime numbers. Consider the state
S = h{prime(2), prime(3), prime(4), prime(5), prime(X)}; X=6i.
There are three possible rule applications, removing the non-prime numbers 4 and twice 6:
/ ⊤, X=6 ∧ N1 =2 ∧ M1 =4),
({prime(N1 )}, {prime(M1 )}, 0,
({prime(N2 )}, {prime(M2 )}, 0,
/ ⊤, X=6 ∧ N2 =2 ∧ M2 =6),
R=
({prime(N3 )}, {prime(M3 )}, 0,
/ ⊤, X=6 ∧ N3 =3 ∧ M3 =6)
We can now perform all three possible rule applications exhaustively parallel, i.e. R = R, resulting in the following
sets:
D = {prime(4), prime(X)}, A = 0,
/
B = (X=6 ∧ N1 =2 ∧ M1 =4) ∧ (X=6 ∧ N2 =2 ∧ M2 =6) ∧ (X=6 ∧ N3 =3 ∧ M3 =6)
This leads to the parallel transition:
S ։R
h{prime(2), prime(3), prime(5)}; X=6 ∧ Bi
Hence, a single parallel step is sufficient to find all prime numbers.
7.2 Example Programs under Exhaustive Parallelism
We examine different algorithms written in CHR and the effect of executing these programs in CHRmp, in particular
with exhaustive parallelism to achieve maximum speed-up.
Filter Programs. Programs that only consist of rules whose body is true can be understood as filtering constraints. They can obviously be executed in constant time with exhaustive parallelism, given enough processors. The
minimum and the prime program fall into this category. The msort rule of merge sort leads to a linear number of
exhaustively parallel steps. It can be rewritten to achieve constant-time complexity. The experiments with the prime
program using massive parallelism (see Section 8) [TORF12] show an run-time improvement of about an order of
magnitude over strong parallelism.
SAT Solving. The SAT formula is given as a tree of its sub-expressions. The tree nodes are of the form eq(Id,B),
where Id is a node identifier and B is either a Boolean variable written v(X) or a Boolean operation (neg, and,
or) applied to identifiers. Additionally, a f(L,[]) constraint is required in the initial state, where L is a list of all n
variables in the SAT formula.
generate : f([X|Xs], A) <=> f(Xs,[true(X)|A]), f(Xs,[false(X)|A]).
assign
: f([],A) \ eq(T,v(X)) <=> true(X) in A | sat(T,A,true).
assign
: f([],A) \ eq(T,v(X)) <=> false(X) in A | sat(T,A,false).
sat(T1,A,S) \ eq(T,neg(T1)) <=> sat(T,A, neg S).
sat(T1,A,S1), sat(T2,A,S2) \ eq(T,and(T1,T2)) <=> sat(T,A, S1 and S2).
sat(T1,A,S1), sat(T2,A,S2) \ eq(T,or(T1,T2)) <=> sat(T,A, S1 or S2).
The generate rule generates, in n parallel steps, 2n f constraints representing all possible truth assignments to variables as a list in its second argument. In the next parallel step (using the assign rules) all n Boolean variables in the
given formula are assigned truth values for each assignment, represented by sat constraints.
The remaining three rules determine the truth values of all sub-expressions of the formula bottom-up. In each
parallel step the truth values of sub-expressions at a certain height of the tree are concurrently computed for all
possible assignments of variables. Therefore, the number of parallel steps in this phase is bound by the depth of the
formula.
A formula is in 3-DNF normal form if it is in disjunctive normal form (a disjunction of conjunctions of literals)
and each clause contains at most 3 literals. Because of its bounded depth, a SAT problem given in 3-DNF normal form
with n variables can be solved in linear time in n with this program under exhaustive parallelism, independent of the
size of the formula.
20
7.3 Properties: Soundness under Deletion-Acyclicity
Soundness of CHRmp is not always possible as the following example shows.
Example 7.2. Consider the following rule that removes one of two differing constraints,
c(N) \ c(M) <=> N=\=M | true.
and the goal c(1), c(2). There are two competing rule instances for application: one matches the two constraints
in the given order, the other in reversed order. So if we apply both rules simultaneously under exhaustive parallelism,
both constraints will be (incorrectly) removed.
In general, computations that allow for mutual removal of constraints are not sound in CHRmp. Soundness requires
that the programs are deletion-acyclic, effectively ruling out mutual removal. A deletion dependency pair (c, d) means
the kept constraint c is required to remove constraint d in a rule of the program. This is the case if c as an instance of
a kept constraint and d is an instance of a removed constraint in the head of the rule.
Definition 7.2 (Deletion Dependency, Deletion-Acyclic). Given a CHRmp state S = hG; Bi. Then deletion dependency D(S) is a binary relation such that (c, d) ∈ D if and only if there exist (H1 , H2 , Bc , Bb , B′ ) ∈ R(S) and
c′ ∈ H1 , d ′ ∈ H2 such that c′ ∧ B′ → c and d ′ ∧ B′ → d.
A CHRmp program P is deletion-acyclic if and only if for all S such that S ։R T the transitive closure D(S)+ is
irreflexive.
In a deletion-acyclic program, we can simulate the CHRmp computation steps by a sequence of sequential rule
applications in multiset semantics, provided we initially have enough copies of the user-defined constraints and can
remove them when needed. The latter is accomplished by so-called set-rules of the form
set-rule: c(X1,...Xn) \ c(X1,...Xn) <=> true.
for each CHR constraint c/n in the given program. These rules remove multiple occurrences of the same constraint.
The following soundness theorem requires a deletion-acyclic program and set-rules [RF10]. Let be a sequential
transition in a suitable variant of the usual multiset CHR semantics.
Theorem 7.1 (Soundness). Let P be a deletion-acyclic CHRmp program and P ′ be the CHR program P extended
with set-rules. If S = hG; Bi ։P T , then there exists a multiset G′ with c ∈ G′ ⇔ c ∈ G such that S′ = hG′ ; Bi ∗P ′ T ′ ,
where c ∈ T ′ ⇔ c ∈ T .
Example 7.3. Consider the initial goal a and the program
a <=> b,c.
a <=> b,d.
b,c,d <=> true.
Exhaustive parallelism leads to the set-based computation
a ։ b,c,d ։ true.
The sequential correspondence in the multiset CHR program extended with set-rules is
a,a b,b,c,d b,c,d true.
The example can also be used to show that Serializability in general does not hold for massively-parallel set-based
CHR. There is not sequential computation in CHRmp that can simulate the exhaustively parallel computation, since the
first rule application will remove a, so either b,c or b,d can be produced sequentially, but not their union. Similarly,
monotonicity does not hold.
21
8 Parallel Hardware Implementations of CHR
The work reported in [TORF12, Tri11] investigates the compilation of CHR to specialized hardware. The implementation follows the standard scheme for translating CHR into procedural languages. The compiler translates the CHR
code into the low-level hardware description language VHDL, which in turn creates the necessary hardware using
Field Programmable Gate Array (FPGA) technology. FPGA is a hardware consisting of programmable multiple arrays of logic gates. We also implement a hybrid CHR system consisting of a software component running a CHR
system for sequential execution, coupled with hardware for parallel execution of dedicated rules in the program. The
resulting hardware system is typically an order of magnitude faster than the fastest software implementation of CHR
(in C).
8.1 Basic Compilation of CHR to Procedural Languages
As preliminaries, we give the basic implementation scheme for Ground CHR in procedural languages like C and Java,
but also VHDL. This translation scheme applies throughout this section. In Ground CHR, we do not need to wake-up
constraints, because all variables are ground at run-time. A CHR rule can be translated into a procedure using the
following simple scheme:
procedure(kept head constraints, removed head constraints) {
if (head constraints not marked removed && head matching && guard check)
then {remove removed head constraints; execute body constraints;}
}
The parameter list references the head constraints to be matched to the rule. In the procedure, we first check that
the constraints have not been marked as removed. Then head matching is explicitly performed and then the guard is
checked. If all successful, one removes the removed head constraints, executes the built-in constraints and then adds the
body CHR constraints. Added constraints may overwrite removed head constraints for efficiency. Constraints that are
removed and not overwritten are marked as deleted. Such a rule procedure is executed on every possible combination
of constraints from the store, typically through a nested loop (that can be parallelized). This basic translation scheme
corresponds to the abstract semantics, since it does not distinguish between active and suspended CHR constraints. It
needs to be refined to be practical [VW10].
8.2 Compiling CHR to Parallel Hardware
Our compiler translates the CHR code into the low-level hardware description language VHDL, which in turn creates
the necessary hardware using FPGAs. The architecture of FPGA hardware is basically divided into three parts: the
internal computational units called configuration logic blocks, the Input/Output (I/O) blocks that are responsible for
the communication with all the other hardware resources outside the chip, and the programmable interconnections
among the blocks called routing channels. In addition, there can be complex hardware blocks designed to perform
higher-level functions (such as adders and multipliers), or embedded memories, as well as logic blocks that implement
decoders or mathematical functions.
CHR Fragment with Non-Increasing Rules. We assume Ground CHR. Since the hardware resources can only
be allocated at compile time, we need to know the largest number of constraints that can occur in the constraint store
during the computation. In non-increasing rules, the number of body CHR constraints added is not greater than the
number of head constraints removed. Thus the number of constraints in the initial goal provides an upper bound on
the number of constraints during the computation. Hence we only allow for non-increasing simpagation rules.
CHR Compilation Hardware Components. A Program Hardware Block (PHB) is a collection of Rule Hardware Blocks (RHBs), each corresponding to a rule of the CHR program. A Combinatorial Switch (CS) assigns the
constraints to the PHBs. In more detail:
Rule Hardware Block (RHB) In VHDL the rule is translated into a single clocked process following the transformation scheme described above. Here, the parameters are input signals for each argument of the head constraints.
Each signal is associated with a validity signal to indicate if the associated constraint has been removed. A
concrete example is given below.
22
Program Hardware Block (PHB) The PHB makes sure that the RHBs keep applying themselves until the result
remains unchanged for two consecutive clock cycles. Each rule is executed by one or more parallel processes
that fire synchronously every clock cycle. The initial goal is directly placed in the constraint store from which
several instances of the PHB concurrently retrieve the constraints.
Combinatorial Switch (CS) The CS sorts, partitions and assigns the constraints to the PHBs, ensuring that the entire
constraint store gets exposed to the rule firing hardware. It acts as a synchronization barrier, allowing the faster
PHBs to wait for the slower ones, then communicating the results between the blocks. It also reassigns the input
signals to make sure that all constraint combinations have been exposed to the rule head matching.
Strong Parallelism with Overlap. For a given kept constraint, multiple RHBs are used to try rules with all
possible partner constraints. For the case of simpagation rules with one kept and one removed constraint, we introduce
a hardware block that consists of a circular shift register which contains all the initial goal constraints. The first
register cell contains the kept constraint and it is connected to the first input of all the RHBs, the rest of the register
cells contain the potential partner constraints and are each connected to the second input of one RHB. Every time
the PHBs terminate their execution, the new added constraints replace the removed ones. They registers shift until a
non-removed constraint is encountered.
Example 8.1. Consider the rule for the greatest common divisor:
r :
gcd(N) \ gcd(M) <=> M>=N | gcd(M-N).
In Figure 5 we give an excerpt of the VHDL code produced for the above rule. There are two processes executed in
parallel, one for each matching order, that correspond to two RHBs called r 1 and r 2. The input parameters gcd1 and
gcd2 are byte signals holding the numbers. valid1s and valid2s are bit signals. They are set to 0 if the associated
constraint is removed. The shared variable flag is a bit. It is used to control the application of the two processes.
Massive Parallelism. The set-based semantics CHRmp (see Section 7) allows multiple simultaneous removals
of the same constraint. Our implementation eliminates the conflicts in the constraint removals by allowing different
rule instances to work concurrently on distinct copies of the constraints. We provide all possible combinations of
constraints to distinct parallel PHB instances in a single step. So the same constraint will be fed to several PHBs.
Valid constraints are collected. A constraint is valid if no PHB has removed it. This is realized in hardware by AND
gates. The improvement due to massive parallelism is about an order of magnitude for goals with a low number of
constraints and it decreases with higher numbers of constraints. This is due to reaching the physical bounds of the
hardware.
Experimental Results. A few experiments were performed including the programs for Minimum, Prime Numbers, GCD, Merge Sort, Shortest-Path and Preflow-Push[TORF12, Tri11]. Unfortunately, no tables with concrete
performance numbers are given, just log-scale diagrams. From them we can see the following. The FPGA implementations of CHR are at least one order of magnitude faster than the fastest software implementations of CHR. In the
experiments, Shortest-Path and Preflow-Push showed a consistent parallel speed-up. Strong parallelism improves the
performance, and massive parallelism improves it further by up to an order of magnitude for the Prime example. In
the examples, the code produced by the CHR-to-FPGA compiler is slower but within the same order of magnitude as
handcrafted VHDL code.
Translation into C++ for CUDA GPU. Graphical Processing Units (GPUs) consist of hundreds of small cores to
provide massive parallelism. Similar to the work on parallel CHR FPGA hardware, the preliminary work in [ZFG12]
transforms non-increasing Ground CHR rules to C++ with CUDA in order to use a GPU to fire the rules on all combinations of constraints. As proof of concept, the scheme was encoded by hand for some typical CHR examples. The
constraint store is implemented as an array of fixed length consisting of the structures that represent CHR constraints.
A CHR rule can be translated into a function in C++ using the basic procedural translation scheme. The rule is executed on every possible combination of constraints using nested for-loops. Finally, the code is rewritten for the CUDA
library. The outer for-loop is parallelized for the thread pools of the GPU.
23
r_1: process (..., gcd1s, gcd2s, valid1s, valid2s)
begin
if ...
% checking and setting flags and parameters
if (valid1s=1 and valid2s=1) then
if gcd2s>=gcd1s then
gcd2s <= gcd2s-gcd1s;
flag := 1;
else
flag := 0;
end if;
end if;
end if;
end process r_1;
r_2: process (..., gcd1s, gcd2s, valid1s, valid2s)
begin
if ...
% checking and setting flags and parameters
if (valid1s=1 and valid2s=1) then
if flag=0 then
if gcd1s>=gcd2s then
gcd1s <= gcd1s-gcd2s;
end if;
end if;
end if;
end if;
end process r_2;
Figure 5: Excerpt of VHDL Code for GCD Rule
9 Distribution in CHR
Before we introduce a full-fledged distributed refined semantics for CHR and its implementation, we set the stage
by describing a distributed but sequential implementation of set-based CHR. This system is successfully employed
in a verification system for concurrent software. Both semantics work with a syntactic subset of CHR where head
constraints in rules must share variables in specific ways to enable locality of computations. Both semantics feature
propagation rules, but they use different mechanisms to avoid their repeated re-application.
9.1 Distributed Set-Based Goal Stores in CHRd
CHRd [SSR07] is an implementation of a sequential set-based refined semantics for CHR with propagation rules.
CHRd features a distributed constraint store.
Termination of Propagation Rules. There are basically two ways to avoid repeated application of propagation
rules: Either they are not applied a second time to the same constraints or they do not add the same constraints a
second time. Since we can remove constraints in CHR, usually the first option is chosen: we store the sequence of
CHR constraint identifiers to which a propagation rule has been applied. It can be garbage-collected if one of the
constraints is removed. This information is called a propagation history. CHRd replaces the check on the propagation
history by an occurrence check on the constraint store. This can be justified by the set-based semantics.
Set-Based Refined Semantics. Our set-based semantics closely follows the standard refined semantics [DSGH04].
The essential differences are as follows:
24
• The propagation history is dropped from the states.
• There is an additional transition to ensure a set-based semantics. It removes a constraint from the goal store
before its activation, if it is already in the constraint store.
• There are additional transitions to avoid immediate re-application of a propagation rule. In the first transition,
all head matching substitutions where the active constraint is kept are computed at once and all corresponding
rule instances are added to the goal store. These rule instances are called conditional activation events.
• When a conditional activation event is processed, it is checked if the matching head constraints are still in the
constraint store. If not, a second transition removes the event from the goal store. Otherwise, a third transition
applies the rule instance by adding its body constraints to the goal store.
The semantics does not model the distribution of the CHRd constraint store.
Our set-based semantics is not always equivalent to the standard refined semantics. In the semantics a propagation
rule may fire again on a constraint that has been re-activated (woken). In the refined multiset semantics, it will not be
fired again. So a CHR program may not terminate with the set-based semantics, but with the refined semantics.
Distributed Local Constraint Stores by Variable Indexing. Finding the partner constraints in head matching
efficiently is crucial for the performance of a CHR system. If variables are shared among head constraints, we can
use the corresponding arguments of the constraints for indexing. If the argument is an unbound variable at run-time,
we store (a pointer to) the constraint as attribute of that variable. If the argument becomes bound (or even ground) at
run-time, the constraint can be accessed from a hash table instead.
A conjunction of constraints is direct-indexed (connected) if all subsets of constraints share variables with the
remaining constraints. In other words, it is not possible to split the constraints in two parts that do not share a variable.
Definition 9.1. The matching graph of a set C of constraints is a labeled undirected graph G = (V, E) where V = C,
and E is the smallest set such that ∀c1 , c2 ∈ V, vars(c1 ) ∩ vars(c2 ) 6= {} → (c1 , c2 ) ∈ E where vars(c) returns the set of
variables in a constraint c. A rule R in a CHR program is said to be direct-indexed (connected) if the matching graph
for its head constraints is connected. A CHR program is direct-indexed if all its rule heads are direct-indexed.
Clearly, head matching is significantly improved for direct-indexed programs. Instead of combinatorial search
for matching partner constraints, constant-time lookups are possible with indexing. CHRd requires direct-indexed
programs that only index on unbound variables. This permits the constraint store to be represented in a distributed
fashion as a network of constraints on variables.
Any CHR program can be trivially translated to a direct-indexed program. We just have to add an argument to
each CHR constraint that always contains the same shared variable. For example, the direct-index rule for minimum
is:
min(X,N) \ min(X,M) <=> N=<M | true.
With the help of the new variable, we can distinguish between different minima. In general, this technique can be used
to localize computations.
Implementation and Experimental Results. We have an implementation of ground CHRd in the Datalog fragment of Prolog, where terms are constants only. Our implementation has been integrated into XSB, a Prolog programming system with tabling. It can be obtained online with a free download from http://xsb.sourceforge.net.
CHRd performs significantly better on programs using tabling, and shows comparable results on non-tabled benchmarks. This indicates that constraint store occurrence checks can be done as efficiently as propagation history checks
while avoiding the maintenance of a propagation history.
Verification of Multi-Threaded Applications. The paper [SSSD07] describes an approach for checking for
deadlocks in multi-threaded applications based on the concurrency framework SynchroniZation Units MOdel (Szumo)
[SS08a]. The framework associates each thread with a synchronization contract that governs how it must synchronize
with other threads. At run-time, schedules are derived by negotiating contracts among threads.
The Szumo system includes a constraint solver written in CHRd encoding the synchronization semantics of thread
negotiation. The verification system performs a reachability analysis: it constructs execution paths incrementally until
either a deadlock is detected or further extending the path would violate a synchronization contract.
25
With Szumo, we analyzed an implementation of the dining philosophers problem, where no deadlock was found.
We verified the in-order message delivery property of an n-place FIFO buffer. We also analyzed Fischers protocol, a
mutual-exclusion protocol that is often used to benchmark real-time verification tools. There we employed CHRd to
specify a solver for the clock constraints.
9.2 Distributed Parallel CHRe and its Syntax
The paper [LC13] introduces a decentralized distributed execution model consisting of an ensemble of computing entities, each with its own local constraint store and each capable of communicating with its neighbors: in
CHRe, rules are executed at one location and can access the constraint stores of its immediate neighbors. We
have developed a prototype implementation of CHRe in Python with MPI (Message Passing Interface) as a proof
of concept and demonstrated its scalability in distributed execution. It is available online for free download at
https://github.com/sllam/msre-py.
Syntax of CHRe. We assume Ground CHR. CHRe introduces locations.
Definition 9.2. All user-defined constraints in a program must be explicitly localized. A location l is a term (typically
an unbound variable or constant) that annotates a CHR constraint c, written as [l]c. A location l is directly connected
to a location l ′ if there is a constraint [l]c at location l such that l ∈ vars(c).
We are interested in rules that can read data from up to n of their immediate neighbors, but can write to arbitrary
neighbors. We therefore define n-neighbor restricted (star-shaped) rules (which are a subclass of direct-indexed rules
introduced in CHRd). The rule head refers to directly connected locations in a star topology. At the center of the star
is the primary location.
Definition 9.3. A CHR rule with n + 1 head constraints is n-neighbor restricted (star-shaped) if and only if there is
a dedicated location called primary location and n —em neighbor locations in the rule head satisfying the following
conditions:
• The primary location is directly connected to each of its n neighbor locations.
• If a variable is shared between constraints at different locations, it also must occur in the primary location.
• Each constraint in the guard shares variables with at most one neighbor location.
This definition ensures that computation can be structured and distributed by considering interactions between the
primary location and each neighboring location separately.
Example 9.1. This variant of the Floyd-Warshall algorithm computes all-pair shortest paths of a directed graph in a
distributed manner.
base : [X]arc(Y,D) ==> [X]path(Y,D).
elim : [X]path(Y,D1) \ [X]path(Y,D2) <=> D1<D2 | true.
trans : [X]arc(Y,D1), [Y]path(Z,D2) ==> X\=Z | [X]path(Z,D1+D2).
We distinguish between arcs and paths. [X]path(Y,D) denotes a path of length D from X to Y. The rules base and
elim are 0-neighbor restricted (local) rules because their left-hand sides involve constraints from exactly one location.
Rule trans is a 1-neighbor restricted rule since its left-hand side involves X and a neighbor Y. We see that X is the
primary location of this rule because it refers to location Y in an argument.
9.3 Refined Semantics of CHRe
Before we discuss the refined semantics, we shortly mention the abstract semantics of CHReto introduce the basic
principles.
Abstract Distributed CHRe Semantics for n-Neighbor Restricted Rules. Each location has its own goal store.
Based on the standard abstract CHR semantics, we introduce abstract ensemble states, which are sets of local stores
26
Gk where G is a goal and k a unique location name. In the adapted (Apply) transition, each of the locations in an
n-neighbor rule provides a partial match in their stores. If the matchings can be combined and if the guard holds,
we add the rule body goals to their respective stores. We show soundness with respect to the standard CHR abstract
semantics, where locations are encoded as an additional argument to each CHR constraint.
Refined Distributed CHRe Semantics for 0-Neighbor Restricted Rules. We extend the standard CHR refined
semantics to support decentralized incremental multiset matching for 0-neighbor restricted rules.
~ S̄; H̄ik ,
~ G;
Localized States. In CHRe, an ensemble Ω is a set of localized states. A localized state is a tuple hU;
where
~ is a queue of CHR constraints that have been sent to a location,
• the Buffer U
~ is a stack of the constraints to be executed,
• the Goal Store (Execution Stack) G
• the Constraint Store S̄ is a set of identified constraints to be matched,
• the Propagation History H̄ is a set of sequences of identifiers of constraints that matched the head constraints of
a rule,
• the state is at location k.
To add a further level of refinement, an active occurrenced CHR constraint c(x̄)#i: j is an identified constraint that is
only allowed to match with the j-th occurrence of the constraint predicate symbol c in the head of a rule of a given
CHR program P.
To simplify the presentation of the semantics, we assume static locations: for all locations occurring in a computation, there is a localized state (possibly with empty components) in the ensemble.
Localized Sequential Transitions. Figure 6 shows the sequential transitions for a single location.
• The (Flush) transition step applies if the goal store is empty and the buffer is non-empty. It moves all buffer
constraints into the goal store.
The transitions (DropLoc) and (MoveLoc) apply if the first constraint in the goal store of location k is one for
location [k′ ]c. They deliver constraint [k′ ]c to location k′ .
• The (MoveLoc) transition applies if k′ is distinct from k and there exists a location k′ . It it strips the location [k]
away and sends constraint c to the buffer of k′ .
• The (DropLoc) transition applies if k′ is the same as k. The location [k] is dropped.
The remaining transitions apply to a location as to a state in the standard refined semantics. Buffers are ignored and
remain unchanged. The transitions model the activation of a constraint, the application of rules to it, and its suspension
if no more rule is applicable. These transitions are as in the standard refined semantics of CHR, except that here we
take care of locations and handle a propagation history.
• In the (Activate) transition, a CHR constraint c becomes active (with first occurrence 1) and is also introduced
as identified constraint into the constraint store.
• The (Remove) transition applies a rule where the active constraint is removed. There is a substitution θ under
which constraints from the constraint store match the head of the rule and satisfy its guard (written |= θ ∧ G).
The auxiliary function DropIds removes the identifiers from identified constraints.
• The (Keep) transition is like the (Remove) transition except that the active constraint c matches a kept constraint
and it is checked if the application of the resulting rule instance has not been recorded in the propagation history.
If so, the active constraint is kept and remains active. The propagation history is therefore updated. (It remains
unchanged in all other transitions.) The function Ids returns the identifiers of identified constraints.
• In the (Suspend) transition, the active constraint cannot be matched against its occurrence in the rule head. One
proceeds to the next occurrence in the rules of the program. This makes sure that rules are tried in the order
given in the program.
27
(Flush)
(MoveLoc)
(DropLoc)
~ 6= {}
U
~ {}; S̄; H̄ik 7→ Ω, h{}; U;
~ S̄; H̄ik
Ω, hU;
~ S̄; H̄ik , h(U,
~ S̄; H̄ik′
~ S̄; H̄ik , hU;
~ S̄; H̄ik′ 7→ Ω, hU;
~ G;
~ [c]); G;
~ ([k′ ]c, G);
~ G;
Ω, hU;
~ S̄; H̄ik 7→ Ω, hU;
~ S̄; H̄ik
~ ([k]c, G);
~ (c, G);
Ω, hU;
(Activate)
d is a fresh identifier
~ S̄; H̄ik 7→ Ω, hU;
~ (S̄, c#d); H̄ik
~ (c, G);
~ (c#d : 1, G);
Ω, hU;
(Remove)
Variant of (r : [l]HP′ \[l]HS′ <=> C | B) ∈ P such that |= φ (C) k = φ (l)
φ (HP′ ) = DropIds(HP ) φ (HS′ ) = {c} ∪ DropIds(HS)
~ (S̄, HP , HS , c#d); H̄ik 7→ Ω, hU;
~ (S̄, HP ); H̄ik
~ (c#d : i, G);
~ (φ (B), G);
Ω, hU;
(Keep)
Variant of (r : [l]HP′ \[l]HS′ <=> C | B) ∈ P such that |= φ (C) k = φ (l)
φ (HS′ ) = DropIds(HS ) φ (HP′ ) = {c} ∪ DropIds(HP) h = (r, Ids(HP , HS )), h 6∈ H̄
~ (S̄, HP , HS , c#d); H̄ik 7→ Ω, hU;
~ (S̄, HP , c#d); (H̄, h)ik
~ (c#d : i, G);
~ (φ (B), c#d : i, G);
Ω, hU;
(Suspend)
(Remove) and (Keep) do not apply for c#d : i, occurrence i exists
~ S̄; H̄ik 7→ Ω, hU;
~ S̄; H̄ik
~ (c#d : i, G);
~ (c#d : (i + 1), G);
Ω, hU;
(Drop)
i is not an occurrence in the program P
~ S̄; H̄ik 7→ Ω, hU;
~ S̄; H̄ik
~ (c#d : i, G);
~ G;
Ω, hU;
Figure 6: The Sequential Part of the Refined CHRe Semantics for 0-Neighbor Restricted Rules
• The (Drop) transition, if there is no more occurrence to try, removes the active constraint the goal store, but it
stays suspended in the constraint store.
Localized Parallel Transitions. Figure 7 shows the parallel transitions. They are particularly simple. As usual,
the transition (Intro-Par) says that any sequential transition is a parallel transition. Transition (Parallel-Ensemble)
allows to combine two independent transitions on non-overlapping parts of the state (ensembles, i.e. sets of disjoint
locations) into one parallel transition. This means that computation steps on different localized states can be executed
in parallel.
9.4 Properties: Monotonicity, Soundness and Serializability
In the refined CHRe semantics, monotonicity holds with respect to locations, this means computations can be repeated
in any larger context of more locations. Serializability holds in that every parallel CHRe computation can be simulated
using sequential CHRe transitions. We also prove soundness of the refined CHRe semantics with respect to the abstract
CHRe semantics.
We say that a CHRe program is locally quiescent (terminating) if given a reachable state, we cannot have any
infinite computation sequences that do not include the (Flush) transition. Hence local quiescence guarantees that each
28
(Intro-Par)
Ω 7→ Ω′
Ω Z⇒ Ω′
(Parallel-Ensemble)
(Ω1 , Ω2 ) Z⇒ (Ω′1 , Ω2 )
(Ω1 , Ω2 ) Z⇒ (Ω1 , Ω′2 )
(Ω1 , Ω2 ) Z⇒ (Ω′1 , Ω′2 )
Figure 7: The Parallel Part of the Refined CHRe Semantics for 0-Neighbor Restricted Rules
location will eventually process the constraints delivered to its buffer.
Serializability and soundness of the encoding holds for quiescent programs: computations between commit-free
states of 0-neighbor restricted encodings have a mapping to computations of the original 1-neighbor restricted program.
The corresponding theorems and their detailed proofs can be found in the appendix of [LC13].
9.5 Encoding 1- and n-Neighbor Rules in Local Rules
We give an encoding of the more general 1-neighbor restricted rules into local, i.e. 0-neighbor restricted rules. We
can do the same for n-neighbor restricted rules. In this way, a programmer can use n-neighbor rules while the translation generates the necessary communication and synchronization between locations. The encodings are a block-free
variation of a two-phase commit consensus protocol between locations.
Two-Phase-Commit Consensus Protocol. The protocol consists of two phases:
• Commit-Request Phase (Voting Phase). The coordinator process informs all the participating processes about
the transaction and to vote either commit or abort. The processes vote.
• Commit Phase. If all processes voted commit, the coordinator performs its part of the transaction, otherwise
aborts it. The coordinator notifies all processes. The processes then act or abort locally.
The standard protocol can block if a process waits for a reply. Not so in the variation we use.
Encoding 1-Neighbor Restricted Programs. According to the following scheme, we translate each 1-neighbor
restricted rule of the form
r : [X]Px, [X]Px’, [Y]Py \ [X]Sx, [Y]Sy <=> Gx,Gy | Body.
In the head, Px are the persistent constraints and Px’ are the non-persistent constraints. Constraints are persistent if
they are not removed by any rule in the program. In the guard, Gx contains only variables from location X. In the rule
scheme below, XYs contains all variables from the rule head, and Xs only the variables from location x.
request : [X]Px,[X]Sx ==> Gx | [Y]r_req(Xs).
vote : [Y]Py,[Y]Sy \ [Y]r_req(Xs) <=> Gy | [X]r_vcom(XYs). % if Sx non-e.
vote : [Y]Py,[Y]Sy, [Y]r_req(Xs) ==> Gy | [X]r_vcom(XYs). % if Sx empty
commit : [X]Px \ [X]Px’,[X]Sx, [X]r_vcom(XYs) <=> [Y]r_commit(XYs).
act
: [Y]Py \ [Y]Sy, [Y]r_commit(XYs) <=> [X]Px’, Body.
The rule scheme uses different vote rules depending on the emptiness of Sx. If Sx is empty, it should be possible to
remove several instances of Sy with the same request. Note that the rule scheme requires a refined semantics where
rules are tried in the given order, because we have to make sure that rule act is tried before the abort rule abort.
The rule scheme implements an asynchronous and optimistic consensus protocol between two locations of the
ensemble. It is asynchronous because neither primary nor neighbor location ever block or busy-wait for responses.
29
Rather they communicate asynchronously via the protocol constraints, while potentially interleaving with other computations. The temporary removal of non-persistent constraints in the rule scheme ensures that the protocol cannot
be interfered with. It is optimistic because non-protocol constraints are only removed after both locations have independently observed their part of the rule head instance. It is possible that some protocol constraints are left if the
transaction did not commit, but these can be garbage-collected.
We can generalize the above encoding to n-neighbor restricted rules.
CoMingle. This new programming language can be characterized as an extension of CHRe for distributed logic
programming [LCF15, CLE16]. There is a prototype on the Android operating system for mobile devices, see
https://github.com/sllam/CoMingle. One application was built both using CoMingle and by writing traditional
code: the former was about one tenth of the size of the latter without a noticeable performance penalty.
10 Models of Concurrency in CHR
Theoretical and practical models of concurrency have been encoded in CHR. Such an effective and declarative embedding holds many promises: It makes theoretical models executable. It can serve as executable specification of
the practical models. One can toy with alternative design choices. The implementations can be formally verified and
analyzed using standard and novel CHR analysis techniques. Last but not least it allows to compare different models
on a common basis.
We will shortly introduce some common models of concurrency by their implementation in CHR: Software Transactional Memory, Colored Petri Nets, Actors and Join-Calculus. Typically soundness and completeness results will
prove the correctness of these embeddings.
10.1 Software Transactional Memory STM
We have already seen the description of STM and its use to implement parallel CHR in Haskell in Section 6. Now
we do it the other way round. For the STM model, as a starting reference see [ST97], for a high-level description see
[GK08]. The paper [SC08] gives a rule-based specification of Haskell’s Software Transactional Memory in parallel
CHR which naturally supports the concurrent execution of transactions.
We classify CHR constraints once more into operation constraints and data constraints. We assume CHR rules
where the head contains exactly one operation constraint and the body contains at most one operation constraint.
Shared Memory Operations. We first model shared memory and its associated read and write operations in CHR.
read : cell(L,V1) \ read(L,V2) <=> V1=V2.
write : cell(L,V1), write(L,V2) <=> cell(L,V2).
L is a location identifier and V1 and V2 are values. cell is a data constraint, read and write are operation constraints.
The write rule performs a destructive assignment to update the value of the cell. With indexing and in-place constraint
updates, the compiled rule can run in constant time.
STM run-time manager in CHR. The effects of an STM transaction are reads and writes to shared memory. The
STM run-time must guarantee that all reads and writes within a transaction happen logically at once. In case transactions are optimistically executed in parallel the STM run-time must take care of any potential read/write conflicts.
The STM run-time must ensure that in case of conflicts at least one transaction can successfully commit its updates
whereas the other transaction is retried.
To accomplish this behavior, we use for each transaction a read log and a write log. Before we can commit the
write log and actually update the memory cell, we first must validate that for each cell whose value is stored in the
read log, the actual value is still the same.
In Figure 8, we specify the STM manager via CHR rules. It has been slightly simplified in this survey. Besides
locations and values, we introduce an identifier for transactions T. The operation constraints are read and write and
the protocol constraints are validate, commit and rollback, retry. The data constraint CommitRight acts as a
token a committing transaction has to acquire in order to avoid concurrent writes. The constraint validate is issued
at an end of the transaction if the CommitRight is available. Rules for rollback and retry of transactions are not shown
here for space reasons.
30
% Execution phase
% Read from write
r1 : WLog(t,l,v1)
r2 : RLog(t,l,v1)
r3 : Cell(l,v1) \
--or read log, create read log otherwise
\ Read(t,l,v2) <=> v1=v2.
\ Read(t,l,v2) <=> v1=v2.
Read(t,l,v2) <=> v1=v2, RLog(t,l,v1).
% Write to write log, create write log otherwise
w1 : WLog(t,l,v1), Write(t,l,v2) <=> WLog(t,l,v2).
w2 : Write(t,l,v) <=> WLog(t,l,v).
% Validation phase --% Check and remove read log, rollback on read log conflict
v1 : Cell(l,v1), Validate(t) \ RLog(t,l,v2) <=> v1=v2 | True.
v2 : Cell(l,v1) \ Validate(t), RLog(t,l,v2) <=> v1=\=v2 | Rollback(t).
% Start commit phase by acquiring CommitRight, otherwise rollback
s1 : CommitRight, Validate(t) <=> Commit(t).
s2 : Validate(t) <=> Rollback(t).
% Commit phase
% write update
c1 : Commit(t)
c2 : Commit(t)
--cells, then return CommitRight
\ Cell(l,v1), WLog(t,l,v2) <=> Cell(l,v2).
<=> CommitRight.
Figure 8: STM Run-Time Manager in CHR
Soundness and Correctness. Our implementation guarantees atomicity, isolation and optimistic concurrency. It
is therefore sound. It is correct: if a transaction commits successfully, the store reflects correctly all the reads/writes
performed by that transaction.
10.2 Colored Petri Nets CPN
Petri nets are diagrammatic formalism to describe and reason about concurrent processes. They consist of labelled
places ( ) in which tokens (•) reside. Tokens can move along arcs passing through transitions (
) from one place
to another. A transition may have several incoming arcs and several outgoing arcs. A transition can only fire if all
incoming arcs present a token. On firing, all incoming tokens will be removed and a token will be presented on each
outgoing arc. Colored Petri Nets (CPN) [Jen87] significantly generalize Petri nets. Tokens are colored and places are
typed by the colors they allow. Transitions can have conditions on tokens and equations that compute new tokens from
old ones.
The paper [Bet07] shows that (Colored) Petri nets can easily be embedded into CHR. When CPNs are translated
to CHR, color tokens are encoded as numbers. Place labels are mapped to CHR constraint symbols, tokens at a place
to instances of CHR constraints, transitions and their arcs to simplification rules. Incoming arc places form the rule
head, outgoing arc places form the rule body, and the transition conditions as well as equations form the rule guard.
Example 10.1. For simplicity, we consider the dining philosophers problem with just three philosophers as CPN in
Figure 9. Each philosopher (and fork) corresponds to a colored token, given as a number from 0 to 2. Two philosophers
x and y are neighboring if y = (x+1) mod 3. Places are think, eat and fork, transitions are eat-think and and think-eat.
The CPN of Figure 9 translates into the following two CHR rules
31
✲
✲
y = (x+1) mod 3
think-eat
x
x,y
✤✜
✤✜
think
fork
0✐
0✐
✐
✐
✐
1 2
1 2✐
✣✢✣✢
{0, 1, 2} ✻ {0, 1, 2} ✻
x
x,y
x
❄
✤✜
eat
✣✢
{0, 1, 2}
y = (x+1) mod 3
x
✛
eat-think
Figure 9: The Three Dining Philosophers Problem as Colored Petri Net
think_eat : think(X), fork(X), fork(Y) <=> Y =:= (X+1) mod n | eat(X).
eat_think : eat(X) <=> Y =:= (X+1) mod n | think(X), fork(X), fork(Y).
Soundness and Completeness. For both classical and Colored Petri nets, these correctness theorems are proven
for the translation into CHR.
10.3 Actor Model
In the Actor Model [Agh86], one coordinates concurrent computations by message passing. Actors communicate by
sending and receiving messages. Sending is a non-blocking asynchronous operation. Each sent message is placed in
the actors mailbox (a message queue). Messages are processed via receive clauses which perform pattern matching and
guard checks. Receive clauses are tried in sequential order. The receive operation is blocking. If none of the receive
clauses applies the actor suspend until a matching message is delivered. Receive clauses are typically restricted to a
single-headed message pattern. That is, each receive pattern matches at most one message.
In [SLVW08], we extend the Actor Model with receive clauses allowing for multi-headed message patterns. Their semantics is inspired by their translation into CHR. We have implemented a prototype in Haskell
https://code.google.com/archive/p/haskellactor/.
Example 10.2. In the Santa Clause problem, Santa sleeps until woken by either all of his nine reindeer or by three of
his ten elves. If woken by the reindeer, he harnesses each of them to his sleigh, delivers toys and finally unharnesses
them. If woken by three elves, he shows them into his study, consults with them on toys and finally shows them out.
Here is a solution using the proposed multi-head extension:
santa sanActor =
receive sanActor of
Deer x1, Deer x2, ..., Deer x8, Deer x9 -> harness, deliver, unharness.
Elf x1, Elf x2, Elf x3 -> enter_study, consult, leave_study.
This straightforward solution avoids the clumsiness of explicitly counting deers and elves in the mailbox. There is an
obvious direct embedding of the matching receive clauses into CHR simplification rules.
32
Semantics of Actors with Multi-Headed Message Patterns. We study two possible semantics for this extension,
inspired by the standard refined semantics of CHR:
• The first-match semantics provides a conservative extension of the semantics of single-headed receive clauses.
This semantics guarantees monotonicity: any successful match remains valid if further messages arrive in the
actors mailbox.
• The rule-order-match semantics guarantees that rule patterns are executed in textual order. In this semantics,
newly arrived messages can invalidate earlier match choices.
It will depend on the application which semantics is the better choice.
10.4 Join-Calculus and Join-Patterns
In Join-Calculus [FG02], concurrency is expressed via multi-headed declarative reaction rules that rewrite processes
or events. The (left-hand side of a) rule is called join-pattern. They provide high-level coordination of concurrent
processes. The thesis [Lam11b] extends join-patterns with guards and describes a prototype implementation in parallel
CHR compiled to Haskell, see http://code.haskell.org/parallel-join.
Join-Calculus with Guarded Join-Patterns. A concurrent process (or event), say P, has the form of a predicate.
We introduce guards into these rules:
Guarded Reaction Rule P1 , . . . Pn if Guard ⇒ P1′ , . . . Pm′
The Join-Calculus semantics is defined by a chemical abstract machine (CHAM). This model specifies transformations
using a chemical reaction metaphor. The CHAM can be embedded in CHR, see Chapter 6 in [Frü09].
Example 10.3. A print job is to be executed on any available printer where it fits. So print jobs have a size, and
printers have a certain amount of free memory. This behavior is captured by the following guarded reaction rule:
ReadyPrinter(p,m), Job(j,s) if m>s => SendJob(p,j)
There is an obvious direct translation into CHR simplification rules.
Implementation and Experimental Results. Standard CHR goal-based lazy matching is a suitable model for
computing the triggering of join-patterns with guards: each process (CHR goal) essentially computes only its own
rule head matches asynchronously and then proceeds immediately. We conducted experiments of our parallel JoinCalculus implementation with examples for common parallel programming problems. They show consistent speed-up
as we increase the number of processors.
11 Discussion and Future Work
We now present common topics and issues that we have identified as a result of this survey and that lead to research
questions for future work.
Syntactic Fragments of CHR. The parallel and distributed semantics surveyed are concerned with expressive
Turing-complete fragments of CHR. Their properties are summarized in Table 1. Except for the distributed semantics
(CHRd and CHRe) they do not allow for terminating propagation rules. In the distributed semantics of CHRd and
CHRe one restricts rule heads to be sufficiently connected by shared variables, requiring direct-indexed and n-neighbor
(star-shaped) rules, respectively. The former is no real restriction, the latter is.
Software implementations always presume Ground CHR (and so does CHRt). Hardware implementations in
addition rely on non-size-increasing rules which are still Turing complete.
Sometimes the notion of constraints is too abstract, and one differentiates between data and operation constraints.
Operation constraints update data constraints. This dichotomy clarifies programs like Blocks World and Union-Find,
is essential in the semantics of CHR with transactions (CHRt) and in the concurrency model of Software Transactional
Memory when encoded in CHR.
33
CHR Semantics
Abstract Par.
Refined Par.
CHRmp
CHRt
CHRd
CHRe
Syntactic Restriction
propagation rules do not terminate
no propagation rules
no propagation rules
ground data and operation constraints
direct-indexed rule heads
ground star-shaped rule heads
Monotonicity Soundness Serializability
yes
yes
soundness for deletion-acyclic programs
yes
yes for ground confluent programs?
for quiescent programs
Table 1: Syntactic Restrictions and Properties of CHR Parallel and Distributed Semantics
All example programs in the survey and in general many other sequential CHR programs can still be run in parallel
without modification, since the syntactic restrictions are observed as they cover expressive subsets of CHR. However,
changes are necessary if the program is not ground, for parallel execution if the program contains propagation rules,
and for distributed execution if the rule heads are not sufficiently connected. This need for program modifications
weakens the promise of declarative parallelism, and therefore (semi-)automatic methods of program transformation
should be investigated. Note that such transformations would be purely syntactical and do not require to come up with
any scheduling for parallelism.
Propagation Rules. Surprisingly, while propagation rules seem perfect for parallelization (because they do not
remove any constraints), they are currently only supported in distributed CHRd and CHRe (see Table 1). (In the
abstract parallel semantics, they are allowed, but do not terminate.) On the other hand it seems possible to extend
the refined parallel semantics with propagation rules, either using the propagation history of CHRe or the occurrence
check approach of CHRd to avoid their trivial non-termination. The former seems to come with some implementation
overhead, since the data structure needs to be updated in parallel. The latter approach does not work in all cases, but it
could be applicable to set-based semantics like CHRmp. As for a third possibility, in the literature on optimizing CHR
implementations one can find program analyses that detect if propagation rules can be executed without any checks.
Ground CHR is a good candidate for avoiding checks altogether, because constraints cannot be re-activated.
Semantics Properties: Monotonicity, Serializability and Soundness. These properties have been proven for all
parallel CHR semantics based on multisets, for distributed CHRe with the restriction to quiescent programs. Surprisingly, these properties do not hold in general for the set-based semantics of distributed CHRd and massively-parallel
CHRmp. The papers on CHRd do not fully investigate these properties, while CHRmp is sound for deletion-acyclic
programs. Clearly, set-based semantics for CHR have to be studied more deeply. There seems to be a mismatch
between their elegance of the concept and its actual behavior.
Program Analysis. We should re-examine CHR program analysis for parallel and distributed CHR to see how
they carry over. Termination corresponds to quiescence in the concurrent context. There is a vast literature on (non)termination and complexity analysis of CHR programs. Confluence is an essential desirable property of sequential CHR programs. It already plays a role in parallel CHR for sound removal of transactions and seems trivial in
exhaustively-parallel CHRmp. Confluence seems strongly related to soundness and serializability properties of concurrent CHR semantics. Semi-automatic completion generates rules to make programs confluent. This method has
been used in parallelizing the Union-Find algorithm and can be used for translating away CHR transactions. When
transactions are involved, confluence seems to avoid deadlocks. We also think that the property of deletion-acyclicity
of CHRmp has a broader application in rule-based systems. It seems related to confluence and we think can be
expressed as a termination problem.
Software and Hardware Implementations. All software implementations surveyed are available online for free
download, the links have been given. The implementations cover parallel CHR, set-based CHRd and distributed
CHRe as well as CoMingle. All implementations restrict themselves to the ground subset of CHR. A full-fledged
widely used stable implementation of parallel CHR is still missing. It could serve as a basis to foster further research
and applications, as does the K.U. Leuven platform for sequential CHR. With CoMingle, the situation seems better
in the case of distributed CHR. In any case, more evidence in the form of experimental results is needed to further
confirm the promise of declarative concurrency made by CHR.
Models of Concurrency in CHR. Embedding models of concurrency in CHR is promising for understanding,
analyzing and extending models, but still in its infancy. It is appealing because of the lingua franca argument for
34
CHR: different embeddings can be compared on its common basis and fertilize each other. Conversely, the striking
similarity of some models when encoded in CHR leads one to speculate about a generic concurrency model that is a
suitable fragment of CHR which could then be mapped to many existing models, yielding a truly unified approach.
12 Conclusions
We have given an exhaustive survey of abstract and more refined semantics for parallel CHR as well as distributed
CHR. Most of them have been proven correct. These semantics come with several implementations in both software
and hardware. All software implementations are available online for free download. We presented non-trivial classical
example programs and promising experimental results showing parallel speed-up. Last but not least we reviewed
concurrency models that have been encoded in CHR to get a better understanding of them and sometimes to extend
them. Most of these embeddings have been proven correct, i.e. sound and complete. Some embeddings are available
online.
In the discussion, we identified the following main topics for future work: Including propagation rules into the
parallel semantics and providing program transformations into the expressive syntactic fragments for distributed CHR,
investigate set-based semantics and the deletion-acyclic programs, provide a full-fledged implementation of parallel
CHR, apply the wealth of existing program analyses for sequential CHR to distributed and parallel CHR programs
and the embedding of concurrency models, and explore similarities of the concurrency models embedded in CHR as
lingua franca to come up with unified models.
On a more general level, it should be investigated how the research surveyed here carries over to related languages
like constraint logic programming ones and the other rule-based approaches that have been embedded in CHR. Overall, the CHR research surveyed here should be related to more mainstream research in concurrency, parallelism and
distribution.
Acknowledgements. We thank the anonymous referees for their helpful, detailed and demanding suggestions on
how to improve this survey.
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38
| 6 |
1
Integrated PV Charging of EV Fleet Based on
Dynamic Energy Prices and Offer of Reserves
Gautham Ram Chandra Mouli, Student Member, IEEE, Mahdi Kefayati, Member, IEEE,
Ross Baldick, Fellow, IEEE, Pavol Bauer, Senior Member, IEEE
Fig. 1. Design of solar powered EV charging station
PEVrc
xe+t,v
Power (W)
Abstract—Workplace charging of electric vehicles (EV) from
photovoltaic (PV) panels installed on an office building can
provide several benefits. This includes the local production and
use of PV energy for charging the EV and making use of dynamic
tariffs from the grid to schedule the energy exchange with the
grid. The long parking time of EV at the workplace provide the
chance for the EV to support the grid via vehicle-to-grid
technology, the use of a single EV charger for charging several
EV by multiplexing and the offer of ancillary services to the grid
for up and down regulation. Further, distribution network
constraints can be considered to limit the power and prevent the
overloading of the grid. A single MILP formulation that
considers all the above applications has been proposed in this
paper for a charging a fleet of EVs from PV. The MILP is
implemented as a receding-horizon model predictive energy
management system. Numerical simulation based on market and
PV data in Austin, Texas have shown 31% to 650% reduction in
the cost of EV charging when compared to immediate and
average rate charging policies.
Smart charging
PV generation
Immediate charging
Average rate
xe(ar)v
I. INTRODUCTION
Electric vehicles (EV) provide a zero-emission, low noise
and highly efficient mode of transportation. The current
estimate for the USA is that there will be 1.2 million EV by
2020 [1]. Electric vehicles are, however, sustainable only if
the electricity used to charge them comes from sustainable
sources. Electricity generated from a fuel mix that is largely
dominated by fossil fuels does not eliminate the emissions but
mostly moves it from the vehicle to the power plant [2], [3].
While this can have environmental advantages, complete
elimination of emissions is contingent on utilizing nonemitting resources for electricity production. It is here that the
phenomenal growth in the use of photovoltaic (PV) system for
distributed generation and its falling cost over the years can
have a direct impact.
EVs used to commute to work are parked at the workplace
for long hours during the day when the sun is shining.
Workplaces like industrial sites and office buildings harbor a
great potential for PV panels with their large surfaces on flat
Gautham Ram and Pavol Bauer are with the Department of Electrical
Sustainable Energy, Delft University of Technology, 2628 CD Delft, The
Netherlands. (e-mail: P.Bauer@tudelft.nl, G.R.Chandamouli@tudelft.nl).
Ross Baldick and Mahdi Kefayati are with the Department of Electrical
and Computer Engineering, The University of Texas at Austin, Austin, TX
78712. (e-mail: baldick@ece.utexas.edu, kefayati@utexas.edu).
This work was supported by the TKI Switch2SmartGrids grant,
Netherlands and the Partners for International Business(PIB) program of the
Rijksdienst voor Ondernemend (RVO), Netherlands.
0
Tva
Hour of day (h)
T vd
24
Fig. 2. Immediate, average rate and smart charging of EV
roofs. This potential is largely unexploited today. Energy
generated from PV array installed at the workplace can hence
be used for charging EVs as shown in Fig. 1. This has several
benefits namely:
1. EV battery doubles up as an energy storage for the PV
2. Negative impact of large-scale PV and EV integration on
distribution network is mutually reduced [4], [5]
3. Long parking time of EV paves way for implementation
of Vehicle-to-grid (V2G) technology where the EV can
offer energy and ancillary services to the grid [6], [7].
4. Cost of EV charging from solar is cheaper than charging
from the grid and net CO2 emission is lowered [2], [8].
A. Immediate and average rate charging
Today, when an EV arrives at the workplace and is
connected to an electric vehicle supply equipment (EVSE), the
EV typically starts charging immediately at the nominal EVSE
power rating 𝑃𝑐𝐸𝑉𝑟 . The charging continues at a constant power
till the battery is full1. This is referred to as immediate
charging (IMM) or uncontrolled charging [9]. This is the
simplest form of charging requiring no information from the
user or communication infrastructure and results in the lowest
2
charging time. However, IMM typically results in a huge
demand on the grid, depending on the EVSE capability, as
shown in Fig. 2.
EVs parked at the workplace usually have long parking
times and this offers the flexibility in scheduling the charging
in terms of both charging power and duration. This means that
EVs can be charged at a much lower power than the EVSE
nominal rating if the EV user arrival time, 𝑇𝑣𝑎 , departure time,
𝑇𝑣𝑑 and required energy demand, 𝑑𝑣 are known. One approach
is the “Average Rate” (AR) charging policy [9].
𝑥𝑣𝑒(𝑎𝑟) = 𝑀𝑖𝑛. {
𝑑𝑣
≤ 𝑥𝑣𝑢𝑏 , 𝑃𝑐𝐸𝑉𝑟 }
𝑇𝑣𝑑 − 𝑇𝑣𝑎
∀ 𝑡 ∈ {𝑇𝑣𝑎 , 𝑇𝑣𝑑 }
(1)
𝑥𝑣𝑒(𝑎𝑟)
Here, the charging power
is the minimum of the
EVSE capacity and the ratio of the energy demand divided by
the parking time of the EV1. The advantage of the AR policy
is that the charging of the fleet is spread through the day
instead of being concentrated in the arrival time (typically
early morning), as seen in Fig. 2.
However, both IMM and AR strategies are not ‘smart’ as it
has no correlation to local renewable generation, distribution
network capacity constraints and/or energy prices.
B. Smart charging
The optimal way to charge EVs is hence to use an energy
management system (EMS) that can schedule the charging of
an EV fleet by taking into consideration the EV user
preferences, local renewable generation and energy prices
from the market. Fig. 2 shows the example of a smart charging
where the EV charging follows the PV generation. Further,
EVs can have extremely fast ramp up and ramp down rates.
Chademo and Combo EV charging standards for DC charging
stipulate response time of 200ms [10]. This makes EVs ideal
candidates for providing ancillary services in the form of
reserve capacity to the grid [6], [7], [11], [12].
Following the formulation in [12], [13], an Energy Services
Company (ESCo) company acts as an intermediary between
the wholesale market operated by the Independent System
Operator (ISO) and the EV end-users. The ESCo operates at
the workplace where employees drive to the office with an EV
and the building has overhead PV installation or a solar
carport. The motive of the ESCo is to schedule the charging of
the EV and feeding of PV power to the grid in such a way that
charging costs are lowered, regulation services are offered to
the ISO and at the same time, the income from PV are
increased. ESCo achieves this motive by using an EMS to
schedule the EV based on a multitude of inputs namely:
1. Information from the EV user about arrival and departure
times, the state of charge (SOC) and energy demand.
2. 𝑝𝑡𝑒(𝑏𝑢𝑦) , 𝑝𝑡𝑒(𝑠𝑒𝑙𝑙) are the settlement point prices for buying
and selling electricity from the grid at time t.
3. 𝑝𝑡𝑟(𝑢𝑝) , 𝑝𝑡𝑟(𝑑𝑛) are the clearing prices for capacity for
offering reserves to the ISO for up and down regulation.
4. Distribution network constraints 𝑃𝑡𝐷𝑁+ , 𝑃𝑡𝐷𝑁− which are the
1
The expression does not consider the duration in the constant-voltage
(CV) charging mode, which occurs typically when EV battery is above 80%
SOC and the maximum charging power is limited [28].
upper limits for drawing and feeding power between the
EV car park and the grid at time t. These values can be
adjusted to implement demand side management (DSM).
5. Solar forecast information: 𝑃𝑡𝑃𝑉(𝑓𝑐) is the generation
forecast for a 1kWp PV system installed at the workplace.
The use of solar forecast data in the EMS will help in
reducing the uncertainties due to variability in PV
generation on diurnal and seasonal basis [14].
C. Literature review and overview of contributions
Several earlier works have formulated the optimization
problem to charge EV based on renewable generation, energy
prices and offer of ancillary services.
Fuzzy logic is used to optimize the EV charging based on
PV generation forecast, energy prices in [15] and V2G
frequency regulation, grid energy exchange in [16]. The
disadvantage is that the use of fuzzy logic without
optimization techniques does not guarantee that the obtained
solution is optimal.
In [12], [13], linear programming (LP) is used to find the
optimal EV strategy for charging and offering reserves based
on market prices. In [17], LP is used to reduce the cost of
charging EV from PV based on time of use tariffs and PV
forecasting. Cost reduction of 6% and 15.2% compared to the
base case are obtained for simulation for 12 EV powered from
a 50kW PV system. The LP formulation in [18] and heuristic
methods used in [19] aim to achieve the two goals: increasing
the PV self-consumption in a micro-grid by charging of EVs
and reducing the dependency on the grid. However, there is no
consideration for time of use tariffs without which there is no
incentive to achieve the two goals. In [20], LP is used for
planning EV charging based on renewable power forecasting,
spinning reserve and EV user requirements in a micro-grid
Stochastic programming is used in [21] to charge EV and
offer regulation services based on day-ahead and intraday
market prices. For a case study with 50 EV, cost reduction in
the tune of 1% to 15% was achieved.
Main contributions of this work include:
Proposing a single comprehensive model that captures
charging of EV from PV, use of dynamic grid prices,
implementation of V2G for grid support, using EV to offer
ancillary services and considering distribution network
capacity constraints as a single MILP problem. Earlier
works have considered these as separate optimization
problems or as a combination of two or three applications.
The disadvantage of the earlier approach is that each
optimization problem gives a different optimized EV
charging profile and all these profiles cannot be
implemented on the same EV at the same time! The best
approach is to combine them into one formulation which
will then yield a single optimized EV charging profile
Demonstrating that the formulation of the abovementioned
five aspects into one MILP formulation results in large
cost savings, which is much higher than what has been
achieved earlier.
With a large number of EV parked at the workplace with
long parking times, multiplexing a few EVSE to a larger
3
ISO
e(buy)
e(sell)
pt
pt
P t
Pr(dn)T PDN+t PDN-t
Solar Forecast
Cpv , Cv2x
r(up)
PPV(fc)t,c
Copt
Energy
Management
System [EMS]
xe-t,v , xe+t,v
r(up)
x t,v , xr(dn)t,v
a
Nchc Pcconv ηcconv
d
T v,T V
Bav , dv
xlbv , xubv
Bmaxv ,Bminv
ηchv , ηv2xv
Power (kW)
EV & Chr Data
PV
panels
Pt,c
PV
DC link
PV MPPT
Inverter
converter
(DC/AC)
(DC/DC)
EV charger
EV charger
(DC/DC)
(DC/DC)
xe-t,v
Kv,c N
conn
c
Car park
AC Grid
Pt,c
draw
Pt,cfeed
0
Min{PcEVr , xub
v }
xr(up)
t,v
xe+
t,v
V2X
CH
xe-t,v
xr(dn)
t,v
Min{PcEVr ,xlbv }
xr(+-)t,v
xe+t,v
Bt,v
𝒄𝒐𝒏𝒏
Fig. 3. (Left) Schematic of the Energy Management System (EMS) for the solar powered EV parking garage. 𝑵𝒄𝒉
EV connected to each
𝒄 of the total 𝑵𝒄
𝒓(𝒖𝒑) 𝒓(𝒅𝒏)
𝒄𝒉
𝒄𝒐𝒏𝒏
EV-PV charger can be simultaneously charged or discharged, where 𝑵𝒄 ≤ 𝑵𝒄
(Right) offer of reserve power capacity 𝒙𝒕,𝒗 , 𝒙𝒕,𝒗 for up and down
regulation during charging (CH) and discharging (V2G) of EV.
number of EVs is a cost effective strategy [22], [23]. These
EVSE could offer simultaneous charging of several EV or
charge one EV at a time. The scheduling of the
multiplexing is formulated for the first time in this work
using an MILP formulation.
Implementation of the optimization using C# code, SQL
server and Microsoft solver foundation making it ready for
hardware implementation.
D. Structure of the paper
Section II describes the layout and parameters of the EMS
and the EV-PV car park infrastructure. In section III, the
MILP formulation of the EMS is explained and the
parameters, constraints and objective function are elaborated.
Section IV uses PV generation data and market data for
Austin, TX to estimate the optimized cost of charging an EV
fleet from PV. The costs are compared to immediate and
average rate charging policies to evaluate the cost reduction.
II. PRELIMINARIES AND INPUTS
A. Layout of the EMS
The schematic of the EV-PV charger and the EMS used by
the ESCo to optimize the EV charging is shown in Fig. 3.
1) EV and user input
If v is the index of the EV and total number of EVs is V,
each EV arrives at the car park with a state of charge 𝐵𝑣𝑎 at
time 𝑇𝑣𝑎 and is parked at one of the several EV-PV chargers.
The EV owners provide the information to the EMS about
their expected departure time 𝑇𝑣𝑑 and charging energy demand
𝑑𝑣 . This means that the departure SOC of the vehicle 𝐵𝑣𝑑 is:
𝐵𝑣𝑑 = 𝐵𝑣𝑎 + 𝑑𝑣
(2)
If the required SOC is not reached by the departure time,
the EV owner will be compensated by the ESCo at the rate of
Cpv $/kWh. The users can enter the maximum and minimum
allowed SOC of the EV 𝐵𝑣𝑚𝑖𝑛 , 𝐵𝑣𝑚𝑎𝑥 and the maximum
charging and discharging power 𝑥𝑣𝑢𝑏 , 𝑥𝑣𝑙𝑏 respectively. By
setting 𝑥𝑣𝑙𝑏 to a non-zero value, the users can choose to
participate in V2G services. The efficiency of the EV battery
for charging and discharging is 𝜂𝑣𝑐ℎ , 𝜂𝑣𝑣2𝑥 and is either
obtained from the EV or stored in a database within the EMS
for different EV models.
2) EV-PV charger
The ‘EV-PV charger’ as the term used here is an integrated
power converter that consists of three ports to connect to the
EVs, PV and the AC grid, as shown in Fig. 3 [14], [22]. Each
EV-PV charger is connected to a PV array of rated power
𝑃𝑐𝑃𝑉𝑟 via a maximum power point tracking (MPPT) DC/DC
converter [24]. The output of the DC/DC PV converter is
connected to an internal DC-link. The DC-link is connected to
the grid via a DC/AC inverter of rated power 𝑃𝑐𝑐𝑜𝑛𝑣 , such that
𝑃𝑐𝑃𝑉𝑟 ≤ 𝑃𝑐𝑐𝑜𝑛𝑣 . There are 𝑁𝑐𝑐ℎ number of isolated DC/DC
converter for EV charging that are connected to the DC-link
and each have a rated power 𝑃𝑐𝐸𝑉𝑟 . All power exchanges
between any of the three ports namely PV, EV, grid happens
via the DC-link.
This integrated converter provides several benefits
compared to using separate converters for PV and EV
connected over the 50Hz AC grid. First, direct interconnection
of the PV and EV over a DC-link is more efficient than an AC
interconnection [25], [26]. Second, the integrated converter
requires one common inverter to the AC grid instead of
separate inverters for PV and EV. This reduces the component
count and cost of the converter [22]. Third, by making the
isolated DC/DC converter for the EV bidirectional, the EV can
4
now offer V2G services via the integrated converter.
Due to the long parking times of EVs at the workplace, it is
economical to use a single EVSE that can be multiplexed to
several EVs, with the possibility to charge the EVs
simultaneously or sequentially as shown in Fig. 3. Therefore,
𝑁𝑐𝑐𝑜𝑛𝑛 EVs can be connected to each EV-PV charger via DC
isolators. The binary variable 𝐾𝑣,𝑐 = 1 indicates the physical
connection of vth EV with cth charger and a zero value
indicates otherwise.
Each EV-PV charger has 𝑁𝑐𝑐ℎ number of isolated DC/DC
converters, where 𝑁𝑐𝑐ℎ ≤ 𝑁𝑐𝑐𝑜𝑛𝑛 . As per the EV charging
standards [27], each EV must be connected to separate power
converter and isolated from all power sources. This means that
𝑁𝑐𝑐ℎ of the total 𝑁𝑐𝑐𝑜𝑛𝑛 EVs connected to each EV-PV charger
can be simultaneously charged or discharged. In the simple
case where 𝑁𝑐𝑐ℎ = 1, 𝑁𝑐𝑐𝑜𝑛𝑛 =2 and 𝑃𝑐𝑐𝑜𝑛𝑣 = 𝑃𝑐𝐸𝑉𝑟 , two EVs are
connected to one EV-PV charger and one out of the two can
(dis)charge at any time up to a power of 𝑃𝑐𝑐𝑜𝑛𝑣 . The binary
𝑐
variable 𝑎𝑡,𝑣
indicates which of the 𝑁𝑐𝑐𝑜𝑛𝑛 EV connected to an
EV-PV charger is actively (dis)charging at time t.
𝑐𝑜𝑛𝑛
∀𝑐
∑𝑣=𝑉
(3)
𝑣=1 𝐾𝑣,𝑐 ≤ 𝑁𝑐
𝑣=𝑉
𝑐
∀𝑐
(4)
∑𝑣=1 𝐾𝑣,𝑐 𝑎𝑡,𝑣 ≤ 𝑁𝑐𝑐ℎ
𝑓𝑒𝑒𝑑
𝑑𝑟𝑎𝑤
Each EV-PV charger feeds 𝑃𝑡,𝑐 or draws 𝑃𝑡,𝑐
power
from the EV car park as determined by the EMS. Different
EV-PV chargers can exchange power within the car park and
these are ‘intra-park’ power exchanges. When the net ‘intrapark’ energy exchanges are non-zero, the EV park imports or
𝑔(𝑖𝑚𝑝)
exports power with the external grid referred to as 𝑃𝑡
,
𝑔(𝑒𝑥𝑝)
𝑃𝑡
respectively.
B. Trading energy and reserves in the energy market
The ESCo uses the EMS to control the solar powered EV
car park for energy trading with the grid. Since 𝑃𝑡𝑔(𝑖𝑚𝑝) , 𝑃𝑡𝑔(𝑒𝑥𝑝)
are small relative to the power traded in the market, the ESCo
is a price taker and does not influence the market clearing
prices. It uses the market settlement point prices 𝑝𝑡𝑒(𝑏𝑢𝑦) ,
𝑒(𝑠𝑒𝑙𝑙)
𝑝𝑡
at time instant t for buying and selling power; and
reserve capacity prices 𝑝𝑡𝑟(𝑢𝑝) , 𝑝𝑡𝑟(𝑑𝑛) for up regulation and
down regulation for offer of ancillary services. The prices are
used as inputs to the EMS for trading (buying and selling)
energy between the car park and the grid. Markets like the
Electric Reliability Council of Texas (ERCOT) provide
different prices for offering capacity reserves for up and down
regulation (𝑝𝑡𝑟(𝑢𝑝) ≠𝑝𝑡𝑟(𝑑𝑛) ). However, other US markets such as
PJM trade up and down regulation as a single product. In
order to make the EMS flexible and work with both type of
markets, it is designed to take different inputs for 𝑝𝑡𝑟(𝑢𝑝) and
𝑟(𝑑𝑛)
𝑝𝑡
and allow for a requirement that up and down regulation
quantities could be equal.
The amount of reserves offered by the EV depends on
whether the user enables V2G option or not, i.e. if 𝑥𝑣𝑙𝑏 =0 or
not. When an EV is connected to a bidirectional charger and
𝑥𝑣𝑙𝑏 ≠0, even an idle EV that is not charging can offer down
regulation and up regulation up to 𝑥𝑣𝑢𝑏 and 𝑥𝑣𝑙𝑏 respectively.
With a unidirectional charger, an idle EV that is not charging
can only offer down regulation up to 𝑥𝑣𝑢𝑏 .
Power generated by PV panels can be ramped down by
moving out of the maximum power point of the PV array. This
can be achieved by controlling the DC/DC converter in the
𝑃𝑉
EV-PV charger that is connected to the PV array. If 𝑃𝑡,𝑐
is the
th
power generated by the PV at time t at the c charger, this
power can also be offered for down regulation services.
C. Receding horizon model predictive control
There are two sources of variability in the EV-PV system.
The first is the diurnal and seasonal variation in PV generation
due to changes in weather. The EMS uses solar forecast
information as an input to predict the PV variation. 𝑃𝑡𝑃𝑉(𝑓𝑐) is
power generation forecast for an optimally orientated 1kWp
PV array at the car park location with a maximum uncertainty
𝑓𝑐
in forecast of 𝑦𝑃𝑉 . The second is the variation in the arrival
and departure patterns of the EV user and the EV parameters
like charging powers limits, efficiency of the battery and SOC.
The EMS is implemented as a receding horizon model
predictive control with a time step ∆𝑇 to manage these two
variations. The horizon for the model is from 00:00AM to
23:59 PM at midnight. This means for every ∆𝑇 time, the
EMS gathers all the inputs in real time, performs the
optimization and plans the EV charging for the rest of the day.
This means the model inaccuracies with respect to PV forecast
or SOC estimation will be corrected by the EMS for every ∆𝑇.
III. MILP FORMULATION
This section describes the objective function and constraints
for the MILP formulation of the EMS. It is important to note
that all optimization variables considered are positive.
A. Acceptance criteria
When an EV arrives at the EV car park, it is connected to
one of the C number of EV-PV chargers. As mentioned
earlier, each EV-PV charger can have up to 𝑁𝑐𝑐𝑜𝑛𝑛 number of
EV connected to it. The user links to the EMS and the EMS
instructs the user on which EV-PV charger he/she must
connect to, based on two ‘acceptance criteria’. The first
criteria is that the energy demand 𝑑𝑣 and parking time,
(𝑇𝑣𝑑 − 𝑇𝑣𝑎 ) of all the EVs connected to one EV-PV charger
must be such that it is able to meet the energy demand within
the stipulated time and within the power limits of the charger,
(5). The second criteria is that the arrival SOC of the vehicle
must be above the minimum SOC as set by the user (6).
𝑉
𝐶
∑ ∑ 𝐾𝑣,𝑐
𝑣=1 𝑐=1
𝑑𝑣
≤ 𝑀𝑖𝑛. {𝑁𝑐𝑐ℎ 𝑃𝑐𝐸𝑉𝑟 , 𝑃𝑐𝑐𝑜𝑛𝑣 }
𝑇𝑣𝑑 − 𝑇𝑣𝑎
𝐵𝑣𝑚𝑖𝑛 ≤ 𝐵𝑣𝑎
∀ 𝑣, 𝑐
(5)
∀𝑣
(6)
B. Constraints: EV and user inputs
𝑒+
The EMS controls the charging power 𝑥𝑡,𝑣
and discharging
𝑒−
power 𝑥𝑡,𝑣 , up and down regulation reserve capacity
𝑟(𝑢𝑝)
𝑟(𝑑𝑛)
𝑥𝑡,𝑣 , 𝑥𝑡,𝑣 of each EV and the power extracted from the PV
𝑃𝑉
system 𝑃𝑡,𝑐
of each charger at time t. Equations (7) and (8) are
5
used to set the charging power of the EV to zero before the
arrival (𝑡 < 𝑇𝑣𝑎 ) and after the departure of the EV (𝑡 ≥ 𝑇𝑣𝑑 ).
𝑐
The binary variable 𝑎𝑡,𝑣
indicates if the EV is connected to
the isolated DC/DC converter for charging/discharging and
can offer regulation services or not. Since an EV cannot
simultaneously charge and discharge, a second binary variable
𝑐ℎ_𝑣2𝑥
𝑎𝑡,𝑣
is used to ensure that only one of the two variables
𝑐ℎ_𝑣2𝑥
𝑒−
𝑒+
𝑥𝑡,𝑣 , 𝑥𝑡,𝑣 has a non-zero value for a given t. 𝑎𝑡,𝑣
is set to 1
𝑒−
𝑒+
for charging and to 0 for V2G. 𝑥𝑡,𝑣 , 𝑥𝑡,𝑣 have to be within the
power limits of the power converter 𝑃𝑐𝐸𝑉𝑟 and the charging
and discharging power limits 𝑥𝑣𝑢𝑏 , 𝑥𝑣𝑙𝑏 as set by the EV
respectively, as shown in equations (9)-(14).
The maximum charging and discharging powers are also
dependent on the SOC of the EV battery as shown in (15) and
(16). For example, fast charging of EV battery cannot be done
beyond 80% SOC of the battery [28]. Here it is assumed that
the maximum charging power linearly reduces from 𝑥𝑣𝑢𝑏 to
zero when the battery is charged beyond 80% SOC till 100%
(𝑆𝑐ℎ =0.9). Similarly the maximum discharging power reduces
linearly from 𝑥𝑣𝑙𝑏 to zero when the battery is discharged below
10% SOC till 0% (𝑆𝑣2𝑥 =0.1). Even though the dependence of
battery power on the SOC is non-linear, this is not considered
here as it is beyond the scope of the paper and would prevent
us from casting the problem into an MLIP formulantion.
𝑟(𝑢𝑝) 𝑟(𝑑𝑛) 𝑐
𝑒−
𝑒+
𝑥𝑡,𝑣
, 𝑥𝑡,𝑣
, 𝑥𝑡,𝑣 , 𝑥𝑡,𝑣 , 𝑎𝑡,𝑣
=0
(7)
∀ 𝑡 < 𝑇𝑣𝑎
𝑟(𝑢𝑝)
𝑒−
𝑒+
𝑥𝑡,𝑣
, 𝑥𝑡,𝑣
, 𝑥𝑡,𝑣
𝑟(𝑑𝑛)
, 𝑥𝑡,𝑣
𝑐
, 𝑎𝑡,𝑣
=0
𝑒+
𝑐
𝑥𝑡,𝑣
≤ 𝑥𝑣𝑢𝑏 (𝑎𝑡,𝑣
)
𝑒+
𝑐ℎ−𝑣2𝑥
𝑢𝑏
𝑥𝑡,𝑣 ≤ 𝑥𝑣 (𝑎𝑡,𝑣
)
𝑒−
𝑐
𝑙𝑏
𝑥𝑡,𝑣 ≤ −𝑥𝑣 (𝑎𝑡,𝑣 )
𝑒−
𝑐ℎ−𝑣2𝑥
𝑥𝑡,𝑣
≤ −𝑥𝑣𝑙𝑏 (1 − 𝑎𝑡,𝑣
)
𝑒−
𝑒+
𝐸𝑉𝑟
𝑥𝑡,𝑣 , 𝑥𝑡,𝑣 ≤ 𝑃𝑐
𝑐
𝑎𝑡,𝑣
,
𝑐ℎ_𝑣2𝑥
𝑎𝑡,𝑣
,
𝑢𝑏
−𝑥𝑣
𝑑_𝑓
𝑎𝑡,𝑐
∈ {0,1}
∀ 𝑡 ≥ 𝑇𝑣𝑑
(8)
∀ 𝑡, 𝑣
∀ 𝑡, 𝑣
∀ 𝑡, 𝑣
∀ 𝑡, 𝑣
∀ 𝐾𝑣,𝑐 =1
(9)
(10)
(11)
(12)
(13)
∀ 𝑡, 𝑐, 𝑣
(14)
𝐵𝑡,𝑣
(
− 1)
(15)
∀ 𝑡, 𝑣
(1 − 𝑆𝑐ℎ ) 𝐵𝑣𝑚𝑎𝑥
−𝑥𝑣𝑙𝑏 𝐵𝑡,𝑣
𝑒−
𝑥𝑡,𝑣
≤
(
)
(16)
∀ 𝑡, 𝑣
𝑆𝑣2𝑥 𝐵𝑣𝑚𝑎𝑥
Equations (17)-(22) are used to set the initial SOC of the
EV battery and estimate the SOC of the battery 𝐵𝑡,𝑣 based on
the charging and discharging efficiency 𝜂𝑣𝑐ℎ , 𝜂𝑣𝑣2𝑥 and power
𝑒+
𝑒−
𝑥𝑡,𝑣
, 𝑥𝑡,𝑣
respectively. The EMS restricts the SOC to be
within the limits 𝐵𝑣𝑚𝑖𝑛 , 𝐵𝑣𝑚𝑎𝑥 as set by the EV and/or user. It is
assumed that the net energy delivered/absorbed by the EV
over one time period due to offer of reserves is zero [12], [13].
𝑟(𝑢𝑝) 𝑟(𝑑𝑛)
Hence, 𝑥𝑡,𝑣
, 𝑥𝑡,𝑣 do not appear in (22) for SOC estimation.
𝐵𝑡,𝑣 = 0
(17)
∀ 𝑡 < 𝑇𝑣𝑎
𝑎
𝑎
𝐵𝑡,𝑣 = 𝐵𝑣
(18)
∀ 𝑡 = 𝑇𝑣
𝐵𝑡,𝑣 ≤ 𝑑𝑣 + 𝐵𝑣𝑎
(19)
∀ 𝑡 = 𝑇𝑣𝑑
𝑚𝑖𝑛
𝑎
𝐵𝑡,𝑣 ≥ 𝐵𝑣
(20)
∀ t ≥ 𝑇𝑣
𝐵𝑡,𝑣 ≤ 𝐵𝑣𝑚𝑎𝑥
(21)
∀ t ≥ 𝑇𝑣𝑎
𝑒−
𝑥𝑡,𝑣
𝑒+ 𝑐ℎ
𝐵𝑡+1,𝑣 = 𝐵𝑡,𝑣 + ∆𝑇 (𝑥𝑡,𝑣
𝜂𝑣 − 𝑣2𝑥 )
(22)
∀ 𝑡, 𝑣
𝜂𝑣
𝑒+
𝑥𝑡,𝑣
≤
C. Constraints: EV–PV charger and car park
Under normal operation, the EMS extracts maximum power
from the PV array using MPPT as shown in right side of
equation (23). The PV power is dependent on the scaling
factor 𝐾𝑐𝑃𝑉 which scales the installation characteristics (e.g.
azimuth, tilt, module parameters) of the PV array connected to
the charger c with respect to the 1kWp reference array used
𝑃𝑉(𝑓𝑐)
for the forecast data 𝑃𝑡
. The EMS implements PV
curtailment if it is uneconomical to draw PV power or if there
are distribution network constraints for feeding to the grid.
𝑃𝑉
This means that the actual PV power extracted 𝑃𝑡,𝑐
can be
lower than the MPPT power of the array, as shown in (23).
The DC-link is used for power exchanges between the three
ports of the converter and (24) is the power balance equation
for the EV-PV converter. It is assumed that each of the power
converters within the EV-PV charger operates with an
𝑓𝑒𝑒𝑑
𝑑𝑟𝑎𝑤
efficiency 𝜂𝑐𝑐𝑜𝑛𝑣 . 𝑃𝑡,𝑐
, 𝑃𝑡,𝑐 are limited by the power limit
𝑑_𝑓
of the inverter port 𝑃𝑐𝑐𝑜𝑛𝑣 . The binary variable 𝑎𝑡,𝑐 is used to
ensure that only one of the two variables has a finite value for
a given t as shown in (25)-(26).
𝑃𝑉(𝑓𝑐)
𝑃𝑉
𝑃𝑡,𝑐
≤ 𝐾𝑐𝑃𝑉 𝑃𝑐𝑃𝑉𝑟 𝑃𝑡
(23)
∀ 𝑡, 𝑐
𝑣=𝑉
𝑃𝑉
𝑑𝑟𝑎𝑤
{𝑃𝑡,𝑐
+ 𝑃𝑡,𝑐
+∑
𝑓𝑒𝑒𝑑
= {𝑃𝑡,𝑐
𝑣=1
𝑣=𝑉
+∑
𝑣=1
𝑒−
(𝐾𝑣,𝑐 𝑥𝑡,𝑣
)} η𝑐𝑜𝑛𝑣
𝑐
∀ 𝑡, 𝑐, 𝑣 (24)
𝑒+
(𝐾𝑣,𝑐 𝑥𝑡,𝑣
)} /η𝑐𝑜𝑛𝑣
𝑐
𝑑_𝑓
𝑑𝑟𝑎𝑤
𝑃𝑡,𝑐
≤ 𝑃𝑐𝑐𝑜𝑛𝑣 (𝑎𝑡,𝑐 )
𝑓𝑒𝑒𝑑
𝑃𝑡,𝑐
∀ 𝑡, 𝑐
(25)
𝑑_𝑓
𝑎𝑡,𝑐 )
𝑃𝑐𝑐𝑜𝑛𝑣 (1
≤
−
(26)
∀ 𝑡, 𝑐
The intra car-park power exchanges between different EVPV chargers are related to the power exchanged with the
external grid 𝑃𝑡𝑔(𝑖𝑚𝑝) , 𝑃𝑡𝑔(𝑒𝑥𝑝) using (27). They should be
within the distribution network capacity 𝑃𝑡𝐷𝑁+ , 𝑃𝑡𝐷𝑁− as shown
in (28)-(29). Both 𝑃𝑡𝑔(𝑖𝑚𝑝) , 𝑃𝑡𝑔(𝑒𝑥𝑝) do not have finite values at
the same time because of the way the objective function is
𝑒(𝑏𝑢𝑦)
formulated and because 𝑝𝑡
≥ 𝑝𝑡𝑒(𝑠𝑒𝑙𝑙) at all times
𝑐=𝐶
∑
𝑐=1
𝑓𝑒𝑒𝑑
𝑑𝑟𝑎𝑤
(𝑃𝑡,𝑐
− 𝑃𝑡,𝑐
𝑔(𝑖𝑚𝑝)
) = 𝑃𝑡
𝑔(𝑒𝑥𝑝)
− 𝑃𝑡
∀ 𝑡
(27)
𝑔(𝑖𝑚𝑝)
≤ 𝑃𝑡𝐷𝑁+
∀ 𝑡
(28)
𝑔(𝑒𝑥𝑝)
𝑃𝑡
𝑃𝑡𝐷𝑁−
∀ 𝑡
(29)
𝑃𝑡
≤
𝑟(𝑢𝑝) 𝑟(𝑑𝑛)
𝑥𝑡,𝑣 , 𝑥𝑡,𝑣
Finally, each of the EV offers reserve capacity
for up and down regulation. From an EV perspective, the
regulation power offered is restricted by the power limitations
of the EV (𝑥𝑣𝑢𝑏 , 𝑥𝑣𝑙𝑏 ) and the EV charger port 𝑃𝑐𝐸𝑉𝑟 as shown
in Fig. 3. From the EV-PV charger perspective, the regulation
power offered depends on the power rating of the inverter port
𝑓𝑒𝑒𝑑
𝑑𝑟𝑎𝑤
𝑃𝑐𝑐𝑜𝑛𝑣 and the power exchanged with the grid 𝑃𝑡,𝑐
, 𝑃𝑡,𝑐 .
This is summarized in equations (30)-(35). While asymmetric
𝑟(𝑢𝑝)
𝑟(𝑑𝑛)
reserve offers are assumed here (𝑥𝑡,𝑣
≠𝑥𝑡,𝑣 ), symmetric
𝑟(𝑢𝑝)
𝑟(𝑑𝑛)
reserves can be achieved by adding 𝑥𝑡,𝑣
=𝑥𝑡,𝑣
to the
constraints.
𝑣=𝑉
∑
𝑣=1
𝑟(𝑢𝑝)
𝐾𝑣,𝑐 𝑥𝑡,𝑣
𝑓𝑒𝑒𝑑
+ 𝑃𝑡,𝑐
≤ 𝑃𝑐𝑐𝑜𝑛𝑣
∀ 𝑡, 𝑐, 𝑣
(30)
6
𝑣=𝑉
∑
𝑣=1
𝑟(𝑑𝑛)
𝐾𝑣,𝑐 𝑥𝑡,𝑣
𝑟(𝑢𝑝)
𝑒−
𝑥𝑡,𝑣
+ 𝑥𝑡,𝑣
𝑑𝑟𝑎𝑤
+ 𝑃𝑡,𝑐
≤ 𝑃𝑐𝑐𝑜𝑛𝑣
𝑐
≤ 𝑃𝑐𝐸𝑉𝑟 (𝑎𝑡,𝑣
)
𝑟(𝑢𝑝)
+ 𝑥𝑡,𝑣 ≤ 𝑥𝑣𝑢𝑏
𝑟(𝑑𝑛)
𝑒+
𝑐
𝑥𝑡,𝑣
+ 𝑥𝑡,𝑣 ≤ 𝑃𝑐𝐸𝑉𝑟 (𝑎𝑡,𝑣
)
𝑟(𝑑𝑛)
𝑒+
𝑥𝑡,𝑣
+ 𝑥𝑡,𝑣 ≤ −𝑥𝑣𝑙𝑏
𝑒−
𝑥𝑡,𝑣
∀ 𝑡, 𝑐, 𝑣
(31)
∀ 𝐾𝑣,𝑐 =1
(32)
∀ 𝑡, 𝑣
(33)
∀ 𝐾𝑣,𝑐 =1
(34)
∀ 𝑡, 𝑣
(35)
D. Objective function
𝑝
𝑀𝑖𝑛. 𝐶 𝑜𝑝𝑡 =
(𝐵𝑣𝑎 + 𝑑𝑣 − 𝐵 𝑇𝑣𝑑 ,𝑣 )𝐶𝑣
𝑇
𝑔(𝑖𝑚𝑝) 𝑒(𝑏𝑢𝑦)
𝑝𝑡
+∆𝑇 ∑ ( 𝑃𝑡
𝑡=1
𝑇
(36)
𝑔(𝑒𝑥𝑝) 𝑒(𝑠𝑒𝑙𝑙)
𝑝𝑡
)
− 𝑃𝑡
𝐶
𝑉
𝑟(𝑢𝑝) 𝑟(𝑢𝑝)
𝑝𝑡
𝑓𝑐
− ∆𝑇 (1 − 𝑦𝑃𝑉 )(𝜂𝑐𝑐𝑜𝑛𝑣 )2 ∑ ∑ ∑ 𝐾𝑣,𝑐 { 𝑥𝑡,𝑣
𝑇
𝑡=1 𝑐=1 𝑣=1
𝑉
𝑒−
+∆𝑇 ∑ ∑ 𝑥𝑡,𝑣
𝑡=1 𝑣=1
𝐶
𝑉2𝑋
+
𝑇
𝑟(𝑑𝑛) 𝑟(𝑑𝑛)
𝑝𝑡
}
+ 𝑥𝑡,𝑣
𝐶
𝑃𝑉(𝑓𝑐)
∆𝑇 ∑ ∑ 𝑃𝑐𝑃𝑉𝑟 𝑃𝑡
𝐶 𝑃𝑉
𝑡=1 𝑐=1
The objective function is to minimize the total costs 𝐶 𝑜𝑝𝑡 of
EV charging, feeding PV power and offering reserves. The
formulation is such that the 𝐶 𝑜𝑝𝑡 can be positive or negative. It
has five components respectively:
The penalty to be paid to the user if the energy demand 𝑑𝑣
𝑝
is not met by the departure time 𝑇𝑣𝑑 . 𝐶𝑣 is EV user specific
and the penalty can be different for each user based on EV
battery size, tariff policy and customer ‘loyalty’ program.
The cost of buying and selling energy from the grid based
on the settlement point prices 𝑝𝑡𝑒(𝑏𝑢𝑦) , 𝑝𝑡𝑒(𝑠𝑒𝑙𝑙) . The market
dynamics will ensure that 𝑝𝑡𝑒(𝑏𝑢𝑦) ≥ 𝑝𝑡𝑒(𝑠𝑒𝑙𝑙)
Income 𝑆 𝑎𝑠 obtained from offering reserve capacity
𝑟(𝑢𝑝) 𝑟(𝑑𝑛)
𝑥𝑡,𝑣 , 𝑥𝑡,𝑣 to the ISO. (𝜂𝑐𝑐𝑜𝑛𝑣 )2 indicates the energy losses
in the two step conversion between the EV and grid port of
the EV-PV charger. Since the reserves offered to the grid
have to be guaranteed and the uncertainty in the PV
𝑓𝑐
𝑓𝑐
forecast is 𝑦𝑃𝑉
, only a fraction (1 − 𝑦𝑃𝑉
) of the available
reserves are guaranteed and sold to the ISO.
EV battery capacity degrades due to the additional life
cycles caused by the V2G operation and EV user is
compensated for this loss. Typical value of
𝐶 𝑉2𝑋 =4.2¢/kWh based on analysis in [29], [30].
PV power that is used to charge the EV need not always be
free of cost. If the PV is installed by a third-party, it can be
obtained at a pre-determined contractual cost of 𝐶 𝑃𝑉 .
E. MILP implementation
The EMS engine is implemented in C# leveraging
Microsoft Solver Foundation for algebraic modeling in
Optimization Modeling Language (OML). MS SQL Server
database is used to warehouse system inputs, namely the EV,
charger, network and market data as well as the decision
outputs that is sent to the EV-PVs in the field. The MILP
formulation is solved using branch-and-bound (B&B)
algorithm using ‘LPsolve’ open source solver. One of the main
advantages of the B&B algorithm is that, given enough
computation time, it guarantees global optimality despite the
non-convex nature of the problem. The EV-PV chargers will
be interfaced with the output database to implement the
optimal power profiles.
IV. SIMULATION RESULTS
Simulations are performed to test the validity of the
proposed MILP formulation and to quantify the reduction in
costs of EV charging from PV with respect to AR and IMM.
A. Simulation parameters
Settlement point prices (SPP) and prices for reserve
capacity (REGUP, REGDN) are obtained from the ERCOT
day-ahead market (DAM) for Austin, Texas for 2014 for load
zone LZ_AEN. These are wholesale energy prices with a data
resolution of 1hr. Since separate values for 𝑝𝑡𝑒(𝑠𝑒𝑙𝑙) was not
available, it is assumed that 𝑝𝑡𝑒(𝑠𝑒𝑙𝑙) =0.98*𝑝𝑡𝑒(𝑏𝑢𝑦) .
For 2014, the largest values observed for 𝑝𝑡𝑒(𝑏𝑢𝑦) , 𝑝𝑡𝑟(𝑢𝑝) ,
𝑟(𝑑𝑛)
𝑝𝑡
were 136.47¢/kWh, 499.9¢/kWh and 31¢/kWh
respectively while the average values were 3.9¢/kWh,
1.25¢/kWh, 0.973¢/kWh. It can be clearly seen than energy
prices are normally much higher than regulation prices, but
there are several instances where it is otherwise.
The PV generation data is obtained from the Pecan Street
Project database for a house in the Mueller neighborhood with
an 11.1 kW PV system [31]. The data resolution is 1min. The
power output is scaled down for a 1kW system for use as
𝑃𝑉(𝑓𝑐)
𝑃𝑡
with 𝑦 𝑃𝑉(𝑓𝑐) =10%. It is assumed that the PV
installation at the car park is owned by the workplace and
hence 𝐶 𝑃𝑉 =0
The EV arrival and departure times and SOC requirements
are listed in TABLE I for 6 EVs. The EV data imitates the
capacity of a Tesla Model S, BMW i3 and a Nissan Leaf. For
all the EVs, 𝐵𝑣𝑚𝑖𝑛 =5kWh, 𝑥𝑣𝑢𝑏 =50kW, 𝑥𝑣𝑙𝑏 =(-10kW),
𝜂𝑣𝑐ℎ = 𝜂𝑣𝑣2𝑥 =0.95, Cpv=1$/kWh, 𝐶 𝑉2𝑋 = 4.2¢/kWh. The penalty
Cpv is approximately 25 times the average wholesale ERCOT
electricity price of 3.9¢/kWh.
There are 4 EV-PV chargers and the TABLE I shows the
connections of the 6 EVs to the 4 chargers in ‘Chr conn.’.
10kWp PV is connected to each of chargers 1,2,4 and no PV is
connected to charger 3. Chargers 1,4 have two EV connected
to them. 𝑁𝑐𝑐ℎ =1 for all chargers, which means that only one of
the two EV can be charged at a time for chargers 1,4. The
following parameters are used: 𝜂𝑐𝑐𝑜𝑛𝑣 =0.96, 𝑃𝑐𝐸𝑉𝑟 =𝑃𝑐𝑐𝑜𝑛𝑣 =10
kW. ∆𝑇=15min for all simulation. 𝑃𝑡𝐷𝑁+ = 𝑃𝑡𝐷𝑁+ =40kW.
B. Simulation results
1) Average rate and immediate charging
The net costs of EV charging and PV sales for average rate
TABLE I
EV AND EV-PV CHARGER DATA
v
𝑇𝑣𝑎
1
2
3
4
5
6
900
830
930
900
830
930
𝑇𝑣𝑑
d
𝐵𝑣𝑎
𝐵𝑣𝑚𝑎𝑥
Chr
conn.
𝑃𝑐𝐸𝑉𝑟
𝑃𝑐𝑐𝑜𝑛𝑣
1700
1630
1730
1700
1630
1730
40
30
10
40
30
10
(kWh)
20
20
5
20
20
5
85
60
24
85
60
24
1
1
2
3
4
4
(kW)
10
10
10
10
10
10
(h)
7
𝐶 𝑎𝑟 and immediate charging 𝐶 𝑖𝑚𝑚 is estimated using (1), (37).
𝐶 𝑎𝑟 , 𝐶 𝑖𝑚𝑚 = 𝐶 𝑒𝑣 − 𝑆𝑃𝑉
𝑇
= ∆𝑇 ∑
∆𝑇 ∑
𝑇
𝑡=1
𝑡=1
∑
𝑣=𝑉
∑
𝐶
𝑐=1
𝑣=1
2
𝑒+ 𝑒(𝑏𝑢𝑦)
(𝜂𝑐𝑜𝑛𝑣
) 𝑥𝑡,𝑣
𝑝𝑡
−
𝑐
2
𝑃𝑉(𝑓𝑐)
(𝜂𝑐𝑜𝑛𝑣
) 𝑃𝑐𝑃𝑉𝑟 𝑃𝑐
𝑐
(37)
(𝑝𝑡𝑒(𝑠𝑒𝑙𝑙) − 𝐶 𝑃𝑉 )
where 𝐶 𝑒𝑣 is the EV charging costs and 𝑆 𝑃𝑉 the revenues
𝑒+
𝑒+
from PV sales. For AR, 𝑥𝑡,𝑣
=𝑥𝑣𝑒(𝑎𝑟) and for IMM, 𝑥𝑡,𝑣
=𝑃𝑐𝐸𝑉𝑟 .
With AR and IMM, there is no provision to provide V2G,
regulation services or multiplexing of chargers due to the
absence of communication with an EMS. The peak power for
the car park for IMM would be 60kW and 20kW for AR
charging for 6EV based on (1).
Fig. 4 and TABLE II shows the net costs 𝐶 𝑎𝑟 , 𝐶 𝑖𝑚𝑚
estimated for 2014 with the corresponding mean and standard
deviation (SD). Three vital observations can be made. First,
there is a large variation in net costs, ranging between {1.35$,
24.17$} and {-19.58$, 40.43$} for AR and IMM respectively.
This is mainly due to the varying energy prices in ERCOT.
The costs went negative for IMM on certain days indicating
that the ESCo got paid by the ISO! It must be remembered
that PV sales 𝑆 𝑃𝑉 for both strategies is the same as shown in
TABLE II. Second, IMM charging was found to be better than
AR in summer and vice versa in winter, with IMM charging
net costs being cheaper than AR for 233 days. Third, the
average net cost per day for 2014 for AR and IMM was found
to be 3.79$ and 2.90$, with IMM being cheaper than AR by
31.7%. This is because, EVs are charged in morning for IMM
when ERCOT prices are generally lower when compared to
prices in the afternoon.
2) Optimized charging costs
Using the MILP formulation of the optimized charging
(OPT) described in section III, the net costs 𝐶 𝑜𝑝𝑡 are
determined for each day of 2014 and shown in Fig. 4 and
TABLE II. The benefits of the MILP optimization can be
clearly seen in the figure, where the optimized net costs are
much lower than IMM and AR. 𝐶 𝑜𝑝𝑡 range is {-42.91$,
11.56$}, which is much lower than IMM and AR. Due to the
large penalty Cpv=1$/kWh, EVs were always charged up to the
required departure SOC.
EV charging costs 𝐶 𝑒𝑣 (not net cost!) are estimated
separately for AR, IMM and OPT and shown in TABLE II. It
can be seen that mean value of 𝐶 𝑒𝑣 is not that different
between IMM and OPT. The reason is that the objective
function is not optimized to reduce charging costs alone but
increase the sale of PV power and reserves as well.
The percentage reduction in costs 𝐶%𝑖𝑚𝑚 , 𝐶%𝐸𝑉−𝑃𝑉 is estimated
based on AR charging costs 𝐶 𝑎𝑟 using (38)-(39) and shown in
Fig. 5. 𝐶 𝑎𝑟 was chosen as a reference as the costs never go
negative and don’t have values close to zero.
𝑖𝑚𝑚
𝐶%
= 100(𝐶 𝑎𝑟 − 𝐶𝑖𝑚𝑚 )/𝐶𝑎𝑟
(38)
𝑜𝑝𝑡
𝑜𝑝𝑡
𝑎𝑟
𝑎𝑟
(39)
𝐶% = 100(𝐶 − 𝐶 )/𝐶
As can be seen, the proposed optimized charging results in a
𝑜𝑝𝑡
cost reduction 𝐶% in the range of 31.74% to 650.81%, with a
mean of 158.63% with respect to AR charging. A reduction of
>100% results in the net cost to be negative. This means that
the EV car park receives money for the EV charging and sale
of PV and reserves rather than having to pay overall.
The large cost reduction is a result of aggregating the multiaspect PV and EV problems into a single MILP formulation.
This results in the sale of PV and V2G power when prices are
high, buying of EV charging power when prices are low and
continuous sale of ancillary services. The current MILP
formulation is such that IMM or AR will be a special case of
optimized charging as dictated by the PV forecast and market
prices. Further, the sharing of a single charger to charge
several EVs results in a reduction of charging infrastructure
cost. While these costs have not be included in the estimate,
they can be up to 15,000$ for 10kW chargers with 𝑁𝑐𝑐𝑜𝑛𝑛 =4.
MILP solve times were in the range of 11.2-17.3s with a
relative MILP gap of 0.015%. The mean solve-time was
13.05s with a standard deviation of 1.09s. A Windows PC
with Intel Xeon 2.4Ghz CPU and 12GB RAM was employed.
It must be kept in mind that even though wholesale DAM
prices and small EV fleet have been used in this simulation,
the formulation is generic to be used with large EV fleet, realtime market (RTM) and retail electricity prices as well.
V. CONCLUSIONS
EV charging from PV can be controlled to achieve several
motives – to take advantage of time of use tariffs, provide
ancillary services or follow the PV production. However, the
common approach is that each of these applications are solved
as separate optimization problems which leads to several EV
charging profiles. This is impractical, as a single EV cannot be
controlled at the same time with different charging profiles.
TABLE II
CHARGING COSTS, PV SALES AND NET COSTS - MEAN, SD ($)
[Mean, SD]
AR
IMM
OPT
4.41, 2.81
4.41, 2.81
𝑆 𝑃𝑉
8.21, 3.21
7.32, 3.87
7.30, 1.92
𝐶 𝑒𝑣
2.90, 42.01
-1.53, 3.92
𝐶 𝑎𝑟 , 𝐶 𝑖𝑚𝑚 , 𝐶 𝑜𝑝𝑡 3.79, 2.13
𝑜𝑝𝑡
31.72, 61.26 158.63, 87.88
𝐶%𝑖𝑚𝑚 , 𝐶% (%)
Fig. 4. Cost of charging the EV fleet by average rate, immediate and the
proposed optimized charging strategy (top); zoomed view (bottom)
Fig. 5. Percentage reduction in the charging cost for the proposed charging
strategy and immediate charging with respect to average rate charging.
8
Hence it is vital to make a single problem formulation that
bundles several applications together so that one optimal EV
charging profile is obtained.
In this paper, an MILP formulation has been proposed for
charging of an EV fleet from PV that has several application
built into one - charging of EV from PV, using time of use
tariffs to sell PV power and charge EV from the grid,
implementation of V2G for grid support, using EV to offer
ancillary services in the form of reserves and considering
distribution network capacity constraints. The scheduling of
the connection of a single EVSE to several EV has been
formulated for the first time in this work. This provides the
ability to use lower capacity EVSE at workplaces resulting in
substantial reduction in the cost of EV infrastructure.
The MILP optimization has been implemented as a receding
horizon model predictive control and operates with a fixed
time period. Using 2014 data from Pecan Street Project and
ERCOT market, simulations were performed for an EV fleet
of six connected to four chargers. The formulation of five
applications into one resulted in large savings in the range of
31.74% to 650.81% with respect to average rate charging. The
MILP formulation is generic and can be adapted to different
energy and ancillary markets, EV types, PV array installations
and EVSE.
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| 3 |
Alternating Direction Graph Matching
D. Khuê Lê-Huu1, 2
1
Nikos Paragios1, 2, 3
CentraleSupélec, Université Paris-Saclay
2
Inria
3
TheraPanacea
arXiv:1611.07583v4 [cs.CV] 23 Feb 2018
{khue.le, nikos.paragios}@centralesupelec.fr
Abstract
Graph matching has been an active research topic in the
computer vision field for the past decades. In the recent
literature, [13] proposed a graduated assignment algorithm
to iteratively solve a series of convex approximations to the
matching problem. In [22], a spectral matching based on
the rank-1 approximation of the affinity matrix (composed
of the potentials) was introduced, which was later improved
in [9] by incorporating affine constraints towards a tighter
relaxation. In [23], an integer projected fixed point algorithm that solves a sequence of first-order Taylor approximations using Hungarian method [18] was proposed, while
in [28] the dual of the matching problem was considered
to obtain a lower-bound on the energy, via dual decomposition. In [6], a random walk variant was used to address
graph matching while [32] factorized the affinity matrix into
smaller matrices, allowing a convex-concave relaxation that
can be solved in a path-following fashion. Their inspiration
was the path-following approach [29] exploiting a more restricted graph matching formulation, known as KoopmansBeckmann’s QAP [17]. Lately, [7] proposed a max-pooling
strategy within the graph matching framework that is very
robust to outliers.
In this paper, we introduce a graph matching method
that can account for constraints of arbitrary order, with arbitrary potential functions. Unlike previous decomposition
approaches that rely on the graph structures, we introduce
a decomposition of the matching constraints. Graph matching is then reformulated as a non-convex non-separable optimization problem that can be split into smaller and mucheasier-to-solve subproblems, by means of the alternating
direction method of multipliers. The proposed framework
is modular, scalable, and can be instantiated into different variants. Two instantiations are studied exploring pairwise and higher-order constraints. Experimental results on
widely adopted benchmarks involving synthetic and real examples demonstrate that the proposed solutions outperform
existing pairwise graph matching methods, and competitive
with the state of the art in higher-order settings.
1. Introduction
The task of finding correspondences between two sets
of visual features has a wide range of applications in computer vision and pattern recognition. This problem can be
effectively solved using graph matching [28], and as a consequence, these methods have been successfully applied to
various vision tasks, such as stereo matching [14], object
recognition and categorization [1, 12], shape matching [1],
surface registration [31], etc.
The general idea of solving feature correspondences via
graph matching is to associate each set of features an attributed graph, where node attributes describe local characteristics, while edge (or hyper-edge) attributes describe
structural relationships. The matching task seeks to minimize an energy (objective) function composed of unary,
pairwise, and potentially higher-order terms. These terms
are called the potentials of the energy function. In pairwise
settings, graph matching can be seen as a quadratic assignment problem (QAP) in general form, known as Lawler’s
QAP [19]. Since QAP is known to be NP-complete [5, 27],
graph matching is also NP-complete [13] and only approximate solutions can be found in polynomial time.
Recently, researchers have proposed higher-order graph
matching models to better incorporate structural similarities and achieve more accurate results [11, 30]. For solving such high-order models, [30] viewed the matching problem as a probabilistic model that is solved using an iterative
successive projection algorithm. The extension of pairwise
methods to deal with higher-order potentials was also considered like for example in [11] through a tensor matching
(extended from [22]), or in [31] through a third-order dual
decomposition (originating from [28]), or in [21] through
a high-order reweighted random walk matching (extension
of [6]). Recently, [26] developed a block coordinate ascent
algorithm for solving third-order graph matching. They
lifted the third-order problem to a fourth-order one which,
after a convexification step, is solved by a sequence of linear
or quadratic assignment problems. Despite the impressive
performance, this method has two limitations: (a) it cannot
be applied to graph matching of arbitrary order other than
third and fourth, and (b) it cannot deal with graph matching
1
where occlusion is allowed on both sides, nor with manyto-many matching.
In this paper, a novel class of algorithms is introduced
for solving graph matching involving constraints with arbitrary order and arbitrary potentials. These algorithms rely
on a decomposition framework using the alternating direction method of multipliers.
The remainder of this paper is organized as follows. Section 2 provides the mathematical foundations of our approach while in Section 3 the general decomposition strategy is proposed along with two instantiations of this framework to a pairwise and higher-order approach. Section 4
presents in-depth experimental validation and comparisons
with competing methods. The last section concludes the paper and presents the perspectives.
2. Mathematical background and notation
Let us first provide the elementary notation as well as the
basic mathematical foundations of our approach. In the first
subsection we will give a brief review of tensor, which will
help us to compactly formulate the graph matching problem, as will be shown in the subsequent subsection.
2.1. Tensor
A real-valued Dth -order tensor F is a multidimensional
array belonging to Rn1 ×n2 ×···×nD (where n1 , n2 , . . . , nD
are positive integers). We denote the elements of F by
Fi1 i2 ...iD where 1 ≤ id ≤ nd for d = 1, 2, . . . , D. Each
dimension of a tensor is called a mode.
We call the multilinear form associated to a tensor F the
function F : Rn1 × Rn2 × · · · × RnD → R defined by
F (x1 , . . . , xD ) =
n1
X
i1 =1
nD
X
···
Fi1 i2 ...iD x1i1 x2i2 · · · xD
iD
(1)
where xd = (xd1 , xd2 , . . . , xdnd ) ∈ Rnd for d = 1, 2, . . . , D.
A tensor can be multiplied by a vector at a specific mode.
Let v = (v1 , v2 , . . . , vnd ) be an nd dimensional vector. The
mode-d product of F and v, denoted by F ⊗d v, is a (D −
1)th -order tensor G of dimensions n1 × · · · × nd−1 × nd+1 ×
· · · × nD defined by
Gi1 ...id−1 id+1 ...iD =
2.2. Graph and hypergraph matching
A matching configuration between two graphs G1 =
(V1 , E1 ) and G2 = (V2 , E2 ) can be represented by a assignn ×n
ment matrix X ∈ {0, 1} 1 2 where n1 = |V1 | , n2 = |V2 |.
An element x(i1 ,i2 ) of X equals 1 if the node i1 ∈ V1 is
matched to the node i2 ∈ V2 , and equals 0 otherwise.
Standard graph matching imposes the one-to-(at most)-one
constraints, i.e. the sum of any row or any column of X
must be ≤ 1. If the elements of X are binary, then X obeys
the hard matching constraints. When X is relaxed to take
real values in [0, 1], X obeys the soft matching constraints.
In this paper we use the following notations: vec(V)
denotes the column-wise vectorized replica of a matrix V;
mat(v) is the n1 × n2 reshaped matrix of an n-dimensional
vector v, where n = n1 n2 ; X ∈ Rn1 ×n2 the assignment
matrix and x = vec(X) ∈ Rn the assignment vector; M∗
(respectively M) is the set of n1 × n2 matrices that obey
the hard (respectively, the soft) matching constraints.
Energy function. Let xi = x(i1 ,i2 ) be an element of X representing the matching of two nodes i1 and i2 . Suppose that
matching these nodes requires a potential Fi1 ∈ R. Simi2
larly, let Fij
denote the potential for matching two edges
3
for matching two (third-order)
(i1 , j1 ), (i2 , j2 ), and Fijk
hyper-edges (i1 , j1 , k1 ), (i2 , j2 , k2 ), and so on. Graph
matching can be expressed as minimizing
X
X
X
2
3
Fi1 xi +
Fij
xi xj +
Fijk
xi xj xk + · · · (5)
i
iD =1
nd
X
Nb
By convention, F d=a xd = F if a > b.
In this work, we are interested in tensors having the same
dimension at every mode, i.e. n1 = n2 = . . . = nD = n.
In the sequel, all tensors are supposed to have this property.
Fi1 ...id ...iD vid .
(2)
ij
ijk
The above function can be re-written more compactly using
tensors. Indeed, let us consider for example the third-order
3
3
has three indices, (Fijk
)1≤i,j,k≤n
potentials. Since Fijk
can be seen as a third-order tensor belonging to Rn×n×n
and its multilinear form (c.f . (1)) is the function
F 3 (x, y, z) =
n X
n X
n
X
3
Fijk
xi yj zk
(6)
i=1 j=1 k=1
id =1
With this definition, it is straightforward to see that the multilinear form (1) can be re-written as
F (x1 , x2 , . . . , xD ) = F ⊗1 x1 ⊗2 x2 · · · ⊗D xD . (3)
Nb
Let us consider for convenience the notation d=a to denote a sequence of products from mode a to mode b:
F
b
O
d=a
defined for x, y, z ∈ Rn . Clearly, the third-order terms
in (5) can be re-written as F 3 (x, x, x). More generally,
Dth -order potentials can be represented by a Dth -order tensor F D and their corresponding terms in the objective function can be re-written as F D (x, x, . . . , x), resulting in the
following reformulation of graph matching.
Problem 1 (Dth -order graph matching) Minimize
xd = F ⊗a xa ⊗a+1 xa+1 · · · ⊗b xb .
(4)
F 1 (x) + F 2 (x, x) + · · · + F D (x, x, . . . , x)
(7)
subject to x ∈ M∗ , where F d (d = 1, . . . , D) is the multilinear form of a tensor F d representing the dth -order potentials.
In the next section, we propose a method to solve the
continuous relaxation of this problem, i.e. minimizing (7)
subject to x ∈ M (soft matching) instead of x ∈ M∗ (hard
matching). The returned continuous solution is discretized
using the usual Hungarian method [18].
3. Alternating direction graph matching
3.1. Overview of ADMM
We briefly describe the (multi-block) alternating direction method of multipliers (ADMM) for solving the following optimization problem:
Minimize φ(x1 , x2 , . . . , xp )
subject to A1 x1 + A2 x2 + · · · + Ap xp = b,
xi ∈ Xi ⊆ Rni
(8)
∀1 ≤ i ≤ p,
where Xi are closed convex sets, Ai ∈ Rm×ni ∀i, b ∈ Rm .
The augmented Lagrangian of the above problem is
Lρ (x1 , x2 , . . . , xp , y) = φ(x1 , x2 , . . . , xp )
!
p
p
X
ρ X
>
Ai x i − b
+y
Ai xi − b +
2 i=1
i=1
2
,
(9)
2
where y is called the Lagrangian multiplier vector and ρ >
0 is called the penalty parameter.
In the sequel, let x[a,b] denote (xa , xa+1 , . . . , xb ) (by
convention, if a > b then x[a,b] is ignored). Standard
ADMM solves problem (8) by iterating:
1. For i = 1, 2, . . . , p, update xi :
k
k
xk+1
= argmin Lρ (xk+1
i
[1,i−1] , x, x[i+1,p] , y ).
(10)
x∈Xi
2. Update y:
yk+1 = yk + ρ
p
X
!
Ai xk+1
i
− bk+1
.
(11)
i=1
The algorithm converges if the following residual converges
to 0 as k → ∞:
k
r =
p
X
i=1
2
Ai xki
k
−b
+
2
p
X
Ai xki
−
2
Ai xik−1 2
.
i=1
(12)
We will discuss the convergence of ADMM in Section 3.4.
3.2. Graph matching decomposition framework
Decomposition is a general approach to solving a problem by breaking it up into smaller ones that can be efficiently addressed separately, and then reassembling the results towards a globally consistent solution of the original
non-decomposed problem [2, 4, 10]. Clearly, the above
ADMM is such a method because it decomposes the large
problem (8) into smaller problems (10).
In computer vision, decomposition methods such as
Dual Decomposition (DD) and ADMM have been applied
to optimizing discrete Markov random fields (MRFs) [15,
16, 20, 25] and to solving graph matching [28]. The main
idea is to decompose the original complex graph into simpler subgraphs and then reassembling the solutions on these
subgraphs using different mechanisms. While in MRF inference, this concept has been proven to be flexible and
powerful, that is far from being the case in graph matching,
due to the hardness of the matching constraints. Indeed,
to deal with these constraints, [28] for example adopted a
strategy that creates subproblems that are also smaller graph
matching problems, which are computationally highly challenging. Moreover, subgradient method has been used to
impose consensus, which is known to have slow rate of
convergence [2]. One can conclude that DD is a very slow
method and works for a limited set of energy models often
associated with small sizes and low to medium geometric
connectivities [28].
In our framework, we do not rely on the structure of the
graphs but instead, on the nature of the variables. In fact, the
idea is to decompose the assignment vector x (by means of
Lagrangian relaxation) into different variables where each
variable obeys weaker constraints (that are easier to handle).
For example, instead of dealing with the assignment vector
x ∈ M, we can represent it by two vectors x1 and x2 ,
where the sum of each row of mat(x1 ) is ≤ 1 and the sum
of each column of mat(x2 ) is ≤ 1, and we constrain these
two vectors to be equal. More generally, we can decompose
x into as many vectors as we want, and in any manner, the
only condition is that the set of constraints imposed on these
vectors must be equivalent to x1 = x2 = · · · = xp ∈
M where p is the number of vectors. As for the objective
function (7), there is also an infinite number of ways to rewrite it under the new variables x1 , x2 , . . . , xp . The only
condition is that the re-written objective function must be
equal to the original one when x1 = x2 = · · · = xp = x.
For example, if p = D then one can re-write (7) as
F 1 (x1 ) + F 2 (x1 , x2 ) + · · · + F D (x1 , x2 , . . . , xD ). (13)
Each combination of (a) such a variable decomposition and
(b) such a way of re-writing the objective function will yield
a different Lagrangian relaxation and thus, produce a different algorithm. Since there are virtually infinite of such combinations, the number of algorithms one can design from
them is also unlimited, not to mention the different choices
of the reassembly mechanism, such as subgradient methods [2, 4], cutting plane methods [2], ADMM [3], or others.
We call the class of algorithms that base on ADMM Alternating Direction Graph Matching (ADGM) algorithms. A
major advantage of ADMM over the other mechanisms is
that its subproblems involve only one block of variables, regardless of the form the objective function.
As an illustration of ADGM, we present below a particular example. Nevertheless, this example is still general
enough to include an infinite number of special cases.
Thus, let cst be a constant independent of x, we have:
k
k
k >
Lρ (xk+1
[1,d−1] , x, x[d+1,D] , y ) = (pd ) x
ρ
Ad x + skd
+ (yk )> (Ad x + skd ) +
2
2
2
+ cst, (23)
and the subproblems (19) are reduced to minimizing the following quadratic functions over Md (d = 1, 2, . . . , D):
>
1 > >
1 > k
> k
k
x Ad Ad x + Ad sd + (Ad y + pd )
x. (24)
2
ρ
Problem 2 (Decomposed graph matching) Minimize
F 1 (x1 ) + F 2 (x1 , x2 ) + · · · + F D (x1 , x2 , . . . , xD ) (14)
subject to
A1 x1 + A2 x2 + · · · + AD xD = 0,
(15)
xd ∈ Md
(16)
∀ 1 ≤ d ≤ D,
where (Ad )1≤d≤D are m × n matrices, defined in such a
way that (15) is equivalent to x1 = x2 = · · · = xD , and
(Md )1≤d≤D are closed convex subsets of Rn satisfying
M1 ∩ M2 ∩ · · · ∩ MD = M.
(17)
It is easily seen that the above problem is equivalent to
the continuous relaxation of Problem 1. Clearly, this problem is a special case of the standard form (8). Thus, ADMM
can be applied to it in a straightforward manner. The augmented Lagrangian of Problem 2 is
D
X
Lρ (x1 , x2 , . . . , xD , y) =
F d (x1 , . . . , xd )
d=1
+y
D
X
>
!
Ad x d
D
ρ X
Ad x d
+
2
d=1
d=1
2
. (18)
2
The y update step (11) and the computation of the residual (12) is trivial. Let us focus on the x update step (10):
k
k
xk+1
= argmin Lρ (xk+1
d
[1,d−1] , x, x[d+1,D] , y ).
(19)
In summary, an ADGM algorithm has three main steps:
1) choose (Ad )1≤d≤D and (Md )1≤d≤D satisfying the conditions stated in Problem 2, 2) update xk+1
by minimizd
ing (24) over Md , and 3) update yk+1 using (11) (and repeat 2), 3) until convergence).
3.3. Two simple ADGM algorithms
Let us follow the above three steps with two examples.
Step 1: Choose (Ad )1≤d≤D and (Md )1≤d≤D . First,
(Md )1≤d≤D take values in one of the following two sets:
Mr = {x : sum of each row of mat(x) is ≤ 1} ,
(25)
Mc = {x : sum of each column of mat(x) is ≤ 1} , (26)
such that both Mr and Mc are taken at least once. If no
occlusion is allowed in G1 (respectively G2 ), then the term
“≤ 1” is replaced by “= 1” for Mr (respectively Mc ). If
many-to-many matching is allowed, then these inequality
constraints are removed. In either case, Mr and Mc are
closed and convex. Clearly, since Mr ∩ Mc = M, condition (17) is satisfied.
Second, to impose x1 = x2 = · · · = xD , we can for example choose (Ad )1≤d≤D such that
x1 = x2 ,
x1 = x3 , . . . ,
x1 = xD
(27)
xD−1 = xD .
(28)
or alternatively
x∈Md
x1 = x2 ,
Denote
skd =
d−1
X
Ai xk+1
+
i
i=1
pkd =
D
X
i=d
D
X
Aj xkj ,
(20)
j=d+1
Fi
d−1
O
j=1
xk+1
j
i
O
xkl . (see (4))
(21)
l=d+1
It can be seen that (details given in the supplement)
D
X
i=d
k
>
F i (xk+1
[1,d−1] , x, x[d+1,i] ) = (pk ) x.
(22)
x2 = x3 , . . . ,
It is easily seen that the above two sets of constraints can
be both expressed under the general form (15). Each choice
leads to a different algorithm. Let ADGM1 denote the one
obtained from (27) and ADGM2 obtained from (28).
Step 2: Update xk+1
. Plugging (27) and (28) into (15),
d
the subproblems (24) are reduced to (details given in the
supplement)
xk+1
d
= argmin
x∈Md
1
2
>
kxk2 − cd x ,
2
(29)
where (cd )1≤d≤D are defined as follows, for ADGM1:
1
c1 =
D−1
D
X
xkd
d=2
D
D
d=2
d=1
d
1 X k 1 X dO k
−
yd −
F
xi
ρ
ρ
i=2
!
,
(30)
D
d−1
i
1 k 1 X i O k+1 O k
y
−
F
x
cd = xk+1
+
xj
1
i
ρ d ρ
i=1
i=d
j=d+1
(31)
for 2 ≤ d ≤ D, and for ADGM2:
D
c1 =
xk2
d
1 X dO k
1
F
xi ,
− y2k −
ρ
ρ
i=2
(32)
d=1
D−1
1 k
1 D O k+1
cD = xk+1
+
y
F
xi ,
−
D−1
ρ D ρ
i=1
1
1 k+1
k
(xd−1 + xkd+1 ) + (ydk − yd+1
)
2
2ρ
i
D
d−1
1 X i O k+1 O k
−
xj
F
xi
2ρ
i=1
(33)
cd =
i=d
(34)
j=d+1
for 2 ≤ d ≤ D − 1.
Step 3: Update yk+1 . From (27) and (28), it is seen that
this step is reduced to ydk+1 = ydk + ρ(xk+1
− xk+1
) for
1
d
k+1
k+1
k+1
k
ADGM1 and yd = yd + ρ(xd−1 − xd ) for ADGM2.
Remark. When D = 2 the two algorithms are identical.
Note that (29) means xk+1
is the projection of cd
d
onto Md . Since (Md )1≤d≤D obey only row-wise or
column-wise constraints, the projection becomes row-wise
or column-wise and can be solved based on the projection onto a simplex [8]. We show how to do that and give
sketches of the above algorithms in the supplement.
3.4. Convergent ADGM
Note that the objective function in Problem 2 is neither
separable nor convex in general. Convergence of ADMM
for this type of functions is unknown. Indeed, our ADGM
algorithms do not always converge, especially for small values of the penalty parameter ρ. When ρ is large, they are
likely to converge. However, we also notice that small
ρ often (but not always) achieves better objective values.
Motivated by these observations, we propose the following
adaptive scheme that we find to work very well in practice:
starting from a small initial value ρ0 , the algorithm runs
for T1 iterations to stabilize, after that, if no improvement
of the residual rk is made every T2 iterations, then we increase ρ by a factor β and continue. The intuition behind
this scheme is simple: we hope to reach a good objective
value with a small ρ, but if this leads to slow (or no) convergence, then we increase ρ to put more penalty on the
consensus of the variables and that would result in faster
convergence. Using this scheme, we observe that our algorithms always converge in practice. In the experiments, we
n
.
set T1 = 300, T2 = 50, β = 2 and ρ0 = 1000
4. Experiments
We adopt the adaptive scheme in Section 3.4 to two
ADGM algorithms presented in Section 3.3, and denote
them respectively ADGM1 and ADGM2. In pairwise settings, however, since these two algorithms are identical,
we denote them simply ADGM. We compare ADGM and
ADGM1/ADGM2 to the following state of the art methods:
Pairwise: Spectral Matching with Affine Constraint
(SMAC) [9], Integer Projected Fixed Point (IPFP) [23],
Reweighted Random Walk Matching (RRWM) [6], Dual
Decomposition with Branch and Bound (DD) [28] and
Max-Pooling Matching (MPM) [7]. We should note that
DD is only used in the experiments using the same energy
models presented in [28]. For the other experiments, DD
is excluded due to the prohibitive execution time. Also, as
suggested in [23], we use the solution returned by Spectral
Matching (SM) [22] as initialization for IPFP.
Higher-order: Probabilistic Graph Matching (PGM) [30],
Tensor Matching (TM) [11], Reweighted Random Walk
Hypergraph Matching (RRWHM) [21] and Block Coordinate Ascent Graph Matching (BCAGM) [26]. For BCAGM,
we use MPM [7] as subroutine because it was shown
in [26] (and again by our experiments) that this variant of
BCAGM (denoted by “BCAGM+MP” in [26]) outperforms
the other variants thanks to the effectiveness of MPM. Since
there is no ambiguity, in the sequel we denote this variant
“BCAGM” for short.
We should note that, while we formulated the graph
matching as a minimization problem, most of the above
listed methods are maximization solvers and many models/objective functions in previous work were designed to
be maximized. For ease of comparison, ADGM is also converted to a maximization solver (by letting it minimize the
additive inverse of the objective function), and the results
reported in this section are for the maximization settings
(i.e. higher objective values are better). In the experiments,
we also use some pairwise minimization models (such as
the one from [28]), which we convert to maximization problems as follows: after building the affinity matrix M from
the (minimization) potentials, the new (maximization) affinity matrix is computed by max(M) − M where max(M)
denotes the greatest element of M. Note that one cannot
simply take −M because some of the methods only work
for non-negative potentials.
And last, due to space constraints, we leave the reported
running time for each algorithm in the supplement (except
for the very first experiment where this can be presented
compactly). In short, ADGM is faster than SMAC [9]
Methods
Error
(%)
Global
opt. (%)
Time
(s)
House
MPM
RRWM
IPFP
SMAC
DD
ADGM
42.32
90.51
87.30
81.11
0
0
0
0
0
0
100
100
0.02
0.01
0.02
0.18
14.20
0.03
Hotel
MPM
RRWM
IPFP
SMAC
DD
ADGM
21.49
85.05
85.37
71.33
0.19
0.19
44.80
0
0
0
100
100
0.02
0.01
0.02
0.18
13.57
0.02
(a) 20 pts vs 30 pts (10 outliers)
(b) MPM 15/20 (352.4927)
(c) RRWM 15/20 (352.4927)
(d) IPFP 5/20 (316.9299)
Table 1: Results on House and Hotel se(e) SMAC 12/20 (315.0426)
(f) ADGM 18/20 (353.3569)
quences using the pairwise model A, described in Section 4.1 and previously pro- Figure 1: House matching using the pairwise model B described in Section 4.1. Ground-truth value is 343.1515. (Best viewed in color.)
posed in [28].
(in pairwise settings) and ADGM1/ADGM2 are faster than
TM [11] (in higher-order settings) while being slower than
the other methods.
4.1. House and Hotel
The CMU House and Hotel sequences1 have been widely
used in previous work to evaluate graph matching algorithms. It consists of 111 frames of a synthetic house and
101 frames of a synthetic hotel. Each frame in these sequences is manually labeled with 30 feature points.
Pairwise model A. In this experiment we match all possible
pairs of images in each sequence, with all 30 points (i.e. no
outlier). A Delaunay triangulation is performed for these 30
points to obtain the graph edges. The unary terms are the
distances between the Shape Context descriptors [1]. The
pairwise terms when matching (i1 , j1 ) to (i2 , j2 ) are
2
Fij
= η exp δ 2 /σl2 + (1 − η) exp α2 /σa2 − 1 (35)
where η, σl , σa are some weight and scaling constants and
−−→
−−→
δ, α are computed from d1 = ki1 j1 k and d2 = ki2 j2 k as
−−→ −−→ !
|d1 − d2 |
i1 j1 i2 j2
δ=
, α = arccos
·
.
(36)
d1 + d2
d1
d2
This experiment is reproduced from [28] using their energy
model files2 . It should be noted that in [28], the above unary
potentials are subtracted by a large number to prevent occlusion. We refer the reader to [28] for further details. For
ease of comparison with the results reported in [28], here we
also report the performance of each algorithm in terms of
1 http://vasc.ri.cmu.edu/idb/html/motion/index.html
2 http://www.cs.dartmouth.edu/ lorenzo/Papers/tkr_
˜
pami13_data.zip.
overall percentage of mismatches and frequency of reaching the global optimum. Results are given in Table 1. One
can observe that DD and ADGM always reached the global
optima, but ADGM did it hundreds times faster. Even the
recent methods RRWM and MPM performed poorly on this
model (only MPM produced acceptable results). Also, we
notice a dramatic decrease in performance of SMAC and
IPFP compared to the results reported in [28]. We should
note that the above potentials, containing both positive and
negative values, are defined for a minimization problem.
It was unclear how those maximization solvers were used
in [28]. For the reader to be able to reproduce the results,
we make our software publicly available.
Pairwise model B. In this experiment, we match all possible pairs of the sequence with the baseline (i.e. the separation between frames, e.g. the baseline between frame 5 and
frame 105 is 100) varying from 10 to 100 by intervals of
10. For each pair, we match 10, 20 and 30 points in the first
image to 30 points in the second image. We set the unary
terms to 0 and compute the pairwise terms as
−−→
−−→
2
Fij
= exp − ki1 j1 k − ki2 j2 k /σ 2 ,
(37)
where σ 2 = 2500. It should be noted that the above pairwise terms are computed for every pair (i1 , j1 ) and (i2 , j2 ),
i.e. the graphs are fully connected. This experiment has
been performed on the House sequence in previous work,
including [6] and [26]. Here we consider the Hotel sequence as well. We report the average objective ratio (which
is the ratio of the obtained objective value over the groundtruth value) and the average accuracy for each algorithm in
Figure 2. Due to space constraints, we only show the results
for the harder cases where occlusion is allowed, and leave
the other results in the supplement. As one can observe,
0.7
20
60
80
40
60
80
20
0.2
0
40
60
80
100
Baseline
(a) House: 20 pts vs 30 pts
0.9
100
MPM
RRWM
IPFP
SMAC
ADGM
20
20
80
20
80
0.6
0.4
100
40
MPM
RRWM
IPFP
SMAC
ADGM
20
80
100
80
100
0.6
0.4
MPM
RRWM
IPFP
SMAC
ADGM
0.2
0
40
Baseline
60
80
100
20
40
Baseline
(b) House: 10 pts vs 30 pts
60
Baseline
1
0.8
0
60
MPM
RRWM
IPFP
SMAC
ADGM
100
Baseline
0.2
40
0.8
0.6
60
0.8
MPM
RRWM
IPFP
SMAC
ADGM
1
0.9
0.7
40
1
0.6
0.4
1
0.7
0.8
MPM
RRWM
IPFP
SMAC
ADGM
1.1
0.8
Baseline
1
Accuracy
Accuracy
0
MPM
RRWM
IPFP
SMAC
ADGM
20
Baseline
0.6
0.2
0.8
100
0.8
0.4
0.9
0.7
40
1
1
1.1
Accuracy
0.8
MPM
RRWM
IPFP
SMAC
ADGM
1.2
Objective
0.9
1.1
Accuracy
Objective
Objective
1
Objective
1.1
60
Baseline
(c) Hotel: 20 pts vs 30 pts
(d) Hotel: 10 pts vs 30 pts
1.4
1.2
0.8
0.7
0.6
0.5
PGM
TM
RRWHM
BCAGM
ADGM1
ADGM2
20
where fijk is a feature vector composed of the angles of
the triangle (i, j, k), and γ is the mean of all squared distances. We report the results for House sequence in Figure 3 and provide the other results in the supplement. One
can observe that ADGM1 and ADGM2 achieved quite similar performance, both were competitive with BCAGM [26]
while outperformed all the other methods.
80
100
20
Baseline
0.2
0
40
60
80
100
80
100
Baseline
1
0.8
0.6
0.4
PGM
TM
RRWHM
BCAGM
ADGM1
ADGM2
0.6
0.2
60
0.8
Accuracy
(38)
1
0.8
0.4
40
1
2
3
Fijk
= exp − kfi1 j1 k1 − fi2 j2 k2 k2 /γ ,
Objective
1
0.9
Accuracy
ADGM produced the highest objective values in almost all
the tests.
Third-order model. This experiment has the same settings as the previous one, but uses a third-order model proposed in [11]. We set the unary and pairwise terms to 0 and
compute the potentials when matching two triples of points
(i1 , j1 , k1 ) and (i2 , j2 , k2 ) as
Objective
Figure 2: Results on House and Hotel sequences using the pairwise model B described in Section 4.1.
PGM
TM
RRWHM
BCAGM
ADGM1
ADGM2
20
0.6
PGM
TM
RRWHM
BCAGM
ADGM1
ADGM2
0.4
0.2
0
40
60
80
100
20
Baseline
40
60
Baseline
(a) 20 pts vs 30 pts
(b) 10 pts vs 30 pts
Figure 3: Results on House sequence using the third-order
model described in Section 4.1.
4.2. Cars and Motorbikes
The Cars and Motorbikes dataset was introduced in [24]
and has been used in previous work for evaluating graph
matching algorithms. It consists of 30 pairs of car images
and 20 pairs of motorbike images with different shapes,
view-points and scales. Each pair contains both inliers (chosen manually) and outliers (chosen randomly).
Pairwise model C. In this experiment, we keep all inliers in
both images and randomly add outliers to the second image.
The number of outliers varies from 0 to 40 by intervals of 5.
We tried the pairwise model B described in Section 4.1 but
obtained unsatisfactory matching results (showed in supplementary material). Inspired by the model in [28], we propose below a new model that is very simple yet very suited
for real-world images. We set the unary terms to 0 and com-
pute the pairwise terms as
2
Fij
= ηδ + (1 − η)
1 − cos α
,
2
(39)
where η ∈ [0, 1] is a weight constant and δ, α are computed
−−→
−−→
from d1 = ki1 j1 k and d2 = ki2 j2 k as
δ=
|d1 − d2 |
,
d1 + d2
cos α =
−−→ −−→
i1 j1 i2 j2
·
.
d1
d2
(40)
2
Intuitively, Fij
computes the geometric agreement be−−→
−−→
tween i1 j1 and i2 j2 , in terms of both length and direction. The above potentials measure the dissimilarity between the edges, as thus the corresponding graph matching
1
0.4
0.2
0.2
0
MPM
RRWM
IPFP
SMAC
ADGM
10
20
30
40
0
Outliers
1
10
20
30
0
MPM
RRWM
IPFP
SMAC
ADGM
0.2
0
0
0.4
MPM
RRWM
IPFP
SMAC
ADGM
0.2
0
10
20
30
40
Outliers
(a) Cars
(c) RRWM 6/46 (988.0872)
10
0
10
0
30
40
Outliers
(b) Motorbikes
5
10
15
Outliers
1
0.8
0.6
PGM
TM
RRWHM
BCAGM
ADGM1
ADGM2
0.4
0
20
PGM
TM
RRWHM
BCAGM
ADGM1
ADGM2
0.6
15
Outliers
0.2
Figure 4: Results on Cars and Motorbikes using the pairwise model C described in Section 4.2.
(a) 46 pts vs 66 pts (20 outliers)
5
1
0.6
0.8
0.4
0.8
Accuracy
Accuracy
0.4
0.6
40
0.8
0.6
PGM
TM
RRWHM
BCAGM
ADGM1
ADGM2
0.4
Outliers
1
0.8
0.8
Objective
MPM
RRWM
IPFP
SMAC
ADGM
0.6
1
1
Accuracy
0.6
Objective
0.8
0.4
Accuracy
1.2
0.8
Objective
Objective
1
0
0.6
PGM
TM
RRWHM
BCAGM
ADGM1
ADGM2
0.4
0.2
0
5
10
15
0
5
Outliers
(a) Cars
10
15
Outliers
(b) Motorbikes
Figure 6: Results on Cars and Motorbikes using the thirdorder model described in Section 4.2.
(a) 25 pts vs 36 pts (9 outliers)
(b) PGM 4/25 (337.8194)
(c) RRWHM 3/25 (1409.832)
(d) BCAGM 15/25 (1713.487)
(b) MPM 13/46 (966.2296)
(d) IPFP 35/46 (1038.3965)
(e) ADGM1 25/25 (2161.5354) (f) ADGM2 25/25 (2161.5354)
(e) SMAC 11/46 (1028.7961)
(f) ADGM 46/46 (1043.0687)
Figure 7: Car matching using the third-order model described in Section 4.2. (Best viewed in color.)
Figure 5: Motorbike matching using the pairwise model C
described in Section 4.2. (Best viewed in color.)
problem is a minimization one. Pairwise potentials based on
both length and angle were previously proposed in [24, 28]
and [32]. However, ours are the simplest to compute. In this
experiment, η = 0.5.
We match every image pair and report the average in terms
of objective value and matching accuracy for each method
in Figure 4. One can observe that ADGM completely outperformed all the other methods.
Third-order model. This experiment has the same settings
as the previous one, except that it uses a third-order model
(the same as in House and Hotel experiment) and the number of outliers varies from 0 to 16 (by intervals of 2). Results
are reported in Figure 6 and a matching example is given in
Figure 7. ADGM did quite well on this dataset. On Cars,
both ADGM1 and ADGM2 achieved better objective val-
ues than BCAGM in 7/9 cases. On Motorbikes, ADGM1
beat BCAGM in 5/9 cases and had equivalent performance
in 1/9 cases; ADGM2 beat BCAGM in 8/9 cases.
5. Conclusion and future work
We have presented ADGM, a novel class of algorithms
for solving graph matching. Two examples of ADGM were
implemented and evaluated. The results demonstrate that
they outperform existing pairwise methods and competitive
with the state of the art higher-order methods. In future
work, we plan to adopt a more principled adaptive scheme
to the penalty parameter, and to study the performance of
different variants of ADGM. A software implementation of
our algorithms are available for download on our website.
Acknowledgements. We thank the anonymous reviewers
for their insightful comments and suggestions.
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| 1 |
A Reduction for Optimizing Lattice Submodular Functions with
Diminishing Returns
arXiv:1606.08362v1 [cs.DS] 27 Jun 2016
Alina Ene
∗
Huy L. Nguyen†
June 28, 2016
Abstract
A function f : ZE
+ → R+ is DR-submodular if it satisfies f (x+χi )−f (x) ≥ f (y+χi )−f (y) for
all x ≤ y, i ∈ E. Recently, the problem of maximizing a DR-submodular function f : ZE
+ → R+
subject to a budget constraint kxk1 ≤ B as well as additional constraints has received significant
attention [6, 7, 5, 8].
In this note, we give a generic reduction from the DR-submodular setting to the submodular
setting. The running time of the reduction and the size of the resulting submodular instance
depends only logarithmically on B. Using this reduction, one can translate the results for
unconstrained and constrained submodular maximization to the DR-submodular setting for
many types of constraints in a unified manner.
1
Introduction
Recently, constrained submodular optimization has attracted a lot of attention as a common abstraction of a variety of tasks in machine learning ranging from feature selection, exemplar clustering
to sensor placement. Motivated by the use cases where there is a large budget of identical items,
a generalization of submodular optimization to integer lattice is proposed by [6]. Previously, submodular functions has been generalized to lattices via the lattice submodular property. A function
E
f : ZE
+ → R+ is lattice submodular if for all x, y ∈ Z+ ,
f (x) + f (y) ≥ f (x ∨ y) + f (x ∧ y)
In the generalization due to [6, 7], a function f is DR-submodular if it satisfies
f (x + χi ) − f (x) ≥ f (y + χi ) − f (y)
for all x ≤ y, i ∈ E (diminishing return property), where χi is the vector in {0, 1}E that has a 1 in
the coordinate corresponding to i and 0 in all other coordinates.
It can be shown that any DR-submodular function is also lattice submodular (but the reverse direction is not necessarily true). Similar to submodular functions, the applications can be
formulated as maximizing a DR-submodular function f subject to constraints, such as a budget
constraint max{f (x) : x ∈ ZE
+ , kxk1 ≤ B}. While it is straightforward to reduce optimization of
∗
†
Department of Computer Science and DIMAP, University of Warwick, A.Ene@warwick.ac.uk.
Toyota Technological Institute at Chicago, hlnguyen@cs.princeton.edu.
1
DR-submodular function with budget constraint B to optimization of submodular function with
B · E items, the goal of [6] is to find algorithms for this setting with running time logarithmic in B
rather than polynomial in B, which follows from the straightforward reduction. Following [6], there
have been several works extending problems involving submodular functions to the DR-submodular
setting [7, 5, 8].
In this note, we give a generic reduction from the DR-submodular setting to the submodular
setting. The running time of the reduction and the size of the resulting submodular instance depends
only logarithmically on B. Using this reduction, one can translate the results for unconstrained and
constrained submodular maximization to the DR-submodular setting for many types of constraints
in a unified manner.
2
The Reduction
Lemma 1. For any n, there is a decomposition n = a1 + a2 + . . . + at with t ≤ 2 log n + 1 so that
for any q ≤ n, there is a way to express q as the sum of a subset of the multiset {a1 , . . . , at }.
Proof. Let n = b0 20 + b1 21 + . . . + bm 2m be the binary representation of n with bm = 1. Let
bc1 , bc2 , ..., bcp be all the non-zeroes among b0 , . . . , bP
2i−2 for 2 ≤ i ≤ m + 1,
m−1 . Let a1 = 1, ai =
P
c
m
and am+1+j = bcj 2 j for 1 ≤ j ≤ p. It is clear that i≤m+1 ai = 2 and i ai = n.
Consider an arbitrary number 1 < q < n. Let j be the largest bit that is 1 for n but it is 0 for
q (j must exist because q < n). Let r be the number that agrees with n on all bits larger or equal
to j and has all 0s for the smaller bits. We can form r from a1 , . . . , am+1 (which sum up to 2m )
and the additional numbers from {am+2 , . . . , am+1+p } corresponding to the bits equal to 1 from j
to m − 1 in the binary representation of r. Notice that r − q < 2m and it can be written as a sum
of numbers from a2 , . . . , am+1 (just (r − q)’s binary representation). By removing those numbers
from the representation of q above, we obtain a subset of the ai ’s that sums to q.
Corollary 2. For any n, there is a way to write n = a1 + a2 + . . . + at with t ≤ 2 log n + 1 + 1/ǫ so
that ai ≤ ǫn ∀i and for any q ≤ n, there is a way to express q as the sum of a subset of the multiset
{a1 , . . . , at }.
Proof. We start with the decomposition of the above lemma and refine it until the condition ai ≤
ǫn ∀i is satisfied. As long as there exists some ai > ǫn, replace ai with two new numbers ai − ǫn and
ǫn. Each replacement step produces a new term equal to ǫn so the number of replacement steps is
at most 1/ǫ. Thus, the number of terms in the decomposition is at most 2 log n + 1 + 1/ǫ.
The reduction. Suppose we need to optimize f over the domain [B1 ] × [B2 ] × · · · × [BE ]. By
the above lemma, we can write Bi = ai,1 + . . . + ai,ti with ti ≤ 2 log B
Pi + 1 and any number at
most Bi can be written as a sum S
of a subset of the {ai,j }j ’s. Let t = i ti . Consider a function
′
g defined P
on the ground set E ′ = i∈E {(i, 1), . . . , (i, ti )} defined as follows. Consider y ∈ {0, 1}E .
Let xi = j yi,j aj and we define g(y) := f (x).
P
By Lemma 1, for any vector x, there is a vector y such that xi = j yi,j ai,j for all i. Thus, the
set {g(y)}y captures all of {f (x)}x . Next, we show that g is submodular.
Lemma 3. The function g is submodular.
2
′
′ for all i, j. Consider an arbitrary
Proof. Consider 2 vectors y, y′ ∈ {0, 1}E such that yi,j ≤ yi,j
P
′
′
element
x′ defined as x′i =
P ′ (i0 , j0 ) ∈ E that is not in y . Let x defined as xi = j yi,j ai,j and
′
′
j yi,j ai,j . We have g(y + χ(i0 ,j0 ) ) − g(y) = f (x + ai0 ,j0 χi0 ) − f (x) and g(y + χ(i0 ,j0 ) ) − g(y ) =
′
′
f (x + ai0 ,j0 χi0 ) − f (x ). By the diminishing return property, we have
f (x + ai0 ,j0 χi0 ) − f (x) ≥ f (x′ + ai0 ,j0 χi0 ) − f (x′ ).
3
Modeling Constraints
We are interested in maximizing f (x) subject to constraints. In this section, we show how to
translate constraints on x to constraints for maximizing g(y).
Cardinality constraint.
P
The constraint i xi ≤ K with K > 1/ǫ can be translated to
X
ai,j yi,j ≤ K.
i,j
By applying Corollary 2, we map from a cardinality constraint to a knapsack constraint where all
weights are at most an ǫ fraction of the budget.
Knapsack constraint.
P
The knapsack constraint i ci xi ≤ K can be translated to
X
ci ai,j yi,j ≤ K.
i,j
General constraints. Consider the problem max{f (x) : x ∈ I}, where I ⊆ [B1 ]×[B2 ]×. . .×[BE ]
denotes the set of all solutions that satisfy the constraints.
We can apply algorithmic frameworks from the submodular setting — such as the frameworks
based on continuous relaxations and rounding [9, 3] — to the DR-submodular setting as follows.
Let P ⊆ RE
+ be a relaxation of I that satisfies the following conditions:
• P is downward-closed: if x ≤ z and z ∈ P then x ∈ P.
• There is a separation oracle for P: given x, there is an oracle that either correctly decides
that x ∈ P or otherwise returns a hyperplane separating x from P, i.e., a vector v ∈ RE and
D ∈ R such that hv, xi ≥ D and hv, zi < D for all z ∈ P.
We apply Lemma 1 (or Corollary
x,
P2) to obtain the multiset {ai,j }i,j such that, for any vector
′
there is a vector y such that xi = j yi,j ai,j for all i. Define the linear function M : RE → RE
P
′
where x = M (y) is computed according to xi = j yi,j ai,j ∀i. Let g : 2E → R+ be the submodular
′
function given by the reduction. Let G : [0, 1]E → R+ be the multilinear extension of g:
G(y) = E[g(R(y))],
where R(y) is a random set that contains each element e ∈ E ′ independently at random with
probability ye .
3
′
Thus we obtain the following fractional problem: max{G(y) : y ∈ [0, 1]E , M (y) ∈ P}. As
shown in the following lemma, we can use the separation oracle for P to maximize a linear objective
′
′
hw, yi, where w ∈ RE , subject to the constraints y ∈ [0, 1]E and M (y) ∈ P.
Lemma 4. Using the separation oracle for P and an algorithm such as the ellipsoid method, for
′
′
any vector w ∈ RE , one can find in polynomial time a vector y ∈ RE that maximizes hw, yi subject
′
to y ∈ [0, 1]E and M (y) ∈ P.
Proof. It suffices to verify that the separation oracle for P allows us to separate over {y : y ∈
′
′
′
[0, 1]E , M (y) ∈ P}. To this end, let y be a vector in RE . Separation for the constraint y ∈ [0, 1]E
can be done trivially by checking if every coordinate of y is in [0, 1]. Thus, we focus on separation
for the constraint M (y) ∈ P. Using the separation oracle for P, we can check whether M (y) ∈ P.
If yes, then we are done. Otherwise, the oracle returns v ∈ RE and D ∈ R such that hv, M (y)i ≥ D
′
and hv, zi < D for all z ∈ P. Let v′ ∈ RE be v′ = M ∗ (v), where M ∗ is the adjoint of M i.e.
′ = a v ∀i, j. Then hv′ , yi = hM ∗ (v), yi = hv, M (y)i ≥ D. Now let y′ be a vector in RE ′ such
vi,j
i,j i
that M (y′ ) ∈ P. We have hv′ , y′ i = hM ∗ (v), y′ i = hv, M (y′ )i < D. Thus (v′ , D) is a hyperplane
separating y from {y′ : M (y′ ) ∈ P}.
′
′
Since we can solve max{hw, yi : y ∈ [0, 1]E , M (y) ∈ P}, where w ∈ RE , we can approximately
′
solve the fractional problem max{G(y) : y ∈ [0, 1]E , M (y) ∈ P} using the (measured) Continuous
Greedy algorithm or local search [9, 3, 4].
We note that in some settings, such as when P is a polymatroid polytope1 , we can round the
resulting fractional solution to the problem max{G(y) : M (y) ∈ P} and obtain an integral solution
(similarly to [8]); in this case, the rounding preserves the value of the fractional solution and thus
we obtain an α-approximation for the problem max{f (x) : x ∈ I}, where α = 1 − 1/e for monotone
functions and α = 1/e for non-monotone functions. The detailed proof is in Theorem 7.
Some examples of results. Using the reduction above, we immediately get algorithms for maximizing DR-submodular functions subject to various types of constraints. We include a few examples
below.
Theorem 5. There is a 1/2 approximation
algorithm for unconstrained DR-submodular maximizaP
tion with running time O(n + i log Bi ).
Proof. By the reduction
using Lemma 1, we need to solve an unconstrained submodular maximizaP
tion with O(n + i log Bi ) items. The result follows from applying the Double Greedy algorithm
of [2] to the resulting instance of unconstrained submodular maximization.
Theorem 6. There is a 1 − 1/e − ǫ approximation algorithm for maximizing a monotone DRsubmodular P
function subject to a cardinality constraint B with running time O(m log(m/ǫ)/ǫ) where
m = n/ǫ + i log Bi .
Proof. If B ≤ 1/ǫ, the result follows via the trivial reduction of making B copies of every item. Next,
we consider the case B > 1/ǫ. By the reduction using Corollary 2, we need to solve a submodular
maximization problem with
P a knapsack constraint where all weights are at most ǫ times the budget
and there are O(n/ǫ + i log Bi ) items. The result follows from applying the Density Greedy
algorithm with either descending thresholds or lazy evaluation [1].
Let ρ : 2E → Z+ be P
a monotone submodular function with ρ(∅) = 0. The polymatroid associated with ρ is the
polytope P = {x ∈ RE
∀S ⊆ E}.
+:
i∈S xi ≤ ρ(S)
1
4
Theorem 7. There is an α approximation algorithm for maximizing a DR-submodular
function
P
subject to a polymatroid constraint with running time that is polynomial in n and i log Bi , where
α = 1 − 1/e if the function is monotone and α = 1/e otherwise.
Proof. Let P be the polymatroid polytope. We apply Lemma 1 (or CorollaryP2) to obtain the
multiset {ai,j }i,j such that, for any vector x, there is a vector y such that xi = j yi,j aj for all i.
′
′
Let g : 2E → R+ be the submodular function given by the reduction. Let G : [0, 1]E → R+ be the
multilinear extension of g.
Since we can separate over P in polynomial time using a submodular minimization algorithm,
it follows from Lemma 4 that we can optimize any linear (in y) objective over x ∈ P and xi ≤ Bi
for all i ∈ E, where x = M (y). Therefore, using the measured Continous Greedy algorithm, we
can find an α-approximate fractional solution to the problem max{G(y) : x ∈ P, xi ≤ Bi ∀i ∈ E},
where α = 1 − 1/e for monotone functions and α = 1/e for non-monotone functions. Similarly
to [8], we can round the resulting fractional solution without any loss in the approximation. Let
z ∈ ZE be defined as zi = ⌊xi ⌋. Define H : [0, 1]E → R as H(v) = ER [f (z + R(v))] for any v ∈ RE .
Note that H is the multilinear extension of a submodular function agreeing with H on {0, 1}E .
Let v ∈ RE be defined as vi = xi − ⌊xi ⌋. First one can show that H(v) ≥ G(y) via a hybrid
argument. Let z(i) ∈ ZE be a random integral vector whose first i coordinates are distributed
according to M (R(y)) (that is, constructing a randomized rounding of y and then converting it to
an integral vector in ZE ) and the last |E| − i coordinates are picked randomly among {zi , zi + 1}
so that the expectation is xi . Note that E[f (z(0) )] = H(v) and E[f (z(|E|) )] = G(y) and we will
(i−1)
(i)
are identically distributed
show that E[f (z(i−1) )] ≥ E[f (z(i) )]. Indeed, for all j 6= i, zj and zj
(i)
(i−1)
∀j 6= i. Let w be z(i) with the ith coordinate
so we can couple the randomness so that zj = zj
zeroed out and define a single variable function g : Z → R where g(x) = f (w + x · ei ). Define
g′ : R → R be the piecewise linear function agreeing with g on integral points and g ′ does not
have any break points other than integral points. By the DR property of f , we have that g ′ is
(i)
(i)
concave. Thus, E[g ′ (zi )] ≤ g′ (E[zi ]). On the other hand, because g ′ is linear in [zi , zi + 1], we
(i−1)
(i−1)
(i)
have E[g ′ (zi
)] = g′ (E[zi
]) = g ′ (E[zi ]).
Next as done in [8, Lemma 13], one can show that the constraints v ∈ [0, 1]E , z + v ∈ P are
equivalent to a matroid polytope. Thus, one can round v to an integral vector without losing any
value in H using strategies such as pipage rounding or swap rounding.
4
The DR-Submodular Cover Problem
We conclude this note with some remarks on the DR-submodular cover problem. In this problem,
the goal is to minimize the cost c(x) subject to the constraint f (x) ≥ α where f is a DR-submodular
function and the cost c is a modular function. This problem has been considered in [7] (in fact,
they even consider a more general setting where c is a subadditive function) but there is a mistake
in the proof. The proof of [7, Claim 5.2] uses the equality f (x∗ ) = f (xL ), which is incorrect since
it could happen that f (xL ) > f (x∗ ). Note that this equality is correct for the usual special case
where α = maxx∈P f (x) but not for the general case considered in [7].
Algorithm 1 in [7] in fact fails even for the case where both c and f are modular. For instance,
consider E = {1, 2}, f (∅) = 0, f ({1}) = M, f ({2}) = 1, f ({1, 2}) = M + 1, c({1}) = M, c({2}) = 1,
α = 1. The algorithm first picks element 1 as it has the highest density ratio, and then breaks the
5
loop as it realizes f is already exceeding α = 1. However, the cost of the solution is M , which is
arbitrarily larger than the optimal cost, which is 1 for the solution {2}.
Nevertheless, it is known that submodular cover can be reduced to submodular maximization
with a knapsack constraint so we can obtain algorithms for DR-submodular objective via the reduction in the previous section.
References
[1] Ashwinkumar Badanidiyuru and Jan Vondrák. Fast algorithms for maximizing submodular
functions. In Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 1497–1514, 2014.
[2] Niv Buchbinder, Moran Feldman, Joseph Naor, and Roy Schwartz. A tight linear time (1/2)approximation for unconstrained submodular maximization. SIAM J. Comput., 44(5):1384–
1402, 2015.
[3] Chandra Chekuri, Jan Vondrák, and Rico Zenklusen. Submodular function maximization via the
multilinear relaxation and contention resolution schemes. SIAM J. Comput., 43(6):1831–1879,
2014.
[4] Moran Feldman, Joseph Naor, and Roy Schwartz. A unified continuous greedy algorithm for
submodular maximization. In Proceedings of the 52nd Annual IEEE Symposium on Foundations
of Computer Science (FOCS), pages 570–579, 2011.
[5] Takanori Maehara, Akihiro Yabe, JP NEC, and Ken-ichi Kawarabayashi. Budget allocation
problem with multiple advertisers: A game theoretic view. In Proceedings of the 32nd International Conference on Machine Learning (ICML), pages 428–437, 2015.
[6] Tasuku Soma, Naonori Kakimura, Kazuhiro Inaba, and Ken-ichi Kawarabayashi. Optimal budget allocation: Theoretical guarantee and efficient algorithm. In Proceedings of The 31st International Conference on Machine Learning (ICML), pages 351–359, 2014.
[7] Tasuku Soma and Yuichi Yoshida. A generalization of submodular cover via the diminishing
return property on the integer lattice. In Advances in Neural Information Processing Systems
(NIPS), pages 847–855, 2015.
[8] Tasuku Soma and Yuichi Yoshida. Maximizing monotone submodular functions over the integer
lattice. In International Conference on Integer Programming and Combinatorial Optimization
(IPCO), pages 325–336, 2016.
[9] Jan Vondrák. Optimal approximation for the submodular welfare problem in the value oracle
model. In Proceedings of the 40th Annual ACM Symposium on Theory of Computing (STOC),
pages 67–74, 2008.
6
| 8 |
Prophet Inequalities Made Easy:
Stochastic Optimization by Pricing Non-Stochastic Inputs
arXiv:1612.03161v2 [cs.GT] 9 Jul 2017
Paul Dütting∗
Michal Feldman†
Thomas Kesselheim‡
Brendan Lucier§
July 11, 2017
Abstract
We present a general framework for stochastic online maximization problems with combinatorial feasibility constraints. The framework establishes prophet inequalities by constructing
price-based online approximation algorithms, a natural extension of threshold algorithms for
settings beyond binary selection. Our analysis takes the form of an extension theorem: we
derive sufficient conditions on prices when all weights are known in advance, then prove that
the resulting approximation guarantees extend directly to stochastic settings. Our framework
unifies and simplifies much of the existing literature on prophet inequalities and posted price
mechanisms, and is used to derive new and improved results for combinatorial markets (with
and without complements), multi-dimensional matroids, and sparse packing problems. Finally,
we highlight a surprising connection between the smoothness framework for bounding the price
of anarchy of mechanisms and our framework, and show that many smooth mechanisms can be
recast as posted price mechanisms with comparable performance guarantees.
1
Introduction
A concert is being held in a local theatre, and potential audience members begin calling to reserve
seats. The organizer doesn’t know individuals’ values for seats in advance, but has distributional
knowledge about their preferences. Some need only a single seat, others require a block of seats.
Some think seats are very valuable, others are only willing to attend if tickets are very cheap.
Some prefer front-row seats, some prefer to sit a few rows back, and some prefer the balcony. The
organizer needs to decide which seats, if any, to allocate to each individual as they call. The goal
is to maximize the total value (i.e., social welfare) of the seating arrangement.
Such stochastic online optimization problems have been studied for decades. A common goal
is to attain “prophet inequalities” that compare the performance of an online algorithm to that of
an omniscient offline planner. A classic result is that if the goal is to choose exactly one element
(i.e., there is only a single seat to allocate), then a simple threshold strategy—choosing the first
value higher than a certain pre-computed threshold—yields at least half of the expected maximium
∗
Department of Mathematics, London School of Economics, Houghton Street, London WC2A 2AE, UK, Email:
p.d.duetting@lse.ac.uk.
†
Blavatnic School of Computer Science, Tel Aviv University, P.O.B. 39040, Ramat Aviv, Tel Aviv, Israel. Email:
mfeldman@tau.ac.il
‡
TU
Dortmund,
Otto-Hahn-Str.
14,
44221
Dortmund,
Germany.
Email:
thomas.kesselheim@cs.tu-dortmund.de. This work was done while the author was at Max Planck Institute
for Informatics and Saarland University, supported in part by the DFG through Cluster of Excellence MMCI, and
while he was visiting Simons Institute for the Theory of Computing.
§
Microsoft Research, 1 Memorial Drive #1, Cambridge, MA 02142, USA. Email: brlucier@microsoft.com
1
value [34, 35, 44]. This solution has the appealing property that it corresponds to posting a takeit-or-leave-it price and allocating to the first interested buyer. A natural question is whether more
complex allocation problems (like the concert example above) can be approximated by posting
prices and allowing buyers to select their preferred outcomes in sequence.
Driven in part by this connection to posted prices, prophet inequalities have seen a resurgence
in theoretical computer science. Recent work has established new prophet inequalities for a variety
of allocation problems, including matroids [13, 33], unit-demand bidders [13, 3], and combinatorial
auctions [23]. In this paper we develop a framework for proving prophet inequalities and constructing posted-price mechanisms. Our framework, which is based on insights from economic theory,
unifies and simplifies many existing results and gives rise to new and improved prophet inequalities
in a host of online settings.
1.1
Example: Combinatorial Auctions
To introduce our framework we will consider a combinatorial auction problem. There is a set M of m
items for sale and n buyers. Each buyer i has a valuation function vi : 2M → R≥0 that assigns nonnegative value to every subset of at most d items.1 Valuations are non-decreasing and normalized
so that vi (∅) = 0, butP
otherwise arbitrary. The goal is to assign items to buyers to maximize total
value. Write v(x) = ni=1 vi (xi ) for the total value of allocation x = (x1 , . . . , xn ), where xi ⊆ M
for all i. There is a simple O(d)-approximate greedy algorithm for this problem and a lower bound
of Ω(d/ log d) assuming P 6= N P [46]. Our goal is to match this O(d) approximation as a prophet
inequality with posted item prices. That is, given distributions over the valuations, compute prices
for the items so that, when buyers arrive in an arbitrary order and each chooses his most-desired
bundle from among the unsold items, the expected total value is an O(d) approximation to the
expected optimum.2
Let’s first consider the simpler full information case where all valuations are known in advance.
This problem is still non-trivial, and in fact there may not exist prices that lead to the optimal
allocation.3 Intuitively, what we need for an approximation result are prices that balance between
two forces. They should be small enough that high-valued buyers are willing to purchase their
optimal bundles if available, but also large enough that those items will not first be scooped up
by bidders with much lower values. Such “balanced” prices can be obtained as follows: Given
valuation profile v, consider the welfare-maximizing allocation x∗ (which we can assume allocates
all items). Then for each item j, say j ∈ x∗i , set the price of j to pj = vi (x∗i )/2|x∗i |. These prices
are low enough that the total price of all items is at most 1/2 · v(x∗ ), which is significantly less
than the total value of x∗ . At the same time, prices are high enough that, for any set of goods
S, the total price of S is at least 1/2d of the value of allocations in the optimal allocation x∗ that
intersect S. So, in particular, a bidder that purchases S must have value at least that high.
1
Alternatively, we can suppose that there is a cardinality constraint that no buyer can receive more than d items.
There is a straightforward lower bound of Ω(d) on the approximation of any posted item prices. Suppose there
are d items and two agents. The first agent is unit-demand and has value 1 for any single item. The second agent
values the set of all d items for value d, and has value 0 for any subset. If all items have price greater than 1, then
neither agent purchases anything. If any item has price less than 1, then the unit-demand agent (who chooses first)
will purchase the cheapest single item and the other agent will purchase nothing, generating a total value of 1 whereas
the optimum is d. One can avoid issues of tie-breaking by perturbing the values by an arbitrarily small amount.
3
For example, suppose there are three items and four single-minded bidders. The first three bidders each have
value 2 for a different pair of items, and the last bidder has value 3 for the set of all three items, so at most one bidder
can get positive value, and it is optimal to allocate all items to the last bidder. However, at any item prices where
the last bidder is willing to purchase, one of the other bidders will purchase first if arriving before the last bidder.
This leads to a 3/2 approximation in the worst arrival order.
2
2
To see why these prices yield an O(d) approximation, let x denote the purchase decisions of the
players and let I ⊆ N be the set of players i such that x∗i intersects with x. The welfare achieved
by x is equal to the revenue generated plus the sum of buyer utilities. The revenue
P is the sum of
prices of the items sold, and since prices are “balanced” this is at least (1/2d) · i∈I vi (x∗i ). Also,
each buyer i 6∈ I could have chosen to purchase x∗i , and therefore must get at least as much utility
∗
as they would by purchasing x∗i , which is vi (x∗i ) minus the
prices are
Pprice of∗ xi . Again, since
∗
balanced, this means the sum of buyer utilities is at least i6∈I vi (xi ) − 1/2 · v(x ). Multiplying
this by 1/2d and adding the revenue gives an O(d) approximation.4 .
The argument above was for the full information case. Perhaps surprisingly, the existence of
sufficiently “balanced” prices for full information instances also establishes an O(d)-approximate
prophet inequality for the general stochastic problem, where one has only distributional knowledge about valuations. Our main result is this reduction from the stochastic setting to the full
information setting, which holds for a broad class of allocation problems.
1.2
A Framework for Prophet Inequalities
Consider a more general combinatorial allocation problem, where the cardinality constraint d is
replaced with an arbitrary downward-closed feasibility constraint F and each vi is drawn independently from an arbitrary distribution Di . While our framework applies for more general outcome
spaces (see Sections 2 and 3), combinatorial allocation problems provide a sweet spot between
expressiveness and clarity. Our key definition is the following notion of balanced prices for fullinformation instances. For P
each x ∈ F we write OPT(v | x) for the optimal residual allocation: the
allocation that maximizes i vi (x′i ) over x′ ∈ F with x, x′ disjoint and x ∪ x′ ∈ F. Given a fixed
to buyer
valuation profile v, a pricing rule defines a price pvi (xi ) for every bundle that we can assignP
v
i. For example, the item prices described in Section 1.1 define a pricing rule pi (xi ) = j∈xi pj .
Below we also extend the definition to dynamic prices, i.e., prices that depend on which allocations
have already been made.
Key Definition (special case) ((α, β)-balanced prices). Let α, β > 0. A pricing rule pv =
(pv1 , . . . , pvn ) defined by functions pvi : 2M → R≥0 is (α, β)-balanced with respect to valuation profile
v if for all x ∈ F and all x′ ∈ F with x, x′ disjoint and x ∪ x′ ∈ F,
P v
1
(a)
i pi (xi ) ≥ α (v(OPT(v)) − v(OPT(v | x))) ,
P v ′
(b)
i pi (xi ) ≤ β v(OPT(v | x)) .
The first condition formalizes what it means that prices are high enough: the sum of prices for
x should partially cover the welfare lost due to allocating x. The second condition formalizes “low
enough:” the sum of prices for any x′ that is still feasible “after” allocating x should not be much
higher than the optimal residual welfare.
Our main result is that the existence of balanced prices for full information instances directly
implies a price-based prophet inequality for the stochastic setting. The idea to choose balanced
prices is a natural one and has appeared in the prophet inequality literature before, most explicitly
in the notion of balanced thresholds of Kleinberg and Weinberg [33]. Previous definitions, however,
applied to the stochastic setting directly, which made the construction and analysis of balanced
thresholds inherently probabilistic. A main advantage of our framework is that it suffices to reason
about the simpler full-information setting.
P
∗
∗
For simplicity we assumed here that
i ) − 1/2 · v(x ) ≥ 0. More generally, since utilities are noni6∈I vi (x
P
∗
∗
negative,
P the sum of buyer utilities is at least max{ i6∈I vi (xi ) − 1/2 · v(x ), 0}. If the maximum is attained at 0,
then i∈I vi (x∗i ) > 1/2 · v(x∗ ) and the revenue alone exceeds (1/4d) · v(x∗ ), as desired.
4
3
Main Theorem (informal). Consider the setting where valuations are drawn from product distribution D. Suppose that the pricing rule pv is (α, β)-balanced with respect to valuation profile v.
Then posting prices
α
Eṽ∼D pṽi (xi )
pi (xi ) =
1 + αβ
achieves welfare at least
1
1+αβ E[v (OPT(v))].
In other words, to construct appropriate prices for a stochastic problem instance, it suffices
to construct balanced prices for the full-information instances in its support and then post the
expected values of those prices, scaled by an appropriate factor. The proof of our main theorem is
similar in spirit to proofs in the Price of Anarchy literature [41, 45] or for establishing algorithmic
stability [29], in that it uses “ghost samples.” It is, however, considerably more involved because
of the sequential, online aspect of our problem.
Remark 1.1 (Weakly Balanced Prices). We also define a notion of weakly balanced prices, in which
it suffices to upper bound the prices by βv(OPT(v)). In this case, we can show that posting an
appropriately scaled version of the expected prices yields a 1/4αβ-approximate prophet inequality.
Remark 1.2 (Computation). It is sometimes easier to compute prices that are balanced with
respect to an approximation algorithm ALG rather than OPT. Our result still applies in this
case, with OPT replaced by ALG in the welfare guarantee. We also note that if the price rule
p in the main theorem is perturbed to some p̂ with ||p − p̂||∞ < ǫ, then the welfare guarantee
degrades by at most an additive O(nǫ) term. This robustness is desirable in itself, and also implies
that appropriate prices can be computed for bounded values with poly(n, m, 1/ǫ) samples using
standard concentration bounds, as has been observed for various posted price settings [12, 23].
Remark 1.3 (Static vs. Dynamic, Anonymous vs. Discriminatory, Bundle vs. Item Pricing). We
have described our framework for static, discriminatory, bundle prices. In general, our construction
has the property that if the full-information balanced prices pv are dynamic, anonymous, and/or
take the form of item prices, then the derived prices for the stochastic setting will have these
properties as well. For example, our result holds also for dynamic prices, replacing pi (xi ) and
pi (x′i ) with pi (xi | x[i−1] ) and pi (x′i | x[i−1] ) where the conditioning on x[i−1] indicates that the price
to player i may depend on the purchase decisions of players that precede him. See Sections 2 and
3 for details.
Remark 1.4 (Arrival Order). Balancedness can depend on player arrival order. In the applications
we consider, our results hold even if the arrival order is chosen by an adaptive adversary that
observes previous realized values and purchase decisions before selecting the next player to arrive.
Let’s return to our example from Section 1.1. We established the existence of weakly (d, 1)balanced prices (simply undo the scaling by 1/2), so our main result implies a O(d)-approximate
prophet inequality. What about computation? We can compute prices in polynomial time by
basing them on the O(d)-approximate greedy algorithm (rather than the optimal allocation), but
then we only get a O(d2 )-approximate solution. It turns out that we can further improve this
to O(d) in polynomial time, as we hoped for in Section 1.1, by applying our main theorem to a
fractional relaxation of the auction problem. See Section 4 for more details.
Composition We also show that balanced prices “compose”, as was shown for mechanism
smoothness in [45]. This means that to derive a prophet inequality for a complex setting it often suffices to show balancedness for a simpler problem. See Appendix C.
4
Feasibility
Constraint
Valuation
Class
Pricing Model
Upper Bound
Query
Model
Combinatorial
Auction
XOS
Static, anonymous item prices
2e
e−1
[23]
2 [this work]
XOS,
Demand
Combinatorial
Auction
MPH-k
Static, anonymous item prices
O(k2 ) [23]
4k − 2 [this work]
MPH,
Demand
Matroid
Submodular Dynamic prices
2 (existential)
4 (computational)
Value
Knapsack
Additive
Static, anonymous prices
3
Explicit
d-Sparse PIPs
Additive
Static, anonymous prices
8d
Explicit
Table 1: Overview of applications. Results are computational unless otherwise stated. The query
model refers to the valuation access needed for the computational upper bounds, where “explicit”
indicates that valuations can be described explicitly. All results are order oblivious (see Section 2).
1.3
Unification of Existing Prophet Inequality Proofs
Our framework unifies and simplifies many of the existing prophet inequality proofs. We list some
representative examples below. We discuss the first example in more detail in Appendix A. The
other two examples are covered in Appendices D and F.
Example 1.1 (Classic Prophet Inequality, [34, 35]). The goal is to pick the single highest-value
element vi . The pricing rule pv defined by pvi (xi ) = maxi vi for all i is (1, 1)-balanced.
Example 1.2 (Matroids, [33]). The goal is to pick a maximum weight independent set in a matroid.
Encode sets S by n-dimensional vectors x over {0, 1} such that xi = 1 if i ∈ S. Then one can define
a dynamic pricing rule pv by pi (xi | y) = v(OPT(v | y)) − v(OPT(v | y ∪ xi )) for all i, where y is
the set of previously-selected elements. This pricing rule is (1, 1)-balanced.
Example 1.3 (XOS Combinatorial Auctions, [23]). The goal is to assign m goods to n buyers with
XOS valuations5 . Let x∗ = OPT(v) and let a1 , . . . , an be the corresponding additive supporting
functions. Set item prices pj = ai (j) for j ∈ x∗i . This pricing rule is (1, 1)-balanced.
The final example illustrates the power of our composition results (see Appendix C): the existence of (1, 1)-balanced prices for XOS combinatorial auctions, and hence a 2-approximate prophet
inequality, follows directly from the existence of (1, 1)-balanced prices for a single item, despite
being significantly more complex. It also yields a O(log m)-approximate prophet inequality for
subadditive valuations by approximating subadditive valuations with XOS valuations [17, 9].
1.4
New and Improved Prophet Inequalities
We also establish new prophet inequalities using our framework; see Table 1. Our first result is a
poly-time (4k − 2)-approximate prophet inequality for MPH-k combinatorial auctions6 .
5
A valuation v is XOS if there is a collection of additive functions a1 (·), . . . , ak (·), such that for every set S,
v(S) = max1≤i≤k ai (S). This is a generalization of submodular valuations [37].
6
The maximum over positive hypergraphs-k (MPH-k) hierarchy of valuations [21] is an inclusive hierarchy, where
k measures the degree of complementarity.
5
Theorem 1.1 (Combinatorial auctions with MPH-k valuations). For combinatorial auctions with
MPH-k valuations, a (4k − 2 + ǫ)-approximate posted-price mechanism, with static item prices, can
be computed in poly(n, m, 1/ǫ) demand and MPH-k queries.
Theorem 1.1 improves the poly-time result of [23] from O(k 2 ) to O(k). We note two interesting
special cases. First, combinatorial auctions with bundle size d (from Section 1.1) belong to MPH-d,
so Theorem 1.1 captures the polytime O(d) approximation discussed above. Second, XOS valuations
coincide with MPH-1, so Theorem 1.1 improves the previously best known poly-time result of [23]
from 2e/(e − 1) to 2, matching the existential lower bound. See Section 4 and Appendix D.
The second set of new results includes Knapsack feasibility constraints and d-sparse Packing
Integer Programs (PIPs), for which we obtain a constant- and a O(d)-approximation, respectively.
These settings are presented in Sections 4 and E, respectively.
Theorem 1.2 (Knapsack). For Knapsack constraints, a factor (5 + ǫ)-approximate posted-price
mechanism, with static prices, can be computed in poly(n, 1/ǫ). This improves to a (3 + ǫ) approximation if no individual demands more than half of the total capacity.
Theorem 1.3 (Sparse PIPs). For d-sparse Packing Integer Programs (PIPs) with constraint matrix
A ∈ Rm×n
≥0 where aj,i ≤ 1/2 for all i, j and unit capacities, a factor (8d+ǫ)-approximate posted-price
mechanism, with static prices, can be computed in time poly(n, m, 1/ǫ).
To the best of our knowledge, Theorems 1.2 and 1.3 are the first prophet inequalities for these
settings. We note that [24] derived a prophet inequality for closely-related fractional knapsack
constraints, with approximation factor ≈ 11.657. We obtain an improved prophet inequality for this
fractional setting: a corollary of Theorem 1.1 (with k = 1) is that one can obtain a 2-approximation
for a fractional knapsack constraint using a static per-unit price, even when knapsack weights are
private and arbitrarily correlated with buyer values. See Section 4 for more details.
Finally, in Appendix F we generalize the matroid prophet inequalities of Kleinberg and Weinberg
[33] to settings where players make choices regarding multiple elements of a matroid, and have
submodular preferences over subsets of elements.
Theorem 1.4 (Multi-Dimensional Matroids). For matroid feasibility constraints and submodular
valuations, there is a (4 + ǫ)-approximate posted-price mechanism, with dynamic prices, that can
be computed in poly(n, 1/ǫ) value queries.
1.5
From Price of Anarchy to Prophet Inequalities
In the proof sketch in Section 1.1, we derived a lower bound on buyer utility by considering a
deviation to a certain purchasing decision. This deviation argument, which appears in the proof of
our main result, is also useful for establishing Price of Anarchy bounds [41, 45]. There is a subtle
but important difference, however. In smoothness proofs one considers deviations against a fixed
strategy profile, while the prophet inequality problem is inherently temporal and agents deviate at
different points in time. As it turns out, many smoothness proofs have a built-in charging scheme
(which we refer to as outcome smoothness) that, under the assumption that critical payments are
monotonically increasing, implies prophet inequalities with the same (asymptotic) approximation
guarantee. We provide examples showing that both outcome smoothness and monotonicity are
necessary for this result to hold. We also provide two “black-box reductions” for binary singleparameter settings, where PoA guarantees of O(γ) established by (normal) smoothness imply O(γ 2 )approximate prophet inequalities. See Section 5.
6
Theorem 1.5 (informal). For general multi-parameter problems, if the first-price (i.e., pay-yourbid) mechanism based on declared welfare maximization has a Price of Anarchy of O(γ) provable
via outcome smoothness, and critical payments are monotonically increasing, then posting a scaled
version of the critical payments yields a O(γ)-approximate price-based prophet inequality.
Using these results we can, for example, rederive the classic prophet inequality [34, 35] from
the smoothness of the first-price single-item auction [45] or the matroid prophet inequality [33]
from the smoothness of the pay-your-bid, declared welfare maximizing mechanism for selecting a
maximum-weight basis [39].
1.6
Further Related Work
Prophet inequalities and their applicability as posted-price mechanisms were (re-)discovered in theoretical computer science by [28]. Subsequently, threshold-based prophet inequalities and postedprice mechanisms were developed for matroids and matroid intersection [13, 33, 7], polymatroids
[20], unit-demand bidders [13, 3], and combinatorial auctions [3, 23].
Not all prophet inequalities in the literature are based on explicit thresholds. Examples include
prophet inequalities for the generalized assignment problem [4, 5], matroids and matroid intersection [26], and for general binary feasibility constraints [42]. On the other hand, many posted-price
mechanisms from the literature are constructed either without explicit reference to prophet inequalities or via different techniques. Chawla et al. [12] developed approximately-optimal (revenue-wise)
posted-price mechanisms for unit-demand buyers. Posted-price mechanisms have subsequently been
developed for a variety of other auction settings [18, 8, 11, 6]. Dynamic posted prices that give
optimal welfare for unit-demand buyers were established in [15]. Recently, dynamic posted prices
for various online settings have been considered, including k-server on the line and metrical task
systems [14], and makespan minimization for scheduling problems [27].
Most recently, and in parallel to this work combinatorial prophet inequalities were developed
in [43] and [10]. The former, amongst others, proves prophet inequalities for subadditive CAs, but
considers a different allocation model and is therefore imcomparable. The latter, in turn, focuses
on revenue and not welfare as we do here. Finally, [1] re-considers the classic prophet inequality
setting, but in a large market setting and assuming random or best order.
The notion of smooth games was introduced by Roughgarden [41] as a tool for bounding the
price of anarchy, which measures the inefficiency that can be incurred in equilibrium. This notion
has been extended to mechanisms by Syrgkanis and Tardos [45]. Notions of outcome smoothness
were considered in [16, 40].
2
General Model and Notation
Problem Formulation There is a set N of n agents. For each agent i ∈ N there is an outcome
space Xi containing a null outcome ∅. We write X = X1 × . . . × Xn for the joint outcome space.
Given outcome profile x ∈ X and a subset of agents S ⊆ N , we will write xS for the outcome in
which each i ∈ S receives xi and each i 6∈ S receives ∅. Specifically, we will write x[i−1] for allocation
x with the outcomes of agents i, . . . , n set to ∅. There is a subset F ⊆ X of feasible outcomes. We
will assume that F is downward-closed, so that if x ∈ F then also xS ∈ F for all S ⊆ N .
A valuation function for agent i is a function vi : Xi → R≥0 . We will assume values are
bounded, and without loss of generality scaled to lie in [0, 1]. Each agent i’s valuation vi is drawn
independently from a publicly known distribution Di . We write D = D1 × · · · × Dn for the product
distribution over the set V = V1 × · · · × Vn of valuation profiles. We often suppress dependence on
7
D from our notation when clear from context. Agent utilities are quasilinear: if agent i receives
outcome xi and makes a payment πi , his utility
is ui = vi (xi ) − πi .
P
The welfare of outcome x is v(x) =
v
(x
i
i ). An outcome rule ALG maps each valuation
i
profile to a feasible outcome. ALGi (v) denotes the outcome of agent i on input v. We will write
OPT(v, F) = arg maxx∈F {v(x)} for the welfare-maximizing outcome rule for F, omitting the
dependence on F when it is clear from context.
Pricing Rules and Mechanisms A pricing rule is a profile of functions p = (p1 , . . . , pn ) that
assign prices to outcomes. We write pi (xi | y) for the (non-negative) price assigned to outcome
xi ∈ Xi , offered to agent i, given partial allocation y ∈ F. Define pi (xi ) = pi (xi | ∅) for convenience.
We require that pi (xi | y) = ∞ for any xi such that (xi , y−i ) 6∈ F. A pricing rule is said to be
monotone non-decreasing if pi (xi | y) ≥ pi (xi | yS ) for all i, xi ∈ Xi , y ∈ X, (xi , y−i ) ∈ F, and
S ⊆ N . In general, we allow prices to be dynamic and discriminatory. We refer to prices that
do not depend on the partial allocation (apart from feasibility) as static and to prices that do not
depend on the identity of the agent as anonymous.
A posted-price mechanism is defined by a pricing rule p and an ordering over the agents. This
pricing rule can, in general, depend on the distributions D. The agents are approached sequentially.
Each agent i is presented the menu of prices determined by pi , given all previous allocations, and
selects a utility-maximizing outcome. A posted-price mechanism is order-oblivious if it does not
require the agents to be processed in a specific order. In all of the applications we consider, the
mechanisms we construct are order-oblivious. It is well-known that every posted-price mechanism
is truthful [13].
Online Allocations and Prophet Inequalities We consider stochastic allocation algorithms
that can depend on the value distributions D. That is, an allocation algorithm A maps a value
profile and distribution to a feasible outcome. We say A is an online allocation algorithm if Ai (v, D)
does not depend on the entries of v that occur after i in some ordering over the indices. Extending
the notion of competitive ratio from the worst-case analysis of online algorithms, we’ll say the
(stochastic) competitive ratio of online allocation algorithm A is
max
D
Ev∼D [v(OPT(v))]
.
Ev∼D [v(A(v, D))]
We somtimes refer to a competitive ratio using its inverse, when convenient. A prophet inequality
for constraint F is an upper bound on the stochastic competitive ratio of an online allocation
algorithm for F. We note that a posted-price mechanism describes a particular form of an online
allocation algorithm.
3
A Framework for Prophet Inequalities
In this section we state and prove our main result, which reduces prophet inequalities to finding
balanced prices for the simpler full information setting. We say that a set of outcome profiles
H ⊆ X is exchange compatible with x ∈ F if for all y ∈ H and all i ∈ N , (yi , x−i ) ∈ F. We call a
family of sets (Fx )x∈X exchange compatible if Fx is exchange compatible with x for all x ∈ X.
Definition 3.1. Let α > 0, β ≥ 0. Given a set of feasible outcomes F and a valuation profile v,
a pricing rule p is (α, β)-balanced with respect to an allocation rule ALG, an exchange-compatible
family of sets (Fx )x∈X , and an indexing of the players i = 1, . . . , n if for all x ∈ F
8
pi (xi | x[i−1] ) ≥ α1 · v(ALG(v)) − v(OPT(v, Fx ) , and
P
′
(b) for all x′ ∈ Fx :
i∈N pi (xi | x[i−1] ) ≤ β · v(OPT(v, Fx )).
(a)
P
i∈N
The definition provides flexibility in the precise choice of Fx . As Fx becomes larger (more
permissive), both inequalities become easier to satisfy since v(OPT(v, Fx )) increases. On the other
hand, a larger set Fx means that the second condition must be satisfied for more outcomes x′ ∈ Fx .
We say that a collection of pricing rules (pv )v∈V is (α, β)-balanced if there exists a choice of (Fx )x∈X
such that, for each v, the pricing rule pv is balanced with respect to (Fx )x∈X .
The definition of (α, β)-balancedness captures sufficient conditions for a posted-price mechanism
to guarantee high welfare when agents have a known valuation profile v. Our interest in (α, β)balanced pricing rules comes from the fact that this result extends to Bayesian settings.
Theorem 3.1. Suppose that the collection of pricing rules (pv )v∈V for feasible outcomes F and
valuation profiles v ∈ V is (α, β)-balanced with respect to allocation rule ALG and indexing of the
α
players i = 1, . . . , n. Then for δ = 1+αβ
the posted-price mechanism with pricing rule δp, where
1
ṽ
· Ev [v(ALG(v))] when approaching
pi (xi | y) = Eṽ [pi (xi | y)], generates welfare at least 1+αβ
players in the order they are indexed.
Proof. We denote the exchange-compatible family of sets with respect to which the collection of
pricing rules (pv )v∈V is balanced by (Fx )x∈X . We will first use Property (b) to show a lower
bound on the utilities of the players, and Property (a) to show a lower bound on the revenue of the
posted-price mechanism. We will then add these together to obtain a bound on the social welfare.
We will write x(v) for the allocation returned by the posted-price mechanism on input valuation
profile v and x′ (v, v′ ) = OPT(v′ , Fx(v) ) for the welfare-maximizing allocation with respect to
valuation profile v′ under feasibility constraint Fx(v) .
Utility bound: We obtain a lower bound on the expected utility of a player as follows. We
′ ), F
sample valuations v′ ∼ D. Player i now considers buying OPTi ((vi , v−i
x(vi′ ,v−i ) ) at price
′
δ·pi (OPTi ((vi , v−i ), Fx(vi′ ,v−i ) ) | x[i−1] (v)). Taking expectations and exploiting that x[i−1] (v) does
not depend on vi we obtain
i
h
′
′
), Fx(vi′ ,v−i ) ) x[i−1] (v)
), Fx(vi′ ,v−i ) ) − δ · pi OPTi ((vi , v−i
E[ui (v)] ≥ E ′ vi OPTi ((vi , v−i
v
v,v
i
h
= E ′ vi′ x′i (v, v′ ) − δ · pi x′i (v, v′ ) x[i−1] (v) .
v,v
Summing the previous inequality over all agents we get
"
#
"
#
"
#
X
X
X
δ · pi x′i (v, v′ ) x[i−1] (v)
vi′ x′i (v, v′ ) − E ′
ui (v) ≥ E ′
E
v
v,v
i∈N
v,v
i∈N
i∈N
"
#
i
h
X
′
′
′
′
δ · pi xi (v, v ) x[i−1] (v) .
= E ′ v OPT(v , Fx(v) ) − E ′
v,v
v,v
(1)
i∈N
We can upper bound the last term in the previous inequality by using Property (b). This gives
h
i
X
δ · pi x′i (v, v′ ) x[i−1] (v) ≤ δβ · E ṽ OPT(ṽ, Fx(v) )
ṽ
i∈N
pointwise for any v and v′ , and therefore also
#
"
h
i
X
δ · pi x′i (v, v′ ) x[i−1] (v) ≤ δβ · E ṽ OPT(ṽ, Fx(v) ) .
E′
v,v
v,ṽ
i∈N
9
(2)
Replacing v′ with ṽ in Inequality (1) and combining it with Inequality (2) we obtain
#
"
h
i
X
ui (v) ≥ (1 − δβ) · E ṽ OPT(ṽ, Fx(v) ) .
E
v
v,ṽ
i∈N
(3)
Revenue bound: The second step is a lower bound on the revenue achieved by the posted-price
mechanism. Applying Property (a) we obtain
i
X
X h
δ · pi (xi (v) | x[i−1] (v)) = δ ·
E piṽ (xi (v) | x[i−1] (v))
i∈N
i∈N
ṽ
h
i
δ
≥ · E ṽ(ALG(ṽ)) − ṽ(OPT(ṽ, Fx(v) )) .
α ṽ
Taking expectation over v this shows
#
"
X
δ
δ
δ · pi (xi (v) | x[i−1] (v)) ≥ · E [ṽ(ALG(ṽ))] − · E ṽ(OPT(ṽ, Fx(v) )) .
E
v
α ṽ
α ṽ,v
(4)
i∈N
Combination: It remains to show how the two bounds can be combined so that they imply the
approximation guarantee. By quasi-linearity we can rewrite the expected social welfare that is
achieved by the posted-price mechanism as the sum of the expected utilities plus the expected
revenue. Using δ = α/(1 + αβ) and Inequalities (3) and (4), this gives
#
"
"
#
#
"
X
X
X
δ · pi xi (v) | x[i−1] (v)
ui (v) + E
vi (xi (v)) ≥ E
E
v
i∈N
v
v
i∈N
i∈N
δ
δ
≥ (1 − δβ) E ṽ(OPT(ṽ, Fx(v) )) + E [ṽ(ALG(ṽ))] −
E ṽ(OPT(ṽ, Fx(v) ))
v,ṽ
α ṽ
α ṽ,v
1
E [ṽ(ALG(ṽ)] .
=
1 + αβ ṽ
In what follows, we provide an alternative definition of balancedness, in which Property (b) is
refined. This definition will be useful for some applications, as exemplified in Section 4.
Definition 3.2. Let α > 0, β1 , β2 ≥ 0. Given a set of feasible outcomes F and a valuation profile
v, a pricing rule p is weakly (α, β1 , β2 )-balanced with respect to allocation rule ALG, an exchangecompatible family of sets (Fx )x∈X , and an indexing of the players i = 1, . . . , n if, for all x ∈ F,
P
1
(a)
i∈N pi (xi | x[i−1] ) ≥ α · v(ALG(v)) − v(OPT(v, Fx ) , and
P
′
(b) for all x′ ∈ Fx :
i∈N pi (xi | x[i−1] ) ≤ β1 · v(OPT(v, Fx )) + β2 · v(ALG(v)).
The following theorem specifies the refined bound on the welfare that is obtained by weakly
(α, β1 , β2 )-balanced pricing rules. Its proof appears in Appendix B.
Theorem 3.2. Suppose that the collection of pricing rules (pv )v∈V for feasible outcomes F and
valuation profiles v ∈ V is weakly (α, β1 , β2 )-balanced with respect to allocation ALG and indexing
1
the posted-price
of the players i = 1, . . . , n with β1 + β2 ≥ α1 . Then for δ = β1 +max{2β
2 ,1/α}
mechanism with pricing rule δp, where pi (xi | y) = Eṽ [pṽi (xi | y)], generates welfare at least
1
α(2β1 +4β2 ) · Ev [v(ALG(v))] when approaching players in the order they are indexed.
10
4
New and Improved Prophet Inequalities
We have already argued that our framework unifies and simplifies many of the existing prophet
inequality proofs. In this section we show how it can be used to derive new and improved bounds
on the approximation ratio that can be obtained via price-based prophet inequalities. We highlight
two results: the new poly-time O(d)-approximation for combinatorial auctions with bundle size at
most d, and the new poly-time constant-approximation for knapsack problems. Additional results
include combinatorial auctions with MPH-k valuations (see Appendix D), d-sparse packing integer
programs (see Appendix E) and multi-dimensional matroids (where the result follows from the
Rota exchange theorem [36, Lemma 2.7] and our composition results, see Appendix F).
Combinatorial Auctions with Bounded Bundle Size An existential O(d)-approximate pricebased prophet inequality is presented in Section 1.1. Combined with the O(d)-approximation greedy
algorithm for this setting, it gives a poly-time O(d2 )-approximate price-based prophet inequality
(as shown in [23]). In what follows we use the flexibility of our framework to work directly with a
relaxation of the allocation problem, thereby improving the approximation of the prophet inequality
from O(d2 ) to O(d). This is a special case of Theorem 1.1, which is proved in Appendix D.
Theorem 4.1. For combinatorial auctions where every agent can get at most d items, there exist
weakly (1, 1, d − 1)-balanced item prices that are static, anonymous, and order oblivious. Moreover,
a (4d − 2 − ǫ)-approximate posted-price mechanism can be computed in poly(n, m, 1/ǫ) demand
queries, where ǫ is an additive error due to sampling.
Proof. Consider the canonical fractional relaxation of the combinatorial auction problem:
a feasible
P
allocationPis described by values xi,S ∈ [0, 1] for all i ∈ N and S ⊆ M such that S xi,S ≤ 1 for
all i and i,S∋j xi,S ≤ 1 for all j ∈ M . TakePF to be all such fractional allocations, and
P Fx to be
the set of fractional allocations y such that i,S∋j (xi,S + yi,S ) ≤ 1 for all j ∈ M , and S yi,S ≤ 1
for all i. As usual, we think of Fx as the set of allocations that remain feasible given a partial
allocation x.
Consider the following pricing rule for fractional allocations. Given valuation
v, let x∗
P profile
P
be the welfare-maximizing fractional allocation. Then for each item j, set pj = i S∋j x∗i,S vi (S).
We claim that these prices are (1, 1, d − 1)-balanced P
with respect to the optimal allocation rule.
For Property (a), fix some x ∈ F. Write xj = i,S∋j xi,S . Consider the following allocation
y ∈ Fx : for each S, choose jS ∈ arg maxj∈S {xj }. Set yi,S = (1 − xjS ) · x∗i,S . We think of y as the
optimal allocation x∗ adjusted downward to lie in Fx . We then have that
X
X
XX
X
X
pi (xi ).
v(x∗ ) − v(y) =
xjS · x∗i,S · vi (S) =
xj
x∗i,S · vi (S) ≤
xj · p j =
i
j
S
j
i,S : j=jS
i
Property (a) follows since v(y) ≤ v(OPT(v, Fx )). For Property (b), fix x ∈ F and x′ ∈ Fx . Then
X
X
X
X
X
X
pi (x′i ) ≤
(1 − xj )pj =
(1 − xj )
x∗i,S · vi (S) =
x∗i,S · vi (S)
(1 − xj )
i
j
=
j
i,S∋j
i,S
j∈S
X
X
X
xj .
(|S| − 1)x∗i,S · vi (S) +
x∗i,S · vi (S) · 1 −
i,S
i,S
j∈S
The first expression on the RHS is at most (d − 1)v(OPT(v)), since |S| ≤ d whenever x∗i,S > 0.
For the second expression, note that it is at most the welfare of the allocation y defined by yi,S =
11
P
x∗i,S · (1 − j∈S xj )+ . Moreover, this allocation y is in Fx . So the second expression is at most
v(OPT(v, Fx )), giving Property (b).
Theorem 3.2 therefore yields prices that guarantee a (4d − 2) approximation for the fractional
allocation problem, and an ǫ-approximation to those prices can be computed via sampling. To
complete the proof, note that for every agent i, if all previous agents have selected integral outcomes,
then agent i also has a utility-maximizing outcome that is integral. This is because any fractional
allocation can be interpreted as a convex combination of integral allocations. These same prices
therefore guarantee a (4d − 2 − ǫ) approximation even if the mechanism prohibits non-integral
allocations from being purchased.
The more general Theorem 1.1 also improves the best-known polytime prophet inequality for
XOS valuations from 2e/(e − 1) to 2 (which is tight [23]) and for MPH-k valuations it improves the
best known polytime bounds from O(k2 ) to O(k).
Knapsack In the knapsack allocation problem, there is a single divisible unit of resource and each
agent has a private value vi ≥ 0 for receiving at least si ≥ 0 units. Assume for now that si ≤ 1/2
for all i. We allow both vi and si to
P be private information, drawn from a joint distribution. In
our notation: Xi = [0, 12 ], F = {x | i xi ≤ 1}, and vi (xi ) = vi if xi ≥ si and vi (xi ) = 0 otherwise.
Based on an arbitrary allocation algorithm ALG, we design anonymous, static prices by setting
pi (xi | y) = xi · v(ALG(v)) if xi can feasibly be added and ∞ otherwise. The following restates the
second half of Theorem 1.2.
Theorem 4.2. For the knapsack allocation problem in which no single agent can request more than
half of the total capacity, the prices above are (1, 2)-balanced with respect to ALG. This implies a
(3 + ǫ)-approximate polytime posted-price mechanism with a single static anonymous per-unit price.
Proof. The polytime claim follows from Theorem 3.1 with ALG set to the classic
for knapP FPTAS
1
sack [31], so it suffices to prove balancedness. For any x ∈ F, let Fx = F if i xi < 2 , and Fx = ∅
otherwise.
Note that Fx is exchange compatible with x since, for any x′ ∈ Fx and any agent k,
P
′
xk + i xi ≤P1. To establish balancedness with respect to (Fx )x , we consider two cases based on
the value of i xi .
P
1
Case 1:
Property (a) is trivially fulfilled because v(ALG(v)) − v(OPT(v, Fx )) ≤
i xi < 2 .
v(OPT(v)) − v(OPT(v, Fx )) = 0. For Property b, note that for any x′ ∈ Fx , we have
X
X
pi (x′i | x[i−1] ) =
x′i · v(ALG(v)) ≤ v(ALG(v)) ≤ v(OPT(v)) = v(OPT(v, Fx )).
i
Case 2:
X
i
i
P
i xi
≥ 21 . Property (b) is vacuous since Fx = ∅. For Property (a), we have
pi (xi | x[i−1] ) =
X
i
1
1
xi · v(ALG(v)) ≥ v(ALG(v)) = (v(ALG(v)) − v(OPT(v, Fx ))).
2
2
We can remove the restriction that si ≤ 1/2 as follows, completing the proof of Theorem 1.2.
Consider the contribution to the expected optimal welfare separated into welfare from agents with
si ≤ 1/2, and agents with si > 1/2. The posted-price mechanism described above obtains a
3-approximation to the former. For the latter, a mechanism that treats the unit of resource as
indivisible, and posts the best take-it-or-leave-it price for the entire unit, is a 2-approximation.
This is because at most one agent with si > 1/2 can win in any realization. Thus, for any
distribution profile, one of these two mechanisms must be a 5-approximation to the unrestricted
12
knapsack problem.7 One can therefore obtain a (5 + ǫ)-approximate price-based prophet inequality
by estimating the expected welfare of each pricing scheme (via sampling) and selecting the better of
the two. In Appendix E we show how to generalize the result for the knapsack problem to d-sparse
packing integer programs.
Finally, consider the fractional version of the knapsack problem, where agents obtain partial
value for receiving a portion of their desired allocation: vi (xi ) = vi · min{xi /si , 1}. If we restrict
allocations xi to be multiples of some δ > 0, this is a special case of a submodular combinatorial
auction with ⌈1/δ⌉ identical items. Since Theorem 1.1 implies that a fixed per-item price yields
a 2-approximation for any δ, we can infer by taking the limit as δ → 0 that for any ǫ > 0 there
is a (2 + ǫ)-approximate polytime posted-price mechanism for the fractional knapsack problem,
with a single static anonymous per-unit price, even if each agent’s size si is private and arbitrarily
correlated with their value. As mentioned in Section 1.4, this improves the previously best-known
prophet inequality of ≈ 11.657 due to [24].
5
From Price of Anarchy to Prophet Inequalities
In this section we explore the connection between balanced prices and mechanism smoothness.
While generally smoothness does not suffice to conclude the existence of a posted-price mechanism
with comparable welfare guarantee (see Appendix G), we will show that this is the case for typical
smoothness proofs and present pretty general reductions from the problem of proving prophet
inequalities to mechanism smoothness.
We first recall the definition of a smooth mechanism. A (possibly indirect) mechanism Mπ for
an allocation problem π is defined by a bid space B = B1 × · · · × Bn , an allocation rule f : B → F,
and a payment rule P : B → Rn≥0 . We focus on first-price mechanisms, where Pi (b) = bi (f (b)).
Typically, mechanisms are defined for a collection of problems Π, in which case we will simply refer
to the mechanism as M.
Definition 5.1 (Syrgkanis and Tardos [45]). Mechanism Mπ is (λ, µ)-smooth for λ, µ ≥ 0 if for
any valuation profile v ∈ V and any bid profile b ∈ B there exists a bid b′i (v, bi ) ∈ Bi for each
player i ∈ N such that
X
X
ui (b′i , b-i ) ≥ λ · v(OPT(v)) − µ ·
Pi (b).
i∈N
i∈N
A mechanism M that is (λ, µ)-smooth has a Price of Anarchy (with respect to correlated and
Bayes-Nash equilibria) of at most max{µ, 1}/λ [45].
The following formal notion of a residual market will be useful for our further analysis. For
any x ∈ F we define the contraction of F by x, F/x, as follows. Let N + (x) = {i ∈ N | xi 6= ∅}.
Then F/x = {z = (zj )j∈N \N + (x) | (z, xN + (x) ) ∈ F}. That is, F/x contains allocations to players
who were allocated nothing in x, that remain feasible when combined with the allocations in x.
We think of the contraction by x as a subinstance on players N \ N + (x) with feasibility constraint
F/x, and refer to it as the subinstance induced by x. We say that a collection of problems Π is
subinstance closed if for every π ∈ Π with feasible allocations F and every x ∈ F the subinstance
induced by x is contained in Π. The contraction by x also naturally leads to an exchange feasible
set Fx by padding the allocations z ∈ F/x with null outcomes. We refer to this Fx as the canonical
exchange-feasible set.
7
The worst case is when both mechanisms achieve the same expected welfare, which occurs if 3/5 of the expected
welfare is due to agents with si ≤ 1/2. The expected welfare of each mechanism is then 13 · 35 = 51 of the optimum.
13
5.1
Warm-up: Binary, Single-Parameter Problems with Monotone Prices
We begin with a simple result that serves to illustrate the connection between balancedness and
smoothness. We will show that if a binary, single-parameter problem has the property that the
welfare-maximizing mechanism is (λ, µ)-smooth and its critical prices τi ( · | y) are non-decreasing
in y,8 then there exists a pricing rule that is (α, β)-balanced, where αβ = O(max{µ, 1}/λ). In
particular, this implies that the welfare guarantee due to Theorem 3.1 is within a constant factor
of the Price of Anarchy of the mechanism implied by smoothness.
Theorem 5.1. Consider a subinstance-closed collection of binary, single-parameter problems such
that the first-price mechanism based on the welfare maximizing allocation rule OPT is (λ, µ)-smooth.
If the critical prices τi ( · | y) are non-decreasing in y then setting pi (1 | y) = max{vi , τi (v-i | y)}
)-balanced with respect to OPT and the canonical exchange-feasible
and pi (0 | y) = 0 is (1, µ+1+λ
λ
sets (Fx )x∈X .
Proof. Fix any y and x ∈ Fy . Observe that by definition of the prices, it holds that
pi (xi | y) ≥ v(OPT(v, F(∅,y−i ) )) − v(OPT(v, F(xi ,y−i ) )).
(5)
To see this, first note that both sides of the inequality are equal to 0 if xi = 0. If xi = 1 and
vi ≥ τi (v-i | y), then agent i is allocated in OPT(v, F(∅,y−i ) ) and hence both sides of the inequality
are equal to vi . If xi = 1 and vi < τi (v-i | y), then agent i is not allocated in OPT(v, F(∅,y−i ) ),
and hence the right-hand side of the inequality is at most the externality imposed by forcing an
allocation to agent i, which is at most τi (v-i | y) = pi (xi | y).
We are now ready to prove balancedness. To verify Condition (a), choose x ∈ F and note that
n
X
i=1
n
X
v(OPT(v, Fx[i−1] )) − v(OPT(v, Fx[i] )) = v(OPT(v)) − v(OPT(v, Fx ))
pi (xi | x[i−1] ) ≥
i=1
as required, where the inequality follows from Equation (5), and the equality follows by a telescoping
sum. For Condition (b), we get
X
τi (v-i | x[i−1] ) ≤
i∈x′
X
τi (v-i | x) ≤
i∈x′
µ+1
v(OPT(v, Fx )),
λ
(6)
where the first inequality follows by the monotonicity of critical prices, and the second inequality
follows by a known implication of smoothness [19] (see Appendix I.1). Therefore, for any x′ ∈ Fx ,
X
X
X
pi (x′i | x[i−1] ) ≤
vi +
τi (v-i | x[i−1] )
i
i∈x′
i∈x′
µ+1
v(OPT(v, Fx ))
λ
µ+1+λ
≤
v(OPT(v, Fx )),
λ
≤ v(x′ ) +
where the first inequality follows by replacing the maximum in the definition of the prices by a
sum, the second inequality follows by Equation (6), and the last inequality follows by v(x′ ) ≤
v(OPT(v, Fx )), since x′ ∈ Fx .
8
The critical price τi (v-i | y) is the infimum of values vi such that the mechanism allocates 1 to agent i on input
(vi , v-i ), in the problem subinstance induced by y.
14
5.2
General Problems and Outcome Smoothness
We proceed to show an implication from smoothness to prices that works in more general settings.
It is based on the observation that many smoothness proofs proceed by showing that agent i could
bid b′i to get some target outcome x∗i . We capture proofs that proceed in this manner through the
following notion of outcome smoothness. Similar but different notions were considered in [16, 40].
Definition 5.2. A mechanism is (λ, µ)-outcome smooth for λ, µ ≥ 0 if for all valuation profiles
v ∈ V there exists an outcome x′ (v) ∈ F such that for all bid profiles b ∈ B,
X
X
′
′
Pi (b).
Pi (bi , b-i ) ≥ λ · v(OPT(v)) − µ ·
vi (xi ) − ′
inf
′
′
i∈N
bi : fi (bi ,b-i )xi
i∈N
We show that if a first-price, declared welfare maximizing mechanism (i.e., a mechanism with
allocation rule f (b) = OPT(b)) is (λ, µ)-outcome smooth and has non-decreasing critical prices,
then the critical prices for that mechanism (from the definition of outcome smoothness) can be
used as posted prices that yield an O(λ/µ) approximation to the optimal welfare. Recall that these
critical prices are different from the first-price payments that make up the mechanism’s payment
rule. This result has a mild technical caveat: we require that the mechanism continues to be smooth
in a modified problem with multiple copies of each bidders. An allocation is feasible in the modified
feasibility space F ′ if it corresponds to a feasible allocation x ∈ F, with each xi being partitioned
between the copies of agent i.
Theorem 5.2. Fix valuation space V and feasibility space F, and suppose F ′ is an extension of F
as defined above. Suppose that the first-price mechanism based on the declared welfare maximizing
allocation rule for valuation space V and feasibility space F ′ has non-decreasing critical prices,
and is (λ, µ)-outcome smooth for every F ′ /z. Then there is a collection of exchange-feasible sets
(Fx )x∈X , and an allocation rule ALG that returns the welfare-maximization allocation with probability λ, such that for every v ∈ V there exists a pricing rule that is (λ, µ/λ)-balanced with respect
to ALG and (Fx )x∈X .
Theorem 5.2 implies that posting (an appropriately scaled version of) the critical prices from
the outcome smooth mechanism yields a welfare approximation of O(λ/µ), matching the Price of
Anarchy guarantee of the original mechanism. The proof of Theorem 5.2 appears in Appendix H.
5.3
Binary, Single-Parameter Problems
We conclude with two general “black-box reductions” for binary single-parameter settings, in which
agents can either win or lose, which show how to translate PoA guarantees of O(γ) provable via
(regular) smoothness into O(γ 2 )-approximate posted-price mechanisms. Proofs appear in Appendix
I. The key to both these results is a novel, purely combinatorial implication of smoothness for the
greedy allocation rule proved in Lemma I.2.
Theorem 5.3. Suppose that the first-price mechanism based on the greedy allocation rule GRD
has a Price of Anarchy of O(γ) provable via smoothness, then there exists a O(γ 2 )-approximate
price-based prophet inequality.
Theorem 5.4. Suppose that the first-price mechanism based on the declared welfare maximizing
allocation rule OPT has a Price of Anarchy of O(γ) provable via smoothness, then there exists a
O(γ 2 )-approximate price-based prophet inequality.
15
We note that Theorem 5.1 applied to matroids (using known smoothness results for pay-yourbid greedy mechanisms over matroids [39]) implies the existence of (1, 3)-balanced prices and hence
a 4-approximate prophet inequality. A strengthening of Theorem 5.4 for monotonically increasing
critical prices (discussed in Remark I.2) leads to an improved factor of 2, matching the prophet
inequality for matroids shown in Kleinberg and Weinberg [33]. This also captures the classic singleitem prophet inequality, as a special case.
6
Conclusions and Open Problems
We introduced a general framework for establishing prophet inequalities and posted price mechanisms for multi-dimensional settings. This work leaves many questions open.
A general class of questions is to determine the best approximation guarantee that a prophet
inequality can achieve for a particular setting. For example, even for the intersection of two matroids
there is a gap between the trivial lower bound of 2 and the upper bound of 4k − 2 = 6. Similarly,
in subadditive combinatorial auctions, the best-known upper bound is logarithmic in the number
of items m [23], but again the best-known lower bound is 2, inherited from the case of a single
item. Notably, the price of anarchy for simultaneous single-item auctions is known to be constant
for subadditive valuations [22], but the proof does not use the smoothness framework and hence
our results relating posted prices to smooth mechanisms do not directly apply.
A related question is whether there exist prophet inequalities that cannot be implemented
using posted prices. Interestingly, we are not aware of any separation between the two so far.
More generally, one could ask about the power of anonymous versus personalized prices, item
versus bundle prices, static versus dynamic prices, and so on. For example, to what extent can
static prices approximate the welfare under a matroid constraint, an intersection of matroids, or
an arbitrary downward-closed feasibility constraint?
Regarding the pricing framework itself, it would be interesting to extend the notion of (α, β)balancedness to allow randomization in a dynamic pricing rule, and to understand the additional
power of randomization. One could also generalize beyond feasibility constraints to more general
seller-side costs for allocations. For the connection between smoothness and balancedness, we leave
open the question of removing the price-monotonicity condition from Theorem 5.2, or whether the
approximation factors can be improved for our single-parameter reductions (Theorems 5.3 and 5.4).
Finally, recent work has shown that smoothness guarantees often improve as markets grow large [25];
is there a corresponding result for balancedness?
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120–129, 2016.
[41] T. Roughgarden. Intrinsic robustness of the price of anarchy. Journal of the ACM, 62(5):32,
2015.
[42] A. Rubinstein. Beyond matroids: Secretary problem and prophet inequality with general
constraints. In Proceedings of the 48th ACM Symposium on Theory of Computing, 2016. 324–
332.
[43] A. Rubinstein and S. Singla. Combinatorial prophet inequalities. In Proceedings of the 28th
ACM-SIAM Symposium on Discrete Algorithms, pages 1671–1687, 2017.
[44] E. Samuel-Cahn. Comparison of threshold stop rules and maximum for independent nonnegative random variables. Annals of Probability, 12:1213–1216, 1984.
[45] V. Syrgkanis and É. Tardos. Composable and efficient mechanisms. In Proceedings of the 45th
ACM Symposium on Theory of Computing, pages 211–220, 2013.
[46] L. Trevisan. Non-approximability results for optimization problems on bounded degree instances. In Proceedings of the 33rd ACM Symposium on Theory of Computing, pages 453–461,
2001.
19
A
Classic Prophet Inequality via Balanced Prices
In this appendix we show how to re-derive the classic prophet inequality via our framework in
Section 3. Specifically, we show the existence of a (1, 1)-balanced pricing rule as defined in Definition
3.1. Theorem 3.1 then shows the factor 2-approximation.
P
In the classic setting we have Xi = {0, 1} for all i and F = {x | i xi ≤ 1}. We set pi (1 | x) =
maxℓ vℓ if x does not allocate the item, ∞ otherwise; pi (0 | x) = 0 for all x. This corresponds to a
fixed posted price of maxℓ vℓ on the item.
Claim A.1. These prices are (1, 1)-balanced with respect to OPT and Fx defined by Fx = F if x
does not allocate the item and Fx = ∅ otherwise.
Proof. Let x be an arbitrary allocation profile. If x allocates the item, then Fx = ∅ and thus
Condition (b) is trivially P
fulfilled. For Condition (a), we observe that v(OPT(v, Fx )) = 0 and that
v(OPT(v)) = maxℓ vℓ = i pi (xi | x[i−1] ), because exactly one buyer pays maxℓ vℓ . If x does not
allocate the item, then v(OPT(v, Fx )) = v(OPT(v)), making Condition (a) trivial.
For Condition
P
′
(b), we use that in x′ at most one buyer is allocated the item. Therefore,
p
(x
i i i | x[i−1] ) ≤
maxℓ vℓ = v(OPT(v)) = v(OPT(v, Fx )).
B
Proof of Theorem 3.2
Our proof will follow the same steps as the one for Theorem 3.1. As in that poof, denote the
exchange-compatible family of sets with respect to which the collection of pricing rules (pv )v∈V is
balanced by (Fx )x∈X , and let x(v) be the allocation returned by the posted-price mechanism on
input valuation profile v. Let x′ (v, v′ ) = OPT(v′ , Fx(v) ) be the allocation that maximizes welfare
with respect to valuation profile v′ over feasibility constraint Fx(v) .
Utility bound: Again sample valuations v′ ∼ D. By the same reasoning as in the proof of
Theorem 3.1, we obtain
"
#
"
#
X
X
′ ′
.
(7)
δ · pi x′i (v, v′ ) x[i−1] (v)
ui (v) = E ′ v (x (v, v′ )) − E ′
E
v
i∈N
v,v
v,v
i∈N
We upper bound the last term in the previous inequality by using Property (b). This gives pointwise
for any v and v′
X
′
′
δ · pi xi (v, v ) x[i−1] (v) ≤ δβ1 · E ṽ(OPT(ṽ, Fx(v) )) + δβ2 · E [ṽ(ALG(ṽ))] ,
ṽ
i∈N
and therefore also
"
X
δ · pi x′i (v, v′ )
E′
v,v
i∈N
ṽ
#
≤ δβ1 · E ṽ(OPT(ṽ, Fx(v) ) + δβ2 · E [ṽ(ALG(ṽ)] .
x[i−1] (v)
v,ṽ
v,ṽ
Replacing v′ with ṽ in Inequality (7) and combining it with Inequality (8) we obtain
#
"
X
ui (v) ≥ (1 − δβ1 ) · E ṽ(OPT(ṽ, Fx(v) ) − δβ2 · E [ṽ(ALG(ṽ)] .
E
v
i∈N
v,ṽ
v,ṽ
20
(8)
(9)
Revenue bound: Again, applying Property (a) we obtain
#
"
X
δ
δ
δ · pi (xi (v) | x[i−1] (v)) ≥ · E [ṽ(ALG(ṽ))] − · E ṽ(OPT(ṽ, Fx(v) ) .
E
v
α ṽ
α ṽ,v
(10)
i∈N
Combination: To combine our bounds, we distinguish whether β2 ≥
1
2α .
1
We use that point-wise for every i ∈ N we have ui (v) ≥ 0. Therefore, ui (v) ≥
Case 1: β2 ≥ 2α
1
1
ρui (v) for all 0 ≤ ρ ≤ 1. Using δ = β1 +2β
, ρ = 2αβ
, and Inequalities (9) and (10), we get
2
2
E
v
"
X
i∈N
#
vi (xi (v)) ≥ ρ · E
v
"
X
#
ui (v) + E
i∈N
v
"
X
i∈N
δ · pi xi (v) | x[i−1] (v)
≥ ρ(1 − δβ1 ) · E ṽ(OPT(ṽ, Fx(v) ) − ρδβ2 · E [ṽ(ALG(ṽ)]
v,ṽ
=
Case 2: β2 <
E
v
1
2α
"
i∈N
#
v,ṽ
δ
δ
+ · E [ṽ(ALG(ṽ))] − · E ṽ(OPT(ṽ, Fx(v) )
α ṽ
α ṽ,v
1
· E [ṽ(ALG(ṽ)] .
α(2β1 + 4β2 ) ṽ
Now, we use δ =
X
#
vi (xi (v)) ≥ E
v
"
1
β1 +1/α .
X
i∈N
Inequalities (9) and (10) yield
#
ui (v) + E
v
"
X
i∈N
δ · pi xi (v) | x[i−1] (v)
#
≥ (1 − δβ1 ) · E ṽ(OPT(ṽ, Fx(v) ) − δβ2 · E [ṽ(ALG(ṽ)]
v,ṽ
+
v,ṽ
δ
δ
· E [ṽ(ALG(ṽ))] − · E ṽ(OPT(ṽ, Fx(v) )
α ṽ
α ṽ,v
1 − αβ2
· E [ṽ(ALG(ṽ)]
1 + αβ1 ṽ
1
E [ṽ(ALG(ṽ)] ,
≥
α(2β1 + 4β2 ) ṽ
=
where the last step uses that β1 + β2 ≥
C
1
α
and β2 <
1
2α .
Composition Results
In this section we show that balanced prices are composable, in the sense that balanced prices
for separate markets remain balanced when the markets are combined. We consider two forms of
composition. The first is a composition of preferences: it shows how to extend from a class of
valuations V to any maximum over valuations from V . The second shows how to compose allocations across different markets, where agents have additive preferences across markets. Together
these two composition results capture XOS composition, in the sense of [45]. Our theorems apply
to balancedness with respect to OPT, and they extend to general approximation algorithms ALG
under mild conditions.
21
Closure under Maximum Given an arbitrary valuation space Vi for player i, we consider
its extension Vimax , which contains all functions vimax : Xi → R for which there is a finite set
{vi1 , . . . , vim } ∈ Vi such that vimax (xi ) = maxℓ viℓ (xi ) for all xi ∈ Xi . We say that a valuation profile
ṽ ∈ V is a supporting valuation profile for allocation x and valuation profile v ∈ V max if ṽi ≤ vi
and ṽi (xi ) = vi (xi ) for all i.
Definition C.1. Allocation rule ALG is consistent if for every v ∈ V max and corresponding supporting valuation profile ṽ ∈ V for ALG(v), we have ṽ(ALG(ṽ)) ≥ ṽ(ALG(v)).
Lemma C.1. The optimal allocation rule OPT is consistent.
Proof. Since OPT(v) is a feasible outcome under ṽ, it holds that ṽ(OPT(ṽ)) ≥ ṽ(OPT(v)) by
optimality.
Theorem C.1. Suppose that for each v ∈ V there exists a pricing rule pv that is (α, β)-balanced
with respect to v, consistent allocation rule ALG, and the exchange-compatible family of sets (Fx )x∈X .
For each v ∈ V max let ṽ ∈ V be a supporting valuation profile for ALG(v). Then the pricing rule
pṽ is (α, β)-balanced with respect to v, ALG, and (Fx )x∈X .
Proof. We first establish Property (a). We use Property (a) of the original pricing rules, plus the
definition of a supporting valuation profile to conclude that
X
pṽi (xi | x[i−1] ) ≥
i∈N
1
· (ṽ(ALG(ṽ)) − ṽ(OPT(ṽ, Fx ))
α
1
· ṽ(ALG(v)) − v(OPT(ṽ, Fx )
α
1
≥ · v(ALG(v)) − v(OPT(v, Fx ) .
α
≥
Next we establish Property (b). By Property (b) of the original pricing rule and the definition
of a supporting valuation profile,
X
pṽi (x′i | x[i−1] ) ≤ β · ṽ(OPT(ṽ, Fx ))
i∈N
≤ β · v(OPT(ṽ, Fx ))
≤ β · v(OPT(v, Fx )).
Closure under Addition Suppose there are m separate allocation problems, each with feasibility constraint F ℓ over allocation space X ℓ = X1ℓ × . . . × Xnℓ . The joint problem is then defined over
the product allocation space X = X 1 × . . . × X m , with feasibility constraint F = F 1 × . . . × F m . We
say that a valuation vi : X → R is additive if it is defined by a (not necessarily additive) valuation
function viℓ for each P
outcome xℓi ∈ Xiℓ , and for an outcome xi = (x1i , . . . , xm
i ) ∈ Xi , the value of xi
ℓ (xℓ ).
is given by vi (xi ) = m
v
ℓ=1 i i
Theorem C.2. Suppose that v is additive over a set of allocation problems F 1 , . . . , F m , and
ℓ
for each individual allocation problem there exists a pricing rule pv that is (α, β)-balanced with
respect to vℓ , allocation rule ALGℓ on F ℓ , and the exchange-compatible family of sets (Fxℓ ℓ )xℓ ∈X ℓ .
Pm v ℓ
is (α, β)-balanced with respect to v and ALG on F, where
Then the pricing rule p =
ℓ=1 p
ALG(v) = (ALG1 (v1 ), . . . , ALGm (vm )).
22
Q
ℓ
Proof. Set Fx = ( m
i=1 Fxℓ )x∈X . We first verify Condition (a). We use the definition of the
joint pricing rule, Condition (a) of the component pricing rules together with additivity across
subproblems to conclude that
X
pi (xi | x[i−1] ) =
i∈N
≥
m
XX
ℓ
pvi (xℓi | xℓ[i−1] )
i∈N ℓ=1
m
X
ℓ=1
=
1
· vℓ (ALGℓ (vℓ )) − vℓ (OPT(vℓ , Fxℓ )
α
1
· (v(ALG(v)) − v(OPT(v, Fx ))) .
α
Next we verify Condition (b). We use the definition of the joint pricing rule, Condition (b) of
the component pricing rules together with additivity across subproblems to conclude that
X
pi (x′i
| x[i−1] ) =
i∈N
≤
m X
X
ℓ
pvi ((x′ )ℓi | xℓ[i−1] )
ℓ=1 i∈N
m
X
β · vℓ (OPT(vℓ , Fxℓ ))
ℓ=1
= β · v(OPT(v, Fx )).
We note that both Theorem C.1 and Theorem C.2 can be generalized so that they apply to
weakly balanced pricing rules.
D
Proof of Theorem 1.1
In this appendix we establish our new prophet inequality results for combinatorial auctions with
MPH-k valuations. In particular, we establish the existence of (1, 1)-balanced prices for XOS
combinatorial auctions.
Combinatorial Auctions with MPH-k Valuations The maximum over positive hypergraphs
(MPH) hierarchy of valuations [21] is an inclusive hierarchy (i.e., it is expressive enough to include
all valuations), which subsumes many interesting classes of valuations as special cases.
To formalize this valuation class, we first need a few preliminaries. A hypergraph
P representation
M
w of valuation function v : 2 → R≥0 is a set function that satisfies v(S) = T ⊆S w(T ). Any
valuation function v admits a unique hypergraph representation and vice versa. A set S such that
w(S) 6= 0 is said to be a hyperedge of w. Pictorially, the hypergraph representation can be thought
as a weighted hypergraph, where every vertex is associated with an item in M , and the weight of
each hyperedge e ⊆ M is w(e). Then the value of the function for any set S ⊆ M , is the total
value of all hyperedges that are contained in S. The rank of a hypergraph representation w is the
cardinality k of the largest hyperedge. The rank of v is the rank of its corresponding w and we refer
to a valuation function v with rank k as a hypergraph-k valuation. If the hypergraph representation
of v is non-negative, i.e. for any S ⊆ M , w(S) ≥ 0, then we refer to function v as a positive
hypergraph-k function (PH-k) [2].
Definition D.1 (Maximum Over Positive Hypergraph-k (MPH-k) class [21]). A monotone valuation function v : 2M → R≥0 is Maximum over Positive Hypergraph-k (MPH-k) if it can be expressed
23
as a maximum over a set of PH-k functions. That is, there exist PH-k functions {vℓ }ℓ∈L such that
for every set S ⊆ M ,
v(S) = maxℓ∈L vℓ (S),
(11)
where L is an arbitrary index set.
An important special case of MPH-k are XOS valuations, which are defined as the maximum
over additive functions, and therefore coincide with MPH-1.
Existential O(k) Result Given an arbitrary allocation algorithm ALG and valuation profile
v, write ṽ for the supporting Hypergraph-k valuations
Pfor allocation ALG(v). Write wi for the
hypergraph representation of ṽi , so that ṽi (ALG(v)) = T ⊆ALGi (v) wi (T ).
Then, for each item j, we define
X
pv ({j}) =
wi (T ).
(12)
T ∋j
T ⊆ALGi (v)
That is, item prices are determined by adding up the weights of each supporting hyperedges an
item
of goods x, set pv (x) =
P isvcontained in. These prices then extend linearly: For each set
v
v
j∈x p ({j}). Finally, for each agent i and allocation xi , we will have pi (xi | z) = p (xi ) whenever
xi is disjoint from the sets in z and ∞ otherwise.
Theorem D.1. The pricing rule defined in Equation 12, extended linearly to sets of items, is
weakly (1, 1, k − 1)-balanced with respect to an arbitrary allocation rule ALG and MPH-k valuation
profile v.
S
S
Proof. We will show balancedness with respect to Fx = {y ∈ F : ( i yi ) ∩ ( i xi ) = ∅}. Observe
that we can lower-bound the value of OPT(v, Fx ) by removing all items that are allocated by x
from the allocation ALG(v). This gives us
X
X X
[
v(OPT(v, Fx )) ≥
vℓ (ALGℓ (v) \ ( xi )) ≥
wℓ (T ).
(13)
i
ℓ∈N
ℓ T ⊆ALGℓ (v)
∀i : T ∩xi =∅
Furthermore, note first that pv is a fixed pricing scheme, meaning that the price of an outcome
does not change unless it becomes infeasible. Therefore, for every allocation y ∈ Fx , we have
X
X
pi (yi | x)
pi (yi | x[i−1] ) =
i
i
=
XX
i
=
j∈yi
XX
i
=
pv ({j})
pv ({j})
ℓ j∈ALGℓ (v)∩yi
XX
ℓ
X
i
X
j∈ALGℓ (v)∩yi
24
X
T ∋j
T ⊆ALGℓ (v)
wℓ (T ).
(14)
For condition (a), by Equation (14),
X
X
pvi (xi | x[i−1] ) ≥
i
ℓ
=
X
wℓ (T )
T ⊆ALGℓ (v)
∃i : T ∩xi 6=∅
X
X
wℓ (T ) −
ℓ T ⊆ALGℓ (v)
X
ℓ
X
T ⊆ALGℓ (v)
∀i : T ∩xi =∅
wℓ (T )
≥ (v(ALG(v)) − v(OPT(v, Fx ))),
where the first inequality follows by changing the order of summation over j and T , and the second
inequality is given in Equation 13.
For condition (b), note that for all x and all x′ ∈ Fx , by Equation (14), after splitting the sum
depending on whether the respective set T intersects with any of the bundles in x or not
X
XX
X
X
X
XX
X
wℓ (T ).
wℓ (T ) +
pi (x′i | x[i−1] ) =
i
ℓ
i
j∈ALGℓ (v)∩x′i
T ∋j
T ⊆ALG
S ℓ (v)
T ∩( i′ xi′ )=∅
i
ℓ
j∈ALGℓ (v)∩x′i
T ∋j
T ⊆ALG
S ℓ (v)
T ∩( i′ xi′ )6=∅
Observe that in the first sum for a fixed set T , the term wℓ (T ) occurs at most |T | times. In the
second sum, it can even occur only |T | − 1 times because the intersection of x and x′ is empty but
x intersects with T . By applying that |T | ≤ k whenever wℓ (T ) > 0, this gives us
X
X
X
X
X
(|T | − 1)wℓ (T )
|T |wℓ (T ) +
pi (x′i | x[i−1] ) ≤
i
ℓ
≤k
X
ℓ
=
X
ℓ
ℓ
T ⊆ALG
S ℓ (v)
T ∩( i′ xi′ )=∅
X
T ⊆ALG
S ℓ (v)
T ∩( i′ xi′ )6=∅
wℓ (T ) + (k − 1)
ℓ
T ⊆ALG
S ℓ (v)
T ∩( i′ xi′ )=∅
X
X
wℓ (T ) + (k − 1)
X
X
wℓ (T )
T ⊆ALG
S ℓ (v)
T ∩( i′ xi′ )6=∅
X
wℓ (T )
ℓ T ⊆ALGℓ (v)
T ⊆ALG
S ℓ (v)
T ∩( i′ xi′ )=∅
≥ v(OPT(v, Fx )) + (k − 1)v(ALG(v)).
Note that Theorem D.1 when specialized to XOS valuations shows the existence of weakly
(1, 1, 0)-balanced prices, or equivalently, (1, 1)-balanced prices.
Computational O(k) Result We now show how to obtain a polytime (4k − 2)-approximate
price-based prophet inequality for MPH-k valuations. For this result we will assume access to the
following kind of MPH-k oracle. Suppose that valuation function v is MPH-k, with supporting
PH-k functions {vℓ }ℓ∈L . The query for v takes as input a set of items S, and returns a value oracle
to access the PH-k function vℓ for which v(S) = vℓ (S). That is, for every T ⊆ M , we can query
the value of vℓ (T ) in a second step.
The following linear problem, known as the configuration LP for combinatorial auctions, computes a fractional allocation that maximizes the social welfare among all fractional allocations.
25
max
X
vi (S) · xi,S
i,S
s.t.
X
xi,S ≤ 1 for every i ∈ N
S
X
xi,S ≤ 1 for every j ∈ M
i,S:j∈S
xi,S ∈ [0, 1] for every i ∈ N, S ⊆ M
We extend the definition of F to include all fractional allocations that fulfill the above
P LP; Fx
is extended to all fractional allocations that use at most a fractional equivalent of 1 − i,S:j∈S xi,S
of every item j ∈ M . Our first observation will be that Theorem D.1 holds even for the fractional
version of the MPH-k combinatorial auction problem. That is, if the set of feasible outcomes is
extended to include all fractional allocations, then an appropriate extension of the prices described
above remain weakly (1, 1, k − 1)-balanced. In particular, given a fixed valuation profile v, we
define prices based on the optimal LP solution x∗ . For every agent i and every bundle S, let
wi,S be the
agent
of S, which is
P PH-k representation in the support that maximizes P
P i’s∗valuation
P
vP
=
w
(T
).
Then,
price
item
j
as
follows:
p({j})
=
x
i (S)
i,S
i
S i,S
T :j∈T,T ⊆S wi,S (T ) =
P ∗ T ⊆S
x
(w
(S)
−
w
(S
\
{j})).
This
sum
can
be
computed
in
polynomial
time given oracle
i,S
i,S
i
S i,S
∗
access as described above because xi,S 6= 0 only for polynomially many S. Then, extend this pricing
linearly to prices over sets of items, and to prices
allocations by simple scaling, i.e.,
Pover fractional
P
for every x ∈ F and x′ ∈ Fx , we let p(x′i | x) = S x′i,S j∈S p({j}).
It is well known that an optimal fractional solution to the configuration LP can be computed in
polynomial time given access to demand queries (using demand queries to implement a separation
oracle, as is standard). Therefore, for the space of fractional allocations, prices that are weakly
(1, 1, k − 1)-balanced with respect to the optimal fractional solution can be computed in polynomial
time, given access to demand and MPH-k oracles.
We can therefore compute prices that give a (4k − 2)-approximation to the optimal fractional
allocation, in a posted price mechanism, where agents are free to purchase any fractional allocation
at the posted prices. This, however, seems unsatisfactory. After all, we do not wish to allow agents
to purchase infeasible sets, and the analysis only applies if every agent can purchase a set in her
demand correspondence. The following observation comes to our help: for every agent i, if all
previous agents have selected integral outcomes from the posted price mechanism, then agent i has
a utility-maximizing outcome in their demand correspondence that is integral. This is because any
fractional allocation can be interpreted as a convex combination of integral allocations. Since our
approximation guarantee holds regardless of the demanded set chosen by each agent, it will hold
even if we restrict agents to only select integral outcomes in the posted price mechanism. This
means that we only need to price integral allocations and the analysis follows. We conclude that
the prices computed using the configuration LP, as described above, actually generate a (4k − 2)
approximation for the (non-fractional) MPH-k combinatorial auction problem.
E
Proof of Theorem 1.3
In this appendix we prove our polytime price-based prophet inequalities for sparse linear packing
programs. We consider programs with and without integrality constraints. That is, the possible
outcomes for agent i are either [0, 1] (fractional solutions) or {0, 1} (integral solutions). We assume
26
that valuations are linear, i.e., vi (xi ) = vi · xi . In both cases, the feasibility constraints F are given
such that x ∈ F if and only if A · x ≤ c. Without loss of
by a constraint matrix A ∈ Rm×n
≥0
generality, let cj = 1 for all j ∈ [m].
We assume that aj,i ≤ 12 for all i, j and that the column sparsity is bounded by d, meaning that
for each i there are at most d choices of j such that aj,i > 0.
such that
Theorem E.1. For d-sparse linear packing programs with constraint matrix A ∈ Rm×n
≥0
1
aj,i ≤ 2 for all i, j and unit capacities, there exist weakly (1, 0, d)- and (2, 0, d)-balanced prices, for
fractional and integral solutions, respectively, with respect to arbitrary allocation algorithms ALG.
The prices can be computed by running ALG once.
We will define static prices pi (xi | y) P
as follows. Let x∗ = ALG(v). For each constraint
P j, we
define a per-unit price ρj by setting ρj = i∈N :aj,i >0 vi x∗i . Now, we define pi (xi | y) = j aj,i xi ρj
for every quantity xi that can feasibly be added to y. The claim will now follow by the two lemmas
below.
Lemma E.1. For F being all fractional solutions, the devised pricing scheme is weakly (1, 0, d)balanced with respect to ALG.
Proof. Let Fx = {z | A(x + z) ≤ 1}. To verify Condition (a), we derive a lowerPbound on
v(OPT(v, Fx )) as follows. Given x, let z be defined by setting zi = x∗i (1 − maxj:aj,i >0 i′ aj,i′ xi′ ).
We have z ∈ Fx because for every constraint j we have
X
X
X
X
X
aj,i xi +
aj,i zi ≤
aj,i xi +
aj,i x∗i (1 −
aj,i′ xi′ ) ≤ 1.
i
i
i
i′
i
P
Note that furthermore, by this definition, x∗i′ ( i aj,i xi ) ≥ x∗i′ − zi′ for all i′ and all j.
Now, it follows that
XX
X
pi (xi | x[i−1] ) =
aj,i xi ρj
i
i
j
=
XX
=
X
i
aj,i xi
X
vi′ x∗i′
i′ :aj,i′ >0
j
j
X
i′ :aj,i′ >0
X
aj,i xi )
vi′ x∗i′ (
≥
X
X
vi′ (x∗i′ − zi′ )
≥
X
j
i
i′ :aj,i′ >0
vi′ (x∗i′ − zi′ )
i′
≥ v(ALG(v)) − v(OPT(v, Fx )).
For Condition (b), we simply observe that
X
XX
pi (x′i | x[i−1] ) =
aj,i x′i ρj
i
i
=
j
X
ρj
j
j
≤
X
X
ρj
j
27
aj,i x′i
=
X
j
=
X
vi x∗i
i∈N :aj,i >0
X X
i
≤d
vi x∗i
j:aj,i >0
X
vi x∗i = d · v(ALG(v)).
i
Lemma E.2. For F being all integral solutions, the devised pricing scheme is weakly (2, 0, d)balanced with respect to ALG.
P
Proof. To define Fx , let c be the vector such that cj = 1 if i aj,i xi ≤ 21 and cj = 0 otherwise.
That is, cj = 1 if and only if constraint j still has capacity at least 21 after adding x. Let Fx be the
set of integral solutions y that fulfill Ay ≤ P
c.
with aj,i > 0 and let zi = 0
Let z be defined such that zi = x∗i if i′ aj,i′ xi′ ≤ 12 for all j P
∗
otherwise.
By this definition, we have zi ≥ xi (1 − 2 maxj:aj,i >0 i′ aj,i′ xi′ ). For this reason,
P
x∗i′ ( i aj,i xi ) ≥ 12 (x∗i′ − zi′ ) for all i′ and all j.
From here on, we can follow the exact same calculations as above. For Condition (a), we have
X
XX
pi (xi | x[i−1] ) =
aj,i xi ρj
i
i
j
=
XX
=
X
j
i′ :aj,i′ >0
≥
X
X
i
j
≥
X
aj,i xi
vi′ x∗i′
i′ :aj,i′ >0
j
X
i′ :aj,i′ >0
X
aj,i xi )
vi′ x∗i′ (
i
1
vi′ (x∗i′ − zi′ )
2
1X
vi′ (x∗i′ − zi′ )
2 ′
i
1
≥ v(ALG(v)) − v(OPT(v, Fx )) .
2
For Condition (b), we observe that again
X
XX
pi (x′i | x[i−1] ) =
aj,i x′i ρj
i
i
=
j
X
ρj
X
j
=
X
≤d
X
ρj
j
vi x∗i
i∈N :aj,i >0
X X
i
aj,i x′i ≤
j
j
=
X
vi x∗i
j:aj,i >0
X
vi x∗i = d · v(ALG(v)).
i
Pricing Integral Problems based on Fractional Solutions The way we described the pricing schemes above was to use an offline allocation algorithm ALG for the respective problem. This
28
algorithm has to solve an NP-hard problem and as we show any approximation guarantee is preserved in the process. However, it is also possible to use the respective optimal fractional solution
instead of ALG. The pricing schemes defined this way then achieve the described approximation
guarantees with respect to the fractional optimum. To show this result, we have to slightly extend our framework and allow Fx to contain distributions over outcome profiles that are exchange
compatible.
Specifically, for the integral part of Theorem E.1 we would define Fx as the
Pset of distributions
1
such that the expected consumption in the jth constraint is at most 1 if
i aj,i xi ≤ 2 and 0
otherwise. Note that this extends the definition of Fx in the proof of Lemma E.2 to distributions.
Following the steps above, z is now
based on the optimal fractional solution x∗ . We
P randomized
1
∗
let zi = 1 with probability xi if i′ aj,i′ xi′ ≤ 2 for all j with aj,i > 0 and let zi = 0 otherwise. In
the remainder, zi′ is replaced by its expectation.
F
Proof of Theorem 1.4
We consider a matroid feasibility constraint M = (E, I), where E is a ground set of elements and
I ⊆ 2E is a family of feasible subsets. Set E is partitioned into subsets E1 , . . . , En , corresponding
to agents, and for each i, the outcome space Xi contains all subsets of Ei . One can think of Ei
as
S the set of elements from which agent i can choose. An allocation9 x is feasible if and only if
Given a valuation profile
i xi ∈ I; that is, if the chosen elements form an independent set.
v, let OPT(v) ∈ I denote a social-welfare maximizing allocation. Furthermore, given S ∈ I, let
OPT(v | S) denote the set T ∈ I maximizing v(T ) such that S ∩ T = ∅ and S ∪ T ∈ I. Theorem
F.1 asserts the existence of balanced prices for matroid settings with additive, submodular, and
XOS valuations.
Theorem F.1. There exist prices for multi-dimensional matroid settings in which agents have
additive, submodular, or XOS valuations that are (1, 1)-balanced with respect to OPT.
As a special case, we obtain (1, 1)-balanced prices for the single-dimensional matroid setting of
[33], where each agent controls exactly one element of the ground set.
We prove the
P theorem for additive valuations. That is, there are non-negative numbers vi,j such
that vi (xi ) = j∈xi vi,j . The proof for submodular and XOS Valuations then follows directly from
Theorem C.1 (composition theorem: closure under maximum).
Additive Valuations Given S ∈ I, i ∈ N , and a valuation profile v, define
(
S
S
S
v(OPT(v | j yj )) − v(OPT(v | ( j yj ∪ xi ))) , if ( j yj ∪ xi ) ∈ I
v
pi (xi | y) =
∞
, otherwise.
(15)
The following lemma shows that the prices in Equation 15 are monotonically increasing. We
then prove in Theorem F.2 that these prices are (1, 1)-balanced.
Lemma F.1. Consider the pricing rule pv . For an arbitrary profile v, feasible outcome profiles
y, y′ ∈ F such that yj ⊆ yj′ for all j, agent i, and allocation xi ,
pvi (xi | y) ≤ pvi (xi | y′ ).
9
Kleinberg and Weinberg [33] also provide bounds for a multi-dimensional matroid setting. Their result, which
has implications for revenue and welfare, applies when buyers have unit-demand preferences with independent values
across elements. Our approach is different, and provides a welfare bound for a broader class of valuations.
29
Proof. Let S =
Otherwise,
S
j
yj , T =
S
j
yj′ . If S ∪xi 6∈ I, then T ∪xi 6∈ I, and so pvi (xi | y) = pvi (xi | y′ ) = ∞.
pvi (xi | y) = v(OPT(v | S)) − v(OPT(v | (S ∪ xi )))
≤ v(OPT(v | T )) − v(OPT(v | (T ∪ xi )))
≤ pvi (xi | y′ ),
where the equation follows by the definition of pvi (xi | y), and the inequality follows from the
submodularity of the function f (U ) = v(OPT(v | U )) (cf. [32, Lemma 3]).
Theorem F.2. The pricing rule defined in Equation 15 is (1, 1)-balanced with respect to the optimal
allocation rule OPT.
S
S
Proof.SWe define
F
as
the
set
of
all
outcome
profiles
y
∈
E
×
.
.
.
×
E
such
(
y
)
∩
(
x
1
n
i
i
S
Si xi ) = ∅
and ( i yi ) ∪ ( i xi ) ∈ I. In other words, we consider the matroid contracted by the set i xi .
We first show that Condition (a) holds for α = 1. For every agent i ∈ N and every x ∈ F by a
telescoping-sum argument,
i
i−1
[
X
X
[
v(OPT(v |
xj ))
pvi (xi | x[i−1] ) =
xj )) − v(OPT(v |
i
i
j=1
j=1
= v(OPT(v)) − v(OPT(v |
[
xj ))
j
= v(OPT(v)) − v(OPT(v, Fx )).
We next showSthat the second condition holds for β = 1. Consider some arbitrary x ∈ F,
x′ ∈ Fx . Let S = i xi . Note that OPT(v | S) is precisely a maximum-weight basis of the matroid
contracted with S. By the generalized Rota exchange theorem [36, Lemma 2.7], for each i there
exists some set Ri ⊆ OPT(v | S) (which may not be contained in Ei ) such that each element of
OPT(v | S) appears in exactly one Ri , and (OPT(v | S) \ Ri ) ∪ x′i is an independent set in the
matroid after contracting S. We therefore have
X
X
pvi (x′i | x)
pvi (x′i | x[i−1] ) ≤
i∈N
i∈N
=
X
v(OPT(v | S)) − v(OPT(v | S ∪
i∈N
≤
X
i∈N
=
X
x′i ))
v(OPT(v | S)) − v(OPT(v | S) \ Ri )
v(Ri )
i∈N
= v(OPT(v | x)),
where the first inequality follows from the monotonicity of the prices (Lemma F.1), and the second
inequality follows by observing that (OPT(v) \ Ri ) ∪ x′i is feasible in the contracted matroid (see
the Rota exchange argument above), and thus v(OPT(v) \ Ri ) ≤ v(OPT(v | S ∪ x′i )). The final
equality follows by additivity. We conclude that the suggested prices are (1, 1)-balanced.
30
Computational Aspects Theorem F.1 establishes the existence of a price-based 2-approximate
prophet inequality for additive, submodular, and XOS preferences. For additive valuations the
construction is polytime, as the greedy algorithm is optimal. For submodular and XOS valuations
computing the optimal allocation is NP-hard.
We claim that the result for submodular valuations can turn into a computational one, by
basing the prices on an approximately optimal allocation. Namely, let GRD be the algorithm that
allocates items greedily by value, always choosing the item that locally increases v(x) the most
subject to the matroid constraint. We show that GRD is consistent (see Definition C.1) in this
setting.
Lemma F.2. The greedy algorithm GRD is consistent for XOS valuations over matroid feasibility
constraints.
Proof. Fix an XOS valuation profile v, and let ṽ be a supporting valuation profile for the allocation
GRD(v). Note that as the supporting valuation profile is additive, GRD returns the optimal solution.
As GRD(v) is another feasible solution, we have ṽ(GRD(ṽ)) ≥ ṽ(GRD(v)).
Theorem C.1 (closure under maximum) now implies that the prices given in (15) for an additive
valuation supporting GRD(v) are (1, 1)-balanced with respect to the greedy allocation. We can
compute these prices in polynomial time by simulating GRD and then determining the supporting
additive valuation. Since GRD is a 2-approximation to OPT for submodular valuations, Theorem 3.1
then implies that we can compute prices that yield a 4-approximation to the optimal expected
welfare, less an additive sampling error.
G
A Smooth Mechanism Without Good Posted Prices
In this appendix we show that there are allocation problems that admit constant-factor smooth
mechanisms, but for which no posted-price mechanism can guarantee more than a linear fraction
of the optimal social welfare.
Proposition G.1. There exists a downward-closed welfare maximization problem that admits a
(1, 0)-smooth mechanism, but for which any posted price mechanism has approximation factor Ω(n).
Proof. Let k be a positive integer to be fixed later. In the allocation problem we consider, Xi =
[k]n ∪ {∅} for each i ∈ [n]. That is, each agent is allocated either ∅ or a sequence of n integers
between 1 and k. For xi ∈ Xi \{∅}, write xi = (xi1 , . . . , xin ), where each xij ∈ [k]. The set of
feasible allocations is F = {x : (xi 6= xj ) =⇒ ((xi = ∅) ∨ (xj = ∅))}. That is, all agents who
receive a non-empty allocation must receive the same allocation.
Agents have the following valuations. Each agent i has some desired value zi ∈ [k]. The value
of an allocation is vi (xi ) = 1 if xii = zi , and vi (xi ) = 0 otherwise. Let Di be the distribution over
such valuations in which zi is chosen uniformly from [k].
For this feasibility constraint and space of valuations, consider the mechanism that returns the
welfare-optimal allocation and charges payments of 0. This mechanism simultaneously satisfies all
desires by allocating (z1 , . . . , zn ) to every agent, where zi is the desired value reported by agent i.
This mechanism is (1, 0)-smooth, with truth-telling being the required deviation.
On the other hand, consider any posted-price mechanism. Whichever is the first agent to obtain
a non-∅ outcome, say agent i purchasing allocation xi , each subsequent agent j can obtain positive
value only if zj = xij , which occurs with probability 1/k. We therefore have that the expected
welfare of any posted price mechanism is at most 1 + n−1
k . Taking k = n and noting that the
31
optimal welfare is n, we conclude that no posted price mechanism can obtain more than an Ω(n)
approximation to the optimal welfare.
H
Proof of Theorem 5.2
To prove Theorem 5.2, we will first introduce the notion of weak outcome smoothness, then
show how to construct the prices, and finally establish the two conditions necessary for (α, β)balancedness separately.
H.1
Weak Outcome Smoothness and Its Properties
We begin by defining a more general notion of outcome smoothness, corresponding to the notion
of weak smoothness.
Definition H.1. A mechanism is weakly (λ, µ1 , µ2 )-outcome smooth for λ, µ1 , µ2 ≥ 0 if for all
valuation profiles v ∈ V there exists an outcome x′ (v) ∈ F such that for all bid profiles b ∈ B,
X
X
′
′
,
b
)
≥
λ
·
v(OPT(v))
−
µ
·
P
(b
Pi (b) − µ2 · b(f (b)).
vi (xi ) − ′
inf
i
-i
1
i
′
′
bi : fi (bi ,b-i )xi
i∈N
i∈N
The following observation will facilitate our discussion in what follows. Under a mild scale
invariance assumption, we can without loss of generality consider (λ, 0, µ)-outcome smooth mechanisms.
Proposition H.1. Consider a (λ, µ1 , µ2 )-outcome smooth mechanism with allocation rule f and
payment rule P such that for every δ > 0, we have x′ (v/δ) = x′ (v). For every player i ∈ N and
all bid profiles b ∈ B define
b(f (b))
, 1 · Pi (b).
P̃i (b) = min P
i∈N Pi (b)
Then the mechanism with allocation rule f and payment rule P̃ is (λ, 0, µ1 + µ2 )-outcome smooth.
Proof. Given a valuation profile v, we claim that the re-defined mechanism is (λ, 0, µ1 +µ2 )-outcome
smooth with respect to the same outcome x′ (v). n
o
Fix a bid profile b ∈ B and let δ = min Pb(f (b))
,
1
. Note that δ ≤ 1. Using outi∈N Pi (b)
come smoothness of the original mechanism for valuation profile v/δ and corresponding outcome
x′ (v/δ) = x′ (v) we obtain
X
X 1
1
′
′
P
(b
,
b
)
≥
λ
·
· vi (xi ) − ′
inf
·
v(OPT(v))
−
µ
·
Pi (b) − µ2 · b(f (b)).
i
-i
1
i
δ
δ
bi : f (b′i ,b-i )x′i
i∈N
i∈N
Multiplying by δ, we have
X
inf
vi (x′i ) − δ · ′
′
X
′
P
(b
,
b
)
≥
λ
·
v(OPT(v))
−
µ
·
δ
·
Pi (b) − µ2 · δ · b(f (b)).
i
-i
1
i
′
bi : f (bi ,b-i )xi
i∈N
i∈N
P
From the definition of P̃ and δ we know that δ · Pi (b′i , b-i ) = P̃i (b′i , b-i ), δ · i∈N Pi (b) ≤ b(f (b)),
and δ · b(f (b)) ≤ b(f (b)). This yields
X
′
′
P̃i (bi , b-i ) ≥ λ · v(OPT(v)) − (µ1 + µ2 ) · b(f (b)).
vi (xi ) − ′
inf
′
i∈N
bi : f (bi ,b-i )xi ′
32
Another observation is that the function x′ of a weakly (λ, µ1 , µ2 )-outcome smooth mechanism
is indeed a λ-approximation of social welfare, which also implies that its range λ-approximates the
space of all outcome profiles.
Lemma H.1. Suppose Y is the range of x′ . Then
X
X
max
vi (yi ) ≥
vi (x′ (v)) ≥ λ · v(OPT(v)).
y∈Y
i∈N
i∈N
Proof. We use outcome smoothness setting the valuation profile to v and the bid profile to 0. Then
for y = x′ (v),
X
′
Pi (bi , 0) ≥ λ · v(OPT(v)) .
vi (yi ) − ′ inf
′
bi : f (bi ,0)yi
i∈N
As we assume that payments are never negative, this implies
X
vi (yi ) ≥ λ · v(OPT(v)) .
i∈N
H.2
Constructing the Prices
We first have to define pi (xi | z) for all i ∈ N , z ∈ F, xi ∈ Xi . To this end, we create two additional
copies of each agent, implying that we have 3n agents overall. The different incarnations of agent
i, denoted by i, i + n, and i + 2n correspond to different roles when setting the prices. They
particularly ensure that agent i competes against itself.
In more detail, for i ∈ [n], the roles will be as follows in defining pi (xi | z). Agent i is used
to represent vi , agent i + n is used to represent zi and agent i + 2n tries to buy outcome xi in
the outcome-smooth mechanism. We will frequently map outcome profiles z ∈ F from the original
space of n agents to the agents n + 1, . . . , 2n. This operation we denote by z. That is z i = zi−n for
i ∈ {n + 1, . . . , 2n} and z i = ∅ otherwise.
We will adopt the more general notion of weak outcome smoothness, from Appendix H.1. We
assume that outcome smoothness holds in every subinstance F ′ /z of the extended outcome space
F ′ . To avoid any possible confusion of the numbering of the agents, we denote by Gz , the outcome
space F ′ /z padded with ∅ wherever zi 6= ∅. That is, Gz is isomorphic to F ′ /z but uses the exact
same indices as F ′ . We will denote the allocation rule of the outcome smooth mechanism in this
space by f ( · | z) and its payment rule by P ( · | z).
Outcome smoothness now ensures that for all z ∈ F ′ and valuation profiles v ∈ V there exists
an outcome x′ (v | z) ∈ Gz such that for all bid profiles b ∈ B,
3n
X
i=1
vi (x′i )
−
inf
b′i : fi ((b′i ,b-i )|z)x′i
Pi ((b′i , b-i )
| z) ≥ λ · v(OPT(v, Gz )) − µ · b(f (b | z)).
Given these definitions, we define prices based on the payments in the outcome-smooth mechanism by
Pi (b′i+2n , v−(i+2n) ) z .
pvi (xi | z) =
inf
′
′
bi+2n : fi ((bi+2n ,v−(i+2n) )|z)xi
The meaning is as follows: Agents n + 1, . . . , 2n incorporate the allocation z and agents 1, . . . , n
represent the part of v(OPT(v)) that is not yet taken away by z. Now, another copy of agent i is
added to the system and competes with agents 1, . . . , n for the outcome.
33
H.3
Showing Balancedness
Let ALG denote the algorithm that returns OPT(v) with probability λ. We will show that the
constructed prices are (λ, µ/λ)-balanced with respect to ALG if they are monotone. To this end,
we define Fx as the range of x′ ( · | z) on agents 2n + 1, . . . , 3n. Formally,
Fx = {y ∈ X1 × . . . × Xn | ∃v : x′ (v | x) = y},
where y is the outcome profile on 3n agents that sets y2n+i = yi for i ∈ [n] and yi = ∅ for i ≤ 2n.
Lemma H.2. Suppose (f ( · | z), P ( · | z)) is (λ, 0, µ)-outcome smooth on every Gz such that x′ (v |
z) = x′ (ǫ · v | z) for every ǫ > 0, pvi is monotonically increasing, then pvi satisfies Condition (b) of
(α, β)-balancedness with β = µ/λ, for the choice of (Fx )x∈X described above.
Proof. Consider x and y ∈ Fx . By definition of Fx , we can find some v′ such that x′ (v′ ) = y.
Recall that, in allocation y, agents 1 through 2n obtain the empty allocation, and agents 2n + 1
through 3n receive allocation profile y.
By assumption for all ǫ > 0 we have x′ (ǫ · v′ ) = y . Outcome smoothness for problem subinstances, with valuation profile ǫ · v′ and bid profile v implies
!
3n
X
Pi ((b′i , v-i ) | x) ≥ λ · ǫ · v′ (OPT(ǫ · v′ , Gx )) − µ · v(f (v | x)) .
ǫ · vi′ (y i ) −
inf
b′i : fi ((b′i ,v-i )|x)y i
i=1
As payments Pi are never negative, this implies
!
3n
3n
X
X
′
′
′
ǫ·vi′ (y i ) .
−
inf
Pi ((bi , v-i ) | x) ≥ λ·ǫ·v (OPT(ǫ·v , Gx ))−µ·v(f (v | x))−
i=2n+1
b′i : fi ((b′i ,v-i )|x)y i
i=1
This implies
n
X
i=1
pvi (yi | x) =
3n
X
i=2n+1
b′i :
inf
fi ((b′i ,v-i )|x)y i
≤ µ · v(f (v | x)) − ǫ
Pi ((b′i , v-i ) | x)
3n
X
vi′ (y i ) −
′
′
!
λ · v (OPT(ǫ · v , Gx )) .
i=1
As this holds for all ǫ > 0, we also have
n
X
pvi (yi | x) ≤ µ · v(f (v | x)) ≤ µ · v(OPT((v1 , . . . , vn , 0, . . . , 0), Gx )) ≤
i=1
µ
v(OPT(v, Fx )) ,
λ
where the last step uses Lemma H.1.
Lemma H.3. Suppose (f ( · | z), P ( · | z)) is (λ, 0, µ)-outcome smooth on every Gz such that x′ (v |
z) = x′ (ǫ · v | z) for every ǫ > 0, f is the declared welfare maximizer, and P is the first-price
payment rule. Then pvi satisfies Condition (a) of (α, β)-balancedness with α = λ, for the choice of
ALG and (Fx )x∈X described above.
Proof. Consider an arbitrary player i and arbitrary outcomes x ∈ F. To bound pi (xi | x[i−1] ),
we consider player 2n + i in the outcome-smooth mechanism. Let b2n+i be a bid such that
f2n+i (b2n+i , v−(2n+i) | x[n+i−1] ) xi . We show that for any such b2n+i , we have
Pi (b2n+i , v−(2n+i) | x[n+i−1] ) ≥ v(OPT(v, Gx[n+i−1] )) − v(OPT(v, Gx[n+i] )) .
34
As this holds for all b2n+i , this gives also a lower bound on the infimum.
In the following, we keep the allocation x[n+i−1] = (∅, . . . , ∅, x1 , . . . , xi−1 , ∅, . . . ∅) fixed and com(i−1)
pare the two possible feasible solutions a := f (b2n+i , v−(2n+i) | x[n+i−1] ) and q := OPT(v, Gx[n+i−1] ).
As a maximizes (b2n+i , v−(2n+i) ) when keeping x[n+i−1] fixed, we have
b2n+i (a2n+i ) +
3n
X
vj (aj ) ≥ b2n+i (q2n+i ) +
j=1
j6=2n+i
3n
X
vj (qj ) .
j=1
j6=n+i
As vj = 0 for all j > n, this is equivalent to
b2n+i (a2n+i ) ≥ b2n+i (q2n+i ) +
n
X
vj (qj ) −
j=1
n
X
vj (aj ) .
j=1
Now, observe that if we replace a2n+i by xi in allocation a, then the modified vector a′ is still
of players 2n + i and n + i, this also
feasible in the space that keeps x[n+i−1] fixed. By symmetry
P
means that (a1 , . . . , an , ∅, . . . , ∅) ∈ Gx[n+i] . This implies nj=1 vj (aj ) ≤ v(OPT(v, Gx[n+i] )).
Overall, we get
pvi (xi | x[i−1] ) ≥ v(OPT(v, Gx[n+i−1] )) − v(OPT(v, Gx[n+i] )) .
Summing up the prices for all players i and using a telescoping sum, this implies a lower bound of
n
X
pvi (xi | x[i−1] ) ≥ v(OPT(v, Gx∅ )) − v(OPT(v, Gx[n] ))
i=1
= v(OPT(v)) − v(OPT(v, Gx )) .
Finally, we have v(OPT(v)) = λ1 v(ALG(v)) and v(OPT(v, Gx )) ≤ λ1 v(OPT(v, Fx )) by Lemma H.1.
So, in combination
n
X
pvi (xi | x[i−1] ) ≥
i=1
I
1
(v(ALG(v)) − v(OPT(v, Fx ))) .
λ
Proofs of Theorem 5.3 and Theorem 5.4
In this appendix we describe how to obtain balanced prices from smooth mechanisms in binary,
single-parameter settings. In these settings players can either “win” or ”lose”, and have a value
vi ∈ R≥0 for winning. Feasible solutions x ∈ F ⊆ {0, 1}n are subsets of players that can win
simultaneously. For ease of notation we identify the vectors x ∈ F with the subset of players i ∈ N
for which xi = 1. This lets us write i ∈ x if xi = 1 and i 6∈ x otherwise.
I.1
Permeable Allocation Rules
We begin by defining the permeability of an allocation rule f , and by showing that (λ, µ)-smoothness
implies a bound on permeability.
An algorithm f for a binary, single-parameter problem is monotone if for every player i ∈ N ,
any two bids b′i ≥ bi , and any bid vector b-i ,
fi (bi , b-i ) = 1
⇒
fi (b′i , b-i ) = 1.
For monotone allocation rules the critical value for player i is the smallest bid that ensures that
player i wins against bids b-i . That is, τif (b-i ) = inf{bi | fi (bi , b-i ) = 1}.
35
Definition I.1 (Dütting and Kesselheim [19], see also [38, 45, 30]). A monotone allocation algorithm f for a binary, single parameter problem F is γ-permeable if γ ≥ 1 is the smallest multiplier
such that for all bid vectors b and all feasible allocations x ∈ F it holds that
X
τif (b-i ) ≤ γ · b(f (b)).
(16)
i∈N : xi =1
Theorem I.1 (Dütting and Kesselheim [19]). Suppose that the first-price mechanism M based on
allocation f is (λ, µ)-smooth, then f is γ-permeable with γ ≤ (µ + 1)/λ.
P
Proof. Given a bid vector b, we have to show that i: xi =1 τif (b-i ) ≤ µ+1
λ · b(f (b)). To this end,
consider fixed x ∈ F and b. Let ǫ > 0 and let v be defined by vi = max{bi , τif (b-i )}. By smoothness
of M, there are b′i such that
X
X
ui (b′i , b-i ) ≥ λ · v(OPT(v)) − µ ·
Pi (b).
i∈N
i∈N
τif (b-i )
ui (b′i , b-i )
Observe that if i 6∈ f (b), then
≤ 0 because
> bi and this means that i 6∈ f (b′i , b-i )
results in positive utility. Furthermore, for i ∈ f (b), we have
unless b′i > vi . None of these choices
P
ui (b′i , b-i ) ≤ vi = bi . Therefore i∈N ui (b′i , b-i ) ≤ b(f (b)).
Next, we can lower-bound v(OPT(v)) by the value of the feasible solution x, which gives us
P
P
P
v(OPT(v)) ≥ i: xi =1 vi ≥ i: xi =1 (τif (b-i ) −Pǫ) ≥ i: xi =1 τif (b-i ) − nǫ.
Finally, as M is first-price, we also have i∈N Pi (b) = b(f (b)).
In combination this yields
!
X f
τi (b-i ) − nǫ − µ · b(f (b)),
b(f (b)) ≥ λ ·
i: xi =1
which implies
X
i: xi =1
τif (b-i ) ≤
µ+1
· b(f (b)) + nǫ.
λ
As this holds for all ǫ > 0, this shows the claim.
Applying Theorem I.1 to each problem in a collection of problems Π, we see that if a mechanism
M is (λ, µ)-smooth for Π then it is (µ + 1)/λ-permeable for Π.
Remark I.1. While the definition of permeability requires γ to be the smallest multiplier for which
inequality (16) is satisfied, all our results can be derived from any upper bound on this multiplier
at the cost of slightly worse guarantees.
I.2
Proof of Theorem 5.3
In this subsection we prove Theorem 5.3, which shows that (λ, µ)-smoothness of the greedy allocation rule for a subinstance-closed closed collection of binary, single-parameter problems Π implies
the existence of a weakly ((µ + 1)/λ, 0, (µ + 1)/λ)-balanced pricing rule. By Theorem I.1, in order
to show this result, it suffices to show the following theorem.
Theorem I.2. Let ALG be any allocation rule. Suppose that the greedy allocation rule GRD is
γ-permeable for a subinstance-closed collection of binary, single-parameter feasibility problems Π.
Then for every v ∈ V there exists a pricing rule that is weakly (γ, 0, γ)-balanced with respect to
ALG and the canonical exchange-feasible sets (Fx )x∈X .
We first describe the pricing rule that achieves this result. Afterwards, we show that this pricing
rule has the desirable properties.
36
I.2.1
Construction of the Prices
We set the price pi (zi | x) for player i ∈ N and outcome zi ∈ {0, 1} for arbitrary but fixed valuations
v and allocation x ∈ F through Algorithm 1. For this let GRD(v | x) ∈ Fx denote the allocation
that results if we go through the players in order of non-increasing value but only add a player if
he is not in x and feasible together with x and the previously accepted players.
We generally set pi (0 | x) = 0. That is, the price for losing is always zero. To determine the
price pi (1 | x) for winning we first compute a sequence of reference allocations r(0) ≥ · · · ≥ r(n) and
a sequence of reference valuations v(0) ≥ · · · ≥ v(n) . We then set pi (1 | x) = vi if i ∈ r(n) , i.e., the
price of player i is that player’s valuation if he is part of the final reference allocation. Otherwise,
(n)
we set pi (1 | x) = inf{vi′ : i ∈ GRD(vi′ , v−i | x)}, i.e., we set the price to the player’s critical value
against the players in the final reference allocation.
While we need to define prices pi (zi | x) for any possible allocation x ∈ F, the prices that player
i will actually see are the ones where x is set to the purchase decisions of the players j = 1, . . . , i − 1
that precede player i in the ordering. Note that in this case xi = xi+1 = xn = 0 and therefore
(i−1)
r(n) = · · · = r(i−1) and v(n) = · · · = v(i−1) . We use the shorthand τiGRD (v−i | x[i−1] ) := inf{vi′ :
(i−1)
i ∈ GRD(vi′ , v−i
| x[i−1] )}.
Algorithm 1: Pricing Rule Derived from GRD (Parametrized by ALG)
Input: zi ∈ {0, 1}, v, x ∈ F
Output: pi (zi | x)
if zi = 0 then
// In this case the price is simply zero
pi (zi | x) = 0
else
// First determine reference allocation and valuations
(0)
(0)
r(0) ← ALG(v), vk ← vk if k ∈ r(0) and vk ← 0 otherwise
for j ← 1 to n do
(j)
(j−1)
(j)
= vk if k ∈ r(j) and vk ← 0 else
r(j) ← GRD(v(j−1) | x[j] ), vk ← vk
// Now determine the price
if i ∈ r(n) then
// If player i is part of the reference allocation he pays his valuation
pi (zi | x) ← vi
else
// Otherwise he pays the critical value against the players in the reference allocation
(n)
pi (zi | x) ← inf{vi′ : i ∈ GRD(vi′ , v−i | x)}
return pi (zi | x)
I.2.2
Proof of Theorem I.2
We prove the theorem in two steps. We first use permeability of the greedy allocation rule to
establish Condition (a) (in Lemma I.1). We then show Condition (b). For this we first prove a
novel combinatorial implication of permeability of the greedy allocation rule (in Lemma I.2) by
considering valuations that are either zero or one. We then use this property in a careful layering
37
argument to establish Condition (b) (in Lemma I.3).
Lemma I.1. Let ALG be any allocation rule. Suppose that the greedy allocation rule GRD is γpermeable for a subinstance-closed collection of binary, single-parameter problems Π. Then the
pricing rule described in Algorithm 1 fulfills Condition (a) of Definition 3.2 with α = γ with respect
to allocation rule ALG and the canonical exchange-feasible sets (Fx )x∈X .
Proof. Let x ∈ F. We will show that pi (xi | x[i−1] ) ≥ γ1 · (v(r(i−1) ) − v(r(i) )). By a telescoping-sum
argument, this then implies
X
X1
(i−1)
(i)
pi (xi | x[i−1] ) ≥
· v(r
) − v(r )
γ
i∈N
i∈N
1
1
(0)
(n)
= · v(r ) − v(r ) ≥ · v(ALG(v)) − v(OPT(v, Fx )) ,
γ
γ
where the last step follows from the fact that r(0) = ALG(v) and r(n) ∈ Fx .
So, it only remains to show pi (xi | x[i−1] ) ≥ γ1 · (v(r(i−1) ) − v(r(i) )). Observe that if xi = 0,
we have r(i−1) = r(i) and this claim follows trivially. So, consider an arbitrary player i for which
xi = 1. If i ∈ r(i−1) then r(i−1) \ r(i) = {i}. So pi (1 | x[i−1] ) = vi , while v(r(i−1) ) − v(r(i) ) = vi and
the claim is true.
Otherwise, i 6∈ r(i−1) , and we will first use smoothness with respect to subinstances to bound
(i−1)
the size of the set r(i−1) \ r(i) . For a fixed ǫ > 0, define v′ by setting vi′ = τiGRD (v−i | x[i−1] ) + ǫ,
vj′ = vj for j ∈ r(i−1) \ r(i) , and vj′ = 0 for all other j.
Now player i ∈ GRD(v′ | x[i−1] ∪r(i) ) by definition of v′ and vi′ in particular, while for each player
j ∈ r(i−1) \ r(i) we have j 6∈ GRD(v′ | x[i−1] ∪ r(i) ) because it cannot be added to x[i−1] ∪ r(i) ∪ {i} =
x[i] ∪ r(i) by definition of r(i) . Hence, the greedy critical values of each player j ∈ r(i−1) \ r(i) must
be at least τjGRD (v′ | x[i−1] ∪ r(i) ) ≥ vi′ .
Since both player i and the set of players r(i−1) \ r(i) are feasible extensions to x[i−1] ∪ r(i) , we
can use γ-permeability of the greedy allocation rule in the subinstance in which we hold x[i−1] ∪ r(i)
fixed to obtain,
X
|r(i−1) \ r(i) | · vi′ ≤
τjGRD (v′ | x[i−1] ∪ r(i) ) ≤ γ · v′ GRD(v′ | x[i−1] ∪ r(i) ) = γ · vi′ .
j∈r(i−1) \r(i)
Cancelling vi′ shows that |r(i−1) \ r(i) | ≤ γ.
To show the claim it now suffices to observe that the greedy critical value of player i in the
subinstance where we hold x[i−1] fixed under the original valuations is the highest value of a player
j ∈ r(i−1) \ r(i) . Namely,
pi (1 | x[i−1] ) =
max
j∈r(i−1) \r(i)
vj ≥
1
·
γ
X
vj = v(r(i−1) ) − v(r(i) ),
j∈r(i−1) \r(i)
which concludes the proof.
Lemma I.2. Suppose that the greedy allocation rule GRD is γ-permeable for a subinstance-closed
collection of problems of binary, single-parameter problems Π. Consider any problem π ∈ Π with
feasibility structure F. Furthermore, let B0 ⊇ B1 ⊇ . . . ⊇ Bn and A0 ⊆ A1 ⊆ . . . ⊆ An with
Bt ∪ At ∈ F for all t. Consider a set C that fulfills C ∪ An ∈ F and for every i ∈ C there is a
t ∈ {0, 1, . . . , n} with i ∈ Bt or {i} ∪ Bt ∪ At 6∈ F. Then we have |C| ≤ γ · |B0 |.
38
Proof. Set Bn+1 = ∅ and define Cn+1 = C. Furthermore, define Ct for 0 ≤ t ≤ n recursively as a
maximal subset of the players in Ct+1 \ Bt such that Ct ∪ At ∪ Bt ∈ F.
We will show that for all t ∈ {0, 1, . . . , n},
|Ct+1 \ Ct | ≤ γ · |Bt \ Bt+1 |.
Consider some fixed t and define D := Bt ∩ Ct+1 .
Now a crucial observation is that the set Ct+1 \(Ct ∪D) is feasible holding E := Bt+1 ∪At ∪Ct ∪D
fixed. This is because
Ct+1 \ (Ct ∪ D) ∪ Bt+1 ∪ At ∪ Ct ∪ D = Bt+1 ∪ At ∪ Ct+1 ⊆ Bt+1 ∪ At+1 ∪ Ct+1 ∈ F.
Further note that by the way we have chosen Ct we know that Bt \ (Bt+1 ∪ D) is another feasible
extension to E because
(Bt \ (Bt+1 ∪ D)) ∪ Bt+1 ∪ At ∪ Ct ∪ D = Bt ∪ At ∪ Ct ∈ F.
To apply γ-permeability in the subinstance where we hold E fixed, define a valuation profile v̄
by setting v̄i = 1 for i ∈ Bt \ (Bt+1 ∪ D) and 0 otherwise. Now, for every i ∈ Ct+1 \ (Ct ∪ D), we
have
τiGRD (v̄ | E) = 1.
This is due to the maximality of Ct : If for some i ∈ Ct+1 \ (Ct ∪ D) this value is 0, then also
{i} ∪ (Bt \ (Bt+1 ∪ D)) ∪ Bt+1 ∪ At ∪ Ct ∪ D ∈ F.
So by permeability,
X
|Ct+1 \ (Ct ∪ D)| =
τiGRD (v̄ | E) ≤ γ · v̄(GRD(v̄ | E)) = γ · |Bt \ (Bt+1 ∪ D)|,
i∈Ct+1 \(Ct ∪D)
and therefore
|Ct+1 \ Ct | = |D| + |Ct+1 \ (Ct ∪ D)| ≤ |D| + γ · |Bt \ (Bt+1 ∪ D)| ≤ γ · |Bt \ Bt+1 |.
We now obtain the desired bound on the size of the set C by summing the previous inequality
over all t and using that it becomes a telescoping sum
|C| + |C0 | = |Cn+1 | + |C0 | =
n
X
|Ct+1 \ Ct | ≤ γ ·
n
X
|Bt \ Bt+1 | = γ · (|B0 | − |Bn+1 |) = γ · |B0 |.
t=0
t=0
It remains to show that all players in C will be covered (i.e., that C0 = ∅). This follows from the
fact that for each player i ∈ C by the definition of C there exists a t such that i ∈ Bt or i does not
fit into Bt ∪ At and, thus, in either case i 6∈ Ct .
Lemma I.3. Let ALG be any allocation rule. Suppose that the greedy allocation rule GRD is
γ-permeable for a subinstance-closed collection of binary, single-parameter problems Π. Then the
pricing rule described in Algorithm 1 fulfills Condition (b) of Definition 3.2 with β1 = 0 and β2 = γ
with respect to allocation rule ALG and the canonical exchange-feasible sets (Fx )x∈X .
39
v(5)
v(4)
v(3)
v(2)
v(1)
Figure 1: Our proof that the pricing rule derived from the greedy allocation rule satisfies Condition
(b), relies on the fact that we can chop the valuation and price space into discrete layers, which
reduces the problem to 0/1-valuations.
Proof. Consider an arbitrary feasible x ∈ F and an arbitrary feasible extension x′ ∈ Fx . We want
to show that
X
pi (x′i | x[i−1] ) ≤ γ · v(ALG(v)).
i∈N
We will prove this claim through a layering argument; as depicted in Figure 1. To this end, let
v(j) be the j-th highest value of v1 , . . . , vn ; furthermore v(n+1) = 0. For each j ∈ [n], let S j denote
the set of players with value at least v(j) and let T j denote the set of players with x′i = 1 that see
a price pi (1 | x[i−1] ) of at least v(j) .
We now apply Lemma I.2 for each j ∈ [n] by setting At = x[t] , Bt = r(t) ∩S j , C = T j . Note that
C ∪ An ∈ F and for every i ∈ C there is a t ∈ {0, 1, . . . , n} such that i ∈ Bt or {i} ∪ Bt ∪ At 6∈ F.
We obtain |T j | ≤ γ · |r(0) ∩ S j |.
We conclude that
X
pi (x′i | x[i−1] ) =
i∈N
=
≤
n
XX
1i∈T j · (v(j) − v(j+1) )
i∈N j=1
n
X
j
|T | · (v(j) − v(j+1) )
j=1
n
X
γ · |r(0) ∩ S j | · (v(j) − v(j+1) )
j=1
= γ · v(ALG(v)) ,
where the first equality holds by definition of the sets T j , the second equality is basic calculus, the
inequality follows from Lemma I.2 as argued above, and the final equality holds by definition of
r(0) = ALG(v) and the sets S j .
I.3
Proof of Theorem 5.4
In this subsection we prove Theorem 5.4, which claims that (λ, µ)-smoothness of the pay-yourbid mechanism based on the welfare-maximizing allocation rule for subinstance-closed collection
of binary, single-parameter problems Π implies the existence of a weakly (1, (µ + 1)/λ)-balanced
pricing rule. By Theorem I.1 it suffices to show the following theorem.
Theorem I.3. Let ALG be any allocation rule. Suppose that the welfare-maximizing allocation
rule OPT is γ-permeable for a subinstance-closed collection of binary, single-parameter feasibility
40
problems Π. Then there exists a pricing rule that is weakly (1, 0, γ 2 )-balanced with respect to ALG
and the canonical exchange-feasible sets (Fx )x∈X .
As in the case of greedy we first describe the construction of the prices, and then we show that
these prices are balanced.
I.3.1
Construction of the Prices
We define the price pi (zi | x) for player i ∈ N and outcome zi ∈ {0, 1} and arbitrary but fixed
valuation profile v and allocation x ∈ F through Algorithm 2. For this section, define OPT(v | x)
as the allocation that results by padding the welfare-maximizing allocation for valuation profile v
over F/x with empty allocations.
As in the the case of the greedy allocation rule, we again set pi (0 | x) = 0 and we compute
pi (1 | x) via reference allocations and reference valuations. We again define the initial reference
(0)
allocation as r(0) = ALG(v) and the initial reference valuations by setting vj = vj for j ∈ r(0) and
(0)
vj
= 0 otherwise. The subsequent reference allocations and valuations are defined recursively as
(i)
r (i) = OPT(v (i−1) | x[i] ) and vj
(i−1)
= vj
(i)
= vj for j ∈ r(0) and vj
inf{vi′
r(n)
= 0 otherwise. We then set
(n)
i ∈ OPT(vi′ , v-i | x)}
(i−1)
= v(r
) − v(r(i) ).
pi (1 | x) = vi if i ∈
and pi (1 | x) =
|
otherwise. Note that this
definition immediately implies that pi (xi | x[i−1] )
By substituting all occurrences of n with i − 1 we obtain the formula for the price pi (1 | x[i−1] ).
(i−1)
We use the shorthand τ OPT (v-i
(i−1)
| x[i−1] ) := inf{vi′ | i ∈ OPT(vi′ , v-i
) | x[i−1] )}.
Algorithm 2: Pricing Rule Derived from OPT (Parametrized by ALG)
Input: zi ∈ {0, 1}, v, x ∈ F
Output: pi (zi | x)
if zi = 0 then
// In this case the price is simply zero
pi (zi | x) = 0
else
// First determine reference allocation and valuations
(0)
(0)
r(0) ← ALG(v), vk ← vk if k ∈ r(0) and vk ← 0 otherwise
for j ← 1 to n do
(j)
(j−1)
(j)
r(j) ← OPT(v(j−1) | x[j]), vk ← vk
= vk if k ∈ r(j) and vk ← 0 else
// Now determine the price
if i ∈ r(n) then
// If player i is part of the reference allocation he pays his valuation
pi (zi | x) ← vi
else
// Otherwise he pays the critical value against the players in the reference allocation
(n)
pi (zi | x) ← inf{vi′ : i ∈ OPT(vi′ , v−i | x)}
return pi (zi | x)
41
I.3.2
Proof of Theorem I.3
We again proceed in two steps. We first show Condition (a) (in Lemma I.4 below). Afterwards
we show that the permeability of OPT provides an upper bound on the permeability of GRD and
that the critical prices with respect to OPT are not much higher than those with respect to GRD
(in Lemmas I.5 and I.6). This allows us to bound Condition (b) using the same machinery that we
used in the previous section (in Lemma I.7)
Lemma I.4. Let ALG be any allocation rule. Suppose that the welfare-maximizing allocation rule
OPT is γ-permeable for a subinstance-closed collection of problems of binary, single-parameter
problems Π. Then the pricing rule described in Algorithm 2 fulfills Condition (a) of Definition 3.2
with α = 1 with respect to allocation rule ALG and the canonical exchange-feasible sets (Fx )x∈X .
Proof. Consider x ∈ F. We defined prices exactly so that pi (xi | x[i−1] ) = v(r(i−1) ) − v(r(i) ).
Therefore, using a telescoping-sum argument, we get
X
pi (xi | x[i−1] ) = v(r(0) ) − v(r(n) ).
i∈N
The claim now follows from the fact that r(0) = ALG(v) and r(n) ∈ Fx .
Lemma I.5. If the greedy allocation rule GRD is γ GRD -permeable and the welfare-maximizing allocation rule OPT is γ OPT -permeable for a subinstance-closed collection of of binary, single-parameter
problems Π, then γ OPT ≥ γ GRD .
Proof. We only have to show that for all x ∈ F, x′ ∈ Fx and all v, we have
X
τiGRD (v-i | x) ≤ γ OPT · v(GRD(v | x)).
i∈x′
Let y = x′ ∩ GRD(v | x). Observe that because y ⊆ GRD(v | x), we have GRD(v | x ∪ y) =
GRD(v | x). Define a valuation profile v′ by setting vi′ = vi for all i ∈ x′ \ Q and vi′ = 0
otherwise. By loser independence of greedy, GRD(v | x ∪ y) = GRD(v′ | x ∪ y). Furthermore,
OPT(v′ | x ∪ y) = GRD(v′ | x ∪ y).
We claim that for i ∈ x′ \ y we have
′
′
τiGRD (v-i | x) ≤ τiGRD (v-i
| x ∪ y) ≤ τiOPT (v-i
| x ∪ y) .
For the first inequality we use that τiGRD (v-i | x) is the value vj of some j that has to be outbid
by player i. By fixing another set y, this value can only go up because the options are limited
further. Also, when fixing x ∪ y, replacing the valuations by v′ has no influence because the set of
players that are selected remains unchanged.
The second inequality holds because under v′ player i in order to win when we hold x ∪ y fixed
′ | x ∪ y) = OPT(v′ | x ∪ y) out of the solution. Under
has to force some subset z ⊆ GRD(v-i
-i
OPT his payment is the sum of the respective players’ valuations, under GRD it is just the highest
valuation of any such player.
On the other hand, for players i ∈ y, because y ⊆ GRD(v | x), the greedy critical value
GRD
τi (v-i | x) is at most vi .
42
Using these two bounds on τiGRD (v-i | x) we obtain,
X
X
X
τiGRD (v-i | x) ≤
vi +
τiGRD (v-i | x)
i∈x′
i∈x′ \y
i∈y
≤
X
vi +
≤
′
τiOPT (v-i
| x ∪ y)
i∈x′ \y
i∈y
X
X
vi + γ OPT · v′ (OPT(v′ | x ∪ y))
i∈y
=
X
vi + γ OPT · v′ (GRD(v′ | x ∪ y))
i∈y
≤ γ OPT ·
X
vi + γ OPT · v′ (GRD(v′ | x ∪ y))
i∈y
=γ
OPT
· v(OPT(v | x) ,
where the third inequality use γ OPT -permeability of the welfare-maximizing allocation rule OPT in
the subinstance in which we hold x ∪ y fixed, the subsequent equality holds by the definition of v′ ,
the fourth inequality uses that γ OPT ≥ 1, and the final equality holds by the definition of y and
v′ .
Lemma I.6. If the greedy allocation rule GRD is γ GRD -permeable for a subinstance-closed collection
of of binary, single-parameter problems Π, then
τiOPT (v(i−1) | x[i−1] ) ≤ γ GRD · τiGRD (v(i−1) | x[i−1] )
Proof. Note that for valuations v(i−1) both GRD and OPT over F/x return the same set of players.
The same is true if we drop any player j from v(i−1) . Dropping player i we can define y =
(i−1)
(i−1)
GRD(v-i
| x[i−1] ) = OPT(v-i
| x[i−1] ).
(i−1)
(i−1)
Now consider player i bidding b′i = τiGRD (v-i
| x[i−1] ) + ǫ. Then i ∈ GRD(b′i , v-i
| x[i−1] ).
The addition of player i causes the removal of a (possibly empty) subset of players z ⊆ y. That is,
(i−1)
GRD(b′i , v-i
| x[i−1] ) = (y \ z) ∪ {i}.
(i−1)
Define valuations v′ by setting vi′ = b′i , vj′ = vj
for j ∈ z, and vj′ = 0 for every other player
j. Consider the subinstance in which we hold y \ z fixed. Since both player i and the set of players
z are feasible extensions we can apply γ GRD -permeability of the greedy allocation rule to obtain
X
|z| · b′i =
τjGRD (v′ | y \ z) ≤ γ GRD · v′ (GRD(v′ | y \ z)) = γ GRD · b′i ,
j∈z
and therefore |z| ≤ γ GRD .
The final step is now to observe that the critical value τiOPT (v(i−1) | x[i−1] ) of player i under
the welfare-maximizing allocation rule is at most |T | times the critical value pGRD
(v(i−1) | x[i−1] ) of
i
player i under the greedy allocation rule. To see this let z′ ⊆ y be the set with the smallest value
such that (y \ z′ ) ∪ {i} ∈ F. Then,
X
X
τiOPT (v(i−1) | x[i−1] ) =
vj ≤
vj ≤ |T | · max vj = |z| · τiGRD (v(i−1) | x[i−1] ),
j∈z′
j∈z
j∈z
where we used that z is some subset of y such that (y \ z) ∪ {i} ∈ F and so its combined value can
only be larger than that of z′ .
43
Lemma I.7. Let ALG be any allocation rule. Suppose that the welfare-maximizing allocation rule
OPT is γ-permeable for a subinstance-closed collection of binary, single-parameter problems Π.
Then the pricing rule described in Algorithm 2 fulfills Condition (b) of Definition 3.2 with β1 = 0
and β2 = γ 2 with respect to allocation rule ALG and the canonical exchange-feasible sets (Fx )x∈X .
Proof. Consider an arbitrary allocation x ∈ F and feasible extension x′ ∈ Fx . We want to show
that
X
pi (x′i | x[i−1] ) ≤ (γ OPT )2 · v(ALG(v)).
i∈N
We will again show this claim through layering. This time, however, we need the layering to
arbitrarily fine-grained. We will specify the granularity by ǫ > 0. For each j ∈ N, let S j denote the
set of players with value at least j · ǫ. Let T j denote the set of players with x′i = 1 that see a price
pOPT
(x′i | x[i−1] ) of at least γ GRD · j · ǫ.
i
For any fixed j ∈ N we now bound |T j | using Lemma I.2. We set At = x[t] , Bt = r(t) ∩ S j ,
C = T j . Note that C ∪ An ∈ F. We claim that for every i ∈ C we have i ∈ Bt or {i} ∪ Bt ∪ At 6∈ F
for t = i−1. To see this, consider the two options how pOPT
(x′i | x[i−1] ) can be set. If i ∈ r(i−1) , then
i
pOPT
(x′i | x[i−1] ) = vi . As by definition pOPT
(x′i | x[i−1] ) ≥ γ GRD · j · ǫ ≥ j · ǫ, this implies vi ≥ j · ǫ
i
i
(i−1)
and so i ∈ Bi−1 = r(i−1) ∩ S j . Otherwise, if i 6∈ r(i−1) , then pOPT
(x′i | x[i−1] ) = τiOPT (v-i
i
In this case, we can apply Lemma I.6 to get
(i−1)
pOPT
(x′i | x[i−1] ) = τiOPT (v-i
i
(i−1)
| x[i−1] ) ≤ γ GRD · τiGRD (v-i
| x[i−1] ).
| x[i−1] ).
(i−1)
So, we know that τiGRD (v-i
| x[i−1] ) ≥ j · ǫ and therefore {i} ∪ x[i−1] ∪ (r(i−1) ∩ Y j ) 6∈ F.
So, by Lemma I.2, we get |T j | ≤ γ GRD · |r(0) ∩ Y j |.
We conclude that
X
pOPT
(x′i | x[i−1] ) ≤ nǫ +
i
i∈x′
∞
XX
1i∈T j · γ GRD · ǫ
i∈x′ j=1
= nǫ + γ GRD ·
≤ nǫ + γ GRD ·
∞
X
j=1
∞
X
|T j | · ǫ
γ GRD · |r(0) ∩ Y j | · ǫ
j=1
≤ (γ
GRD 2
) · v(ALG(v)) + 2nǫ.
As this argument holds for any ǫ > 0, we also have
By Lemma I.5, γ GRD ≤ γ OPT and the claim follows.
P
i∈x′
pOPT
(x′i | x[i−1] ) ≤ (γ GRD )2 · v(ALG(v)).
i
Remark I.2. When the prices defined by Algorithm 2 are non-decreasing then γ-permeability
(i−1)
(i−1)
of OPT immediately implies that τiGRD (v−i
| x[i−1] ) ≤ τiGRD (v−i
| x) ≤ γ · v(OPT(v, Fx ))
implying that the pricing rule is (α, β)-balanced with β = γ. In this case Theorem I.3 can be
strengthened to show the existence of a (1, γ)-balanced pricing rule.
44
| 8 |
Computational homogenization of non-stationary
transport processes in masonry structures
arXiv:1110.2055v1 [cs.CE] 10 Oct 2011
Jan Sýkoraa , Tomáš Krejčı́a , Jaroslav Kruisa , Michal Šejnohaa,∗
a
Department of Mechanics, Faculty of Civil Engineering, Czech Technical University in
Prague, Thákurova 7, 166 29 Prague 6, Czech Republic
Abstract
A fully coupled transient heat and moisture transport in a masonry structure is examined in this paper. Supported by several successful applications
in civil engineering the nonlinear diffusion model proposed by Künzel [1] is
adopted in the present study. A strong material heterogeneity together with
a significant dependence of the model parameters on initial conditions as well
as the gradients of heat and moisture fields vindicates the use of a hierarchical modeling strategy to solve the problem of this kind. Attention is limited
to the classical first order homogenization in a spatial domain developed
here in the framework of a two step (meso-macro) multi-scale computational
scheme (FE2 problem). Several illustrative examples are presented to investigate the influence of transient flow at the level of constituents (meso-scale)
on the macroscopic response including the effect of macro-scale boundary
conditions. A two-dimensional section of Charles Bridge subjected to actual
climatic conditions is analyzed next to confirm the suitability of algorithmic
format of FE2 scheme for the parallel computing.
Keywords: Computational homogenization, Masonry, Coupled heat and
moisture transport, Parallel computing
Corresponding author. Tel.: +420-2-2435-4494; fax +420-2-2431-0775
Email addresses: jan.sykora.1@fsv.cvut.cz (Jan Sýkora),
krejci@cml.fsv.cvut.cz (Tomáš Krejčı́), jk@cml.fsv.cvut.cz (Jaroslav Kruis),
sejnom@fsv.cvut.cz (Michal Šejnoha)
∗
Preprint submitted to Elsevier
February 13, 2018
1. Introduction
Historical masonry structures all over the world enjoy constant attention by many entities including technical audience and public authorities.
When subject to restoration these structures naturally invite complex analyses combining both experimental work [2] and numerical simulations [3].
Even after closing the scheduled reconstruction steps a continuation of insitu monitoring is now becoming almost standard allowing the engineers not
only to evaluate the current state of the structure but also to improve the
predictive capability of theoretical models being supported by up to date
material data and instantaneous measurements of the state of stress and deformation. A particularly vivid example of this line of inquiry is the Charles
Bridge information system [4] integrating detailed geometrical description
of the bridge including changes resulting from previous reconstructions, historical and contemporary materials forming the bridge as well as novel materials and technologies used to improve its durability and serviceable life.
The available results of long-term measurements of temperature and moisture fields, time-dependent displacement data, complemented by advanced
methods of computational analysis then open the way to incorporate multiphysics, multi-scale, time-dependent and three-dimensional aspects of the
problem to create a realistic computational model of such a complex structure.
These issues have been supported by our recent study [5] devoted to the
homogenization of masonry walls with emphases on the effect of imperfect
hydraulic contact. On the one hand, it has been demonstrated that the introduction of interface transition zone at the mesostructural level, considerably
complicating the computational matter, is essentially negligible for the prediction of effective properties. On the other hand, an accompanied parametric study has shown a substantial dependence of the homogenized properties
on both the initial and loading conditions. Keeping the format of a simplified uncoupled multi-scale scheme with several successful applications particularly in civil engineering [3, 6, 7, to cite a few] would require performing the
homogenization analysis for various values of water content and certain referenced initial values of temperature and relative humidity to construct the
homogenized macro-scale retention curves. Utilizing these curves in an independent macroscopic study would considerably increase the computational
efficiency since avoiding a time consuming down scaling for the parameters
update at every macroscopic time step. Unfortunately, an observed strong
2
dependence of the effective properties on the applied macroscopic gradients
may lead to an enormous database of these response functions essentially
loosing the advantage over a full-fledged coupled multi-scale framework. This
issue together with the possibility of including the mesostructural morphology and mesostructural material behavior in the macro-level, where typical
structures are analyzed, without the need for assigning the fine-scale details
to the entire structure thus motivated the developments in this paper towards
an iterative FE2 algorithm much similar to that presented in [8].
Owing to the finite size of the representative volume element on mesoscale we adopt a variationally consistent homogenization developed in [8] to
reflect a certain size dependency of distributions of macroscopic fields due to
higher order terms appearing on the left hand side of macroscopic balance
equations when a non-linear transient flow is assumed on both the macro
and meso-scale. A short numerical study of this topic is presented in Section 4.1 preceded by the theoretical formulation in Section 3. Section 4.2
then investigates the influence of macroscopic finite element mesh and application of the associated loading conditions in FE2 scheme on the accuracy
of evolution of macroscopic moisture and temperature fields. The obtained
results are then utilized in Section 4.3 devoted to the parallel format of the
underlying multi-scale analysis performed for a two-dimensional section of
Charles Bridge. Summary of the essential findings is available in Section 5.
To keep the paper self-contained a short overview of the adopted constitutive
model is provided in Section 2.
In the following text, a and A denote a vector and a symmetric secondorder tensor, respectively. The symbol ∇ = {∂/∂x, ∂/∂y, ∂/∂z}T stands for
the gradient representation. All materials are assumed locally isotropic.
2. Material model
The literature offers a manifold of material models that allow for the description of coupled heat and moisture transport. An extensive overview
of transport models is presented in the monograph by Černý and Rovnanı́ková [9]. Among others the non-linear diffusion model proposed by
Künzel holds a great potential for an accurate description of transport processes in building engineering including masonry structures and will be adopted
here to follow up our previous works in this area [10, 5].
Künzel derived the coupled system of energy and mass balance equations
based on concepts put forward by Krischer and Kiessl, see e.g. [11, 1]. In [12]
3
Krischer identified two transport mechanisms for material moisture, one being the vapor diffusion and the other being described as a capillary water
movement. In other words, Krischer introduced the gradient of partial pressure in the air as the driving force for the water vapor transport and the
gradient of liquid moisture content as the driving force for the water transport. Kiessl further extended the diffusion model of Krischer and developed
in [13] his own original version. The unification for the description of moisture transport in the hygroscopic ϕ ≤ 0.9 and overhygroscopic ϕ > 0.9 range
(ϕ is the relative humidity) was achieved with the help of moisture potential,
which brought several advantages particularly a very simple expression for
the moisture transport across interfaces. On the other hand, the definition of
moisture potential in the overhygroscopic range was too artificial, and Kiessl
introduced it without any theoretical background, see [9].
For the description of simultaneous water and water vapor transport
Künzel neglected the liquid water and water vapor convection driven by
gravity and total pressure as well as enthalpy changes due to liquid flow and
choose the relative humidity ϕ as the only moisture potential for both hygroscopic and overhygroscopic range. He also divided overhygroscopic region
into two sub-ranges - capillary water region and supersaturated region, where
different conditions for water and water vapor transport are considered. In
comparison with Kiessl’s or Krischer’s model Künzel’s model introduces several simplifications. Nevertheless, the proposed model describes all substantial phenomena of the heat and moisture transport in building materials and
the predicted results comply well with the experimentally obtained data [5].
Employing the classical Fick’s law for the description of water vapor diffusion, Kelvin’s law to simulate the transport of liquid water and Fourier’s
law to account for the flow of heat energy the Künzel model integrates into
the energy balance equation
dH dθ
= ∇T [λ∇θ] + hv ∇T [δp ∇{ϕpsat (θ)}],
dθ dt
(1)
and the conservation of mass equation
dw dϕ
= ∇T [Dϕ ∇ϕ] + ∇T [δp ∇{ϕpsat (θ)}] ,
dϕ dt
(2)
where H is the enthalpy of the moist building material, w is the water content of the building material, λ is the thermal conductivity, Dϕ is the liquid
4
conduction coefficient, δp is the water vapor permeability, hv is the evaporation enthalpy of the water, psat is the water vapor saturation pressure, θ is
the temperature and ϕ is the relative humidity. The material parameters,
both measured and functionally derived, that enter the above equations are
summarized for the sake of completeness in Appendix A, see also [11, 14]
for more detailed discussion on this subject. Note that the second term on
the right hand side of Eq. (1) represents the change of enthalpy due to phase
transition being considered the only heat source or sink.
Figure 1: A two dimensional region
Application of principal of virtual work then yields the weak form of these
two balance equations
Z
Z
dpsat
dH dθ
T
dΩ + {∇δθ} {λ∇θ} + hv δ p ϕ
∇θ dΩ +
δθ
dθ dt
dθ
Ω
Ω
Z
Z
T
δθ q̄ν dΓ = 0,
(3)
+ {∇δϕ} [hv {δ p psat ∇ϕ}] dΩ −
Γq̄θ
Ω
Z
dw dϕ
dΩ + + {∇δϕ}T [{Dϕ ∇ϕ} + {δ p psat ∇ϕ}] dΩ +
δϕ
dϕ
dt
Ω
Ω
Z
Z
dpsat
T
+ {∇δθ}
δpϕ
∇θ dΩ −
δϕ ḡν dΓ = 0,
(4)
dθ
Ω
Γḡϕ
Z
to be solved numerically for the prescribed boundary and initial conditions,
see Fig. 1 for the definition of prescribed boundary terms and domain representation. Owing to a strong nonlinear dependence of material parameters
on both temperature and moisture fields, recall Appendix A, the NewtonRaphson method is generally needed to solve the resulting discretized system
of equations.
5
3. First order homogenization of non-stationary coupled heat and
moisture transport
The present section derives the governing equations of the coupled heat
and moisture transport in the framework of coupled two-scale analysis of
FE2 type. In this regard it is presumed that the homogenized macro-scale
fields are found from the solution of a certain sub-scale (meso-scale) problem
performed on a representative volume element (RVE) being identical, at least
in a statistical sense, to a real meso-structure from both the geometrical and
material composition point of view. Such an RVE is then usually termed
the statistically equivalent periodic unit cell (SEPUC) [15]. Examples of
such SEPUCs for both regular as well as irregular masonry walls adopted
herein are plotted in Fig. 6(d)(e). It has been advocated in [8] that for
a finite size RVE the assumption of transient flow on both the macro and
meso-scale introduces certain non-local, size dependent, terms in equations
governing the macroscopic response. Some numerical simulations addressing
this issue are presented in Section 4.1, whilst the theoretical grounds are
provided next following closely, although in more abbreviated format, the
variationally consistent homogenization outlined in detail in [8].
To introduce this subject suppose that a local field a can be replaced by
a spatially homogenized one hai such that
Z
Z
Z
Z
1
a dΩ ≈
hai dΩ =
a dΩ dΩ,
(5)
|Ω | Ω
Ω
Ω
Ω
Z
Z
Z
Z
1
a dΓ ≈
hai dΓ =
a dΓ dΓ,
(6)
|Γ | Γ
Γ
Γ
Γ
where Ω and Γ represent the internal and boundary parts of the SEPUC.
In what follows, owing to the space limitation, we shall treat only the energy
balance equation (3) which upon employing Eqs. (5) and (6) becomes
Z
Z
dH dθ
dpsat
T
δθ
dΩ +
{∇δθ} {λ∇θ} + hv δ p ϕ
∇θ
dΩ +
dθ dt
dθ
Ω
Ω
Z D
Z
E
T
hδθ q̄ν i dΓ = 0.
(7)
+
{∇δϕ} [hv {δ p psat ∇ϕ}] dΩ −
Ω
Γq̄θ
In the spirit of the first order homogenization it is assumed that the
macroscopic temperature and relative humidity vary only linearly over the
6
SEPUC. This can be achieved by loading its boundary by the prescribed temperature Θhom and relative humidity Φhom derived from the uniform macroscopic temperature ∇Θ and relative humidity ∇Φ gradients. In such a case
the local temperature and relative humidity inside the SEPUC admit the
following decomposition
θ(x) = Θ(X 0 ) + {∇Θ}T {x − X 0 } + θ∗ (x) = Θhom (x) + θ∗ (x), (8)
ϕ(x) = Φ(X 0 ) + {∇Φ}T {x − X 0 } + ϕ∗ (x) = Φhom (x) + ϕ∗ (x), (9)
where θ∗ (x) and ϕ∗ (x) are the fluctuations of local fields superimposed onto
linearly varying quantities Θhom (x) and Φhom (x) . The temperature Θ(X 0 )
and the moisture Φ(X 0 ) at the reference point X 0 are introduced to link the
local fields to their macroscopic counterparts. For convenience the SEPUC is
typically centered at X 0 . Henceforth, the local fluctuations will be demanded
to be periodic, i.e. the same values are enforced on the opposite sides of a
rectangular SEPUC.
Next, substituting Eqs. (8) and (9) into Eq. (7) and collecting the terms
corresponding to δΘhom , δΦhom and δθ∗ , δϕ∗ splits the original problem (3)
into the homogenized (macro-scale) problem
Z
Z
dpsat
hom dH dθ
hom T
{δΘ }
dΩ +
{∇δΘ } {λ∇θ} + hv δ p ϕ
∇θ
dΩ +
dθ dt
dθ
Ω
Ω
Z D
Z
E
hom T
+
{∇δ(Φ )} [hv {δ p psat ∇ϕ}] dΩ −
{δΘhom } q̄ν dΓ = 0,
(10)
Γq̄θ
Ω
and the local sub-scale (meso-scale) problem
Z
Z
dpsat
∗ T
∗ dH dθ
dΩ +
{∇δθ } {λ∇θ} + hv δ p ϕ
∇θ
dΩ +
{δθ }
dθ dt
dθ
Ω
Ω
Z D
Z
E
∗ T
h{δθ∗ } q̄ν i dΓ =
+
{∇δϕ } [hv {δ p psat ∇ϕ}] dΩ −
(11)
Ω
|
Γq̄θ
{z
=0 due to periodicity
Z
Ω
1
|Ω |
Z
∂Ωq̄
{δθ∗ } q̄ν d(∂Ω ) dΩ = 0,
}
which is satisfied identically owing to the assumed periodic boundary conditions. Solving Eq. (11) for the prescribed increments of ∇Θ and ∇Φ provides
the instantaneous effective properties and storage terms that appear in the
7
macro-scale equation (10). Because of a strong non-linearity the two equations must be solved iteratively in a certain nested loop, see [16, 8] for further
reference.
Since details on the solution of Eq. (11) are available in our preceding
paper [5] we limit our attention to the macro-scale problem and write the
first term of Eq. (10) with the help of Eqs. (8) and (9) as
Z
Z
dH dθ
0
hom dH dθ
T
{δΘ }
dΩ =
{δ(Θ + {∇Θ} {x − X })}
dΩ =
dθ dt
dθ dt
Ω
Ω
Z
dH dθ
0 dH dθ
T
=
{δΘ}
dΩ,
(12)
+ {δ∇Θ} {x − X }
dθ dt
dθ dt
Ω
thus clearly identifying the solution dependence on the actual size of the
SEPUC through the second term in the integral (12). We may now substitute
from Eq. (12) into Eq. (10) to get
Z
Z
dH dθ
0 dH dθ
T
T
−
{δΘ}
{δ∇Θ} {x − X }
dΩ −
dΩ −
dθ dt
dθ dt
Ω
Ω
{z
} |
{z
}
|
Cθθ
drθ
dt
′
Cθθ
drθ
dt
dpsat
T
−
{δ∇Θ} {λ∇θ} + hv δ p ϕ
∇θ
dΩ −
dθ
Ω
{z
}
|
Kθθ rθ
Z D
Z D
E
E
T
−
{δ∇Φ} [hv {δ p psat ∇ϕ}] dΩ +
{δΘ}T q̄ν dΓ = 0. (13)
q̄
|Ω
{z
} | Γθ
{z
}
Z
Kθϕ rϕ
qext
An analogous approach can be applied also to the moisture transport equation (4) to arrive, after classical finite element discretization, into a discretized system of coupled macroscopic heat and moisture equations
dr θ
= q ext ,
dt
dr ϕ
′
= g ext ,
Kϕθ r θ + Kϕϕ rϕ + (Cϕϕ + Cϕϕ )
dt
′
Kθθ r θ + Kθϕ r ϕ + (Cθθ + Cθθ )
(14)
(15)
which have to be properly integrated in the time domain adopting for example
the Crank-Nicolson integration scheme. Details on the numerical implementation are available in [14].
8
4. Examples
Several illustrative example problems were analyzed to address the nonlinear transient coupled heat and moisture transport assumed on both scales,
the influence of the way of prescribing the macroscopic loading conditions
closely related to the macro-scale finite element mesh and finally the solution
strategy exploiting the parallel computation. The same material data were
adopted in all analyses. These were obtained from a set of experimental
measurements providing the hygric and thermal properties of mortars and
bricks/stones, which have been used in the reconstructions works of historical
buildings in the Czech Republic including Charles Bridge, see [17]. The
measured material parameters of individual masonry phases listed in Table. 1
then served to derive the non-measurable transport coefficients presented in
Eqs. (1) and (2), see also Appendix A.
parameter
wf [kgm−3 ]
w80 [kgm−3 ]
λ0
[Wm−1 K−1 ]
btcs [−]
ρs
[kgm−3 ]
µ
[−]
A
[kgm−2 s−0.5 ]
cs
[Jkg−1 K−1 ]
free water saturation
water content at ϕ = 0.8 [-]
thermal conductivity
thermal conductivity supplement
bulk density
water vapor diffusion resistance
water absorption coefficient
specific heat capacity
brick mortar
229.30 160.00
141.68 22.72
0.25
0.45
10
9
1690
1670
16.80
9.63
0.51
0.82
840
1000
Table 1: Material parameters of individual phases.
4.1. Influence of transient flow at meso-level
This section supports through numerical simulations the theoretically predicted size dependence of the homogenized response first suggested for the
case of a non-linear single variable diffusion problem in [8] and also established here in Section 3 for the special case of the coupled heat and moisture
problem in the framework of Künzel’s constitutive model.
In doing so we considered three particular units cells in Fig. 2(a) varying
in size from millimeters to decimeters. Each cell was loaded by the same
constant gradients of temperature and moisture along the x-direction. The
resulting evolutions of the fluctuation part of the local temperature at the
9
center of individual cells appear in Fig. 2(b) clearly manifesting the influence of the size of the cell which necessary projects into the prediction of
the homogenized properties and thus evolution of the predicted macroscopic
response. For the largest cell the steady state was reached in about 80 [h].
(a)
(b)
Figure 2: (a) Investigated periodic unit cells, (b) resulting evolution of fluctuation temperature
4.2. Influence of macrostructural finite element mesh
One of the concerns of the implementation of FE2 scheme is the application of boundary conditions on the macro-scale keeping in mind the scale
transition requirement and periodic boundary conditions imposed on the
meso-scale. This may become important particularly with a relatively large
periodic unit cells which may even exceed the size of macroscopic elements
in the vicinity of outer boundary where the macro-elements should be fine
enough to ensure a smooth and accurate evolution of driving variables from
the outside into the inner parts of a structure.
To address this issue we studied two types of macro-scale discretizations
adopted in the two-scale (meso-macro) analysis. The corresponding macroscale finite element meshes appear in Figs. 3(a),(b). The case when the finescale details are assigned to the entire structure is plotted in Fig. 3(c). This
mesh, consisting of 7050 triangular finite elements, served to evaluate the accuracy of the two former discretizations. Fig. 3(a) shows 108 macro-elements
each representing a single meso-problem with assigned periodic boundary
conditions, whereas the case in Fig. 3(b) assumes the outer boundary being
fully discretized (note that only one-directional flow is considered). There,
10
only the inner part consisting of 72 elements is subject to multi-scale analysis
whilst the outer part is modeled as a structure with a real masonry bonding
consisting of 3888 finite elements. A multi-point constrains were introduced
to account for an incompatible discretization along the common interface.
This latter case is, therefore, expected to heal inaccuracies in the estimation
of temperature and moisture fields in the region close to the surface layer.
(a)
(b)
(c)
Figure 3: Different finite element representations of masonry wall (lx = 1.92 [m], ly =
1.80 [m]) - (a) full multi-scale scheme, (b) semi multi-scale scheme, (c) full fine-scale discretization
The following boundary conditions were imposed: on the right-hand side
(interior) a constant temperature of 24 [◦ C] and a constant relative humidity 0.5 [−] were maintained, while on the left-hand side (exterior) the real
climatic data collected over the entire year were prescribed, see Fig. 4.
(a)
(b)
Figure 4: Annual loading conditions - (a) temperature, (b) moisture
11
The results appear in Fig. 5 showing variation of the temperature and
moisture along the mid section of the wall after the duration of load of 10 [h]
and 100 [h], respectively, derived for the macroscopic time step equal to 1 [h].
Clearly, the notable difference between the exact (full fine-scale discretization) and full multi-scale scheme can be observed in surface layers only and
this difference almost disappears with a sufficiently long duration of time. It
thus appears that the refined representation of the surface layer through the
semi multi-scale scheme, although more accurate compare to full multi-scale
scheme, does not bring any particular advantage. This is supported by the
calculated average and absolute errors stored in Table 2 taking into account
all nodal macroscopic temperatures and moistures in the domain over all
time integration steps. Note that for the sake of comparison the fine-scale
variables (solution employing the mesh in Fig. 3(c)) were averaged over the
cell basically covered by two macro-elements in Fig. 3(a).
(a)
(b)
(c)
(d)
Figure 5: Comparison of different macrostructural computations - (a) temperature profile
after t = 10 [h], (b) moisture profile after t = 10 [h] (c) temperature profile after t = 100 [h],
(d) moisture profile after t = 100 [h]
The presented results thus promote the more accurate semi multi-scale
12
scheme only for calculations demanding higher accuracy of local results especially in initial stages of computation and/or examples with fast changing
boundary conditions.
type of comparison
avg. relative error
[%]
avg. absolute error
[◦ C]/[−]
2.62
0.71
0.11
0.03
0.18
0.05
0.001
0.001
calculation of temperature
- fine-scale vs. multi-scale
- fine-scale vs. semi multi-scale
calculation of relative humidity
- fine-scale vs. multi-scale
- fine-scale vs. semi multi-scale
Table 2: Averaged relative and absolute errors
4.3. Parallel computation
The essential request by the contractor when studying the mechanical response of Charles Bridge to provide the basis for reconstruction works was a
full scale three-dimensional analysis of the bridge. Performing such an analysis in a fully coupled format on a single computer would be computationally
unfeasible thus creating the need for a parallel computing. Concentrating on
the implementation part of the parallel version of FE2 scheme we limit our
attention to a two-dimensional section of Charles Bridge subjected, however,
to real climatic data displayed already in Fig. 4. Extension to a fully threedimensional problem is under current investigation and will be presented
elsewhere.
As already discussed in the previous section, the present FE2 based multiscale analysis assumes each macroscopic integration point be connected with
a certain mesoscopic problem represented by an appropriate periodic unit
cell. The solution of a meso-scale problem then provides instantaneous effective data needed on the macro-scale. Such an analysis is particularly suitable
for a parallel computing because the amount of transferred data is small. In
this regard, the master-slave strategy can be efficiently exploited. To that
end, the macro-problem is assigned to the master processor while the solution at the meso-level is carried out on slave processors. At each time step
the current temperature and moisture together with the increments of their
13
gradients at a given macroscopic integration point are passed to the slave processor (imposed onto the associated periodic cell), which, upon completing
the small scale analysis, sends the homogenized data (effective conductivities,
averaged storage terms and fluxes) back to the master processor.
If the meso-scale problems are large enough, the ideal solution is to assign one meso-problem to one slave processor. Clearly, even for very small
macro-problems with a few thousands of finite elements, the hardware requirements would be in such a case excessive. On the other hand, if the
meso-problems are relatively small, i.e. they contain small number of finite
elements, the corresponding analysis might be even shorter than the data
transfer between the processors. Then, the computational time associated
with the data transfer between the master processor and many slave processors may grow excessively. It is worth mentioning that this time consists of
two contributions. The first one represents the latency time (the processors
make connection) which is independent of the amount of transferred data
whilst the second contribution clearly depends on the amount of data being
transferred. For small meso-problems it is therefore reasonable to assign several of them to a single slave processor. The master processor then sends a
larger package of data from many macroscopic integration points at the same
time to each slave processor so that the latency time does not play a crucial
role. This approach was adopted hereinafter.
Fig. 6(a) displays a three-dimensional segment of Charles Bridge examined in the original three-dimensional static calculation [3]. A two-dimensional
cut through the mid part of a Charles Bridge arch examined for the parallel computation appears in Fig. 6(b) together with four material regions.
These are associated with two heterogeneity systems of the bridge, one representing a regular sand stone masonry of side walls, fence and arches and the
other corresponding to an irregular quarry masonry made of arenaceous marl
blocks and sand and black lime mortar filling the inner part of the bridge.
For simplicity, the bridge deck was assigned the regular pattern. The corresponding periodic unit cells employed for the meso-scale analysis are plotted
in Figs. 6(d) and 6(e), respectively.
The finite element mesh used at the macro-level is evident from Fig. 6(c)
featuring 7, 081 nodes and 13, 794 triangular elements with a single integration point thus amounting to the solution of 13, 794 meso-problems at each
macroscopic time step. This figure also shows decomposition of the macroproblem into 12 slave processors. The numbers of elements in individual
sub-domains being equal to the number of meso-problems handled by the
14
(a)
(b)
(d)
(c)
(e)
Figure 6: (a) Three-dimensional view of Charles Bridge with a two-dimensional A-A
section analyzed, (b) Analyzed section showing material regions with assigned meso-scale
unit cells, (c) Macrostructural mesh with identified loading conditions and decomposition
into sub-domains representing individual slave processors (lx = 10.40 [m], ly = 6.82 [m]),
(d) Mesostructural mesh of regular bonding of masonry (SEPUC 1: lx = 0.45 [m], ly =
0.44 [m]), (e) Mesostructural mesh of irregular quarry masonry (SEPUC 2: lx = 0.45 [m],
ly = 0.35 [m]).
assigned slave processor are listed in Table 3. It should be noted that the assumed decomposition of the macro-problem is not ideal. In comparison with
domain decomposition methods, the macro-problem has to be split with respect to the heterogeneity of the material resulting in the variation of number
of elements in individual sub-domains between 1046 and 1748.
The number of elements in the two meso-problems amounts to 265 (160
nodes) for SEPUC 1 and to 414 (239 nodes) for SEPUC 2, respectively.
15
Processor No.
No. of
meso-problems
1
9
1218
1046
2
10
1748
1052
3
11
1046
1054
4
12
1052
1048
5
1214
-
6
1210
-
7
1052
-
8
1054
-
Table 3: Decomposition of the macro-problem into sub-domains.
Similarly to the macro-problem, the meso-problems have to account for the
material heterogeneity. Clearly, the ideal speedup and load balancing are
obtained when the decomposition of the macro-problem reflects the mesoproblem meshes. However, this is considerably more difficult when compared
to the classical mesh decomposition.
The actual analysis was performed on a cluster built at our department.
Each node of the cluster is a single processor personal computer Dell Optiplex
GX620 equipped with 3.54 GB of RAM. The processors are Intel Pentium
with the frequency 3.4 GHz. The cluster is based on Debian linux 5.0 and
32-bit architecture.
(a)
(b)
Figure 7: Evolution of macroscopic (a) temperature, (b) moisture
Although various mesoscopic heterogeneity patterns were properly accounted for, the material data of individual constituents (bricks or stones
and mortar) were taken the same for both the outer and inner part of the
bridge, recall Table 1 for specific values. The initial conditions on the macroscale were set equal to ϕ = 0.5 [-] and θ = 20 [◦ C] and the year round
variation of moisture and temperature in Fig. 4 was imposed onto all outer
surfaces of the bridge, see Fig. 6(c). Owing to the computational demands
16
the macroscopic time increment was set equal to 10 [h], which agreed with
the real computational time equal to 2.08 minute for each time step. In
view of the results presented in Section 4.2, recall Fig. 5, this justifies, although at the loss of accuracy at the initial stage of computation, the use of
full multi-scale scheme. One particular example of the resulting evolutions
of macroscopic temperature and moisture at the selected nodes labeled in
Fig. 6(c) is seen in Fig. 7.
5. Conclusions
A fully coupled multi-scale analysis of simultaneous heat and moisture
transport in masonry structures was implemented in the framework of FE2
computational strategy. Two particular issues were addressed: the influence
of the finite size of SEPUC when running the transient analysis on both scales
and the way of introducing loading on the macro-scale. While the former one
plays a significant role in the estimates of macroscopic response, the latter
one proves important only in the initial stages of loading.
Special attention was accorded to the implementation of FE2 concept
in the parallel format employing the master-slave approach. Although not
qualitatively fully acceptable, the two-dimensional example of Charles Bridge
raised a number of questions to the solution efficiency particularly with reference to a proper subdivision of the analyzed macro-domain and local finite
element mesh of individual meso-scale SEPUCs. The present findings summarized in Section 4.3 will be utilized in a fully three-dimensional analysis
being the subject of our current research effort.
Acknowledgment
This outcome has been achieved with the financial support of the Czech
Science Foundation, project No. 105/11/0411, by the Ministry of Industry
and Trade of the Czech Republic though FR-TI1/381 project, and partially
also by the research project CEZ MSM 6840770003.
Appendix A.
The list of material parameters to be obtained experimentally are stored
in Table 1. The transport coefficients that enter Eqs. (1) and (2) are provided
by
17
• w - water content [kgm−3 ],
w = wf
(b − 1)ϕ
,
b−ϕ
(A.1)
where wf is the free water saturation and b is the approximation factor, which must always be greater than one. It can be determined
from the equilibrium water content (w80 ) at 0.8 [-] relative humidity by
substituting the corresponding numerical values in equation (A.1).
• δp - water vapor permeability [kgm−1 s−1 Pa−1 ],
δp =
δ
,
µ
(A.2)
where µ is the water vapor diffusion resistance factor and δ is the vapor
diffusion coefficient in air [kgm−1 s−1 Pa−1 ] given by
1.81
2.306 · 10−5 pa
θ + 273.15
δ=
,
(A.3)
Rv (θ + 273.15) p
273.15
with p set equal to atmospheric pressure pa = 101325 [Pa] and Rv =
R/Mw = 461.5 [Jkg−1 K−1 ]; R is the gas constant (8314.41 [Jmol−1 K−1 ])
and Mw is the molar mass of water (18.01528 [kgmol−1 ]).
• Dϕ - liquid conduction coefficient [kgm−1 s−1 ],
Dϕ = Dw
dw
,
dϕ
(A.4)
where Dw is the capillary transport coefficient given by
2
A
Dw = 3.8
· 103w/(wf −1) ,
wf
where the derivative of the moisture storage function
(A.5)
dw
is obtained
dϕ
by differentiating Eq. (A.1).
• λ - thermal conductivity [Wm−1 K−1 ],
btcs w
,
λ = λ0 1 +
ρs
(A.6)
where λ0 is the thermal conductivity of dry building material, ρs is the
bulk density and btcs is the thermal conductivity supplement.
18
• psat - water vapor saturation pressure [Pa],
aθ
psat = 611 exp
,
θ0 + θ
where
(A.7)
a = 22.44 θ0 = 272.44 [◦C] θ < 0 [◦ C]
(A.8)
a = 17.08 θ0 = 234.18 [ C] θ ≥ 0 [ C]
◦
◦
• hv - evaporation enthalpy of water [Jkg−1 ]
hv = 2.5008 · 10
6
273.15
θ
(0.167+3.67·10−4 θ)
.
(A.9)
References
[1] H. M. Künzel, K. Kiessl, Calculation of heat and moisture transfer in
exposed building components, International Journal of Heat and Mass
Transfer 40 (1997) 159–167.
[2] R. Přikryl, A. Št́astná, Contribution of clayey-calcareous silicite to the
mechanical properties of structural mortared rubble masonry of the medieval Charles Bridge in Prague (Czech Republic), Engineering Geology
115 (3–4) (2010) 257–267. doi:10.1016/j.enggeo.2010.06.009.
[3] J. Novák, J. Zeman, M. Šejnoha, J. Šejnoha, Pragmatic multi-scale and
multi-physics analysis of Charles Bridge in Prague, Engineering Structures 30 (11) (2008) 3365–3376.
[4] Charles bridge IS. the information system based on Charles Bridge reconstruction research, http://iskarluvmost.fsv.cvut.cz/homepage/.
[5] J. Sýkora, M. Šejnoha, J. Šejnoha, Homogenization of coupled
heat and moisture transport in masonry structures including interfaces, Applied Mathematics and Computation 0 (2011) 0–0.
doi:10.1016/j.amc.2011.02.050.
[6] R. Valenta, M. Šejnoha, J. Zeman, Macroscopic constitutive law for
mastic asphalt mixtures from multiscale modeling, International Journal
for Multiscale Computational Engineering 8 (1) (2010) 131–149.
19
[7] M. Šejnoha, J. Vorel, R. Valenta, J. Zeman, Virtual experiments and
statistically equivalent rves towards macroscopic constitutive laws, in:
B. Topping (Ed.), Proceedings of the The Thirteenth International Conference on Civil, Structural and Environmental Engineering Computing,
Chania, Crete, Greece 6-9 September 2011, Civil-Comp Press, 2011.
[8] F. Larsson, K. Runesson, F. Su, Variationally consistent computational
homogenization of transient heat flow, International Journal for Numerical Methods in Engineering 81 (2010) 1659–1686.
[9] R. Černý, P. Rovnanı́ková, Transport Processes in Concrete, London:
Spon Press, 2002.
[10] J. Sýkora, J. Vorel, T. Krejčı́, M. Šejnoha, J. Šejnoha, Analysis of coupled heat and moisture transfer in masonry structures, Materials and
Structures 42 (8) (2009) 1153–1167.
[11] H. M. Künzel, Simultaneous Heat and Moisture Transport in Building
Components, Tech. rep., Fraunhofer IRB Verlag Stuttgart (1995).
[12] O. Krischer, W. Kast, Die wissenschaftlichen Grundlagen der Trocknungstechnik, Dritte Auflage, Berlin: Springer, 1978.
[13] K. Kiessl, Kapillarer und dampfförmiger feuchtetransport in
mehrschichtlichen bauteilen, Ph.D. thesis, Universität in Essen
(1983).
[14] J. Sýkora, Multiscale modeling of transport processes in masonry structures, Ph.D. thesis, Czech Technical University in Prague (2010).
[15] J. Zeman, M. Šejnoha, From random microstructures to representative
volume elements, Modelling and Simulation in Materials Science and
Engineering 15 (4) (2007) S325–S335.
[16] I. Özdemir, W. A. M. Brekelmans, M. G. D. Geers, Computational homogenization for heat conduction in heterogeneous solids, International
Journal for Numerical Methods in Engineering (2) (2008) 185–204.
[17] M. Pavlı́ková, Z. Pavlı́k, R. Černý, Hygric and thermal properties of
materials of historical masonry, in: Proceedings of the 8th Symposium
on Building Physics in the Nordic Countries, 2008.
20
| 5 |
arXiv:1611.02801v1 [math.AC] 9 Nov 2016
The resultants of quadratic binomial complete
intersections
T. Harima, Niigata University
Department of Mathematics Education, Niigata, 950-2181 Japan
∗
A. Wachi, Hokkaido University of Education
Department of Mathematics,
Kushiro, 085-8580 Japan
J. Watanabe, Tokai University
Department of Mathematics, Hiratsuka, 259-1292 Japan
December 31, 2017
Abstract
We compute the resultants for quadratic binomial complete intersections. As an
application we show that any quadratic binomial complete intersection can have the
set of square-free monomials as a vector space basis if the generators are put in a normal
form.
1
Introduction
Consider the homogeneous polynomial in 5 variables of degree 5
F = 12vwxyz + t1 + t2 ,
where
t1 = −2(p1 v 3 yz + p2 vw3 z + p3 vwx3 + p4 wxy 3 + p5 xyz 3 ),
t2 = p1 p3 v 3 x2 + p2 p4 w3 y 2 + p3 p5 x3 z 2 + p1 p4 v 2 y 3 + p2 p5 w2 z 3 .
The polynomial F , as the Macaulay dual generator of an Artinian Gorenstein algebra
in the Macaulay’s inverse system, defines a flat family of Gorenstein algebras with Hilbert
function (1 5 10 10 5 1). If p1 p2 p3 p4 p5 + 1 6= 0, then the ideal which F defines is 5 generated,
and if p1 p2 p3 p4 p5 +1 = 0, it is 7 generated. All the members in this family possess the strong
Lefscehtz property. This is known from the computation of the 1st and the 2nd Hessians of
the form F . On the other hand there exists a similar, but slightly more complicated form G
(see §6), involving 5 parameters p1 , . . . , p5 , which has generic Hilbert function (1 5 10 10 5 1),
but if p1 p2 p3 p4 p5 + 1 = 0 then the Hilbert function reduces to (1 5 5 5 5 1). It seems
remarkable that the second Hessian of G is divisible by (p1 p2 p3 p4 p5 + 1)5 . A striking fact is
that it is precisely equal to the resultant of the complete intersection as the generic member
of this family.
In this paper we do not pursue the Hessians; instead, we study the resultant for quadratic
complete intersections. Generally speaking the resultant of n forms in n variables is a very
∗ Supported
by JSPS KAKENHI Grant 15K04812.
1
complicated polynomial in the coefficients of the forms. The meaning of the resultant is that
it does not vanish if and only if the ideal generated by the n forms is a complete intersection.
Thus if the n forms are monomials, the resultant is easy to describe.
Through the investigation of the strong Lefschetz property of complete intersections, we
found it necessary to acquire the skill to calculate the resultant of complete intersections.
The theory of the resultants has a long history, but a method to obtain it is described
completely in the book [2]. Generic computation is impossible except in a few cases because
of the huge number of variables involved and the high degree of the resultant. Hence, if
one wishes to deal with the resultant in an arbitrary number of variables, it is reasonable to
restrict the attention to a special class of homogeneous polynomials.
The purpose of this paper is to describe the resultant of quadratic binomials in n variables. We restricted our attention only to quadratic forms. It is because, though it seems
needless to say, the degree-two condition makes the computation simple. However, in addition to it, we can expect that a lot of information can be obtained from the knowledge
of quadratic complete intersections. Indeed the authors [6] showed that a certain family of
complete intersections is obtained as subrings of quadratic complete intersections. In addition, McDaniel [4] showed that many complete intersections could be embedded in quadratic
complete intersections. We can expect, from the manner their results were proved, a vast
amount of complete intersections to be obtained as subrings of quadratic complete intersections sharing the socle and general linear elements.
Suppose that I = (f1 , . . . , fn ) is an ideal generated by n homogeneous polynomials of
the same degree λ in the polynomial ring R. Then I is a complete intersection if and only
if Ind−n+1 = Rnλ−n−d+1 Id is the whole homogeneous space Rnλ−n+1 . With the aid of
an idea of Gelfand et. al [2], it is possible to pick up certain polynomials from among the
elements in Iλn−n+1 such that their span is the whole space. We apply their method to
the particular situation where the generators are quadratic binomials. It turns out that the
same method can be used to obtain the conditions which guarantees that, for smaller values
of λ, the elements in Iλ generate the subspace of Rλ complementary to the space spanned
by the square-free monomials. Thus our computation unexpectedly gives us a proof that
the complete intersections defined by quadratic binomials can have square free monomials
as a vector space basis. We state it as Theorem 3.2. This follows from Theorem 4.3, which
is the main result of this paper.
This paper is organized as follows. In §2 we show that any n-dimensional quadratic
space can have a normal form. This is an elementary observation but it seems that it has
never been used systematically before. It reduces the amount of the computation of the
resultant considerably even in the generic case. In §3 we investigate some properties of the
determinants of square matrices of the “binomial type”, which we need in the sequel. In §4,
we introduce matrices consisting of rows as coefficients of the polynomials in the ideal Iλ .
Then we apply the result of §2 to finding the determinants of these matrices. In §5 we give
some examples of quinternary quintics and in §6 we briefly discuss the relation between the
2nd Hessian of the Macaulay dual and the resultant of the quadratic binomials. We refer
the reader to [5, Definition 3.75] or [3] for the definition of the 2nd Hessian.
2
The normal form of a quadratic vector space
Definition 2.1. A binomial is a polynomial of the form f = αM + βN , where M, N are
distinct monomials in certain variables and α, β are some elements in a field.
Notation . We denote by R = K[x1 , x2 , . . . , xn ] the polynomial ring in n variables over a
field K. We denote by Rλ the homogeneous space of degree λ of R.
2
Proposition 2.1. Let V ⊂ R2 be an n-dimensional vector subspace of R2 . Then by a linear
transformation of the variables we may choose a basis f1 , f2 , . . . , fn for V such that
fi = x2i + linear combination of square-free monomials , i = 1, 2, . . . , n.
Proof. We induct on n. If n = 1, then the assertion is trivial. Assume that n > 1. Suppose
that V is spanned by
f1 , f2 , . . . , fn .
Let M = (mij ) be the matrix defined by
mij = the coefficient of x2j in fi .
By a linear transformation of variables if necessary, we may assume that f1 , . . . , fn−1 are
linearly independent by reduction xn = 0. Then by the induction hypothesis we may assume
that M takes the form:
1 0 0 0 ∗
0 1 0 0 ∗
..
M =
.
.
0 0 ··· 1 ∗
∗ ∗ ··· ∗ ∗
First assume det M 6= 0. Define
f1
g1
f2
g2
.. = M −1 .. .
.
.
fn
gn
Then g1 , . . . , gn are a desired basis for V . Next assume that det M = 0. This means that
we may make the last row of M a zero vector after subtracting a linear combination of
f1 , . . . , fn−1 from fn . Consider the linear transformation of the variables
xi 7→ xi + ξi xn , for i = 1, 2, . . . , n − 1,
xn 7→ xn .
With this transformation only the last column of M is affected and non-zero element appears
as the (n, n) entry. (This is because fn contains some square-free monomial with non-zero
coefficient.) Thus we are in the situation as det M 6= 0.
Definition 2.2. Suppose that f1 , . . . , fn ∈ R2 are linearly independent. We will say that
these elements are in a normal form as a basis for a quadratic vector space if
fi = x2i + linear combination of square-free monomials , i = 1, 2, . . . , n.
Example 2.1. Consider R = K[x, y, z], V = hx2 , y 2 , xyi. Then we have the matrix expression:
2
x
2
y2
x
1 0 0 0 0 0
2
y 2 = 0 1 0 0 0 0 z
xy
xy
0 0 0 1 0 0
xz
yz
3
Make the translation σ ∈ GL(3) : x 7→ x + ξz, y 7→ y + ηz and z 7→ z. Then we have the
isomorphism of vector spaces V ∼
= hf, g, hi, where
2
x
y2
2
f
1 0 ξ
0 2ξ 0 2
z
2
g = 0 1 η
0 0 2η
xy .
h
0 0 ξη 1 η
ξ
xz
yz
Furthermore we have the equality of vector
′
1
f
g′ =
0
h′
0
spaces: hf, g, hi = hf ′ , g ′ , h′ i, where
0 −ξ
f
η
g .
1 −η
ξ
1
h
0 ξη
So the vector space V is transformed to hf ′ , g ′ , h′ i, where
′
1
f
g′ =
0
h′
0
0 0
1 0
0 1
−ξ
η
−η
ξ
1
ξη
ξ
−η 2
ξ
1
ξ
2
x
y2
−ξ 2
2
η
z
η
xy .
1
xz
η
yz
It follows that we have the isomorphism of the algebras K[x, y, z]/(V ) ∼
= K[x, y, z]/(f ′g ′ , h′ )
′ ′
′
with the elements f , g , h are put in a normal form.
For a later purpose we prove some propositions in linear algebra.
Proposition 2.2. Let A := {a1 , a2 , . . . , aN } and B := {b1 , b2 , . . . , bN } be two independent sets of variables. Suppose that P is an N × N square matrix satisfying the following
conditions:
1. The ith row of P contains ai and bi as entries and all the other entries are zero.
2. Any column of P contains an element in A and an element in B.
Then the following conditions are equivalent.
(1) det P is an irreducible polynomial as an element in the polynomial ring
R := K[a1 , . . . , aN , b1 , . . . , bN ].
(2) The matrix P is irreducible, i.e., does not split to blocks like
P1
O
O
P2
by permutation
of rows and columns.
(3) det P = ±(a1 a2 · · · aN + (−1)N +1 b1 b2 · · · bN ).
Proof. Assume that P splits into blocks. Then obviously det P is reducible. This proves
(1) implies (2). Assume that P is irreducible. Permuting the columns do not affect the
condition of P . So we may assume that the diagonal entries of P are (a1 , a2 , . . . , aN ).
Furthermore we may conjugate P by a permutation matrix so that a1 , . . . , aN are diagonal
entries and elements of B are distributed in the super-diagonal entries and at the (N, 1)position. This enables us to compute the determinant of P as (3). Thus we have proved
4
that (2) implies (3). It remains to show that (3) implies (1). In other words we have to
show that d := a1 · · · aN ± b1 · · · bN is an irreducible polynomial in R. It is easy to see that
we have the isomorphism
K[a1 , . . . , aN , b1 , . . . , bN ]/(d, b1 − 1, b2 − 1, . . . , bN − 1) ∼
= K[a1 , . . . , aN ]/(a1 . . . aN ± 1),
and that this algebra is an integral domain of Krull dimension N − 1. This shows that
d, b1 − 1, . . . , bN − 1
is a regular sequence in R and furthermore R/(d) is an integral domain.
In the next proposition we slightly weaken the conditions on P and prove similar properties.
Proposition 2.3. Let A := {a1 , a2 , . . . , aN } and B := {b1 , b2 , . . . , bN } be independent
sets of variables. Suppose that P is an N × N square matrix which satisfies the following
conditions.
1. The ith row of M contains ai and bi as entries and all the other entries are zero.
2. Any column of M does not contain two elements in A.
Then we have:
(1) The variable ai divides det P if ai is the only non-zero element in the column that
contains ai .
(2) det P is independent of bi if ai is the only non-zero element in the column that contains
ai .
(3) det P factors into a product of a monomial in a1 , . . . , aN and binomials of the form
ai1 ai2 · · · air ± bi1 bi2 · · · bir .
(4) If a binomial ai1 ai2 · · · air ± bi1 bi2 · · · bir is a factor of det P , then we can arrange the
order of the variables so that aij and bij are in the same row and aij and bij−1 are in
the same column. (We let bi0 = bir .)
For proof we use a digraph associated to the matrix P as defined in the next Definition.
Definition 2.3. In the same notation of Proposition 2.3, we put X = A ⊔ B and call it the
set of vertices. We define the set E of arcs to be the union of the two sets
{ai → bj |ai and bj are in the same row}
and
{bj → ai | bj and ai are in the same column}.
We may call (X, E) the digraph associated to P . (Since we have assumed that ith row
contains ai and bi , there is an arc ai → bj if and only if i = j.)
Suppose {aj1 , aj2 , . . . , ajr } ⊂ A and {bj1 , bj2 , . . . , bjr } ⊂ B are subsets both consisting of
r elements.
We call them a circuit of M if aji and bji are contained in one row and bji and aji+1
are in one column for all i = 1, 2, . . . , r. (We let ajr+1 = aj1 .) A circuit will be represented
as
aj1 → bj1 → aj2 → bj2 → · · · → ajr → bjr → aj1 .
If we drop the last term from a circuit, we call it a chain of M . A chain is maximal if it
cannot be embedded into a circuit or a longer chain.
5
Proof of Proposition 2.3. (1) Suppose that ai is the only non-zero entry of a column. Then
obviously ai divides det P .
(2) Recall that ai and bi are in the ith row of P . If ai is the single non-zero element in
a column, then we may make a column operation to clear bi . Hence det P does not
involve bi .
(3) Let (X, E) be the digraph introduced in Definition 2.3. It is easy to see that (X, E)
decomposes into the disjoint union of circuits and maximal chains. The first element
in a maximal chain is an element of A as a single non-zero entry of the column. Indeed
any element in B can be prepended by an element in A and in addition, an element in
A cannot be prepended by an element of B if and only if it is a single non-zero entry
of the column. If an element ai appears in a row as a single non-zero element of the
column, we have det P = ai det P ′ , where det P ′ is the matrix P with the row and
column deleted that contains ai . Thus it is enough to treat P in which all columns have
two non-zero elements as well as rows. In this case (X, E) decomposes as a disjoint
union of circuits. A circuit in (X, E) like
a(1) → b(1) → a(2) → b(2) → · · · → a(k−1) → b(k−1) → a(k) → b(k) → a(1) .
gives us a binomial as a divisor of det P . This completes the proof for (3).
(4) This follows immediately from (3).
Example 2.2.
a1
0
det
0
0
3
0
a2
0
0
0
0
a3
b4
b1
b2
= a1 a2 (a3 a4 − b3 b4 ).
b3
a4
The quadratic binomial complete intersections
In this section we denote by E = Khx1 , x2 , . . . , xn i the graded vector space spanned by
the square-free monomials in the variables x1 , x2 , . . . , xn over a field K. As well as Rλ ,
we denote by Eλ = Khx1 , x2 , . . . , xn iλ the homogeneous space spanned by the square-free
monomials of degree λ over K.
Definition 3.1. For all positive integers λ = 1, 2, 3, . . . , we define the sets of monomials
M1 (λ), M2 (λ), . . ., Mn (λ) of degree λ as follows:
M1 (λ) =
{xλ1 1 xλ2 2 xλ3 3
· · · xλnn |
M2 (λ) = {xλ1 1 xλ2 2 x3λ3 · · · xλnn |
M3 (λ) = {xλ1 1 xλ2 2 xλ3 3 · · · xλnn |
n
X
j=1
n
X
j=1
n
X
λj = λ},
λj = λ, λ1 < 2},
λj = λ, λ1 < 2, λ2 < 2},
j=1
..
.
Mn (λ) = {xλ1 1 xλ2 2 xλ3 3 · · · xλnn |
n
X
j=1
λj = λ, λ1 < 2, λ2 < 2, · · · , λn−1 < 2}.
6
Proposition 3.1.
(1) The span of M1 (λ) is K[x1 , x2 , . . . , xn ]λ .
(2) M1 (λ) ⊃ M2 (λ) ⊃ · · · ⊃ Mn−1 (λ) ⊃ Mn (λ).
F
(3) For all λ ≥ 1, the set nj=1 Mj (λ) ⊗ ej is in one-to-one correspondence with M1 (λ +
2) \ Khx1 , x2 , . . . , xn iλ+2 . ({e1 , . . . , en } is a set of indeterminantes used to separate
the monomials.)
Fn
(4) For all λ ≥ n − 1, the set j=1 Mj (λ) ⊗ ej is in one-to-one correspondence with the
set of all the monomials in Rλ+2 .
Proof.
(1) and (2) are obvious.
F
(3) Consider the correspondence nj=1 Mj (λ) ⊗ ej → M1 (λ + 2) defined by Mj (λ) ⊗ ej ∋
m ⊗ ej 7→ mx2j . We can make the inverse map as follows. Let m := xλ1 1 · · · xλnn ∈
K[x1 , . . . , xn ]λ+2 \ Khx1 , . . . , xn iλ+2 . Then some exponent λj is greater than 1. Let j
be the smallest index such that λj ≥ 2. Then we let m 7→ (m/x2j ) ⊗ ej ∈ Mj (λ) ⊗ ej .
(4) Since λ ≥ n − 1, we have λ + 2 ≥ n + 1. For such degrees λ, the homogeneous part of
Khx1 , . . . , xn i does not exist. So (1) and (3) imply (4).
Remark 3.1.
(1) We put M1 (0) = · · · = Mn (0) = {1}.
(2) If λ = 1, then M1 (λ) = · · · = Mn (λ) = {x1 , x2 , . . . , xn }.
(3) For all λ ≥ 1, Mj (λ)xn ⊂ Mj (λ + 1) for all j = 1, 2, . . . , n. (This will play a crucial
role in the proof of Proposition 4.2.)
Theorem 3.2. Let R = K[x1 , x2 , . . . , xn ] be the polynomial ring over a field K and A = R/I
a quadratic binomial complete intersection where the generators of I are put in a normal
form. Then the set of square-free monomials are a basis of A.
Proof. We fix the generators for I as follows.
f1 = a1 x21 + b1 m1 ,
f2 = a2 x22 + b2 m2 ,
..
.
fn = an x2n + bn mn .
(mj is a square-free monomial.) First we assume that a1 , . . . , an , b1 , . . . , bn are indeterminates. This means we work over K = π(a1 , . . . , an , b1 , . . . , bn ), where π is a prime field.
Since the growth of the dimension of Iλ is the same as the monomial ideal (x21 , x22 , . . . , x2n ),
we have
n
µ(mλ−2 I) = dimK Iλ = dimK Rλ −
.
λ
(µ denotes the number of generators of an ideal.) Put N := µ(mλ−2 I). We want to specify
N polynomials in mλ−2 I suitable for our purpose. For a minimal set of generators for mλ−2 I
we can choose the set of polynomials
S := M1 (λ − 2)f1 ∪ M2 (λ − 2)f2 ∪ · · · ∪ Mn−1 (λ − 2)fn−1 ∪ Mn (λ − 2)fn .
Note that these unions are in fact disjoint unions and furthermore these elements are linearly
independent. To see this set b1 = b2 = · · · = bn = 0. Then one sees easily that S contains all
7
the monomials in (x21 , . . . , x2n )∩Rλ . These are dimK Rλ − nλ in number. On the other hand,
the number of elements |S| can be as large as this number only if the union is the disjoint
union. If we drop the condition b1 = · · · = bn = 0, the linear independence should be easier
to prove. We rewrite the
set S as S = {g1 , g2 , . . . , gN }. Index the monomials in Rλ in such
a way that the last nλ are square-free monomials. Recall that dimK (Iλ ) + nλ = dim Rλ .
Let C ′ = (cij ) be the matrix consisting of the coefficients of the polynomials in S. So C ′
satisfies the following equality:
w1
w2
g1
..
.
g2
wN
.. = C ′
,
.
wN +1
gN
.
..
wN ′
where w1 , w2 , . . . , wN ′ are all the monomials in Rλ and N ′ = N + nλ . Let C be the
submatrix of C ′ consisting of rows 1, 2, . . . , N and columns 1, 2, . . . , N . By Theorem 4.3,
′
which we prove in the next section, we have det C 6= 0. Define the polynomials g1′ , g2′ , . . . , gN
as follows:
′
g1
g1
w1
g2′
g2
w2
.. = C −1 .. = C −1 C ′ .. .
.
.
.
′
gN
gN
wN ′
This matrix notation shows that
gk′ − wk =
linear combination of square-free monomials
for all k = 1, 2, . . . , N .
′
Since the elements g1′ , . . . , gN
are a K-basis for Iλ , it follows that any element of Rλ can
be expressed, mod Iλ , as a linear combination of square-free monomials of degree λ. Now
the proof is complete for the generic case. Next we assume R is the polynomial ring over an
arbitrary field K. Suppose that I is an ideal of R obtained by substituting elements of K
for the variables ai , bi . The ideal I is a complete intersection if and only if the resultant is
non-zero.
Note that we have established a rewriting rule which assigns any monomial m ∈ Rλ mod
I to a linear combination of square-free monomials in Rλ , for every λ = 2, . . . , n + 1.
For each λ ≥ 2, we have used the matrices C ′ and C. So we index them as C ′ (λ) and
C(λ), λ = 2, 3, . . . , n + 1. Define the matrix C ′ (2) as the coefficient matrix for f1 , . . . , fn
and C(2) is the first n × n submatrix of C ′ (2), which is automatically the diagonal matrix
with diagonal entries (a1 , a2 , . . . , an ). By Theorem 4.3, if the resultant is non-zero, then it
implies all C(3), C(4), . . . , C(n + 1) are invertible. Hence the proof is complete.
Remark 3.2.
1. It seems conceivable that any quadratic complete intersection defined by
quadrics put in a normal form can have the set of square-free monomials as a vector
space basis. Theorem 3.2 should be regarded as a case where this can be verified.
2. If each of the quadrics fi is a product of linear forms, the elements fi are in a normal
form if we adopt the variables x1 , . . . , xn as linear factors of f1 , . . . , fn respectively.
Abedelfatah [1] proved that the Artinian algebra defined by such forms can have the
square-free monomials as a vector space basis.
8
Remark 3.3. In Notation 3.1 we introduced the sets of monomials M1 (λ), M2 (λ), . . . , Mn (λ)
for all λ ≥ 1 and used them to define the set S in the proof of Theorem 3.2. Suppose that
1 2 ··· n
σ=
1 ′ 2 ′ · · · n′
is a permutation of the indices. If we use the order
1 ′ < 2 ′ < · · · < n′ ,
for the definition of Mi (λ), instead of the natural order
1 < 2 < · · · < n,
we have the different sets of monomials
M1′ (λ), M2′ (λ), . . . , Mn′ (λ).
The flag of subspaces
M1′ (λ) ⊃ M2′ (λ) ⊃ · · · ⊃ Mn′ (λ)
is different from
M1 (λ) ⊃ M2 (λ) ⊃ · · · ⊃ Mn (λ).
In this case we should adopt the set S as
S ′ = M1′ (λ − 2)f1′ ∪ M2′ (λ − 2)f2′ ∪ · · · ∪ Mn′ (λ − 2)fn′ .
The consideration of the set S ′ is important in the definition of the resultant of f1 , . . . , fn for
the binomial complete intersection. See the proof of Theorem 4.3(4). Mk′ (λ) should not be
confused with Mk′ (λ). It is important that Mk′ (λ − 2)fk′ contains the polynomial xλ−2
k ′ fk ′
for all k.
4
Some remarks on the coefficient matrices of generic
complete intersections
We work with the generic complete intersection generated by fi = ai x2i + bi mi , where mi is
a square free monomial of degree 2. Recall that we have defined the matrices
C ′ (λ) and C(λ)
for
λ = 2, 3, . . . , n + 1.
To define them we used the subsets Mi (λ) ⊂ Rλ of monomials for i = 1, 2, . . . , n for all
λ = 2, 3, . . . , n + 1.
Actually it is possible to define these sets for all λ > n + 1, although we do not need
them. From now on C(λ) are defined for all λ ≥ 2.
Lemma 4.1. Let A = {a1 , . . . , an }, B = {b1 , . . . , bn }. The matrices C(λ) and C ′ (λ) have
the following property.
(1) C(λ) is a square matrix of size dimK Rλ − nλ .
(2) Each row of C ′ (λ) contains exactly one element in the set A and one element in B.
9
(3) Each row of C(λ) contains exactly one element in the set A and at most one element
in B.
(4) A column of C(λ) contains exactly one element in the set A. (It may contain none of
or many of the elements in B.)
(5) For any i and λ ≥ 2, ai divides det C(λ).
Proof. (1) was proved earlier. Recall that a row of the matrix C ′ (λ) is defined as the
coefficients of wk fi for some i with some monomial wk of degree λ − 2. This proves (2).
Recall that the last nλ columns of C ′ (λ) are indexed by square-free monomials. On the
other hand ai can appear only in the columns indexed by monomials which are divisible by
x2i for some i. This proves (3). Suppose that ai appears in a column indexed by a monomial
w with w = xλ1 1 xλ2 2 · · · xλnn . It implies that
λ1 < 2, λ2 < 2, · · · , λi−1 < 2 ≤ λi .
Hence ak cannot appear in this column if k 6= i. This proves (4). The column of C(λ)
indexed by xλi , regarded as a monomial, contains ai and all other entries are 0. This proves
(5).
We define a circuit in C(λ) in the same way as P considered in Proposition 2.3. Namely,
a circuit in C(λ) is a sequence of elements in A ⊔ B
a(1) → b(1) → a(2) → b(2) → · · · → a(r) → b(r) → a(1)
such that a(k) and b(k) are in the same row and b(k) and a(k+1) and are in the same column
of C(λ). (The matrix C(λ) differs from P in that the same element occurs in different rows.
Thus the repetition may exit in a circuit.) It is easy to see that if there is a circuit in C(λ),
it gives us a binomial
αi αi
αi αi
α
α
ai1 1 ai2 2 · · · airir ± bi1 1 bi2 2 · · · birir .
as a factor in the determinant of C(λ). We will say that two circuits are the same if
they give the same determinant. Note that a submatrix of C(λ) whose determinant is a
binomial is another name for a circuit. In this sense C(λ) and C(λ′ ), λ 6= λ′ , can have the
same circuit.
Proposition 4.2. Suppose that a circuit exists in C(λ). Then C(λ + 1) contains the same
circuit.
Proof. Recall that the columns of C(λ) are indexed by certain monomials. By the definition of the elements {g1 , g2 , . . . , gN } to be the set S, we may index the rows of C(λ)
by the elements of Mj (λ − 2) ⊗ ej . If we multiply the elements (as indices) by xn , they
remain as indices for C(λ + 1), since the multiplication by xn does not change the exponents of monomials except the exponent of xn itself. (See Remark 3.1(3).) Suppose that
{U1 , U2 , . . . , Ur } are indices of rows and {W1 , W2 , . . . , Wr } are indices of columns which
gives us a circuit of C(λ). Then the submatrix of C(λ + 1) consisting of rows and columns
indexed by {U1 xn , U2 xn , . . . , Ur xn } and {W1 xn , W2 xn , . . . , Wr xn } gives us the same circuit
in C(λ + 1). This proves the assertion.
We denote by Res(f1 , . . . , fn ) the resultant of f1 , . . . , fn . The resultant is a polynomial
in the variables of the coefficients a1 , . . . , an , b1 , . . . , bn , but even after the coefficients are
substituted for elements in K, we call it the resultant. The ideal I obtained by substitution
is a complete intersection if and only if the resultant does not vanish. For details see Gelfand
et. al [2].
Define ∆λ := det C(λ). Now we can prove the main theorem of this paper.
10
Theorem 4.3.
(0) a1 a2 · · · an | ∆λ for any λ ≥ 2.
(1) a1 a2 · · · an | Res(f1 , . . . , fn ).
(2) Res(f1 , . . . , fn ) | ∆n+1 .
√
p
√
√
(3) ∆2 | ∆3 | · · · | ∆n+1 . ( ∆ means that the exponents are replaced by 1 in the
factorization of ∆.)
(4) A circuit that appears in some ∆(λ) is a factor of Res(f1 , . . . , fn ).
p
p
(5)
Res(f1 , f2 , · · · , fn ) = ∆n+1 .
Proof.
(0) This is proved in Lemma 4.1(5).
(1) It is easy to see that if we set ai =0 for some i, the ideal (f1 , . . . , fn ) cannot be a
complete intersection. Therefore a1 · · · an divides Res(f1 , . . . , fn ).
(2) See [2, p.429].
(3) It is easy to see that ∆2 = a1 a2 · · · an . For λ > 2, it is also easy to see that ai
divides ∆λ . Suppose that a binomial is a factor of ∆λ . Then it is a factor of ∆λ+1 by
Proposition 4.2.
(4) In §3 we constructed the sets {Mj (λ)}. They depend on the order of the indices
1 < 2 < . . . < n. Suppose that
1 2 ··· n
σ :=
1 ′ 2 ′ · · · n′
is a permutation of indices. Then with the order 1′ < 2′ < · · · < n′ we can obtain
another sequence of determinants:
∆σ2 , ∆σ3 , . . . , ∆σn+1 .
It is known that the GCD of {∆σn+1 }, where σ runs over all cycles of length n
1
2
··· n− 1
n
σ=
,
k k + 1 ··· k − 2 k − 1
gives us the resultant Res(f1 , . . . , fn ). (See [2, 429].) Thus to prove the claim it suffices
to show that if a circuit appears in one of ∆λ , with λ ≤ n, then that circuit also appears
in ∆′n+1 which is obtained based on a permutation σ, whatever the permutation is.
This is easy to see since C ′ (n + 1) = C(n + 1) up to permutation of rows and columns,
every circuit in C ′ (λ) for λ ≤ n is contained in C(n + 1).
√ √
(5) Put r = Res(f1 , . . . , fn ). By (2) we have r | ∆n + 1. A factor of ∆n+1 is either a
factor of a1 · · · an or a circuit in C(n + 1). We have seen that each ai divides r. On
the other hand if a circuit divides C(n + 1), it divides r by (4). This completes the
proof.
11
5
Some examples
In this section we set K = π(a1 , . . . , a5 ,
p1 , . . . , p5 ), the rational
function field over the prime
1 2 3 4 5
field π, and R = K[x1 , . . . , x5 ]. By σ =
, we denote the cyclic permutation
2 3 4 5 1
of the indices. If f ∈ R, we denote by f σ the polynomial obtained from f by substituting
the indices i by σ(i). In Example 5.1, the polynomials fi are determined by f1 by the rule
σ
fi = fi−1
for i = 2, 3, 4, 5.
Example 5.1.
1. If f1 = a1 x21 +p1 x1 x2 , we have Res(f1 , . . . , f5 ) = (a1 a2 a3 a4 a5 )14 (a1 a2 a3 a4 a5 +p1 p2 p3 p4 p5 )
2. If f1 = a1 x21 + p1 x2 x3 , Res(f1 , . . . , f5 ) = (a1 a2 a3 a4 a5 )5 (a1 a2 a3 a4 a5 + p1 p2 p3 p4 p5 )11
3. If f1 = a1 x21 + p1 x2 x5 , Res(f1 , . . . , f5 ) = (a1 a2 a3 a4 a5 )11 (a1 a2 a3 a4 a5 + p1 p2 p3 p4 p5 )5
There are 10 square free monomials in R5 . In all cases the resultant is one of the above
three types. It is known that the resultant is a polynomial of degree 80. If all fj factor into
two linear forms, the resultant can be computed by [2, Chapter 13, Propsotion 1.3]. This
was also computed by Abedelfatah [1] without referring to the resultant.
In the following table α, β are the integers such that
Res(f1 , . . . , f5 ) = (a1 a2 a3 a4 a5 )α (a1 a2 a3 a4 a5 + p1 p2 p3 p4 p5 )β .
monomial in f1
x1 x2
x1 x3
x1 x4
x1 x5
x2 x3
x2 x4
x2 x5
x3 x4
x3 x5
x4 x5
f1
a1 x21 + p1 x1 x2
a1 x21 + p1 x1 x3
a1 x21 + p1 x1 x4
a1 x21 + p1 x1 x5
a1 x21 + p1 x2 x3
a1 x21 + p1 x2 x4
a1 x21 + p1 x2 x5
a1 x21 + p1 x3 x4
a1 x21 + p1 x3 x5
a1 x21 + p1 x4 x5
α
15
15
15
15
5
5
11
11
5
5
β
1
1
1
1
11
11
5
5
11
11
Example 5.2. In this example we chose the square-free monomials rather randomly.
Put
f1 = a1 x21 + p1 x2 x3 ,
f2 = a2 x22 + p2 x3 x5 ,
f3 = a3 x23 + p3 x4 x5 ,
f4 = a4 x24 + p4 x1 x3 ,
f5 = a5 x25 + p5 x1 x2 .
Then
7 7 8 10 5 9
7 8 10 5 9
Res(f1 , . . . , f5 ) = a91 a82 a63 a11
4 a5 (a1 a2 a3 a4 a5 + p1 p2 p3 p4 p5 ).
12
6
Relevance to the 2nd Hessian
Example 6.1. Let K and R be the same as in the previous section. We use the notation
v = x1 , w = x2 , . . . , z = x5 interchangeably. We consider the polynomial
G = 120vwxyz + s1 + s2 + s3 + s4 + s5 ,
where
s1 = −(p31 p3 p4 v 5 + p32 p4 p5 w5 + p1 p33 p5 x5 + p1 p2 p34 y 5 + p2 p3 p35 z 5 ),
s2 = −20(p1 v 3 wz + p2 vw3 x + p3 wx3 y + p4 xy 3 z + p5 vyz 3 ),
s3 = 20(p21 p3 p4 v 3 xy + p22 p4 p5 w3 yz + p1 p23 p5 vx3 z + p1 p2 p24 vwy 3 + p2 p3 p25 wxz 3 ),
s4 = 30(p1 p3 v 2 wx2 + p2 p4 w2 xy 2 + p3 p5 x2 yz 2 + p1 p4 v 2 y 2 z + p2 p5 vw2 z 2 ),
s5 = −30(p1 p2 p4 v 2 w2 y + p2 p3 p5 w2 x2 z + p1 p3 p4 vx2 y 2 + p2 p4 p5 wy 2 z 2 + p1 p3 p5 v 2 xz 2 ).
We consider G as a polynomial in the polynomial ring R = K[v, w, x, y, z]. G was
obtained as the Macaulay dual generator of the complete intersection I = (f1 , f2 , f3 , f4 , f5 ),
where
f1 = v 2 + p1 wz,
f2 = w2 + p2 xv,
f3 = x2 + p3 yw,
f4 = y 2 + p4 zx,
f5 = z 2 + p5 vw.
As we said in the introduction, the resultant of these elements is
(p1 p2 p3 p4 p5 + 1)5 .
(The polynomial G was computed by the computer algebra system Mathematica [7].)
It is not difficult to verify that AnnR (F ) ⊃ (f1 , . . . , f5 ). If p1 p2 p3 p4 p5 + 1 6= 0, then since
AnnR G is a Gorenstein ideal containing f1 , . . . , f5 , it follows that they coinside: AnnR (G) =
(f1 , . . . , f5 ) and A := K[v, w, x, y, z]/AnnR (G) has the Hilbert function (1 5 10 10 5 1).
If p1 p2 p3 p4 p5 +1 = 0, then we can calculate that the algebra A = K[v, w, x, y, z]/AnnR (G)
has the Hilbert function (1 5 5 5 5 1).
Since we know that the square free monomials are linearly independent, the second
Hessian matrix of G is, in this case, computed as the 10 × 10 matrix
∂ 4 (G)
2
.
H (G) =
∂xi ∂xj ∂k ∂l (1≤i<j≤5),(1≤k<l≤5)
For details of higher Hessians, see [3]. Let hess2 (G) be the determinant of H 2 (G). It is a
polynomial in v, . . . , z, p1 , . . . , p5 . We may regard hess2 as a polynomial in v, w, x, y, z with
coefficients in π[p1 , p2 , p3 , p4 , p5 ]. Let P be the ideal in the polynomial ring π[p1 , p2 , p3 , p4 , p5 ]
generated by the coefficients of hess2 , where π is a prime field. It has many complicated
generators but surprisingly enough, it turns out that the ideal P is a principal ideal generated
by of (1 + p1 p2 p3 p4 p5 )5 . The computation was done also by Mathematica [7].
Example 6.2. Let F be the polynomial in the first paragraph of Introduction. F was in
fact obtained as the Macaulay dual generator fo the complete intersection I = (f1 , . . . , f5 ),
where
f1 = v 2 + p1 wx,
13
f2 = w2 + p2 xy,
f3 = x2 + p3 yz,
f4 = y 2 + p4 vz,
f5 = z 2 + p5 vw.
As in the previous example, let H 2 (F ) be the second Hessian of F . Then the coefficient ideal
in K[p1 , . . . , p5 ] turns out to be the unit ideal. Hence the algebra π[p1 , . . . , p5 ][v, w, x, y, z]/I
gives a flat family of Artinian Gorenstein algebras over K = π[p1 , . . . , p5 ]. The fiber is a
complete intersection if and only if p1 p2 p3 p4 p5 + 1 6= 0, and otherwise it is a Gorenstein
algebra defined by a 7-generated ideal. (This is a computational result.)
Acknowledgement
The third author would like to thank K. Yanagawa for a helpful conversation for quadratic
binomial complete intersections.
References
[1] A. Abedelfatah, On the Eisenbud–Green–Harris conjecture, Proc. of AMS 143, (2014),
no. 1, 105–115.
[2] I. M. Gelfand, M. M. Kaplanov, A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional determinants, Birkhäuser, 1995.
[3] T. Maeno and J. Watanabe, Lefschetz elements of Artinian Gorenstein algebras and
Hessians of homogeneous polynomials, Illinois J. Math. 53, (2009), no.2, 591–603.
[4] C.
McDaniel,
Some
remarks
arXiv:1603.09401v1[math.AC](2016).
on
Watanabe’s
bold
conjecture,
[5] T. Harima, T. Maeno, H. Morita, Y. Numata, A. Wachi, and J. Watanabe, The Lefschetz
Properties, Springer Lecture Notes 2080, Springer-Verlag, 2013.
[6] T. Harima, A. Wachi and J. Watanabe, The quadratic complete intersections associated
with the symmetric group, Illinois J. Math. 59 (2015), no. 1, 99-113.
[7] S. Wolfram, Mathematica 10.4, Wolfram Research, Inc., Mathematica, Version 10.4,
Champaign, IL (2016).
14
| 0 |
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
WITH CHARACTERISTIC-BASED FLUX PARTITIONING∗
arXiv:1510.05751v2 [cs.CE] 14 Apr 2016
DEBOJYOTI GHOSH† ‡ AND EMIL M. CONSTANTINESCU† §
Abstract. This paper presents a characteristic-based flux partitioning for the semi-implicit time
integration of atmospheric flows. Nonhydrostatic models require the solution of the compressible Euler equations. The acoustic time scale is significantly faster than the advective scale, yet it is typically
not relevant to atmospheric and weather phenomena. The acoustic and advective components of the
hyperbolic flux are separated in the characteristic space. High-order, conservative additive RungeKutta methods are applied to the partitioned equations so that the acoustic component is integrated
in time implicitly with an unconditionally stable method, while the advective component is integrated explicitly. The time step of the overall algorithm is thus determined by the advective scale.
Benchmark flow problems are used to demonstrate the accuracy, stability, and convergence of the
proposed algorithm. The computational cost of the partitioned semi-implicit approach is compared
with that of explicit time integration.
Key words. atmospheric flows, nonhydrostatic, compressible, Euler equations, implicit-explicit
time integration, characteristic-based splitting
AMS subject classifications. 65M-06, 86A-10, 76N-15
1. Introduction. The simulation of mesoscale and limited-area atmospheric
flows requires the solution to the compressible Euler equations, of which several
formulations are used by operational weather prediction codes [28, 29]. Expressing the governing equations in terms of the Exner pressure and potential temperature [18, 30, 32, 33, 67] do not conserve mass, momentum, and energy. Alternatively, the equations are expressed as the conservation of mass, momentum, and
potential temperature [3, 27, 59, 62, 68] by assuming adiabatic flows [15]. Recent
efforts [2, 10, 24, 28, 55] proposed solving the conservation laws for mass, momentum, and energy [41]. If discretized by a conservative numerical method, this approach
yields a truly conservative algorithm and allows for the specification of the true viscous
terms. The Euler equations are characterized by two temporal scales—the acoustic
and the advective scales. Atmospheric flows are often low-Mach flows where the acoustic scale is significantly faster than the advective scale [11]. The fluid velocities vary
from stationary to ∼ 30 m/s within the troposphere [64], resulting in Mach numbers
lower than ∼ 0.1. In addition, the acoustic modes do not affect weather phenomena
significantly.
Explicit time integration methods are inefficient because the largest stable time
step is restricted by the physically inconsequential acoustic time scale. Implicit time
integration methods can be unconditionally stable; however, they have rarely been
applied to atmospheric flows [47, 61, 68]. One of their drawbacks is that they require
the solution of either a nonlinear system of equations or a linearized approximation
that introduces an error in the overall discretization. An alternative approach is an
operator-split method, where the flux operator is split into its fast (acoustic) and slow
(advective) components and each component is integrated in time separately. Splitexplicit methods have been proposed and applied to atmospheric flows [39, 66, 38, 58,
∗ This material is based upon work supported by the U.S. Department of Energy, Office of Science,
Advanced Scientific Computing Research, under contract DE-AC02-06CH11357
† Mathematics & Computer Science Division, Argonne National Laboratory, Lemont, IL 60439
‡ ghosh@mcs.anl.gov
§ emconsta@mcs.anl.gov
1
2
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
65, 34, 64]. These methods are a form of decoupled multirate methods [20, 13, 54].
In this paper, we consider semi-implicit or implicit-explicit (IMEX) approaches
that stabilize the fast modes by integrating them implicitly in time; time-step sizes
are thus dictated by the slow scales. Semi-implicit methods for the primitive meteorological equations were introduced [40, 11] where the terms involving pressure and
gravitational forces are integrated implicitly. A split-step semi-implicit method for the
Euler equations expressed in terms of the primitive flow variables was proposed [19];
the prognostic variables are perturbations to the hydrostatic mean profile, and the
acoustic modes are separated by decomposing the velocity into its anelastic, curl-free,
and harmonic components. Partially implicit peer methods were applied to the Euler
equations expressed in terms of the velocity and perturbations to the density and potential temperature [35]. Multistep IMEX methods based on the Adam’s method and
backward differencing were applied to the compressible Boussinesq equations [17, 16].
Other notable algorithms include a semi-Lagrangian semi-implicit method [9], all-scale
models [60, 8], and a split-step algorithm [63]. Drawbacks of these efforts include lack
of conservation (due to the form of the governing equations, the operator splitting
for semi-implicit time integration, or the choice of the implicit and explicit methods
in the semi-implicit time integration) and lack of higher-than-second-order accuracy.
An operator splitting was introduced for the governing equations expressed as perturbations to the hydrostatic mean [29, 27]; and integrated in time by using multistep
and multistage semi-implicit methods to yield a conservative, high-order accurate algorithm. In addition to scale separation between the acoustic and advective modes,
splitting by dimension is possible, leading to horizontally explicit, vertically implicit
algorithms [55, 62, 27].
This paper presents a characteristic-based partitioning of the hyperbolic flux for
the semi-implicit time integration of limited-area and mesoscale atmospheric flows.
Our motivation is the development of a conservative, high-order accurate atmospheric
flow solver based on the Euler equations expressed as the conservation of mass, momentum, and energy, with no other assumptions. The equations are not expressed
as perturbations to a hydrostatic mean profile, and a well-balanced algorithm [24] is
used to ensure numerical accuracy; we thus avoid any assumptions or manipulations
specific to atmospheric flows. In contrast to previous approaches, we define the fast
and slow components of the hyperbolic flux by partitioning it in the characteristic
space. The discretized equations thus comprise scale-separated terms; eigenvalues
of the fast term correspond to the acoustic mode, and the eigenvalues of the slow
term correspond to the advective mode. In the context of implicit time integration
methods, characteristic-based partitioning has been previously applied to selectively
precondition the stiff characteristic modes of a hyperbolic system [48]. We linearize
the partitioning such that the solution to a linear system of equations is required;
in contrast, implicit time integration requires the solution to a nonlinear system of
equations. Moreover, we show that this linearization does not introduce an error in
the overall discretization. The partitioned equations are integrated in time with semiimplicit additive Runge-Kutta (ARK) methods [37, 27] implemented in the Portable,
Extensible Toolkit for Scientific Computing (PETSc) [6, 7]. We show that this partitioning of the flux allows time step sizes determined by the advective speeds. We also
verify that the overall algorithm is conservative and achieves its theoretical orders
of convergence. Although atmospheric flows are low-speed flows, they often develop
strong gradients, and stabilizing mechanisms are required [3, 28, 62, 45]. In this paper, we use the fifth-order weighted essentially nonoscillatory (WENO) [44, 36] and
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
3
the compact-reconstruction WENO (CRWENO) [22, 23, 26] schemes for the spatial
discretization. The algorithm described here is implemented in HyPar [1], an opensource conservative finite-difference solver for hyperbolic-parabolic partial differential
equations (PDEs).
The paper is organized as follows. Section 2 describes the governing equations,
and Section 3 outlines the overall numerical method, including the spatial discretization. The characteristic-based flux partitioning is introduced in Section 4. Section 5
describes the semi-implicit time integration and the implementation of the linearized
characteristic-based partitioning with multistage ARK methods. The extension to
two-dimensional flows is presented in Section 6. The proposed algorithm is tested for
small problems in Section 7 and applied to atmospheric flow problems in Section 8.
Section 9 contains concluding remarks.
2. Governing Equations. The governing equations for limited-area and mesoscale
nonhydrostatic atmospheric flows are the Euler equations [41], with the addition of
gravitational force as a source term. They are expressed as
∂ρ
+ ∇ · (ρu) = 0,
∂t
∂ (ρu)
+ ∇ · (ρu ⊗ u + pId ) = −ρg,
∂t
∂e
+ ∇ · (e + p) u = −ρg · u,
∂t
(2.1)
(2.2)
(2.3)
where ρ is the density, u is the velocity vector, p is the pressure, and g is the gravitational force vector (per unit mass). Id denotes the identity matrix of size d, where
d is the number of space dimensions, and ⊗ represents the Kronecker product. The
energy is given by
e=
p
1
+ ρu · u,
γ−1 2
(2.4)
where γ = 1.4 is the specific heat ratio. The equation of state relates the pressure,
density, and temperature as p = ρRT , where R is the universal gas constant and T is
the temperature. Two additional quantities of interest in atmospheric flows are the
Exner pressure π and the potential temperature θ, defined as
π=
p
p0
γ−1
γ
and θ =
T
,
π
(2.5)
respectively. The pressure at a reference altitude is denoted by p0 . We consider oneand two-dimensional flows (d = 1, 2) in this paper. The governing equations share the
same form as (2.1)–(2.3) when expressed in terms of nondimensional variables [24],
and thus these equations are used for both dimensional and nondimensional problems.
3. Numerical Methodology. The numerical discretization of the governing
equations is described in one spatial dimension, and it can be trivially extended to
multiple dimensions. Equations (2.1)–(2.3) (with d = 1) can be expressed as a system
of hyperbolic PDEs,
∂q ∂f (q)
+
= s (q) ,
∂t
∂x
(3.1)
4
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
Fig. 3.1. Illustration of a one-dimensional domain and the grid on which (3.1) is discretized.
where
ρ
ρu
0
q = ρu , f = ρu2 + p , and s = −ρg .
e
(e + p)u
−ρug
(3.2)
Equation (3.1) is discretized in space with a conservative finite-difference formulation.
Figure 3.1 shows a one-dimensional domain of unit length, discretized by N + 1 grid
points. The cell centers and interfaces are shown. The resulting semi-discrete ODE
in time is given by
dQ
= F̂ (Q) + Ŝ (Q) ,
dt
(3.3)
where Q = [qj ; j = 1, · · · , N − 1] is the solution vector of the state variable at the
cell centers (excluding boundary points), Ŝ is the discretized source term, and the
discretized hyperbolic flux at a grid point is given by
F̂j = −
1
f̂j+1/2 − f̂j−1/2 .
∆x
(3.4)
The numerical flux f̂ is an approximation to the primitive of f (q) at the cell interfaces
xj±1/2 .
Equation (3.1) represents a hyperbolic balance law that admits equilibrium states
where the pressure gradient is balanced by the gravitational force. The spatially discretized ODE, (3.3), must preserve this balance on a finite grid to machine precision;
failure to do so will result in inaccurate solutions since atmospheric phenomena are
often small perturbations around this balanced equilibrium state. We use a wellbalanced formulation to evaluate the source term Ŝ [24]. The description of this
is omitted because it is independent of the time integration aspects discussed here;
however, it is a necessary component of the overall algorithm.
The numerical flux at the cell interfaces f̂j±1/2 in (3.4) is computed by using the
Rusanov upwinding scheme [51, 43],
1 L
R
L
,
(3.5)
− max ν
−
q̂
f̂j+1/2 + f̂j+1/2
q̂R
f̂j+1/2 =
j+1/2
j+1/2
j,j+1
2
where the superscripts L and R indicate the left- and right-biased
p interpolations, respectively. The dissipation factor is ν = a + |u|, where a = γp/ρ is the speed of
L,R
sound. The left- and right-biased flux f̂j+1/2
and solution q̂L,R
j+1/2 at the interfaces
are computed by using the fifth order WENO [36] and CRWENO [22] schemes. The
following paragraphs describe a left-biased reconstruction; the corresponding expressions for the right-biased reconstruction can be trivially obtained. The description
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
5
below applies to a scalar flux function, and it is extended to the vector flux in (3.3)
through a componentwise approach.
The WENO schemes use a solution-dependent interpolation stencil selection [44]
to achieve high-order accuracy where the solution is smooth and to avoid oscillations
across discontinuities. The fifth-order WENO scheme [36] is constructed by three
third-order interpolation schemes:
1
7
11
1
1
fˆj+1/2
= fj−2 − fj−1 + fj , c1 =
,
3
6
6
10
5
1
6
1
2
,
= − fj−1 + fj + fj+1 , c2 =
fˆj+1/2
6
6
3
10
5
1
3
1
3
.
= fj + fj+1 − fj+2 , c3 =
fˆj+1/2
3
6
6
10
(3.6)
(3.7)
(3.8)
Multiplying (3.6)–(3.8) with their optimal coefficient ck , k = 1, 2, 3, and then adding
them results in the following fifth-order accurate interpolation scheme:
1
13
47
27
1
fˆj+1/2 =
fj−2 − fj−1 + fj + fj+1 − fj+2 .
30
60
60
60
20
(3.9)
Solution-dependent weights are computed based on the local solution smoothness as
αk
ck
ωk = P
; αk =
p ; k = 1, 2, 3,
α
(ǫ
+
βk )
k k
(3.10)
where ǫ = 10−6 is a small number to prevent division by zero and βk are the smoothness indicators for the stencils, given by
13
(fj−2 − 2fj−1 + fj )2 +
12
13
β2 =
(fj−1 − 2fj + fj+1 )2 +
12
13
and β3 =
(fj − 2fj+1 + fj+2 )2 +
12
β1 =
1
(fj−2 − 4fj−1 + 3fj )2 ,
4
1
(fj−1 − fj+1 )2 ,
4
1
(3fj − 4fj+1 + fj+2 )2 .
4
(3.11)
(3.12)
(3.13)
The fifth-order WENO (WENO5) scheme is obtained by multiplying (3.6)–(3.8) by
the solution-dependent weights ωk (instead of the optimal coefficients ck ) and then
adding them. It can be expressed as
1
1
ω1
fˆj+1/2 = fj−2 − (7ω1 + ω2 )fj−1 + (11ω1 + 5ω2 + 2ω3 )fj
3
6
6
ω3
1
fj+2 .
+ (2ω2 + 5ω3 )fj+1 −
6
6
(3.14)
If the solution is locally smooth, ωk → ck , k = 1, 2, 3, and (3.14) is equivalent to (3.9).
The CRWENO scheme [22] applies the WENO concept of solution-dependent
interpolation stencils to compact finite-difference methods [42]. The fifth-order CRWENO scheme [22, 23] is constructed by considering three third-order compact interpolation schemes:
2ˆ
fj−1/2 +
3
1ˆ
fj−1/2 +
3
2ˆ
fj+1/2 +
3
1ˆ
1
2
fj+1/2 = (fj−1 + 5fj ) ; c1 =
,
3
6
10
2ˆ
1
5
fj+1/2 = (5fj + fj+1 ) ; c2 =
,
3
6
10
1ˆ
1
3
fj+3/2 = (fj + 5fj+1 ) ; c3 =
.
3
6
10
(3.15)
(3.16)
(3.17)
6
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
80
3
80
F
60
60
40
40
20
20
FF
FS
Imaginary
Imaginary
1
0
Imaginary
2
0
0
-20
-20
-40
-40
-60
-60
-1
-2
-3
-2
-1.5
-1
-0.5
Real
(a) Spatial discretization D
0
-80
-80 -60 -40 -20
0
Real
(b) Right-hand-side operator
-80
-80 -60 -40 -20
0
Real
∂ F̂
∂Q
(c) Split operators
∂ F̂F,S
∂Q
Fig. 4.1. Eigenvalues of the spatial discretization operator corresponding to WENO5, the Jacobian of the right-hand side of (4.2), and the Jacobians of the fast and slow partitioned terms of
(4.13). Note that the eigenvalues shown in (b) are those shown in (a) scaled by {u, u ± a} /∆x.
Multiplying (3.15)–(3.17) with their optimal coefficients (ck , k = 1, 2, 3) and adding
them results in a fifth-order compact scheme:
6
1
1
19
1
3 ˆ
fj−1/2 + fˆj+1/2 + fˆj+3/2 =
fj−1 + fj + fj+1 .
10
10
10
30
30
3
(3.18)
Replacing the optimal coefficients ck with solution-dependent weights ωk yields the
fifth-order CRWENO scheme (CRWENO5):
2
1
2
1
1
ω1 + ω2 fˆj−1/2 +
ω1 + (ω2 + ω3 ) fˆj+1/2 + ω3 fˆj+3/2
3
3
3
3
3
ω1
5(ω1 + ω2 ) + ω3
ω2 + 5ω3
=
fj−1 +
fj +
fj+1 . (3.19)
6
6
6
The weights ωk are computed by (3.10) and (3.11)–(3.13). If the solution is locally
smooth, ωk → ck , k = 1, 2, 3, and (3.19) is equivalent to (3.18). The left-hand side
of (3.19) represents a tridiagonal system with solution-dependent coefficients that
needs to be solved at each time-integration step or stage. An efficient and scalable
implementation of the CRWENO5 scheme [25] is used in this study.
4. Characteristic-Based Flux Partitioning. The separation of the acoustic
and advective components of the hyperbolic flux is described by considering (3.1) and
7
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
its semi-discrete form (3.3), without the source terms. The one-dimensional Euler
equations, although nonlinear, satisfy the following property [41],
f (q) = A (q) q, A (q) =
∂f
,
∂q
(4.1)
where A is the flux Jacobian. This property, though not essential to the flux partitioning, is useful as a tool to describe it. Equation (3.3) (without the source term)
can be expressed as
dQ
= F̂ (Q) ≡ [D ⊗ A] Q,
dt
(4.2)
where D represents a finite-difference operator for a scalar function φ (x) on a grid,
− [φx,j ] = [D] [φj ] + O (∆xr ) , 0 < j < N, φj = φ (xj ) , φx,j = φx (xj )
(4.3)
with r being the spatial order of accuracy. The WENO5 and CRWENO5 schemes,
described in the preceding section, can be expressed in this form [21]. Therefore, the
eigenvalues of the right-hand side (RHS) operator of (4.2) are the products of the
eigenvalues of the discretization operator D and the eigenvalues of the flux Jacobian
that are the characteristic wave speeds of the Euler equations,
!
∂ F̂
= λ (D) ∗ λ (A) ,
(4.4)
λ
∂Q
where ∗ denotes the following operation between two sets A and B:
A ∗ B = {(ab) |a ∈ A, b ∈ B} .
(4.5)
The flux Jacobian has three real eigenvalues [41],
λ (A) = {u, u + a, u − a} ,
(4.6)
where u is the flow velocity and a is the local speed of sound. Figure 4.1(a) shows
the eigenvalues of the finite-difference operator D representing the WENO5 scheme,
computed by using a linear spectral analysis [23]. Figure 4.1(b) shows the eigenvalues
of the Jacobian of F̂ evaluated on a periodic domain of unit length, discretized by
a grid with 40 points and the WENO5 scheme,
p with ρ = 1 + 0.1 sin (2πx), u = 0.2,
p = 1/γ. The mean speed of sound is a∞ = γp∞ /ρ∞ = 1, and therefore the mean
Mach number is M∞ = u∞ /a∞ = 0.2. The Jacobian of F̂ is computed by using finite
differences. The eigenvalues in Figure 4.1(b) form three distinct sets that correspond
to the eigenvalues of D (in Figure 4.1(a)) multiplied by each of the characteristic wave
speeds of the Euler equations. The smallest ring represents the advective mode (u)
where the eigenvalues of D are scaled by u/∆x. The two larger rings represent the
acoustic modes (u ± a) where the eigenvalues of D are scaled by (u ± a) /∆x. The
separation in magnitude of the acoustic and advective eigenvalues is a function of the
Mach number M = u/a; lower Mach numbers result in a larger separation.
The flux term f (q) is partitioned into its slow and fast components as follows:
f (q) = A (q) q = AF (q) q + AS (q) q = fF (q) + fS (q) ,
(4.7)
8
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
where A = AF + AS , and the subscripts F and S denote “fast” and “slow” time
scales, respectively. The partitioned flux Jacobians AF,S are defined as
0
u
, ΛS =
,
u+a
0
AF,S = X ΛF,S X −1 , ΛF =
(4.8)
u−a
0
where X is the matrix with the right eigenvectors as its columns and X −1 is the
matrix with the left eigenvectors as its rows. ΛF,S represent the fast (acoustic) and
slow (advective) characteristic modes and satisfy
ΛF + ΛS = diag [u, u − a, u + a] = X −1 AX .
(4.9)
The flux Jacobian A, and the matrices X , X −1 for the one-dimensional Euler equations
are provided in [41], and the resulting expressions for the slow and fast flux fS,F (q) =
AS,F q are
γ−1
1
ρu
ρu
γ
γ
γ−1
1
2
2
,
f
(q)
=
(4.10)
ρu
ρu
+
p
fS (q) =
.
F
γ
γ
1 γ−1
ρu3
ρu3
(e + p) u − 12 γ−1
2
γ
γ
The corresponding partitioning for the RHS operator F̂ of (4.2) is expressed as follows:
F̂ (Q) = [D ⊗ A] Q = [D ⊗ (AF + AS )] Q
= [D ⊗ AF ] Q + [D ⊗ AS ] Q = F̂F + F̂S ,
(4.11)
where F̂F,S are the spatially discretized terms corresponding to the partitioned flux
fF,S . The fast term F̂F represents only the acoustic modes, while the slow term F̂S
represents the advective mode. Figure 4.1(c) shows the eigenvalues of the Jacobians
of the partitioned terms F̂F,S for the same flow as in Figure 4.1(b). The partitioning
results in a clear separation of the advective and acoustic eigenvalues; the eigenvalues
of the slow term are significantly smaller in magnitude than those of the fast term.
We note that the eigenvalues of the partitioned terms F̂F,S do not correspond exactly
to the eigenvalues of F̂ because of the nonlinearity of the Euler equations,
#
"
#
"
#
"
∂fF,S
∂ F̂S
∂ F̂
∂ F̂F,S
∂ F̂F
∪Λ
6= Λ
. (4.12)
6= AF,S ⇒
6= D ⊗ AF,S ⇒ Λ
∂q
∂Q
∂Q
∂Q
∂Q
Atmospheric flows are low-speed flows where the advective mode is significantly slower
than the acoustic modes (u ≪ a). The separation of the two time scales is useful in
the context of semi-implicit time integration, discussed in the next section.
With the partitioning defined as (4.11), (3.3) can be expressed as
o
dQ n
(4.13)
= F̂F (Q) + F̂S (Q) + Ŝ (Q) .
dt
We note that (4.11) holds true if and only if both the slow and fast flux terms fF,S are
discretized by the same finite-difference operator D. In the context of the nonlinear
WENO5 and CRWENO5 schemes, this implies that the same solution-dependent coefficients for the interpolation operators (3.14) or (3.19) need to be used for discretizing
fS and fF . In our implementation, the WENO coefficients (3.10) are computed based
on f (q).
9
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
Table 5.1
List of time integration methods and their orders and number of stages.
Name
ARK 2c
ARK 3
ARK 4
RK 2a
RK 3
RK 4
Type
Semi-implicit
Semi-implicit
Semi-implicit
Explicit
Explicit
Explicit
Order
2
3
4
2
3
4
Stages (s)
3
4
6
2
3
4
Comments/Reference
[27]
[37]
[37]
Explicit midpoint method
Kutta’s third-order method
Classical fourth-order method
5. Time Integration. Equation (4.13) is integrated in time by using semiimplicit additive Runge-Kutta (ARK) methods [5, 37, 46] implemented in the time
integration module (TS) of PETSc [6, 7]. These methods apply two different integrators for the slow and the fast terms; the fast terms are integrated in time implicitly,
and thus the largest stable time step size of the algorithm is determined by the eigenvalues of the slow term. ARK methods can be represented with the following Butcher
tableaux [12]:
ci
c̃i
aij
,
bj
s
s
X
X
ãij
ãij ,
aij , c̃i =
; i, j = 1, · · · , s , ci =
b̃j
j=1
(5.1)
j=1
where aij , bj , ci define the explicit integrator for the slow term, ãij , b̃j , c̃i define the
implicit integrator for the fast term, and s is the number of stages. The coefficients
satisfy aij = 0, j ≥ i and ãij = 0, j > i. The ARK methods applied to (4.13) and
using coefficients (5.1) result in the following:
Stage computations i = 1, · · · , s :
i
i−1
o
n
X
X
ãij F̂F Q(j) + Ŝ Q(j) , (5.2a)
aij F̂S Q(j) + ∆t
Q(i) = Qn + ∆t
j=1
j=1
Step completion :
s
s
o
n
X
X
b̃i F̂F Q(j) + Ŝ Q(i) ,
bi F̂S Q(j) + ∆t
Qn+1 = Qn + ∆t
i=1
(5.2b)
i=1
where Qn is the solution at the current time step and Qn+1 is the solution at the
next time step. The gravitational source term is treated implicitly in time.
Three high-order ARK methods are considered in this study: a second-order
(three-stage) method (ARK 2c) constructed in [27] and defined by
0√
0
0√
0√
2− 2 2− 2
2 − 2 1 − √12 1 − √12
0
,
,
1
1
1
√
√
√
1
1
−
1
1 − a3,2 a3,2
0
2 2
2 2
2
1
1
1
1
1
√
√
1 − √2
√
√
1 − √12
2 2
2 2
2 2
2 2
(5.3)
with a3,2 = 21 , a third-order (four-stage) method (ARK 3), and a fourth-order (sixstage) method (ARK 4) constructed in [37]. The implicit parts of the ARK methods
10
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
used here are ESDIRK (explicit first-stage, single-diagonal coefficient) and L-stable.
The performance of the ARK methods is compared with that of the explicit RK
methods: second-order, two-stage RK 2a, third-order, three-stage RK 3, and the
classical fourth-order four-stage RK 4. Table 5.1 summarizes the time integration
methods used in this paper.
5.1. Linearization. The stage calculations (5.2a) require the solution of a nonlinear system of equations. We modify the partitioning of the RHS such that only a
linear system needs to be solved instead. The fast term is linearized, and the implicit
integrator is applied on this linear part. The slow term, redefined as total RHS with
the linearized fast term subtracted from it, is treated explicitly. The linearized fast
term removes the stiffness from the original RHS and reduces the computational cost
of solving the implicit part. We note that the linearization does not introduce an
error in the overall discretized equations.
Ignoring the source term for now, we rewrite (5.2a) as the following nonlinear
system of equations for an implicit ARK stage:
i−1 n
o
X
aij F̂S Q(j) + ãij F̂F Q(j) ,
Q(i) − σ F̂F Q(i) = Qn + ∆t
j=1
⇒ [I − σD ⊗ AF ] Q(i)
i−1 n
o
X
aij F̂S Q(j) + ãij F̂F Q(j) ,
= Qn + ∆t
(5.4)
j=1
where σ = ∆tãii . The nonlinearity of (5.4) arises from two sources: the fast Jacobian
AF = AF (Q) and the WENO5/CRWENO5 finite-difference operator D = D (ω),
where ω = ω (f (q)) are the solution-dependent weights given by (3.10).
The fast Jacobian is evaluated at the beginning of the step and kept fixed for all
the stages. The partitioning of the flux at stage i is modified as follows:
fF Q(i) = AF (Qn ) Q(i) ,
fS Q(i) = f Q(i) − fF Q(i) .
(5.5)
The corresponding expressions for spatially discretized partitioned flux terms are
F̂F Q(i) = [D ⊗ AF (Qn )] Q(i) ,
oi
n
h
F̂S Q(i) = F̂ Q(i) − F̂F Q(i) = D ⊗ A Q(i) − AF (Qn ) Q(i) . (5.6)
Equation (5.6) satisfies F̂F Q(i) + F̂S Q(i) = F̂ Q(i) exactly. Therefore, the linearized partitioning is consistent with the unpartitioned RHS and does not introduce
an error in the overall algorithm.
The nonlinear finite-difference operator D (ω) is linearized by computing and fixing the solution-dependent
weights (3.10) at the beginning of each stage. The com
putation of F Q(i) and FF Q(i) during the iterative solution of (5.4) does not
recalculate the weights ω based on the smoothness of the current guess for Q(i) . We
define the finite-difference operator at stage i as
ω f Q(i−1)
i>1
D̄ = D (ω̄) , where ω̄ =
.
(5.7)
ω (f (Qn ))
i=1
Thus, during the stage computation, the interpolation coefficients in (3.14) or (3.19)
are constant, and the resulting operators are linear.
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
11
Inspection of (3.1) shows that the source term is linear if the gravitational forces do
not depend on the solution (this is true for our application). As previously mentioned,
a well-balanced formulation [24] is used to evaluate it on the discretized domain; this
formulation preserves its linearity. Denoting S = ∂S/∂Q as the Jacobian of the
source term, we apply (5.6) and (5.7) to (5.2a) to obtain the following linear system
of equations for the implicit ARK stages:
I − σ D̄ ⊗ AF (Qn ) + S Q(i)
i−1 n
o
X
aij F̂S Q(j) + ãij D̄ ⊗ AF (Qn ) + S Q(j) , (5.8)
= Qn + ∆t
j=1
where F̂S is defined by (5.6). Equation (5.8) is solved iteratively by using the generalized residual method (GMRES) [52, 53] implemented in the Krylov solver module
(KSP) of PETSc, and a Jacobian-free approach is adopted where the Jacobian
J ≡ I − σ D̄ ⊗ AF (Qn ) + S
(5.9)
is specified as its action on a vector. The stopping criterion for the iterative solver is
specified as krk+1 − rk k2 ≤ max (τr kr0 k2 , τa ), where τa and τr are the absolute and
relative tolerances, respectively; r is the residual given by
(i)
rk = I − σ D̄ ⊗ AF (Qn ) + S Qk
i−1 n
o
X
− Qn + ∆t
aij F̂S Q(j) + ãij D̄ ⊗ AF (Qn ) + S Q(j) ;
(5.10)
j=1
and the subscript k denotes the kth guess for the stage solution Q(i) .
5.2. Modified Upwinding. The interpolated flux at a grid interface is computed by using (3.5), which can be written for the total and the fast flux terms as
follows:
i
1h L
L
R
f̂j+1/2 =
,
(5.11)
− δj+1/2 q̂R
f̂j+1/2 + f̂j+1/2
j+1/2 − q̂j+1/2
2
i
1h L
L
R
F
,
(5.12)
q̂R
f̂F,j+1/2 =
f̂F,j+1/2 + f̂F,j+1/2
− δj+1/2
j+1/2 − q̂j+1/2
2
where δ, δ F are the diffusion coefficients for the upwinding scheme and f̂ , f̂F are the
reconstructed numerical total and fast flux terms at the grid interfaces, related to
F̂, F̂F in (5.6) through (3.4). Subtracting (5.12) from (5.11) results in the following
expression for the slow flux at a grid interface:
i
1 h L
R
L
R
f̂S,j+1/2 =
− f̂F,j+1/2
+ f̂F,j+1/2
f̂j+1/2 + f̂j+1/2
2
i
1 h
L
F
.
(5.13)
−
q̂R
δj+1/2 − δj+1/2
j+1/2 − q̂j+1/2
2
If the same diffusion coefficient for the upwinding scheme is used for both total and
the fast flux,
F
δj+1/2 = δj+1/2
= max ν,
j,j+1
(5.14)
12
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
we obtain a central discretization of the slow flux term (5.13) with no diffusion and
purely imaginary eigenvalues. This is undesirable with respect to the ARK time
integrator, as explained in Sec. 5.3. To avoid this, we modify the upwinding method
to apply the diffusion specifically along the characteristic fields that the flux term
represents. The diffusion coefficients are expressed as
µ̄
0
h i
h i
X −1 ,
X −1 ,
ν̄
ν̄
δ̃
=X
=X
δ̃ F
(5.15)
j+1/2
j+1/2
ν̄
ν̄
where
ν̄ = max (|u| + a) ,
j,j+1
µ̄ = max |u| ,
(5.16)
j,j+1
and the equations to compute the flux at the grid interfaces from their left- and
right-biased interpolated values are
h i
1 L
R
L
R
,
(5.17)
+ f̂j+1/2 − δ̃
f̂
f̂j+1/2 =
q̂j+1/2 − q̂j+1/2
2 j+1/2
j+1/2
h i
1 L
L
R
q̂R
−
q̂
.
(5.18)
f̂F,j+1/2 + f̂F,j+1/2
− δ̃ F
f̂F,j+1/2 =
j+1/2
j+1/2
2
j+1/2
We then obtain the following expression for the slow term:
h i
1 L
L
R
,
− δ̃ S
−
q̂
f̂S,j+1/2 =
q̂R
f̂F,j+1/2 + f̂F,j+1/2
j+1/2
j+1/2
2
j+1/2
(5.19)
where
h i
δ̃ S
j+1/2
h i
= δ̃
j+1/2
h i
− δ̃ F
j+1/2
µ̄
=X
0
0
X −1 .
(5.20)
The modified upwinding applies the diffusion to the fast term only along the acoustic
modes and to the slow term only along the advective mode; it does not add any
additional diffusion compared with the spatial discretization of the unsplit flux. This
modified upwinding scheme resembles the Roe upwinding scheme [49].
5.3. Linear Stability Considerations. We analyze the linear stability of the
semi-implicit time integration method (5.2) by considering a linear test problem,
Q′ (t) = λQ(t) + µQ(t) ,
(5.21)
where λ, µ ∈ C represent eigenvalues of the nonstiff (slow) and stiff (fast) components
[14], respectively, and C is the set of complex numbers. In our application, λQ(t)
represents the slow component F̂S (Q), and µQ(t) represents the fast component
F̂F (Q) + Ŝ (Q). This problem provides useful insights into the stability behavior for
the nonlinear problem. A time step can be expressed as
Qn+1 = R(λ∆t, µ∆t) Qn ,
(5.22)
where R is the stability function of the method. The stability region S of the semiimplicit method is then defined by
S = {λ∆t ∈ C, µ∆t ∈ C : |R(λ∆t, µ∆t)| ≤ 1} ⊂ C × C.
(5.23)
13
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
3
2.5
∂ Sλ(µa ∆ t)
3
µa
2.5
∂ S (µ ∆ t)
2
µ
λ
b
2
1.5
µ
λ
Im(λ) ∆ t
Im(λ) ∆ t
b
∂ S (µ ∆ t)
c
c
1.5
1
1
0.5
0.5
0
−3
−2.5
−2
−1.5
−1
−0.5
0
0
−5
−4.5
−4
−3.5
Re(λ) ∆ t
(a) a3,2 =
1
(3
6
−3
−2.5
−2
−1.5
−1
−0.5
0
Re(λ) ∆ t
√
+ 2 2)
(b) a3,2 =
1
2
Fig. 5.1. The (explicit) stability regions for method (5.3) for three fixed stiff eigenvalues sets
with different values for a3,2 coefficients. The stability region is degraded more with method coefficients set in 5.1(a) than with 5.1(b) as the implicit eigenvalues are set to be pure imaginary.
The high dimensionality of the stability region makes its analysis difficult. To simplify
it, we fix the stiff stability region Sµ as the set of stiff eigenvalues
Sµ = {µ1 ∆t, µ2 ∆t, . . . , µk ∆t} ,
(5.24)
where k is the total number of eigenvalues, and focus on the nonstiff stability region
Sλ ⊂ C. The method is stable for all λ∆t ∈ Sλ for the nonstiff component subject to
Sµ .
The condition ℜ (λ) → 0 imposes tight restrictions on the classes of methods
that can be used in practice because of the stability properties of the time integration
methods. It is challenging to construct methods whose Sλ has a large overlap with the
imaginary axis in the complex plane. Moreover, the overlap of Sλ with the imaginary
axis is negatively impacted for ℜ (µ) → 0 [14], as is the case in our application. Semiimplicit methods with explicit imaginary stability that are less dependent on the
implicit operator have been constructed [37, 17, 27]; however, relaxing this restriction
allows for more efficient methods.
Figure 5.1 illustrates
the behavior of the stability region for scheme (5.3) us√
ing a3,2 = 16 (3 + 2 2) and a3,2 = 12 for different sets of implicit eigenvalues. The
stiff eigenvalues µ(·) and the boundaries of the corresponding explicit stability regions ∂Sλ µ(·) ∆t are shown; the subscript a refers to the case where µ are purely
imaginary, the subscript b refers to the case where µ has both real and imaginary components, and the subscript c refers to the case where the real part of µ is larger than
its imaginary part. Overall, the size of the stability region reduces as the imaginary
components of the stiff eigenvalues increase. In addition, the overlap of Sλ µ(·) ∆t
with the imaginary axis is largest for µc and smallest for µa . The degradation of Sλ
is more pronounced in Fig. 5.1(a), and thus we choose a3,1 = a3,2 = 21 in (5.3). A
more detailed discussion is presented in [27]. This brief analysis demonstrates the
importance of avoiding imaginary eigenvalues λ for the nonstiff component F̂S (Q)
since ℜ (µ) → 0 holds true for several eigenvalues of F̂F (Q) + Ŝ (Q).
5.4. Preconditioning. The block Jacobi preconditioner [52], implemented in
the preconditioning module (PC) of PETSc, is used in the current work. Although
14
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
the Jacobian of the implicitly treated operator is specified in a matrix-free way (5.9),
an approximation to the Jacobian is provided as a sparse matrix. The approximate
Jacobian for the preconditioner is defined as
Jp ≡ I − σ D̄1st ⊗ AF (Qn ) + S ≈ J ,
(5.25)
where D1st represents a first-order upwind discretization operator. This results in
a block tridiagonal matrix for the one-dimensional system and will result in block
penta- and septa-diagonal systems for two- and three-dimensional flows, respectively.
Development of more advanced preconditioning techniques for the algorithm presented
here is beyond the scope of this paper and will be studied in the future.
6. Extension to Two-Dimensional Flows. The two-dimension Euler equations with gravitational source terms can be expressed as the following hyperbolic
conservation law:
∂q ∂f (q) ∂h (q)
+
+
= s (q) ,
∂t
∂x
∂y
where
ρu
ρ
ρu2 + p
ρu
q=
ρv , f = ρuv
(e + p)u
e
0
ρv
−ρg
· î
ρuv
,s =
,h =
2
ρv + p
−ρg · ĵ
(e + p)v
− ρug · î + ρvg · ĵ
(6.1)
.
The Cartesian unit vectors along x and y are denoted by î and ĵ, respectively, and u, v
are the velocity components along x, y. The spatial discretization described in the
preceding sections is extended to the two-dimensional equations through a dimensionby-dimension approach, where the derivatives along one dimension are computed independently of the other dimension. This paper considers only problems solved on
Cartesian grids. The eigenvalues of the two-dimensional system are given by
∂ (f , h)
= {(u, v) , (u, v) , (u, v) − a, (u, v) + a} ,
(6.2)
Λ
∂q
and they are split into their advective and acoustic components as
ΛS = {(u, v) , (u, v) , 0, 0} , ΛF = {0, 0, (u, v) − a, (u, v) + a} .
(6.3)
The slow and fast Jacobians are obtained by using the similarity transformations given
by (4.8), and the partitioned flux and its spatially discretized counterpart are then
computed. The left and right eigenvectors for the two-dimensional Euler equations
are provided in [31, 50], and we use these in this paper. The resulting semi-discrete
ODE can be expressed as
o n
o
dQ n
= F̂F (Q) + ĤF (Q) + F̂S (Q) + ĤS (Q) + Ŝ (Q) ,
(6.4)
dt
where ĤF,S denotes the spatially discretized partitioned fluxes along the y-direction.
Equation (6.4) is integrated in time by using
an ARK
o method given by (5.1), where
n
the fast flux terms and the source term F̂F + ĤF + Ŝ are treated implicitly and
o
n
the slow flux terms F̂S + ĤS are treated explicitly.
15
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
200
FF(u)
FS(u)
150
150
100
100
50
20
0
0
-50
-20
Imaginary
Imaginary
200
-20
-100
-10
0
50
2
0
0
-50
-2
-2
-100
-150
-1
0
-150
-200
-250
FF(u)
FS(u)
-200
-200
-150
-100
-50
0
-250
-200
-150
Real
-100
-50
0
Real
(a) M∞ = 0.1
(b) M∞ = 0.01
Fig. 7.1. Eigenvalues of Jacobians of the partitioned flux terms F̂F,S for the one-dimensional
density wave advection at two Mach numbers. The CRWENO5 scheme is used, and the problem is
discretized on a grid with 80 points. The insets are magnified plots of the eigenvalues of the slow
flux term F̂S .
1e-06
1e-07
1e-08
1e-08
1e-09
1e-09
||ε||2
||ε||2
1e-07
1e-06
ARK 2c
ARK 3
RK 2a
RK 3
1e-10
1e-10
1e-11
1e-11
1e-12
1e-12
1e-13
ARK 2c
ARK 3
RK 2a
RK 3
1e-13
0.1
1
σa
(a) M∞ = 0.1
10
0.1
1
10
100
σa
(b) M∞ = 0.01
Fig. 7.2. Density wave advection: L2 norm of the error as a function of the acoustic CFL
number for the ARK and explicit RK methods.
7. Numerical Tests. The performance of the semi-implicit time integrators
with characteristic-based flux partitioning is tested in this section with two simple
flow problems. The tests verify that the integration of the acoustic modes in time by
using an implicit method results in a largest stable time step that is determined by
the advective scale. In addition, the accuracy and convergence of the time integration
methods are demonstrated. The two problems solved in this section are formulated
in terms of nondimensional flow variables.
7.1. Density Wave Advection. This one-dimensional test problem involves
the advection of a sinusoidal density wave over a periodic domain. The exact solution
16
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
is given by
ρ (x, t) = ρ∞ + ρ̂ sin [2π (x − u∞ t)] , u (x, t) = u∞ , p (x, t) = p∞ .
(7.1)
With this solution, the Euler equations are equivalent to the linear advection equation.
The mean density and pressure are taken as ρ∞ = 1 and p∞ = 1/γ, respectively,
resulting in the mean speed of sound as a∞ = 1. We consider two values for the mean
Mach number (given by M∞ = u∞ /a∞ ): 0.1 and 0.01. The domain is x ∈ [0, 1], and
periodic boundary conditions are applied at the boundaries.
Figure 7.1 shows the eigenvalues of the partitioned flux Jacobians for the CRWENO5 scheme on a grid with 80 points. The Jacobians are evaluated from the
discretized operators F̂F,S through finite differences. The eigenvalues for the case
with mean Mach number 0.1 is shown in Figure 7.1(a), with the magnified subplot
showing the eigenvalues of the slow flux term F̂S . As shown earlier, the flux partitioning results in a separation of the advective and acoustic modes. The eigenvalues
of the slow flux correspond to the advective mode, and they are smaller in magnitude than those of the fast flux by an approximate factor of 10 (the inverse of the
Mach number). Figure 7.1(b) shows the eigenvalues for the case with a mean Mach
number of 0.01. At this smaller Mach number, the separation between the advective
and acoustic scales is larger. The magnitudes of the eigenvalues of the slow flux are
−1
smaller than those of the fast flux by an approximate factor of M∞
= 100. The two
acoustic modes are characterized by the wave speeds u ± a; and thus, as the Mach
number decreases, they converge to a.
Figure 7.2 shows the error as a function of the acoustic Courant-Friedrichs-Lewy
(CFL) for the second- and third-order ARK methods, ARK 2c and ARK 3, as well
as the two explicit Runge-Kutta (RK) methods of the same orders, RK 2a and RK
3. The solutions are obtained with the tolerances for the iterative solver specified as
τr = τa = 10−10 . The final times for both cases correspond to one cycle over the
periodic domain (10 for M∞ = 0.1 and 100 for M∞ = 0.01). The time step sizes are
increased from a small value until they reach a value for which the solution blows up,
thus indicating the largest stable time step size of that time integrator. The error and
the acoustic CFL are defined as
ǫ = Q (x, t) − Qexact (x, t) ,
σa = a∞
∆t
,
∆x
(7.2)
where Qexact is given by (7.1). The ARK methods converge at their theoretical orders
for all the cases. The case with M∞ = 0.1 is shown in Figure 7.2(a), and the largest
stable time steps for the ARK methods are observed to be larger than those of the
−1
explicit RK methods by a factor of approximately M∞
= 10. The explicit RK
methods are restricted in their time step size by the acoustic mode, while the implicit
treatment of the acoustic modes in the ARK method allows time step sizes restricted
by the advective mode. Figure 7.2(b) shows the case with M∞ = 0.01. The advective
eigenvalues are smaller in magnitude for this lower Mach number, and therefore larger
time step sizes are possible. The largest time step sizes for the ARK methods are again
−1
approximately M∞
= 100 times larger than those of the explicit RK methods. This
demonstrates that the stability limits for the ARK methods are determined by the
advective time scale because of the characteristic-based flux partitioning.
7.2. Isentropic Vortex Convection. The convection of an isentropic vortex [56] is used to test the flux partitioning in two dimensions. The flow involves
the inviscid convection of a vortex over a periodic domain and tests the ability of
17
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
10
1
FF + HF
FS + HS
8
FF + HF
FS + HS
6
0.5
4
Imaginary
Imaginary
2
0
-2
0
-4
-0.5
-6
-8
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
-1
-0.6
1
-0.5
-0.4
-0.3
Real
-0.2
-0.1
0
Real
(a) Eigenvalues
(b) Magnified plot of the advective eigenvalues
Fig. 7.3. Eigenvalues of Jacobians of the partitioned flux terms
F̂F + ĤF and F̂S + ĤS
in (6.4) for the isentropic vortex convection case. The WENO5 scheme is used, and the problem is
discretized on a grid with 322 points.
1
0.999
7
6.5
0.999
0.998
0.998
0.997
y
5.5
0.997
5
4.5
0.996
4
0.995
3.5
ρ|y=5
6
0.996
0.995
0.994
RK4 αa ≈ 0.8
ARK4 αa ≈ 7.6
0.993
3
3
4
5
6
7
0.994
0
2
(a) ARK 4, σa ≈ 7.6
4
6
8
10
x
x
(b) Cross-sectional density (y = 5)
Fig. 7.4. Density contours of the isentropic vortex convection case after 2 cycles over the
periodic domain obtained with the WENO5 scheme on a grid with 642 points, and the cross-sectional
density profile at y = 5; σa is the acoustic CFL number.
the numerical method to preserve the shape and strength of the vortex. We modify the original test case by reducing the Mach number. The domain is specified as
2
(x, y) ∈ [0, 10] , and the mean (freestream) flow is ρ∞ = 1, u∞ = 0.1, v∞ = 0, and
p∞ = 1. A vortex is introduced in the flow, whose density and pressure are specified
as
(γ − 1) b2 1−r2
e
ρ= 1−
8γπ 2
1
γ−1
, p = ργ ,
(7.3)
18
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
1e-02
1e-06
1e-03
1e-04
1e-08
1e-05
1e-10
εc
||ε||2
1e-06
ARK 2c (mass)
ARK 2c (momentum)
ARK 2c (energy)
ARK 3 (mass)
ARK 3 (momentum)
ARK 3 (energy)
1e-07
1e-08
1e-12
1e-09
ARK 2c
ARK 3
ARK 4
RK 2a
RK 3
RK 4
1e-10
1e-11
1e-12
0.1
1
σa
1e-14
1e-16
10
(a) Error vs. acoustic CFL
0.1
1
σa
10
(b) Conservation error vs. acoustic CFL
Fig. 7.5. Solution error (ǫ) and conservation error (ǫc ) as a function of the acoustic CFL σa
for the isentropic vortex convection. The solutions are obtained on a 322 grid with the WENO5
scheme at a final time of 100 (one cycle over the domain).
and thus ρ, p → ρ∞ , p∞ as r → ∞. The velocity field is
b 21 (1−r2 )
b 21 (1−r2 )
e
e
(y − yc ) , v = v∞ +
(x − xc ) ,
(7.4)
2π
2π
1/2
where b = 0.5 is the vortex strength and r = (x − xc )2 + (y − yc )2
is the distance
from the vortex center (xc , yc ) = (5, 5). Periodic boundary conditions are applied
at all boundaries. As the solution is evolved in time, the vortex convects over the
periodic domain with a time period of Tp = 100.
Figure 7.3 shows the eigenvalues of the partitioned Jacobians for the WENO5
scheme on a grid with 322 points. The freestream Mach number for this example is
M∞ ≈ 0.08, and thus we see a significant separation in the magnitudes of the advective
and acoustic eigenvalues. Figure 7.3(b) is a magnified plot of the advective eigenvalues. This demonstrates that the extension of the characteristic-based partitioning to
two dimensions, as described in Section 6, works as expected. Figure 7.4(a) shows
the density contours of the flow for the solution obtained with the ARK 4 method at
an acoustic CFL number of ∼ 7.6 on a grid with 642 points and the WENO5 scheme.
The final time is 200, corresponding to 2 cycles over the periodic domain. The horizontal cross-sectional density profile through y = 5 for these solutions is shown in
Figure 7.4(b). The solution obtained with ARK 4 agrees well with that obtained with
the explicit RK 4 scheme at an acoustic CFL number of 0.8.
The error as a function of the acoustic CFL is shown in Figure 7.5(a). The
solutions are obtained on a grid with 322 points with the WENO5 scheme after one
cycle over the periodic domain. The tolerances for the GMRES solver are specified
as τa = τr = 10−10 . We start the tests with an initially small time step and increase
it until it reaches the stability limit of the time integrator being used. The error and
the acoustic CFL are defined as
u = u∞ −
ǫ = Q (x, y, t) − Qref (x, y, t) ,
σa = a∞
∆t
,
min (∆x, ∆y)
(7.5)
19
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
2.5
2
F +H
F +H
S
S
3
S
S
ARK 3 (IMEX)
ARK 3 (Expl)
ARK 2c (IMEX)
ARK 2c (Expl)
2
1.5
1
Im(λ) ∆ t
Im(λ) ∆ t
1
0.5
0
−0.5
0
−1
−1
−1.5
−2
−2
−3
−2.5
−5
−4.5
−4
−3.5
−3
−2.5
−2
Re(λ) ∆ t
−1.5
(a) ARK 2c, σa ≈ 7.6
−1
−0.5
0
−4.5
−4
−3.5
−3
−2.5
−2
Re(λ) ∆ t
−1.5
−1
−0.5
0
(b) ARK 3, σa ≈ 11.3
Fig. 7.6. Eigenvalues of the slow partitioned term F̂S + ĤS multiplied by the time step ∆t,
and the stability regions of the explicit components of the ARK time integration methods. “IMEX”
denotes
the stability
region of the explicit method when the implicit method handles the eigenvalues
of F̂F + ĤF , and “Expl” denotes its stability region when it is used by itself as an explicit time
integrator.
where Qref (x, y, t) is the reference solution obtained with the explicit RK 4 time
integration method with a very small time step of 0.0005. The ARK methods converge
at their theoretical orders for acoustic CFL numbers less than 1; at higher CFL
numbers, the acoustic mode is not resolved, and thus convergence is only second order.
However, the absolute errors for a higher-order ARK method (say, ARK 4) are smaller
than those for a lower-order ARK method (say, ARK 2c). The largest stable time step
for the ARK methods are larger than those of the explicit RK methods by a factor of
−1
approximately M∞
, thus demonstrating that the time step size is determined
by the
advective scale. Figure 7.6 shows the eigenvalues of the slow operator F̂S + ĤS
scaled by the time step ∆t and the stability regions of the explicit components of
the ARK 2c and ARK 3 methods. The time step ∆t corresponds to acoustic CFL
numbers of ∼ 7.6 for ARK 2c and ∼ 11.3 for ARK 3. These are close to the observed
largest stable CFL numbers for these methods in Figure 7.5(a). At these time step
sizes, the advective eigenvalues have started spilling out of the respective stability
regions. Comparison of the stability regions of the explicit method by itself (denoted
by “Expl”) and when
of an ARK method with the implicit part handling
it is a part
the eigenvalues of F̂F + ĤF (denoted by “IMEX”) shows significant reduction in
the imaginary stability [14].
Figure 7.5(b) shows the conservation errors ǫc for mass (ρ), momentum (ρu), and
energy (e) as a function of the acoustic CFL, for the ARK 2c and ARK 3 methods.
The conservation error is defined as
Z
1 k
Q̄ (t) − Q̄k (0) , Q̄k (t) =
kQk (x, y, t) k2 dV,
(7.6)
ǫc = k
Q̄ (0)
V
where Q̄ is the volume integral over the domain, V denotes the two-dimensional
domain, and the superscript k denotes the component (k = 1 for mass, k = 2, 3 for
momentum, and k = 4 for energy). The conservation errors are on the order of roundoff errors with the specified GMRES tolerances, for both the methods and at all the
20
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
10000
3.0e−03
8000
1.5e−03
y
6000
4000
0.0e+00
2000
0
0
0.5
1
1.5
x
2
2.5
3
−1.5e−03
5
x 10
Fig. 8.1. Inertia-gravity waves: Potential temperature perturbation ∆θ at t = 3000 s, obtained
with the CRWENO5 scheme and the ARK 4 time integrator on a grid with 1200 × 50 points. The
time step is ∆t = 12 s, corresponding to an acoustic CFL number of σa ≈ 20.8.
0.003
0.002
∆θ
0.001
0.000
-0.001
NUMA
σa ≈ 20.8
-0.002
0.0e+00
5.0e+04
1.0e+05
1.5e+05
2.0e+05
2.5e+05
3.0e+05
x
Fig. 8.2. Inertia-gravity waves: Cross-sectional potential temperature perturbation ∆θ at y =
5000 m and t = 3000 s, obtained with the CRWENO5 scheme and the ARK 4 time integrator on a
grid with 1200 × 50 points. “NUMA” refers to the reference solution obtained with a spectral element
solver [28].
CFL numbers considered. In addition, they do not show any trends with respect to
the CFL number. This demonstrates that the partitioned semi-implicit algorithm is
conservative.
8. Application to Atmospheric Flows. In this section, the algorithm is applied to atmospheric flows, which are governed by the two-dimensional Euler equations with a gravitational source term. Two benchmark flow problems are solved—
the inertia-gravity wave and the rising thermal bubble. The flow solver used in this
study has been previously verified for atmospheric flows with explicit Runge-Kutta
schemes [24]; therefore, the focus of this section is to demonstrate the accuracy, stability, and numerical cost of the ARK methods. We note that the problems solved in
this section are in terms of dimensional quantities, unlike the previous section where
all quantities were nondimensional.
8.1. Inertia-Gravity Waves. The inertia-gravity wave [57, 28] involves the
evolution of a potential temperature perturbation. The domain is a channel with
dimensions 300, 000 m × 10, 000 m. The initial flow consists of a perturbation introduced into a hydrostatically balanced (stratified) atmosphere. The mean flow is the
stratified atmosphere with a specified Brunt-Väisälä frequency (N ). The potential
21
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
1
1e-04
FF + HF + S
FS + HS
1e-06
1e-08
||ε||2
Imaginary
0.5
0
1e-10
-0.5
ARK 2c
ARK 3
ARK 4
RK 2a
RK 3
RK 4
1e-12
-1
-1.6 -1.4 -1.2
1e-14
-1
-0.8 -0.6 -0.4 -0.2
Real
(a) Eigenvalues
0
0.1
1
10
σa
(b) Solution error as a function of acoustic CFL
Fig. 8.3.
Inertia-gravity
waves: Eigenvalues of Jacobians of the partitioned terms
F̂F + ĤF + Ŝ and F̂S + ĤS in (6.4), and the solution error (ǫ) as a function of the acoustic
CFL σa .
temperature and Exner pressure are given by
2
N2
(γ − 1)g 2
N
exp −
y ,π = 1 +
y −1 ,
θ = T0 exp
g
γRT0 N 2
g
(8.1)
and the density and pressure are
γ/(γ−1)
(γ − 1)g 2
N2
p = p0 1 +
exp −
y −1
,
γRT0 N 2
g
1/(γ−1)
N2
(γ − 1)g 2
N2
exp
−
y 1+
y
−
1
.
ρ = ρ0 exp −
g
γRT0 N 2
g
(8.2)
(8.3)
The initial velocity components are u = 20 m/s and v = 0 m/s. Periodic boundary
conditions are applied on the left (x = 0 m) and right (x = 300, 000 m) boundaries,
while inviscid wall boundary conditions are applied on the bottom (y = 0 m) and
top (y = 10, 000 m) boundaries. The Brunt-Väisälä frequency is N = 0.01 /s, and
the gravitational force per unit mass is 9.8 m/s2 along the y-direction. The reference
pressure (p0 ) and temperature (T0 ) at y = 0 m are 105 N/m2 and 300 K, respectively,
and the reference density is computed from the equation of state p0 = ρ0 RT0 . The
universal gas constant R is 287.058 J/kg K. The perturbation is added to the potential
temperature, specified as
"
2 #−1
x − xc
πc y
1+
,
(8.4)
∆θ (x, y, t = 0) = θc sin
hc
ac
where θc = 0.01 K is the perturbation strength, hc = 10, 000 m is the height of the
domain, ac = 5, 000 m is the perturbation half-width, xc = 100, 000 m is the horizontal
location of the perturbation, and πc ≈ 3.141592654 is the Archimedes (trigonometric)
constant. The evolution of the perturbation is simulated until a final time of 3000 s.
22
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
√
The reference speed of sound is a0 = γRT0 = 347.22 m/s, and the reference Mach
number for this flow is approximately 0.06. Figure 8.3(a) shows the eigenvalues of the
slow and the fast operators for the problem discretized on a 300 × 10-point grid with
the CRWENO5 scheme; the fast operator includes the gravitational source term.
Figure 8.1 shows the potential temperature perturbation ∆θ = (θ − θ0 ) contours
for the solution obtained with the CRWENO5 scheme on a grid with 1200 × 50 points.
The ARK 4 method is used for time integration with a time step of ∆t = 12 s, corresponding to an acoustic CFL number of σa ≈ 20.8. The tolerances for the GMRES
solver are specified as τa = τr = 10−6 . A good agreement is observed with results in
the literature [3, 57, 4, 68]. Figure 8.2 shows the cross-sectional potential temperature
perturbation at an altitude of y = 5, 000 m for this solution. The reference solution
“NUMA” refers to the solution obtained with a spectral-element solver [28], with
10th-order polynomials, 3rd-order explicit RK time integration, and 250 m effective
grid resolution. The solution obtained with the partitioned semi-implicit approach
agree well with the reference solution.
Figure 8.3(b) shows the L2 norm of the solution error as a function of the acoustic
CFL for solutions obtained with the CRWENO5 scheme on a 600 × 20 grid. The error
and the acoustic CFL are as defined in (7.5). The reference speed of sound a0 is used
to compute the acoustic CFL, and the reference solution is obtained with the explicit
RK 4 time integrator and a very small time step of 0.005. The tolerances for the
GMRES solver are specified as τa = τr = 10−10 . The errors for the partitioned ARK
methods are shown, as well as the explicit RK 2a, RK 3, and RK 4 methods. The
ARK methods converge at their theoretical orders of convergence, and the largest
−1
stable time steps are observed to be approximately M∞
≈ 15 times larger than those
of the explicit RK methods. The mass conservation errors are on the order of round-off
errors for all the solutions and at all CFL numbers considered.
8.2. Rising Thermal Bubble. The two-dimensional rising thermal bubble [28]
simulates the dynamics of a warm bubble. A square domain of size 1000 m × 1000 m
is specified with inviscid wall boundary conditions on all sides. The initial solution is
a warm bubble introduced in a hydrostatically balanced atmosphere. The mean flow
is the stratified atmosphere with a constant potential temperature θ = T0 = 300 K;
and the density, pressure, and velocity are respectively
ρ = ρ0
(γ − 1)gy
1−
γRθ
1/(γ−1)
γ/(γ−1)
(γ − 1)gy
, p = p0 1 −
, u = v = 0.
γRθ
(8.5)
The reference pressure is 105 N/m2 , and the reference density is computed from the
equation of state p0 = ρ0 RT0 . The universal gas constant R is 287.058 J/kg K. A
constant gravitation force per unit mass of 9.8 m/s2 is specified along the y-direction.
The warm bubble is added as a potential temperature perturbation,
(
0 i r > rc
p
h
∆θ (x, y, t = 0) =
, r = (x − xc )2 + (y − yc )2 ,
θc
πc r
r
≤
r
1
+
cos
c
2
rc
(8.6)
where θc = 0.5 K is the perturbation strength, (xc , yc ) = (500, 350) m is the initial
location at which the bubble is centered, rc = 250 m is the radius of the bubble, and
πc is the trigonometric constant. The flow is simulated to a final time of 400 s.
Figure 8.4 shows the initial solution at t = 0 s and the solution at t = 400 s. The
warm bubble rises as a result of buoyancy and deforms as a result of the temperature
23
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
1000
1000
0.500
0.500
800
0.400
600
0.300
600
0.300
y
0.400
y
800
0.200
400
0.200
400
0.100
0.100
200
200
0.000
0
0
200
400
600
800
0.000
0
0
1000
200
400
x
600
800
1000
x
(a) t = 0 s
(b) t = 400 s
Fig. 8.4. Rising thermal bubble: Potential temperature perturbation ∆θ for the solution obtained
with the WENO5 scheme and the ARK 4 time integrator on a grid with 2012 points. The time step
is ∆t = 2 s, corresponding to an acoustic CFL number of σa ≈ 139.
0.35
1e-04
0.3
1e-06
0.25
1e-08
||ε||2
∆θ
0.2
0.15
1e-10
0.1
0.05
RK 4 σa ≈ 0.7
ARK 4 σa ≈ 139
0
200
ARK 2c
ARK 3
ARK 4
RK 2a
RK 3
RK 4
1e-12
300
400
500
x
600
700
800
(a) Cross-sectional potential temperature perturbation at y = 550 m
1e-14
0.01
0.1
1
10
100
1000
σa
(b) Error vs. acoustic CFL
Fig. 8.5. Rising thermal bubble: Comparison of cross-sectional solution profile (2012 -points
grid), and solution error (ǫ) as a function of the acoustic CFL σa (512 -points grid) at a final time
of 400 s.
and velocity gradients. The potential temperature perturbation ∆θ = θ − θ0 is shown.
The solution is obtained with the WENO5 scheme and the ARK 4 time integrator
on a grid with 2012 points. The GMRES solver tolerances are τa = τr = 10−6 . The
time step size is 2 s, which results in an acoustic CFL number of approximately 139.
The acoustic CFL is given by (7.5), and the reference speed of sound a0 is used. The
flow is initially at rest, and thus the advective eigenvalues are all zero. As the bubble
rises, it induces a velocity field; at t = 400 s, the maximum velocity magnitude in the
domain is approximately 2.1 m/s, corresponding to a maximum local Mach number of
approximately 0.006. Thus, the disparity between the advective and acoustic scales
is very large, and the semi-implicit approach allows time steps that are much larger
24
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
than those allowed by an explicit time integrator. Figure 8.5(a) compares the crosssectional profiles of ∆θ along x at y = 550 m for this solution and that obtained with
the explicit RK 4 method at an acoustic CFL number of ∼ 0.7, and an excellent
agreement is observed.
The L2 norm of the solution error is shown in Figure 8.5(b) as a function of the
acoustic CFL. The error, defined in (7.5), is computed with respect to a reference
solution that is obtained on the same grid with the same spatial discretization and
with the explicit RK 4 time integrator with a very small time step size of 10−4 . The
figure shows the errors for the ARK 2c, ARK 3, and ARK 4 methods, as well as the
explicit RK 2a, RK 3, and RK 4 methods. The tolerances specified for the GMRES
solver are τa = τr = 10−10 . In the region where the acoustic waves are resolved
and the explicit methods are stable, all the methods converge at their theoretical
orders of accuracy. At higher CFL numbers, the acoustic mode is not resolved, and
thus the errors for the ARK methods (relative to the reference solution, in which
the acoustic mode is resolved) converge toward a similar value. This behavior has
been previously analyzed and discussed for the semi-implicit time integration of the
perturbation form of the governing equations [27]. The mass conservation error are
zero to machine precision for all the solutions at all the CFL numbers considered.
8.3. Numerical Cost. The main objective of using semi-implicit time integration is to obtain well-resolved solutions at a lower computational cost than with
explicit time integrators. These methods allow time step sizes that step over the fast
acoustic scales; however, they require the solution of a system of equations. Thus,
their performance depends on the cost and accuracy of the linear solver. In this section, we compare the computational cost of the ARK methods with the explicit RK
methods in terms of the minimum wall time and the number of function calls required
to obtain a stable and resolved solution. In the following discussion, the number of
function calls (nFC ) refers to the total number of calls to the functions that compute
the partitioned flux components F̂F or F̂S . Since a matrix-free implementation of the
Jacobian is used, nFC is the sum of the total number of time iterations (nT ) times
the number of stages s (of the time integration method), and the total number of
GMRES iterations. It is thus an estimate of the total computational cost; however,
it does not include the cost of assembling and inverting the preconditioning matrix.
The algorithm is implemented in serial; its performance and scalability on parallel
platforms are being investigated. The reported simulations are run on a 2200 MHz
AMD Opteron processor.
Table 8.1 shows the wall times (in seconds), the number of function calls, and
the L2 norm of the error ǫ for the inertia-gravity wave problem, solved on a grid with
1200 × 50 points with the CRWENO5 scheme. The tolerances for the GMRES solver
are τa = τr = 10−6 . The ARK 2c and ARK 4 methods are compared with the explicit
RK 2a and RK 4. The time steps for the explicit RK methods are chosen close to
their stability limits; thus, the reported wall times are the fastest time to solution
for the explicit methods. The final row for each ARK method reports the cost with
the largest stable time step and thus represents their fastest time to solution. The
cost of the ARK methods decreases as the time step size increases (both the number
of function calls and the wall times). ARK 2c is the fastest method among those
considered. The acoustic scale is approximately 17 times faster than the advective
scale for this problem. While the ARK 2c is faster than the explicit methods by 25%,
the ARK 4 is generally slower at all the CFL numbers except at the largest CFL,
where its cost is comparable. The solution errors are consistent with those reported
SEMI-IMPLICIT TIME INTEGRATION OF ATMOSPHERIC FLOWS
25
Table 8.1
Inertia-gravity waves: L2 norm of the error and computational cost as a function of time step
size and acoustic CFL number of the ARK and RK methods for solutions on a grid with 1200 × 50
points discretized in space with the CRWENO5 scheme. The final time is 3000 s. Boldfaced rows
indicate the performance at the largest stable time step for the ARK methods.
Method
RK 2a
RK 4
ARK 2c
ARK 4
∆t
0.15
0.30
2.0
4.0
8.0
4.0
8.0
12.0
15.0
kǫk2
1.4 × 10−8
1.7 × 10−9
4.6 × 10−7
1.3 × 10−6
9.1 × 10−7
1.9 × 10−8
2.0 × 10−7
5.1 × 10−7
9.2 × 10−7
nT
20, 000
10, 000
1, 500
750
375
750
375
250
200
σa
0.26
0.45
3.47
6.94
13.89
6.94
13.89
20.83
26.04
nFC
40, 000
40, 000
57, 121
34, 530
21, 164
80, 479
47, 258
34, 900
29, 556
Wall time (s)
12, 353
12, 072
23, 180
14, 086
8, 797
33, 296
19, 875
14, 398
12, 608
Table 8.2
Rising thermal bubble: L2 norm of the error and computational cost as a function of time step
size and acoustic CFL number of the fourth-order ARK and RK methods for solutions on a grid
with 2012 points discretized in space with the WENO5 scheme. The final time is 400 s. Boldfaced
rows indicate the performance at the largest stable time step for the ARK method.
Method
RK 4
ARK 4
∆t
0.01
0.10
0.50
2.00
kǫk2
7.5 × 10−8
1.5 × 10−7
1.6 × 10−6
1.9 × 10−6
nT
40, 000
4, 000
800
200
σa
0.69
6.94
34.72
138.89
nFC
160, 000
360, 016
111, 824
45, 969
Wall time (s)
30, 154
73, 111
22, 104
8, 569
in Figure 8.3(b), and thus the larger tolerances for the GMRES solver used for the
performance tests (τa,r ) do not degrade the accuracy of the overall solution. Since we
are considering large time step sizes, a more relaxed tolerance suffices to ensure that
the error in solving the implicit stages remains small with respect to the truncation
error of the time integration scheme.
Table 8.2 shows the cost of the ARK 4 method and the L2 norm of the numerical
error for the rising thermal bubble, solved on a grid with 2012 points with the WENO5
scheme. The tolerances for the GMRES solver are τa = τr = 10−6 . The cost of the
explicit RK 4 method is used as a reference. The separation between the acoustic and
advective scales is very large; the flow is initially at rest, with the Mach number at the
final time being ∼ 0.006. The semi-implicit method is thus able to take much larger
time steps. The cost of the ARK method decreases as the time step size increases;
and for CFL numbers greater than ∼ 30, the ARK 4 is faster than the RK 4. At
the largest stable time step, the ARK 4 is faster than the RK 4 method by a factor
of approximately 3.5. The numerical errors are consistent with those reported in
Figure 8.5(b) thus ensuring that the relaxed tolerance for the GMRES solver does not
compromise the accuracy of the time integration.
The results reported here are obtained with basic preconditioning of the linear
system, as described in Section 5.4. The primary focus of this paper is to introduce a
flux partitioning for semi-implicit time integration based on the governing equations
expressed as (2.1)–(2.3). Improving the efficiency of the time integrator by developing suitable preconditioning techniques for the GMRES solver is currently being
26
DEBOJYOTI GHOSH AND EMIL M. CONSTANTINESCU
investigated.
9. Conclusion. This paper presents a characteristic-based partitioning of the
hyperbolic flux in the compressible Euler equations for semi-implicit time integration.
The acoustic and the advective modes are separated; the former is integrated in time
implicitly because of its stiffness, while the latter is integrated explicitly. The stiff
term is linearized, and thus the semi-implicit algorithm requires only the solution to a
linear system. The nonstiff term, defined as the total nonlinear flux with the linearized
stiff term subtracted from it, is treated explicitly. High-order additive Runge-Kutta
methods are applied to the partitioned equations, and the WENO and CRWENO
schemes are used for the spatial discretization. We note that Rosenbrock schemes are
also viable time stepping alternatives.
We test this approach on simple inviscid flow problems at low Mach numbers. The
results show that the largest stable time step is determined by the advective scale.
The algorithm is then applied to atmospheric flows where the acoustic modes are
much faster than the advective mode but are not physically relevant. The accuracy
and convergence of the algorithm are demonstrated for benchmark problems, and the
results show that the partitioned semi-implicit approach is conservative. Moreover,
the computational cost is assessed and compared with that of explicit time integrators.
The extension of this algorithm to parallel platforms and the development of more
effective preconditioning techniques are areas of current research.
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| 5 |
1
Continuous-Domain Solutions of Linear Inverse
Problems with Tikhonov vs. Generalized TV
Regularization
arXiv:1802.01344v1 [cs.IT] 5 Feb 2018
Harshit Gupta, Julien Fageot, and Michael Unser
Abstract—We consider linear inverse problems that are formulated in the continuous domain. The object of recovery is a
function that is assumed to minimize a convex objective functional. The solutions are constrained by imposing a continuousdomain regularization. We derive the parametric form of the
solution (representer theorems) for Tikhonov (quadratic) and
generalized total-variation (gTV) regularizations. We show that,
in both cases, the solutions are splines that are intimately related
to the regularization operator. In the Tikhonov case, the solution
is smooth and constrained to live in a fixed subspace that
depends on the measurement operator. By contrast, the gTV
regularization results in a sparse solution composed of only a
few dictionary elements that are upper-bounded by the number
of measurements and independent of the measurement operator.
Our findings for the gTV regularization resonates with the
minimization of the `1 norm, which is its discrete counterpart and
also produces sparse solutions. Finally, we find the experimental
solutions for some measurement models in one dimension. We
discuss the special case when the gTV regularization results in
multiple solutions and devise an algorithm to find an extreme
point of the solution set which is guaranteed to be sparse.
Index Terms—Linear inverse problem, representer theorem,
regularization, spline, total variation, L2 , quadratic regularization.
I. I NTRODUCTION
In a linear inverse problem, the task is to recover an
unknown signal from a finite set of noisy linear measurements.
To solve it, one needs a forward model that describes how
these measurements are acquired. Generally, this model is
stated as the continuous-domain transform of a continuousdomain signal. For example, MRI data is modeled as the
samples of the Fourier transform of a continuous-domain
signal. The traditional approach to state this inverse problem
is to choose an arbitrary but suitable basis {ϕn } and to write
that the reconstructed signal is
f (x) =
N
X
fn ϕn (x),
(1)
n=1
where H : RM × RN has elements [H]m,n = hhm , ϕn i. The
analysis functions {hm }M
m=1 specify the forward model which
encodes the physics of the measurement process. Term I in
(2) is the data fidelity. It ensures that the recovered signal
is close to the measurements. Term II is the regularization,
which encodes the prior knowledge about the signal. The
regularization is imposed on some transformed version of the
signal coefficients using the matrix L. Various linear [1], [2]
and iterative algorithms [3], [4], [5] have been developed to
solve Problem (2). In recent years, the notion that the realworld signals are sparse in some basis (e.g., wavelets) has
become popular. This prior is imposed by using the sparsitypromoting `1 -regularization norm [6], [7] and results in the
minimization problem
f ∗ = arg min kz − Hf k22 + λkLf k1 .
(3)
f ∈RN
The solutions to (2), (3), and their variants with generalized
data-fidelity terms are well known [8], [9], [10], [11].
While those discretization paradigms are well studied and used
successfully in practice, it remains that the use of a prescribed
basis {ϕn }, as in (1), is somewhat arbitrary.
In this paper, we propose to bypass this limitation by reformulating and solving the linear inverse problem directly in the
continuous domain. To that end, we impose the regularization
in the continuous domain, too, and restate the reconstruction
task as a functional minimization. We show that this new
formulation leads to the identification of a natural basis for the
solution; this results in an exact discretization of the problem.
Our contributions are summarized as follows:
M
• Given z ∈ R , we formalize the inverse problem in the
continuous domain as
fR = arg min kz − H{f }k22 + λR(f ) ,
(4)
f ∈X |
{z
}
JR (z|f )
N
where f = (f1 , . . . , fN ) ∈ R . Given the measurements z ∈
RM , the task then is to find the expansion coefficients f by
minimizing
f ∗ = arg min kz − Hf k22 +λ kLf k22 ,
| {z }
| {z }
f ∈RN
I
(2)
II
The authors are with the Biomedical Imaging Group, École polytechnique
fédérale de Lausanne, Lausanne 1015, Switzerland. This project has been
funded by H2020-ERC, Grant agreement No. 692726-GlobalBioIm.
•
where f is a function that belongs to a suitable function
space X . Similarly to the discrete regularization terms
kLf k2`2 and kLf k`1 in (2) and (3), we focus on their
continuous-domain counterparts R(f ) = kLf k2L2 and
R(f ) = kLf kM , respectively. There, L and H are the
continuous-domain versions of L and H, while kLf kM is
the proper continuous-domain counterpart of the discrete
`1 norm. We show that the effect of these regularizations
is similar to the effect of their discrete counterparts.
We provide the parametric form of the solution (representer theorem) that minimizes JR (z|f ) in (4) for
2
the Tikhonov regularization R(f ) = kLf k2L2 and the
generalized total-variation (gTV) regularization R(f ) =
kLf kM . Our results underline how the discrete regularization resonates with the continuous-domain one. The
optimal solution for the Tikhonov case is smooth, while
it is sparse for the gTV case. The optimal bases in the
two cases are intimately connected to the operators L and
H.
• We present theoretical results that are valid for any
convex and lower-semicontinuous data-fidelity term. This
includes the case when the data-fidelity term is kz −
H{f }k22 .
• We propose an exact discretization scheme to minimize
JR (z|f ) in the continuous domain. Even though the
minimization of JR (z|f ) is an infinite-dimensional problem, the knowledge of the optimal basis of the solution
makes the problem finite-dimensional: it boils down to
the search for a set of optimal expansion coefficients.
• We devise an algorithm to find a sparse solution when the
gTV solution is non-unique. For this case, the optimization problem turns out to be a LASSO [9] minimization
with non-unique solution. We introduce a combination
of FISTA [12] and the simplex algorithm to find a sparse
solution which we prove to be an extreme point of the
solution set.
The paper is organized as follows: In Sections 2 and 3,
we present the formulation and the theoretical results of the
inverse problem for the two regularization cases. In Section 4,
we compare the solutions of the two cases. We present our
numerical algorithm in Section 5 and illustrate its behavior
with various examples in Section 6. The mathematical proofs
of the main theorems are given in the appendices and the
supplementary material.
A. Related Work
The use of R(f ) = kLf k2L2 goes back to Tikhonov’s theory
of regularization [1] and to kernel methods in machine learning
[13]. In the learning community, representer theorems (RT)
as in [14], [15] use the theory of reproducing-kernel Hilbert
spaces (RKHS) to state the solution of the problem for the
restricted case where the measurements are samples of the
function. For the generalized-measurement case, there are also
tight connections between these techniques and variational
splines and radial-basis functions [16], [17], [18]. These
representer theorems, however, either have restrictions on the
empirical risk functional or on the class of measurement
operators.
Specific spline-based methods with quadratic regularization
have been developed for inverse problems. In particular, [19],
[20] used variational calculus. Here, we strengthen these
results by proving the unicity and existence of the solution
of (4) for R(f ) = kLf k2L2 . We revisit the derivation of the
result using the theory of RKHS.
Among more recent non-quadratic techniques, the most
popular ones rely on (TV) regularization which was introduced
as a noise-removal technique in [21] and is widely used in
computational imaging and compressed sensing, although always in discrete settings. Splines as solutions of TV problems
for restricted scenarios have been discussed in [22]. More
recently, a RT for the continuous-domain R(f ) = kLf kM
in a general setting has been established in [23], extending
the seminal work of Fisher and Jerome [24]. The solution has
been shown to be composed of splines that are directly linked
to the differential operator L. Other recent contributions on
inverse problems in the space of measures include [25]–[29].
In particular, in this paper, we extend the result of [23] to an
unconstrained version of the problem.
II. F ORMULATION
In our formulation of a linear inverse problem, the signal f
is a function of the continuous-domain variable x ∈ R. The
task is then to recover f from the vector of measurements
z = H{f } + n ∈ RM , where n is an unknown noise
component that is typically assumed to be i.i.d. Gaussian.
In the customary discrete formulation, the basis of the recovered function is already chosen and, therefore, all that
remains is to recover the expansion coefficients of the signal
representation (1). In this scenario, one often includes matrices
H and L that directly operate on these coefficients. However,
for our continuous-domain formulation, the operations have to
act directly on the function f . For this reason, we also need
the continuous-domain counterparts of the measurement and
regularization operators. The entities that enter our formulation
are described next.
A. Measurement Operator
The system matrix H in (2) and (3) is henceforth replaced
by the operator H : X → RM that maps the continuousdomain functions living in the space X to the linear measurements z ∈ RN . This operator is described as
H{f } = (hh1 , f i, . . . , hhM , f i) = (z1 , . . . , zM ) = z, (5)
R
where hh, gi = R h(x)g(x) dx. For example, the components
of the measurement operator that samples a function at the
locations x1 , . . . , xM are modeled by hm = δ(· − xm ).
Similarly, Fourier measurements at pulsations ω1 , . . . , ωM are
obtained by taking hm = e−jωm (·) .
B. Data-Fidelity Term
As extension of the conventional quadratic data-fidelity term
kz − Hf k22 , we consider the general convex cost functional
E : RM ×RM → R+ ∪{∞} that measures the discrepancy between the measurements z and the values H{f } predicted from
the reconstruction. A relevant example is the Kullback-Leibler
(KL)-divergence, which is often used as the data-fidelity term
when the measurements are corrupted by Poisson noise [30].
Alternatively, when the measurements are noiseless, we use
the indicator function
(
0, z 0 = H{f }
I(z 0 , H{f }) =
(6)
∞, z 0 6= H{f },
which imposes an exact fit. We assume that E is a convex
lower semi-continuous function with respect to its arguments.
This will enable us to state the existence of a solution and use
convex optimization techniques to find the minimum of the
objective functional.
3
C. Regularization Operator
Since the underlying signal is continuously defined, we need
to replace the regularization matrix L in (2) and (3) by a
regularization operator L : X → Y, where X and Y are
appropriate function spaces to be defined in Section II-E. The
typical example that we have in mind is the derivative operator
L = D = ddx . The continuous-domain regularization is then
imposed on Lf . We assume that the operator L is admissible
in the sense of defintion 1.
Definition 1. The operator L : X → Y is called splineadmissible if
• it is linear and shift-invariant;
• its null space NL = {p ∈ X : Lp = 0} is finitedimensional;
• it admits the Green’s function ρL : R → R with the
property that LρL = δ.
b is the frequency response of L, the Green’s
Given that L
function can be calculated
through the inverse Fourier trans1
−1
. For example, if L = D, then
form ρL = F
b
L
1
ρD (x) = 2 sign(x).
D. Regularization Norms
Since the optimization is done in the continuous domain, we
also have to specify the proper counterparts of the `2 and `1
norms, as well as the corresponding vector spaces.
i) Quadratic (or Tikhonov) regularization: RTik (f ) =
kLf k2L2 , where
Z
kwk2L2 :=
|w(x)|2 dx.
(7)
R
ii) Generalized total variation: RgTV (f ) = kLf kM , where
kwkM :=
sup
hw, ϕi.
(8)
ϕ∈C0 (R),kϕk∞ =1
There, C0 (R) is the space of continuous functions that
decay to 0 at infinity. Moreover, M = {w : R →
R | kwkM < ∞}. In particular, when w ∈ L1 ⊂ M,
we have that
Z
kwkM = |w(x)| dx = kwkL1 .
(9)
R
Yet, we note that M is slightly larger than L1 since it
also includes the Dirac distribution δ with kδkM = 1.
The popular TV norm is recovered by taking kf kTV =
kDf kM [23].
E. Search Space
The Euclidean search space RN is replaced by spaces of
functions, namely,
X2 ={f : R → R | kLf kL2 < +∞},
X1 ={f : R → R | kLf kM < +∞}.
(10)
(11)
In other words, our search (or native) space is the largest space
over which the regularization is well defined. It turns out that
X2 and X1 are Hilbert and Banach spaces, respectively. This
means that there exists a well defined inner product h·, ·iX2 on
X2 and a norm k · kX1 on X1 . The structure of these spaces
has been studied in [23] and is recalled in the supplementary
material.
As we shall in Section III, the solution of (4) will be composed
of splines; therefore, we also review the definition of the
splines.
Definition 2 (Nonuniform L-spline). A function f : R → R
is called a nonuniform L-spline with spline knots (x1 , . . . , xK )
and weights (a1 , . . . , aK ) if
Lf =
K
X
k=1
ak δ(· − xk ).
(12)
By solving the differential equation in (12), we find that the
generic form of the nonuniform spline f is
f = p0 +
K
X
k=1
ak ρL (· − xk ),
(13)
where p0 ∈ NL . Note that ρL (· − xk ) = L−1 {δ(· − xk )},
where L−1 : f 7→ L−1 f = ρL ∗ f , is the shift-invariant inverse
of L.
III. T HEORETICAL R ESULTS
To state our theorems, we need some technical assumptions.
Assumption 1.
i) The bounded vector-valued functional
H : X → RM gives the linear measurements f 7→
H{f } = (hh1 , f i, . . . , hhM , f i).
ii) The functional E : (RM × RM ) → R+ ∪ {∞} is convex
and lower semi-continuous.
iii) The regularization operator L : X → Y is splineadmissible. Its finite-dimensional null space NL has the
basis p = (p1 , . . . , pN0 ).
iv) The inverse problem is well posed over the null space.
This means that, for any pair p1 , p2 ∈ NL , we have that
H{p1 } = H{p2 } ⇔ p1 = p2 .
(14)
In other words, different null-space functions result in
different measurements.
In particular Condition iv) implies that NL ∩ NH = {0},
where NH is the null space of the vector-valued measurement
functional. This property is essential to make the optimization
problem (4) well posed. This kind of requirement is common
to every regularization scheme.
We now state our two main results. Their proofs are given
in Appendix C and Appendix D.
A. Inverse Problem with Tikhonov/L2 Regularization
Theorem 3. Let Assumption 1 hold for the search space X =
X2 and regularization space Y = L2 . Then, the minimizer
f2 = arg min E(z, H(f )) + λkLf k2L2
(15)
f ∈X2
is unique and admits a parametric solution of the form
f2 (x) =
M
X
m=1
am ϕm (x) +
N0
X
n=1
bn pn (x),
(16)
4
n
o
bm
where ϕm = F −1 |hL|
= (L∗ L)−1 hm , a = (a1 , . . . , aM ),
b2
and b = (b1 , . . . , bN0 ) are expansion coefficients such that
M
X
m=1
am hhm , pn i = 0
(17)
where b1 ∈ ND is a constant.
We contrast (21) with the gTV version
M
X
f1 = arg min
|zm − f (xm )|2 + λ kDf kM . (23)
f ∈X1
{z
}
|
m=1
kf kTV
for all n ∈ {1, . . . , N0 }.
B. Inverse Problem with gTV Regularization
Theorem 4. Let Assumption 1 hold for the search space X =
X1 and regularization space Y = M. Moreover, assume that
H is weak*-continuous (see Supplementary Material). Then,
the set
V = arg min (E(z, Hf ) + λkLf kM )
(18)
f ∈X1
of minimizer is nonempty, convex, weak*-compact, and its
extreme points are nonuniform L-splines of the form
In this scenario, the term kDf kM is the total variation of the
function f . It penalizes solutions that vary too much from one
point to the next.
One readily checks that ρD = 1+ is a Green’s function of
D since it satisfies D{1+ } = δ. Based on Theorem 4, any
extreme point of (23) is of the form
f1 (x) = b1 +
K
X
k=1
a0k 1+ (x − τk ),
(24)
Theorem 5. Let Assumption 1 hold where X is the search
space and Y is the regularization space. Then, every member
of the solution set
V = arg min (E(z, H{f }) + λR(f )) ,
(20)
which is a piecewise constant function composed of a constant
term b1 and K ≤ (M − 1) unit steps (Heaviside functions)
located at {τk }K
k=1 . These knots are not fixed a priori and
usually differ from the measurement points {xm }M
m=1 .
The two solutions and their basis functions are illustrated
in Figure 1 for specific data. This example demonstrates that
the mere replacement of the L2 penalty with the gTV norm
has a fundamental effect on the solution: piecewise-linear
functions having knots at the sampling locations are replaced
by piecewise-constant functions with a lesser number of
adaptive knots. Moreover, in the gTV case, the regularization
has been imposed on the derivative of the
(kDf kM ),
Pfunction
K
0
which uncovers the innovations Df1 = k=1 ak δ(· − τk ). By
contrast, when R(f ) = kDf k2L2 = hD∗ Df, f i, the recovered
PM
solution is such that D∗ Df2 = m=1 am δ(· − xm ), where
D∗ = −D is the adjoint operator of D. Thus, in both cases,
the recovered functions are composed of the Green’s function
of the corresponding active operators: D vs. D∗ D = −D2 .
where R is either RTik or RgTV , has the same measurement
z 0 given that the problem has at least one solution.
IV. C OMPARISON
f1 (x) =
K
X
k=1
ak ρL (x − xk ) +
N0
X
bn pn (x)
(19)
n=1
for some K ≤ (M − N0 ). The parameters of the solution are
the unknown knots (x1 , . . . , xK ) and the expansion coefficients
a = (a1 , . . . , aK ), b = (b1 , . . . , bN0 ). The solution set V is the
convex hull of these extreme points and kLf kM = kak1 .
The existence and nature of the solution set in these 2 cases
is stated jointly in Theorem 5. The proof is given in Appendix
A.
f ∈X
Theorem 5 implies that, for the gTV case when E is strongly
convex, the elements of the solution set V map to the unique
point z V = H{f }, ∀ f ∈ V.
C. Illustration with Ideal Sampling
Here, we discuss the regularized case where noisy data points
((x1 , z1 ), . . . , (xM , zM )) are fitted by a function. The measurement functionals in this case are the shifted Dirac impulses
hm = δ(·−xm ) whose Fourier transform is b
hm (ω) = e−jωxm .
We choose L = D and E = kz−H{f }k22 . For the L2 problem,
we have that
!
M
X
f2 = arg min
|zm − f (xm )|2 + λkDf k2L2 . (21)
f ∈X2
m=1
As given in Theorem
n 3, f2ois unique and has the basis function
−1 e−j(·)xm
(x) = 12 |x − xm |. The resulting
ϕm (x) = F
|j(·)|2
solution is piecewise linear. It can be expressed as
f2 (x) = b1 +
M
X
1
am |x − xm |,
2
m=1
(22)
We now discuss and contrast the results of Theorems 3
and 4. In either case, the solution is composed of a primary
component and a null-space component whose regularization
cost vanishes.
A. Nature of the Primary Component
1) Shape and Dependence on Measurement Functionals:
The solution for the gTV regularization is composed of atoms
within the infinitely large dictionary {ρL (· − τ )}, ∀τ ∈ R,
whose shapes depend only on L. In contrast, the L2 solution
is composed of fixed atoms {ϕm }M
m=1 whose shapes depend
on both L and H. As the shape of the atoms of the gTV
solution does not depend on H, this makes it easier to inject
prior knowledge in that case.
2) Adaptivity: The weights and the location of the atoms
of the gTV solution are adaptive and found through a datadependent procedure which results in a sparse solution that
turns out to be a nonuniform spline. By contrast, the L2
solution lives in a fixed finite-dimensional space.
5
convolved with the measurement functions, whose temporal
support is not localized. This allows us to say that the gTV
solution is sparser than the Tikhonov solution.
(a) f1 (x) and f2 (x).
(b) ρD (x) and ρD∗ D (x).
Fig. 1: Reconstructions of a signal from nonuniform samples
for L = D: (a) Tikhonov (L2 ) vs. gTV solution, and (b)
Corresponding basis functions ρD vs. ρD∗ D . Note that the gTV
solution is non-unique since, for example, any nondecreasing
piecewise-constant interpolation between the fourth and the
fifth measurement has the same arc-length as the solution
shown.
V. D ISCRETIZATION AND A LGORITHMS
We now lay down the discretization procedure that translates
the continuous-domain optimization into a more tractable
finite-dimensional problem. Theorems 3 and 4 imply that
the infinite-dimensional solution lives in a finite-dimensional
space that is characterized by the basis functions {ϕm }M
m=1
N0
for L2 and {ρL (· − τk )}K
k=1 for gTV, in addition to {pn }n=1
as basis of the null space. Therefore, the solutions can be
uniquely expressed with respect to the finite-dimensional parameter a ∈ RM or a ∈ RK , respectively, and b ∈ RN0 .
Thus, the objective functional JR (z|λ, f ) can be discretized to
get the objective functional JR (z|λ, a, b). Its minimization is
done numerically, by expressing H{f } and kLf k2L2 or kLf kM
in terms of a and b. We discuss the strategy to achieve
JR (z|λ, a, b) and its minima for the two cases.
A. Tikhonov Regularization
For the L2 regularization, given λ > 0, the solution
f2 = arg min E(z, H{f }) + λkLf k2L2
f ∈X2 |
{z
}
J2 (z|λ,f )
can be expressed as
f2 =
B. Null-Space Component
The second component in either solution belongs to the null
space of the operator L. As its contribution to regularization
vanishes, the solutions tend to have large null-space components in both instances.
C. Oscillations
The modulus of the
Fourier transform of the basis function
n o
1
of the gTV case,
typically decays faster than that of
n
oLb
b
hm
the L2 case,
. Therefore, the gTV solution exhibits
b2
|L|
weaker Gibbs oscillations at edges.
D. Unicity of the Solution
Our hypotheses guarantee existence. Moreover, the minimizer of the L2 problem is unique. By contrast, the gTV
problem can have infinitely many solutions, despite all having
the same measurements. The solution set in this case is convex
and the extreme points are nonuniform splines with fewer
knots than the number (M − N0 ) of measurements. When
the gTV solution is unique, it is guaranteed to be an L-spline.
E. Nature of the Regularized Function
One of the main differences between the reconstructions f2
and f1 is their sparsity. Indeed, Lf1 uncovers Dirac impulses
situated
PM −1 at (M − 1) locations for the gTV case, with Lf1 =
m=1 am δ(· − τm ). In return, Lf2 is a nonuniform L-spline
(25)
M
X
am ϕm +
m=1
N0
X
bn pn .
(26)
n=1
Recall that ϕm = (L∗ L)−1 hm , so that
L∗ Lf2 =
M
X
am hm .
(27)
m=1
The corresponding J2 (z|λ, a, b) is then found by expressing
H{f2 } and kLf2 k2L2 in terms of a and b. Due to the linearity
of the model,
H{f2 } =
M
X
m=1
am H{ϕm } +
= Va + Wb,
N0
X
n=1
bn H{pn }
(28)
where [V]m,n = hhm , ϕn i and [W]m,n = hhm , pn i. Similarly,
* M
+
X
∗
hLf2 , Lf2 i = hL Lf2 , f2 i =
am hm , f2
(29)
m=1
= aT Va + aT Wb = aT Va,
(30)
where (29) uses (27) and where (30) uses the orthogonality
property (17), which we can restate as aT W = 0. By substituting these reduced forms in (25), the discretized problem
becomes
f2 = arg min E(z, Va + Wb) + λaT Va .
(31)
a,b |
{z
}
J2 (z|λ,a,b)=J2 (z|λ,f2 )
Due to Assumption 1.ii), this problem is convex. If E is
differentiable with respect to the parameters, the solution can
6
be found by gradient descent.
When E(z, H{f }) = kz − H{f }k22 , the problem is reduced
to
arg min kz − (Va + Wb)k22 + λaT Va
(32)
a,b |
{z
}
Similarly to the L2 case, J1 (z|λ, a, b) is found by expressing H{f1,∆ } and kLf1,∆ kM in terms of a and b. For this,
we use the properties that LρL = δ, kδkTV = 1, and Lpn = 0
for n ∈ [1 . . . N0 ]. This results in
H{f1,∆ } = Pa + Qb,
J2 (z|λ,a,b)
which is very similar to (2). This criterion is convex with
respect to the coefficients a and b. Enforcing that the gradient
of J2 vanishes with respect to a and b and setting the gradient
to 0 then yields M linear equations with respect to the M +
N0 variables, while the orthogonality property (17) gives N0
additional constraints. The combined equations correspond to
the linear system
V + λI W a
z
=
.
(33)
b
0
WT
0
The system matrix so obtained can be proven to be positive
definite due to the property of Gram matrices generated in
an RKHS and the admissibility condition of the measurement
functional (Assumption 1). This ensures that the matrix is
always invertible. The consequence is that the reconstructed
signal can be obtained by solving a linear system of equation, for instance by QR decomposition or by simple matrix
inversion. The derived solution is the same as the least-square
solution in [20].
kLf1,∆ kM = kak1 ,
f1 = arg min (E(z, (Pa + Qb)) + λkak1 ) .
a,b |
{z
}
When E is differentiable with respect to the parameters, a
minimum can be found by using proximal algorithms where
the slope of kak1 is defined by a Prox operator. We discuss
the two special cases when E is either an indicator function
or a quadratic data-fidelity term.
1) Exact Fit with E = I(z 0 , H{f }): To perfectly recover
the measurements, we impose an infinite penalty when the
recovered measurements differ from the given ones. In view
of (38) and (39), this corresponds to solving
subject to Pa + Qb = z. (41)
We then recast Problem (41) as the linear program
In the case of gTV regularization, the problem to solve is
f1 = arg min (E(z, H{f }) + λkLf kM ) .
f ∈X2 |
{z
}
(34)
(a∗ , u∗ , b∗ ) = min
a,u,b
J1 (z|λ,f )
N
X
n=1
un subject to u + a ≥ 0,
According to Theorem 4, an extreme-point solution of (34) is
f1 (x) =
k=1
and satisfies
ak ρL (x − τk ) +
Lf1 = w1 =
K
X
k=1
N0
X
bn pn (x)
(35)
n=1
ak δ(· − τk )
(36)
with K ≤ (M − N0 ). Theorem 4 implies that we only have
to recover ak , τk , and the null-space component p to recover
f1 .
Since we usually know neither K nor τk beforehand, our
solution is to quantize the x-axis and look for τk in the
range [0, T ] on a grid with N K points. We control
the quantization error with the grid step ∆ = T /N . The
discretized problem is then to find a ∈ RN with fewer than
(M − N0 ) nonzero coefficients and b ∈ RN0 such that
f1,∆ (x) =
N
−1
X
n=0
(40)
J1 (z|λ,a,b)=J1 (z|λ,f1 )
a,b
K
X
(39)
where a = (a0 , . . . , aN −1 ), [P]m,n = hhm , ρL (· − n∆)i for
n ∈ [0 . . . N − 1], [Q]m,n = hhm , pn i for n ∈ [1 . . . N0 ],
PN
kak1 =
n=1 |an |, and where N is the initial number
of Green’s functions of our dictionary. The new discretized
objective functional is
(a∗ , b∗ ) = arg min kak1
B. gTV Regularization
(38)
an ρL (x − n∆) +
N0
X
bn pn (x)
(37)
n=1
satisfies (34), with K ≤ (M − N0 ) N nonzero coefficients
an . When the discretization step ∆ goes to 0 (or when N is
large enough), we recover the solution of the original problem
(34).
u − a ≥ 0,
Pa + Qb = z,
(42)
where the inequality x ≥ y between any 2 vectors x ∈ RN
and y ∈ RN means that xn ≥ yn for n ∈ [1 . . . N ]. This
linear program can be solved by a conventional simplex or a
dual-simplex approach [31], [32].
2) Least Squares Fit with E = kz − H{f }k22 : When E is
a quadratic data-fidelity term, the problem becomes
(a∗ , b∗ ) = arg min kz − (Pa + Qb) k22 + λkak1 , (43)
a,b
which is more suitable when the measurements are noisy.
The discrete version (43) is similar to (3), the fundamental
difference being in the nature of the underlying basis function.
The problem is converted into a LASSO formulation [9] by
decoupling the computation of a∗ and b∗ . Suppose that a∗ is
fixed, then b∗ is found by differentiating (43) and equating
the gradient to 0. This leads to
−1 T
b∗ = QT Q
Q (z − Pa∗ ).
(44)
Upon substitution in (43), we get that
a∗ = arg min kQ0 z − Q0 Pak22 + λkak1 ,
(45)
a
−1 T
where Q0 = I − Q QT Q
Q
and I is the (M × M )
identity matrix. Problem (45) can be solved using a variety
7
of optimization techniques such as interior-point methods
or proximal-gradient methods, among others. We employ
the popular iterative algorithm FISTA [12], which has an
O(1/t2 ) convergence rate with respect to its iteration number
t. However, in our case, the system matrices are formed by
the measurements of the shifted Green’s function on a fine
grid. This leads to high correlations among the columns and
introduces two issues.
• If LASSO has multiple solutions, then FISTA can converge to a solution within the solution set, whose sparsity
index is greater than M .
• If LASSO has a unique solution, then the convergence
to the exact solution can be slow. The convergence rate
is inversely proportional to the Lipschitz constant of the
gradient of a quadratic loss function max Eig HT H ,
which is typically high for the system matrix obtained
through our formulation.
We address these issues by using a combination of FISTA
and simplex, governed by the following Lemma 6 and Theorem 7. The properties of the solution of the LASSO problem
have been discussed in [33], [34], [35]. We quickly recall one
of the main results from [33].
Lemma 6 ( [33, Lemma 1 and 11]). Let z ∈ RM and H ∈
RM ×N , where M < N . Then, the solution set
αλ = arg min kz − Hak22 + λkak1
(46)
a∈RN
has the same measurement Ha∗ = z 0 for any a∗ ∈ αλ .
Moreover, if the solution is not unique, then any two solutions
(1)
(2)
a
n , a ∈ αλ are
such
that
o their mth element satisfies
(1)
(2)
sign am sign am ≥ 0 for m ∈ [1 . . . M ]. In other
words, any two solutions have the same sign over their
common support.
We use Lemma 6 to infer Theorem 7, whose proof is given
in Appendix 7.
Theorem 7. Let z ∈ RM and H ∈ RM ×N , where M <
N . Let z 0,λ = Ha∗ , ∀a∗ ∈ αλ , be the measurement of the
solution set αλ of the LASSO formulation
a∗ = arg min kz − Hak22 + λkak1 .
(47)
a∈RN
lution is shown in Figure 2.b. In this case, FISTA converges to
a non-sparse solution with kaF k > M , shown as solid stems.
This implies that it is not an extreme point of the solution set.
The simplex algorithm is then deployed to minimize the `1
norm such that the measurement z 0 = HaF is preserved. The
final solution shown as dashed stems is an extreme point with
the desirable level of sparsity. The continuous-domain relation
of this example is discussed later.
The solution of the continuous-domain formulation is a convex
set whose extreme points are composed of at most M shifted
Green’s functions. To find the position of these Green’s functions, we discretize the continuum into a fine grid and then run
the proposed two-step algorithm. If the discretization is fine
enough, then the continuous-domain function that corresponds
to the extreme point of the LASSO formulation is a good
proxy for the actual extreme point of the convex-set solution
of the original continuous-domain problem. This makes the
extreme-point solutions of the LASSO a natural choice among
the solution set. For the case when there is a unique solution
but the convergence is too slow owing to the high value
of the Lipschitz constant of the gradient of the quadratic
loss, the simplex algorithm is used after the FISTA iterations
are stopped using an appropriate convergence criterion. For
FISTA, the convergence behavior is ruled by the number of
iterations t as
C
,
(49)
F (at ) − F (a∗ ) ≤
(t + 1)2
where F is the LASSO functional and
C = 2ka0 − a∗ k22 max Eig HT H
(50)
(see [12]). This implies that an neighborhood
p of the minima
of the functional is obtained in at most t = C/ iterations.
However, there is no direct relation between the functional
value and the sparsity index of the iterative solution. Using the
simplex algorithm as the next step guarantees the upper bound
M on the sparsity index of the solution. Also, F (aSLP ) ≤
F (aF ). This implies that an -based convergence criterion, in
addition to the sparsity-index-based criterion like aF ≤ M , can
be used to stop FISTA. Then, the simplex scheme is deployed
to find an extreme point of the solution set with a reduced
sparsity index.
a∗SLP
Then, the solution
(obtained using the simplex algorithm)
of the linear program corresponding to the problem
a∗SLP
= arg min kak1
subject to Ha = z 0,λ
is an extreme point of αλ . Moreover,
ka∗SLP k0
(48)
≤ M.
Theorem 7 helps us to find an extreme point of the solution
set αλ of a given LASSO problem in the case when its solution
is non-unique. To that end, we first use FISTA to solve the
LASSO problem until it converges to a solution aF . By setting
z 0,λ = HaF , Lemma 6 then implies that Ha = z 0,λ , ∀a ∈
αλ . We then run the simplex algorithm to find
aSLP = arg min kak1
VI. I LLUSTRATIONS
subject to Ha = HaF ,
which yields an extreme point of αλ by Theorem 7.
An example where the LASSO problem has a non-unique so-
We discuss the results obtained for the cases when the
measurements are random samples either of the signal itself or
of its continuous-domain Fourier transform. The operators of
interest are L = D and L = D2 . The test signal f is solution
of the stochastic differential equation Lf = w [36] for the two
cases when w is
•
Impulsive Noise. Here, the innovation w is a sum of
Dirac impulses whose locations follow a compoundPoisson distribution and whose amplitudes follow a Gaussian distribution. The corresponding process s has then
the particularity of being piecewise smooth [37]. This
case is matched to the regularization operator kLf kM
8
regularization operator kLf kL2 . Unlike the impulsive
∗
noise, wL
= LfL∗ 2 is not localized to finite points
2
and therefore is a better model for the realization of a
Gaussian white noise.
In all experiments, we also constrain the test signals to
be compactly supported. This can be achieved by putting
linear constraints on the innovations of the signal. In Sections
VI-A and VI-C, we confirm experimentally that matched
regularization recovers the test signals better than non-matched
regularization. While reconstructing the Tikhonov and gTV
solutions when the measurements are noisy, the parameter λ
in (33) and (43) is tuned using a grid search to give the best
recovered SNR.
(a)
a∗F and a∗SLP
A. Random Sampling
In this experiment, the measurement functionals are Dirac
impulses with the random locations {xm }M
m=1 . The regularization operator is L = D2 . It corresponds to ρD2 (x) = − 12 |x|
and ϕD2 (x) = (ρL∗ L ∗ hm ) (x) = |x − xm |3 /12. The null
space is ND2 = span{1, x} for this operator. This means
that the gTV-regularized solution is piecewise linear and that
the L2 -regularized solution is piecewise cubic. We compare
in Figures 3.a and 3.b the recovery from noiseless samples
of a second-order process, referred to as ground truth (GT).
It is composed of sparse (impulsive Poisson) and non-sparse
(Gaussian) innovations, respectively [38]. The sparsity index—
the number of impulses or non-zero elements—for the original
sparse signal is 9. The solution for the gTV case is recovered
with ∆ = 0.05 and N = 200. The sparsity index of the
gTV solution for the sparse and Gaussian cases are 9 and 16,
respectively. As expected, the recovery of the gTV-regularized
reconstruction is better than that of the L2 -regularized solution
when the signal is sparse. For the Gaussian case, the situation
is reversed.
(b)
(c)
Fig. 2: Illustration of inability of FISTA to deliver a sparse
∗
for
solution : (a) comparison of solutions, fF∗ vs. fSLP
continuous-domain gTV problem, (b) signal innovations with
sparsity index 64 (> M ) and 21 (< M ), respectively, and (c)
derivative of the two solutions. The two signal innovations in
(b) are solutions of the same Lasso problem, but only aSLP
is an extreme point of the solution set. The original signal
is a second-order process (L = D2 ) and the measurements
are M = 30 nonuniform noisy samples (SNR = 40 dB). The
1
parameters are λ = 0.182, N = 400, and grid step ∆ = 80
.
and is covered by Theorem 4 which states that the minima
∗
fgTV
for this regularization case is such that
∗
∗
wgTV
= LfgTV
=
K
X
k=1
•
ak δ(· − xk ),
(51)
which is a form compatible with a realization of an
impulsive white noise.
Gaussian White Noise. This case is matched to the
B. Multiple Solutions
We discuss the case when the gTV solution is non-unique.
We show in Figure 2.a examples of solutions of the gTVregularized random-sampling problem obtained using FISTA
alone (fF ) and FISTA + simplex (linear programming, fSLP ).
In this case, M = 30, L = D2 , and λ = 0.182. The
continuous-domain functions fF and fSLP have basis functions
whose coefficients are the (non-unique) solutions of a given
LASSO problem, as shown in Figure 2.b. The `1 norms of the
corresponding coefficients are the same. Also, it holds that
kD2 fF kM = kD2 fSLP kM = kDfF kTV = kDfSLP kTV,
(52)
which implies that the TV norm of the slope of fF and fSLP
are the same. This is evident from Figure 2.c. The arc-length
of the two curves are the same. The signal fSLP is piecewise
linear (21 < M ), carries a piecewise-constant slope, and is by
definition, a non-uniform spline of degree 1. By contrast, fF
has many more knots and even sections whose slope appears
to be piecewise-linear.
Theorem 4 asserts that the extreme points of the solution set
of the gTV regularization need to have fewer than M knots.
9
0.8
100
0.6
80
0.4
60
0.2
40
0
20
-0.2
0
-0.4
-20
0
2
4
6
8
10
(a) Sparse Signal
0
2
4
6
8
10
(b) Gaussian Signal
Fig. 3: Recovery of sparse (a) and Gaussian (b) second-order processes (GT) using L = D2 from their nonuniform samples
corrupted with 40 dB measurement noise.
Remember that fSLP is obtained by combining FISTA and
simplex; this ensures that the basis coefficients of fSLP are
the extreme points of the solution set of the corresponding
LASSO problem (Theorem 7) and guarantees that the number
of knots is smaller than M .
This example shows an intuitive relationship between the
continuous-domain and the discrete-domain formulations of
inverse problems with gTV and `1 regularization, respectively.
The nature of the continuous-domain solution set and its
extreme points resonates with its corresponding discretized
version. In both cases, the solution set is convex and the
extreme points are sparse.
C. Random Fourier Sampling
Let now the measurement functions be hm (x) =
rect Tx e−jωm x , where T is the window size. The samples are
thus random samples of the continuous-domain Fourier transform of a signal restricted to a window. For the regularization
operator L = D, the Green’s function
is ρD (x) = 1+ (x) and
the basis is ϕD,m (x) = 12 | · | ∗ hm (x).
Figure 4.a and 4.b correspond to a first-order process with
sparse and Gaussian innovations, respectively. The grid step
∆ = 0.05, M = 41, and N = 200. The sparsity index of
the gTV solution for the sparse and Gaussian cases is 36
and 39, respectively. For the original sparse signal (GT), it is
7. The oscillations of the solution in the L2 -regularized case
are induced by the sinusoidal form of the the measurement
functionals. This also makes the L2 solution intrinsically
smoother than its gTV counterpart. Also, the quality of the
recovery depends on the frequency band used to sample.
In Figures 4.c and 4.d, we show the zoomed version of
the recovered second-order process with sparse and Gaussian innovations, respectively. The grid step is ∆ = 0.05,
M = 41 and N = 200. The operator L = D2 is used for
the regularization. This corresponds
to ρD2 (x) = x+ and
1
| · |3 ∗ hm (x). The sparsity index of the
ϕD2 ,m (x) = 12
gTV solution in the sparse and Gaussian cases is 10 and 36,
respectively. For the original sparse signal (GT), it is 10. Once
again, the recovery by gTV is better than by L2 when the
signal is sparse. In the Gaussian case, the L2 solution is better.
The effect of sparsity on the recovery of signals from their
noiseless and noisy (40 dB SNR) Fourier samples are shown
in Table 1. The sample frequencies are kept the same for all
the cases. Here, M = 41, N = 200, T = 10, and the grid step
∆ = 0.05. We observe that reconstruction performances for
random processes based on impulsive noise are comparable
to that of Gaussian processes when the number of impulses
increases. This is reminiscent of the fact that generalizedPoisson processes with Gaussian jumps are converging in law
to corresponding Gaussian processes [39].
VII. C ONCLUSION
We have shown that the formulation of continuous-domain
linear inverse problems with Tikhonov- and total-variationbased regularizations leads to spline solutions. The nature
of these splines is dictated by the Green’s function of the
regularization operator L and (L∗ L) for Tikhonov and total
variation, respectively. The former is better to reconstruct
smooth signals; the latter is an attractive choice to reconstruct
signals with sparse innovations. Representer theorems for the
two cases come handy in the numerical reconstruction of the
solution. They allow us to reformulate the infinite-dimensional
optimization as a finite-dimensional parameter search. The
formulations and the results of this paper are summarized in
Figure 5.
A PPENDIX A
P ROOF OF T HEOREM 5
Let J ∗ be the minimum value attained by the solutions. Let
f1 and f2 be two solutions. Let E1 , E2 be their corresponding
E functional value and let R1 , R2 be their corresponding
regularization functional value. Since the cost function is
convex, any convex combination f12 = βf1 + (1 − β)f2 is
also a solution for β ∈ [0, 1] with functional value J ∗ . Let us
assume that H{f1 } =
6 H{f2 }. Since E is strongly convex and
R is convex, we get that
J(f )=E(z, H{βf1 + (1 − β)f2 }) + λR(βf1 + (1 − β)f2 )
<βE1 + (1 − β)E2 + βR1 + (1 − β)R2 .
{z
}
|
J∗
10
0.5
120
0
100
80
-0.5
60
-1
40
-1.5
20
-2
0
-2.5
-5
-20
0
5
-5
0
(a) Sparse Signal
5
(b) Gaussian Signal
130
-2.3
128
126
-2.35
124
122
-2.4
120
118
-2.45
0.9
1
1.1
1.2
1.3
1.4
1.5
-2.6
-2.5
(c) Sparse Signal
-2.4
-2.3
-2.2
-2.1
-2
-1.9
(d) Gaussian Signal
Fig. 4: Recovery of first-order (first row) and second-order (second row) processes from their random noiseless Fourier samples.
In all the cases, M = 41 and N = 200. In the interest of clarity, (c) and (d) contain the zoomed versions of the actual signals.
No. of
impulses
10
100
2000
-
Sparsity
Strong
Medium
Low
Gaussian
D
TV
19.60
16.58
14.45
14.30
L2
15.7
16.10
16.14
16.32
D2
TV
52.08
41.91
39.68
40.05
L2
41.54
41.26
41.40
41.23
No. of
impulses
10
100
2000
-
Sparsity
Strong
Medium
Low
Gaussian
D
TV
17.06
13.24
10.61
10.40
L2
11.52
10.94
11.13
11.10
D2
TV
25.55
24.44
25.80
24.95
L2
24.60
24.24
26.19
25.48
TABLE I: Comparison of TV and L2 recovery from their (left table) noiseless and (right table) noisy (with 40 dB SNR)
random Fourier samples. The results have been averaged over 40 realizations.
This is a contradiction. Therefore, H{f1 } = H{f2 } = H{f12 }.
∗
f =
A PPENDIX B
A BSTRACT R EPRESENTER T HEOREM
Theorem 8. Let X be a Hilbert space equipped with the inner
product h·, ·iX and a set of linear functionals h1 , . . . , hM ∈
X 0 . Let C ∈ RM be a feasible convex compact set, meaning
that there exists at least a function f ∈ X such that H{f } ∈ C.
Then, the minimizer
f ∈X
M
X
am h∗m
(54)
m=1
The result presented in this section is preparatory to Theorem 3. It is classical for Hilbert spaces. We give its proof for
the sake of completeness.
f ∗ = arg min kf k2X s.t. H{f } ∈ C
exists, is unique, and can be written as
(53)
∗
0
for some {am }M
m=1 ∈ R, where hm = Rhm and R : X → X
is the Riesz map of X .
Proof. Let CX = H−1 (C) = {f ∈ X , H{f } ∈ C} ∈ X ,
assumed to be nonempty. Since H is linear and bounded and
since C is convex and compact, its preimage CX is also convex
and closed. By Hilbert’s projection theorem [40], the solution
f ∗ exists and is unique as the projection of the null function
onto CX . Let the measurement of this unique point f ∗ be
H{f ∗ } = z 0 .
The Riesz representation theorem states that hhm , f i =
11
L2
(kLf k2 )
Regularization
Norms
Functional
Analysis
TV
(kLf kM )
(kLf k2 < 1) 2.a
1.a
Regularization
4.a
3.a (kLf kM < 1)
Regularization
Operator
L
Ex. D, D2 ,
D ↵I, D
L{⇢L } =
Ex. 1+ , x+
Reproducing
Kernel
Hilbert Space
Search
Space
Unique &
smooth
Representer
1.b
Theorem
p0 +
Parametric
Solution
M
X
Non-unique &
Sparse
3.b
2.b
a m 'm
Defines
4.b
Banach Space
p0 +
m=1
K
X
ak ⇢L (·
⌧k )
Ex. |x|, |x|3
Basis
functions
4.c
KM
N0
4.d
Effect
Linear
Equations
Numerical
Solution
Null Space: NL
L{p} = 0 8 p 2 NL
k=1
p0 2 N L
Convex
1.c
Optimization
L⇤ L{'m } = hm
3.c
2.c
H
FISTA +
Simplex
Iterative
Optimization
Easy
Optimization
Measurement
Functional
4.e
Ex. Sampling
(Interpolation
machine learning),
Fourier
sampling (MRI),
Line integral
(Tomography), etc.
Fig. 5: Summary of the whole scheme. The regularization operator with a given norm {4.a} defines the search space for the
solution{1.a, 4.b}. Representer theorems then give the parametric representation of the solution {1.b}. The numerical solution
is then recovered by optimizing over the parameters to minimize JR (z|f ) {1.c}.
hh∗m , f iX for every f ∈ X , where h∗m ∈ X is the unique Riesz
conjugate of the functional
hm . We then uniquely decompose
PM
f ∗ as f ∗ = f ⊥ + m=1 am h∗m , where f ⊥ is orthogonal to
the span of the h∗m with respect to the inner product on X .
The orthogonality implies that
kf ∗ k2X
=
2
f⊥ X
+
M
X
2
am h∗m
m=1
.
(55)
X
This means that the minimum norm is reached P
when f ⊥ = 0,
M
∗
implying that the form of the solution is f = m=1 am h∗m .
A PPENDIX C
P ROOF OF T HEOREM 3
The proof of Theorem 3 has two steps. We first show that
there exists a unique solution. Then, we use Theorem 8 to
deduce the form of the solution.
Existence and Unicity of the Solution. As is classical in
convex optimization, it suffices to show that the functional
J2 (z|·) is coercive and strictly convex. We start with the
coercivity. The measurement operator H is continuous and
linear from X2 to RM ; hence, there exists a constant C such
that
kH{f }k2 ≤ Ckf kX2
(56)
for every f ∈ X2 . Likewise, the condition H{p} = H{q} ⇒
p = q for p, q ∈ NL implies the existence of B > 0 such that
[23, Proposition 8]
kH{p}k2 ≥ BkpkNL
(57)
for every p ∈ NL . Any f ∈ X2 can be uniquely decomposed
as f = L−1 w + p with w ∈ L2 (R) and p ∈ NL . Then, we
remark that kf − pkX2 = kwkL2 .
Putting (56) and (57) together, we deduce with the triangular
inequality that
kH{f }k2 = kH{p} + H{f − p}k2
(58)
≥ kH{p}k2 − kH{f − p}k2
≥ BkpkNL − Ckf − pkX2 = BkpkNL − CkwkL2 .
(59)
Assume that kf kX2 → ∞. It means that kpkNL or kwkL2
are unbounded. If kpkNL is significantly larger than kwkL2 ,
then kH{f }k → ∞ according to (59); hence, J2 (z|f ) ≥
E(z, H{f }) → ∞ using the coercivity of E. Otherwise, it
means that kwkL2 is dominating and J2 (z|f ) ≥ λkwkL2 →
∞. In both cases, J2 (z|f ) → ∞ and J2 (z|·) is coercive.
For the strict convexity, we first remark that J2 (z|·) is convex.
For β ∈ (0, 1), f1 , f2 ∈ X2 , we denote f12 = βf1 +
(1 − β)f2 . Then, the equality case J2 (z|f12 ) = βJ2 (z|f1 ) +
(1 − β)J2 (z|f2 ) implies that E(z|f12 ) = βE(z|f1 ) + (1 −
β)E(z|f2 ) and kLf12 kL2 = βkLf1 kL2 + (1 − β)kLf2 kL2 ,
since the two parts of the functional are themselves convex.
The strict convexity of E(z|·) and the norm k·k2 then implies
that
Lf1 = Lf2 and H{f1 } = H{f2 }
(60)
and, therefore, (f1 − f2 ) ∈ NL ∩ NH . Hence, f1 = f2 and
the strict convexity is demonstrated. The functional J2 (z|·) is
coercive and strictly convex and, therefore, admits a unique
minimizer f ∗ ∈ X .
Form of the Minimizer. Let z 0 = H{f ∗ }. One decomposes
12
again X2 as the direct sum X2 = H + NL , where
H = {f ∈ X2 , hf, pi = 0, ∀p ∈ NL }
is the Hilbert space with norm kL·kL2 . In particular, we have
that f ∗ = h∗ + p∗ with h∗ ∈ H and p∗ ∈ NL . Consider the
optimization problem
arg min kLgk2L2 s.t. H{g} = (z 0 − H{p∗ }),
(61)
g∈H
which is well-posed because the measurements hm are in
X20 ⊂ H0 . According to P
Theorem 8, this problem admits a
M
∗
∗
unique minimizer g ∗ =
m=1 am hm , where hm ∈ H. By
∗
∗
definition, the function h also satisfies H{h } = (z 0 −
H{p∗ }). Moreover, kLh∗ k2L2 ≤ kLg ∗ k2L2 ; otherwise, the
function f˜ = g ∗ +p∗ ∈ X2 would satisfy J2 (z|f˜) < J2 (z|f ∗ ),
which is impossible. This means P
that f˜ is minimizing (61). By
M
unicity, one has that h∗ = g ∗ = m=1 am h∗m .
PM
So far, we have shown that f ∗ = p∗ + m=1 am h∗m . The
Riesz map R : H0 → H is given for h ∈ H0 by
Z
R{h}(x) =
ρL∗ L (x − y)h(y)dy = (ρL∗ L ∗ h)(x), (62)
R
∗
where ρL∗ L is the Green’s function of the operator (L L) (see
Definition 1). This is easily seen from the form of the norm
kL·kL2 over H and the characterization of the Riesz map as
hRf, giH = hf, gi. This implies that h∗m = ρL∗ L ∗ hm = ϕm
and f ∗ has the form (16).
We conclude by remarking that the condition
P Rh ∈ H for
every h ∈ H0 implies
in
particular
that
m am hm ∈ H,
P
or, equivalently, that m am hhm , pi = 0 for every p ∈ NL ,
which proves (17).
A PPENDIX D
P ROOF OF T HEOREM 4
As for the L2 case, the proof has two steps: We first show
that the set of minimizers is nonempty. We then connect the
optimization problem to the one studied in [23, Theorem 2]
to deduce the form of the extreme points. The functional to
minimize is J1 (z|f ) = E(z, H{f }) + λkLf kM , defined over
f in the Banach space X1 .
Existence of Solutions. We first show that V =
{arg minf ∈X1 J1 (z|f )} is nonempty. We use the results of
Theorem 9, which can be found in [41, Section 3.6].
Theorem 9. Let F : X → R+ be a functional on the Banach
space X with norm k·k.
i) A convex and lower semi-continuous functional on X is
weakly lower semi-continuous.
ii) The norm k·k is weakly lower semi-continuous in X .
iii) A weakly lower semi-continuous and coercive functional
on X reaches its infimum.
According to Theorem 9, the existence of solutions is
guaranteed if J1 (z|·) is weakly lower semi-continuous and
coercive. The coercivity is deduced exactly in the same way
we did for Theorem 3. The continuity is obtained as follows:
The function E(z|·) is convex and lower semi-continuous in
RM and, therefore, weakly lower semi-continuous by Theorem
9. Moreover, H is weak*-continuous by assumption. Hence,
it is continuous for the norm topology. (Indeed, the weak*topology being weaker than the norm topology on X1 , it is
less restrictive to be continuous for the norm topology, that has
more open sets, than for the weak*-topology.) It implies that
E(z|H{·}) is weakly lower semi-continuous by composition.
Moreover, the norm k·kX1 is lower semi-continuous on X1 by
Theorem 9. Finally, J1 (z|·) is lower semi-continuous as the
sum of two lower semi-continuous functionals.
Form of the Extreme Points. Theorem 5 implies that all minimizers of J1 (z|·) have the same measurement H{f ∗ } = z 0 .
The set of minimizers is thus equal to
V = arg min kLf kM , s.t. H{f } = z 0 .
(63)
f ∈X1
Since V is nonempty, the condition H{f } = z 0 is feasible.
We can therefore apply Theorem 2 of [23] to deduce that V
is convex and weak*-compact, together with the general form
(19) of the extreme-point solutions.
A PPENDIX E
P ROOF OF T HEOREM 7
We first state two propositions that are needed for the proof.
Their proofs are given in the supplementary material.
Proposition 10 (Adapted from [11, Theorem 5]). Let z ∈
RM and H ∈ RM ×N , where M < N . Then, the solution
set αλ of
a∗ = arg min kz − Hak22 + λkak1
(64)
a∈RN
is a compact convex set and kak0 ≤ M, ∀a ∈ αE,λ , where
αE,λ is the set of the extreme points of αλ .
Proposition 11. Let the convex compact set αλ be the solution
set of Problem (46) and let αE,λ be the set of its extreme
points. Let the operator T : αλ → RN be such that
Ta = u with um = |am |, m ∈ [1 . . . N ]. Then, the operator is
linear and invertible over the domain αλ and the range Tαλ
is convex compact such that the image of any extreme point
aE ∈ αE,λ is also an extreme point of the set Tαλ .
The linear program corresponding to (48) is
(a∗ , u∗ ) = min
a,u
N
X
n=1
un , subject to u + a ≥ 0,
u − a ≥ 0,
Pa = z.
(65)
By putting u + a = s1 and (u − a) = s2 , the standard form
of this linear program is
!
N
X
(s∗1 , s∗2 ) = min
s1n + s2n , s.t. s1 ≥ 0,
s1 ,s2
n=1
s2 ≥ 0,
Ps1 − Ps2 ≤ z
−Ps1 + Ps2 ≤ −z.
(66)
Any solution a∗ of (65) is equal to (s∗1 −s∗2 ) for some solution
pair (66). We denote the concatenation of any two independent
points sr1 , sr2 ∈ RN by the variable sr = (sr1 , sr2 ) ∈ R2N .
13
Then, the concatenation of the feasible pairs sf = sf1 , sf2
that satisfies the constraints of the linear program (66) forms
a polytope in R2N . Given that (66) is solvable, it is known
that at least one of the extreme points of this polytope is
also a solution. The simplex algorithm
is devised such that
its solution s∗SLP = s∗1,SLP , s∗2,SLP is an extreme point
of this polytope [32]. Our
remaining task is to prove that
a∗SLP = s∗1,SLP − s∗2,SLP is an extreme point of the set αλ ,
the solution set of the problem (46).
Proposition 10 claims that the solution set αλ of the LASSO
problem is a convex set with extreme points αE,λ ∈ RN .
As αλ is convex and compact, the concatenated set ζ =
{w ∈ R2N : w = (a∗ , u∗ ) , a∗ ∈ αλ } is convex and
compact by Proposition 11. The transformation (a∗ , u∗ ) =
(s∗1 − s∗2 , s∗1 + s∗2 ) is linear and invertible. This means that the
solution set of (66) is convex and compact, too. The simplex
solution corresponds to one of the extreme points of this
convex compact set.
Since the map (a∗ , u∗ ) = (s∗1 − s∗2 , s∗1 + s∗2 ) is linear and
invertible, it also implies that an extreme point of the solution
set of (66) corresponds to an extreme point of ζ. Proposition
11 then claims that this extreme point of ζ corresponds to an
extreme point aSLP ∈ αλ,E .
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14
have different signs for their mth element. This means that
the following statements are true:
S UPPLEMENTARY M ATERIAL
A. Structure of the Search Spaces
Decomposition of X1 and X2 . The set X1 is the search space,
or native space, for the gTV case. It is defined and studied in
[23, Section 6], from which we recap the main results. Note
that the same construction is at work for X2 , which is then a
Hilbert space.
Let p = (p1 , . . . , pN0 ) be a basis of the finite-dimensional
null space of L. If φ = (φ1 , . . . , φN0 ) and p = (p1 , . . . , pN0 )
form a biorthonormal system such P
that hφn1 , pn2 i = δ[n1 −
N0
hf, φn ipn is a welln2 ], and if φn is in X 1 , then Pf = n=1
defined projector from X1 to NL . The finite-dimensional null
space of L is a Banach (and even a Hilbert) space for the norm
!1/2
N0
X
2
hp, φn i
.
(67)
kpkNL =
n=1
Moreover, f ∈ X1 is uniquely determined by w = Lf ∈
M(R) and p = Pf ∈ NL . More precisely, there exists a
right-inverse operator L−1
φ of L such that [23, Theorem 4]
f = L−1
φ w + p.
(68)
In other words, X1 is isomorphic to the direct sum M(R) ⊕
NL , from which we deduce that it is a Banach space for the
norm [23, Theorem 5]
kf kX1 = kLf kM + kPf kNL = kwkM + kpkNL .
(69)
Predual of X1 . The space M(R) is the topological dual of
the space C0 (R) of continuous and vanishing functions. The
space X1 inherits this property: It is the topological dual of
CL (R), defined as the image of C0 (R) by the adjoint L∗ of
L according to [23, Theorem 6].
We can therefore define a weak*-topology on X1 : It is the
topology for which fn → 0 if hfn , ϕi → 0 for every ϕ ∈
CL (R). The weak*-topology is crucial to ensure the existence
of solutions of (18); see [23] for more details.
B. Proof of Proposition 10
Using Lemma 6, it is clear that αλ is also a solution set of
αλ = arg min kak1
s.t.
Ha = z 0,λ
(70)
for some z 0,λ . The solution of the problem akin to (70) has
been discussed in [11] and is proven to be convex and compact
such that the extreme points αE,λ of the convex set αλ satisfy
kak0 ≤ M for any a ∈ αE,λ .
C. Proof of Proposition 11
Proof. Let β and γ be such that βm
=
min (0, mina∈αλ sign(am ))
∈
{−1, 0} and γm
=
max (0, maxa∈αλ sign(am )) ∈ {0, 1} for m ∈ [1 . . . N ].
Lemma 6 claims that no two solutions from the solution set
{βm sign(am ) ≥ 0, γm sign(am ) ≥ 0, ∀a ∈ αλ }
{βm 6= 0 ⇒ γm = 0, γm 6= 0 ⇒ βm = 0}
{βm + γm = 0 ⇒ βm = 0, γm = 0 ⇒ sign(am ) = 0, ∀a ∈ αλ }
(71)
{∀a ∈ αλ , am 6= 0 ⇒ βm + γm = sign(am )}
{∀a ∈ αλ , |am | = (βm + γm )am }.
(72)
(73)
Statement (73) shows that, for any a ∈ αλ , Ta = Ra,
where R ∈ RN ×N is a diagonal matrix with entries Rmm =
βm + γm . Thus, the operation of T is linear in the domain
αλ . Also, a = RRa for a ∈ α implies that the operator T is
invertible.
This ensures that the image of the convex compact set Tαλ
is also convex compact and the image of any extreme point
aE ∈ αE,λ is also an extreme point of the set Tαλ . Similarly,
it can be proved that the concatenated set ζ = {w ∈ R2N :
w = (a, Ta) , a ∈ αλ } is the image of a linear and invertible
concatenation operation on α. Thus, it is convex and compact,
and the image of any extreme point through the inverse
operation of the concatenation wE ∈ ζE,λ is also an extreme
point of αλ .
| 7 |
A CHARACTERIZATION OF LOCALLY QUASI-UNMIXED RINGS
arXiv:1607.07641v1 [math.AC] 26 Jul 2016
SIMIN MOLLAMAHMOUDI, ADELEH AZARI AND REZA NAGHIPOUR∗
Abstract. Let I¯ denote the integral closure of an ideal in a Noetherian ring R. The
main result of this paper asserts that R is locally quasi-unmixed if and only if, the
topologies defined by I n and I hni , n ≥ 1, are equivalent. In addition, some results about
the behavior of linearly equivalent topologies of ideals under various ring homomorphisms
are included.
1. Introduction
Let R denote a commutative Noetherian ring, and I an ideal of R. The interesting
concept of quintasymptotic prime ideals of I was introduced by McAdam [3]. A prime
ideal p of R is called a quintasymptotic prime ideal of I if there exists z ∈ mAssR∗p Rp∗ such
that Rad(IRp∗ + z) = pRp∗ . The set of quintasymptotic prime ideals of I is denoted by
Q̄∗ (I), and it is a finite set. Also, in [10], Ratliff, Jr., introduced set of associated primes
Ā∗ (I) := AssR R/I n for large n, called the presistent prime ideals of I, and he showed
that this finite set has some nice properties in the theory of asymptotic prime divisors;
here for any ideal J of R, J¯ denotes the integral closure of J in R, i.e., J¯ is the ideal of
R consisting of all elements x ∈ R which satisfy an equation xn + r1 xn−1 + · · · + rn = 0,
where ri ∈ J i , i = 1, . . . , n.
In his famous paper [11], D. Rees showed that a local ring (R, m) is analytically unramified if and only if the topology defined by I n , n > 1, is equivalent to the I-adic topology
for an m-primary ideal I of R. In [9], L.J. Ratliff, Jr., proved corresponding results in order to characterize reduced unmixed local rings. (Recall that a local ring (R, m) is called
analytically unramified (resp. unmixed), if the m-adic completion, R∗ , of R is reduced
(resp. all the prime ideals of AssR∗ R∗ have the same dimension)). The main theorem
of this paper gives a characterization of locally quasi-unmixed Noetherian rings, which
is closely related to Ress’ result [11]. Since such rings occur in many investigations in
commutative algebra and algebraic geometry, it is desirable to know as many properties
of such ring as possible. This characterization gives one such property, and that such
rings have this property is a new result, and until now was not know to hold even in a
regular local ring. More precisely we shall show that:
Key words and phrases. Associated primes, ideal topologies, integral closure, locally quasi-unmixed
ring, Rees ring.
2010 Mathematics Subject Classification: 13D45, 14B15, 13E05.
This research was in part supported by a grant from IPM.
∗
Corresponding author: e-mail: naghipour@ipm.ir (Reza Naghipour).
1
2
SIMIN MOLLAMAHMOUDI, ADELEH AZARI AND REZA NAGHIPOUR∗
Theorem 1.1. Let R denote a commutative Noetherian ring. Then the following conditions are equivalent:
(i) R is locally quasi-unmixed.
(ii) For every ideal I of the principal class in R, the topologies defined by I n and I hni ,
n ≥ 1, are linearly equivalent.
(iii) For every ideal I of the principal class in R, the topologies defined by I n and I hni ,
n ≥ 1, are equivalent.
This is closely related to Ress’ result [11] for characterization S
quasi-unmixed local rings.
Here I hni denotes the union (I n :R s), where s varies in R\ {p ∈ mAssR R/I}. See
Theorem 2.5 for the proof of Theorem 1.1.
One of our tools for proving Theorem 1.1 is the following, which is a characterization
of the equivalence between the topologies defined by the filtration I n and I hni , n > 1.
Proposition 1.2. Let I denote an ideal in a commutative Noetherian ring R. Then the
topologies defined by I n and I hni , n ≥ 1, are equivalent (resp. linearly equivalent) if and
only if Q̄∗ (I) (resp. Ā∗ (I)) is equal to mAssR R/I.
Pursuing this point of view further we prove some results about the behavior of the
linearly equivalent topologies of ideals under various ring homomorphisms. In connection
to this we derive the following consequence of Proposition 1.2.
Corollary 1.3. Let R be a Noetherian ring and T be a finitely generated integral ring
extension of R such that every minimal prime of T lies over a minimal prime of R. If
the topologies I n and I hni , n > 1, are linearly equivalent, then the topologies defined by
(IT )n and (IT )hni , n > 1, are also linearly equivalent; and the converse holds whenever
T is faithfully flat.
Throughout this paper, for any commutative Noetherian ring R with nonzero identity,
and for any ideal I of R, we denote by mAssR R/I the set of minimal prime ideals over
I. If (R, m) is local, then R∗ denotes the completion of R with respect to the m-adic
topology. Then R is said to be quasi-unmixed ring if for every p ∈ mAssR∗ R∗ , the
condition dim R∗ /p = dim R is satisfied. More generally, if R is not necessarily local, R
is a locally quasi-unmixed ring if for any p ∈ Spec(R), Rp is a local quasi-unmixed
ring.
L nn
For any ideal I of R, we denote by R the graded Rees ring R[u, It] :=
I t of R with
n∈Z
respect to I, where t is an indeterminate and u = t−1 . Also, the radical of I, denoted by
Rad(I), is defined to be the set {x ∈ R | xn ∈ I for some n ∈ N}. Finally, if (R, m) is
local, then the analytic spread of I is defined to be ℓ(I) := dim R/(m, u)R (see [7]). For
any unexplained notation and terminology we refer the reader to [1] or [5].
2. Locally quasi-unmixed rings and comparison of topologies
The purpose of this section is to establish a characterization of locally quasi-unmixed
Noetherian rings, which is closely related to Ress’ result [11]. The main goal is Theorem
2.5. The following lemmas are needed in the proof of that theorem.
A CHARACTERIZATION OF LOCALLY QUASI-UNMIXED RINGS
3
Lemma 2.1. Let I be an ideal of a Noetherian ring R. Then the following conditions are
equivalent:
(i) Q̄∗ (I) = mAssR R/I.
(ii) The topologies defined by I n and I hni , n ≥ 1, are equivalent.
Proof. The assertion follows easily from [3, Theorem 1.5] and the fact that mAssR R/I ⊆
Q̄∗ (I).
Lemma 2.2. Let R be a Noetherian ring such that the topologies defined by I n and
I hni , n > 1, are equivalent for all ideals I of the principal class in R. Then for every
prime ideal p of R and every ideal J of the principal class in Rp , the topologies defined by
J n and J hni , n > 1, are equivalent.
Proof. Let p ∈ Spec(R) and let J be an ideal of the principal class in Rp . Then in view
of [12, Lemma 5.1], there exists an ideal I of R of the principal class such that J = IRp .
Now, in view of Lemma 2.1, it is enough for us to show that Q̄∗ (J) = mAssRp Rp /J. To do
this, let qRp ∈ Q̄∗ (J). That is qRp ∈ Q̄∗ (IRp ). Then, by [3, Proposition 1.1], q ∈ Q̄∗ (I),
and so by Lemma 2.1, q ∈ mAssR R/I. Therefore, qRp ∈ mAssRp Rp /IRp , as required.
The next Lemma was proved by McAdam and Ratliff in [4].
Lemma 2.3. Let I be an ideal of a locally quasi-unmixed Noetherian ring R such that
ℓ(IRp ) = height(IRp ) for all p ∈ Ā∗ (I). Then Ā∗ (I) = mAssR R/I.
Proof. See [4, Lemma 5.4].
Lemma 2.4. Let I denote an ideal in a Noetherian ring R. Then the following conditions
are equivalent:
(i) Ā∗ (I) = mAssR R/I.
(ii) The topologies defined by I n and I hni , n ≥ 1, are linearly equivalent.
Proof. The result follows from [3, Corollary 1.6] and the fact that mAssR R/I ⊆ Ā∗ (I).
We are now ready to state and prove the main theorem of this section which is a
characterization of locally quasi-unmixed Noetherian rings in terms of the equivalence
(resp. linearly equivalence) between the topologies induced by I n and I hni , n ≥ 1, for the
principal class ideals I of R. Recall that an ideal I of R is called of the principal class if
I is generated by height I elements.
Theorem 2.5. Let R denote a commutative Noetherian ring. Then the following conditions are equivalent:
(i) R is locally quasi-unmixed.
(ii) For every ideal I of the principal class in R, the topologies defined by I n and I hni ,
n ≥ 1, are linearly equivalent.
SIMIN MOLLAMAHMOUDI, ADELEH AZARI AND REZA NAGHIPOUR∗
4
(iii) For every ideal I of the principal class in R, the topologies defined by I n and I hni ,
n ≥ 1, are equivalent.
Proof. First we show (i) =⇒ (ii). If R is locally quasi-unmixed, then in view of Lemmas
2.3 and 2.4, it is enough for us to show that, for all p ∈ Ā∗ (I), ℓ(IRp ) = height(IRp )
for every ideal I of the principal class. To this end, in view of [2, Proposition 4.1],
height(p) = ℓ(IRp ). Now, since at least ℓ(a) elements are needed to generate a, for any
ideal a in a commutative Noetherian ring A, and as IRp is an ideal of the principal class
in Rp , it follows that ℓ(IRp ) ≤ height(IRp ). Furthermore, since I ⊆ p, it yields that
height(IRp ) ≤ height(pRp ) = height(p),
and so
ℓ(IRp ) ≤ height(IRp ) ≤ height(p) = height(IRp ).
Hence ℓ(IRp ) = height(IRp ), as required.
Now, because of the implication (ii) =⇒ (iii) is trivially true, so in order to complete
the proof we have to show that (iii) =⇒ (i). Let p ∈ Spec(R). We need to show that
Rp is a quasi-unmixed ring. To do this, in view of Lemma 2.2 and [8, Remark 2.9],
without loss of generality we may assume that (R, m) is a local ring. Now, for proving
the quasi-unmixedness of R, there are two cases to consider.
Case 1. Suppose that mR∗ ∈ mAssR∗ R∗ . Then height(mR∗ ) = 0, and so dim R∗ = 0.
Hence, R is a quasi-unmixed ring, as required.
Case 2. Now, suppose that mR∗ ∈
/ mAssR∗ R∗ , and let q ∈ mAssR∗ R∗ . We need to
∗
show that dim R /q = dim R. To this end, as mR∗ ∈
/ mAssR∗ R∗ , we have dim R∗ /q := n,
where n > 0. Therefore in view of [6, Proposition 3.5], there exists an ideal a of R of the
principal class of height n and Rad(aR∗ + q) = mR∗ . Whence, m ∈ Q̄∗ (a). Moreover, as a
is the principal class, it follows from assumption (iii) and Lemma 2.1 that m ∈ mAssR R/a.
Consequently, height(m) = n, and so dim R∗ /q = dim R, as required.
The following corollary gives us a characterization of locally quasi-unmixed Noetherian
rings in terms of quintasymptotic and presistent prime ideals of I.
Corollary 2.6. Let R be a commutative Noetherian ring. Then the following conditions
are equivalent:
(i) R is locally quasi-unmixed.
(ii) Ā∗ (I) = mAssR R/I, for every ideal I of the principal class of R.
(iii) Q̄∗ (I) = mAssR R/I, for every ideal I of the principal class of R.
Proof. The assertion follows from Theorem 2.5, Lemma 2.1 and [3, Lemma 2.1].
As the final result of this section, we construct an example to show that the Theorem
2.5 is not true, if I is not ideal of the principal class. The following lemma is needed in
the proof of the Example 2.8.
Lemma 2.7. Let R be a Noetherian ring such that dim R > 0. Let I ⊆ p be ideals of R
with p ∈ Spec(R). Then the following conditions are equivalent:
A CHARACTERIZATION OF LOCALLY QUASI-UNMIXED RINGS
5
(i) p ∈ Ā∗ (I).
(ii) p ∈ Ā∗ (xI) for any element x not contained in any minimal prime of R.
Proof. See [2, Proposition 3.26].
Example 2.8. Let k be a field and let R = k[x, y](x,y) . Set m = (x, y)R and I = xm.
Then m ∈ Ā∗ (I) and m ∈
/ Q̄∗ (I).
Proof. Since x is not contained in any minimal prime of R and m ∈ Ā∗ (m), it follows from
Lemma 2.7 that m ∈ Ā∗ (I). Now, we need to show that m ∈
/ Q̄∗ (I). Suppose, the contrary,
∗
∗
that m ∈ Q̄ (I). Then, there exists z ∈ mAssR∗ R such that Rad(IR∗ + z) = mR∗ . Since
I = xm, it yields that Rad(xR∗ + z) = mR∗ . Hence, mR∗ /z is minimal over x(R∗ /z), and
so in view of Krull’s Principal Ideal Theorem, height(mR∗ /z) ≤ 1. On the other hand, as
R∗ is a Cohen-Macaulay ring, it follows that
height(mR∗ /z) = height(mR∗ ) − height(z),
and so height(mR∗ /z) = 2, which provides a contradiction.
3. Lineally equivalent topologies
Our aim of this section is to obtain some results about the behavior of the lineally
equivalent topologies of ideals under various ring homomorphisms.
Proposition 3.1. Let R be a Noetherian ring and let I be an ideal of R such that the
topologies defined by I n and I hni , n ≥ 1, are linearly equivalent. Let T be a finitely
generated integral ring extension of R such that every minimal prime of T lies over a
minimal prime of R. Then the topologies defined by (IT )n and (IT )hni , n ≥ 1, are
linearly equivalent.
Proof. In view of Lemma 2.4, it is enough to show that Ā∗ (IT ) = mAssT T /IT . To
this end, let p ∈ Ā∗ (IT ) and we show that p ∈ mAssT T /IT . Suppose the contrary
that p ∈
/ mAssT T /IT . Then, there exists q ∈ mAssT T /IT such that q $ p. Now as
p, q ∈ Ā∗ (IT ), ( note that mAssT T /IT ⊆ Ā∗ (IT )), it follows from [10, Theorem 3.3] that
p ∩ R and q ∩ R are contained in Ā∗ (I). Hence, in view of Lemma 2.4, p ∩ R and q ∩ R
are contained in mAssR R/I, and so p ∩ R = q ∩ R. Therefore, by [1, Theorem 9.3], p = q
which is a contradiction.
Proposition 3.2. Let R be a Noetherian ring and let I be an ideal of R. Let T be a
faithfully flat ring extension of R such that the topologies defined by (IT )n and (IT )hni , n ≥
1, are linearly equivalent. Then the topologies defined by I n and I hni , n ≥ 1, are linearly
equivalent.
SIMIN MOLLAMAHMOUDI, ADELEH AZARI AND REZA NAGHIPOUR∗
6
Proof. In view of Lemma 2.4, it is enough to show that Ā∗ (I) = mAssR R/I. To do
this, let p ∈ Ā∗ (I). Then in view of [9, Corollary 6.9], there exists p∗ ∈ Ā∗ (IT ) such
that p∗ ∩ R = p. Now by virtue of Lemma 2.4, p∗ ∈ mAssT T /IT . Hence, by the Going
Down property between T and R (cf. [1, Theorem 9.5]), we see that p ∈ mAssR R/I, as
required.
Theorem 3.3. Let R be a Noetherian ring and let I be an ideal of R. Then the topologies
defined by I n and I hni , n ≥ 1, are linearly equivalent if only if the topologies defined by
(IR[x])n and (IR[x])hni , n ≥ 1, are linearly equivalent
Proof. Since R[x] is a faithfully flat ring extension of R, the sufficiency follows from
Proposition 3.1. For necessity, in view of Lemma 2.4, it is enough to show that
Ā∗ (IR[x]) = mAssR[x] R[x]/IR[x].
To this end, let pR[x] ∈ Ā∗ (IR[x]), note that by [2, Proposition 3.21],
Ā∗ (IR[x]) = {pR[x] | p ∈ Ā∗ (I)}.
Then p ∈ Ā∗ (I), and so by Lemma 2.4, p ∈ mAssR R/I. Now, it easy to see that
pR[x] ∈ mAssR[x] R[x]/IR[x], as required.
Proposition 3.4. Let R be a Noetherian ring and let I be an ideal of R such that
the topologies defined by I n and I hni , n ≥ 1, are linearly equivalent. Then for any
z ∈ mAssR R/I, the topologies defined by (I + z/z)n and (I + z/z)hni , n ≥ 1, are linearly equivalent
Proof. In view of Lemma 2.4, it suffices to show that
Ā∗ (I + z/z) = mAssR/z ((R/z)/(I + z/z)).
To do this, let p/z ∈ Ā∗ (I + z/z). Then in view of [9, Corollary 6.3], p ∈ Ā∗ (I). Hence,
by Lemma 2.4, p ∈ mAssR R/I. Now, it easy to see that
p/z ∈ mAssR/z ((R/z)/(I + z/z)),
as required.
Acknowledgments
The authors would like to thank Professor Monireh Sedghi for reading of the first draft
and valuable discussions. Finally, the authors would like to thank from the Institute for
Research in Fundamental Sciences (IPM), for the financial support.
References
[1] H. Matsumura, Commutative Ring Theory, Cambridge Studies in Advanced Mathematics, Cambridge University Press, 1986.
[2] S. McAdam, Asymptotic Prime Divisors, Lecture Notes in Math. 1023, Springer-Verlag, New York,
1983.
A CHARACTERIZATION OF LOCALLY QUASI-UNMIXED RINGS
7
[3] S. McAdam, Quintasymptotic primes and four results of Schenzel, J. Pure Appl. Algebra 47 (1987),
283-298.
[4] S. McAdam and L.J. Ratliff, Jr., On the asymptotic cograde of an ideal, J. Algebra 87 (1984) 36–52.
[5] M. Nagata, Local Rings, Interscience, New York, 1961.
[6] R. Naghipour, Locally unmixed modules and ideal topologes, J. Algebra 236 (2001), 768-777.
[7] D. G. Northcott and D. Rees, Reductions of ideals in local rings, Proc. Cambridge Philos. Soc. 50
(1954), 145-158.
[8] L. J. Ratliff, Jr., Asymptotic sequences, J. Algebra 85 (1983), 337-360.
[9] L. J. Ratliff, Jr., On asymptotic prime divisors, Pacific J. Math. 111 (1984), 395-413.
[10] L. J. Ratliff, Jr., Asymptotic prime divisors and integral extension rings, J. Algebra 95 (1985),
409-431.
[11] D. Rees, A note on analytically unramified local rings, J. London Math. Soc. 36 (1961), 24-28.
[12] J. K. Verma, On ideals whose adic and symbolic topologies are linearly equivalent, J. Pure Appl.
Algebra 47 (1987), 205-212.
Department of Mathematics, University of Tabriz, Tabriz, Iran, and, School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box: 19395-5746,
Tehran, Iran.
E-mail address: mahmoudi.simin@yahoo.com (Simin Mollamahmoudi)
E-mail address: adeleh azari@yahoo.com (Adeleh Azari)
E-mail address: naghipour@ipm.ir (Reza Naghipour)
E-mail address: naghipour@tabrizu.ac.ir (Reza Naghipour)
| 0 |
Derived supersymmetries of determinantal varieties
Steven V Sam
arXiv:1207.3309v1 [math.RT] 13 Jul 2012
July 13, 2012
Abstract
We show that the linear strands of the Tor of determinantal varieties in spaces of symmetric
and skew-symmetric matrices are irreducible representations for the periplectic (strange) Lie
superalgebra. The structure of these linear strands is explicitly known, so this gives an explicit
realization of some representations of the periplectic Lie superalgebra. This complements results
of Pragacz and Weyman, who showed an analogous statement for the generic determinantal
varieties and the general linear Lie superalgebra. We also give a simpler proof of their result. Via
Koszul duality, this is an odd analogue of the fact that the coordinate rings of these determinantal
varieties are irreducible representations for a certain classical Lie algebra.
Contents
1 Lie superalgebras.
1.1 General linear Lie superalgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Periplectic superalgebras. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
4
5
2 Z-graded representations and 2-sided complexes.
2.1 General linear case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Periplectic case. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
6
3 (Skew-)symmetric minors.
7
3.1 JPW complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Trace and evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 Existence of representation structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Generic minors.
13
4.1 Lascoux complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.2 Trace and evaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Existence of representation structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
Introduction.
In this paper, we are interested in the symmetries of three classes of varieties known as determinantal varieties. Let E, F be vector spaces. We consider V
the space of generic matrices Hom(E, F ),
2
symmetric matrices S E, and skew-symmetric matrices 2 E. These carry natural actions of general linear groups GL(E) × GL(F ), GL(E), and GL(E), respectively, via change of basis of the
vector spaces. The orbits under these group actions are classified by the rank of the matrix, and
the orbit closures are the determinantal varieties. The algebraic and geometric properties of these
1
varieties have been intensely studied in the past (see [BV] for a general reference), and the group
action is incredibly useful as a computational and theoretical tool since one gets induced actions
on all “functorial” constructions, such as the coordinate ring, the minimal free resolution, the local
cohomology, etc.
These induced actions are an “obvious symmetry” and the point of departure of this paper is
that there are additional “hidden symmetries”. Before explaining our results, we highlight some
of the previous literature. First, the group actions can be replaced by infinitesimal actions of
their Lie algebras gl(E) × gl(F ) and gl(E). For the coordinate ring of a determinantal variety in
Hom(E, F ), there is an action of Hom(E, F )∗ = Hom(F, E), which is multiplication by linear forms,
and Hom(F, E) is the lower triangular part of the block decomposition of the larger Lie algebra
gl(E ⊕ F ), while gl(E) × gl(F ) forms the block diagonal part. It was shown by Levasseur and
Stafford [LS] that (after a suitable character twist of the action of gl(E) × gl(F )) one can extend
the above two actions to an action of the whole Lie algebra gl(E ⊕ F ) by having Hom(E, F ) act by
certain differential operators on the coordinate ring of a determinantal variety. Furthermore, the
coordinate ring becomes a (non-integrable) irreducible highest weight representation of gl(E ⊕ F ).
[LS] also handled determinantal varieties in the space of symmetric or skew-symmetric matrices
(the large Lie algebra gl(E ⊕ F ) is replaced by either a symplectic or orthogonal Lie algebra,
respectively). These varieties are related to the classical Hermitian symmetric pairs. The results
were extended in [Tan] to all Hermitian symmetric pairs, and explicit formulas for the differential
operators and highest weights are given.
Enright and Willenbring [EW] showed that the action of the large Lie algebra carries over to
the entire minimal free resolution of the determinantal varieties (it is an analogue of the Bernstein–
Gelfand–Gelfand resolution), and the extension to all Hermitian symmetric pairs appears in [EH].
One could think of this as a “vertical” hidden symmetry. In fact, there is an additional “horizontal”
hidden symmetry on the minimal free resolution of a determinantal variety. More specifically, on the
linear strands of the resolution, there is an action of Hom(E, F ) given by applying the differentials.
This time, one interprets Hom(E, F ) as the lower-triangular part of the block decomposition of
the Lie superalgebra gl(E|F ). Again, gl(E) × gl(F ) forms the block diagonal part. It was shown
by Pragacz and Weyman [PW] (see also [AW1, AW2, AW3] for further developments) that one
can extend the above two actions to an action of the whole Lie superalgebra gl(E|F ) (again after
a suitable character twist). The linear strands tensored with the residue field (i.e., Tor) become
irreducible highest weight representations of gl(E|F ), with the exception of the degenerate case of
rank 0 matrices, i.e., the Koszul complex.
We remark that some other interactions between Lie superalgebras and free resolutions (related
to degeneracy loci) appear in [Pra, §A.6] and [Sam].
The goal of this paper is to give a simpler proof of the existence of this superalgebra action
(Theorem 4.3) on the Tor of the determinantal varieties in Hom(E, F ) as well as to construct an
analogous action (Theorem 3.4) for the Tor of the (skew-)symmetric determinantal varieties (it
turns out these two cases essentially collapse to one) as well as some other modules supported
in the determinantal varieties which are related to classical invariant theory (Remark 3.3). The
simplest non-trivial case is given in Example 3.5. The superalgebra gl(E|F ) is replaced by the
periplectic Lie algebra, which is a super analogue of the symplectic Lie algebra. A foreshadowing of
this result and some small cases are contained in [JPW], but as in [PW], the word “superalgebra”
never appears.
The main idea of the paper is to use a variant of “Weyl’s construction” for representations
of the classical groups [FH, §17.3, §19.5]. The idea is to start with the vector representation of a
classical group, apply a Schur functor to it, and then mod out by the image of a map from a smaller
Schur functor to obtain an irreducible representation. By semisimplicity, one could instead map to
2
a smaller Schur functor and take the kernel. For the superalgebras of interest in this paper, the
main point is that the vector representations, and their duals, can be interpreted as 2-term chain
complexes, and we can construct Schur complexes (see [ABW] or [Wey, §2.4]) from them. One point
of difficulty is that these superalgebras do not have a semisimple representation theory, so we have
to combine the two approaches to Weyl’s construction. The Schur complexes give super analogues
of most of the rich combinatorics and invariant theory that one associates with Schur functors.
In particular, they are the irreducible polynomial representations for the superalgebra gl(E|F ),
which were studied in [BR]. In fact, the category of polynomial representations is semisimple.
In our construction, we mix the Schur complexes on the vector representation with the Schur
complexes on the dual vector representation, so we leave the polynomial category. Also, the vector
representation of the periplectic Lie superalgebra is isomorphic to its dual up to grading shift. So
the Weyl construction above is a transition from the classical semisimple setting.
We finish the introduction with an outline of the paper. In §1, we give concrete realizations of
the two classes of Lie superalgebras that we will discuss. In particular, they are Z-graded, and in §2,
we introduce some notation to think about Z-graded representations in terms of chain complexes.
In §3 we state and prove that the linear strands of the Tor of the (skew-)symmetric determinantal
varieties are irreducible representations for the periplectic superalgebra. In §4 we discuss the case
of generic determinantal varieties. The proofs are similar, and the result in this case is already
known, so we only mention the differences from §3.
Notation.
• Z denotes the set of all integers
• We work over a field K of characteristic 0 throughout. Unless
otherwise stated, all multilinear
V
symmetric
operations are taken with respect to K. We use SkL
and k to denote
Vk and exterior
V• L
•
k
.
= k≥0
powers, respectively. We also use the notation S = k≥0 S and
• We use det to denote the top exterior power of a vector space. The dual of a vector space E is
denoted E ∗ . Given two vector spaces E, F , we use Hom(E, F ) to denote the space of all linear
maps E → F . Given a matrix ϕ, we use ϕT to denote the transpose matrix. The general linear
group is GL(E), and its Lie algebra is gl(E).
• P
A partition λ = (λ1 , λ2 , . . . ) is a weakly decreasing sequence of nonnegative integers with |λ| :=
i λi < ∞. We use ℓ(λ) to denote the largest r such that λr > 0. We visualize partitions
by Young diagrams: left-justified collections of boxes such that there are λi boxes in the row i
(counted from top to bottom). The transpose partition λ† is the partition obtained by counting
column lengths of the Young diagram of λ. We will also use exponential notation for partitions,
i.e., sd denotes the sequence (s, s, . . . , s) (d times). We also make use of skew Young diagrams:
if the diagram of µ is a subset of the diagram of λ, i.e., µi ≤ λi for all i, then λ/µ is the set
complement of these diagrams.
• Given a partition λ, Sλ denotes the Schur functor associated to λ. This follows the notation Kλ
in [Wey, §2.1]. In particular, if ℓ(λ) = 1, then Sλ = S|λ| is a symmetric power, and if ℓ(λ† ) = 1,
V
then Sλ = |λ| is an exterior power. The functor Lλ in [Wey, §2.1] is isomorphic to Sλ† since we
work over a field of characteristic 0. We will also use skew Schur functors Sλ/µ which can also
be found in [Wey, §2.1].
• Given a graded K-algebra A, and vector spaces U, V , we will use the notation
U ⊗ A(−i) → V ⊗ A
to denote a map between free A-modules such that in matrix form, the entries consist of homogeneous elements in A of degree i. We might also write U ⊗ A(−i − j) → V ⊗ A(−j) if we need
3
to compose such maps.
• All complexes F• are graded in homological notation, i.e., the differential d : Fi → Fi−1 lowers
the degree of the index. The dual complex F∗• has grading F∗i = F−i . We denote homological
grading shift by [i], i.e., F[i]j = Fi+j .
• A superspace is a Z/2-graded vector space V , and we will write it as V0 ⊕ V1 . The dimension of
V is (dim V0 | dim V1 ). We will use V [1] to denote the superspace V1 ⊕ V0 .
• Given a complex F• or a superspace V , we can define Schur complexes Sλ (F• ) or super analogues
of Schur functors Sλ V . The construction is analogous in both cases, and references for this
construction are [ABW, §V] and [Wey, §2.4]. Skew versions will also be used.
Acknowledgements.
I thank Jerzy Weyman for suggesting this problem. I thank Andrew Snowden and Jerzy Weyman
for carefully reading a draft of this article.
The author was supported by an NDSEG fellowship while this work was done.
1
Lie superalgebras.
We will not give the general definitions of Lie superalgebras. Rather, in this section, we just give
concrete realizations of the superalgebras that will appear in this paper. For general background,
we refer the reader to [Ka1, Ka2].
1.1
General linear Lie superalgebras.
Fix positive integers n, m and consider the space of (n + m) × (n + m) block matrices
A B
gl(n|m) =
,
C D
where A is n × n, B is n × m, C is m × n and D is m × m.
We put a Z-grading on gl(n|m) by
0 0
A 0
gl(n|m)−1 =
,
gl(n|m)0 =
,
C 0
0 D
gl(n|m)1 =
0 B
0 0
and for homogeneous elements X, Y ∈ gl(n|m) of homogeneous degrees deg(X), deg(Y ), we define
a bracket
[X, Y ] = XY − (−1)deg(X) deg(Y ) Y X.
Then gl(n|m) is a Lie superalgebra via the bracket [, ].
We can also define gl(n|m) in a basis-free way. Let V = E ⊕ F be a Z/2-graded vector space
of dimension (n|m). Then gl(n|m) is identified with the space of endomorphisms of V with the
natural Z-grading, and we can write
gl(V )−1 = Hom(F, E),
gl(V )0 = gl(E) × gl(F ),
The Lie bracket is GL(E) × GL(F )-equivariant.
4
gl(V )1 = Hom(E, F ).
1.2
Periplectic superalgebras.
We define the periplectic Lie superalgebra as the subalgebra of gl(n|n) of matrices of the form
A
B
T
T
pe(n) =
B = −B , C = C
.
C −AT
It is straightforward to check that pe(n) is closed under the Lie bracket [, ] and that pe(n) inherits
the Z-grading from gl(n|n).
We can also define pe(n) as the subalgebraof gl(n|n)which preserves the odd skew-symmetric
0
In
bilinear form represented by the matrix ω =
. Let us pause to say what exactly this
−In 0
means since one has to be careful with signs. We define the supertranspose on gl(n|m) by
ST T
A B
A
−C T
=
.
B T DT
C D
Then a homogeneous element X ∈ gl(n|n) preserves ω if
X ST ω + (−1)deg(X) ωX = 0.
This definition is an odd analogue of the definition of the symplectic Lie algebra.
We can also define pe(n) in a basis-free way. First write gl(n|n) as gl(E ⊕ F ) from the previous
section. The bilinear form ω defines an isomorphism E ∼
= F ∗ , and the Z-grading of pe(V ) becomes
pe(V )−1 = S2 E,
pe(V )0 = gl(E),
pe(V )1 =
2
^
E∗.
The Lie bracket is GL(E)-equivariant.
2
Z-graded representations and 2-sided complexes.
Although everything in the language of superalgebras is only Z/2-graded, we have seen that for
the two algebras of interest in this paper, gl(V ) and pe(V ), the Z/2-grading can be lifted to a Zgrading. Here we will develop some notation for thinking about those representations which carry
a compatible Z-grading.
We will reinterpret the Z-grading on a representation as a certain pair of complexes. Before
beginning, let us elaborate on why using complexes is relevant for studying Z-graded representations. For both of our Lie superalgebras g, the bracket of any two elements in g1 is 0 (similarly
for two elements in g−1 ). Explicitly, this means that they anticommute with one another in any
representation. On the level of the universal enveloping algebra, this gives one an action of an
exterior algebra, and this is the object which acts on complexes (via the differential).
The idea behind this section comes from [JPW].
2.1
General linear case.
The vector representation of gl(V ) is V = E ⊕ F with the action of matrix multiplication. We can
view V as a 2-term complex in the following way. First, we have maps
F ⊗ Hom(F, E) → E
E ⊗ Hom(E, F ) → F
5
given by evaluation, and this coincides with the action of gl(V )−1 and gl(V )1 on V .
Now let A = S• (Hom(F, E)∗ ) be the (graded) coordinate ring of the vector space Hom(F, E).
By adjunction, we can rewrite the evaluation map F ⊗ Hom(F, E) → E as F → E ⊗ Hom(F, E)∗ .
We can extend this map A-linearly to get
Φ : F ⊗ A(−1) → E ⊗ A,
If we choose bases for E and F , then Φ can be represented by a matrix of linear forms on Hom(F, E),
and to recover the action of X ∈ Hom(F, E), we simply evaluate Φ(X). Of course, along with the
map Φ, we also have the map
Φ′ : E ⊗ A′ (−1) → F ⊗ A′
where A′ = S• (Hom(E, F )∗ ), and the maps Φ and Φ′ satisfy certain compatibility relations coming
from the fact that they come from the action of the Lie superalgebra gl(V ). We encode this
structure in the next definition.
Definition 2.1. A 2-sided gl(V )-complex is a sequence of gl(E) × gl(F )-modules (Fi )i with
equivariant maps Φi : Fi ⊗ A(−1) → Fi−1 ⊗ A and Φ′i−1 : Fi−1 ⊗ A′ (−1) → Fi ⊗ A′ such that for
all X ∈ Hom(F, E) and Y ∈ Hom(E, F ),
1. Φi+1 (X)Φ′i (Y ) − Φ′i−1 (Y )Φi (X) : Fi → Fi coincides with the action of [X, Y ] ∈ gl(E) × gl(F ),
2. Φi+1 (X)Φi (X) = 0, and
3. Φ′i (Y )Φ′i−1 (Y ) = 0.
Condition 2 implies that for all X, X ′ ∈ Hom(F, E), we have Φi+1 (X)Φi (X ′ ) = −Φi+1 (X ′ )Φi (X)
(apply it to X + X ′ ). Similar remarks for condition 3 hold also.
We put F0 = E and F1 = F for the vector representation.
There is an obvious notion of morphisms between 2-sided gl(V )-complexes and the tensor product of two 2-sided gl(V )-complexes. The following is immediate.
Proposition 2.2. The tensor category of 2-sided gl(V )-complexes is equivalent to the tensor category of Z-graded representations of gl(V ).
The advantage of this viewpoint is that we will be able to compare certain tensor constructions
on Φ with the linear strands of certain free resolutions over the polynomial ring A.
2.2
Periplectic case.
The same kinds of definitions can be made for pe(V ). The vector representation of pe(V ) is V =
E ⊕ E ∗ , again with the action of matrix multiplication. As before, we have evaluation maps
E ∗ ⊗ S2 E → E,
E⊗
2
^
E∗ → E∗,
and this Vcoincides with the action of pe(V )−1 and pe(V )1 on E ⊕ E ∗ . Let B = S• (S2 E ∗ ) and
B ′ = S• ( 2 E). As before, we get maps
Φ : E ∗ ⊗ B(−1) → E ⊗ B,
Φ′ : E ⊗ B ′ (−1) → E ∗ ⊗ B ′ .
6
Definition 2.3. A 2-sided pe(V )-complex is a sequence of gl(E)-modules (Fi )i with equivariant
maps Φi : Fi ⊗ B(−1) → Fi−1 ⊗ B and Φ′i−1 : Fi−1 ⊗ B ′ (−1) → Fi ⊗ B ′ such that for all X ∈ S2 E
V
and Y ∈ 2 E ∗ ,
1. Φi+1 (X)Φ′i (Y ) − Φ′i−1 (Y )Φi (X) : Fi → Fi coincides with the action of [X, Y ] ∈ gl(E),
2. Φi+1 (X)Φi (X) = 0, and
3. Φ′i (Y )Φ′i−1 (Y ) = 0.
We put F0 = E and F1 = E ∗ for the vector representation.
As before, the following is immediate from the definitions.
Proposition 2.4. The tensor category of 2-sided pe(V )-complexes is equivalent to the tensor category of Z-graded representations of pe(V ).
When the context about which algebra we are discussing is clear, we will simply use the phrase
“2-sided complex”.
3
(Skew-)symmetric minors.
V
We continue to use the notation B = S• (S2 E ∗ ) and B ′ = S• ( 2 E) from §2.2.
3.1
JPW complexes.
Choose nonnegative integers s and r so that dim E > s + r. Given a partition α with ℓ(α) ≤ s,
define the partition
Pr,s (α) = (s + α1 , . . . , s + αs , sr , α†1 , . . . , α†α1 ),
(3.1)
which we visualize as follows:
s×s
α
r×s
α†
Set
JPWr,s
i =
M
SPr,s (α) E ∗
(3.2)
|α|=i
which naturally carries an action of gl(E). Up to a homological grading shift, the sequences
JPWr,s
• ⊗ B can be realized as the linear strands of certain minimal free resolutions over the
polynomial ring B. More specifically, when s is even, they appear in the minimal free resolution of
the ideal of (r + 2) × (r + 2) minors of the generic symmetric matrix S2 E [Wey, Theorem 6.3.1(c)].
When s is odd and r is odd, JPWr,s
• ⊗ B is a linear strand in the minimal free resolution of the
module M2 mentioned on [Wey, p. 180]. The case of s odd and r even is not mentioned in [Wey],
but in this case, JPWr,s
• ⊗ B can be realized as the linear strand in the minimal free resolution of
the module obtained from M2 via the localization trick in [Wey, §6.3, part 3]. The construction of
these complexes first appears in [JPW].
7
Remark 3.3. We pause to point out the significance of the module M2 mentioned above. Consider
a vector space W equipped with a (split) orthogonal form, and let SO(W ) and O(W ) be the
special orthogonal and general orthogonal groups. The ring of O(W )-invariant polynomials on
W ⊕N is naturally isomorphic to the coordinate ring R of the determinantal variety of symmetric
N × N matrices with rank at most dim W [LR, Theorem 10.4.0.3]. The ring of SO(W )-invariant
polynomials is a degree 2 extension of R, and as an R-module, it is R ⊕ M2 . This direct sum
decomposition is the eigenspace decomposition of the action of Z/2 ∼
= O(V )/SO(V ).
As a consequence of the above discussion, we get B-linear maps
r,s
Φi : JPWr,s
i ⊗ B(−1) → JPW i−1 ⊗ B.
The main result in this section is that Φ can be completed to a 2-sided complex (Φ, Φ′ ).
Theorem 3.4. There exist B ′ -linear maps
r,s
′
′
Φ′i : JPWr,s
i ⊗ B (−1) → JPW i+1 ⊗ B
so that (Φ, Φ′ ) is a 2-sided pe(V )-complex. In particular, JPWr,s
• affords an action of pe(V ).
r,s
Moreover, JPW• is an irreducible pe(V )-representation.
The proof will be given in §3.3.
Example 3.5. Before handling the general case, we illustrate the case of s = 1.
Set n = dim E and k = n − 1 − r. Start with the vector representation V , which corresponds
to the 2-sided complex
Φ : E ∗ ⊗ B(−1) → E ⊗ B,
Φ′ : E ∗ ⊗ B ′ ← E ⊗ B ′ (−1),
V
and consider the representation W = k V ⊗ det E ∗ , which corresponds to the 2-sided complex
∗
k
∗
det E ⊗ S E ⊗ B(−k) →
det E ∗ ⊗ Sk E ∗ ⊗ B ′ ←
n−1
^
n−1
^
∗
E ⊗S
k−1
∗
E ⊗ B(−k + 1) → · · · →
E ∗ ⊗ Sk−1 E ∗ ⊗ B ′ (−1) ← · · · ←
r+1
^
r+1
^
E ∗ ⊗ B,
E ∗ ⊗ B ′ (−k).
V
So Wi = r+1+i E ∗ ⊗ Si E ∗ . From this description, we can find a pe(V )-subrepresentation of W .
First note that Wi is an irreducible gl(E)-module for i = 0 or i = k. Otherwise we have
Wi = S(i,1r+1+i ) E ∗ ⊕ S(i+1,1r+i ) E ∗
by Pieri’s rule [Wey, Corollary 2.3.5]. Again by Pieri’s rule, we see that S(i,1r+i+1 ) E ∗ is not a direct
summand of S(i,1r+i−1 ) E ∗ ⊗ S2 E ∗ . Since Φ and Φ′ are gl(E)-equivariant, we see that in the map
Φ : Wi → Wi−1 ⊗ S2 E ∗ , the summand S(i,1r+1+i ) E ∗ can only map to S(i−1,1r+i ) E ∗ . Similarly, in the
map Φ′ : Wi → Wi+1 , the summand S(i,1r+1+i ) E ∗ can only map to S(i+1,1r+2+i ) E ∗ .
So we get a subrepresentation W ′ ⊂ W given by Wi′ = S(i,1r+1+i ) E ∗ . We also see that W/W ′ =
r,1
JPWr,1
• , so we deduce that JPW • can be given the structure of a 2-sided complex.
Remark 3.6. The quotient W → JPWr,1
• → 0 from Example 3.5 can be extended to a long exact
sequence
k−4
^
k−2
^
k
^
V ⊗ det E ∗ → JPWr,1
• →0
V
where the differentials are induced by the trace map K[1] → 2 V , which is defined in Proposition 3.7.
··· → (
V )[2] ⊗ det E ∗ → (
V )[1] ⊗ det E ∗ →
8
3.2
Trace and evaluation.
V
Proposition 3.7. We have nonzero pe(V )-equivariant maps K[1] → 2 V and S2 V → K[1],
where K denotes the trivial 1-dimensional module. We call these maps trace and evaluation,
respectively.
These maps also appeared in [JPW, §4].
Proof. In terms of 2-sided complexes, we can write
2
∗
V2
V as
∗
S E ⊗ B(−2) → E ⊗ E ⊗ B(−1) →
S2 E ∗ ⊗ B ′ ← E ⊗ E ∗ ⊗ B ′ (−1) ←
2
^
2
^
E ⊗ B,
E ⊗ B ′ (−2).
∗
There is an invariant
we see that
V2 K ⊂ E ⊗ E . Since the complexes above• are2 GL(E)-equivariant,
∗
E ⊗ B in the first complex since B = S (S E ). Similarly, K maps to 0 in
K maps to 0 in
S2 E ∗ ⊗ B ′ . So K[1] is a subcomplex ofVboth of the above complexes, and this shows the existence
of the pe(V )-equivariant map K[1] → 2 V .
The existence of the map S2 V → K[1] is similar and is obtained by showing that K[1] is a
quotient complex of the corresponding 2-sided complexes.
We can use the trace and evaluation maps to define a larger class of nonzero pe(V )-equivariant
morphisms, which we will need later.
Proposition 3.8. Let λ be a partition. There are pe(V )-equivariant morphisms
(Sλ/(1,1) V )[1] → Sλ V,
Sλ V → (Sλ/(2) V )[1]
which are nonzero in homological degree 0.
Proof. By [Wey, §2.4], we can define Sα/β V as a quotient of
^
Consider the diagram
†
α†α1 −βα
1
α†1 −β1†
α/β
V :=
^
^
V ⊗ ··· ⊗
V
( λ/(1,1) V )[1]
/
Vλ
V.
V
(Sλ/(1,1) V )[1]
Sλ V
where the horizontal map is
λ†1 −2
^
trace
V ⊗ K[1] −−−→
λ†1 −2
^
V ⊗
2
^
†
m
V −→
λ1
^
V
V ◦
(m is the multiplication map) tensored with the identity on λ where λ◦ is the diagram of λ with
the first column removed. We claim that this horizontal map descends to a map which completes
the diagram. By [Wey, §2.4], the kernels of the vertical maps (for general α/β) are spanned by the
subspaces
†
α†α1 −βα
1
α†1 −β1†
^
V ⊗ · · · ⊗ Ra,a+1 V ⊗ · · · ⊗
9
^
V
(1 ≤ a ≤ α1 − 1)
†
V † †
V †
where Ra,a+1 V ⊂ αa −βa V ⊗ αa+1 −βa+1 V is defined as the span of the images of the maps, for
all u + v < α†a+1 − βa† ,
u
^
†
α†a −βa† −u+α†a+1 −βa+1
−v
^
V ⊗
u
^
V ⊗
†
α†a −β
^a −u
V
1⊗∆⊗1
†
α†a+1 −βa+1
−v
^
V ⊗
α†a^
−βa†
V ⊗
v
^
V ⊗
v
^
V
m12 ⊗m34
†
α†a+1 −βa+1
^
V ⊗
V.
Here ∆ is comultiplication, and mij denotes multiplication on the ith and jth factor. So we just
V
V
have to check that each such subspace in λ/(1,1) V is mapped to another such subspace in λ V .
These relations only involve two consecutive columns, and ν and λ have the same columns except
for the first one, so we only need to check the claim for a = 1. In this case, it becomes a matter of
verifying that the following diagram commutes (mij is multiplication on the ith and jth factors)
u
^
†
†
^
λ1 −2−u+λ2 −v
V ⊗
V ⊗
u
^
V ⊗
†
λ2 −v
V ⊗
†
λ1 −2
^
V [1]
trace
/
u
^
†
V ⊗
†
^
V ⊗
v
^
V [1]
trace
/
u
^
V ⊗
^
V ⊗
V [1]
/
^
V ⊗
V ⊗
v
^
m23
V
/
u
^
†
^
^
V ⊗
2
^
V ⊗
V ⊗
v
^
V
m24
/
u
^
2
^
V ⊗
^
V ⊗
^
V
^
†
^
V ⊗
V ⊗
m12 ⊗m34
†
V ⊗
λ2
^
V
The commutativity of the left-hand side squares is clear since the maps do not interact in a nontrivial way. The commutativity of the top-right square follows from the compatibility between
multiplication and comultiplication in a bialgebra, and the commutativity of the bottom-right
square follows from associativity of multiplication.
Dually, recall that we can also define Sλ V as a submodule of Sλ V := Sλ1 V ⊗ · · · ⊗ Sλℓ(λ) V .
Consider the diagram
Sλ/(2) V [1]
Sλ V
Sλ V
/ Sλ/(2) V [1]
where the bottom horizontal map is induced by the evaluation map S2 V → K[1]. We claim that
this descends to a map which completes the diagram. This diagram is dual to one using the Weyl
functor presentation for V ∗ instead of V (see [Wey, §2.1]; the definition of Weyl functor can easily
be extended to “Weyl complex”). The relations defining Weyl functors are analogous to the ones
defining the Schur functors. So the proof that the map descends reduces to the commutativity of
a similar 9-term diagram. We only need that multiplication and comultiplication make the divided
power algebra into a bialgebra.
10
V
1⊗∆⊗1
†
λ1
/
v
^
λ2 −v
m12
V ⊗
†
λ1 −u
m12 ⊗m35
†
λ2
†
λ1 −u+λ2 −v
1⊗∆⊗1⊗1
†
λ1 −2
trace
2
^
†
λ2 −v
m12 ⊗m34
^
V ⊗
†
λ1 −2−u
†
λ2
V ⊗
^
λ1 −2−u+λ2 −v
1⊗∆⊗1
†
λ1 −2−u
^
v
^
v
^
V
Example 3.9. When λ = (2, 1), the composition of trace and evaluation is an isomorphism. If
{e1 , . . . , en } is a basis for E, then the trace map is
ej 7→
n
n
X
X
(e∗i ⊗ ei ) ⊗ ej .
(ei ⊗ e∗i ) ⊗ ej +
i=1
i=1
The evaluation map pairs the first and third tensor factors, so the first sum becomes 0 and the
second sum becomes ej . Similarly, the composition maps all dual basis vectors to themselves. As
a consequence, V [1] is a pe-equivariant direct summand of S2,1 V .
We make the following definitions:
kλ (V ) = ker(Sλ V → (Sλ/(2) V )[1]),
iλ (V ) = image((Sλ/(1,1) V )[1] → Sλ V ),
(3.10)
S[λ] V = kλ (V )/(kλ (V ) ∩ iλ (V )).
3.3
Existence of representation structure.
Recall the definition of Pr,s (α) from (3.1). Consider the partition Pr,s (∅) = (ss+r ). Then we have
Sλ E = S(ss+r ) E ∗ ⊗ (det E)s where
λ = (sdim E−s−r )
[Wey, Exercise 2.18]. We will repeatedly use the fact that if µ = (ab ) is a rectangular shape, and
ν ⊆ µ, then Sµ/ν = Sη where η = (a − νb , a − νb−1 , . . . , a − ν1 ). This follows by showing that they
have the same character using semistandard Young tableaux [Wey, Proposition 2.1.15].
We will also make use of the following simple consequence of the Littlewood–Richardson rule
[Wey, Theorem 2.3.4]: if η 1 , . . . , η d are partitions, then Sη1 +···+ηd U appears with multiplicity 1 in
Sη1 U ⊗ · · · ⊗ Sηd U . We define this to be the Cartan product.
Recall that B = S• (S2 E ∗ ). We will use the 2-sided complex interpretation of Sλ V and S[λ] V to
treat them as complexes over B. We define
e λ V = Sλ V ⊗ (det E ∗ )s ,
S
e [λ] V = S[λ] V ⊗ (det E ∗ )s ,
S
e [λ] V )0 = (S
e λ V )0 = S(ss+r ) E ∗ ⊗ B. We use (Φ, Φ′ ) to denote the 2-sided complex structure
so that (S
e λ V and S
e [λ] V .
on S
e λ V )|α| and
Lemma 3.11. If s + r + α1 ≤ dim E, then SPr,s (α) E ∗ appears with multiplicity 1 in (S
e
also in (S[λ] V )|α| .
Proof. The inequality just means that SPr,s (α) E ∗ 6= 0. L
The ith term of the Schur complex Sµ (W0 ⊕ W1 ) is ν⊆µ, |ν|=i Sµ/ν W0 ⊗ Sν † W1 [Wey, Theorem
2.4.5]. When µ = λ = (sdim E−s−r ) and W0 = E, we see from the above discussion that Sλ/ν E ∼
=
S(ss+r ,ν) E ∗ .
A simple consequence of the Littlewood–Richardson rule [Wey, Theorem 2.3.4] is that if Sη U
appears in Sη′ U ⊗ Sη′′ U , then η ′ ⊆ η and η ′′ ⊆ η. In particular, if we choose ν of size |α| as
above, we can only get SPr,s (α) E ∗ ⊂ Sλ/ν E ⊗ Sν † E ∗ if ν = α† . Taking the Cartan product of
e λ V )|α| with multiplicity 1.
Sλ/α† E = S(sr+s ,α† ) E ∗ and Sα E ∗ gives that SPr,s (α) E ∗ appears in (S
∗
We also see that SPr,s (α) E does not appear in either Sλ/(2) V nor Sλ/(12 ) V . This shows that
e [λ] V )|α| with multiplicity 1.
SPr,s (α) E ∗ also appears in (S
11
Lemma 3.12. Pick α with ℓ(α) ≤ s, and pick β ⊂ α such that |α| − 1 = |β|. Then SPr,s (β) E ∗
e λ V )|β| . Similarly, SP (α) E ∗ is in the image of
is in the image of Φ : SPr,s (α) E ∗ ⊗ S2 E → (S
r,s
V
e λ V )|α| .
Φ′ : SPr,s (β) E ∗ ⊗ 2 E ∗ → (S
Proof. We only
about Φ since the proof for Φ′ is similar.
Vrprove the statement
e λ V is Wr . It was analyzed in
V ⊗ det E ∗ . For the case s = 1, the Schur complex S
Set Wr =
Example 3.5, and the lemma holds by inspection in this case.
e λ V [Wey, §2.4]. The first s column
For the general case, consider the quotient map π : Wr⊗s → S
lengths of Pr,s (α) are c1 := r + s + α1 , . . . , cs := r + s + αs . By [Wey, Exercise 2.18], we have
dim^
E−ci
αi
∗
E⊗S E =
ci
^
E ∗ ⊗ Sαi E ∗ ,
which is naturally a subspace of (Wr )V
αi [Wey, Theorem 2.4.5].
e λ V . To see this, first
We claim that the product of the ci E ∗ ⊗ Sαi E ∗ generates SPr,s (α) E ∗ in S
∗
replace V1 = E Vwith an arbitrary vector space F . Then repeating
we are considering
Vci ∗ the above,
ci ∗
∗
α
i
E is Sµ E where µ consists of the
E ⊗ S F . The Cartan product of the
the product of
e λ V which contains a
first s columns of Pr,s (α). Also, Sµ E ∗ ⊗ Sα F is the unique summand in S
∗
Sµ E -isotypic component, so this must be in the image of π. Finally, note that SPr,s (α) E ∗ is the
Cartan product of Sµ E ∗ and Sα E ∗ . This proves the claim.
To go from α to β, we decrease ci by 1, where i is the unique column index in which α and β
differ. Consider the map on Wr⊗s which is Φ on the ith factor and the identity elsewhere. Descending
e λ V , we see that SP (β) E ∗ is in the image of Φ restricted to SP (α) E ∗ ⊗ S2 E ∗ .
this map to S
r,s
r,s
Proof of Theorem 3.4. In §3.1, we discussed that the sequences JPWr,s
• ⊗ B can be given the
structure of linearly exact B-complexes. Let JPWr,s
denote
this
B-complex.
We will construct a
•
e [λ] V → JPWr,s . The presentation for H0 (JPWr,s ) is
map of complexes S
•
•
S(s+1,ss−1+r ,1) E ∗ ⊗ B(−1) → S(ss+r ) E ∗ ⊗ B.
Recall the definitions from (3.10). By Proposition 3.8, we have
e λ V )1 = S(s+1,ss−1+r ,1) E ∗ ⊕ S(ss+r ,2) E ∗ ⊕ S(ss+r ,1,1) E ∗ ,
(S
(kλ V ⊗ (det E ∗ )s )1 = S(s+1,ss−1+r ,1) E ∗ ⊕ S(ss+r ,1,1) E ∗ ,
(iλ V ⊗ (det E ∗ )s )1 = S(s+1,ss−1+r ,1) E ∗ ⊕ S(ss+r ,2) E ∗ ,
e [λ] V )1 = S(s+1,ss−1+r ,1) E ∗ .
(S
e [λ] V )1 → (S
e [λ] V )0 are unique up to a choice of scalar, so it only matters
The possible maps (S
e [λ] V ) → H0 (JPWr,s ).
if it is zero or nonzero. In either case, we have a natural surjection f−1 : H0 (S
•
r,s
r,s
e
e
For notation, set (S[λ] V )−1 = H0 (S[λ] V ) and JPW−1 = H0 (JPW• ). We will construct maps
e [λ] V → JPWr,s by induction on i satisfying
fi : S
−1
• fi is gl(E)-equivariant,
• fi is surjective,
• the fj for j ≤ i form a morphism f≤i of the truncated complexes,
• for all x ∈ ker fi−1 and Y ∈ pe(V )1 , we have Φ′ (Y )(x) ∈ ker fi
These conditions hold for i = −1, which handles the base case of our induction. Assuming that
we have constructed fi , we show how to construct fi+1 . First pick x ∈ ker fi and Y ∈ pe(V )1 . Since
ker fi is gl(E)-equivariant and
[Φ(X), Φ′ (Y )] = Φ(X)Φ′ (Y ) + Φ′ (Y )Φ(X) ∈ pe(V )0 = gl(E)
12
for all X ∈ pe(V )−1 , we have
(Φ(X)Φ′ (Y ) + Φ′ (Y )Φ(X))(x) ∈ ker fi .
Since f≤i is a morphism of complexes, Φ(X)(x) ∈ ker fi−1 , which implies that Φ′ (Y )Φ(X)(x) ∈
ker fi by our conditions on f≤i . In particular, we also have Φ(X)Φ′ (Y )(x) ∈ ker fi . Hence the
subspace
Ui+1 = {Φ′ (Y )(x) | Y ∈ pe(V )1 , x ∈ ker fi }
is sent to 0 under the composition
fi
Φ e
e [λ] V )i+1 −
(S
→ (S
→ JPWr,s
[λ] V )i −
i .
Again since f≤i is a morphism of complexes, the image of this map is in the kernel of the differential
r,s
JPWr,s
i → JPWi−1 . More specifically, the generators map to the degree 1 piece of this kernel. Since
r,s
JPW• is linearly exact, we can find a lift
e [λ] V )i+1 /Ui+1 → JPWr,s .
(S
i+1
Since everything above is gl(E)-equivariant, we can choose this lift to also be gl(E)-equivariant by
e [λ] V )i+1 → JPWr,s by composing with
invoking the semisimplicity of gl(E). We define fi+1 : (S
i+1
the quotient map. Using Lemma 3.11, Lemma 3.12, and the explicit definition (3.2), we deduce
that fi+1 is surjective from the fact that fi is surjective. The other conditions on fi+1 follow by
construction.
So we have constructed a surjective B-degree 0 map of complexes
e [λ] V → JPWr,s .
f: S
•
(3.13)
e [λ] V , we conclude that JPWr,s has the structure of a
Since ker f is a pe(V )-subrepresentation of S
•
2-sided complex, and hence that JPWr,s
is
a
representation
of
pe(V
).
All
that remains is to show
•
that this action makes JPWr,s
an
irreducible
representation.
So
let
F
be
any
nonzero submodule
•
•
r,s
of JPW• and consider its preimage under f . Using Lemma 3.12, we conclude that this preimage
contains all SPr,s (α) E ∗ , so the same is true for F• , and hence F• = JPWr,s
• .
Conjecture 3.14. The map f from (3.13) is an isomorphism.
Remark 3.15. For r odd, we have linear maps
JPWr,s
i
⊗
2
^
E → JPW r,s
i−1
which come from the minimal free resolutions of Pfaffian ideals [Wey, §6.4]. If we had defined the
periplectic superalgebra as the stabilizer of an odd symmetric bilinear form rather than a skewsymmetric one in §1.2, we would get an isomorphic algebra with the Z-grading reversed (and the
roles of E and E ∗ swapped). We could have used this other direction to define an action of pe(V )
on JPWr,s
• .
4
Generic minors.
Recall that we defined A = S• (Hom(F, E)∗ ) and A′ = S• (Hom(E, F )∗ ) in §2.1.
13
4.1
Lascoux complexes.
Choose integers s ≥ 0 and r > 0. Given partition α, β with ℓ(α) ≤ s and β1 ≤ s, define the
partitions
Pr,s (α, β) = (s + α1 , . . . , s + αs , sr , β1 , . . . , βℓ(β) ),
Qr,s (α, β) = (s + β1† , . . . , s + βs† , sr , α†1 , . . . , α†α1 ),
(4.1)
which we visualize as follows:
s×s
α
s×s
r×s
β†
r×s
β
α†
Set
Lar,s
i =
M
SPr,s (α,β) E ∗ ⊗ SQr,s (α,β) F
(4.2)
|α|+|β|=i
which naturally carries an action of gl(E) × gl(F ). Up to a homological grading shift, the sequences
Lar,s
• ⊗ A can be realized as the linear strands of the ideal of (r + 1) × (r + 1) minors of the generic
matrix Hom(F, E) [Wey, §6.1] (we disallowed the case r = 0 because it corresponds to the Koszul
complex, which is a degenerate case). This definition of this complex was given by Lascoux [Las].
As a consequence, we get A-linear maps
r,s
Φi : Lar,s
i ⊗ A(−1) → Lai−1 ⊗ A.
The main result in this section is that Φ can be completed to a 2-sided complex (Φ, Φ′ ).
Theorem 4.3. There exist A′ -linear maps
r,s
′
′
Φ′i : Lar,s
i ⊗ A (−1) → Lai+1 ⊗ A
so that (Φ, Φ′ ) is a 2-sided gl(V )-complex. In particular, Lar,s
• affords an action of gl(E). Moreover,
r,s
La• is an irreducible gl(V )-representation.
The proof will be given in §4.3.
4.2
Trace and evaluation.
Let e1 , . . . , en be a basis for E and let f1 , . . . , fm be a basis for F . The degree 0 piece of V ⊗ V ∗ is
(E ⊗ E ∗ ) ⊕ (F ⊗ F ∗ ), and we define the trace map
K → (E ⊗ E ∗ ) ⊕ (F ⊗ F ∗ )
X
X
α 7→ α
ei ⊗ e∗i − α
fj ⊗ fj∗ .
i
j
14
We also define the evaluation map
X
i,j
(E ⊗ E ∗ ) ⊕ (F ⊗ F ∗ ) → K
X
ai,j ei ⊗ e∗j +
bk,ℓ fk ⊗ fℓ∗ 7→ a1,1 + · · · + an,n + b1,1 + · · · + bm,m .
k,ℓ
These maps also appeared in [PW].
Proposition 4.4. Trace and evaluation give nonzero gl(V )-equivariant maps K → V ⊗ V ∗ and
V ⊗ V ∗ → K, where K denotes the trivial 1-dimensional module.
Proof. It is straightforward to check that trace is gl(E) × gl(F )-equivariant and that the image is
in the kernel of the map
1⊗Φ(X)+Φ(X)∗ ⊗1
(E ⊗ E ∗ ) ⊕ (F ⊗ F ∗ ) −−−−−−−−−−−−→ (E ⊗ F ∗ )
for any X ∈ Hom(F, E), and is in the kernel of the map
Φ′ (Y )⊗1+1⊗Φ′ (Y )∗
(E ⊗ E ∗ ) ⊕ (F ⊗ F ∗ ) −−−−−−−−−−−−→ F ⊗ E ∗
for any Y ∈ Hom(E, F ). Using the 2-sided complex interpretation, this shows that the image of K
is a gl(V )-submodule of V ⊗ V ∗ .
The analysis of the evaluation map is similar.
Given two partitions λ and µ, we can define maps
(Sλ/1 V ⊗ Sµ/1 (V ∗ [1]))[1] → Sλ V ⊗ Sµ (V ∗ [1]) → (Sλ/1 V ⊗ Sµ/1 (V ∗ [1]))[1]
(4.5)
which are induced by trace and evaluation. The construction is analogous to the one in §3.2.
Proposition 4.6. If dim E − λ†1 + λ1 = dim F − µ†1 + µ1 , then the composition (4.5) is 0 in
homological degree 1.
Example 4.7. When λ = µ = (1), this is saying that the composition of trace and evaluation is 0
when dim E = dim F , which is easily seen.
Proof. Using the standard basis of [Wey, Proposition 2.1.15(b)], an element in homological degree
1 in the left hand side of (4.5) is a sum of pairs of tableaux (Te , Tf ) of shapes (λ/1, µ/1) and whose
∗ }, respectively. So fix such a pair. Now we apply
entries are filled with {e1 , . . . , en } and {f1∗ , . . . , fm
the map (4.5).
The trace map says to sum over the tableaux we get by inserting ei ⊗ e∗i and −fj ⊗ fj∗ into the
empty boxes of (Te , Tf ). When we insert v into the empty box of Te , we denote it by vTe (same for
Tf ). So we can write the trace map as
X
X
(Te , Tf ) 7→
(ei Te , e∗i Tf ) −
(fj Te , fj∗ Tf )
i
j
The next part of the map tells us to antisymmetrize all columns. In detail, for each pair of
permutations (σ, ρ) of the boxes of λ and µ, we sum over those which preserve the columns, and
multiply by the sign of the permutation. In symbols:
X
(ei Te , e∗i Tf ) 7→
sgn(σ)sgn(ρ)(σ · ei Te , ρ · e∗i Tf ).
σ,ρ
15
Fix a term in this sum, we will show that its image is 0.
To each term in this sum, we symmetrize rows (i.e., interpret them as monomials in symmetric
powers) and then pick one element from the first row of both tableaux in all possible ways and
evaluate them against one another. When we do this to (ei Te , e∗i Tf ), we can only get a nonzero result
if we pick e∗i in the first row of e∗i Tf (this might not be possible depending on the signed term we
picked from the antisymmetrization). When we pick this e∗i , then the sum of all possible evaluations
is 1 plus the number of times that ei appears in the first row of the fixed antisymmetrization of
Te . The only i that can contribute are those such that ei does not appear in the first column of Te
(otherwise ei Te would have two instances of ei in the same column and be identically 0). So summing
over all i such that i is not in the first column of Te , we get (dim E−λ†1 +1)+(λ1 −1) = dim E−λ†1 +λ1
contributions. Similarly, considering (fj Te , fj∗ Tf ), we get dim F −µ†1 +µ1 contributions, each having
coefficient −1. So the total coefficient is 0.
We make the following definitions:
kλ;µ (V ) = ker(Sλ V ⊗ Sµ (V ∗ [1]) → (Sλ/1 V ⊗ Sµ/1 (V ∗ [1]))[1])
iλ;µ (V ) = image((Sλ/1 V ⊗ Sµ/1 (V ∗ [1]))[1] → Sλ V ⊗ Sµ (V ∗ [1]))
(4.8)
S[λ;µ] V = kλ;µ (V )/(kλ;µ (V ) ∩ iλ;µ (V )).
4.3
Existence of representation structure.
Proof of Theorem 4.3. Recall the definitions of Pr,s (α, β) and Qr,s (α, β) from (4.1). Take λ =
(sdim E−r−s ) and µ = (sdim F −r−s ). Then
Sλ E = SPr,s (∅,∅) E ∗ ⊗ (det E)s
Sµ F ∗ = SQr,s (∅,∅) F ⊗ (det F ∗ )s
[Wey, Exercise 2.18]. In §4.1, we discussed that Lar,s
• ⊗ A can be realized as a linearly exact Ar,s
linear complex, which we denote by La• . We now interpret Sλ V , Sµ (V ∗ [1]), and S[λ;µ] V as 2-sided
complexes. The first two terms of Sλ V ⊗ (det E ∗ )s are (write (α; β) in place of Sα E ∗ ⊗ Sβ F )
((ss+r , 1); 1) → (ss+r ; ∅)
Similarly, the first two terms of Sµ (V ∗ [1]) ⊗ (det F )s are
(1; (ss+r , 1)) → (∅; ss+r )
So the first two terms of Sλ V ⊗ Sµ (V ∗ [1]) ⊗ (det E ∗ )s ⊗ (det F )s are
((ss+r , 1); (s + 1, ss+r−1 ))⊕
((ss+r , 1); (ss+r , 1))⊕
→ (ss+r ; ss+r )
((ss+r , 1); (ss+r , 1))⊕
((s + 1, ss+r−1 ); (ss+r , 1))
Also, ((ss+r , 1); (ss+r , 1)) is the 0th term of Sλ/1 V ⊗ Sµ/1 (V ∗ [1]) ⊗ (det E ∗ )s ⊗ (det F )s . Our choice
of λ and µ satisfies Proposition 4.6, so neither instance of ((ss+r , 1); (ss+r , 1)) remains when we
e [λ;µ] V := S[λ;µ] V ⊗ (det E ∗ )s ⊗ (det F )s .
pass to S
Hence we get a surjection
e [λ;µ] V ) → H0 (Lar,s
H0 (S
• ).
The rest of the proof is essentially the same as the proof of Theorem 3.4, so we omit the details.
16
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Steven V Sam, Department of Mathematics, University of California, Berkeley, CA
svs@math.berkeley.edu, http://math.berkeley.edu/~svs/
17
| 0 |
CSVideoNet: A Real-time End-to-end Learning Framework for
High-frame-rate Video Compressive Sensing
arXiv:1612.05203v5 [cs.CV] 28 Jan 2018
Kai XU
Fengbo Ren
School of Computing, Informatics, and Decision Systems Engineering
Arizona State University, Tempe AZ 85281
{kaixu, renfengbo}@asu.edu
Abstract
This paper addresses the real-time encoding-decoding
problem for high-frame-rate video compressive sensing (CS).
Unlike prior works that perform reconstruction using iterative optimization-based approaches, we propose a noniterative model, named “CSVideoNet”, which directly learns
the inverse mapping of CS and reconstructs the original input in a single forward propagation. To overcome the limitations of existing CS cameras, we propose a multi-rate CNN
and a synthesizing RNN to improve the trade-off between
compression ratio (CR) and spatial-temporal resolution of
the reconstructed videos. The experiment results demonstrate
that CSVideoNet significantly outperforms state-of-the-art
approaches. Without any pre/post-processing, we achieve a
25dB Peak signal-to-noise ratio (PSNR) recovery quality at
100x CR, with a frame rate of 125 fps on a Titan X GPU.
Due to the feedforward and high-data-concurrency natures
of CSVideoNet, it can take advantage of GPU acceleration
to achieve three orders of magnitude speed-up over conventional iterative-based approaches. We share the source code
at https://github.com/PSCLab-ASU/CSVideoNet.
1. Introduction
High-frame-rate cameras are capable of capturing
videos at frame rates over 100 frames per second (fps).
These devices were originally developed for research purposes, e.g., to characterize events that occur at a rate that
traditional cameras are incapable of recording in physical and biological science. Some high-frame-rate cameras,
such as Photron SA1, SA3, are capable of recording high
resolution still images of ephemeral events such as a supersonic flying bullet or an exploding balloon with negligible motion blur and image distortion artifacts. However,
due to the complex sensor hardware designed for high sampling frequency, these types of equipment are extremely
expensive (over tens of thousand dollars for one camera).
The high cost limits the field of their applications. Furthermore, the high transmission bandwidth and the large storage space associated with the high frame rate challenges
the manufacture of affordable consumer devices. For example, true high-definition-resolution (1080p) video cameras
at a frame rate of 10k fps can generate about 500 GB data
per second, which imposes significant challenges on existing transmission and storage techniques. Also, the high
throughput raises energy efficiency a big concern. For example, “GoPro 5” can capture videos at 120 fps with 1080p
resolution. However, the short battery life (1-2 hours) has
significantly narrowed their practical applications.
Traditional video encoder, e.g., H.264/MPEG-4, is composed of motion estimation, frequency transform, quantization, and entropy coding modules. From both speed and
cost perspectives, the complicated structure makes these
video encoder unsuitable for high-frame-rate video cameras. Alternatively, compressive sensing (CS) is a much
more hardware-friendly acquisition technique that allows
video capture with a sub-Nyquist sampling rate. The advent of CS has led to the emergence of new image devices,
e.g., single-pixel cameras [6]. CS has also been applied
in many practical applications, e.g., accelerating magnetic
resonance imaging (MRI) [13]. While traditional signal
acquisition methods follow a sample-then-compress procedure, CS could perform compression along with sampling. The novel acquisition strategy has enabled lowcost on-sensor data compression, relieving the pain for
high transmission bandwidth and large storage space. In
the recent decade, many algorithms have been proposed
[3, 16, 1, 4, 22, 2, 12] to solve the CS reconstruction problem.
Generally, these reconstruction algorithms are based on either optimization or greedy approaches using signal sparsity as prior knowledge. As a result, they all suffer from
high computational complexity, which requires seconds to
minutes to recover an image depending on the resolution.
Therefore, these sparsity-based methods cannot satisfy the
real-time decoding need of high-frame-rate cameras, and
they are not appropriate for the high-frame-rate video CS
y = Φf
f = Ψθ
Synthesize DL
y
f
θ
Analysis DL
Compressed
Domain
?
Signal
Domain
θ = Ωf
Coefficient
Domain
Figure 1: Illustration of domain transformations in CS. This
work bridges the gap between compressed and signal domains.
application.
The slow reconstruction speed of conventional CS approaches motivates us to directly model the inverse mapping from the compressed domain to original domain,
which is shown in Figure 1. Usually, this mapping is extremely complicated and difficult to model. However, the
existence of massive unlabeled video data gives a chance
to learn such a mapping using data-driven methods. In
this paper, we design an enhanced Recurrent convolutional
neural network (RCNN) to solve this problem. RCNN has
shown astonishingly good performance for video recognition and description [5, 23, 25, 21]. However, conventional
RCNNs are not well suited for video CS application, since
they are mostly designed to extract discriminant features
for classification related tasks. Simultaneously improving compression ratio (CR) and preserving visual details
for high-fidelity reconstruction is a more challenging task.
To solve this problem, we develop a special RCNN, called
“CSVideoNet”, to extract spatial-temporal features, including background, object details, and motions, to significantly improve the compression ratio and recovery quality
trade-off for video CS application over existing approaches.
The contributions of this paper are summarized as follows:
• We propose an end-to-end and data-driven framework for video CS. The proposed network directly learns the inverse mapping from the compressed videos to the original input without additional
pre/post-processing. To the best of our knowledge,
there has been no published work that addresses this
problem using similar methods.
• We propose a multi-level compression strategy to improve CR with the preservation of high-quality spatial resolution. Besides, we perform implicit motion
estimation to improve temporal resolution. By combining both spatial and temporal features, we further
improve the compression ratio and recovery quality trade-off without increasing much computational
complexity.
• We demonstrate CSVideoNet outperforms the reference approaches not only in recovery quality but also
in reconstruction speed because of its non-iterative
nature. It enables real-time high-fidelity reconstruction for high-frame-rate videos at high CRs. We
achieve state-of-the-art performance on the largescale video dataset UCF-101. Specifically, CSVideoNet
reconstructs videos at 125 fps on a Titan X GPU and
achieves 25dB PSNR at a 100x CR.
2. Related work
There have been many recovery algorithms proposed
for CS reconstruction, which can be categorized as follows:
Conventional Model-based CS Recovery: In [18],
the authors model the evolution of scenes as a linear dynamical system (LDS). This model comprises two submodels: the first is an observation model that models
frames of video lying on a low-dimensional subspace; the
second predicts the smoothly varied trajectory. The model
performs well in stationary scenes, however, inadequate
for non-stationary scenes.
In [27], the authors use Gaussian mixture model (GMM)
to recover high-frame-rate videos, and the reconstruction
can be efficiently computed as an analytical solution. The
hallmark of the algorithm is that it adapts temporal compression rate based upon the complexity of the scene. The
parameters in GMM are trained off-line and tuned during
the recovery process.
In [19], the authors propose a multi-scale video recovery framework. It first obtains a low-resolution video preview with very low computational complexity, and then
it exploits motion estimates to recover the full-resolution
video by solving an optimization problem. In a similar
work [8], the authors propose a motion-compensated and
block-based CS reconstruction algorithm with smooth projected Landweber (MC-BCS-SPL). The motion vector is estimated from a reference and a reconstructed frame. The
reconstructed video is derived from the combination of the
low-resolution video and the estimated motion vector. The
drawback of the two work is the requirement of specifying the resolution at which the preview frame is recovered, which requires prior knowledge of the object speed.
Also, the recovery performance is highly dependent on the
quality of motion estimation. To accurately estimate motion vector is a challenging task especially in high-framerate scenarios. The high computational cost further makes
this model inadequate for reconstructing high-frame-rate
videos.
Deep Neural Network (DNN) Based CS Recovery:
In [15], the authors propose a stacked autoencoder to learn
a representation of the training data and to recover test
data from their sub-sampled measurements. Compared to
the conventional iterative approaches, which usually need
hundreds of iterations to converge, the feed-forward deep
neural network runs much faster in the inference stage.
In [11], the authors propose a convolutional neural network, which takes CS measurements of an image as input
and outputs an intermediate reconstruction. The intermediate output is fed into an off-the-shelf denoiser to obtain
the final reconstructed image. The author shows the network is highly robust to sensor noise and can recover visually higher quality images than competitive algorithms
at low CRs of 10 and 25. Both [15] and [11] are designed
for image reconstruction, which only focus on spatial feature extraction. For video applications, temporal features
between adjacent frames are also important. Therefore, the
overlook of temporal correlation makes the image reconstruction algorithms inadequate for video applications.
In [9], the authors propose a Video CS reconstruction
algorithm based on a fully-connected neural network. This
work focuses on temporal CS where multiplexing occurs
across the time dimension. A 3D volume is reconstructed
from 2D measurements by a feed-forward process. The
author claims the reconstruction time for each frame can
be reduced to about one second. The major drawback of
this work is that the algorithm is based on a plain fullyconnected neural network, which is not efficient in extracting temporal features.
3. Methodology
3.1. Overview of the proposed framework for video
CS
Two kinds of CS cameras are being used today. Spatial multiplexing cameras (SMC) take significantly fewer
measurements than the number of pixels in the scene to
be recovered. SMC has low spatial resolution and seeks to
spatially super-resolve videos. In contrast, temporal multiplexing cameras (TMC) have a high spatial resolution but
low frame-rate sensors. Due to the missing of inter frames,
extra computation is needed for motion estimation. For
these two sensing systems, either spatial or temporal resolution is sacrificed for achieving a better spatial-temporal
trade-off. To solve this problem, we propose a new sensing
and reconstruction framework, which combines the advantage of the two systems. The random video measurements
are collected by SMC with very high temporal resolution.
To compensate for the low spatial resolution problem in
SMC, we propose a multi-CR strategy. The first key frame
in a group of pictures (GOP) is compressed with a low CR,
and the remaining non-key frames are compressed with a
high CR. The spatial features in the key frame are reused
for the recovery of the entire GOP due to the high interframe correlation in high-frame-rate videos. The spatial
resolution is hence improved. The RNN extrapolates motion from high-resolution frames and uses it to improve the
temporal resolution. Therefore, a better compression ratio
and spatial-temporal resolution trade-off are obtained by
the proposed framework.
The overall architecture of the proposed video CS reconstruction framework is shown in Figure 2. The network contains three modules: 1) an encoder (sensing matrix) for simultaneous sampling and compression; 2) a dedicated CNN for spatial features extraction after each compressed frame; 3) an LSTM for motion estimation and video
reconstruction. As mentioned earlier, to improve the spatial resolution, the random encoder encodes the key frame
in a GOP with more measurements and the remaining with
less. Also, a recent research [26] shows that sensing matrix can be trained with raw data to better preserve the
Restricted Isometry Property (RIP). Therefore, the encoder
can also be integrated into the entire model and trained
with the whole network to improve reconstruction performance. Besides, as the proposed algorithm eliminates
the sparsity prior constraint, the direct optimization of RIP
preservation in [26] is not necessary. Instead, we can use
the reconstruction loss to train the sensing matrix along
with the model. For simplicity, we still use a random
Bernoulli matrix for information encoding in the experiment. Different from the prior work that extracts motion
from low-resolution previews, the proposed LSTM network infers motion from high-resolution frames generated
by multi-rate CNNs. The resolution of the reconstructed
video is further improved with the incorporation of highquality motion estimation.
3.1.1
Multi-rate CNN Encoder for compression ratio
enhancement
Typical CNN architectures used for recognition, classification, and segmentation that map input to rich hierarchical visual features is not applicable to the reconstruction
problem. The goal of the CNN is not only to extract spatial visual features but also to preserve details as much as
possible. Therefore, we eliminated the pooling layer which
causes information loss. Also, we discard the convolutiondeconvolution architecture (widely used in segmentation
tasks [17]), which first encodes salient visual features into
low-dimension space and then interpolates the missing information to generate a high-resolution image. Instead, we
design a special CNN suitable for CS reconstruction, which
has the best recovery performance among all the tested
structures mentioned above. The overall network structure
is shown in Figure 3. All feature maps have the same dimension as the reconstructed video frames, and the number of feature maps decreases monotonically. This process
resembles the sparse coding stage in CS, where a subset of
dictionary atoms is combined to form the estimation of the
original input. There is a fully-connected (FC) layer, denoted in gray color in Figure 3, which converts vectorized
m-dimensional video data to 2D features maps. To reduce
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Key Frame 1
Random Encoder
CR=M
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Frame T
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.
..
Random Encoder
CR=N (N>>M)
Random Encoder
CR=N (N>>M)
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3
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16*32*32
6*32*32
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Re 3 Re
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16*32*32
6*32*32
6*32*32
1*32*32
T
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Synthesizing LSTM
Non-key CNN
Figure 2: Overall architecture of the proposed framework. The compressed video frames are acquired by compressive
sensing. In a length T GOP, the first one frame and the remaining (T-1) frames are compressed with a low and high CR,
respectively. The reconstruction is performed by the CSVideoNet that is composed of a key CNN, multiple non-key CNNs,
and a synthesizing LSTM.
the latency of the system and to simplify the network architecture, we use video blocks as input and set the block
size n to 32×32. All convolutional layers are followed by
a ReLU layer except the final layer. We pre-train an eightlayer key CNN to process the key frame that is compressed
with a low CR. For other non-key frames compressed with
a high CR, we use 3-layer non-key CNNs to handle them
since they carry information of low entropy. All weights of
the non-key CNNs are shared to reduce the requirement of
storage. Hence the proposed framework can be easily generalized to other high-frame-rate video applications that
require a larger number of non-key frames. It should be
noted that the pre-training of the key CNN is critical for improving the reconstruction performance. In the case where
the whole network is trained from scratch without any pretraining, the convergence performance is bad. The reason
is partly due to the vanishing gradients, since we have a
long path from the CNNs to the LSTM. The pre-training
greatly alleviate this problem.
3.1.2
Motion-estimation synthesizing LSTM Decoder for spatial-temporal resolution enhancement
The proposed framework is end-to-end trainable, computationally efficient, and requires no pre/post-processing.
This is achieved by performing motion estimation implicitly, which is different from prior works [19, 27, 8]. We utilize an LSTM network to extract motion features that are
critical for improving temporal resolution from the CNN
output. Since the information flows from the first LSTM
node to the remaining, the LSTM will implicitly infers representations for the hidden motion from the key frame to
the non-key frames. Therefore, the recovery quality of the
GOP is improved by the aggregation of motion and spatial visual features. That is why we call this network the
motion-estimation synthesizing LSTM. For simplicity, each
input LSTM node in the experiment accepts input data with
equal length. In fact, since the non-key frames carry less
information than the key frame, the LSTM network can
be designed to accept inputs with variable lengths. Hence,
we can further reduce the model size and get a faster reconstruction speed. From the experiment results, we find
the utilization of the LSTM network is critical to improving recovery fidelity. As a result, our model outperforms
the competitive algorithms by a significant margin.
The update of the LSTM units is as follows:
it
= σ (Wxi xt + Whi ht−1 + Wci ct−1 + bi ) ,
ft
= σ (Wxf xt + Whf ht−1 + Wcf ct−1 + bf ) ,
ct
= ft ct−1 + it tanh (Wxc xt + Whc ht−1 + bc ) ,
ot
= σ (Wxo xt + Who ht−1 + Wco ct + bo ) ,
ht
= ot tanh(ct ),
where xt is the visual feature output of the CNN encoder.
The detailed information flow and the output dimension
at each LSTM node is shown in Figure 2. The number on
the LSTM nodes denotes the dimension of the output features. Specifically, the output feature map of each CNN
has a dimension of 16x32x32. All these feature maps are
directly fed into the input nodes of the LSTM. The LSTM
has two hidden layers, the dimension of the output of each
hidden layer is 6x32x32. The dimension of the final output
is 1x32x32.
3.2. Learning algorithm
Given the ground-truth video frames x{1,··· ,T } and the
corresponding compressed frames y{1,··· ,T } , we use mean
square error (MSE) as the loss function, which is defined
as:
T
1 X
kf (yi ; W, b) − xi k22 ,
(1)
L(W, b) =
2N i
where W, b are network weights and biases, respectively.
Using MSE as the loss function favors high PSNR. PSNR
is a commonly used metric to quantitatively evaluate recovery quality. From the experiment results, we illustrate
that PSNR is partially correlated to the perceptual quality.
To derive a better perceptual similarity metric will be a future work. The proposed framework can be easily adapted
to a new loss function.
Three training algorithms, i.e., SGD, Adagrad [7] and
Adam [10] are compared in the experiment. Although consuming most GPU memory, Adam converges towards the
best reconstruction results. Therefore, Adam is chosen to
optimize the proposed network.
4. Experiment
As there is no standard dataset designed for video CS,
we use UCF-101 dataset introduced in [20] to benchmark
the proposed framework. This dataset consists of 13k
clips and 27 hours of video recording data collected from
YouTube, which belong to 101 action classes. Videos in
the dataset are randomly split into 80% for training, 10%
for validation and the remaining for testing. Videos in the
dataset have a resolution of 320×240 and are sampled at
25 fps. We retain only the luminance component of the
extracted frames and crop the central 160×160 patch from
each frame. These patches are then segmented into 32×32
non-overlapping image blocks. We get 499,760 GOPs for
training and testing in total.
We set three test cases with CRs of 25, 50 and 100, respectively. Since the CR for key and non-key frames are
different in the proposed method, we derive and define the
CR for a particular GOP as follows. Let m1, m2 denotes
the dimension of compressed key and non-key frame, respectively. Let n denotes the dimension of raw frames. T
is the sequential length of a GOP.
CR1 =n/m1, CR2 = n/m2,
CR1 × 1 + CR2 × (T − 1)
.
(2)
T
In the experiment, the CR of each key frame is m1=5,
and the CR of non-key frames in each test case is m2=27,
55, and 110, respectively. Therefore, the averaged CR for
each test case is about 25, 50, and 100, respectively.
The dimension of data for pre-training the key CNN is
(N × C × H × W ), where N =100 is the batch size, C=1
is the channel size, and W, H=(32, 32) is the height and
width of each image block, respectively. The dimension of
the data used for training the entire model is (N 0 ×T ×C ×
H × W ), where T =10 is the sequence length for one GOP,
and N 0 =20 is the batch size. The other dimensions are the
same. We shrink the batch size here because of the GPU
memory limitation. In every ten consecutive video frames,
we define the first one as the key frame, and the remaining
as non-key frames.
CR =
Table 1: Summary of major differences between the proposed approach and all baselines.
Iterative Based
Image CS
Non-iterative Based
Iterative Based
Video CS
Non-iterative Based
Denoising-based
approximate message passing
Stacked denoising autoencoder
Convolutional neural network
Motion-compensated blockbased CS with smooth
projected Landweber
Gaussian mixture model
Fully-connected neural network
Proposed approach
D-AMP [14]
SDA [15]
ReconNet [11]
MC-BCS-SPL [8]
GMM [27]
VCSNet [9]
CSVideoNet
4.1. Comparison with the state-of-the-art
We compare our algorithm with six reference work for
CS reconstruction: [27, 8, 15, 14, 11, 9]. We summarize all
baseline approaches and our approach in Table 1. For a
fair comparison, we also re-train reference algorithms using UCF-101 dataset. Three metrics: Peak signal-to-noise
ratio (PSNR), structural similarity (SSIM) [24], and pixelwise mean absolute error (MAE) are applied for performance evaluation. Note that MAE is the averaged absolute error of each pixel value within the range of [0,255],
which gives a straightforward measure of the pixel-wise
distortion. The authors of VCSNet only offer a pre-trained
model with CR of 16, without providing sufficient training
details to reproduce the experiment at present. Therefore,
we train the proposed model and compare it with CVSNet
at a single CR of 16.
4.1.1
Comparison with image CS approaches
We first compare with the algorithms used for image CS
reconstruction. D-AMP is a representative of the conventional iterative algorithms developed for CS, e.g., matching pursuit, orthogonal mating pursuit, iterative hardthresholding. It offers state-of-the-art recovery performance and operates tens of times faster compared to
other iterative methods [14]. Both SDA and ReconNet
are DNN-based reconstruction approaches for images proposed recently. Specifically, ReconNet is based on CNN
and achieves state-of-the-art performance among all image CS reconstruction algorithms [11]. In the experiment,
we tested both frame-based and block-based D-AMP that
reconstructs an entire frame and an image block at a time,
respectively. For other approaches, we test them in a blockbased pattern to reduce the difficulty for training the models. The quantized results of average PSNR, SSIM, and MAE
for each method under different CRs are shown in Table 2.
It is shown that CSVideoNet outperforms the reference approaches on all three metrics by a meaningful margin, especially at the CR of 100. The MAE of CSVideoNet is 4.59 at
a 100x CR which means the averaged pixel-wise distortion
is only 4.59/255 = 1.2% compared to the ground-truth
video. The PSNR drop from the CR of 25 to 100 is also cal-
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Key CNN
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ReLU
ReLU
Random Encoder
CR=M
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3
3
3
3
ReLU
3
ReLU
32
16
32
32
3
3
ReLU
16
Figure 3: Pre-training of the key CNN.
Table 2: Performance comparison with image CS reconstruction approaches.
PSNR
SSIM
MAE
PSNR↓
CR
25
50
100
25
50
100
25
50
100
25 → 100
D-AMP(F)
25.34
12.49
7.17
0.76
0.08
0.03
4.65
64.30
92.12
72%
D-AMP(B)
15.1494
9.1719
8.0942
0.0934
0.0249
0.0067
24.92
81.67
86.04
13%
SDA
23.39
21.96
20.40
0.69
0.65
0.61
5.76
6.60
8.50
47%
ReconNet
24.27
22.47
20.44
0.73
0.67
0.61
5.02
5.67
7.42
16%
CSVideoNet
26.87
25.09
24.23
0.81
0.77
0.74
3.38
4.31
4.59
10%
Table 3: Performance comparison with video CS reconstruction approaches.
PSNR
SSIM
MAE
PSNR↓
culated in Table 2. We found the proposed approach suffers
from the least performance degradation. This is partly due
to the feature sharing between the key and non-key frames
when the compressed input carries limited information.
For visual quality assessment purpose, we list the reconstructed frame by each approach in Figure 4. The reconstructed frame is the middle (fifth) frame in a GOP. We
find all the reconstructed non-key frames have homogeneous recovery quality, and the key frame has slightly better reconstruction quality than the non-key frames. As the
proportion of key and non-key frames is 1:9, and the reconstruction quality of the video is dominated by that of
the non-key frames. Therefore, the middle frame (a nonkey frame) shown in Figure 4 well represents the average
reconstruction quality.
For all the numerical results, we calculate all the quality metrics, including PSNR, SSIM, and MAE, by averaging the results over all frames in a GOP. We can see
that CSVideoNet provides the finest details among all approaches. The edges produced by CSVideoNet is much
sharper, while such details are no longer preserved by
other methods after reconstruction. This comparison
demonstrates that the temporal correlation is critical for
video reconstruction, the overlook of such features will significantly degrade the recovery quality of videos. Therefore, the conventional image CS approaches are not suitable for video applications.
4.2. Comparison with video CS approaches
We compare the proposed CSVideoNet with existing
video CS approaches. MC-BCS-SPL estimates motion directly from the current and the reference frame. GMM
models the spatial-temporal correlation by assuming all
CR
25
50
100
25
50
100
25
50
100
25 → 100
MC-BCS-SPL
22.41
20.59
19.67
0.37
0.30
0.19
11.88
16.03
28.86
26%
GMM
23.76
21.26
19.64
0.72
0.61
0.54
5.14
7.50
9.37
17%
CSVideoNet
26.87
25.09
24.23
0.81
0.77
0.74
3.38
4.31
4.59
10%
pixels within a video patch are drawn from a GMM distribution. GMM has the state-of-the-art performance among
conventional model-based video CS approaches [27]. To
the best of our knowledge, [9] is the only DNN-based work
proposed for video CS. The quantized results of average
PSNR, SSIM, and MAE for each method under different
CRs are shown in Table 3. It is observed that the proposed approach improves PSNR by 3 to 5dB over the reference methods. Specifically, we find MC-BCS-SPL and
GMM have similar performance and perform much better
than the model-based image CS approach, D-AMP. However, their performance are similar to SDA and ReconNet,
which are designed for processing images. This implies
that the conventional model-based methods suffer from
limited performance due to the limited model capacity
when dealing with large-scale problem. Even though they
consider the temporal correlation among video frames, the
model capacity is insufficient for visual patterns. To improve performance, one could increase the size of the conventional models. However, the computational complexity
forof these meods will also increase substantially, inhibiting their application to video CS.
DNN provides a viable solution. Both CSVideoNet and
VCSNet are designed for video CS reconstruction. For reasons explained earlier, we compare the two approaches at
a CR of 16. The results are shown in Table 4 and Figure 5.
Both the two approaches achieve high recovery quality
compared to other baselines. However, VCSNet is a plain
fully-connect network that has limited capability for processing sequential data. As a result, it suffers from a lowquality motion estimation, which explains why it has inferior performance compared to the proposed solution.
28.49dB
21.43dB
22.04dB
23.57dB
23.43dB
28.86dB
24.52dB
18.26dB
20.52dB
23.59dB
21.19dB
27.92dB
5.78dB
5.78dB
17.04dB
18.05dB
23.49dB
18.43dB
27.84dB
D-AMP
SDA
MC-BCS-SPL
GMM
CSVideoNet
(a)
(b)
(c)
ReconNet
Figure 4: Illustration of reconstruction results for each method at the CR of (a) 25, (b) 50, and (c) 100, respectively.
27.14dB
29.46dB
Table 5: Structures of CNN1 and CNN2.
# Layer
CNN1
CNN2
*
Ground Truth
VCSNet
CSVideoNet
Figure 5: Illustration of reconstruction results at the CR of
16.
Table 4: Performance comparison with VCSNet at the CR
of 16.
PSNR
SSIM
MAE
VCSNet
25.07704
0.817669
3.887867
CSVideoNet
28.078
0.8431
2.9452
To illustrate that the performance improvement of the
proposed approach comes from integrating temporal features through the LSTM network rather than simply increasing the model size, we set another experiment, in
which we compare the performance of two CNNs with different sizes. The structure of the two CNNs are shown in
1
1
1
2
128
512
3
64
256
4
32
256
5
32
128
6
16
128
7
16
64
8
1
64
9
10
11
12
13
32
32
16
16
1
CNN1 is used in CSVideoNet. The dimension of all feature maps in both CNNs are 32×32.
Table 5, and the performance comparison is shown in Table 7. We can see that simply increasing the size of CNN
does not provide meaningful improvement for reconstruction. This, wh be explained by the incapability of CNN
to capture temporal features. The incorporation of the
LSTM network improves the PSNR by up to 4 dB, which
represents more than twice of error reduction. Specifically, the performance improvement increases with thealong wiachieves theits maximum wheR is 100. This explains that the implicit motion estimation by LSTM is critical to the video CS reconstruction especially at high CRs.
4.3. Performance under noise
To demonstrate that the robustness of CSVideoNet to
sensor noise, we conduct a reconstruction experiment with
input videos contaminated by random Gaussian noise. In
this experiment, the architecture of all DNN-based frameworks remains the same as in the noiseless case. We test
the performance at three levels of SNR - 20dB, 40dB, and
Table 6: Runtime comparison for reconstructing a 160×160
video frame at different CRs.
CR = 25
27
26
Model
D-AMP(F)
D-AMP(B)
SDA
ReconNet
MC-BCS
GMM
CSVideoNet
CR=25
38.37
8.4652
0.0278
0.064
7.17
8.87
0.0094
CR=50
41.20
8.5498
0.027
0.063
8.03
10.54
0.0085
CR=100
31.74
8.4433
0.023
0.061
9.00
18.34
0.0080
Table 7: Performance comparison with CNN methods.
PSNR
SSIM
MAE
CR
25
50
100
25
50
100
25
50
100
CNN1
24.27
22.47
20.44
0.73
0.67
0.61
5.02
5.67
7.42
CNN2
23.74
22.17
20.10
0.69
0.65
0.58
6.46
6.23
8.92
CSVideoNet
26.87
25.09
24.23
0.81
0.77
0.74
3.38
4.31
4.59
60dB. For each noise level, we evaluate all approaches at
three CRs of 25, 50, and 100. The average PSNR achieved
by each method at different CRs and noise levels are shown
in Figure 6. It can be observed that CSVideoNet can reliably achieve a high PSNR across at different noise levels
and outperform the reference methods consistently.
25
24
23
22
20
We benchmark the runtime performance of different
methods. Due to the iterative nature of conventional CS algorithms (D-AMP, MC-BCS-SPL, GMM), they suffer from
high data-dependency and low parallelism, which is not
suitable for GPU acceleration. Due to the lack of GPU
solvers, we run these reference algorithms on an octacore Intel Xeon E5-2600 CPU. Benefiting from the feedforward data-path and high data concurrency of DNN-based
approaches, we accelerate CSVideoNet and other DNNbased baselines using a Nvidia GTX Titan X GPU. The time
cost for fully reconstructing a video frame in the size of
(160×160) are compared in Table 6. CSVideoNet consumes
8 milliseconds (125 fps) to reconstruct a frame at the CR of
100. This is three orders of magnitude faster than the reference methods based on iterative approaches. The time cost
of VCSNet and CSVideoNet at the CR of 16 is 3.5 and 9.7
milliseconds, respectively. Through further hardware optimization, we believe CSVideoNet has the potential to be
integrated into CS cameras to enable the real-time reconstruction of high-frame-rate video CS.
CR = 50
60
Inf
25
20
15
20
40
CR = 100
60
Inf
25
20
15
10
20
4.4. Time complexity
40
40
D-AMP
MC-BCS-SPL
60
SDA
GMM
Inf
ReconNet
CSVideoNet
Figure 6: PSNR comparison at different SNRs.
multi-rate CNN variant and a synthesizing LSTM network are developed to jointly extract spatial-temporal features. This is the key to enhancing the compression
ratio and recovery quality trade-off. The magnificent
model capacity of the proposed deep neural network allows to map the inverse mapping of CS without exploiting any sparsity constraint. The feed-forward and highdata-concurrency natures of the proposed framework are
the key to enabling GPU acceleration for real-time reconstruction. Through performance comparison, we demonstrate that CSVideoNet has the potential to be extended as
a general encoding-decoding framework for high-framerate video CS applications. In the future work, we will exploit the effective learning methods to decode high-level
information from compressed videos, e.g., object detection,
action recognization, and scene segmentation.
6. Acknowledgement
5. Conclusion
In this paper, we present a real-time, end-to-end, and
non-iterative framework for high-frame-rate video CS. A
This work is supported by NSF grant IIS/CPS-1652038.
The research infrastructure used by this work is supported
by NSF grant CNS-1629888.
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| 1 |
EMBEDDINGS OF CANONICAL MODULES AND RESOLUTIONS OF
CONNECTED SUMS
arXiv:1704.03072v1 [math.AC] 10 Apr 2017
ELA CELIKBAS, JAI LAXMI, AND JERZY WEYMAN
Abstract. For an ideal Im,n generated by all square-free monomials of degree m in a
polynomial ring R with n variables, we obtain a specific embedding of a canonical module
of R/Im,n to R/Im,n itself. The construction of this explicit embedding depends on a
minimal free R-resolution of an ideal generated by Im,n . Using this embedding, we give a
resolution of connected sums of several copies of certain Artinian k-algebras where k is a
field.
1. Introduction
For a Cohen-Macaulay ring S with a canonical module ωS , it is well-known that, if S is
generically Gorenstein (e.g. S is reduced), then ωS can be identified with an ideal of S,
that is, ωS embeds into S; see, for example [2, 3.3.18]. In this paper we give an explicit
construction of such an embedding for a certain ring. More precisely, if R is a polynomial
ring in n variables over a field k, Im,n is the ideal of R generated by all square-free monomials
of degree m and ωR/Im,n is the canonical module of R/Im,n , then, in Theorem 5.5, we establish
an explicit standard graded embedding of ωR/Im,n into R/Im,n . Our motivation for this study
comes from obtaining minimal free resolutions of connected sums of Gorenstein rings. As
given in [1], a connected sum of several Gorenstein rings Si is a Gorenstein ring S that is
a special quotient of the fiber product (pullback) of Si ’s. Indeed, as a consequence of our
argument, we give a construction of a resolution of a connected sum of several copies of
Si := k[x]/(xei +1 ) over a field k; see Corollary 6.3.
In order to construct a specific embedding from ωR/Im,n to R/Im,n , we use generators
of the R/Im,n -module HomR (ωR/Im,n , R/Im,n ). In section 3, we give a set of generators of
HomR (R/Im,n , R/Im,n ) in Theorem 3.2. Moreover, as an immediate result of Theorem 3.2,
we get a presentation of HomR (ωR/Im,n , R/Im,n ). Section 4 deals with the computation of
Hilbert-Poincaré functions of R/Im,n and ωR/Im,n .
The main result of this paper is Theorem 5.5 which gives a specific standard graded
embedding of a canonical module
ψ := ψm,n : ωR/Im,n −→ R/Im,n .
In Corollary 5.6, the image of ψm,n is identified with an ideal of R/Im,n generated by maximal
minors of a certain Vandermonde-like matrix D. We use Theorem 5.5 and Corollary 5.6 to
Date: April 12, 2017.
2010 Mathematics Subject Classification. Primary 13D02, 13D40, 13H10, 20C30; Secondary 13F55.
Key words and phrases. squarefree monomial ideal, canonical module, Gorenstein ring, connected sum.
Jerzy Weyman was partially supported by NSF grant DMS-1400740.
1
2
ELA CELIKBAS, JAI LAXMI, AND JERZY WEYMAN
get a resolution of a Gorenstein ring obtained from an embedding of a canonical module of
R/Im,n in Corollary 5.7.
In section 6 we specialize to m = 2. In this case, in Theorem 6.1, we give another, Nn graded embedding of a canonical module of R/I2,n into the ring R/I2,n . As a corollary of
this theorem, a canonical module of R/I2,n is identified with an Nn -graded ideal of R/I2,n .
The mapping cone of the map of free resolutions over R covering the embedding ωR/Im,n −→
R/Im,n gives a minimal free resolution of the connected sum of algebras Si := k[x]/(xei +1 ).
Section 2 contains known results regarding the main tools used in the rest of the paper
including the definition of connected sums, resolutions of the ideals generated by square-free
monomials of a given degree and of corresponding Stanley-Reisner rings.
2. Preliminaries
2.1. Notation.
a) For a positive integer n, let [n] = {1, . . . , n}. If σ ⊂ [n], then |σ| denotes the number of
elements contained in σ.
b) Let R = k[x1 , . . . , xn ] be a polynomial ring in n variables over a field k with x1 > . . . > xn .
We order the monomials in R with graded lexicographic order.
c) Let m and n be positive integers with m ≤ n, then Im,n denotes an ideal generated by
all square-free monomials of degree m in n variables. Furthermore, ωR/Im,n denotes a
canonical module of R/Im,n .
d) For an R-module M, ℓ(M) and µ(M) denote the length and the minimal number of
generators of M, respectively.
e) For a commutative Noetherian ring T , dim(T ) denotes the Krull dimension of T .
f) Let M = ⊕i≥0 Mi be a graded
The Hilbert-Poincaré function of M is the
P R-module.
i
formal power series HM (t) = i≥0 ℓ(Mi )t .
g) Let (T, m, k) be an Artinian local ring. Then the socle of T is soc(T ) = (0 :T m).
h) For a Noetherian local ring T and a T -module M, a finite presentation of M is an exact
sequence T ⊕m → T ⊕n → M → 0 with m, n positive integers.
2.2. Connected Sums.
Definition 2.1. Let Si = k[xi ]/(xei i +1 ), soc(Si ) = (xei i ), and J = hxi xj , xei i − xe11 |1 ≤ i ≤ ni
where ei ≥ 1 be the ideal in R := k[x1 , . . . , xn ] defining the connected sum S1 #k . . . #k Sn of
the algebras Si (compare [1] for the definition of connected sums). Therefore we have
S1 #k . . . #k Sn = R/J.
Remark 2.2. With notation in Definition 2.1, S1 #k . . . #k Sn is Gorenstein by [1].
2.3. Specht Modules and Free Resolution of the Ring R/Im,n. We recall the definition
q
of Specht module S (p,1 ) associated to a hook partition (p, 1q ) of n where p, q are nonnegative
integers. We follow [3, Section 7.4]. Let n = p + q and let Sn be a symmetric group on [n].
Let (p, 1q ) be a hook partition of n. An oriented column tabloid of shape (p, 1q ) is filling
of Young diagram of (p, 1q ) with positive integers 1, 2, . . . , n, with each number appearing
once, which is skew-symmetric in the first column and symmetric in the remaining rows.
q
The Specht module S (p,1 ) is the k-vector space generated by the equivalence classes [T ] of
oriented column tabloids of shape (p, 1q ) with entries in [n] modulo the following relations:
EMBEDDINGS OF CANONICAL MODULES AND RESOLUTIONS OF CONNECTED SUMS 3
a) Alternating columns: σ[T ] = sign(σ)[T ] for all σ ∈ Sn fixing the columns of T (so in the
case of hook, just permuting
P the numbers in the first column).
b) Shuffling relations: [T ] = [T ′ ], where sum is over all T ′ acquired from T by exchanging
the element of the second column of [T ] with one of the element of the first column of T .
We recall some facts about Specht modules associated to a hook partition (p, 1q ).
q
a) The Specht module S (p,1 ) is a k-vector space. By using hook length formula from [5,
Theorem 20.1], we have
p+q−1
(p,1q )
.
dim(S
)=
q
q
b) The symmetric group Sn acts on S (p,1 ) by permuting the numbers in oriented column
tabloids.
c) An oriented column tabloid of shape (p, 1q ) is called standard tableau of shape (p, 1q )
if the entries in each row and column are increasing (in the case of hooks it means the
entries in the first column are increasing and the first entry in the first column is 1).
q
The equivalence classes of standard tableaux of shape (p, 1q ) form a k-basis of S (p,1 ) .
Let SY T ([p + q], (p, 1q )) denote the set of standard tableaux of shape (p, 1q ) with entries
1, 2, . . . , p + q (each number appearing once).
In the remaining part of this subsection, we state the results from [4].
Definition 2.3. (a) Let n, m and i be integers. For 1 ≤ m ≤ n and 0 ≤ k ≤ n − m,
(1)
k
Ukm,n := IndSSnm+k ×Sn−m−k (S (m,1 ) ⊗k S (n−m−k) ).
q
Here S (n−m−k) is the Specht module S (p,1 ) with p := n−m−k, q := 0. The right hand side of
k
Equation 1 is the k[Sn ]-module induced by the k[Sm+k × Sn−m−k ]-module S (m,1 ) ⊗k S (n−m−k) .
If any of the inequalities involving n, m, and i are violated, then we set Uim,n := 0.
(b) A k[Sn ]-module Fkm,n is defined as
Fkm,n := Ukm,n ⊗k R(−m − k).
(2)
Remark 2.4. Let n, m, and k be positive integers with 1 ≤ m ≤ n and 0 ≤ k ≤ n − m.
(a) The module Ukm,n is generated by the equivalence classes of oriented column tabloids of
shape (m, 1k ), filled with numbers 1, 2, . . . , n without repetitions. Moreover, the equivalence classes of standard tableaux of shape (m, 1k ) form a k-basis of Ukm,n .
(b) The module Fkm,n is a free R-module generated by the equivalence classes of oriented
column tabloids of shape (m, 1k ), filled with numbers 1, 2, . . . , n without repetitions.
(c) The equivalence classes of standard tableaux of the shape (m, 1k ) with entries in [n]
(without
repetitions)
form an R-basis of Fkm,n and the rank of Fkm,n is βk := rank(Fkm,n ) =
n
m+k−1
.
m+k
k
We define an R-linear map
m,n
∂km,n : Fkm,n −→ Fk−1
by setting
∂km,n ([T ]) :=
k
X
p=0
(−1)k−p xip [T \ ip ]
4
ELA CELIKBAS, JAI LAXMI, AND JERZY WEYMAN
where [T ] is an oriented column tabloid of shape (m, 1k ), and T \ ip is an oriented column
tabloid of shape (m, 1k−1 ) obtained from T by omitting the number ip in position p − 1 in
the first column of T .
Proposition 2.5. Let n, m and k be positive integers with m ≤ n and 0 ≤ k ≤ n − m.
m,n
Then (Fm,n
• , ∂• ) is a complex of free R-modules which is a minimal free resolution of the
R-module R/Im,n . This complex is Sn -equivariant, where Sn acts on Fkm,n diagonally (the
action on R just permutes the variables xi ).
2.4. Simplicial Complex and Stanley-Reisner Rings.
Definition 2.6. [2, Definition 5.1.1] Let V = {v1 , . . . , vn } be a finite set.
(1) A non-empty set ∆ of subsets of V with the property that τ ∈ ∆ whenever τ ⊂ σ
for some σ ∈ ∆ is called a simplicial complex on the vertex set V . The elements of
∆ are called faces, and the dimension, dim σ, of a face σ is the number |σ| − 1. The
dimension of the simplicial complex ∆ is dim(∆) = max{dim σ : σ ∈ ∆}.
(2) Let k be a field. The Stanley-Reisner ring of the complex ∆ is the homogeneous
k-algebra k[∆] = k[x1 , . . . , xn ]/I∆ , where I∆ is the ideal generated by all monomials
xi1 . . . xis such that {vi1 , . . . , vis } 6∈ ∆. The Krull dimension of the Stanley-Reisner
ring k[∆] is dim(∆) + 1.
Lemma 2.7. Let I∆ = Im,n , then k[∆] is Cohen-Macaulay.
Proof. If I∆ = Im,n , then all monomials xi1 . . . xim−1 ∈ I∆ , so {vi1 , . . . , vim−1 } 6∈ ∆. Hence,
dim(∆) = m − 2. Then dim(k[∆]) = m − 1.
By Proposition 2.5, projective dimension of k[∆] is n − m + 1. By graded AuslanderBuchsbaum formula, depth(k[∆]) = m − 1. Thus, k[x1 , . . . , xn ]/Im,n is Cohen Macaulay.
m,n
Remark 2.8. By Proposition 2.5, (Fm,n
) is a minimal free resolution of R/Im,n . Let
• , ∂•
m,n
m,n
G• = HomR (F• , R) be the dual complex. Then Gm,n
is a minimal free R-resolution of
•
ωR/Im,n .
3. Generators of Hom
The goal of this section is to find the generators (Theorem 3.2) and a presentation (Corollary 3.3) of the R-module HomR (ωR/Im,n , R/Im,n ). We start with the example n = 4, m = 2.
Example 3.1. Let R = k[x
"1 , x2 ,#x3 , x4 ] and I2,4
" = hx
# 1 x2 , x1 x3 , x1 x4 , x"2 x3 , x#2 x4 , x3 x4 i be an
ideal of R. Let [T[4]\{2} ] =
1 2
3
4
, [T[4]\{3} ] =
1 3
2
4
and [T[4]\{4} ] =
1 4
2
3
. The formulas
EMBEDDINGS OF CANONICAL MODULES AND RESOLUTIONS OF CONNECTED SUMS 5
for differentials in the complex F4,2
• are
2,4
3 2
∂2 ([T[4]\{2} ]) = x1 4
− x3 14
2 3
− x1 23
= x1 4
2,4
2 3
∂2 ([T[4]\{3} ]) = x1 4
− x2 14
2,4
2 4
− x2 13
∂2 ([T[4]\{4} ]) = x1 3
2
4
3
4
+ x4
− x3
+ x4
+ x3
1 2
3
1 2
4
+ x4
.
4
.
1 2
3
.
1 3
2
1
2
Let P be the matrix of ∂22,4 with respect to the bases of standard tableaux in modules F24,2
and F14,2 .
0
0
x4
0
x4
0
0
0 −x3
x3
0
0
P =
0 −x2
0
−x2
0
0
0
x1
x1
x1
0 −x1
Columns are listed in order T[4]\{4} =
are listed in order
1 2
3
,
1 3
2
,
1 2
4
,
1 4
2
1 4
2
3
,
, T[4]\{3} =
1 3
4
,
1 4
3
,
1 3
2
4
2 3
4
, and T[4]\{2} =
2 4
3
, and
1 2
3
4
, and rows
.
Then the transpose of P , denoted by P T , gives a matrix presentation of ωR/I2,4 . For
2 ≤ i ≤ 4, let f{i} : ωR/I2,4 → R/I2,4 be defined as
(
xi , if i = j
(3)
f{i} ([T[4]\{j} ]) =
0, if i 6= j.
In order to show that f{i} is well defined, it is enough to prove that f{i} satisfies the relations
of P T . Note that the entry xi is missing in the column corresponding to ∂22,4 ([T[4]\{i} ]) in P ,
T
hence f{i} satisfies the relations
"
#of P .
The tableau [T[4]\{1} ] =
2 1
3
4
"
2 1
3
4
is expressed in terms of standard tableaux such as
#
=
"
1 2
3
4
#
−
"
1 3
2
4
#
+
"
1 4
2
3
#
.
Let
(4)
f{1} ([T[4]\{2} ]) = −f{1} ([T[4]\{3} ]) = f{1} ([T[4]\{4} ]) = x1 .
Since ∂22,3 ([T[4]\{1} ]) = ∂22,4 ([T[4]\{2} ])−∂22,4 ([T[4]\{3} ])+∂22,4 ([T[4]\{4} ]), there is no term involving
x1 in ∂22,3 ([T[4]\{1} ]). Hence, f{1} is well defined.
6
ELA CELIKBAS, JAI LAXMI, AND JERZY WEYMAN
Now suppose ψ : ωR/I2,4 → R/I2,4 satisfies ψ([T[4]\{i} ]) = c + u for some c ∈ k, and
u ∈ hx1 , . . . , xn i. Then ψ satisfies the relations of P T , which implies, cxi = 0, hence c = 0.
Then, taking into account relations given by the first six columns of P T , we can write
X (e) (e) X (e) (e)
ψ([T[4]\{4} ]) =
a1 x1 +
a4 x4 ,
e≥1
ψ([T[4]\{3} ]) =
X
e≥1
(e) (e)
b1 x1 +
e≥1
ψ([T[4]\{2} ]) =
X
(e)
bi ,
(e)
ci
(e) (e)
b3 x3 ,
e≥1
(e) (e)
c1 x1
+
e≥1
(e)
ai ,
X
X
(e) (e)
c2 x2 ,
e≥1
and
are all in k.
where
(e)
(e)
(e)
Using the relations from last two columns of P T , we get a1 = c1 = −b1 for each e.
This means
X (e) (e−1)
X (e) (e−1)
X (e) (e−1)
X (e) (e−1)
c2 x2 )f2 + (
b3 x3 )f3 + (
a4 x4 )f4 .
c1 x1 )f1 + (
ψ=(
e≥1
e≥1
e≥1
e≥1
This shows that {f{1} , f{2} , f{3} , f{4} } is a minimal generating set of HomR (ωR/I2,4 , R/I2,4 ).
In the light of the example above, the following theorem gives a general description of a
minimal generating set of HomR (ωR/Im,n , R/Im,n ).
Theorem 3.2. For 1 < j1 < . . . < jm−1 ≤ n, let Θ = {j1 , . . . , jm−1 }. Suppose fj1 ,...,jm−1 and
f1,j2 ,...,jm−1 are maps from
( ωR/Im,n to R/Im,n defined as
xj1 xj2 . . . xjm−1 , if Γ = Θ
and
fj1 ,...,jm−1 ([T[n]\Γ]) =
0
, otherwise.
(
x1 xj2 . . . xjm−1 , if Γ = Θ or Γ = (Θ \ {j1 }) ∪ {l} for 1 6= l ∈ [n] \ Θ,
f1,j2 ,...,jm−1 ([T[n]\Γ ]) =
0
, otherwise.
Then {fj1 ,...,jm−1 , f1,j2 ,...,jm−1 : 1 < j1 < . . . < jm−1 ≤ n} is a minimal generating set of
HomR (ωR/Im,n , R/Im,n ).
Proof. First note that by Remark 2.4, Bk := {[T ] : T ∈ SY T ((m, 1k ), [n])} is a basis of Fkm,n .
For 1 = i0 < j1 < . . . < jm−1 ≤ n and 1 = i0 < i1 < i2 < . . . < in−m ≤ n, we set standard
tableau of shape (m, 1n−m ) as
i0
j1
j2
. . . jm−1
i1
(5)
[T[n]\{j1 ,...,jm−1 } ] = [T1,i1 ,...,in−m ] =
i2
..
.
in−m
m,n
m,n
m,n
By Proposition 2.5, we get the differential ∂n−m
: Fn−m
→ Fn−m−1
as
EMBEDDINGS OF CANONICAL MODULES AND RESOLUTIONS OF CONNECTED SUMS 7
(6)
m,n
∂n−m
([T1,i1 ,...,in−m ]) =
n−m
X
(−1)n−m−k xik [T1,i1 ,...,in−m \ ik ].
k=0
m,n
∂n−m
Let P be the matrix of
with respect to the bases Bn−m and Bn−m−1 . Then P T , the
transpose of P , is a presentation of ωR/Im,n by Remark 2.8.
In order to show that fj1 ,...,jm−1 and f1,j2 ,...,jm−1 are well defined, it is enough to see that
fj1 ,...,jm−1 and f1,j2 ,...,jm−1 satisfy the relations of P T . Since the column with respect to
m,n
∂n−m
([T1,i1 ,...,in−m ]) does not involve xjk , the corrensponding row in P T has no xjk as well.
Thus fj1 ,...,jm−1 satisfies the relations of P T . A non-standard tableau [T[n]\{1,j2,...,jm−1 } ] can be
expressed in terms of standard tableaux as
[T[n]\{1,j2 ,...,jm−1 } ] = [T[n]\{j1 ,...,jm−1 } ] +
n−m
X
(−1)k [T[n]\{ik ,j2 ,...,jm−1 } ].
k=0
m,n
By direct computation one sees that column with respect to ∂n−m
([T[n]\{1,j2 ,...,jm−1 } ]) does
not involve x1 and xjk for k = 2, . . . , m − 1. Therefore, f{1,j2 ,...,jm−1 } satisfies the relations of
P T , hence f{1,j2 ,...,jm−1 } is well defined.
We now claim that {fτ : τ ⊂ [n], |τ | = m−1} is a generating set of HomR (ωR/Im,n , R/Im,n ).
Let ϕ ∈ HomR (ωR/Im,n , R/Im,n ). For 1 < l1 < . . . < lm−1 ≤ n, let σ = {l1 , . . . , lm−1 } ⊂ [n].
Since ϕ([T[n]\σ ]) ∈ R/Im,n , we can write
X
X
npm−1
mp
mp
np
+
bτ xp1 1 . . . xpk k
ϕ([T[n]\σ ]) =
aτ xp1 1 . . . xpm−1
τ ={p1 ,...,pk }⊂[n],k<m−1
τ ={p1 ,...,pm−1 }⊂[n]
where npk , mpl ≥ 0 and aτ , bτ ∈ k. The fact that ϕ satisfies the relations of P T implies bτ = 0
and aτ = 0 provided τ 6= σ or τ 6= {1, l2 , . . . , lm−1 }. Thus we get
(7)
ϕ([T[n]\σ ]) = cσ fσ ([T[n]\σ ]) + c(σ\{l1 })∪{1} f(σ\{l1 })∪{1} ([T[n]\σ ])
n −1
nl
n −1
nl
m−1
m−1
where cσ = aσ xl1l1 . . . xlm−1
and c(σ\{l1 })∪{1} = a(σ\{l1 })∪{1} x1n1 −1 xl2l2 . . . xlm−1
.
For every Γ ⊂ [n] with 1 6∈ Γ and |Γ| = m − 1, the equivalence class of a standard tableau
[T[n]\Γ ] is in Bn−m . By Equation 7, for cτ ∈ R/Im,n , we get
X
ϕ([T[n]\Γ ]) =
cτ fτ ([T[n]\Γ ]).
−1
−1
τ ⊂[n],|τ |=m−1
P
Since [T[n]\Γ ] ∈ Bn−m , we have ϕ([T ]) = τ ⊂[n],|τ |=m−1 cτ fτ ([T ]) for every oriented column
tabloid [T ]. Hence ϕ ∈ hfτ : τ ⊂ [n], |τ | = m−1i. This proves that hfτ : τ ⊂ [n], |τ | = m−1i
is a generating set of HomR (ωR/Im,n , R/Im,n ).
If hfτ : τ ⊂ [n], |τ
P| = m − 1i is not a minimal generating set, then for some Γ ⊂ [n] with
|Γ| = m − 1, fΓ = τ ⊂[n],|τ |=m−1,τ 6=Γ aτ fτ where aτ ∈ k. Then for 1 ∈ γ and |γ ∩ Γ| = m − 2,
xΓ = aγ xγ which is not possible.
As a consequence of Theorem 3.2, we get a finite presentation as stated below.
Corollary 3.3. Let S = R/Im,n , σ ⊂ [n], and |σ| = m − 1. Suppose fσ : ωS → S is a map
defined in Theorem 3.2. Let C = {eσ : σ ⊂ [n]} and D = {pσ∪{i} : σ ⊂ [n], i 6∈ σ} be bases of
8
ELA CELIKBAS, JAI LAXMI, AND JERZY WEYMAN
n
n
S (m−1) and S m(m) , respectively. Then
n
n
φ
µ
− HomR (ωS , S) → 0
− S (m−1) →
S m(m) →
with φ(eσ ) = fσ and µ(pσ∪{i} ) = xi eσ is a finite presentation of HomR (ωS , S).
3.2 is an R/Im,n -module
Proof. Observe that fσ : ωR/Im,n → R/Im,n defined in Theorem
n
homomorphism and HomR (ωS , S) is generated by m−1 elements. Then we show that
n
ker(φ) = hx e : σ ⊂ [n]i. Since there is a surjective map φ : S (m−1) → Hom (ω , S)
R
i σ
S
defined by φ(eσ ) = fσ , by Theorem 3.2, xi fσ = 0 for each σ ⊂ [n] and i ∈
/ σ. Thus
φ(xi eσ ) = 0 and hence xi eσ ∈ ker(φ).
P
Assume to the contrary that we have a relation σ aσ fσ = 0 with aσ being a polynomial
depending only on the variables xi with i ∈ σ. We need to show that each aσ = 0. Let us
fix a subset τ and let us apply the zero homomorphism to the tableau T[n]\τ . We get
Y
X Y
aσ
xi fσ (T[n]\τ ).
0 = aτ
xi +
i∈τ
σ6=τ
i∈σ
But the first summand cannot cancel with any other summand since it is the only monomial
containing precisely the variables from τ . This shows that aτ = 0. Therefore
n
n
µ
φ
− S (m−1) →
− HomR (ωS , S) → 0
S m(m) →
is an exact sequence.
4. Hilbert-Poincaré function
In this section, we compute the Hilbert-Poincaré functions of R/Im,n and ωR/Im,n by using
Stanley-Reisner rings.
Lemma 4.1. Let R/Im,n be a standard graded ring. The Hilbert-Poincaré function of R/Im,n
is of the form
Pm−1 j
n−m+j
j=0 hj t
HR/Im,n (t) =
.
, where hj =
j
(1 − t)m−1
Pm−1 i
n−i−1
i=0 αi t
Moreover, HωR/Im,n (t) =
.
, where αi =
m−i−1
(1 − t)m−1
Proof. Suppose ∆ is a simplicial complex on the vertex set V = {v1 , . . . , vn } such that
{vi1 , . . . , vim } 6∈ ∆ for each 1 ≤ i1 < . . . < im ≤ n. Then the Stanley-Reisner ring of the
complex ∆ is the homogeneous k-algebra
k[∆] = k[x1 , . . . , xn ]/Im,n
where In,m is the ideal generated by all monomials of degree m.
Let fi denote the number of i-dimensional faces of ∆. Then fi−1
By a known combinatorial identity, we get
n
for 0 ≤ i ≤ m−1.
=
i
X
j
n
n−m+j
j−i m − 1 − i
.
(−1)
=
i
j−i
j
i=0
EMBEDDINGS OF CANONICAL MODULES AND RESOLUTIONS OF CONNECTED SUMS 9
Pm−1
hj tj
n−m+j
.
Then, by [2, Lemma. 5.1.8], we have HR/In,m (t) =
, where hj =
(1 − t)m−1
j
Now one can see that
Pm−1 m−1−j
Pm−1
j
j=0 hj t
j=0 hm−1−j t
−1
m−1
HωR/Im,n (t) = (−1)
HR/Im,n (t ) =
=
(1 − t)m−1
(1 − t)m−1
Pm−1
j
j=0 αj t
by [2, Corollary. 4.4.6]. Thus, for αj = hm−1−j , we get HωR/Im,n (t) =
.
(1 − t)m−1
j=0
Let us fix an n-tuple (e1 , . . . , en ) of integers greater than 1. Let r1 , . . . , rn be defined as
ri = e1 . . . êi . . . en where êi denotes the missing term in the product, and e = e1 . . . en . For
2 ≤ i ≤ n we set deg(xi ) = ri which makes R an Nn -graded ring. Let H̃M (t) denote the
Hilbert function of a module M in this new grading.
In the following remark, we give the Hilbert-Poincaré functions of R/I2,n and R/L.
Remark 4.2. Suppose deg(xi ) = ri and L2,n = hxei i −xenn : 1 ≤ i ≤ n−1i, and L = I2,n +L2,n .
Then
n
n
X
X
tri − te
tri
and
H̃
(t)
=
1
+
+ te .
H̃R/I2,n (t) = 1 +
R/L
r
r
i
i
1−t
1−t
i=1
i=1
5. N-Graded Embedding of A Canonical Module
Throughout this section we fix 1 ≤ m < n and the integers d = (d1 , . . . , dm−1 ) satisfying
1 < d1 < . . . < dm−1 .
In this section, for each d, an explicit embedding ψ := ψ(d) of ωR/Im,n into R/Im,n is
constructed in Theorem 5.5. We also prove that ωR/Im,n is identified with an N-graded ideal
of R/Im,n for each 1 < d1 < . . . < dm−1 where di ∈ N. In order to do that, we order
monomials in graded lexicographic order and all initial ideals are taken with respect to that
order.
Definition 5.1. Let I be an ideal of R and 0 6= f ∈ R. The initial ideal of I, denoted in(I),
is defined as
in(I) = {in(f )|f ∈ I \ {0}}
where in(f ) is the largest monomial appearing in f .
Setup 5.2. Let m − 1 ∤ char(k). For 1 < d1 < d2 < . . .
Consider m × n matrices B and D of the form
1
1
···
1
xd1 −1
xd21 −1 · · · xdn1 −1
1
B = ..
,
D
=
..
.
..
.
.
.
.
.
d
x1m−1
−1
d
x2m−1
−1
· · · xndm−1 −1
< dm−1 , let d = d1 + . . . + dm−1 .
1
xd11
..
.
1
xd21
..
.
d
d
···
···
..
.
1
xdn1
..
.
x1m−1 x2m−1 · · · xndm−1
Let βΛ be an m × m minor of B involving columns Λ where Λ ⊂ [n] and |Λ| = m. Let
Jm,n := Jm,n (d) = hδi1 ,...,im |1 ≤ i1 < . . . < im ≤ ni where δi1 ,...,im is an m × m minors of D
and J := J(d) = Im,n + Jm,n (d).
One can observe the relations between m × m minors of B and D as given in the remark
below.
10
ELA CELIKBAS, JAI LAXMI, AND JERZY WEYMAN
Remark 5.3. With notation in Setup 5.2, we see that δΛ = βΛ
The following lemma is crucial in the proof of Theorem 5.5.
P
τ ⊂Λ,|τ |=m−1 xτ
in R/Im,n .
Lemma 5.4. Assume Setup 5.2. Let σ ⊂ [n], |σ| = m − 1, and fσ : ωR/Im,n → R/Im,n be
the map defined in Theorem 3.2. Let ψ : ωR/Im,n → R/Im,n be the map given by
ψ=
X
Λ⊂[n]
|Λ|=m
X
βΛ fσ .
σ⊂Λ
|σ|=m−1
Then J/Im,n ⊂ im(ψ).
Proof. Let 1 < j1 < . . . < jm−1 ≤ n. For all Λ1 , Λ2 ⊂ [n] and |Λ1 | = |Λ2 | = m, we consider
σ = Λ1 ∩ Λ2 with |σ| = m − 1. If σ ⊂ {1, j1 , . . . , jm−1 }, then we see that
βΛ1 xσ = βΛ1 fσ ([T[n]\{j1 ,...,jm−1 } ]) = βΛ2 fσ ([T[n]\{j1 ,...,jm−1 } ]) = βΛ2 xσ
in R/Im,n . Hence,
X
ψ([T[n]\{j1 ,...,jm−1 } ]) = (m − 1)βΛ
fτ ([T[n]\{j1 ,...,jm−1 } ])
τ ⊂Λ
|τ |=m−1
where {j1 , . . . , jm−1 } ⊂ Λ. By Theorem 3.2, fτ ([T[n]\{j1 ,...,jm−1 } ]) = xτ provided τ ⊂ Λ and
|τ | = m − 1. Thus, we get ψ([T[n]\{j1 ,...,jm−1 } ]) = (m − 1)δΛ by Remark 5.3. Hence, for every
Λ ⊂ [n], we have δΛ ∈ im(ψ). This proves J/Im,n ⊂ im(ψ).
We are now ready to state and prove the main theorem in this paper.
Theorem 5.5. With notation in Setup 5.2, let ψ : ωR/Im,n → R/Im,n be the map stated in
Lemma 5.4. Then ψ is injective and im(ψ) = J/Im,n .
Proof. Let S = R/Im,n . Consider a short exact sequence of the form
ψ
(8)
0 → ker(ψ) → ωS (−d) −
→ S → S/ im(ψ) → 0.
k
By Lemma 5.4, we get J/Im,n ⊂ im(ψ), and hence HS/ im(ψ) (t) ≤ HR/J (t). Set Pm,n
as
k
Pm,n
= hin(xn−k+1 xn−k . . . xn−1 xn δi1 ,i2 ,...,im−k−1 ,n−k,n−k+1,...,n )|1 ≤ i1 < . . . < im−k−1 < n − k − 1i
d
d
k+1
k +1
xdn−k+1
. . . xdn1 +1 |1 ≤ i1 < i2 < . . . < im−k−1 < n − k − 1i.
= hxi1m−1 . . . xim−k−1
P
k
Then Q = Im,n + m−1
k=0 Pm,n is an ideal of R such that Q ⊂ in(J). Futhermore, the fact that
HR/J (t) = HR/in(J) (t) implies
(9)
HS/ im(ψ) (t) ≤ HR/J (t) ≤ HR/Q (t).
To prove the theorem, it is enough to see that HR/Q (t) ≤ HS/ im(ψ) (t).
Let Ai1 ,...,im−k−1 ,k = k[xi1 , . . . , xim−k−1 , xn−k+1, . . . , xn ] and let the k-linear maps gi1 ,...,im−k−1 :
Ai1 ,...,im−k−1 ,k → Q/Im,n be defined as
d
d
k+1
k +1
xdn−k+1
. . . xdn1 +1 .
gi1 ,...,im−k−1 ,k (1) = xi1m−1 . . . xim−k−1
EMBEDDINGS OF CANONICAL MODULES AND RESOLUTIONS OF CONNECTED SUMS11
By the universal property of coproduct, we have
m−1
M
Q/Im,n ≃
Ai1 ,...,im−k−1 ,k (−d − k).
1≤i1 <...<im−k−1 <n−k−1,k=0
Pm−1
αk tk
n−k−1
where
α
=
. Therefore, by Lemma 4.1, we get
k
m−k−1
(1 − t)m−1
HQ/Im,n (t) = td HωS (t). By Equation 8, one can see that
d
Hence, HQ/Im,n (t) = t
k=0
HR/Q (t) = HS (t) − HQ/Im,n (t) ≤ HS/ im(ψ) (t).
Then, by Equation 9, we have HS/ im(ψ) (t) = HR/J (t), hence im(ψ) = J/Im,n and Q = in(J).
Moreover, we get
HR/J (t) = HS/ im(ψ) (t) = HS (t) − td HωS (t).
By Equation 8, Hker(ψ) (t) = 0, hence ker(ψ) = 0. Thus, ψ is injective.
In the following corollary, we see that ωR/Im,n is identified with an N-graded ideal of R/Im,n
for each 1 < d1 < . . . < dm−1 .
Corollary 5.6. Assume Setup 5.2. Then ωR/Im,n (−d) ≃ J/Im,n .
Proof. The following diagram is commutative:
0
/
J/Im,n
R/Im,n
/
O
ψ
0
/
ωR/Im,n (−d)
R/J
/
/
R/Im,n
0
/
0
≃
≃
ψ
/
O
O
/
R/J
By the Snake Lemma, ωR/Im,n (−d) ≃ J/Im,n .
Corollary 5.7. There are infinitely many N-graded embeddings of ωR/Im,n into R/Im,n .
As a consequence of Theorem 5.5, each specific embedding of canonical module ωR/Im,n to
R/Im,n produces a Gorenstein ring.
Proposition 5.8. Let J = Im,n + Jm,n and d = d1 + . . . + dm−1 . Then R/J is a Gorenstein
ring with dim(R/J) = m − 2.
Proof. Let ψ : ωR/Im,n → R/Im,n be the map in Theorem 5.5. Then Cone(ψ) is a minimal
m,n
free resolution of R/J with Cone(ψ)i = (Gm,n
, where Fm,n
and Gm,n
are minimal
•
•
n−m−i+1 )⊕Fi
free resolutions, given in Proposition 2.5 and Remark 2.8, of R/Im,n and ωR/Im,n , respectively.
Then we have pdim(R/J) = n − m + 2.
By Corollary 5.6, we have ωR/Im,n (−d) ≃ J/Im,n , and hence J/Im,n is an ideal with finite
resolution. Now note that J/Im,n contains a R-regular element by [2, Corollary 1.4.7]. Since
dim(R/Im,n ) = m − 1, we have dim(R/J) ≤ m − 2. By the graded Auslander-Buchsbaum
formula, we get depth(R/Im,n ) = m − 2.
Therefore R/J is Cohen-Macaulay. Proposition 2.5
n
m+i−1
implies that βi (R/Im,n ) = m+i
, and hence βi (R/J) = βn−m+2−i (R/J). This proves
i
that R/J is Gorenstein.
12
ELA CELIKBAS, JAI LAXMI, AND JERZY WEYMAN
6. Nn -Graded Embedding and Connected Sums
In this section we specialize to m = 2. In this situation we define even more embeddings
of ωR/I2,n in R/I2,n and all of these embeddings are even Nn -graded. These embeddings are
closely related to connected sums of several copies of certain Artinian k-algebras.
Throughout the rest of the section we fix an n-tuple (e1 , . . . , en ) of integers bigger than 1.
Let e = e1 . . . en and ri = e1 . . . êi . . . en where êi denotes the missing term in the product.
For 2 ≤ i ≤ n we set deg(xi ) = ri which makes R an Nn -graded ring. In this setup we have
the following.
Theorem 6.1. The map ψ : ωR/I2,n → R/I2,n defined by
ψ([T[n]\{i} ]) = xei i − xe11
is an Nn -graded embedding.
Proof. By Theorem 3.2, {f{i} : {i} ⊂ [n]} is a generating set of HomR (ωR/I2,n , R/I2,n ) where
f{i} ([T[n]\{i} ]) = xi and f{1} ([T[n]\{i} ]) = x1 for i 6= 1. Let A be a 2 × n matrix of the form
1
1
···
1
A = e1 −1 e2 −1
x1
x2
· · · xenn −1
and α1i = xei i −1 − x1e1 −1 be a 2 × 2 minor of A for each i. Then, for i 6= 1,
ψ([T[n]\{i} ]) = α1i (f{i} − f{1} )([T[n]\{i} ]) = xei i − xe11 .
Now consider ideals of the form L2,n := L2,n (e1 , . . . , en ) = hxei i − xe11 : 2 ≤ i ≤ ni and
L = I2,n + L2,n . There is a short exact sequence of the form
(10)
ψ
0 → ker(ψ) → ωR/I2,n (−e) −
→ R/I2,n → R/L → 0.
Next we show that H̃ker(ψ) (t) = 0. By Remark 4.2,
n
h
X
1 i
e
H̃R/I2,n (t) − H̃R/J (t) = −t 1 +
= −te H̃R/I2,n (t−1 ).
ri − 1
t
i=1
By [2, Corollary. 4.4.6], we get H̃ωI2,n (t) = −H̃R/I2,n (t−1 ), hence
H̃R/I2,n (t) − H̃R/L (t) = te H̃ωI2,n (t).
The short exact sequence in (10) implies H̃ker(ψ) (t) = 0. Therefore ker(ψ) = 0. This proves
ψ is injective.
As an immediate consequence of Theorem 6.1, ωR/Im,n is identified with an Nn -graded
ideal of R/Im,n for a specific embedding.
Corollary 6.2. With notation as in Theorem 6.1, let L2,n = hxei i − xe11 : 2 ≤ i ≤ ni and
L = I2,n + L2,n . Then ωR/I2,n (−e) ≃ L/I2,n .
Proof. By Theorem 6.1, the following commutative diagram is
0
/
L/I2,n
/
R/I2,n
O
/
ωR/I2,n (−e)
ψ
/
R/I2,n
/
0
/
0
O
≃
≃
ψ
0
R/L
/
O
/
R/L
EMBEDDINGS OF CANONICAL MODULES AND RESOLUTIONS OF CONNECTED SUMS13
By the Snake Lemma, ωR/I2,n (−e) ≃ L/I2,n .
Now we state an application of Theorem 6.1 which gives a minimal free resolution of a
connected sum of Artinian rings of embedding dimension one.
Corollary 6.3. Let Ri = k[xi ]/hxei i +1 i, L2,n = hxei i − xe11 : 2 ≤ i ≤ ni, and L = I2,n + L2,n .
Suppose ψ : ωR/I2,n → R/I2,n satisfies Theorem 6.1. Then R/L is a Gorenstein Artin ring
such that R/L ≃ R1 #k . . . #k Rn . Furthermore, the mapping cone Cone(ψ) is a minimal free
R-resolution of R/L.
Proof. First note that soc(Ri ) = hxei i i. By Definition 2.1, we have R1 #k . . . #k Rn ≃ R/L, and
hence R/L is Gorenstein by Remark 2.2. Using Corollary 6.2, we get ωR/Im,n (−e) ≃ L/Im,n .
Let Fm,n
and Gm,n
be minimal free resolutions of R/Im,n and ωR/Im,n , respectively as in
•
•
Proposition 2.5 and Remark 2.8. Then Cone(ψ) is a minimal free resolution of R/L.
References
[1] H. Ananthnarayan, E. Celikbas, Jai Laxmi, Z. Yang, Decomposing Gorenstein rings as connected sums,
arXiv:1406.7600.
[2] W. Bruns and J. Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39,
Cambridge University Press, Cambridge, 1993.
[3] W. Fulton, Young tableaux, London Mathematical Society Student Texts, vol. 35, Cambridge University
Press, Cambridge, 1997.
[4] F. Galetto, On the Ideals Generated by all Squarefree Monomials of a Given Degree, arXiv:1609.06396.
[5] G. D. James, The Representation Theory of the Symmetric Groups, vol. 682, Lectures Notes in Mathematics, Springer, Berlin.
Department of Mathematics, West Virginia University, Morgantown, WV 26506.
E-mail address: ela.celikbas@math.wvu.edu
Department of Mathematics, I.I.T. Bombay, Powai, Mumbai 400076.
E-mail address: jailaxmi@math.iitb.ac.in
Department of Mathematics, University of Connecticut, Storrs, CT 06269.
E-mail address: jerzy.weyman@uconn.edu
| 0 |
Overcommitment in Cloud Services – Bin packing
with Chance Constraints
Maxime C. Cohen
Google Research, New York, NY 10011, maxccohen@google.com
Philipp W. Keller
arXiv:1705.09335v1 [cs.DS] 25 May 2017
Google, Mountain View, CA 94043, pkeller@google.com
Vahab Mirrokni
Google Research, New York, NY 10011, mirrokni@google.com
Morteza Zadimoghaddam
Google Research, New York, NY 10011, zadim@google.com
This paper considers a traditional problem of resource allocation, scheduling jobs on machines. One such
recent application is cloud computing, where jobs arrive in an online fashion with capacity requirements
and need to be immediately scheduled on physical machines in data centers. It is often observed that the
requested capacities are not fully utilized, hence offering an opportunity to employ an overcommitment
policy, i.e., selling resources beyond capacity. Setting the right overcommitment level can induce a significant
cost reduction for the cloud provider, while only inducing a very low risk of violating capacity constraints.
We introduce and study a model that quantifies the value of overcommitment by modeling the problem as
a bin packing with chance constraints. We then propose an alternative formulation that transforms each
chance constraint into a submodular function. We show that our model captures the risk pooling effect and
can guide scheduling and overcommitment decisions. We also develop a family of online algorithms that are
intuitive, easy to implement and provide a constant factor guarantee from optimal. Finally, we calibrate
our model using realistic workload data, and test our approach in a practical setting. Our analysis and
experiments illustrate the benefit of overcommitment in cloud services, and suggest a cost reduction of 1.5%
to 17% depending on the provider’s risk tolerance.
Key words : Bin packing, Approximation algorithms, Cloud computing, Overcommitment
1.
Introduction
Bin packing is an important problem with numerous applications such as hospitals, call centers,
filling up containers, loading trucks with weight capacity constraints, creating file backups and
more recently, cloud computing. A cloud provider needs to decide how many physical machines to
purchase in order to accommodate the incoming jobs efficiently. This is typically modeled as a bin
packing optimization problem, where one minimizes the cost of acquiring the physical machines
subject to a capacity constraint for each machine. The jobs are assumed to arrive in an online
fashion according to some vaguely specified arrival process. In addition, the jobs come with a
specific requirement, but the effective job size and duration are not exactly known until after the
1
2
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
actual scheduling has occurred. In practice, job size and duration can be estimated from historical
data. One straightforward way to schedule jobs is to assume that each job will fully utilize its
requirement (e.g., if a job requests 32 CPU cores, the cloud provider allocates this exact amount for
the job). However, there is empirical evidence, that most of the virtual machines do not use the full
requested capacity. This offers an opportunity for the cloud provider to employ an overcommitment
policy, i.e., to schedule sets of jobs with the total requirement exceeding the respective capacities
of physical machines. On one hand, the provider faces the risk that usage exceeds the physical
capacity, which can result in severe penalties (e.g., acquiring or reallocating machines on the fly,
canceling and rescheduling running jobs, mitigating interventions, etc.). On the other hand, if
many jobs do not fully utilize their requested resources, the provider can potentially reduce the
costs significantly. This becomes even more impactful in the cloud computing market, which has
become increasingly competitive in recent years as Google, Amazon, and Microsoft aim to replace
private data centers. “The race to zero price” is a commonly used term for this industry, where
cloud providers have cut their prices very aggressively. According to an article in Business Insider
in January 2015: “Amazon Web Services (AWS), for example, has cut its price 44 times during
2009-2015, while Microsoft and Google have both decreased prices multiple times to keep up with
AWS”. In January 2015, RBC Capital’s Mark Mahaney published a chart that perfectly captures
this trend and shows that the average monthly cost per gigabyte of RAM, for a set of various
workloads, has dropped significantly: AWS dropped prices 8% from Oct. 2013 to Dec. 2014, while
both Google and Microsoft cut prices by 6% and 5%, respectively, in the same period. Other
companies who charge more, like Rackspace and AT&T, dropped prices even more significantly.
As a result, designing the right overcommitment policy for servers has a clear potential to increase
the cloud provider profit. The goal of this paper is to study this question, and propose a model
that helps guiding this type of decisions. In particular, we explicitly model job size uncertainty to
motivate new algorithms, and evaluate them on realistic workloads.
Our model and approaches are not limited to cloud computing and can be applied to several
resource allocation problems. However, we will illustrate most of the discussions and applications
using examples borrowed from the cloud computing world. Note that describing the cloud infrastructure and hardware is beyond the scope of this paper. For surveys on cloud computing, see, for
example Dinh et al. (2013) and Fox et al. (2009).
We propose to model the problem as a bin packing with chance constraints, i.e., the total load
assigned to each machine should be below physical capacity with a high pre-specified probability.
Chance constraints are a commonly used modeling tool to capture risks and constraints on random variables (Charnes and Cooper (1963)). Introducing chance constraints to several continuous
optimization problems was extensively studied in the literature (see, e.g., Calafiore and El Ghaoui
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
3
(2006) and Delage and Ye (2010)). This paper is the first to incorporate capacity chance constraints
in the bin packing problem, and to propose efficient algorithms to solve the problem. Using some
results from distributionally robust optimization (Calafiore and El Ghaoui (2006)), we reformulate
the problem as a bin packing with submodular capacity constraints. Our reformulations are exact
under the assumption of independent Gaussian resource usages for the jobs. More generally, they
provide an upper bound and a good practical approximation in the realistic case where the jobs’
usages are arbitrarily distributed but bounded.
Using some machinery from previous work (see Goemans et al. (2009), and Svitkina and Fleischer
(2011)), we show that for the bin packing problem with general monotone submodular constraints,
it is impossible to find a solution within any reasonable factor from optimal (more precisely,
√
N
,
ln(N )
where N is the number of jobs). In this paper, we show that our problem can be solved using a class
of simple online algorithms that guarantee a constant factor of 8/3 from optimal (Theorem 2). This
class of algorithms includes the commonly used Best-Fit and First-Fit heuristics. We also develop
an improved constant guarantee of 9/4 for the online problem (Theorem 4), and a 2-approximation
for the offline version (Theorem 6). We further refine our results to the case where a large number
of jobs can be scheduled on each machine (i.e., each job has a small size relative to the machine
capacity). In this regime, our approach asymptotically converges to a 4/3 approximation. More
importantly, our model and algorithms allow us to draw interesting insights on how one should
schedule jobs. In particular, our approach (i) translates to a transparent recipe on how to assign
jobs to machines; (ii) explicitly exploits the risk pooling effect; and (iii) can be used to guide an
overcommitment strategy that significantly reduces the cost of purchasing machines.
We apply our algorithm to a synthetic but realistic workload inspired by historical production workloads in Google data centers, and show that it yields good performance. In particular,
our method reduces the necessary number of physical machines, while limiting the risk borne by
the provider. Our analysis also formalizes intuitions and provides insights regarding effective job
scheduling strategies in practical settings.
1.1.
Contributions
Scheduling jobs on machines can be modeled as a bin packing problem. Jobs arrive online with
some requirements, and the scheduler decides how many machines to purchase and how to schedule
the jobs. Assuming random job sizes and limited machine capacities, one can formulate the problem
as a 0/1 integer program. The objective is to minimize the number of machines required, subject
to constraints on the capacity of each machine. In this paper, we model the capacity constraints as
chance constraints, and study the potential benefit of overcommitment. The contributions of the
paper can be summarized as follows.
4
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
• Formulating the overcommitment bin packing problem.
We present an optimization formulation for scheduling jobs on machines, while allowing the provider
to overcommit. We first model the problem as Bin Packing with Chance Constraints (BPCC). Then,
we present an alternative Submodular Bin Packing (SMBP) formulation that explicitly captures
the risk pooling effect on each machine. We show that the SMBP is equivalent to the BPCC under
common assumptions (independent Gaussian usage distributions), and that it is distributionally
robust for usages with given means and diagonal covariance matrix). Perhaps most importantly
from a practical perspective, the SMBP provides an upper bound and a good approximation under
generic independent distributions over bounded intervals (see Proposition 1). This last setting is
most common in today’s cloud data centers, where virtual machines are sold as fixed-size units.
• Developing simple algorithms that guarantee a constant factor approximation from optimal.
We show that our (SMBP) problem can be solved by well-known online algorithms such as First-Fit
and Best-Fit, while guaranteeing a constant factor of 8/3 from optimal (Theorem 2). We further
refine this result in the case where a large number of jobs can be scheduled on each machine, and
obtain a 4/3 approximation asymptotically (Corollary 1). We also develop an improved constant
guarantee of 9/4 for the online problem using First-Fit (Theorem 4), and a 2 approximation for
the offline version (Theorem 6). We then use our analysis to infer how one should assign jobs to
machines, and show how to obtain a nearly optimal assignment (Theorem 5).
• Using our model to draw practical insights on the overcommitment policy.
Our approach translates to a transparent and meaningful recipe on how to assign jobs to machines
by clustering similar jobs in terms of statistical information. In addition, our approach explicitly
captures the risk pooling effect: as we assign more jobs to a given machine, the “safety buffer”
needed for each job decreases. Finally, our approach can be used to guide a practical overcommitment strategy, where one can significantly reduce the cost of purchasing machines by allowing a
low risk of violating capacity constraints.
• Calibrating and applying our model to a practical setting.
We use realistic workload data inspired by Google Compute Engine to calibrate our model and test
our results in a practical setting. We observe that our proposed algorithm outperforms other natural
scheduling schemes, and realizes a cost saving of 1.5% to 17% relative to the no-overcommitment
policy.
1.2.
Literature review
This paper is related to different streams of literature.
In the optimization literature, the problem of scheduling jobs on virtual machines has been
studied extensively, and the bin packing problem is a common formulation. Hundreds of papers
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
5
study the bin packing problem including many of its variations, such as 2D packing (e.g., Pisinger
and Sigurd (2005)), linear packing, packing by weight, packing by cost, online bin packing, etc.
The basic bin packing problem is NP-hard, and Delorme et al. (2016) provide a recent survey of
exact approaches. However, several simple online algorithms are often used in practice for largescale instances. A common variation is the problem where jobs arrive online with sizes sampled
independently from a known discrete distribution with integer support and must be immediately
packed onto machines upon arrival. The size of a job is known when it arrives, and the goal is
to minimize the number of non-empty machines (or equivalently, minimize the waste, defined as
the total unused space). For this variation, the sum-of-squares heuristic represents the state-ofthe-art. It is almost distribution-agnostic, and nearly universally optimal for most distributions by
achieving sublinear waste in the number of items seen (see, Csirik et al. (2006)). In Gupta and
Radovanovic (2012), the authors propose two algorithms based on gradient descent on a suitably
√
defined Lagrangian relaxations of the bin packing linear program that achieve additive O( N )
waste relative to the optimal policy. This line of work bounds the expected waste for general classes
of job size distribution in an asymptotic sense.
Worst-case analysis of (finite, deterministic) bin packing solutions has received a lot of attention
as well. For deterministic capacity constraints, several efficient algorithms have been proposed.
They can be applied online, and admit approximation guarantees in both online and offline settings. The offline version of the problem can be solved using (1 + )OP T + 1 bins in linear time
(de La Vega and Lueker (1981)). A number of heuristics can solve large-scale instances efficiently
while guaranteeing a constant factor cost relative to optimal. For a survey on approximation algorithms for bin packing, see for example Coffman Jr et al. (1996). Three such widely used heuristics
are First-Fit (FF), Next-Fit (NF) and Best-Fit (BF) (see, e.g., Bays (1977), Keller et al. (2012) and
Kenyon et al. (1996)). FF assigns the newly arrived job to the first machine that can accommodate
it, and purchase a new machine only if none of the existing ones can fit the new job. NF is similar to
FF but continues to assign jobs from the current machine without going back to previous machines.
BF uses a similar strategy but seeks to fit the newly arrived job to the machine with the smallest
remaining capacity. While one can easily show that these heuristics provide a 2-approximation
guarantee, improved factors were also developed under special assumptions. Dósa and Sgall (2013)
provide a tight upper bound for the FF strategy, showing that it never needs more than 1.7OP T
machines for any input. The offline version of the problem also received a lot of attention, and
the Best-Fit-Decreasing (BFD) and First-Fit-Decreasing (FFD) strategies are among the simplest
(and most popular) heuristics for solving it. They operate like BF and FF but first rank all the
jobs in decreasing order of size. Dósa (2007) show that the tight bound of FFD is 11/9OP T + 6/9.
6
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
Our problem differs as our goal is to schedule jobs before observing the realization of their size. In
this case, stochastic bin packing models, where the job durations are modeled as random variables,
are particularly relevant. Coffman et al. (1980) consider the problem and study the asymptotic and
convergence properties of the Next-Fit online algorithm. Lueker (1983) considers the case where
the job durations are drawn uniformly from intervals of the form [a, b], and derive a lower bound
on the asymptotic expected number of bins used in an optimum packing. However, unlike this and
other asymptotic results where the jobs’ sizes are known when scheduling occurs, we are interested
in computing a solution that is feasible with high probability before observing the actual sizes.
Our objective is to assign the jobs to as few machines as possible such that the set of jobs assigned
to each machine satisfies the capacity constraint with some given probability (say 99%). In other
words, we are solving a stochastic optimization problem, and studying/analyzing different simple
heuristic solutions to achieve this goal. To make the difference with the worst case analysis clear,
we note that the worst case analysis becomes a special case of our problem when the objective
probability threshold is set to 100% (instead of 99%, or any other number strictly less than 1). The
whole point of our paper is to exploit the stochastic structure of the problem in order to reduce
the scheduling costs via overcommitment.
In this paper, we consider an auxiliary deterministic bin packing problem with a linear cost but
non-linear modified capacity constraints. In Anily et al. (1994), the authors consider general cost
structures with linear capacity constraints. More precisely, the cost of a machine is assumed to
be a concave and monotone function of the number of jobs in the machine. They show that the
Next-Fit Increasing heuristic provides a worst-case bound of no more than 1.75, and an asymptotic
worst-case bound of 1.691.
The motivation behind this paper is similar to the overbooking policy for airline companies and
hotels. It is very common for airlines to overbook and accept additional reservations for seats on a
flight beyond the aircraft’s seating capacity1 . Airline companies (and hotels) employ an overbooking
strategy for several reasons, including: (i) no-shows (several passengers are not showing up to their
flight, and the airline can predict the no-show rate for each itinerary); (ii) increasing the profit
by reducing lost opportunities; and (iii) segmenting passengers (charging a higher price as we get
closer to the flight). Note that in the context of this paper, the same motivation of no-shows
applies. However, the inter-temporal price discrimination is beyond the scope of our model. Several
academic papers in operations research have studied the overbooking problem within the last forty
years (see, e.g., Rothstein (1971), Rothstein (1985), Weatherford and Bodily (1992), Subramanian
et al. (1999) and Karaesmen and Van Ryzin (2004)). The methodology is often based on solving
1
http://www.forbes.com/2009/04/16/airline-tickets-flights-lifestyle-travel-airlines-overbooked.html
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
7
a dynamic program incorporating some prediction of the no-show rate. In our problem, we face a
large-scale bin packing problem that needs to be solved online. Rather than deciding how many
passengers (jobs) to accept and at what price, cloud providers today usually aim to avoid declining
any reasonable workloads at a fixed list price2 .
This paper is also related to the robust optimization literature, and especially to distributionally
robust optimization. The goal is to solve an optimization problem where the input parameter distribution belongs to a family of distributions that share some properties (e.g., all the distributions
with the same mean and covariance matrix) and consider the worst-case within the given family
(concrete examples are presented in Section 2.4). Examples of such work include: Ghaoui et al.
(2003), Bertsimas and Popescu (2005), Calafiore and El Ghaoui (2006) and Delage and Ye (2010).
That work aims to convert linear or convex (continuous) optimization problems with a chance constraint into tractable formulations. Our paper shares a similar motivation but considers a problem
with integer variables. To the best of our knowledge, this paper is the first to develop efficient algorithms with constant approximation guarantees for the bin packing problem with capacity chance
constraints.
Large-scale cluster management in general is an important area of computer systems research.
Verma et al. (2015) provide a full, modern example of a production system. Among the work
on scheduling jobs, Sindelar et al. (2011) propose a model that also has a certain submodular
structure due to the potential for sharing memory pages between virtual machines (in contrast
to the risk-pooling effect modeled in this paper). Much experimental work seek to evaluate the
real-world performance of bin packing heuristics that also account for factors such as adverse
interactions between jobs scheduled together, and the presence of multiple contended resources
(see for example Rina Panigrahy (2011) and Alan Roytman (2013)). While modeling these aspects
is likely to complement the resource savings achieved with the stochastic model we propose, these
papers capture fundamentally different efficiency gains arising from technological improvements
and idiosyncratic properties of certain types (or combinations) of resources. In this paper, we limit
our attention to the benefit and practicality of machine over-commitment in the case where a single
key resource is in short supply. This applies directly to multi-resource settings if, for example, the
relatively high cost of one resource makes over-provisioning the others worthwhile, or if there is
simply an imbalance between the relative supply and demand for the various resources making one
of the resources scarce.
Structure of the paper. In Section 2, we present our model and assumptions. Then, we
present the results and insights for special cases in Section 3. In Section 4, we consider the general
2
The ”spot instances” provided by Amazon and other heavily discounted reduced-availability services are notable
exceptions.
8
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
case and develop a class of efficient approximation algorithms that guarantee a constant factor
from optimal. In Section 5, we exploit the structure of the problem in order to obtain a nearly
optimal assignment and to draw practical insights. In Sections 6 and 7, we present extensions and
computational experiments using realistic data respectively. Finally, our conclusions are reported
in Section 8. Most of the proofs of the Theorems and Propositions are relegated to the Appendix.
2.
Model
In this section, we present the model and assumptions we impose. We start by formulating the
problem we want to solve, and then propose an alternative formulation. As we previously discussed,
job requests for cloud services (or any other resource allocation problem) come with a requested
capacity. This can be the memory or CPU requirements for virtual machines in the context of
cloud computing, or job duration in more traditional scheduling problems where jobs are processed
sequentially3 . We refer to Aj as the size of job j and assume that Aj is a random variable. Historical
data can provide insight into the distribution of Aj . For simplicity, we first consider the offline
version of the problem where all the jobs arrive simultaneously at time 0, and our goal is to pack
the jobs onto the minimum possible number of machines. Jobs cannot be delayed or preempted.
The methods we develop in this paper can be applied in the more interesting online version of
the problem, as we discuss in Section 4. We denote the capacity of machine i by Vi . Motivated by
practical problems, and in accordance with prior work, we assume that all the machines have the
same capacity, i.e., Vi = V ; ∀i. In addition, each machine costs ci = c, and our goal is to maximize
the total profit (or equivalently, minimize the number of machines), while scheduling all the jobs
and satisfying the capacity constraints. Note that we consider a single dimensional problem, where
each job has one capacity requirement (e.g., the number of virtual CPU cores or the amount of
memory). Although cloud virtual machine packing may be modeled as a low-dimensional vector bin
packing problem (see for example, Rina Panigrahy (2011)), one resource is often effectively binding
and/or more critical so that focusing on it offers a much larger opportunity for overcommitment4 .
3
Although there is also a job duration in cloud computing, it is generally unbounded and hence, even less constrained
than the resource usage from the customer’s perspective. The duration is also less important than the resource usage,
since most virtual machines tend to be long-lived, cannot be delayed or pre-empted, and are paid for by the minute.
In contrast, over-allocating unused, already paid-for resources can have a large impact on efficiency.
4
Insofar as many vector bin packing heuristics are actually straightforward generalizations of the FF, NF and BF
rules, it will become obvious how our proposed algorithm could similarly be adapted to the multi-resource setting in
Section 4, although we do not pursue this idea in this paper.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
2.1.
9
Bin packing problem
For the case where Aj is deterministic, we obtain the classical deterministic bin packing problem:
B = min
xij ,yi
s.t.
N
X
yi
i=1
N
X
j=1
N
X
Aj xij ≤ V yi
∀i
(DBP)
xij = 1
∀j
i=1
xij ∈ {0, 1}
yi ∈ {0, 1}
∀i, j
∀i
For the offline version, we have a total of N jobs and we need to decide which machines to
use/purchase (captured by the decision variable yi that is equal to 1, if machine i is purchased
and 0 otherwise). The solution is a B-partition of the set {1, 2, . . . , N } that satisfies the capacity
constraints. The decision variable xij equals one if job j is assigned to machine i and zero otherwise.
As we discussed in Section 1.2, there is an extensive literature on the DBP problem and its many
variations covering both exact algorithms as well and approximation heuristics with performance
bounds.
The problem faced by a cloud provider is typically online in nature since jobs arrive and depart
over time. Unfortunately, it is not possible to continually re-solve the DBP problem as the data is
updated for both practical and computational reasons. Keeping with the majority of prior work,
we start by basing our algorithms on static, single-period optimization formulations like the DBP
problem, rather than explicitly modeling arrivals and departures. The next section explains how,
unlike prior work, our single-period optimization model efficiently captures the uncertainty faced
by a cloud provider. We will consider both the online and offline versions of our model.
We remark that, while our online analysis considers sequentially arriving jobs, none of our results
explicitly considers departing jobs. This is also in line with the bin-packing literature, where results
usually apply to very general item arrival processes {Aj }, but it is typically assumed that packed
items remain in their assigned bins. In practice, a large cloud provider is likely to be interested
in a steady-state where the distribution of jobs in the systems is stable over time (or at least
predictable), even if individual jobs come and go. Whereas the online model with arrivals correctly
reflects that the scheduler cannot optimize to account for unseen future arrivals, it is unclear if
and how additionally modeling departures would affect a system where the overall distribution of
jobs remains the same over time. We therefore leave this question open. Note that several works
consider bin-packing with item departures (see, e.g., Stolyar and Zhong (2015) and the references
therein). In this work, the authors design a simple greedy algorithm for general packing constraints
and show that it can be asymptotically optimal.
10
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
2.2.
Chance constraints
The DBP problem suffers from the unrealistic assumption that the jobs’ sizes Aj are deterministic. In reality, jobs’ requirements (or durations) can be highly unpredictable and quite volatile,
especially from the perspective of a cloud provider with no control over the software executed in
a virtual machine. Ensuring that the capacity constraints are satisfied for any realization of Aj
generally yields a conservative outcome. For example, if the jobs’ true requirements are Bernoulli
random variables taking on either 0.3 or 1.0 with equal probability, one needs to plan as if each
job consumes a capacity of 1.0. By overcommitting resources, the provider can reduce the cost
significantly. Caution is required however, since overcommitting can be very expensive if not done
properly. Planning according to the expected value (in the previous simple example, 0.65), for
instance, would result in capacity being too tight on many machines. Specifically, for large machines,
the realized requirements could exceed capacity up to half of the time. Depending on the specific
resource and the degree of violation, such performance could be catastrophic for a cloud service
provider. Concretely, sustained CPU contention among virtual machines would materially affect
customers’ performance metrics, whereas a shortage of available memory could require temporarily
“swapping” some data to a slower storage medium with usually devastating consequences on performance. Other mitigations are possible, including migrating a running virtual machine to another
host, but these also incur computational overhead for the provider and performance degradation for
the customer. In the extreme case where overly optimistic scheduling results in inadequate capacity
planning, there is even a stock-out risk where it is no longer possible to schedule all customers’ jobs
within a data center. With this motivation in mind, our goal is to propose a formulation that finds
the right overcommitment policy. We will show that by slightly overcommitting (defined formally
in Section 2.3), one can reduce the costs significantly while satisfying the capacity constraints with
high probability.
While not strictly required by our approach, in practice, there is often an upper bound on Aj ,
denoted by Āj . In the context of cloud computing, Āj is the requested capacity that a virtual
machine is not allowed to exceed (32 CPU cores, or 128 GB of memory, say). However, the job
may end up using much less, at least for some time. If the cloud provider schedules all the jobs
according to their respective upper bounds Āj , then there is no overcommitment. If the cloud
provider schedules all the jobs according to some sizes smaller than the Āj , then some of the
machines may be overcommitted.
We propose to solve a bin packing problem with capacity chance constraints. Chance constraints
are widely used in optimization problems, starting with Charnes and Cooper (1963) for linear
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
11
programs, and more recently in convex optimization (see, e.g., Nemirovski and Shapiro (2006)) and
in finance (see, e.g., Abdelaziz et al. (2007)). In this case, the capacity constraints are replaced by:
P
N
X
Aj xij ≤ V yi ≥ α,
(1)
j=1
where α represents the confidence level of satisfying the constraint (α = 0.999, say) and is exogenously set by the cloud provider depending on considerations such as typical job’s running time
and contractual agreements. Note that when α = 1, this corresponds to the setting with no overcommitment, or in other words, to the worst-case solution that covers all possible realizations of
all the Aj ’s. One of our goals is to study the trade-off between the probability of violating physical
capacity and the cost reduction resulting from a given value of α.
The problem becomes the bin packing with chance constraints, parameterized by α:
B(α) = min
N
X
xij ,yi
s.t.
yi
i=1
P
N
X
Aj xij ≤ V yi ≥ α
∀i
j=1
N
X
xij = 1
(BPCC)
∀j
i=1
xij ∈ {0, 1}
yi ∈ {0, 1}
2.3.
∀i, j
∀i
Overcommitment
One can define the overcommitment level as follows. Consider two possible (equivalent) benchmarks. First, one can solve the problem for α = 1, and obtain a solution (by directly solving the
IP or any other heuristic method) with objective B(1). Then, we solve the problem for the desired
value α < 1. The overcommitment benefit can be defined as 0 < B(α)/B(1) ≤ 1. It is also interesting
to compare the two different jobs assignments.
The second definition goes as follows. We define the overcommitment factor as the amount of
sellable capacity divided by the physical capacity of machines in the data center, that is:
P
j Āj
OCF (α) , P
.
i V yi
Since we assume that all the machines have the same capacity and cost, we can write:
P
P
Āj
j Āj
OCF (α) =
≥
= OCF (1).
V B(α) V B(1)
j
12
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
Note that OCF(1) is (generally strictly) less than one, as the bin packing overhead prevents the
sale of all resources5 . Then, we have:
OCF (1) B(α)
=
.
OCF (α) B(1)
For illustration purposes, consider the following simple example with N = 70 1-core jobs. The jobs
are independent and Bernoulli distributed with probability 0.5. In particular, the jobs are either
high usage (i.e., fully utilize the 1 core), or low usage (in this case, idle). Each machine has a
capacity V = 48 cores. Without overcommitting, we need 2 machines, i.e., B(1) = 2. What happens
if we schedule all the jobs in a single machine? In this case, one can reduce the cost (number of
machines) by half, while satisfying the capacity constraint with probability 0.9987. In other words,
B(0.99) = 1. The overcommitment benefit in this simple example is clear. Our goal is to formalize
a systematic way to overcommit in more complicated and realistic settings.
Note that overcommitment may lead to Service Level Agreement (SLA) violations. This paper
does not discuss in detail the SLAs (with some possible associated metrics), and the corresponding
estimation/forecast procedures as they are usually application and resource specific. Instead, this
research treats a general Virtual Machine (VM) scheduling problem. More precisely, our context
is that of a cloud computing provider with limited visibility into the mix of customer workloads,
and hard SLAs. While the provider does track numerous service-level indicators, they are typically
monotonic in the resource usage on average (we expect more work to translate to worse performance). Therefore, we believe that it is reasonable to rely on resource utilization as the sole metric
in the optimization problem.
2.4.
A variant of submodular bin packing
In this section, we propose an alternative formulation that is closely related to the (BPCC) problem.
Under some mild assumptions, we show that the latter is either exactly or approximately equivalent
to the following submodular bin packing problem:
BS (α) = min
xij ,yi
s.t.
N
X
yi
i=1
PN
j=1 µj xij + D(α)
N
X
qP
N
j=1 bj xij
≤ V yi
∀i
(SMBP)
xij = 1
∀j
i=1
xij ∈ {0, 1}
yi ∈ {0, 1}
5
∀i, j
∀i
Technical note: other production overheads such as safety stocks for various types of outages and management
overheads, are generally also included in the denominator. For the purpose of this paper, we omit them.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
13
The difference between the (BPCC) and the (SMBP) problems is the way the capacity constraints
are written. Here, we have replaced each chance constraint with a linear term plus a square root
term. These constraints are submodular with respect to the vector x. The variable µj denotes the
expected value of Aj . In what follows, we will consider different definitions of bj and D(α) in three
different settings. The first two are concrete motivational examples, whereas the third one is a
generalization. In each case, we formally show the relation between the (BPCC) and the (SMBP)
problems.
1. Gaussian case: Assume that the random variables Aj are Gaussian and independent. In this
PN
case, the random variable Z = j=1 Aj xij for any given binary vector x is Gaussian, and therefore,
one can use the following simplification:
P
N
X
Aj xij ≤ V yi = P Z ≤ V yi ≥ α.
j=1
For each machine i, constraint (1) becomes:
N
X
v
u N
uX
−1
σj2 xij ≤ V yi ,
µj xij + Φ (α) · t
(2)
j=1
j=1
where Φ−1 (·) is the inverse CDF of a normal N (0, 1), µj = E[Aj ] and σj2 = Var(Aj ). Note that we
have used the fact that x is binary so that x2ij = xij . Consequently, the (BPCC) and the (SMBP)
problems are equivalent with the values bj = σj2 and D(α) = Φ−1 (α).
When the random variables Aj are independent but not normally distributed, if there are a
large number of jobs per machine, one can apply the Central Limit Theorem and obtain a similar
approximate argument. In fact, using a result from Calafiore and El Ghaoui (2006), one can extend
this equivalence to any radial distribution6 .
2. Hoeffding’s inequality: Assume that the random variables Aj are independent with a finite
support [Aj , Aj ], 0 ≤ Aj < Aj with mean µj . As we discussed, one can often know the value of Aj
and use historical data to estimate µj and Aj (we discuss this in more detail in Section 7). Assume
PN
that the mean usages fit on each machine, i.e.,
j=1 xij µj < yi Vi . Then, Hoeffding’s inequality
states that:
P
N
X
Aj xij ≤ V yi ≥ 1 − e
P
2
−2[V yi − N
j=1 µj xij ]
PN
2
(A −Aj )
j=1 j
.
j=1
Equating the right hand side to α, we obtain:
PN
−2[V yi − j=1 µj xij ]2
= ln(1 − α),
PN
b
x
j
ij
j=1
6
Radial distributions include all probability densities whose level sets are ellipsoids. The formal mathematical definition can be found in Calafiore and El Ghaoui (2006).
14
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
where bj = (Aj − Aj )2 represents the range of job j’s usage. Re-arranging the equation, we obtain
for each machine i:
N
X
v
u N
uX
µj xij + D(α)t
bj xij ≤ V yi ,
j=1
where in this case, D(α) =
p
(3)
j=1
−0.5 ln(1 − α). Note that in this setting the (BPCC) and the (SMBP)
problems are not equivalent. We only have that any solution of the latter is a feasible solution for
the former. We will demonstrate in Section 7 that despite being very conservative, this formulation
based on Hoeffding’s inequality actually yields good practical solutions.
The next case is a generalization of the last two.
3. Distributionally robust formulations: Assume that the random variables Aj are independent with some unknown distribution. We only know that this distribution belongs to a family
of probability distributions D. We consider two commonly used examples of such families. First,
we consider the family D1 of distributions with a given mean and (diagonal) covariance matrix,
µ and Σ, respectively. Second, we look at D2 , the family of generic distributions of independent
random variables over bounded intervals [Aj , Aj ].
In this setting, the chance constraint is assumed to be enforced robustly with respect to the
entire family D of probability distributions on A = (A1 , A2 , . . . , AN ), meaning that:
inf P
A∼D
N
X
Aj xij ≤ V yi ≥ α.
(4)
j=1
In this context, we have the following result.
Proposition 1. Consider the robust bin packing problem with the capacity chance constraints
(4) for each machine i. Then, for any α ∈ (0, 1), we have:
• For the family D1 of distributions with a given mean and diagonal covariance matrix, the
p
robust problem is equivalent to the (SMBP) with bj = σj2 and D1 (α) = α/(1 − α).
• For the family D2 of generic distributions of independent random variables over bounded inter-
vals, the robust problem can be approximated by the (SMBP) with bj = (Aj − Aj )2 and D2 (α) =
p
−0.5 ln(1 − α).
The details of the proof are omitted for conciseness. In particular, the proof for D1 is analogous to
an existing result in continuous optimization that converts linear programs with a chance constraint
into a linear program with a convex second-order cone constraint (see Calafiore and El Ghaoui
(2006) and Ghaoui et al. (2003)). The proof for D2 follows directly from the fact that Hoeffding’s
inequality applies for all such distributions, and thus for the infimum of the probability.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
15
We have shown that the (SMBP) problem is a good approximation for the bin packing problem
with chance constraints. For the case of independent random variables with a given mean and
covariance, the approximation is exact and for the case of distributions over independent bounded
intervals, the approximation yields a feasible solution. We investigate practical settings in Section
7, and show that these approximate formulations all yield good solutions to the original problem.
From now on, we consider solving the (SMBP) problem, that is repeated here for convenience:
BS (α) = min
xij ,yi
s.t.
N
X
yi
i=1
PN
j=1 µj xij + D(α)
N
X
qP
N
j=1 bj xij
≤ V yi
∀i
(SMBP)
xij = 1
∀j
i=1
xij ∈ {0, 1}
yi ∈ {0, 1}
∀i, j
∀i
As discussed, the capacity constraint is now replaced by the following equation, called the modified
capacity constraint:
N
X
j=1
v
u N
uX
µj xij + D(α)t
bj xij ≤ V yi .
(5)
j=1
One can interpret equation (5) as follows. Each machine has a capacity V . Each job j consumes
capacity µj in expectation, as well as an additional buffer to account for the uncertainty. This
buffer depends on two factors: (i) the variability of the job, captured by the parameter bj ; and (ii)
the acceptable level of risk through D(α). The function D(α) is increasing in α, and therefore we
impose a stricter constraint as α approaches 1 by requiring this extra buffer to be larger.
Equation (5) can also be interpreted as a risk measure applied by the scheduler. For each machine
PN
i, the total (random) load is j=1 Aj xij . If we consider that µj represents the expectation and bj
qP
PN
N
corresponds to the variance, then j=1 µj xij and
j=1 bj xij correspond to the expectation and
the standard deviation of the total load on machine i respectively. As a result, the right hand side of
equation (5) can be interpreted as an adjusted risk utility, where D(α) is the degree of risk aversion
of the scheduler. The additional amount allocated for job j can be interpreted as a safety buffer to
account for the uncertainty and for the risk that the provider is willing to bear. As we discussed,
this extra buffer decreases with the number of jobs assigned to the same machine. In Section 4, we
develop efficient methods to solve the (SMBP) with analytical performance guarantees.
16
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
2.5.
Two naive approaches
In this section, we explore the limitations of two approaches that come to mind. The first attempt
is to rewrite the problem as a linear integer program: the decision variables are all binary and the
non-linearity in (SMBP) can actually be captured by common modeling techniques, as detailed
in Appendix A. Unfortunately, solving this IP is not a viable option. Similarly as for the classical
deterministic bin packing problem, solving even moderately large instances with commercial solvers
takes several hours. Moreover, applying the approach to smaller, specific toy instances provides
little insight about the assignment policy, and how the value of α affects the solution. Since our
goal is to develop practical strategies for the online problem, we chose not to further pursue exact
solutions.
The second potential approach is to develop an algorithm for a more general problem: the
bin packing with general monotone submodular capacity constraints. Unfortunately, using some
machinery and results from Goemans et al. (2009) and Svitkina and Fleischer (2011), we next show
that it is in fact impossible to find a solution within any reasonable factor from optimal.
Theorem 1. Consider the bin packing problem with general monotone submodular capacity constraints for each machine. Then, it is impossible to guarantee a solution within a factor better than
√
N
ln(N )
from optimal.
The proof can be found in Appendix A. We will show that the (SMBP) problem that we consider
is more tractable as it concerns only a specific class of monotone submodular capacity constraints
that capture the structure of the chance-constrained problem. In the next session, we start by
addressing simple special cases in order to draw some structural insights.
3.
Results and insights for special cases
In this section, we consider the (SMBP) problem for some given µj , bj , N and D(α). Our goals are
to: (i) develop efficient approaches to solve the problem; (ii) draw some insights on how to schedule
the different jobs and; (iii) study the effect of the different parameters on the outcome. This will
ultimately allows us to understand the impact of overcommitment in resource allocation problems,
such as cloud computing.
3.1.
Identical distributed jobs
We consider the symmetric setting where all the random variables Aj have the same distribution,
such that µj = µ and bj = b in the (SMBP) problem. By symmetry, we only need to find the number
of jobs n to assign to each machine. Since all the jobs are interchangeable, our goal is to assign as
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
17
many jobs as possible in each machine. In other words, we want to pick the largest value of n such
that the constraint (5) is satisfied, or equivalently:
√
nµ + D(α) nb ≤ V
[V − nµ]2
D(α)2 ≤
.
nb
For a given value of α, this is the largest integer smaller than:
n(α) =
p
1
V
+ 2 bD(α)2 − b2 D(α)4 + 4bD(α)2 V µ .
µ 2µ
(6)
• For a given value of α, the number of jobs n(α) increases with V /µ. Indeed, since µ represents
the expected job size, increasing the ratio V /µ is equivalent to increasing the number of ”average”
jobs a machine can host. If the jobs are smaller or the machines larger, one can fit more jobs per
machine, as expected.
• For a given value of V /µ, n(α) is a non-increasing function of α. When α increases, it means
that we enforce the capacity constraint in a stricter manner (recall that α = 1 corresponds to the
case without overcommitment). As a result, the number of jobs per machine cannot increase.
• For given values of α and V /µ, n(α) is a non-increasing function of b. Recall that the parameter
b corresponds to some measure of spread (the variance in the Gaussian setting, and the range for
distributions with bounded support). Therefore, when b increases, it implies that the jobs’ resource
usage is more volatile and hence, a larger buffer is needed. Consequently, the number of jobs cannot
increase when b grows.
• For given values of α and V , n(α) is non-increasing with
√
b/µ. The quantity
√
b/µ represents
the coefficient of variation of the random job size in the Gaussian case, or a similarly normalized
measure of dispersion in other cases. Consequently, one should be able to fit less jobs, as the
variability increases.
The simple case of identically distributed jobs allows us to understand how the different factors
affect the number of jobs that one can assign to each machine. In Figure 1, we plot equation (6)
for an instance with A = 1, A = 0.3, µ = 0.65, V = 30 and 0.5 ≤ α < 1. The large dot for α = 1
in the figure represents the case without overcommitment (i.e., α = 1). Interestingly, one can see
that when the value of α approaches 1, the benefit of allowing a small probability of violating the
capacity constraint is significant, so that one can increase the number of jobs per machine. In this
case, when α = 1, we can fit 30 jobs per machine, whereas when α = 0.992, we can fit 36 jobs,
hence, an improvement of 20%. Note that this analysis guarantees that the capacity constraint is
satisfied with at least probability α. As we will show in Section 7 for many instances, the capacity
constraint is satisfied with an even higher probability.
18
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
Figure 1
Parameters: A = 1, A = 0.3, µ = 0.65, V = 30
Alternatively, one can plot α as a function of n (see Figure 2a for an example with different
values for V /µ). As expected, the benefit of overcommitting increases with V /µ, i.e., one can fit a
larger number of jobs per machine. In our example, when V /µ = 25, by scheduling jobs according
to A (i.e., α = 1, no overcommitment), we can schedule 14 jobs, whereas if we allow a 0.1%
violation probability, we can schedule 17 jobs. Consequently, by allowing 0.1% chance of violating
the capacity constraint, one can save more than 20% in costs.
(a) Parameters: A = 1, A = 0.3, µ = 0.65
Figure 2
(b) Parameters: V = 30
Example for identically distributed jobs
We next discuss how to solve the problem for the case with a small number of different classes
of job distributions.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
3.2.
19
Small number of job distributions
We now consider the case where the random variables Aj can be clustered in few different categories.
For example, suppose standard clustering algorithms are applied to historical data to treat similar
jobs as a single class with some distribution of usage. For example, one can have a setting with
four types of jobs: (i) large jobs with no variability (µj is large and bj is zero); (ii) small jobs with
no variability (µj is small and bj is zero); (ii) large jobs with high variability (both µj and bj are
large); and (iv) small jobs with high variability (µj is small and bj is high). In other words, we
have N jobs and they all are from one of the 4 types, with given values of µj and bj . The result for
this setting is summarized in the following Observation (the details can be found in Appendix C).
Observation 1. In the case where the number of different job classes is not too large, one can
solve the problem efficiently as a cutting stock problem.
The resulting cutting stock problem (see formulation (14) in Appendix C) is well studied in many
contexts (see Gilmore and Gomory (1961) for a classical approach based on linear programming,
or the recent survey of Delorme et al. (2016)). For example, one can solve the LP relaxation of
(14) and round the fractional solution. This approach can be very useful for cases where the cloud
provider have enough historical data, and when the jobs can all be regrouped into a small number
of different clusters. This situation is sometimes realistic but not always. Very often, grouping all
possible customer job profiles into a small number of classes, each described by a single distribution
is likely unrealistic in many contexts. For example, virtual machines are typically sold with 1, 2, 4,
8, 16, 32 or 64 CPU cores, each with various memory configurations, to a variety of customers with
disparate use-cases. Aggregating across these jobs is already dubious, before considering differences
in their usage means and variability. Unfortunately, if one decides to use a large number of job
classes, solving a cutting stock problem is not scalable. In addition, this approach requires advance
knowledge of the number of jobs of each class and hence, cannot be applied to the online version
of our problem.
4.
Online constant competitive algorithms
In this section, we analyze the performance of a large class of algorithms for the online version of
problem (SMBP). We note that the same guarantees hold for the offline case, as it is just a simpler
version of the problem. We then present a refined result for the offline problem in Section 6.1.
4.1.
Lazy algorithms are 38 -competitive
An algorithm is called lazy, if it does not purchase/use a new machine unless necessary. The formal
definition is as follows.
Definition 1. We call an online algorithm lazy if upon arrival of a new job, it assigns the job
to one of the existing (already purchased) machines given the capacity constraints are not violated.
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Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
In other words, the algorithm purchases a new machine if and only if non of the existing machines
can accomodate the newly arrived job.
Several commonly used algorithms fall into this category, e.g., First-Fit, Best-Fit, Next-Fit,
greedy type etc. Let OP T be the optimal objective, i.e., the minimum number of machines needed
to serve all the jobs {1, 2, · · · , N }. Recall that in our problem, each job 1 ≤ j ≤ N , has two characteristics: µj and bj which represent the mean and the uncertain part of job j respectively. For a
qP
P
set of jobs S, we define the corresponding cost Cost(S) to be j∈S µj +
j∈S bj . Without loss of
generality, we can assume (by normalization of all µj and bj ) that the capacity of each machine is
1 and that D(α) is also normalized to 1. We call a set S feasible, if its cost is at most the capacity
limit 1. In the following Theorem, we show that any lazy algorithm yields a constant approximation
for the (SBBP) problem.
Theorem 2. Any lazy algorithm ALG purchases at most 83 OP T machines, where OP T is the
optimum number of machines to serve all jobs.
Proof.
Let m be the number of machines that ALG purchases when serving jobs {1, 2, · · · , N }.
For any machine 1 ≤ i ≤ m, we define Si to be the set of jobs assigned to machine i. Without loss
of generality, we assume that the m machines are purchased in the order of their indices. In other
words, machines 1 and m are the first and last purchased ones respectively.
For any pair of machines 1 ≤ i < i0 ≤ m, we next prove that the set Si ∪ Si0 is infeasible for the
qP
P
modified capacity constraint, i.e., j∈Si ∪S 0 µj +
j∈Si ∪S 0 bj > 1. Let j be the first job assigned
i
i
to machine i0 . Since ALG is lazy, assigning j to machine i upon its arrival time was not feasible,
i.e., the set {j } ∪ Si is infeasible. Since we only assign more jobs to machines throughout the course
of the algorithm, and do not remove any job, the set Si0 ∪ Si is also infeasible.
In the next Lemma, we lower bound the sum of µj + bj for the jobs in an infeasible set.
Lemma 1. For any infeasible set T , we have
P
j∈T (µj
+ bj ) > 34 .
qP
P
Proof. For any infeasible set T , we have by definition j∈T µj +
j∈T bj > 1. We denote
qP
P
x = j∈T µj , and y =
j∈T bj . Then, y > 1 − x. If x is greater than 1, the claim of the lemma
holds trivially. Otherwise, we obtain:
1
3 3
x + y 2 > x + (1 − x)2 = x2 − x + 1 = (x − )2 + ≥ .
2
4 4
We conclude that
P
j∈T (µj
+ bj ) > 43 .
As discussed, for any pair of machines i < i0 , the union of their sets of jobs Si ∪ Si0 is an infeasible
set that does not fit in one machine. We now apply the lower bound from Lemma 1 for the infeasible
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
set Si ∪ Si0 and imply
P
j∈Si ∪Si0 (µj
+ bj ) > 34 . We can sum up this inequality for all
m
2
21
pairs of
machines i and i0 to obtain:
X
X
1≤i<i0 ≤m j∈Si ∪Si0
3 m
(µj + bj ) >
.
4 2
We claim that the left hand side of inequality (7) is equal to (m − 1)
(7)
PN
j=1 (µj
+ bj ). We note that
for each job j ∈ Sk , the term µj + bj appears k − 1 times in the left hand side of inequality (7) when
i0 is equal to k. In addition, this term also appears m − k times when i is equal to k. Therefore,
every µj + bj appears k − 1 + m − k = m − 1 times, which is independent of the index k of the
machine that contains job j. As a result, we obtain:
N
X
(µj + bj ) >
j=1
m
3m
3
.
=
4(m − 1) 2
8
On the other hand, we use the optimal assignment to upper bound the sum
PN
j=1 (µj +bj ),
and relate
m to OP T . Let T1 , T2 , · · · , TOP T be the optimal assignment of all jobs to OP T machines. Since Ti is
qP
P
P
a feasible set, we have j∈Ti µj +
j∈Ti bj ≤ 1, and consequently, we also have
j∈Ti (µj + bj ) ≤ 1.
PN
POP T P
Summing up for all the machines 1 ≤ i ≤ OP T , we obtain: j=1 (µj + bj ) = i=1
j∈Ti (µj + bj ) ≤
OP T . We conclude that:
OP T ≥
N
X
j=1
This completes the proof of m <
8OP T
3
(µj + bj ) >
3m
.
8
.
Theorem 2 derives an approximation guarantee of 8/3 for any lazy algorithm. In many practical
settings, one can further exploit the structure of the set of jobs, and design algorithms that achieve
better approximation factors. For example, if some jobs are usually larger relative to others, one
can incorporate this knowledge into the algorithm. We next describe the main intuitions behind the
8/3 upper bound. In the proof of Theorem 2, we have used the following two main proof techniques:
P
• First, we show a direct connection between the feasibility of a set S and the sum j∈S (µj + bj ).
P
In particular, we prove that j∈S (µj + bj ) ≤ 1 for any feasible set, and greater than 3/4 for any
infeasible set. Consequently, OP T cannot be less than the sum of µj + bj for all jobs. The gap of
4/3 between the two bounds contributes partially to the final upper bound of 8/3.
• Second, we show that the union of jobs assigned to any pair of machines by the lazy algorithm
is an infeasible set, so that their sum of µj + bj should exceed 3/4. One can then find m/2 disjoint
pairs of machines, and obtain a lower bound of 3/4 for the sum µj + bj for each pair. The fact
that we achieve this lower bound for every pair of machines (and not for each machine) contributes
another factor of 2 to the approximation factor, resulting to
4
3
× 2 = 83 .
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Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
Note that the second loss of a factor of 2 follows from the fact that the union of any two machines
forms an infeasible set and nothing stronger. In particular, all machines could potentially have a
cost of 1/2 + for a very small , and make the above analysis tight. Nevertheless, if we assume
that each machine is nearly full (i.e., has Cost close to 1), one can refine the approximation factor.
Theorem 3. For any 0 ≤ ≤ 0.3, if the lazy algorithm ALG assigns all the jobs to m machines
such that Cost(Si ) ≥ 1 − for every 1 ≤ i ≤ m, we have m ≤ ( 34 + 3)OP T , i.e., a ( 43 + 3) approximation guarantee.
P
To simplify the analysis, we denote β to be 1 − . For a set Si , we define x = j∈Si µj
qP
and y =
j∈Si bj . Since Cost(Si ) is at least β, we have x + y ≥ β. Assuming x ≤ β, we have:
Proof.
2β − 1 2
1
1 3
(µj + bj ) = x + y 2 ≥ x + (β − x)2 = x −
+ β − ≥ β − = − ,
2
4
4 4
j∈S
X
i
where the first equality is by the definition of x and y, the second inequality holds by x + y ≥ α,
P
and the rest are algebraic manipulations. For x > β, we also have j∈Si (µj + bj ) ≥ x > β > 43 − .
PN
PN
We conclude that j=1 (µj + bj ) ≥ m × ( 34 − ). We also know that OP T ≥ j=1 (µj + bj ), which
implies that m ≤
OP T
3/4−
≤ OP T ( 43 + 3) for ≤ 0.3.
A particular setting where the condition of Theorem 3 holds is when the capacity of each machine
p
is large compared to all jobs, i.e., max1≤j≤N µj + bj is at most . In this case, for each machine
i 6= m (except the last purchased machine), we know that there exists a job j ∈ Sm (assigned to the
last purchased machine m) such that the algorithm could not assign j to machine i. This means
that Cost(Si ∪ {j }) exceeds one. Since Cost is a subadditive function, we have Cost(Si ∪ {j }) ≤
Cost(Si ) + Cost({j }). We also know that Cost({j }) ≤ which implies that Cost(Si ) > 1 − .
Remark 1. As elaborated above, there are two main sources for losses in the approximation
factors: non-linearity of the cost function that can contribute up to 4/3, and machines being only
partially full that can cause an extra factor of 2 which in total implies the 8/3 approximation
guarantee. In the classical bin packing case (i.e., bj = 0 for all j), the cost function is linear, and
the non-linearity losses in approximation factors fade. Consequently, we obtain that (i) Theorem
2 reduces to a 2 approximation factor; and (ii) Theorem 3 reduces to a (1 + ) approximation
factor, which are both consistent with known results from the literature on the classical bin packing
problem.
Theorem 3 improves the bound for the case where each machine is almost full. However, in
practice machines are often not full. In the next section, we derive a bound as a function of the
minimum number of jobs assigned to the machines.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
4.2.
23
Algorithm First-Fit is 94 -competitive
So far, we considered the general class of lazy algorithms. One popular algorithm in this class
(both in the literature and in practice) is First-Fit. By exploring the structural properties of
allocations made by First-Fit, we can provide a better competitive ratio of
9
4
< 83 . Recall that
upon the arrival of a new job, First-Fit purchases a new machine if the job does not fit in any of
the existing machines. Otherwise, it assigns the job to the first machine (based on a fixed ordering
such as machine IDs) that it fits in. This algorithm is simple to implement, and very well studied
in the context of the classical bin packing problem. First, we present an extension of Theorem 2
for the case where each machine has at least K jobs.
Corollary 1. If the First-Fit algorithm assigns jobs such that each machine receives at least
K jobs, the number of purchased machines does not exceed
4
(1
3
+
1
)OP T ,
K
where OP T is the
optimum number of machines to serve all jobs.
One can prove Corollary 1 in a similar fashion as the proof of Theorem 2 and using the fact that
jobs are assigned using First-Fit (the details are omitted for conciseness). For example, when
K = 2 (resp. K = 5), we obtain a 2 (resp. 1.6) approximation. We next refine the approximation
factor for the problem by using the First-Fit algorithm.
Theorem 4. The number of purchased machines by Algorithm First-Fit for any arrival order
of jobs is not more than 94 OP T + 1.
The proof can be found in Appendix D. We note that the approximation guarantees we developed
in this section do not depend on the factor D(α), and on the specific definition of the parameters
µj and bj . In addition, as we show computationally in Section 7, the performance of this class of
algorithm is not significantly affected by the factor D(α).
5.
Insights on job scheduling
In this section, we show that guaranteeing the following two guidelines in any allocation algorithm
yields optimal solutions:
• Filling up each machine completely such that no other job fits in it, i.e., making each machine’s
Cost equal to 1.
• Each machine contains a set of similar jobs (defined formally next).
We formalize these properties in more detail, and show how one can achieve optimality by
qP
P
satisfying these two conditions. We call a machine full if j∈S µj +
j∈S bj is equal to 1 (recall
that the machine capacity is normalized to 1 without loss of generality), where S is the set of jobs
assigned to the machine. Note that it is not possible to assign any additional job (no matter how
small the job is) to a full machine. Similarly, we call a machine -full, if the the cost is at least 1 − ,
24
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
i.e.,
P
j∈S µj +
qP
j∈S
bj ≥ 1 − . We define two jobs to be similar, if they have the same b/µ ratio.
Note that the two jobs can have different values of µ and b. We say that a machine is homogeneous,
if it only contains similar jobs. In other words, if the ratio bj /µj is the same for all the jobs j
assigned to this machine. By convention, we define bj /µj to be +∞ when µj = 0. In addition, we
introduce the relaxed version of this property: we say that two jobs are δ-similar, if their b/µ ratios
differ by at most a multiplicative factor of 1 + δ. A machine is called δ-homogeneous, if it only
contains δ-similar jobs (i.e., for any pair of jobs j and j 0 in the same machine,
bj /µj
b0j /µ0j
is at most
1 + δ).
Theorem 5. For any ≥ 0 and δ ≥ 0, consider an assignment of all jobs to some machines
with two properties: a) each machine is -full, and b) each machine is δ-homogeneous. Then, the
number of purchased machines in this allocation is at most
OP T
.
(1−)2 (1−δ)
The proof can be found in Appendix E.
In this section, we proposed an easy to follow recipe in order to schedule jobs to machines. Each
arriving job is characterized by two parameters µj and bj . Upon arrival of a new job, the cloud
provider can compute the ratio rj = bj /µj . Then, one can decide of a few buckets for the different
values of rj , depending on historical data, and performance restrictions. Finally, the cloud provider
will assign jobs with similar ratios to the same machines and tries to fill in machines as much as
possible. In this paper, we show that such a simple strategy guarantees a good performance (close
to optimal) in terms of minimizing the number of purchased machines while at the same time
allowing to strategically overcommit.
6.
Extensions
In this section, we present two extensions of the problem we considered in this paper.
6.1.
Offline 2-approximation algorithm
Consider the offline version of the (SMBP) problem. In this case, all the N jobs already arrived,
and one has to find a feasible schedule so as to minimize the number of machines. We propose
the algorithm Local-Search that iteratively reduces the number of purchased machines, and
also uses ideas inspired from First-Fit in order to achieve a 2-approximation for the offline
problem. Algorithm Local-Search starts by assigning all the jobs to machines arbitrarily, and
then iteratively refines this assignment. Suppose that each machine has a unique identifier number.
We next introduce some notation before presenting the update operations. Let a be the number of
machines with only one job, A1 be the set of these a machines, and S1 be the set of jobs assigned to
these machines. Note that this set changes throughout the algorithm with the update operations.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
25
We say that a job j ∈
/ S1 is good, if it fits in at least 6 of the machines in the set A1 7 . In addition,
we say that a machine is large, if it contains at least 5 jobs, and we denote the set of large machines
by A5 . We say that a machine is medium size, if it contains 2, 3, or 4 jobs, and we denote the set
of medium machines by A2,3,4 . We call a medium size machine critical, if it contains one job that
fits in none of the machines in A1 , and the rest of the jobs in this machine are all good. Following
are the update operations that Local-Search performs until no such operation is available.
• Find a job j in machine i (i is the machine identifier number) and assign it to some other
machine i0 < i if feasible (the outcome will be similar to First-Fit).
• Find a medium size machine i that contains only good jobs. Let j1 , · · · , j` (2 ≤ ` ≤ 4) be the
jobs in machine i. Assign j1 to one of the machines in A1 that it fits in. Since j1 is a good job,
there are at least 6 different options, and the algorithm picks one of them arbitrarily. Assign j2 to
a different machine in A1 that it fits in. There should be at least 5 ways to do so. We continue
this process until all the jobs in machine i (there are at most 4 of them) are assigned to distinct
machines in A1 , and they all fit in their new machines. This way, we release machine i and reduce
the number of machines by one.
• Find a medium size machine i that contains one job j that fits in at least one machine in A1 ,
and the rest of the jobs in i are all good. First, assign j to one machine in A1 that it fits in. Similar
to the previous case, we assign the rest of the jobs (that are all good) to different machines in A1 .
This way, we release machine i and reduce the number of purchased machines by one.
• Find two critical machines i1 and i2 . Let j1 and j2 be the only jobs in these two machines
that fit in no machine in A1 . If both jobs fit and form a feasible assignment in a new machine,
we purchase a new machine and assign j1 and j2 to it. Otherwise, we do not change anything
and ignore this update step. There are at most 6 other jobs in these two machines since both are
medium machines. In addition, the rest of the jobs are all good. Therefore, similar to the previous
two cases, we can assign these jobs to distinct machines in A1 that they fit in. This way, we release
machines i1 and i2 and purchase a new machine. So in total, we reduce the number of purchased
machines by one.
We are now ready to analyze this Local-Search algorithm that also borrows ideas from
First-Fit. We next show that the number of purchased machines is at most 2OP T + O(1), i.e., a
2-approximation.
Theorem 6. Algorithm Local-Search terminates after at most N 3 operations (where N is
the number of jobs), and purchases at most 2OP T + 11 machines.
7
The reason we need 6 jobs is technical, and will be used in the proof of Theorem 6.
26
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
The proof can be found in Appendix F.
We conclude this section by comparing our results to the classical (deterministic) bin packing
problem. In the classical bin packing problem, there are folklore polynomial time approximation
schemes (see Section 10.3 in Albers and Souza (2011)) that achieve a (1 − )-approximation factor
by proposing an offline algorithm based on clustering the jobs into 1/2 groups, and treating them
as equal size jobs. Using dynamic programming techniques, one can solve the simplified problem
with 1/2 different job sizes in time O(npoly(1/) ). In addition to the inefficient time complexity of
these algorithms that make them less appealing for practical purposes, one cannot generalize the
same ideas to our setting. The main obstacle is the lack of a total ordering among the different
jobs. In the classical bin packing problem, the jobs can be sorted based on their sizes. However,
this is not true in our case since the jobs have the two dimensional requirements µj and bj .
6.2.
Alternative constraints
Recall that in the (SMBP) problem, we imposed the modified capacity constraint (5). Instead, one
can consider the following family of constraints, parametrized by 0.5 ≤ p ≤ 1:
N
X
j=1
µj xij + D(α)
N
X
bj xij
p
≤ V yi ,
(8)
j=1
Note that this equation is still monotone and submodular in the assignment vector x, and captures
some notion of risk pooling. In particular, the “safety buffer” reduces with the number of jobs
already assigned to each machine. The motivation behind such a modified capacity constraint lies
in the shape that one wishes to impose on the term that captures the uncertain part of the job. In
one extreme (p = 1), we consider that the term that captures the uncertainty is linear and hence,
as important as the expectation term. In the other extreme case (p = 0.5), we consider that the
term that captures the uncertainty behaves as a square root term. For a large number of jobs per
machine, this is known to be an efficient way of handling uncertainty (similar argument as the
central limit theorem). Note also that when p = 0.5, we are back to equation (5), and when p = 1
we have a commonly used benchmark (see more details in Section 7). One can extend our analysis
and derive an approximation factor for the online problem as a function of p for any lazy algorithm.
Corollary 2. Consider the bin packing problem with the modified capacity constraint (8).
Then, any lazy algorithm ALG purchases at most
2
OP T
f (p)
machines, where OP T is the optimum
number of machines to serve all jobs and f (p) is given by:
1
f (p) = 1 − (1 − p)p p −1 .
The proof is in a very similar spirit as in Theorem 2 and is not repeated due to space limitations.
p
PN
PN
Intuitively, we find parametric lower and upper bounds on
in terms of j=1 bj xij .
j=1 bj xij
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
27
Note that when p = 0.5, we recover the result of Theorem 2 (i.e., a 8/3 approximation) and as p
increases, the approximation factor converges to 2. In Figure 3, we plot the approximation factor
as a function of 0.5 ≤ p ≤ 1.
Figure 3
Approximation factor
2
f (p)
as a function of p
Finally, one can also extend our results for the case where the modified capacity constrain is
PN
PN
given by: j=1 µj xij + D(α) log 1 + j=1 bj xij ≤ V yi .
7.
Computational experiments
In this section, we test and validate the analytical results developed in the paper by solving
the (SMBP) problem for different realistic cases, and investigating the impact on the number of
machines required (i.e., the cost). We use realistic workload data inspired by Google Compute
Engine, and show how our model and algorithms can be applied in an operational setting.
7.1.
Setting and data
We use simulated workloads of 1000 jobs (virtual machines) with a realistic VM size distribution
(see Table 1). Typically, the GCE workload is composed of a mix of CPU usages from virtual
machines belonging to cloud customers. These jobs can have highly varying workloads, including
some large ones and many smaller ones.8 More precisely, we assume that each VM arrives to the
cloud provider with a requested number of CPU cores, sampled from the distribution presented in
Table 1.
8
The average distribution of workloads we present in Table 1 assumes small percentages of workloads with 32 and
16 cores, and larger percentages of smaller VMs. A “large” workload may consist of many VMs belonging to a single
customer whose usages may be correlated at the time-scales we are considering, but heuristics ensure these are spread
across different hosts to avoid strong correlation of co-scheduled VMs. The workload distributions we are using are
representative for some segments of GCE. Unfortunately, we cannot provide the real data due to confidentiality.
28
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
Table 1
Example distribution of VM sizes in one Google data center
Number of cores 1
2
4
8
16 32
% VMs
36.3 13.8 21.3 23.1 3.5 1.9
In this context, the average utilization is typically low, but in many cases, the utilization can be
highly variable over time. Although we decided to keep a similarl VM size distribution as observed
in a production data center, we also fitted parametric distributions to roughly match the mean and
the variance of the measured usage. This allows us to obtain a parametric model that we could vary
for simulation. We consider two different cases for the actual CPU utilization as a fraction of the
requested job size: we either assume that the number of cores used has a Bernoulli distribution, or
a truncated Gaussian distribution. As discussed in Section 2.4, we assume that each job j has lower
and upper utilization bounds, Aj and Aj . We sample Aj uniformly in the range [0.3, 0.6], and Aj
in the range [0.7, 1.0]. In addition, we uniformly sample µ0j and σj0 ∈ [0.1, 0.5] for each VM to serve
as the parameters for the truncated Gaussian (not to be confused with its true mean and standard
deviation, µj and σj ). For the Bernoulli case, µ0j = µj determines the respective probabilities of the
realization corresponding to the lower or upper bound (and the unneeded σj0 is ignored).
For each workload of 1000 VMs generated in this manner, we solve the online version of the
(SMBP) problem by implementing the Best-Fit heuristic, using one of the three different variants
for the values of D(α) and bj . We solve the problem for various values of α ranging from 0.5 to
0.99999. More precisely, when a new job arrives, we compute the modified capacity constraint
in equation (5) for each already-purchased machine, and assign the job to the machine with the
smallest available capacity that can accommodate it9 . If the job does not fit in any of the already
purchased machines, the algorithm opens a new machine. We consider the three variations of the
(SMBP) discussed earlier:
• The Gaussian case introduced in (2), with bj = σj2 and D(α) = Φ−1 (α). This is now also
an approximation to the chance constrained (BPCC) formulation since the true distributions are
truncated Gaussian or Bernoulli.
• The Hoeffding’s inequality approximation introduced in (3), with bj = (Aj − Aj )2 and D(α) =
p
−0.5 ln(1 − α). Note hat the distributionally robust approach with the family of distributions D2
is equivalent to this formulation.
• The distributionally robust approximation with the family of distributions D1 , with bj = σj2
p
and D1 (α) = α/(1 − α).
9
P
Note that we clip the value of the constraint at the effective upper bound
j xij Aj , to ensure that no trivially
feasible assignments are excluded. Otherwise, the Hoeffding’s inequality-based constraint may perform slightly worse
relative to the policy without over-commitment, if it leaves too much free space on the machines.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
7.2.
29
Linear benchmarks
We also implement the following four benchmarks which consist of solving the classical (DBP)
problem with specific problem data. First we have:
• No overcommitment – This is equivalent to setting α = 1 in the (SMBP) problem, or solving
the (DBP) problem with sizes Aj .
Three other heuristics are obtained by replacing the square-root term in constraint (5) by a linear
term, specifically we replace the constraint with:
N
X
µj xij + D(α)
j=1
N
X
p
j=1
bj xij =
N
X
p
µj + D(α) bj xij ≤ V yi
(9)
j=1
to obtain:
• The linear Gaussian heuristic that mimics the Gaussian approximation in (2).
• The linear Hoeffding’s heuristic that mimics the Hoeffding’s approximation in (3).
• The linear robust heuristic that mimics the distributionally robust approach with the family
of distributions D1 .
Notice that the linearized constraint (9) is clearly more restrictive for a fixed value of α by concavity
of the square root, but we do of course vary the value of α in our experiments. We do not expect
these benchmarks to outperform our proposed method since they do not capture the risk-pooling
effect from scheduling jobs concurrently on the same machine. They do however still reflect different
relative amounts of ”padding” or ”buffer” above the expected utilization of each job allocated due
to the usage uncertainty.
The motivation behind the linear benchmarks lies in the fact that the problem is reduced to the
standard (DBP) formulation which admits efficient implementations for the classical heuristics.
For example, the Best-Fit algorithm can run in time O(N log N ) by maintaining a list of open
machines sorted by the slack left free on each machine (see Johnson (1974) for details and linear time approximations). In contrast, our implementation of the Best-Fit heuristic with the non-linear
constraint (5) takes time O(N 2 ) since we evaluate the constraint for each machine when each new
job arrives. Practically, in cloud VM scheduling systems, this quadratic-time approach may be
preferred anyway since it generalizes straightforwardly to more complex “scoring” functions that
also take into account additional factors besides the remaining capacity on a machine, such as
multiple resource dimensions, performance concerns or correlation between jobs (see, for example,
Verma et al. (2015)). In addition, the computational cost could be mitigated by dividing the data
center into smaller “shards”, each consisting of a fraction of the machines, and then trying to assign
each incoming job only to the machines in one of the shard. For example, in our experiments we
found that there was little performance advantage in considering sets of more than 1000 jobs at
30
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
a time. Nevertheless, our results show that even these linear benchmarks may provide substantial
savings (relative to the no-overcommitment policy) while only requiring very minor changes to
classical algorithms: instead of Aj , we simply use job sizes defined by µj , bj and α.
7.3.
Results and comparisons
We compare the seven different methods in terms of the number of purchased machines and show
that, in most cases, our approach significantly reduces the number of machines needed.
We consider two physical machine sizes: 32 cores and 72 cores. As expected, the larger machines
achieve a greater benefit from modeling risk-pooling. We draw 50 independent workloads each
composed of 1000 VMs as described above. For each workload, we schedule the jobs using the BestFit algorithm and report the average number of machines needed across the 50 workloads. Finally,
we compute the probability of capacity violation as follows: for each machine used to schedule each
of the workloads, we draw 5000 utilization realizations (either from a sum of truncated Gaussian
or a sum of Bernoulli distributions), and we count the number of realizations where the total CPU
usage of the jobs scheduled on a machine exceeds capacity.
The sample size was chosen so that our results reflect an effect that is measurable in a typical
data center. Since our workloads require on the order of 100 machines each, this corresponds to
roughly 50 × 100 × 5000 = 25, 000, 000 individual machine-level samples. Seen another way, we
schedule 50 × 1000 = 50, 000 jobs and collect 5000 data points from each. Assuming a sample is
recorded every 10 minutes, say, this corresponds to a few days of traffic even in a small real data
center with less than 1000 machines10 . The sample turns out to yield very stable measurements,
and defining appropriate service level indicators is application-dependent and beyond the scope of
this paper, so we do not report confidence intervals or otherwise delve into statistical measurement
issues. Similarly, capacity planning for smaller data centers may need to adjust measures of demand
uncertainty to account for the different scheduling algorithms, but any conclusions are likely specific
to the workload and data center, so we do not report on the variability across workloads.
In Figure 4, we plot the average number of machines needed as a function of the probability
that a given constraint is violated, in the case where the data center is composed of 72 CPU core
machines. Each point in the curves corresponds to a different value of the parameter α. Without
overcommitment, we need an average of over 54 machines in order to serve all the jobs. By allowing
10
The exact time needed to collect a comparable data set from a production system depends on the data center size
and on the sampling rate, which should be a function of how quickly jobs enter and leave the system, and of how
volatile their usages are. By sampling independently in our simulations, we are assuming that the measurements from
each machine are collected relatively infrequently (to limit correlation between successive measurements), and that
the workloads are diverse (to limit correlation between measurements from different machines). This assumption is
increasingly realistic as the size of the data center and the length of time covered increase: in the limit, for a fixed
sample size, we would record at most one measurement from each job with a finite lifetime, and it would only be
correlated with a small fraction of its peers.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
31
a small chance of violation, say a 0.1% risk (or equivalently, a 99.9% satisfaction probability), we
only need 52 machines for the Bernoulli usage, and 48 machines for the truncated Gaussian usage.
If we allow a 1% chance of violation, we then only need 50 and 46 machines, respectively. The
table of Figure 6 summarizes the relative savings, which are roughly 4.5% and 11.5% with a 0.1%
risk, and roughly 8% and 14% with a 1% risk, for the Bernoulli and truncated Gaussian usages,
respectively. In terms of the overcommitment factor defined in Section 2.3, the reported savings
translate directly to the fraction of the final capacity that is due to overcommitment,
B(1) − B(α) OCF (α) − OCF (1)
=
.
B(1)
OCF (α)
Figure 4 shows that all three variations of our approach (the Gaussian, Hoeffding’s, and the
distributionally robust approximations) yield very similar results. This suggests that the results
are robust to the method and the parameters. The same is true for the corresponding linear
benchmarks, though they perform worse, as expected. We remark that although the final performance tradeoff is nearly identical, for a particular value of the α parameter, the achieved violation
probabilities vary greatly. For example, with α = 0.9 and the truncated normal distribution, each
constraint was satisfied with probability 0.913 when using the Gaussian approximation, but with
much higher probabilities 0.9972 and 0.9998 for the Hoeffding and robust approximations, respectively. This is expected, since the latter two are relatively loose upper bounds for a truncated
normal distribution, whereas the distributions N (µj , σj ) are close approximations to the truncated
Gaussian with parameters µ0j and σj0 . (This is especially true for their respective sums.) Practically,
the normal approximation is likely to be the easiest to calibrate and understand in cases where the
theoretical guarantees of the other two approaches are not needed, since it would be nearly exact
for normally-distributed usages.
In Figure 5, we repeat the same tests for smaller machines having only 32 physical CPU cores. The
smaller machines are more difficult to overcommit since there is a smaller risk-pooling opportunity,
as can be seen by comparing the columns of Table 6. The three variations of our approach still
yield similar and significant savings, but now they substantially outperform the linear benchmarks:
the cost reduction is at least double with all but the largest values of α. We highlight that with
the “better behaved” truncated Gaussian usage, we still obtain a 5% cost savings at 0.01% risk,
whereas the linear benchmarks barely improve over the no-overcommit case.
As mentioned in Section 2.2, the value of α should be calibrated so as to yield an acceptable
risk level given the data center, the workload and the resource in question. Any data center has
a baseline risk due to machine (or power) failure, say, and a temporary CPU shortage is usually
much less severe relative to such a failure. On the other hand, causing a VM to crash because of a
memory shortage can be as bad as a machine failure from the customer’s point of view. Ultimately,
32
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
(a) Bernoulli usage
Figure 4
(b) Truncated Gaussian usage
Average number of 72-core machines needed to schedule a workload, versus the probability that any
given machine’s realized load exceeds capacity
(a) Bernoulli usage
Figure 5
(b) Truncated Gaussian usage
Results for 32 core machines
the risk tolerance will be driven by technological factors, such as the ability to migrate VMs or
swap memory while maintaining an acceptable performance.
7.4.
Impact
We conclude that our approach allows a substantial cost reduction for realistic workloads. More
precisely, we draw the following four conclusions.
• Easy to implement: Our approach is nearly as simple to implement as classical bin packing
heuristics. In addition, it works naturally online and in real-time, and can be easily incorporated
to existing scheduling algorithms.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
Figure 6
33
Percentage savings due to overcommitment for two CPU usage distributions, using the three
proposed variants of the chance constraint. The linear Gaussian benchmark is shown for comparison
• Robustness: The three variations we proposed yield very similar results. This suggests that
our approach is robust to the type of approximation. In particular, the uncertain term bj and the
risk coefficient D(α) do not have a strong impact on the results. It also suggests that the method
is robust to estimation errors in the measures of variability that define bj .
• Significant cost reduction: With modern 72-core machines, our approach allows a 8-14%
cost savings relative to the no overcommitment policy. This is achieved by considering a manageable
risk level of 1%, which is comparable to other sources of risk that are not controllable (e.g., physical
failures and regular maintenance operations).
• Outperforming the benchmarks: Our proposals show a consistent marked improvement
over three different “linear” benchmarks that reduce to directly apply the classical Best-Fit heuristic. The difference is most substantial in cases where the machines are small relative to the jobs
they must contain, which is intuitively more challenging. Although our approach does not run in
O(n log n) time, ”sharding” and (potentially) parallelization mitigate any such concerns in practice.
8.
Conclusion
In this paper, we formulated and practically solved the bin-packing problem with overcommitment. In particular, we focused on a cloud computing provider that is willing to overcommit when
allocating capacity to virtual machines in a data center. We modeled the problem as bin packing
with chance constraints, where the objective is to minimize the number of purchased machines,
while satisfying the physical capacity constraints of each machine with a very high probability.
We first showed that this problem is closely related to an alternative formulation that we call the
SMBP (Submodular Bin Packing) problem. Specifically, the two problems are equivalent under
the assumption of independent Gaussian job sizes, or when the job size distribution belongs to the
distributionally robust family with a given mean and (diagonal) covariance matrix. In addition, the
34
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
bin packing problem with chance constraints can be approximated by the SMBP for distributions
with bounded supports.
We first showed that for the bin packing problem with general monotone submodular capacity
constraints, it is impossible to find a solution within any reasonable factor from optimal. We then
developed simple algorithms that achieve solutions within constant factors from optimal for the
SMBP problem. We showed that any lazy algorithm is 8/3 competitive, and that the First-Fit
heuristic is 9/4 competitive. Since the First-Fit and Best-Fit algorithms are easy to implement and
well understood in practice, this provides an attractive option from an implementation perspective.
Second, we proposed an algorithm for the offline version of the problem, and showed that it
guarantees a 2-approximation. Then, we used our model and algorithms in order to draw several
useful insights on how to schedule jobs to machines, and on the right way to overcommit. We
convey that our method captures the risk pooling effect, as the “safety buffer” needed for each job
decreases with the number of jobs already assigned to the same machine. Moreover, our approach
translates to a transparent and meaningful recipe on how to assign jobs to machines by naturally
clustering similar jobs in terms of statistical information. Namely, jobs with a similar ratio b/µ
(the uncertain term divided by the expectation) should be assigned to the same machine.
Finally, we demonstrated the benefit of overcommitting and applied our approach to realistic
workload data inspired by Google Compute Engine. We showed that our methods are (i) easy to
implement; (ii) robust to the parameters; and (iii) significantly reduce the cost (1.5-17% depending
on the setting and the size of the physical machines in the data center).
Acknowledgments
We would like to thank the Google Cloud Analytics team for helpful discussions and feedback. The first
author would like to thank Google Research as this work would not have been possible without a one year
postdoc at Google NYC during the year 2015-2016. The authors would also like to thank Lennart Baardman,
Arthur Flajolet and Balasubramanian Sivan for their valuable feedback that has helped us improve the
paper.
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Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
Appendix A:
Details for Section 2.5
IP formulation
By taking the square on both sides of the submodular capacity constraint (5), we obtain:
V 2 yi +
N
X
µj xij
2
− 2V yi
j =1
N
X
µj xij ≥ D(α)2 ·
j =1
N
X
bj xij .
j =1
Note that since yi is binary, we have yi2 = yi . We next look at the term: yi
PN
j =1
µj xij . One can linearize this
term by using one of the following two methods.
1. Since yi = 1 if and only if at least one xij = 1, we have the constraint:
PN
j =1
xij ≤ M yi , for a large
positive number M (actually, one can take M = N ). Consequently, one can remove the yi in the above term.
2. One can define a new variable tij , yi xij and add the four following constraints:
Next, we look at the term:
tij ≤ yi ; tij ≤ xij ; tij ≥ 0; tij ≥ xij + yi − 1.
2
N
µ
x
. Since x2ij = xij , we remain only with the terms xij · xik for k > j.
j
ij
j =1
P
One can now define a new variable for each such term, i.e., zijk , xij · xik with the four constraints as before:
zijk ≤ xij ; zijk ≤ xik ; zijk ≥ 0; zijk ≥ xij + xik − 1.
The resulting formulation is a linear integer program. Note that the decision variables tij and zijk are
continuous, and only xij and yi are binary.
Appendix B:
Proof.
Proof of Theorem 1
In this proof, we make use of the submodular functions defined by Svitkina and Fleischer (2011)
for load balancing problems. Denote the jobs by 1, 2, · · · , N , and for every subset of jobs S ⊆ [N ], let f (S) be
the cost of the set S (i.e., the capacity cost induced by the function f ). We use two submodular functions f
and f 0 (defined formally next) which are proved to be indistinguishable with a polynomial number of value
oracle queries (see Lemma 5.1 of Svitkina and Fleischer (2011)). Let denote x = ln(N ). Note that Svitkina
and Fleischer (2011) require x to be any parameter such that x2 dominates ln(N ) asymptotically and hence,
√
5 N
x
, α0 =
N
m0
2
, and β0 = x5 . We choose N such
P
that m0 takes an integer value. Define f (S) to be min{|S|, α0 }, and f 0 (S) to be min{ i min{β0 , |S ∩ Vi |}, α0 }
includes the special case we are considering here. Define m0 =
0
where {Vi }m
i=1 is a random partitioning of [N ] into m0 equal sized parts. Note that by definition, both set
functions f and f 0 are monotone and submodular.
As we mentioned, it is proved in Svitkina and Fleischer (2011) that the submodular functions f and
0
f cannot be distinguished from each other with a polynomial number of value oracle queries with high
probability. We construct two instances of the bin packing problem with monotone submodular capacity
constraints by using f and f 0 as follows. In both instances, the capacity of each machine is set to β0 .
In the first instance, a set S is feasible (i.e., we can schedule all its jobs in a machine) if and only if
f (S) ≤ β0 . By definition, f (S) is greater than β0 if |S| is greater than β0 . Therefore, any feasible set S in
this first instance consists of at most β0 jobs. Consequently, in the first instance, any feasible assignment of
jobs to machines requires at least
N
β0
machines.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
39
We define the second instance of the bin packing problem based on the submodular function f 0 . A set S
is feasible in the second instance, if and only if f 0 (S) ≤ β0 . Since f 0 (Vj ) is at most β0 for each 1 ≤ j ≤ m0 ,
each set Vj is a feasible set in this second instance. Therefore, we can assign each Vj to a separate machine
to process all jobs, and consequently, m0 machines suffice to do all the tasks in the second instance. We note
that with our parameter setting, m0 is much smaller that
N
β0
. We then conclude that the optimum solutions
of these two instances differ significantly.
We next prove the claim of Theorem 1 by using a contradiction argument. Assume that there exists a
polynomial time algorithm ALG for the bin packing problem with monotone submodular capacity constraints
√
with an approximation factor better than
N
ln(N )
. We next prove that by using ALG, we can distinguish
0
between the two set functions f and f with a polynomial number of value oracles, which contradicts the
result of Svitkina and Fleischer (2011). The task of distinguishing between the two functions f and f 0 can
be formalized as follows. We have value oracle access to a set function g, and we know that g is either the
same as f or the same as f 0 . The goal is to find out whether g = f or g = f 0 using a polynomial number of
value oracle queries. We construct a bin packing instance with N jobs, capacity constraints g, and ask the
algorithm ALG to solve this instance. If ALG uses less than
N
β0
machines to process all jobs, we can say
that g is the same as f 0 (since with capacity constraints f , there does not exist a feasible assignment of all
jobs to less than
N
β0
machines). On the other hand, if ALG uses at least
N
β0
machines, we can say that g is
0
equal to f . This follows from the fact that if g was equal to f , the optimum number of machines would have
√
been at most m0 . Since ALG has an approximation factor better than
√
by ALG should have been less than m0 ×
N
ln(N )
=
√
5 N
x
√
×
N
ln(N )
=
N
β0
N
ln(N )
, the number of machines used
. Therefore, using at least
N
β0
machines
by ALG is a sufficient indicator of g being the same as f . This argument implies that an algorithm with an
√
approximation factor better than
N
ln(N )
for the bin packing problem with monotone submodular constraints
yields a way of distinguishing between f and f 0 with a polynomial number of value oracle queries (since
ALG is a polynomial time algorithm), which contradicts the result of Svitkina and Fleischer (2011).
Appendix C:
Details related to Observation 1
For ease of exposition, we first address the case with two job classes. Classes 1 and 2 have parameters (µ1 , b1 )
and (µ2 , b2 ) respectively. For example, an interesting special case is when one class of jobs is more predictable
relative to the other (i.e., µ1 = µ2 = µ, b2 = b and b1 = 0). In practice, very often, one class of jobs has low
variability (i.e., close to deterministic), whereas the other class is more volatile. For example, class 1 can
represent loyal recurring customers, whereas class 2 corresponds to new customers.
We assume that we need to decide the number of machines to purchase, as well as how many jobs of types
1 and 2 to assign to each machine. Our goal is to find the right mix of jobs of classes 1 and 2 to assign to
each machine (note that this proportion can be different for each machine). Consider a given machine i and
denote by n1 and n2 the number of jobs of classes 1 and 2 that we assign to this machine. We would like to
ensure that the chance constraint is satisfied in each machine with the given parameter α. Assuming that
V > n1 µ1 + n2 µ2 , we obtain:
[V − n1 µ1 − n2 µ2 ]2
= D(α)2 .
n 1 b1 + n 2 b2
(10)
40
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
For a given α, one can find the value of n1 as a function of n2 that satisfies equation (10):
q
V − n2 µ2
1
n1 (n2 ) =
+ 2 b1 D(α)2 − b21 D(α)4 + 4b1 D(α)2 (V − n2 µ2 )µ1 + 4µ21 n2 b2 D(α)2 .
µ1
2µ1
(11)
As we discussed, an interesting special case is when both classes of jobs have the same expectation, i.e.,
µ1 = µ2 = µ but one type of jobs is much more predictable (i.e., smaller range or variance). In the extreme
case, one can assume that class 1 jobs are deterministic, (i.e., b1 = 0). In this case, equation (11) becomes:
√
V − n2 µ
2n2 b2 B
n1 (n2 ) =
−
.
(12)
µ
2µ
Alternatively, by directly looking at the modified capacity constraint (5) for this special case, we obtain:
p
V = (n1 + n2 )µ + D(α) n2 b2 .
(13)
Equation (13) can be interpreted as follows. Any additional job of type 1 takes µ from the capacity budget
V , whereas any additional job of type 2 is more costly. The extra cost depends on both the uncertainty of
the job (through b2 ) and the overcommitment policy (through D(α)). The higher is one of these two factors,
the larger is the capacity we should plan for jobs of type 2 (i.e., “safety buffer”). Note that the submodular
nature of constraint (5) implies that this marginal extra cost decreases with the number of jobs n2 . In other
words, when n2 becomes large, each additional job of type 2 will converge to take a capacity of µ, as the
central limit theorem applies. More generally, by taking the derivative of the above expression, any additional
√ √
job of type 2 will take µ + 0.5D(α) b2 / n2 (where n2 here represents how many jobs of type 2 are already
assigned to this machine).
In Figure 7, we plot equation (12) for a specific instance with V = 30 and different values of α. As expected,
if n2 = 0, we can schedule n1 = 50 jobs of class 1 to reach exactly the capacity V , no matter what is the
value of α. On the other hand, for α = 0.99, if n1 = 0, we can schedule n2 = 38 jobs of class 2. As the value
of n2 increases, the optimal value of n1 decreases. For a given value of α, any point (n1 , n2 ) on the curve
(or below) guarantees the feasibility of the chance constraint. The interesting insight is to characterize the
proportion of jobs of classes 1 and 2 per machine. For example, if we want to impose n1 = n2 in each machine,
what is the optimal value for a given α? In our example, when α = 0.99, we can schedule n1 = n2 = 21 jobs.
If we compare to the case without overcommitment (i.e., α = 1), we can schedule 18 jobs from each class.
Therefore, we obtain an improvement of 16.67%. More generally, if the cost (or priority) of certain jobs is
higher, we can design an optimal ratio per machine so that it still guarantees to satisfy the chance constraint.
To summarize, for the case when V = 30, µ = 0.65, A = 1, A = 0.3 and α = 0.99, one can schedule either 50
jobs of class 1 or 38 jobs of class 2 or any combination of both classes according to equation (12). In other
words, we have many different ways of bin packing jobs of classes 1 and 2 to each machine.
We next consider solving the offline problem when our goal is to schedule N1 jobs of class 1 and N2 jobs of
class 2. The numbers N1 and N2 are given as an input, and our goal is to minimize the number of machines
denoted by M ∗ , such that each machine is assigned a pair (n1 , n2 ) that satisfies equation (11). Since n1 and
n2 should be integer numbers, one can compute for a given value of α, all the feasible pairs that lie on the
curve or just below (we have at most K = mini=1,2 max ni such pairs). In other words, for each k = 1, 2, . . . , K,
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
Figure 7
41
Parameters: A = 1, A = 0.3, µ = 0.65, V = 30.
we compute a pair of coefficients denoted by (βk , γk ). The optimization problem becomes a cutting stock
problem:
M ∗ = min
zk
s.t.
K
X
zk
k=1
K
X
βk zk ≥ N1
(14)
k=1
K
X
γ k z k ≥ N2
k=1
zk ≥ 0, integer
∀k.
The decision variable zk represents the number of times we use the pair (βk , γk ), and M ∗ denotes the optimal
number of machines. As a result, assuming that we have only two classes of jobs (with different µs and bs),
one can solve the deterministic linear integer program in (14) and obtain a solution for problem (SMBP).
Note that the above treatment can easily be extended to more than two classes of jobs.
Appendix D:
Proof.
Proof of Theorem 4
Let n1 be the number of machines purchased by First-Fit with only a single job, and S1 be the
set of n1 jobs assigned to these machines. Similarly, we define n2 to be the number of machines with at least
two jobs, and S2 be the set of their jobs. The goal is to prove that n1 + n2 ≤ 49 OP T + 1. We know that
any pair of jobs among the n1 jobs in S1 does not fit in a single machine (by the definition of First-Fit).
Therefore, any feasible allocation (including the optimal allocation) needs at least n1 machines. In other
words, we have OP T ≥ n1 . This observation also implies that the sum of µj + bj for any pair of jobs in S1 is
greater than
3
4
(using Lemma 1). If we sum up all these inequalities for the different pairs of jobs in S1 , we
P
have: (n1 − 1) j∈S1 (µj + bj ) > n21 43 . We note that the n1 − 1 term on the left side appears because every job
j ∈ S1 is paired with n1 − 1 other jobs in S1 , and the n21 term on the right side represents the total number
P
of pairs of jobs in S1 . By dividing both sides of this inequality by n1 − 1, we obtain j∈S1 (µj + bj ) > 3n8 1 .
P
We also lower bound
j∈S2 (µj + bj ) as a function of n2 as follows. Let m1 < m2 < · · · < mn2 be the
machines that have at least two jobs, and the ordering shows in which order they were purchased (e.g.,
42
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
m1 was purchased first). Define Mi to be the set of jobs in machine mi . By definition of First-Fit, any
job j in machine mi+1 could not be assigned to machine mi because of the feasibility constraints for any
1 ≤ i < n2 . In other words, the set of jobs Mi with any job j ∈ Mi+1 form an infeasible set. Therefore, we
P
have µj + bj + j 0 ∈Mi (µj 0 + bj 0 ) > 43 . For each 1 ≤ i < n2 , one can pick two distinct jobs j1 and j2 from Mi+1 ,
and write the following two inequalities:
X
µj1 + bj1 +
(µj 0 + bj 0 ) >
j 0 ∈Mi
3
4
and
µj2 + bj2 +
X
j 0 ∈Mi
Summing up these two inequalities implies that: µj1 + bj1 + µj2 + bj2 + 2
3
(µj 0 + bj 0 ) > .
4
P
j 0 ∈Mi
(µj 0 + bj 0 ) > 32 . Since j1 and
j2 are two distinct jobs in Mi+1 , we have:
X
(µj + bj ) + 2
X
j 0 ∈Mi
j∈Mi+1
3
(µj 0 + bj 0 ) > .
2
Now we sum up this inequality for different values of i ∈ {1, 2, · · · , n2 − 1} to achieve that:
n2 −1
X
2
(µj + bj ) + 3
X X
i=2 j∈Mi
j∈M1
Pn2 P
X
(µj + bj ) +
(µj + bj ) >
j∈Mn2
3
× (n2 − 1),
2
3
2
P
× (n2 − 1). This is equivalent to j∈S2 (µj + bj ) >
P
1
× (n2 − 1). Combining both inequalities, we obtain: j∈S1 ∪S2 (µj + bj ) > 38 n1 + 12 (n2 − 1). On the other
2
P
hand, OP T is at least j∈S1 ∪S2 (µj + bj ). We then have the following two inequalities:
and consequently, we have 3
i=1
j∈Mi
(µj + bj ) >
OP T ≥ n1 ,
1
3
OP T > n1 + (n2 − 1).
8
2
We can now multiply (15) by
1
4
and (16) by 2, and sum them up. We conclude that
(15)
(16)
9
OP T
4
+ 1 is greater
than n1 + n2 , which is the number of machines purchased by Algorithm First-Fit.
Appendix E:
Proof of Theorem 5
We first state the following
√ Lemma that provides a lower bound on OP T . For any a, b ≥ 0, we define the
2a+b+ b(4a+b)
.
function f (a, b) =
2
Lemma 2. For any feasible set of jobs S, the sum
P
µj
bj
j∈S
f (µj , bj ) is at most 1.
and b̄ = j∈S
. Since the function f is concave with respect to both a and
|S|
P
b, using Jensen’s inequality we have j∈S f (µj , bj ) ≤ |S|f (µ̄, b̄). Since S is a feasible set, Cost(S) = |S|µ̄ +
p
p
|S|b̄ ≤ 1. The latter is a quadratic inequality with variable x = |S|, so that we we can derive an upper
√
bound on |S| in terms of µ̄ and b̄. Solving the quadratic form µ̄X + b̄X = 1 yields:
p
√
− b̄ ± b̄ + 4µ̄
X=
.
2µ̄
√
√
p
4µ̄+b̄− b̄
4q
1
We then have |S| ≤
= q 2 √ . Therefore, |S| ≤
= f (µ̄,
, where the equality
2µ̄
b̄)
4µ̄+b̄+b̄+2 b̄(4µ̄+b̄)
(4µ̄+b̄)+ b̄
P
follows by the definition of the function f . Equivalently, |S|f (µ̄, b̄) ≤ 1, and hence j∈S f (µj , bj ) ≤ 1.
Proof.
Define µ̄ =
j∈S
P
P
|S|
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
43
Proof of Theorem 5.
For any arbitrary allocation of jobs, applying Lemma 2 to all the machines implies
PN
that the number of machines is at least j =1 f (µj , bj ). So it suffices to upper bound the number of machines
PN
as a function of j =1 f (µj , bj ). For any given machine, we prove that the sum of f (µj , bj ) for all jobs j
assigned to this machine is at least 1 − O( + δ). Consider the set of jobs S assigned to a given machine.
P
Similar to the proof of Lemma 2, we define µ̄ =
j∈S
µj
|S|
P
and b̄ =
j∈S
|S|
bj
P
. Define r =
P j∈S
j∈S
bj
µj
= µ̄b̄ . We start by
lower bounding f (µj , bj ) as a function of f (µj , rµj ). Recall that each machine is δ-homogeneous, i.e., for all
pairs of jobs in the same machine, the ratios
other. Hence, any ratio
bj
µj
f (µj , bj ) ≥ f
r
is at least
(1+δ )
bj
µj
are at most a multiplicative factor of 1 + δ away from each
. Consequently, we have bj ≥
rµj
1+δ
which implies that:
µj
rµj
1
µj
, bj ≥ f
,
=
f (µj , rµj ) ≥ (1 − δ)f (µj , rµj ).
1+δ
1+δ 1+δ
1+δ
The first and second inequalities follow from the monotonicity of the function f , the equality follows from
P
the definition of f , and the last inequality holds since δ ≥ 0. It now suffices to lower bound j∈S f (µj , rµj )
P
so as to obtain a lower bound for j∈S f (µj , bj ). We note that for any η ≥ 0, we have f (ηµ, ηb) = ηf (µ, b)
by the definition of f . Applying η = µj , µ = 1 and b = r, we obtain f (µj , rµj ) = µj f (1, r) which implies:
X
X
f (µj , rµj ) =
µj f (1, r) = f (A, B),
j∈S
where A =
P
A0 =
and B 0 =
µj , and B = r
P
µj =
j∈S
P
bj . The last equality above follows from f (ηµ, ηb) = ηf (µ, b)
√
with η = A. Recall that each machine is -full, i.e., Cost(S) ≥ 1 − or equivalently, A + B ≥ 1 − . Let
A
(1−)2
j∈S
B
(1−)2
j∈S
j∈S
. Then, we have:
√
√
A
+
B
B
A
+
≥
≥ 1.
A + B0 =
(1 − )2 1 −
1−
0
√
√
Since B = rA, we also have B 0 = rA0 , and the lower bound can be rewritten as follows: A0 + rA0 ≥ 1, which
is the same quadratic form as in the proof of Lemma 2. Similarly, we prove that A0 ≥
is equal to
1
.
f (1,r )
0
0
0
0
0
4
√
4+2r+2
r (4+r )
We also know that f (A , B ) = f (A , rA ) = A f (1, r). We already proved that A ≥
which
1
f (1,r )
,
so we have f (A0 , B 0 ) ≥ 1. By definition of A0 and B 0 , we know thatf (A, B) = (1 − )2 f (A0 , B 0 ) ≥ (1 − )2 .
P
We conclude that the sum j∈S f (µj , bj ) ≥ (1 − δ)f (A, B) ≥ (1 − δ)(1 − )2 . As a result, for each machine
the sum of f (µj , bj ) is at least (1 − δ)(1 − )2 . Let m be the number of purchased machines. Therefore, the
PN
sum j =1 f (µj , bj ) is lower bounded by (1 − δ)(1 − )2 m, and at the same time upper bounded by OP T .
Consequently, m does not exceed
Appendix F:
Proof.
OP T
(1−δ )(1−)2
and this concludes the proof.
Proof of Theorem 6
We first prove that the algorithm terminates in a finite number of iterations. Note that all the
update operations (except the first one) reduce the number of purchased machines, and hence, there are
no more than N of those. As a result, it suffices to upper bound the number of times we perform the first
update operation. Since we assign jobs to lower id machines, there cannot be more than N 2 consecutive first
update operations. Consequently, after at most N × N 2 = N 3 operations, the algorithm has to terminate.
Next, we derive an upper bound on the number of purchased machines at the end of the algorithm. Note
that all the machines belong to one of the following four categories:
• Single job machines, i.e., the set A1 .
44
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
• Medium machines with only one non-good job – denoted by the set B.
• Medium machines with at least two non-good jobs – denoted by the set C.
• Large machines, i.e., the set A5 .
Let a, b, c, and d be the number of machines in A1 , B, C, and A5 respectively. Since no update operation is
possible (as the algorithm already terminated), the b non-good jobs assigned to the machines in the set B
do not fit in any of the single job machines, and no pair of them fit together in a new machine. Consider
these b non-good jobs in addition to the a jobs in the machines of the set A1 . No pair of these a + b jobs fit
in one machine together and therefore, OP T ≥ a + b.
PN
We also know that OP T ≥ j =1 (µj + bj ). Next, we derive a more elaborate lower bound on OP T by
writing the sum of µj + bj as a linear combination of the sizes of the sets A1 , B, C, and A5 . For each machine
i, let Mi be the set of jobs assigned to this machine. Let i1 < i2 < · · · < id be the indices of machines in the
set A5 , where d = |A5 |. Since we cannot perform the first update operation anymore, we can say that no
P
job in machine i`+1 fits in machine i` for any 1 ≤ ` < d. Therefore, µj + bj + j 0 ∈Mi (µj 0 + bj 0 ) > 34 for any
`
j ∈ Mi`+1 (using Lemma 1). We write this inequality for 5 different jobs (arbitrarily chosen) in Mi`+1 (recall
that there are at least 5 jobs in this machine), and for all the values of 1 ≤ ` < d. If we sum up all these
5(d − 1) inequalities, then the right hand side would be 5(d − 1) × 43 . On the other hand, the term µj + bj for
every job in these d machines appears on the left hand side at most 5 + 1 = 6 times. Therefore, by summing
P
P
1)
1)
= 5(d−
.
up these inequalities, we obtain: i∈A5 j∈Mi (µj + bj ) > 34 × 5(d−
6
8
Each machine in A5 has at least 5 jobs. Therefore, the term
5
6
appears in the lower bound. With a similar
argument, each machine in either B or C has at least 2 jobs and hence, this term is now replaced by 32 . The
P
P
1)
= b−2 1 , and
inequalities for every pair of machines in B and C are then: i∈B j∈Mi (µj + bj ) > 34 × 2(b−
3
P
P
2(c−1)
3
= c−2 1 .
i∈C
j∈Mi (µj + bj ) > 4 ×
3
P
P
Next, we lower bound i∈A1 ∪C j∈Mi (µj + bj ) as a function of a and c in order to complete the proof.
Recall that each machine in C has at least two non-good jobs. If we pick one of these non-good jobs j, and a
random machine i from A1 , with probability at least
a−5
a
, job j does not fit in machine i. This follows from
the fact that a non-good job fits in at most 5 machines in A1 and hence, a random machine in A1 would
P
not be be able to fit job j with probability at least a−a 5 . Therefore, µj + bj + j 0 ∈Mi (µj 0 + bj 0 ) ≥ 34 with
probability at least
a−5
a
. We next consider two different cases depending on the value of c.
If c is at least a2 , we pick a random machine in C, and two of its non-good jobs j1 and j2 arbitrarily. We
also pick a random machine in A1 . For each of these two jobs, the sum µj + bj of the non-good job and the
single job in the selected machine in A1 is greater than
3
4
with probability at least
a−5
a
. Summing up these
two inequalities, we obtain:
i 1hX X
i a−5 3
2h X X
(µj + bj ) +
(µj + bj ) >
× .
a i∈A j∈M
c i∈C j∈M
a
2
1
i
(17)
i
The left hand side of the above equation is composed of two terms. The first term is obtained through
picking a random machine in A1 (i.e., with probability a1 ), and once the machine is picked, we sum up both
equations so we obtain
2
a
. For the second term, every machine in the set C is chosen with probability
1
c
.
Cohen, Keller, Mirrokni and Zadimoghaddam: Overcommitment in Cloud Services - Bin packing with Chance Constraints
45
When the machine is picked, we sum up on all the jobs and hence get an upper bound. As we have shown,
P
P
(µj + bj ) ≥ c−2 1 . Combining these two inequalities leads to (using c ≥ a2 ):
i∈C
j∈Mi
X
X
(µj + bj ) >
i∈A1 ∪C j∈Mi
a
c − 1 3a 15 c a 1 a + c
a 3(a − 5)
×
+ (1 − ) ×
≥
−
+ − − =
− 4.25.
2
2a
2c
2
4
4
2 4 2
2
PN
By combining the three different bounds (on A1 ∪C, B and A5 ), we obtain j =1 (µj +bj ) ≥ a+2b+c + 58d −5.875.
PN
Since OP T ≥ j =1 (µj + bj ), we conclude that the number of purchased machines a + b + c + d is no more
than 2OP T + 11.
In the other case, we have c < a2 . Note that inequality (17) still holds. However, since the coefficient 1 − 2ac
becomes negative, we cannot combine the two inequalities as before. Instead, we lower bound the sum µj + bj
of jobs in A1 . We know that there is no pair of a jobs in the A1 machines that fit together in one machine.
P
P
Therefore, i∈A1 j∈Mi (µj + bj ) ≥ 38a . Next, we multiply inequality (17) by c, and combine it with this new
P
P
lower bound on i∈A1 j∈Mi (µj + bj ), to obtain (using c < a2 ):
X
X
i∈A1 ∪C j∈Mi
(µj + bj ) > c ×
3(a − 5)
2c
3a 3c 5 3a 3c 3a 3c 5
+ (1 − ) ×
>
− +
−
=
+
− .
2a
a
8
2
4
8
4
8
4
4
Combining this inequality with similar ones on the sets B and A5 , we obtain OP T ≥
3a
8
PN
j =1
(µj + bj ) >
+ 2b + 34c + 58d − 19
. Finally, combining this with OP T ≥ a + b leads to a + b + c + d ≤ 85 OP T + 52 OP T + 19
=
8
5
2OP T + 3.75, which concludes the proof.
| 8 |
Approximating Cycles in Directed Graphs:
Fast Algorithms for Girth and Roundtrip Spanners
Jakub Pachocki
arXiv:1611.00721v1 [cs.DS] 2 Nov 2016
Roei Tov
§
∗
Liam Roditty
†
Aaron Sidford
Virginia Vassilevska Williams
‡
¶
Abstract
The girth of a graph, i.e. the length of its shortest cycle, is a fundamental graph parameter.
Unfortunately all known algorithms for computing, even approximately, the girth and girthrelated structures in directed weighted m-edge and n-node graphs require Ω(min{nω , mn}) time
(for 2 ≤ ω < 2.373). In this paper, we drastically improve these runtimes as follows:
• Multiplicative Approximations in Nearly Linear Time: We give an algorithm that
e
e
in O(m)
time computes an O(1)-multiplicative
approximation of the girth as well as an
e
e
O(1)-multiplicative
roundtrip spanner with O(n)
edges with high probability (w.h.p).
• Nearly Tight Additive Approximations: For unweighted graphs and any α ∈ (0, 1)
1−α
e
we give an algorithm that in O(mn
) time computes an O(nα )-additive approximation
α
of the girth as well as an O(n )-additive roundtrip spanner with Õ(n2−α ) edges w.h.p.
We show that the runtime of our algorithm cannot be significantly improved without a
breakthrough in combinatorial Boolean matrix multiplication, and that unconditionally
the size of our spanner is essentially optimal.
Our main technical contribution to achieve these results is the first nearly linear time algorithm for computing roundtrip covers, a directed graph decomposition concept key to previous
roundtrip spanner constructions. Previously it was not known how to compute these significantly faster than Ω(min{nω , mn}) time. Given the traditional difficulty in efficiently processing
directed graphs, we hope our techniques may find further applications.
∗
Harvard University, meret@seas.harvard.edu
Bar Ilan University, liamr@macs.biu.ac.il
‡
Stanford University, sidford@stanford.edu
§
Bar Ilan University, roei81@gmail.com
¶
Stanford University, virgi@cs.stanford.edu
†
1
Introduction
The girth of a graph G is the length of the shortest cycle in G. It is a natural and fundamental
graph parameter that has been extensively studied (see Diestel’s book [Die00] for a discussion) with
research on its computation dating back to the 1960s. Perhaps, the most straightforward algorithm
for the girth is simply to compute All-Pairs Shortest Paths (APSP). Surprisingly, this simple
relationship leads to the best known algorithm for girth in a n-node graph with nonnegative
√ weights:
log n) time.
3
Θ(
the breakthrough APSP algorithm of Williams [Wil14] can compute the girth in n /2
For sparse weighted graphs with m edges, an O(mn) runtime was recently obtained by Orlin [Orl17],
improving upon an O(mn + n2 log log n) runtime that follows from using the best known sparse
APSP algorithm [PR05].
When the graph can be dense, all known girth algorithms run in n3−o(1) time (unless the weights are
e nω ) time is known [RW11]).
small integers bounded in absolute value by M in which case an O(M
Vassilevska W. and Williams [VW10] explained this by showing that the girth in weighted graphs
is equivalent to APSP in the sense that if one of the two problems has a “truly subcubic”, O(n3−ε )
time algorithm for some ε > 0, then both do. As it is a longstanding open problem whether
APSP has a truly subcubic time algorithm, computing the girth in O(n3−ε ) time would be a huge
breakthrough.
For unweighted graphs, in the 1970s Itai and Rodeh [IR78] showed that the girth can be computed
in O(nm) time via BFS, or in O(nω ) ≤ O(n2.373 )-time using fast matrix multiplication [CW90,
Vas12, Gal14]. These are still the best runtimes for the problem. Similarly to the relationship to
APSP, [VW10] showed that the girth in unweighted graphs is subcubically equivalent to Boolean
Matrix Multiplication (BMM). A large open question in BMM is whether there exist truly subcubic
“combinatorial” algorithms, that can avoid the sophisticated but often impractical tools for Strassenlike fast matrix multiplication (e.g. [CW90, Vas12, Gal14]). The reduction from [VW10] shows
that either both BMM and girth have truly subcubic combinatorial algorithms, or neither of them
does.
Because of these cubic barriers for exact computation, efficient approximation algorithms are of
interest. Fast approximations for girth in undirected graphs are possible and have been studied extensively. It is well known ([ADD+ 93]) that for any integer k ≥ 1, every weighted undirected n-node
graph G contains an O(n1+1/k ) edge (2k − 1)-spanner, i.e. a subgraph that (2k − 1)-approximates
1/k ) time [TZ05], and
e
all pairwise distances in G. Such (2k − 1)-spanners can be computed in O(mn
immediately imply efficient approximation algorithms for girth in undirected weighted graphs. In
unweighted undirected graphs even better results are known. Itai and Rodeh [IR78] gave an O(n2 )
time additive 1-approximation algorithm, and follow-up work [RW12, LL09] developed even more
efficient, combinatorial, truly subquadratic, approximation algorithms.
Although impressive approximate girth algorithms are possible for undirected graphs, they all
exploit properties particular to undirected graphs. In particular, not only do sparse spanners exist
in undirected graphs, but it is also known [BS74] that undirected graphs with Ω(n1+1/k ) edges must
contain a 2k-cycle. This fact is at the heart of obtaining fast girth approximation algorithms.
In contrast, directed graphs do not always contain sparse spanners and dense digraphs might not
have any cycles at all (a directed bipartite clique is an example of each). It is completely unclear
what (if any) structure there is to exploit in directed graphs to obtain fast girth approximation
algorithms. Due to the close relationship between girth and APSP [VW10], a-priori it could be that
the girth problem in directed graphs suffers from the same problem as APSP in directed graphs
and no finite approximation is even possible without resolving a major open problem about BMM.
1
A potential saving point is that while directed graphs may not contain sparse spanners, they do
contain sparse roundtrip spanners. That is, if one uses d(u, v) + d(v, u) for the distance between
u and v instead of d(u, v), then one can obtain a very similar result for directed graphs as in the
undirected case: for all k ≥ 1 and ε > 0, every n node directed graph G contains a (2k+ε)-roundtrip
spanner on O(k2 /εn1+1/k ) edges. Unfortunately, while such roundtrip spanners have been known
to exist for over a decade, the fastest algorithms for computing them run in O(mn) time, essentially
the time to solve APSP.
1.1
Our Results
In this paper we provide the first non-trivial approximation algorithms for computing the girth and
related properties on directed graphs that run substantially faster than the roughly Ω(min{nω , mn})
time currently needed to solve APSP on a n-node, m-edge directed graph with non-negative weights.
e
We show how to compute O(1)-multiplicative
approximations to the girth and construct multiplicative roundtrip spanners in nearly linear time (See Section 1.1.1), and we show how to compute
additive approximations to the girth and construct additive roundtrip spanners with bounds that
are nearly tight under standard assumptions (See Section 1.1.2). To achieve these results we provide the first nearly linear time algorithms for computing roundtrip covers, a natural directed graph
decomposition notion in prior work on roundtrip spanners [CW04, RTZ08] (See Section 1.2).
1.1.1
Multiplicative Approximations in Nearly Linear Time
[VW10] showed that the girth problem in weighted graphs is subcubically equivalent to APSP in
general graphs. Thus, obtaining a truly subcubic algorithm for girth in weighted graphs would
imply a major breakthrough, and is a daunting task for current techniques. In fact, no nontrivial
combinatorial approximation algorithms were previously known even for the restricted case of
unweighted directed graphs. On the other hand, we show how to obtain an O(log n) approximation
to girth in weighted graphs in slightly super-linear time (See Theorem 1.1). Setting k := log n in
this theorem allows us to compute an O(log2 n) approximation to the girth in nearly linear time.
Theorem 1.1 (Multiplicative Girth Approximation) For any n-node, m-node directed graph
with nonnegative integer edge weights, with unknown girth g and integer k ≥ 1, in time O(mn1/k log5 n)
we can compute an estimate ḡ such that g ≤ ḡ ≤ O(k log n) · g with high probability.
Using our new directed graph decomposition algorithm we also show how to compute multiplicative roundtrip spanners in nearly linear time. A spanner is a sparse subgraph that preserves the
distances of the original graph with some multiplicative or additive approximation. Since even
preserving the asymmetric reachability structure of directed graphs may require Ω(n2 ) edges (e.g.
the complete directed bipartite graph), no sparse spanner yielding a finite multiplicative approximation is possible. Instead, we consider spanners under the roundtrip distance metric, i.e. roundtrip
spanners.
Given vertices u and v the roundtrip distance between u and v is the distance from u to v plus the
distance from v to u. Roundtrip distances were studied by Cowen and Wagner [CW04] in the context of routing. Later Roditty, Thorup and Zwick [RTZ08] obtained roundtrip spanners for directed
graphs that are almost as good in terms of their sparsity/approximation tradeoff as the spanners
of undirected graphs [ADD+ 93]: (2k − ε)-multiplicative approximation with Õ((k2 /ε)n1+1/k ) edges
for any integer k ≥ 1 and ε ∈ (0, 1). However, the construction of these spanners requires precomputing the roundtrip distances between all pairs of vertices, resulting in a running time of roughly
Ω(min{mn, nω }). We show how to nearly match this size/approximation tradeoff while decreasing
the time need to construct them to nearly linear in the number of edges in the graph.
2
Theorem 1.2 (Multiplicative Roundtrip Spanners) Given any n-node, m-edge directed graph
with nonnegative integer edge weights and any k ≥ 1 in time O(mn1/k log5 n) we can compute an
O(k log n)-multiplicative roundtrip spanner with O(n1+1/k log2 n) edges with high probability.
Our techniques are inherently parallelizable, and we provide the first work-efficient parallel algorithms for computing both the approximate girth and the strongly connected components of an
unweighted directed graph with depth linear in the diameter of the computed objects (See Section 5.2).
1.1.2
Nearly Tight Additive Approximations
Our techniques can also be used to obtain fast combinatorial algorithms that achieve additive
approximations of the girth on unweighted graphs as follows. Let a ∈ (0, 1) be any constant and
suppose the girth g is < na / log n. Then, the algorithm from Theorem 1.1 will return w.h.p. in
Õ(m) time a cycle of length O(na ), which is (trivially) an additive O(na ) approximation of g. If
on the other hand g ≥ na / log n, then if we take a random sample S of Cn1−a log2 n nodes for
large enough constant C, then w.h.p. S will contain a vertex of the shortest cycle. Then, running
BFS from each node of S will find the shortest cycle containing a node of S and hence compute g
exactly:
Corollary 1.3 (Additive Girth Approximation) For any unweighted n-node, m-edge directed
graph with unknown girth g and a ∈ (0, 1) in time Õ(mn1−a ) we can compute an estimate ḡ such
that g ≤ ḡ ≤ g + O(na ) with high probability.
Our algorithms for Theorem 1.1 and Corollary 1.3 are combinatorial, but randomized. It is unclear
whether they can be derandomized without incurring a large runtime cost. In particular, our
algorithms use sampling to crudely estimate the sizes of reachability sets for all vertices in the
graph. As far as we know, there are no faster deterministic ways to do this in the worst case
than explicitly computing the reachability sets which requires Ω(min{nω , mn}) time. We partially
derandomize Corollary 1.3 using different techniques:
Theorem 1.4 (Deterministic Additive Girth Approximation) There is a deterministic combinatorial algorithm that for any unweighted n-node m-edge directed graph with unknown girth g and
parameters a, ǫ ∈ (0, 1) computes in Õ(ǫ−2 mn1−a ) time an estimate ḡ such that g ≤ ḡ ≤ g + O(nα )
if g ≤ na and g ≤ ḡ ≤ (1 + ǫ)g if g > na .
A natural question is whether the Õ(mn1−a ) runtime for na -additive approximation is necessary.
Surprisingly, we show that when it comes to combinatorial algorithms, Theorem 1.4 and Corollary 1.3 are optimal up to constant factors in the additive error, barring a breakthrough in BMM
algorithms:
Theorem 1.5 (Hardness for Improving Additive Running Time) Suppose there is a combinatorial algorithm for some ε > 0 and a = 1/2 that computes an additive na − 1 approximation
to the girth of any unweighted n-node m-edge directed graph in O(mn1−a−ε ) time. Then for some
constant δ > 0 there is an O(n3−δ ) time combinatorial algorithm for n × n BMM.
Finally, we note that these algorithm can also be extend to produce additive roundtrip spanners:
Corollary 1.6 (Additive Roundtrip Spanners) Given any unweighted n-node, m-edge directed
graph and any a ∈ (0, 1) in time Õ(mn1−a ) we can compute an O(na )-additive roundtrip spanner
with Õ(n2−a ) edges edges with high probability.
Another natural questions is whether the O(n2−a ) edges needed to create a na -additive roundtrip
spanner is necessary. We give a lower bound construction that this is indeed the case:
3
Theorem 1.7 (Bounds on Additive Spanner Size) For all a ∈ (0, 1), there exist (arbitrarily
large) n-vertex directed graphs where all na -additive roundtrip spanners have Ω(n2−a ) edges.
1.2
Algorithmic Techniques : Roundtrip Covers in Nearly Linear Time
Our key technical contribution towards achieving the majority of our algorithmic results is the
first nearly linear time algorithm for computing roundtrip covers of directed graphs. Informally, a
roundtrip cover is a decomposition of a directed into an overlapping collection of balls, i.e. roundtrip
distance induced subgraphs. It is required that the radius, or maximum roundtrip distance, in each
ball be bounded and that any pair of vertices of bounded roundtrip distance appear together in some
ball (See Section 5 for formal definition). Computing such covers naturally yields multiplicative
roundtrip spanners and has been considered in previous work [CW04, RTZ08].
Unfortunately, all known roundtrip cover computation algorithms prior to this paper ran in at least
Ω(min{nω , mn}) time. It was not clear how to efficiently manipulate the roundtrip metric for the
purposes of computing such decompositions. It seemed that, in the worst case, one would have to
compute almost the entire roundtrip metric explicitly, i.e. solve APSP.
We overcome this difficulty through a careful application of a few techniques. The first is natural:
cluster the graph by growing balls of exponentially distributed radii or using exponential distribution based clustering techniques. (Similar ideas were used recently for parallel algorithms for
undirected graph decomposition [MPX13] and directed maximum flow [EMPS16]). This does not
distort too many roundtrip distances, but may fail to produce clusters of significantly smaller size.
The second is our key insight: we show that if we carefully seed such a clustering routine we can
ensure that we either find a large cluster with small roundtrip diameter or we break the graph into
significantly smaller pieces while sufficiently preserving roundtrip distances. Unfortunately, naively
implementing such a procedure would be expensive (i.e. involve solving APSP). To circumvent this,
we use another trick: a known sampling based approach to estimate the fraction of vertices that
each vertex can reach or is reachable by within a given distance and show this suffices to pick seeds
for clustering.
In short, we achieve our multiplicative approximations via a delicate combination of several powerful tools that have been defined and used before: (1) low diameter graph decompositions first
introduced in [Awe85], (2) using the exponential distribution for decomposition (e.g. in [LS91,
Bar96, MPX13, EMPS16]), (3) recursive graph decomposition (e.g. in [Bar96, FRT04, EMPS16]),
and (4) sampling based reachability set estimation ([Coh97]). However, despite the prevalence of
this machinery, it was an open question whether or not it could be leveraged to yield any running
time improvement for the directed problems we consider. It was unclear a-priori if there was structure to exploit to quickly decompose the roundtrip metric and if the problems we consider were as
hard as APSP.
Our key contribution is to show that this is not the case and there is in fact a way to rapidly reveal
non-trivial directed graph structure sufficient to achieve our results. There are several pitfalls that
occur when naively applying standard machinery to this problem and we believe the strength of our
result is to show how to methodically overcome them (see Section 3). The lack of fast combinatorial
primitives for directed graphs is occasionally referenced as indicative of the gap between recent
progress on approximate undirected network optimization problems [CKM+ 11, Mad10, LRS13,
She13, KLOS14, Pen14, She16] and directed problems [Mad13, LS14, CMSV16]. We hope our
results and the insights that underlie them may find future use.
4
1.3
Additional Related Work
For girth in undirected unweighted graphs, besides Itai and Rodeh’s [IR78] original O(n2 ) time ade 3 /m)-time
ditive 1-approximation algorithm, Roditty and Vassilevska W. [RW12] presented an O(n
additive 3-approximation algorithm. The additive 1-approximation of [IR78] is also a multiplicative 4/3-approximation. Lingas and Lundell [LL09] presented the first algorithm that breaks the
quadratic time bound of [IR78], at the price of a weaker approximation: their algorithm runs in
e 3/2 ) time and returns a multiplicative 8/3-approximation. Roditty and Vassilevska W. [RW12]
O(n
e 5/3 )-time deterministic multiplicative 2-approximation algorithm. They also
presented an O(n
showed how to obtain a less than 2 multiplicative approximation in truly subquadratic time for
triangle-free graphs.
The history of combinatorial algorithms for BMM of n × n matrices is as follows. Bansal and
Williams [BW12] obtained an O(n3 / log2.25 n) time combinatorial algorithm improving on the 40year record of O(n3 / log2 n) by the Four-Russians Algorithm [ADKF70]. The result of [BW12]
was further improved by Chan [Cha15] to O((n3 / log3 n) logO(1) log n) time and most recently by
Yu [Yu15] to Ô((n3 / log4 n) logO(1) log n).
Obtaining a truly subcubic time algorithm for APSP is among the most studied longstanding
open problems in graph algorithms. In the 1970s Fredman [Fre76] showed that the O(n3 ) time
classical Floyd-Warshall algorithm is not optimal by giving an O(n3 (log log n/ log n)1/3 ) time running time. Many polylogarithmic improvements followed, the last being O(n3 log3 log n/ log2 n) by
Chan [Cha07]. Two years ago, Williams [Wil14] used techniques from circuit complexity, namely
the polynomial
method, to shave all polylogs, thus obtaining the current best bound for APSP,
√
3
Θ(
log
n)
n /2
.
Spanners in undirected weighted graphs were first studied by Awerbuch [Awe85] and Peleg and
Schäffer [PS89]. Althöfer et al. [ADD+ 93] showed that for every integer k ≥ 1, every n-node graph,
even if it is weighted, contains a multiplicative (2k − 1)-spanner on O(n1+1/k ) edges. This result is
optimal, conditioned on a well-known (and partially proven [Wen91]) conjecture by Erdös [Erd63]
about the existence of graphs of high girth.
1.4
Organization
The remainder of the paper is structure as follows. We introduce notation in Section 2, provide
an overview of our approach in Section 3, and show how to compute roundtrip covers in Section 4.
We then provide our algorithms for multiplicative approximations in Section 5 and our algorithms
for additive approximations in Section 6. We conclude with our lower bounds in Section 7.
2
Preliminaries
Here we introduce various terminology we use throughout the paper.
Graphs: Throughout this paper we let G = (V, E, l) denote a directed graph with vertices V , edges
E ⊆ V × V , and non-negative edges lengths l ∈ RE
≥0 . At times we consider unweighted graphs, that
is graphs in which le = 1 for all e ∈ E and in this case we will omit the l altogether.
Distances: We let dG (u, v) denote the (shortest path) distance from u to v in G and we abbreviate
this as d(u, v) when G is clear from context. At times we consider shortest path distances over edge
subgraphs F ⊆ G and write dF (u, v) to denote the length of the shortest path from u to v using
only the edges in F . In all cases we define d(u, v) = ∞ if there is not a path from u to v.
def
Roundtrip Spanners: For a, b ∈ V we refer to d(a ⇄ b) = d(a, b) + d(b, a) as the roundtrip
5
distance between a and b. We call a subgraph S ⊆ E an α-multiplicative roundtrip spanner if
dS (a ⇄ b) ≤ α · (dG (a ⇄ b)) for all a, b ∈ V and we instead call it an α-additive roundtrip spanner
if dS (a ⇄ b) ≤ dG (a ⇄ b) + α.
Distance Measures: For a directed graph G = (V, E, l) we call minv∈V maxv′ ∈V d(v, v ′ ) the radius
of G. We call maxv,v′ ∈V d(v, v ′ ) the diameter of G.
Balls: For a given metric, a ball of radius r around v is the set of vertices within distance r of
v. We generally use the term ‘ball’ to refer to balls in the roundtrip metric. For a directed graph
G, we use inballG (v, r) and outballG (v, r) to denote the subsets of vertices of G that can reach v
within distance r or be reached from v within distance r, respectively.
Trees: Given a directed graph G = (V, E, l) we call a Tout ⊆ E an out-tree with root r ∈ V if
the edges form an undirected tree and are all oriented away from r (i.e. there is a r to v path for
every node v in the tree). Similarly, we call Tin an in-tree with root r ∈ V if the edges form an
undirected tree and are all oriented towards r ∈ V (i.e. there is a v to r path for every node v in
the tree).
Paths and Cycles: A directed (simple) path P = hu = v1 , v2 , . . . , vk = vi ⊆ V from u to v is
an ordered set of vertices, where for every i ∈ {1, . . . , k − 1}, (vi , vi+1 ) ∈ E. A cycle C = hu =
v1 , v2 , . . . , vk = vi is a direct (simple) path with an additional requirement that (vk , v1 ) ∈ E. If P
is a path and u, v ∈ P such that u precedes v in P then we denote by P (u, v) the subpath of P
from u to v. If P1 and P2 are paths in G, then we denote by P1 · P2 the concatenation of P1 and
P2 .
e
e (n)) = O(f (n) logc f (n)).
Running Times: We use O-notation
to hide logarithmic factors, i.e. O(f
Probability: We use with high probability (w.h.p) to denote that an event happens with probability
at least 1 − 1/O(poly(n)) where n is the size of the input to the problem.
3
Overview of the Approach
Our approach for computing multiplicative roundtrip spanners is broadly inspired by the following
simple general strategy for computing spanners in undirected unweighted graphs:
1. Repeat until there are no vertices left:
• Grow a ball of random radius from a vertex.
• Add the edges in the computed shortest path tree to the spanner.
• Remove all vertices in the ball from the graph.
2. Recurse on the subgraph induced by the edges that have endpoints in different balls.
If the radii are chosen appropriately one can show that the shortest path trees approximately
preserve the distance between the endpoints of all edges inside a ball and that not too many edges
are cut (i.e. have endpoints in different balls). While there is a great body of work on efficiently
constructing spanners with many desirable properties [BKMP05, Coh99, DHZ00, EP04, TZ05], this
simple strategy suffices to provide a polylogarithmic multiplicative spanner in nearly linear time.
Unfortunately there are two serious issues that prevent us from easily extending this approach to
compute roundtrip spanners for directed graphs:
• Recursing on cut edges does not work (or even make sense).
• There may be problematic vertices that are at a small distance to (or from) all the vertices,
but have a large roundtrip distance to every vertex.
We derive our algorithm by carefully addressing these two issues.
6
Issue #1: How to Recurse?
The first issue is immediate. If we grow multiple balls in a directed graph and attempt to recurse
on cut edges, it may be the case that we disconnect the graph and the roundtrip distances for the
cut edges become infinite. Consequently, if we recurse on the cut edges alone we simply do not
have enough information to recover the path information we lost. Therefore, it is not clear if there
is any reasonable way to recurse on the edges that we may distort.
To alleviate this issue we instead build a randomized scheme where we reason directly about the
probability of cutting or distorting the roundtrip distance between any particular pair of vertices.
Building on previous work on graph decomposition [LS91, Bar96, MPX13] and directed maximum
flow [EMPS16] we use the fact that if we grow a ball where the radius are chosen from an exponential
distribution then we can directly reason about the probability that removing that ball cuts a
cycle. We then proceed to repeatedly grow balls of exponential radii, removing them in each
iteration. Since the exponential distribution is memoryless, we can show that the probability that
this approach cuts a cycle depends only on the parameter of the exponential distribution we use.
For completeness, in Appendix C we prove formally that repeated exponential ball growing works.
For our algorithms we instead use a slightly more sophisticated clustering technique described in
Section 4.1. This scheme works for similar reasons, but is more easily parallelizable. Ultimately,
both techniques allow us to reason about the probability of distorting a cycle (rather than an edge)
and repeat until all cycles corresponding the roundtrip distances between vertex pairs are preserved
with high probability. The difficulty remains in ensuring that we can actually terminate such a
procedure in a small number of iterations (i.e. Issue #2).
Issue #2: How to Avoid Problematic Vertices?
The second issue seems even more troubling. Suppose that there is a graph with non-trivial cycle
structure we would like to approximate. To create a harder instance one could simply create a new
graph by adding many new vertices each of which has one short length edge to every original vertex
and one long length edge from every vertex. Clearly these new vertices do not affect the cycle
structure of the original graph that we wish to approximate. However, any shortest path query
from these new vertices will quickly explore the entire graph. Consequently, starting any sort of
clustering from these vertices could be quite expensive in terms of running time, yet reveals very
little information about the graph’s cycle structure.
Even if we take a different approach and simply attempt to improve the running time of constructing
the roundtrip spanners from prior work [RTZ08], a similar issue arises. Here the immediate issue
is that the algorithm computes balls in the roundtrip metric. However, to do this, again we need
to explore many vertices at a large distance in the roundtrip metric for analogous reasons.
To alleviate this issue, we use sampling to estimate, in near linear time, the fraction of vertices in
|V | of O(r)-balls around all nodes, up to a small additive error ǫ. This can be done in nearly linear
time due to a clever technique of Cohen [Coh97] (For completeness, see Appendix however, we
give a self-contained construction tailored to our purposes in Section A.) This allows us to find the
problematic vertices that can reach many vertices at distance r yet are reachable by few at distance
r (or vice versa). By using this technique to find problematic vertices we can better bias the seeds
of our decomposition routines and make more progress in nearly linear time. This is crucial to our
algorithm.
7
Building the Algorithm
Combining our ideas to deal with these two issues yields our algorithm. We estimate for every
vertex the number of vertices at distance O(r) both from and to it. In one case there is a vertex
that can both reach and is reachable by many vertices. In this case, we can compute a large enough
ball (in the roundtrip metric) that contains vertices of small roundtrip distance, and then we recurse
on the rest of the vertices outside the ball. In the other case, there are many vertices that either do
not reach or are not reached by too many vertices at distance O(r) and we can grow clusters from
them and recurse on all the clusters. In either case we show that we do not need to recurse too
many times and that ultimately, with constant probability (see Section 4.3), any particular pair
of vertices at small roundtrip distance is together in a cluster. Repeating this procedure multiple
times yields our nearly linear time roundtrip cover algorithm (see Section 4).
With our our roundtrip cover algorithm in hand, nearly we use them to compute multiplicative
roundtrip spanners and obtain multiplicative estimates of the girth. A naive application of our
procedure would yield a logarithmic dependence on the range of lengths in the graph. To avoid
this, we show how to break a directed graph into smaller graphs reducing to subproblems where
lengths vary only polynomially in the number vertices. Furthermore, we show that our algorithm
is inherently parallel and we obtain new work / depth tradeoffs for these problems. As discussed,
this also yields faster additive approximation for the girth, though new insights are needed to
obtain deterministic results (see Section 5). As discussed in the introduction, our roundtrip cover
algorithm is also used to compute additive approximations to the girth and additive roundtrip
spanners as discussed earlier (see Section 6).
4
Roundtrip Covers
In this section we provide our main results on graph partitioning. In particular we show how to
efficiently construct roundtrip covers, first introduced in [RTZ08].
Definition 4.1 (Roundtrip Covers, definition 2.4 in [RTZ08]) A collection C of balls is a
(k, R)-roundtrip-cover of a directed graph G = (V, E, l) if and only if each ball in C is of radius at
most kR, and for every u, v ∈ V such that dG (u ⇄ v) ≤ R, there is a ball B ∈ C such that u, v ∈ B.
The main result of this section is the following theorem, stating that we can construct a (O(k log n), R)1/k ) time.
e
roundtrip-cover with high probability in O(mn
Theorem 4.2 (Fast Roundtrip Cover) The algorithm Fast-Roundtrip-Cover(G, k, R), returns a collection C that is an (O(k log n), R)-roundtrip-cover of directed graph G = (V, E, l), w.h.p.
in time O(mn1/k log4 n). Moreover, every vertex v ∈ V belongs to O(n1/k log n) elements of C.
Note that if G has integer edge lengths between 0 and U we can immediately apply Theorem 4.2
for a value of R that is a power of 2 and obtain O(k log n)-multiplicative roundtrip spanners of
with O(n1+1/k log2 n log U ) edges in time O(mn1/k log4 n log U ) as well as compute an O(k log n)
multiplicative approximation to the girth in the same running time. Consequently, proving Theorem 4.2 encapsulates much of the difficulty in achieving our desired algorithmic results. However,
in Section 5 we show how a more careful application of Theorem 4.2 yields even stronger results,
completely removing the dependence on U .
The remainder of this section is dedicated to providing the algorithm Fast-Roundtrip-Cover
and proving Theorem 4.2. First in Section 4.1 we provide our main graph clustering tool, then
in Section 4.2 we provide our technique for estimating the fraction of vertices reachable to and
from each vertex at some radius. Finally, in Section 4.3 we put these tools together to provide
Fast-Roundtrip-Cover and prove Theorem 4.2.
8
4.1
Clustering
Here we provide the primary clustering/partitioning technique we use for our algorithm. We provide
an algorithm that partitions the vertices into regions of bounded radius of our choice centered
around a chosen subset of the vertices so that the probability of separating any two vertices of
bounded roundtrip distance is small. The ability to control the radii and choose the starting
vertices is key to deriving our algorithm.
As discussed in the introduction we use an exponential-distribution-based clustering procedure
so that we can argue directly about the probability of cutting any particular cycle. This allows
us to apply this procedure multiple times and argue by union and Chernoff bounds that with
high probability we do not cut any cycle that we want to approximate, and thus obtain a good
approximation of any relevant cycle. However, rather than simply growing balls of exponentially
distributed radius and repeating (as discussed in Section 3), we provide a different scheme in the
flavor of [MPX13, EMPS16] that better parallelizes. For completeness we complement our analysis
with a proof that this sequential ball growing scheme also works in Appendix C.
Our algorithms, Cluster-Out and Cluster-In are given in Figure 1. Given a graph G = (V, E, l),
a set of vertices S ⊆ V and a target radius r, the algorithm uses the exponential distribution to
assign vertices in G to clusters for each of the v ∈ S. The assignment is done in a way that ensures
that these clusters each have bounded radius. By our choice of assignment rule and distribution
we formally show that the probability that two vertices of small roundtrip distance are not in the
same cluster is sufficiently small.
(V1 , . . . , Vt ) = Cluster-Out(-In)(G, S, r), where G = (V, E, l) is a directed graph, S ⊆ V
and r > 0.
1. Set β := log(n)/r.
2. For every vertex v ∈ S, pick xv ∼ Exp(β).
3. For each vertex u ∈ V , assign u to the cluster rooted at the vertex v ∈ S which
minimizes −xv + dG (v, u), unless that quantity is positive; in that case, do not assign
u to any cluster. (use dG (u, v) for Cluster-In)
4. Let V1 , . . . , Vt−1 be the clusters
produced by the above step.
S
5. Return (V1 , . . . , Vt−1 , V \ i Vi ).
Figure 1: The clustering algorithm.
In the remainder of this section we formally analyze this algorithm proving Lemma 4.3. The analysis
we present is very similar to that of [MPX13] and uses a subset of the ideas of [EMPS16]. The main
difference is that we start only from a subset of the vertices S. Our analysis makes use of several
facts regarding the exponential distribution which for completeness we prove in Appendix B.
Lemma 4.3 Let (V1 , V2 , . . . , Vt ) be the partition of V returned by Cluster-Out(G, S, r) (analogously of Cluster-In). Then, for any c ≥ 1 we have
1. with probability at least 1 − n1−c for all i < t, the radius of the tree corresponding to Vi is at
most c · r,
2. for any pair of vertices u, v at roundtrip distance at most R in G, they are in the same set Vi
with probability at least exp(− log(n)R/r).
Furthermore, the algorithm runs in time O(m log n).
Proof:
We prove the lemma for Cluster-Out (the proof for Cluster-In is analogous). We use
9
various facts about the exponential distribution though this proof (See Section B for their proof).
Note that the maximum radius of any cluster, Vi is upper bounded by maxi xi by design. For every
i ∈ 1, . . . , t, we have
P r [xi ≥ c · r] ≤ exp(−c · βr) = n−c .
By a union bound the maximum radius of is at most c · r with probability at least 1 − n1−c .
To prove the remainder of the lemma, fix u, v ∈ V with roundtrip distance at most R in G.
Assume s ∈ S is the vertex minimizing −xs + min(dG (s, u), dG (s, v)), and that this quantity is less
than 0 (otherwise we have u, v ∈ Vt ). Let T be the second smallest value of this quantity, or 0,
whichever is smaller. Condition on the values of s and T and assume without loss of generality that
dG (s, u) ≤ dG (s, v). Then u is assigned to the cluster rooted at s, and u and v can be separated
only if −xs + dG (s, v) > T . By the triangle inequality, this would imply −xs + dG (s, u) + R > T .
By assumption, we have −xs + dG (s, u) < T , or equivalently xs > dG (s, u) − T . By the memoryless
property of the exponential distribution (See Lemma B.1), we see that the probability that the
cluster rooted at s contains both vertices u and v is at least
h
i
h
i
Pr xs > dG (s, u) − T + R | xs > dG (s, u) − T = Pr Exp(β) ≥ R = exp(−βR),
yielding the desired result.
4.2
Estimating Ball Sizes
To compute part of a roundtrip cover, ideally we would just partition the graph using the decomposition scheme of the previous graph and repeat until the clusters have good roundtrip diameter.
Unfortunately, as discussed in Section 3 this approach fails as there may be problematic vertices
that have a large low-radius ball in one direction, and a small low-radius ball in the other direction.
In other words, calls to Cluster-In and Cluster-Out with the wrong set S might only yield
trivial partitions, i.e. V1 = V .
To alleviate this issue, we use a fast sampling approach to estimate the sizes of the O(r)-balls of
all vertices, that allows us to identify these problematic vertices efficiently.
Lemma 4.4 ([Coh97]) For all ǫ ∈ (0, 1) there is an algorithm Estimate-Balls(G, r, ǫ) that in
O(mǫ−2 log2 n) time computes n-length vectors sout , sin , with the following property. For any vertex
u, let s̄out
u be the fraction of vertices in V such that dG (u, vi ) ≤ r. Then, w.h.p., for all vertices u,
out
out corresponding to u. An analogous statement
|s̄out
−
sout
u
u | ≤ ǫ, where su is the component of s
holds for sin .
For completeness, we provide a self-contained proof of Lemma 4.4 in Appendix A.
4.3
Fast Roundtrip Covers
Combining the techniques of Section 4.1 and Section 4.2 here we provide our efficient algorithm for
constructing roundtrip covers, i.e. Fast-Roundtrip-Cover, and prove the main theorem of this
section, Theorem 4.2, analyzing this algorithm.
We push much of the work of this algorithm to a subroutine Probabilistic-Cover that performs
the simpler task of constructing probabilistic roundtrip cover: that is a partition of the vertex set
such that any two vertices close enough in the roundtrip metric are in the same cluster with at least
some fixed probability. Our main roundtrip cover construction is then simply a union of sufficiently
many probabilistic roundtrip covers computed by Probabilistic-Cover.
10
{B1 , B2 , . . .} = Probabilistic-Cover(G, r), where G = (V, E, l) is a directed graph and
r > 0.
1. Set c > 1 to a sufficiently large constant (for high probability bounds).
2. If V is empty, return ∅.
3. Let sout , sin := Estimate-Balls(G, c · r, 18 ).
3
in := {v ∈ V : sin ≥ 3 }.
4. Let S out := {v ∈ V : sout
v ≥ 4 }, S
v
4
5. If S out ∩ S in 6= ∅:
(a) Choose an arbitrary vertex u ∈ S out ∩ S in .
(b) (failure) If |S out ∩ S in | < 41 · |V |, return {V }.
(c) Pick rB uniformly at random in [2c · r, 2(c + 1) · r].
(d) Let B be a ball of radius rB in the roundtrip metric around u.
(e) Let G′ be the graph induced by G on V \ B.
(f) Return {B} ∪ Probabilistic-Cover(G′ , r).
6. If |S out | ≤ 21 · |V |:
(a) Let (V1 , . . . , Vt ) := Cluster-Out(G, V − S out , r).
Otherwise (we have |S in | ≤ 21 · |V |):
(b) Let (V1 , . . . , Vt ) := Cluster-In(G, V − S in , r).
7. (failure) If maxi |Vi | > 87 · |V |, return {V }.
8. For i = 1, . . . , t, let Gi be the graph induced by G on Vi .
9. Return Probabilistic-Cover(G1 , r) ∪ . . . ∪ Probabilistic-Cover(Gt , r).
Figure 2: Single pass of cover construction.
The statement and analysis of Probabilistic-Cover are the most technically involved results of
this section. Our algorithm, Probabilistic-Cover, takes as input a directed graph G, a target
radius r, and proceeds as follows. First we use Estimate-Balls from Section 4.2 to estimate the
fraction of vertices in all balls of radius O(r) up to an additive 1/8. Then we consider two cases.
In the first case we find that there is some vertex that can reach a large fraction of the vertices at
distance O(r) and can be reached by a large fraction of the vertices at distance O(r). In this case
we know that many vertices have a small roundtrip distance to this vertex so we simply output
a roundtrip metric ball around this vertex and recurse on the remaining vertices. Otherwise, we
know that there are many vertices that do not reach (or are not reachable by) many vertices at
distance O(r) and we can cluster to or from these vertices using Cluster-Out(-In) analyzed in
Section 4.1 and recurse on the clusters. In either case we recurse on subsets of vertices that are a
constant fraction of the original size and hence only need to recurse a for a logarithmic number of
iterations. We formally analyze this algorithm and prove that it has the desired properties in the
following Lemma 4.5.
Lemma 4.5 Let C := Probabilistic-Cover(G, r). Then:
1. each B ∈ C is a ball of radius O(r) in the roundtrip metric, w.h.p.,
2. any pair of vertices u, v at roundtrip distance at most R in G are in the same element of C
with probability at least exp(−6 log2 (n)R/r), and
3. every vertex v ∈ V belongs to exactly one element of C.
Furthermore, the algorithm runs in time O(m log3 n).
Proof:
Property 3. is easily verified.
To prove property 1., first note that for large enough c w.h.p. all calls to the subroutines Cluster-In, Cluster-Out
11
and Estimate-Balls yield the guarantees described in Lemmas 4.3 and 4.4. Conditioning on this
event, we show the failures in steps 5(b) and 7 of Probabilistic-Cover never occur; this is
enough to show the thesis. First assume that there exists a vertex u ∈ S out ∩ S in . Then, by
assumption, we have |outballG (u, c · r)| ≥ ( 34 − 18 ) · |V | ≥ 85 · |V | and |inballG (u, c · r)| ≥ 85 · |V |.
Hence |outballG (u, c · r) ∩ inballG (u, c · r)| ≥ 14 · |V |, which implies |S out ∩ S in | ≥ 41 · |V |. Hence
the failure in step 5(b) cannot occur. Now assume that |S out | ≤ 21 · |V | (the case |S in | ≤ 21 · |V |
is analogous). By assumption, the radii of all balls grown in calls to Cluster-Out are at most
c · r, and so the sizes of the clusters constructed are at most 87 · |V | (by assumption on accuracy
of Estimate-Balls). The last cluster cannot be larger than 21 · |V | by construction. Hence the
failure in step 7 cannot occur.
To prove property 2., first note that in every recursive call, the size of the vertex set is multiplied by
at most 7/8. Therefore, there are at most ⌈log8/7 n⌉ levels of recursion. Now fix two vertices u and
v at roundtrip distance at most R ≤ r in G. In each level of recursion, the vertex set is partitioned
by a call to Cluster-In or Cluster-Out, or growing a roundtrip metric ball of radius chosen
uniformly at random from [2c · r, 2(c + 1) · r]. For the first two cases, the probability u and v are not
separated if they have not been separated previously is at least exp(− log(n)R/r)) by Lemma 4.3.
For the last case, the probability is easily seen to be at least 1 − R/(2 · r) ≥ exp(− log(n)R/r).
Hence, the probability that u and v are not separated at all is at least
exp(− log(n)R/r)⌈log8/7 n⌉ ≥ exp(−6 log2 (n)R/r).
Note that computing the ball in the roundtrip metric in step 5(d) reduces to two single source
shortest path computations from u. Consequently, the running time is dominated by the O(log n)
calls to Estimate-Balls.
With Probabilistic-Cover in hand, we are ready to present the complete efficient algorithm for
constructing roundtrip covers. The algorithm, Fast-Roundtrip-Cover is given in Figure 3 and
we conclude with its analysis, i.e. the proof of Theorem 4.2.
{C1 , C2 , . . .} = Fast-Roundtrip-Cover(G, k, R), where G = (V, E, l) is a directed graph
and k, R > 0.
1. Let r := 6Rk log n.
2. Let c > 1 to a sufficiently large constant (for high probability bounds).
3. Let C0 := ∅.
4. For i = 1, . . . , c · ⌈n1/k ⌉ · ⌈log n⌉:
Ci := Ci−1 ∪ Probabilistic-Cover(G, r).
5. Return C.
Figure 3: The fast roundtrip cover algorithm.
Proof of Theorem 4.2: By Lemma 4.5, w.h.p. all balls in C have the desired radius properties
and the size assertions are easily verified.
It remains to show that w.h.p. any two vertices at short roundtrip distance share at least one
cluster in C. Fix two vertices u and v at roundtrip distance at most R in G. By Lemma 4.5, the
probability they are in the same cluster in any single cover pass is at least exp(−6 log2 (n)R/r) =
exp(− log(n)/k). Hence, the probability they are separated in ⌈n1/k ⌉ = ⌈exp(log(n)/k)⌉ inde-
12
pendent passes is at most exp(−1). Therefore, the probability they are separated in all the of
c · ⌈n1/k ⌉ · ⌈log n⌉ passes is at most exp(−c log n) = n−c .
5
Roundtrip Spanners and More
The analysis in Section 4 as encompassed by Theorem 4.2 yields our best result for unweighted
graphs G; the union of Fast-Roundtrip-Cover(G, k, R) for R = 20 , 21 , . . . , 2⌈log 2 n⌉ is w.h.p. an
O(k log n)-multiplicative roundtrip-spanner of G. More generally, for weighted graphs we obtain
the following corollary:
Corollary 5.1 Given a directed graph G = (V, E, l) we can construct w.h.p. a O(k log n)-roundtripspanner of G with O(n1+1/k log n log(nW )) edges in O(mn1/k log4 n log(nW )) time, where W is the
ratio between the largest and the smallest length in G.
The first aim of this section is to remove the dependence on W from both the size of the spanner and
the running time (Section 5.1). This allows us to prove the following result on spanner construction,
and Theorem 1.1 as its corollary.
Theorem 5.2 The algorithm Fast-Roundtrip-Spanner(G, k) in O(mn1/k log4 n) time computes
w.h.p. an O(k log n)-roundtrip-spanner of G of size O(n1+1/k log2 n).
Then we shall investigate parallel algorithms that result from our scheme, obtaining the following
results (Section 5.2).
Theorem 5.3 Given an unweighted directed graph G and an upper bound R on the maximum
diameter of any strongly connected component of G, we can w.h.p. compute the strongly connected
e
e
components of G in O(m)
work and O(R)
depth.
Theorem 5.4 Given an unweighted directed graph G, we can w.h.p. compute an O(k log n) ap1/k ) work and O(girth(G))
e
e
proximation to the girth of G in O(mn
depth.
5.1
Removing the Dependence on Edge Lengths
Our algorithm for constructing the spanner will remain based on the idea of taking a union of
(O(k log n), R)-roundtrip-covers over R ∈ R, for some set R such that every roundtrip distance in
G is a constant factor smaller than some element of R.
The main idea in removing the dependence on the lengths of the edges is that for a fixed value of
R, we do not have to consider all the edges in G when constructing a (O(k log n), R)-roundtripcover. First, note that we can remove all the edges longer than R, as that does not change
any roundtrip distance smaller than R. Simultaneously, for any strongly connected component
of edges shorter than R/n, we can replace it by a single vertex. Indeed, uncontracting all such
vertices after obtaining a roundtrip cover will increase the length of any path found by only an
additive R. Finally, we can remove all the edges that do not participate in any strongly connected
component, as that does not impact any roundtrip distances. The idea is similar to those given
in [CMP+ 14, EMPS16]; the main difference from the scheme of [EMPS16] is in preserving only
edges that are parts of connected components of edges shorter than R. The described process is
formalized in Definition 5.5.
Definition 5.5 Let G = (V, E, l) be a directed graph and xL , xR ∈ R be such that 0 < xL < xR .
We construct G collapsed to [xL , xR ] by:
• merging any vertices that can reach each other while following only edges of length at most xL ,
• removing all edges longer than xR ,
13
• removing all edges whose endpoints cannot reach each other while following only edges of length
at most xR , and
• removing all vertices of degree 0 remaining after the above operations.
To simplify notation, we define the L∞ -roundtrip distance d∞
G (u, v):
Definition 5.6 For a directed graph G = (V, E, l) and a pair of vertices u, v ∈ V , we define the
L∞ -roundtrip distance between u and v, denoted d∞
G (u, v), as the minimum value of d such that
there is a cycle C containing u and v such that le ≤ d for all e ∈ C.
We now show that performing the process for all R ∈ {2t : t ∈ Z} results in a collection of graphs
with a bounded size.
Lemma 5.7 Let G = (V, E, l) be a directed graph. For every t ∈ Z, let G(t) be G collapsed to
[2t /n, 2t ]. Then the total number of edges in all G(t) is O(m log n), and the total number of vertices
in all G(t) is O(n log n).
Proof: Fix an edge e = (u, v) ∈ E and t such that e is an edge in G(t) . Note that since e is not
t
contracted, we must have d∞
G (u, v) > 2 /n. Simultaneously, since e is part of a strongly connected
t
component in G consisting of edges of length at most 2t , we must have d∞
G (u, v) ≤ 2 . Hence
(t)
t − log2 d∞
G (u, v) ∈ [0, log 2 n). Therefore e is included in O(log n) of the graphs G .
Assume that u is a vertex of G(t) . By construction, it must be part of a nontrivial strongly con′
nected component in G(t) , and so it is merged with another vertex in G(t ) for all t′ ≥ t + log2 n.
Since there are only O(n) possible vertices that can result from merging vertices in V , and each of
them appears in O(log n) graphs G(t) , we obtain the thesis.
If we can construct all the graphs G(t) efficiently, we can simply run Fast-Roundtrip-Cover on
each of them to obtain a spanner for G. Following the idea of the proof of Lemma 5.7, to construct
all of G(t) , is enough to compute for each edge (u, v) ∈ E the value of d∞
G (u, v). This is obtained
by the algorithms Roundtrip-L∞ -Spanner and Find-Collapse-Times, described below. The
algorithm Find-Collapse-Time computes d∞
G (u, v) for every edge (u, v) in G, assuming that all
the edges have distinct weights from 1 to m.
s = Find-Collapse-Times(G, (eL , . . . , eR )), where G = (V, E) is a directed graph, L, R ∈
N with 1 ≤ L ≤ R.
1. If L = R, set se = L for all e ∈ E and return s.
2. Let M = ⌊(L + R)/2⌋.
3. Let
E ′ := {e ∈ E|e is contained inside a SCC of the graph (V, {eL , . . . , eM })}.
4.
5.
6.
7.
Let V ′ be V with edges in E ′ contracted.
Let s′ := Find-Collapse-Times((V, E ′ ), (eL , . . . , eM )).
Let s′′ := Find-Collapse-Times((V ′ , E \ E ′ ), (eM +1 , . . . , eR )).
Return s′ merged with s′′ .
Figure 4: The recursive algorithm for computing collapse times.
Lemma 5.8 Let G = (V, E) be a directed graph and (eL , . . . , eR ) be a sequence of edges on V .
Assume that every edge in E is contained in a strongly connected component of (V, {eL , . . . , eR }).
Let s = Find-Collapse-Times(G, (eL , . . . , eR )). Then for every e ∈ E, it holds that se is the min14
imum i such that e is contained in a strongly connected component of (V, {eL , . . . , ese }). Moreover,
the algorithm runs in O((|V | + |E| + (R − L + 1)) log(R − L + 1)) time.
Proof: Correctness is easily proven by induction. To bound the running time, it is enough to
observe that every recursive call halves (R − L + 1), and every edge in E is only passed to one
recursive call.
The algorithm Roundtrip-L∞-Spanner constructs an O(n)-sized subset F of the edges of G that
preserves the L∞ -roundtrip distances between vertices of G. It also returns a tree T containing all
the vertices that can result from collapsing cycles of maximum edge length lower than some bound,
with edges of T describing the hierarchical structure on them. Lowest common ancestor queries on
T enable us to efficiently compute d∞
G (u, v) for any u, v.
(F, T ) = Roundtrip-L∞ -Spanner(G), where G = (V, E, l) is a directed graph.
1. Remove from G any edges that are not part of a strongly connected component.
2. Let e1 , . . . , em be the edges of G, ordered by increasing length.
3. Let s := Find-Collapse-Times(G, (e1 , . . . , em )).
4. Let V0 = V, F0 = ∅.
5. Let T = (V, ∅).
6. For i in 1, . . . , m:
(a) Let Ei be the set of edges e for which se = i.
(b) Let Ei′ be union of any out- and in-trees for the strongly connected components
of (Vi−1 , Ei ).
(c) Let Fi := Fi−1 ∪ Ei′ .
(d) Let Vi be the set of vertices obtained from Vi−1 by contracting all of Ei (equivalently Ei′ ).
(e) Label every vertex of Vi \ Vi−1 by l(ei ).
(f) Add all vertices of Vi \ Vi−1 to T .
(g) For every vertex u ∈ Vi−1 that was contracted into a vertex v ∈ Vi \ Vi−1 , add
an edge between v to u to T .
7. Return (Fm , T ).
Figure 5: The recursive algorithm for computing collapse times.
Lemma 5.9 Let G = (V, E, l) be a directed graph. Let (F, T ) = Roundtrip-L∞ -Spanner(G).
Then:
1. F ⊆ E is such that for any pair of vertices u, v contained in a cycle in G with maximum edge
length R, there exists a cycle in (V, F ) containing u and v with maximum edge length R,
2. |F | = O(n), and
3. for any two vertices u, v ∈ V , the label of the lowest common ancestor of u and v in T is equal
to d∞
G (u, v).
Moreover, the algorithm works in O(m log n) time.
Proof: By Lemma 5.8, the application of Find-Collapse-Times computes for each edge (u, v) ∈
E the value of d∞
G (u, v). The claims of the lemma follow by construction.
Finally, we describe our complete algorithm for computing roundtrip distance spanners of weighted
graphs. The algorithm first computes all graphs G(t) using a call to Roundtrip-L∞ -Spanner and
15
the ideas of the proof of Lemma 5.7. It then invokes Fast-Roundtrip-Cover for each of G(t) and
returns the union of the results, together with a L∞ -roundtrip distance spanner for G to account
for the collapsed clusters in G(t) .
F = Fast-Roundtrip-Spanner(G, k), where G = (V, E, l) is a directed graph and k ≥ 1.
1. Let (F0 , T ) := Roundtrip-L∞ -Spanner(G).
2. For all t ∈ Z, let G(t) be G collapsed to [2t /n, 2t ].
3. Let i := 0.
4. For every t such that G(t) is nonempty:
(a) Ci := Fast-Roundtrip-Cover(G(t) , k, 2t ).
(b) Fi+1 := Fi ∪ shortest path trees to and from roots of each ball in Ci .
(c) i := i + 1.
5. Return Fi .
Figure 6: The roundtrip spanner algorithm.
Theorem 5.2 The algorithm Fast-Roundtrip-Spanner(G, k) in O(mn1/k log4 n) time computes
w.h.p. an O(k log n)-roundtrip-spanner of G of size O(n1+1/k log2 n).
Proof of Theorem 5.2:
First note that for each t, F0 provides a low-cost spanner for every
(t)
collapsed vertex of G . By uncollapsing the collapsed vertices of G(t) and adding in edges from
F0 , the length of any path in the roundtrip cover at radius 2t increases by at most an additive 2t .
Since edges larger than 2t have no influence on roundtrip distances not larger than 2t , we see that
the roundtrip covers computed for each G(t) are also roundtrip covers for G (after adding the edges
of F0 ).
To obtain the claimed running time, we need to show that the nonempty graphs G(t) can be computed efficiently. Following the idea of the proof of Lemma 5.7, we see that it is sufficient to
compute for every edge (u, v) the value of d∞
G (u, v). By Lemma 5.9, this is easily done using lowest
common ancestor queries on T .
Theorem 1.1 is an immediate corollary of Theorem 5.2.
Theorem 1.1 (Multiplicative Girth Approximation) For any n-node, m-node directed graph
with nonnegative integer edge weights, with unknown girth g and integer k ≥ 1, in time O(mn1/k log5 n)
we can compute an estimate ḡ such that g ≤ ḡ ≤ O(k log n) · g with high probability.
It is sufficient to execute Fast-Roundtrip-Spanner(G, k). The
Proof of Theorem 1.1:
smallest diameter of any cluster computed in calls to Fast-Roundtrip-Cover will be no larger
than O(k log n) · girth(G).
5.2
Parallel Strongly Connected Components and Girth Estimation
Our main subroutine, Fast-Roundtrip-Cover, is inherently parallelizable. This enables us to
obtain a new parallel algorithm for computing strongly connected components in nearly linear work,
and depth proportional to the maximum diameter of a strongly connected component (assuming
access to a known upper bound). To our knowledge, no previous guarantees of this type have
been known, despite the classical status of analogous guarantees for problems such as parallel u-v
reachability in directed graphs.
Theorem 5.3 Given an unweighted directed graph G and an upper bound R on the maximum
16
diameter of any strongly connected component of G, we can w.h.p. compute the strongly connected
e
e
components of G in O(m)
work and O(R)
depth.
To prove this result, we first formally state the parallel runtime guarantees of Fast-Roundtrip-Cover.
Lemma 5.10 For unweighted graphs, a parallel version of Fast-Roundtrip-Cover(G, k, R) can
1/k ) work and O(R)
e
e
be implemented to work in O(mn
depth.
Proof: Since Probabilistic-Roundtrip-Cover has only O(log n) levels of recursion, and the
separate calls to it can be made in parallel, the bottleneck for depth is computing shortest paths.
Since for unweighted graphs any paths computed in calls to Estimate-Balls, Cluster-Out and
e
Cluster-In are of length O(R),
the thesis follows by employing standard parallel breadth first
search (cf. [MPX13]).
We now proceed to prove Theorem 5.3.
Proof of Theorem 5.3:
We start by computing C := Fast-Roundtrip-Cover(G, log n, R).
Now note that w.h.p., for any pair of vertices u and v, they are part of the same cluster in C if and
only if they are in the same strongly connected component of G. Hence, it is enough to compute
weakly connected components of the relation of being part of the same cluster in C; this is achieved
with classical parallel algorithms [SV82, RS92].
Another corollary is that we can parallelize our girth estimation algorithm for unweighted graphs.
Theorem 5.4 Given an unweighted directed graph G, we can w.h.p. compute an O(k log n) ap1/k ) work and O(girth(G))
e
e
proximation to the girth of G in O(mn
depth.
It suffices to invoke Fast-Roundtrip-Cover(G, k, R) for R ∈
Proof of Theorem 5.4:
20 , 21 , . . . until it returns a nonempty result. The work and depth bounds follow from Lemma 5.10.
6
Additive Approximations
6.1
Additive Roundtrip Spanners
We are given an unweighted directed graph G on n nodes and we want to show that for any
0 < a < 1 we can find in Õ(mn1−a ) time a subgraph H ⊆ G on Õ(n2−a ) edges such that for every
pair of vertices u, v ∈ V , we have that dH (u, v) + dH (v, u) ≤ dG (u, v) + dG (v, u) + na .
The algorithm will proceed using the idea from the introduction about how to approximate the
girth. First run the algorithm from Corollary 5.1 to obtain an O(log n) roundtrip spanner H ′ of G
on O(n2−a ) edges1 . Consider any u, v ∈ V with dG (u, v) + dG (v, u) ≤ na /(k log n) for k = 1/(1 − a).
Then H ′ approximates the roundtrip distance between u and v to an additive factor of na . On
the other hand, if dG (u, v) + dG (v, u) > na /(k log n), then we can select O(n1−a log2 n) nodes S
uniformly at random. S will hit any of the ≤ n2 roundtrip cycles that are longer than na /(k log n)
w.h.p. Thus, adding in- and out- BFS trees to and from all nodes of S will cover every such
roundtrip cycle. Thus adding ≤ 2|S|n = O(n2−a log2 n) edges to H ′ to form H gives an additive
na roundtrip spanner of G. The running time is Õ(mn1−a ).
Now we consider whether the sparsity of our roundtrip spanner can be improved. Surprisingly we
show that the answer is no.
1
One can set 2 − a = 1 + 1/k and get that k = 1/(1 − a) is a constant.
17
Theorem 6.1 For every a ∈ (0, 1), there exists a directed graph Gn on n nodes and Θ(n2−a ) edges,
such that the only additive na roundtrip spanner of Gn is Gn .
Proof:
Let ℓ ≥ 1 be any integer s.t. k = ℓa/(1−a) is also an integer. Let n = ℓk and m = kℓ2 .
We create a directed graph Gn on n nodes and m edges as follows. The vertex set of Gn consists
of k partitions V1 , . . . , Vk such that |Vi | = ℓ for each i. The edge set is as follows: for every i, for
every u ∈ Vi and for every v ∈ Vi+1 mod k , add a directed edge (u, v). That is, the Vi form a virtual
directed k-cycle each edge of which is a complete bipartite ℓ × ℓ graph.
Suppose that some edge (u, v) of Gn is omitted, where u ∈ Vi and v ∈ Vi+1 . The distance from u to
v becomes at least k + 1: the best you can do is go from u to some other v ′ ∈ Vi+1 and then around
the virtual cycle to some other node u′ ∈ Vi and then take the edge (u′ , v). Thus no additive k
roundtrip spanner can omit any edge.
Let’s see how m and k relate to n. As k = ℓa/(1−a) and n = kℓ, we get that n = ℓ1/(1−a) and k = na .
We also get that m = kℓ2 = ℓ2+a/(1−a) = ℓ(2−a)/(1−a) = n2−a .
6.2
Additive Approximation for the Girth
As discussed in the introduction, combining Theorem 1.1 with the BFS computation of the lengths
of shortest cycles through all nodes in a random sample of size Õ(n1−a ), yields the following
corollary:
Reminder of Corollary 1.3 For all a ∈ (0, 1), there is an Õ(mn1−a ) time combinatorial algorithm that w.h.p returns an O(na ) additive approximation to the girth of an unweighted directed
graph.
It is unclear whether the algorithm from the above corollary can be derandomized. The algorithm
uses randomization in many places: (1) it uses a random sample to hit long cycles that we don’t
have a handle on otherwise, (2) it uses sampling quite heavily to obtain estimates of the sizes of
reachability sets of all vertices, (3) it grows random neighborhoods according to an exponential
distribution.
We are not aware of any deterministic approach that achieves running time O(mn1−ε ) for ε > 0 for
any of the above cases. In fact, as far as we know, the only way to achieve (2) deterministically is
to compute the reachability trees explicitly. Despite this, we show that the result can be partially
derandomized using different techniques:
Reminder of Theorem 1.4 Let G = (V, E) be a directed unweighted graph with unknown girth g,
and let 0 < a, ε < 1 be parameters. There is a deterministic combinatorial algorithm that computes
in Õ((1/ε2 )mn1−a ) time a cycle whose length is
1. an O(na ) additive approximation of g if g ≤ na , and
2. a (1 + ε) multiplicative approximation of g if g > na .
In the the reminder of this section we prove Theorem 1.4.
Roughly speaking, our algorithm works in iterations, where each iteration takes Õ((1/ε)m) time.
Let C be a shortest cycle in G = (V, E). The idea of the algorithm is as follows. In each iteration
we consider a shortest path of ⌈ε · na ⌉ vertices. If no such path exists, then the diameter of G must
be smaller than εna , and we can pick any cycle and return it as our approximation. Assume now
that there is a shortest path P with at least ⌈ε · na ⌉ vertices. Either P ∩ C 6= ∅, or we can remove
P from G and recurse on the remaining graph. If P ∩ C 6= ∅, our algorithm finds an approximation
for C by constructing a new weighted graph G′ and a shortest path P ′ between two nodes s and t
18
in G′ whose second shortest simple path length is a good approximation to the length of C.
If we could compute this second shortest path exactly, then we would be done. Unfortunately,
the fastest known algorithm for second shortest path takes O(mn + n2 log log n) time [GL09], and
moreover Vassilevska W. and Williams [VW10] showed that the problem is subcubically equivalent
to APSP, so that a truly subcubic algorithm for it would be a breakthrough. Our goal is to obtain
an almost linear time algorithm, however, since we might need to repeat the procedure n1−a times
(removing na nodes in each iteration).
Fortunately, Bernstein[Ber10] developed an Õ(m/ε) time algorithm that computes a (1 + ε) multiplicative approximation for the second shortest simple path in directed weighted graphs. We use
this algorithm for our Õ(mn1−a ) mixed approximation algorithm.
Before we formally describe our algorithm, we note that a cycle in a directed graph must be
contained in a strongly connected component (SCC). We can assume that G is strongly connected,
as otherwise we compute in O(m)-time its SCCs and run the algorithm in every non-singleton SCC.
If all SCCs are singletons, then the graph is a directed acyclic graph and has no cycles.
We start by taking an arbitrary vertex z of G and using BFS in O(m) to find the longest shortest
paths Qin , Qout , in and out of z, respectively. Let Q be the longer of Qin and Qout . By the triangle
inequality, Q must have length at least half of the diameter of G (notice that since G is stronglyconnected, the diameter is well-defined). Let d = ⌈ε · na ⌉. If the length of Q is < d, then the
diameter is < 2d, and any vertex of G is on a cycle of length at most 2d: take the edge (x, y) on a
shortest cycle C̃ through x; the length of C̃ is 1 + d(y, x) ≤ 1 + (2d − 1) = 2d. Therefore, by running
a BFS from an arbitrary vertex and stopping when the first backward edge is detected, we find a
cycle that is an O(εna ) additive approximation to the shortest cycle.
Otherwise, the diameter is at least 2d. Let P = hvd , . . . , v0 i be a portion of d edges from the path
that we have computed. We construct a new directed weighted graph G′ as follows.
1. Initialize G′ to be G. Set all weights to 1.
2. Add the following vertices and edges to G′ .
(a) For each vi ∈ P , where i ∈ {0, . . . , d}, create new nodes ui and u′i .
(b) For each i ∈ {0, . . . , d} add an edge from ui to u′i , and for each i ∈ {0, . . . , d − 1} add an
edge from u′i to ui+1 . All edges are of weight 1.
3. For each vi ∈ P , where i ∈ {0, . . . , d}, add the following new edges to G′ .
(a) Add a new edge of weight 4d − 3i from ui to vi .
(b) For each outgoing edge (vi , x) ∈ E of vi , add a new edge of weight 4d − 3i from ui to x.
(c) For each incoming edge (y, vi ) ∈ E of vi , add a new edge of weight 3i from y to u′i .
From the above construction it follows that there is a path P ′ = hu0 , u′0 , u1 , u′1 , . . . , ud , u′d i in G′ of
length 2d + 1. Moreover, P ′ is the shortest path from u0 to u′d . To see this, notice first that there
is no edge from u to v, where u, v ∈ P ′ are not consecutive. Therefore, any path from u0 to u′d
other than P ′ , contains a vertex v ∈
/ P ′ . The length of such a path is at least 4d > 2d + 1 since
it must use an edge of weight 4d − 3i to leave P ′ and an edge of weight 3j to return P ′ , where
0 ≤ i ≤ j ≤ d.
Next we apply Bernstein’s algorithm to find a second shortest path for P ′ in G′ . Similarly to
prior work on replacement paths, given a shortest path Q we say that a path D(u, v) is a hu, videtour of Q if D(u, v) is a simple path for which D(u, v) ∩ Q = {u, v} and u precedes v on Q.
It is easy to show that the second shortest path Q′ of Q = hv0 , . . . , vk i has the following form:
19
Q′ = Q(v0 , u) · D(u, v) · Q(v, vk ), where Q(v0 , u) and Q(v, vk ) are the subpaths of Q from v0 to u
and from v to vk , respectively, and D(u, v) is a hu, vi-detour of Q (e.g. see Lemma 2.1 in [Ber10]).
The following fact follows easily from the construction of G′ and P ′ above.
Fact 6.2 If P ′ has a hui , u′j i-detour then it has the following structure: (ui , x) · Q′ · (y, u′j ), where
Q′ is a path from x to y in G, and x is an out-neighbor of ui in G and y is an in-neighbor of vj in
G. Notice Q′ might be an empty path.
In the next lemmas we establish the relationship between a shortest cycle that intersects P in G
and a second shortest path for P ′ in G′ .
Lemma 6.3 Let 0 ≤ i ≤ j ≤ d. If P ′ has a hui , u′j i-detour D(ui , u′j ), then there is a cycle in G
that contains P (vj , vi ) and has length ≤ |D(ui , u′j )| + |P (vj , vi )| = |D(ui , u′j )| + (j − i).
Furthermore, if G has a simple cycle C that contains P (vj , vi ), then P ′ has a hui , u′j i-detour of
length at most |C| − |P (vj , vi )| + 1.
Proof: Let Q be a hui , u′j i-detour of P ′ . From the construction of G′ it follows that Q = (ui , x) ·
Q′ · (y, u′j ) where Q′ is a path from x to y in G2 .
We show that C = (vi , x) · Q′ · (y, vj ) · P (vj , vi ) is a cycle in G. From the construction of G′ it follows
that the edges (ui , x) and (y, u′j ) in G′ correspond to the edges (vi , x) and (y, vj ) in G, respectively,
and since the path Q′ is also in G it follows that C is a cycle in G.
Let C be a simple cycle such that P (vj , vi ) ⊆ C for some i, j (possibly equal). If C = hvj , . . . , vi i
(i.e. (vi , vj ) ∈ E), then i 6= j and we have the following hui , u′j i-detour : D(ui , u′j ) = hui , vi , u′j i.
Otherwise, we have a shortest path B from vi to vj , such that B ∩ P = {vi , vj } and B 6= {vi , vj }.
Let x be the vertex that follows vi in B and y the vertex the precedes vj in B (it might be that
x = y), then we have the following hui , u′j i-detour : D(ui , u′j ) = (ui , x) · B(x, y) · (y, u′j ).
Lemma 6.4 Let C ∗ be a shortest cycle in G. If P (vj , vi ) ⊆ C ∗ , where j ≥ i, then the length of a
second shortest path of P ′ is at most 6d − 2 + |C ∗ |.
Proof: It follows from Lemma 6.3 that P ′ has a hui , u′j i-detour. From Fact 6.2 it has the following
structure: (ui , x)·Q′ (x, y)·(y, u′j ). Consider the path P ′ (u0 , ui )·(ui , x)·Q′ (x, y)·(y, u′j )·P ′ (u′j , u′d ). Its
length is 2i+(4d−3i)+dG (x, y)+3j+2(d−j) = 6d+(j−i)+dG (x, y) = 6d−2+dG (x, y)+2+(j−i) ≤
6d − 2 + dG (vi , vj ) + dG (vj , vi ) = 6d − 2 + |C ∗ |.
According to Lemma 6.3, a second shortest path implies a cycle C in G consisting of the detour
of the second shortest path together with the path in G corresponding to the subpath of P ′ that
was circumvented. Notice it is easy to derive from C a simple cycle in G, which might be shorter.
Denote by L the length of a second shortest path. The length of C is then at most d + L.
Let C ∗ be a shortest cycle in G. If P ∩ C ∗ 6= ∅, according to Lemma 6.4, L ≤ 6d − 2 + |C ∗ |. Since
we are using a (1 + ε)-approximation for the second shortest path (ε < 1), we get a cycle of length
at most:
d + (1 + ε)L = d + (1 + ε)(6d − 2 + |C ∗ |) = 7d − 2 + ε(6d − 2) + (1 + ε)|C ∗ | ≤ O(d) + (1 + ε)|C ∗ |.
If g ≤ na , then we found a cycle of size O(na ). If g > na , then since d ≤ O(εna ), we have a 1 + O(ε)
2
Notice it is possible that x = y, and then hvi , xi · Q′ · hy, vj i is actually hvi , x, vj i. It is also possible, in addition,
vi = x = y, and then hvi , xi · Q′ · hy, vj i is actually hvi , vj i; this is the reason for adding the edges (ui , vi ) in G′ . For
simplicity of the presentation, we assume the concatenation notation subsume all these cases.
20
multiplicative approximation for the girth.
Thus, if an approximate second shortest path of length ≤ 16d is found, we can conclude that
L ≤ 16d and hence |C ∗ | ≤ O(d), so we can stop and return. Otherwise, we can conclude that none
of the vertices of P are on cycles of length ≤ d in G, as otherwise the algorithm would return an
approximate second shortest path of length 7d − 2 + ε(6d − 2) + (1 + ε)|C ∗ | < 13d − 4 + 2d < 16d.
We can thus remove all the vertices of P from G and repeat the process above on the new graph.
Consider the first iteration in which P ′ contains a vertex of C ∗ . Since up to this iteration no vertices
of P ′ are removed, the graph will contain a detour corresponding to the portion of C ∗ not on P ,
and the approximate cycle returned will be of length ≤ O(d) + (1 + ε)|C ∗ |, as argued above. If the
girth is ≤ 2d, the approximation is additive O(d), and otherwise it is multiplicative 1 + O(ε).
The correctness of the algorithm follows from the discussion above. The runtime is as follows. The
time to decompose the graph to its strongly-connected components is O(m + n) [Tar72]. The time
to construct G′ is O(m), and the running time of Bernstein’s algorithm for the second shortest path
on G′ is Õ(m/ε). We conclude that the running time for a single iteration is Õ(m/ε). The number
of iterations we have in the algorithm is at most ⌈(1/ε)n1−a ⌉, since we are removing ⌈ε · na ⌉ vertices
from G in each iteration. It follows that the total running time of the algorithm is Õ((1/ε2 )mn1−a ),
thus proving Theorem 1.4.
7
Lower Bounds
In this section we provide a conditional lower bound for the problem of computing additive approximations for the girth of a directed unweighted graph.
Let us begin with our plausible hypothesis:
Hypothesis 7.1 Any combinatorial (possibly randomized) algorithm for triangle detection in nnode m-edge graphs with m = Θ(n2 ) requires (expected) n3−o(1) time.
Combinatorial algorithms informally do not use Strassen-like matrix multiplication, and hopefully
do not hide high constants in the big-O. The current best combinatorial algorithms for triangle
detection run in time min{O(n3 / log4 n), O(m3/2 )} time [Yu15, IR78]. It is a major open problem
to design a truly subcubic, i.e. an O(n3−ε ) time combinatorial algorithm for constant ε > 0 for
triangle detection. Triangle detection is known [VW10] to be subcubically equivalent to Boolean
Matrix Multiplication (BMM) under combinatorial fine-grained reductions, and thus the above
hypothesis is equivalent to the hypothesis that combinatorial BMM of n × n matrices requires
n3−o(1) time.
We now state our result:
Theorem 7.2 Under Hypothesis 7.1, any combinatorial algorithm that computes an additive n1/2 −
1 approximation to the girth of all directed n-node, m = O(n)-edge graphs requires mn1/2−o(1) time.
Proof: Let G = (V, E) be an n-node, m-edge directed graph for m = Θ(n2 ), so that we want to
detect the presence of a 3-cycle in G. We now create a new directed H as follows:
• H has n2 vertices: for every v ∈ V we create n copies v1 , . . . , vn .
• For every edge (u, v) ∈ E of G, we create directed edges (un , v1 ) and (ui , vi+1 ) for all i ∈ {1, 2}.
• For every vertex v ∈ V , create directed edges (vi , vi+1 ) for all i ∈ {3, . . . , n − 1}.
Every triangle a1 → a2 → a3 → a1 in G is represented by an n-cycle in H: a1n → a21 → a32 → a13 →
a14 → . . . → a1n .
Every n-cycle in H must correspond to a 3-cycle in G, as there is a path from v3 to wn if and
21
only if v = w. Moreover, any cycle in H has length that is a multiple of n as each cycle must
go through all n partitions of the graph over and over until it lands at the same node. The girth
of H is thus either n if G has a 3-cycle, or at least 2n otherwise. H has N = n2 vertices and
M ≤ 3m + n2 = Θ(n2 ) edges.
Suppose that there is some constant ε > 0 such that for all a there is an O(M N 1/2−ε ) time algorithm
that achieves an additive N 1/2 − 1 approximation to the girth of M -edge, N -node directed graphs.
Let’s apply this algorithm to H. If it finds an additive N 1/2 − 1 = n − 1 -approximation to the girth
of H, it will be able to detect whether G contains a triangle. The running time of the algorithm
would be
O(M N 1/2−ε ) = O(n2 · n1−2ε ) = O(n3−2ε ),
which contradicts Hypothesis 1.
Considering multiplicative approximation for the girth in directed unweighted graphs, it is known
that any truly subcubic combinatorial algorithm that computes a 2 − ε approximation (0 < ε < 1)
for the girth in directed unweighted graphs, implies a truly subcubic time combinatorial algorithm
for triangle detection. This is formalized in the next probably folklore theorem. A formal proof of
it appears in [RW12].
Theorem 7.3 (Folklore) Under Hypothesis 7.1, any combinatorial algorithm that for ǫ ∈ (0, 1)
computes a multiplicative 2 − ε approximation for the girth of a directed n-node, m-edge graph
requires n3−o(1) time.
Acknowledgments
Jakub Pachocki is partially supported by ONR grant N00014-15-1-238 and NSF award 1065106.
Virginia Vassilevska Williams is supported by NSF awards 1417238, 1528078, and 1514339, and
BSF Grant 2012338. Jakub Pachocki was a student at Carnegie Mellon University and Aaron
Sidford was a postdoctoral researcher at Microsoft Research New England while part of this work
was done. Part of this work was done while some of the authors were visiting the Simons Institute
for the Theory of Computing, UC Berkeley.
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7
A
Ball Size Estimation
Here we present a routine that can estimate the sizes of neighborhoods of all vertices. The approach
is similar to that of [Coh97]. We provide the algorithm, Estimate-Balls, that when invoked on
a graph G with radius parameter r and error parameter ǫ computes w.h.p. for every vertex the
fraction of vertices that it can reach at distance at r and the fraction of vertices that can reach it.
(sout , sin ) = Estimate-Balls(G, r, ǫ), where G = (V, E, l) is a directed graph and r, ǫ > 0.
1. Sample t = ⌈20 · log n/ǫ2 ⌉ vertices v1 , . . . , vt independently uniformly at random from
V with replacement.
2. Compute the distances between every vertex in V and each of v1 , . . . , vt .
3. For each u ∈ V , let sout
u be the fraction of v1 , . . . , vt such that dG (u, vi ) ≤ r.
in
4. For each u ∈ V , let su be the fraction of v1 , . . . , vt such that dG (vi , u) ≤ r.
5. Return (sout , sin ).
Figure 7: The algorithm for estimating the sizes of out- and inballs at radius r for a given graph.
Our algorithm simply samples vertices with replacement and computes distances to and from them
to estimate the ball sizes. The analysis of Estimate-Balls reduces to a simple application of
Chernoff bounds and union bound. We prove that it works in Lemma A.1.
Lemma A.1 Let sout , sin be the output of Estimate-Balls(G, r, ǫ). For any vertex u, let s̄out
u be
the fraction of vertices in V such that dG (u, vi ) ≤ r. Then, whp., for all vertices u it holds that
out
in
−2 log 2 n).
|s̄out
u − su | ≤ ǫ. An analogous statement holds for s . The algorithm runs in time O(mǫ
Proof:
By a standard Chernoff bound, we have
out
2
−40
Pr[|s̄out
.
u − su | > ǫ] ≤ 2 exp(−2tǫ ) ≤ 2 exp(−40 log n) = 2n
An analogous bound holds for sin .
B
Exponential Distributions
Here we recall some basic facts about the exponential distribution we use in the paper.
Lemma B.1 (Exponential Distribution Facts) We let Exp(α) denote the exponential distribution with parameter α. This distribution is supported on R≥0 with PDF given by p(x) = α·exp(−αx).
This distribution has the following properties:
• CDF: Pr[Exp(α) ≤ x] = 1 − exp(−αx) for x ≥ 0.
• Expected Value: EExp(α) =
1
α
• Memoryless: Pr [Exp(α) ≥ s + t | Exp(α) ≥ s] = Pr [Exp(α) ≥ t]
• High Probability: The maximum of n independent r.v.s drawn from Exp(α) is O( logα n ) with
high probability.
Proof:
Direct calculation reveals that
Z x
α exp(−αx) = − exp(−αx) + exp(0) = 1 − exp(−αx)
Pr [Exp(α) ≤ x] =
−∞
26
giving the formula for the CDF. Furthermore, integration by parts yields that
Z ∞
Z ∞
∞
− exp(−αx)dx
αx exp(−αx)dx = [−x exp(−αx)] |0 −
EExp(α) =
0
0
1
1
1
= − exp(−α∞) + exp(−α0) =
α
α
α
giving the expected value formula. Direct calculation again yields
Pr[exp(α) ≥ s + t]
exp(−α(s + t))
=
Pr [exp(α) ≥ s]
exp(−αs)
= exp (−αt) = Pr [Exp(α) ≥ t]
Pr [Exp(α) ≥ s + t | Exp ≥ s] =
proving the memoryless property. Finally the high probability bound is immediate from the CDF
and the definition of high probability.
C
Sequential Clustering Algorithm
Here we formalize and prove the alternative approach to clustering described in Section 3.
(V1 , V2 , . . .) = Sequential-Cluster-Out(-In)(G, (v1 , . . . , vt−1 ), r), where G = (V, E, l) is
a directed graph, v1 , . . . vt−1 ∈ V , and r > 0.
1. Set β := log(n)/r.
2. Let G0 := G.
3. For i in 1, . . . , t:
(a) If vi is not in Gi−1 , let Gi := Gi−1 , Vi := ∅ and continue the loop. Otherwise:
(b) Pick xi ∼ Exp(β).
(c) Let Vi := outballGi−1 (vi , xi ). (inballGi−1 (vi , xi ) for Cluster-In)
(d) Let Gi := Gi−1 with Vi S
and incident edges removed.
4. Return (V1 , V2 , . . . , Vt−1 , V \ i Vi )
Figure 8: The sequential clustering algorithm.
Lemma C.1 Let (V1 , V2 , . . . , Vt ) = Sequential-Cluster-Out(G, (v1 , . . . , vt ), r0 , r) (analogously
of Sequential-Cluster-In). Then for any c ≥ 1 we have
1. with probability at least 1 − n1−c for all i < t, the radius of the tree corresponding to Vi is at
most c · r, whp.,
2. for any pair of vertices u, v at roundtrip distance at most R in G, they are in the same set Vi
with probability at least exp(− log(n)R/r).
Proof: Note that the maximum radius of any constructed tree is upper bounded by maxi xi . For
every i ∈ 1, . . . , t, we have
P r [xi ≥ c · r] ≤ exp(−c · βr) = n−c ,
and so by union bound the maximum radius is at most c · r with probability at least 1 − n1−c .
To prove the remainder of the lemma, fix two vertices u and v at roundtrip distance at most R in G.
Assume the i-th cluster is the first one to contain an element of the set {u, v}. By the memoryless
property of the exponential distribution we see that conditioned on this event the probability that
27
cluster i contains both vertices u and v is at least
Pr [Exp(β) ≥ R] = exp(−βR),
yielding the desired result.
28
| 8 |
COMPUTATION OF EXTENDED ROBUST KALMAN FILTER FOR REAL-TIME
ATTITUDE AND POSITION ESTIMATION
Gaurav R. Yengera∗, Roberto S. Inoue†, Mundla Narasimhappa‡, Marco H. Terra‡
∗
Department of Electrical Engineering, Indian Institute of Technology (Banaras Hindu University),
Varanasi and also with Department of Electrical Engineering, University of São Paulo at São Carlos,
São Paulo, Brazil
†
Department of Electrical Engineering, Federal University of São Carlos at São Carlos, São Paulo,
Brazil
‡
Department of Electrical Engineering, University of São Paulo at São Carlos, São Paulo, Brazil
arXiv:1801.04386v1 [cs.SY] 13 Jan 2018
Emails: g.yengera@gmail.com, rsinoue@ufscar.br, mr.narasimha08@gmail.com,
terra@sc.usp.br
Abstract— This paper deals with the implementation of the extended robust Kalman filter (ERKF) which
was developed considering uncertainties in the parameter matrices of the underlying state-space model. A key
contribution of this work is the demonstration of a method for real-time computation of the filter on parallel
computing devices. The solution of the filter is expressed as a set of simultaneous linear equations, which can
then be evaluated based on QR decomposition using Givens rotation. This paper also presents the application
of the ERKF in the development of an attitude and position reference system for a cargo transport vehicle. This
work concludes by analyzing the performance of the ERKF and verifying the validity of the Givens rotation
method.
Keywords—
1
Extended Robust Kalman Filter, Givens Rotation, QR Decomposition, Localization.
Introduction
The Kalman filter (Kalman, 1960; Anderson and
Moore, 1979; Kailath et al., 2000) has played
an important role in solving estimation problems
appearing in navigation, economics, communications, control and other areas. The real-time
computation of the Kalman filter has been an
important and recurring feature in many of its
applications. A fundamental assumption in the
Kalman filter is that the underlying state-space
model is accurate and does not contain uncertainties. When this condition is violated, the
performance of the filter could deteriorate drastically (Sayed, 2001). As a result robust estimation is necessary in several real world applications
and certain algorithms have been developed for
this purpose. However, very little research has
been carried out on the real time implementation
of these robust estimation algorithms on parallel
computing devices such as FPGAs and GPUs.
A detailed explanation of the robust optimal filtering approach utilized in the ERKF has
been presented in (Ishihara et al., 2015; Inoue
et al., 2016). Also its advantages over other robust filtering approaches have been discussed in
those papers. To summarize the important features of the ERKF: it does not require any auxiliary parameter to be tuned while for linear systems, stability and convergence are guaranteed for
all steady-state estimates. Thereby this filtering
approach is preferable for real time implementation as no offline computations are necessary. Additionally the ERKF assumes the existence of uncertainties in all parameter matrices of the state-
space model.
FPGAs have been a popular choice for the
real time implementation of the Kalman filter. The most significant prior work on floatingpoint FPGA implementations have been developed based on direct mapping of the equations on
the FPGA which either involves explicit matrix inversion, see for instance (Bonato et al., 2007; Lee
and Salcic, 1997), or representation of the equations in Schur complement form and then applying Fadeev’s algorithm as done in (Chen and
Guo, 2005).
This paper will focus on the implementation
of the ERKF presented in (Inoue et al., 2016).
For the evaluation of the ERKF, a matrix of particularly large dimensions needs to be inverted.
Hence, the matrix inversion approach would be
computationally intensive and not ideal for real
time applications. It will be shown that the
proposed method is computationally more efficient than the conventional matrix inversion approach. As compared to Fadeev’s algorithm, it
is more straightforward to evaluate the filter as
a set of simultaneous linear equations by applying QR decomposition using Givens rotation, see
(Francis, 1962; Givens, 1958), followed by back
substitution to obtain desired state vector and
state covariance matrix.
The application of the ERKF in an attitude
and position reference system for a cargo transport vehicle utilizing a global positioning system
(GPS) along with an inertial measurement unit
(IMU) is presented. To accommodate for the
lower measurement update frequency of the GPS,
as discussed in (Farrel, 2008), an inertial naviga-
tion system based on attitude estimates is used to
estimate position in the absence of GPS measurements. At the same time the IMU measurements
contain residual errors even after calibration and
the ERKF is used to ensure that state estimates
are robust to such errors or uncertainties. In
(Inoue et al., 2016) the performance improvement
of the ERKF over the extended Kalman filter in
the presence of uncertainties has been discussed.
The role of the ERKF is especially crucial when
working with low-cost IMUs which tend to possess
substantial uncertainties . A method to model
the uncertainties present in the IMU rate gyros
and accelerometers measurements, and incorporate them into the ERKF has been shown. This
application is of general importance for developing
more sophisticated navigation systems and it also
highlights the necessity for real time computation
of the ERKF. This paper concludes by using sensor data collected from this experimental setup to
verify the Givens rotation based computation approach.
The organization of the paper is as follows:
the extended robust Kalman filter is presented in
Section 2. In Section 3, the proposed algorithm
and its computational complexity are discussed.
The vehicle attitude and position estimation system is presented in Section 4. In Section 5, the
experimental setup and results are discussed. Finally, Section 6 presents the conclusion of the paper.
problem:
min max{Jkµ (xk , wk , vk , xk+1 , δk )}, (4)
xk ,xk+1
where δk := {δFk , δGk , δKk , δHk }.
The
cost function Jkµ (xk , wk , vk , xk+1 , δk ) is given in
(Inoue et al., 2016).
The ERKF is essentially in the form of a predicted estimator filter where the measurement update is computed first and the time update step
later. Thereby the filter recursively computes
bk+1|k and Pk+1|k from x
bk|k−1 and Pk|k−1 respecx
tively.
The optimal estimates for linear systems are
obtained by substituting µ → ∞. The recursive
algorithm for obtaining the optimal filtered estimates is presented in Table 1.
The ERKF is applied to nonlinear system
bk|k−1
−Fk x
models by defining bk =
and
zk − h(b
xk|k−1 )
is used
models by defining
with linear system
bk|k−1
−Fk x
bk =
.
bk|k−1
zk − Hk x
3
k|k−1
Extended Robust Kalman Filter
For the development of the robust Kalman filter,
the underlying state-space model of the dynamic
discrete-time system is modified to incorporate
parametric uncertainties:
xk+1 = (Fk + δFk ) xk + (Gk + δGk ) wk ,
zk = (Hk +δHk ) xk +(Kk +δKk ) vk ,
(1)
for k ≥ 0, where xk ∈ Rn is the state vector, zk ∈
Rp is the measurement vector, wk and vk are the
noise vectors corresponding to the state update
and measurement equations respectively. Typically, x0 , wk and vk are considered as mutually independent zero-mean Gaussian random variables
with respective variances E{x0 xT0 } = Π0 ≥ 0,
E{wk wTk } = Qk ≥ 0 and E{vk vTk } = Rk ≥ 0.
Fk ∈ Rn×n , Gk ∈ Rn×m , Hk ∈ Rp×n , Kk ∈ Rp×m
are nominal parameter matrices, and δFk ∈ Rn×n ,
δGk ∈ Rn×m , δHk ∈ Rp×n , δKk ∈ Rp×m are uncertainty matrices which are modeled as:
δFk
δHk
δGk =
δKk =
M1k ∆1 NFk
M2k ∆2 NHk
NGk
(2)
NKk
(3)
where ||∆1 || < 1 and ||∆2 || < 1 are arbitrary
contractions. Matrices M1k , M2k , NFk , NGk ,
NHk , and NKk are known a-priori.
The ERKF is developed considering the solution of the following unconstrained optimization
Computation using Givens Rotation
The ERKF given in Table 1 can be rewritten as a
linear system Ay = b, as shown in (5). The µ and
λ terms are values we are not interested in, while
all the other terms are as defined in Table 1.
P
2
δk
|
0
0
0
I
0
0
0
Rk
0
0
0
I
0
0
0
0
0
FkT
GkT
EkT
0
0
0
0
T
NF
k
T
NG
k
NET
k
{z
A1
I
0
Fk
NFk
0
0
0
0
I
Gk
NGk
0
0
0
0
λ1
0
λ2
Ek
λ3
NEk
λ4
0 x
bk|k−1
bk|k − x
b
ν
0
k|k
bk+1|k
x
0
}|
{z
y1
µ1
µ2
µ3
µ4
∗
∗
0
0
0
0
0
bk
0 ,
= Nbk
0
0
0
0
Pk+1|k
0
−I
{z
}
|
}
b1
(5)
To solve for only the required elements of mabk+1|k and state covariance
trix y, i.e. state vector x
matrix Pk+1|k , each system of linear equations corresponding to an individual column of matrix y is
evaluated one at a time using QR decomposition
based on Givens rotation. And only the required
values are calculated using back-substitution. The
following equations describe the proposed solution:
Ayl = bl ,
(6)
QR × yl = bl ,
(7)
R × yl = Q−1 × bl = Zl ,
(8)
where yl and bl are the lth columns of y and b
respectively, and A = QR is the result of QR decomposition of matrix A.
The matrices R and Zl can be obtained by the
following equation:
[R, Zl ] = Θ[A, bl ] = ΘMl ,
(9)
where Θ is any sequence of Givens rotations which
result in the upper triangularization of matrix A.
Table 1: Recursive Algorithm for Extended Robust Kalman filter
Uncertain Model: Consider (1) with Π0 0, Qk 0, and Rk 0.
b0|−1 = 0.
Step 0: (Initial Conditions) P0|−1 = Π0 , x
bk+1|k ; Pk+1|k } from {zk ; x
bk|k−1 ; Pk|k−1 } as follows:
Step k: Given zk , update {b
xk|k ; x
bk|k
x
∗
bk|k−1
x
0
0 0 0 0 I
0 0
ν
bk|k
∗ = 0
0
0 + 0 0 0 0 0 I
0 0 0 0 0 0 I
bk+1|k
0
0
x
Pk+1|k
P
0
0
0
I
0
0 −1 0
k|k−1
0
Rk
0
0
0
I
0
0
0
0
0
0
0
0
Fk
Gk
Ek
0
bk
0
0
0
0
NFk
NGk
NEk
N
0
,
bk
I
T
0
0
FkT
NF
0
0
0
0
k
T
0
0
I
GkT
NG
0
0
0
0
k
T
T
0
−I
0
0
Ek
NE
0
0
0
k
b k|k
w
Fk
Gk
0
−I
bk|k =
, Fk =
, Gk =
, Ek =
,
ν
bk|k
v
H
0
K
k
k
0
NGk
0
NFk
0
, NGk =
, NEk =
,
NFk =
NHk
N
0
0
Kk
bk|k−1
−NFk x
bk|k−1
−Fk x
Qk
0
, Rk =
bk =
, Nbk =
bk|k−1
bk|k−1
zk − Hk x
−NHk x
0
Rk
Since matrix R is a triangular matrix, the elements of vector yl , specifically the ones correbk+1|k or state covariance
sponding to state vector x
matrix Pk+1|k , can be calculated from (8) using
bk+1|k and
back-substitution. All the elements of x
Pk+1|k are obtained after (9) and (8) have been
carried out for all the columns of y.
The number of floating point operations
(FLOPS) required by this method, where one
FLOP is counted as any individual floating point
operation; has been calculated referring to (Golub
and Loan, 1996) and considering the following
general case: x ∈ Rn×1 , P ∈ Rn×n , A ∈ Rm×m . It
can be noticed that in (5), the matrix A is a square
matrix and m n. Thereby, y ∈ Rm×(n+1) ,
b ∈ Rm×(n+1) and Ml ∈ Rm×(m+1) .
Givens rotation, in (9), requires FLOPS of an
order of magnitude equal to 3(m + 1)2 (m − m+1
3 ).
Since m 1, this can be approximated as 2m3 .
Back-substitution is applied in (8) to calculate the
bottom n elements of vector yl as only these corbk+1|k or Pk+1|k .
respond to elements of either x
Hence, the FLOPS required in (8) is of the order
of magnitude n2 .
Remembering that (9) and (8) need to be carried out n+1 times, once for each column of matrix
y, the total number of FLOPS required is given
by:
Total FLOPS = (n + 1) × (2m3 + n2 ) ∝ 2nm3 .
(10)
For evaluating the ERKF directly as presented in Table 1, explicitly calculating the inverse
of matrix A using Gaussian elimination would
involve applying Gaussian elimination and backsubstitution m times. The number of FLOPS re4
quired would be of the order of magnitude 2m
3 .
bk+1|k and Pk+1|k after having obTo only obtain x
tained the inverse, requires matrix multiplication
considering the bottom n rows of A−1 alone. The
number of FLOPS for the matrix multiplication
step would be 2nm(n + 1). The computationally
intensive step in this method is the matrix inversion step.
Comparing the computational cost of the ma4
trix inversion approach, 2m
3 , with (10) and noticing from the structure of matrix A that n < m
3;
it can be concluded that the Givens rotation approach is computationally more efficient.
An important remark to be added here is that
instead of the QR decomposition approach, the
more efficient LU decomposition approach to solve
the linear system given in (5) would appear to be
a better choice. However, it was seen that the result of the LU decomposition method was unstable and did not converge with the results obtained
from the matrix inversion approach. It can further be noticed that matrix A is sparse and that
is the reason for Givens rotation being preferred
over Householder reflections. Additionally, it is
important to note that Givens rotation based QR
decomposition easily lends itself to parallel implementations (Wang and Leeser, 2009). Hence, the
computation speed can further be increased on a
parallel computing device such as an FPGA.
4
Vehicle State Determination
In this section the vehicle state determination system (attitude, position and velocity) is presented.
The system is composed of an IMU and a GPS
module. The IMU is based on uncertain output models of rate gyros and accelerometers ωg ,
aa . In Figure 1 the model of the IMU considers
rate gyro bias bg , accelerometer bias ba , Gaussian
white noise in the rate gyros and accelerometers,
wg and wa , respectively, and uncertain terms due
to scale factor and axes misalignment of the rate
gyros, accelerometers, δωg and δaa , respectively.
The IMU also measures the tilt angles φIM U and
θIM U , as well as the yaw angle ψIM U . The GPS
module provides geodetic position pGP S and yaw
angle ψGP S .
The estimation of the state is split in two fil-
is Gaussian white noise of the rate gyros bias; and
Ω(φ, θ, ψ) is the transformation matrix between
angular velocities, which is given by:
1 sin φ tan θ cos φ tan θ
cos φ
− sin φ .
Ω(φ, θ, ψ) = 0
0 sin φ sec θ cos φ sec θ
(14)
Figure 1: Cargo transport vehicle experimental
setup.
ters: (1) an attitude estimator and (2) a position estimator, in order to obtain a trade-off between accuracy and processing power (Bijker and
Steyn, 2008). In Figure 2, the method of obtaining attitude and position estimates has been illustrated. This is essentially a pictorial representation of the systems described in Sections 4.1 and
4.2. Further, Figure 2 shows how the inertial navigation system updates position estimates in the
absence of GPS measurements by integrating accelerometer readings. The states of the attitude
ba , and
and position system are represented by x
p
b , respectively.
x
The filters presented in this paper are based
on discrete-time systems. In this regard, Equations (11) - (13) are represented in the form of (1)
after being linearized and discretized considering
a sample time T . The terms of (1) are chosen
as; xa = [φ θ ψ bTg ]T ∈ R6×1 is the state vector, wa = [wTg wTbg ]T ∈ R6×1 is the vector Gaussian process with zero mean and covariance Qa ,
za = [φIM U θIM U ψIM U + δψ ]T ∈ R3×1 is the
measurement vector, δψ = ψGP S − ψIM U is the
yaw error between GPS and IMU computed when
GPS is available, va ∈ R3×1 is the vector Gaussian process with zero mean and covariance Ra of
the measured angles in za , Fka is the state transition matrix, Gak is the input noise matrix, and
Hka = [I3×3 03×3 ] is the measurement matrix.
4.2
Position system
The dynamic equations of the position model are
given by (Farrel, 2008; Bijker and Steyn, 2008):
[λ̇ ϕ̇ ḣ]T = Ψ(λ, ϕ, h)[υN υE υD ]T , (15)
[υ̇N υ̇E υ̇D ]T = ge + AT (φ, θ, ψ)a,
(16)
a = aa + δaa − ba − wa ,
1
ḃa = − ba + wba ,
τa
Figure 2: Estimation Diagram.
4.1
Attitude system
The dynamic equations of the attitude model are
given by (Farrel, 2008; Kim, 2011):
[φ̇ θ̇ ψ̇]T = Ω(φ, θ, ψ)[p q r]T ,
[p q r]T = ωg + δωg − bg − wg ,
1
ḃg = − bg + wbg ,
τg
(12)
Ψ(λ, ϕ, h) =
where φ and θ are the tilt angles roll and pitch,
respectively; ψ is the yaw angle, p, q and r are
the angular velocities in the body frame; τg is the
correlation time of the Gauss Markov process; wbg
(18)
where p = [λ ϕ h]T are geodetic positions in the
LLA (Latitude, Longitude and Altitude) frame ;
υ = [υN υE υD ]T are the velocities in the
NED (North, East and Down) frame; Rλ is the
radius of meridian curvature at a given latitude;
Rφ is the transverse radius of curvature, ge is the
Earth’s gravity vector; a is the actual linear acceleration; A(φ, θ, ψ) is the rotation matrix from
Inertia frame to Body frame; τa is the correlation
time of the Gauss Markov process; wba is Gaussian white noise of the accelerometer bias; and
Ψ(λ, ϕ, h) is the transformation matrix between
linear velocities, which is given by:
(11)
(13)
(17)
1
Rλ +h
0
0
0
1
(Rφ +h)cosλ
0
0
0 . (19)
−1
Equations (15)-(18) are written in the
state-space form of (1) after being linearized
and discretized, such that state vector xp =
[pT υ T bTa ]T ∈ R6×1 , wp = [wTa wTba ]T ∈ R6×1 is
the vector Gaussian process with zero mean and
covariance Qp , zp = [pTGP S ]T ∈ R3×1 is the measurement vector, vp ∈ R3×1 is the vector Gaussian
process with zero mean and covariance Rp of the
measured position and velocity in zp , Fkp is the
state transition matrix, Gpk is the input noise matrix, and Hkp = [I3×3 03×6 ] is the measurement
matrix.
5
5.1
Experimental Results
Description of Experimental Setup
An IMU and GPS were used to track the attitude
and position of the cargo transport vehicle shown
in Figure 1. The IMU used was the Xsens MTi300-AHRS-2A5G4, which utilized MEMS based
sensors. It comprised of a 3-axial accelerometer, 3axial gyroscope and a 3-axial magnetometer. The
update rate of the IMU was 400 Hz and it had
an in-built algorithm which computed orientation,
angular velocity and linear velocity from sensor
readings. For this experimental setup, orientation
was measured in terms of Euler angles and both
the angular and linear velocities were measured
about the body reference frame of the vehicle.
Septentrio AsteRx2eH PRO GPS was used in
the cargo transport vehicle. The update rate of
the GPS was 10 Hz and this was considerably
slower than the IMU update rate. The measurements provided by the GPS were latitude, longitude, altitude, linear velocity as well as heading
or yaw angle.
The weighting matrices Qs and Rs for the
ERKF were chosen based on the method described
in (Xing and Gebre-Egziabher, 2008), where s =
s
a, p. And the parameter matrices NFsk , NG
, and
k
s
NHk are modeled in a manner to attenuate the
uncertain terms δωg presented in (12), δaa presented in (17) and others sources of uncertainties
occurring in the matrices Fks , Gsk , and Hks . This
is done by taking the average of each uncertain
position present in rows i along the corresponding
columns l of matrices Fks , Gsk , and Hks ; see (Inoue
et al., 2016). The obtained matrices are as follows:
NFsk = 102 f1s . . . fnss ,
s
s
NG
= 102 g1s . . . gm
,
(20)
s
k
s
s
NHk = 0 . . . 0 , NKk = 0 . . . 0 ,
5.2.1
Attitude Estimates
The majority of variation in attitude is seen in the
yaw angle as shown in Figure 5(c), while the total
variation in roll, Figure 5(a), and pitch, Figure
5(b), are considerably lower. This is as expected
for a ground vehicle.
The yaw angle estimates provided by the filter, Figure 5(c), show greater certainty in GPS
measurements by following them more closely
than IMU measurements, which are observed to
have errors due to the presence of uncertainties.
It should be noted that due to the ERKF the yaw
estimates are robust to the uncertainties present
in the IMU measurements.
5.2.2
Position Estimates
The position estimates are a good fit with GPS
readings as shown in Figure 6 as well as in Figure 3, which is a 3 dimensional plot showing the
route followed by the cargo transport vehicle. An
important observation to be made here is that by
implementing an inertial navigation system when
GPS measurements are not available, the position
estimates are updated at a frequency of 400 Hz,
which is greater than the 10 Hz update rate of the
GPS. This is depicted in Figure 4.
Figure 3: 3D Plot of Vehicle Position.
where ns is the number of variables in the state
vector xs ; ms is the of variables in the noise vector
ws ; na = 6; ma = 6; np = 9; mp = 6; fls =
s
s
F k (i,l)
, F k = Fks − Ins ×ns , for l
nP
s
ns
Gp (i,l)
gls = i=1ns k
, for l = 1, . . . , ms .
Pns
i=1
5.2
= 1, 2, .., ns ;
Plot of Attitude and Position Estimates
In this section, the graphs for the attitude and
position estimates are presented. Figure 5 shows
the attitude estimates while Figure 6 shows the
position estimates.
Figure 4: Update Rates of GPS and Estimates.
Figure 5: Attitude estimates of the ERKF: roll (φ), pitch (θ) and yaw (ψ).
Figure 6: Position estimates of the ERKF: latitude, longitude and altitude.
5.3
Numerical analysis
The maximum and minimum singular values of
the state covariance matrix of the attitude system are shown in Figure 7. The ERKF has been
implemented through the conventional matrix inversion approach as well as the proposed Givens
rotation approach. The corresponding maximum
and minimum singular values of the state covariance matrix of the position system are shown in
Figure 8.
These figures clearly show that the singular
values of the covariance matrices obtained from
the standard matrix inversion and Givens rotation
implementations are very nearly the same. It was
seen that absolute value of the difference between
the singular values was smaller than 10−13 when
64 bit precision floating point arithmetic was used.
6
Conclusions
In this paper we have presented an attitude and
heading reference system, based on IMU and GPS
data, using the ERKF. The yaw estimates provided by the ERKF followed the accurate GPS
measurements. The position estimates were obtained at a higher update frequency, due to the
utilization of an inertial navigation system based
on attitude estimates. And the ERKF ensured
that the estimates are robust to uncertainties in
both system models.
Additionally, we have presented and verified
a method for computing the ERKF in real-time
based on QR decomposition using Givens rotation. The increased computational efficiency of
this method over the conventional matrix inversion approach has been discussed.
Acknowledgement
This work was supported by grants #2014/084320, #2014/50851-0 and #2015/18085-8, São Paulo
Research Foundation (FAPESP) and by grants
#484095/2013-7 and 465755/2014-3, Brazilian
National Council for Scientific and Technological
Development (CNPq).
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Farrel, J. A. (2008). Aided navigation GPS with
high rate sensors, The McGRaw-Hill Companies, New York.
Figure 7: Maximum and minimum singular values of state covariance matrix P of the attitude system
model.
Figure 8: Maximum and minimum singular values of state covariance matrix P of the position system
model.
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Embed. Comput. Syst. 9(1): 3:1–3:17.
Inoue, R. S., Terra, M. H. and Cerri, J. P. (2016).
Extended robust Kalman filter for attitude
estimation, IET Control Theory Applications
10(2): 162–172.
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Ishihara, J. Y., Terra, M. H. and Cerri, J. P.
(2015). Optimal robust filtering for systems subject to uncertainties, Automatica
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Kailath, T., Sayed, A. H. and Hassibi, B. (2000).
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Lee, C. R. and Salcic, Z. (1997).
Highperformance FPGA-based implementation of
| 3 |
Design, Generation, and Validation of
Extreme Scale Power-Law Graphs
Jeremy Kepner1 , Siddharth Samsi1 , William Arcand1 , David Bestor1 , Bill Bergeron1 , Tim Davis2 ,
Vijay Gadepally1 , Michael Houle1 , Matthew Hubbell1 , Hayden Jananthan1,3 , Michael Jones1 , Anna Klein1 ,
Peter Michaleas1 , Roger Pearce4 , Lauren Milechin1 , Julie Mullen1 , Andrew Prout1 ,
Antonio Rosa1 , Geoff Sanders4 , Charles Yee1 , Albert Reuther1
arXiv:1803.01281v1 [cs.DC] 4 Mar 2018
1
Massachusetts Institute of Technology, 2 Texas A&M, 3 Vanderbilt University, 4 Lawrence Livermore National Laboratory
Abstract—Massive power-law graphs drive many fields:
metagenomics, brain mapping, Internet-of-things, cybersecurity,
and sparse machine learning. The development of novel algorithms and systems to process these data requires the design,
generation, and validation of enormous graphs with exactly
known properties. Such graphs accelerate the proper testing
of new algorithms and systems and are a prerequisite for
success on real applications. Many random graph generators
currently exist that require realizing a graph in order to know
its exact properties: number of vertices, number of edges, degree
distribution, and number of triangles. Designing graphs using
these random graph generators is a time-consuming trial-anderror process. This paper presents a novel approach that uses
Kronecker products to allow the exact computation of graph
properties prior to graph generation. In addition, when a real
graph is desired, it can be generated quickly in memory on
a parallel computer with no-interprocessor communication. To
test this approach, graphs with 1012 edges are generated on a
40,000+ core supercomputer in 1 second and exactly agree with
those predicted by the theory. In addition, to demonstrate the
extensibility of this approach, decetta-scale graphs with up to
1030 edges are simulated in a few minutes on a laptop.
I. I NTRODUCTION
Power-law (or heavy-tail) [1], [2] graphs are found throughout a wide range of applications [3], [4]. In such graphs, there
are a small number of vertices with a large number of edges
and a large number of vertices with a small number of edges.
Specific domains where such graphs are important include
genomics [5]–[10], brain mapping [11], computer networks
[12]–[15], social media [16], [17], cybersecurity [18], [19],
and sparse machine learning [20]–[24].
Many graph processing systems are currently under development. These systems are exploring innovations in algorithms
[25]–[35], software architecture [36]–[44], software standards
[45]–[49], and parallel computing hardware [50]–[56]. The
development of novel algorithms and systems to process these
data requires the design, generation, and validation of enormous graphs with known properties. Such graphs accelerate
the proper testing of new algorithms and systems and are a
prerequisite for success on real applications.
This material is based in part upon work supported by the NSF under
grant number DMS-1312831. Any opinions, findings, and conclusions or
recommendations expressed in this material are those of the authors and do
not necessarily reflect the views of the National Science Foundation.
Many random graph generators currently exist that require
creating a graph in order to know its exact properties, such
as the number of vertices, number of edges, degree distribution, and number of triangles. Perhaps the most wellknown and scalable power-law graph generator is used in
the Graph500.org [57]–[59] and GraphChallenge.org [60]–
[62] benchmarks. This generator, often referred to as RMAT, is based on randomly sampling recursive Kronecker
graphs. Other highly scalable graph generators are based on
randomly specified degree distributions [63]–[65]. Designing
graphs using these random graph generators is an iterative
process whereby the graph designer selects the parameters of
the graph generator, randomly creates the graph with those
parameters, and then measures the desired properties. Such a
process places certain natural limits on the ability of the graph
designer to explore enormous graphs and know prior to graph
generation the exact properties of the graph.
This paper presents a complementary approach using Kronecker products that allows the exact computation of graph
properties prior to graph generation. In addition, when a real
graph is desired, it can be generated quickly in memory on a
parallel computer with no interprocessor communication. The
paper begins with a review of the relevant properties of Kronecker products. Next, the types of constituent matrices that
are suited for generating power-law graphs are described. Various mathematical properties of power-law Kronecker graphs
are then derived. Subsequently, a parallel algorithm for rapidly
generating large graphs is provided. A variety of performance
results and specific examples of various graphs generated using
this approach are presented. Finally, the conclusions and a
discussion of further research are given.
II. K RONECKER P RODUCTS
The Kronecker product of two square matrices is defined as
follows [66]
C=A⊗B=
A(1, 1) ⊗ B
A(2, 1) ⊗ B
..
.
A(mA , 1) ⊗ B
A(1, 2) ⊗ B
A(2, 2) ⊗ B
..
.
...
...
..
.
A(1, mA ) ⊗ B
A(2, mA ) ⊗ B
..
.
A(mA , 2) ⊗ B
...
A(mA , mA ) ⊗ B
The Kronecker
is defined
as fol
generation
is used inproduct
testing graphs
algorithm
0 data with models
and to compare real graph
A(1, 1) ⌦ B
A
where A, B, and C matrices of scalar values S convenient and well-defined matrix operatio
B A(2, 1) ⌦ B
A
A∈S
B
of graphsCfrom
a parameters
= Aa⌦few
B=
B =
B∈S
.. [Chakrab
@ is defined
C∈S
.
The Kronecker product
as fol
More explicitly, the Kronecker product can be written as
A(NA, 1)! ⌦ B A
0
!
!
C (i −1)m +i , (j −1)m +j = A(i , j )⊗B(i , j )
A
B A(1, 1)
C⌦ B
A
The element-wise multiply operation ⊗ can be a variety of
B A(2, 1) ⌦ B
functions so long as the resulting operation obeys the standard
A
i
B
rules of element-wise multiplication, such as 0 being the
C = A ⌦ B = B =! .
multiplicative annihilator for any value of s ∈ S
..
@
0⊗s=s⊗0=0
mA ×mA
mB ×mB
mA mB ×mA mB
A
A
B
A
A
B
A
A
B
B
P!
i
Furthermore, if element-wise multiplication and addition obey
the conditions of a semiring [67]–[69], then the Kronecker
product has many of the same desirable properties, such as
associativity
i
A(NA, 1) ⌦ B A
Fig. 1. Kronecker product of the adjacency matrix of two bipartite graphs A
P
and B results in a graph C with two bipartite sub-graphs. The = notation is
used to indicate that the adjacency matrix C has been permuted so that the
two bipartite sub-graphs are more apparent.
(A ⊗ B) ⊗ C = A ⊗ (B ⊗ C)
and element-wise distributivity over addition
i
A ⊗ (B ⊕ C) = (A ⊗ B) ⊕ (A ⊗ C)
Finally, one unique feature of the Kronecker product is its
relation to the matrix product. Specifically, the matrix product
of two Kronecker products is equal to the Kronecker product
of two matrix products
(A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD)
where matrix multiply
C = AB = A ⊕.⊗ B
is given by
C(i, j) =
M
A(i, k) ⊗ B(k, j)
k
III. G ENERATING P OWER -L AW G RAPHS
Generating graphs is a common operation in a wide range
of graph algorithms. Graph generation is used in the testing of
graph algorithms, in creating graph templates to match against,
and for comparing real graph data with models. Given a graph
adjacency matrix A, if
A(i, j) = 1
edges within its own set of vertices. The Kronecker product
of such graphs was first looked at by Weischel [72], who
observed that the Kronecker product of two bipartite graphs
resulted in a new graph consisting of two bipartite sub-graphs
(see Figure 1).
The essence of a power-law graph is that it has a degree
distribution vector n(d) with non-zero entries that follows the
relation
1
n(d) ∝ α
d
where d is the number of edges in the vertex of a graph, n(d)
is the number vertices with a specific degree d, and α > 0 is
the slope of the power law when it is plotted using logarithmic
axes [65]. If the graph is represented as an adjacency matrix,
then the degree of a vertex is the number of non-zero (nnz)
entries in the corresponding row and column in the matrix.
A star graph is a bipartite graph where one set has only one
vertex. Star graphs are always a power-law graph. If a star
graph has m vertices, then the number of points in the star is
given by
m̂ = m − 1
with a corresponding degree distribution of
n(1) = m̂
then there exists an edge going from vertex i to vertex j [70],
[71]. Likewise, if
A(i, j) = 0
then there is no edge from i to j. The Kronecker product of
two graph adjacency matrices is a convenient, well-defined
matrix operation that can be used for generating a wide range
of graphs from a few parameters [57], [58]. The relation of the
Kronecker product to graphs is easily illustrated in the context
of bipartite graphs. Bipartite graphs have two sets of vertices,
and every vertex has an edge to the other set of vertices but no
n(m̂) = 1
which agrees with the power-law relation α given by
α=
log(n(1))
log(m̂)
=
=1
log(dmax )
log(m̂)
where dmax is the degree of the vertex with the most edges.
The Kronecker product of two star graphs can, under certain
conditions, produce another power-law graph. In Figure 1, the
The Kronecker product is defined as fol
0 data with models
and to compare real graph
A(1, 1) ⌦ B
A
graph of the Kronecker product of two star graphs
with
m̂
=
convenient and well-defined
matrix operatio
B A(2,
5 and m̂ = 3 has a degree distribution of
1) ⌦ B
A
B
of
graphs
from
a
few
a
parameters
[Chakrab
n(1) = 15
C=A ⌦ B=B =
..
n(3) = 5
@
The Kronecker product is defined
as fol
.
n(5) = 3
n(15) = 1
0 A(NA, 1) ⌦ B A
which are all points on the curve
A(1, 1) ⌦ B
A
15
n(d) =
d
B A(2, 1) ⌦ B
A
i The Kronecker product of star graphs can be used to build C = A ⌦ B = B
B =
up extremely large power-law graphs. The degree distributions
..
will follow the power-law relation as long as all of the products
@
.
isUused
Sgeneration
UPERCOMP
T I N Gin
Ctesting
E N T E R graphs algorithm
A
B
i
i
of the corresponding m̂ are unique.
It is worth noting that real-world graphs often have approximate power-law distributions when plotted simply, as in this
case, or when plotted with logarithmic degree binning, but
rarely both. It is possible to use Kronecker products to produce
power-law graphs under logarithmic degree binning by placing
additional constraints on the values of m̂.
IV. P ROPERTIES OF K RONECKER G RAPHS
i
LLSC- 2
The most powerful feature of Kronecker
graphs is that many
of their properties can be computed from their constituent
matrices without ever having to form the full matrix. It is thus
possible to design and analyze extremely large graphs quickly
and only actually form the full graph when it is needed.
Let the adjacency matrix of graph A be constructed by the
following Kronecker product
A=
⊗ Nk=1 Ak
k
where Ak are each adjacency matrices of the smaller constituent graphs. The number of vertices in the graphs is equal
to the number of rows in A (or columns since Ak are square),
which can be computed from
Y
mA =
mAk
k
Likewise, the number of edges in the graph is equal to the
number of non-zero entries in A and is given by
Y
nnz(A) =
nnz(Ak )
k
The degree distribution nA (d) can be computed from the
Kronecker product of the degree distributions nAk (d)
nA (d) =
⊗ Nk=1 nA
k
(d)
k
A. Triangles
The number of vertices, number of edges, and degree distribution are good examples of the core properties of Kronecker
products. A more sophisticated example is computing the
number of triangles in a graph [73]–[76]. Triangles are an
important feature of a graph, and counting triangles is a basic
property of many graph analysis systems. The total number
A(NA, 1) ⌦ B A
Fig. 2. (top) Kronecker product of two star graphs with self-loops on the
central vertex. The resulting graph has 15 triangles. (bottom) Kronecker
product of two star graphs with self-loops on a leaf node. The resulting graph
has 3 triangles.
of triangles in a graph can be computed from the following
formula
1
Ntri (A) = 1T (AA ⊗ A)1
6
where 1 is a column vector of all 1’s and ⊗ is the element-wise
product. The same properties of Kronecker products apply
to counting triangles, and the number of triangles can be
computed from the component matrices via
1Y T
Ntri (A) =
1 (Ak Ak ⊗ Ak )1
6
k
B. Case 1: Many Triangles
Bipartite graphs have no triangles, so the Kronecker product
of star graphs will produce a large graph with zero triangles,
which can be a useful test case. Fortunately, it is possible to
simply modify the Ak to create a graph with a rich triangle
structure. Specifically, if a self-loop is put on the central vertex
of the star, the resulting graph will have a large number of
triangles. If the central vertex in the star is denoted by vertex
1, then a self-loop can be created in every constituent graph
by setting
Ak (1, 1) = 1
Removal of the self-loop in the final graph is accomplished
by setting a single value back to zero
A(1, 1) = 0
The number of vertices is unmodified by the inclusion of the
self-loops. The number of edges is computed from the Ak as
before, followed by subtracting 1 from the total to account for
the removal of the self-loop
nnz(A) − 1
Likewise, the degree distribution is computed from the Ak as
before with the following adjustments
n(dmax − 1) = 1
a graph with out-vertex incidence matrix Eout and in-vertex
incidence matrix Ein , the corresponding adjacency matrix is
[69], [77]
A = ET
out Ein
n(dmax ) = 0
The triangle count is computed from the Ak as before with
the following correction
1
1
Ntri (A) − mA +
2
3
Figure 2 (top) shows an example of a graph with 15 triangles
produced using this method.
C. Case 2: Some Triangles
Kronecker products can also be used to construct incidence
matrices that satisfy the above adjacency matrix equation.
Specifically, let Ek,out and Ek,in be incidence matrices corresponding to Ak . The incidence matrices can then be constructed by
Eout =
⊗ Nk=1 Ek,out
Ein =
⊗ Nk=1 Ek,in
k
and
A more modest number of triangles can be generated if one
self-loop is put on one of the point vertices of each star, for
example by setting
Ak (mAk , mAk ) = 1
Removal of the self-loop in the final graph is accomplished
by setting a single value back to zero
It is worth noting that the order of edges in the incidence
matrices is not uniquely determined. Different realizations of
an incidence matrix are only equivalent when comparing their
resulting adjacency matrices.
V. PARALLEL G ENERATION
A(mA , mA ) = 0
The number of vertices is unmodified by the inclusion of the
self-loops. The number of edges is computed from the Ak as
before, followed by subtracting 1 from the total to account for
the removal of the self-loop
nnz(A) − 1
Likewise, the degree distribution is computed from the Ak as
before with the following adjustments
k
Kronecker products allow the properties of a graph to be
determined in advance, thus avoiding the iterative approach of
other methods. Once the desired graph properties have been
determined, Kronecker products also allow large graphs to be
generated quickly on a parallel processor. The overall approach
is to split the constituent matrices into two matrices B and C
⊗ Nk=1 Ak
N
N
= ⊗ k=1 Ak ⊗ ⊗ k=N
A=
k
B
Nk
n(2
− 1) = 1
n(2Nk ) = 0
The triangle count is computed from the Ak as before with
the following correction
1
1
Ntri (A) − 2Nk +
2
3
Figure 2 (bottom) shows an example of a graph with 1 triangle
produced using this method.
D. Incidence Matrix
An incidence, or edge, matrix E uses the rows to represent
every edge in the graph, and the columns represent every
vertex. There are a number of conventions for denoting an
edge in an incidence matrix. One such convention is to use
two incidence matrices
Eout (e, i) = 1
and Ein (e, j) = 1
to indicate that edge e is a connection from i to j. Incidence
matrices are useful because they can easily represent multigraphs and hyper-graphs. These complex graphs are difficult
to capture with an adjacency matrix. One of the most common
uses of matrix multiplication is to construct an adjacency
matrix from an incidence matrix representation of a graph. For
C
B −1
Ak
=B⊗C
The matrices B and C are designed so that both can fit in the
memory of any one processor. Let the parallel computer have
Np processors, and each processor is given an identifier p [78],
[79]. Each processor reads in B and C and extracts the triples
of the non-zero element B into three vectors i, j, and s, each
of length nnz(B). Each processor then selects a nnz(B)/Np
of the triples ip , jp , and sp . If the underlying sparse storage
of the matrices is compressed sparse columns (CSC), then the
minimum value of jp is subtracted from jp and a new matrix
Bp is formed from these triples. Each processor can then form
the submatrix Ap of the overall matrix A via the Kronecker
product
Ap = Bp ⊗ C
The resulting Ap matrices will have the same number of nonzero entries on each processor. In addition, the resulting graph
is free of many of the problematic vertices and edges, such
as empty vertices and self-loops, that are found in randomly
generated graphs. These problematic vertices and edges often
require randomly generated graphs to be reindexed before their
properties can be computed.
0
0
10
1.E+12
1012
1.E+11
1011
1010
1.E+10
0
10 0
10
10
degree count, n(d)
2
Rate (edges generated/second)
0
1.E+13
1013
power-law
predicted
measured
8
10 6
10 4
109
1.E+09
10 2
108
1.E+08
10
10 4
107
1.E+07
2
10 6
10 0
0
10
10 8
10
2
10
4
vertex degree, d
10
6
10 10
10
8
10
10
10
12
vertex degree, d
1
10
100
1000
10000
100000
Number of Processor Cores
Fig. 3. Edge generation rate vs. number of processor cores. Performance
scales linearly with processor cores and achieves a peak rate of over 1 trillion
edges generated per second on over 40,000 processor cores.
VI. R ESULTS
This section presents a variety of scalability results to
demonstrate the properties of the proposed Kronecker graph
generation method. Figure 3 shows the rate of graph edge
generation as a function of the number of processing cores
used in the parallel graph generation technique described in
the previous section. In this example, B is a 530,400 vertex
graph with 13,824,000 edges constructed from the Kronecker
product of star graphs with m̂ = {3, 4, 5, 9, 16}. Likewise, C
is a 21,074 vertex graph with 82,944 edges constructed from
the Kronecker product of star graphs with m̂ = {81, 256}.
The Kronecker product of B and C, produces a graph A with
11,177,649,600 vertices and 1,146,617,856,000 edges and zero
triangles. This graph construction was run in parallel on a
supercomputer consisting of 648 compute nodes, each with at
least 64 Xeon processing cores, for a total of 41,472 processing
cores. Using the entire system, the trillion edge graph was
generated in 1 second.
Computing the degree distribution of the generated graph
can be used to verify that a generated graph agrees with the
theory. Figure 4 shows the measured and predicted degree
distribution of a graph produced using the parallel graph
generation technique. In this example, B is a 530,400 vertex
graph with 22,160,060 edges constructed from the Kronecker
product of star graphs with m̂ = {3, 4, 5, 9, 16} and selfloops on the central vertices of the stars. Likewise, C is a
21,074 vertex graph with 83,618 edges constructed from the
Kronecker product of star graphs with m̂ = {81, 256} and
self-loops on the central vertices of the stars. The Kronecker
product of B and C produces a graph A with 11,177,649,600
vertices and 1,853,002,140,758 edges and 6,777,007,252,427
triangles. This calculation confirms that the predicted and
Fig. 4. Trillion-edge (1012 ) power-law Kronecker graph showing the exact
agreement between the predicted and measured degree distribution. The
resulting graph has exactly 11,177,649,600 vertices, 1,853,002,140,758 edges,
and 6,777,007,252,427 triangles.
measured graph are in exact agreement.
Kronecker products can allow the rapid design of very large
graphs suitable for the world’s largest computers. Figures 5
and 6 show the degree distribution for two graphs with over
1015 edges. Both graphs are generated from star graphs with
m̂ = {3, 4, 5, 9, 16, 25, 81, 256, 625} and 6,997,208,649,600
vertices. Figure 5 has 1,433,272,320,000,000 edges and zero
triangles, and the degree distribution exactly follows the
power-law degree formula. Figure 6 is generated with selfloops on the central vertices producing 2,318,105,678,089,508
edges, 12,720,651,636,552,426 triangles, with the degree distribution that follows the power-law degree formula with small
deviations above and below the line.
Kronecker products can also enable the exact analysis
of graphs that are far beyond the scale of any current
or planned computing system. Figures 7 shows the
degree distribution of a graph with over 1030 edges.
The graph was generated from star graphs with m̂ =
{3, 4, 5, 7, 11, 9, 16, 25, 49, 81, 121, 256, 625, 2401, 14641}
and a self-loop on one point vertex of each star. The resulting
graph has exactly 144,111,718,793,178,936,483,840,000
vertices, 2,705,963,586,782,877,716,483,871,216,764 edges,
and 178,940,587 triangles. Most of the points follow the
power-law degree line, but there are many points that deviate
from this. This degree distribution was computed on a
standard laptop computer in a few minutes.
VII. C ONCLUSION
Emerging data in metagenomics, brain mapping, Internetof-things, cybersecurity, and sparse machine learning produce
massive power-law graphs and are driving the development
of novel algorithms and systems to process these data. The
scale and distribution of these data makes validation of graph
processing systems a significant challenge. The ability to
14
10
12
10
10
10 30
10 25
degree count, n(d)
degree count, n(d)
10
10 8
10
6
10
4
10
2
10
0
10 0
10 2
10 4
10 6
10 8
10 10
10 12
10 14
10 20
10
15
10
10
10
5
10
0
10
0
10
5
10
10
10
15
10
20
10
25
10
30
vertex degree, d
vertex degree, d
Fig. 5. Quadrillion-edge (1015 ) power-law Kronecker graph predicted degree
distribution. The resulting graph has exactly 6,997,208,649,600 vertices,
1,433,272,320,000,000 edges, and zero triangles.
Fig. 7.
Predicted degree distribution of a decetta-edge (1030 )
power-law Kronecker graph. The resulting graph is predicted
to
have
exactly
144,111,718,793,178,936,483,840,000
vertices,
2,705,963,586,782,877,716,483,871,216,764
edges,
and
178,940,587
triangles.
degree count, n(d)
10 14
10
12
10
10
10
8
10
6
responding properties of the constituent matrices. The ability
to compute the properties of large graphs using only small
graphs allows the graph designer to find these prior to creating
the actual graph. Furthermore, real graphs can be created
using Kronecker products on a parallel computer with no
interprocessor communication. The resulting graphs will have
the same number of edges on each processor. In addition,
the graph avoids many of the difficulties, such as empty
vertices and self-loops, that are found in other graph generators
that rely random sampling. These problematic vertices and
edges often require randomly generated graphs to be reindexed
before their properties can be computed.
10 4
10
2
10 0
10 0
10 2
10 4
10 6
10 8
10 10
10 12
10 14
vertex degree, d
Fig. 6. Quadrillion-edge (1015 ) power-law Kronecker graph predicted degree
distribution. The resulting graph has exactly 6,997,208,649,600 vertices,
2,318,105,678,089,508 edges, and 12,720,651,636,552,426 triangles.
create enormous graphs with exactly known properties can
significantly accelerate the design, generation, and validation
of new graph processing systems. Many current graph generators produce random graphs whose exact properties, such as
number of vertices, number of edges, degree distribution, and
number of triangles, can only be computed after the graph has
been generated. Thus, designing graphs using these random
graph generators is a time-consuming trial-and-error process.
Kronecker products of the adjacency matrices of star graphs
are a powerful way to create large power-law graphs. The
properties of Kronecker products allow many properties of a
larger graph to be computed by simply combining the cor-
To test this approach, graphs with 1012 edges are generated
on a 40,000+ core supercomputer in 1 second and exactly
agree with those predicted by the theory. In addition, in order
to demonstrate the extensibility of this approach, decetta-scale
graphs with up to 1030 edges are simulated in a few minutes
on laptop. These results indicate that the proposed method
can be a powerful tool for enabling the design, generation,
and validation of new graph processing systems.
This paper has presented formulas for a number of properties of Kronecker graphs. There are many additional properties that could be computed in future research, such as
eigenvectors, iso-parametric ratios, betweenness centrality, and
triangle enumeration. The parallel Kronecker graph generator
is ideally suited to the GraphBLAS.org software standard and
the creation of a high performance version using this standard
is a future goal. Finally, the ability to reason about graphs
that are beyond any current or planned computer opens up
new possibilities for the theoretical study of phenomena on
these large graphs.
ACKNOWLEDGMENTS
The authors wish to acknowledge the following individuals
for their contributions and support: Alan Edelman, Charles
Leiserson, Steve Pritchard, Michael Wright, Bob Bond, Dave
Martinez, Sterling Foster, Paul Burkhardt, and Victor Roytburd.
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| 8 |
What
the
F-‐measure
doesn’t
measure…
Features,
F laws,
F allacies
a nd
F ixes
David M.W. Powers, Beijing University of Technology, China & Flinders University, Australia
Technical Report KIT-14-001 Computer Science, Engineering & Mathematics, Flinders University
The F-measure or F-score is one of the most commonly used “single number” measures
in Information Retrieval, Natural Language Processing and Machine Learning, but it is
based on a mistake, and the flawed assumptions render it unsuitable for use in most
contexts! Fortunately, there are better alternatives…
What
the
F-‐measure
is!
F-measure, sometimes known as F-score or (incorrectly) the F1 metric (the β=1 case of
the more general measure), is a weighted harmonic mean of Recall & Precision (R & P).
There are several motivations for this choice of mean. In particular, the harmonic mean is
commonly appropriate when averaging rates or frequencies, but there is also a settheoretic reason we will discuss later. The most general form, F, allows differential
weighting of Recall and Precision but commonly they are given equal weight, giving rise
to F1 but as it is so ubiquitous this is often understood when referring to F-measure.
Who’s
the
F-‐measure
for?
F-measure comes from Information Retrieval (IR) where Recall is the frequency with
which relevant documents are retrieved or ‘recalled’ by a system, but it is known
elsewhere as Sensitivity or True Positive Rate (TPR). Precision is the frequency with
which retrieved documents or predictions are relevant or ‘correct’, and is properly a form
of Accuracy, also known as Positive Predictive Value (PPV) or True Positive Accuracy
(TPA). F is intended to combine these into a single measure of search ‘effectiveness’.
Figure 1. Medium Accuracy with High Precision (left) or High Trueness (right) but
not both together - stolen from Wikipedia (redraw)
More generally, precision refers to the concept of consistency, or the ability to
group well, while accuracy refers to how close we are to the target, and trueness refers to
how close we are to a specific target on average (Figure 1)1.
High precision and low accuracy is possible due to systematic bias. One of the
problems with Recall, Precision, F-measure and Accuracy as used in Information
Retrieval is that they are easily biased. To better understand the relationships between
these measures it is useful to give their formulae in two forms, one form related to the
raw counts, and one related to normalized frequencies (Equation 1 and Table 1).
These statistics are all appropriate when there is one class of items that is of
interest or relevance out of a larger set of N items or instances. This single class of
interest we call the positive class, or the real positives (RP). More generally we name
multiple classes and refer to the proportion of the total each represent as its prevalence
(for RPs, Prev=ρ=rp=RP/N). IR also assumes a retrieval mechanism that gives rise to the
predicted positives, and also represents the bias of this “classifier” (Bias=π=pp=PP/N).
In this systematic notation, we use upper case initialisms to refer to counts of items (e.g.
RP), and lower case equivalents to refer to the corresponding probabilities or proportions
(rp). It is also common to use Greek equivalents for probabilities in a mnemonic way (ρ).
How’s
the
F-‐measure
defined?
We can now formally define Recall (R or Rec) and Precision (P or Prec) in terms of true
positives (tp=TP/N) and the +ve prevalence (Prev) and Bias, using either their count or
probability forms. Table 1 is provided to explain the notation in our equations, where we
also include definitions of Accuracy (A or Acc) and F using two different means:
R = TP / RP = tp / rp
(1)
P = TP / PP = tp / pp
(2)
A = [TP+TN]/N = tp+tn
(3)
F = tp/am(rp,pp) = hm(R,P)
(4)
Table 1. Systematic notation underlying (1-4) defining F as a harmonic mean (hm) and showing how
this corresponds to referencing to the putative distribution given by the arithmetic mean (am).
+R
−R
+P
tp
fp
pp
−P
fn
tn
pn
rp
rn
1
+R
−R
+P
TP
FP
PP
−P
FN
TN
PN
RP
RN
N
Note that A, R and P are themselves probabilities or proportions. Accuracy is the
probability that a randomly chosen instance (positive or negative, relevant or irrelevant)
will be correct. Recall is the probability that a randomly chosen relevant instance will be
predicted (positive). Precision is the probability that a randomly chosen predicted
instance (positive) will be relevant. Accuracy can also shown to be the average of Recall
and Inverse Recall (viz. with positives and negatives inverted) weighted by Prevalence.
Accuracy is also the average of Precision and Inverse Precision weighted by Bias.
Why
the
F-‐measure
is
used!
We can now see another, set theoretic, way of looking at the definition of F-measure, and
this is how it was originally defined in the equally weighted version (Figure 2)2. In fact,
what was defined was E = 1-F which is a reinvention of the Dice semimetric3 that was
normalized by dividing by the average size of the compared sets. This is how we come to
have F as a harmonic mean, as the TP intersection size is divided by the arithmetic mean
of the RP and PP cardinalities. Moreover the Dice distance measure, which failed to
satisfy the triangle inequality, was a faulty reinvention of the Jaccard metric4 - Jaccard
normalizes more appropriately by the size of the union of the two sets (without double
counting), and we note that it indeed satisfies the mathematical definition of a metric.
Figure 2. Set theoretic interpretation of F-measure. Stolen from van Rijsbergen (1979) Fig 7.11
(redraw). F-measure corresponds to the number of items in the white intersection divided by the
average size of the A and B groups being the Real and Predicted items.
F1 is also a reinvention of the statistical measure Positive Specific Agreement,
which is designed to compare the agreement of two raters under the assumption that their
ratings come from the same distribution (e.g. they are both native speakers of the same
dialect with the same understanding of nouns and verbs). Thus the average of the two
separate sample distributions is taken as an approximation of the underlying but unknown
distribution. This is hardly appropriate when one distribution is a real distribution based
on human judgements, and the other comes from a system being evaluated. We will
however meet this assumption again when we discuss chance-corrected measures.
The F-Measure was popularized by the MUC and TREC5 competitions based on a
presentation in the 2nd edition of the early van Rijsbergen textbook on Information
Retrieval2 that recapitulated his 1974 paper. However, the function defined there was E
for Effectiveness (E=1-F) and made use of a different function F. But F has stuck, and it
is now virtually impossible to publish work in Information Retrieval or Natural Language
Processing without including it. But that is a big mistake…
When
the
F-‐measure
is
wrong!
We have already noted in passing four flaws with F-measure that emerged from
theoretical considerations, and we add three further practical issues:
•
F-measure (like Accuracy, Recall and Precision) focuses on one class only
•
F-measure (like Accuracy, Recall and Precision) is biased to the majority class
•
F-measure as a probability assumes the Real and Prediction distributions are identical
•
E-measure (1-F) is not technically a metric as it does not satisfy a triangle inequality
•
F-measures don’t average well across real classes or predicted labels or runs
•
F-measure doesn’t in general take into account the True Negatives (TN)
•
F-measure gives different optima from other approaches and tradeoffs.
One-‐Class
The one-class issue is one of usage. If you are only interested in one class, then Fmeasure is not unreasonable. But note that the number of true negatives (TN) can change
arbitrarily without changing F-measure (as we discuss further below), and that there is no
easy way to form a macro-average across class or prediction distributions – it is assessing
performance relative to a mixture of the real and prediction distribution, and if we did
perform the calculations to average over that fictitious distribution, we would get Rand
Accuracy (3). With Recall (1), if we macro-average weighted by the prevalence of each
class, we also get Rand Accuracy (3). With Precision (2), if we macro-average weighted
by the bias towards predicting each label, we also get Rand Accuracy (3).
Bias
The bias issue is arguably the most serious, and affects both Recall and Precision as well
as Accuracy. In fact, van Rijsbergen2 reviewed techniques that dealt with bias and chance
(effecively ROC and Informedness, which we discuss below). He however started with
the goal of finding a way of combining Recall and Precision into a single measure, and
concluded that for the purposes of Information Retrieval the additional complexity of the
proposals wasn’t warranted as the studies of the other methods did not conclusively show
they were optimal. The E-measures that we know in complementary form as the F1measure and the more general F-measure, were in fact special cases of a far more general
approach to fitting intuitions about effectiveness and choosing the point to optimize in
relation to the Recall-Precision tradeoff. But bias impacts both Recall and Precision
themselves, and no form of averaging is going to get rid of it, so it is clearly suboptimal.
But van Rijsbergen also notes that his general F-formula for combining them (not the Fmeasure we know) could be useful with measures other than Recall and Precision.
The damning example of bias in F-measure that brought this to our attention came
from real life examples in Natural Language Processing (parsing and tagging)6. Here we
found that a common tuning approach to increase the F-score was to dumb down the
system by getting it to guess rather than use principled statistical techniques. There is a
real problem here as a better system can actually get a worse F-score.
One example concerned the word ‘water’, which was almost always a noun
(roughly 90% of the time) and much more rarely a verb (say 10%) and we can assume for
simplicity that any apparent adjectival uses are in fact nominal. What the systems did was
simply say water is always a noun. This particular form of guessing (always guessing
noun) delivers 100% Recall, and 90% Precision, and 90% Accuracy, while any of the
arithmetic, geometric or harmonic means is somewhere around halfway between the 90%
and 100% effectiveness estimates, with arithmetic being the highest, the harmonic mean
more conservative, and the arithmetically weighted average Accuracy is the minimum.
The arithmetic, geometric and harmonic means correspond to the Lp means for
p = +1, 0 and –1. The geometric mean is actually the geometric means of the paired Lp
means for p=±p and is thus is in a sense the middle mean and is usually closest to the
mode and least affected by outliers. In application to Recall and Precision, this gives rise
to the G-measure (which also satisfies the general form of 1-E-measure). Accuracy is
based on a weighted arithmetic mean, but even though we get 0% Inverse Recall and
10% Inverse Precision, the low Inverse Bias (0%) and Prevalence (10%) mean that the
weighted average doesn’t help either, and this also applies for any other weighting in
[0,1] that might be applied to Recall and Precision, in the most general form of the Fmeasure. Furthermore the limits for p=±∞ correspond to max and min, and min is the
most conservative of the Lp family, but all these Lp means lie in the [90,100]% range,
irrespective of weighting. Thus no choice or weighting can avoid the bias problem.
Distribution
The distribution problem is more subtle, but when it comes down to it the correct number
of positives to predict is the actual number of real positives. In the closely related PPV
and Dice measures, there is an explicit assumption that two different raters’ class labels
are drawn from the same distribution without any expectation that one is more reliable.
This does not apply here, if we truly believe our ‘Gold Standard’ is ground truth, and that
the system we are evaluating is the one that is at fault if there are discrepancies. This is
clearly a problem for Machine Learning and Intelligent Systems in general.
For Information Retrieval, the situation is a little different, as there is really only
one class of interest. The number of irrelevant documents is sufficiently large to be
beyond our comprehension, but the number of relevant documents may also be very
large, and indeed not completely known. Early in the TREC5 competitions, the set of
relevant documents was initially seeded with the documents returned by any of the
systems, and only these were evaluated for relevance. Later better systems being
evaluated were marked wrong, reducing Precision, for returning a relevant document that
none of the seeding systems discovered. This has been recognized by use of alternate
terminology such as Coverage where only a subset of relevant documents is known.
Moreover the number of documents returned is often limited by practical considerations,
often to a fixed ceiling (and search engines may allow users to chose it). Furthermore,
search engines will allow further blocks of hits and users will often decide to take another
block two or three times, if it seems like it is worthwhile (e.g. sufficient promising leads
to keep going but nothing good enough a match to stop).
Two alternatives to F-measure that became common in IR in recent years through
TREC5 are MAP (Mean Average Precision) and R-Precision. MAP is a kind of area
under the Recall-Precision curve that averages over the different bias points
corresponding to this “keeping going” possibility. MAP tends to correlate well with the
simpler R-Precision that effectively evaluates at the point where Bias = Prevalence or PP
= RP. Here of course Prevalence or RP represents the number of documents you expect
which in an experimental context is the number you know you have.
Interestingly, van Rijsbergen’s derivation2 hinges on the assumption that the user
will be interested in a particular tradeoff of Recall vs Precision – that is how much
increase in Recall will be traded off for a decrease in Precision. This is expressed in
terms of the ratio P/R and directly leads both to the weighting factor and the choice of the
harmonic mean in his derivation. However, given the definitions of Recall and Precision
(1&2), this corresponds to Prev/Bias (π/ρ=RP/PP). On the other hand, setting Prev=Bias
corresponding to P/R=1, or any other constant value, doesn’t allow for the fact that
different levels of Precision mean you need different numbers of hits returned to ensure
you get any desired number of relevant documents, D: D = PP/Precision. But for a
particular system and user (and set of topics searched) it may be that Precision
approximates a constant and the P/R ratio is thus meaningful, setting the desired Recall.
Setting P/R can also be viewed as setting a tradeoff between False Positives (FP)
and False Negatives (FN), if we expand out the denominators of (1&2). In a more general
Intelligent Systems context, this corresponds to setting the relative costs of False
Positives and Negatives, and this is also a feature of ROC curves which trade off TPR
(aka Recall) and FPR (aka Fallout = 1 – Inverse Recall).
Setting 0<P/R<∞ also forces P>0 and R>0, which in turn implies and requires
TP>0. However, setting the weighting for F-measure does not achieve this, and does not
solve the bias problem although it does affect the distribution problem. P/R ≠ 1. A choice
of F-measure other than F1 (β=1) does imply a bias away from the Bias = Prevalence
constraint. This however considers only the mean of the distributions, and it is possible
that the FP and FN errors have different distributions, both from each other and for
different queries. In particular they can have different variances (as covered by van
Rijsbergen in his review of ROC-related measures2), and this means that any setting of
P/R or β other than 1 cannot be effective.
Metricity
Whether a measure is a metric or not may seem rather academic. F is set up as a
similarity measure – we want it to be 1. But E was set up as a distance or dissimilarity or
error measure – we want it to be 0. This difference is similar to the difference between
use of the Cosine or Correlation type measures we discuss later, which we want to be 1
for maximum similarity, versus the Euclidean distance measure, which we want to be 0.
This in turn reflects a Sine function versus a Cosine function of the divergence of the
vectors being compared. For purposes of graphical visualization well behaved measures
are desirable, and the ability to convert sensibly into distances is desirable for similarity
measures.
In terms of our intuitions, we expect things to add together in certain ways. The
triangle equality is the missing element that the E and F-measures fail on. This is the idea
that the shortest distance between two points is the direct line between them. For the Emeasure, the sets {R} and {P}are 1/3 away from {R,P} but 1 away from each other, so it
is closer to go from {R} to {P} via {R,P} than directly (where R and P here represent
individual documents)!
If such measures are used for clustering it leads to very confusing, non-monotonic
results.
Averaging
We saw earlier that averaging Recall with Prevalence weighting and averaging Precision
with Bias weighting both give Accuacy. This means that averaging generalizes from the
two class case we have been considering to the multiclass case.
F-measure’s harmonic mean essentially means we should be averaging over the
putative expected (real or predicted) population, and in practice this means that macroaveraging in a principled way is not practical, and that macro-averaging based on either
equal weighting or prevalence weighting, as many systems do, is not meaningful. Some
systems even do different kinds of averaging in different places and get inconsistent
results.
This ‘apples vs pears’ principle applies when we are averaging over multiple runs
or multiple queries or multiple datasets as well as multiple classes. It is important always
to average using weights reflecting the appropriate units. If we are talking Recall we are
talking proportions relative to the actual members of a class (per real positive). If we are
talking Precision we are talking proportions relative to the predictions (per predicted
positive). If we are talking Accuracy we are talking about all instances (relative to N if N
differs between datasets). If we are talking F-measure, we don’t know what we are
talking about! Results are relative to some fictitious intermediate distribution.
Negatives
In Information Retrieval the negatives, the irrelevant documents, do not concern us at all.
There are so many of them that the Inverse Precision and Inverse Recall are near enough
to 0 to not be a significant factor, and thus F-measure has been argued to be a useful
simplification for the sake of efficiency and comprehensibility.
Nonetheless for other Intelligent Systems, both (or multiple) classes tend to be
significant for us, but neither Recall nor Precision take the TN cell of the contingency
table into account. TN can be incremented from 0, to include almost all examples,
without affecting the Precision or Recall, and thus without affecting F.
With this view, based on counts, we are adding new examples to the system,
increasing N, and it is getting them all wrong.
It has however been claimed that the F-measure does in fact take TN into account
if we consider RP, RN and N to be fixed, and N known. This is based on dividing by TP
to recover RP and PP, and then subtracting from N to recover RN and PN, and thus all
cells of the contingency table are determined. Technically this does not hold in general,
specifically in the case where TP=0, and it is thus ill-conditioned as tp→0. Moreover,
even when it is implicitly specified it is reflected only indirectly in the denominator in
cancellation against N (which also does not appear explicitly).
It may be thought that we can explore the F-measure both for the positive and
negative class, reversing the labels. But this can give very different answers (consider the
water as noun or verb example versus the search for documents about water). In the end it
comes back to how to average, and some systems do macro-average F-measure
inappropriately across multiple classes (it depends on prediction as well as class).
Tradeoffs
F-measure is about finding one number with which to compare systems and find a
winner. Unfortunately this often succeeds in optimizing the wrong thing when there are
more than one class of interest, because of the issues we have already discussed: Fmeasure is specifically seeking a trade off between Recall and Precision, but there is
another pair of measures we commonly trade off in ROC: Recall (TPR) and Fallout
(FPR). There have been claims PR and ROC are essentially doing the same tradeoff (see
Fig. 3). In some practical contexts, this can even be true. In particular, if Bias =
Prevalence (diagonals in Fig. 3) all positive measures considered become equivalent:
Recall=Precision=Accuracy=F, as do the ROC and chance-corrected measures we will
discuss later.
One specific claim is that 1–Precision acts as a surrogate for Fallout (1– Inverse
Precision), and hence Recall-Precision is just as useful a tradeoff to consider as RecallFallout (ROC) since RP and RN are constant. Indeed, the relationship is quite clear as
Precision = TP/PP while Fallout = FP/RN ∝ PP–TP. This however makes the tradeoff
relationship quite different, although the Shannon Information conveyed by Precision,
–log(Precision), interestingly gives rise to the same linear form. It also means that the
areas under the curves are different and the maxima will not correspond. We can also see
that taking the reciprocal for the harmonic mean of the F-measure is essentially trading
off Bias (pp) against Prevalence (rp) linearly. On the other hand, in ROC, Recall versus
Fallout is trading off TP against FP, normalized by constants (RP and RN) that imply
differential costs for the positive and negative cases.
Where
the
F-‐measure
can
be
improved
on!
We consider again our list of issues: one-class, bias, distribution, metricity, averaging,
negatives and tradeoffs.
One-‐Class
If we are dealing with more than one class, then the answer is simply to calculate Rand
Accuracy directly (3) or via macro-averaging of Recall (2), as it is complex to macroaverage (4) correctly, and the result would still be Rand Accuracy. Weighting F-measure
simply by the size of each class (or the number of predictions of each class) enshrines a
bias when these are different, and means a better result can be achieved by changing the
bias towards the more prevalent classes (and some learning algorithms do this).
Bias
There are many approaches to dealing with bias. In statistics these include regression and
correlation techniques, as well as more ad hoc chance-correction techniques that attempt
to subtract off the chance component and restore the statistic to the form of a probability.
F-measure is similarly designed to retain the form of a probability, for a fictitious
distribution. The tradeoff technique of Receiver Operating Characteristics (ROC) also
gives rise to a method of controling for bias, and indeed there are strong relationships
between all of these techniques, and we introduce them briefly now and in detail below.
Kappa: This is the ad hoc approach that subtracts off a chance estimate and then
renormalizes to a [0,1] range by dividing by the expected error, that is the room for
improvement over chance, as shown in equation (4). It was originally designed to
compare human raters, but has recently been applied in Machine Learning to rate
classifiers (higher kappa = better classifier), and in Classifier Fusion as a measure of
Diversity (higher kappa = less diversity). The use as a measure of Diversity is closer to
the original use for comparing human raters.
Distribution
There are quite a few different versions of kappa7 (5), the most common being
Cohen Kappa, which is the one people in Machine Learning tend to know and use. But
Fleiss Kappa is the one that corresponds most closely with F-measure in its distributional
assumptions, and neither reflects a probability in or relative to a well defined distribution.
KappaX = [AccX – ExpAccX] / ExpErrX
(5)
Cohen Kappa (KappaC) assumes that the two marginal distributions are
independent in estimating the expected values of the contingency table due to chance – it
multiples the marginal probabilities, the prevalences and biases in our context, on the
assumption that they are independent distributions. Fleiss Kappa (KappaF) makes the
same assumption as F-measure and assumes the margins actually derive from the same
distribution. Like F-measure, instead of using the actual Bias and Prevalence (and Inverse
Bias and Inverse Prevalence) Fleiss replaces them with their arithmetic means before
performing the same calculation as Cohen Kappa.
In these calculations Acc and ExpAcc (4) are the Accuracy A (3) and the expected
value by adding the expected versions of tp and tn, etp and etn, while ExpErr = 1–A is the
sum of the expected version of fp and fn, namely efp and efn. For Cohen Kappa we have
etp = pp*rp and fp = pp*rn. For Fleiss Kappa etp = epp*erp where epp = erp = (pp+rp)/2.
These can be applied not just to the two class situation we have focussed on here, but to
multiple classes, and indeed to multiple raters or classifiers.
Informedness and Markedness8 are principled Kappa forms derived from Recall
and Precision and their respective inverses, but have been derived in many different
ways, being also closely related to Gini and ROC.
Informedness = R + InvR – 1
(6)
Markedness = P + InvP – 1
(7)
Informedness corresponding to your edge when betting on races or speculating in
the stock market, the method of marking multiple choice exams that leads to an expected
mark of zero for guessing, the distance of the contingency table in ROC space above the
chance line, or the Gini function of the area under the curve (AUC) subtended by that
single point in ROC space9. Under the guise of Youden’s J or DeltaP’, Informedness
represents a regression coefficient, and Markedness or DeltaP represents one for the
opposite direction of prediction.
Informedness, KappaI based on Recall, is the probability of an informed decision.
Markedness, KappaM based on Precision, is the probability of a decision variable being
marked by the real class. The G-measure associated with these derivatives of Recall and
Precision, consistent with an Information theoretic instantiation of the original F function
of van Rijsbergen, is Matthews Correlation7.
Metricity
Of these measures, the ones that have straightforward inversion from similarity measures
(1 is best) to metric distances (0 is best) are Correlation (equivalent to Cosine measure),
Informedness and Markedness (both corresponding to linear regression coefficients and
interpretable as Cosines).
Averaging
Informedness is correctly averaged weighted by Bias (since each component is
the Informedness of a prediction) and Markedness is correctly averaged weighted by
Prevalence (since it is the Markedness of a class). As with F-measure and G-measure,
Correlation is not meaningfully macroaveraged, and if required should be calculated from
the Multiclass Informedness and Markedness. If a single number is required against a
Gold Standard, Multiclass Informedness is it. If you are interested in information flow in
the other direction, then Markedness gives you that, and for an unsupervised comparison
with the same number of classes, Correlation is recommended. But the unsupervised
case gets more complicated as the number of classes not constrained to match10.
Negatives
For the dichotomous case of Positive-Negative, the same result is achieved whichever
class is designated positive, unlike Recall, Precision and F-measure. Informedness tells
you the probability that you have made an informed decision, as opposed to guessing –
the ability to bias based on Prevalence that we exploited with F-measure, for the water
example… It’s been eliminated!
Tradeoffs
In regard to the relationship between Precision-Recall (PR) and Receiver
Operating Characteristics (ROC) curves, the same constraint that TP, Recall and
Precision are nonzero is necessary to show relationships (this means that thresholds or
other parameters must be constrained to avoid this case or these points eliminated from
the curve, although the zero points are traditionally shown). This has been studied
comprehensively by Davis and Goadrich11 who show that the ROC curve and PR curve
for a given algorithm contain the same points, that one ROC curve dominates another in
ROC space if and only if the corresponding PR curves display the same dominance.
Furthermore there is an analog in PR space of the well known convex hull of ROC space,
which we call an achievable PR curve. The similar linear relationships expressed by
Precision Information and Fallout mean that the points in the curve correspond in a deep
way, with the result that the operating points interpolated and omitted in a ROC Convex
Hull correspond to the achievable and omitted points on the corresponding smoothing of
the PR curve, although the interpolation is no longer linear in PR space (see Fig. 3).
Constant F-curves are indeed curves in PR space, but isocost curves in ROC space
are linear and parallel, with the default cost equating the value of the full set of true
positives and the full set of real negatives.
Would
the
F-‐measure
ever
be
the
best
measure?
No! There is always something better, but sometimes the error in using F-measure is
small, and at times it can even vanish – just don’t depend on this!
Under the constraint that Bias = Prevalence, or equivalently at the break-even
point where we constrain Recall = Precision, the question of which measure to use
becomes academic: P = R = F, and all the Kappa, Regression and Correlation variants
also coincide. This constraint is thus a useful heuristic and corresponds to the default
assumption that getting all the positives right is of equal value to getting all the negatives
right, so you need to work equally hard on each class. Any other bias upsets this, and as
extreme cases, Bias = 0 will get all the negatives right, but none of the positives (F=0),
while Bias = 1 will get all the positives right, but none of the negatives
(Precision=F=Prevalence, Recall=Bias=1), and the Inverse Prevalence, IPrev = 1–Prev, is
thus the room for improvement or expected error for KappaI. For optimal performance we
need to get both right, and Informedness and Markedness expressed in Kappa form
capture this normalization for Recall and a corresponding one for Precision:
KI = Informedness = [R–Bias] / IPrev
(8)
KM = Markedness = [P–Prev] / IBias
(9)
For supervised learning or other systems where there is a ‘Gold Standard’, in the
absence of further information about the cost or probability distribution of cases,
Informedness8 is the appropriate measure to use, and as it is the height above the chance
line in ROC (Fig. 3), Receiver Operating Characteristics is the appropriate graphical
representation to use for assessing tradeoff and resilience to changes in the prevalence
conditions9. Moreover, ROC does also have the flexibility to explicitly manage the cost
tradeoff just as the general form of F-measure aims to do with its β tradeoff parameter.
For unsupervised contexts or where each side has equal status as opposed to one
being a Gold Standard (which are usually rather tarnished), Matthews Correlation, the
geometric mean of Informedness and Markedness, is in general an appropriate measure,
providing the number of classes match. To the extent that Bias tracks Prevalence,
Correlation = Informedness = Markedness is the probability of information flow in each
direction. To the extent that Bias is independent of Prevalence, the coefficient of
determination, Correlation² is the joint probability of informed determination in both
directions.
If unsupervised techniques such as clustering do not satisfy the constraint that the
number of categories on one side equals the number of classes on the other, then some
heuristic, e.g. a greedy approach to equate classes, can be applied to allow the use of
Informedness or Correlation8. For this case a variety of modifed or alternate techniques
are available.10
ROC
PRA
PRF
PRG
Figure 3. Comparison of Receiver Operating Characteristics (ROC) equal Informedness isobars with
Precision-Recall (PR) with Arithmetic (A), Harmonic (F) and Geometric (G) Mean isobars using MultiClassifier Fusion for Facial Expression Recognition on the Cohn-Kanade image set12. True [Positive]
Rate results are shown for anger, disgust, fear, happiness, sadness and surprise based on 10-fold
Cross Validation with the key showing number of images per real class. Blue isobars are equal cost
(default Recall=Fallout+constant in ROC) or equal score (X-am(Recall,Fallout)=constant in PRX).
Red break-even lines also shown in the PRX curves, corresponding to Bias=Prevalence and
Informedness=Kappa=Correlation and Precision=Recall=Accuracy and is an appropriate constraint
when variance (noise) is similar for both positives and negatives near the decision boundary).
The appropriate linear, logarithmic or reciprocal scalings are used to permit isobars to be linear:
X-axis is Fallout (FRemo) for ROC and Precision (PRemo) for PRA (linear scale) and PRG (log scale);
and Y-axis is reciprocal of Recall (TRemo) and X-axis reciprocal of Precision (PRemo) for PRF.
Note that we 6 classes, so 1/6 or 16.7% is chance Recall and Precision, without distributional data.
Therefore for PRF, the axes representing reciprocal of Recall and Precision truncate at 2x this level,
and K=6F represents that we are doing Kx better relative to this naive 1/6 baseline chance level.
The average angle of transition from predicting positives to predicting negatives at the threshold
approximates that of the default weighting, for all of the measures, across all six of the curves.
The ROC curve is smoother and thus tuning for costs seems more appropriate than for any PRX.
The sharper the elbow, the less tuning the β for F or the costs for ROC will affect the optimum.
References
(seminal
and
culminal)
BS ISO 5725-1: "Accuracy (trueness and precision) of measurement methods and reults - Part 1: General
principles and definitions." (1994)
2 C.J. van Rijsbergen (1979) Information Retrieval, 2nd ed., Butterworths, or the original research paper
C.J. van Rijsbergen (1974), 'Foundations of evaluation', Journal of Documentation, 30, 365-373.
3 L.R. Dice (1945). Measures of the Amount of Ecologic Association Between Species. Ecology 26 (3): 297–
302.
4 P. Jaccard (1901), "Étude comparative de la distribution florale dans une portion des Alpes et des Jura",
Bulletin de la Société Vaudoise des Sciences Naturelles 37: 547–579. or in English (1912), "The
distribution of the flora in the alpine zone", New Phytologist 11: 37–50.
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REtrieval Conference (TREC) http://trec.nist.gov retrieved Mon 9 June 2014
6 J. Entwisle, D.M.W. Powers (1998), The present use of statistics in the evaluation of NLP parsers, Joint
Conferences on New Methods in Language Processing and Computational Natural Language Learning,
pp. 215-214.
7 D.M.W. Powers (2012), The problem with kappa, Conference of the European Chapter of the Association
for Computational Linguistics, pp.345-355 and the related paper focussed on area under the curve
D.M.W. Powers (2012), International Conference on Information Science and Technology, pp. 567-573.
8 D.M.W. Powers (2003), Recall & Precision versus The Bookmaker, International Conference on Cognitive
Science, pp.529-534, or in expanded form as Flinders University technical report SIE-07-001 or full paper
D.M.W. Powers (2011), Evaluation: from precision, recall and F-measure to ROC, informedness,
markedness & correlation, Journal of Machine Learning Technologies 2(1):37-63.
9 N. Lavrac, P.A. Flach, and B. Zupan (1999). Rule evaluation measures: A unifying view.
International Workshop on Inductive Logic Programming, pp. 174–185, or in a more relevant context,
P.A. Flach (2003). The Geometry of ROC Space: Understanding Machine Learning Metrics through ROC
Isometrics International Conference on Machine Learning, pp. 226-233.
10 D. Pfitzner, R.E. Leibbrandt and D.M.W. Powers (2009), Characterization and evaluation of similarity
measures for pairs of clusterings, Knowledge and Information Systems 19(3):361-394
11 J. Davis and M. Goadric (2006) The Relationship Between Precision-Recall and ROC Curves, ICML.
12 X.B. Jia, Y.H. Zhang, D.M.W. Powers and H.B. Ali (2014), Multi-classifier fusion based facial expression
recognition approach, KSII Transactions on Internet & Information Systems 8(1):196-212. a b
1
| 9 |
1
Trajectory Optimization for Completion Time
Minimization in UAV-Enabled Multicasting
arXiv:1708.06478v1 [cs.IT] 22 Aug 2017
Yong Zeng, Member, IEEE, Xiaoli Xu, and Rui Zhang, Fellow, IEEE
Abstract—This paper studies an unmanned aerial vehicle
(UAV)-enabled multicasting system, where a UAV is dispatched
to disseminate a common file to a number of geographically
distributed ground terminals (GTs). Our objective is to design
the UAV trajectory to minimize its mission completion time, while
ensuring that each GT is able to successfully recover the file with
a high probability required. We consider the use of practical
random linear network coding (RLNC) for UAV multicasting, so
that each GT is able to recover the file as long as it receives
a sufficiently large number of coded packets. However, the
formulated UAV trajectory optimization problem is non-convex
and difficult to be directly solved. To tackle this issue, we first
derive an analytical lower bound for the success probability of
each GT’s file recovery. Based on this result, we then reformulate
the problem into a more tractable form, where the UAV trajectory
only needs to be designed to meet a set of constraints each on the
minimum connection time with a GT, during which their distance
is below a designed threshold. We show that the optimal UAV
trajectory only needs to constitute connected line segments, thus it
can be obtained by determining first the optimal set of waypoints
and then UAV speed along the lines connecting the waypoints.
We propose practical schemes for the waypoints design based on
a novel concept of virtual base station (VBS) placement and by
applying convex optimization techniques. Furthermore, for given
set of waypoints, we obtain the optimal UAV speed over the
resulting path efficiently by solving a linear programming (LP)
problem. Numerical results show that the proposed UAV-enabled
multicasting with optimized trajectory design achieves significant
performance gains as compared to benchmark schemes.
Index Terms—UAV communication, multicasting, trajectory
optimization, network coding.
I. I NTRODUCTION
Wireless communication systems have gradually evolved
to aim not only for high throughput, but also for ultrareliability, low energy consumption, and supporting highly
diversified applications with heterogeneous quality-of-service
(QoS) requirements [1]. To this end, research efforts in
the past have mainly focused on conventional networking
architectures typically with fixed infrastructures such as
ground base stations (BSs), access points, and relays, which
fundamentally limit their capability to meet the increasingly
multifarious service requirements cost-effectively. To address
this issue, there have been growing interests in providing
wireless connectivity from the sky, by utilizing various
airborne platforms such as balloons [2], helikites [3], and
Y. Zeng and R. Zhang are with the Department of Electrical and Computer
Engineering, National University of Singapore, Singapore 117583 (e-mail:
{elezeng, elezhang}@nus.edu.sg). X. Xu is with the School of Electrical
and Electronic Engineering, Nanyang Technological University, Singapore
639801. (email: xu0002li@e.ntu.edu.sg).
Part of this work has been submitted to the IEEE Wireless Communications
and Networking Conference (WCNC), Barcelona, Spain, April 15-18, 2018.
unmanned aerial vehicles (UAVs) [4], [5]. In particular,
wireless communications by leveraging the use of low-altitude
UAVs (typically at altitude within one kilometer above the
ground) are appealing due to their many advantages, such as
the ability of on-demand and swift deployment, high flexibility
with fully-controllable mobility, and high probability of having
line-of-sight (LoS) communication links with the ground
terminals (GTs) [5]. Therefore, with the continuous cost
reduction and endurance improvement of UAVs, together with
the device miniaturization of communication equipment, it is
anticipated that UAV-enabled communications will play an
increasingly more important role in future wireless systems.
Depending on the practical applications, UAVs in wireless
communication systems could either be deployed quasistationarily at predetermined locations, or fly contiguously
over the served GTs by following certain trajectories. In
the former case, one typical application is UAV-enabled
ubiquitous coverage, where UAVs are deployed to assist the
existing ground BSs, if any, to ensure seamless wireless
coverage for the GTs within a service area [6], [7]. In
this case, the UAVs resemble all essential functionalities of
the conventional terrestrial BSs, but typically at a much
higher altitude. Some practical scenarios for this application
include UAV-enabled offloading in hot spot areas and fast
communication service recovery after natural disasters. Along
this direction, significant research efforts have been devoted
to optimizing the UAV placement in two dimensional (2D)
or 3D space [8]–[13], by exploiting the unique channel
characteristics of the UAV-ground links. On the other hand,
in the case with flying UAVs for applications such as UAVenabled mobile relaying [14] and UAV-enabled information
dissemination or data collection [15], the fully controllable
mobility of UAVs offers new degrees of freedom in the
system design. This can help to significantly enhance the
performance compared to conventional systems with fixed
relays/BSs on the ground, by dynamically adjusting the UAV
positions according to the locations of the served GTs and
their communication requirements [5]. For instance, for UAVenabled data collection in Internet of Things (IoT) [16] and
machine type communications, the UAV can fly close to each
of the GTs sequentially so as to shorten their link distance
for more energy-efficient data gathering [15], [17]. For such
applications, the system performance critically depends on the
UAV trajectories, which need to be optimally designed.
Trajectory design or path planning has been a major
research area in the existing literature on UAVs. However,
prior works mainly focus on UAV navigation applications to
ensure its safe fly between a pair of predetermined initial
2
v
Fig. 1: UAV-enabled information multicasting.
and final locations, under various practical constraints such as
collision avoidance with other UAVs and/or terrain obstacles
[18]–[21]. There have been a handful of works recently
on the UAV trajectory design dedicated to optimizing the
communication performance. For example, by assuming that
the UAV is equipped with multiple antennas and flies with
a constant speed, the authors in [22] proposed an algorithm
to dynamically adjust the UAV’s heading to maximize the
ergodic sum rate of the uplink communications from the
GTs to the UAV. In [14], for UAV-enabled mobile relaying
systems, a design framework for jointly optimizing the
communication power/rate allocation and the UAV trajectory,
including both the flying direction and speed, was proposed
to maximize the communication throughput. For the nonconvex UAV trajectory optimization, [14] proposed the use
of successive convex optimization technique to find efficient
suboptimal solutions. This technique has been later adopted
for UAV trajectory optimization in various other setups,
including the energy efficiency maximization for UAV-enabled
communication [23], throughput maximization for UAVenabled multi-user downlink communication [24], and sensor
energy minimization in UAV-enabled data collection [15].
In this paper, we study a new UAV-enabled multicasting
system as shown in Fig. 1, where a UAV is dispatched
to disseminate a common file to a set of geographically
distributed GTs [25]. UAV-enabled information dissemination
or multicasting is one important use case of UAV-enabled
communication systems [5], with a variety of applications
such as for public safety and emergency responses [26], video
streaming [27], [28], and intelligent transportation systems
[29]. Different from the conventional multicasting with static
transmitters (e.g., terrestrial BSs), where the multicasting
performance is fundamentally limited by the bottleneck link
of the user that is most far away from the transmitter,
UAV-enabled multicasting is able to overcome this issue by
exploiting its high mobility via adaptive trajectory design,
which is the main focus of this work.
Specifically, under a general flat-fading channel model
between the UAV and GTs, our objective is to design the
UAV trajectory to minimize its mission completion time,
while ensuring that each GT is able to recover the file
with a success probability no smaller than a given target.
Mission completion time minimization is a desirable goal
in practice due to the limited UAV on-board energy and
hence endurance time. We consider the use of random linear
network coding (RLNC) [30] for UAV multicasting, since it
is known to be a robust practical coding technique for such
applications with random packet erasures and without the need
of dedicated receiver feedback for ARQ (Automatic Repeat
reQuest). With RLNC, each GT is able to successfully recover
the file as long as it can reliably receive a sufficiently large
number of coded packets, whose probability critically depends
on the UAV trajectory design. Due to the fundamentally
different setups and design objectives, existing UAV trajectory
designs (in e.g., [14], [23]), which are typically for throughput
maximization with independent messages for the GTs under
a given mission time constraint, are no longer applicable for
the new problem considered in this paper, thus calling for new
problem formulation and solutions. The main contributions of
this paper are summarized as follows.
First, for UAV-enabled multicasting systems with RLNC,
we formulate the optimization problem to minimize the
mission completion time, while ensuring that each GT
is able to successfully recover the file with a targeting
probability, subject to the UAV’s maximum speed constraint.
The formulated problem is difficult to be directly solved, since
the file recovery probability of each GT is a complicated
function of the UAV trajectory. To tackle this issue, we derive
an analytical lower bound for the file recovery probability of
each GT by introducing an auxiliary distance parameter D.
The main idea is to ignore the portion of the UAV flight
time during which the horizontal distance with each GT of
interest is greater than D, hence incurring relatively higher
packet loss probabilities than a threshold value (specified by
D). As a result, the UAV trajectory design is reformulated to
meet a corresponding constraint on the minimum connection
time with each GT, during which their distance is below the
critical distance D.
Next, we show that for the reformulated problem, the
optimal UAV trajectory only needs to constitute connected
line segments. Thus, the problem is further reduced to finding
a set of optimal waypoints for the UAV trajectory, and
then optimizing the instantaneous UAV speed along the lines
connecting these waypoints. However, finding the optimal
waypoints is challenging since it is a generalization of the
classic Travelling Salesman Problem (TSP) [31]–[33], which
is known to be NP hard. We thus propose two practical
waypoints design schemes based on a novel concept of
virtual base station (VBS) placement and by applying convex
optimization techniques. Furthermore, for a given waypoint
design, we obtain the optimal UAV speed over the resulting
path efficiently by solving a linear programming (LP) problem.
Finally, numerical results are provided to validate the
performance of the proposed designs. It is shown that
compared to the heuristic benchmark waypoint designs,
the proposed designs can significantly reduce the required
mission completion time. Furthermore, as compared to the
conventional multicasting setup with a static transmitter, the
proposed UAV-enabled multicasting with optimized trajectory
achieves significant performance gains in terms of file recovery
probability and/or mission completion time. This demonstrates
the great potential of UAV-enabled information multicasting in
3
future wireless systems.
The rest of this paper is organized as follows. Section II
presents the system model and problem formulation. In
Section III, the lower bound of the file recovery probability
is derived, based on which the optimization problem is
reformulated. In Section IV, the proposed UAV trajectory
designs are presented. Section V provides the numerical
results, and finally we conclude the paper in Section VI.
Notations: In this paper, scalars are denoted by italic letters.
Boldface lower-case letters denote vectors. RM×1 denotes the
space of M -dimensional real-valued vectors. For a vector
a, kak represents its Euclidean norm. log2 (·) denotes the
logarithm with base 2. E[·] denotes the statistical expectation
and Pr(·) represents the probability. Bern(p) represents the
Bernoulli distribution with success probability p, B(N, p)
denotes the binomial distribution with N independent trials
each with success probability p, and N (µ, v 2 ) denotes the
Gaussian distribution with mean µ and variance v 2 . For a timedependent function q(t), q̇(t) denotes the first-order derivative
with respect to time t. For a set M, |M| denotes its cardinality.
For two sets M1 and M2 , M1 ⊂ M2 denotes that M1 is a
subset of M2 .
II. S YSTEM M ODEL AND P ROBLEM F ORMULATION
As shown in Fig. 1, we consider a wireless communication
system consisting of K GTs denoted by the set K =
{1, · · · , K}, with the location of GT k denoted as wk ∈ R2×1 ,
k ∈ K. The GTs’ locations are assumed to be known for the
UAV trajectory design. A UAV flying at a constant altitude
H above the ground is dispatched to disseminate a common
information file of total size W bits to all the K GTs. Note
that in practice, H could correspond to the minimum altitude
to ensure safe UAV operations, e.g., for obstacle avoidance
without frequent aircraft ascending or descending.
A. Random Linear Network Coding
We assume that RLNC [30] is employed for the UAV
transmission, where the information file is linearly coded in
the packet level. Specifically, denote the size of each packet
as Rp bits/pakcet. Then the total number of information
packets is N ′ = W/Rp , which are linearly combined with
randomly generated coding coefficients from a finite field
to obtain N > N ′ coded packets.1 These coded packets
are then broadcasted by the UAV’s transmitter to the GTs
along its flight trajectory. As the randomly generated coding
coefficients in RLNC are linearly independent almost surely
for a sufficiently large field size, each GT will be able to
recover the information file as long as any N ′ out of the N
coded packets are successfully received. Note that for assisting
the file recovery based on the network coded packets at the
GTs, only the seeds used for generating the random coding
coefficients need to be appended with the payload of each
packet, and hence the network coding overhead is negligible.
Denote by R the transmission rate in bits/second (bps),
which is assumed to be predetermined and remain constant.
1 For
convenience, we assume that W is an integer multiple of Rp .
Then the time required to complete one packet transmission
is Tp = Rp /R in second. As a result, the mission completion
time, or the total time required to complete the transmission
of the N coded packets, is given by
W N
.
(1)
T = N Tp =
R N′
B. Channel Model
Within the mission completion time, denote by q(t) ∈
R2×1 , 0 ≤ t ≤ T , the UAV’s flying trajectory projected
onto the ground. Further denote by Vmax the maximum UAV
speed in meter/second (m/s). We then have the constraint
kq̇(t)k ≤ Vmax , ∀t. The time-dependent distance between the
UAV and the GTs can then be written as
p
dk (t) = H 2 + kq(t) − wk k2 , 0 ≤ t ≤ T, ∀k ∈ K. (2)
For a general flat-fading channel model for the UAV-toGT links, with the N coded packets transmitted by the UAV
during the time horizon T , the probability that each of the
K GTs reliably receives at least N ′ packets to successfully
recover the information file critically depends on the UAV’s
trajectory q(t), 0 ≤ t ≤ T . Our objective in this paper
is to optimize q(t) so as to minimize the total mission
completion time T , or equivalently the total number of coded
packets N that need to be transmitted, while ensuring that
each of the K GTs is able to recover the information file
with a success probability no smaller than a given target P̄ .
Note that for practical information multicasting systems, a
subsequent device-to-device (D2D) packet sharing phase could
be employed, so that those GTs who fail to recover the file will
receive additional packets from their peers until they can also
successfully recover the file [5]. By increasing the targeting
threshold P̄ for the UAV multicasting phase, in general less
packets need to be shared in the D2D phase. In this paper,
we focus on the UAV multicasting phase, whereas a joint
investigation of the UAV multicasting and D2D file sharing
would be an interesting problem for future research.
For the ease of exposition, the time horizon T is discretized
into M equally spaced time slots, i.e., T = M δt , with δt
denoting the elemental slot length, which is appropriately
chosen so that the distance between the UAV and the GTs can
be assumed to be approximately constant within each slot.
For instance, δt might be chosen such that δt Vmax ≪ H.
Thus, the UAV trajectory q(t) over the time horizon T can be
approximated by the M -length sequence {q[m]}M
m=1 , where
q[m] , q(mδt ) denotes the UAV’s horizontal location at
time slot m. Furthermore, the UAV speed constraint can be
expressed as
q[m] − q[m − 1] ≤ Ṽmax , δt Vmax , m = 2, · · · , M. (3)
The distance between the UAV and the GTs in (2) can be
discretized as
p
dk [m] = H 2 + kq[m] − wk k2 , 1 ≤ m ≤ M, k ∈ K. (4)
The average channel power gain from the UAV to GT k at
slot m can be modeled as
β0
βk [m] = β0 d−α
,
(5)
k [m] =
(H 2 + kq[m] − wk k2 )α/2
4
TABLE I: List of parameters.
Information file size
Packet size
Number of information packets
Number of network coded packets
UAV transmission rate
Time for transmitting one packet
Mission completion time
Time slot length
Number of time slots
Number of transmitted packets per
slot
W bits
Rp bits
N ′ = W/Rp
N > N′
R bits/second
Tp = Rp /R seconds
T = N Tp = W
R
seconds
δt seconds
M = T /δt
L = δt /Tp = N/M
Thus, the probability that GT k can successfully receive the
(m, l)th packet can be expressed as
N
N′
where β0 denotes the channel power gain at the reference
distance of d0 = 1m, and α ≥ 2 is the path loss exponent.
With the slot duration fixed to δt , the number of packets
that can be transmitted by the UAV during each time slot
is L = δt /Tp = Rδt /Rp . For convenience, we assume that
L ≥ 1 is an integer. It then follows that the total number
of transmitted packets by the UAV is related to M as N =
M L. The relationship between the different parameters of the
considered system is summarized in Table I.
We assume quasi-static fading channels, where the
instantaneous channel coefficients between the UAV and GTs
remain unchanged for each packet duration of Tp seconds, and
may vary across different packets. Therefore, the instantaneous
channel gains between the UAV and GT k can be modeled as
p
hk [m, l] = βk [m]gk [m, l], m = 1, · · · , M, l = 1, · · · , L,
(6)
where hk [m, l] denotes the channel coefficient between the
UAV and GT k during the transmission of the lth packet
in time slot m, βk [m] is the large-scale channel coefficient
that depends on the distance between the UAV and GT
k as given in (5), and gk [m, l] is a random variable with
E[|gk [m, l]|2 ] = 1 accounting for the small-scale fading of
the UAV-to-GT channel, which is independent and identically
distributed (i.i.d.) for different k, m, l. Note that (6) includes
the LoS UAV-GT channel as a special case, for which gk [m, l]
is deterministic with unit magnitude, i.e., |gk [m, l]| = 1.
C. Problem Formulation
Denote by P the transmission power of the UAV. The
achievable rate in bps between the UAV and GT k during
the transmission of the (m, l)th packet is given by
!
2
P hk [m, l]
Ck [m, l] = B log2 1 +
σ2 Γ
!
2
P βk [m] gk [m, l]
,
(7)
= B log2 1 +
σ2 Γ
where B denotes the channel bandwidth in Hertz (Hz), σ 2
represents the power of the additive white Gaussian noise
(AWGN) at the GT receivers, and Γ > 1 is the signalto-noise ratio (SNR) gap between the practical modulation
schemes and the theoretical Gaussian signaling. With the
UAV’s transmission rate fixed to R, the (m, l)th packet can
be successfully received by GT k if and only if Ck [m, l] ≥ R.
pk [m, l] = Pr Ck [m, l] ≥ R
γth
2
2 α/2
2
= Pr gk [m, l]| ≥
H + kq[m] − wk k
γ̄0
α/2
γth
,
(8)
H 2 + kq[m] − wk k2
=F
γ̄0
where γth , 2R/B − 1 is the SNR threshold for successful
packet reception, γ̄0 , P β0 /(σ 2 Γ) is the average received
SNR at the reference distance of 1m, and F (x) denotes the
complementary cumulative distribution function (ccdf) of the
random variable |gk [m, l]|2 , which, by definition, is a nonincreasing function with respect to x for any given fading
distribution. We assume that F (x) is known in this paper.
Define a distance parameter D∗ for the UAV-GT horizontal
separation such that the average received SNR at D∗ equals
γth , i.e., the resulting argument in (8) equals 1, we then have
q
2/α
D = (γ̄0 /γ̄th )
− H 2.
∗
(9)
For the special case of deterministic LoS channel such that
|gk [m, l]| = 1, we have F (x) = 1 if x ≤ 1 and 0 otherwise.
In this case, we have pk [m, l] = 1 if kq[m] − wk k ≤ D∗
and 0 otherwise. In other words, for the special case of LoS
channel, a packet is guaranteed to be successfully received if
the UAV-GT horizontal distance is no greater than D∗ and it
will be lost otherwise. Note that for practical fading channels,
all the L packets transmitted by the UAV within the same time
slot experience i.i.d. fading for any given GT k and time slot
m, since its distance from the UAV is assumed to be constant
in each slot. Thus, pk [m, l] in (8) is independent of l but only
depends on the slot number m.
Let Zk [m, l], m = 1, ..., M, l = 1, ..., L, be a random
variable indicating whether the (m, l)th packet is successfully
received by GT k, which follows the Bernoulli distribution
with success probability pk [m, l], denoted as Zk [m, l] ∼
Bern(pk [m, l]). The total number of packets that can be
successfully received by GT k, denoted as Nk , is then a
random variable given by
Nk =
M X
L
X
m=1 l=1
Zk [m, l], k ∈ K.
(10)
Since Zk [m, l] are independent Bernoulli random variables
with possibly different success probabilities, Nk follows the
Poisson binomial distribution [34].
Recall that with N = M L network coded packets
transmitted by the UAV, each GT is able to recover the
information file as long as any N ′ out of the N packets are
successfully received, whose probability can be written as
Pk,succ , Pr Nk ≥ N ′ , k ∈ K.
(11)
5
Trajectory
D
GT k
pk [m, l ] ³ pD
Fig. 2: Illustration of the lower bound derivation for Pk,succ .
Thus, the problem to minimize the mission completion
time via trajectory optimization while ensuring a targeting file
recovery probability P̄ for all GTs can be formulated as
(P1) : min M
q[m],M
s.t. Pk,succ ≥ P̄ , ∀k ∈ K,
(12)
q[m] − q[m − 1] ≤ Ṽmax , m = 2, · · · , M.
(13)
III. L OWER B OUND OF Pk,succ AND P ROBLEM
R EFORMULATION
Problem (P1) is difficult to be directly solved. One major
difficulty lies in that the successful file recovery probability
Pk,succ in (11) is related to the UAV trajectory {q[m]}
in a rather implicit and complicated manner. In fact, even
with a given UAV trajectory, and hence with known success
probability pk [m, l] for each of the N = M L transmitted
packets, the complexity for evaluating the probability mass
function (pmf) of Nk is exponential with respect to N . This
makes it quite challenging to obtain the optimal solution
to (P1). In this paper, we propose an efficient approximate
solution to (P1). To this end, we first derive an analytical lower
bound for Pk,succ and transform the constraint (12) into a
more tractable form in terms of the minimum connection time
between the UAV and each GT, during which their distance is
below a certain threshold. We then propose effective trajectory
designs for the reformulated optimization problem.
with the UAV trajectory can be revealed more explicitly. As
illustrated in Fig. 2, the main idea is to introduce an auxiliary
distance parameter D, and ignore the portion of the UAV flight
time during which the horizontal distance with each GT of
interest is greater than D, hence incurring relatively higher
packet loss probabilities than a threshold value (specified by
D). Furthermore, for the considered time slots, the packet
success probabilities are guaranteed to be no smaller than that
corresponding to D, based on which a lower bound on the file
recovery probability can be obtained. The detailed derivations
are given as follows.
For any given auxiliary distance parameter D ≥ 0, denote
as pD the probability that a packet transmitted by the UAV is
successfully received by a GT that has a horizontal distance
D from the UAV. Based on (8), for any channel model with
known ccdf of the small-scale fading given by F (·), pD can
be expressed as
γth
2
2 α/2
.
(14)
H +D
pD = F
γ̄0
Furthermore, for any UAV trajectory {q[m]}M
m=1 , define the
set Mk,D ⊂ {1, · · · , M } for GT k as the subset of all time
slots such that the horizontal distance between the UAV and
GT k is no greater than D, i.e.,
Mk,D , {m : kq[m] − wk k ≤ D}.
(15)
For any given D, if m ∈ Mk,D , we deem that the UAV and
GT k are in connection at time slot m; otherwise, they are not
connected. Then the cardinality of Mk,D , denoted as |Mk,D |,
is referred to as the number of connection time slots between
UAV and GT k. Since F (·) is a non-increasing function by
definition, based on (8), the following inequality holds for any
given D,
pk [m, l] ≥ pD , ∀m ∈ Mk,D .
(16)
Theorem 1. For any given D ≥ 0, the successful file recovery
probability for GT k defined in (11) is lower-bounded as
Pk,succ ≥ Pk,lb , Pr N̂k ≥ N ′ ,
(17)
where N̂k ∼ B Mk,D L, pD is a binomial random variable
with |Mk,D |L independent trials each with success probability
pD .
Proof: To prove Theorem 1, we need the following result.
A. Lower Bound of Pk,succ
As can be seen from (8), (10) and (11), the successful file
recovery probability Pk,succ for each GT k is determined by
the pmf of Nk , which in turn implicitly depends on the UAV
trajectory q[m] via the successful packet reception probability
pk [m, l]. Due to UAV mobility, the packets transmitted by
the UAV in different time slots in general experience nonidentically distributed channels, i.e., pk [m, l] 6= pk [m′ , l],
m 6= m′ . This makes it challenging to find an explicit
expression for Pk,succ in terms of the UAV trajectory q[m] via
directly deriving the pmf of Nk in (10). To overcome this issue,
we derive a lower bound for Pk,succ in (11), whose relationship
Lemma 1. Let Xn ∼ Bern(pn ), n = 1, · · · , N , be
N independent Bernoulli random variables withP success
N
probability p1 , · · · , pN , respectively. Then X ,
n=1 Xn
follows a Poisson binomial distribution. Furthermore, let X̂
be a binomial random variable with X̂ ∼ B(N, p̂) whose
success probability satisfies p̂ ≤ pn , ∀n. Denote the ccdf of X
and X̂ as FX (x) , Pr(X ≥ x) and FX̂ (x) , Pr(X̂ ≥ x),
respectively. We then have
FX (x) ≥ FX̂ (x), x = 0, 1, · · · , N.
Proof: Please refer to Appendix B.
(18)
6
By substituting (10) into (11), Pk,succ can be expressed as
!
M X
L
X
′
Pk,succ , Pr
Zk [m, l] ≥ N
(19)
m=1 l=1
L
X X
≥ Pr
Zk [m, l] ≥ N ′
(20)
m∈Mk,D l=1
≥ Pr N̂k ≥ N ′ , Pk,lb ,
(21)
where (20) holds since Mk,D ⊂ {1, · · · , M } for any D ≥ 0,
and (21) is obtained by applying Lemma 1 together with the
inequality (16).
B. Problem Reformulation
With Theorem 1, for any chosen D, by replacing Pk,succ in
(12) with its lower bound Pk,lb , (P1) is recast into
(P2) : min M
min T = δt M
q[m],M
s.t. Pk,lb ≥ P̄ , ∀k ∈ K,
(22)
q[m] − q[m − 1] ≤ Ṽmax , m = 2, · · · , M.
(23)
Note that if (22) is satisfied, then (12) is guaranteed to be
satisfied as well due to the lower bound in (17), but the reverse
is not true in general. Therefore, for any given D, the optimal
objective value of (P2) provides an upper bound to that of
(P1). Thus, by solving (P2) for some appropriately chosen
values for D, (P1) can be approximately solved. As will be
discussed in Section V, one reasonable choice of D is given
by (9). In the following, we focus on solving (P2) for any
given value of D.
To obtain a more tractable form for the constraint (22), note
that with moderately large |Mk,D |L, the binomial random
variable B(|Mk,D |L, pD ) defined in Theorem 1 can be well
approximated by Gaussian random variable N (µ, v 2 ), where
µ = |Mk,D |LpD and v 2 = |Mk,D |LpD (1 − pD ). As a result,
the lower bound Pk,lb defined in (17) can be approximated as
!
N ′ − |Mk,D |LpD
,
(24)
Pk,lb ≈ Q p
|Mk,D |LpD (1 − pD )
R∞
2
where Q(x) , 0 e−u /2 du is the Gaussian Q-function.
Therefore, by substituting (24) into constraint (22) and solving
for |Mk,D |, we get
|Mk,D | ≥ Mmin , A2 /L,
where
1
A, √
2 pD
case when D is sufficiently small such that pD → 1. In this
case, it follows from (25) that we have Mmin = N ′ /L. In other
words, if D is small so that each packet transmitted by the
UAV can be successfully received almost surely by those GTs
in connection with the UAV, then the UAV only needs to stay
in connection with each GT for N ′ /L time slots to transmit
N ′ packets, as expected. On the other hand, if D is chosen to
be large such that pD → 0, it then follows from (25) and (26)
that we have Mmin ∝ 1/pD , i.e., the minimum number of
connection time slots Mmin increases inversely proportional
with pD .
Define the following indicator function
(
1, if kq[m] − wk k ≤ D,
Ik,D [m] =
(27)
0, otherwise.
PM
Then |Mk,D | =
m=1 Ik,D [m]. Therefore, (P2) can be
reformulated as
q[m],M
s.t. |Mk,D | ≥ Mmin, ∀k ∈ K,
(28)
q[m] − q[m − 1] ≤ Ṽmax , m = 2, · · · , M. (29)
When the time slot length δt is chosen to be sufficiently small,
then the above problem can be written in its continuous-time
format as
(P3) : min T
q(t),T
s.t. Tk,D ,
Z
0
T
Ik,D (t)dt ≥ Tmin , ∀k ∈ K
kq̇(t)k ≤ Vmax , 0 ≤ t ≤ T,
where Tmin , Mmin δt and
(
1, if kq(t) − wk k ≤ D,
Ik,D (t) =
0, otherwise.
(30)
(31)
(32)
In the next section, we focus on solving the trajectory
optimization problem (P3).
IV. P ROPOSED T RAJECTORY D ESIGN
The main challenge for optimally solving (P3) lies in the
non-convex constraint (30), which involves time-dependent
indicator functions (32) in terms of the UAV trajectory. To
solve (P3), we first show the following result.
(25)
Theorem 2. Without loss of optimality to (P3), the UAV
trajectory can be assumed to constitute only connected line
q
segments.
p
−1
′
−1
2
4N + (1 − pD )(Q (P̄ )) − Q (P̄ ) 1 − pD , Proof: Please refer to Appendix C.
(26)
with Q−1 (·) denoting the inverse Gaussian Q-function.
In other words, for any given D, the constraint (22) on the
success file recovery probability is equivalent to the constraint
that the number of connection time slots |Mk,D | between the
UAV and each GT should be no smaller than the minimum
threshold Mmin, where Mmin is a constant determined by pD ,
P̄ and N ′ . To gain more insights for (25), consider the special
Theorem 2 implies that finding the optimal solution to (P3)
is equivalent to finding the optimal set of ordered waypoints
Qwp , which contains the locations representing the starting
and ending points of each line segment, as well as optimizing
the instantaneous UAV speed along the path connecting the
waypoints. However, finding the optimal set of waypoints
Qwp is a challenging problem in general. In fact, for the
extreme case when D = 0, the constraint (30) reduces to that
7
the UAV needs to sequentially visit all the K GTs and stay
stationary on top of each for at least Tmin seconds. In this case,
finding the optimal waypoints to (P3) reduces to determining
the visiting order of all the K GTs so as to minimize the
total UAV travelling distance, which is essentially equivalent
to the classic TSP [31]–[33]. The only difference is that
different from the standard TSP, the traveller/UAV in our
considered problem does not need to return to the origin
where it starts the tour. Note that TSP is an NP-hard problem
in combinatorial optimization. However, various heuristic and
high-quality approximation algorithms have been developed.
A brief overview on TSP and its variations are given in
Appendix A. On the other hand, for the general case with D >
0, (P3) seems to be similar to the TSP with neighborhoods
(TSPN) [35]. However, as existing algorithms for TSPN such
as [36] assume that the neighborhoods are disjoint disks and
do not have the minimum connection time constraints, they
cannot be directly applied for solving problem (P3). In the
following, for (P3) with the general D ≥ 0, we will first
present a simple benchmark scheme by taking the GTs as the
waypoints, and then propose two more efficient schemes for
waypoints design based on a novel concept of VBS placement
and by applying convex optimization techniques. Furthermore,
for any given waypoints design, the optimal UAV speed over
time will be efficiently obtained via solving an LP problem.
A. Waypoint Design
(1) Scheme 1 (benchmark): GTs as Waypoints. Note that
a feasible UAV trajectory to (P3) needs to ensure that the
minimum connection time constraints in (30) are satisfied with
the designed waypoints. For any D ≥ 0, one straightforward
approach to ensure the feasibility of (30) is to let the
UAV sequentially visit (i.e., stay on top of) all GTs. More
specifically, Qwp is determined by simply applying the TSP
algorithm over all the K GTs (without the need of returning
to the origin as discussed in Appendix A). In this case, since
each GT is guaranteed to be in connection with the UAV when
it is just above the GT, the constraints in (30) can be met
by appropriate UAV speed optimization, as will be studied in
Section IV-B.
(2) Scheme 2 (proposed): VBSs as Waypoints. It is intuitive
to see that for a given D > 0, it is in general unnecessary
for the UAV to fly over all the GTs since at one location,
the UAV could be in connection with more than one GTs
simultaneously. Thus, the number of waypoints that the UAV
needs to visit to ensure the feasibility of (30) could be much
less than K, especially when D is large and the GTs are
densely distributed. Therefore, in this subsection, we propose
an alternative waypoint design based on a new idea of VBS
placement.
Specifically, given the GT locations {wk } and the UAV
threshold coverage range D, the VBS placement problem aims
to find a minimum number of VBSs and their respective
locations, so that each GT is covered by at least one VBS.
This problem resembles the standard BS placement problem
for ensuring user coverage with a given coverage distance
D, where several efficient algorithms have been proposed,
D
GT1
VBS1
D
VBSs as waypoints
Alternative waypoints
GT2
VBS2
Fig. 3: A toy example for illustrating the inefficacy of directly
using VBSs as waypoints.
such as the spiral BS placement algorithm proposed in [11].
Let G ≤ K be the minimum number of VBSs obtained by
applying the BS placement algorithm, and their locations are
denoted as vg ∈ R2×1 , g = 1, · · · , G. An efficient waypoint
design to ensure the feasibility of (30) is to let the UAV
sequentially visit these VBSs by following the path obtained
by the TSP algorithm applied over {vg }G
g=1 . In this case, the
number of waypoints that the UAV needs to travel is G, which
is in general less than K.
(3) Scheme 3 (proposed): Waypoints Based on VBS
Placement and Convex Optimization. Traversing over all the
G VBSs, though providing a feasible waypoints design to
(P3), may not always be desirable. This is illustrated by
a toy example shown in Fig. 3, where there are two GTs,
each covered by one VBS that is placed in essentially the
same location as the GT. It is observed that traversing over
both VBSs in fact leads to unnecessarily longer trajectory
than the alternative design shown in Fig. 3. To overcome
this limitation, in this subsection, we propose a more efficient
waypoint design based on the placed VBSs and by applying
convex optimization techniques.
Specifically, with VBS placement and TSP algorithm
applied over the obtained G VBSs, the GTs in K are essentially
partitioned into G ordered clusters Sg , g = 1, · · · , G, where
Sg ⊂ K denotes the subset of GTs that are covered by the gth
VBS while applying the VBS placement algorithm. For the
gth ordered cluster with GTs Sg , define the following set
Cg , {q ∈ R2×1 : kq − wk k ≤ D, ∀k ∈ Sg }.
(33)
In other words, Cg is the set of all possible UAV locations
ensuring that all GTs in Sg are simultaneously in connection
with the UAV. It is obvious that Cg is non-empty (since the
VBS g with location vg belongs to this set) and a convex set
(since it is an intersection of |Sg | convex sets). As a result, as
long as the UAV sequentially visits all the G convex regions
Cg , the constraints in (30) can be met by appropriate UAV
speed optimization. In the following, the waypoints in each of
the convex region Cg is optimized.
Without loss of generality, let sg , fg ∈ Cg be the starting
and ending points of the UAV trajectory intersecting with the
region Cg , respectively. Note that since Cg is a convex set, all
points on the line segment between sg and fg are also in Cg ,
i.e., they ensure that all the GTs in Sg are in connection with
the UAV. Given the UAV’s maximum flying speed Vmax , the
minimum time required for the UAV to travel within the region
kf −s k
Cg , i.e., from sg to fg , is Vgmaxg . On the other hand, to ensure
the minimum connection time constraint (30), one viable
approach is to ensure that the UAV remains in Cg for at least
8
Tmin seconds. Thus, the minimum
time required
for the UAV
o
n
kfg −sg k
.
Furthermore,
the
,
T
to travel within Cg is max
min
Vmax
minimum time required for the UAV to travel between Cg
ks
−fg k
and Cg+1 is g+1
. As a result, the waypoints {sg , fg }G
g=1
Vmax
could be designed by solving the following problem
G
X
kfg − sg k
(P4) : min
max
, Tmin +
Vmax
{sg ,fg }G
g=1 g=1
G−1
X
g=1
ksg+1 − fg k
Vmax
s.t. sg , fg ∈ Cg , ∀g.
(34)
Note that the cost function of (P4) is the total mission
completion time with waypoints {sg , fg }, which is a convex
function with respect to {sg , fg }. Furthermore, all the
constraints in (P4) are convex. Thus, (P4) is a convex
optimization problem, which can be efficiently solved by
standard convex optimization techniques or existing software
such as CVX [37].
Note that as compared to the previous scheme by directly
taking the VBSs as waypoints, the new waypoints obtained in
(P4) avoid the unnecessary traveling to the VBSs, and thus are
expected to achieve better performance, as will be numerically
verified in Section V.
B. UAV Speed Optimization
For any given set of feasible waypoints Qwp , the UAV
path is determined by sequentially connecting the waypoints
Qwp with line segments. As a result, problem (P3) reduces to
finding the optimal instantaneous UAV speed over time along
the path connecting these waypoints. To this end, we discretize
the UAV path with the infinitesimal displacement δd (instead
of over time) to get J UAV sampled locations on the path,
denoted by {qj }Jj=1 . As a result, the corresponding value of
the indicator function in (32) can be obtained, which is denoted
as Ikj , k ∈ K, j = 1, · · · , J. That is, Ikj = 1 represents that
the UAV is in connection with GT k when it is at location j.
Denote by τj ≥ 0 the time for the UAV to travel from location
qj to qj+1 , with the speed Vj = δτdj . Note that since δd is
set sufficiently small, Vj well approximates the instantaneous
UAV speed, and we must have δτdj ≤ Vmax . For any given set
of feasible waypoints, (P3) reduces to optimizing the UAV
speed Vj or equivalently the time duration τj , j = 1, · · · , J,
which is formulated as
(P5) : min
{τj }
s.t.
J
X
τj
j=1
J
X
j=1
Ikj τj ≥ Tmin, ∀k ∈ K,
τj ≥
δd
Vmax
, j = 1, · · · , J.
(35)
(36)
Note that (P5) is feasible if and only if ∀k ∈ K, there exists at
least one UAV location j such that Ikj = 1. This is guaranteed
based on the three waypoint designs presented in Section IV-A.
TABLE II: System setup for numerical simulations.
UAV altitude
Maximum UAV speed
UAV transmission power
Bandwidth
Noise power
SNR gap
Information file size
Packet size
Minimum number of packets
required for file recovery
UAV transmission rate
Time for transmitting one packet
Time slot length
Number of transmitted packets per
slot
Channel gain at reference distance
Path loss exponent
Rician factor
Target probability for file recovery
H = 100m
Vmax = 50m/s
P = 10dBm
B = 1 MHz
σ2 = −109dBm
Γ = 10 dB
W = 2 Mbits
Rp = 104 bits/packet
N ′ = 200
R = 1 Mbits/second
Tp = 0.01 seconds
δt = 0.1 seconds
L = 10
β0 = −40 dB
α = 2.6
Kc = 2
P̄ = 0.9
(P5) is a standard LP problem, which can be efficiently solved
via e.g. [37].
V. N UMERICAL R ESULTS
In this section, numerical results are provided to evaluate the
performance of our proposed trajectory designs. We assume
that the K GTs are randomly and uniformly distributed in a
square area of side length equal to 3000m. For UAV-to-ground
channels, we adopt the practical Rician fading channel model,
which is characterized by the Rician factor Kc representing the
power ratio between the LoS signal component to the scattered
component. In this case, the small-scale fading coefficients
gk [m, l] in (6) can be explicitly modeled as
r
r
Kc
1
ḡ +
g̃
(37)
gk [m, l] =
Kc + 1
Kc + 1
s
p
√
1
=
(38)
2Kc ḡ + 2g̃ ,
2(Kc + 1) |
{z
}
Y
where ḡ denotes the deterministic LoS channel component
with |ḡ| = 1, and g̃ represents the random scattered
component, which is a zero-mean unit-variance circularly
symmetric complex Gaussian (CSCG) random variable. With
Y defined in (38), |Y |2 follows the non-central chi-square
distribution with two degrees of freedom (DoF) and noncentrality parameter λ = 2Kc, denoted as |Y |2 ∼ χ′2
2 (2Kc ).
Thus, the ccdf of |gk [m, l]|2 in (8) can be explicitly written as
F (z) , Pr(|gk [m, l]|2 ≥ z) = Pr |Y |2 ≥ 2(Kc + 1)z
p
p
2Kc , 2(Kc + 1)z ,
(39)
= Q1
where Q1 (a, b) is the standard Marcum-Q-function. Unless
otherwise stated, the numerical setup of the following
simulations is given in Table II.
For the proposed waypoint designs with VBSs placement,
we use the spiral BS placement algorithm proposed in [11] to
obtain the VBSs. Furthermore, since the TSP problem involved
2500
2500
2000
2000
y (m)
y (m)
9
1500
1500
1000
1000
500
500
0
500
1000
1500
2000
2500
3000
0
500
1000
x (m)
2500
3000
(b) GTs as waypoints, dtr = 14.8km, T = 295.9s.
3000
3000
2500
2500
2000
2000
y (m)
y (m)
2000
x (m)
(a) Strip-based waypoints, dtr = 13.9km, T = 279.0s.
1500
1500
1000
1000
500
500
0
-500
1500
0
0
500
1000
1500
2000
2500
3000
3500
x (m)
(c) VBSs as waypoints, dtr = 8.7km, T = 186.3s.
-500
0
500
1000
1500
2000
2500
3000
3500
x (m)
(d) Optimized waypoints, dtr = 8.1km, T = 173.5s.
Fig. 4: Comparison of the UAV trajectories with different waypoint designs. Small circles denote GTs and squares represent VBSs.
in our design does not require the UAV to return to the starting
point, we apply the strategy by adding a dummy node as
described in Appendix A. The resulting TSP is solved by
using the existing Matlab codes available in [33]. Note that
by applying the corresponding TSP variations as discussed
in Appendix A, our proposed UAV trajectory design can be
directly applied to the case when the UAV’s initial and/or final
locations are predetermined. Such extensions are omitted for
brevity. Besides the three waypoint design schemes presented
in Section IV-A, we also consider another benchmark scheme,
called “strip-based waypoints”, where the UAV’s trajectory is
designed to ensure full area coverage. Specifically, for any
given realization of the GT locations and chosen distance
parameter D, the UAV first obtains the smallest rectangle that
contains all the K GTs, and then partitions this rectangular
area into rectangular strips each with width 2D. The UAV
then sequentially travels along the center of the rectangular
strips, as shown in Fig. 4(a). Note that such a trajectory
design ensures that all locations within the rectangular area
are covered by the UAV. For all the four trajectory design
schemes, the UAV’s instantaneous speed is optimized based
on the LP problem (P5), given their respective waypoints.
A. Trajectory Comparison and Lower Bound Verification
By choosing the auxiliary distance parameter as D = 400m,
Fig. 4 compares the different UAV trajectories with the four
considered waypoint designs for one specific realization of
the GT locations with K = 50. The corresponding total UAV
traveling distances dtr and the mission completion time T
are also shown in the figure. It is observed that for both
benchmark schemes with strip-based waypoints and GTs as
waypoints, the UAV needs to travel longer distances and hence
require larger mission completion time, as compared to the
proposed designs as shown in Fig. 4(c) and Fig. 4(d). This
is expected since compared to the two benchmark schemes,
the proposed designs jointly utilize the information of the GT
locations and the coverage distance D via VBS placement
and convex optimization. Furthermore, by comparing Fig. 4(c)
and Fig. 4(d), it is observed that by solving the convex
optimization problem (P4) based on the obtained VBSs, the
UAV can further reduce its required traveling distance and
mission completion time by avoiding the unnecessary visit to
all the VBSs.
For the proposed UAV trajectory shown in Fig. 4(d),
Fig. 5 plots the actual file recovery probability Pk,succ and
our derived lower bound Pk,lb , where Pk,succ is obtained
numerically via Monte Carlo simulations over 104 random
channel realizations. Note that for better visualization, only
the results for 10 of the GTs are shown in the figure. It
is observed that with the proposed UAV trajectory design,
the constraints in (22) based on the lower bound of the
file recovery probability are satisfied with strict equality for
10
1
700
Strip−based waypoints
GTs as waypoints
VBSs as waypoints
Optimized waypoints
0.9
650
0.8
600
0.7
550
Mission completion time (s)
Probability
0.6
0.5
0.4
0.3
500
450
400
350
0.2
Pk,succ
Pk,lb
P̄
0.1
0
40
41
42
43
44
45
46
300
250
47
48
49
50
GT index
Fig. 5: Numerical verification of the lower bound for the succuss
file recovery probability.
200
150
0
100
200
300
400
500
600
700
D (m)
Fig. 6: Mission completion time versus D.
B. Effect of Auxiliary Distance Parameter D
Next, we study the effect of the auxiliary distance parameter
D on the system performance. Fig. 6 plots the total mission
completion time versus D for the four UAV trajectory
design schemes, with the GT locations same as Fig. 4. It
is observed that for all schemes, the mission completion
time has the general trend of firstly decreasing and then
increasing with D. This is expected since the value of D
affects the UAV trajectory design in two different ways. On
one hand, increasing D leads to lower successful packet
reception probability pD in (14), which in turn requires that
each GT to keep in connection with the UAV for a longer
duration in order to ensure the same file recovery probability.
From this perspective, the mission completion time tends to
increase with D. On the other hand, as D increases, there will
be more GTs that are simultaneously in connection with the
UAV. As a result, the UAV in general needs to travel shorter
distances if larger D is chosen. From this perspective, the
mission completion time tends to decrease with the increasing
of D. Thus, for any given GT locations, there exists an optimal
threshold distance D that balances the above two conflicting
effects and achieves the minimum mission completion time.
To our best effort, it is challenging to find the optimal value D
analytically. However, as illustrated in Fig. 6, one good choice
for D is such that the average received SNR when the GT and
UAV are separated by horizontal distance D is equal to the
threshold SNR γ̄th , in which case D is given by D∗ in (9). For
the setup under consideration, we have D∗ = 430.3m, which
gives the near optimal choice based on Fig. 6.
C. Performance Comparison
Fig. 7 compares the average mission completion time versus
the number of GTs K, where the average is taken over 100
450
400
Average mission completion time (s)
some of the GTs, as expected. Furthermore, it is found
that with the optimized UAV trajectory, all GTs are able to
successfully recover the file almost surely, i.e., with actual
success probability almost equal to 1. This verifies the
proposed lower bound and also shows the effectiveness of the
proposed trajectory design.
Strip−based waypoints
GTs as waypoints
VBSs as waypoints
Optimized waypoints
350
50%
300
250
30%
200
150
100
10
20
30
40
50
60
Number of GTs, K
70
80
90
100
Fig. 7: Average mission completion time versus the number of GTs.
random realizations of the GT locations. For all schemes, the
auxiliary distance parameter D is set as D∗ = 430.3m. It
is first observed that for small or moderate number of GTs,
all the three trajectories with the waypoints designs given in
Section IV-A significantly outperform the benchmark stripbased trajectory. This is expected since when the GTs are
sparsely distributed, utilizing the location information of the
GTs more wisely is beneficial for the UAV trajectory design.
As K increases or the GTs are more densely deployed, the
trajectory design by simply taking the GTs as the waypoints
performs worse than the other benchmark scheme with stripbased waypoints, since it becomes time wasteful for the UAV
to visit all the GTs even when many of them are near to each
other. For all the K values considered, both proposed designs
with the VBSs as waypoints or with the optimized waypoints
significantly outperform the two benchmark schemes. For
instance, for K = 80, the mission completion time with the
two proposed trajectory designs is reduced by around 50% as
compared to the benchmark scheme with GTs as waypoints,
and by 30% than the strip-based waypoints design.
Last, to illustrate the performance gain by exploiting
11
23
demonstrated significant performance gains of the proposed
designs over various benchmark schemes.
22
Number of successfull GTs
21
A PPENDIX A
OVERVIEW OF T RAVELLING S ALESMAN P ROBLEM
VARIATIONS
20
19
18
17
16
15
14
13
0
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
Total transmission time (s)
Fig. 8: Number of successful GTs versus transmission time for the
benchmark scheme with a static transmitter.
the high UAV mobility, we consider another benchmark
multicasting scheme with static transmitter, i.e., the horizontal
projection of the transmitter (e.g., a static UAV) is fixed
at the geometric center of the GTs. For K = 100 GTs,
Fig. 8 shows the number of successful GTs (i.e., those
with successful file recovery probability no smaller than P̄ )
for the benchmark scheme with a static transmitter as the
transmission time increases. It is observed that although the
number of successful GTs increases with the transmission
time, or equivalently with the number of transmitted coded
packets, the increasing rate is very slow. For example, even
with the transmission time increased to 104 s, only 23 GTs
are able to achieve the targeting file recovery probability. This
is expected since with the transmitter fixed in location, the
GTs that have a long distance with the transmitter suffer from
high packet loss probabilities. On the other hand, with UAVenabled multicasting with the proposed trajectory design, only
about 210s is needed to ensure that all the 100 GTs satisfy
the file recovery requirement, as can be seen from Fig. 7. This
demonstrates the dramatic performance gain by exploiting the
high mobility of UAVs for wireless multicasting.
VI. C ONCLUSION
This paper studied the trajectory design problem for a
UAV-enabled multicasting system to minimize the mission
completion time, while ensuring that each GT is able to
successfully recover the file with a high target probability.
We first converted the formulated optimization problem into
a more tractable form based on the derived analytical lower
bound of the successful file recovery probability, so that its
complicated constraint for each GT is simplified to one on
its minimum connection time with the UAV. We showed that
the optimal UAV trajectory only needs to constitute connected
line segments, which can be determined by finding the optimal
set of waypoints and then the optimal speed over time along
the path connecting the waypoints. We proposed two practical
waypoints design schemes and applied the LP to find the
optimal traveling speed given waypoints. Numerical results
AND I TS
In this section, we give a brief description on the classic
TSP [31]–[33] and discuss its variations. The standard TSP
is described as follows. Given a set of K cities and the
distances between each pair of the cities, a traveler wishes
to start and end at the same city and visit each other city
exactly once. The problem is to find the route (or sequence
of visiting) such that the total traveling distance is minimized.
TSP is an NP-hard problem in combinatorial optimization and
hence is difficult to be optimally solved. Various heuristic and
approximation algorithms have been proposed to give efficient
high-quality solutions [31], [38]. In particular, the TSP can be
formulated as a binary integer programming, and an efficient
high-quality solution can be obtained by using the existing
Matlab optimization toolbox (Matlab version 2014 onwards).
The complete Matlab codes and one illustrative example can
be found in [33]. On the other hand, for many applications,
different variations of the TSP need to be considered. In the
following, we discuss five of these variations depending on
whether the traveler needs to return to the origin and whether
the origin/end city is predetermined.
1) Return-GivenOrigin: In this setup, the traveler needs
to return to the origin city, and the origin/end city
is predetermined. This is essentially the same as the
standard TSP, which would return a closed tour so that
any city can be regarded as the origin city.
2) NoReturn-ArbitraryOriginAndEnd: In this setup, the
traveler does not need to return to the origin city, and the
origin and end cities are not predetermined and hence
can be optimized. The optimal solution can be found as
follows [32]. First, add a dummy city whose distances to
all the existing K cities are 0 (this is a virtual node that
does not exist physically). Then solve the standard TSP
problem for the K + 1 cities, and then remove the two
edges associated with the dummy city. It can be shown
by contradiction that such a solution is optimal.
3) NoReturn-GivenOriginAndEnd: In this setup, the
traveler does not need to return to the origin city,
and the origin and end cities are both predetermined,
denoted as A and B, respectively. To solve this problem,
we similarly add a dummy city, with its distance to
both A and B set to 0, whereas that to all other K − 2
cities set to a sufficiently large number (so as to avoid
the traveling from the dummy city to all other cities
except A and B). By solving the standard TSP problem
for the K + 1 cities, and then removing the two edges
associated with the dummy city, we obtain the optimal
solution.
4) NoReturn-GivenOrigin-ArbitraryEnd: In this setup, the
traveler does not need to return to the origin city, and
only the origin city is predetermined, denoted as A.
To solve this problem, we similarly add a dummy city
12
whose distance to A is set to 0, whereas that to all other
K − 1 cities are set to an identical arbitrary positive
value. By solving the standard TSP problem for the
K +1 cities, and then removing the two edges associated
with the dummy city, we obtain the optimal solution.
5) NoReturn-ArbitraryOrigin-GivenEnd: In this setup, the
traveler does not need to return to the origin city, and
only the end city is predetermined. This problem can be
solved similarly as the previous one.
A PPENDIX B
P ROOF OF L EMMA 1
Lemma 1 can be shown by induction. We start by
considering the special case with N = 1. In this case, by
definition, we have
(
(
p1 , x = 1
p̂, x = 1
FX (x) =
FX̂ (x) =
(40)
1, x = 0.
1, x = 0.
Since p1 ≥ p̂, the inequality FX (x) ≥ FX̂ (x) in (18) is
satisfied for N = 1. Next, by assuming that Lemma 1 is
true for N = N̄ , we need to show that it also holds for
N = N̄ + 1. For notational convenience, for N = N̄ , denote
N̄
N̄
the ccdf of X and X̂ as FX
(x) and FX̂
(x), respectively. Then
N̄
N̄
by assumption, we have FX (x) ≥ FX̂ (x), x = 0, 1, · · · , N̄ .
As N increases from N̄ to N̄ + 1, ∀x ∈ {1, · · · , N̄ }, the
following relationships can be obtained,
N̄ +1
N̄
N̄
(x),
(x − 1) + (1 − pN̄ +1 )FX
FX
(x) = pN̄ +1 FX
N̄ +1
FX̂
(x)
=
N̄
p̂FX̂
(x
− 1) + (1 −
N̄
p̂)FX̂
(x).
(41)
(42)
By subtracting (42) from (41) and after some manipulations,
we have
N̄ +1
N̄+1
N̄
N̄
FX
(x) − FX̂
(x) = (1 − p̂) FX
(x) − FX̂
(x)
N̄
N̄
+ p̂ FX
(x − 1) − FX̂
(x − 1)
N̄
N̄
(x − 1) − FX
(x)
+ pN̄+1 − p̂ FX
≥ 0.
(43)
N̄
N̄
Note that the inequality in (43) holds since FX
(x) ≥ FX̂
(x),
N̄
N̄
pN̄ +1 ≥ p̂, and FX (x−1) ≥ FX (x). Thus, ∀x ∈ {1, · · · , N̄ },
N̄ +1
N̄ +1
the inequality FX
(x) ≥ FX̂
(x) holds. For x = 0 or
x = N̄ + 1, the same result can be shown similarly. This
completes the proof of Lemma 1.
A PPENDIX C
P ROOF OF T HEOREM 2
Theorem 2 can be shown by construction. Specifically,
suppose that (q⋆ (t), T ⋆ ) is the optimal solution to (P3), and
the trajectory q⋆ (t) contains at least one curved segment.
Then we show that there always exists an alternative solution
(q̂(t), T̂ ) to (P3) such that q̂(t) contains only line segments
and T̂ ≤ T ⋆ , as follows.
For any given optimal UAV trajectory q⋆ (t), 0 ≤ t ≤ T ⋆ ,
define
K(t) , {k ∈ K : kq⋆ (t) − wk k ≤ D}.
(44)
In other words, for any time t ∈ [0, T ⋆ ], K(t) ⊂ K denotes
the subset of the K GTs that are in connection with the UAV
at time t, given the optimal UAV trajectory q⋆ (t). Since the
total number of subsets of K is 2K (including the empty set),
K(t) can be regarded as a time-dependent function with 2K
discrete values.
Let t1 , t2 , · · · , tL ∈ (0, T ⋆ ) be the L critical time instances
when the subset of connecting GTs changes, i.e., tl is the time
instance such that K(tl − ǫ) 6= K(tl ) with any arbitrarily small
ǫ. Then the optimal UAV trajectory q⋆ (t) can be partitioned
into L + 1 portions, with the subset of connecting GTs
remaining unchanged within each portion. Specifically, the lth
portion constitutes the time interval t ∈ [tl−1 , tl ] with total
duration
Tl , tl − tl−1 , l = 1, · · · , L + 1. We thus have
PL+1
T ⋆ = l=1 Tl . For the lth portion of the UAV trajectory, let
K(t) = Kl , tl−1 ≤ t ≤ tl ,
⋆
⋆
q̂l−1 = q (tl−1 ), q̂l = q (tl ).
(45)
(46)
Then we show in the following that without loss of optimality
to (P3), each of the lth portion of the UAV trajectory q⋆ (t),
tl−1 ≤ t ≤ tl , can be replaced by the line segment connecting
q̂l−1 and q̂l . We show this by addressing the two different
cases with Kl = ∅ or Kl 6= ∅, separately.
Case 1: Kl = ∅. In this case, no GT is in connection with
the UAV for the lth portion of the UAV trajectory. As a result,
this portion does not contribute to the left hand side (LHS) of
the minimum connection time constraint (30). Thus, replacing
this trajectory portion with a line segment from q̂l−1 to q̂l
does not alter the feasibility of (30). Furthermore, since line
segment gives the shortest distance for any two given points,
it is always feasible for the UAV to travel along this new
segment within the time duration T̂l ≤ Tl while satisfying
the maximum speed constraint (31). Thus, such a replacement
ensures the feasibility of (P3) and at least achieves the same
minimum mission completion time as T ⋆ .
Case 2: Kl 6= ∅. In this case, those GTs in Kl are in
connection with the UAV, i.e., the lth portion of the UAV
trajectory contributes to the LHS of (30) for those GTs in Kl .
Define Ql , {q ∈ R2×1 : kq − wk k ≤ D, ∀k ∈ Kl }, i.e., Ql
denotes the set of all possible UAV locations ensuring that all
the GTs in Kl are in connection with the UAV. Note that Ql is
the intersection of |Kl | convex sets, and hence is also convex
[39]. As a result, since both q̂l−1 and q̂l belong to the convex
set Ql , then any point on the line segment connecting q̂l−1
and q̂l must also belong to Ql . In other words, by replacing
the original curved trajectory portion q⋆ (t), t ∈ [tl−1 , tl+1 ],
with the line segment connecting q̂l−1 and q̂l , the subset
of connecting GTs Kl remains unchanged, while the UAV
needs to travel a shorter distance for this portion. Thus, such
a replacement ensures the feasibility of (P3) and at least
achieves the same minimum mission completion time as T ⋆ .
In summary, for any given optimal solution (q⋆ (t), T ⋆ ) to
(P3) with curved UAV trajectory, we can always construct an
alterative optimal trajectory to (P3) by sequentially connecting
the critical locations q̂0 , q̂1 , · · · , q̂L+1 with line segments,
which achieves at least the same minimum mission completion
time as T ⋆ . This thus completes the proof of Theorem 2.
13
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| 7 |
Fast Exact k-Means, k-Medians and Bregman Divergence
Clustering in 1D
arXiv:1701.07204v3 [cs.DS] 30 Jun 2017
Allan Grønlund∗
Kasper Green Larsen†
Jesper Sindahl Nielsen
§
Stefan Schneider
Alexander Mathiasen‡
¶
Mingzhou Song
k
Abstract
The k-Means clustering problem on n points is NP-Hard for any dimension d ≥ 2, however, for
the 1D case there exist exact polynomial time algorithms. Previous literature reported an O(kn2 ) time
dynamic programming algorithm that uses O(kn) space. We present a new algorithm computing the
optimal√clustering in only O(kn) time using linear space. For k = Ω(lg n), we improve this even further
to n2O( lg lg n lg k) time. We generalize the new algorithm(s) to work for the absolute distance instead of
squared distance and to work for any Bregman Divergence as well.
∗ Aarhus University. Email: jallan@cs.au.dk. Supported by MADALGO - Center for Massive Data Algorithmics, a Center
of the Danish National Research Foundation.
† Aarhus University. Email: larsen@cs.au.dk. Supported by MADALGO, a Villum Young Investigator Grant and an AUFF
Starting Grant.
‡ Aarhus University. Email: alexander.mathiasen@gmail.com. Supported by MADALGO and an AUFF Starting Grant.
§ Aarhus University. Email: jasn@cs.au.dk. Supported by MADALGO.
¶ University of California, San Diego. Email: stschnei@cs.ucsd.edu. Supported by NSF grant CCF-1213151 from the
Division of Computing and Communication Foundations. Any opinions, findings and conclusions or recommendations expressed
in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.
k New Mexico State University. Email: joemsong@cs.nmsu.edu
1
Introduction
Clustering is the problem of grouping elements into clusters such that each element is similar to the elements
in the cluster assigned to it and not similar to elements in any other cluster. It is one of, if not the, primary
problem in the area of machine learning known as Unsupervised Learning and no clustering problem is as
famous and widely considered as the k-MeansPproblem: Given a multiset X = {x1 , ..., xn } ⊂ Rd find k
centroids M = {µ1 , ..., µk } ⊂ Rd minimizing x∈X minµ∈M ||x − µ||2 . Several NP-Hardness results exist
for finding the optimal k-Means clustering in general, forcing one to turn towards heuristics. k-Means is
NP-hard even for k = 2 and general dimension [4] and it is also NP-hard for d = 2 and general k [16]. Even
hardness of approximation results exist [15, 8]. In [8] the authors show there exists a ε > 0 such that it is
NP-hard to approximate k-Means to within a factor 1 + ε of optimal, and in [15] it is proved that ε ≥ 0.0013.
On the upper bound side the best known polynomial time approximation algorithm for k-Means has an
approximation factor of 6.357 [3]. In practice, Lloyd’s algorithm is a popular iterative local search heuristic
that starts from some random or arbitrary clustering. The running time of Lloyd’s algorithm is O(tknd)
where t is the number of rounds of the local search procedure. In theory, if Lloyd’s algorithm is run to
convergence to a local minimum, t could be exponential and there is no guarantee on how well the solution
found approximates the optimal solution [6, 20]. Lloyd’s algorithm is often combined with the effective
seeding technique for selecting initial centroids due to [7] that gives an expected O(lg k) approximation ratio
for the initial clustering, which can then improved further by Lloyd’s algorithm.
For the one-dimensional case, the k-Means problem is not NP-hard. In particular, there is an O(kn2 )
time and O(kn) space dynamic programming solution for the 1D case, due to work by [22]. The 1D kMeans problem is encountered surprisingly often in practice, some examples being in data analysis in social
networks, bioinformatics and retail market [5, 13, 18].
It is only natural to try other reasonable distance measures for the data considered and define different
clustering problems. There are many other choices than the sum of squares of the Euclidian distances that
define k-Means. For instance, one could use any Lp norm instead. The special case of p = 1 is known as
k-Medians clustering and has also received considerable attention. The k-Medians problems is also NP-hard
and the best polynomial time approximation algorithms has an approximation factor of 2.633 [8].
In [9] the authors consider and define clustering with Bregman Divergences. Bregman Divergence generalize squared Euclidian distance and thus Bregman Clusterings include the k-Means problem, as well as a
wide range of other clustering problems that can be defined from Bregman Divergences like e.g. clustering
with KullbackLeibler divergence as the cost. Interestingly, the heuristic local search algorithm for Bregman
Clustering [9] is basically the same approach as Lloyd’s algorithm for k-Means. Clustering with Bregman
Divergences is clearly NP-Hard as well since it includes k-Means clustering. We refer the reader to [9] for
more about the general problem. For the 1D version of the problem, [17] generalized the algorithm from
[22] to the k-Medians problems and Bregman Divergences achieving the same O(kn2 ) time and O(kn) space
bounds.
1.1
Our Results
In this paper we give theoretically and practically efficient algorithm for 1D clustering problems, in particular
k-Means.
The k-Means clustering problem in 1D is defined as follows. Given X = {x1 , ..., xn } ⊂ R and k, find
centroids M = {µ1 , ..., µk } ⊂ R minimizing the cost
X
min (x − µ)2
x∈X
µ∈M
The main results of this paper are new fast algorithms for 1D k-Means. First we give an algorithm that
computes the optimal k-Means clustering that runs in O(n lg n + kn) time using optimal O(n) space, or
O(kn) time if the input is already sorted. The algorithm also computes the cost of the optimal clustering
using k ′ clusters for all k ′ ≤ k. This is relevant for instance for model selection of the right k. This is an
1
improvement by a factor of n in time and k in space compared to the existing solution (which also supports
computing the cost for all k ′ ≤ k. The constant factors hidden by the O-notation are small and we expect this
algorithm
to be very efficient in practice. Second, we show how to compute an optimal k-Means clustering in
√
n2O( lg lg n lg k) time using O(n) space. This algorithm is mainly of theoretical interest as we expect constants
to be rather large. As opposed to the O(kn) time algorithm, this algorithm does not compute the optimal
′
′
costs for using
√ k clusters for all k ≤ k.
O( lg lg n lg k)
time algorithm relates to a natural regularized version of k-Means clustering where
The n2
instead of specifying the number of clusters beforehand, we instead specify a cost of using an extra cluster
and then minimize the cost of the clustering plus the cost of the number of clusters used. Formally, the
problem is as follows: Given X = {x1 , ..., xn } ⊂ R and λ, compute the optimal regularized clustering:
X
min (x − µ)2 + λk
arg min
k,M={µ1 ,...,µk } x∈X µ∈M
We show that this problem is solvable in O(n) time if the input is sorted.
In 1D, Lloyd’s algorithm can be implemented to run in O(n lg n + tk lg n) time where t is the number of
rounds, and we expect that our algorithm can compute the optimal clustering for reasonable k in essentially
the same time as Lloyd’s algorithm can approximate it.
The k-Medians problem is to compute a clustering that minimize the sum of absolute distances to the
centroid, i.e. compute M = {µ1 , ..., µk } ⊂ R that minimize
X
min |x − µ|
x∈X
µ∈M
Our algorithms generalize naturally to solve this problem in the same time bounds as for the k-means
problem.
Let f be a differentiable real-valued strictly convex function. The Bregman Divergence Df induced by f
is defined as
Df (x, y) = f (x) − f (y) − ∇f (y)(x − y)
Notice that the Bregman Divergence induced from f (x) = x2 , gives squared Euclidian Distance (k-Means).
Bregman divergences are not metrics since they are not symmetric in general and the triangle inequality is
not necessarily satisfied. They do however have many redeeming qualities, for instance Bregman Divergences
are convex in the first argument, albeit not the second, see [9, 10] for a more comprenhensive treatment.
The Bregman Clustering problem as defined in [9] is to find k centroids M = {µ1 , ..., µk } that minimize
X
min Df (x, µ)
x∈X
µ∈M
where Df is a Bregman Divergence. For our case, where the inputs x, y ∈ R, we assume that computing a
Bregman Divergence, i.e. evaluating f and its derivative, takes constant time. We show that our algorithms
naturally generalize to 1D clustering using any Bregman Divergence to define the cluster cost while still
maintaing the same running time as for k-Means.
Implementation. An independent implementation of the O(n lg n + kn) time algorithm is available in the
R package Ckmeans.1d.dp [21]. The implementation is for k-Means clustering, and uses O(kn) space.
1.2
Outline
In Section 2 we describe the existing O(kn2 ) time algorithm for 1D k-Means clustering that uses O(kn)
space. In Section 3 we show how to compute the same output as the old algorithm
using only O(kn) time
√
and O(n) space. Then we show how to improve the running time to O(n2O( lg lg n lg k) ). Finally, in Section
4 we show how our new algorithms generalizes to different cluster costs than squared Euclidian distance.
2
2
The O(kn2 ) Dynamic Programming Algorithm
In this section, we describe the previous O(kn2 ) time and O(kn) space algorithm presented in [22]. We also
introduce the definitions and notation we use in our new algorithm. We will always assume sorted input
x1 ≤ ... ≤ xn ∈ R. If the input is not sorted, we start by sorting it in O(n lg n) time. We also remark
that there could be many ways of partitioning the point set and computing centroids that achieve the same
cost. This is for instance the case if the input is n identical points. The task at hand is to find any optimal
solution.
P
Let CC(i, j) = jℓ=i (xℓ − µi,j )2 be the cost of grouping xi , ..., xj into one cluster with the optimal choice
Pj
1
of centroid, µi,j = j−i+1
ℓ=i xℓ , the arithmetic mean of the points.
Lemma 1. There is an O(n) space data structure that can compute CC(i, j) in O(1) time for any i ≤ j
using O(n) time preprocessing.
Proof. This is a standard application of prefix sums and works as follows. By definition,
CC(i, j) =
j
X
(xℓ − µi,j )2 =
ℓ=i
j
X
x2ℓ + µ2i,j − 2xℓ µi,j = (j − i + 1)µ2i,j + µi,j
ℓ=i
j
X
ℓ=i
xℓ +
j
X
x2ℓ .
ℓ=i
With access to prefix sum arrays of x1 , . . . , xn and x21 , . . . , x2n both the centroid µi,j and the cost is easily
computed in constant time .
2.1
Algorithm Sketch
The algorithm computes the optimal clustering using i clusters for all prefixes of input points x1 , . . . , xm ,
for m = 1, . . . , n, and for all i = 1, . . . , k using Dynamic Programming as follows.
Let D[i][m] be the cost of optimally clustering x1 , ..., xm into i clusters. For i = 1 the cost of optimally
clustering x1 , ..., xm into one cluster is the cluster cost CC(1, m). That is, D[1][m] = CC(1, m) for all m.
This can be computed in O(n) time by Lemma 1.
For i > 1
m
D[i][m] = min D[i − 1][j − 1] + CC(j, m)
(1)
j=1
Notice that D[i − 1][j − 1] is the cost of optimally clustering x1 , ..., xj−1 into i − 1 clusters and CC(j, m)
is the cost of clustering xj , ..., xm into one cluster. This makes xj the first point in the last and rightmost
cluster. Let T [i][m] be the argument that minimizes (1)
m
T [i][m] := arg min D[i − 1][j − 1] + CC(j, m)
j=1
(2)
It is possible there exists multiple j obtaining same minimal value for (1). To make the optimal clustering
C˜(m) unique, such ties are broken in favour of smaller j.
Notice xT [i][m] is the first point in the rightmost cluster of the optimal clustering. Thus, given T one can
find the optimal solution by standard backtracking:
X̃k = {xT [k][n] , ..., xn },
X̃k−1 = {xT [k−1][T [k][n]−1] , ..., xT [k][n]−1 }
..
.
Here X̃i is the i’th cluster in the optimal clustering. One can naively compute each entry of D and T using
(1) and (2). This takes O(n) time for each cell, thus D and T can be computed in O(kn2 ) time using O(kn)
space. This is exactly what is described in [22].
3
3
New Algorithms
The idea of the first new algorithm is simply to compute the tables D and T faster, by reducing the time to
compute each row of D and T to O(n) time instead of O(n2 ) time. This improvement exploits a monotonicity
property of the values stored in a row of T . This is explained in Section 3.1, resulting in an O(kn) time and
O(kn) space solution, assuming sorted inputs. Section 3.2 then shows how to reduce the space usage to just
O(n) while retaining O(kn) running time.
In Section 3.3 we show that the same property allows us to solve
√
O( lg lg n lg k)
1D k-Means for k = Ω(lg n) in, n2
time and linear space, and solve the regularized version of
1D k-Means in O(n) time.
3.1
Faster Algorithm From Monotone Matrices
In this section we reduce the problem of computing a row of D and T to searching an implicitly defined
n × n matrix of a special form, which allows us to compute each row of D and T in linear time.
Define Ci [m][j] as the cost of the optimal clustering of x1 , . . . , xm using i clusters, restricted to having
the rightmost cluster (largest cluster center) contain the elements xj , . . . , xm . For convenience, we define
Ci [m][j] for j > m as the cost of clustering x1 , . . . , xm into i − 1 clusters, i.e. the last cluster is empty. This
means that Ci satisfies:
Ci [m][j] = D[i − 1][min{j − 1, m}] + CC(j, m)
where by definition CC(j, m) = 0 when j > m (which is consistent with the definition in Section 2). We
have that D[i][m] relates to Ci as follows:
D[i][m] = min Ci [m][j]
j
where ties are broken in favor of smaller j (as defined in Section 2.1).
This means that when we compute a row of D and T , we are actually computing minj Ci [m][j] for all
m = 1, . . . , n. We think of Ci as an n × n matrix with rows indexed by m and columns indexed by j. With
this interpretation, computing the i’th row of D and T corresponds to computing for each row r in Ci ,
the column index c that corresponds to the smallest value in row r. In particular, the entries D[i][m] and
T [i][m] correpond to the value and the index of the minimum entry in the m’th row of Ci respectively. The
problem of finding the minimum value in every row of a matrix has been studied before [1]. First we need
the definition of a monotone matrix.
Definition 1. [1] Let A be a matrix with real entries and let arg min(i) be the index of the leftmost column
containing the minimum value in row i of A. A is said to be monotone if a < b implies that arg min(a) ≤
arg min(b). A is totally monotone if all of its submatrices are monotone.1
In [1], the authors showed the following:
Theorem 1. [1] Finding arg min(i) for each row i of an arbitrary n× m monotone matrix requires Θ(m lg n)
time, whereas if the matrix is totally monotone, the time is O(m) when m > n and is O(m(1 + lg(n/m)))
when m < n.
The fast algorithm for totally monotone matrices is known as the SMAWK algorithm and we will refer
to it by that (cool) name.
Let’s relate this to the 1D k-Means clustering problem. That Ci is monotone means that if we consider the
optimal clustering of the points x1 , . . . , xa with i clusters, then if we start adding more points xa+1 ≤ · · · ≤ xb
after xa , then the first (smallest) point in the last of the i clusters can only increase (move right) in the new
optimal clustering of x1 , . . . , xb . This sounds like it should be true for 1D k-Means and it turns out it is.
Thus, applying the algorithm for monotone matrices, we can fill a row of D and T in O(n lg n) time leading
to an O(kn lg n) time algorithm for 1D k-Means, which is already a great improvement.
1 In
[1] the authors use the maximum instead of the minimum
4
However, as we show below, the matrix Ci induced by the 1D k-Means problem is in fact totally monotone:
Lemma 2. The matrix Ci is totally monotone.
Proof. As [1] remarks, a matrix A is totally monotone if all its 2 × 2 submatrices are monotone. To prove
that Ci is totally monotone, we thus need to prove that for any two row indices a, b with a < b and two
column indices u, v with u < v, it holds that if Ci [a][v] < Ci [a][u] then Ci [b][v] < Ci [b][u].
Notice that these values correspond to the costs of clustering elements x1 , . . . , xa and x1 , . . . , xb , starting
the rightmost cluster with element xv and xu respectively. Since Ci [m][j] = D[i−1][min{j−1, m}]+CC(j, m),
this is the same as proving that
D[i − 1][min{v − 1, m}] + CC(v, a) < D[i − 1][min{u − 1, m}] + CC(u, a) ⇒
D[i − 1][min{v − 1, m}] + CC(v, b) < D[i − 1][min{u − 1, m}] + CC(u, b)
which is true if we can prove that CC(v, b) − CC(v, a) ≤ CC(u, b) − CC(u, a). Rearranging terms, what we
need to prove is that for any a < b and u < v, it holds that:
CC(v, b) + CC(u, a) ≤ CC(u, b) + CC(v, a).
(3)
This is the property known as the concave (concave for short) property [24, 12, 23] and has been used to
significantly speed up algorithms, including Dynamic Programming algorithms, for for other problems. We
start by handling the special case where v > a. In this case, we have by definition that CC(v, a) = 0, thus
we need to show that CC(v, b) + CC(u, a) ≤ CC(u, b). This is the case since any point amongst xu , . . . , xb
is included in at most one of xv , . . . , xb and xu , . . . , xa (since a < v). Thus CC(v, b) + CC(u, a) is the cost
of taking two disjoint and consecutive subsets of the points xu , . . . , xb and clustering the two sets using the
optimal choice of centroid in each. Clearly this cost is less than clustering all the points using one centroid.
We now turn to the general case where u < v ≤ a < b. Let µv,a be the mean of xv , . . . , xa and
µu,b be the mean
P of xu , . . . , xb and assume that µv,a ≤ µu,b (the other case is symmetric). Finally, let
CC(w, z)µ = zℓ=w (xℓ − µ)2 denote the cost of grouping the elements xw , . . . , xz in a cluster with centroid
µ. Split the cost CC(u, b) into the cost of the elements xu , . . . , xv−1 and the cost of the elements xv , . . . , xb
as
CC(u, b) =
b
X
(xℓ − µu,b )2 =
ℓ=u
v−1
X
(xℓ − µu,b )2 +
ℓ=u
b
X
(xℓ − µu,b )2 = CC(u, v − 1)µu,b + CC(v, b)µu,b .
ℓ=v
We trivially get CC(v, b)µu,b ≥ CC(v, b) since CC(v, b) is the cost using the optimal centroid. Secondly,
CC(u, v − 1)µu,b + CC(v, a) ≥ CC(u, v − 1)µv,a + CC(v, a) = CC(u, a)µv,a ≥ CC(u, a)
since µv,a ≤ µu,b and all elements xu , . . . , xv−1 are less than or equal to µv,a (since µv,a is the mean of points
xv , . . . , xa that all are greater than xu , . . . , xv−1 ). Combining the results, we see that:
CC(v, b) + CC(u, a) ≤ CC(v, b)µu,b + CC(u, v − 1)µu,b + CC(v, a) = CC(u, b) + CC(v, a).
This completes the proof.
Theorem 2. Computing an optimal k-Means clustering of a sorted input of size n for takes O(kn) time.
By construction the cost of the optimal clustering is computed for all k ′ ≤ k. If we store the T table the
cluster centers for any k ′ ≤ k can be extracted in O(k ′ ) time.
5
3.2
Reducing Space Usage
In the following, we show how to reduce the space usage to just O(n) while maintaining O(kn) running time
using a space reduction technique of Hirschberg [11]. First observe that each row of T and D only refers to
the previous row. Thus one can clearly “forget”row i − 1 when we are done computing row i. The problem
is that if we don’t store all of T , we cannot backtrack and find the optimal solution. In the following, we
present an algorithm that avoids the table T entirely.
Our key observation is the following: Assume k > 1 and that for every prefix x1 , . . . , xm , we have
computed the optimal cost of clustering x1 , . . . , xm into ⌊k/2⌋ clusters. Note that this is precisely the set
of values stored in the ⌊k/2⌋’th row of D. Assume furthermore that we have computed the optimal cost of
clustering every suffix xm , . . . , xn into k − ⌊k/2⌋ clusters. Let us denote these costs by D̃[k − ⌊k/2⌋][m] for
m = 1, . . . , n. Then clearly the optimal cost of clustering x1 , . . . , xn into k clusters is given by:
n
D[k][n] = min D[⌊k/2⌋][j] + D̃[k − ⌊k/2⌋][j + 1].
j=1
(4)
Our main idea is to first compute row ⌊k/2⌋ of D and row k − ⌊k/2⌋ of D̃ using linear space. From these two,
we can compute the argument j minimizing (4). We can then split the reporting of the optimal clustering
into two recursive calls, one reporting the optimal clustering of points x1 , . . . , xj into ⌊k/2⌋ clusters, and one
call reporting the optimal clustering of xj+1 , . . . , xn into k − ⌊k/2⌋ clusters. When the recursion bottoms
out with k = 1, we can clearly report the optimal clustering using linear space and time as this is just the
full set of points.
From Section 3.1 we already know how to compute row ⌊k/2⌋ of D using linear space: Simply call
SMAWK to compute row i of D for i = 1, . . . , ⌊k/2⌋, where we throw away row i − 1 of D (and don’t
even store T ) when we are done computing row i. Now observe that table D̃ can be computed by taking
our points x1 , . . . , xn and reversing their order by negating the values. This way we obtain a new ordered
sequence of points X̃ = x̃1 ≤ x̃2 ≤ · · · ≤ x̃n where x̃i = −xn−i+1 . Running SMAWK repeatedly for
i = 1, . . . , k − ⌊k/2⌋ on the point set X̃ produces a table D̂ such that D̂[i][m] is the optimal cost of clustering
x̃1 , . . . , x̃m = −xn , . . . , −xn−m+1 into i clusters. Since this cost is the same as clustering xn−m+1 , . . . , xn
into i clusters, we get that the (k − ⌊k/2⌋)’th row of D̂ is identical to the i’th row of D̃ if we reverse the
order of the entries.
To summarize our algorithm for reporting the optimal clustering, do as follows: Let L be an initially
empty output list of clusters. If k = 1, append to L a cluster containing all points. Otherwise (k > 1),
use SMAWK on x1 , . . . , xn and −xn , . . . , −x1 to compute row ⌊k/2⌋ of D and row k − ⌊k/2⌋ of D̃ using
linear space (by evicting row i − 1 from memory when we have finished computing row i) and O(kn) time.
Compute the argument j minimizing (4) in O(n) time. Evict row ⌊k/2⌋ of D and row k − ⌊k/2⌋ of D̃ from
memory. Recursively report the optimal clustering of points x1 , . . . , xj into ⌊k/2⌋ clusters (which appends
the output to L). When this terminates, recursively report the optimal clustering of points xj+1 , . . . , xn into
k − ⌊k/2⌋ clusters. When the algorithm terminates, L contains the optimal clustering of x1 , . . . , xn into k
clusters.
At any given time, our algorithm uses only O(n) space. To see this, first note that we evict all memory
used to compute the value j minimizing (4) before recursing. Furthermore, we complete the first recursive
call (and evict all memory used) before starting the second. Finally, for the recursion, we don’t need to
make a copy of points x1 , . . . , xj . It suffices to remember that we are only working on the subset of inputs
x1 , . . . , xj .
Now let F (n, k) denote the time used by the above algorithm to compute the optimal clustering of n
sorted points into k clusters. Then there is some constant C > 0 such that F (n, k) satisfies the recurrence:
F (n, 1) ≤ Cn,
and for k > 1:
n
F (n, k) ≤ max F (j, ⌊k/2⌋) + F (n − j, k − ⌊k/2⌋) + Cnk.
j=1
6
We claim that F (n, k) satisfies F (n, k) ≤ 3Ckn. We prove the claim by induction in k. The base case k = 1
follows trivially by inspection of the formula for F (n, 1). For the inductive step k > 1, we use the induction
hypothesis to conclude:
F (n, k) ≤
n
max 3Cj⌊k/2⌋ + 3C(n − j)(k − ⌊k/2⌋) + Cnk
j=1
n
≤
max 3Cj⌈k/2⌉ + 3C(n − j)⌈k/2⌉ + Cnk
=
3Cn⌈k/2⌉ + Ckn.
j=1
For k > 1, we have that ⌈k/2⌉ ≤ (2/3)k, therefore:
F (n, k) ≤ 3Cn(2/3)k + Ckn
= 3Ckn.
Which is what we needed to prove.
Theorem 3. Computing an optimal k-Means clustering of a sorted input of size n takes O(kn) time and
uses O(n) space.
Note to compute the cost of the optimal clustering for all k ′ ≤ k we ensure that we never delete the last
column of the cost matrix D which requires an additional O(k) = O(n) space.
3.3
Even Faster Algorithm
In this section we show that the concave property we proved for the cluster costs
yields and algorithm for
√
computing the optimal k-Means clustering for one given k = Ω(lg n) in n2O( lg lg n lg k) time. The result
follows almost directly from [19]. In [19] Schieber gives an algorithm with the aforementioned running time
for the problem of finding the shortest path of fixed length k in a directed acyclic graph with nodes 1, . . . , n
where the weights, w(i, j), satisfy the concave property and are represented as a function that returns the
weight of a given edge in constant time.
Theorem 4 ([19]). Computing a minimum weight path of length k between any
two nodes in a directed
√
acyclic graph of size n where the weights satisfy the concave property takes n2O( lg lg n lg k) time using O(n)
space.
We reduce the 1D k-Means problem to a directed graph problem as follows. Sort the input in O(n lg n)
time and let x1 ≤ x2 ≤ . . . xn denote the sorted input sequence. For each input xi we associate a node
vi and add an extra node vn+1 . Now define the weight of the edge from vi to vj as the cost of clustering
xi , . . . , xj−1 in one cluster, which is CC(i, j − 1). Each edge weight is computed in constant time and by the
proof of Lemma 2, particularly Equation 3, the edge weights satisfy the monge concave property. Finally, to
compute the optimal clustering we use Schiebers algorithm to compute the lowest weight path with k edges
from v1 to vn+1 .
Theorem
5. Computing an optimal k-Means clustering of an input of size n for given k = Ω(lg n) takes
√
O( lg lg n lg k)
time using O(n) space.
n2
It is relevant to briefly consider parts of Schiebers algorithm and how it relates to k-Means clustering, in
particular a regularized version of the problem. Schiebers algorithm relies crucially on algorithms that given
a directed acyclic graph where the weights satisfy the concave property computes a minimum weight path
in O(n) time [23, 14]. Note the only difference in this problem compared to above, is that the search is not
restricted to paths of k edges only.
7
3.3.1
Regularized Clustering
Consider a regularized version of the k-Means clustering problem where we instad of providing the number
of clusters k specify the cost of a cluster and ask to minimize the cost of the clustering plus the penalty λ
for each cluster used. For simplicity, assume all the input points are distinct.
If we set λ = 0 the optimal clustering has cost zero and use a cluster for each input point. If we let λ
increase towards infinity, the optimal number of clusters used in the optimal solution monotonically decrease
towards one (zero clusters is not well defined). Let dmin be the smallest distance between points in the input.
The optimal cost of using n − 1 clusters is then d2min /2. When λ > λn = d2min /2 it is less costly to use
only n − 1 clusters since the added clustering of using one less cluster is smaller than the cost of a cluster.
Letting λ increase again will inevitably lead to a miminum value λn−1 < λn such that for λ > λn−1 only
n − 2 clusters is used in the optimal solution. Following the same pattern λn−1 is the difference between
the optimal cost using n − 2 clusters and n − 1 clusters. Continuing this yields the very interesting event
sequence 0 < λn < · · · < λ1 that encodes the only relevant choices for the regularization parameter. Note
that the O(nk) algorithm actually yields λ1 , . . . , λk since it computes the optimal cost for all k ′ ≤ k.
In the reduction to the directed graph problem, adding a cost of λ for each cluster used corresponds to
adding λ to the weight of each edge. Note that the edge weights clearly still satisfy the concave property.
Thus, solving the regularized version of k-Means clustering correpoonds to finding the shortest path (of any
length) in a directed acyclic graph where the weights satisfy the concave property. By the algorithms in
[23, 14] this takes O(n) time.
Theorem 6. Computing an optimal regularized 1D k-Means clustering of a sorted input of size n takes O(n)
time.
Now notice if we actually use λk−1 as the cost per cluster, or any λ ∈ [λk−1 , λk [ there is an optimal
solution using k clusters which is an optimal k-Means clustering. This means that if the inputs are integers,
we can solve the 1D k-Means problem by a simple application of binary search in O(n lg U ) time where U is
the universe size [2].
4
Extending to More Distance Measures
In the following we show how to generalize our algorithm to Bregman Divergences and sum of absolute
distances while retaining the same running time and space usage.
4.1
Bregman Divergence and Bregman Clustering
In this section we show how our algorithm generalizes to any Bregman Divergence. First, let us remind
ourselves what a Bregman Divergence and a Bregman Clustering is. Let f be a differentiable real-valued
strictly convex function. The Bregman Divergence Df defined by f is defined as
Df (x, y) = f (x) − f (y) − ∇f (y)(x − y)
Bregman Clustering. The Bregman Clustering problem as defined in [9], is to find a clustering, M =
{µ1 , ..., µk }, that minimize
X
min Df (x, µ)
x∈X
µ∈M
Notice that the cluster center is the second argument of the Bregman Divergence. This is important since
Bregman Divergences are not in general symmetric.
For the purpose of 1D clustering, we mention two important properties of Bregman Divergences. For any
Bregman Divergence, the unique element that minimizes the summed distance to a multiset of elements is
8
the mean of the elements, exactly as it was for squared Euclidian distance. This is in one sense the defining
property of Bregman Divergences [9].
The second important is the linear separator property, which is very important for clustering with Bregman Divergences but also very relevavant to Bregman Voronoi Diagrams [9, 10].
Linear Separators For Bregman Divergences. For all Bregman divergences, the locus of points that
are equidistant to two fixed points µ1 , µ2 in terms of a Bregman divergence is given by
{x ∈ X | Df (x, p) = Df (x, q)} = {x ∈ X | x(∇f (µ1 ) − ∇f (µ2 )) = f (µ1 ) − µ1 ∇f (µ1 ) − f (µ2 ) + µ2 ∇f (µ2 )}
which corresponds to a hyperplane. Also, the points µ1 , µ2 sits on either side of the hyperplane and the
Voronoi cells defined using Bregman divergences are connected.
This means, in particular, that between any two points in 1D, µ1 < µ2 , there is a hyperplane (point) h
with µ1 < h < µ2 and all points smaller than h are closer to µ1 and all points larger than h are closer to µ2 .
We capture what we need from this observation in a simple “distance” lemma:
Lemma 3. Given two fixed real numbers µ1 < µ2 , then for any point xr ≥ µ2 , we have Df (xr , µ1 ) >
Df (xr , µ2 ), and for any point xl ≤ µ1 we have Df (xl , µ1 ) < Df (xl , µ2 )
Computing Cluster Costs for Bregman Divergences. Since the mean minizes Bregman Divergences,
the centroids used in optimal clusterings are unchanged compared to the k-Means case. The prefix sums idea
used to implement the data structure used for Lemma 1 generalizes to Bregman Divergences as observed
in [17] (under the name Summed Area Tables). The formula for computing the cost of grouping the points
Pj
1
xi , . . . , xj in one cluster is as follows. Let µi,j = j−i+1
ℓ=i xℓ be the arithmetic mean of the points
xi , . . . , xj , then
CC(i, j) =
j
X
Df (xℓ , µi,j ) =
=
ℓ=i
f (xℓ ) − f (µi,j ) − ∇f (µi,j )(xℓ − µi,j )
ℓ=i
ℓ=i
j
X
j
X
!
f (xℓ )
− (j − i + 1)f (µi,j ) − ∇f (µi,j )
j
X
ℓ=i
xℓ
!
− (j − i + 1)µi,j
!
It follows that the Bregman Divergence cost of a consecutive subset of input points and the centroid can be
computed in in constant time with stored prefix sums for x1 , . . . , xn and f (x1 ), . . . , f (xn ).
Monge Concave - Totally Monotone Matrix. The only properties we used in Section 3.1 to prove the
monge concave property and that the matrix Ci is totally monotone, is that the mean is the minimizer of
the sum of distances to a multiset of points, and that
CC(u, v − 1)µu,b + CC(v, a) ≥ CC(u, v − 1)µv,a + CC(v, a) = CC(u, a)µv,a
when µv,a ≤ µu,b and all elements in xu , . . . , xv−1 ≤ µv,a . This is clearly still true by Lemma 3.
It follows that the algorithms we specified for 1D k-Means generalize to any Bregman Divergence.
4.2
k-Medians - Clustering with sum of absolute values
For the k-Medians problem we replace the the sum of squared Euclidian distances with the sum of absolute
distances. Formally, the k-Medians problem is to compute a clustering, M = {µ1 , ..., µk }, minimizing
X
min |x − µ|
x∈X
µ∈M
Note that in 1D, all Lp norms are the same and reduce to this case. Also note that the minimizing centroid
for a cluster is no longer the mean of the points in that cluster, but the median. To solve this problem, we
9
change the centroid to be the median, and if there an even number of points, we fix the median to be the
exact middle point between the two middle elements, making the choice of centroid unique.
As for Bregman Divergences, we need to show that we can compute the cost CC(i, j) with this new cost
in constant time. Also, we need to compute the centroid in constant time and argue that the cost is monge
moncave which implies the implicit matrix Ci is totally monotone. The arguments are essentially the same,
but for completeness we briefly cover them below.
Computing Cluster Costs for Absolute Distances. Not surprisingly, using prefix sums still allow
x⌊mi,j ⌋ +x⌈mi,j ⌉
constant time computation of CC(i, j). Let mi,j = j+i
2 , and compute the centroid as µi,j =
2
CC(i, j) =
j
X
ℓ=i
|xℓ − µi,j | =
⌊mi,j ⌋
X
ℓ=i
µi,j − xℓ +
j
X
xℓ − µi,j
ℓ=1+⌊mi,j ⌋
which can be computed in constant time with access to a prefix sum table of x1 , . . . , xn . This was also
observed in [17].
Monge Concave - Totally Monotone Matrix. The monge concave and totally monotone matrix argument above for Bregman Divergences (and for squared Euclidian distance) remain valid since first of all,
we still have xu , . . . , xv−1 ≤ µv,a as µv,a is the median of points all greater than xu , . . . , xv−1 . Furthermore,
it still holds that when µv,a ≤ µu,b and all elements xu , . . . , xv−1 are less than or equal to µv,a , then:
CC(u, v − 1)µu,b + CC(v, a) ≥ CC(u, v − 1)µv,a + CC(v, a) = CC(u, a)µv,a
It follows that the algorithms we specified for 1D k-Means generalize to the 1D k-Median problem.
Acknowledgements
We wish to thank Pawel Gawrychowski for pointing out important earlier work on concave property.
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| 2 |
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
arXiv:1712.08913v1 [math.RT] 24 Dec 2017
MARC CABANES
Abstract. We review old and new results about the modular representation theory of finite reductive groups with a strong emphasis on local methods. This includes subpairs, Brauer’s Main Theorems, fusion, Rickard equivalences. In the defining characteristic we describe the relation between p-local
subgroups and parabolic subgroups, then give classical consequences on simple modules and blocks, including the Alperin weight conjecture in that case.
In the non-defining characteristics, we sketch a picture of the local methods pioneered by Fong-Srinivasan in the determination of blocks and their
ordinary characters. This includes the relationship with Lusztig’s twisted induction and the determination of defect groups. We conclude with a survey of
the results and methods by Bonnafé-Dat-Rouquier giving Morita equivalences
between blocks that preserve defect groups and the local structures.
Contents
1. p-local subgroups and parabolic subgroups
1.A. Parabolic subgroups and Levi subgroups: reductive groups.
1.B. Parabolic subgroups and Levi subgroups: finite groups
1.C. p-local subgroups and simple groups of characteristic p type
1.D. Consequences on fusion
1.E. Consequences on p-blocks
2. Yokonuma-Hecke algebras in characteristic p
2.A. Self-injective endomorphism algebras, a theorem of Green
2.B. Yokonuma-Hecke algebras: a presentation
2.C. Yokonuma-Hecke algebras: simple modules
3. Simple FG-modules and p-blocks
3.A. Simple FG-modules, the Steinberg module
3.B. Relation with weight modules
3.C. Alperin weight conjecture
4. Rational series and ℓ-blocks
4.A. Blocks and central functions
4.B. Uniform and p-constant functions
4.C. Rational series and Broué-Michel’s theorem
4.D. Jordan decomposition and ℓ-blocks
5. Local methods for blocks of finite quasi-simple groups
5.A. Subpairs and local structure of an ℓ-block
5.B. Brauer’s second Main Theorem
5.C. Centric or self-centralizing subpairs
5.D. Two main theorems of Brauer and blocks of quasi-simple groups
5.E. The symmetric group: characters
5.F. The symmetric group: blocks
Date: October 20, 2017.
2010 Mathematics Subject Classification. Primary 20C15.
1
4
5
6
7
9
9
11
11
13
13
14
14
16
16
17
18
18
20
22
25
25
26
26
27
27
29
2
MARC CABANES
5.G. The symmetric group: Chuang-Rouquier’s theorems
6. Local methods for unipotent blocks: the strategy
6.A. Generalized d-Harish-Chandra theory
6.B. The theorem
7. Local methods: unipotent blocks and d-Harish-Chandra theory
7.A. The main subpair inclusion
7.B. φd -tori and ℓ-subgroups
7.C. Defect groups
7.D. Non-unipotent characters of unipotent blocks
7.E. Unipotent blocks are non-exotic
7.F. A theorem of Broto-Møller-Oliver
8. Some applications
8.A. Abelian defect
8.B. Brauer’s height zero conjecture
8.C. Nilpotent blocks
8.D. Broué’s abelian defect conjecture when ℓ divides q − 1
9. Bonnafé-Dat-Rouquier’s theorems
9.A. Etale topology and sheaves
9.B. Broué’s reduction
9.C. Bonnafé-Rouquier (2003)
9.D. Bonnafé-Dat-Rouquier (2017)
10. Recreation: Blocks of defect zero
References
Index
31
33
34
36
37
37
40
41
42
42
44
45
45
46
47
48
50
51
53
54
56
61
63
68
Introduction
This survey aims at presenting in an almost self contained fashion some key
results in the representation theory of finite quasi-simple groups that can be related
to some global-local principle. For finite group theorists local information means
information relating to normalizers of nilpotent subgroups. The typical situation is
when given a finite group G and a prime number p, one wants to guess information
about G from information of the same kind about subgroups N normalizing a non
normal p-subgroup of G. Those N are sometimes called p-local subgroups. One
has N
G so the process looks like somehow reducing the questions we might
have about G to questions about more tractable subgroups. This is particularly
apparent in the classification of finite simple groups (CFSG, 1955–1980) where, at
least in the earliest stages, 2-local subgroups were systematically used to sort out
simple groups by the structure of centralizers of involutions.
But what is the relevance of all that to representations, in particular of quasisimple groups ? We try to give very concrete answers here.
It is clear that in the years of the classification it was strongly believed that the
p-local information on G should determine many aspects of linear representations
of G in characteristic p. A short textbook by J.L. Alperin appeared in 1986
with the title “Local representation theory” [Alper]. The main themes: Green’s
vertex theory, Brauer’s morphism and defect groups, the case of cyclic defect and
its consequence on the module category B-mod of the block B. On the other
hand, the theme of “simple groups and linear representations” was at that time
recalling mainly the spectacular applications of character theory (both modular
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
3
and ordinary) to CFSG, especially through Glauberman’s Z*-theorem. This was
exemplified by the influential survey [Da71] or the textbooks [Feit], [Nava].
Today the perspective has changed a little. CFSG and the wealth of knowledge on representations of finite groups of Lie type (see the survey [Ge17] of this
volume) or symmetric groups make that representations of (quasi-)simple groups
are becoming the main subject. The development of combinatorial representation
theory and the recent interpretations in terms of categorification (see [ChRo08],
[DuVV15], [DuVV17]), seem to hint at a situation where p-blocks of finite group
algebras are classified regarding their module categories mod and other associated
categories (derived Db , homotopy Hob or stable) before the relevant p-local information is known. The latter can even possibly be a consequence of the equivalence
of blocks as rings (see Puig’s theorem (Theorem 9.16 below) and its use in [BoDaRo17]). On the other hand, the notion of fusion systems and the topological
questions or results it provides (see for instance [AschKeOl], [Craven]), have given
a new perspective to the determination of local structure both for groups and
blocks.
We try here to sum up the relevance of local methods in representations of
quasi-simple groups, essentially for groups of Lie type. In the defining characteristic we describe the relation between p-local subgroups and parabolic subgroups,
then we give classical consequences on simple modules and blocks, including the
Alperin’s weight conjecture. In the non-defining characteristics, we sketch a picture of the local methods pioneered by Fong-Srinivasan in the determination of
blocks and their ordinary characters. On the method side one will find Brauer’s
three Main Theorems, Alperin-Broué subpairs, both revolving around the Brauer
morphism which will reappear also when discussing Rickard equivalences. On the
side of results, we describe the relationship between blocks and Lusztig’s twisted
induction including the determination of defect groups. We also recall applications to Brauer’s height zero conjecture (Kessar-Malle) and Broué’s abelian defect
conjecture. In all cases we try to give many proofs at least for “main cases” (leaving aside bad primes). We conclude with a survey of the results and methods of
Bonnafé-Dat-Rouquier ([BoRo03], [BoDaRo17]).
The exposition follows a route prescribed by the groups we study. Abstract
methods on blocks are only introduced when needed. The basics about p-local
subgroups and fusion are in sections 1.C-D, p-blocks appear for the first time in
1.E with Brauer’s first and third Main Theorems, Alperin’s weight conjecture is
recalled in 3.C, sections 5.A-D recall the general strategy to find the splitting
of Irr(G) (G a finite group) into blocks and the defect groups as an application
of Brauer’s second Main Theorem. Categorifications are evoked in 5.E, Rickard
equivalences in 9.C.
This text grew out of the course and talks I gave in July and September 2016
during the program “Local representation theory and simple groups” at CIB Lausanne. I heartily thank the organizers for giving me the opportunity to speak in
those occasions and publish in this nice proceedings volume.
On background and notation. We use freely the standard results and notation
of basic module theory (see first chapter of [Benson]). For characters and block
theory we refer to [NagaoTsu] and [AschKeOl, Ch. IV] but restate most theorems
used with references. For categories and homological algebra, we refer to the first
part of [Du17] whose notations we follow. For varieties, algebraic groups and
finite groups of Lie type, our notations are the ones of [DigneMic] and [CaEn].
We borrow as much as possible from [Ge17] and [Du17], but since their algebraic
4
MARC CABANES
group is denoted G and G respectively, we felt free to stick to our good old G. I
also thank Lucas Ruhstorfer for his careful reading, suggestions and references.
I. DEFINING CHARACTERISTIC
We first construct the finite groups GF that will be the main subject of this
survey. Symmetric groups are also evoked in Sections 5.E-G and 10. The groups
GF are commonly called finite groups of Lie type or finite reductive groups. In
order to simplify the exposition we will not try to cover the Ree and Suzuki groups,
nor speak of finite BN-pairs. We will even sometimes assume that F induces no
permutation of the roots (“untwisted groups”) and refer to the bibliography for
the original theorems in their full generality.
1. p-local subgroups and parabolic subgroups
The groups and subgroups we will study are defined as follows (see [CaEn],
[Carter2], [DigneMic], [MalleTe], [Sri], [Spr]).
Let p be a prime and F := Fp the algebraic closure of the field with p elements.
Let G be a connected algebraic group over F. We assume that it is defined over
a finite subfield Fq (q a power of p) thus singling out a Frobenius endomorphism
F : G → G. The group of fixed points
GF = {g ∈ G | F (g) = g}
is a finite group.
Remark 1.1. Our way of defining things may be less concrete than saying that
G is a subgroup of some GLn (F) (n ≥ 1) defined by polynomial equations (on
the matrix entries) with all coefficients in the finite subfield Fq . This is indeed
equivalent to the definition we gave, but the more intrinsic definition is generally
preferred and also leads to a more compact notation. Subgroups of G that are
F -stable are also very important.
Example 1.2. (a) The group GLn (F) is such a group G. It is defined over any
finite subfield and the map F : G → G raising matrix entries to the q-th power
gives GF = GLn (Fq ). Note that any element of GLn (F) has finite order and that
the Jordan decomposition g = gu gss of matrices coincides with the decomposition
g = gp gp′ into p-part and p′ -part. This also defines a notion of unipotent/semisimple elements and Jordan decomposition inside any algebraic group G over
F.
(b) The group Un (F) consisting of upper triangular unipotent matrices is clearly
defined over Fp and is stable under F defined in (a). Note that any element of
Un (F) has finite order a power of p.
(c) The group Dn (F) ∼
= (F× )n consists of invertible diagonal matrices. Every
element there has order prime to p.
(d) Groups of type GF are rarely finite simple groups. For instance, SLn (Fq ) is
such a group with G = SLn (F) but in general it is not possible to find a connected
group G such that GF is isomorphic to PSLn (Fq ). Even factoring out the center
of SLn (F) would produce a PGLn (F) whose subgroup of fixed points under F is ∼
=
PGLn (Fq ), a non-simple group! But realizing SLn (Fq ), a perfect central extension
of our simple group, is preferable for our representation theoretic purposes. Of
course any representation of PSLn (Fq ) identifies with a representation of SLn (Fq )
trivial on its center.
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
5
The unipotent radical Ru (H) of an algebraic group H is the maximal connected
normal unipotent subgroup of H (see [Hum1, 19.5]).
The groups G we study are assumed to be reductive, i.e. Ru (G) = {1}. This
implies essentially that Z(G) consists of semi-simple elements and Gad := G/Z(G)
is a direct product of abstract simple groups [MalleTe, §8.4]. The factors in the
direct product are in fact taken in a list obtained by the classification of simple
algebraic groups, due to Chevalley and depending on root systems in the usual
list.
We now introduce some subgroups of fundamental importance.
1.A. Parabolic subgroups and Levi subgroups: reductive groups. Each
group G as above contains closed subgroups B = Ru (B)T ≥ T called a Borel
subgroup and a maximal torus. Borel means connected solvable and maximal
as such. Torus means isomorphic to some Dn (F) (n ≥ 0) as in Example 1.2.(c).
Moreover the normalizer of T is such that the Weyl group
WG (T) := NG (T)/T
is finite and
S := {s ∈ WG (T) | B ∪ BsB is a subgroup }
generates WG (T). When w ∈ WG (T) the expression BwB means the set of
products b1 xb2 with bi ∈ B and x ∈ w where the latter is a class mod T. Since
T ≤ B, BwB is a single double coset with regard to B. The pair (WG (T), S)
satisfies the axioms of Coxeter systems (see [Hum2]).
One has the Bruhat decomposition
[
BwB (a disjoint union).
(1)
G=
w∈WG (T)
One classically defines the root system Φ(G, T) as a finite subset of the Zlattice X(T) := Hom(T, F× ) (algebraic morphisms). It is stable under the action
of WG (T). The actual definition of roots refers to the Lie algebra of G and
roots also define certain unipotent subgroups. One has a family of so-called root
subgroups
(Xα )α∈Φ(G,T)
ranging over all minimal connected unipotent subgroups of G normalized by T.
One has w Xα = Xw(α) for any α ∈ Φ(G, T), w ∈ W (G, T).
There is a basis of the root system ∆ ⊆ Φ(G, T) of cardinality the rank
of Φ(G, T) in R ⊗Z X(T). One has Φ(G, T) = Φ(G, T)+ ⊔ Φ(G, T)− where
Φ(G, T)+ = Φ(G, T) ∩ R+ ∆, Φ(G, T)− = −Φ(G, T)+ .
One has Xα ≤ B if and only if α ∈ Φ(G, T)+ (which defines Φ(G, T)+ and
therefore ∆ from B).
+
When α ∈ Φ(G, T) , s ∈ S, the condition X−α ≤ B ∪ BsB implies α ∈ ∆.
This establishes a bijection
δ: S → ∆
s 7→ δs .
(2)
The commutator formula in a simplified version says the following for any
linearly independent α, β ∈ Φ(G, T)
[Xα , Xβ ] ≤ hXiα+jβ | i, j ∈ Z>0 , iα + jβ ∈ Φ(G, T)i .
(3)
One calls parabolic subgroups of G the ones containing a conjugate of B.
Denoting WI := hIi ≤ W (G, T) for I ⊆ S, the subgroups of G containing B are
6
MARC CABANES
in bijection with subsets of S by the map
I 7→ PI := BWI B.
(4)
Note that P∅ = B, PS = G.
One has a semi-direct decomposition called the Levi decomposition
PI = Ru (PI ) ⋊ LI
(5)
where LI := T hXα | α ∈ Φ(G, T) ∩ Rδ(I)i, a reductive group with same maximal
torus as G, Borel subgroup B ∩ LDI and root system Φ(G, T) ∩ Rδ(I).
E
One denotes UI := Ru (PI ) = Xα | α ∈ Φ(G, T)+ , α 6∈ Rδ(I) .
Example 1.3. The case of G = GLn (F). Then B = TU is the group of upper
triangular matrices, U = Ru (B) the group of upper unipotent matrices (see Example 1.2.(b)). It is not difficult to see that NG (T) is the subgroup of monomial
matrices (each row and column has a single non-zero entry) and NG (T)/T identifies with the subgroup of permutation matrices ∼
= Sn , where S corresponds to the
set of transpositions of consecutive integers {s1 := (1, 2), . . . , sn−1 := (n − 1, n)}.
The roots Φ(G, T) = {α(i,j) | 1 ≤ i, j ≤ n, i 6= j} are defined as elements of X(T)
by
α(i,j) : T → F× , diag(t1 , . . . , tn ) 7→ ti t−1
(6)
j .
+
The elements of Φ(G, T) , resp. ∆, are defined by the condition i < j, resp.
j = i + 1.
When α ∈ Φ(G, T) corresponds to (i, j) then Xα is the subgroup of matrices
idn +λEi,j (λ ∈ F) where Ei,j is the elementary matrix with 1 as (i, j) entry and
0 elsewhere.
If I ⊆ S, let us write S\I = {sn1 , sn1 +n2 , . . . , sn1 +n2 +···+nk−1 } with n1 , n2 , . . . , nk−1 ≥
1 and define nk = n − (n1 + n2 + · · · + nk−1 ). Then the elements of PI = UI LI
decompose as
∗
∗
∗
∗ 0 0 0
idn1
n1
∗ ∗ ∗ ∗
0
n2
idn2 ∗
∗
0 ∗ 0 0
0 ∗ ∗ ∗
=
..
.
.
.
..
0 0 ..
0
0
. 0 0 ..
0 0 0 ∗
0
0
0 idnk
0 0 0 ∗
nk
∼
Note that LI = GLn (F) × GLn (F) × · · · × GLn (F).
2
1
k
1.B. Parabolic subgroups and Levi subgroups: finite groups. All the
above can be taken F -stable: F (B) = B, F (T) = T. Then one denotes B = BF ,
T = TF , N = NG (T)F and W = N/T = (WG (T))F . The later is generated by
the set
S := {wω | ω ∈ S/ hF i} ←→ S/ hF i
where ω ranges over F -orbits in S and if I ⊆ S, wI denotes the element of maximal
S-length in WI . From (1) one gets a Bruhat decomposition
[
G=
BwB , a disjoint union.
(7)
w∈W
For J ⊆ S corresponding to an F -stable subset J ⊆ S, the subgroups PJ , LJ are
F
F
F
F -stable, PJ := PF
J = BWJ B and LJ := LJ . Moreover UJ := UJ = Op (PJ ).
One has
PJ = UJ ⋊ LJ .
The roots are also acted upon by F and the quotient set Φ(G, T ) has properties similar to Φ(G, T). Similar ideas allow to associate to them p-subgroups
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
7
(Xα )α∈Φ(G,T ) that satisfy consequently an analogue of the commutator formula
(3) seen above.
The relevance to simple groups starts with the following (see [MalleTe, 12.5]).
Theorem 1.4. Assume S has no non-trivial partition into commuting subsets.
Assume G is perfect (i.e. [G, G] = G). Then G/Z(G) is simple.
Recall that a quasi-simple group is a finite perfect group H such that H/Z(H)
is simple. A universal covering of a simple group V is a quasi-simple group H
of maximal order such that H/Z(H) ∼
=V.
The classification of finite simple groups (CFSG) (see [GoLySo], [Asch,
§47]) tells us that finite non-abelian simple groups are either
• alternating groups An (n ≥ 5),
• groups of Lie type GF /Z(GF ) as above,
• or among the 26 so-called sporadic groups.
Remarkably enough, simple groups of Lie type have universal coverings that
are of type GF (short of 17 exceptions, see [GoLySo, 6.1.3]).
When dealing with finite groups GF , an important tool is Lang’s theorem. It
tells us that if C is a connected closed F -stable subgroup of G, then x 7→ x−1 F (x)
is surjective from C to itself.
1.C. p-local subgroups and simple groups of characteristic p type. The
proof of the classification of finite simple groups makes crucial use of the notions
of 2-local subgroups and of simple groups of characteristic 2 type, this last one to
separate simple groups of even and odd characteristic. The notions have also been
defined for any prime (see [Asch, Ch. 48]).
We fix here a prime p.
Definition 1.5. Let H be a finite group. A p-local subgroup of H is any normalizer NH (Q) where 1 6= Q ≤ H is a non trivial p-subgroup of H.
Definition 1.6. Let p be a prime and H a finite group. A radical p-subgroup of
H is any p-subgroup Q of H such that
Q = Op (NH (Q)).
Note that a Sylow p-subgroup of H is always p-radical.
Example 1.7. Let G = GF as in the last section, let I ⊆ S. Then UI defined in
1.B satisfies NG (UI ) = PI and UI = Op (PI ). Both properties are a consequence
of the commutator formula. This proves that the UI ’s are p-radical subgroups.
Proposition 1.8. The maximal p-local subgroups of a finite group H satisfying
Op (H) = {1} are normalizers of radical p-subgroups.
Proof. For any subgroup M ≤ H, we clearly have NH (Op (M )) ≥ M . Applying
this to our maximal p-local subgroup M we get that either M = NH (Op (M )) or
Op (M ) ⊳ H and therefore Op (M ) = {1}. But the second case is impossible by the
definition of p-local subgroups.
Definition 1.9. Let H be a finite simple group and p be a prime. Then H is
said to be of characteristic p type, if and only if
CX (Op (X)) ≤ Op (X)
for any p-local subgroup X of H.
This is equivalent to (8) holding for any maximal p-local subgroup.
(8)
8
MARC CABANES
The second statement in the above definition is a non trivial one. We refer to
the proof of [Asch, 31.16], using among other things Thompson’s A×B lemma.
One will of course check here that our groups GF give rise to simple groups of
characteristic p type, see [Asch, 47.8.(3)].
We take G = GF as in Sect. 1.B above. Recall the subgroups PI = UI ⋊ LI for
I ⊆ S.
Theorem 1.10 ([BoTi65]).
(1) The p-radical subgroups of G are the {g (UI ) |
g ∈ G, I ⊆ S} with UI = Op (PI ), PI = NG (UI ).
(2) If g ∈ G and I, J ⊆ S are such that g (UI ) = UJ , then I = J and g ∈ PI .
(3) If S ) I and G/Z(G) is simple then CG (UI ) ≤ Z(G)UI .
Corollary 1.11. If G/Z(G) is simple then it has characteristic p type.
We finish this subsection by giving some ideas in the proof of Theorem 1.10.
First the theorem has an equivalent in G as follows (Platonov 1966, see [Hum1,
30.3]).
Lemma 1.12. In G, if V is a closed subgroup of U, then the sequence V0 = V ,
Vi+1 := Vi Ru (NG (Vi )) is an ascending sequence stabilizing at some Ru (P(V ))
where P(V ) is a parabolic subgroup of G.
Note that if V is F -stable then all Vi ’s and therefore P(V ) itself are F stable. Once written as g PI for g ∈ G and I ⊆ S, using F -stability one gets
g−1 F (g)
PF (I) = PI . By the argument we are going to use for (2) of the Theorem,
this implies F (I) = I and g −1 F (g) ∈ PI . Lang’s theorem then allows to assume
′
that g = g ′ h where g ′ ∈ G and h ∈ PI , so that P(V ) = g PI with F (I) = I and
g ′ ∈ G.
Assume moreover V p-radical in G. The inclusions V ≤ Ru (P(V )) and NG (V ) ≤
P(V ) imply NRu (P(V ))F (V ) ⊳ NG (V ). But Ru (P(V ))F is a p-subgroup of G, so
p-radicality of V implies NRu (P(V ))F (V ) = V . But V ≤ Ru (P(V ))F is an inclusion
of p-groups so we must have indeed V = Ru (P(V ))F . Using the above this gives
′
V = g UI , hence the claim (1).
For the claim (2), writing g ∈ BwB thanks to Bruhat decomposition (7) and
using that B normalizes both UI and UJ , one finds that w UI = UJ . Assume for
simplicity that F acts trivially on W (G, T) and S. Our equality implies on roots
that
+
+
+
w(Φ(G, T) \ Φ(G, T)I ) ⊆ Φ(G, T) .
But a basic property of Weyl groups acting on roots tells us that any element of
+
+
W (G, T) decomposes as w = w′ v where v ∈ hIi and w′ (Φ(G, T)I ) ⊆ Φ(G, T) .
+
+
′
′
But then w (Φ(G, T) ) = Φ(G, T) , therefore w = 1, w = v ∈ hIi and g ∈ PI .
(3) Using (7) again and arguing on roots it is easy to show that CG (UI ) ≤ B.
We then check that under our assumptions, CB (UI ) ≤ Z(G). We show it for I = ∅
and refer to [Asch, 47.5.3] for the general case. Given the semi-direct product
structure B = U ⋊ T with U the Sylow p-subgroup of B, it is not difficult to see
that our claim about CB (U ) reduces to the inclusion CT (U ) ≤ Z(G). For s ∈ S,
let Cs = CLs (Us ). It is normalized by Xs , Us (hence U ), but also by s and we
have seen Cs ≤ B. So
Cs ≤ Ls ∩ B ∩ B s = Ls ∩ T Us = T.
So Cs = CT (Us ) normalizes U , hence centralizes it since U ∩ Cs = {1}. So
Cs = CT (U ). We deduce that CT (U ) is normalized by any s ∈ S and by T , hence
by N . On the other hand B = T U ≤ CG (CT (U )), so the latter is a parabolic
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
9
subgroup normalized by N , hence normal in G. By our hypothesis, it has to equal
G, hence the inclusion CT (U ) ≤ Z(G).
1.D. Consequences on fusion. The problem of p-fusion in finite groups is as
follows. Let Q be a Sylow p-subgroup of a finite group H. One wants to know
when subsets of Q can be H-conjugate.
More generally one defines the “fusion system” FQ (H) as follows
Definition 1.13 ([AschKeOl, I.1.1]). For Q a Sylow p-subgroup of H, the fusion
system FQ (H) has objects the subgroups of Q and if Q1 , Q2 ≤ Q, one defines
HomFQ (H) (Q1 , Q2 ) ⊆ Hom(Q1 , Q2 ),
the former consisting of maps
adh,Q1 ,Q2 : Q1
x
→ Q2
7→ hxh−1
for h ∈ H with h Q1 ≤ Q2 .
A theorem by Alperin (1967) first showed that this category is generated by
certain elementary operations, see [Asch, 38.1].
A tame intersection of Sylow p-subgroups of H is a p-subgroup of type Q1 ∩Q2
with Q1 , Q2 both Sylow p-subgroups of H and NQ1 (Q1 ∩ Q2 ), NQ2 (Q1 ∩ Q2 ) both
Sylow p-subgroups of NH (Q1 ∩ Q2 ).
Theorem 1.14 ([Al67]). Let h ∈ H and A ⊆ Q such that Ah ⊆ Q. Then
there exist Sylow p-subgroups Q1 , . . . , Qn and elements hi ∈ NH (Q ∩ Qi ) for i =
1, . . . , n − 1 such that
(i) h = h1 . . . hn−1 ,
(ii) for any i = 1, . . . , n, Q ∩ Qi is a tame intersection, and
(iii) A ⊆ Q ∩ Q1 , Ah1 ⊆ Q ∩ Q2 , . . . , Ah1 ...hn−1 ⊆ Q ∩ Qn .
This can be summed up as saying that normalizers of tame intersections Q ∩ Q′
(Q another Sylow p-subgroup of H) generate FQ (H).
′
Remark 1.15. A tame intersection Q1 ∩ Q2 of Sylow subgroups is a p-radical
subgroup (see Definition 1.6). Indeed Op (NH (Q1 ∩ Q2 )) is included in both
NQ1 (Q1 ∩ Q2 ) and NQ2 (Q1 ∩ Q2 ) by the tame intersection hypothesis, so included
in Q1 ∩ Q2 . In the case of groups G = GF it means that they are G-conjugates of
subgroups UI (I ⊆ S) thanks to Theorem 1.10.
Alperin’s theorem has been strengthened by Goldschmidt so as to find a minimal
family of normalizers of so-called essential p-subgroups (see [AschKeOl, I.3.2])
which generates FQ (H). In the case of groups G = GF , it gives the following
[Puig76, Appendix I]. Recall that U := UF is a Sylow p-subgroup of G.
Theorem 1.16. The fusion system FU (G) is generated by minimal parabolic subgroups P{s} = B ∪ BsB for s ranging over S.
1.E. Consequences on p-blocks. We show that the condition of being of characteristic p type has strong consequences on the p-blocks of our simple group.
Blocks and the Brauer morphism. Let us recall what are p-blocks of a finite
group H.
10
MARC CABANES
We keep F the algebraic closure of Fp and consider the group algebra FH. As
for any finite dimensional algebra over a field, one has a maximal decomposition
FH ∼
(9)
= B0 × B1 × · · · × Bν
as a direct product of F-algebras. The corresponding two-sided ideals Bi of FH
are called the p-blocks of H, one denotes Bl(H) = {B0 , B1 , . . . , Bν }. The unit
element bi of each Bi is a primitive idempotent of the center Z(FH) and one
has a bijection between Bl(H) and the primitive idempotents of Z(FH) since any
such idempotent b defines the block FHb. Any FH-module M decomposes as
M = ⊕i Bi M as F H-module. So each indecomposable module has only one block
acting non-trivially on it. This induces a partition IBr(H) = ⊔νi=0 IBr(Bi ) of the
isomorphism classes of simple FH-modules. The principal block B0 (H) is by
definition the one corresponding to the trivial FH-module, i.e. the line F with H
acting as identity, often denoted as F or 1.
When Q is a p-subgroup of H, the Brauer morphism
BrQ : Z(FH) → Z(FCH (Q))
X
X
λh h
λh h 7→
h∈H
(10)
(11)
h∈CH (Q)
is a morphism of commutative algebras. The defect groups of a given block Bi
are the p-subgroups Q of H maximal for the property that BrQ (bi ) 6= 0. For a
given Bi they form a single H-conjugacy class. For D ≤ H a given p-subgroup of
H, one denotes Bl(H | D) the subset of Bl(H) consisting of blocks admitting D
as defect group.
The principal block has defect group any Sylow p-subgroup of H.
A block Bi has defect group {1} if and only if Bi is a semi-simple algebra (in
fact simple with | IBr(Bi )| = 1), this is called a block of defect zero (defect was
first defined as a numerical invariant corresponding to the exponent d such that
|D| = pd ).
Brauer’s first and third so-called Main Theorems are as follows. One keeps H
a finite group.
Theorem 1.17. Let Q be a p-subgroup of H.
(i) The Brauer morphism BrQ induces bijections
Bl(H | Q) ←→ Bl(NH (Q) | Q) ←→ Bl(QCH (Q) | Q)/NH (Q)- conj .
(ii) Through the above, the principal blocks of H, NH (Q) and CH (Q) correspond.
Blocks in the defining characteristic. Let us return to our finite reductive
groups G = GF , or more generally simple groups of characteristic p type (see 1.C).
Proposition 1.18 (Dagger-Humphreys, see [Hum3, §8.5]). Assume H is a finite
simple group of characteristic p type. Then the non principal p-blocks of H all
have defect {1}.
Proof. Let D be a defect group 6= {1} of a p-block B of H. By (i) of the above
theorem, Bl(DCH (D) | D) 6= ∅. The condition that H has characteristic p type
implies that CX (Op (X)) ≤ Op (X) for X = NH (D). But we clearly have
DCH (D) ≤ Op (X)CX (Op (X)),
so DCH (D) is a p-group. A p-group has only one simple module over F (see
[Benson, 3.14.1]), hence only one p-block, the principal block. So by (ii) of the
above theorem, B is the principal block of H.
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
11
We will see later that blocks of defect {1} actually exist in this case, corresponding to the so-called Steinberg module, see Theorem 3.3.
2. Yokonuma-Hecke algebras in characteristic p
Iwahori-Hecke algebras are algebras similar to the group algebras of Coxeter
groups (W, S), only the quadratic relations s2 = 1 (s ∈ S) have been there replaced
by an equation
s2 = (q − 1)s + q
where q is a scalar. See [GePf00]. This models the endomorphism algebras of
F
induced representations IndG
B 1 for G = G as before with Weyl group W and q
the order of root subgroups Xα .
Yokonuma-Hecke algebras are a bit larger and serve as model for the endomorphism algebra of induced representations IndG
U 1.
2.A. Self-injective endomorphism algebras, a theorem of Green. We first
present a general theorem of J.A. Green about certain A-modules where A is a
finite dimensional F-algebra. Green’s theorem shows that if Y is an A-module such
that EndA (Y ) is self-injective then EndA (Y )-modules give a lot of information on
A-modules, in particular the simple submodules of Y . This will be applied to
G
A = FG (F and G as in Sect. 1.B), Y = IndG
U F, so that EndFG (IndU F) is a
Yokonuma-Hecke algebra in characteristic p.
In the following Y is a finitely generated left module over the finite dimensional
F-algebra A, E := EndA (Y )opp and one considers the functor HY sending an
A-module M to the E-module HY (M ) := HomA (Y, M ), E acting through composition by elements of EndA (Y ) on the right. Note that HY (Y ) = E the regular
left module.
Theorem 2.1. Assume
(1) there is a linear map λ : E → F such that for any x ∈ E, x 6= 0, one has
λ(xE) 6= 0 6= λ(Ex), and
(2) any simple A-module is both a submodule and a quotient of Y .
Then the functor HY sends simple A-modules to simple E-modules and this induces
a bijection between isomorphism types of simple modules for both algebras.
The relevance of self-injectivity (implied by the slightly stronger hypothesis (1)
above, see [Benson, §1.6]) essentially lies in the following lemma where we keep
the same assumptions.
P
Lemma 2.2. Let V ⊆ E = EndA (Y ) an E-submodule. Denote V.Y := f ∈V f (Y ) ⊆
Y . Then HY (V.Y ) = V by taking the image of the latter inclusion.
Proof. We clearly have V ⊆ HomA (Y, V.Y ) = HY (V.Y ) as subspaces of HomA (Y, Y ) =
HY (Y ). If the inclusion is strict, since it is an inclusion of E-modules, there is
V ( U ⊆ HY (V.Y ) ⊆ E,
E-modules with U/V simple. Hypothesis (1) implies that projective and injective
E-modules coincide (see [NagaoTsu, 2.8.11]), so any finitely generated module
injects into a free one and any simple module into the regular one. So we have a
map φ : U → E of E-modules such that φ(U ) 6= 0 = φ(V ). By injectivity of the
regular module E the map φ extends into φb : E → E.
12
MARC CABANES
0
/ U
/E
φ
φb
E
Such a map φb is clearly of the form HY (φ′ ) for some φ′ : Y → Y a map of
b ) = 0 6= φ(U
b ) implies φ′ (V.Y ) = 0 6= φ′ (U.Y ). But on the
A-modules. Now φ(V
other hand V.Y = U.Y since V.Y ⊆ U.Y ⊆ HY (V.Y ) ⊆ V.Y . This contradiction
finishes the proof.
The proof of the Theorem goes as follows. Let M be a simple A-module.
Then M is a factor and a submodule of Y by (2), so E = HY (Y ) ⊇ HY (M ) =
HomA (Y, M ) 6= 0. Let now 0 6= V ( HY (M ) ⊆ E an E-submodule. By simplicity
of M , V.Y = M , but the lemma tells us that V = HY (V.Y ) = HY (M ). This
shows that HY (M ) is simple.
Moreover, every simple E-module V is obtained that way since V embeds in
E = HY (Y ) as seen before, thus allowing to form V.Y and the Lemma gives
V = HY (V.Y ). If M is a simple submodule of V.Y , then
0 6= HY (M ) ⊆ HY (V.Y ) = V
so indeed V = HY (M ).
Eventually, if M, M ′ are simple A-modules and HY (M ) ∼
= HY (M ′ ), then M
and M ′ can be assumed to be submodules of Y , so that HY (M ) and HY (M ′ ) are
seen as submodules of E. Now the isomorphism HY (M ′ ) → HY (M ) extends to
some map E → E that writes HY (φ) for φ : Y → Y . The restriction of φ to M
gives a non zero map M → M ′ , and therefore M ∼
= M ′.
/ E = HY (Y )
/ HY (M )
0
6=0
0
/ HY (M ′ )
/ E = HY (Y )
Example 2.3. Assume now that H is a finite group and X a subgroup, let k be
any commutative ring. The kH-module Y = IndH
X k = kH ⊗X k is the permutation module on the set of cosets {hX | h ∈ H}. Denote ω ∈ Y the element
corresponding to the coset X or 1 ⊗ 1 ∈ kH ⊗X k. If M is a kH-module, one
denotes by M X the space of fixed points under X. By Frobenius reciprocity, one
can identify explicitly
∼
HomkH (Y, M ) −
→ M X , f 7→ f (ω).
This can serve first to give a basis of EndkH (Y ) ∼
= Y X as a vector space. One
′
′
has EndkH (Y ) = ⊕n∈X\H/X k.an where an is the kH-linear map Y → Y defined
by
X
X
a′XhX (ω) =
y=
xhω.
(12)
y∈XhXω
x∈X/X∩h X
One has EndkH (Y ) ∼
= EndkH (Y )opp by a′n 7→ an := a′n−1 .
Moreover, through the identification above the action of aXhX on M X is by
X
xh−1 m.
(13)
m 7→ aXhX (m) =
x∈X/X∩X h
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
13
2.B. Yokonuma-Hecke algebras: a presentation. As said before we will apply Green’s theorem to A = FG, Y = IndG
U F in the notations of Sect. 1.B. We
recall the subgroups B = BF , U = Op (B), T = TF , W = W (G, T)F , etc.. In
order to simplify a bit we assume that F acts trivially on W (G, T), so that also
S = S.
One writes U− = U wS where wS is the longest element of W with regard to S.
G
Definition 2.4. Let HP
F (G, U ) = EndkG (IndU F). For n ∈ N , let an : Y → Y
−1
defined by an (1 ⊗ 1) = u∈U∩U n un ⊗ 1.
−
For s ∈ S corresponding to some δ ∈ ∆ through (2), one defines
Ts := T ∩ hXδ , X−δ i
and one can find some representative
ṡ ∈ N ∩ Xδ X−δ Xδ
(see [CaEn, 6.3.(i)]). Moreover for s1 , s2 ∈ S and r the order of the product s1 s2
in W , one has
ṡ1 ṡ2 · · · = ṡ2 ṡ1 . . . (r terms on each side).
(14)
Theorem 2.5. Let n, n′ ∈ N , s ∈ S.
(1) an an′ = ann′ as soon as lS (nn′ T ) = lS (nT ) + lS (n′ T ).
(2) The at ’s for t ∈ T P
generate a semi-simple subalgebra ∼
= FT .
(3) (aṡ )2 = −|Ts |−1 aṡ t∈Ts at .
(4) HF (G, U ) is presented as an algebra by symbols an (n ∈ N ) subject to the
relations (1) and (3) above.
′
′
′
nn
n
n n
Proof. (1) The additivity of lengths implies that U ∩ U−
= U ∩ U−
.(U ∩ U−
)
with uniqueness by considerations on roots. Note that this is the same argument
as for the corresponding equation in Iwahori-Hecke algebras EndCG (IndG
B 1). The
equality an an′ = ann′ then follows by the definition of the an ’s.
(2) Clear from the first point, noting that T has order prime to p.
s
(3) Let δ ∈ ∆ correspond to s by (2), so that Xδ = U ∩ U−
. The Bruhat
decomposition (7) in Ls implies that hXδ , X−δ i = Xδ Ts ∪ Xδ Ts ṡXδ . For v ∈
Xδ \ {1} one denotes t(v) ∈ Ts such that
ṡ−1 v ṡ−1 ∈ Xδ t(v)−1 ṡ−1 Xδ .
From Definition 2.4, one gets clearly
X
ṡ−2 u ⊗ 1 +
(aṡ )2 (1 ⊗ 1) =
u∈Xδ
X
aṡt(u) (1 ⊗ 1).
u∈Xδ \{1}
The first term is 0 since each u acts trivially on 1 ⊗U 1. The second term gives
what is claimed once we check that the cardinality |ṡXδ ṡ ∩ Xδ ṡtXδ | is the same for
any t ∈ Ts . This is an easy check in the group hXδ , X−δ i which in our hypotheses
is a quotient of ∼
= SL2 (q).
(4) The proof is similar to the one for Iwahori-Hecke algebras [CurtisRei, §67].
2.C. Yokonuma-Hecke algebras: simple modules.
Proposition 2.6. Let nS be an element of N whose class mod T is the element
wS ∈ W of largest S-length. Let
λ : HF (G, U ) → F
14
MARC CABANES
the F-linear map sending anS to 1 and an for n ∈ N , n 6= nS to 0. Then λ
vanishes on no non-zero left or right ideal of HF (G, U ).
′
Proof. From Theorem
P 2.5 it is clear that when n, n ∈′′ N , the product′′an an′
is always in ann′ + n′′ Fan′′ where
P the sum is over n ∈ N with lS (n T ) <
lS (nT ) + lS (n′ T ). Now if 0 6= x = n∈N µn an with µn ∈ F, let n0 be such that
µn0 6= 0, with lS (n0 T ) maximal as such. Then an0 an−1 nS = anS n−1 an0 = anS and
0
0
therefore λ(xan−1 nS ) = λ(anS n−1 x) = µn0 6= 0.
0
0
×
Definition 2.7. For θ : T → F a group morphism, let Sθ := {s ∈ S | θ(Ts ) = 1}.
One calls admissible pair any pair (θ, I) where θ∈ Hom(T, F× ) and I ⊆ Sθ .
Theorem 2.8. The simple HF (G, U )-modules are one-dimensional. Seen as maps
HF (G, U ) → F, they are of the form ψ(θ,I) where (θ, I) is an admissible pair and
ψ(θ,I) is defined by
(a) ψ(θ,I) (at ) = θ(t) for any t ∈ T
(b) ψ(θ,I) (aṡ ) = −1 for s ∈ I, 0 otherwise.
Proof. Let V be a simple HF (G, U )-module. The subalgebra ⊕t∈T Fat being commutative, semi-simple with F algebraically closed, V decomposes as a sum of
lines stable under the at ’s. Let L ⊆ V be such a line and n0 ∈ N such that
an0 .L 6= 0 and lS (n0 T ) is maximal as such. One shows that Fan0 .L is stable under HF (G, U ). For t ∈ T , one has at an0 .L = atn0 .L = an0 atn0 .L = an0 .L. For
s ∈ S, if lS (sn0 T ) = lS (n0 T )+ 1 then Theorem 2.5.(1) and maximality of n0 imply
aṡ an0 L = aṡn0 L = 0. If lS (sn0 T ) = lS (n0 T ) − 1 then Theorem 2.5.(1) and (3)
imply aṡ an0 = aṡ aṡ aṡ−1 n0 ∈ an0 (⊕t∈T at ) hence aṡ an0 L ⊆ an0 L. We get our claim
by noting that the at ’s and the aṡ ’s generate HF (G, U ) by Theorem 2.5.(4).
The form of the F-algebra morphisms HF (G, U ) → F is easy to deduce from
Theorem 2.5.(4).
3. Simple FG-modules and p-blocks
As announced before, we now apply Theorem 2.1 and the information gathered
on HF (G, U ) to simple FGF -module. This theory is due to J.A. Green (see [Gre78],
[Tin79], [Tin80]). This provided a more conceptual framework to a classification
of simple modules of split BN-pairs due to Curtis-Richen [Cu70], [Ri69] (see also
[CaLu74]).
Among other properties of the simple FGF -modules we show the existence of
a block of defect zero (see Proposition 1.18) associated to the so-called Steinberg
module.
The notations are the same as in the preceding chapter.
3.A. Simple FG-modules, the Steinberg module.
Theorem 3.1. For any simple FG-module M , the subspace of fixed points M U is
a line. Moreover M = FU− .M U .
Theorem 3.2. There is a bijection between the isomorphism types of simple FGmodules and the set of admissible pairs (see Definition 2.7).
Let the simple FG-module M correspond to the pair (θ, I) then
(i) T acts by θ on the line M U
P
(ii) for s ∈ S associated with δ ∈ ∆ and m ∈ M U one has u∈Xδ uṡ.m = −m
if s ∈ I, 0 if s ∈ S \ I.
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
15
(iii) (Smith 1982 [Sm82]) if J ⊆ S and UJ LJ is the Levi decomposition of the
parabolic subgroup PJ , then M UJ is a simple FLJ -module associated with
the admissible pair (θ, J ∩ I) of LJ .
Theorem 3.3. Keep M a simple FG-module associated with the admissible pair
(θ, I). Then M is projective if and only if I = S.
If moreover G is perfect and G := G/Z(G) is simple, then G has only two
p-blocks
• the principal block
• the block whose unique simple module is the FG-module corresponding to
the simple FG-module associated with the admissible pair (1, S).
The projective simple module associated with the admissible pair (1, S) is called
the Steinberg module.
Proof of the Theorems. One applies Theorem 2.1 to A = FG, Y = IndG
U F. The
condition (1) of the theorem is satisfied by Proposition 2.6. The condition (2)
comes from the fact that if M is simple HomFG (Y, M ) ∼
= M U 6= 0 since U is a
p-group and Y is isomorphic with its F-dual. The first statement of Theorem 3.1
along with Theorem 3.2.(i) and (ii) then come from Theorem 2.1 and the explicit
description we made of the functor HY in our case, see (13).
For the equality M = FU− .M U , it is enough to show that FU− .M U is stable
under G. The latter is generated by U− , T and S, so we just check stability under
T and S. First FU− .M U is T -stable since T normalizes U and U− . Let now s ∈ S
corresponding to δ ∈ ∆. Then
s
sU− M U = s(U− ∩ U−
)X−δ M U ⊆ U− sX−δ M U
so it suffices to show that sxM U ⊆ FU− M U for any x ∈ X−δ . Using the Bruhat
decomposition in Ls = T Xδ ∪ Xδ sT Xδ , one gets x ∈ Xδ sT Xδ or x = 1. In the
first case
sxM U ⊆ X−δ T Xδ M U = X−δ M U ⊆ U− M U .
P
On the other hand ( x∈X−δ x)sM U ⊆ M U by (ii) of Theorem 3.2, so the case
x = 1 can be deduced from the first case just treated.
(iii) The weaker statement that FLJ .M U is a simple FLJ -module is enough for
our purpose. Indeed if M ′ ⊆ FLJ .M U is a simple FLJ -submodule, it has fixed
points under the p-subgroup U ∩ LJ , but M ′ ⊆ M UJ since UJ is normalized by
LJ , so M ′U∩LJ ⊆ M UJ (U∩LJ ) = M U by the Levi decomposition. Since M U is a
line, M ′ must contain it and therefore M ′ ⊇ FLJ .M U .
On Theorem 3.3. M is projective if and only if its restriction to the Sylow
p-subgroup U− is projective (see for instance [Benson, 3.6.9] or (17) below), i.e.
free as an FU− -module. By Theorem 3.1, the restriction to U− is FU− /V where
V = {v ∈ FU− | v.M U = 0}, and FU− /V is free if andPonly if V = 0. Since the only
simple submodule of FU− is the line Fσ where σ = u∈U− u, one gets that M is
P
projective if and only if ( u∈U− u)M U 6= 0. Let n0 := ṡ1 ṡ2 . . . ṡl where s1 s2 . . . sl
is a decomposition of the lS -longest element of W . By the Theorem 3.2.(ii) giving
the action of HF (G, U ) on the line M U , one has
X
n−1
0 (
u∈U−
u)M
U
=(
X
u∈U−
U
u)n−1
0 M
= ψθ,I (an0 )M
U
=
l
Y
i=1
By the definition of ψθ,I and since {s1 , . . . , sl } = S, one has
and only if I = S.
ψθ,I (aṡ ) M U . (15)
Ql
i=1
ψθ,I (aṡ ) 6= 0 if
16
MARC CABANES
Now concerning the p-blocks. We know from Proposition 1.18 and Corollary 1.11 that G has only one block of defect 6= {1}. Let’s see that there is only one
block of defect {1}. This would correspond to a simple projective FG-module. On
the other hand Z(G) is a p′ -group (since U ∩ U− = {1}), so we get an FG-module
simple and projective. By Theorem 3.2 it has to be associated with an admissible
pair (θ, S). It is not too difficult to check that when G is perfect the condition
Sθ = S implies θ = 1 (show first that the associated M is one-dimensional as a
consequence of Theorem 3.2.(iii)).
3.B. Relation with weight modules. Assume that our pair (G, F ) from Sect.
1.B is such that F acts trivially on the Weyl group NG (T)/T. The simple FGF modules have been classified in the following way by R. Steinberg in the 60’s (see
[St63], [Hum3, §2.11]). First the irreducible rational representations
G → GLn (F)
are classified by the subset of so-called dominant weights X(T)+ ⊆ X(T) where
λ ∈ X(T) is dominant if and only if (λ, δ ∨ ) ≥ 0 for any fundamental coroot
δ ∨ ∈ Φ(G, T)∨ , δ ∈ ∆. Let us denote by M(λ) the corresponding G-module.
Most of the features described in 3.A above are also present regarding the
rational modules M(λ). The link between the two situations is provided by the
following.
Theorem 3.4. The M(λ) for λ such that 0 ≤ (λ, δ ∨ ) ≤ q − 1 have irreducible
restrictions to GF . This gives all simple FGF -modules only once when G =
[G, G].
Among the properties of the M(λ)’s is the fact that M(λ) has a line of fixed
points under U. The torus T acts by λ on that line, and one proves easily the
following relation with the description given before
Proposition 3.5. The admissible pair associated to ResG
GF M(λ) is (θ, I) where
T
θ = ResTF λ and I ⊆ S is in bijection by (2) with the fundamental roots δ such
that (λ, δ ∨ ) = q − 1.
3.C. Alperin weight conjecture. For F an algebraic closure of Fp and H a finite
group, let us recall Alp(H) the set of H-conjugacy classes of pairs (Q, π) where Q
is a p-subgroup of H and π is a simple projective F(NH (Q)/Q)-module. Alperin
conjectured that
| Alp(H)| equals the number of simple FH-modules
(16)
for all finite groups H and primes p [Al87].
We take G = GF as before.
Theorem 3.6. G satisfies Alperin’s weight conjecture (16) for the prime p. Moreover there is a map M 7→ (Q, π) inducing a bijection IBr(G) → Alp(G) and such
that
(i) the bijection is Aut(G)-equivariant
(ii) π, seen as an FNG (Q)-module, is a submodule of M Q .
Proof. Let us first note that for any (Q, π) in Alp(G), the subgroup Q is p-radical,
or equivalently that Op (L) = {1} for L := NG (Q)/Q. Indeed L has an FL-module
π that is simple and projective. Then π Op (L) is a non trivial FL-submodule, so
Op (L) acts trivially on π. On the other hand π remains projective when restricted
to Op (L), so it is a free FOp (L)-module. This is possible only if Op (L) = {1}.
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
17
Now by Theorem 1.10.(1) and (2), Alp(G) is in bijection with the set of pairs
(UI , π) where I ⊆ S and π is an FLI -module that is simple and projective. From
Theorem 3.2 we know that any such module is associated to an admissible pair
(θ, I) of LI , i.e., θ : T → F× has to be such that θ(Tδ ) = {1} for any δ ∈ ∆
corresponding to an element of I.
By Theorem 3.2 we then see that our sets IBr(G) and Alp(G) are both in
bijection with the admissible pairs for G.
Moreover if M ↔ (θ, I) ↔ (Q, π) = (UI , MLI (θ, I)), one has that Q is the smallest p-radical subgroup of G normal in U such that FNG (Q)M U is a simple projective module for NG (Q)/Q (use Theorem 3.2.(iii)). Then π = FNG (Q)M U . This
intrinsic definition of the map shows that it is equivariant for automorphisms that
preserve U . The latter being a Sylow p-subgroup and inner automorphisms acting
trivially on both IBr(G) and Alp(G), we actually get equivariance for Aut(G).
Remark 3.7. (a) Let us recall Green’s notion of vertex of an indecomposable
kH-module M (see [NagaoTsu, §4.3]). It is a subgroup V of H minimal for the
V
property that M is isomorphic to a direct summand of IndG
V ResG M . It is easy
to see that of
V
M is a direct summand of IndG
V ResG M for V a Sylow p-subgroup of H.
(17)
Consequently the vertex of an indecomposable module is always a p-subgroup.
On the other hand, Alperin’s conjecture asks for associating to each simple FHmodule a conjugacy class of p-subgroups of H. When H is p-solvable, the vertex
of the given simple FH-module provides such a correspondence (see [Oku81, 4.1],
[IsNa95]). This can’t be a solution for our groups G = GF since there, when
G/Z(G) is simple non abelian, vertices of simple modules are Sylow p-subgroups
except for simple projective modules (Dipper, see [Di80], [Di83]).
(b) In this case of the defining characteristic a more suggestive definition for Q
associated with a simple FG-module M is as follows. The module IndG
U F decomposes as a sum Y1 ⊕ · · · ⊕ Yv where the Yi are indecomposable. Then M is the
quotient of a single Yi0 , and Q is the vertex of that Yi0 .
(c) In our case, all p-radical subgroups of G are present in Alp(G). Whether this
is a general fact relates strongly with the question of quasi-simple groups having
blocks of central defect (case of Q = Op (G)). Quasi-simple groups GF have such
blocks for all primes. For the primes 2 and 3, there are infinitely many alternating
groups An without block of defect zero. For all that, see below Theorem 10.1.
II. NON-DEFINING CHARACTERISTIC (ℓ 6= p)
4. Rational series and ℓ-blocks
From now on we will be looking at modular aspects of the representations of
our groups GF (see Sect. 1.B) with regard to a prime ℓ different from the defining
prime p. So we assume essentially that a prime ℓ 6= p has been chosen and also
that we have a so-called ℓ-modular system (O, K, k) where O is a complete
valuation ring with ℓ ∈ J(O), K its fraction field, k = O/J(O). One assumes that
O contains |H|-th roots of 1 for any finite group H we encounter (see [NagaoTsu,
§3.6]).
18
MARC CABANES
4.A. Blocks and central functions. Let us recall the notion of ℓ-blocks of a
finite group H as decomposing the group algebra kH = B0 × B1 × · · · × Bν as in
(9) where Bi = kHbi with bi primitive idempotents of the center Z(kH). The field
k having enough roots of unity with regard to H, the bi belong in fact to kH (one
can also impose that k = k, see [AschKeOl, Sect. IV.4.2]). On the other hand it
is easy to see that reduction mod J(O) induces an epimorphism
Z(OH) → Z(kH)
(18)
which implies through the lifting of idempotents that the blocks of OH and the
p-blocks of H identify by OHei 7→ kHbi where bi is the reduction mod J(O)
of ei . Now the p-blocks of H also induce a partition of Irr(H), whose elements
are seen as isomorphism types of simple KH-modules,
namely Irr(H) = ⊔i Irr(Bi )
Q
corresponding to the decomposition KH = i KHei . We will also need to look at
character values, that is we see the elements of Irr(H) as central functions H → K.
Noting that the elements of Irr(H) take values in the subring of K generated by
the |H|-th roots of 1, we even see Irr(H) as a C-basis of CF(H) the complex vector
space of central functions H → C, hence with the decomposition
CF(H) = ⊕i CF(H | Bi ).
Each element χ ∈ Irr(H) defines some central idempotent eχ :=
Z(KH) and
X
eχ .
ei =
(19)
χ(1)
|H|
P
h∈H
χ(h−1 )h ∈
(20)
χ∈Irr(Bi )
We are later interested in “union of blocks” in Irr(H). It is an easy exercice to
prove the following.
Proposition 4.1. Let A ⊆ Irr(H). The three statements below are equivalent.
(i) A
P is a union of subsets Irr(Bi ).
(ii)
χ∈A eχ ∈ OH.
(iii) The projection prA : CF(H) → CF(H) associated to A, sends the regular
character regH to a central function with values in |H|O = |H|ℓ O, namely
prA (regH )(h) ∈ |H|ℓ O for all h ∈ H.
(21)
4.B. Uniform and p-constant functions. We return to our groups G = GF
keeping the same notation except for the basic pair T ≤ B of F -stable maximal
torus and Borel, that we rename T0 ≤ B0 since we will now allow our maximal
tori (even when F -stable) not to be included in F -stable Borel subgroups.
We give some elements of Deligne-Lusztig’s theory on Irr(GF ) in a very quick
fashion. We refer to the contributions by Geck and Dudas for a more in-depth
introduction (see [Ge17], [Du17]).
There are basically two very important facts about ordinary characters of finite
groups of Lie type. The functors RG
L and the existence of rational series. A
third basic feature – a so-called Jordan decomposition of characters – will be
seen later (see Sect. 4.D).
F
F
The functor RG
L : Z Irr(L ) → Z Irr(G ) is defined as follows.
One takes P = LRu (P) a Levi decomposition of a parabolic subgroup of G and
one assumes that L (and not necessarily P) is F -stable. Then the variety
YP = {gRu (P) | g −1 F (g) ∈ Ru (P)F (Ru (P))}
F
F
(22)
is clearly acted on by L on the right and G on the left. Understandably any
cohomology theory of that object would produce modules acted on by those finite
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
19
groups on those sides. One denotes by Hic (YP ) the i-th cohomology group defined
by ℓ-adic cohomology with compact support of YP tensored by C (see more details
opp
in Sect. 9.A below), so as to give a CGF × LF -module.
F
F
Definition 4.2. One defines RG
L : Z Irr(L ) → Z Irr(G ) by
X
[M ] 7→
(−1)i [Hic (YP ) ⊗CLF M ]
i∈Z
F
where [M ] is the class of a CL -module M in N Irr(LF ). One denotes by
∗
F
F
RG
L : Z Irr(G ) → Z Irr(L )
the adjoint map for the usual scalar product of central functions.
Remark 4.3. (a) The maps defined above are independent of the choice of P for a
given L in many cases in particular when L is a torus (see [DigneMic, Ch. 11]). The
independence in the general case relates with the validity of a reasonable Mackey
formula similar to the one known for induction/restriction of characters of finite
groups. The most complete result about such a formula is due to Bonnafé-Michel
[BoMi11] (see also [Tay17]).
(b) When P is F -stable, YP identifies with the finite set GF /Ru (P)F and the
functor RG
L with the so-called parabolic or Harish-Chandra induction, making
any representation of LF into a representation of PF (through trivial action of
F
Ru (P)F ) then applying ordinary induction IndG
PF . The formula
X
∗ G
DG =
(23)
(−1)|I| RG
LI RLI
I⊆S
involves only Harish-Chandra induction and is called Alvis-Curtis duality.
(c) In order to see how many more functors Deligne-Lusztig theory constructs
as opposed to usual functors defined from subgroups of GF , let us focus on the
case of L a torus. In the finite group G = GF there is only one conjugacy class
of subgroups one would call a maximal torus, the one denoted T = TF
0 at the
beginning of this section. Allowing any group TF for T an F -stable maximal
torus brings a lot more to the picture. Starting from our reference T0 , the GF conjugacy classes of F -stable maximal tori are in bijection
gT0 g −1 7→ g −1 F (g)T0 ∈ W (G, T0 )
(24)
with W = W (G, T0 ) mod the relation w ∼F vwF (v)−1 for any w, v ∈ W . The
element g −1 F (g)T0 , or its ∼F -class is called the type of the F -stable maximal
torus gT0 g −1 . This is a classical consequence of Lang’s theorem (see [Ge17, 2.3]).
For GLn (F) with F the usual Frobenius on matrix entries, this gives as many
classes of tori as the number of conjugacy classes of Sn .
An important fixed point property of étale cohomology implies the following
character formula ([DL76, §3-4]).
Proposition 4.4. If (u, v) ∈ GF × LF is assumed to be unipotent, let
X
QG
(−1)i tr((u, v), Hic (YP )).
L (u, v) =
i∈Z
If su is the Jordan decomposition of an element of GF and f ∈ CF(LF ), then
X
X
C◦ (s)
F −1 ◦
−1
RG
|CG (s)F |−1
|C◦g L (s)F |
QCG
)f (g −1 svg).
◦ (s) (u, v
L (f )(su) = |L |
g
L
{g∈GF |s∈g L}
F
v∈C◦
g L (s)u |
20
MARC CABANES
One lists below some consequences of the character formula that prove useful
in the proof of Broué-Michel’s theorem on ℓ-blocks.
Definition 4.5. We call uniform functions the elements of CF(GF ) that are
F
C-linear combinations of RG
T θ for T an F -stable maximal torus and θ ∈ Irr(T ).
Some f ∈ CF(GF ) is called p-constant if and only if f (su) = f (s) for any
Jordan decomposition su ∈ G.
Lemma 4.6. Let f ∈ CF(G) be p-constant.
(i) f is uniform.
′
(ii) If L is an F -stable Levi subgroup of G and f ′ ∈ CF(LF ), then RG
L (f )f =
F
G
′
RG
L (f ResLF f ).
(iii) If χ ∈ CF(G) then DG (f χ) = f DG (χ).
(iv) |G|−1
p′ DG (regG ) is the characteristic function of the set of unipotent elements (i.e. p-elements) of G.
4.C. Rational series and Broué-Michel’s theorem. An important case of
the functor RG
L is when L is a maximal torus. It allows an important partition of
Irr(GF ).
One defines first a group G∗ dual to G. This means that a choice has been
made of maximal tori T0 ≤ G, T∗0 ≤ G∗ such that Hom(T0 , F× ) ∼
= Hom(F× , T∗0 )
×
∗
× ∼
and Hom(T0 , F ) = Hom(F , T0 ) in a way that is compatible with roots and
coroots. One also assumes that all those are stable under compatible Frobenius
endomorphisms F : G → G, F ∗ : G∗ → G∗ . The interest of groups in duality is in
the parametrization of characters of finite tori TF . Indeed for w ∈ W (G, T0 ) iden∼ ∗ w∗ F ∗ (where
tified with w∗ ∈ W (G∗ , T∗0 ), one has isomorphisms Irr(TwF
0 ) = T0
the notation wF stands for F followed by conjugation by w).
One also gets a bijection
{GF -conjugacy classes of pairs (T, θ) where T is an F -stable maximal torus of
G and θ ∈ Irr(TF )}
l
∗F ∗
∗
(25)
∗
∗
{G
-classes of pairs (T , s) where T is an F -stable maximal torus of G∗
∗
and s ∈ T∗F }.
∗
Theorem 4.7 (Deligne-Lusztig). For s ∈ G∗F a semi-simple element, one defines E(GF , s) the set of irreducible components of generalized characters RG
T θ for
(T, θ) corresponding to some (T∗ , s) through the above correspondence.
One gets a partition
Irr(GF ) = ⊔s E(GF , s)
(26)
∗
where s ranges over semi-simple classes of G∗F .
∗
The subsets E(GF , s) for s ∈ G∗ss F are called the rational series of Irr(GF ).
The proof of the Theorem, given in [DigneMic, Ch. 14] is quite indirect, going
through the intermediate notion of geometric series and using a regular embedding
e (see [Ge17, §6], [CaEn, §15.1]) with connected Z(G).
e
G ⊆ G
It is easier to
F
F
G
show that RL sends E(L , s) into ZE(G , s) via the correspondence between Levi
subgroups of G and G∗ [CaEn, 15.7]. This implies in particular that Alvis-Curtis
duality (see Remark 4.3.(b) above) satisfies
∗
DG (E(G, s)) ⊆ ZE(G, s) for any semi-simple s ∈ G∗F .
(27)
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
21
∗
Theorem 4.8 (Broué-Michel). For s ∈ G∗F a semi-simple element of order
prime to ℓ, one defines
[
Eℓ (GF , s) :=
E(GF , st).
t∈CG∗ (s)F
ℓ
∗
This is a union of ℓ-blocks.
Proof. We abbreviate GF = G. We show that the projection pr : CF(G) → CF(G)
associated with the subset Eℓ (G, s) ⊆ Irr(G), satisfies pr(regG )(G) ⊆ |G|ℓ O. This
will give our claim by Proposition 4.1. For π a set of primes, one denotes by δπ
the characteristic function of π-elements of G.
Note that
δπ ∈ O Irr(G) as soon as ℓ ∈ π
(28)
thanks to a classical consequence of Brauer’s characterization of generalized characters (see [NagaoTsu, 3.6.15.(iii)]). Note also that δπ is p-constant as soon as
p ∈ π.
We also prove
pr(δℓ′ f ) = δℓ′ pr(f ) for any uniform f ∈ CF(G).
(29)
To show (29) it suffices to show the equality with f = RG
T θ for some F -stable
maximal torus T and θ ∈ Irr(TF ). The claim then reduces to showing that
∗ ′
′
′
δℓ′ RG
T θ ∈ CEℓ (G, s) when (T, θ) ↔ (T , s ) (see (25)) for some s ∈ with sℓ′ = s.
′
Note that this can be done for any semi-simple ℓ -element, conjugate or not to
F
G (T )
the s we are given. By Lemma 4.6.(ii), we have δℓ′ RG
θ). On the
T (θ) = RT (δℓ′
P ′
(TF )
′
F
other hand δℓ′ θ = θ′ θ where the sum is over θ ∈ Irr(T ) with θℓ′ ′ = θℓ′
(we consider Irr(TF ) as a multiplicative group). But it is easy to check from the
identifications of duality that if s′ℓ′ = s then (T, θ′ ) ↔ (T∗ , s′′ ) with s′′ℓ′ = s. This
gives δℓ′ RG
T θ ∈ CEℓ (G, s).
Now the proof of the Theorem goes as follows. From Lemma 4.6.(i) we know
that δ{p,ℓ} is uniform and (29) gives now
δℓ′ pr(δ{p,ℓ} ) = pr(δ{p} ).
(30)
The image of the right hand side by DG is DG ◦ pr(δp ) = pr ◦ DG (δp ) =
|G|−1
p′ pr(regG ) thanks to (27) and Lemma 4.6.(iv). Using (iii) of the same lemma,
the image by DG of (30) now gives
|G|p−1
′ pr(regG ) = δℓ′ .pr(DG (δ{p,ℓ} )).
(31)
We have seen (28) that δ{p,ℓ} hence also DG (δ{p,ℓ} ) ∈ O Irr(G). So the right hand
side of (31) takes values in O. Then indeed pr(regG ) takes values in |G|p′ O = |G|ℓ O
as claimed.
The sum of blocks of OG corresponding to the theorem is denoted as follows.
P
Definition 4.9. One denotes eℓ (GF , s) := χ∈Eℓ (GF ,s) eχ ∈ Z(OGF ), eℓ (GF , s)
its image in Z(kGF ), and Bℓ (GF , s) := OGF eℓ (GF , s).
Combining (29) and the fact that multiplication by δℓ′ preserves ℓ-blocks (see
below Brauer’s “second Main Theorem”) one easily gets
Proposition 4.10 (Hiss). For every p-block B of GF such that Irr(B)∩Eℓ (GF , s) 6=
∅, one has Irr(B) ∩ E(GF , s) 6= ∅.
22
MARC CABANES
4.D. Jordan decomposition and ℓ-blocks. We keep G, F, G∗ , F ∗ , etc... as
before.
Definition 4.11. The elements of E(GF , 1) are called unipotent characters.
Similarly ℓ-blocks B of GF such that Irr(B) ∩ E(GF , 1) 6= ∅ are called unipotent
blocks.
The set of unipotent characters tends to be sensitive only to the root system
of G and the action of F on it. In particular one has bijections (see [DigneMic,
13.20])
E(GF , 1) ↔ E([G, G]F , 1) ↔ E((G/Z(G))F , 1)
(32)
∗
m
and E(GF , 1) ↔ E(G∗ F , 1) but also E(GF , 1) ↔ E(GF , 1) (m ≥ 1) when F acts
trivially on the root system of G. However the last bijection relates characters of
different degree, though the degree is the same polynomial in various powers of q,
see [Carter2, Sect. 13.8].
Example 4.12. We consider the case of GF = GLn (Fq ), see Example 1.3. For
w ∈ Symn let Tw denote an F -stable torus of type w with regard to the diagonal
torus in the sense of Remark 4.3.c. The set of unipotent characters is in bijection
with Irr(Sn ) by the map
X
χ(w)RG
χ 7→ Rχ = n!−1
Tw 1
w∈Sn
F
which takes values in ±E(G , 1) and with adequate signs gives indeed a bijection
Irr(Sn ) → E(GF , 1) (see for instance [DigneMic, §15.4]).
It is customary to call Jordan decomposition of Irr(GF ) any bijection
∗
E(GF , s) ↔ E(CG∗ (s)F , 1)
∗
where s ∈ (G∗ )F
p′ . However the definition we gave of unipotent characters applies
∗
only to connected groups G, so the set E(CG∗ (s)F , 1) above would be defined as
C
the set of constituents of induced characters IndCG
◦
∗ (s)
G∗
F∗
(s)F ∗
∗
ζ for ζ ∈ E(C◦G∗ (s)F , 1).
The existence of such a Jordan decomposition compatible with the RG
L functors
has been shown by Lusztig [Lu88], here again in a quite indirect way, the results
being a lot more complete in the cases where Z(G) is connected, which in turn
ensures that CG∗ (s) is then connected. A basic idea is that Jordan decomposition
should behave like a RG
C functor for a suitable C.
For the following, see [DigneMic, 13.25].
Theorem 4.13 (Lusztig). Assume L∗ is an F ∗ -stable Levi subgroup of G∗ such
that CG∗ (s) ≤ L∗ . Then L := (L∗ )∗ can be seen as a Levi subgroup of G and there
is a sign ǫL,G ∈ {1, −1} such that ǫL,G RG
L induces a bijection
F
F
ǫL,G RG
L : E(L , s) → E(G , s).
In this situation and with s being an ℓ′ -element it is not difficult to prove that
ǫL,G RG
L also induces a bijection
F
F
ǫL,G RG
L : Eℓ (L , s) → Eℓ (G , s).
Theorem 4.14 (Broué). The above bijection preserves the partitions induced by
ℓ-blocks. Moreover two ℓ-blocks that thus correspond have defect groups of the same
order.
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
23
About the proof. We sketch the main ideas of the proof, based on Broué’s notion
of perfect bi-characters (see [Bro90a]). For finite groups H, L, a bi-character
µ ∈ Z Irr(H × L) is called perfect if and only if for all (h, l) ∈ H × L, µ(h, l) ∈
|CH (h)|O ∩ |CL (l)|O and whenever µ(h, l) 6= 0 then h ∈ Hℓ′ if and only if l ∈ Lℓ′ .
This is a Z-submodule of Z Irr(H × L) and the trace character of an OH × Lopp bimodule which is projective on each side is perfect. This last property is very
important since it gives an arithmetic test for bicharacters that could come from
a Morita equivalence of blocks over O or even a derived equivalence since by a
theorem of Rickard such equivalences are induced by complexes of bi-projective
bimodules [Rick89].
Broué shows that if
a) I ⊆ Irr(H) and J ⊆ Irr(L) are unions of ℓ-blocks, with
b) σ : J → I a bijection and
P
c) one has signs (ǫχ )χ∈J such that χ∈J ǫχ σ(χ) ⊗ χ is perfect,
then
(i) σ preserves the partition of I and J into ℓ-blocks and
(ii) corresponding blocks have defect groups of the same order and same number of simple modules over k = O/J(O).
In order to apply this to our situation H = GF , L = LF , I = Eℓ (LF , s) and the
bijection of Theorem 4.13, it just remains to show that our bi-character is perfect.
This is a consequence of what has been said about bi-projective modules producing
perfect trace characters and the fact the action of GF ×LF on the variety YP is free
on each side. This last fact translates into a related property of ℓ-adic cohomology
opp
groups as OGF × LF -modules, thanks to a result of Deligne-Lusztig ([DL76,
3.5]) which is also key to the proof of Proposition 4.4 above.
It is important to notice that in the above the isometry of characters is without
signs as would happen in the case of a Morita equivalence. Indeed Broué made
the conjecture that this corresponds to a Morita equivalence
∼
Bℓ (LF , s)-mod −
→ Bℓ (GF , s)-mod
between the module categories of Bℓ (GF , s) and Bℓ (LF , s).
This was proved by Bonnafé-Rouquier [BoRo03]. In Sect. 9 below, we try to
give an idea of their proof which goes very deep into the definition of RG
L functors.
The result was completed recently by Bonnafé-Dat-Rouquier [BoDaRo17] into a
statement showing isomorphism of defect groups and local structure.
In order to get a Morita equivalence from an isometry of characters, one needs
essentially to have the latter induced by a bi-projective module thanks to the
following
Lemma 4.15 (Broué [Bro90b]). Assume H, L are finite groups, B, C are sums
of blocks of OH, OL respectively. Assume M is a B ⊗ C opp -bimodule that is biprojective (i.e., projective on restricting to the subalgebras B ⊗ O and O ⊗ C).
Then
M ⊗OL − : C-mod → B-mod
is an equivalence of categories if and only if M ⊗ K induces a bijection of ordinary
characters Irr(C) → Irr(B).
Proof. Let N := HomO (M, O). This is a C ⊗ B opp -bimodule, projective on each
side.
24
MARC CABANES
We have
M⊗ −
N⊗ −
C-mod −−−−C−→ B-mod and B-mod −−−B−→ C-mod are left and right adjoint.
(33)
Indeed the classical (left) adjoint for the tensor product functor M ⊗C − is
HomB (M, −). But the B-projectivity of M allows to identify HomB (M, −) with
HomB (M, B)⊗B −. On the other hand, the algebra B is symmetric over O, namely
the restriction to B of the evaluation of the coordinate at 1 in the group algebra
yields a linear map λ : B → O inducing an isomorphism between B and its O-dual
and such that λ(bb′ ) = λ(b′ b) for all b, b′ ∈ B (compare with assumption (1) in
Theorem 2.1 above). A basic property is then that
HomB (M, B) ∼
(34)
= N by the map f 7→ λ ◦ f.
. This and exchanging the roles of B and C gives (33).
Using the subscript K to denote tensoring by K for B, C, M, N , we have the
same as (33) for the semi-simple algebras BK and CK . The assumption on MK
implies that MK ⊗CK − and NK ⊗BK − are inverse functors and therefore
(35)
M K ⊗ C NK ∼
= CK as bi-modules.
= BK and NK ⊗B MK ∼
K
K
On the other hand BB and (M ⊗C N )B are projective as right B-modules
thanks to the bi-projectivity of M and N for the second. But (35) above tells us
that they are isomorphic once tensored with K as BK -modules. It is well-known
that two projective OH modules are isomorphic if and only if they are so when
tensored with K, see for instance [Du17, §4.4]. So we get
(M ⊗C N )B ∼
= CC
= CC and C (N ⊗B M ) ∼
= B B, (N ⊗B M )C ∼
= BB , B (M ⊗C N ) ∼
(36)
by the symmetry of the situation.
The adjunction between the functors M ⊗C − and N ⊗B − mentioned above
provide natural transformations of the composites into identity functors. In the
case of tensor products functors, this means we have bimodule maps
ǫ : C → N ⊗B M
and
η : M ⊗C N → B.
Note that they can be made explicit by following the steps used above, for instance
η(m ⊗ n) = λ∗ (n)(m) where λ∗ is the inverse of the map (34). The basic property
of adjunctions (see [McLane, IV.1]) implies that the composite
ǫ⊗id
idN ⊗η
→ N ⊗B M ⊗C N −−−−→ N
N −−−−N
(37)
is the identity. Keeping only the action of B on the right, the three modules
are all isomorphic thanks to the first statement in (36) and the maps are inverse
isomorphisms. So the maps in (37) are indeed isomorphisms. But then, tensoring
by M on the right gives an isomorphism
ǫ⊗idN ⊗M
N ⊗B M −−−−−−→ N ⊗B M ⊗C N ⊗B M.
By the last statement of (36) this means that ǫ was an isomorphism in the first
place. We also get the same for η and this is enough to conclude that our functors
M ⊗C − and N ⊗B − induce inverse (Morita) equivalences.
A first application of the lemma is to show that the map of Theorem 4.13 is
induced by a Morita equivalence in a special case.
Corollary 4.16. Assume the hypotheses of Theorem 4.13 with moreover that
L is a Levi subgroup of an F -stable parabolic subgroup P. Then the functor
OGF /Ru (P)F ⊗LF − induces a Morita equivalence Bℓ (LF , s)-mod −
→ Bℓ (GF , s)-mod
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
25
The proof simply consists in noting that the functor given induces on characters the Harish-Chandra induction which coincides with RG
L in our case (see
Remark 4.3.b), hence a bijection by Theorem 4.13, and on the other hand this
bimodule is clearly bi-projective
since one may write it OGF e where e is the
P
idempotent |Ru (P)F |−1 u∈Ru (P)F u.
5. Local methods for blocks of finite quasi-simple groups
We give more material on general methods for blocks of finite groups. We
then illustrate them with the case of symmetric groups. We conclude with a brief
discussion of Chuang-Rouquier theorems [ChRo08].
5.A. Subpairs and local structure of an ℓ-block. We go back to H some abstract finite group, and ℓ a prime, with (K, O, k) an associated ℓ-modular system.
An ℓ-subpair in H is any pair (Q, bQ ) where Q is an ℓ-subgroup of H and bQ
is a primitive idempotent of Z(OCH (Q)). Recall (see Sect. 1.E above) that for C
a finite group we have bijections (where prid stands for primitive idempotents)
blocks of OC ↔ prid(Z(OC)) ↔ prid(Z(kC)) ↔ blocks of kC
where the middle map is i 7→ i (reduction mod J(O)) whose inverse is given by
idempotent lifting.
We identify all four kinds of objects above and thus extend the notations Irr(B),
CF(H | B) already seen.
We already introduced the Brauer morphism BrQ : Z(kH) → kCH (Q) in Sect.
1.E, but in fact it can be defined on a bigger algebra. Denoting by (kH)Q the fixed
point subalgebra for the conjugacy action of Q, one has an algebra morphism
BrQ : (kH)Q → kCH (Q)
X
X
λh h
λh h 7→
h∈H
h∈CH (Q)
One defines an order relation ≤ on ℓ-subpairs of H by transitive closure of the
following
Definition 5.1 (Alperin-Broué). (Q′ , b′ ) ⊳ (Q, b) if and only if
′
• Q normalizes Q′ and b′ (so that b ∈ (kH)Q ) and
′
• BrQ (b )b = b.
The ℓ-blocks of H itself can be seen as ℓ-subpairs of type ({1}, b1). An inclusion
({1}, b1 ) ⊳ (Q, b) would exist if and only if BrQ (b1 ) 6= 0 which is the criterion we
have seen to define defect groups (see 1.E).
Theorem 5.2 (Alperin-Broué).
(i) If (Q, b) is an ℓ-subpair in H and Q′ is
some subgroup of Q, then there is a single subpair with (Q′ , b′ ) ≤ (Q, b).
(ii) If ({1}, b1 ) is an ℓ-subpair of H, the ≤-maximal subpairs of H containing
it are all H-conjugate and are of type (D, b) where D is a defect group of
the ℓ-block kHb1 .
To an ℓ-block B of H with defect group D ≤ H, one can associate a finite
category similar to the fusion system of Definition 1.13, see [AschKeOl, IV.2.21].
26
MARC CABANES
Definition 5.3. Let (D, bD ) be a maximal ℓ-subpair containing ({1}, B). Then
F(D,bD ) (B) is the category whose objects are the subgroups of D and if D1 , D2 ≤
D, one defines
HomF(D,bD ) (B) (D1 , D2 )
as the set of maps D1 → D2 of the form x 7→ ch (x) = hxh−1 where h ∈ H is such
that one has ℓ-subpair inclusions
h
(D1 , b1 ) ≤ (D2 , b2 ) ≤ (D, bD ) ≥ (D1 , b1 ).
Like FQ (H) from Definition 1.13 on Q, the above defines a fusion system in
the sense of [AschKeOl] on the ℓ-group D. The “local structure” of the ℓ-block B
usually means the knowledge of F(D,bD ) (B), which of course does not depend on
the choice of the maximal subpair (D, bD ).
5.B. Brauer’s second Main Theorem. We need first to define the generalized decomposition map dx (x an ℓ-element) on central functions. We already
had a glimpse of the ordinary decomposition map (when x = 1) in the form of
multiplication by the function denoted by δℓ′ in the proof of Theorem 4.8.
Definition 5.4. For x ∈ Hℓ let
dx : CF(G) → CF(CH (x))
defined by dx (f )(y) = f (xy) if y ∈ CH (x)ℓ′ , dx (f )(y) = 0 otherwise.
Theorem 5.5 (Brauer 1959). Let x ∈ Hℓ . Let ({1}, b1), (hxi , bx ) be ℓ-subpairs
of H. Let χ ∈ Irr(b1 ) and assume dx (χ) ∈ CF(CH (x)) has non-zero projection on
CF(CH (x) | bx ). Then
({1}, b1) ≤ (hxi , bx ).
5.C. Centric or self-centralizing subpairs.
Definition 5.6. Let (Q, bQ ) be an ℓ-subpair of H. Then it is called centric if and
only if bQ has defect group Z(Q) in CH (Q). Then there is a single ζ ∈ Irr(bQ ) with
Z(Q) in its kernel, this is called the canonical character of the centric subpair.
It is easy to show the uniqueness of ζ above, using that kCH (Q)bQ has a single
simple module, hence a single projective indecomposable module. One can recover
bQ from ζ by the formula
bQ =
ζ(1)
|CH (Q)|
X
ζ(h)h−1 .
(38)
h∈CH (Q)ℓ′
Theorem 5.7 (Brauer). Let (Q, b), (Q′ , b′ ) some centric ℓ-subpairs of H, with
Q′ ⊳ Q. Let ζ ∈ Irr(b), ζ ′ ∈ Irr(b′ ) the canonical characters. Then (Q′ , b′ ) ≤ (Q, b)
C (Q′ ) ′
ζ is in N \ ℓN.
if and only if ζ ′ is Q-stable and the multiplicity of ζ in ResCH
H (Q)
In practice, centric subpairs lead easily to maximal subpairs.
Proposition 5.8. A subpair inclusion (Q1 , b1 ) ≤ (Q2 , b2 ) with centric (Q1 , b1 )
implies that (Q2 , b2 ) is also centric and Z(Q2 ) ≤ Z(Q1 ). A subpair (Q, bQ ) is
maximal if and only if it is centric and NH (Q, bQ )/QCH (Q) is an ℓ′ -group.
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
27
5.D. Two main theorems of Brauer and blocks of quasi-simple groups.
Assume we are given a finite group H and a prime ℓ. We assume we have some
information on Irr(H) and some character values, especially in the form of algorithms reducing to the related questions for smaller groups of the same type.
Using local methods we want to determine the splitting of Irr(H) into sets
Irr(B) for B the ℓ-blocks of H, along with the ℓ-subpairs of H (which includes
determining the defect groups of ℓ-blocks).
For χ ∈ Irr(H), let us denote by bH (χ) the ℓ-block such that χ ∈ Irr(bH (χ)).
Starting with χ ∈ Irr(H), two possiblities occur. Either there is some 1 6= x ∈
Hℓ such that dx χ 6= 0 or χ(H \ Hℓ′ ) = 0.
In the second case it is classical that bH (χ) has defect {1} and is the only
character of that block. In the first case it’s often the case that such an x can be
found non-central. Then Brauer’s second Main Theorem allows to get an inclusion
({1}, bH (χ)) ≤ (hxi , b′ ).
If now H ′ := CH (x) has a similar structure as H, or say we know Irr(H ′ ) just
′
as well, and if the maps dx : Irr(H ′ ) → CF(CH ′ (x′ )) are also not too difficult to
compute, we can do for H ′ the same as above.
This subpair enlargement process will provide us with an inclusion
({1}, bH (χ)) ≤ (A, bA )
where A is an abelian ℓ-subgroup and bA has central defect group in CH (A). So
we can assume that (A, bA ) is centric.
Using now Theorem 5.7, including (A, bA ) into other centric subpairs is a relatively classical problem of character restrictions. One then gets to a maximal
subpair (D, bD ) ≥ ({1}, bH (χ)). By conjugacy of maximal subpairs, this solves the
problem of saying when two characters χ, χ′ of H belong to the same block. One
has bH (χ) = bH (χ′ ) if and only if the corresponding pairs (D, bD ) and (D′ , bD′ )
are conjugate.
This is not precisely the pattern followed by Brauer-Robinson to determine the
blocks of symmetric groups first conjectured by Nakayama (see [Naka41b],[Br47])
but it was used by others (see [MeTa76] and Sect. 5.F below) and by FongSrinivasan for the blocks of finite classical groups ([FoSr82] and [FoSr89]).
Remark 5.9. Note that we have avoided the question of characters that would
vanish on H \Z(H)Hℓ′ but are not in an ℓ-block of central defect. This can happen
only if ℓ | |Z(H)|. If we have started with a quasi-simple group H, this means that
ℓ divides the order of the Schur multiplier of a simple group. Indeed for H the
double cover of alternating or symmetric groups, the 2-blocks of faithful characters
had to be determined by other methods (Bessenrodt-Olsson [BeOl97]). But such
a phenomenon seems a bit isolated and not present in finite groups of Lie type.
5.E. The symmetric group: characters. Let us recall the parametrization of
Irr(Sn ) and the formula of Murnaghan-Nakayama giving the character values. We
refer for instance to [JamesKer] for the classical theory while [Klesh] gives a very
direct approach to a more general setting.
For n ≥ 0, one defines P(n) = {λ | λ ⊢ n} the set of integer partitions of n,
λ = (λ1 ≥ λ2 ≥ · · · ≥ λk ) with λk > 0 and λ1 + λ2 + · · · + λk = n. One also
denotes |λ| = n. This includes ∅ ⊢ 0.
One has a bijection
P(n) → Irr(Sn ),
λ 7→ ζλ .
28
MARC CABANES
The trivial character corresponds to the partition (n) through this bijection.
We don’t give its definition but go to the Murnaghan-Nakayama rule that allows
to compute inductively the character values.
Let d ≥ 1. To each λ ⊢ n is associated the set hookd (λ) of its hooks of length
d and for each τ ∈ hookd (λ) there is a removal operation producing λ − τ ⊢ n − d.
Each hook has a height h(τ ) ∈ N. The Murnaghan-Nakayama rule is as
follows.
Assume 1 ≤ d ≤ n and x ∈ Sn writes as x = x′ c where x′ ∈ Sn−d and c is a
cycle of order d on {n − d + 1, . . . , n}. Let λ ⊢ n. Then [JamesKer, 2.4.7]
ζλ (x) =
X
(−1)h(τ ) ζλ−τ (x′ ).
(39)
τ ∈hookd (λ)
Let’s be more explicit on hooks and the removal process. Partitions are often
represented by Young diagrams, where λ = (λ1 ≥ λ2 ≥ · · · ) is represented by
rows of boxes of sizes λ1 , λ2 , etc.. The rows are aligned on the left and all boxes
are identical. Below is the diagram for the partition (4, 3, 1, 1) ⊢ 9. The rim of
the diagram consists of the boxes such that no box is at the same time under and
on the right of them. On the first diagram below the rim of 7 boxes is dotted.
A hook is an interval in this rim starting and finishing at some box with no box
under or right of it. Its length is the number of boxes it comprises. Its height
is the number of rows affected minus 1. Below are six hooks with length d and
height h indicated. (Exercise: find the four hooks missing.)
. .
.
.
. .
.
.
.
(d, h) = (7, 3)
(1, 0)
. .
. .
.
(2, 0)
(4, 1)
. . .
.
.
(5, 2)
It is clear that removing a hook τ of length d gives a Young diagram with n − d
boxes, hence the meaning of λ − τ ⊢ n − d above. Note that in (39) above d can
be equal to 1. This case of the Murnagan-Nakayama rule gives the restriction of
χ ∈ Irr(Sn ) to Sn−1 and is called the branching rule.
Note that when λ has no d-hook, then (39) gives ζλ (x′ c) = 0. A partition λ is
said to be a d-core if and only if hookd (λ) = ∅. For instance the partition above
is a 6-core.
For a given d, starting with some partition, the hook removal can be iterated
λ 7→ λ − τ1 7→ (λ − τ1 ) − τ2 7→ · · · where τ1 ∈ hookd (λ), τ2 ∈ hookd (λ − τ1 ), etc..
until we get a d-core. This is done below with d = 2, the hook removed next being
dotted.
7−→
. .
7−→
. .
7−→
.
.
It can be proved that given d and λ, this process of hook removal always ends
in the same d-core λ(d) and that the sign ǫλ,d = (−1)h(τ1 )+h(τ2 )+··· also does not
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
29
depend on the path followed. One then gets the following iterated MurnaghanNakayama rule [JamesKer, 2.7.27]
ζλ (x′ c1 c2 . . . cw ) = ǫλ,d Nλ,d ζλ(d) (x′ )
(40)
′
where x ∈ Sn−wd , c1 , c2 , . . . , cw are disjoint cycles of order d on {n−wd+1, . . . , n},
and Nλ,d is the number of ways to go from λ to λ(d) by successive d-hook removals.
Remark 5.10. Young diagrams were essentially created to fill the boxes with
additional information. Working with hooks and cores is made easier by using
β-numbers instead of partitions. One replaces the partition λ = (λ1 ≥ λ2 ≥
· · · ≥ λk ) by the set β := {λ1 + k − 1, λ2 + k − 2, . . . , λk }. A hook of length d is
then replaced by a pair {a, a − d} such that a ∈ β and 0 ≤ a − d 6∈ β. The removal
λ 7→ λ − τ becomes β 7→ β \ {a} ∪ {a − d}. The iteration with a fixed d is then
easy to control and uniqueness of the outcome is quite clear. The height of hook
is the number of elements of β between a and a − d, so that the sign (−1)h(τ ) can
be interpreted as the signature of a cycle and the product of signs at the end of
the process is clearly independent of the path followed. This makes clear how to
get (40) from (39).
The following fact is also made trivial by working with β-numbers.
If λ ⊢ n and hookd (λ) = ∅ then hookdd′ (λ) = ∅ for any d′ ≥ 1.
(41)
5.F. The symmetric group: blocks. We give here the classification of blocks
of symmetric groups by use of local methods. The approach to Irr(Sn ) described
in [Klesh] gives more generally the blocks of all Iwahori-Hecke algebras of type A
(see [Klesh, 9.6.2]).
Let n ≥ 1 and ℓ be a prime.
Theorem 5.11 (Brauer-Robinson). The ℓ-blocks of Sn are parametrized
κ 7→ Bκ
by the ℓ-cores κ such that κ ⊢ n − wℓ for some w ≥ 0. One has ζλ ∈ Irr(Bκ ) if
and only if λ(ℓ) = κ.
The Sylow ℓ-subgroups of Sn−|κ| are defect groups of Bκ .
Lemma 5.12. If λ ⊢ n is an ℓ-core, then ζλ vanishes outside ℓ′ -elements of Sn .
To prove the above, let x ∈ Sn with xℓ 6= 1. Then it has in its cycle decomposition a cycle of order t a multiple of ℓ, x = x′ c with c a cycle of order t and x′
fixing any element in the support of c. Then hookt (λ) = ∅ by (41) above and the
Murnaghan-Nakayama rule (39) gives ζλ (x′ c) = 0 as claimed.
We now prove Theorem 5.11. Let λ ⊢ n with λ(ℓ) ⊢ n − wℓ, w ≥ 0. Let c a
product of w disjoint cycles of order ℓ on {n − wℓ + 1, . . . , n}. Then CSn (c) =
Sn−wℓ ×W where W is isomorphic to the centralizer of a product of w disjoint
cycles of order ℓ in Swℓ . We have
CW (Op (W )) ≤ Op (W ).
(42)
If c = c1 . . . cw is the product of our disjoint cycles of order ℓ, it is clear that the
ci ’s are permuted by any element of W , so C := hc1 , . . . , cw i ∼
= (Z/ℓZ)w is normal
in W . On the other hand CW (C) = C since a permutation centralizing C has to
stabilize the support of each ci and the centralizer of a cycle of order ℓ in Sℓ is
clearly the cyclic subgroup this cycle generates. Thus (42).
Note that (42) implies that
W has a single ℓ-block B0 (W )
(43)
30
MARC CABANES
Indeed the defect group of an ℓ-block of W must contain Op (W ) (see for instance
[NagaoTsu, 5.2.8.(i)]) and then we can argue as in the proof of Proposition 1.18.
Let B := bSn−wℓ (ζλ(ℓ) ).B0 (W ) ∈ Bl(Sn−wℓ ×W ). We prove
({1}, bSn (ζλ )) ⊳ (hci , B).
(44)
By Brauer’s second Main Theorem (Theorem 5.5), it suffices to prove that dc ζλ
has non zero projection on CF(Sn−wℓ ×W | B). Since W has only one block
CF(Sn−wℓ ×W | B) = CF(Sn−wℓ | bSn−wℓ (ζλ(ℓ) )) × CF(W ). On the other hand,
Lemma 5.12 implies that ζλ(ℓ) is the only irreducible character in its ℓ-block (see
for instance [NagaoTsu, 3.6.29]) so CF(Sn−wℓ ×W | B) = Cζλ(ℓ) × CF(W ). If the
S
×W
projection of dc ζλ were 0 on it, we would have ResSn−wℓ
dc ζλ ∈ C(Irr(Sn−wℓ )\
n−wℓ
{ζλ(ℓ) }). Using the usual inner product on central functions, this would give
X
ζλ (xc)ζλ(ℓ) (x−1 ) = 0.
(45)
x∈(Sn−wℓ )ℓ′
But we have
X
ζλ (xc)ζλ(ℓ) (x−1 ) =
x∈(Sn−wℓ )ℓ′
=
X
ζλ (xc)ζλ(ℓ) (x−1 ) by Lemma 5.12
x∈Sn−wℓ
ǫλ,ℓ Nλ,ℓ
X
ζλ(ℓ) (x)ζλ(ℓ) (x−1 ) by (40)
x∈Sn−wℓ
=
ǫλ,ℓ Nλ,ℓ | Sn−wℓ | 6= 0, a contradiction.
Note that having (44) proves at once that bSn (ζλ ) = bSn (ζµ ) as soon as λ, µ ⊢ n
have same ℓ-core κ ⊢ n − wℓ. This gives the map κ 7→ Bκ announced. It is also
easy to include the second subpair of (44) into a maximal one. Let D be a Sylow ℓsubgroup of S ′ , the symmetric group on {n−wℓ+1, . . . , n}. Assume D contains the
cycles of which c is a product. Then the centralizer of D in Sn is Sn−wℓ ×CS ′ (D)
and we define BD = bSn−wℓ (ζκ ).B0 (CS ′ (D)) where B0 (CS ′ (D)) is the principal
block of CS ′ (D). Using Theorem 5.7 and Proposition 5.8, one gets inclusions
({1}, Bκ ) ≤ (hci , B) and (hci , B) ≤ (D, BD )
the latter being maximal.
(46)
Remark 5.13. It is easy to check that (42) above is true for any centralizer of an
ℓ-subgroup of Sm having no fixed point on {1, . . . , m}. Let P be an ℓ-subgroup
of Sn . We assume that its fixed points in {1, . . . , n} are {1, . . . , nP }, so that
CSn (P ) = SnP ×WP where WP has only one ℓ-block by (43). Let Bκ an ℓ-block
(n)
of Sn as in Theorem 5.11, let bκ ∈ Z(k Sn ) the corresponding central idempotent
in the group algebra of characteristic ℓ. From the above, one computes easily the
Brauer morphism
(
(n )
bκ P ⊗ 1kWP , if nP ≥ |κ|,
(n)
BrP (bκ ) =
(47)
0,
otherwise.
This shows that the fusion system of ℓ-subpairs of Bκ (see Definition 5.3) is isomorphic with the fusion system of ℓ-subgroups of Sn−|κ| (Broué-Puig, see [Bro86,
2.B.4]).
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
31
5.G. The symmetric group: Chuang-Rouquier’s theorems. We keep ℓ a
prime number.
Theorem 5.14 (Chuang-Rouquier, 2008). If two ℓ-blocks A ⊆ O Sn and A′ ⊆
O Sn′ have isomorphic defect groups then
Db (A-mod) ∼
= Db (A′ -mod).
The proof introducing to representations of finite groups the notion of categorifications certainly opens a new chapter of representation theory. See [Mazor]
for a very complete introduction. [ChRo08] appeared on arXiv in 2004 and categorification was then a very recent notion. At that time MathSciNet had only 10
papers with the word in their title, the first in 1998; there are now on average 20
per year.
The common feature of the various instances of categorification is for A a ring
the existence of exact endofunctors of an abelian category C such that their images
in the endomorphism ring of the Grothendieck group give a representation
A → End(K0 (C)).
This is generally verified by using a presentation of A and checking that our
endofunctors induce endomorphisms of K0 (C) satisfying the relations presenting
A. The phrase “categorical action” of A is also used (see [DuVV15]).
Concerning blocks of symmetric groups, it was known for some time that the
size of the defect group of a block of Sn determined all its numerical invariants
(see [En90]). The sum
G := ⊕n≥0 K0 (k Sn )
of the Grothendieck groups of all symmetric groups had been considered by various
authors (see [Ze81]) in connection with the branching rule. More recently, LascouxLeclerc-Thibon had shown an action of affine Lie algebras on it related with JucysMurphy elements Li = (1, i) + (2, i) + · · · + (i − 1, i). See also [Klesh, §9]. The
Li ’s pairwise commute and the algebra they generate compares with the Cartan
subalgebra of a Lie algebra, thus bringing to symmetric groups a key feature of
Lie theory.
Recall the Lie algebra sln over Z of n × n matrices with trace 0. It can be
presented by generators and relations satisfying the Chevalley-Serre relations. In
the case of n = 2, we get a Lie algebra sl2 = ZE ⊕ ZF ⊕ ZH defined by the
relations [E, F ] = H, [H, E] = 2E, [H, F ] = −2F .
b n is generated by the elements E0 , . . . , En−1 , F0 , . . . , Fn−1 ,
The affine Lie algebra sl
H0 , . . . , Hn−1 subject to the relations
[Ei , Fj ] = δi,j Hi , [Hi , Ej ] = Ci,j Ej , [Hi , Fj ] = −Ci,j Fj
1−Ci,j
(adEi )
1−Ci,j
(Ej ) = (adFi )
(Fj ) = 0 for i 6= j.
(48)
(49)
b n−1 .
where Ci,j is the Cartan matrix of the affine root system of type A
Let a ∈ Fℓ . For M a k Sn -module one denotes
Fa,n (M ) = {v ∈ M | av = (1, n) + (2, n) + · · · + (n − 1, n) .v}
the eigenspace of the n-th Jucys-Murphy element. This is Sn−1 -stable. This gives
a decomposition of the additive restriction functor
n
ResS
Sn−1 = ⊕a∈Fℓ Fa,n : k Sn -mod → k Sn−1 -mod.
n
Analogously one gets a decomposition IndS
Sn−1 = ⊕a∈Fℓ Ea,n with corresponding
adjunctions. One defines
Ea = ⊕n≥1 Ea,n , Fa = ⊕n≥1 Fa,n : ⊕n≥1 k Sn -mod → ⊕n≥1 k Sn -mod.
(50)
32
MARC CABANES
Theorem 5.15 (Lascoux-Leclerc-Thibon).
(i) The action of the above E0 , . . . ,
b
Eℓ−1 , F0 , . . . , Fℓ−1 induces an action of slℓ on the Grothendieck group G.
(ii) The decomposition of G induced by ℓ-blocks corresponds to a decomposition
into weight spaces (for the subalgebra generated by the Ha = [Ea , Fa ]’s).
(iii) Two ℓ-blocks have same defect group if and only if they are in the same
b ℓ.
orbit under the action of the Weyl group of sl
For each pair a ∈ Fℓ , the above situation restricts to actions of sl2 . This is
called more generally by Chuang-Rouquier a weak sl2 -categorification. A strong
sl2 -categorification is defined as follows. We give the version actually used for
blocks of symmetric groups, the one in [ChRo08] uses a parameter q which is 1
here.
Definition 5.16 (Chuang-Rouquier). Let A be a k-linear abelian category with
finiteness properties (satisfied in the application given). A strong sl2 -categorification
is the data of a ∈ k, exact functors
E, F : A → A
and natural transformations
X : E → E , T : E2 → E2
such that
(1) (E, F ) is an adjoint pair and F is isomorphic to a left adjoint of E,
(2) E and F induce on the Grothendieck group K0 (A) a locally finite representation of sl2 ,
(3) the simple objects of A are weight vectors for the above in K0 (A),
(4) (idE T ) ◦ (T idE ) ◦ (idE T ) = (T idE ) ◦ (idE T ) ◦ (T idE ) as natural transformations E 3 → E 3 ,
(5) T 2 = idE 2 and T ◦ (idE X) ◦ T = X idE −T as natural transformations
E2 → E2,
(6) X − a idE is locally nilpotent.
A very important feature is of course the role of the endomorphisms of functors X and T and the relations they satisfy. Categorification techniques lead to
consider functors as objects and natural transformations as morphisms. Note that
in equations like (4) and (5) above, ◦ denotes the classical composition of natural transformations of functors. Meanwhile, an expression like idE T means the
endomorphism of EE 2 obtained functorially from endomorphisms of E and E 2 .
Note that in the case of module categories (or categories closely related with due
adaptations), functors are mostly tensor products by bi-modules which in turn are
easy to consider as objects of an abelian category as in the proof of Lemma 4.15.
In practice, one will define several sl2 -categorifications that come from a strucb ℓ where ℓ is the characteristic of k. On top of the “repreture involving a whole sl
sentations” of sl2 one gets, the various X’s and T ’s will contribute to controlling
the modules produced through the action of (affine) Hecke algebras.
In the setting of Definition 5.16, Chuang-Rouquier prove the following fundamental theorem (see [Du17, Sect. 1.2-3] for the categories Hob and Db ).
Theorem 5.17 ([ChRo08, 6.4]). There is an equivalence of categories
Θ : Hob (A) → Hob (A)
inducing the action of the reflection of the Weyl group of sl2 on the Grothendieck
group K0 (A).
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
33
We have skipped the (difficult) definition of Θ (originally due to Rickard for
blocks of symmetric groups). A key point is to find an equivalent of the divided
powers em /m!, f m /m! (e and f being the images of E, F ∈ sl2 in a representation
over Q) that are necessary to define the action exp(−f ) exp(e) exp(−f ) of the
Weyl group on a representation. See [ChRo, 5.13, 6.1] for the model proposed.
Then the proof of invertibility of Θ is another challenge, where a key step is to
show invertibility of the induced functor Db (A) → Db (A), see [ChRo, 6.4, 6.6] and
proofs.
This theorem, with an additional parameter q ∈ k × , has several applications
in [ChRo08] beyond symmetric groups, namely blocks of Ariki-Koike algebras, of
finite general linear groups, or the so-called category O (see [ChRo08, §7]).
With Theorem 5.17 in hand, the proof of Theorem 5.14 consists then in constructing a strong sl2 -categorification for each a ∈ Fℓ with A = ⊕n≥1 k Sn -mod.
The functors Ea , Fa have been seen above. The natural transformations
Xa : Ea → Ea , Ta : Ea2 → Ea2
are defined as follows. The functor Ea = ⊕n≥1 Ea,n is such that
Ea,n : k Sn−1 -mod → k Sn -mod
is a direct summand of induction, hence induced by a direct summand of the
k Sn−1 ⊗k Sopp
n -bimodule k Sn . One defines Xa there as the right multiplication by (1, n) + (2, n) + · · · + (n − 1, n) on the bimodule. On the other hand
Ea2 = ⊕n≥2 Ea,n Ea,n−1 where the n-th term is a direct summand of the functor
k Sn−2 -mod → k Sn -mod induced by the k Sn−2 ⊗k Sopp
n -bimodule k Sn . The
natural transformation Ta is then defined by right multiplication by (n − 1, n).
One proves that this provides a strong sl2 -categorification. Working with A =
⊕n≥1 k Sn -mod, Theorem 5.17 then allows to deduce an equivalence
Hob (A) → Hob (A) restricting to Hob (A-mod) → Hob (A′ -mod)
(51)
for each pair (A, A′ ) of ℓ-blocks of symmetric groups such that A′ is the image of
A by a fundamental reflection in Theorem 5.15.(iii). Using the integral nature of
all functors involved one can lift that to algebras over O or even Zℓ . Using Theorem 5.15.(iii) one can iterate this strengthened version of (51) to get equivalences
Hob (A-mod) → Hob (A′ -mod) for each pair of ℓ-blocks A ⊆ O Sn and A′ ⊆ O Sn′
with isomorphic defect groups.
Using the knowledge of the Brauer morphism (see Remark 5.13), ChuangRouquier prove an even stronger equivalence of the blocks A, A′ concerned, namely
a Rickard equivalence (see Definition 9.15 below) that basically preserves the local
structures of the blocks and whose image by the Brauer morphism induces similar
equivalences at the local level [ChRo08, 7.2].
6. Local methods for unipotent blocks: the strategy
In view of a possible Jordan decomposition of characters inducing a strong
equivalence of ℓ-blocks (see Sect. 9 below) it may make sense to give details about
local methods only for unipotent blocks (see Definition 4.11). This is what we do
in Sections 6 to 8.
34
MARC CABANES
6.A. Generalized d-Harish-Chandra theory. We keep G and F : G → G as
before (Ch. 4).
We have until now defined Levi subgroups as complements in a decomposition of
a parabolic subgroup P = Ru (P) ⋊ L. A more intrinsic definition is by saying that
they are centralizers of tori (not necessarily maximal), see for instance [DigneMic,
1.22].
We will need to speak of cyclotomic polynomials. So, if d ≥ 1, recall that
φd (x) ∈ Z[x] denotes the d-th cyclotomic polynomial, whose complex roots are the
roots of unity of order d.
Definition 6.1. Any F -stable torus S of G has a so-called polynomial order
PS,F ∈ Z[x] defined by
m
|SF | = PS,F (q m )
for some a ≥ 1 and any m ∈ 1 + aN. Moreover PS,F is a product of cyclotomic
polynomials PS,F = Πd≥1 φnd d , (nd ≥ 0).
We give below a fundamental example.
Example 6.2. Let T0 be an F -stable maximal torus of a group (G, F ) such
that F acts on Y (T0 ) by q. Note that this is the case as soon as the coroots
Φ(G, T0 )∨ generate a lattice of finite index of Y (T0 ) and F fixes them. Let
w ∈ W(G, T0 ) and assume T is an F -stable maximal torus of type w in the sense
of Remark 4.3.(c). Then the pair (T, F ) is made isomorphic to (T0 , wF ) through
m
(wF )m
conjugation by g ∈ G such that g −1 F (g)T0 = w. Therefore |TF | = |T0
|
m
(wF ) ∼
m
Y
(T
)/(1
−
(wF
)
)Y
(T
)
for any m ≥ 1. It is an elementary fact that T0
=
0
0
see [DigneMic, 13.7]. Such a quotient is finite if and only if the endomorphism
1 − (wF )m of the lattice Y (T0 ) has non zero determinant and the cardinality of
Y (T0 )/(1−(wF )m )Y (T0 ) is | det(1−(wF )m )|. In our case we get the characteristic
polynomial of w−1 at q m . So if a is the order of w and m ∈ 1 + aN, we actually
m
get |TF | = PT,F (q m ) for PT,F the characteristic polynomial of w on Y (T0 ). It
is a product of cyclotomic polynomials since it is a monic polynomial whose zeroes
are roots of unity, w having finite order.
In all cases of interest, F induces a map of the form qφ on Y (T0 ) where φ is
an automorphism of finite order. Then the above applies almost unchanged.
The polynomial orders of tori have many properties of orders of abelian groups,
only cyclotomic polynomials play now the role of prime divisors.
Proposition 6.3. Let S be an F -stable torus of G. If PS,F = Πd≥1 φnd d , then for
any d ≥ 1, there is a unique subtorus Sd ≤ S such that PSd ,F = φnd d .
Indeed an F -stable torus S of G is essentially characterized as a subtorus of a
maximal one T by the F -stable pure sublattice Y (S) of Y (T).
Definition 6.4. A φd -torus of G is an F -stable torus whose polynomial order is
a power of φd .
A d-split Levi subgroup is any CG (S) where S is a φd -torus of G.
Example 6.5. (a) For d = 1, the 1-split Levi subgroups are the one that are
complements of F -stable parabolic subgroups, hence GF -conjugate to the standard
Levi subgroups LI , for I ⊆ S, F (I) = I.
(b) Let G = GLn (F) with F the raising of matrix entries to the q-th power.
Let T1 the diagonal torus. By (24) the GF -classes of maximal tori are indexed by
conjugacy classes of Sn , or equivalently partitions of n. For λ ⊢ n, denote by Tλ
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
35
an F -stable maximal torus in the corresponding class. If λ = (λ1 ≥ λ2 . . . ) then
the polynomial order of Tλ is (xλ1 − 1)(xλ2 − 1) · · · by Example 6.2. One calls
T(n) a Coxeter torus. This maximal torus T(n) is the only n-split proper Levi
subgroup of G up to GF -conjugation. To see this, note that its polynomial order
xn − 1 is the only polynomial order of an F -stable maximal torus divisible by φn .
Let d ≥ 1, m ≥ 0 such that md ≤ n. Let S(d) be a Coxeter torus of GLd (F).
Let L(m) be GLn−md (F) × (S(d) )m embedded in G = GLn (F) via the diagonal
subgroup GLn−md (F) × (GLd (F))m . Then L(m) is d-split thanks to the above. A
maximal d-split proper Levi subgroup L of G is isomorphic to (GLm )d × GLn−md
with LF ∼
= GLm (q d ) × GLn−md (q).
For finite group theorists, Harish-Chandra theory consists in relating two
elements of Irr(GF ) whenever they are constituents of the same RG
L ζ for ζ ∈
Irr(LF ) and L an F -stable Levi subgroup of an F -stable parabolic subgroup.
This leads quickly to the notion of cuspidal characters, i.e. characters that are
G
F
in no RG
L ζ as above unless L = G. From the fact that RL ζ ∈ N Irr(G ) it is
F
easy to see that each set in our partition of Irr(G ) then coincides with the set of
F
components of some RG
L ζ with ζ a cuspidal character of L .
The idea of Broué-Malle-Michel [BrMaMi93] is to generalize that to d-split Levi
subgroups in the place of the 1-split ones considered in Harish-Chandra theory.
Definition 6.6. One writes (L1 , χ1 ) ≤d (L2 , χ2 ) when Li are d-split Levi subL2
groups in G, χi ∈ E(LF
i , 1) and χ2 is a component of RL1 χ1 .
F
A character χ ∈ E(G , 1) is said to be d-cuspidal if a relation (L, ζ) ≤d (G, χ)
is possible only with L = G. A pair (L, ζ) with L a d-split Levi subgroup and a
d-cuspidal ζ ∈ E(LF , 1) is called a unipotent d-cuspidal pair of GF .
The following is due to Broué-Malle-Michel, building on observations made by
Fong-Srinivasan [FoSr86] in non-exceptional types. One keeps G, F as before, and
d ≥ 1.
Theorem 6.7 ([BrMaMi93, 3.2]).
(i) ≤d is transitive among pairs of the type
considered in Definition 6.6
(ii) If (L, ζ) is a unipotent d-cuspidal pair of GF , then for any component χ
of RG
L ζ one has
X
∗ G
g
RL χ = N
ζ
(52)
g∈NG (L)F /NGF (L,ζ)
where N = hχ, RG
L ζiGF 6= 0.
F
(iii) E(GF , 1) = ∪˙ (L,ζ) Irr(GF | RG
L ζ) where (L, ζ) ranges over G -conjugacy
classes of unipotent d-cuspidal pairs and Irr(GF | RG
L ζ) denotes the set of
irreducible components of the generalized character RG
L ζ.
It is fairly clear that (ii) and (iii) above are easy consequences of each other
and both consequences of the first point. The proof of the theorem is by a case by
case analysis and relies in fact on the explicit description of the sets of unipotent
characters and the computation of each Lusztig functor
F
F
RG
L : E(L , 1) → ZE(G , 1).
Such a computation was done by Asai for classical types ([As84a] and [As84b])
and by Broué-Malle-Michel for exceptional types (see [BrMaMi93, Tables 1,2]).
Broué-Malle-Michel also give a parametrization of Irr(GF | RG
L ζ) much in the
spirit of McKay’s conjecture (see [Sp17, 3.A]) on character degrees
36
MARC CABANES
Theorem 6.8 (Broué-Malle-Michel). If (L, ζ) is a unipotent d-cuspidal pair of
GF , then one has a bijection
Irr(NG (L, ζ)F /LF ) → Irr(GF | RG
L ζ)
with good equivariance properties.
Example 6.9. Let us describe the partition of generalized d-Harish-Chandra theory in the case of G = GLn (F), GF = GLn (q). We have seen in Example 4.12 the
parametrization
X
χ(w)RG
χ 7→ Rχ = ±n!−1
Tw 1
w∈Sn
of E(GF , 1) by Irr(Sn ).
Let L(m) = GLn−dm (F) × (S(d) )m as in Example 6.5.
Let us compose the above parametrization of E(GF , 1) with the parametrization
F
of Irr(Sn ) by partitions of n (see § 5.E above), thus giving λ 7→ χG
λ ∈ E(G , 1).
Let λ ⊢ n, then
X
(1)
∗ G
(53)
(−1)h(τ ) χL
RL(1) (χG
λ−τ
λ )=
τ ∈hookd (λ)
where the notations about partitions and hooks is the one of § 5.E and where we
identify E(L(1)F , 1) = E(GLn−d (q), 1).
The proof of (53) relies on computing each ∗ RG
(RG
Tw 1) by means of a Mackey
L(1)
type formula (see Remark 4.3 above) and the observation that L(1) can contain
a GF -conjugate of Tw only if w is conjugate in Sn to w′ c where w′ ∈ Sn−d
and c = (n − d + 1, n − d + 2, . . . , n). Then (53) is just a consequence of the
Murnaghan-Nakayama rule (39).
Like in the case of the symmetric group, (53) can be iterated as long as d-hooks
can be removed and one gets the equivalent of (40), namely
∗
G
L
RG
L(m) (χλ ) = N χκ
(m)
(54)
where κ ⊢ n − md is the d-core of λ and N is a non-zero integer.
(m)
It is not too difficult to show that χκL
is d-cuspidal.
So indeed we get enough unipotent d-cuspidal pairs (L, ζ) with any χ ∈ E(GF , 1)
being in one of the disjoint sets Irr(GF | RG
(ζ)).
L(m)
More work with (39) as main ingredient would tell us that the above are all the
unipotent d-cuspidal pairs and that Theorem 6.7 holds.
6.B. The theorem. The relation between ℓ-blocks and d-Harish-Chandra theory
is given by the following kind of theorem.
Theorem 6.10 (Cabanes-Enguehard). Let G be a reductive group defined over
the finite field Fq , and let F : G → G be the associated Frobenius map. Assume ℓ
is a prime ≥ 7, not dividing q. Let d the (multiplicative) order of q mod ℓ. Then
there is a bijection
(L, ζ) 7→ BGF (L, ζ)
between GF -classes of unipotent d-cuspidal pairs and unipotent blocks (see Definition 4.11). One has
(i) Irr(BGF (L, ζ)) ∩ E(GF , 1) = Irr(GF | RG
L ζ),
(ii) the Sylow ℓ-subgroups of C◦G ([L, L])F are defect groups of BGF (L, ζ).
The theorem has many precursors, first of all by Fong-Srinivasan ([FoSr82] and
[FoSr89]) who treat all blocks (not just unipotent) for classical groups. Note that it
is possible to show that just like unipotent characters are insensitive to the center
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
37
of the group, unipotent blocks are basically the same for all groups of same type
and rank (see [CaEn, §17]), so the above could be deduced from Fong-Srinivasan’s
work in many cases. We have chosen the statement for its simplicity and its
relatively straightforward proof sketched in the next section.
The theorem essentially relates the splitting of E(GF , 1) into ℓ-blocks with
the Lusztig functor. More theorems of the same kinds were given by CabanesEnguehard (all ℓ-blocks, ℓ ≥ 5 [CaEn99]), Enguehard (unipotent blocks for all
primes [En00]) and recently by Kessar-Malle (all blocks and primes [KeMa15]).
Note that given Bonnafé-Dat-Rouquier’s theorem showing equivalence of blocks in
a very strong sense with blocks of generally smaller groups, the above is interesting
only for blocks in Eℓ (GF , s) (see Definition 4.9) where C◦G∗ (s) can’t be embedded
in a proper Levi subgroup of G∗ (”isolated” series), which brings us close to
unipotent blocks.
The statement by Kessar-Malle is as follows. Here G is assumed to be an F stable Levi subgroup of a simple simply connected group. One keeps ℓ a prime
not dividing q and d the order of q mod ℓ when ℓ is odd, while d is the order of
q mod 4 when ℓ = 2. One denotes E(GF , ℓ′ ) the union of rational series E(GF , s)
∗
with s ∈ G∗F semi-simple of order prime to ℓ.
Theorem 6.11 ([KeMa15, Th. A]).
(i) For any d-Jordan-cuspidal pair (L, λ)
of G such that λ ∈ E(LF , ℓ′ ), there exists a unique ℓ-block bGF (L, λ) of
GF such that all irreducible constituents of RG
L (λ) lie in bGF (L, λ).
(ii) The map
(L, λ) 7→ bGF (L, λ)
(55)
is a surjection from the set of GF -conjugacy classes of d-Jordan-cuspidal
pairs (L, λ) of G such that λ ∈ E(LF , ℓ′ ) to the set of ℓ-blocks of GF .
(iii) The map (55) restricts to a surjection from the set of GF -conjugacy classes
of d-Jordan quasi-central cuspidal pairs (L, λ) of G such that λ ∈ E(LF , ℓ′ )
to the set of ℓ-blocks of GF .
(iv) For ℓ ≥ 3 the map (55) restricts to a bijection between the set of GF conjugacy classes of d-Jordan quasi-central cuspidal pairs (L, λ) of G with
λ ∈ E(LF , ℓ′ ) and the set of ℓ-blocks of GF .
(v) The map (55) is bijective if ℓ ≥ 3 is a good prime for G, and ℓ 6= 3 if GF
has a factor 3 D4 (q).
Here the notion of d-Jordan cuspidal character (or pair) is adapted from the
unipotent case through Jordan decomposition. Quasi-central means belonging to
a block of LF covering a block of [L, L]F of central defect (see [KeMa15, §2]).
7. Local methods: unipotent blocks and d-Harish-Chandra theory
The proofs of Theorems 6.10, 6.11 follow the pattern described in § 5.D above
through subpair enlargement and use of Brauer’s second main Theorem.
7.A. The main subpair inclusion.
◦
Lemma 7.1 (see [DigneMic, 13.15.(i)]). If x ∈ GF
p′ then CG (x)/CG (x) has exponent dividing the order of x and injects into Z(G∗ )/Z◦ (G∗ ).
The following compatibility between generalized decomposition maps dx and
functors is of crucial importance. It was first spotted by Fong-Srinivasan
[FoSr82, (2C)].
∗
RG
L
38
MARC CABANES
Proposition 7.2. Let P = Ru (P)L be a Levi decomposition in G with F -stable
◦
L. Let ℓ a prime 6= p, let x ∈ LF
ℓ . Then CG (x) is an F -stable reductive group
◦
◦
◦
and CP (x) = CRu (P) (x)CL (x) the Levi decomposition of a parabolic subgroup.
Moreover
◦
F
∗ CG (x)
x,GF
dx,L ◦ ∗ RG
L⊆P = RC◦ (x)⊆C◦ (x) ◦ d
L
P
on CF(GF , K).
Proof. The group theoretic part of the proposition is classic and was already used
in our statement of the character formula (Proposition 4.4). The composition
◦
F
F
◦
∗ CG (x)
RC◦ (x)⊆C◦ (x) ◦ dx,G makes sense thanks to the inclusion CG (x)F
ℓ′ ⊆ CG (x)
L
P
ensured by Lemma 7.1. The formula itself is an easy consequence of the character
formula.
Though we will apply this property mainly to unipotent blocks, it is fundamental to the proof of a theorem of Broué-Michel on general sums of blocks eℓ (GF , s)
(see § 4.C, Definition 4.9).
We keep G, F as before and ℓ some prime 6= p.
∗
Theorem 7.3 (Broué-Michel [BrMi89]). Let s ∈ (G∗ )F
ℓ′ a semi-simple element
and eℓ (GF , s) the central idempotent of OGF associated (see Definition 4.9). Denote eℓ (GF , s) its image in kGF . Let x ∈ GF
ℓ . Then
X
F
Brx (eℓ (G , s)) =
eℓ (C◦G (x)F , t),
t | ix (t)=s
∗
where ix is a map associating conjugacy classes of semi-simple elements of G∗F
∗
to conjugacy classes of semi-simple elements of C◦G∗ (x)∗F through pairs (T∗ , t) →
(T, θ) → (T∗1 , s) using (25) above.
Proof. Through Brauer’s second Main Theorem it is easy to see that the main
statement is equivalent to checking that
X
F
(C◦ (x)F )
Pt G
(γ1 )
(56)
(dx ◦ Ps(G ) )(γGF .x ) =
t | ix (t)=s
F
(G )
where Ps
: CF(GF ) → CF(GF , Bℓ (GF , s)) is the projection and γGF .x is the
function being 1 on the conjugacy class of x and 0 elsewhere. One has γ1 =
dx (γGF .x ) and γGF .x is uniform (apply Lemma 4.6.(i)), so it is easy to reduce (56)
to the following
X
F
h F
C◦
◦
F −1
G (x)
dx,G ◦ RG
Rh T
◦ dx, T ◦ adh
(57)
T = |CG (x) |
h∈GF | x∈h T
where adh is the conjugation by h of central functions. This in turn can be deduced
from Proposition 7.2 by taking adjoints. Note that, for x an ℓ-element of a finite
group H, the adjoint of dx : CF(H, K) → CF(CH (x), K) is the map sending the
central function f : CH (x) → K to f ′ : H → K defined by
X
f ′ (h) = |CH (x)|−1
f (x−1 hv ).
v∈H | hℓ =vxv −1
Now for unipotent blocks and with the aim of proving Theorem 6.10, the main
step is achieved by the following
Theorem 7.4. Let L be an F -stable Levi subgroup of G and ζ ∈ E(LF , 1). Assume
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
39
F
(a) LF = CG (Z(L)F
ℓ ) , and
C◦ (A)
F
(b) for all A ≤ Z(L)ℓ and χA an irreducible component of RL G ζ, denoting
H = C◦G (A), one has
X
∗ H
g
RL χA = h∗ RH
ζ.
L χA , ζiLF
g∈NH (L)F /NH (L,ζ)F
Then for any irreducible component χ of RG
L ζ one has an inclusion of ℓ-subpairs
({1}, bGF (χ)) ≤ (Z(L)F
ℓ , bLF (ζ)).
Proof. One uses an induction on |GF /LF |. Everything is clear when GF = LF .
Assume LF 6= GF , so that, thanks to (a), one can pick z ∈ Z(L)F
ℓ non central
in GF . Let H := C◦G (z) ⊇ L. Let us show
F
hdz,G χ, RH
L ζiHF 6= 0.
(58)
F
F
One has indeed dz,G χ ∈ CF(HF , K) since CG (z)F
ℓ′ ⊆ H thanks to Lemma 7.1.
We have
F
hdz,G χ, RH
L ζiHF
=
=
hdz,L
F
∗
RG
L χ, ζiLF by Proposition 7.2
X
F
G
hRL ζ, χiGF
hdz,L ζ ′ , ζiLF by (b) with A = {1}
ζ ′ ∈NG (L)F .ζ
=
X
hRG
L ζ, χiGF
F
hd1,L ζ ′ , ζiLF since z ∈ Z(L)F ≤ ker(ζ)
ζ ′ ∈NG (L)F .ζ
X
F
F
hd1,L ζ ′ , d1,L ζiLF
=
hRG
L ζ, χiGF
=
F
−1
hRG
hf, f iLF
L ζ, χiGF |NG (L) .ζ|
ζ ′ ∈NG (L)F .ζ
P
F
for f := ζ ′ ∈NG (L)F .ζ d1,L ζ ′ . But f ∈ CF(LF , K) is clearly a central function
such that f (x−1 ) is the complex conjugate of f (x) for any x ∈ LF and f 6= 0 by
the value at 1. So hf, f iLF 6= 0 and we get (58) from the above.
Now (58) implies that there is an irreducible component χH of RH
L ζ such that
z,GF
F
d
χ has a non-zero projection on CF(H | bHF (χH )).
One may apply the induction hypothesis to H, L, ζ replacing G, L, ζ since (a)
and (b) are clearly satisfied there. The fact that χH is a component of RH
L ζ implies
the subpair inclusion
({1}, bHF (χH )) ≤ (Z, bLF (ζ)) in HF
F
where we abbreviate Z = Z(L)F
= CGF (z). Then it is easy to
ℓ . Assume H
deduce from the above the subpair inclusion
(hzi , bHF (χH )) ≤ (Z, bLF (ζ))
in
GF .
(59)
F
On the other hand the fact that dz,G χ has a non-zero projection on CF(HF |
bHF (χH )) implies that we have
({1}, bGF (χ)) ≤ (hzi , bHF (χH ))
in GF
(60)
thanks to Brauer’s second Main Theorem. We then get our claim from (59) and
(60) by transitivity of subpair inclusion.
We have assumed for simplification that HF = CGF (z). In general we only
have HF ⊳ CGF (z) with index a power of ℓ thanks to Lemma 7.1. Then it is easy
to define the unique block b′ of CGF (z) covering bHF (χH ) and prove the analogues
of (59) and (60) with it.
40
MARC CABANES
7.B. φd -tori and ℓ-subgroups. We keep G, F as before over Fq , and ℓ a prime
∤ q. We also assume now that ℓ ≥ 7.
Note that ℓ divides φm (q) if and only if mℓ′ = d (see for instance [Serre, § II.3.2]).
Proposition 7.5. Assume ℓ divides φm (q) but neither |Z(G)F /Z◦ (G)F | nor
|Z(G∗ )F /Z◦ (G∗ )F |. Let S be a φm -torus (see Definition 6.4), L := CG (S),
Z := Z(L). Then
◦
F
(i) L = C◦G (SF
ℓ ) = CG (Zℓ ) and
◦
F
F F
(ii) L = CG (Zℓ ) = CGF (ZℓF ).
Proof. (i) It suffices to check the first equality. We show it by induction on the
dimension of G. Let π : G → Gad := G/Z(G) the reduction mod Z(G).
By a classical argument we have an exact sequence
1 → π(SF ) → π(S)F → [S, F ] ∩ Z(G)/[Z(G), F ] → 1.
By Lang’s theorem, [Z(G), F ] ⊇ Z◦ (G), so [S, F ] ∩ Z(G)/[Z(G), F ] is a section of Z(G)/Z◦ (G) on which the action of F is trivial. But ℓ does not divide
F
|Z(G)F /Z◦ (G)F | so ([S, F ] ∩ Z(G))/[Z(G), F ] is ℓ′ ; thus π(S)F
ℓ ⊆ π(S ). Moreover, if s is of finite order, then π(sℓ ) = π(s)ℓ . This implies
F
π(S)F
ℓ = π(Sℓ ) .
(61)
C◦G (SF
ℓ ).
Now denote C :=
The fact that ℓ ≥ 7 eliminates some exceptional
behaviour (“bad” primes, see [GeHi91, 2.1] or [CaEn, §13.2]) and ensures that C
is a Levi subgroup of G. One has clearly L ⊆ C. If C 6= G, then the induction
hypothesis gives L = C, that is our claim.
F
Assume C = G, that is π(SF
ℓ ) = {1}. By (61), this implies π(S)ℓ = {1}. But
π(S) is a φm -torus of Gad whose number of fixed points under F is a power of
φm (q). This is prime to ℓ only if this exponent is 0, that is S ⊆ Z◦ (G). This
implies L = G and our claim is trivial.
(ii) The first equality comes from (i). For the second we have an inclusion
C◦G (ZℓF )F ⊳ CGF (ZℓF ). But the factor group is trivial thanks to Lemma 7.1 and
the hypothesis on ℓ with regard to G∗ .
Corollary 7.6. Let ℓ be a prime ≥ 7 and 6= p. Let d be the order of q mod ℓ. Let
(L, ζ) be a unipotent d-cuspidal pair of G. Then
(i) LF = CGF (ZℓF ) and
(ii) ζ(1)ℓ = |LF /Z(L)F |ℓ = |LF /Z◦ (L)F |ℓ .
Proof. (i) To deduce this from Proposition 7.5.(ii) we essentially have to show that
the condition ℓ ∤ |Z(G)F /Z◦ (G)F |.|Z(G∗ )F /Z◦ (G∗ )F | can be assumed. Since ℓ is
large, this concerns chiefly groups of type An−1 with ℓ dividing q − ǫ with ǫ = 1
or −1 according to the action of F on roots being trivial or not, respectively (see
[CaEn, 13.11]). Let T1 be the diagonal maximal torus. That is the one whose
image in Gad is such that F acts trivially on the associated Weyl group. Then we
have
L = T1 and LF = CGF (LF
(62)
ℓ ).
Indeed one can then assume d = 1 or 2 according to ǫ = 1 or −1. On the other
hand it is well-known that E(GF , 1) is the set of components of RG
T1 1, so (T1 , 1)
is the only unipotent d-cuspidal pair. This forces L = T1 . The second statement
is an easy verification in PGL.
(ii) The degrees of d-cuspidal characters are known from [BrMaMi93] and, up
to integral scalars involving only bad primes, they are polynomials in q where the
power of each φdℓa is the same as in the order of the group.
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
41
7.C. Defect groups. We finish to review the proof of Theorem 6.10 whose hypotheses we keep.
We have a unipotent d-cuspidal pair (L, ζ) and we have seen that if (L, ζ) ≤d
(G, χ) (see Definition 6.6) then one has LF = CGF (Z(L)F
ℓ ) and the inclusion of
ℓ-subpairs
({1}, bGF (χ)) ≤ (Z(L)F
(63)
ℓ , bLF (ζ)).
This is obtained by applying Corollary 7.6.(i) and Theorem 7.4 above. Note that
this already implies that we can define BGF (L, ζ) as the ℓ-block B such that Irr(B)
contains all irreducible components of RG
L ζ. Concerning defect groups we prove
Proposition 7.7. Let D be a Sylow ℓ-subgroup of CG ([L, L])F containing Z(L)F
ℓ .
F
Then ResL
ζ
is
irreducible
and
CGF (D)
F
L
(Z(L)F
ℓ , bLF (ζ)) ≤ (D, bCG (D)F (ResC
GF
(D)
ζ)).
Both subpairs are centric and the second is a maximal subpair.
Proof. For the first statement notice that [L, L]F ≤ CGF (D) ≤ CGF (Z(L)F
ℓ ) =
LF . Unipotent characters of LF restrict irreducibly to [L, L]F thanks to (32)
above, hence irreducibly to CGF (D). From Corollary 7.6 we know that (Z(L)F
ℓ , bLF (ζ))
is a centric subpair. The subpair inclusion of the proposition is then easily checked
F
by applying Theorem 5.7. The maximality of the subpair (D, bCG (D)F (ResL
CGF (D) ζ))
is not too difficult, see also Remark 7.9 below.
We have now proved almost all of Theorem 6.10. There remains to show that
if two unipotent d-cuspidal pairs (L1 , ζ1 ), (L2 , ζ2 ) are such that BGF (L1 , ζ1 ) =
BGF (L2 , ζ2 ) then they are GF -conjugate. In such a case the maximal subpairs
given by Proposition 7.7 would be GF -conjugate by Theorem 5.2.(ii). But since in
Proposition 7.7, one has [L, L] ≤ CG (D) ≤ L and therefore [L, L] = [CG (D), CG (D)],
F
i
one gets easily that the pairs ([Li , Li ], ResL
[Li ,Li ] ζi ) are G -conjugate. The lemma
L2
1
below shows that one may assume (L1 , ResL
[L1 ,L1 ] ζ1 ) = (L2 , Res[L2 ,L2 ] ζ2 ). But
then ζ1 = ζ2 by (32).
Lemma 7.8. If two d-split Levi subgroups L1 , L2 of G have same derived subgroup, then they are C◦G ([L1 , L1 ])F -conjugate.
To show this one notices first that C := C◦G ([Li , Li ]) is a reductive group
where Z◦ (Li ) is a maximal torus. Moreover Z◦ (Li )φd is a maximal φd -torus in
C. Both properties are by computing the centralizers in C and remembering that
Li = CG (Z◦ (Li )φd ) by definition. But then the Sylow theorem for maximal φd -tori
(see [BrMa92], [CaEn, 13.18]) implies our claim.
Remark 7.9. Using Theorem 6.10 the principal ℓ-block of GF is described in the
following fashion. It corresponds to (L, ζ) with L = CG (S) where S is a maximal
φd -torus of (G, F ) and ζ = 1 is the trivial character of LF . With the hypothesis we
have on ℓ (which may be loosened to include ℓ = 5) one can prove that the defect
group D may be taken normalizing Z := Z(L)F
ℓ with the additional property that
Z is the unique maximal abelian normal subgroup of D
(64)
(see [Ca94]). This gives a quite handy property of Sylow ℓ-subgroups of finite
groups of Lie type for the transversal primes. The exceptions for the primes 2, 3
are given in [Ma07, 5.14, 5.19]. Indeed the subgroup Z is therefore characteristic
in D. This can help conclude about the maximality of the subpair proposed in
Proposition 7.7 (see [KeMa15]).
42
MARC CABANES
One also has M := NG (S)F = NGF (Z) ≥ NGF (D) and any automorphism
of GF preserving D will obviously preserve M thanks to (64). This explains
why, paving the way for future checkings in finite groups of Lie type, the inductive
conditions for McKay or Alperin-McKay conjectures consider a possible overgroup
for the normalizer of the defect group involved in the original statement of the
conjectures (see [IsMaNa07, §10 (2)], [Sp13, 7.2]).
7.D. Non-unipotent characters of unipotent blocks. Brauer’s second Main
Theorem can also be used to give a complete description of Irr(BGF (L, ζ)) (see
Theorem 6.10 above) in terms of Lusztig’s functor.
We keep the context of Theorem 6.10 where ℓ is a prime ≥ 7 different from the
defining characteristic of GF . We assume (G, F ) is in duality with some (G∗ , F ∗ ).
∗
◦
∗
Let t ∈ (G∗ )F
ℓ . Then CG∗ (t) is a Levi subgroup of G , which by duality yields
an F -stable Levi subgroup G(t) of G and a linear character b
t : G(t)F → C× . A
quite elementary generalization of Theorem 4.13 (see [CaEn, 15.10]) shows that
there is a sign ǫG,t ∈ {1, −1} such that we get a map
ψt : E(G(t)F , 1) → NE(GF , t)
χt
7→
by
b
ǫG,t RG
G(t) (tχt ).
Theorem 7.10 (See [CaEn, §23.1). Keep the hypotheses of Theorem 6.10. Let
∗
F
(L, ζ) be a unipotent d-cuspidal pair in G. Let t ∈ (G∗ )F
ℓ , χt ∈ E(G(t) , 1).
Then ψt (χt ) has components in Irr(BGF (L, ζ)) if and only if there is a unipotent
d-cuspidal pair (Lt , ζt ) in G(t) such that
(i) (Lt , ζt ) ≤d (G(t), χt ) in G(t), with
L
t
(ii) [L, L] = [Lt , Lt ] and ResL
[Lt ,Lt ] ζt = Res[L,L] ζ.
Then all components of ψt (χt ) are in Irr(BGF (L, ζ)).
Note that condition (ii) above implies that t must be in the centralizer of [L∗ , L∗ ]
for L∗ an F ∗ -stable Levi subgroup of G∗ corresponding to L by duality. This
condition looks like dual to the condition for an element of GF of being in the
defect group of BGF (L, ζ), see Theorem 6.10.(ii).
The proof of Theorem 7.10 is by using Proposition 7.2 with x = 1. One gets
d1 ψt (χt ) = ±d1 (RG
G(t) χt )
which by Brauer’s second Main Theorem must have a non-zero projection on
BGF (L, ζ). This reduces the theorem to a question about unipotent characters. It
is solved by studying a bit more the relation between d-split Levi subgroups and
centralizers of ℓ-subgroups beyond what has been seen in §7.B above (see [CaEn,
§23.1]).
7.E. Unipotent blocks are non-exotic. One of the main questions about blocks
of quasi-simple groups in relation with fusion systems is to relate their fusion systems (see Definition 5.3) with the ones of finite groups, or equivalently principal
blocks (Open problems 1 and 3 in [AschKeOl, Sect. IV.7]). Fusion systems on
a p-group that are not isomorphic to a FQ (H) (see Definition 1.13) for a Sylow
p-subgroup Q of a finite group H are called exotic. See Remark 5.13 above for
the case of symmetric groups; a similar result is also known for general linear and
unitary groups [Bro86, 3.8].
We show here the same in the context of Theorem 6.10. In other words unipotent ℓ-blocks (ℓ ≥ 7) are non-exotic (sorry). This builds on an earlier theorem
[CaEn99b] showing control of fusion in the sense of [Thev, §49], a slightly weaker
statement.
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
43
Theorem 7.11. Keep the assumptions and notations of Theorem 6.10. Let (L, ζ)
be a unipotent d-cuspidal pair and BGF (L, ζ) the associated ℓ-block of which a
Sylow ℓ-subgroup D of C◦G ([L, L])F is a defect group.
F
(i) There exists a subgroup H ≤ NGF ([L, L], ResL
[L,L]F ζ) such that
(a) D is a Sylow ℓ-subgroup of H, and
F
(b) H[L, L]F = NGF ([L, L], ResL
[L,L]F ζ) .
(ii) For any H satisfying the above, the fusion system of BGF (L, ζ) is isomorphic to FD (H).
Proof. The first point is purely group theoretic. The proof uses basically considerations in NG (T), where T is a maximally split torus of [L, L]C◦G ([L, L]), see
[CaEn, Ex 23.1], [CaEn99b, 6-7] for all details. We now prove (ii).
◦
F
Denote Z := Z(L)F
ℓ , C := CG ([L, L]). Note that C ⊳ H. Recall from (63)
and Proposition 7.7 that we have subpair inclusions in GF
({1}, BGF (L, ζ)) ≤ (Z, bLF (ζ)) ≤ (D, bD )
(65)
where the middle one is centric and (D, bD ) is maximal with
F
bD = bCGF (D) (ResL
CGF (D) ζ).
For X ≤ D denote (X, bX ) the unique subpair of GF such that
(X, bX ) ≤ (D, bD ).
Our category isomorphism will be (see Definition 5.3)
F(D,bD ) (bGF (L, ζ))
→ FD (H),
(X, bX ) 7→ X.
From the theory of fusion systems, essentially the fact that “F -essential objects
are F -centric” see [AschKeOl, §I.3] and the identification of centric objects in those
categories with what we have called so until now [AschKeOl, IV.3.20], it suffices
to check for X ≤ D
NGF ((X, bX )) = NH (X)CGF (X) if X in H or (X, bX ) in GF is centric.
(66)
Note that CF ⊳ H with ℓ′ index by assumption (i.a), so a subgroup of D is
centric in H if and only if it is centric in CF .
Let us recall the decomposition G = Ga Gb associated to a pair (G, F ) and a
prime ℓ (see [CaEn, 22.4]). In the decomposition of
[G, G] = G1 . . . Gm
as a central product of F -stable closed subgroups, one defines Ga = Z◦ (G)G′a
mi
∼
)
where G′a is the subgroup generated by the Gi ’s such that (Gi )F
ad = PGLni (ǫi q
mi
with ℓ dividing q − ǫi . The other Gi ’s generate by definition Gb . From the
properties of |Z(Gsc )F | according to the type of (G, F ) it is easy to see that
F
′
Z(Gb )F and GF /GF
a Gb are abelian ℓ -groups.
(67)
An important (but easy) consequence for centric ℓ-subgroups is the following
[CaEn, 22.5.(ii)], where Y denotes an ℓ-subgroup of GF :
If Z(CGF (Y ))ℓ ⊆ Ga then Y ⊆ Ga .
Ga C◦G (Z(D)).
(68)
Arguing as in the proof of Proposition 7.5,
Let us define K :=
one sees that K is a Levi subgroup of G such that K ⊇ L = SKb , where S is a
diagonal torus of Ka (therefore [L, L] = Kb ), and D ⊆ Ka .
44
MARC CABANES
By (32), restriction maps induce bijections
F
F
F
∼
E(KF , 1) ∼
= E(KF
a , 1) × E(Kb , 1) = E(Ka , 1) × E(L , 1).
We then define ζe ∈ E(KF , 1) corresponding to (1KFa , ζ) in the last product.
Assume X ≤ D is either centric in CF or (X, bX ) is centric in GF . By Proposition 5.8, Z(D) ⊆ Z ∩ Z(X) and therefore K contains CGF (X), and C◦G (X).
Iterating the above (68) it is easy to see that X ⊆ Ka .
F
F
e
Let ζX := ResK
CGF (X) ζ whose restriction to [L, L] is of central defect by applying for instance Theorem 6.10 to [L, L]. Note that bD is the block of ζD . By
a slight variant of Theorem 5.7 (see [CaEn, 5.29]) one gets the subpair inclusion
(X, bCGF (X) (ζX )) ≤ (D, bD ) and therefore
bX = bCGF (X) (ζX ).
(69)
If X is assumed centric in CF , then (X, bX ) is centric, or equivalently ζX (1)ℓ ≥
|CGF (X)/Z(X)|ℓ because
|CGF (X)/Z(X)|ℓ
=
|(CKa (X)Kb )F /Z(X)|ℓ
=
|CKa (X)F /Z(X)|ℓ .|KF
b |ℓ by (67)
=
F
◦
F
|KF
b |ℓ (X centric in Ka ⊆ CG (Kb ) )
=
ζ(1)ℓ , see above.
Now assume conversely the weaker assumption that (X, bX ) is centric. First ζX
is the canonical character of bX because it has Z(X) ∈ KF
a in its kernel. Moreover
C◦G (X) = C◦Ka (X)Kb has its first term of a-type (an easy check by induction on
|X| in groups of type A) so
C◦G (X)b = [L, L] = Kb .
(70)
The restriction of ζe to C◦G (X)F (or any (MKb )F with M an F -stable connected
reductive subgroup of Ka ) is a unipotent character, it is the unique one whose
restriction to [L, L]F = KF
b is the restriction of ζ.
So we get
C F (X)
◦
F
◦
◦
F
(iii) ResCG
◦ (X)F ζX ∈ Irr(CG (X) ) is the only unipotent character ζX ∈ E(CG (X) , 1)
G
C◦ (X)F
G
such that Res[L,L]
F
F
◦
ζX
= ResL
[L,L]F ζ.
Let’s keep (X, bX ) centric. If g ∈ GF normalizes it, the above implies that
g normalizes [L, L] while the canonical character of bX restricts to [L, L]F as
F
LF
F
ResL
[L,L]F ζ. Then g normalizes ([L, L], Res[L,L]F ζ) and therefore g ∈ H[L, L] ⊆
HCGF (X) by assumption (i.b).
F
◦
Conversely, if h normalizes X and ([L, L], ResL
[L,L]F ζ), it normalizes CG (X)
C (X)F
F
and sends ResCG
is
◦ (X)F (ζX ) to a unipotent character whose restriction to [L, L]
G
h
C (X)F
F
C (X)F
F
L
G
G
(ζX ) = h ResL
Res[L,L]
F
[L,L]F ζ = Res[L,L]F ζ by (iii) above. So h fixes ResC◦ (X)F (ζX ).
G
C (X)F
(ResCG
◦ (X)F (ζX ))
G
By [NagaoTsu, 5.5.6], bX is the unique block covering b
since the index is a power of ℓ (use Lemma 7.1). So bX is fixed by h. By assumption
(i.b) this applies to any h ∈ NH (X). This completes the proof of (66).
F
C◦
G (X)
7.F. A theorem of Broto-Møller-Oliver. Until now we have compared only
fusion systems of ℓ-blocks of groups GF in the same defining characteristic p.
Broto-Møller-Oliver [BrMO12] have proved a very impressive theorem showing
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
45
equivalence of fusion systems of ℓ-subgroups for groups GF of various defining
characteristics.
We give the theorem in a simplified form (the original one is stronger, see [BMO,
Th. A]).
Theorem 7.12. Let G a reductive group over F. Let H a reductive group over
a field of K of characteristic r. Assume that for some maximal tori T ≤ G, S ≤
H, the two groups have same quadruple Hom(T, F× ) ⊇ Φ(G, T), Hom(F× , T) ⊇
Φ(G, T)∨ . Let F : G → G a Frobenius endomorphism acting trivially on the root
system, let q the power of p associated (for instance q = |XF
α | for all α ∈ Φ(G, T)).
Let F ′ : H → H a similar endomorphism for H and q ′ the corresponding power of
r.
Assume ℓ is a prime 6∈ {2, p, r}, assume q, q ′ have same multiplicative order d
mod ℓ and that (q d − 1)ℓ = (q ′d − 1)ℓ .
′
Then GF and HF have isomorphic fusion systems of ℓ-subgroups.
The proof would be too long to sketch here, see also [AschKeOl, Sect. III.1.7].
Let’s say just that it uses all the strength of the topological methods developed by
Broto-Levi-Oliver along with an old theorem of Friedlander [Fr82, 12.2] on étale
homotopy of the algebraic groups G and a less old one by Martino-Priddy (see
[Mis90], [MaPr96]).
Remark 7.13. With the elementary group theoretical methods used in the proof
of Theorem 6.10 (Sylow φd -tori and their normalizers) and under the same assumptions about ℓ, it is easy to describe the Sylow ℓ-subgroups of GF as semi-direct
products
Z ⋊N
(see [Ca94, 4.4]) where
• Z = Z(CG (S))F for S a Sylow φd -torus of G (see also Remark 7.9),
• N is a Sylow ℓ-subgroup of (WCG (S) (T)⊥ )F , where T is an F -stable
maximal torus of CG (S) and WCG (S) (T)⊥ is the subgroup of the Weyl
group WG (T) generated by reflections through roots orthogonal to any
α ∈ Φ(G, T) with α(S) = 1
• the action of N on Z comes from the inclusion Z ≤ TF .
All the above can be read in the “root datum” quadruple of the pair (G, F ).
′
This would imply that the two finite groups GF and HF of the Theorem above
have isomorphic Sylow ℓ-subgroups. Comparing the fusion systems needs to find
the essential subgroups of ZN in the sense of [AschKeOl, §I.3] and the action
of their normalizers. This has been determined in many cases by Jianbei An as
a by-product of his program to determine radical subgroups (see Definition 1.6)
in finite groups of Lie type and check Alperin’s weight conjecture (16) for those
groups. See [AnDi12, §3] for essential ℓ-subgroups of finite classical groups. See
[AnDiHu14] and the references given there for many exceptional types.
8. Some applications
8.A. Abelian defect. When the defect group of some block B defined by Theorem 6.10 is assumed to be abelian, the description of Irr(B) simplifies a lot. One
keeps the same hypotheses on G, F , ℓ, d, (L, ζ).
46
MARC CABANES
Theorem 8.1. Assume the defect ℓ-groups of BGF (L, ζ) are abelian. Then
[
b
Irr(RG
Irr(BGF (L, ζ)) =
G(t) (tχt ))
t,χt
where t and χt are subject to the following conditions
∗
(a) t ∈ (G∗ )F
,
ℓ
(b) L ⊆ G(t) where the latter is a Levi subgroup in duality with C◦G∗ (t) ,
G(t)
(c) χt is an irreducible component of RL ζ.
Proof. By Corollary 7.6 and Proposition 7.7 we know that the defect group can be
F
abelian only if the centric subpair (Z(L)F
ℓ , bLF (ζ)) is maximal and Z(L)ℓ is a Sylow
◦
F
ℓ-subgroup of CG ([L, L]) . By Corollary 7.6.(ii) this means that the polynomial
order of (C◦G ([L, L]), F ) has not more powers of cyclotomic polynomials φm with
mℓ′ = d than its (maximal) torus (Z◦ (L), F ). This property can be written entirely
in the groups X(T0 ) and Y (T0 ) of G, so they transfer to the same property in the
∗
∗
dual, namely C◦G∗ ([L∗ , L∗ ])F has a Sylow ℓ-subgroup in Z◦ (L∗ )F . Then when
∗
∗
imposing the condition that t commutes with [L , L ] from Theorem 7.10, one
may assume that t ∈ Z◦ (L∗ ) and therefore L∗ ⊆ C◦G∗ (t). Then one may choose
G(t) ⊇ L. The last point is then clear from Theorem 7.10 by use of Lemma 7.8.
8.B. Brauer’s height zero conjecture. The description of Theorem 8.1, along
with the parametrization of Theorem 6.8, leads quickly to check the degrees in
Irr(BGF (L, ζ)) when the unipotent block BGF (L, ζ) has abelian defect (see [BrMaMi93, 5.15]), keeping the restrictions on ℓ of Theorem 6.10. In particular, χ(1)ℓ
takes only one value for χ ∈ Irr(BGF (L, ζ)), thus confirming Brauer’s height
zero conjecture (BHZC)
D is abelian if and only if |{χ(1)ℓ | χ ∈ Irr(B)}| = 1
(71)
where B is an ℓ-block of a finite group with defect group D.
Kessar-Malle have proven
Theorem 8.2 (see [KeMa13], [KeMa17]). The equivalence of (71) is true for all
blocks of finite quasi-simple groups.
Given past knowledge about alternating groups, sporadic groups and blocks
of finite reductive groups for “good” primes recalled above, Kessar-Malle’s proof
concentrates mostly on ℓ-blocks of groups of Lie type for ℓ ≤ 5 where the challenge
is still remarkably difficult.
This type of verification in groups of Lie type is important because of the
reduction theorems of Berger-Knörr [BeKn88] and Navarro-Späth [NaSp14].
Theorem 8.3 (Berger-Knörr). Let ℓ be a prime number. If for any ℓ-block B of
a quasi-simple group with abelian defect group, (χ(1)ℓ )χ∈Irr(B) is constant, then it
is the case for any ℓ-block with abelian defect of any finite group, i.e. (BHZ1), the
“only if” part of (71), holds.
Corollary 8.4 (see [KeMa13]). If an ℓ-block B of a finite group has abelian defect
groups, then (χ(1)ℓ )χ∈Irr(B) is constant.
The converse should be checked through Navarro-Späth’s reduction theorem.
Theorem 8.5 (Navarro-Späth). If all blocks of finite quasi-simple groups satisfy
the inductive Alperin-McKay condition of [Sp13, 7.2], then (71) holds.
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
47
The reduction theorems for the two directions of Brauer’s height zero conjecture
are proven with very different methods pointing possibly to problems of quite
different nature. While the proof of Theorem 8.3 uses module theoretic methods
and a theorem of Knörr on Green vertices of simple modules (see [Kn79]), the
proof of Theorem 8.5 uses mainly the techniques described in [Sp17].
8.C. Nilpotent blocks. A nilpotent ℓ-block is one such that any of its defect
groups controls the fusion of its subpairs (see [Thev, Sect. 49], [AschKeOl, Sect.
IV.5.6]). Namely
Definition 8.6. An ℓ-block B of a finite group H is called a nilpotent block if
and only if for any B-subpair (P, bP ) in H the quotient
NH (P, bP )/CH (P )
is an ℓ-group.
As in all statements about the fusion system of subpairs, the condition above
can be loosened to be asked only for centric B-subpairs (P, bP ). Note that the
above condition about ℓ-subgroups instead of subpairs would give the well-known
local characterization of ℓ-nilpotent groups (i.e., H/Oℓ′ (H) is an ℓ-group) due to
Frobenius [Asch, 39.4].
The main structure theorem about nilpotent blocks is due to Puig (see [Thev,
§49-51], see also [Kuls, 15.3] for the easier version over a finite field).
Theorem 8.7. Let B be a nilpotent ℓ-block seen as a subalgebra of OH. Let D
one of its defect group. Then there is an integer m such that
B∼
= Matm (OD).
This of course implies the same over the finite field k = O/J(O) and therefore
the very important property
| IBr(B)| = 1.
(72)
Determining nilpotent blocks of quasi-simple groups H was achieved by An-Eaton.
Their result implies
Theorem 8.8 ([AnEa11, 1.1, 1.2], [AnEa13, 1.1, 1.3]). Let B be an ℓ-block of a
finite quasi-simple group H. Then B is nilpotent if and only if | IBr(bP )| = 1 for
any B-subpair (P, bP ). Moreover B has abelian defect groups.
We prove below a slightly stronger statement concentrating on the property
(72) again in the framework of unipotent ℓ-blocks with ℓ not too bad.
We keep GF , ℓ ≥ 5 a prime good for G (see [GeHi91, 2.1] or [CaEn, §13.2]) not
dividing q, and BGF (L, ζ) a unipotent ℓ-block of GF as in Theorem 6.10.
Proposition 8.9. Assume BGF (L, ζ) has just one Brauer character. Then BGF (L, ζ)
is a nilpotent block and its defect groups are abelian.
Proof. By [CaEn, 14.6], the restrictions of the elements of E(GF , 1) to the set GF
ℓ′
of ℓ-regular elements of GF are distinct and linearly independent central functions.
Since Brauer characters are a basis for the central functions on GF
ℓ′ , our hypothesis
implies that E(GF , 1)∩Irr(BGF (L, ζ)) has a single element. By Theorem 6.10, this
implies that RG
L (ζ) is a multiple of a single irreducible character. By Theorem 6.8,
this implies in turn that Irr(NGF (L, ζ)/LF ) has a single element and therefore
NGF (L, ζ) = LF .
Then the centric subpair (Z(L)F
ℓ , bLF (ζ)) of Proposition 7.7 above is maximal
(and the only centric subpair up to conjugacy). This can be seen by applying
48
MARC CABANES
Proposition 5.8 and noting that (Z(L)F
ℓ , bLF (ζ)) is normal in no other subpair
F
F
since NGF (Z(L)ℓ , bLF (ζ)) = L = CGF (Z(L)F
ℓ ). This proves at the same time
that the defect groups are abelian and that the block is nilpotent.
8.D. Broué’s abelian defect conjecture when ℓ divides q−1. Broué’s abelian
defect conjecture [Bro90a, 6.2] is as follows.
Let H be a finite group, (O, K, k) an associated ℓ-modular system, B a block
of OH, D its defect group and BD its Brauer correspondent (see Theorem 1.17.(i)
above) viewed as a subalgebra of ONH (D). When D is abelian, Broué’s abelian
defect conjecture says that the derived categories of B and BD should be equivalent
(73)
Db (B-mod) ∼
= Db (BD -mod)
later strengthened to the requirement that
Hob (B-mod) ∼
= Hob (BD -mod)
(74)
by a Rickard equivalence (see Definition 9.15 below), that is an equivalence of the
homotopy categories with a strong compatibility with fusion. Note that here one
does not expect consequences on the fusion systems of the blocks involved since
in this case it is very simply the one of BD as a classical consequence of abelian
defect.
In the case of principal blocks, Craven and Rouquier have proved a reduction
theorem to simple groups [CrRo13]. The conjecture for arbitrary blocks with
abelian defect has been checked in many cases. For the defining prime and SL2 (q)
it was proved by Okuyama in the influential preprint [Oku00]. Chuang-Kessar
showed it for certain blocks of symmetric groups [ChKe02]. This combined with
Theorem 5.14 allow Chuang-Rouquier to also check it for blocks of symmetric
groups [ChRo08, 7.6]. The same paper shows it for GLn (q) for ℓ ∤ q as a consequence of the Rickard equivalences they prove between blocks of GLn (q)’s and
theorems of Turner [Tu02] supplying results similar to [ChKe02] for those groups.
Dudas-Varagnolo-Vasserot in [DuVV15] and [DuVV17] have also checked Broué’s
conjecture (and Rickard equivalences similar to Theorem 5.14) for certain unipotent blocks of finite reductive groups of types 2 A, B and C through categorifications
they build for certain affine Lie algebras. For the application to Broué’s conjecture, some work of Livesey is also used to spot nicer representatives among Rickard
equivalent blocks (see [Li15]).
We just prove here a very elementary case yet substantial where the equivalence
is in fact a quite explicit Morita equivalence. The following is a simplification of
a more general statement by Puig with a different proof [Puig90].
We keep (G, F ) defined over Fq .
Theorem 8.10. Let ℓ ≥ 7 be a prime dividing q − 1. Let D be a Sylow ℓ-subgroup
of GF . Assume D is abelian. Then the principal ℓ-blocks over O of GF and
NGF (D) are Morita equivalent:
B0 (GF )-mod ∼
= B0 (NGF (D))-mod.
We will prove more concretely
Proposition 8.11. Let T ⊆ B both F -stable in (G, F ) as above. Let ℓ be a
prime dividing q − 1 and such that NG (T)F /TF is an ℓ′ -group. Let U := Ru (B)F ,
GF
T ′ = TF
ℓ′ . Let M := IndUT ′ O. Then
EndOGF M ∼
= O(NG (T)F /TF
ℓ′ )
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
49
by an isomorphism mapping any t ∈ TF
ℓ to the endomorphism of M sending 1 ⊗ 1
to t−1 ⊗ 1.
Let’s see first how this will give Theorem 8.10. We will essentially apply
Lemma 4.15 and Theorem 8.1.
Let’s note that
F
Irr(B0 (GF )) = Irr(IndG
(75)
UT ′ 1) .
This is indeed easy to deduce from Theorem 8.1 and the fact that (T, 1) is the
1-cuspidal pair satisfying (GF , 1) ≥ (T, 1).
Let us denote
F
A = EndOGF (IndG
UT ′ O).
By Proposition 8.11, A and B0 (GF ) are both blocks of finite groups. Moreover
M is a bi-projective B0 (GF ) ⊗ Aopp -module.
P Indeed right projectivity is ensured
−1
by writing M = OGF e for e = |BF
|
′
ℓ
x∈BF′ x. For the right projectivity it
ℓ
suffices to check the restriction to a Sylow ℓ-subgroup of NG (T)F /T ′ through the
isomorphism of Proposition 8.11. By the assumption on NG (T)F /TF this is TF
ℓ
whose action on the right is said to be through the left action of TF , so has already
been checked.
By Broué’s lemma (Lemma 4.15), in order to get Theorem 8.10 it suffices to
show that K ⊗O M induces a bijection between simple K ⊗O A-modules and
Irr(B0 (GF )). One has K ⊗O A = EndKGF (K ⊗O M ), so K ⊗O M bijects the
simple K ⊗O A-modules and the Irr(K ⊗O M ). Then (75) gives our claim.
Let us now prove Proposition 8.11. We abbreviate G = GF , N = NG (T)F ,
T = TF , W = N/T . Using again that W is an ℓ′ -group, one has
N/T ′ ∼
= Tℓ ⋊ W
(76)
by a map leaving unchanged the elements of Tℓ .
On the other hand by Example 2.3, A has a basis (an )n∈UT ′ \G/UT ′ defined
by (13). Note that one can take n ∈ N/T ′ by Bruhat decomposition (7). This
contains Tℓ for which the action of at (t ∈ Tℓ ) is by multiplication by t. One has
clearly
an at = ant = an t an for any n ∈ N/T ′ , t ∈ Tℓ .
(77)
Let us consider the map
G
M = IndG
UT ′ O → IndUT O defined by 1 ⊗UT ′ 1 7→ 1 ⊗UT 1.
One sees easily that the kernel of this map is stable under A, so any endomorphism
of M induces an endomorphism of IndG
UT O seen as a quotient. The corresponding
morphism between endomorphism rings is (notations of Example 2.3)
an 7→ aw
(78)
for n ∈ N/T ′ and w = nT ∈ W . The algebra on the right, EndOG (IndG
UT O) is the
well-known Iwahori-Hecke algebra whose generators satisfy (as )2 = (qs − 1)as + qs
for s ∈ S, qs := |U/U ∩ U s | (a power of q) and aw aw′ = aww′ when the lS lengths add (see for instance [CurtisRei, 67.3] or deduce it from the proof of
Theorem 2.5). By the assumption on ℓ, by reduction mod. J(O) the above
relations become the defining relations of W . So composing (78) with reduction
mod J(O). EndOG (IndG
UT O) gives a ring morphism
ρ : A → kW such that ρ(an ) = nT ∈ W .
(79)
50
MARC CABANES
The kernel of ρ is clearly J(OTℓ ) where we identify ⊕t∈Tℓ Oat with OTℓ as said
before. So we get an exact sequence of O-modules
ρ
0 → J(OTℓ )A → A −
→ kW → 0.
(80)
Note that J(OTℓ )A ⊆ J(A) (in fact an equality) thanks to (77), so that an ∈ A×
for any n ∈ N/T ′ . Let Γ ≤ A× the group generated by the an ’s (n ∈ N/T ′ ). So
(80) yields an exact sequence of groups
ρ
1 → Γ ∩ (1 + J(OTℓ )A) → Γ −
→W →1
(81)
where the second term acts trivially on Tℓ . If the above had been done with
O/J(O)m (m ≥ 1) instead of O we would get some Γm an extension of the ℓ′ group W by a finite ℓ-group, so (81) would split. In the general case we consider
the J(A)-adic topology on A for which ρ is continuous. We have an exact sequence
of groups
ρ
1 → C1+J(OTℓ )A (Tℓ ) → C1+J(OTℓ )A (Tℓ ).Γ −
→ W → 1.
(82)
The sequence splits by a classical lemma about lifting of J(A)-closed subgroups
(see [CaEn, 23.18]), thus giving some W ′ ≤ C1+J(OTℓ )A (Tℓ ).Γ isomorphic to W
by ρ and acting the same on Tℓ . Then A = OTℓ W ′ by Nakayama’s lemma and
the equality OTℓ W ′ + J(A) = A implied by ρ(OTℓ W ′ ) = kW . This shows that
A∼
= O(Tℓ ⋊ W ) as claimed.
Remark 8.12. A typical example of Morita equivalence between algebras A, B
over O that are sums of blocks over finite groups is when
∼ Matn (A)
B=
for some integer n ≥ 1. This is equivalent to our Morita equivalence inducing a
bijection of characters
Irr(A ⊗O K) → Irr(B ⊗O K) ; χ 7→ χ′
(∗)
′
where the ratio of degrees χ (1)/χ(1) is a constant integer n (see [CaEn, Ex. 9.6]).
Examples are Theorem 8.7 and the equivalences of Bonnafé-Dat-Rouquier, see
below Theorem 9.1.
In the case of Theorem 8.10 above it is generally not the case. For instance
when GF is SL2 (q) for q a power of 2 and ℓ ≥ 7 is a prime divisor of q − 1,
then NG (T)F is a dihedral group of order 2(q − 1) whose principal ℓ-block has
((q−1)ℓ −1)/2 ≥ 2 characters of degree 2. On the other hand the whole Irr(GF ) has
only one character of even degree, the Steinberg character (see for instance [DiMi,
§15.9]). This makes impossible any bijection as in (∗) with the ratio χ′ (1)/χ(1)
being always an integer, even depending on χ.
9. Bonnafé-Dat-Rouquier’s theorems
The main theorem of [BoDaRo17] is about the situation of Theorem 4.13 above
where (G, F ) is defined over Fq with dual (G∗ , F ∗ ), ℓ is a prime not dividing q,
∗
∗
∗
∗
s ∈ (G∗ )F
ℓ′ is a semi-simple element and L is an F -stable Levi subgroup of G
such that
(83)
CG∗ (s) ⊆ L∗
a condition sometimes weakened to
C◦G∗ (s) ⊆ L∗ .
(84)
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
51
Bonnafé-Dat-Rouquier’s main theorem [BoDaRo17] in that situation is the following.
Theorem 9.1 ([BoDaRo17, Th. 1.1]). Let L∗ minimal for the condition (84), let
L an F -stable Levi subgroup of G associated with L∗ by duality, so that E(LF , s)
and eℓ (LF , s) (see Definition 4.9) make sense. Let N be the stabilizer of eℓ (LF , s)
in NG (L)F . Then one has a Morita equivalence
ON eℓ (LF , s) −→ OGF eℓ (GF , s).
Moreover if two ℓ-blocks are related by the above they have isomorphic defect groups
and fusion systems in the sense of Definition 5.3.
9.A. Etale topology and sheaves. We refer to [CaEn, Ch. A2] for the basic
notions about algebraic varieties.
Sheaves on topological spaces. (See [KaSch].) If a topological space is given
by the datum of the set OpenX of open subsets of a certain set X, OpenX can
be considered as a category with Hom(O, O′ ) = {→} (a single element) when
O′ ⊆ O, Hom(O, O′ ) = ∅ otherwise. A presheaf on this topological space is then
any functor
F : OpenX → Set or F : OpenX → A-Mod
(85)
to the category of sets or the category of A-modules for A a ring. An example
is the constant presheaf. When O′ ⊆ O in OpenX and s ∈ F (O) one denotes
s|O′ := F (→)(s) ∈ F (O′ ). One also generally denotes
Γ(X, F ) = F (X)
(global sections). A sheaf is a presheaf F such that if (Oi )i is a family of elements
of OpenX and si ∈ F (Oi ) is a family such that si|Oi ∩Oj = sj|Oi ∩Oj for any i, j, then
there is a unique s ∈ F (∪i Oi ) such that s|Oi = si for any i. There is a canonical
way, called “sheafification”,
F 7→ F +
(86)
to associate a sheaf with a presheaf, adjoint to the forgetful functor F 7→ F . The
constant sheaf is the sheafification of the constant presheaf. For M in A-Mod, the
associated constant sheaf MX on X satisfies MX (U ) = M π0 (U) where π0 (U ) is the
set of connected components of U . When f : X → X ′ is continuous and F , F ′ are
sheaves on X, X ′ respectively the formulas
f∗ F (U ′ ) = F (f −1 (U ′ )) , f ∗ F ′ (U ) =
lim
U ′ ⊇f (U)
F ′ (U ′ )
define direct and inverse images of sheaves that are obvious presheaves. They
are made into sheaves by (86) keeping the same notation. For the map σX : X →
{•}, one gets
(σX )∗ F = Γ(X, F ).
(87)
′
When j : X → X is an open immersion (i.e. a homeomorphism between X and
j(X) ∈ OpenX ′ )then one defines a presheaf by
(
F (U ′ ) if U ′ ⊆ j(X)
′
j! F (U ) =
(88)
0
otherwise.
This is also made into a sheaf by (86).
Most sheaves of interest are deduced from locally constant sheaves by those
operations. Assume X is pathwise connected and locally simply connected. Let
π1 (X, x0 ) its fundamental group (homotopy classes of loops based at a given x0 ).
The topological relevance of sheaves is partly contained in the elementary fact
52
MARC CABANES
that locally constant sheaves with values in sets and some additional finiteness
condition (finite stalks) are in bijection with continuous finite π1 (X, x0 )-sets.
The category ShA (X) of sheaves F : OpenX → A-Mod has enough injectives.
When f : X → X ′ is continuous, we can right-derive the left exact functor f∗ : ShA (X) →
ShA (X ′ ) into
Rf∗ : D+ (ShA (X)) −→ D+ (ShA (X ′ )).
In the case of (87) one writes
RΓ(X, F ) ∈ D+ (A-Mod)
(89)
since ShA ({•}) = A-Mod. The i-th cohomology A-module of F is by definition
Hi (X, F ) := Hi (RΓ(X, F )).
Étale cohomology. (See [Milne], [CaEn, A3], [Du17, §2].) Let X be a variety
over F. The sheaves for the étale topology on X and their cohomology are roughly
defined as follows from the topological model sketched above. The topology on X
is not the Zariski topology but a Grothendieck topology where OpenX is replaced
by the category Xet whose objects are étale maps of varieties over F with codomain
X
U→X
and morphisms are given by commutative triangles. Presheaves are defined with
values in A-Mod for A a ring that is generally finite of characteristic prime to p.
A lot of adaptations are needed to define substitutes to intersections (pullbacks),
coverings, sheaves, etc... One defines a certain category of sheaves ShA (Xet )
(finiteness and constructibility assumptions) to which the homological constructions of above can apply. The map σX : X → {•} used above is replaced by the
structural map σX : X → Spec(F). This leads to RΓ(X, F ) ∈ D+ (A-mod) and
the corresponding cohomology modules. One has also a notion of cohomology
with compact support. Assume one has a compactification j : X ֒→ X (an open
embedding with X complete), then
RΓc (X, F ) := R(σX )∗ j! F ∈ D+ (A-mod)
(90)
with corresponding homology groups Hic (X, F ) := Hi (RΓc (X, F )).
The notion of ℓ-adic cohomology (here with compact support) is defined as
follows. Denote O(n) := O/J(O)n (recall O is a finite extension of Zℓ ). An ℓ-adic
sheaf is a projective system F = (F (n) )n≥1 of sheaves where F (n) ∈ ShO(n) (X)
and F (n) = F (n+1) ⊗ (O(n) ) . Then
Hic (X, F ) := lim Hic (X, F (n) ) ∈ O-mod.
←
−
n
For instance the module Hic (YP ) defining the functor RG
L of Deligne-Lusztig
(n)
i
i
theory in Definition 4.2 is Hc (YP ) := C ⊗O limn Hc (YP , OYP ).
←−
Compactifications give rise to the notion of ramification. The context is
roughly as follows. Assume one has a compactification j : X → X with smooth X
and complement X \ X = D1 ∪ D2 ∪ . . . a smooth divisor with normal crossings.
For each irreducible component Dm let
Xm = X \ ∪i6=m Di
and
jm
i
m
− Dm \ ∪i|i6=m (Dm ∩ Di )
X −−→ Xm ←−
the associated open and closed immersions.
Definition 9.2. One says that F ramifies along Dm when F is not of the form
∗
jm
Fm for Fm a sheaf on Xm .
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
53
Then by results from [SGA4.5], [SGA5] (see also the survey in [CaEn, A3.19])
one gets that the above condition is equivalent to
i∗m R(jm )∗ F = 0
(otherwise F =
∗
jm
Fm
for Fm
(91)
:= (jm )∗ F ).
9.B. Broué’s reduction. In the context of the functor RG
L , one starts in general
with an F -stable Levi subgroup L complement in a parabolic subgroup P. From
(22) recall the varieties
G/P ⊇ XP := {gP | g −1 F (g) ∈ PF (P)}
and
YP := {gRu (P) | g −1 F (g) ∈ Ru (P)F (Ru (P))}
and the actions of GF and LF on them. One has π : YP → XP the LF -quotient
map sending gRu (P) to gP.
Definition 9.3. Let L and s be like in Theorem 4.14. Recall O(n) := O/J(O)n .
(n)
Then let Fs = (Fs )n≥1 where
(n)
Fs(n) := π∗ (OYP )eℓ (LF , s)
recalling that π∗ sends sheaves of O(n) -modules to sheaves of O(n) LF -modules.
For simplicity we assume that XP is affine. This is conjectured in general and
known in many cases (see [Du17, ]). However what follows can be proven knowing
just that it is quasi-affine, which is the case (see [CaEn, 7.15]).
j
i
− XP \ XP
→ XP ←
Theorem 9.4 ([Br90b]). If there exists a compactification XP −
such that
i∗ Rj ∗ Fs(n) = 0
for all n ≥ 1, then
YP
(YP , O(n) )eℓ (LF , s)
lim Hdim
←− c
n
induces a Morita equivalence
Bℓ (LF , s)-mod → Bℓ (GF , s)-mod.
(n)
Proof. A first consequence of (91) for Fs is that
j! F (n) ∼
= Rj∗ F (n) .
s
This is seen by applying to
(n)
Fs
(92)
s
the open-closed exact sequence
0 → j! → j∗ → i∗ i∗ j∗ → 0
suitably right-derived (see [Du17, 2.6]) into a distinguished triangle (note that j! ,
i∗ and i∗ are right exact).
We omit the subscripts P from now on. We denote σ : X → Spec(F) and
σ = σ ◦ j : X → Spec(F) the structure morphisms of X and X. Then (92) and the
definition of RΓ and RΓc allow to write
RΓ(X, Fs(n) ) = Rσ∗ Fs(n) = Rσ ∗ ◦ Rj∗ Fs(n) = Rσ ∗ ◦ j! Fs(n) = RΓc (X, Fs(n) ). (93)
(n)
Since X is affine of dimension the same d everywhere, RΓ(X, Fs ) has cohomology
in degrees only within the interval [0, d] (see [Du17, §2.1]). But by Poincaré(n)
Verdier duality (see [Du17, 2.4]) since X is smooth, RΓc (X, Fs ) has cohomology
(n)
in degrees ∈ [d, 2d]. So (93) implies that RΓc (X, Fs ) = RΓc (Y, O(n) ).eℓ (GF , s)
has cohomology in degree d only. Let’s call H (n) this cohomology O(n) -module.
54
MARC CABANES
One can prove that it is O(n) -free. Moreover the groups GF and LF act on Y with
stabilizers that are finite unipotent groups of order invertible in O(n) (trivial in the
case of LF ). So applying for instance [Du17, 2.4] one gets that both restrictions
of H (n) to O(n) GF and O(n) LF are projective. So the same is true for H ∞ the
limit over n. By definition C ⊗O limn RΓc (Y, O(n) ) is the bimodule inducing the
←−
opp
∞
functor RG
is actually a bi-projective OGF ⊗OLF -module such that
L⊆P , so H
∞
K ⊗O H induces the bijection between ordinary characters
Eℓ (LF , s) = Irr(KLF eℓ (LF , s)) → Eℓ (GF , s) = Irr(KGF eℓ (GF , s))
thanks to Theorem 4.13. Now we have everything to apply Lemma 4.15 and get
our claim.
Remark 9.5. When L is a torus, Deligne-Lusztig have shown the existence of
an X such that (91) is satisfied [DeLu76, 9.14]. So the Morita equivalence holds
in that case [Br90b, 3.6]. Note however that in that case methods similar to
Sect. 7 above allow to show that Bℓ (GF , s) (see Definition 4.9) is a single block
which is nilpotent of defect LF
ℓ . The Morita equivalence is then a consequence of
Theorem 8.7 which gives the structure of nilpotent blocks in general.
9.C. Bonnafé-Rouquier (2003). In view of Theorem 9.4, the main objective of
[BoRo03] is to prove
j
→
Theorem 9.6 ([BoRo03, 11.7]). There exists a smooth compactification XP −
i
∗
∗
− XP \ XP such that i Rj Fs = 0.
XP ←
As a consequence the authors get
Theorem 9.7 ([BoRo03, Theorem B’]). Assume CG∗ (s)F
Morita equivalence
∗
⊆ L∗ . One has a
OLF eℓ (LF , s)-mod → OGF eℓ (GF , s)-mod.
The construction of the smooth compactification for varieties XP with P a Borel
subgroup extends the one of Bott-Samelson-Demazure-Hansen for Schubert varieties (which are obtained by removing the condition involving F in what follows).
Let B0 , T0 a pair of F -stable Borel and torus as before. Let
[
Σ :=
(S ∪ {1})m
m≥0
the set of finite sequences of elements of S ∪ {1}. One recalls that a lifting S →
NG (T0 ), denoted s 7→ ṡ has been chosen satisfying the braid relations of the Weyl
group (see (14) above). For w = (s1 , . . . , sr ) ∈ Σ, let
X(w) := {(X1 , . . . , Xr ) ∈ (G/B0 )r |
−1
X1−1 X2 ∈ B0 s1 B0 , . . . , Xr−1
Xr ∈ B0 sr−1 B0 , Xr−1 F (X1 ) ∈ B0 sr B0 }
r
:=
Y(w)
{(Y1 , . . . , Yr ) ∈ (G/U0 ) |
−1
Y1−1 Y2 ∈ U0 ṡ1 U0 , . . . , Yr−1
Yr ∈ U0 ṡr−1 U0 , Yr−1 F (Y1 ) ∈ U0 ṡr U0 }.
F
:=
Both are acted on by G on the left, the first is also acted on by TwF
0
s1 ...sr F
T0
on the right. The reduction mod B0 gives a finite quotient
πw : Y(w) → X(w) ∼
= Y(w)/TwF .
0
Let
X(w) =
[
X(w′ )
(94)
w ′ ≤w
where w′ ≤ w means that w′ ∈ {1, s1 } × · · · × {1, sr }. This is smooth just like
B0 ∪ B0 si B0 is smooth, being an algebraic group.
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
55
Bonnafé-Rouquier define the set ∇ of pairs (w, θ) where w = (s1 , . . . , sr ) ∈ Σ
and
θ : TwF
= T0s1 ...sr F → k ×
0
is a group morphism ([BoRo03, §4.4]). For such a pair they define wθ = (s′1 , . . . , s′r )
by
(
1
if si 6= 1 and θ ◦ Ns1 ...sr (s1 . . . si−1 (δi∨ )) = 1
′
si =
(95)
si otherwise,
where δi∨ ∈ Φ(G, T0 )∨ is the fundamental coroot corresponding to si and Nv : Y (T0 ) →
TvF
for v ∈ W (G, T0 ) is the norm map used in the classical identification
0
Y (T0 )/(1 − vF )Y (T0 ) ∼
= TvF
0 (see for instance [DigneMic, 13.7]).
They also define F(w,θ) and S(w,θ) as follows.
Definition 9.8. Let bθ ∈ kTwF
the primitive idempotent such that θ(bθ ) 6= 0.
0
Let
F(w,θ) = (πw )∗ kY(w) .bθ
a sheaf on X(w) with values in k-vector spaces. Since RΓ(Y(w), kY(w) ) is repreopp
sented by a complex of kGF ⊗ kTwF
-modules, we can define
0
S(w,θ) := RΓ(Y(w), kY(w) )bθ ∈ Db (kGF -mod).
One proves
w
′
Theorem 9.9 ([BoRo03, 7.7]). For w′ ≤ w, let jw
′ : X(w ) → X(w) the inclusion
w
w ∗
from (94). Then R(jw )∗ F(w,θ) is annihilated by (jw′ ) unless wθ ≤ w′ .
Theorem 9.10 ([BoRo03, Th. A]). The subcategory of Db (kGF -mod) generated
(through shifts, direct sums, direct summands and mapping cones) by the S(w,θ)
for (w, θ) ∈ ∇ contains the regular module kGF [0].
Those two theorems, of a quite different nature, both concern only varieties
associated to Borel subgroups, not parabolic subgroups. The proof of Theorem 9.9 needs a particularly deep study of the sheaves and tori actions involved,
see [BoRo03, §4]. See also [BoRo09] on a related question. For the proof of Theorem 9.10, see [Du17, §3.5]. Note that [BoDaRo17, 1.2] gives a strengthened version
of that theorem (see also [Du17, 3.12]).
Let’s sketch briefly how Theorem 9.6 is deduced from those two theorems (proof
of [BoRo03, 10.7]). The pair (L, F ) can be changed into (LI , v̇F ) for some I ⊆ S
and v̇ ∈ NG (T0 ) with vF (I)v −1 = I through conjugation by an element of G.
Then the varieties X and Y of Theorem 9.6 become XI,v = {gPI | g −1 F (g) ∈
PI v̇F (PI )} and YI,v = {gUI | g −1 F (g) ∈ UI v̇F (UI )} with evident Lv̇F
I -quotient
map π : YI,v → XI,v . Abbreviating L = Lv̇F
,
one
has
to
prove
I
i∗ Rj∗ (π∗ k ⊗kL kLeℓ (L, s)) = 0
(96)
where we have kept the notation i, j for the immersions associated with XI,v ⊆
XI,v the later being the Zariski closure in the complete variety G/PI . By the
generation property of Theorem 9.10 (applied to LI ) it suffices to check
LI ,v̇F
i∗ Rj∗ (π∗ k ⊗kL S(w,θ)
)=0
(97)
for any (w, θ) ∈ ∇LI ,v̇F relating to s by duality.
Let dv ∈ S lS (v) be a reduced expression of v and w ∪ dv be the concatenation in
Σ. Let τ : X(w ∪ dv ) → XI,v defined by (g1 B0 , . . . ) 7→ g1 PI . By basic properties
56
MARC CABANES
of (derived) direct image functors and an isomorphism of varieties related to the
transitivity of Deligne-Lusztig induction, one gets
LI ,v̇F
π∗ k ⊗kL S(w,θ)
= Rτ∗ F(w∪dv ,θ) .
(98)
w∪dv
where τ : X(w∪dv ) → XI,v is (g1 B0 , . . . ) 7→
On the other hand we have jτ = τ jw∪d
v
g1 PI , a proper morphism. So now (97) reduces to
w∪dv
)∗ F(w∪dv ,θ) = 0.
i∗ Rτ ∗ R(jw∪d
v
(99)
One now applies base change (see for instance [CaEn, A3.5]) and gets
i∗ Rτ ∗ = Rτ∗′ ◦ i∗v
where
iv :
[
(100)
X(w′ ∪ v ′ ) = X(w ∪ dv ) \ τ −1 (XI,v ) −→ X(w ∪ dv )
(101)
w ′ ≤w,v ′ dv
is the open immersion and τ ′ is the restriction of τ . In view of (99) and (100) it
then suffices to prove that
w∪dv
i∗v R(jw∪d
)∗ F(w∪dv ,θ) = 0.
v
(102)
The situation is now close to the one covered by Theorem 9.9 for each inclusion
X(w′ ∪ v ′ ) → X(w ∪ dv ). One checks that (w ∪ dv )θ = wθ ∪ dv 6≤ w′ ∪ v ′ for
w∪dv
)∗ F(w∪dv ,θ) = 0
pairs (w, θ) relating to s. Theorem 9.9 then tells us i∗w′ ∪v′ R(jw∪d
v
′
′
for each iw′ ∪v′ : X(w ∪ v ) → X(w ∪ dv ) involved in (101). This implies (102) by
checking stalks.
9.D. Bonnafé-Dat-Rouquier (2017). Among many results (see also [Du17,
3.12]) the paper [BoDaRo17] shows that the situation of Theorem 9.7 implies more
than a Morita equivalence. The hypothesis is also slightly strengthened assuming
just (84).
∗
One takes G, G∗ in duality, s a semi-simple ℓ′ -element of G∗ F . One lets
∗
L∗s := CG∗ (Z◦ (C◦G∗ (s))) ⊲ C◦G∗ (s) and N∗s = CG∗ (s)F L∗s
so L∗s is the smallest Levi subgroup of G∗ containing C◦G∗ (s). Let Ls be an F stable Levi subgroup of G in duality with L∗s . Note that E(LF
s , s) makes sense.
Let Ns ≤ NG (Ls ) such that Ns /Ls identifies with N∗s /L∗s through duality. It
is then F -stable and
F
NF
s = NGF (Ls , E(Ls , s))
F
so that eℓ (LF
s , s) is a central idempotent of ONs .
The following establishes a Morita equivalence for the blocks in characteristic
ℓ.
Theorem 9.11 ([BoDaRo17, 7.5]). Let P = Ru (P)Ls be a parabolic subgroup
having Ls as Levi subgroup. We have:
opp
opp
.
(i) The action of GF × (LF
on Hcdim YP (YP , k) extends to GF × (NF
s)
s )
(ii) The resulting bimodule induces a Morita equivalence
F
F
F
kNF
s eℓ (Ls , s)-mod −→ kG eℓ (G , s)-mod.
A. Independence of the parabolic P. A first step in proving Theorem 9.11.(i)
opp
is to show that the kGF ⊗ kLF
-module Hcdim YP (YP , k) is invariant under the
s
F
F
F
F
action of G × Ns (through automorphisms of GF × LF
s induced by G × Ns ).
F
F
F opp
Only the action of some x ∈ Ns needs to be checked. The action of G × Ls
twisted by (1, x) on YP = {gRu (P) | g −1 F (g) ∈ Ru (P)F (Ru (P))} is clearly the
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
57
opp
action of GF × LF
on YP · x = YPx . The equivariance of étale cohomology
s
(here the automorphism is even a homeomorphism for Zariski topology) implies
YP
that Hdim
(YP , k)(1,x) ∼
= Hcdim YP (YPx , k) and therefore
c
(1,x)
∼
Hcdim YP (YP , k)eℓ (LF
= Hcdim YP (YPx , k)eℓ (LF
s , s)
s , s).
The parabolic subgroups P and Px have both Ls as a Levi complement, so the
invariance sought is a consequence of the following.
Theorem 9.12 ([BoDaRo17, 6.5, 6.7]). If P and P′ are parabolic subgroups of G
admitting Ls as Levi complement then
Hdim YP (YP , k)eℓ (LF , s) ∼
= Hdim YP′ (YP′ , k)eℓ (LF , s)
c
s
c
s
opp
as kGF ⊗ kLF
-modules.
s
Remark 9.13. Note that this answers the question of the dependence of the
Morita equivalence from Theorem 9.7 on the parabolic subgroup used. Note that
the corresponding statement on characters was known for long [DigneMic, 13.28].
In the case of F -stable P, P′ , one has
O(GF /Ru (P)F ) ∼
= O(GF /Ru (P′ )F )
as bimodules (Dipper-Du, Howlett-Lehrer [CaEn, 3.10]). In general one does not
have Theorem 9.12 without projecting on eℓ (LF
s , s). Theorem 9.12 has also been
used by Dat in a study of representations of p-adic groups [Dat16].
The proof is quite delicate ([BoDaRo17, Sect. 5-6], see additional explanations
and perspective in [Dat15]). It involves “intermediate” varieties of type
YP,P′ := {(gRu (P), g ′ Ru (P′ )) | g −1 g ′ ∈ Ru (P)Ru (P′ ) , g ′−1 F (g) ∈ Ru (P′ )F (Ru (P))}
and maps from their cohomology complexes (up to a shift) to our RΓc (YP , k)
or RΓc (YP′ , k). Then getting quasi-isomorphism once those are projected on the
sum of blocks Bℓ (LF
s , s) needs quite a lot of additional considerations in the spirit
of the proof of Theorem 9.9.
B. Extendibility. The next problem is to extend Hcdim YP (YP , k) into a k(GF ×
opp
NF
)-module.
s
YP
One actually shows that the obstruction to extending Hdim
(YP , k) is the
c
same in G as it would be in an overgroup with connected center, where this
obstruction does not exist. One considers a regular embedding (see [CaEn, §15.1]),
that is an inclusion of algebraic groups
e = GZ(G)
e
G ֒→ G
e
e This induces a
with connected Z(G).
One may assume that F extends to G.
∗
∗
e
surjection σ : G → G with connected central kernel. Denote
e s.
e := Z(G)P
e
e s = Z(G)L
e s, N
e s := Z(G)N
P
, L
∗
e ∗ F ∗ be a representative system for G
e ∗F ∗ Definition 9.14. Let J ⊆ σ −1 (s)F
ℓ′ ⊆ Ls
∗
conjugacy in σ −1 (s)F
ℓ′ . Let
X
eF , e
eF
e :=
eℓ (L
s t) ∈ Z(k Ls ).
e
t∈J
e F × (G
e F )opp
One also defines the following subgroups of G
e F × (L
e F )opp ⊳ N
e := G
e F × (N
e F )opp ,
Le := G
s
s
58
MARC CABANES
opp
e F ⊳ N := (GF × (NF )opp )∆N
eF
L := (GF × (LF
)∆L
s )
s
s
s
−1
F
e
(where ∆H = {(h, h ) | h ∈ H} for a given subgroup H ≤ G ).
∗
F ∼
∗
∗ F∗
e /Le ∼
Note that N
≤ (CG∗ (s)/C◦G∗ (s))F , an ℓ′ -group by
= NF
s /Ls = (Ns /Ls )
Lemma 7.1.
From the definition of YP , it is clear that it is acted on by L, so we may consider
e
L
f
M := Hdc (YP , k)eℓ (LF
s , s) , M = IndL M.
e
The variety YP
e ⊆ G/Ru (P) is defined with regard to the same unipotent subgroup as YP so it has same dimension as YP and one has (see for instance [CaEn,
12.15.(iii)])
F
f∼
(103)
M
= Hdc (YP
e , k)eℓ (Ls , s).
Some mild considerations in the dual groups show that
X
X
eF,e
eℓ (GF , s) =
eℓ (G
t) and eℓ (LF
ex .
(104)
s , s) =
F
x∈NF
s /Ls
e
t∈J
−1
e one has C e ∗ (e
e ∗ , so Bonnafé-Rouquier’s theorem
In G,
(C◦G∗ (s)) ⊆ L
s
G t) = σ
(Theorem 9.7 above) implies
fe ⊗ e F − induces a Morita equivalence k L
e F e-mod −→ k G
e F eℓ (GF , s)-mod.
M
s
Ls
(105)
f
e
So M e is a direct sum of pairwise non-isomorphic indecomposable k L-modules.
e
e
N
f∼
f
The same applies to M
= ResN
e M e.
e IndL
L
The next step is to deduce from the above that M extends to N . When the
quotient N /L is cyclic this is enough to extend the action of L on M into an
action of N (see for instance [Da84, 4.5]). For the general case, see Remark 9.24
below. One writes
′
′
M = ResN
L M for some kN -module M .
(106)
e
′
It is not too difficult to deduce from (105) that IndN
N M induces a Morita equivalence
e F eℓ (GF , s)-mod.
e F eℓ (LF , s)-mod −→ k G
(107)
kL
s
s
′
Then one shows that M induces the sought Morita equivalence
F
F
F
kNF
s eℓ (Ls , s) −→ kG eℓ (G , s).
F
′
For instance the canonical map kNF
s eℓ (Ls , s) → EndkGF (M ) is indeed an
e
N
F
F
′
e eℓ (L , s) → End e F (Ind M ) is one by (107) and one
isomorphism since k N
s
s
N
kG
e
N
′ ∼
F
e
has EndkG
(Ind
M
)
End
F
(M
)
⊗
F
N
.
=
eF
kG
Ns
N
s
This finishes the proof of Theorem 9.11.
C. Rickard equivalence and local structure. Bonnafé-Dat-Rouquier prove
then that Theorem 9.11 can be strengthened to a Rickard equivalence preserving
the local structure of the blocks of GF and NF
s that are related through this
equivalence.
Recall [BoDaRo17, 2.A]
Definition 9.15. A Rickard equivalence between sums of block algebras A, A′
over Λ ∈ {O, k} is an equivalence
Hob (A) → Hob (A′ )
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
59
induced by a complex C of bi-projective A′ ⊗Λ Aopp -modules such that the canonical maps A → End•A′ (C) and A′ → End•Aopp (C) are isomorphisms in Hob (A ⊗Λ
Aopp -mod) and Hob (A′ ⊗Λ A′opp -mod) respectively (notations of [Du17, §1.2]).
This coincides with the usual definition requiring additional properties of the
Green vertices of summands of C by results of Rouquier [Rou01]. On the other
hand those ℓ-subgroups of the product of the two finite groups involved serve as a
bridge between the local structures of blocks so related. While Rickard’s original
paper [Rick96] had the assumption that blocks involved have same defect group,
one can prove that a Rickard equivalence in the above sense implies a strong
relation at the level of local subgroups. The following is due to Puig.
Theorem 9.16 ([Puig99, 19.7]). If two ℓ-block algebras A, A′ over O are Rickard
equivalent then the defect groups are isomorphic D ∼
= D′ and the associated fusion
systems (see Definition 5.3) on D and D′ are equivalent.
Recall a theorem of Rickard (see [Du17, 2.2]).
Theorem 9.17 ([Rick95], [Rou02]). The element RΓc (YP , O) of Db (OGF ⊗
opp
OLF
) is represented by a well-defined element GΓc (YP , O) of Hob (OGF ⊗
s
opp
GF ×LF
s
opp
OLF
) whose terms are direct summands of modules of type IndQ
s
opp
where Q is an ℓ-subgroup of GF × LF
such that (YP )Q 6= ∅.
s
O
The main result of [BoDaRo17] can then be stated as follows.
Theorem 9.18 ([BoDaRo17, 7.7]). In the framework of Theorem 9.11 for G,
s, P ≥ Ls ⊳ Ns the complex GΓc (YP , O)eℓ (LF
s , s) induces a Rickard equivalence
F
,
s).
e
(L
between OGF eℓ (GF , s) and ONF
s
s ℓ
We sum up some of the main features of the proof ([BoDaRo17, §7.D]).
One works first over k. Denote
C = GΓc (YP , O)eℓ (LF
s , s) ⊗ k.
The main step is to prove the following.
b
F opp
Proposition 9.19. End•kGF (C) ∼
)
= EndDb (kGF ) (C)[0] in Ho (kLF
s × Ls
The proof of that Proposition leads to checking the following about the action
opp
of GF × LF
on YP .
s
Lemma 9.20 ([BoDaRo17, 3.5]). Assume P = Ru (P)L is a Levi decomposition
with F (L) = L. If Q is an ℓ-subgroup of GF × LF opp with fixed points on YP ,
then Q is GF × LF opp -conjugate to a subgroup of ∆(LF ) := {(x, x−1 ) | x ∈ LF } ⊆
GF × LF opp .
Let us now recall that for H a finite group, an ℓ-permutation kH-module is by
definition any direct summand of a permutation module. For Q an ℓ-subgroup of
H and M an ℓ-permutation kH-module one denotes
BrQ (M ) := M Q /(M Q ∩ J(kQ)M ) in kCH (Q)-mod
(108)
the image of the Q-fixed points of M in the cofixed points (see also [Du17, §2.3]).
This induces an additive functor from ℓ-permutation kH-modules to ℓ-permutation
kCH (Q)-modules. Note that if Ω is a set acted upon by H, then BrQ (kΩ) = k(ΩQ )
which allows to identify our first definition (10) of the Brauer morphism with a
special case of the above.
The following, chiefly due to Bouc, is very useful to check homotopic equivalence
locally.
60
MARC CABANES
Lemma 9.21 ([Bouc98, 6.4, 6.9]). Let E be a bounded complex of ℓ-permutation
kH-modules. Assume that for any ℓ-subgroup Q ≤ H, BrQ (E) has homology in
degree 0 only. Then
b
E∼
= H0 (E)[0] in Ho (kH-mod).
F opp
This will be applied to H = LF
and E := End•kGF (C).
s × Ls
Lemma 9.20 somehow shows that the relevant ℓ-subgroups to check are of the
form ∆Q for Q an ℓ-subgroup of LF
s . By a theorem of Rickard (see [Du17, 2.11])
Br∆Q (RΓc (YP , k)) identifies with RΓc ((YP )∆Q , k). In [BoDaRo17, §3.A] it is
shown that (YP )∆Q is to be considered as a variety YCP (Q) in the (possibly nonconnected) reductive group CG (Q), which in turn gives sense to and establishes
(C (Q))
Br∆Q (C) = GΓc (YCPG(Q) , k) BrQ (eℓ (LF
s , s)).
(109)
Then the authors show for BrQ (eℓ (LF
s , s)) a formula [BoDaRo17, 4.14] generalizing the one of Broué-Michel seen before (Theorem 7.3) for cyclic subgroups Q.
This allows to identify the right hand side of (109) with a sum of complexes of
the same type as C itself in the local subgroup CG (Q)F . One applies to them
Bonnafé-Rouquier’s theorem (Theorem 9.7) thus getting that their homology is in
one single degree. This essentially gives Proposition 9.19 thanks to Lemma 9.21.
Let us comment that the above adaptations to the case of non-connected reductive
groups needs indeed a lot of work [BoDaRo17, Sect. 3-4].
The next steps go through the following propositions and are less difficult.
Remember that one is looking for a complex acted on by NF
s on the right.
Proposition 9.22. One has
GF ×NF opp
EndHob (k(GF ×NFs opp )) (IndGF ×LFs opp (C))
s
∼
=
F
F opp
G ×N
∼
= EndDb (k(GF ×NFs opp )) (IndGF ×LFs opp (C))
s
GF ×NF opp
Endk(GF ×NFs opp ) (IndGF ×LFs opp (Hdc (YP , k))
s
The proof of the following uses Theorem 9.11.(i).
F
F opp
e of IndGF ×NFs opp (C) satisfying
Proposition 9.23. There is a direct summand C
G ×L
s
F
F opp
G ×N
e ∼
(i) ResGF ×LFs opp (C)
= C and
s
•
b
F
F
F opp
∼
∼
e
e
(ii) End F (C) = EndDb (kGF ) (C)[0]
)).
= kNF
s eℓ (Ls , s)[0] in Ho (k(Ns ×Ns
kG
Using relatively standard techniques allowing with extra information to check
only one of the two isomorphisms of Definition 9.15, one now gets a Rickard
equivalence over k. Lifting all that to O as claimed in Theorem 9.18 follows
classical procedures (see [Rick96, 5.2]).
Remark 9.24. Recalling that we have assumed for (106) above that Ns /Ls is
cyclic, one gets at this point Theorem 9.18 in that case. We even get a similar
statement for any F -stable Levi subgroup L∗ (replacing L∗s ) such that L∗ contains
∗
∗
C◦G∗ (s), is normalized by CG∗ (s) and the factor group (CG∗ (s)L∗ )F /L∗F is
F
F
cyclic. Then the two algebras shown to be Rickard equivalent are OG eℓ (G , s)
∗
∗
and ON eℓ (LF , s) where N ≤ NG (L)F is such that N/LF ∼
= (CG∗ (s)L∗ )F /L∗F .
When Ns /Ls is not cyclic, the proof proposed by Bonnafé-Dat-Rouquier in
[BoDaRo17b] consists in several steps described to the present author as follows
(October 2017). First, one reduces the problem to groups G that are simple as
algebraic group (finite center and irrreducible root system) through direct products
and central extension. Once this is done, the cases to care about are when Ns /Ls
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
61
or equivalently CG∗ (s)/C◦G∗ (s) is not cyclic, which by Lemma 7.1 can occur only
in type D2n (n ≥ 2). There are three possibilities for s up to conjugacy, but only
∗
∗
one such that (CG∗ (s)L∗s )F /L∗s F is not cyclic. Then C◦G∗ (s) has type A2n−3 and
one can choose an F -stable Levi subgroup L∗ of type A2n−3 × A1 × A1 satisfying
the above. One then gets the equivalence sought between OGF eℓ (GF , s) and
ON eℓ (LF , s) where N/LF corresponds to a subgroup of order 2 of CG∗ (s)/C◦G∗ (s).
F
Going from ON eℓ (LF , s) to our goal ONF
s eℓ (L , s) can then be done by proving
versions of Theorem 9.12 and (107) in a non-connected group H such that H◦ = L
and H/L covers the missing part of CG∗ (s)/C◦G∗ (s). We refer to [BoDaRo17b] for
more details.
10. Recreation: Blocks of defect zero
Modular group algebras kH where the characteristic of k divides the order of
the finite group H are the typical examples of non semi-simple algebras but they
of course may have blocks that are indeed simple. This may be seen as rather
exceptional and the local structure or representation theory of such blocks is quite
trivial. But on the other hand a statement like Alperin’s weight conjecture (see
3.C above) crucially needs that enough of those situations exist. There are very
few general theorems ensuring that a finite group algebra has such blocks and this
is probably related with how difficult it is to say anything general about Alperin’s
conjecture. Using CFSG, one can see that non abelian simple groups have a lot of
blocks of defect zero.
Theorem 10.1. Let ℓ be a prime and S a finite non-abelian simple group. Then
it has an ℓ-block of defect zero (see 1.D) except in the following cases
(a) ℓ = 2 and S is an alternating group An for n ≥ 7 such that neither n nor
n − 2 is a triangular number, or one of the sporadic groups M12 , M22 ,
M24 , J2 , HS, Suz, Co1 , Co3 , BM .
(b) ℓ = 3 and S is an alternating group An for n ≥ 7 such that (3n + 1)p
is non-square for at least one prime p ≡ −1(3), or S is one of the two
sporadic groups Suz and Co3 .
The checking of this theorem on the character table of a given simple group
is easy since an ℓ-block of defect zero is signaled by an ordinary character of
degree divisible by the highest power of ℓ dividing the order of the group (see
[NagaoTsu, 3.6.29]). This applies to the 26 sporadic groups and the 18 primes
{2, 3, . . . , 43, 47, 59, 67, 71} that divide the order of one of them.
For groups of Lie type, note that the theorem asserts that all have blocks of
defect zero for all primes. This was checked by Michler [Mi86, 5.1] for odd ℓ and
Willems [Wi88] for ℓ = 2. When ℓ is the defining prime, the Steinberg module gives
such a block (see Theorem 3.3 above). Assume now that the defining prime is some
r 6= ℓ. Then the checking for a group S = GF /Z(GF ) basically consists in finding
∗
regular semi-simple elements s ∈ [G∗ , G∗ ]F whose centraliser is a maximal torus
∗
F
F
T such that T /Z(G ) is of order prime to ℓ. Then the corresponding DeligneF
F
Lusztig character ±RG
T θ is irreducible with degree |G /T |r ′ (see [DigneMic,
F
12.9]), has Z(G ) in its kernel and therefore is in an ℓ-block of defect zero of
GF /Z(GF ).
Strangely enough the answer for alternating groups was known after the case
of simple groups of Lie type. The problem reduces to the case of the symmetric
group Sn except for the prime 2. There a 2-block of Sn can restrict to a block
62
MARC CABANES
of defect zero of An if it has defect group 1 or S2 . Theorem 5.11 gives the defect
group of a 2-block in terms of 2-cores. It is easy to see that a Young diagram
has no 2-hook if and only if its rim has the shape of a regular stair. This means
that this is the partition m, m − 1, . . . , 1 of the triangular number m(m + 1)/2.
Similarly the Young diagram can have only one 2-hook if it is as above plus two
more boxes at the first row, or two more boxes at the first column. Hence n − 2
being a triangular number.
For primes ℓ ≥ 3, one gets from Theorem 5.11 that An has an ℓ-block with
defect zero if and only if there exists an ℓ-core κ ⊢ n.
Many things have been known for a long time about cores since their introduction by Nakayama [Naka41a]. If cd (n) denotes the number of d-cores κ ⊢ n, one
has
X
Y (1 − tdn )d
(110)
cd (n)tn =
(1 − tn )
n≥0
n≥1
as seen from the theory of d-quotients [JamesKer, § 2.7.30]. The numbers are
documented in https://oeis.org/A175595.
The general theorem about existence of d-cores was finally reached by GranvilleOno in 1996.
Theorem 10.2 ([GrOn96]). Let n ≥ 1, d ≥ 3. Then cd (n) = 0 if and only if
d = 3 and (3n + 1)ℓ is a non-square for some prime ℓ ≡ −1(3).
For d = 3, Granville-Ono show that
c3 (n) =
X
m|3n+1, m≥1
m
(111)
3
where ( m
3 ) is the Legendre symbol. Using Gauss’ quadratic reciprocity law [Serre,
p.7], this gives the statement about c3 (n). An elementary proof of (111) can be
found in [HiSe09].
For d ≥ 4, one claims that there are d-core partitions of n for any n. Note that
(41) implies that one can restrict the study to d = 4, 6 and odd d ≥ 5.
Using a variant of β-numbers one also shows elementarily that cd (n) 6= 0 if and
only if there is a d-tuple of integers (x1 , . . . , xd ) such that (see [GaKiSt90, Bij. 2])
n=
d
X
d
i=1
2
· x2i + (i − 1) · xi
and
d
X
xi = 0.
(112)
i=1
Granville-Ono [GrOn96] solve the above using modular forms but mainly elementary arguments for primes d ≥ 17. We conclude by giving below an elementary
argument taken from [Ki96]. The number theoretic flavor is quite apparent.
Proposition 10.3. Assume d ≥ 9 is an odd integer and n ≥ 14 d3 + 43 d − 1. Then
cd (n) 6= 0.
Note that n ≤ 14 (d − 1)2 leads to trivial solutions (use Young diagrams included
in a square to get λ ⊢ n such that hookc (λ) = ∅ for any c ≥ d).
Proof. (Kiming) One solves the problem in the form of (112). One will need 8
integers x1 , . . . , x8 with sum 0 to represent n, whence the condition d ≥ 9.
The condition n ≥ 14 d3 + 43 d − 1 implies that the euclidean division of n by d
gives n = dq + r with 4q ≥ d2 − 1 and d − 1 ≥ r ≥ 0. Let’s change slightly the
LOCAL METHODS FOR BLOCKS OF FINITE SIMPLE GROUPS
63
parity of these integers while keeping r2 small. Let
(q, r)
if q ≡ 1 (2) and r 6≡ 0 (4)
(q + 1, r − d)
if q ≡ r ≡ 0 (2)
(q ′ , r′ ) :=
(q
+
2,
r
−
2d)
if q ≡ 1 (2) and r ≡ 0 (4)
(q − ǫ, r + ǫd)
if q ≡ 0 (2) and r ≡ ǫd (4) for ǫ = ±1.
We still have n = dq ′ + r′ but now q ′ is odd and one of the following occurs
(a) r′ is odd and 4q ′ ≥ r′2 (two first cases above).
(b) r′ ≡ 2 (4) and 16q ′ ≥ r′2 .
Let’s look at case (a). Then 0 < 4q ′ − r′2 ≡ 3 (8) so 4q ′ − r′2 can be represented
by a sum of three odd squares (see [Serre, p. 45]). Therefore
4q ′ = r′2 + a2 + b2 + c2
′
(113)
′
with r , a, b, c odd. If necessary, we may change a into −a to ensure that r +a+b+c
is a multiple of 4. One then defines
α = (r′ + a + b + c)/4
γ = (r′ − a + b − c)/4
β = (r′ − a − b + c)/4
δ = (r′ + a − b − c)/4
and (x1 , . . . , x8 ) = (−α, α, −β, β, −γ, γ, −δ, δ).
One has x1 + · · · + x8 = 0 and
d
d
d 2 d 2
x + x + x2 + x23 + 2x3 + · · · + x28 + 7x8
2 1 2 2
2
2
d ′2
(r + a2 + b2 + c2 ) + r′
4′
= dq + r′ = n by (113).
=
This solves the equation (112).
In the case (b), one replaces r′ by r′ /2 to define a, b, c, α, β, γ, δ similarly. Then
one takes (x1 , . . . , x8 ) = (−α, −β, α, β, −γ, −δ, γ, δ).
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[Kuls]
[LaLeTh96]
[Li15]
[Lu84]
[Lu88]
[Ma07]
[MalleTe]
[MaPr96]
[MaSp16]
[Mazor]
[McLane]
[MeTa76]
[Mi88]
[Milne]
[Mis90]
[NagaoTsu]
[Naka41a]
[Naka41b]
[Nava]
[NaSp14]
MARC CABANES
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[Puig90]
[Puig99]
[Ri69]
[Rick89]
[Rick95]
[Rick96]
[Rou01]
[Rou02]
[Serre]
[SGA4.5]
[SGA5]
[Sho85]
[Sm82]
[Sp13]
[Sp17]
[Spr]
[Sri]
[St63]
[Tay17]
[Thev]
[Tin79]
[Tin80]
[Tu02]
[Wi88]
[Ze81]
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Index
(n)
BrQ (M ), 59
YP , 18
B, 6
BGF (L, ζ), 36
Bℓ (GF , s, 21
C, 59
F, 4
HY , 11
N, 6
PS,F , 34
S, 5
Sθ , 14
ShA (X), 52
T, 6
WI , 5
X(T), 5
Alp(H), 16
B, 5
Bl(H | D), 10
CF(H), 18
∆, 5
G, 4
HF (G, U ), 13
Hob , 3
Irr(B), 18
L∗s , 56
LI , 6
Ls , 56
RG
L , 18
N∗s , 56
Ns , 56
OpenX , 51
PI , 5
Φ(G, T)+ , 5
Φ(G, T), 5
Ru (G), 5
RΓ(X, F ), 52
RΓc (X, F ), 52
Σ, 54
T, 5
UI , 6
X(w), 54
Xα , 5
XI,v , 55
Xet , 52
Y(w), 54
YI,v , 55
β-numbers, 29
Fs , 53
Fs , 53
F(D,bD ) (B), 26
F(w,θ), 55
L, 58
N , 58
S(w,θ) , 55
δπ , 21
ṡ, 13
ℓ-adic cohomology, 52
ℓ-modular system, 17
ℓ-subpair, 25
ℓ-permutation module, 59
sln , 31
≤d , 35
Eℓ (GF , s), 21
GΓc (YP , O), 59
∇, 55
S, 6
eℓ (GF , s, 21
X(w), 54
φd -torus, 34
φd , 34
πw , 54
Hic (X, F ), 52
Hic (YP ), 19
regH , 18
∗ G
RL , 19
Db , 3
f, 58
M
e 57
G,
e
Ls , 57
e s , 57
N
e
P, 57
e 58
L,
e , 58
N
a′n , 12
cd (n), 62
d-core, 28
d-cuspidal, 35
d-split Levi subgroup, 34
dx , 26
eχ , 18
eℓ (GF , s, 21
f ∗ , 51
f∗ , 51
i-th cohomology A-module of F , 52
w
jw
′ , 55
68
INDEX
p-constant, 20
w ∪ dv , 55
wθ , 55
(BHZC), 46
étale cohomology., 52
abelian defect conjecture, 48
admissible pair, 14
Alperin weight conjecture, 16
Alvis-Curtis duality, 19
block, 9
Borel subgroup, 5
branching rule, 28
Brauer morphism, 10
Brauer’s first Main Theorem, 10
Brauer’s height zero conjecture, 46
Brauer’s second Main Theorem, 26
Brauer’s third Main Theorem, 10
Bruhat decomposition, 5
canonical character, 26
categorification, 32
categorifications, 31
centric subpair, 26
character formula, 19
characteristic p type, 7
classification of finite simple groups
(CFSG), 7
cohomology with compact support,
52
commutator formula, 5
cuspidal characters, 35
cyclotomic polynomials, 34
defect group, 10
defect zero, 10
direct and inverse images of sheaves,
51
dominant weights, 16
essential p-subgroups, 9
exotic fusion system, 42
finite groups of Lie type, 7
fusion system, 9
good primes, 40
Harish-Chandra induction, 19
Harish-Chandra theory, 35
hooks, 28
Jordan decomposition, 4
69
Jordan decomposition of characters,
22
Jucys-Murphy elements, 31
Lang’s theorem, 7
Levi decomposition, 6
local structure of a block, 26
local subgroup, 7
maximal subpairs, 25
maximal torus, 5
Murnaghan-Nakayama rule, 28
nilpotent block, 47
parabolic subgroups, 5
perfect bi-character, 23
polynomial order, 34
presheaf, 51
principal block, 10
quasi-simple group, 7
radical subgroup, 7
ramification, 52
rational series, 20
reductive group, 5
Rickard equivalence, 58
rim, 28
root subgroups, 5
root system, 5
semi-simple elements, 4
sheaf, 51
Steinberg module, 15
strong sl2 -categorification , 32
tame intersection, 9
torus, 5
trivial module, 10
type of maximal torus, 19
uniform functions, 20
unipotent d-cuspidal pair, 35
unipotent blocks, 22
unipotent characters, 22
unipotent elements, 4
universal covering, 7
vertex, 17
Weyl group, 5
Yokonuma-Hecke algebras, 11
Young diagrams, 28
70
INDEX
CNRS, Institut de Mathématiques de Jussieu-Paris Rive Gauche, Batiment Sophie
Germain, 75205 Paris Cedex 13, France.
E-mail address: marc.cabanes@imj-prg.fr
| 4 |
Rate Optimal Binary Linear Locally Repairable Codes with
Small Availability
Swanand Kadhe and Robert Calderbank
arXiv:1701.02456v2 [cs.IT] 14 Sep 2017
swanand.kadhe@tamu.edu, robert.calderbank@duke.edu
A locally repairable code with availability has the property that every code symbol can be recovered
from multiple, disjoint subsets of other symbols of small size. In particular, a code symbol is said to
have (r,t)-availability if it can be recovered from t disjoint subsets, each of size at most r. A code
with availability is said to be rate-optimal, if its rate is maximum among the class of codes with given
locality, availability, and alphabet size.
This paper focuses on rate-optimal binary, linear codes with small availability, and makes four
contributions. First, it establishes tight upper bounds on the rate of binary linear codes with (r, 2) and
(2, 3) availability. Second, it establishes a uniqueness result for binary rate-optimal codes, showing
that for certain classes of binary linear codes with (r, 2) and (2, 3)-availability, any rate optimal code
must be a direct sum of shorter rate optimal codes. Third, it presents novel upper bounds on the
rates of binary linear codes with (2,t) and (r, 3)-availability. In particular, the main contribution here
is a new method for bounding the number of cosets of the dual of a code with availability, using
its covering properties. Finally, it presents a class of locally repairable linear codes associated with
convex polyhedra, focusing on the codes associated with the Platonic solids. It demonstrates that
these codes are locally repairable with t = 2, and that the codes associated with (geometric) dual
polyhedra are (coding theoretic) duals of each other.
1
Introduction
The enormous growth of data being stored or computed online has encouraged practical distributed
storage systems to migrate from triple replication [1, 2] to erasure coding for handling failures, see,
e.g., [3, 4]. Even though classical erasure codes such as Reed-Solomon codes achieve high storage
efficiency, they are inefficient in handling disk (or node) failures as they usually require to download
large amount of data while repairing a failed node. The conflicting requirements of reliability, storage
efficiency, and repair efficiency in data centers have created a new set of problems for coding theorists.
Two measures of repair efficiency have received particular research attention: (a) repair bandwidth –
the metric is the total number of symbols (or bits) communicated while repairing a failed node, and the
corresponding family of codes is called regenerating codes (see, e.g., [5, 6, 7]); and (b) repair locality
– the metric is the number of nodes participating in the repair process, and the corresponding family of
codes is called locally repairable codes (see, e.g., [8, 9, 10, 11, 12]). We restrict our attention to codes
with locality in this work.
A locally repairable code (LRC) is a code of length n over a finite field F such that every symbol
of a codeword can be recovered by accessing at most r other symbols. The set of symbols participating
in the recovery of a symbol is referred to as a recovering set (or repair group) of the symbol. Codes
with small locality were introduced in [8, 13] (see also [10]). The study of the locality property was
inspired by the pioneering work of Gopalan et al. [9]. One of their key contributions was to establish a
trade-off between the minimum Hamming distance of a code and its locality, analogous to the classical
Shorter version of the paper was presented at ISIT 2017.
Binary LRCs with Availability
2
Singleton bound. In particular, the authors showed that for a (scalar) linear (n, k) code having locality r
for systematic symbols, its minimum distance d is upper bounded as
d ≤ n−k−
k
+ 2.
r
(1)
They also demonstrated that the Pyramid code construction described in [8] achieves this bound. Since
then, a series of papers have extended the distance bound for various types of codes, and have provided
optimal code constructions that achieve the minimum distance bound (see, e.g., [14, 15, 16, 17, 12, 18,
19, 20], and references therein).
In this work, we focus our attention on a class of LRCs with multiple disjoint recovering sets [21, 22,
23, 24]. Providing multiple disjoint recovering groups for symbols enables parallel reads and provides
high availability of data. For this reason, codes with multiple disjoint recovering sets are referred to
as codes with availability. Such codes are particularly attractive for data centers storing hot data, i.e.,
frequently accessed data. Moreover, they are useful in designing coded private information retrieval [25]
and locally rewriteable codes [26].
A code is said to possess (r,t)-availability, if every symbol of a codeword has t disjoint recovering
sets each of size (i.e., locality) at most r. Most of the literature on codes with availability has been
devoted to computing Singleton-like upper bounds on the minimum distance, and constructing codes
with availability and large minimum distance. By comparison, relatively little has been said about bounds
on the code rate, and constructions of high rate codes with availability. However, the authors of [23] (see
also [27]) give an upper bound on the rate of codes with (r,t)-availability, and the authors of [28] give a
field size dependent bound on the size of codes with availability, along the lines of [18]. Very recently,
the authors of [29] presented an improved rate bound for (r,t)-availability, and in particular, for (r, 3)availability.
We are interested in computing tight upper bounds on the rate of LRCs with availability. Note that
as we enhance the availability of the code by increasing the number of disjoint recovering sets, we are
introducing more dependencies amongst the code symbols. Thus, the rate of a code with high availability
cannot be too high, representing tension between high rate and high availability. We first focus our
attention to binary LRCs (i.e., LRCs over F2 ) with availability t = 2, 3. We note that, in practice, small
values of the availability parameter t that are comparable to triple replication are the most interesting. Our
motivation behind considering binary codes is that codes constructed over small finite fields, especially
Galois fields of the form F2m , are preferred in practice for their fast arithmetic [30].
Our main result is a uniqueness result for rate optimal codes for t = 2, 3. In essence, we show that for
certain classes of binary linear codes with (r, 2) and (2, 3)-availability, any rate optimal code must be a
direct sum of shorter rate optimal codes. We note that designing a rate optimal code with availability can
be viewed as a covering problem. In particular, when the i-th symbol of a code has (r,t)-availability, its
dual code must contain t codewords, each of weight at most r + 1, such that their supports intersect only
on {i}. We refer to such codewords as covering codewords. Designing a rate optimal code with (r,t)availability is equivalent to finding a subspace of smallest dimension that contains covering codewords
for all the symbols. It is worth noting that for covering problems, direct sum constructions are known to
give good codes [31].
Next, we obtain upper bounds on the rate of codes with (2,t) and (r, 3)-availability by using covering
properties of their duals. In particular, we develop a novel method to bound the maximum weight of a
coset leader of the dual code, which is known as the covering radius for linear codes (see [31]), by using
the its covering properties. This enables us to bound the number of cosets of the dual and get a rate upper
Kadhe and Calderbank
3
bound. The method of bounding the number of cosets using covering properties may be of independent
interest.
Furthermore, we present a class of codes with t = 2 that are associated with convex polyhedra, in
particular, the Platonic solids. We note that these codes associated with the Platonic solids may be of
independent interest. We outline our contributions in the following section.
1.1 Our Contributions
We highlight our broad contributions in the following.
1. We first consider binary codes associated with convex polyhedra. More specifically, given a convex
polyhedron Γ with e edges, fix an arbitrary labeling of its edges from 1 to e. We define the
code associated with a convex polyhedron1 Γ as a subset C ⊂ Fe2 such that for every vector c ∈
C , the entries corresponding to edges that meet at a vertex of Γ sum to zero over F2 . In other
words, vertices of Γ define parity checks on the codewords of C . We demonstrate that such codes
have t = 2, and that the codes associates with dual polyhedra are duals of each other. Further,
we demonstrate that codes associated with the Platonic solids, namely, tetrahedron, octahedron,
dodecahedron and icosahedron, are near-optimal in terms of their rates.
2. We focus on a class of binary (n, k) codes Cˆ defined as the nullspace of an N × n parity-check
matrix H, where each row has weight r + 1 and each column has weight t, such that nt = N(r + 1).
In addition, supports of any two rows of H intersect in at most one point. We refer to codes in this
class as codes with exact covering.2
(a) First, we consider codes in Cˆ with (r, 2)-availability. We show that, when n ≥ r + 1 and
r
, with equality if and only if C is a direct
r + 1 | 2n, the rate of C is upper bounded as nk ≤ r+2
i
h
(r+1)(r+2)
, (r + 1) codes, each generated by the complete graph on r + 2 points
sum of
2
(Theorem 1).3
(b) Next, we consider codes in Cˆ with (2, 3)-availability. When the block length n is a multiple
of 7, say n = 7m, we show that for any C ∈ Cˆ, we have Rate (C ) ≤ 37 , with equality if and
only if C is a direct sum of m copies of the [7, 3] Simplex code (Theorem 2).
3. We present novel rate upper bounds for codes in Cˆ with (2,t) and (r, 3)-availability (Theorem 3
and Corollary 4). Our bounds for codes with (2,t) and (r, 3)-availability become sharper that the
known bounds as the values of t and r increase, respectively.
1.2 Relationship to Previous Work
Codes with availability: The notion of multiple disjoint recovering sets has been studied in several
works, see e.g. [21, 22, 23, 24, 28, 33, 29].
Rate Bounds: The authors of [23] (see also [27]) show that for an (n, k) code with (r,t)-availability,
the rate is upper bounded as
1
k
.
(2)
≤
n ∏t
1+ 1
j=1
1 More
jr
generally, we can define a code associated with a planar graph in the same way.
2 Codes in this class were also studied very recently in [29], where such codes are referred to as codes with strict availability.
3 The
upper bound of r/(r + 2) has been shown in [32]; see Remark 4 for details.
Binary LRCs with Availability
4
The authors of [21] and [23] also find upper bounds on and the minimum distance for codes with availability. Under suitable divisibility assumptions, these distance bounds can be translated to the rate
bounds. For (r, 2) availability the corresponding rate bounds from [23] and [21], respectively, are
k r3 − r3 1
≤ 3
+ ,
n
r −1 n
and
k 2r − 1
1
≤
+
.
n 2r + 1 n(2r + 1)
(3)
8
2
For (2, 3)-availability these derived rate bounds from [23] and [21] become 15
+ 1n , and 74 + 7n
,
respectively. We note that our rate bounds for (r, 2) and (2, 3) availability strictly improve on these
bounds. In a very recent work, the authors of [29] improve the bounds of [23]. We compare our bounds
with the existing bounds in Section 7.3.
Constructions: It was noted in [21, 23] that direct product codes possess availability property. The
authors of [34] studied the availability property of simplex codes. The authors of [35] present a tensorproduct based construction and a cyclic code construction for r = 2 and t = 3. The authors of [28] analyze
the availability properties of a large number of well-known classical codes. Several code constructions
using combinatorial structures are presented in [22, 24, 33].
LRCs that locally correct multiple erasures: We note that LRCs with (r,t)-availability form a
class of codes that can correct any t erasures locally. There are several classes of LRCs that can locally
correct up to t erasures as outlined below. Rate bounds on LRCs from any of these classes yield upper
bounds on the rate of an LRCs with (r,t)-availability. A discussion on the hierarchy of these classes can
be found in [36].
a) LRCs with strong local codes, wherein every symbol is protected by an (r + t, r,t + 1) local code,
were considered in [20, 17, 15, 37]. For such codes, the rate is upper bounded as
r
k
≤
.
n r+t
(4)
b) LRCs with cooperative local recovery, wherein t erasures can be simultaneously corrected by
reading at most r symbols, are considered in [38]. The rate bound of (4) also applies to this family of
codes.
c) LRCs with multiple repair alternatives are considered in [24], wherein for any subset E ⊂ [n] of
size t, every symbol i ∈ E can be recovered from at most r symbols outside E. The authors present a
family of such codes based on partial geometries, and give lower and upper bounds on the rate of codes
in this family.
d) LRCs that allow sequential (or, successive) repair of t erasures are considered in [32, 36, 39, 40].
The authors of [32] present an upper bound on the rate of an (n, k) code that allows sequential recovery
of t = 2 symbols with locality r as
r
k
≤
.
(5)
n r+2
The authors also present optimal code construction based on Turán graphs for a specific parameter range.
They also demonstrate a code construction based on complete graphs, which has (r, 2)-availability. Our
result shows the uniqueness of such a construction for rate optimal codes with exact covering. For the
sequential recovery of t = 3 erasures with locality r under functional repair model, [36] presents a lower
bound on the length of a code. Under suitable divisibility assumptions, this bound for r = 2 translates to
the rate bound of 4/9. The authors of [39] show a uniqueness result for rate optimal constructions for
the sequential recovery from t = 2 erasures.
Binary locally repairable codes: A number of studies have recently considered LRCs over the
binary field, see e.g., [34, 35, 41, 33, 42, 43, 28]. In [41] (see also [35]), in addition to presenting
Kadhe and Calderbank
5
several code constructions, the authors also establish upper bounds on the rate of binary LRCs for various
parameter regimes when r = 2. In addition, the authors present a direct sum of [7, 3] Simplex codes as an
example of a larger code with (2, 3)-availability. Our result shows rate optimality of such a construction
and its uniqueness for the class of codes with exact covering.
Field size dependent bounds on the code dimension: Field size dependent bounds on the minimum
distance and rate for LRCs are considered in [18, 44]. Simplex codes are shown to be rate optimal
for r = 2 amongst binary codes in [18]. The authors of [28] develop field size dependent bounds to
incorporate the availability.
2
Preliminaries
Notation: We use the following notation. For an integer l, let [l] = {1, 2, . . . , l}. We use x(i) to denote the
i-th coordinate of a vector x, and H(i, j) to denote the element in row i and column j in a matrix H. For
a vector x, Supp (x) denotes its support, i.e., Supp (x) = {i : x(i) 6= 0}. Let wt (x) denote the Hamming
weight of vector x, i.e., wt (x) = |Supp(x) |. For a set of vectors x1 , . . . , xm , hx1 , . . . , xm i denotes their
span; whereas for a matrix H, hHi denotes its row space. For a vector space A , dim (A ) denotes its
dimension. For an [n, k] code C , its rate is denoted as Rate (C ) = nk .
Let C denote a linear (n, k, d) code over F2 with block-length n, dimension k, and minimum distance
d. Let c denote a codeword in C .
We say that a code bit has availability t with locality r if it can be recovered from t disjoint subsets
of size at most r. The formal definition is as follows.
Definition 1. [(r,t)-Availability] We say that the i-th bit of an (n, k, d) code C has (r,t)-availability if
for any codeword c ∈ C , there exist t disjoint subsets R j (i) ⊂ [n] \ {i}, |R j (i) | ≤ r, for 1 ≤ j ≤ t such
that c(i) = ∑l∈R j (i) c(l) for every j ∈ [t]. Each one of such subsets is referred to as a repair group for bit
i. If every bit of C has (r,t) availability, we say that C has (r,t)-availability. We denote such a code as
an (n, k, d, r,t) LRC.
It is worth mentioning that repair groups of a bit can have different sizes, and we denote locality of
the bit as the size of its largest repair group. Next, we focus our attention to codes associated with convex
polyhedra.
3
Codes associated with Convex Polyhedra
In this section, we present codes associated with convex polyhedra, and we focus on Platonic solids.
Given a convex polyhedron, each edge corresponds to an entry of the codeword, and every vertex corresponds to a parity check.
Definition 2. Consider a convex polyhedron Γ with v vertices, e edges, and f faces. Fix an arbitrary
labeling of its edges from 1 through e. Let C be a subset of Fe2 such that for a vector c ∈ C , the entries
corresponding to edges that meet at a vertex sum to zero over F2 . We say that the code C is generated
by Γ, and denote it as C (Γ).
Definition 3. We say that a length-N binary vector v corresponds to a face of Γ if the locations of ones
in v correspond to the edges forming that face.
First, we show that C (Γ) is a linear code generated by the faces of Γ.
Lemma 1. For a convex polyhedron Γ with v vertices, e edges, and f faces, the C (Γ) generated by Γ is
an [e, f − 1] linear code. Further, the vectors corresponding to the faces of Γ span C (Γ).
6
Binary LRCs with Availability
Proof. First, we prove that C is an [e, f − 1] linear code. Let us denote the graph formed by the edges
and vertices of Γ as Γ′ . Note that C is the kernel of the v × e incidence matrix H of Γ′ , i.e., C = {c ∈ Fe2 |
Hc = 0}. Every column of H has two ones, and thus the rows of H sum to zero giving Rank(H) ≤ v − 1.
We show that there is no smaller linear dependency.
Let hi denote the row of H corresponding to vertex i of Γ′ . Suppose, for contradiction, there is a
smaller linear dependency ∑i∈S hi = 0, where S ⊂ [v]. Now, every column of H has exactly two ones
corresponding to an edge of Γ′ . Thus, for any vertex i ∈ S, all of its neighbors should be in S. However,
as Γ′ is connected, S must include all the v vertices. Thus, Rank (H) = v − 1, and dim (C ) = e − v + 1.
For a convex polyhedron, Euler’s formula states that v − e + f = 2. Therefore, C is an [e, f − 1] linear
code.
Next, we show that C is generated by the vectors associated with faces. Let G denote the matrix
containing the vectors associated with the faces of Γ. Observe that each row of G satisfies all the parity
checks. Now, note that every column of G corresponds to an edge of Γ′ , and thus, has exactly two ones
corresponding to the two faces that meet at that edge. Then, by applying the same arguments as in the
case of H, we get that Rank (G) = f − 1, and the result follows.
Recall that the two polyhedra are said to be (geometric) duals of each other if the vertices of one
polyhedron correspond to the faces of the other, and vice-versa. Then, the following result follows from
Lemma 1.
Corollary 1. Let Γ and Γ⊥ be the dual convex polyhedra. Then, the dual code C ⊥ (Γ) of the code
generated by a convex polyhedron Γ is isomorphic to the code generated by its dual polyhedron Γ⊥ , i.e.,
C ⊥ (Γ) ∼
= C (Γ⊥ ).
Lemma 2. The code generated by a convex polyhedron has t = 2 availability.
Proof. The value of the bit indexed by edge {u, v} can be recovered by summing over F2 the entries of
all the other edges incident either on vertex u, or vertex v. The edges incident on u are disjoint from those
incident on v.
Next, we consider the codes associated with Platonic solids. Table 1 summarizes the codes associated
with the Platonic solids. While specifying the parity check and generator matrices for these codes in the
subsequent sections, we omit the zero entries for simplicity.
Remark 1. As we show in the next section, the rate of a binary LRC with (r, 2)-availability is upper
bounded by r/(r + 2). Observe from Table I that the code associated with tetrahedron is rate-optimal,
whereas the rates of codes associated with cube, octahedron, dodecahedron, and icosahedron are nearoptimal.
3.1 Tetrahedron Code
Fig. 1(a) shows the graph of the cube. Following the labeling of edges in Fig. 1(a), the set of parity
checks can be written as
1 1
1
1 1
1
.
H=
(6)
1
1 1
1 1 1
Kadhe and Calderbank
7
Observe from (6) that the tetrahedron code has (2, 2)-availability (see Remark 2). A generator matrix
with rows corresponding to faces is given as
1
1 1
1
1 1
.
G=
(7)
1 1
1
1 1 1
It is easy to verify that Rank (H) = 3, Rank (G) = 3, and GH T = 0. Therefore, the tetrahedron code is a
(6, 3) code with (2, 2)-availability.
One can see that G can be obtained from H by first reordering the rows of H, row 1 → row 2 → row
3 → row 1, and then applying the permutation (16)(24)(35) on the columns. Hence, the tetrahedral code
is equivalent to its dual.
3.2 Cube Code and Octahedron Code
Fig. 1(b) shows the graph of the cube. Following the labeling of edges in Fig. 1(b), the set of parity
checks for the cube code can be written as
1
1 1
1 1
1
1 1
1
1
1
1
.
H =
(8)
1
1
1
1
1 1
1
1 1
1
1 1
From (8), observe that the cube code has (2, 2)-availability (see Remark 2).
A generator matrix composed of vectors associated with the faces of the cube is as follows.
1 1 1 1
1
1 1
1
1
1 1
1
.
G=
1
1 1
1
1 1
1
1
1 1 1 1
(9)
It is easy to verify that Rank (H) = 7, Rank (G) = 5, and GH T = 0. Therefore, the code associated with
the cube is a (12, 5) code with (2, 2)-availability.
Recall that the cube and the octahedron are geometric duals of each other. A parity check matrix H
of the octahedron code is given below. We follow the labeling of edges in as shown in Fig. 1(c).
1 1 1 1
1 1
1
1
1
1
1
1
.
H =
(10)
1
1 1
1
1
1 1
1
1 1 1 1
Binary LRCs with Availability
8
A
5
1
2
D
4
6
C
B
3
(a) Tetrahedron
A
1
A
B
5
6
2
3
9
E
4
F
1
4
12
7
D
2
10
E
10
11
8
6
H
F
G
11
8
9
7
D
12
C
C
B
3
5
(b) Cube
(c) Octahedron
A
A
6
F
1
5
13
12
4
K
5
3
E
Q
2
20 19
18
11
J
25
14
O
25
S
29
30
4
8
N
16
I
C
2
16
24
23
G
M
T
19
F
28
23
17
29
L
22
13
15
P
30 24
21 26
H
G
28
26
20
E
K
27
6
27
R
9
1
12
7
21
J
10
14
22
10
D
I
L
11
15
18
17
H
8
9
B
D
C
7
3
(d) Icosahedron
(e) Dodecahedron
Figure 1: Graphs associated with the Platonic solids.
From (10), observe that the octahedron code has (3, 2)-availability (see Remark 2).
B
Kadhe and Calderbank
9
Polyhedron and its Dual
Tetrahedron
Tetrahedron
Cube
Octahedron
Associated Code
[6, 3] code,
(2, 2)-availability
[6, 3] code,
(2, 2)-availability
[12, 5] code,
(2, 2)-availability
[12, 7] code,
(3, 2)-availability
Dodecahedron
[30, 11] code,
(2, 2)-availability
Icosahedron
[30, 19] code,
(4, 2)-availability
Weight Enumerator
1 + 4z3 + 3z4
1 + 4z3 + 3z4
1 + 6z4 + 16z6 + 9z8
1 + 8z3 + 15z4 + 24z5 + 32z6 + 24z7 + 15z8 + 8z9 + Z 12
1 + 20z3 + 30z4 + 72z5 + 400z6 + 11407 + 2715z8
+6560z9 + 14112z10 + 26280z11 + 42740z12 + 59760z13
+72000z14 + 75912z15 + 70215z16 + 57120z17 + 41440z18
+26820z19 + 15246z20 + 7560z21 + 3120z22 + 900z23 + 125z24
1 + 12z5 + 30z8 + 20z9 + 72z10 + 120z11 + 100z12 + 180z13
+240z14 + 272z15 + 345z16 + 300z17 + 200z18 + 120z19 + 36z20
Table 1: Codes associated with the Platonic solids
From (9) and (10), we see that the cube code and the octahedron code are duals of each other.
A generator matrix of the octahedron code can be given by (8). Note that each row of H in (8)
corresponds to a face of the octahedron.
3.3 Icosahedron Code and Dodecahedron Code
One can easily find a parity check matrix of the icosahedron code following the edge labeling in Fig. 1(d).
Observe that the icosahedron code has (4, 2)-availability. A generator matrix with its rows as the faces
of the icosahedron can be easily computed from Fig. 1(e). One can check that Rank(G) = 11, and the
icosahedron code is a (30, 11) code. Recall that the icosahedron and the dodecahedron are geometric
duals of each other. The dodecahedron code is a (30, 19) code with (2, 2)-availability.
4
Rate-Optimal Codes with Small Availability
We are interested in rate-optimal codes with (r,t)-availability, which are defined as follows.
Definition 4. [Rate Optimality] A code C with (r,t)-availability is said to be rate optimal if its rate is
maximum among all (binary, linear) codes possessing (r,t)-availability.
It is straightforward to see that the (r,t)-availability of a code C imposes certain constraints on its
dual code C ⊥ in the following way.
Remark 2. The i-th bit of a code C has (r,t) availability if and only if its dual code C ⊥ contains t
codewords c̃i,1 , c̃i,2 , . . . , c̃i,t such that for all l ∈ [t], i ∈ Supp(c̃i,l (i)), |Supp (c̃i,l ) | ≤ r + 1, and for all
p, q ∈ [t], p 6= q, Supp (c̃i,p ) ∩ Supp(c̃i,q ) = {i}. We call such t codewords as repair codewords for the
i-th bit.
In other words, the availability requirement of a code places constraints on the supports of certain
codewords in the dual code. Our central idea is to carefully analyze the structure of the dual code to
obtain upper bounds on the rate of the code with availability.
Binary LRCs with Availability
10
For simplicity of notation, we refer to the coordinates as points, and represent every codeword by
its support. In particular, we refer to a weight w codeword as a w-subset of [n] (or just as a subset if its
Hamming weight is clear from the context or if it is not important). For analyzing the structure of the
dual code, we use notions of covering and covering with (r,t)-availability, defined as follows.
Definition 5. [Covering] We say that a w-subset S covers point i, if i ∈ S. Further, we say that a code C
covers point i l times, if C contains l subsets that cover point i.
Definition 6. [Covering with (r,t)-Availability] We say that a code C covers point i with (r,t)-availability,
if C covers point i (at least) t times such that the subsets covering i are of size at most r + 1 and they
intersect only on i. We call such subsets as t-covering subsets (or, simply, as covering subsets).
We can restate Remark 2 in terms of covering as follows.
Remark 3. A code C has (r,t)-availability if and only if its dual code C ⊥ covers each of the n points
with (r,t)-availability.
Finally, we introduce the notion of the code generated by a graph Γ (similar to the code generated by
a convex polyhedron).
Definition 7. Consider a planar graph Γ with v vertices and e edges. Fix an arbitrary labeling of its
edges from 1 through e. Let C be a subset of Fe2 such that for every vector c ∈ C , the entries of c
corresponding to edges that meet at a vertex sum to zero over F2 . We say that the code C is generated
by Γ, and denote it as C (Γ).
Notice that C is a linear code with the incidence matrix of Γ as its parity check matrix.
In the remaining of the paper, we denote the code containing the covering subsets as the primal code
C . Note that its dual code C ⊥ possesses the (r,t)-availability property.
5
Codes with (r, 2)-Availability
Our focus, in this section, is on the codes in which each bit can be recovered from two disjoint recovering
sets each of size at most r + 1. From Remark 3, notice that the primal code C should cover every point
with (r, 2)-availability. From simple counting arguments, it follows that to cover n points with (r, 2)2n
subsets of size up to r + 1. First, we consider the case when C
availability, C should contain at least r+1
2n
contains exactly r+1 2-covering subsets, each of size r + 1.
5.1 Exact Number of Covering Subsets of Size r + 1
Theorem 1. Let n and r be non-negative integers such that n ≥ r + 1 and r + 1 | 2n. Let C be the length-n
2n
primal code spanned by r+1
(r + 1)-subsets that cover every point with (r, 2)-availability. Then, the rate
r
⊥
, with equality if and only if C ⊥ is (equivalent
of its dual code C is upper bounded as Rate C ⊥ ≤ r+2
h
i
to) a direct sum of (r+1)(r+2)
, (r + 1) codes, each of which is the code generated by the complete graph
2
on r + 2 points.
2n
Proof. Let S be the set of N = r+1
covering (r + 1)-subsets. Label them (in arbitrary order) as
S1 , · · · , SN . Form a graph Γ with N vertices, where every vertex corresponds to a covering (r + 1)subset. Join vertices i and j if the corresponding (r + 1)-subsets Si and S j intersect. Observe that a pair
2n
of them. Moreover, since
of covering subsets can intersect in at most one point as there are exactly r+1
S covers every point exactly twice, each vertex in Γ has degree r + 1.
Kadhe and Calderbank
11
If for some T ⊆ S , ∑ j∈T S j = 0, then the vertices of Γ corresponding to subsets in T determine a
connected component of Γ. Note that the size of a connected component in Γ is at least r + 2 as Γ is an
(r + 1)-regular graph.
Now, partition Γ into connected components, and eliminate a vertex from every connected compo2n
these vertices
nent. This yields at least r+1
r+2 N = r+2 vertices such that (r + 1)-subsets corresponding to
2n
are linearly independent. Therefore, dim (C ) ≥ r+2
, and the upper bound on Rate C ⊥ follows. In
2n
addition, we have dim (C ) = r+2
if and only if the connected components of Γ are complete graphs of
of size r + 2. This essentially specifies that the incidence matrix of Γ has a block diagonal structure with
each block being the incidence matrix of the complete graph on r + 2 points. Hence, a rate optimal C ⊥
must be a direct sum of the codes generated by complete graphs on r + 2 points.
Remark 4. The upper bound of r/(r + 2) on the rate of any linear code with (r, 2)-availability has
been established in [32] by considering a broader class of codes that allow sequential recovery of 2
symbols with locality r. Further, the authors note that the code associated with the complete graph on
r + 2 vertices is a rate-optimal code with (r, 2)-availability. Clearly, a direct sum of codes associated
with complete graph on r + 2 vertices is also rate-optimal. Theorem 1 shows the uniqueness of such a
construction for achieving rate-optimality in binary codes with (r, 2)-availability.
5.2 Exact Number of Covering Subsets of Multiple Sizes
Corollary 2. Let n and r be non-negative integers such that n ≥ (r + 1). Let C be the length-n primal
code spanned by N subsets of multiple sizes wth maximum size r + 1, which cover every point exactly
r
twice with availability. Then, the rate of its dual code C ⊥ is upper bounded as Rate C ⊥ < r+2
.
Proof. Let N j be the number of 2-covering j-subsets for 1 ≤ j ≤ r + 1. Since each of the n points is
covered exactly twice, we have
∑r+1
j=1 jN j
n=
.
(11)
2
The proof essentially follows the same argument as the proof of Theorem 1. Form a graph Γ with N
subsets as vertices, wherein a pair of vertices are adjacent if the corresponding subsets intersect.
Now, a minimal linear dependency amongst the covering subsets determines a connected component
of Γ, as every point is covered exactly twice. Partitioning Γ into connected components, and eliminating
a vertex from every connected component, we get a lower bound on the dimension of C as dim (C ) ≥
j
∑r+1
j=1 j+1 N j . This follows since the size of a connected component of Γ containing a vertex corresponding
j
r+1
1
to a j-subset is at least j + 1. Clearly, ∑r+1
j=1 j+1 N j ≥ r+2 ∑ j=1 jN j with strict inequality when there is a
2n
from which the result
covering subset of size less than r + 1. Then, from (11), we have dim (C ) > r+2
follows.
6
Codes with (2, 3)-Availability
In this section, we focus on the codes with r = 2 and t = 3. Simple counting arguments show that to cover
n points with (2, 3)-availability, the primal code C must contain at least n 3-covering subsets of size up
to 3. We consider the case of of exact covering, wherein C contains exactly n 3-covering 3-subsets. For
the case when the block-length is a multiple of 7, we show that the code rate is upper bounded by 3/7,
and prove that any rate optimal code needs to be a direct sum (or tensor-product) style construction. The
statement of the result is as follows.
Binary LRCs with Availability
12
Theorem 2. For a positive integer m, let n = 7m. Let C be the length-n primal code spanned by 7m
3-subsets that cover every point with (2, 3)-availability. Then, we have Rate C ⊥ ≤ 73 , with equality if
and only if C ⊥ is (equivalent to) a direct sum of m copies of the [7, 3] Simplex code.
Remark 5. Simplex codes have been shown to be rate optimal for r = 2 amongst binary codes in [18].
Several constructions based on Simplex codes have been proposed, e.g., [34, 35, 41, 43]. The authors
of [35] present a direct sum of [7, 3] Simplex codes as an example of a code with (2, 3)-availability.
Theorem 2 shows the uniqueness of such a construction for achieving rate optimality in binary codes
with (2, 3)-availability.
6.1 Proof of Theorem 2
The steps involved in the proof are outlined below.
1. First, we show that C must contain at least m pairwise disjoint covering 3-subsets.
2. Next, we prove that dim C ⊥ ≤ 3m, and the equality occurs if and only if the size of a maximum
set of pairwise disjoint covering 3-subsets in C is exactly m. To prove this, we first assume that
there exists a maximum set of pairwise disjoint covering
3-subsets in C of size m + i′ for some
′
⊥
non-negative integer i . Then, we show that dim C is strictly less than 3m if i′ > 0.
3. Finally, we prove that, if dim (C ) = 4m, then the size of a maximum collection of pairwise disjoint
covering 3-subsets in C is exactly m, and C must be (equivalent to) a direct sum of m copies of a
[7, 4] Hamming code.
6.1.1
Step 1
Lemma 3. For a positive integer m, let n = 7m. Let C be the length-n primal code spanned by 7m 3subsets that cover every point exactly thrice with availability. Then, C must contain at least m pairwise
disjoint 3-subsets.
Proof. Label the n covering 3-subsets as S1 , · · · , Sn . Form a graph Γ with n vertices, where every vertex
corresponds to a covering 3-subset. Put an edge between vertices i and j if the corresponding 3-subsets
Si and S j intersect. Since every point is covered exactly thrice, Γ must be a 6-regular graph.
Now, a set of pairwise disjoint covering 3-subsets determine
l man independent set in Γ. For a j-regular
n
(see [45, Theorem 1]), from which the
graph of order n, the size of an independent set is at least j+1
result follows.
6.1.2
Step 2
We begin with establishing the key ingredients that aid in this step.
1. A maximum set of pairwise disjoint 3-subsets in C : Suppose the size of a maximum set of
pairwise disjoint covering 3-subsets in C is m + i′ . We label these subsets as S1 , . . . , Sm+i′ . Let A
′
′
be the set of points covered by these subsets, i.e., A = ∪m+i
j=1 S j . Let A = [n] \ A. See Fig. 2 for the
′
′
′
ease of understanding. Note that |A| = 3m + 3i and |A | = 4m − 3i .
2. Three types of 3-subsets depending on their intersection with A: Let xi be the number of 3subsets that intersect A in i points for 1 ≤ i ≤ 3. (By maximality of S1 , . . . , Sm+i′ , x0 = 0.) Let
Kadhe and Calderbank
13
= A′
. Let A
(S1 ) · · · (Sm+i′ )
(E1 ) · · · (Ex3 )
···
(F1 ) · · · (Fx2 )
···
(T1 ) · · · (Tx1 )
≤
. Let
···
A′1
B′1
A′′
C1′
Figure 2: Schematic depicting the notation for the step 2 in the proof of Theorem 2.
{E j : 1 ≤ j ≤ x3 }, {Fj : 1 ≤ j ≤ x2 }, and {T j : 1 ≤ j ≤ x1 } be the collections of 3-subsets that meet
A in 3, 2, and 1 points, respectively.
1
Let A′′ be the set of points in A′ that are covered by the type T subsets, i.e., A′′ = A′ ∩ ∪xj=1
Tj .
2
Let A′1 be the points in A′ that are covered by the type F subsets, i.e., A′1 = A′ ∩ ∪xj=1
Fj . Let
C1′ ⊆ A′1 be the set of points that are covered only by the type F subsets. These sets are depicted
schematically in Fig. 2.
3. Singletons and pairs of points: Consider the multiset of points in A′1 that are covered by the type
F subsets. We refer to the elements of this multiset as singletons. Note that the size of this multiset
is x2 . Similarly, consider the multiset of points in A′′ that are covered by the type T subset. Every
type T covers two points from A′′ , which are referred to as a pair (of points). There are x1 such
pairs in the multiset.
4. Graph Γ formed on pairs: Form a graph Γ by assigning a vertex corresponding to every point in
A′′ , and adding an edge between two vertices if they correspond to a pair. Note that the number of
vertices of Γ is |A′′ |, and the number of edges in Γ is x1 .
Partition Γ into connected components. Let B′1 be the vertices of the connected components of Γ
that (directly or indirectly) touch A′1 . In other words, B′1 ⊆ A′ \ A′1 be the set of points such that
any vertex corresponding to a point in B′1 is connected to a vertex corresponding to a point in A′1 .
Again, refer to Fig. 2 for a schematic representation of B′1 .
5. Analysis of the singletons and pairs: Suppose that a fraction f x2 of singletons touch connected
components of Γ. Note that these f x2 are the singletons in A′1 \C1′ , and we have
|C1′ | =
(1 − f )x2
.
3
(12)
Next, suppose that a fraction g|A′1 \C1′ | of points in A′1 \C1′ have degree one in Γ and the remaining
(1 − g)|A′1 \C1′ | points have degree two in Γ. Consider the multiset of points with indices in A′1 \C1′
Binary LRCs with Availability
14
that are covered by the type F and type T subsets. There are 3|A′1 \ C1′ | such points, of which,
f x2 are covered by the type F subsets, g|A′1 \ C1′ | are covered once by the type T subsets, and
(1 − g)|A′1 \C1′ | are covered twice by the type T subsets. Therefore,
3|A′1 \C1′ | = f x2 + g|A′1 \C1′ | + 2(1 − g)|A′1 \C1′ |,
which yields
|A′1 \C1′ | =
f
x2
1+g
(13)
For the simplicity of notation, denote the dual code C ⊥ as D. To obtain an upper bound on the
dimension of D, consider the projection of D on A ∪ A′1 ∪ B′1 , denoted as D |A∪A′1 ∪B′1 , and its kernel,
denoted as D ′ , which is the subcode of D that vanishes on A ∪ A′1 ∪ B′1 . Now, by the rank-nullity theorem,
we have
dim (D) = dim D ′ + dim D |A∪A′1 ∪B′1 .
(14)
In the following, we obtain an upper bound on the dimensions of D ′ and D |A∪A′1 ∪B′1 .
Lemma 4. Let D ′ be the subcode of D that vanishes on A ∪ A′1 ∪ B′1 . Then, we have
dim D
′
3i′ 1
≤ m−
−
4
4
1− f
f
+
x2 .
3
1+g
(15)
Proof. Note that the pairs in A′′ act as parity checks for the codewords of D ′ . Thus, the support of any
codeword d ∈ D ′ must be a union of connected components of Γ, otherwise it fails a parity check. Hence,
dim (D ′ ) is at most the number of connected components of the subgraph of Γ formed by the vertices in
A′′ \ (A′1 ∪ B′1 ). We denote such a restriction of Γ to A′′ \ (A′1 ∪ B′1 ) as Γ |A′′ \(A′1 ∪B′1 ) .
Now, note that every vertex of Γ in A′′ \ (A′1 ∪ B′1 ) has degree 3, and thus, the smallest possible
connected component must be a complete graph on four vertices. Hence, the number of connected
|A′′ \(A′1 ∪B′1 )|
.
components in Γ |A′′ \(A′1 ∪B′1 ) is at most
4
Thus, we have
|A′′ \ (A′1 ∪ B′1 )|
dim D ′ ≤
4
4m − 3i′ − |A′1 ∪ B′1 |
≤
4
3i′ 1
− |C1′ | + |A′1 \C1′ |
≤ m−
4
4
3i′ 1 (1 − f )x2
f x2
≤ m−
−
+
,
4
4
3
1+g
(16)
(17)
where (17) follows from (12) and (13), and the result follows from (17).
Lemma 5. Denote by D |A∪A′1 ∪B′1 the projection of D on A ∪ A′1 ∪ B′1 . Then, we have
gf
4
1 1− f
′
+
x2 − x3 .
dim D |A∪A′1 ∪B′1 ≤ 2m + 2i −
3
3
1+g
9
(18)
Kadhe and Calderbank
15
(a) Component of size 2
(b) Component of size 3
Figure 3: Smallest possible connected components in Γ′ corresponding to a minimal linear dependency.
Proof. First, note that dim D |A∪A′1 ∪B′1 = dim D |A∪C1′ . Because, if the dimensions were different,
then there would be a codeword in D |A∪A′1 ∪B′1 that vanishes on A∪C1′ , i.e., it is supported on the connected
components that touch A′1 . This codeword must then be the zero codeword.
Now, for every point in C1′ , arbitrarily choose two of the three type F subsets that cover the point,
and add the subsets to obtain a parity check supported only on A. Label such 4-subsets as F1′ , . . . , F|C′ ′ | .
1
Further, note that any vertex in A′1 \C1′ with degree 1 is covered by two singletons. For each degree
1 vertex in A′1 \C1′ , add the two type F subsets containing the two singletons to produce a parity check
supported only on A. Label such 4-subsets as F|C′ ′ |+1 , . . . , F|C′ ′ |+g|A′ \C′ | .
1
1
1
1
Define a graph Γ′ with x3 type E 3-subsets as blue vertices and |C1′ | + g|A′1 \C1′ | type F ′ 4-subsets as
red vertices. Add l edges between a pair of vertices if the corresponding subsets meet in l points, where
1 ≤ l ≤ 4.
Note that we can view a red vertex as a super-vertex containing two disjoint green vertices, each
corresponding to the pair of points in the type F subset used to obtain a type F ′ subset representing the
red vertex. Further, note that the degree of a green vertex is at most 2, and thus, the degree of a red vertex
is at most 4. On the other hand, the degree of a blue vertex is at most 3.
For any (minimal) linear dependency ∑i Ei + ∑ j Fj′ = 0, the blue vertices corresponding to Ei ’s and
the red vertices corresponding to Fj′ ’s form a connected component in Γ′ such that every blue vertex has
degree 3 and every red vertex has degree 4. Note that the smallest possible size of such a connected component containing all red vertices is 2, while the smallest possible size of such a connected component
containing a blue vertex is 3. Fig. 3 depicts the smallest connected components.
Now, partition Γ′ into connected components, and eliminate one vertex from each connected component in
vertex has degree 3 and every red vertex has degree 4. This yields at least
which every blue
1− f
gf
2
1
3 x3 + 2 1+g x2 + 3 x2 vertices such that the corresponding vectors are linearly independent.
gf
Next, form a matrix M with any 32 x3 + 21 1+g
x2 + 1−3 f x2 linearly independent type E and type F ′
vectors, and reduce the matrix to row echelon form. Whenever there are three diagonal non-zero entries
in M that are are indexed by the same 3-subset S j , delete one of the three rows. Append the resulting
matrix with the vectors S1 , . . . , Sm+i′ . There cannot be any linear dependency in this matrix. Thus, we
Binary LRCs with Availability
16
have
gf
1
1− f
2 2
x3 +
x2 +
x2
.
dim hE1 , . . . , Ex3 , F1′ , . . . , F|C′ ′ |+g|A′ \C′ | , S1 , . . . , Sm+i′ i ≥ m + i′ +
1
1
1
3 3
2 1+g
3
(19)
′
Arbitrarily choose one of the three type F subsets for every point in C1 . Label them as F1 , . . . , F|C1′ | .
None of them can be in the span of type S, type E and type F ′ subsets. Thus, we have
dim hE1 , . . . , Ex3 , F1′ , . . . , F|C′ ′ |+|A′ \C′ | , S1 , . . . , Sm+i′ , F1 , . . . , F|C1′ | i
1
1 1
gf
1
1− f
2 2
x3 +
x2 +
x2
.
(20)
≥ |C1′ | + m + i′ +
3 3
2 1+g
3
This allows us to write
2 2
gf
1
1− f
′
′
′
dim D |A∪C1′ ≤ |A ∪C1 | − |C1 | + m + i +
x3 +
x2 +
x2
,
3 3
2 1+g
3
(21)
from which the result follows noting that |A| = 3m + 3i′ .
Using Lemmas 4 and 5, we get the following corollary.
Corollary 3. We have dim (D) ≤ 3m, with equality if and only if i′ = 0.
Proof. From (14), (15) and (18), we get
5
1 1− f
f
gf
1 1− f
4
dim (D) ≤ 3m + i′ −
+
+
x2 −
x2 − x3 .
4
4
3
1+g
3
3
1+g
9
We want to show that
1 gf
f
1− f
4
5 ′ 1 1− f
i ≤
+
+
x2 +
x2 + x3 .
4
4
3
1+g
3 1+g
9
9
(22)
(23)
It is easy to check that the right hand side (RHS) above is an increasing function of f . We minimize the
RHS by setting f = 0, and, for contradiction, assume that
1
1
5′
4
i >
+
(24)
x2 + x3 .
4
12 9
9
Now, since the number of points in type E, type F and type T subsets is 6m + 6i′ , we have
x1 + 2x2 + 3x3 = 6m + 6i′ .
(25)
Further, as the total number of covering 3-subsets is 7m, we have
x1 + x2 + x3 = 6m − i′ .
(26)
By subtracting (26) from (25), we get
x2 + 2x3 = 7i′ ,
which gives
x2 + 2x3
.
7
5 x2 + 2x3
4
21
x2 + x3 ,
>
4
7
108
9
i′ =
From (24) and (27), we get
(27)
which is a contradiction.
Hence, (23) holds, and thus from (22), we have dim (D) ≤ 3m. Further, the equality can happen if
and only if i = x2 = x3 = 0.
Kadhe and Calderbank
6.1.3
17
Step 3
Lemma 6. If dim (C ) = 4m, then C must be (equivalent to) a direct sum of the m copies of the [7, 4]
Hamming code.
Proof. First note that from Corollary 3, it follows that if dim (C ) = 4m, then the size of a maximum collection of pairwise disjoint covering 3-subsets in C is exactly m. Next, we prove the result by induction
on m.
Basis Step: m = 1. Since no two 3-subsets can be disjoint, every pair of 3-subsets must intersect.
Thus, the 7 3-subsets correspond to the Fano plane. The result follows since the row space of any
incidence matrix of the Fano plane is isomorphic to the [7, 4] Hamming code [46].
Induction Step: m ≥ 2. Consider a maximum collection of pairwise disjoint 3-subsets of size m as
{S1 , · · · , Sm }. Let L be the subset of all 3-subsets that are disjoint from {S1 , · · · , Sm−1 }. Due to exact
covering, each 3-subset intersects six other 3-subsets, and thus, we have |L| ≥ 7. Since Sm ∈ T , and there
are 6 other 3-subsets that intersect Sm , we have |L| = 7. As there are no m + 1 pairwise disjoint 3-subsets,
the 3-subsets in L must intersect pairwise.
Now, pick any subset T ∈ L. The six 3-subsets that intersect T must be the six other 3-subsets in L.
Thus, any 3-subset in L must be disjoint from any 3-subset outside L. Further, the 3-subsets in L must
cover 7 points due to the availability of the points.
Let C1 denote the restriction of C on the points covered by the 3-subsets in L, and C2 denote the
restriction of C on the points covered by the 3-subsets outside L. Then, we have C = C1 ⊕ C2 . Also,
since the 3-subsets in L pairwise intersect, they correspond to the Fano plane and C1 must be equivalent to
the [7, 4] Hamming code. In addition, as dim (C ) = 4m, it must be that dim (C2 ) is a [7(m − 1), 4(m − 1)]
code. Thus, the result follows by induction.
7
Rate Upper Bounds Using Coset Leaders
7.1 Rate Bound for Codes with (2,t)-Availability
First, we present a bound on the rate of a binary code having (2,t)-availability with exact covering. Our
main idea is to bound the maximum weight of a coset leader of its dual code by using the covering
properties imposed by availability constraints. We note that the maximum weight of a coset leader of a
linear code represents its covering radius [31].
Theorem 3. Let C be a length-n code spanned by nt3 3-subsets that cover every point with (2,t)availability. Then, we have
1
⊥
,
(28)
≤ H2
Rate C
t +1
where H2 (·) is the binary entropy function.
Proof. We refer to the nt/3 covering 3-subsets as triples. Let v be a coset leader of a coset of C such
that wt (v) = w. By the minimality of w, every triple should meet v in at most one point.
Let Γ be a graph formed on the complement of v by the wt triples that meet v in one point, defined as
follows. Vertices of Γ are the n − w points in the complement of v, and a pair of vertices are connected by
an edge if the corresponding points belong to a triple. Note that the number of edges in Γ is wt, whereas
the number of vertices in Γ is n − w.
18
Binary LRCs with Availability
Our main goal is to show that w ≤ n/(t + 1). Towards this end, we note the following properties of
Γ. First, Γ does not contain any cycle of odd length. This is because if Γ contains a cycle of odd length,
then the sum of corresponding triples is a codeword of odd weight supported within v. This contradicts
the assumption that v is a coset leader.
Second, the number of edges in Γ is at most the number of vertices in it. If the maximum degree in
Γ is two, then the result follows. Otherwise, let v0 be a vertex in Γ of degree greater than two. Then, in
the following, we show that any neighbor of v0 cannot have degree greater than one.
Let P, Q, and R be any three triples intersecting in the point corresponding to v0 . Denote the points
in P (respectively, Q and R) as Pi (respectively, Qi and Ri ) for i = {1, 2, 3}. Let P1 , Q1 , and R1 be the
points that meet v. Let P2 , Q2 , and R2 correspond to the vertex v0 . Note that P3 , Q3 , and R3 correspond
to the neighbors of v0 .
Suppose, for contradiction, that P3 corresponds to a vertex of degree two or more. Let S be a triple
meeting v that intersects P in P3 . Note that S cannot contain P3 or Q3 , as this would result in a triangle
(which is an odd cycle) in Γ. Let v′ = P + Z + S, where Z is chosen to be either Q or R such that it is
disjoint from S. Then, we have wt (v + v′ ) < wt (v), which contradicts that v is a coset leader. Thus, the
vertex corresponding to P3 cannot have degree greater than one. This proves that every neighbor of a
vertex of Γ of degree greater than two must have degree one. In other words, Γ consists of (even length)
cycles, paths, and stars. Hence, the number of edges in Γ is at most the number of vertices. This yields
that w ≤ n/(t + 1).
Now, we use the bound on w to limit the number of cosets of C , which allows us to lower bound the
dimension of C as follows. Let wmax denote the maximum weight of a coset leader of C . Then, we can
write
2n
,
dim C = log2
Number of cosets of C
!
2n
≥ log2
wmax n ,
∑i=0
i
2n
≥ log2
,
nH2 ( wmax
n )
2
w
max
,
= n 1 − H2
n
1
≥ n 1 − H2
,
(29)
t +1
nH2 ( wmax
max n
n ) ; and the last
where the second inequality follows from the well-known result that ∑wi=0
i ≤2
inequality holds because wmax ≤ n/(t + 1) and H2 (x) is increasing in x for 0 ≤ x ≤ 1/2.
The bound in (28) follows from (29).
Tight Rate Bound for Length-n Codes with 2, n−1
2 -Availability and Optimality of Simplex
Codes: Using the idea of bounding
obtain a tight upper bound
the weight of a coset leader, we can easily
m − 1 for a positive integer m,
-availability.
As
we
will
see,
when
n
=
2
on the rate of codes with 2, n−1
2
this bound is achieved by the (2m − 1, m, 2m−1 ) Simplex code.
3-subsets that cover every point with 2, n−1
Theorem 4. Let C be a length-n code spanned by n(n−1)
6
2 availability. Then, we have
log (n + 1)
2
.
(30)
Rate C ⊥ ≤
n
Kadhe and Calderbank
19
Proof. The proof follows from the observation that the weight of a coset leader of C should be at most
one. This is because every triple must intersect a coset leader in at most one point due to the minimality
of its weight.
Remark 6. It has been observed that the (2m − 1, m, 2m−1 ) Simplex code has (2, 2m−1 )-availability, see,
e.g. [35, 33]. The (2m − 1, m, 2m−1 ) Simplex code achieves the bound in (30). We note that the rate
optimality of Simplex codes amongst binary codes with locality r = 2 has been shown in [18] using their
field size dependent bound. The idea of bounding the weight of a coset leader gives a very simple proof
for this result.
7.2 Bound for Codes with (r, 3)-Availability
The bound in Theorem 3 enables us to obtain, as a corollary, a rate upper bound for binary codes having
(r, 3)-availability with exact covering. The main idea is a simple yet powerful observation from [29],
stated in the following remark.
Remark 7. Let H be a parity-check matrix of an (n, k) code having
(r,t)-availability with exact covering.
nt
T
Then, its transpose H is a parity-check matrix for an r+1 , k code having (t − 1, r + 1)-availability with
exact covering.
Corollary 4. Let C be a length-n code spanned by
availability. Then, we have
3n
r+1
(r + 1)-subsets that cover every point with (r, 3)-
r−2
1
3
Rate C ⊥ ≤
+
H2
,
r+1 r+1
r+2
(31)
where H2 (·) is the binary entropy function.
Proof. Using Remark 7 and (29), we get
3n
1
dim (C ) ≥
1 − H2
,
r+1
r+2
(32)
from which the result follows.
Tight Rate Bound for Codes with (r, 3)-Availability and Length
rate bound using Theorem 4 and Remark 7.
(r+1)(2r+3)
:
3
We get the following
Corollary 5. Let r be a positive integer such that 3 is a divisor of (r + 1)(2r + 3). Let C be a code with
length (r+1)(2r+3)
and (r, 3)-availability with exact covering. Then, we have
3
3
3 log2 (2r + 4)
+
.
Rate C ⊥ ≤ 1 −
r + 1 (r + 1)(2r + 3)
(33)
Remark 8. Consider a 2m − 1, m, 2m−1 Simplex code. Due to its (2, 2m−1 )-availability with exact
m
m−1
covering (see [35, 33]), it has a (2 −1)(23 −1) × (2m − 1) parity-check matrix H with column weight 3
and row weight 2m−1 − 1 such that any pair of rows intersecting in at most one point. The code with H T
as its parity-check matrix has (2m−1 − 2, 3)-availability, and it achieves the bound in (33).
Binary LRCs with Availability
20
1
TBF bound 1
TBF bound 2
BK bound 1
Our bound
0.9
0.8
0.7
rate
0.6
0.5
0.4
0.3
0.2
0.1
0
0
10
20
30
40
50
60
70
80
90
100
t
Figure 4: Rate upper bounds versus t for r = 2.
7.3 Comparison with the Existing Bounds
We compare our bounds with (2) from [23, 27], referred to as TBF bound 1. The authors of [23, 27] also
1
, referred to as TBF
show that the expression on the right hand side of (2) can be upper bounded by √r t+1
bound 2.
We also compare our bound in (28) with the following bound on the rate of a code C with (r,t)availability given in [29].
Rate (C ) ≤ 1 −
t
1
t
+
r + 1 r + 1 ∏r+1 1 +
j=1
1
j(t−1)
.
(34)
We refer to (34) as BK bound 1.
Our bound in (28) is plotted as a function of t in Fig. 4, along with TBF bound 1, TBF bound 2,
and BK bound 1 for r = 2. Observe that our bound gets sharper as t increases crossing TBF bound 1 at
t = 74. This advantage is clarified in Fig. 4, which zooms into the range t = 35 to 100 in Fig. 5.
Next, we compare our bound in (31) with TBF bound 1, TBF bound 2, BK bound 1, and the following
bound from [29] on the rate of a code C with (r, 3)-availability.
3(1 + L1 + L2 )
,
(35)
(r + 1)(3 + L1 + 2L2 )
k
l
m
j
m−3−L′1
1
3n
, and L1 = m − 3 − 2L2 . We refer to (35) as
, L′1 = (2r−1)m
−
1
,
L
=
−
where m = r+1
2
r+1
2
3(r+2)
BK bound 2.
Rate (C ) ≤ 1 −
Kadhe and Calderbank
21
TBF bound 1
TBF bound 2
Our bound
0.18
rate
0.16
0.14
0.12
0.1
0.08
40
50
60
70
80
90
100
t
Figure 5: Magnified version of the rate upper bounds for a range of t when r = 2.
We plot our bound in (31) as a function of r in Fig. 6, along with TBF bound 1, TBF bound 2, BK
bound 1 for t = 3, and BK bound 2 for n = r+3
3 . Our bound is loose for small values of r, but it gets
sharper as r increases, crossing BK bound 1 at r = 72. The gap with BK bound 1 is very small, on the
order of 10−4 , which we clarify in Fig. 6 by zooming into the range r = 40 to 90 in Fig. 7. Note that the
block-length n appears explicitly in the expression of BK bound 2 in (35). We observed the same trend
as shown in Fig. 6 for different values of n, which we do not include for the want of space.
8
Concluding Remarks
We studied availability properties of codes associated with convex polyhedra, focusing on the codes
associated with the Platonic solids. Further, we computed tight upper bounds on the rate of binary linear
codes with (r, 2) and (2, 3)-availability, and showed the uniqueness of direct sum type constructions
for rate optimality. Our main idea is to view the problem of designing a rate-optimal code with (r,t)availability as a covering problem. Since direct sum constructions are known to give good codes for
conventional covering problems [31], we speculate that such a direct sum construction will be present in
rate-optimal codes for other values of r and t. Finally, we presented novel upper bounds on the rates of
binary linear codes with (2,t) and (r, 3)-availability.
Binary LRCs with Availability
22
1
0.9
0.8
rate
0.7
0.6
0.5
0.4
TBF bound 1
TBF bound 2
BK bound 1
0.3
BK bound 2
Our bound
0.2
0
10
20
30
40
50
60
70
80
90
100
r
Figure 6: Rate upper bounds versus r for t = 3.
0.972
TBF bound 1
TBF bound 2
BK bound 1
0.97
BK bound 2
Our bound
rate
0.968
0.966
0.964
0.962
0.96
40
45
50
55
60
65
70
75
80
85
90
r
Figure 7: Magnified version of the rate upper bounds for a range of r when t = 3.
Kadhe and Calderbank
23
Acknowledgment
S. Kadhe would like to thank Anoosheh Heidarzadeh, Krishna Narayanan, and Alex Sprintson for helpful
discussions.
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| 7 |
On Variants of Network Flow Stability
Young-San Lin1 and Thanh Nguyen2
1
arXiv:1710.03091v1 [cs.GT] 9 Oct 2017
2
Computer Science Department, Purdue University, e-mail: lin532@purdue.com
Krannert School of Management, Purdue University, e-mail: nguye161@purdue.edu
Abstract. We present a variant of the stable flow problem. Instead of the traditional flow problem
that obeys Kirchhoff’s law, for each vertex, the outflow is monotone and piecewise linear to the inflow.
In a directed and capacitated network, each vertex has strict preference over their incident edges. A
stable flow assignment does not allow a group of vertices to benefit from privately rerouting along a
path. In this paper, we first show the existence of flow stability by reducing this variant of stable flow
problem to Scarf’s Lemma, then introduce a path augmenting algorithm that runs in polynomial time
to find such a stable flow.
1
Introduction
In the classic stable marriage problem, n men and n women with individual preferences order of the opposite
gender, are to be matched such that no man-woman pair exists who are inclined to abandon their original
partners and marry each other. Gale and Shapley [1] showed the existence of such a stable matching by
the deferred acceptance (DA) algorithm. Since then, the stable marriage problem and the DA algorithm
have become the cornerstones of market design and have changed the way many centralized markets are
organized. (See for example, [2,3].) Motivated by applications in resident matching, school choice, kidney
exchange and supply chain networks, numerous extensions of the problem have been studied. Among them
a trading network with bilateral contracts is perhaps the most general framework. This problem can be
modeled as a directed graph where vertices represent agents and edges represent contracts. The vertex of
the outgoing endpoint of an edge is the seller of that contract while the vertex of the incoming endpoint
is the buyer. Besides, there may be a source vertex as a representative of a producer who only sells and a
sink vertex as a consumer who only buys while other vertices are regarded as intermediate agents. A natural
notion of stability is a configuration of trade such that there is no blocking coalition. A blocking coalition is
a group of agents who will benefit more by cooperating among themselves.
To model agents’ preference, we assume that agents hold preference lists over individual contracts, and
allow capacities over these contracts. One can think of the capacity as the amount of goods traded in this
contract.3 One natural assumption is to require agents to obey Kirchhoff’s law. Namely, the sum of inflow
contracts is the same as the sum of outflow contracts. This standard stable flow problem has been wellstudied. A preflow-push variant of the Gale-Shapley algorithm can be done in pseudo-polynomial time [7]
while a path augmenting variant of the Gale-Shapley algorithm can be done in polynomial time [8]. Besides,
a stable flow instance can be reduced to the stable allocation problem [9] and both stable flow and stable
allocation inherit the lattice structure [10].
However, many applications do not fit into the traditional flow problem that obeys Kirchhoff’s law. For
example, in a supply chain network, one vertex can represent a manufacturing firm that takes raw materials
as input and produces certain part-products while another vertex might correspond to an assembly firm
whose inputs are the part-products and outputs are finished products. Clearly, the Kirchhoff’s law does not
hold for both manufacturing and assembly firms in this example.
3
Another way to model preference is to use choice functions, that is, an agent evaluates a set of contracts C by
specifying a subset of contracts C ′ ⊂ C that is accepted by the agent. [4,5,6] show that in this framework, a stable
solution exists if the choice functions possess some special characteristics. Choice functions are defined over discrete
sets, and it is not clear how it can be generalized to continuous capacities as the problem considered in our paper.
Motivated by this, in this paper we assume a more generalized case. Specifically, each agent can sign any
amount of outgoing contracts under certain threshold when there are no incoming contracts. When there
are incoming contracts, the amount of outgoing contracts is monotone and piecewise linear to the amount
of incoming contracts. If all intermediate agents apply this criteria, then this network is called a monotone
piecewise linear mapping network (MPLM-network).
Finding a stable solution in this general problem is more challenging. We first show the existence of flow
stability by reducing this variant of stable flow problem to Scarf’s Lemma, then introduce a path augmenting
algorithm that runs in polynomial time to obtain such a stable solution.
In section 2, we describe the setting and background definition of this problem including how flow and
stability are defined when agents utilize some special mapping or functions to their inflow and outflow. Besides, we introduce the concept of monotone piecewise linear mapping (MPLM), convex monotone piecewise
linear mapping (CMPLM), and linear mapping (LM).
In section 3, we show the existence of CMPLM-stable-flow by a reduction to Scarf’s Lemma [11,12]. Later
on, as LM is a subset of CMPLM, by reducing MPLM-stable-flow problem to LM-stable-flow problem, we
can show the existence of MPLM-stable-flow.
In section 4, we present a polynomial time path augmenting algorithm that finds an LM-stable-flow for
an acyclic LM-network. The main difference of our approach from [8] is an augmented path may be a σ-cycle,
a path from a source to a vertex that is visited twice. Each iteration in [8] augments a path from source
to sink or a cycle since when augmenting a cycle, the augmented flow from the source to cycle is always
zero. However, this does not apply in our setting because flow conservation property is not guaranteed. An
MPLM-network instance can be reduced to an LM-network instance so an MPLM-stable-flow can be found
in an acyclic MPLM-network. At the end of this section, we show a reduction from cyclic LM-network to an
equivalent acyclic LM-network to enclose the entire problem.
2
2.1
Preliminaries
F -agent, F -network, F -flow, and F -stable-flow
A network is a quadruple (G, s, t, c), where G = (V, E) is a digraph. We abuse a bit of notation, V does not
include s and t but E includes edges with s or t as endpoints. s and t are the source and sink vertices and
c : E → R+ determines the capacity c(e) where e ∈ E. The preference ≻v of a vertex v ∈ V is defined over
edges. e1 ≻v e2 means v prefers e1 to e2 . Note that incoming edges and outgoing edges are ranked strictly
and separately by v. First, we make assumptions over agents in the network:
Definition 1. Let F be a set of functions or mappings that maps from R>0 to R>0 (or from R>0 to a
subset of R>0 ). For v ∈ V , v possesses its own function or mapping gv . If for v ∈ V , gv ∈ F , then v is an
F -agent. If all agents in V are F -agents, then (G, s, t, c) is an F -network.
In a market, each vertex can be regarded as an agent given offers of incoming and outgoing contracts.
They evaluate the quantity of desired outgoing contracts to be signed based on how many incoming contracts
are accepted. Therefore, the feasibility of contract assignment can be defined as the following:
Definition 2. An F -flow of an F -network is a function f : E → R>0 such that:
1. 0 6 f (e) 6 c(e).P
P
2. Let fin (v) =
vu∈E f (vu), then gv (fin (v)) = fout (v) (or fout (v) ∈
uv∈E f (uv) and fout (v) =
gv (fin (v)) if gv (fin (v)) maps to multiple values).
Since agents have preferences over contracts, it is natural to define a scenario when such network flow
in a market is stable or not. The flow is not stable when there exist selfish agents who are willing to work
together without regarding other agents’ benefit. Although each vertex in V has their own preference list, s
and t do not rank their edges, because their preferences are irrelevant with respect to the following definition.
2
Definition 3. An F -stable-flow is an F -flow without a blocking path in an F -network. Given flow f , f
has a blocking path P = (v1 , v2 , ..., vk−1 , vk ) if all the followings hold:
1. There exists a vector VP = (r1 , r2 , ..., rk−1 ) > 0 such that:
(a) ri 6 c(vi vi+1 ) − f (vi vi+1 ) for i = 1, ..., k − 1.
(b) fout (vi ) + ri = gvi (fin (vi ) + ri−1 ) for i = 2, ..., k − 1 (or fout (vi ) + ri ∈ gvi (fin (vi ) + ri−1 ) if gv (fin (v))
maps to multiple values).
2. v1 = s or there is an e = v1 u such that f (e) > 0 and v1 v2 ≻v1 e.
3. vk = t or there is an e = uvk such that f (e) > 0 and vk−1 vk ≻vk e.
An F -flow has a blocking path if the first agent prefers to offer contracts to the second agent to other
agents she had already offered, while intermediate agents still have space for signing contracts, and the
last agent prefers to accept the contracts offered by the penultimate agent to other agents she had already
accepted. Note that when v1 = vk and all the conditions are satisfied, gv1 (fin (v1 ) + rk−1 ) = fout (v1 ) + r1
does not have to hold. The path P is allowed to have duplicate vertices. When there are duplicate vertices,
we just follow Definition 3 regardless of extra incoming or outgoing flow that comes from VP .
We consider the case that there are no parallel edges since one can always add one vertex between each
parallel edge and include the identity function into F . The vertices between parallel edges apply the identity
function. Therefore, whenever we state an edge uv, uv is unique.
Note that when F is just a set of an identity function, this is the standard stable flow problem. We study
the case when agents apply monotone piecewise linear mappings.
2.2
MPLM-network, CMPLM-network, and LM-Network
For a vertex v ∈ V , before defining a monotone piecewise linear mapping (MPLM) gv with kv segments, we
are given the following parameters:
1. av,1 , av,2 , ..., av,kv where av,i > 0.
2. bv > 0.
3. cv,0 , ..., cv,kv where 0 = cv,0 < cv,1 < ... < cv,kv = ∞.
av,i is the slope of segment i, bv is the pseudo starting
of segment i. Now we are able to define gv :
[0, bv ]
gv (x) = av,1 x + bv
av,i x + gv (cv,i−1 )
point of gv , and (cv,i−1 , cv,i ] decides the domain
if x = 0,
if x ∈ (0, cv,1 ],
if x ∈ (cv,i−1 , cv,i ]
(1)
Note that when x = 0, agent v is willing to sign any amount of outgoing contracts without exceeding the
threshold bv . This mapping also depicts that even a bit of incoming contract is an incentive to sign outgoing
contracts beyond certain baseline. If every agent in V applies MPLM, then (G, s, t, c) is an MPLM-network.
If av,i < av,j for any 1 6 i < j 6 kv , then gv is a convex (in R+ ) monotone piecewise linear mapping
(CMPLM). In this case, we can rewrite gv :
(
[0, bv ]
if x = 0,
gv (x) =
(2)
maxi=1,...,kv {gv,i (x)} otherwise
Pi−1
where gv,i = av,i x + bv,i and bv,i = −av,i cv,i−1 + j=1 av,j (cv,j − cv,j−1 ) + bv . Besides, bv,1 = bv and
bv,i − bv,i−1 = −av,i cv,i−1 + av,i−1 (cv,i−1 − cv,i−2 ) + av,i−1 cv,i−2
= −(av,i − av,i−1 )cv,i−1 < 0
(3)
when i > 1, so bv,j < bv,i for any 1 6 i < j 6 kv .
If every agent in V applies CMPLM, then (G, s, t, c) is a CMPLM-network. If kv = 1, then gv is a linear
function when x > 0, we say gv is a linear mapping (LM). Similarly, if for each v ∈ V , kv = 1, then (G, s, t, c)
is an LM-network.
3
3
3.1
Stability of MPLM-networks
Scarf ’s Lemma
Definition 4. Let A be an m × n nonnegative matrix with at least one positive entry in every column and
m
row, b ∈ R+
be a positive vector, and P = {x : x > 0, Ax 6 b}. For each row i of A, there is a strict ranking
≻i over the columns in {1 6 j 6 n : Aij > 0}. k ≻i j means row i prefers column k to column j.
We say x ∈ P dominates column j if there exists a row i such that:
1. Aij > 0 and the constraint i binds, i.e. (Ax)i = bi .
2. k ≻i j for any other k 6= j such that Aik > 0 and xk > 0.
To simplify our notation, we also say row i dominates column j if the above mentioned conditions hold.
Lemma 1 (Scarf ’s Lemma). [12] For any above mentioned A, b, and ≻i , there exists an x∗ ∈ P that
dominates all columns of A.
3.2
The Reductions
Scarf’s Lemma originally appeared as a tool to prove the non-emptiness of the core in an n person game
[11]. Nowadays, it has been universally used for problems about stability.
In our case, before showing the existence of stable-MPLM-flow in an MPLM-network, we first consider
CMPLM-network. MPLM is a more general mapping than CMPLM, so we can reduce MPLM-stable-flow
problem to LM-stable-flow problem, a special case of CMPLM-stable-flow. Although LM is a special case
of CMPLM and may not be needed in the later part of our proofs, we still prove the existence of stableCMPLM-flow in CMPLM-networks by a reduction to Scarf’s Lemma.
Theorem 1. There exists a CMPLM-stable-flow in a CMPLM-network.
Proof. Suppose we are given (G, s, t, c), G = (V, E), ≻v , and gv ∈ CM P LM for any v ∈ V , we would like to
reduce this problem to Scarf’s. First construct matrix A as the following:
1.
2.
3.
4.
For each e ∈ E, create column xe .
For each v ∈ V , create column x1v and x2v .
For each e ∈ E, create row e and set Aexe = 1.
For each gv,i where i = 1, ..., kv , create row v i . For every e = uv ∈ E, set Avi xe = av,i and Avi x1v =
Avi x2v = 1.
5. For each v ∈ V , create row v ′ . For every e = vu ∈ E, set Av′ xe = 1. Also set Av′ x1v = Av′ x2v = 1.
6. Row v i prefers column x1v the most and column x2v the least. Preference of e = uv ∈ E remains the same
as ≻v .
7. Row v ′ prefers column x2v the most and column x1v the least. Preference of e = vu ∈ E remains the same
as ≻v .
P
P
Let q(v) = max( uv∈E c(uv), vu∈E c(vu), bv ) + 1, construct b as the following:
1. Set be = c(e).
2. Set bvi = q(v) − bv,i for i = 1, ..., kv .
3. Set bv′ = q(v).
Note that all the rows of A and b are labeled either e, v i , or v ′ . All the columns of A and rows of x are
labeled either xe , x1v , or x2v .
Let x∗ be a Scarf’s solution of A, b, ≻vi , and ≻v′ , we construct a CMPLM-stable-flow f of (G, s, t, c), gv ,
and ≻v as the following: f (e) = x∗ e . Here we abuse a bit of notation by labeling columns or rows by xe , x1v ,
or x2v and any superscript or subscript of x∗ stands for the value of that entry of x∗ . The rest of the proof
is done in Lemma 2.
⊓
⊔
4
Lemma 2. f is a CMPLM-stable-flow of (G, s, t, c), G = (V, E), ≻v , and gv if and only if x∗ is a Scarf ’s
solution of A, b, ≻vi , and ≻v′ .
Proof. Scarf’s → CMPLM-stable-flow:
Suppose x∗ is a solution of Scarf’s, then x∗ dominates every column of A. Assume there is a row v i that
dominates column x1v , then i = 1 and row v 1 must bind since q(v) − bv,1 is less than q(v) − bv,i for i > 1 by
equation (3). Row v 1 prefers other nonzero entries (column xe where e = uv ∈ E and column x2v ) to column
1
∗2
zero and x∗ 1v = q(v) − bv > 0.
x1v . Row v 1 prefers column
P xv the∗most, so other entries including x′ v must be P
In this case, fin (v) = e=uv∈E x e = 0 and from constraint row v , fout (v) = e=vu∈E x∗ e 6 bv .
If there is no row v i that dominates column x1v , then row v ′ must dominate column x1v so row v ′ binds.
If at the same time row v ′ dominates column x2v , then v ′ prefers other nonzero entries (column xe where
e = vu ∈ E and column x1v ) to column x2v . v ′ prefers x2v the most, so other entries including x1v must be zero
i
1
1
and x∗ 2v = q(v). However, every row v i prefers x1v to x2v and x∗2
v > 0, so row v cannot dominate xv . xv is not
∗
i
2
dominated by x , a contradiction.
column xv and binds. Since
P dominates
P Thus, there are some row∗ 2v that
∗
x
+
x∗ 1v + x∗ 2v . Besides, row v j
=
x
row v ′ also binds, q(v) = av,i P e=uv∈E x∗ e + bv,i + x∗ 1v +
e
e=vu∈E
P v
∗
∗
may not bind for j 6= i, so ai e=uv∈E x e + bv,i > av,j e=uv∈E x e + bv,j which means gv,i maximizes gv
and av,i fin (v) + bv,i = gv,i (fin (v)) = gv (fin (v)) = fout (v).
In both cases whether there is a row v i that dominates column x1v or not, f is a CMLPM-flow.
For stability, suppose f has a blocking path P = (v1 , v2 , ..., vk−1 , vk ) with vector VP = (r1 , ..., rk−1 ), from
the second condition in Definition 3, row v1′ cannot dominate column xe1 where e1 = v1 v2 . If v1 is s, there
are no constraints for these two vertices. For the other case v1 prefers e1 to some other nonzero edges. In
either case row v1′ does not dominate column xe1 since v1 prefers some other nonzero edges to e1 . Besides,
if f (e1 ) > 0, then r1 > 0 since otherwise VP = 0 (the path corresponds to a series of functions in R+ and
the upcoming ri ’s remain unchanged). That is, c(e1 ) > f (e1 ) whenever f (e1 ) > 0 so row e1 cannot dominate
xe1 . This means for some i1 , row v2i1 must dominate column xe1 . Since row v2i1 dominates column xe1 and
column x2v2 is the least preferred, we have x∗ 2v2 = 0. x∗ 1v2 > 0 because row v2i1 must bind and the maximum
sum of either incoming or outgoing flow cannot reach q(v2 ). x∗ 1v2 > 0 indicates that v2′ cannot dominate
xe2 because column x1v2 is row v2i1 ’s least preferred. Similarly, for some i2 , row v3i2 must dominate column
xe2 where e2 = v2 v3 . If we keep on repeating the similar argument, there must be some ik−1 such that row
vk1 must dominate xek−1 where ek−1 = vk−1 vk . However, from the third condition in Definition 3, vk must
prefer ek−1 to some other nonzero edges. This means row vk1 must prefer xek−1 to some other nonzero edges
so x∗ does not dominate column xek−1 , a contradiction.
CMPLM-stable-flow → Scarf’s:
Suppose we have a CMPLM-stable-flow f , first set x∗ e = f (e) for each e ∈ E. If x∗ e = c(e) then row e
dominates column xe . For an unsaturated edge e = uv, exactly one of the following cases holds:
1. u prefers e to some other nonzero edges or u = s.
2. v prefers e to some other nonzero edges or v = t.
3. None of the above.
The first and second cannot hold at the same time or uv is a blocking path.
In the first case, v must prefer all other nonzero edges to e so some row v i dominates column xe . This
leads to x∗ 2v = 0 because xe ≻vi x2v for any i = 1, ...,Pkv . Suppose gv (fin (v)) = gv,i (fin (v)) > gv,j (fin (v)) for
j 6= i, in order to let row v i bind, x∗ 1v = q(v) − av,i e=uv∈E x∗ e − bv,i .
′
Similarly in the second case, u must prefer all other nonzero edges to e so row uP
dominates column xe .
2
1
′
∗
∗1
This leads to x u = 0 because xe ≻u′ xu . In order to let row u bind, x u = q(u) − e=uv∈E x∗ e .
In the third case, column xe is already dominated. The next step is to move on and examine how column
x1v and column x2v can be dominated.
We can classify vertex v into 3 types:
1. v has an incoming unsaturated edge uv and u prefers uv to some other nonzero edges.
2. v has an outgoing unsaturated edge vw and w prefers vw to some other nonzero edges.
3. None of the above.
5
The first and the second case cannot happen at the same time by our construction, since otherwise u
prefers uv to some other nonzero edges and w prefers vw to some other nonzero edges, (u, v, w) forms a
blocking path, a contradiction. Besides, the first case implies x∗ 2v = 0 and the second case implies x∗ 1v = 0.
By our selection of q(v), x∗ 1v = x∗ 2v = 0 cannot happen because the sum of incoming or outgoing flow cannot
reach q(v).
In the third case, if fin (v) = 0 and fout (v) < bv , row v ′ cannot bind and row v 1 binds. In order to
∗2
∗1
∗2
∗1
= q(v) − bv . Otherwise, arbitrarily
dominate column x2v , set x∗ 1v P
P set x v and x v such that x v > 0, x v > 0,
∗2
∗1
and x v + x v = q(v) − av,i uv∈E f (uv) − bv,i = q(v) − vu∈E f (vu) where i is the one that maximizes
gv,i (fin (v)). In this case, both row v i and v ′ bind. Column x2v as the least preferred by row v i is dominated
while column x1v as the least preferred by row v ′ is also dominated.
⊓
⊔
The existence of CMPLM-stable-flow also implies the existence of LM-stable-flow. One can have an
analogous proof for LM-stable-flow by setting kv = 1 for each v ∈ V . In the following proof, we show that
MPLM-stable-flow can be reduced to LM-stable-flow and each MPLM-agent is equivalent to a subnetwork
that consists of LM-agents.
Theorem 2. There exists a MPLM-stable-flow in an MPLM-network.
Proof. We can reduce MPLM-stable-flow problem to LM-stable-flow problem to show the existence. First
construct (G′ , s′ , t′ , c′ ), G′ = (V ′ , E ′ ), ≻v′ , and gv′ ′ ∈ LM for each v ′ ∈ V ′ as the following:
1. For each v ∈ V , create vin , vout , and v1 , ..., vkv in V ′ .
2. For each uv ∈ E, create uout vin in E ′ and let c′ (uout vin ) = c(uv) (assume sout = s′ and tin = t′ ).
′
′
3. For each vi , create edges vin vi and vi vout in E
c′ (vi vout ) =
P. If i < kv , set c (vin vi ) = ′ cv,i − cv,i−1 and
′
′
′
av,i c (vin vi ). If i = kv , set c (vin vi ) = max{0, uv∈E c(uv) − cv,i−1 } and c (vi vout ) = av,i c (vin vi ).
4. The preference of vin over uout vin is the same as v over uv. Similarly, the preference of vout over vout uin
is the same as v over vu.
5. vin prefers vin vi with smaller i and vout prefers vi vout with smaller i.
6. gv′ in (x) = x. gv′ i (x) = av,i x. gv′ out (0) ∈ [0, bv ] and gv′ out (x) = x + bv if x ∈ R+ .
′
′
′
(vout )) = gv (fin
(vin )) = fout
(vout ) since the sum of
Let f ′ be the LM-stable-flow, it is clear that gv′ out (fin
outgoing flow from vin is the same as the sum of the incoming flow, both vin and vout prioritize flow that
passes through smaller vi ’s and vi output flow weighted by av,i , vi does not have any flow unless vin vi−1 and
′
(vout ) > 0.
vi−1 vout are saturated, and eventually, vout just merely output a flow by adding bv when fin
′
Therefore, assigning f (uv) = f (uout vin ) makes f an MPLM-stable-flow.
⊓
⊔
Note that there are alternative ways to prove the existence of CMPLM-stable-flow and MPLM-stable-flow.
One can first prove the existence of CMPLM-stable-flow and reduce a MPLM-network to a CMPLM-network
via decomposing MPLM into CMPLM segments instead of LM segments in our proof. Another way is to
first prove the existence of LM-stable-flow then reduce both CMPLM-stable-flow and MPLM-stable-flow to
LM-stable-flow.
4
4.1
Path Augmenting Algorithms for MPLM-networks
Acyclic Networks
From the proof of Theorem 2, each MPLM-agent can be transformed as a subnetwork of several LM-agents.
As a consequence, it suffices to design algorithms for LM-stable-flow problem in order to solve the MPLMstable-flow problem. We present a path augmenting algorithm for LM-networks. This algorithm is a slight
modification of the one in [8] which considers the case when network does not have cycles.
6
The Algorithm for Acyclic LM-networks Suppose we are given (G, s, t, c), G = (V, E), ≻v , gv ∈ LM
for any v ∈ V , and G does not have any cycles. In the algorithm, each vertex v ∈ V ∪ {s} is associated with
a state and an edge. The states for vertices are {PROPOSE, REJECT, DONE}. If v is at the PROPOSE state, then
the associated edge is the outgoing edge that v currently prefers the most. v reaches the REJECT state when
v is running out of outgoing edges for proposing. The associated edge at this state is the incoming edge with
positive flow value that v currently prefers the least. v reaches the DONE state when its most preferred edge
is also rejected and there is no associated edge.
Initially, for each v ∈ V ∪ {s}, v.state = PROPOSE, v.edge is set to the most preferred outgoing edge of v.
For s, the order can be arbitrary. We can construct the auxiliary graph H = (VH , EH ) as the following:
1. VH = V .
2. EH = {v.edge : v ∈ V ∪ {s}, v.state = PROPOSE} ∪ {rev(v.edge) : v ∈ V ∪ {s}, v.state = REJECT} where
rev(uv) = vu.
For each uv ∈ EH , the residual capacity cf (uv) based on the current flow f is:
cf (uv) =
(
c(uv) − f (uv) if u.state = PROPOSE and u.edge = uv
f (vu)
if u.state = REJECT and u.edge = uv
Before showing how the algorithm works, we introduce a different path to augment from traditional path
augmenting algorithms for flows.
Definition 5. A σ-cycle is a path P = (s, v1 , v2 , ..., vk ) where all vertices are distinct except vk = s or
vk = vj for some 1 6 j < k.
The algorithm iteratively augments the flow f in H. In each round, one can always augment an s-t path
or a σ-cycle P such that ce (f ) = 0 for some e in P .
After the augmentation, update either the associated edge or the state for each outgoing vertex u of the
saturated edges uv. If u just finished proposing to v, then move on to the next preferred edge. If u is running
out of vertices to propose, then move to the REJECT state, set the associated edge to the least preferred edge
that is currently accepted, and update the vertices that are going to propose to u. If u is running out of
vertices to reject, then u has rejected all the edges, set u.state to DONE.
The algorithm stops when s.state = REJECT. Namely, there are no vertices for s to propose to.
Algorithm 1 : Path Augmenting
1: Initialize the state and associated edge for v ∈ V \ {t} and set f as zero flow.
2: while s.state = PROPOSE do
⊲ assume H and f as global variables
3:
let P be an s-t-path or a σ-cycle in H
4:
augment f by P such that the capacity of at least one edge in H drops to 0
5:
for uv in P do
6:
if cf (uv) = 0 then
7:
Update(u, P )
8: return f
7
1: procedure Update(u, P )
2:
if u ∈ P and was updated then
3:
return
4:
if u.state = PROPOSE then
5:
u.edge ← u’s next prefer edge uv with ”uv ≻v v.edge and v.state = REJECT” or ”v.state = PROPOSE”
6:
if no such v then
7:
u.state = REJECT
8:
if u = s then
9:
return
10:
if u.state = REJECT then
11:
u.edge ← u’s next least prefer edge vu with f (vu) > 0
12:
if no such v then
13:
u.state = DONE
14:
u.edge ← ∅
15:
return
16:
for each w ∈ V ∪ {s} where w.state = PROPOSE do
17:
if wu = w.edge and u.edge ≻u wu then Update(w, P )
Details of Path Augmentation Suppose after running some iterations, the current flow is f and the
auxiliary graph is H. Let P be an s-t-path or a σ-cycle (there must be one, see Lemma 3) and ∆ be a list
of edge values that we are about to augment along P .
If we find an s-t-path P = (s, v1 , ..., vk , t) to augment, let ∆ = (∆0 , ∆1 , ..., ∆k ). We start from sv1 and
set ∆0 = cf (sv1 ), v0 = s, and vk+1 = t, then traverse through P , there are three cases in each step:
1. Push flow: If vi−1 and vi are at PROPOSE state, set ∆i = min{gvi (fin (vi ) + ∆i−1 ) − fout (vi ), cf (vi vi+1 )}.
2. Redirect flow: If vi−1 and vi are at different state, set ∆i = min{∆i−1 , cf (vi vi+1 )}.
(fout (vi )−∆i−1 )−fin (vi ), cf (vi vi+1 )}.
3. Remove flow: If vi−1 and vi are at REJECT state, set ∆i = min{gv−1
i
Once t is reached, we fix the flow by traversing backwards along P :
(fout (vi ) + ∆i ) − fin (vi ).
1. If vi−1 and vi are at PROPOSE state, set ∆i−1 = gv−1
i
2. If vi−1 and vi are at different state, set ∆i−1 = ∆i .
3. If vi−1 and vi are at REJECT state, set ∆i−1 = gvi (fin (vi ) − ∆i ) − fout (vi ).
(fout (vi ) + ∆i ) = 0 and fin (vi ) = 0, ∆i−1 is going to be 0.
In the first case, when fout (vi ) + ∆i < bv , gv−1
i
Similarly, in the third case, when fin (vi ) − ∆i = 0, it must be the case that gvi (fin (vi ) − ∆i ) = fout (vi ), so
∆i−1 is going to be 0.
Note that each vertex cannot be in the DONE state (see Lemma 3). The usage of gv in the push flow case
is well-defined since in the forward stage of traversal along P , ∆0 is positive and at each step, ∆i−1 was
previously set to a positive value and by induction ∆i will also be positive. There exists a bottleneck edge
vj vj+1 that is saturated. The backward traversal keeps the same edge values as in the forward traversal stage
from t to vj and fixes the edge values from vj to s.
If we find a σ-cycle P = (s, v1 , ..., vj , ..., vk ), then let v0 = s, vk = vj for some 0 6 j < k, and set
∆ = (∆0 , ..., ∆k−1 ). There are two cases:
1. The saturated edge e = vi−1 vi and i > j, i.e. e belongs to a cycle.
2. The saturated edge e = vi−1 vi and i 6 j, i.e. e belongs to a path from s to vj .
In order to figure out which is the case, first assume that it is the first case. Regardless of the path from
s to vj , compute only the cycle part of ∆. Next check whether it is possible to augment along the path from
s to vj such that it matches the cycle part of ∆ we computed. If not, then it may be the second case.
To calculate the cycle part of ∆, one can start from vj and do exactly the same as in the P is an s-t-path
case until vk is reached. By traversing the cycle part of P forth and back, the cycle part of ∆ is computed.
However, this may not be a valid augmentation since av,1 ∆k−1 may not be the same as ∆j If one of the
following conditions is met then we are done:
8
1. If j = 0 then we are done since s has no incoming flow and is not an LM-agent.
2. If ∆j = av,1 ∆k−1 , we don’t have to fix anything from s to vj .
If fout (vj )+∆j 6 bv , then ∆k−1 = 0 and ∆j−1 = 0. This cannot happen since after augmenting a positive
value on vj−1 vj , the augmented value from vj to vk must be positive. Therefore, fout (vj ) + ∆j − ∆k−1 > bv
so the outgoing flow of vj so far at least meets the threshold and f (vj−1 vj ) > 0, we have to fix the s to vj
part of ∆. To do this, first apply a similar method as in the s-t path case and traverse forward from s to
(fout (vj ) + ∆j − av,1 ∆k−1 ) − fin (vj ). By
vj to get ∆j−1 . Let the required change of vj ’s outflow be Λ = gv−1
j
comparing ∆j−1 and Λ, there are two cases:
1. If ∆j−1 > Λ, then set ∆j−1 = Λ, traverse backwards from vj to s and apply the similar method as in
the s-t path case to fix the vj to s part of ∆. Note that ∆j−1 will be negative if ∆j − av,1 ∆k−1 < 0.
2. If ∆j−1 < Λ, we have to use the property of LM. If ∆j = 0, then by the property of gvj , ∆j−1 = 0 and
∆k−1 = 0. This cannot happen because we are not augmenting anything along the σ-cycle.
The remaining case is both ∆j−1 and ∆j are positive. vj applies a series of linear functions (in R+ ) to
reach vk . For a vertex vm where j + 1 6 m < k, it either applies gvm for the push flow case, identity
for the remove flow case. As a consequence, ∆j = α∆k−1 for
function for the redirect flow case, or gv−1
m
some α > 0. If α = av,1 , we are in the former case ∆j = av,1 ∆k−1 . If α < av,1 , then ∆j − av,1 ∆k−1 < 0
1
which was also covered earlier. As a result, α > av,1 . Λ = ( av,1
is linear and fin (vj ) =
− α1 )∆j since gv−1
j
αav,1
(f
(v
)).
∆
cannot
reach
the
required
change,
so
we
have
to
set
∆
=
gv−1
out j
j−1
j
j
α−av,1 ∆j−1 and traverse
the cycle part of P and fix the ∆ as the forward traversal in the s-t path case. Next, traverse backwards
from vj−1 to s and fix ∆ by the same method in the s-t path case. The following is an example:
Example 1. Suppose each node is using identity function except for d the outflow is half of the inflow.
The left graph is before augmentation and the right is after augmentation.
v2
s
9/10
0/4
v1
9/9
v2
0/4
v3
9/9
s
t
10/10
2/4
v1
8/9
1/4
v3
9/9
t
The σ-cycle is (s, v1 , v2 , v3 , v1 ). Before augmenting, v3 .state = REJECT, v3 .edge = v1 v3 , v2 .state =
PROPOSE, and v2 .edge = v2 v3 so v3 is the redirecting vertex and v3 v1 ∈ EH . v1 v3 ≻v2 v1 v2 so v1 proposed
to v3 earlier. For the cycle part, we would like to augment (4, 2, 2) along (v1 , v2 , v3 , v1 ). However, this
cannot be done because the net outgoing flow of v1 is 2 and s can only push 1 flow value to v1 . We know
that α = 2 and av1 ,1 = 1, so the augmenting value for v1 v2 must be twice of sv1 . We augment (1, 2, 1, 1)
along (s, v1 , v2 , v3 , v1 ) and obtain the right graph.
Analysis We start from proving the correctness of Algorithm 1. To ensure that Algorithm 1 can be executed
properly in each iteration, we start with the following lemma:
Lemma 3. At the beginning of any iteration in Algorithm 1 line 2, H always contains an s-t-path or a
σ-cycle.
Proof. Consider any v ∈ V , v has an outgoing edge if v is at PROPOSE or REJECT state. When v.state =
DONE, v rejected its last incoming edge. Therefore, it must be the removing flow case (see details of path
augmentation) otherwise v will still be in REJECT state. To remove the last flow, v was rejected by some
vertex w then v rejected some vertex u with f (uv) > 0 such that fin (v) drops to 0. Once f is updated after
a path augmentation that involved the removal of f (uv), there are no incoming edges of v in EH . Hence v
does not have incoming and outgoing edges if and only if v.state = DONE.
Before finding an augmenting path, s.state = PROPOSE so s has an outgoing edge in H. s always reaches
vertices that is in either the PROPOSE or REJECT state and each such vertex has an outgoing edge. One can
always traverse from s until a vertex is visited twice which forms a σ-cycle or until t is reached.
⊓
⊔
9
Theorem 3. Algorithm 1 computes an LM-stable-flow in polynomial time.
Proof. Suppose there is a blocking path P = (v1 , ..., vk ) in G. Edges in this blocking path must be unsaturated. From the second condition in Definition 3, either v1 = s or there is an edge v1 u such that v1 v2 ≻v1 v1 u
and f (v1 u) > 0.
If v1 6= s, v1 v2 is unsaturated and f (v1 u) > 0 either because v2 .state = REJECT and v2 .edge = v1 v2
or v1 .edge was updated once v2 .state changed to REJECT. In the former case, v1 v2 was once saturated but
f (v1 v2 ) decreased later on. In the later case, v1 did not get the chance to make v1 v2 saturated and v2 .edge
= v1 v2 . If v1 = s, the termination of Algorithm 1 implies that v2 .state = REJECT. In either case, v2 .state =
REJECT and v2 .edge = v1 v2 .
From v2 .state = REJECT and v2 v3 is unsaturated, v2 proposed to all available vertices and either v2 has
proposed to v3 earlier and f (v2 v3 ) decreased later on or v2 .edge was updated once v3 .state changed to REJECT
and v2 did not get the chance to make v2 v3 saturated. In either case v3 .state = REJECT and v2 .edge = v2 v3 .
By continuing analogous arguments, vk .state = REJECT and vk .edge = vk−1 vk . From the third condition
of Definition 3, either vk = t or there is an edge wvk such that vk−1 vk ≻vk wvk and f (wvk ) > 0. For the
former case, t never rejects vertices or forces vertices to update so this cannot happen. For the later case, vk
rejected w earlier and f (wvk ) = 0 since vk .edge = vk−1 vk and vk−1 vk ≻vk wvk , a contradiction.
For time complexity, in each iteration, the residual capacity of at least one edge drops to 0. There are at
most 2|E| iterations since each edge can first be fully saturated then be fully removed. It takes O(|V |) time
to find an s-t path or a σ-cycle in each iteration and deciding the edge augmenting values of the path takes
O(|V |). Therefore the running time of Algorithm 1 is O(|V ||E|).
⊓
⊔
Theorem 4. An MPLM-stable-flow of an acyclic MPLM-network can be computed in polynomial time where
the polynomial involves the number of segments of each vertex.
Proof. First apply the reduction from MPLM-stable-flow to LM-stable-flow in Theorem 2, then run Algorithm 1 on the LM-network.P
For each v ∈ V in the old graph, kv + 2 vertices and 2kv extra edges are created
in the new graph. Let K = v∈V kv , a rough bound for time complexity will be O((|V | + K)(|E| + K)).
The new graph has a special structure. Although the number of iterations is still bounded by |E| + K, the
length of path in each iteration is bounded by 3|V |. The time complexity is therefore O(|V ||E| + K|V |). ⊓
⊔
4.2
A Reduction from LM-Scarf ’s to Acyclic LM-Network
Previously, we designed Algorithm 1 to find MPLM-stable-flow in an acyclic network. With cycles, we cannot
apply Algorithm 1 in certain scenarios. Sometimes there is no proper augmenting path that starts from s as
the following example:
Example 2. In this graph, the outflow is twice of inflow for v1 and v2 . For the preference, v2 v1 ≻v1 sv1 and
v1 v2 ≻v1 v1 t. Algorithm 1 does not work properly for this graph. The first σ-cycle found is (s, v1 , v2 , v1 ).
We would like to augment (∆0 , ∆1 , ∆2 ) along this σ-cycle as the left graph shows. The following must be
satisfied: ∆1 = 2(∆0 + ∆2 ) and ∆2 = 2∆1 . This implies ∆0 = − 23 ∆1 and flows cannot be negative, so it is
impossible to augment this σ-cycle such that one edge is saturated. Besides, it is possible that there is no
outflow from s or there is no inflow to t in a stable flow. The right graph is a stable assignment.
∆0 /1
s
v1
∆1 /2
0/1
0/6
∆2 /4
t
s
v2
v1
2/2
6/6
4/4
t
v2
From Theorem 2, we know that an MPLM-stable-flow always exists. We can first reduce MPLM-stableflow to LM-stable-flow then reduce LM-stable-flow to Scarf’s no matter whether there are cycles or not. If
a Scarf’s instance corresponds to an LM-stable-flow instance, it is an LM-Scarf ’s instance. It turns out that
LM-Scarf’s can be reduced to LM-stable-flow where the LM-network does not contain any cycles.
10
Theorem 5. Any LM-Scarf ’s instance can be reduced to a LM-network that is solvable by Algorithm 1.
Proof. We will use the same notation as in the proof of Theorem 1 except kv = 1 for v ∈ V since we are
only given a network with LM-agents. Construct an LM-network as the following:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
For each column x1v , create vertex vin .
For each column x2v , create vertex vout .
Create s and sout , set c(ssout ) = ∞, and set c(svout ) = bv′ .
Create t and tin , set c(tin t) = ∞, and set c(vin t) = bv1 .
For each e = uv, create a vertex me , set c(uout me ) = be and c(me vin ) = Av1 xe be .
For each v, create vertices m1v and m2v , set c(vout m1v ) = c(vout m2v ) = c(m1v vin ) = c(m2v vin ) = bv′ .
For each v, vout prefers vout m2v the most and vout m1v the least, the preference of vout me for some e = vu
is the same as in row v ′ .
For each v, vin prefers m1v vin the most and m2v vin the least, the preference of me vin for some e = uv is
the same as in row v 1 .
The preference of sout and tin is arbitrary.
The outflow of sout , tin , vin ’s, vout ’s, m1v ’s and m2v ’s are the same as each of their inflow, they apply
identity functions.
The outflow of me where e = vw and w 6= t is Aw1 xe times of its inflow. If w = t then me applies identity
function.
By setting the capacity of an edge in the network, the edge is automatically included in the edge set.
Note that this network has three layers. Starting from s, the first layer has sout and vout ’s, the second layer
has m1v ’s, m2v ’s, and me ’s, and the third layer has tin and vin ’s that eventually merge to t.
Suppose we have an LM-stable-flow f of this network, we show that by setting x∗ e = f (uout me ) for
e = uv, x∗ 1v = f (m1v vin ), and x∗ 2v = f (vout m2v ), we have a solution for FM-Scarf’s. Conversely, the solution
of Scarf’s also corresponds to a LM-stable-flow. The rest of the proof is done in Lemma 4.
⊓
⊔
Lemma 4. f is a LM-stable-flow of the above mentioned network if and only if x∗ is a solution of LMScarf ’s.
Proof. Scarf’s → LM-stable-flow:
Suppose x∗ is a solution of Scarf’s, then x∗ dominates every column of A. We set f as the following:
1.
2.
3.
4.
5.
6.
For e = uv, set f (uout me ) = x∗ e , f (me vin ) = Av1 xe x∗ e .
For each v, set f (vout m1v ) = f (m1v vin ) = x∗ 1v and f (vout m2v ) = f (m2v vin ) = x∗ 2v .
P
P
Set f (svout ) =P e=vu Av′ xe x∗ e + x∗ 1v + x∗ 2v = e=vu x∗ e + x∗ 1v + x∗ 2v .
∗1
∗2
Set f (vin t) = Pe=uv Av1 xe x∗ e + xP
v + x v.
∗
Set f (ssout ) = e=sv Av′ xe x e = e=sv x∗ e .
P
Set f (tin t) = e=vt x∗ e .
We can see that for each vertex, the inflow and outflow satisfy the condition of an LM -flow. The remaining
is to show that f is stable. Assume there is a blocking path P in this network. Observing the structure of
this three-layer network, each vertex me , m1v , or m2v in the second layer only have one incoming and one
outgoing edge. This indicates they cannot be the starting or ending vertex of the blocking path. Besides,
P cannot start from s and end at a vertex in the first layer since each vertex in that layer has only one
incoming edge. Similarly, P cannot start from a vertex in the third layer and end at t as each vertex in the
third layer has only one outgoing edge. The remaining cases of a blocking path are:
1. P starts from s and ends at a vertex vin in the third layer:
This means there exists a vertex uout in the first layer such that row u′ does not bind since suout is
not saturated. For e = uv, me vin and uout me are not saturated and vin prefers me vin to some other
nonzero incoming edges indicate column xe is dominated by neither row e nor row v 1 . Column xe is not
dominated by x∗ , a contradiction.
11
2. P starts from a vertex uout in the first layer and ends at t:
This means there exists a vertex vin in the third layer such that row v 1 does not bind since vin t is
not saturated. For e = uv, me vin and uout me are not saturated and uout prefers uout me to some other
nonzero incoming edges indicate column xe is dominated by neither row e nor row u′ . Column xe is not
dominated by x∗ , a contradiction.
3. P starts from a vertex uout in the first layer and ends at a vertex vin in the third layer:
This means for e = uv, me vin and uout me are not saturated so column xe is not dominated by row e.
uout prefers uout me to some other nonzero incoming edges indicates column xe is not dominated by row
u′ . vin prefers me vin to some other nonzero incoming edges indicates column xe is not dominated by row
v 1 . Column xe is not dominated by x∗ , a contradiction.
4. P starts from s to t:
This means row v 1 and u′ do not bind and for e = uv, me vin and uout me are not saturated which
indicates xe is not dominated by row e. Column xe is not dominated by x∗ , a contradiction.
LM-stable-flow → Scarf’s:
Suppose f is a LM-stable-flow of the network. For e = uv, if me vin and uout me are saturated, then
x∗ e = c(uout me ) = be . Column xe is dominated by row e. Otherwise, the following two cases cannot happen
at the same time or there is a blocking path from uout to vin :
1. uout prefers uout me to some other nonzero incoming edges.
2. vin prefers me vin to some other nonzero incoming edges.
If the first case happens, vin must prefer all nonzero incoming edges to P
me vin and vin t must be saturated
or the path from uout to t is blocking. This forces f (m1v vin ) = c(vin t) − e′ =wvin f (e′ ) which corresponds
to row v 1 binds and dominates column xe .
If the second case happens, uout must prefer all nonzero outgoing edges to uout m
Pe and suout must be
saturated or the path from s to vin is blocking. This forces f (uout m2v ) = c(suout ) − e′ =uout w f (e′ ) which
corresponds to row u′ binds and dominates column xe .
The remaining is to show how f makes column x1v and x2v dominated. If svout is not saturated, then
either m1v vin and vout m1v are saturated or the flow of incoming edges of vin except m1v vin have to be zero.
Otherwise, s to vin forms a blocking path since vin prefers m1v vin the most. However, m1v vin and vout m1v
cannot be saturated since bv1 = c(vin t) 6 c(vout m1v ) = c(m1v vin ) = c(svout ) = bv′ and now f (vout m1v ) =
f (m1v vin ) 6 f (svout ) < c(svout ) so f (vout m1v ) < c(vout m1v ) and f (m1v vin ) < c(m1v vin ). This also forces vin t
saturated otherwise path s to t is blocking. Consequently, f (vout m1v ) = f (m1v vin ) = c(vin t) corresponds to
row v 1 binds and dominates column x1v and x2v .
If svout is saturated, then either m2v vin and vout m2v are saturated or vin t is saturated or vout to t forms
a blocking path since vout prefers vout m2v the most. If m2v vin and vout m2v are saturated, then vin t is also
saturated since f (vin t) > f (m1v vin ) = c(m1v vin ) and f (vin t) 6 bv1 = c(vin t) 6 c(m1v vin ) = c(svout ) = bv′
implies f (vin t) = c(vin t). In either case, vin t must be saturated. This corresponds to row v 1 binds and
dominates x2v and row v ′ binds and dominates x1v .
⊓
⊔
4.3
Cyclic Networks
The standard stable flow problem can be reduced to the stable allocation problem [9] according to [10].
Conversely, the stable allocation problem can also be reduced to the stable flow problem. For cyclic networks,
one can always reduce stable flow to stable allocation, then reduce the obtained stable allocation instance
to stable flow. It turns out that the new network is acyclic. The number of vertices is still O(|V |) and the
number of edges is still O(|E|). Applying the path augmentation algorithm in [8] to the new flow still takes
O(|V ||E|) time.
Inspired by this observation, we transform a cyclic LM-network to a new acyclic LM-network then apply
Algorithm 1. MPLM-stable-flow can be reduced to LM-stable-flow by Theorem 2 so one can always transform
a cyclic MPLM-network to an acyclic LM-network.
First we simplify the transformation from cyclic LM-networks to acyclic LM-networks.
12
Theorem 6. For an LM-network (G, s, t, c), G = (V, E), ≻v , and gv ∈ LM for any v ∈ V with cycles, there
is an equivalent LM-network (G′ , s′ , t′ , c′ ), G′ = (V ′ , E ′ ), ≻v′ , and gv′ ′ ∈ LM for any v ′ ∈ V ′ without cycles.
Proof. This is a P
direct result by
Pcombining Theorem 1 and Theorem 5. To wrap everything up, for each v ∈ V ,
let q(v) = max( uv∈E c(uv), vu∈E c(vu), bv ) + 1, we construct the new LM-network as the following:
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
Create s′out ∈ V ′ and t′in ∈ V ′ .
For each v ∈ V , create vin , vout , m1v , and m2v in V ′ .
For each vout ∈ V ′ , set c(s′ vout ) = q(v) (by setting capacity of e′ , e′ is automatically added to E ′ ).
For each vin ∈ V ′ , set c(vin t′ ) = q(v) − bv .
Set c(s′ s′out ) = c(t′in t′ ) = ∞.
For each e = uv ∈ E, set c(uout me ) = c(uv) and c(me vin ) = av,1 c(uv).
Set c(vout m1v ) = c(vout m2v ) = c(m1v vin ) = c(m2v vin ) = q(v).
For each vout ∈ V ′ , vout prefers vout m2v the most and vout m1v the least, the preference of vout me for some
e = vu is the same as in row v ′ .
For each vin ∈ V ′ , vin prefers m1v vin the most and m2v vin the least, the preference of me vin for some
e = uv is the same as in row v 1 .
The preference of sout and tin is arbitrary.
′
′
Set gs′ out (x) = x, gt′ in (x) = x. For each v ∈ V , set gv′ in (x) = x, gv′ out (x) = x, gm
1 (x) = x, and gm1 (x) = x.
v
v
′
′
′
′
For each e = uv ∈ E , set gme (x) = av,i x if v 6= t . If v = t , set gme (x) = x.
Suppose f ′ is the LM-stable-flow of the new LM-network, to obtain the LM-stable-flow f for the old
LM-network with cycles, for each e = uv ∈ E, set f (uv) = f ′ (uout me ).
⊓
⊔
Theorem 7. An MPLM-stable-flow of a cyclic network can be computed in polynomial time where the
polynomial involves the number of segments of each vertex.
Proof. We follow a series of reductions:
1. Reduce MPLM-network to LM-network by Theorem 2.
2. Reduce cyclic LM-network to the equivalent acyclic LM-network by Theorem 6.
3. Run Algorithm 1 on the new LM-network.
Suppose the starting MPLM-network has |V | vertices and |E| edges. In the first step, for each v ∈ V in
the MPLM-network,
kv + 2 vertices and 2kv extra edges are created in the LM-network.
P
Let K = v∈V kv , in the second step, the new acyclic LM-network O(|E| + K) vertices and O(|E| + K)
edges.
In the last step, the rough bound for running time of Algorithm 1 is O((|E|+K)2 ). By utilizing the special
structure of the network, the length of the augmenting path is O(|V |) so the running time is O(|V ||E|+K|V |)
which is asymptotically the same as the running time on acyclic MPLM-networks.
⊓
⊔
5
Conclusion
In this paper, we first state a variant of the stable flow problem where agents apply MPLM instead of
identity functions as in traditional flow problems. For a CMPLM-network, a subclass of MPLM-network, the
existence of stable flow can be proved by the reduction to Scarf’s Lemma. For an MPLM-network, each agent
can be transformed into an LM-subnetwork, and the entire network is equivalent to a large LM-network.
The existence of CMPLM-stable-flow also implies the existence of LM-stable-flow, hence there always exists
a stable flow for an MPLM-network.
An acyclic MPLM-network can be reduced to an acyclic LM-network and the stable flow of an acyclic
LM-network can be found by a path augmenting algorithm Algorithm 1. This algorithm augments either an
s-t-path or a σ-cycle in each iteration and runs in polynomial time. For MPLM-networks, the polynomial
involves the number of segments. However, Algorithm 1 is not applicable for cyclic MPLM-networks or
LM-networks. To circumvent this issue, we find an equivalent acyclic LM-network and solve the problem by
Algorithm 1.
13
A monotone continuous mapping (MCM) can output any value below a threshold when inflow is zero
and applies a monotone continuous function when inflow is positive. MPLM can approximate any MCM
by setting a bunch of infinitesimal segments. Therefore, it is natural to assume that there always exists an
MCM-stable-flow in an MCM-network. One can assume that for a vertex v and gv ∈ MCM, if we are given x,
gv (x) can be computed in constant time. However, similar approach as Algorithm 1 may not be applicable to
an acyclic MCM-network. This is because it may not be possible to augment a σ-cycle such that there is one
edge saturated in the auxiliary graph. For acyclic LM-networks, we can do so because of the linear relation
between the change of inflow and outflow. An engrossing open problem is: How to design an algorithm to
find an MCM-stable-flow in an MCM-network? Are there any other special set of functions or mappings F
that makes the augmentation along an σ-cycle computable in an F -network?
The standard stable flow problem can be solved in O(|E| log |V |) time [8] if one utilizes a more sophisticated data structure, the dynamic trees [13]. This technique was also used in [14]. Can we modify dynamic
trees and design faster algorithms for finding stable flow in LM-networks? Another open problem is the
regular flow problem for graphs only depicts contracts with two agent involved. How about contracts with
more agents involved? Contracts with multiple agents can be interpreted by introducing hypergraphs and
there are studies of flow problems for directed hypergraphs [15]. How will this problem be reduced to Scarf’s
Lemma? Is there always a stable assignment for agents?
Based on the properties of the standard stable flow problem such as the lattice structure [10], perhaps
another more accomplishable research direction for MPLM-networks and LM-networks is to examine the
structure of stable solutions. An MPLM-network can be described as a special form of LM-network. It will
be interesting to see how structures of MPLM-stable-flow differs from LM-stable-flow.
Acknowledgement. This research is partly supported by National Science Foundation Grants AST1443965, CMMI 1728165.
References
1. David Gale and Lloyd S Shapley. College admissions and the stability of marriage. The American Mathematical
Monthly, 69(1):9–15, 1962.
2. Alvin E. Roth and Elliott Peranson. The redesign of the matching market for american physicians: Some engineering aspects of economic design. American Economic Review, 89(4):748–780, 1999.
3. Péter Biró and Flip Klijn. Matching With Couples: A Multidisciplinary Survey. International Game Theory
Review, 15(02), 2013.
4. John William Hatfield, Scott Duke Kominers, Alexandru Nichifor, Michael Ostrovsky, and Alexander Westkamp.
Chain stability in trading networks. Technical report, Working paper, 2015.
5. Tamás Fleiner, Zsuzsanna Jankó, Akihisa Tamura, and Alexander Teytelboym. Trading networks with bilateral
contracts. 2016.
6. Michael Ostrovsky. Stability in supply chain networks. The American Economic Review, pages 897–923, 2008.
7. Ágnes Cseh, Jannik Matuschke, and Martin Skutella. Stable flows over time. Algorithms, 6(3):532–545, 2013.
8. Ágnes Cseh and Jannik Matuschke. New and simple algorithms for stable flow problems. arXiv preprint
arXiv:1309.3701, 2013.
9. Mourad Baı̈ou and Michel Balinski. The stable allocation (or ordinal transportation) problem. Mathematics of
Operations Research, 27(3):485–503, 2002.
10. Tamás Fleiner. On stable matchings and flows. In WG, pages 51–62. Springer, 2010.
11. Herbert E Scarf. The core of an n person game. Econometrica: Journal of the Econometric Society, pages 50–69,
1967.
12. Herbert E Scarf. An elementary proof of a theorem on the core of an n person game.
13. Daniel D Sleator and Robert Endre Tarjan. A data structure for dynamic trees. Journal of computer and system
sciences, 26(3):362–391, 1983.
14. Brian C Dean and Siddharth Munshi. Faster algorithms for stable allocation problems. Algorithmica, 58(1):59–81,
2010.
15. Giorgio Gallo, Giustino Longo, Stefano Pallottino, and Sang Nguyen. Directed hypergraphs and applications.
Discrete applied mathematics, 42(2-3):177–201, 1993.
14
A
An Example for Theorem 7
Let the following be an input instance of MPLM-stable-flow:
0/10
s
v1
0/10
if x = 0,
[0, 2]
gv1 (x) = 2x + 2 if x ∈ (0, 2]
x + 4 if x > 2
0/10
0/10
t
vertex v1
in
sv1 ≻v1 v2 v1
out
v1 v2 ≻v1 v1 t
if x = 0,
[0, 1]
gv2 (x) = x + 1 if x ∈ (0, 3]
2x − 2 if x > 3
v2
First reduce this MPLM-network to LM-network by Theorem 2. In this network, the subnetwork of LMagents a, b, c, and d is equivalent to MPLM-agent v1 , and the subnetwork of LM-agents e, f , g, and h is
equivalent to MPLM-agent v2 .
s
0/10
0/2
b
0/4
a
d
0/10
0/18 c 0/18
0/10
0/10
0/3
f
0/3
e
h
0/14 g
0/7
t
vertex a
d
e
h
in
ha ≻a sa bd ≻d cd
f h ≻h gh
out
ab ≻a ac ed ≻d dt ef ≻e eg
v
a
gv (x) x
b
2x
c
x
d
x+2
e
x
f
x
g
2x
h
x+1
Next, reduce this LM-network to an equivalent LM-network without cycle by Theorem 6 and solve it
by Algorithm 1. Note that gmab (x) = 2x and gmeg (x) = 2x, any me that applies identity function is not in
the graph. Because of the space limit, m1v , m2v , vout m1v , vout m2v , m1v vin , and m2v vin are not shown in the graph.
Preference list (from most preferred to least preferred):
vertex aout
bout
cout
dout
to
m2a , mab , cin , m1a m2b , din , m1b m2c , din , m1c m2d , ein , tin , m1d
vertex eout
fout
gout
hout
to
m2e , fin , meg , m1e m2f , hin , m1f m2g , hin , m1g m2h , ain , m1h
vertex ain
bin
cin
din
from m1a , hout , sout , m2a m1b , mab , m2b m1c , aout , m2c m1d , bout , cout , m2d
vertex ein
fin
gin
hin
from m1e , dout , m2e m1f , eout , m2f m1g , meg , m2g m1h , fout , gout , m2h
15
sout 2/10
2/2
aout
2/∞
tin
mab
ain
4/4
10/∞
10/18
21/21
21/21
bout 4/4
bin
5/5
5/5
19/19
s
23/23
cout 10/18
cin
10/10
dout
din
19/19
21/21
t
6/10
11/11
11/11
eout
3/7
4/4
15/15
3/3
ein
4/4
meg
6/14
fout 3/3
fin
18/18
15/15
17/17
gout
10/10
gin
6/14
hout
hin
Rest of the flow:
vertex aout bout cout dout eout fout gout hout
to
m1a m1b m1c m1d m1e m2f m1g m1h
value 9
1 9 7
5 1 9 8
vertex ain bin cin din ein fin gin hin
from m1a m1b m1c m1d m1e m2f m1g m1h
value 9 1 9 7 5 1 9 8
Note that by only changing c(fout m1f ) = c(m1f fin ) = 1 and c(fout m2f ) = c(m2f fin ) = 0 also returns a
stable solution but we list the one returned by Algorithm 1.
16
The corresponding instance of the old LM-network is:
s
2/10
2/2
b
4/4
a
d
10/10
10/18 c 10/18
10/10
6/10
3/3
f
3/3
e
h
6/14 g
3/7
t
vertex a
d
e
h
in
ha ≻a sa bd ≻d cd
f h ≻h gh
out
ab ≻a ac ed ≻d dt ef ≻e eg
v
a
gv (x) x
b
2x
c
x
The corresponding instance of the original MPLM-network is:
if x = 0,
[0, 2]
v1
gv1 (x) = 2x + 2 if x ∈ (0, 2]
2/10
10/10
x + 4 if x > 2
s
6/10
10/10
t
if x = 0,
[0, 1]
v2
gv2 (x) = x + 1 if x ∈ (0, 3]
2x − 2 if x > 3
17
d
x+2
e
x
f
x
g
2x
h
x+1
vertex v1
in
sv1 ≻v1 v2 v1
out
v1 v2 ≻v1 v1 t
| 8 |
Networked SIS Epidemics with Awareness
arXiv:1607.02502v2 [cs.SI] 12 Jul 2016
Keith Paarporn1 , Ceyhun Eksin1,2 , Joshua S. Weitz2,3 , Jeff S. Shamma1,4
Abstract—We study an SIS epidemic process over a static
contact network where the nodes have partial information about
the epidemic state. They react by limiting their interactions
with their neighbors when they believe the epidemic is currently
prevalent. A node’s awareness is weighted by the fraction of
infected neighbors in their social network, and a global broadcast
of the fraction of infected nodes in the entire network. The
dynamics of the benchmark (no awareness) and awareness
models are described by discrete-time Markov chains, from which
mean-field approximations (MFA) are derived. The states of the
MFA are interpreted as the nodes’ probabilities of being infected.
We show a sufficient condition for existence of a “metastable”, or
endemic, state of the awareness model coincides with that of the
benchmark model. Furthermore, we use a coupling technique
to give a full stochastic comparison analysis between the two
chains, which serves as a probabilistic analogue to the MFA
analysis. In particular, we show that adding awareness reduces
the expectation of any epidemic metric on the space of sample
paths, e.g. eradication time or total infections. We characterize
the reduction in expectations in terms of the coupling distribution. In simulations, we evaluate the effect social distancing
has on contact networks from different random graph families
(geometric, Erdős-Renyi, and scale-free random networks).
I. I NTRODUCTION
Mathematical models of epidemic spreading over networks
have been extensively studied. Models characterize how the
spatial features induced by network structure affects epidemic
spread ([1],[2],[3],[4],[5],[6],[7]). The simplest formulation for
such processes is the susceptible-infected-susceptible (SIS)
model, where an individual is either infected or susceptible
to infection. In such models, there is a threshold determining
whether the epidemic eradicates quickly or persists for a
long time. Specifically, δ/β > λmax (A) (β is the disease
transmission rate, δ the healing rate, and λmax (A) the largest
eigenvalue of the network adjacency matrix) is a sufficient
condition for the disease to eradicate exponentially fast. The
opposite strict inequality is a necessary condition for the
disease to persist for a long period of time. The steady state in
this regime is often referred to as the endemic or metastable
state. How to devise control strategies in this regime is both
an important research and policy question.
A commonly studied control strategy is budgeted vaccine
allocation, where the administration of vaccines among central nodes in the network optimally inhibits the epidemic
1 School of Electrical and Computer Engineering, Georgia Institute
of Technology, Atlanta, GA 30332 kpaarporn@gatech.edu,
ceyhuneksin@gatech.edu
2 School
3 School
of Biology, Georgia Institute of Technology, Atlanta, GA
of Physics, Georgia Institute of Technology, Atlanta, GA
jsweitz@gatech.edu
4 Computer, Electrical and Mathematical Sciences and Engineering, King
Abdullah University of Science and Technology (KAUST) Thuwal, Kingdom
of Saudi Arabia jeff.shamma@kaust.edu.sa
([8],[9],[10]). In these situations, a central authority selects individuals to vaccinate and hence is required to have knowledge
of the network structure. In game-theoretic settings, individuals decide for themselves whether or not to vaccinate based on
an assessment of risks and benefits ([11],[12],[13],[14],[15]).
However, these models do not account for social behavior
during the course of an epidemic, which can significantly slow
epidemic spread without the aid of vaccines.
With the widespread availability of social media and news
outlets on the internet and television, individuals may be wellinformed about the current state of ongoing epidemics and how
to take precautionary measures to avoid getting sick. In the
recent 2009 H1N1 Influenza pandemic, people responded to
public service announcements by increasing the frequency of
washing hands, staying at home when they or loved ones were
sick, or avoiding large public gatherings [16]. In the recent
Ebola outbreak in West Africa, a combination of quarantining
and sanitary burial methods were shown to significantly reduce
the rate of virus spread [17]. These precautions and social distancing actions effectively limit epidemic spread. Individuals’
distancing actions depend on the extent of how informed they
are. The dissemination and exchange of information influences
the public’s behavior, affecting the course of the epidemic
itself, in turn affecting the public’s behavior again [18]. This
feedback loop allows epidemic spreading to coevolve with
human social behavior, inducing complex dynamics.
Recent research effort has focused on understanding the
complexities that arise when incorporating human behavioral
elements into existing models of epidemic spreading. A review
of the recent literature can be found in [19]. Such models
present general challenges for characterizing decentralized and
dynamic protection measures and also capture a realistic aspect
of disease spread in society. When individuals take social
distancing actions based on the level of information they have,
they reduce contact with others and the epidemic prevalence
reduces significantly ([20],[21],[22]). They can become aware
of the epidemic by communicating with their social contacts
or by a global broadcast ([23],[24]). Other actions include
switching one’s contact links, giving rise to a coevolving network [25]. Endowing individuals with local prevalence-based
awareness highlights the role of network effects ([26],[27]).
In this work, we study a networked SIS process with dynamically distributed information and social distancing actions.
The information the agents receive comes from their social
contacts and a global broadcast about the current state of the
epidemic. An agent’s social distancing action reduces its contact network interactions, the magnitude of which depends on
how informed it is. We prove awareness reduces the endemic
level, but cannot improve the epidemic threshold for persistence. In addition, we provide a stochastic comparison analysis
between the awareness and benchmark (without awareness)
processes using a coupling technique. This establishes an
inequality between expectations of certain epidemic metrics
(e.g. eradication time, cumulative infected), as well as the
closed-form difference. We are also interested in studying
which combinations of network structure and awareness are
most effective. The results extend prior work by the same
authors in [28],[29], where the mean-field approximation and
coupling technique were initially studied.
The paper is organized as follows. In Section II, we describe
a networked SIS epidemic process in discrete time, which we
modify by incorporating a dynamic form of agent awareness
and social distancing. In Section III, we introduce mean-field
approximations on the probabilities of infection and prove the
epidemic threshold for persistence with distancing remains the
same as without. Section IV provides a stochastic comparison
analysis between the benchmark and awareness Markov chains
through a coupling technique. In Section V, we explore
through simulations which random graph families are effective
at restricting epidemic spread and prevalence when social
distancing is a factor. Section VI gives concluding remarks.
Proofs of some of the results are given in the Appendix.
Notation: Z+ = N ∪ {0} is the set of nonnegative integers.
The all zeros and all ones vector in Rn is written 0n and 1n ,
respectively. For x, y ∈ Rn we write x y if xi ≤ yi for all i,
and x ≺ y if xi < yi . To isolate a particular coordinate i, we
write x = (x−i , xi ) ∈ Rn , where x−i = {xj : j 6= i} ∈ Rn−1 .
P(E) is the power set of some set E. The complement of the
set E is written E c . In probabilistic settings, we write χE (·)
as the indicator on the event E, i.e. χE (x) = 1 if x ∈ E and
zero otherwise.
II. N ETWORKED SIS M ODELS
A. Benchmark SIS Model
We introduce a model of epidemic spread which we refer to
as the benchmark model (studied in [1], [30], and Section 5 of
[6]). Consider the set of nodes N = {1, . . . , n} interconnected
by a set of edges EC . Epidemic spread occurs in discrete time
steps t = 0, 1, . . . over the undirected graph GC = (N , E),
whose n × n adjacency matrix is defined for any i, j ∈ N ,
as [AC ]ij = 1 if (i, j) ∈ EC and 0 otherwise. The graph
GC is called the contact network. An agent i ∈ N is either
susceptible to the disease or infected by it. The epidemic states
are defined as Ω , {0, 1}n . For any s ∈ Ω and i ∈ N ,
either si = 0, meaning agent i is susceptible, or si = 1,
meaning it is infected. A susceptible node i can contract the
disease from neighboring agents in the contact network, NiC ,
{j ∈ N : (i, j) ∈ EC }. When agent i is susceptible in the
epidemic state is s ∈ Ω (si = 0), its probability of getting
infected in the next time step due to an interaction with its
neighbor j ∈ NiC is given by βsj where β ∈ (0, 1) is the
transmission probability of the disease. Hence, an individual
can only contract the disease from an infected neighbor. Agent
i interacts with each of its neighbors independently. Therefore,
i’s probability of not becoming infected in the next time step
is
pi00 (s) , 1 − pi01 (s).
(1)
1−
pi01
δpi00
pi00
si = 1
si = 0
δpi00
(a)
Awareness
Epidemic spread
s(t) → s(t + 1)
j
i
βaj (s)
βai (s)
µ1 (s)
..
.
µn (s)
Distancing actions
a1 (s)
..
.
an (s)
(b)
Fig. 1: (a) Node-level state transition diagram. (b) System-level
diagram
Consequently, the probability i becomes infected is
Y
pi01 (s) , 1 −
(1 − βsj )
(2)
j∈NiC
If i is infected in state s (si = 1), it becomes susceptible
in the next time step with probability δpi00 (s), where δ ∈
(0, 1) is the healing probability. Thus, for an infected node to
become susceptible, it must heal and not get re-infected by its
neighbors. Agent i’s transition probabilities are summarized in
Figure 1a, and described by Pi : Ω × {0, 1} → [0, 1] defined
as
(
Pi (s, 0) = pi00 (s)
If si = 0,
(3)
Pi (s, 1) = pi01 (s)
(
Pi (s, 0) = δpi00 (s)
If si = 1,
(4)
Pi (s, 1) = 1 − δpi00 (s)
For each i ∈ N and s ∈ Ω, the Pi define the benchmark SIS
Markov chain over Ω by the 2n × 2n transition matrix K with
elements
n
Y
K(s, s0 ) ,
Pi (s, s0i ), ∀s, s0 ∈ Ω
(5)
i=1
This chain has one absorbing state, the all-susceptible state
o , {0}n .
B. Awareness SIS Model
We modify the benchmark model to take into account the
agents’ awareness of the current epidemic state. The information agent i receives comes from two sources: the proportion
of infected neighbors in its local social network and a global
broadcast of the proportion of infected nodes in the entire
network. The social network is a graph GI = (N , EI ) with
the same nodes as GC but with different edges, representing
the nodes’ social communication links. The set of i’s neighbors
in GI is written NiI . The information is given by
µi (s) ,
n
1−α X
α X
s
+
sj , ∀s ∈ Ω
j
n j=1
|NiI |
I
(6)
j∈Ni
where α ∈ [0, 1] is a parameter that governs the trust nodes
place in information from their social contacts. Consequently,
node i reduces its interactions with its physical neighbors
through the social distancing action
ai (s) , 1 − µi (s),
(7)
which reduces its susceptible-to-infected probability (2) to
Y
(1 − βai (s)sj ).
(8)
pi01,d (s) , 1 −
j∈NiC
We similarly define pi00,d (s) , 1 − pi01,d (s). An infected
agent’s probability of recovering becomes δpi00,d (s). Note for
all s ∈ Ω, pi01,d (s) ≤ pi01 (s). Combined with the social
distancing behaviors ai , the local awareness spread dynamics
in (6) make the effect of user behavior on its infection
probability endogenous to the benchmark chain model through
a negative feedback loop. We define the Pdi analogously to (3)
and (4):
(
Pdi (s, 0) = pi00,d (s)
(9)
If si = 0,
Pdi (s, 1) = pi01,d (s)
(
Pdi (s, 0) = δpi00,d (s)
(10)
If si = 1,
Pdi (s, 1) = 1 − δpi00,d (s)
Thus, the Pdi define the distancing Markov chain over Ω by
the transition matrix Kd with elements
Kd (s, s0 ) ,
n
Y
i=1
Pdi (s, s0i ), ∀s, s0 ∈ Ω
(11)
whose unique absorbing state is also o, the all-susceptible
state. The feedback between social distancing actions and
epidemic states is illustrated in Figure 1b.
Remark 1. Our model of awareness captures the different
ways an agent may receive information about an ongoing
epidemic from the media. Large media corporations and public
health institutions such as the Centers for Disease Control and
Prevention (CDC) and the World Health Organization (WHO)
often report an estimated total number of people infected
nationwide or globally at a given time, and this information
is disseminated amongst the population. Information is also
exchanged through one’s personalized social links, which can
range beyond a person’s geographic location. Thus, a node’s
awareness is composed of a linear combination of both sources
of information, as given in (6).
III. M EAN - FIELD A PPROXIMATIONS
A. Derivation
We derive a mean-field approximation (MFA) of the Markovian dynamics described in the previous section. The MFA
is a deterministic, discrete-time dynamic system with an ndimensional state space [0, 1]n , which is interpreted to be each
node’s probability of being infected at any given time.
Here, we write st ∈ Ω as the epidemic state at time
t = 0, 1, . . .. Indeed, consider the node-level stochastic state
transition update
st+1
= sti B(1 − δpi00,d (st )) + (1 − sti )B(pi01,d (st ))
i
(12)
of the distancing chain where B(λ) denotes a Bernoulli
random variable with parameter λ ∈ [0, 1]. For shorthand, we
write xti , Pr(sti = 1) for the probability node i is infected at
time t. Taking the probability of both sides equaling one,
xt+1
= Pr(B(1 − δpi00,d (st )) = 1|sti = 1)Pr(sti = 1) + · · ·
i
Pr(B(pi01,d (st )) = 1|sti = 0)Pr(sti = 0)
= xti (1 − δpi00,d (st )) + (1 − xti )pi01,d (st )
= xti (1 − δ) + (1 − (1 − δ)xti )pi01,d (st )
(13)
xt+1
i
Note the expression for
still depends stochastically on
the state st . To obtain a mean-field approximation of xt+1
,
i
we simply replace the state st with xt (the n-vector with
components xti ) in (13). Thus, by redefining xt+1
to obey
i
this approximation and extending the domain of µi (·), ai (·)
and pi01,d (·) from {0, 1}n to [0, 1]n , we have the following
approximate dynamics
xt+1
= xti (1 − δ) + (1 − (1 − δ)xti )pi01,d (xt )
i
(14)
on the time evolution of node i’s probability of being infected.
Stacking the dynamics for each node into a vector, we obtain
a mapping φ : [0, 1]n → [0, 1]n where φi (x) = xi (1 − δ) +
(1 − (1 − δ)xi )pi01,d (x) and xt+1 = φ(xt ). This MFA of the
distancing chain is in contrast to the MFA of the benchmark
chain,
xt+1
= xti (1 − δ) + (1 − (1 − δ)xti )pi01 (xt )
i
(15)
which is studied thoroughly in [6]. It is derived in the same
manner, and is described by the mapping ψ : [0, 1]n → [0, 1]n
with ψi (x) = xi (1 − δ) + (1 − (1 − δ)xi )pi01 (x) and xt+1 =
ψ(xt ).
We make a few remarks on the basic structure of these two
mappings. They are nonlinear, continuous mappings satisfying
φ(x) ≺ ψ(x) whenever x ∈ [0, 1]n \0n and α ∈ [0, 1). Also,
φ(0n ) = ψ(0n ) = 0n . Linearization of φ and ψ about the
origin yields the same Jacobian matrix, βAC + (1 − δ)I, and
hence the same linearized dynamics xt+1 = (βAC + (1 −
δ)I)xt . The linear dynamics serve as an upper bound to both
(14) and (15). Therefore, if λmax (βAC + (1 − δ)I) < 1, the
origin is a globally stable fixed point and it is an unstable fixed
point if λmax (βAC + (1 − δ)I) > 1.
B. Existence of a Non-trivial Fixed Point
We now provide a sufficient condition for the existence of
a non-trivial fixed point ( 0n , the n-vector of zeros) of φ.
Theorem 1. If λmax (βAC + (1 − δ)In ) > 1, there exists a
nontrivial fixed point for φ.
The existence of such a fixed point suggests the epidemic
has an endemic state, where the disease spreads fast enough
to sustain an epidemic in the network. Our condition coincides with the condition for existence, uniqueness, and global
asymptotic stability of the non-trivial fixed point q ∗ of ψ,
which is λmax (βAC + (1 − δ)I) > 1, i.e. when the origin in
the linearized dynamics is unstable (Theorem 5.1, [6]). This
condition incorporates the factors that contribute to the rate
of spreading - δ, β, and the contact network AC . The proof
makes use of the following lemmas.
Lemma 1 (Lemma 3.1, [6]). There exists a vector ν 0n
such that (βAC − δIn )ν 0n if and only if λmax (βAC + (1 −
δ)In ) > 1.
The connectedness assumption for GC is necessary for the
above Lemma because the proof applies the Perron-Frobenius
theorem for nonnegative irreducible matrices. The next result
is an equivalent formulation of Brouwer’s fixed point theorem.
Lemma 2. (Theorem 4.2.3, [31]): Suppose fi : Dn → R, i =
1, . . . , n are continuous mappings, where
Dn = {x ∈ Rn : xi ∈ [`i , ui ], ∀i}
1n−1
u1n−1
Dn
c∗i (x−i )
Ri (εν)
x−i
φ+
i
p∗
ν
φ+
εν
xi
0 c∗i (εν−i )
u
1
Fig. 2: Diagram of the proof of Theorem 1. Here, p∗ denotes
a nontrivial fixed point of φ.
for real numbers `i , ui . We also define the set
D−i = {x−i ∈ Rn−1 : xj ∈ [`j , uj ] ∀j 6= i}
If for every i and for all x−i ∈ D−i ,
fi (x1 , . . . , `i , . . . , xn ) = fi (x−i , `i ) ≥ 0
fi (x1 , . . . , ui , . . . , xn ) = fi (x−i , ui ) ≤ 0,
(16)
(17)
then there exists a point x∗ ∈ Dn such that fi (x∗ ) = 0, for
all i = 1, . . . , n.
The final lemma needed is a technical result for the meanfield mappings φi .
Lemma 3. For each i ∈ N , define the maps fi : [0, 1]n → R
fi (x) , φi (x) − xi
= −δxi + (1 − (1 − δ)xi )pi01,d (x)
(18)
Then for any i ∈ N and x−i ∈ [0, 1]n−1 , the function
fi (x−i , ·) has a unique root c∗i (x−i ) ∈ [0, 1) which depends
continuously on x−i . Furthermore, one can find a sequence
xk−i → 0n−1 s.t. c∗i (xk−i ) is monotonically decreasing to 0.
Proof. For any i ∈ N and x−i ∈ [0, 1]n−1 ,
fi (x−i , 0) =
pi01,d (x−i , 0)
≥ 0.
j∈NiC
(21)
which is a polynomial in xi . The coefficients of the polynomial depend continuously on x−i ∈ [0, 1]n−1 , and the
roots of any polynomial are continuous with respect to its
coefficients. Consequently, for any sequence xk−i → 0n−1 ,
c∗i (xk−i ) → c∗i (0n−1 ) = 0 by continuity. This allows us to
select a subsequence of xk−i such that c∗i is monotonically
decreasing along the subsequence.
We are now ready to prove the main result of this section.
Proof of Theorem 1:. Let the mappings fi , i ∈ N be as in
Lemma 3. We need to verify (16) and (17) hold for all i and
for choices of `i , ui satisfying 0 < `i < ui . This ensures the
awareness dynamic φ has a fixed point other than the origin.
Choose ui = u where u satisfies
max
(19)
max
i∈N x−i ∈[0,1]n−1
c∗i (x−i ) < u < 1.
(22)
Then for all x−i ∈ [0, u]n−1 ,
and
fi (x−i , 1) = δ(pi01,d (x−i , 1) − 1) < 0.
x−i . To see this, observe that c∗i (x−i ) ∈ [0, 1) is a root of the
equation
Y
(1 − (1 − δ)xi ) 1 −
(1 − ai (x−i , xi )βxj ) − δxi = 0,
fi (x−i , u) < 0
(20)
The function fi (x−i , ·) is strictly decreasing: for a, b ∈ [0, 1]
s.t. a < b, fi (x−i , a) − fi (x−i , b) is given by
(b − a)(δ + (1 − δ)pi01,d (x−i , b)) + · · ·
+ (1 − a(1 − δ))(pi01,d (x−i , a) − pi01,d (x−i , b))
> 0.
This follows because pi01,d (x−i , xi ) is decreasing in xi
(xi contributes to global awareness). Hence for every
x−i ∈ [0, 1]n−1 , there is a unique c∗i (x−i ) ∈ [0, 1) s.t.
fi (x−i , c∗i (x−i )) = 0, and c∗i (x−i ) depends continuously on
(23)
c∗i (x−i ).
since u >
Thus, (17) holds, regardless of the choice
of `i . However, it remains to find the `i > 0 s.t. (16) is
satisfied. Let f (x) , [f1 (x), . . . , fn (x)]T and define the sets
n
+
φ+
i , {x ∈ [0, 1] : fi (x) ≥ 0}, φ ,
n
\
φ+
j
(24)
j=1
The Jacobian of f about the origin is (βAC −δIn ). By Lemma
1, there exists a vector ν 0n such that (βAC − δIn )ν 0n .
Consequently for sufficiently small ε > 0,
f (εν) 0n ,
(25)
or εν ∈ φ+ . We also define the set
n
Ri (x−i ) , {y ∈ R : yi ∈
[0, c∗i (x−i )], yj
∈ [xj , u], j 6= i}
(26)
If ενi ≤ c∗i (εν−i ) and Ri (εν) ⊂ φ+
,
then
(16)
is
satisfied,
i
i.e. fi (x−i , ενi ) ≥ 0 on D−i = {εν−i x−i u1n−1 } for
ε sufficiently small. We already have ενi ≤ c∗i (εν−i ) because
+
εν ∈ φ+ ⊂ φ+
i . To show Ri (εν) ⊂ φi , by Lemma 3 we
+
can find a sequence εk ν ∈ φ with εk → 0 s.t. c∗i (εk ν−i ) is
monotonically decreasing to 0. By stopping at a large enough
k, we can take a ε small enough such that
c∗i (εν−i ) =
min
x−i ∈D−i
c∗i (x−i )
(27)
Consequently, Ri (εν) ⊂ φ+
i . Choosing ε small enough to
satisfy (27) for all i ∈ N verifies (16) by using Dn = {x ∈
Rn : xj ∈ [ενj , u]}. See Figure 2 for an illustration of the
proof. By Lemma 2, φ has a fixed point contained in Dn .
The condition of Theorem 1 is independent of the awareness
parameter α and the structure of the information network
GI . Hence, social distancing alone cannot restore stability of
the disease-free equilibrium point. However, social distancing
lowers the overall metastable state of an epidemic.
Corollary 1. If q ∗ 0n is the unique nontrivial fixed point
of ψ, then any nontrivial fixed point p∗ of φ satisfies p∗ ≺ q ∗
whenever α ∈ [0, 1).
Proof. Define the sets ψi+ and ψ + similarly as in (24). It was
shown in [6] that q ∗ is the unique maximal element of ψ + ,
i.e q ∗ q, ∀q ∈ ψ + , q 6= q ∗ . Observe φ(x) ≺ ψ(x) for any
x ∈ [0, 1]n \0n . Let x ∈ φ+ , x 6= 0n . Then x φ(x) ≺ ψ(x),
so x ∈ ψ + . Therefore, φ+ ⊂ ψ + , and since p∗ ∈ φ+ for any
nontrivial fixed point of φ, p∗ ≺ q ∗ .
The mean-field analysis of this section reveals the qualitative dynamics not addressed by an absorbing Markov
chain analysis. Since the all-susceptible state is the unique
absorbing state and accessible from every other state, the
disease eradicates in finite time with probability one. This
answers what happens in the long-run, whereas the MFA
analysis answers what happens before this eventuality. The
MFA analysis concurs with what is observed in simulations of
the Markov chain dynamics - fast convergence to an endemic
“metastable” state that persists for a very long time before
eradication. Numerical simulations suggest stability of one
particular nontrivial fixed point p∗ of φ. A characterization
of these fixed points for different random graph families is
given in Section V.
IV. S TOCHASTIC C OMPARISON B ETWEEN AWARENESS
AND B ENCHMARK M ODEL
In this section, we use the properties of monotone couplings to prove that adding awareness reduces the expectation
of any increasing random variable quantifying an epidemic
metric (e.g. absorption time, total infections), and that the
benchmark chain dominates the distancing chain in terms of
a partial ordering on sample paths. These results serve as
a probabilistic analogue to the conclusions made in Section
III about the strict ordering of the nontrivial fixed points in
the corresponding mean-field approximations. Here, we have a
closed form expression for the reduction whereas in Corollary
1, only an inequality relation is established. First, we provide
relevant definitions and basic preliminary results of monotone
couplings. For a full reference on monotone coupling, see Ch
4 of [32].
A. Monotone couplings
Consider a general countable space X. Recall a partially
ordered set (X, X ) is the set X together with a relation X
among its elements which satisfies for all x, y, and z ∈ X,
•
•
•
x X x
If x X y and y X z, then x X z
If x X y and y X x, then x = y.
Definition 1. Let p1 , p2 be probability measures on a measurable space (X, F) and suppose (X, X ) is a partially ordered
set. A monotone coupling of p1 , p2 is a probability measure p
on (X 2 , F 2 ) such that for all x0 , y 0 ∈ X,
X
X
p(x, y 0 ) = p2 (y 0 ) and
p(x0 , y) = p1 (x0 ). (28)
xX y 0
yX x0
Thus, for any x, y ∈ X s.t x 6X y, p(x, y) = 0, and the
marginals of p are p1 , p2 .
Example 1. For an illustrative example of monotone coupling,
consider two biased coins where the biases qA , qB < 1 for
landing heads satisfy qA < qB . The coupling is a joint
distribution assigned to the pair of coin flips to ensure the
qA coin can never land heads with the qB coin landing
tails, while the marginal coin flip probabilities remain the
same. Specifically, define pX (0) = 1 − qX , pX (1) = qX for
X ∈ {A, B}. Also, define pAB : {0, 1}2 → [0, 1] by
pAB (0, 0) = 1 − qB
p (0, 1) = q − q
AB
B
A
pAB (1, 0) = 0
pAB (1, 1) = qA
(29)
P
Checking
P (28), the marginals are such
P that b≥1 pAB (1, b) =
qP
A,
a≤1 pAB (a, 1) = qB and
b≥0 pAB (0, b) = 1 − qA ,
p
(a,
0)
=
1−q
.
Thus,
p
is a monotone coupling
AB
B
AB
a≤0
of pA , pB .
We say a function Z : X → R is increasing in X if
whenever x X y, Z(x) ≤ Z(y). The next result characterizes
the difference in expectations of increasing random variables
between the marginals of a monotone coupling.
Proposition 1. Keeping the notation of Definition 1, suppose
p is a monotone coupling of p1 , p2 . If a random variable Z :
X → Z+ is increasing in X, then
Ep2 (Z) − Ep1 (Z) =
where Zτ = {x : Z(x) > τ }.
∞
X
τ =0
p(Zτc , Zτ )
(30)
Proof. Consider the following quantities:
X X
p(x, y)
p(Zτ , Zτ ) =
|st |
x∈Zτ y∈Zτ
=
X X
p(x, y) = p1 (Zτ )
(31)
g
x∈Zτ yX x
p(X, Zτ ) = p2 (Zτ )
(32)
The second sum over {y ∈ Zτ } can be replaced with {y X
x} in (31) because 1) for any x ∈ Zτ , we have {y : y X
x} ⊂ Zτ ; and 2) since p is a monotone coupling, for any
y ∈ Zτ s.t. y 6X x, p(x, y) = 0. The last equality of (31)
follows from (28). Since (X, Zτ ) ⊃ (Zτ , Zτ ) we can write
= p((X, Zτ )\(Zτ , Zτ ))
= p(Zτc , Zτ )
Example 2. Consider the biased coins of Example 1. One can
extend this example to sequences of m ≥ 2 flips, {0, 1}m
with the partial order x y if xi ≤ yi , i = 1, . . . , m, for
x, y ∈ {0, 1}m . Define
pX (x) =
pAB (x, y) =
m
Y
k=1
m
Y
pX (xk ), X ∈ {A, B}
(33)
pAB (xk , yk ).
(34)
k=1
m
a monotone coupling of pA , pB . For x ∈ {0, 1} ,
Then pAB isP
m
let Z(x) = i=1 xi be the random variable of the number of
heads for any given toss sequence. Then Z is increasing in
{0, 1}m . By Proposition 1,
EpB (Z) − EpA (Z) =
m
X
pAB (Zτc , Zτ ).
T (h)
T (g)
t
Fig. 3: A pair of sample paths (h, g) drawn from Φπ .
T < ∞ such that g T = o. The set of sample paths is denoted
by Γ.
p2 (Zτ ) − p1 (Zτ ) = p(X, Zτ ) − p(Zτ , Zτ )
Equation (30) immediately follows.
h
The absorption time T : Γ → Z+ of a sample path g is
given by
T (g) , min{t : g t = o}.
(36)
Thus for all g ∈ Γ, T (g) < ∞. Also, g t = o and
K(g t , g t+1 ) = 1 for all t ≥ T (g). Note that Γ is countable
since it is the countable union of the finite sets {g : T (g) = t}
for t = 0, 1, 2, . . ..
The distribution µπ : P(Γ) → [0, 1] on sample paths under
the benchmark SIS chain with starting distribution π ∈ ∆(Ω)
is given, for any A ∈ P(Γ), by
µπ (A) ,
Of course, one could trivially compute the above as m(qB −
qA ) since the distribution of Z is Bernoulli. However, Proposition 1 generalizes the difference for any increasing Z+ -valued
random variable over a partially ordered set.
The notion of stochastic domination is also relevant in our
comparison analysis.
Definition 2. An upper set I is a non-empty subset of (X, X )
that satisfies the following property: if x ∈ I and y X x,
then y ∈ I. Let p1 , p2 be two probability measures on (X, F).
Then p2 stochastically dominates p1 , written as p2 p1 , if
for any upper set I ⊂ X, p1 (I) ≤ p2 (I).
Our comparison between benchmark and distancing chains
falls into the framework of the above analysis.
B. Sample path comparison analysis
Our main result provides a construction of a monotone
coupling between the benchmark and distancing probability
distributions on sample paths.
Definition 3. A sample path is a sequence g = {g t }t∈Z+ such
that g t ∈ Ω and K(g t , g t+1 ) > 0 for all t ≥ 0, and there is a
T (g)−1
π(g 0 )
Y
K(g t , g t+1 )
(37)
t=0
g∈A
and similarly under the distancing chain by
νπ (A) ,
X
T (g)−1
π(g 0 )
Y
Kd (g t , g t+1 ).
(38)
t=0
g∈A
(35)
τ =0
X
Also, note that Γ is defined to exclude the set of sample
paths that are never absorbed, {g : g t 6= o, ∀t ∈ Z+ }.
These are infinite sequences that never terminate, and therefore
are uncountable. The probabilities µπ , νπ , however are welldefined on Γ without such sample paths:
X
µπ (g) =
∞
X
µπ ({g : T (g) = t})
(39)
t=0
g∈Γ
=
X
s∈Ω
=1
π(s)
∞
X
t=0
rs (Qt ) − rs (Qt+1 )
(40)
(41)
Here, Q is the 2n −1×2n −1 sub-stochastic matrix of transition
probabilities between non-absorbing states, and rs (Q) is the
sth row-sum of Q. Hence, rs (Qt )−rs (Qt+1 ) is the probability
a sample path starting from state s is absorbed at time t. The
elements of Qt approach zero as t → ∞.
Remark 2. (Ω, Ω ) is a partially ordered set. For s, s0 ∈ Ω,
s Ω s0 if si ≤ s0i for all i ∈ N .
Remark 3. (Γ, Γ ) is a partially ordered set. For h, g ∈ Γ,
h Γ g if ht Ω g t for all t ∈ Z+ .
Next, we present the main result of this section, which
constructs a monotone coupling distribution of νπ , µπ by
exploiting the differences in node-level transition probabilities.
Theorem 2. Suppose x, y ∈ Ω with x Ω y. For each i ∈ N ,
: {0, 1}2 → [0, 1] according to
define ϕx,y
i
x,y
ϕi (0, 0) = δ(1 − pi01 (y))
ϕx,y (0, 1) = δ(pi (y) − pi (x))
01
i
01,d
xi = yi = 1,
(42)
x,y
ϕ
(1,
0)
=
0
i
x,y
ϕi (1, 1) = 1 − δ(1 − pi01,d (x))
x,y
i
ϕi (0, 0) = 1 − p01 (y)
ϕx,y (0, 1) = pi (y) − pi (x)
01
i
01,d
(43)
xi = yi = 0,
ϕx,y
(1,
0)
=
0
i
x,y
ϕi (1, 1) = pi01,d (x)
x,y
i
ϕi (0, 0) = δ(1 − p01 (y))
ϕx,y (0, 1) = 1 − pi (x) − δ(1 − pi (y))
01
i
01,d
xi = 0, yi = 1,
ϕx,y
(1,
0)
=
0
i
x,y
ϕi (1, 1) = pi01,d (x)
(44)
Also, define ϕx,y : Ω2 → [0, 1] for x Ω y by
ϕx,y (ω, ω 0 ) ,
n
Y
i=1
0
0
ϕx,y
i (ωi , ωi ) ∀ω, ω ∈ Ω.
(45)
Lastly, define Φπ : Γ2 → [0, 1] for any π ∈ ∆(Ω) by
T (g)−1
Y
Φπ (h, g) , χ(h0 = g 0 )π(h0 )
t
ϕh
,g t
(ht+1 , g t+1 ).
t=0
(46)
Then Φπ is a monotone coupling of νπ , µπ .
Proof. See Appendix.
The couplings between node-level transition probabilities
ϕx,y
given in (42)-(44) are used to establish a coupling
i
between benchmark and distancing probability distributions
on sample paths in (46). The form of the ϕx,y
is identical
i
to the method in which two biased coins are coupled in (29).
When the coupling rule is applied to each node’s probability
of infection, it ensures no node can be infected in the distancing chain while being susceptible in the benchmark chain.
Consequently, the monotone coupling Φπ is a distribution on
pairs of sample paths (h, g) satisfying h0 = g 0 , h Γ g (see
Figure 3) and marginally, h ∼ νπ , g ∼ µπ . The next result
characterizes the difference between µπ and νπ -expectations
of any non-negative increasing function on the sample paths
Γ with respect to the coupling distribution Φπ .
Corollary 2. For any increasing Z+ -valued random variable
Z in Γ,
Eµπ (Z) − Eνπ (Z) =
∞
X
Φπ (Zτc , Zτ ).
(47)
τ =0
where Zτ = {x ∈ Γ : Z(x) > τ }.
Proof. Immediate from Theorem 2 and Proposition 1.
One can think of an increasing Z : Γ → Z+ as an epidemic
cost metric. Here, Φπ (Zτc , Zτ ) is simply the probability a
benchmark sample path g incurs a cost greater than τ , while
the corresponding distancing sample path h costs less than
τ , where (h, g) ∼ Φπ . The difference (47) encodes many
complex dependencies on the epidemic parameters δ and β,
the awareness weight α, and the graphs GC and GI .
The following result establishes stochastic domination of
the distancing chain by the benchmark chain.
Corollary 3. The benchmark chain stochastically dominates
the distancing chain on sample paths, i.e. µπ νπ .
Proof. For any upper set I ⊂ Γ, χI (·) is increasing in Γ. By
(47),
µπ (I) − νπ (I) = Eµπ (χI ) − Eνπ (χI )
= Φπ (I c , I) ≥ 0
(48)
The difference in probability (48) confirms the intuition
gained from Corollary 1 that sample paths with consistently
high numbers of infected individuals are more probable under
the benchmark chain. The closed-form differences (47) and
(48) provide a stochastic analogue to Corollary 1, which only
establishes inequality between the mean-field metastable states
of benchmark and distancing models.
Some examples of increasing Z+ -valued random variables
in Γ are
• The absorption time T : Γ → Z+ , defined by (36).
Pm
t
• The social cost up to time m, defined by g 7→
t=0 |g |,
P
where |s| , i∈N si for s ∈ Ω.
• The “epidemic spread”, or how many unique nodes that
contract the disease P
in a given amount of time m. This
is
given
by
g
→
7
i∈N χEi (g), where Ei = {g :
Pm t
g
>
0}.
This
metric
is investigated on different
t=0 i
network structures in the next section.
V. S IMULATIONS ON RANDOM NETWORKS
In this section, we illustrate through numerical simulations
how the structure of the contact network GC affects the
course of an epidemic in the presence of awareness and
social distancing. Extensive analytical and simulation studies
have been conducted without awareness ([1],[2],[3],[5],[33]).
Here, we look at three random graph families - geometric,
Erdős-Renyi, and scale-free. These networks are relevant in
studying epidemic spreading because they exhibit a variety of
qualitative features that reflect real-world networks. Geometric
networks portray people connected by geographic distance.
Erdős-Renyi random networks display a small-world effect
common in many real world networks - e.g neural and social
influence networks. Online social networks and the World
Wide Web are examples of scale-free networks [4].
In our model, the social network GI is generated directly
from GC via a parameter p ∈ (0, 1) through the following
procedure: 1) Select a fraction p of existing edges in EC at
random and remove them from the edge set; 2) For each of
the selected edges, select one of the two end nodes randomly
(e.g with probability 1/2) as the root node; 3) For each of the
Endemic states - Erdos-Renyi
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.8
0.7
0.6
0.5
0.4
0.3
0.7
0.6
0.5
0.4
0.3
0.2
0.2
0.1
0.1
0.1
0
0
5
10
15
20
0
5
10
δ/β
15
Benchmark
distancing (α=1,p=0)
0.9
0.2
0
Endemic states - Scale-free
1
Benchmark
distancing (α=1,p=0)
0.9
Fraction infected
Fraction infected
0.8
Endemic states - Geometric
1
Benchmark
distancing (α=1,p=0)
Fraction infected
1
0
20
0
5
10
δ/β
(a)
15
20
δ/β
(b)
(c)
Fig. 4: Norms of the nontrivial fixed points (solid lines) and long-run fraction of infected in stochastic simulations (diamonds) in the range of
epidemic persistence, δ/β ∈ [0, λmax (AC )], for n = 1000 node networks. The fixed points are computed by iterating the MFA dynamics (14)
and (15) with an arbitrary initial condition until convergence. The stochastic long-run infected fractions are computed by averaging the levels
of epidemic states in the latter half of a sample run of length 200. Vertical dashed lines indicate λmax (AC ). (a) Erdős-Renyi random network
with pER = .01, λmax (AC ) = 11.1. Here, pER > log n/n, the regime where the network is connected with high probability. (b) Geometric
random graph with r = .0564, λmax (AC ) = 16.52. (c) Scale-free generated from the PA algorithm with m = 5, λmax (AC ) = 19.9. The
parameters are chosen such that all networks have the same average degree d ≈ 10.
Epidemic spread:Erdos-Renyi
1000
Epidemic spread:geometric
1000
900
900
800
800
800
700
600
benchmark
α=0
p=1,α=1
p=3/4,α=1
p=1/2,α=1
p=1/4,α=1
p=0,α=1
500
400
300
200
100
0
5
10
15
700
600
benchmark
α=0
p=1,α=1
p=3/4,α=1
p=1/2,α=1
p=1/4,α=1
p=0,α=1
500
400
300
200
100
20
Unique contractions
900
Unique contractions
Unique contractions
1000
0
10
20
time
(a)
30
time
(b)
40
Epidemic spread:scale-free
700
600
benchmark
α=0
p=1,α=1
p=3/4,α=1
p=1/2,α=1
p=1/4,α=1
p=0,α=1
500
400
300
200
100
50
0
5
10
15
20
time
(c)
Fig. 5: Epidemic spreading as a function of time (same networks as Fig. 4). Local contact information (α near 1, p near 0) slows spread
most effectively for (a),(c), and the early stages of (b), whereas global information (α near 0 or α = 1, p near 1) is least effective. Note the
inversion of awareness effectiveness in (b). In these simulations, δ = β = 0.2.
selected root nodes i, select j 6= i uniformly at random and
add the edge (i, j). For p close to one, the resulting graph
GI = (N , EI ) exhibits the small-world effect (small average
shortest path length and small clustering) [34]. When p = 0,
GI = GC .
For the contact networks, geometric random graphs are
generated by placing n points uniformly at random on the
unit torus (unit square with periodic boundary conditions). An
edge exists between any two points if they are less than a
specified distance r ∈ (0, 1) away. Erdős-Renyi random graphs
are constructed by forming an edge between any two nodes
independently with a fixed probability pER ∈ (0, 1). Scale-free
networks are generated by the preferential attachment algorithm [35]: starting with an initial connected graph of m0 ≥ m
nodes, n − m0 additional nodes are added sequentially with
each incoming node establishing links to m existing nodes in
the network. The probability a node receives an incoming link
is proportional to its degree. We performed simulation analysis
on one network from each random graph family. The networks
all have 1000 nodes with an average degree of 10, and hence
the same number of edges.
In Figure 4, the normalized non-trivial fixed points are
characterized for the three random networks in the interval
of epidemic persistence. The norms of these points indicate
the size of the endemic states and they slightly overestimate
the actual long-run infected fraction observed in stochastic
simulations of the Markov chains.
In Figure 5, we quantify the “epidemic spread” by the
number of unique nodes that contract an infection as time progresses when one uniformly random node is initially infected.
This metric is an example of an increasing random variable
over sample paths (Section IV) and is helpful in revealing not
only how fast an epidemic initially spreads in the network, but
also how far-reaching it is. A key observation is that contact
awareness (p = 0, α = 1) slows epidemic spreading better
than any other awareness configuration at the beginning of
an epidemic. This is intuitively clear since contact awareness
provides nodes with the most vital information if they are in
danger of getting infected. As p increases, GI deviates more
from GC and the information nodes receive become less vital.
Erdős-Renyi and scale-free (with m = 5) networks admit
disease spread throughout the entire network in a short amount
of time, even with social distancing (Figure 5a,5c). This is
attributed to small average shortest path lengths (Ch. 8 & 12,
[4]), allowing the epidemic to quickly spread to other parts
of the network. Random geometric networks are characterized
by high clustering and large diameter. Clustering slows the
spread of an epidemic (Figure 5b), but also contributes to
increasing the final epidemic size [7]. The virus stays localized
and spreads slowly. This explains the inversion of awareness
parameters in Figure 5b. By the time the epidemic first reaches
its endemic level around t = 20, many nodes have not yet
been exposed because at this point their local communities are
untouched. Thus, having global or long-range social awareness
(low α or high p) is more beneficial over contact awareness.
Kd (x, ·), K(y, ·):
X
ω 0 Ω ω
i=1
=
n
Y
X
ωi0 ≥ωi
0
ϕx,y
i (ωi , ωi )
Pdi (x, ωi )
(49)
(50)
i=1
= Kd (x, ω)
By
P
(51)
a
completely analogous computation, we obtain
ϕx,y (ω 0 , ω) = K(y, ω).
Also, notice from (46) that Φπ (h, g) > 0 implies h0 = g 0
and h Γ g. Consequently, Φπ is a monotone coupling of
νπ , µπ :
ω 0 Ω ω
X
Φπ (h, g) =
gΓ h
VI. C ONCLUSIONS
ϕx,y (ω, ω 0 ) =
n
Y
T (g)−1
X
0
π(h )
Y
t
ϕh
,g t
(ht+1 , g t+1 ) (52)
t=0
gΓ h
g 0 =h0
T (h)
We modified the benchmark networked SIS epidemic process to include agent awareness, where prevalence-based information comes from social contacts and a global broadcast
of the overall infected fraction. Agents take social distancing
actions based on the level of information received, which
reduces their probabilities of getting infected. We showed
that awareness does not change the epidemic threshold for
persistence by proving existence of a nontrivial fixed point in
the mean-field approximation. Any nontrivial fixed point of
the distancing model is strictly component-wise less than the
unique nontrivial fixed point of the benchmark model.
We provided a full stochastic comparison analysis between
the benchmark and distancing chains in terms of their respective probability distributions on sample paths by constructing a
monotone coupling. The construction relies on exploiting the
differences in node transition probabilities between the two
chains. Consequently, adding awareness reduces the expectation of any increasing random variable on sample paths and we
obtain a closed form expression for the reduction. This implies
the benchmark distribution on sample paths stochastically
dominates the distancing distribution.
In simulations, we showed epidemic spreading heavily
depends on the network structure. In particular, qualitative
features such as small-world effects, clustering, and diameter
explain the results seen in simulations. We also concluded local contact awareness is the most effective at slowing epidemic
spread, and global awareness is the least effective.
A PPENDIX
Proof of Theorem 2. When x Ω y, the ϕx,y
are well-defined
i
probabilities since pi01 (y) − pi01,d (x) ≥ 0 and 1 − pi01,d (x) −
δ(1 − pi01 (y)) > 0. One can see by inspection that ϕx,y
is a
i
monotone coupling of Pdi (x, ·) and Pi (y, ·) defined in (9), (10)
and (3), (4) respectively.
Observe from (45), ϕx,y (ω, ω 0 ) > 0 implies x Ω y and
ω Ω ω 0 . Consequently, ϕx,y is a monotone coupling of
= π(h0 )
Y X
t−1
ϕh
,g t−1
(ht , g t )
(53)
t=1 g t Ω ht
T (h)
= π(h0 )
Y
Kd (ht−1 , ht )
(54)
t=1
= νπ (h)
(55)
The equality (53) is the combinatorial form of writing (52),
and the product terminates at T (h) because 1) g Γ h implies
T (g) ≥ T (h), 2) ht = o for all t ≥ T (h) and 3) for any
t > T (h),
X
X
t−1 t−1
t−1
ϕh ,g (ht , g t ) =
ϕo,g (o, g t ) = Kd (o, o)
g t Ω ht
g t ∈Ω
=1
By an analogous computation,
P
hΓ g
Φπ (h, g) = µπ (g).
ACKNOWLEDGEMENTS
This work is supported by ARO grant #W911NF-14-10402, and supported in part by KAUST. The authors thank
J. Walker Gussler (Georgia Inst. Tech.) for his contribution in
the simulations.
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| 3 |
Are You Talking to Me? Reasoned Visual Dialog Generation
through Adversarial Learning
Qi Wu1 , Peng Wang2 , Chunhua Shen1 , Ian Reid1 , and Anton van den Hengel1
arXiv:1711.07613v1 [cs.CV] 21 Nov 2017
1
Australian Centre for Robotic Vision, The University of Adelaide, Australia
2
Northwestern Polytechnical University, China
Abstract
The Visual Dialogue task requires an agent to engage
in a conversation about an image with a human. It represents an extension of the Visual Question Answering task
in that the agent needs to answer a question about an image, but it needs to do so in light of the previous dialogue
that has taken place. The key challenge in Visual Dialogue is thus maintaining a consistent, and natural dialogue
while continuing to answer questions correctly. We present
a novel approach that combines Reinforcement Learning
and Generative Adversarial Networks (GANs) to generate
more human-like responses to questions. The GAN helps
overcome the relative paucity of training data, and the
tendency of the typical MLE-based approach to generate
overly terse answers. Critically, the GAN is tightly integrated into the attention mechanism that generates humaninterpretable reasons for each answer. This means that the
discriminative model of the GAN has the task of assessing
whether a candidate answer is generated by a human or
not, given the provided reason. This is significant because
it drives the generative model to produce high quality answers that are well supported by the associated reasoning.
The method also generates the state-of-the-art results on the
primary benchmark.
Question
Human-like Responses
Machine-like
Are there any large No tall buildings but large one or Yes there are.
building nearby?
two story buildings, and one clock
is in front of looks like church of.
With the clock does Yes, I think so because it’s made by I don’t know.
it look expensive? stained glass.
Do you see any
signs for church?
Yes, there is a sign with light on, Yes there are.
but not clear enough.
Figure 1: Human-like vs. Machine-like responses in a visual dialog. The
human-like responses clearly answer the questions more comprehensively,
and help to maintain a meaningful dialogue.
a dialogue about an image. This is significant because it demands that the agent is able to answer a series of questions,
each of which may be predicated on the previous questions
and answers in the dialogue. Visual Dialogue thus reflects
one of the key challenges in AI and Robotics, which is to
enable an agent capable of acting upon the world, that we
might collaborate with through dialogue.
Due to the similarity between the VQA and Visual Dialog tasks, VQA methods [19, 40] have been directly applied to solve the Visual Dialog problem. The fact that the
Visual Dialog challenge requires an ongoing conversation,
however, demands more than just taking into consideration
the state of the conversation thus far. Ideally, the agent
should be an engaged participant in the conversation, cooperating towards a larger goal, rather than generating single
1. Introduction
The combined interpretation of vision and language has
enabled the development of a range of applications that
have made interesting steps towards Artificial Intelligence,
including Image Captioning [11, 34, 37], Visual Question
Answering (VQA) [1, 22, 38], and Referring Expressions
[10, 12, 41]. VQA, for example, requires an agent to answer a previously unseen question about a previously unseen image, and is recognised as being an AI-Complete
problem [1]. Visual Dialogue [5] represents an extension to
the VQA problem whereby an agent is required to engage in
1
word answers, even if they are easier to optimise. Figure 1
provides an example of the distinction between the type of
responses a VQA agent might generate and the more involved responses that a human is likely to generate if they
are engaged in the conversation. These more human-like
responses are not only longer, they provide reasoning information that might be of use even though it is not specifically
asked for.
Previous Visual Dialog systems [5] follow a neural translation mechanism that is often used in VQA, by predicting
the response given the image and the dialog history using
the maximum likelihood estimation (MLE) objective function. However, because this over-simplified training objective only focus on measuring the word-level correctness, the
produced responses tend to be generic and repetitive. For
example, a simple response of ‘yes’,‘no’, or ‘I don’t know’
can safely answer a large number of questions and lead to
a high MLE objective value. Generating more comprehensive answers, and a deeper engagement of the agent in the
dialogue, requires a more engaged training process.
A good dialogue generation model should generate responses indistinguishable from those a human might produce. In this paper, we introduce an adversarial learning
strategy, motivated by the previous success of adversarial
learning in many computer vision [3, 21] and sequence generation [4, 42] problems. We particularly frame the task as
a reinforcement learning problem that we jointly train two
sub-modules: a sequence generative model to produce response sentences on the basis of the image content and the
dialog history, and a discriminator that leverages previous
generator’s memories to distinguish between the humangenerated dialogues and the machine-generated ones. The
generator tends to generate responses that can fool the discriminator into believing that they are human generated,
while the output of the discriminative model is used as a
reward to the generative model, encouraging it to generate
more human-like dialogue.
Although our proposed framework is inspired by generative adversarial networks (GANs) [9], there are several
technical contributions that lead to the final success on the
visual dialog generation task. First, we propose a sequential co-attention generative model that aims to ensure that
attention can be passed effectively across the image, question and dialog history. The co-attended multi-modal features are combined together to generate a response. Secondly, and significantly, within the structure we propose the
discriminator has access to the attention weights the generator used in generating its response. Note that the attention
weights can be seen as a form of ‘reason’ for the generated
response. For example, it indicates which region should be
focused on and what dialog pairs are informative when generating the response. This structure is important as it allows
the discriminator to assess the quality of the response, given
the reason. It also allows the discriminator to assess the response in the context of the dialogue thus far. Finally, as
with most sequence generation problems, the quality of the
response can only be assessed over the whole sequence. We
follow [42] to apply Monte Carlo (MC) search to calculate
the intermediate rewards.
We evaluate our method on the VisDial dataset [5] and
show that it outperforms the baseline methods by a large
margin. We also outperform several state-of-the-art methods. Specifically, our adversarial learned generative model
outperforms our strong baseline MLE model by 1.87% on
recall@5, improving over previous best reported results by
2.14% on recall@5, and 2.50% recall@10. Qualitative evaluation shows that our generative model generates more informative responses and a human study shows that 49% of
our responses pass the Turing Test. We additionally implement a model under the discriminative setting (a candidate
response list is given) and achieve the state-of-the-art performance.
2. Related work
Visual dialog is the latest in a succession of vision-andlanguage problems that began with image captioning [11,
34, 37], and includes visual question answering [1, 22, 38].
However, in contrast to these classical vision-and-language
tasks that only involve at most a single natural language
interaction, visual dialog requires the machine to hold a
meaningful dialogue in natural language about visual content. Mostafazadeh et al. [20] propose an Image Grounded
Conversation (IGC) dataset and task that requires a model
to generate natural-sounding conversations (both questions
and responses) about a shared image. De Vries et al. [7]
propose a GuessWhat game style dataset, where one person asks questions about an image to guess which object
has been selected, and the second person answers questions
in yes/no/NA. Das et al. [5] propose the largest visual dialog dataset, VisDial, by pairing two subjects on Amazon
Mechanical Turk to chat about an image. They further formulate the task as a ‘multi-round’ VQA task and evaluate
individual responses at each round in a retrieval or multiplechoice setup. Recently, Das et al. [6] propose to use RL to
learn the policies of a ‘Questioner-Bot’ and an ‘AnswererBot’, based on the goal of selecting the right images that the
two agents are talking, from the VisDial dataset.
Concurrent with our work, Lu et al. [18] propose a
similar generative-discriminative model for Visual Dialog.
However, there are two differences. First, their discriminative model requires to receive a list of candidate responses
and learns to sort this list from the training dataset, which
means the model only can be trained when such information is available. Second, their discriminator only considers the generated response and the provided list of candidate responses. Instead, we measure whether the generated
Weighted
Sum
CNN
Discriminator
+
𝑣
What color are
the jeans?
V: 7 × 7 × 512
Image I
Question Q
What color are the jeans?
Question Q
Weighted
Sum
LSTM
+
𝑞
𝑄: 𝐿 × 512
A woman riding on the back of a white horse.
Does horse have saddle? Yes it does.
How old it woman? I would say maybe 30s or 40s.
MC
search
Is she wearing jeans? Yes she’s wearing jeans.
LSTM
T rounds of History H
((Caption),(𝑸𝟏 , 𝑨𝟏 ),…, (𝑸𝒕−𝟏 , 𝑨𝒕−𝟏 ))
LSTM
Human
or
Machine
Reward
+
𝑢
LSTM
LSTM
F
C
Weighted
Sum
LSTM
Is she wearing boots? Yes she is.
𝑢𝑄𝐴
Answer A
LSTM
What color is saddle? Black and brown.
What color is horses mane and tail? Both are
bright white color.
Decoder
They are black.
Co-attention
Encoder
LSTM
LSTM
𝐹
softmax
U: 𝑇 × 512
Sequential Co-attention Generator
Figure 2: The adversarial learning framework of our proposed model. Our model is composed of two components, the first being a sequential co-attention
generator that accepts as input image, question and dialog history tuples, and uses the co-attention encoder to jointly reason over them. The second
component is a discriminator tasked with labelling whether each answer has been generated by a human or the generative model by considering the attention
weights. The output from the discriminator is used as a reward to push the generator to generate responses that are indistinguishable from those a human
might generate.
response is valid given the attention weights which reflect
both the reasoning of the model, and the history of the dialogue thus far. As we show in our experiments in Sec. 4,
this procedure results in our generator producing more suitable responses.
Dialog generation in NLP Text-only dialog generation
[15, 16, 23, 30, 39] has been studied for many years in
the Natural Language Processing (NLP) literature, and has
leaded to many applications. Recently, the popular ‘Xiaoice’ produced by Microsoft and the ‘Its Alive’ chatbot
created by Facebook have attracted significant public attention. In NLP, dialog generation is typically viewed as
a sequence-to-sequence (Seq2Seq) problem, or formulated
as a statistical machine translation problem [23, 30]. Inspired by the success of the Seq2Seq model [32] in the machine translation, [26, 33] build end-to-end dialog generation models using an encoder-decoder model. Reinforcement learning (RL) has also been applied to train a dialog
system. Li et al. [15] simulate two virtual agents and handcraft three rewards (informativity, coherence and ease of answering) to train the response generation model. Recently,
some works make an effort to integrate the Seq2Seq model
and RL. For example, [2, 31] introduce real users by combining RL with neural generation.
Li et al. in [16] were the first to introduce GANs for
dialogue generation as an alternative to human evaluation.
They jointly train a generative (Seq2Seq) model to produce
response sequences and a discriminator to distinguish between human, and machine-generated responses. Although
we also introduce an adversarial learning framework to the
visual dialog generation in this work, one of the significant differences is that we need to consider the visual content in both generative and discriminative components of
the system, where the previous work [16] only requires textual information. We thus designed a sequential co-attention
mechanism for the generator and an attention memory access mechanism for the discriminator so that we can jointly
reason over the visual and textual information. Critically,
the GAN we proposed here is tightly integrated into the attention mechanism that generates human-interpretable reasons for each answer. It means that the discriminative model
of the GAN has the task of assessing whether a candidate
answer is generated by a human or not, given the provided
reason. This is significant because it drives the generative
model to produce high quality answers that are well supported by the associated reasoning. More details about our
generator and discriminator can be found in Sections 3.1
and 3.2 respectively.
Adversarial learning Generative adversarial networks
[9] have enjoyed great successes in a wide range of applications in Computer Vision, [3, 21, 24], especially in
image generation tasks [8, 43]. The learning process is
formulated as an adversarial game in which the generative
model is trained to generate outputs to fool the discriminator, while the discriminator is trained not to be fooled.
These two models can be jointly trained end-to-end. Some
recent works have applied the adversarial learning to sequence generation, for example, Yu et al. [42] backpropagate the error from the discriminator to the sequence generator by using policy gradient reinforcement learning. This
3. Adversarial Learning for Visual Dialog Generation
In this section, we describe our adversarial learning approach to generating natural dialog responses based on
an image. There are several ways of defining the visual based dialog generation task [7, 20]. We follow the
one in [5], in which an image I, a ‘ground truth’ dialog history (including an image description C) H =
(C, (Q1 , A1 ),...,(Qt−1 , At−1 )) (we define each QuestionAnswer (QA) pair as an utterance Ut , and U0 = C), and the
question Q are given. The visual dialog generation model
is required to return a response sentence  = [a1 ,a2 ,...,aK ]
to the question, where K is the length (number of words)
of the response answer. As in VQA, two types of models may be used to produce the response — generative and
discriminative. In a generative decoder, a word sequence
generator (for example, an RNN) is trained to fit the ground
truth answer word sequences. For a discriminative decoder,
an additional candidate response vocabulary is provided and
the problem is re-formulated as a multi-class classification
problem. The biggest limitation of the discriminative style
decoder is that it only can produce a response if and only if
it exists in the fixed vocabulary. Our approach is based on
a generative model because a fixed vocabulary undermines
the general applicability of the model, but also because it
offers a better prospect of being extensible to the problem
of generating more meaningful dialogue in future.
In terms of reinforcement learning, our response sentence generation process can be viewed as a sequence of
prediction actions that are taken according to a policy defined by a sequential co-attention generative model. This
model is critical as it allows attention (and thus reasoning)
to pass across image, question, and dialogue history equally.
A discriminator is trained to label whether a response is human generated or machine generated, conditioned on the
image, question and dialog attention memories. Considering here that as we take the dialog and the image as a whole
into account, we are actually measuring whether the generated response can be fitted into the visual dialog. The
output from this discriminative model is used as a reward to
the previous generator, pushing it to generate responses that
𝑞1 𝑞2
𝑄
LSTM
𝑄
𝑞𝐿
𝒒
…
෩ 𝒖
)
𝑪𝒐𝑨𝒕𝒕𝒆𝒏𝟑 (𝑸, 𝒗′,
𝑣1 𝑣2
𝐻
𝐼
CNN
𝑈0
LSTM
…
model shows outstanding performance on several sequence
generation problems, such as speech generation and poem
generation. The work is further extended to more tasks such
as image captioning [4, 28] and dialog generation [16]. Our
work is also inspired by the success of adversarial learning,
but we carefully extend it according to our application, i.e.
the Visual Dialog. Specifically, we redesign the generator
and discriminator in order to accept multi-modal information (visual content and dialog history). We also apply an
intermediate reward for each generation step in the generator, more details can be found in Sec. 3.3.
𝑈𝑡−1 LSTM
𝑉
𝑣𝑁
…
𝑢1 𝑢2
𝑈
෩
𝑣′
𝑢𝑇
…
𝑪𝒐𝑨𝒕𝒕𝒆𝒏𝟏 (𝑽, 𝑸, 𝟎)
෩ 𝒖
, 𝒒
)
𝑪𝒐𝑨𝒕𝒕𝒆𝒏𝟒 (𝒗′,
𝒗
෩ 𝑸)
𝑪𝒐𝑨𝒕𝒕𝒆𝒏𝟐 (𝑼, 𝒗′,
𝒖
Figure 3: The sequential co-attention encoder. Each input feature is coattend by the other two features in a sequential fashion, using the Eq.1-3.
The number on each function indicates the sequential order, and the final
attended features ũ,ṽ and q̃ form the output of the encoder.
are more fitting with the dialog history. In order to consider
the reward at the local (i.e. word and phase) level, we use
a Monte Carlo (MC) search strategy and the REINFORCE
algorithm [36] is used to update the policy gradient. An
overview of our model can be found in the Fig. 2. In the
following sections, we will introduce each component of
our model separately.
3.1. A sequential co-attention generative model
We employ the encoder-decoder style generative model
which has been widely used in the sequence generation
problems. In contrast to text-only dialog generation problem that only needs to consider the dialog history, however,
visual dialog generation additionally requires the model to
understand visual information. And distinct from VQA that
only has one round of questioning, visual dialog has multiple rounds of dialog history that need to be accessed and
understood. It suggests that an encoder that can combine
multiple information sources is required. A naive way of
doing this is to represent the inputs - image, history and
question separately and then concatenate them to learn a
joint representation. We contend, however, that it is more
powerful to let the model selectively focus on regions of the
image and segments of the dialog history according to the
question.
Based on this, we propose a sequential co-attention
mechanism [35]. Specifically, we first use a pre-trained
CNN [29] to extract the spatial image features V =
[v1 , . . . , vN ] from the convolutional layer, where N is the
number of image regions. The question features is Q =
[q1 , . . . , qL ], where ql = LST M (wl , ql−1 ), which is the
hidden state of an LSTM at step l given the input word wl
of the question. L is the length of the question. Because
the history H is composed by a sequence of utterance, we
extract each utterance feature separately to make up the dialog history features, i.e., U = [u0 , . . . , uT ], where T is the
number of rounds of the utterance (QA-pairs). And each
u is the last hidden state of an LSTM, which accepts the
utterance words sequences as the input.
Given the encoded image, dialog history and question
feature V, U and Q, we use a co-attention mechanism to
generate attention weights for each feature type using the
other two as the guidance in a sequential style. Each coattention operation is denoted as x̃ = CoAtten(X, g1 , g2 ),
which can be expressed as follows:
Hi
=
αi
softmax(WT Hi ),
PM
=
i=1 αi xi ,
x̃
=
tanh(Wx xi +Wg1 g1 +Wg2 g2 ),
i = 1, . . . ,M,
(1)
(2)
(3)
where X is the input feature sequence (i.e., V , U or Q), and
g1 , g2 ∈ Rd represent guidances that are outputs of previous
attention modules. Here d is the feature dimension. Wx ,
Wg1 , Wg2 ∈ Rh×d and W ∈ Rh are learnable parameters.
Here h denotes the size of hidden layers of the attention
module. M is the input sequence length that corresponding
to the N, L and T for different feature inputs.
As shown in Fig. 3, in our proposed process, the initial
question feature is first used to attend to the image. The
weighted image features and the initial question representation are then combined to attend to utterances in the dialog
history, to produce the attended dialog history (ũ). The attended dialog history and weighted image region features
are then jointly used to guide the question attention (q̃). Finally, we run the image attention (ṽ) again, guided by the
attended question and dialog history, to complete the circle.
All three co-attended features are concatenated together and
embedded to the final feature F :
F = tanh(Weg [ṽ; ũ; q̃])
(4)
where [; ] is a concatenation operator. Finally, this vector
representation is fed to an LSTM to compute the probability of generating each token in the target using a softmax
function, which forms the response Â. The whole generation process is denoted as π(Â|V,U,Q).
3.2. A discriminative model with attention memories
Our discriminative model is a binary classifier that is
trained to distinguish whether the input dialog is generated
by humans or machines. In order to consider the visual
information and the dialog history, we allow the discriminator to access to the attention memories in the generator.
Specifically, our discriminator takes {ṽ, ũ, Q, Â} as the input, where ṽ, ũ are the attended image and dialog history
features produced in the generative model1 , given the question Q. And  is the generated response in the generator.
The Q-Â pair is further sent to an LSTM to obtain a vector
1 we also tested to use the question memory q̃, but we find the discriminator result is not as good as when using the original question input Q.
representation uQÂ . All three features are embedded together and sent to a 2-way softmax function, which returns
the probability distribution of whether the whole visual dialog is human-natural or not:
O
=
tanh(Wed [ṽ; ũ; uQÂ ])
(5)
P
=
softmax(O)
(6)
The probability of the visual dialog being recognised as
a human-generated dialog is denoted as r({ṽ, ũ, Q, Â}).
3.3. Adversarial REINFORCE with an intermediate reward
In adversarial learning, we encourage the generator to
generate responses that are close to human generated dialogs, or, in our case, we want the generated response can
fit into the visual dialog as good as possible. The policy gradient methods are used here to achieve the goal. The probability of the visual dialog being recognised as a humangenerated dialog by the discriminator (i.e., r({ṽ, ũ, Q, Â}))
is used as a reward for the generator, which is trained to
maximize the expected reward of generated response using
the REINFORCE algorithm [36]:
J(θ) = EÂ∼π(Â|V,U,Q) (r({ṽ, ũ, Q, Â})|θ)
(7)
Given the input visual information (V ), question (Q) and
dialog history utterances (U ), the generator generates an response answer  by sampling from the policy. The attended
visual (ṽ) and dialog (ũ) memories with the Q and generated answer  are concatenated together and fed to the discriminator. We further use the likelihood ratio trick [36] to
approximate the gradient of Eq. 7:
∇J(θ) ≈ ∇ log π(Â|V,U,Q) · [r({ṽ, ũ, Q, Â}) − b]
X
=∇
log p(ak |V,U,Q,a1:k−1 ) · [r({ṽ, ũ, Q, Â}) − b]
k
(8)
where p is the probability of the generated responses words,
ak is the k-th word in the response. b denotes the baseline
value. Following [16], we train a critic neural network to
estimate the baseline value b by given the current state under the current generation policy π. The critic network takes
the visual content, dialog history and question as input, encodes them to a vector representation with our co-attention
model and maps the representation to a scalar. The critic
neural network is optimised based on the mean squared loss
between the estimated reward and the real reward obtained
from the discriminator. The entire model can be trained
end-to-end, with the discriminator updating synchronously.
We use the human generated dialog history and answers as
the positive examples and the machine generated responses
as negative examples.
Intermediate reward An issue in the above vanilla REINFORCE is it only considers a reward value for a finished
sequence, and the reward associated with this sequence is
used for all actions, i.e., the generation of each token. However, as a sequence generation problem, rewards for intermediate steps are necessary. For example, given a question
‘Are they adults or babies?’, the human-generated answer is
‘I would say they are adults’, while the machine-generated
answer is ‘I can’t tell’. The above REINFORCE model will
give the same low reward to all the tokens for the machinegenerated answer, but a proper reward assignment way is to
give the reward separately, i.e., a high reward to the token
‘I’ and low rewards for the token ‘can’t’ and ‘tell’.
Considering that the discriminator is only trained to assign rewards to fully generated sentences, but not intermediate ones, we propose to use the Monte Carlo (MC) search
with a roll-out (generator) policy π to sample tokens. An
N-time MC search can be represented as:
{Â11:K , . . . ,ÂN
1:K }
Ân1:k
π
= MC (Â1:k ; N )
(9)
Ânk+1:K
where
= (a1 , . . . ,ak ) and
are sampled based
on the roll-out policy π and the current state. We run the
roll-out policy starting from the current state till the end
of the sequence for N times and the N generated answers
share a common prefix Â1:k . These N sequences are fed to
the discriminator, the average score
rak =
N
1 X
r({ṽ, ũ, Q, Ân1:K })
N n=1
(10)
of which is used as a reward for the action of generating
the token ak . With this intermediate reward, our gradient is
computed as:
∇J(θ) = ∇
X
log p(ak |V,U,Q,a1:k−1 ) · [rak − b] (11)
Algorithm 1 Training Visual Dialog Generator with REINFORCE
Require: Pretrained generator Gen and discriminator Dis
1: for Each iteration do
2:
# Train the generator Gen
3:
for i=1, steps do
4:
Sample (I,H,Q,A) from the real data
5:
Sample (ṽ,ũ,Â) ∼ Genπ (·|I,H,Q)
6:
Compute Reward r for (ṽ,ũ,Q,Â) using Dis
7:
Evaluate ∇J(θ) with Eq. 8 or 11 depends on whether the intermediate reward (Eq. 10) is used
8:
Update Gen parameter θ using ∇J(θ)
9:
Update baseline parameters for b
10:
Teacher-Forcing: Update Gen on (I,H,Q,A) using MLE
11:
# Train the discriminator Dis
12:
Sample (I,H,Q,A) from the real data
13:
Sample (ṽ,ũ,Â) ∼ Genπ (·|I,H,Q)
14:
Update Dis using (ṽ,ũ,Q,A) as positive examples and (ṽ,ũ,Q,Â)
as negative examples
4. Experiments
We evaluate our model on a recently published visual dialog generation dataset, VisDial [5]. Images in Visdial are
all from the MS COCO [17], which contain multiple objects
in everyday scenes. The dialogs in Visdial are collected by
pairing 2 AMT works (a ‘questioner’ and an ‘answerer’)
to chat with each other about an image. To make the dialog measurable, the image remains hidden to the questioner and the task of the questioner is to ask questions
about this hidden image to imagine the scene better. The
answerer sees the image and his task is to answer questions
asked by the questioner. Hence, the conversation is more
like multi-rounds of visual based question answering and it
only can be ended after 10 rounds. There are 83k dialogs in
the COCO training split and 40k in the validation split, for
totally 1,232,870 QA pairs, in the Visdial v0.9, which is the
latest available version thus far. Following [17], we use 80k
dialogs for train, 3k for val and 40k as the test.
k
where we can see the intermediate rewards for each generation action are considered.
Teacher forcing Although the reward returned from the
discriminator has been used to adjust the generation process, we find it is still important to feed human generated
responses to the generator for the model updating. Hence,
we apply a teacher forcing [14, 16] strategy to update the
parameters in the generator. Specifically, at each training
iteration, we first update the generator using the reward
obtained from the sampled data with the generator policy.
Then we sample some data from the real dialog history and
use them to update the generator, with a standard maximum
likelihood estimation (MLE) objective. The whole training
process is reviewed in the Alg. 1.
4.1. Evaluation Metrics
Different from the previous language generation tasks
that normally use BLEU, MENTOR or ROUGE score for
evaluation, we follow [17] to use a retrieval setting to evaluate the individual responses at each round of a dialog.
Specifically, at test time, besides the image, ground truth
dialog history and the question, a list of 100 candidates answers are also given. The model is evaluated on retrieval
metrics: (1) rank of human response, (2) existence of the
human response in top-k ranked responses, i.e., recall@k
and (3) mean reciprocal rank (MRR) of the human response.
Since we focus on evaluating the generalization ability of
our generator, we simply rank the candidates by the generative model’s log-likelihood scores.
Image+Caption
Question
Human Answer
CoAtt-G-MLE
Ours
A bathroom with a white bath tub,
sink and large window.
What color is the bathroom?
Are there any people in there?
Are there towels hanging?
Is there any soap on the sink?
What color are the towels?
What kind of bathtub is it?
Can you see anything out the bathroom window?
Are there curtains on the window?
Is the bathroom light on?
Is there anything else on the sink?
The walls are gray
No
No folded up
I do n’t think so
White
A fancy rectangular
No
No
Yes
No
White
No
No
No soap
White
It ’s a tub
No
No
Yes
No
Most white
No
No, on the floor
I do n’t think so
White
It ’s a shower tub with a shower
No, just the wall
No curtains
Yes
No
What color is the motorcycle?
Is this on a busy street with shops and people?
Is it daylight or night time?
Is the photo in color?
What color are the other cars?
Are there any people walking?
Can you tell what shops businesses they are?
Do you see any traffic lights?
Do you think the motorcycle should be parked on
the sidewalk?
Do you see any signs?
It is black and white
It looks like it is not
Daytime
Yes it is
I see a white van and a blue
Not that i can see
Not really
No, i do not
Yes
One, but only a picture
White and blue
No
It ’s daytime
Yes
white and black
no
i ’m not sure
No i do n’t
No
Yes
It’s black and white
No it is not
It is daytime
Yes
One is blue and the other is white
no, there are no people
I ’m not sure , they are in the background
No i do n’t
No, it looks like it ’s parked
I see a sign on the side of road
Is the photo in color?
How old does the man appear to be?
What color wetsuit?
What color surfboard?
Do the rocks appear to be smooth or sharp?
Is he close to the water?
Does it appear to be a beach or private section?
What color is the water dark or light blue?
Does he have any shoes on?
Does he appear to be wet or dry?
Yes
I would estimate late 30s
Dark blue
White and red
I would guess they are smooth
Moderately close
Private area
It is blurry so it appears black
I ca n’t see his feet
Dry
Yes
20 ’s
Black
White with red
Smooth
No
I ca n’t tell
light blue
I ca n’t see his feet
Dry
Yes
I would say 20 ’s
Black
It ’s white with red
They look smooth
Yes
I ca n’t tell
It ’s light blue
I ca n’t see his feet
He looks dry
A motorcycle, moped and a bus
parked by the street.
A man in a wet suit carrying a
surfboard by some rocks.
Figure 4: Qualitative results of our model (CoAtt-GAN-w/ Rinte -TF) comparing to human ground-truth answer and our baseline model.
4.2. Implementation Details
To pre-process the data, we first lowercase all the texts,
convert digits to words, and remove contractions, before tokenizing. The captions, questions and answers are further
truncated to ensure that they are no longer than 40, 20 and
20, respectively. We then construct the vocabulary of words
that appear at least 5 times in the training split, giving us
a vocabulary of 8845 words. The words are represented
as one-hot vector and 512-d embeddings for the words are
learned. These word embeddings are shared across question, history, decoder LSTMs. All the LSTMs in our model
are 1-layered with 512 hidden states. The Adam [13] optimizer is used with the base learning rate of 10−3 , further decreasing to 10−5 . We use 5-time Monte Carlo (MC) search
for each token. The co-attention generative model is pretrained using the ground-truth dialog history for 30 epochs.
We also pre-train our discriminator (for 30 epochs), where
the positive examples are sampled from the ground-truth dialog, the negative examples are sampled from the dialog
generated by our generator. The discriminator is updated
after every 20 generator-updating steps.
4.3. Experiment results
Baselines and comparative models We compare our
model with a number of baselines and state-of-the-art models. Answer Prior [5] is a naive baseline that encodes answer options with an LSTM and scored by a linear classifier, which captures ranking by frequency of answers in the
training set. NN [5] finds the nearest neighbor images and
questions for a test question and its related image. The op-
Model
Answer Prior [5]
NN [5]
LF [5]
HRE [5]
HREA [5]
MN [5]
HCIAE [18]
CoAtt-G-MLE
CoAtt-GAN-w/o Rinte
CoAtt-GAN-w/ Rinte
CoAtt-GAN-w/ Rinte -TF
MRR
0.3735
0.4274
0.5199
0.5237
0.5242
0.5259
0.5386
0.5411
0.5415
0.5506
0.5578
R@1
23.55
33.13
41.83
42.29
42.28
42.29
44.06
44.32
44.52
45.56
46.10
R@5
48.52
50.83
61.78
62.18
62.33
62.85
63.55
63.82
64.17
65.16
65.69
R@10
53.23
58.69
67.59
67.92
68.17
68.88
69.24
69.75
70.31
71.07
71.74
Mean
26.50
19.62
17.07
17.07
16.79
17.06
16.01
16.47
16.28
15.30
14.43
Table 1: Performance of generative methods on VisDial v0.9. Higher is
better for MRR and recall@k, while lower is better for mean rank.
tions are then ranked by their mean-similarity to answers
to these questions. Late Fusion (LF) [5] encodes the image, dialog history and question separately and later concatenated together and linearly transformed to a joint representation. HRE [5] applies a hierarchical recurrent encoder [27] to encode the dialog history and the HREA [5]
additionally adds an attention mechanism on the dialogs.
Memory Network (MN) [5] maintains each previous question and answer as a ‘fact’ in its memory bank and learns
to refer to the stored facts and image to answer the question. A concurrent work [18] proposes a HCIAE (HistoryConditioned Image Attentive Encoder) to attend on image
and dialog features.
From Table 1, we can see our final generative model
CoAtt-GAN-w/ Rinte -TF performs the best on all the evaluation metrics. Comparing to the previous state-of-the-art
model MN [5], our model outperforms it by 3.81% on R@1.
We also produce better results than the HCIAE [18] model,
Model
LF [5]
HRE [5]
HREA [5]
MN [5]
SAN-QI [40]
HieCoAtt-QI [19]
AMEM [25]
HCIAE-NP-ATT [18]
Ours
MRR
0.5807
0.5846
0.5868
0.5965
0.5764
0.5788
0.6160
0.6222
0.6398
R@1
43.82
44.67
44.82
45.55
43.44
43.51
47.74
48.48
50.29
R@5
74.68
74.50
74.81
76.22
74.26
74.49
78.04
78.75
80.71
R@10
84.07
84.22
84.36
85.37
83.72
83.96
86.84
87.59
88.81
Mean
5.78
5.72
5.66
5.46
5.88
5.84
4.99
4.81
4.47
Table 2: Performance of discriminative methods on VisDial v0.9. Higher
is better for MRR and recall@k, while lower is better for mean rank.
which is the previous best results that without using any
discriminative knowledges. Figure 4 shows some qualitative results of our model. More results can be found in the
supplementary material.
Ablation study Our model contains several components.
In order to verify the contribution of each component, we
evaluate several variants of our model.
• CoAtt-G-MLE is the generative model that uses our
co-attention mechanism shown in Sec. 3.1. This model
is trained only with the MLE objective, without any
adversarial learning strategies. Hence, it can be used
as a baseline model for other variants.
• CoAtt-GAN-w/o Rinte is the extension of above
CoAtt-G model, with an adversarial learning strategy.
The reward from the discriminator is used to guide the
generator training, but we only use the global reward
to calculate the gradient, as shown in Equ. 8.
• CoAtt-GAN-w/ Rinte uses the intermediate reward as
shown in the Equ. 10 and 11.
• CoAtt-GAN-w/ Rinte -TF is our final model which
adds a ‘teacher forcing’ after the adversarial learning.
Our baseline CoAtt-G-MLE model outperforms the previous attention based models (HREA, MN, HCIAE) shows
that our co-attention mechanism can effectively encode the
complex multi-source information. CoAtt-GAN-w/o Rinte
produces slightly better results than our baseline model by
using the adversarial learning network, but the improvement
is limited. The intermediate reward mechanism contributes
the most to the improvement, i.e., our proposed CoAttGAN-w/ Rinte model improves over our baseline by average 1%. The additional Teacher-Forcing model (our final
model) brings the further improvement, by average 0.5%,
achieving the best results.
Discriminative setting We additionally implement a
model for the discriminative task on the Visdial dataset [5].
In this discriminative setting, there is no need to generate
a string, instead, a pre-defined answer set is given and the
problem is formulated as a classification problem. We modify our model by replacing the response generation LSTM
(can be treated as a multi-step classification process) as a
single-step classifier. HCIAE-NP-ATT [18] is the origi-
M1: Percentage of responses that
pass the Turing Test
M2: Percentage of responses that
are evaluated as better or equal to
human responses.
MN [5]
CoAtt-G-MLE
Ours
0.39
0.46
0.49
0.36
0.42
0.45
Table 3: Human evaluation on 1000 sampled responses on VisDial v0.9
nal HCIAE model with a n-pair discriminative loss and a
self-attention mechanism. AMEM [25] applies a more advanced memory network to model the dependency of current question on previous attention. Additional two VQA
models [19, 40] are used for comparison. Table 2 shows
that our model outperforms the previous baseline and stateof-the-art models on all the evaluation metrics.
4.4. Human study
Above experiments verify the effectiveness of our proposed model on the Visdial [5] task. In this section, to check
whether our model can generate more human-like dialogs,
we conduct a human study.
We randomly sample 1000 results from the test dataset
in different length, generated by our final model, our
baseline model CoAtt-G-MLE, and the Memory Network
(MN)2 [5] model. We then ask 3 human subjects to guess
whether the last response in the dialog is human-generated
or machine-generated and if at least 2 of them agree it is
generated by a human, we say it passed the Truing Test.
Table 3 summarizes the percentage of responses in the dialog that passes the Turing Test (M1), we can see our model
outperforms both the baseline model and the MN model.
We also apply our discriminator model in Sec. 3.2 on these
1000 samples and it recognizes that nearly 70% percent of
them as human-generated responses (random guess is 50%),
which suggests that our final generator successfully fool the
discriminator in this adversarial learning. We additionally
record the percentage of responses that are evaluated as better than or equal to human responses (M2), according to the
human subjects’ manual evaluation. As shown in Table 3,
45% of the responses fall into this case.
5. Conclusion
Visual Dialog generation is an interesting topic that requires machine to understand visual content, natural language dialog and have the ability of multi-modal reasoning.
More importantly, as a human-computer interaction interface for the further robotics and AI, apart from the correctness, the human-like level of the generated response is a
significant index. In this paper, we have proposed an adversarial learning based approach to encourage the generator
to generate more human-like dialogs. Technically, by combining a sequential co-attention generative model that can
2 we use the author provided code and pre-trained model provided on
https://github.com/batra-mlp-lab/visdial
jointly reason the image, dialog history and question, and
a discriminator that can dynamically access to the attention
memories, with an intermediate reward, our final proposed
model achieves the state-of-art on VisDial dataset. A Turing
Test fashion study also shows that our model can produce
more human-like visual dialog responses.
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| 1 |
Stability and Transparency Analysis of a Bilateral Teleoperation in
Presence of Data Loss
arXiv:1711.03605v1 [cs.SY] 9 Nov 2017
A. Bakhshi, H.A. Talebi, A.A. Suratgar, and M. Abdeetedal
Abstract— This paper presents a novel approach for stability
and transparency analysis for bilateral teleoperation in the
presence of data loss in communication media. A new model
for data loss is proposed based on a set of periodic continuous
pulses and its finite series representation. The passivity of the
overall system is shown using wave variable approach including
the newly defined model for data loss. Simulation results are
presented to show the effectiveness of the proposed approach.
I. I NTRODUCTION
In a teleoperation system, the operator can perform a task
in a remote environment via slave robot that commanded
by the master robot. Tele-manipulation tasks are done by
sending position, velocity, and/or force information remotely.
The main applications are outer space explorations [1], handling of toxic materials [2] , and minimally invasive surgery
[3]. In bilateral teleoperation, a communication network
structured the connection between master and the slave.
Time delay and data loss occur in communication channel
between the master and slave sites. Wave variable and smith
predictor methods are among the common control schemes
for compensation of time delay and data loss.
Anderson and Spong used scattering transformation, network theory and the passivity to provide the stability of
a teleoperation system in presence of constant time delay
[4]. Niemeyer and Slotine introduced wave variables [5].
In this wave-based framework, the scattered wave variables
are communicated via the delayed transmission lines instead
of the typical power-conjugated variables such as force and
velocity. The use of the wave scattering approach solved the
destabilizing effects incurred in the transmission lines by
passifying the communication channels independent of the
amount of the delay.However, this approach is not readily
applicable to time-varying delays. Moreover there is no
guarantee in performance in the presence of even constant
communication delay.
Since position information does not pass through the communication channels, system faces position drift and raising
tracking error. furthermore the time varying delay causes
distorted velocity signal and results in tracking error. In
order to compensate the effects of time varying delay, wave
variables are sent along their integrations [6] , [7] . In this
method the main signal and its integral are received by the
slave and the position is then calculable. moreover in [7], pre
mentioned approach is extended by sending power signal of
the wave variable along side other signals. In [8], instability
Authors are with the Department of Electrical Engineering, AmirKabir University of Technology, Tehran, Iran. Email:{alibakhshi21,alit,asuratgar,metedal}@aut.ac.ir.
is avoided by defining new gains in communication channels.
These gains result in stability of the system, although cause
poor performance. In [9], delayed position signals are also
used but this strategy causes tracking error as well. Similar
approach has been used in [10] to converge the velocity
error to zero. How ever, it failed to to provide zero position
tracking error.
In [11], a proper state feedback is designed and passive
input/output system is defined such that it includes the
position and velocity information. This approach result in
good performance in presence of time delay. To avoid
the adverse effects of wave distortion due to time varying
delays and data losses, a novel solution based on the digital
reconstruction of the wave variables is proposed in [12]. This
approach introduces the use of buffering and interpolation
scheme that preserves the passivity of the system which
reduces the tracking error under time-varying delays and
packet losses. In [13], the passivity of the system has been
shown in the presence of data loss by considering zero value
Wave in a discrete communication. As long as data loss
causes poor performance, considering data reconstruction
methods is mandatory. In [14], the effects of time delay
and packet loss are compensated by estimating the received
wave variables. This technique is based on Smith predictor
and Kalman filter. Several researches have been done on
compensating the effects of time delay in communication
channels though most of which result in poor performance
in the presence of data loss.
In network control system (NCS) literature, data loss often
is modelled by using three main categories. In [15], data
loss has been modelled as jump linear systems with Markov
chains. In [16], modelling has been done using asynchronous
dynamical system(ADS). In [17], random sample system has
been used for modelling.
In this paper, a robust and high performance teleoperation
system in the presence of data loss and different initial conditions is presented. A new model for data loss is proposed
based on a set of periodic continuous pulses and its finite
series representation and a state feedback control law for
master and slave manipulators is designed. The presented
control laws contains both position and velocity information.
The new architecture, built within the passivity framework,
provides good transparency which is measured in terms
of position tracking abilities of the bilateral system. This
configuration provides robust performance against network
effects such as packet losses and reordering which shows
the ability of the architecture to teleoperate over unreliable
communication networks .
Fig. 2. New teleoperation system according to wave variables in presence
of data loss in communication channel.
Fig. 1.
Periodic data loss modelled as a train pulse
The structure of this paper is as follows. In Section II,
our proposed model for data loss is presented. The data loss
is presented as a set of periodic continuous pulses and its
finite series representation. In Section III, the structure of the
teleoperation system in presence of the data loss is proposed.
In Section IV, the stability of the teleoperation system is
shown by defining a Lyapunov function candidate and obtain
conditions to have acceptable performance. In Section V,
the validation of our control approach is investigated via
simulations and conclusion is presented in Section VI.
II. DATA LOSS MODELLING
In this paper, we propose a new continuous time model
for data loss.
Assumption 2.1: Data loss occurs in periodic manner
which can be written as a train of pulse as shown in Fig.
1.
Where (.)∗ indicates signals which passes through communication channel in which data loss occurs, Tα is loss rate,
α T and L(t) is a periodic function of time and.
Assumption 2.2: L(t) can be written as a finite Fourier
series which is continuous and periodic where :
L(t) =
N
X
Property 1: The inertia matrix M is symmetric positive
definite and there exist some positive constants m1 , m2
such that m1 I < M < m2 I
Property 2: Using the Christoffel symbols, the matrix
Ṁ − 2C is skew-symmetric.
The master and slave control torques are given by
τm = −Mm λq̇m − Cm λqm + gm + τ̄m
τs = −Ms λq̇s − Cs λqs + gs + τ̄s
an cos(nw0 t) + bn sin(nw0 t)
Ms ṙs + Cs rs = −Fe + τ̄s = τ 0 s
2
(1)
∫ L(t) × cos(nw0 t)dt
TT
2
bn = ∫ L(t) × sin(nw0 t)dt
TT
Substituting the loss rate Tα and frequency of the train pulse
signal W0 = 2π
T into (1), we obtain
an =
(2)
n=1
In fact, L(t) is a continues function that describes data loss
as a periodic phenomenon.
III. COORDINATION ARCHITECTURE FOR
BILATERAL TELEOPERATION
The new presented model for data loss is implemented on
the teleoperation structure in [11]. Master and slave dynamics
(5)
Where rm , rs are defined as in (6)
rm = q̇m + λqm
rs = q̇s + λqs .
(6)
In fact, rm , rs are new outputs of the teleoperation system,
where
t
t
T
0 T
∫ τm
rm = ∫ τs0 rs = 0 .
(7)
0
an =
(4)
Where τm , τs are master and slave actuators torques respectively. By substituting (4) in (3) the master and slave dynamic
can be expressed
Mm ṙm + Cm rm = Fh + τ̄m = τ 0 m
n=1
−1
2πnα
sin(
)
πn
T
1
2πnα
bn =
[cos(
) − (−1)n ]
πn
T
N
X
L(t) =
an cos(nw0 t) + bn sin(nw0 ).
are considered as follows :
Mm (qm )q̈m + Cm (qm , q̇m )q̇m + gm (qm ) = J T Fh + τm
Ms (qs )q̈s + Cs (qs , q̇s )q̇s + gs (qs ) = τs − J T Fe
(3)
where qm , qs are n×1 master and slave robots joint variables
vectors, q̇m , q̇s are n × 1 joint velocity vectors, τm , τs are
n × 1 applied torque vectors, Ms , Mm are positive definite
inertial n × n matrices, C is n × n Coriolis matrices and g
is n × 1 gravitational vectors.
It is worth mentioning that some properties of the robot
structures are as fallows
0
Hence according to (7) the new teleoperation system is loss0
less. where (τm
, rm ) and (τs0 , rs ) indicate new input/output
of the system. Since master and salve are passive based
on new input/output definitions hence just the stability of
channel should be proven. As in [11] the new structure of
the teleoperation systems with respects to new definitions is
illustrated in Fig. 2
Where variables in Fig. 2 are as follows:
1
1
um = √ (Fmd + brmd ) Vm = √ (Fmd − brmd )
2b
2b
(8)
1
1
us = √ (Fsd + brsd ) Vs = √ (Fsd − brsd )
2b
2b
Assumption 4.2: The environment’s and operator’s forces
are assumed to be bounded.
Assumption 4.3: rmd ,rsd are assumed to be zero for t < 0
Fig. 3.
Relations between wave variables in presence of data loss.
where rmd , rsd are the signals obtained from scattering
transformation. Control signals Fsd (τ̄s ) and Fmd (−τ̄m ) are
obtained in (9).
Fsd = Ks (rsd − rs )
Fmd = Km (rmd − rm )
(9)
Where b, Ks and Km are positive definite diagonal matrices
which should be chosen properly. According to Fig. 1 Input
signals of communication channel um , vs and output signals
us , vm in presence of data loss are as in Fig. 3 and in 10 :
um ∗ = um L = um
vs ∗ = vs L = vs
N
X
an cos(nw0 t) + bn sin(nw0 ) = us
Theorem 4.1: A teleoperation system described as (5), (8)
and (14) in presence of data loss described as (2) is stable
and errors defined as (15) are bounded.
Proof: Let us define a Lyapunov function candidate as
:
1 T
Mm rm + rsT Ms rs + eTm K1 em + eTs K1 es )+
V = (rm
2
t
t
0
0
T
T
∫ (FeT rs − FhT rm )dt + ∫ (Fmd
rmd − Fsd
rsd )dt
(16)
where V(0)=0 and the first four terms are positive according
to positive definiteness of Ms , Mm , K1 , K2 and property 4.1.
Hence : ∫0t (FeT rs )dt ≥ 0 and− ∫0t (FhT rm )dt ≥ 0 Hence just
should be shown that :
n=1
N
X
t
T
T
V1 = ∫ (Fmd
rmd − Fsd
rsd )dt > 0
an cos(nw0 t) + bn sin(nw0 ) = vm .
0
n=1
(10)
In order to determine Km and Ks , using presented data loss
modelling and substituting (10) into (8) result in (11) .
∗
Fmd
+
∗
brmd
From (8) can be shown :
r
= Fsd + brsd
(11)
Fmd =
r
b
(um + vm ) , Fsd =
2
b
(us + vs )
2
(17)
and by substituting (9) in (11) ,(12) result in.
∗
∗
(b + Ks )rsd = (b − Km )rmd
+ Km rm
+ Ks rs
(12)
rmd =
and similarly (13) is obtained.
∗
(b + Km )rmd = (b − Ks )rsd
+ Ks rs∗ + Km rm
(13)
∗
In (12) and (13) according to the effects of rmd
and
∗
rsd on rsd and rsd respectively and the wave reflecting
phenomenon, Km and, Ks are selected equal to b, which
simplifies (12),(13) to :
Km = Ks = b
∗
2rsd = rm
+ rs
(14)
V1 =
V1 =
es =
(15)
The main goal of the controller is that defined errors are converging to zero which result in highly improve transparency
of the system.
Assumption 4.1: The environment and operator are assumed to be passive with inputs rm ,rs .
√1 (us
2b
− vs )
(18)
1
2
2 (um
2
− vm
)−
Rt
0
1
2
2 (us
− vs2 )dt
(19)
And substituting (10) and (19) :
In order to analyse stability of the system, first tracking
errors are defined as follows :
em =
Rt
0
Zt
IV. STABILITY ANALYSIS
− vm ) , rsd =
Using (17) and (18) :
2rmd = rs∗ + rm .
∗
qm
− qs
∗
qs − qm
√1 (um
2b
1
1 2
(um − v ∗s 2 )dt −
2
2
Zt
0
Zt
=
0
(u∗m 2 − vs2 )dt
0
1 2
1
(um − u∗m 2 )dt +
2
2
Zt
(20)
(vs2 − v ∗s 2 )dt
0
According to data loss model in Fig. 1 it is clear that
u2m − (u∗m )2 > 0 , vs2 − (vs∗ )2 > 0 and therefore the
candidate Lyapunov function is positive definite.The time
derivative of V is :
1 T
T
V̇ =rm
Mm ṙm + rm
Ṁm rm + rsT Ms ṙs
2
1
+ rsT Ṁs rs + ėTm K1 em + ėTs K2 es
2
T
T
+ FeT rs − FhT rm + Fmd
rmd − Fsd
rsd
1
T
T
Ṁm rm
=rm
(−Cm rm + Fh − Fmd ) + rm
2
1
+ rsT (−Cs rs − Fe + Fsd ) + rsT Ṁs rs + ėTm K1 em
2
T
T
+ ėTs K2 es + FeT rs − FhT rm + Fmd
rmd − Fsd
rsd
T
=rm
(Fh − Fmd ) + rsT (−Fe + Fsd ) + ėTm K1 em
T
T
+ ėTs K2 es + FeT rs − FhT rm + Fmd
rmd − Fsd
rsd
= − (rmd − rm )T b(rmd − rm ) − (rs − rsd )T b(rs − rsd )
+ ėTm K1 em + ėTs K2 es
Using (6),(14) and (15) :
rs∗ − rm = q̇s∗ + λqs∗ − q̇m − λqm = ės + λes
∗
∗
rs − rm
= q̇s + λqs − q̇m
− λq ∗ m = ėm + λem
(21)
1
1
V̇ = − (rs∗ − rm )b(rs∗ − rm ) − (rs − r∗ m )b(rs − r∗ m )
4
4
+ėTm K1 em + ėTs K2 es
1
= − (ės + λes )T b(ės + λes )
4
1
− (ėm + λem )T b(ėm + λem ) + ėTm K1 em + ėTs K2 es
4
1
= − (ėTs bės + 2ėTs bλes + eTs λbλes + ėTm bėm
4
+2ėTm bλem + eTm λbλem ) + ėTm K1 em + ėTs K2 es
Since b and λ are positive definite diagonal matrices.
Hence (ėTi bλei )T = ėTi bλei i = m, s and
1
V̇ = − (ėTs bės + 2ėTs bλes + eTs λbλes ) + ėTs K2 es
4
1
− (ėTm bėm + 2ėTm bλem + eTm λbλem ) + ėTm K1 em
4
by choosing K1 = K2 =
V̇ is simplified to
λb
2
, extra terms are omitted and
1
V̇ = − (ėTs bės +eTs λbλes + ėTm bėm +eTm λbλem ) ≤ 0 (22)
4
As V is positive definite and V̇ is negative semi definite
so the system is stable and the errors are bounded. Using
property1, rm and rs remain bounded.
rm
rm = q̇m + λqm ⇒ sqm + λqm = rm ⇒ qm =
s+λ
rs
rs = q̇s + λqs ⇒ sqs + λqs = rs ⇒ qs =
s+λ
(23)
According to (23) and boundedness of rm and rs , boundedness of qm , qs ,q̇m , q̇s are concluded. Hence
qm , qs , rm , rs , q̇m , q̇s is bounded.
(24)
Theorem 4.2: Teleoperation system described with
(3),(4),(5),(6),(8),(9) and (14) which is illustrating in Fig.2,
q̈m , q̈s are bounded if environment and operator are passive.
Proof: From boundedness of rm , rs is bounded we
can conclude the boundedness of rsd , rmd based on (14).
Using (9) Fmd and Fsd remain bounded. Hence Mm , Ms , q̇m
and q̇s are bounded. From (3)
q̈m and q̈s are bounded
(25)
Since the system is non-autonomos, the Barballat’s lemma
[18] is used .
Lemma 1: If V satisfies following three conditions:
• V is lower bounded.
• V̇ ≤ 0
• V̇ is uniformly continuous
Then
V̇ → 0 as t → ∞
At this stage uniformly continuity of V̇ are proven. Using
[18] V̇ is uniformly continuous if V̈ is always bounded.
Hence it should be shown that ës , ëm , ėm , ės in what conditions remain bounded.
Since em , es are bounded , Using (2) and Fig. 1 then lost
∗
= L(t)qm , qs∗ = L(t)qs
signals are qm
Using the boundedness of the first and third term of (26) it
should be proven that L̇(t) × qm is bounded.
∗
em = q m
− qs = L(t)qm − qs
→ ėm = q̇m L(t) + qm L̇(t) + q̇s
(26)
As long as qm is bounded, the boundedness of L̇(t) should
be proven. Hence the train pulse are written by Fourier series
with finite terms and derivative of that it with respect to time
is as follows :
L=
N
X
−1
2πnα
sin(
) cos(nw0 t)
πn
T
n=1
1
2πnα
[cos(
) − (−1)n ] sin(nw0 t)
πn
T
N
X
2
2πnα
2nπ
sin(
) sin(
t)
L̇ =
T
T
T
n=1
+
+
2
2πnα
2nπ
[cos(
) − (−1)n ] cos(
t)
T
T
T
The boundedness of L̇ should be investigated at the point
t = α.
L̇t=α =
N
X
2
2πnα
2nπ
sin(
) sin(
α)
T
T
T
n=1
2
2πnα
2nπ
[cos(
) − (−1)n ] cos(
α)
T
T
T
N
2 X
2nπ
=
1 − (−1)n cos(
α)
T n=1
T
+
Since Fourier series is considered to be written with finite
terms
ėm , ės is bounded
(27)
120
In order to investigate the boundedness of ëm , ës thus
100
∗
em = qm
− qs = L(t)qm − qs
80
ëm =
∗
q̈m
Fh(N)
∗
ėm = q̇m
− q̇s = L̇(t)qm + L(t)q̇m − q̇s
− q̈s = L̈(t)qm + 2L̇(t)q̇m + L(t)q̈m − q̈s
40
20
With using the boundedness of qm , L(t), L̇(t), q̈s , q̈m and q̇m
the boundedness of L̈(t) should be shown hence :
N
X
2πnα
2nπ
−1
sin(
) cos(
t)
πn
T
T
n=1
1
2πnα
2nπ
+ [cos(
) − (−1)n ] sin(
t)
πn
T
T
N
X
2πnα
2nπ
2
L̇ =
sin(
) sin(
t)
T
T
T
n=1
2πnα
2nπ
2
) − (−1)n ] cos(
t)
+ [cos(
T
T
T
N
X
2 2nπ
2πnα
2nπ
L̈ =
sin(
) cos(
t)
T
T
T
T
n=1
−
2 2nπ
2πnα
2nπ
[cos(
) − (−1)n ] sin(
t)
T T
T
T
According to variation of L(t) at t = α it is sufficient to
investigate the boundedness neighbourhood of this point.
L̈ t=α =
N
X
2 2nπ
2πnα
2nπ
sin(
) cos(
α)
T
T
T
T
n=1
2 2nπ
2πnα
2nπ
[cos(
) − (−1)n ] sin(
α)
T T
T
T
N
X
2 2nπ
2nπ
n
=
−
[−(−1) ] sin(
α)
T
T
T
n=1
−
=
N
2nπ
4π X
n
n[(−1) ] sin(
α) → ë is bounded .
2
T n=1
T
(28)
results in converging of this series to zero as α converges
to zero. Hence, L̈ remains bounded. Using (27) and (28),
V̈ remains bounded. Hence V̇ is uniformly continuous,
consequently
V̇ → 0 as t → ∞
V. SIMULATION RESULTS
In this section we simulate our proposed teleoperation
system on a single degree of freedom system with following
dynamics:
Mm q̈m = Fh + τm
Ms q̈s = τs − Fe
With using (3) the dynamics can be simplified as follow
Mm ṙm = Fh − Fmd
Ms ṙs = Fsd − Fe
0
0
10
20
30
40
50
Time(s)
Fig. 4.
apply a pulse to the master in a limited time
70
Master position
Slave position
60
50
40
Position
L=
60
30
20
10
0
−10
−20
0
10
20
30
40
50
Time(s)
Fig. 5. position Tracking and damping all signal after applying a pulse in
a limited time
The environment is set to be mass, spring and damper. The
operator moves the master by inserting forceFh .
Mm = Ms = 1 , b = 1.2
In the first step we investigate the stability of the system. In
order to investigate the dissipation of the channel, we apply a
pulse signal to the master in a limited time as depicted in Fig.
4. All signals converge to zero hence the channel is passive
and stable See Fig. 5. In second step in order to investigate
the performance of the system in the presence of different
initial conditions we consider channel with no data loss ( Tα =
0) and there is just initial condition between master and slave,
see Fig. 6. the performance remains acceptable in presence of
different initial conditions between master and slave position.
Finally to investigate the performance of the system in
the presence of data loss in communication media. Wave
variable signals were lost in channel with period T=10(s)
see Fig7 and Fig8. We assume loss rate Tα = 0.02, 0.05
and different initial conditions between master and slave see
Fig.9 and Fig.10. The resulting performance is acceptable.
Obviously, increasing loss rate results in poor transparency
although the proposed teleoperation system remains stable
with tolerable performance (even in in the presence of data
loss and different initial conditions).
VI. CONCLUSIONS
In this paper we investigated the passivity based traditional
architecture to cover position tracking in the presence of data
loss and offset of initial conditions and proposed a new data
loss model as a set of periodic continues pulses and use a
coordination architecture which uses state feedback to define
a passive output for a teleoperation system. This feedback is
30
containing both position and velocity information and simulation results verified the usefulness of mentioned architecture. Next approach would be investigating this architecture
for a general model of data loss as a stochastic phenomenon
and also improving force tracking in the presence of data
loss.
Master position
Slave position
20
Position
10
0
−10
−20
−30
0
R EFERENCES
20
40
60
80
100
Time(s)
Fig. 6.
α
T
Position tracking with different initial conditions and
=0
50
Wave variable before loss channel
40
30
Wave variable
20
10
0
−10
−20
−30
0
Fig. 7.
20
40
Time(s)
60
80
100
wave variable signal before data loss channel
50
Wave after data loss channel
40
30
Wave variable
20
10
0
−10
−20
−30
0
Fig. 8.
20
40
Time(s)
60
80
100
wave variable signal after data loss channel
30
Master position
Slave position
20
Position
10
0
−10
−20
−30
0
20
40
60
80
100
Time(s)
Fig. 9.
Position tracking with different initial conditions and
α
T
= 0.02
30
Master position
Slave position
20
Position
10
0
−10
−20
−30
0
20
40
60
80
100
Time(s)
Fig. 10.
Position tracking with different initial conditions and
α
T
= 0.05
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[12] P. Berestesky, N. Chopra, and M. W. Spong, ”Discrete time passivity
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| 3 |
Proof of Control of a UAV and a UGV Cooperating
to Manipulate an Object
arXiv:1602.08987v2 [cs.SY] 1 Mar 2016
Tam Nguyen, Emanuele Garone
Abstract—This paper focuses on the control of a system composed of an Unmanned Aerial Vehicle (UAV) and an Unmanned
Ground Vehicle (UGV) which cooperate to manipulate an object.
The two units are subject to actuator saturations and cooperate
to move the object to a desired pose, characterized by its position
and inclination. The paper proposes a control strategy where the
ground vehicle is tasked to deploy the object to a certain position,
whereas the aerial vehicle adjusts its inclination. The ground
vehicle is governed by a saturated proportional-derivative control
law. The aerial vehicle is regulated by means of a cascade control
specifically designed for this problem that is able to exploit the
mechanical interconnection. The stability of the overall system is
proved through Input-to-State Stability and Small Gain theorem
arguments. To solve the problem of constraints satisfaction, a
nonlinear Reference Governor scheme is implemented. Numerical
simulations are provided to demonstrate the effectiveness of the
proposed method.
I. I NTRODUCTION
The use of Unmanned Aerial Vehicles (UAVs) as aerial
manipulators has recently drawn the attention of several researchers around the world [1]–[9]. Early experiments conducted in controlled lab environments have demonstrated the
transportation (control of the position) [1]–[4] and manipulation (control of the position and orientation) [5]–[9] of objects
through UAVs.
Most of the works on this subject concern the transportation
of objects, through single and multiple UAVs, including grasping [1], hovering capture, load stability [2], and cooperative
transportation [3], [4]. For what concerns the manipulation of
objects through UAVs, only a few preliminary works have been
proposed. These works include the manipulation of objects
through a team of UAVs [5] or through a single UAV equipped
with robotic arms [7]. In [6], a triangular object suspended by
cables is manipulated through three UAVs.
The physical interaction between UAVs and Unmanned
Ground Vehicles (UGVs) has recently attracted some interest
and represents a relatively young research topic. Early works
on the subject include the pulling of a cart through one or two
quadrotors [10], the cooperative pose stabilization of a UAV
through a team of ground robots [11], and the modeling [12]
and control [13], [14] of tethered UAVs.
This paper proposes a control framework for the manipulation of an object through a heterogeneous team consisting of
a UAV cooperating with a UGV. Both vehicles are subject to
actuator saturations. The idea of manipulating objects using a
team of autonomous aerial and ground vehicles is, at the best
of the authors’ knowledge, new, and has potential applications
in the world of Autonomous Robotic Construction (ARC)
[15]–[19] as it could allow UAVs to collaborate with ground
vehicles to build structures in human-denied environments by
helping them positioning beams and bars.
The paper is organized as follows. First, the equations of
motion of the UGV-object-UAV system are derived using the
Euler-Lagrange approach. Then, the attainable configurations
of equilibrium of the system are discussed taking into account
the saturations of the actuators. Afterwards, a decentralized
control architecture is proposed, where the stability of the
system is proved through Input-to-State Stability and Small
Gain arguments. In order to ensure constraints satisfaction, the
control law is augmented with a nonlinear Reference Governor
[20], [21]. Numerical simulations are provided to demonstrate
the effectiveness of the proposed solution.
II. N OTATIONS
Definition 1. The saturation function σλ is defined as
σλ (x) := sign(x)min(|x|, λ),
where λ ∈
(1)
R+
0.
Definition 2. The positive saturation function σ0,λ is defined
as
(
σλ (x), if x ≥ 0
σ0,λ (x) :=
(2)
0,
if x < 0,
where λ ∈ R+
0.
III. P ROBLEM S TATEMENT
Consider the planar model of a UGV and of a quadrotor
UAV manipulating a rigid body as depicted in Fig. 1. It is
assumed that the joints between the bodies are ideal and the
center of mass of the UAV coincides with the joint position.
The UAV has mass mu ∈ R+
0 and moment of inertia Iu ∈
+
R0 . The UGV has mass mc ∈ R+
0 . The object has mass
+
+
m b ∈ R+
,
moment
of
inertia
I
∈
R
b
0
0 , and length L ∈ R0 .
+
Its center of mass is at distance dG ∈ R0 from the UGV.
Let the position of the cart x ∈ R, the inclination of the
object α ∈ [0, π], and the attitude of the UAV β ∈ [0, 2π] be
the generalized coordinates of the system. All the angles are
defined with respect to the horizon.
The bodies are subject to the gravity acceleration g. The
UAV propellers generate a total thrust u1 ∈ R+ and a resultant
The objective of this paper is to control the pose of the
object to a desired angle ᾱ and position x̄. To this end, the first
step is to analyze the attainable configurations of equilibrium
considering the saturations of the actuators.
IV. ATTAINABLE C ONFIGURATIONS
Fig. 1.
Planar model of a UAV and a UGV manipulating a rigid body.
torque u2 ∈ R. The UGV motors produce a force u3 ∈ R. The
signs of u1 , u2 , and u3 are defined positive with respect to the
oriented vectors depicted in Fig. 1. The saturations of u1 , u2 ,
and u3 are
0 ≤ u1 ≤ Umax
(3)
−Tmax ≤ u2 ≤ Tmax
−Fmax ≤ u3 ≤ Fmax ,
where Umax , Tmax , Fmax ∈ R+
0.
To derive the equations of motion, the Euler-Lagrange
method is used. To this end, consider the kinetic and potential
energies of the system T and V, respectively, which are
1
1
2
2
2 2
T = 2 mc ẋ + 2 mb (ẋ − 2ẋdG α̇ sin α + dG α̇ )
+ 21 mu (ẋ2 − 2ẋLα̇ sin α + L2 α̇2 ) + 21 Ib α̇2 + 12 Iu β̇ 2
V = mb dG sin αg + mu L sin αg.
Assuming friction forces negligible, the principle of the least
d ∂L
∂L
action dt
∂ q˙i − ∂qi = fi can be used, where L = T − V is
the Langrangian, qi are the generalized coordinates, and fi the
external forces acting on the system. The equations of motion
of the system are
2
Mtot ẍ − M L(sin αα̈ + α̇ cos α) = u3
M (− sin αẍ + cos αg) + I0 α̈
= u1 sin(β − α) (4)
= u2 ,
Iu β̈
where Mtot = mc + mb + mu is the total mass of the system,
M = mbLdG + mu the apparent mass of the UAV and the
m d2G +Ib
+ mu L the moment of inertia of
object, and I0 = b L
the system divided by the length of the object L. For the sake
of simplicity, it is assumed that mc >> mb and mc >> mu
so that the effect of the UAV dynamics on the UGV can be
neglected. Consequently, the system becomes
= u3
mc ẍ
(5)
M (− sin αẍ + cos αg) + I0 α̈ = u1 sin (β − α)
= u2 .
Iu β̈
OF
E QUILIBRIUM
In this section, the attainable configurations of equilibrium
[x̄, ᾱ, β̄]T and the associated steady state input vector ū =
[ū1 , ū2 , ū3 ]T are discussed taking into account the saturations
of the UAV. Setting all the time derivatives of (5) to zero, it
follows that the configurations of equilibrium must satisfy the
system of equations
=0
ū2
(6)
ū3
=0
M g cos ᾱ = ū1 sin (β̄ − ᾱ).
Clearly, the first two equations of (6) give ū2 = 0 and ū3 = 0
as the only input associated to an equilibrium. Moreover, any
x̄ ∈ R is an attainable point of equilibrium since x̄ does not
appear in (6). For what concerns the last equation of (6), due
to the saturation (3) of ū1 , the magnitude of ū1 sin(β̄ − ᾱ) is
maximal when
(
ū1 = Umax
(7)
β̄ = ᾱ ± π/2.
Accordingly, there are two possible cases.
1) If Umax ≥ M g, any ᾱ ∈ [0, π] is an attainable angle of
equilibrium for the object.
2) If Umax < M g, the attainable angles of equilibrium
are restricted to the interval ᾱ ∈ [αmin , αmax ], the
boundaries of which are
(
max
αmin = arccos UMg
(8)
−Umax
αmax = arccos Mg .
Finally note that, for a given steady-state angle ᾱ, the
attainable equilibria for the attitude β̄ are restricted to the
interval β̄ ∈ [βmin , βmax ]. The boundaries of this interval
can be computed solving (6) with ū1 = Umax and are
arcsin Mg cos ᾱ + ᾱ, if αmin ≤ ᾱ < π/2
U
max
βmin =
arcsin −Mg cos ᾱ + ᾱ, if π/2 ≤ ᾱ ≤ αmax
Umax
π − arcsin Mg cos ᾱ + ᾱ, if αmin ≤ ᾱ < π/2
U
max
β
max = π − arcsin −Mg cos ᾱ + ᾱ, if π/2 ≤ ᾱ ≤ α
max .
Umax
(9)
V. C ONTROL A RCHITECTURE
The proposed control architecture consists of two separate
control units in charge of governing the UGV and the UAV.
The UGV control loop generates a control input u3 such that
x(t) asymptotically tends to x̄. The UAV control loop is tasked
to regulate the inclination of the transported object. The proposed UAV controller uses a cascade control approach, where
the inner loop controls the UAV attitude and the outer loop
controls the inclination of the object. The overall asymptotic
where θ̄ := β̄ − α is the desired relative attitude of the UAV
and d := −M ẍ sin α the external disturbance induced by the
UGV. Define ft the tangential force induced by the UAV
ft := u1 sin θ̄.
(12)
I0 α̈ = ft − M g cos α + d.
(13)
Eq. (11) becomes
Fig. 2.
ft can be used as an input to control the dynamics of α. The
proposed control law is a PD with gravity compensation
Decentralized control architecture.
ft = −kp,α (α − ᾱ) − kd,α α̇ + M g cos α,
stability of the system is proved assuming a ProportionalDerivative (PD) controller for the UAV attitude.
For constraints satisfaction, a nonlinear Reference Governor
(RG) is added to the scheme. Whenever necessary, the RG
modifies the references to ensure the non-violation of the
constraints. The complete control architecture is depicted in
Fig. 2. The design of the controllers are detailed in the next
sections.
VI. UGV C ONTROL
The objective of the UGV control loop is to steer the UGV
to a desired position x̄. To this end, the nested saturated PD
control law
u3 = −σλ1 (kd,x ẋ + σλ2 (kp,x (x − x̄)))
(10)
is proposed, where kp,x , kd,x ∈ R+
0 are the parameters to be
tuned and σλ1 , σλ2 are the saturation functions (cf. Definition
1). The choice of λ1 ≤ Fmax ensures the satisfaction of the
saturation constraint on u3 . The following Lemma summarizes
the main properties of this control law.
Lemma 1. Consider the UGV in (5) controlled by the saturated PD (10).
i The closed loop system is Globally Asymptotically Stable
(GAS) for any desired point of equilibrium x̄ and for any
1
kp,x > 0, kd,x > 0, and λ2 < λ1 kd,x .
2
ii The acceleration ẍ is bounded by λ1 /mc and, for a
constant desired point of equilibrium x̄, vanishing in time,
i.e. limt→∞ ẍ = 0.
Proof: The proof can be found in Lemma 1 of [22].
VII. UAV C ONTROL
The objective of the UAV control loop is to ensure that
limt→∞ α(t) = ᾱ. To this end, a cascade strategy approach is
proposed, where the outer loop controller (see Fig. 2) is firstly
designed, assuming the inner loop ideal. Then, the stability of
the system is proved using a PD for the inner control loop.
A. Ideal Attitude Dynamics
Given a desired UAV attitude β̄, assume for the moment
that the attitude dynamics is ideal, and therefore β(t) = β̄ at
each instant t. As a consequence, the second equation of (5)
becomes
I0 α̈ = u1 sin θ̄ − M g cos α + d,
(11)
(14)
R+
0
where kp,α , kd,α ∈
are the parameters to be tuned. Eq.
(13) controlled by (14) is
I0 α̈ = −kp,α (α − ᾱ) − kd,α α̇ + d,
which is a linear system and is therefore Input-to-State Stable
(ISS) with respect to the disturbance d for any kp,α > 0 and
kd,α > 0. Since d is bounded and asymptotically tending to
zero (Lemma 1), limt→∞ α(t) = ᾱ in absence of attitude
dynamics.
At this point, it remains to determine a couple u1 and θ̄
which produces the tangential force ft . In line of principle,
Eq. (12) admits an infinite number of solutions. Rewriting (12)
in the form
ft
,
(15)
u1 =
sin θ̄
the following continuous mapping is proposed in this paper
θ̄ = σπ/2 (γ arctan (ǫft )),
(16)
where σπ/2 is the saturation function limiting the variable
to ±π/2, and γ, ǫ ∈ R+
0 are parameters to be chosen such
that thrust constraints are satisfied. Note that this mapping
always guarantees the positiveness of u1 . In fact, both ft and
sin(σπ/2 (γ arctan (ǫft ))) are odd and monotonically increasing functions. In view of (15), the quotient of two odd and
monotonically increasing functions is always positive. Remark
also that, with the mapping (16), u1 does not present any
1
singularities since limft →0 u1 = .
γǫ
To choose the parameter γ, the saturation (3) on u1 must
be satisfied when ft = Umax . It follows from (15) that, in
this case, θ̄ must be equal to π/2. As a result, following from
(16), γ must satisfy
π
.
(17)
γ=
2 arctan(ǫUmax )
For what concerns the choice of ǫ, as clarified in the following
Lemma, steady-state constraints are always ensured for any
ǫ ∈ R+
0 and therefore, ǫ can be freely chosen as a tuning
parameter.
Lemma 2. For any ǫ ∈ R+
0 , the mapping (16) with γ satisfying
(17) ensures |ū1 | ≤ Umax .
Proof: Consider first ft ∈ [0, Umax ]. In view of (14), the
control input ft at equilibrium must be
ft = M g cos ᾱ,
where ᾱ ∈ [0, π/2]. Define the minimum relative UAV attitude
θ̄min := βmin − ᾱ. Following from (9), θ̄min is
ft
.
θ̄min = arcsin
Umax
Because of the third equation of (6), to ensure |u¯1 | ≤ Umax
for all points of equilibrium, the inequality θ̄ ≥ θ̄min must
be satisfied for ft ∈ [0, Umax ]. Choosing γ as in (17), the
inequality
γ arctan (ǫft ) ≥ arcsin (ft /Umax )
R+
0
since, if restricted to ft ∈ [0, Umax ],
holds true for ǫ ∈
γ arctan (ǫft ) is convex and arcsin(ft /Umax ) is concave (see
Fig. 3). The same arguments hold true for ft ∈ [−Umax , 0],
where θ̄ ≤ θ̄max with θ̄max := βmax − ᾱ, concluding the
proof.
Using the control law (14) in (21), it follows that
I0 α̈ = (−kp,α (α − ᾱ) − kd,α α̇) cos θ̃ + δθ̃ + d,
(22)
where
δθ̃ := u1 cos θ̄ sin θ̃ − M g cos α(1 − cos θ̃),
(23)
which is the disturbance induced by the attitude error θ̃. The
following Lemma states that this disturbance is bounded for
any ft ∈ R.
Lemma 3. The disturbance δθ̃ is bounded and satisfies |δθ̃ | ≤
(2/π| sin θ̃| + M g|1 − cos θ̃|) for any ǫ ∈ R+
0 and any ft ∈ R.
Proof: The proof can be found in Appendix A.
Remark 2. It is clear that the disturbance δθ̃ vanishes when
θ̃ → 0.
Eq. (22) is the outer loop in the presence of an attitude error,
where the states [α, α̇]T are affected by the exogenous inputs
δθ̃ and d. For this system, the following result holds true.
Proposition 1. Consider the outer loop (22) for any kp,α > 0
and kd,α > 0 and for θ̃ ∈ [−θ̃max , θ̃max ] where θ̃max ∈
(−π/2, π/2).
i The system is ISS with restriction θ̃ ∈ [−θ̃max , θ̃max ] with
respect to d.
ii The system is ISS with restriction θ̃ ∈ [−θ̃max , θ̃max ] with
respect to β̃.
iii The asymptotic gain γout between β̃ and β̄˙ is finite.
Proof: The proof can be found in Appendix B.
At this point, consider the inner attitude dynamics described
by the third equation of (5). To control the inner loop, a PD
control law is chosen:
Fig. 3.
Attitude reference θ̄ tuned with ǫ → 0 and ǫ → ∞.
u2 = −kp,β β̃ − kd,β β̇,
Remark 1. For design purposes, the mapping of θ̄ can be
modified with ǫ to tune the response of the UAV with respect
to a change of ft .
B. Presence of Attitude Dynamics
In this subsection, the system dynamics seen in the previous
subsection is analyzed in the case of a non-ideal attitude
dynamics controlled by a PD. Define the attitude error β̃ :=
β − β̄. Clearly, the error on the attitude is equal to the error
on θ
θ̃ = β̃,
(18)
where θ̃ := θ − θ̄. Therefore, in the presence of an attitude
error, Eq. (11) becomes
I0 α̈ = u1 sin (θ̄ + θ̃) − M g cos α + d.
(19)
Developing the sine term, (19) becomes
I0 α̈ = u1 sin θ̄ cos θ̃ + u1 cos θ̄ sin θ̃ − M g cos α + d. (20)
Expressing u1 by the mapping (15), Eq.(20) becomes
I0 α̈ = ft cos θ̃ + u1 cos θ̄ sin θ̃ − M g cos α + d.
(21)
(24)
where kp,β , kd,β ∈ R+
0 are control parameters to be tuned. The
˙
attitude error dynamics β̃ becomes
(
˙
β̃
= β̇ − β̄˙
(25)
Iu β̈ = −kp,β β̃ − kd,β β̇.
System (25) is the inner loop, where the states [β̃, β̇]T are
˙ The following result can
affected by the exogenous input β̄.
be proved.
Proposition 2. The inner loop system (25) is ISS with respect
to β̄˙ for any kp,β > 0 and kd,β > 0. The asymptotic gain γin
between the disturbance β̄˙ and the output β̃ is finite and can
be made arbitrarily small for sufficiently large kp,β > 0 and
kd,β > 0.
Proof: The proof can be found in Appendix C.
Using ISS and Small Gain arguments, it is possible to prove
the asymptotic stability of the overall system.
Proposition 3. Consider (5) controlled by (10), (14)-(16) and
(24). Given a desired position x̄ ∈ R, a desired inclination
ᾱ ∈ [αmin , αmax ], and the resulting steady-state attitude β̄,
the point of equilibrium [x̄, ᾱ, β̄]T is asymptotically stable for
suitably large kp,β and kd,β in absence of the saturations (3)
for any initial condition satisfying
q
p
θ2 (0) + θ̇2 (0)+γin ( α2 (0) + α̇2 (0)) < (1−γin γout )|θ̃max |.
(26)
Proof: From Propositions 1 and 2, γin and γout are
proven to be finite. Since γin can be made arbitrarily small
with sufficiently large kp,β and kd,β , the product γin γout can
be made smaller than one at all times if the initial condition
satisfies (26) since, in this case, the supremum norm of θ̃
satisfies ||θ̃||∞ ≤ θ̃max . Therefore, the Small Gain Theorem
applies and the closed loop system is ISS with respect to
the UGV acceleration ẍ. Since for a constant reference this
acceleration tends to zero, asymptotic stability follows.
In this section, the control law studied in the previous
section will be augmented with the nonlinear RG introduced in
[20] to avoid constraints violation. The RG can be summarized
as follows. Let the desired position and angle references [x̄, ᾱ]
be given, where ᾱ ∈ [αmin + µ, αmax − µ] and µ an arbitrary
(small) positive scalar. If needed, the RG substitutes the
desired set-point [x̄, ᾱ] with a sequence of applied way-points
[ᾱ, x̄]k which do not make the system violate the constraints.
This sequence is computed online as follows. Assume that
at time t = k, the applied reference [x̄, ᾱ]k , if maintained
constant, would not violate the constraints. The RG computes
(at fixed time intervals) the next applied reference
The previous Proposition proves that the system is asymptotically stable in absence of the saturations (3) for all
the points of equilibrium, i.e. for any x̄ ∈ R and ᾱ ∈
[αmin , αmax ]. In the presence of the saturations (3), it can
be proved that all the previous stability results remain valid
for ᾱ ∈ [αmin + µ, αmax − µ], where µ ∈ R+
0 is arbitrarily
small.
by maximizing the scalar c ∈ [0 1] under the condition that
if [x̄, ᾱ]k+1 is kept constant, the system would not violate
constraints at any future time instant. The optimization of c
can be performed using bisection [20] and online simulations
over a sufficiently long prediction horizon. The convexity of
the steady-state admissible equilibria ensures that the waypoint sequence converges to [x̄, ᾱ].
Corollary 1. Consider (5) controlled by (10), (14)-(16) and
(24). For x̄ ∈ R and ᾱ ∈ [αmin + µ, αmax − µ], the point of
equilibrium [x̄, ᾱ, β̄]T is asymptotically stable in presence of
the saturations (3) where the initial condition satisfies (26).
Proof: In presence of the saturations (3), it is enough to
note that, since the control laws are continuous, for any point
of equilibrium ᾱ ∈ [αmin +µ, αmax −µ], there always exists a
suitably small invariant set for which saturations do not occur.
As a consequence, the same results of Proposition 3 apply.
IX. S IMULATIONS
Interestingly enough, it is possible to improve this control
law by substituting (15) with
ft
u1 = σ0,Umax
,
(27)
sin θ
where σ0,Umax is the positive saturation function limiting the
thrust to Umax (cf. Definition 2). The following Proposition
proves that the new control law (27) improves (15) and that
the system is still asymptotically stable.
Proposition 4. Consider (5) controlled by (10),(14),(16),(24)
and (27). For x̄ ∈ R and ᾱ ∈ [αmin + µ, αmax − µ], the
point of equilibrium [x̄, ᾱ, β̄]T is asymptotically stable, where
the initial condition satisfies (26). Moreover, the control law
(27) is equivalent to a feedforward that reduces the gain of
the inner loop γin by delivering a smaller attitude error θ̃f to
the outer loop, i.e. an attitude error that satisfies |θ̃f | ≤ |θ̃|.
Proof: The proof can be found in Appendix D.
In the next section, the control scheme will be improved
by making use of the nonlinear Reference Governor (RG). In
fact, the system can be made asymptotically stable for a larger
set of initial conditions by enforcing the constraints with the
RG.
VIII. C ONSTRAINTS E NFORCEMENT
[x̄, ᾱ]k+1 = (1 − c)[x̄, ᾱ]k + c[x̄, ᾱ]
(28)
Consider a UAV of mass mu = 200[g] and of inertia
Iu = 0.881[g.m2], cooperating with a UGV of mass mc =
2[kg] to manipulate an object of mass mb = 1[kg] and of
inertia Ib = 0.33[kg.m2]. The saturations of the actuators
are Umax = 5[N], Tmax = 1.3[Nm] and Fmax = 10[N].
The system is controlled using (10), (14)-(16) and (24), with
kp,x = 3, kd,x = 3, kp,α = 20, kd,α = 5, kp,β = 0.5,
kd,β = 0.01, and ǫ = 1. The initial condition of the system is
[x(0), ẋ(0), α(0), α̇(0), β(0), β̇(0)]T = [0, 0, π/3, 0, π/4, 0]T
and the desired references for the object are ᾱ = π/2 and
x̄ = 0.3[m]. Fig. 4 depicts the evolution of the states [x, α, θ]T
and of the inputs u1 , u2 and u3 . It is seen that the states are
converging to the desired references and that the UAV thrust
input does not violate the constraint on the thrust positiveness.
Finally, consider the evolution of the system for the
desired reference ᾱ = 2π/3. For the initial condition
[x(0), ẋ(0), α(0), α̇(0), β(0), β̇(0)]T = [0, 0, π/3, 0, π/4, 0]T ,
the system is unstable because the overshoot of α violates the
constraints (see first subplot of Fig. 5 (red-dashed lines)). In
fact, the object goes beyond the constraints and falls down to
α = π since the system cannot recover anymore. This is why,
to enforce the constraints and make the system asymptotically
stable, the RG (28) is implemented with a sampling time of
ts = 0.2[s] (see Fig. 5 (blue continuous line)).
X. C ONCLUSIONS
The paper introduces a control strategy where a UAV and
a UGV collaborate to manipulate an object. In particular, a
scheme is proposed where the UGV is in charge of the position
of the object and the UAV of its inclination. The stability
of this scheme is proved through Input-to-State Stability and
Small Gain theorem arguments. To ensure constraints satisfaction, a nonlinear Reference Governor is added to the control
scheme. Numerical simulations show the effectiveness of the
proposed control strategy. Future works will aim at extending
the results of this paper to the three-dimensional case.
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Fig. 4.
States and inputs evolution for Iu = 0.881[g.m] and ᾱ = π/2.
Fig. 5. States and inputs evolution for Iu = 1.762[g.m] and ᾱ = 2π/3
using the control law (27).
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A PPENDIX A
P ROOF OF L EMMA 3
Following from the Triangular Inequality, Eq. (23) is
bounded by
|δθ̃ | ≤ |u1 cos θ̄|| sin θ̃| + M g| cos α||1 − cos θ̃|.
(29)
First, note that the term M g| cos α||1 − cos θ̃| is clearly
bounded by M g|1 − cos θ̃|. For what concerns the term
|u1 cos θ̄|| sin θ̃| in (29), following from (15) and (16), u1 cos θ̄
is
ft
.
(30)
u1 cos θ̄ =
tan(σπ/2 (γ arctan(ǫft )))
For u1 6∈ [−Umax , Umax ], due to the saturation σπ/2 ,
ft
= 0. As for ft restricted
u1 cos θ̄ = 0 since limθ̄→±π/2 tan
θ̄
to [−Umax , Umax ], it is easy to see that (30) is continuous as
the only potential singularity admits a finite limit, which is
1
ft
= .
(31)
lim
ft →0 tan(γ arctan(ǫft ))
γǫ
Since u1 cos θ̄ is continuous and differentiable in the closed
interval ft ∈ [−Umax , Umax ], the possible extrema of (30)
can be found at the boundaries ft = ±Umax and at the
d
u1 cos θ̄ = 0. In fact, the only
stationary points, where
dft
d
point where
u1 cos θ̄ = 0 is ft = 0. Therefore, since
dft
1
, (30) reaches
u1 cos θ̄
= 0 and u1 cos θ̄
=
γǫ
ft =±Umax
ft =0
1
is strictly
its maximum when ft = 0. In particular, since
γǫ
1
1
decreasing for ǫ ∈ R+
is limǫ→0
=
0 , the maximum of
γǫ
γǫ
2/π. Consequently, |δθ̃ | ≤ (2/π| sin θ̃| + M g|1 − cos θ̃|) for
any ǫ ∈ R+
0 and for any ft ∈ R.
A PPENDIX B
P ROOF OF P ROPOSITION 1
+
+
Definition 3. A function α : R+
0 × R0 → R0 is of class K∞
if it is continuous, positive definite, strictly increasing, and
unbounded.
Define the object inclination error α̃ := α − ᾱ and the state
xα := [α̃, α̇]T . Consider
α̈ = (−kp α̃ − kd α̇) cos θ + δ,
(32)
where kp := kp,α /I0 , kd := kd,α /I0 , and δ := 1/I0 (δθ̃ + d).
Define as a Lyapunov function
˜
1 T (kp + ǫkd ) cos θmax
ǫ
xα ,
(33)
V = xα
ǫ
1
2
where ǫ ∈ (0, kd cos θ̃max ). The square matrix in (33) is
clearly positive definite and, by substituting (32) in (33), the
derivative of V is
ǫkp cθ
(kp + ǫkd )(cθ−θ̃max )
T
xα
V̇ = − xα
2kd cθ − ǫ
(kp + ǫkd )(cθ−θ̃max )
+ (α̇ + ǫα̃)δ
(34)
where cθ−θ̃max = 1/2(cos θ − cos θ̃max ). V̇ can be bounded
by
V̇ ≤ −xα Qxα + (α̇ + ǫα̃)δ,
(35)
ǫkp cos θ̃max
r
and r :=
where Q :=
r
2kd cos θ̃max − ǫ
1
(kp +ǫkd )(1−cos θ̃max ). To ensure that Q is positive definite,
2
the inequality
1
(kp +ǫkd )2 (1−cos θ̃max )2
4
(36)
must be imposed. To simplify the computations, let us denote
kp = ω 2 and kd = 2ξω. Substituting ǫ = 2ξω cos θ̃max ν for
ν ∈ (0, 1) in (36), the inequality becomes
(ǫkp cos θ̃max )(2kd cos θ̃max −ǫ) >
4ξ 2 ν 2 >
1
(1 − cos θ̃max )2
.
(1 + 8νξ 2 + 16ν 2 ξ 4 )
4
cos2 θ̃max
(37)
In view of (35) and (37), given ξ, there exists ν and θ̃max
such that the Lyapunov function V is strictly decreasing for
δ = 0. To prove ISS, it is sufficient to note that
(
(α̇ + ǫα̃)δ = xTα Rδ ≤ ||xTα || ||Rδ||
(38)
xTα Qxα
≥ λQ ||xα ||2 ,
where
R=
ǫ
1
and λQ is the lowest eigenvalue of the positive definite matrix
Q. As a result,
||xα || ≥ ||Rδ||
(39)
implies V̇ ≤ 0, thus proving ISS with restriction θ̃ ∈
[−θ̃max , θ̃max ] with respect to δ. Since the system is ISS with
restriction θ̃ ∈ [−θ̃max , θ̃max ] with respect to δ, it follows ISS
with restriction θ̃ ∈ [−θ̃max , θ̃max ] with respect to δθ̃ and d.
To prove that the system is ISS with restriction θ̃ ∈
[−θ̃max , θ̃max ] with respect to θ̃, it is enough to find a gain
Γ between δθ̃ and θ̃ that is a function of class K∞ . In fact,
following from Lemma 3, the disturbance δθ̃ is bounded by
|δθ̃ | ≤ 2/π| sin θ̃| + M g|1 − cos θ̃|.
(40)
The second member of the inequality is bounded by
2/π| sin θ̃| + M g|1 − cos θ̃| ≤ Γ(θ̃),
(41)
where Γ(θ̃) := 2/π|θ̃| + M g θ̃2 is a function of class K∞ .
As a consequence, since the gain between θ̃ and δθ̃ is a
function of class K∞ and the system is ISS with restriction
θ̃ ∈ [−θ̃max , θ̃max ] with respect to δθ̃ , the system is also ISS
with restriction θ̃ ∈ [−θ̃max , θ̃max ] with respect to β̃ (cf. Eq.
(18)).
It remains to prove that the gain between β̃ and β̄˙ exists
and is finite. The derivative of β̄ is
β̄˙ = θ̄˙ + α̇.
(42)
Since System (22) is ISS with restriction θ̃ ∈ [−θ̃max , θ̃max ]
with respect to β̃, it follows that |α̇| ≤ ξ(β̃) and |α| ≤ ξ(β̃),
where ξ is a function of class K∞ . Consequently, the acceleration of the object in (22) is also bounded and thus
|α̈| ≤ ∆(β̃),
(43)
where ∆ is a function of class K∞ . The derivative of (16) is
θ̄˙ =
γǫf˙t
.
1 + (ǫft )2
Since (ǫft )2 ≥ 0, the inequality
˙ ≤ γǫ|f˙ |
|θ̄|
t
holds true. Expressing the time derivative of ft , it can be said
that
˙ ≤ γǫ| − k α̇ − k α̈ − M sin αα̇g|
|θ̄|
p,α
d,α
≤ γǫ(|kp,α α̇ + kd,α α̈| + |M g α̇|)
(44)
≤ γǫ(kp,α + kd,α + M g)ζ(β̃),
where ζ(β̃) := max(ξ(β̃), ∆(β̃)). Therefore, β̄˙ is bounded
with respect to β̃ in view of (42), (43) and (44). Consequently,
there exists a finite asymptotic gain γout between the distur˙
bance β̃ and the output β̄.
A PPENDIX C
P ROOF OF P ROPOSITION 2
Since (25) is linear and asymptotically stable for any kp,β >
0 and kd,β > 0, it is also ISS and the asymptotic gain γin is the
l1 norm between β̄˙ and β̃. To prove that this gain can be made
arbitrarily small, consider the parameter choice kp,β /Iu = ω 2
and kd,β /Iu = 2ω with ω ∈ R+
0 . It results that the impulsive
response between β̄˙ and β̃ is
0
1
1
1 0 exp
w(t) =
t
−ω 2 −2ω
0
= −(1 + ωt)e−ωt .
By the definition of the l1 norm, the gain γin is
Z ∞
1
|w(t)|dt = ,
γin =
ω
0
which can be made arbitrarily small for a suitably large ω
and therefore for suitably large kp,β /Iu and kd,β /Iu . Finally,
remark that γin also depends on the UAV inertia Iu .
A PPENDIX D
P ROOF OF P ROPOSITION 4
To prove |θ̃f | ≤ |θ̃|, first let ft denote the reference for
the tangential force that is requested by the system. Let fold
and fnew be the actual forces delivered to the object using the
control laws (15) and (27), respectively, which are
ft sin θ
fold :=
sin θ̄
ft sin(θ̄ + θ̃)
(45)
=
sin θ̄
ft
fnew := σ0,Umax
sin θ.
sin θ
The control law (27) can be seen as a feedforward block on
the control law (15), which generates a fictitious attitude error
θ̃f instead of θ̃, where θ̃f is such that
ft
ft sin(θ̄ + θ̃f )
= σ0,Umax
sin θ.
(46)
sin θ
sin θ̄
Under the assumption that ft ≤ Umax , there are three cases:
1) If sign(sin(θ̄ + θ̃)) = −sign(sin θ̄), then fnew = 0,
which corresponds to the case where θ̃f = −θ̄ using
the previous mapping (15) (see Eq. (45) and (46)). Note
that
sign(sin θ) = −sign(sin θ̄)
implies that |θ̃| ≥ |θ̄| in the first law. As a consequence,
|θ̃f | ≤ |θ̃|.
ft
2) If
≥ Umax , then fnew = Umax sin θ. It is worth to
sin θ
remark that
ft sin(θ̄ + θ̃f )
ft sin(θ̄ + θ̃)
≤
≤ ft .
sin θ̄
sin θ̄
The inequalities (47) are equivalent to
sin(θ̄ + θ̃)
sin(θ̄ + θ̃f )
≤
≤ 1.
sin θ̄
sin θ̄
(47)
(48)
As a consequence, 0 ≤ |θ̃f | ≤ |θ̃|.
3) In all the other cases, no saturation occurs and fnew = ft .
In view of (45) and (46), this case is equivalent to the
first control law (15) where θ̃f = 0.
As a result, since |θ̃ | ≤ |θ̃|, the gain between β̄˙ and θ̃ is
f
f
even smaller and all the stability results of Proposition 3 and
Corollary 1 apply.
| 3 |
Deep Learning for Automatic Stereotypical Motor Movement Detection using
Wearable Sensors in Autism Spectrum Disorders
Nastaran Mohammadian Rad 1,2,3,∗, Seyed Mostafa Kia4,5 , Calogero Zarbo1 , Twan van Laarhoven2 , Giuseppe
Jurman1 , Paola Venuti6 , Elena Marchiori2 , Cesare Furlanello1
Fondazione Bruno Kessler, Via Sommarive 18, 38123, Povo, Trento, Italy
arXiv:1709.05956v1 [cs.CV] 14 Sep 2017
Abstract
Autism Spectrum Disorders are associated with atypical movements, of which stereotypical motor movements (SMMs)
interfere with learning and social interaction. The automatic SMM detection using inertial measurement units (IMU)
remains complex due to the strong intra and inter-subject variability, especially when handcrafted features are extracted
from the signal. We propose a new application of the deep learning to facilitate automatic SMM detection using multiaxis IMUs. We use a convolutional neural network (CNN) to learn a discriminative feature space from raw data. We
show how the CNN can be used for parameter transfer learning to enhance the detection rate on longitudinal data. We
also combine the long short-term memory (LSTM) with CNN to model the temporal patterns in a sequence of multi-axis
signals. Further, we employ ensemble learning to combine multiple LSTM learners into a more robust SMM detector.
Our results show that: 1) feature learning outperforms handcrafted features; 2) parameter transfer learning is beneficial
in longitudinal settings; 3) using LSTM to learn the temporal dynamic of signals enhances the detection rate especially
for skewed training data; 4) an ensemble of LSTMs provides more accurate and stable detectors. These findings provide
a significant step toward accurate SMM detection in real-time scenarios.
Keywords: Convolutional Neural Networks, Long Short-Term Memory, Transfer Learning, Ensemble Learning,
Wearable Sensors, Autism Spectrum Disorders
1. Introduction
Autism spectrum Disorder (ASD) is a complex and heterogeneous neuro-developmental disorder. Specific symptoms in ASD are difficulties in social interactions, repetitive or restricted behaviors, and verbal/nonverbal communication difficulties. Prevalence of ASD is reported to
be 1 in 88 individuals [1]. While the majority of studies
have mainly focused on social and communication problems of ASD children, the repetitive and restricted behaviors associated with ASD individuals are also objects
of interest because of their effect on the learning performance and socialization; and also as an indicator of dis∗ Corresponding
author
Email address: nastaran@fbk.eu (Nastaran Mohammadian
Rad )
1 Fondazione Bruno Kessler, Trento, Italy
2 Institute for Computing and Information Sciences, Radboud
University, Nijmegen, The Netherlands
3 Department of Information Engineering and Computer Science,
University of Trento, Trento, Italy
4 Donders Centre for Cognitive Neuroimaging, Donders Institute
for Brain, Cognition and Behaviour, Radboud University, Nijmegen,
The Netherlands
5 Department of Cognitive Neuroscience, Radboud University
Medical Centre, Nijmegen, The Netherlands
6 Department of Psychology and Cognitive Science, University of
Trento, Trento, Italy
Preprint submitted to arxiv
tress [2, 3, 4]. Stereotypical Motor Movements (SMMs) are
the major group of atypical repetitive behaviors in children
with ASD. SMMs occur without evoking stimuli and include hand flapping, body rocking, and mouthing [5, 6, 7].
SMMs have a specific negative effect on the quality of life
of ASD children: they can affect negatively on the performance of children during learning new skills and while
using the learned skills [8]. Furthermore, since these type
of movements are socially abnormal, they cause difficulties in interaction with pairs in the school or other social
settings [9]. In some cases, the severity of SMMs can lead
to a meltdown event and even can cause self-damaging
behaviors [10].
There are three traditional approaches for measuring the
SMMs [11]: 1) paper-and-pencil rating, 2) direct behavioral observation, and 3) video-based methods. Paper-andpencil rating is an interview-based approach which suffers
from the subjectivity in rating. Furthermore, it cannot
accurately detect the intensity, amount, and duration of
SMMs [12]. In the direct behavioral observation approach,
therapists can directly observe and record the sequences of
SMMs. This method is not also a reliable approach due
to the several reasons [11, 13]: first, in high speed movements, it is hard for therapists to accurately observe and
document all SMM sequences instantaneously. Second, determining the start and end time of the SMM sequences
September 19, 2017
is difficult. Third, it is impossible for therapists to concurrently record all environmental conditions and SMMs.
Video-based approaches are based on video capturing, offline coding, and analysis of SMMs. Since multiple coding
sessions of the captured videos are feasible, this observational method is much more accurate than two previous
approaches, but it is time-consuming and unpractical as a
clinical tool[14].
Considering the high prevalence rate of autism in children [15] and the limitations of existing methods for measuring SMMs, it is essential to develop time efficient and
accurate methods for automatic SMM detection. Developing a real-time SMM detection and quantification system
would be advantageous for ASD researchers, caregivers,
families, and therapists. Such a system would provide a
powerful tool to evaluate the adaptation of subjects with
ASD to diverse life contexts within an ecologic approach.
In particular, it helps to mitigate the meltdown behaviors that are anticipated by the increase in atypical behaviors. Any automatic quantification of atypical movements
would indeed help caregivers and teachers to defuse the
mechanism leading to stereotyped behaviors by involving
children in specific activities or social interactions. Such
involvement decreases the frequency of SMMs and gradually alleviates their duration and severity [16, 17]. A
real-time implementation of SMM detection system (see
Figure 1) would help therapists to evaluate the efficacy of
behavioral interventions.
Inertial Measurement Units (IMUs) provide effective
tools for measuring the frequency, intensity, and duration of physical activities over a time period via embedded accelerometer, gyroscope, and magnetometer sensors.
Due to the small size and possibility of embedding in the
mobile phones, IMUs have been adopted as common and
useful sensors for wearable devices to measure the physical activities in either constrained and free-living environments [18, 19, 20]. In recent years, human activity
recognition using IMU sensors has been widely studied.
Most of these studies tried to extract time and frequency
domain features such as mean, standard deviation, skewness, and FFT peaks from raw signals to feed them to
a classifier for activity identification [21, 22]. According
to the achieved results in human activity recognition systems, applying pattern recognition on the collected data
by IMU sensors can reliably and accurately detect physical
activities which are an evidence for the possibility of applying such techniques to automatically detect SMMs in ASD
children [23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36].
Despite meaningful amount of research in this direction,
few challenges for automatic SMM detection using wearable sensors still remain unsolved especially in real-time
applications.
One of the important challenges for accurate SMM detection is to extract a set of effective and robust features
from the IMU signal. As in many other signal processing applications, SMM detection is commonly based on
extracting handcrafted features from the IMU signals. So
far, a wide variety of feature extraction methods have been
used in the literature. Generally two main types of features
are extracted from the accelerometer signal [37]: i) time
domain features, ii) frequency domain features. For time
domain features, some statistical features such as mean,
standard deviation, zero-crossing, energy, and correlation
are extracted from the overlapping windows of the signal.
In the case of frequency features the discrete Fourier transform is used to estimate the power of different frequency
bands. In addition, the Stockwell transform [38] is proposed by [33] for feature extraction from 3-axis accelerometer signals in order to provide better time-frequency resolution for non-stationary signals. In spite of their popularity, manual feature extraction and selection suffer from
subjectivity and time inefficiency [39] that restrict the performance and also the application of SMM detection systems in real-time scenarios.
Another challenge toward developing a real-time SMM
detection system is personalization due to the intra and
inter-subject variability [33, 35]. This challenge, despite
its crucial importance, has been undervalued [33]. Intrasubject variability is mainly due to the high variability
in the intensity, duration, frequency, and topography of
SMMs in each individual with ASD. Inter-subject differences are defined by the same variability across different
individuals. Existence of these two types of variability
within and across ASD persons motivates the necessity
of developing an adaptive SMM detection algorithm that
is capable to adjust to new patterns of behaviors. Feature learning and transfer learning [40] can be considered
as candidate solutions to attack these challenges. To this
end, here we present an extended version of our previous
efforts in [41, 42] with four main contributions: 1) robust
feature learning from multi-sensor IMU signals; 2) enhancing the adaptability of SMM detection system to new data
via parameter transfer learning; 3) improving the detection
rate by incorporating the temporal dynamics of signals in
the feature learning process; and 4) using principles of the
ensemble learning to enhance the detection rate.
To achieve our first goal, we propose a new application of
the deep learning paradigm in order to directly learn discriminating features for detecting SMM patterns. In particular, we use a convolutional neural network (CNN) [43]
to bypass the commonly used feature extraction procedure.
The idea of the CNN is inspired by the visual sensory system of living creatures [44]. Following this idea, LeCun
et al. [45] developed a deep CNN architecture to address
a pattern recognition problem in computer vision. Having fewer connections and parameters due to the weight
sharing property, CNNs are easier to train compared to
other deep neural networks. Currently, CNN solutions are
among the best-performing systems on pattern recognition
systems specifically for the handwritten character [45] and
object recognition [46]. Beyond audio and image recognition systems, CNNs are successfully applied on various
types of signals. Mirowski et al. [47] applied CNN on EEG
signals for seizure detection. In the domain of psychophys2
Data collection using wearable sensors
Data Analysis
Monitoring Children
Figure 1: A real-time automatic SMM detection system. Inertial Measurement Units (IMUs) can be used for data collection. The collected
data can be analyzed locally or remotely to detect SMMs. In case of detecting abnormal movements, an alert is sent to a therapist, caregiver,
or parents.
order to learn the dynamic feature space on the sequence
of IMU data. We further show that combining multiple
LSTM models using an ensemble learning technique can
improve the stability of results. To the best of our knowledge, it is the first time that a recurrent architecture is
used for SMM detection using wearable technologies.
The rest of this paper is organized as follows: in Section 2 we first briefly review the formal definitions and
concepts about CNN, LSTM, parameter transfer learning, and the ensemble of the best base learners approach.
Then using the presented definitions we introduce the proposed CNN and LSTM architectures 1 for SMM detection
on IMU signals. The experimental materials and procedures are explained in this section. Section 3 compares our
experimental results versus the state of the art solutions
in SMM detection. Our results on a simulated dataset
and two real datasets show that feature learning via the
CNN outperforms handcrafted features in SMM classification. Furthermore, it is shown that the parameter transfer learning is beneficial in enhancing the SMM detection
rate when moving to a new dataset. Finally our results
illustrate that including the dynamics of recorded data in
feature learning process improves the classification performance in SMM detection especially when an unbalanced
training set is used in the training phase. In Section 4
we discuss how the proposed deep architecture facilitates
developing real-time SMM detection systems. We further
discuss the main limitation of our method and state the
possible future directions. Section 5 concludes this paper
by summarizing our achievements.
iology, for the first time Martinez et al. [39] proposed a
model based on CNN to predict affective states of fun, excitement, anxiety, and relaxation. Their proposed model
was tested on skin conductance and blood volume pulse
signals. Recent studies show the advantageous of applying
CNN on accelerometer signals for human activity recognition [48, 49, 50].
To fulfill our second goal, we employ the parameter
transfer learning by pre-initializing the parameters of the
CNN [51]. We hypothesize that this capability can be used
to transfer the prior knowledge regarding the distribution
of parameters from one dataset to another dataset that are
collected in a longitudinal study. If successful, our method
can be employed in order to enhance the adaptability of
SMM detection system to new unseen data, thus facilitates
its applications in wild real-world scenarios.
By the definition, SMMs are repetitive behaviors, thus
temporal patterns stored in the sequence of samples are expected to contain valuable information. Our third contribution relies on the fact that the proposed CNN architecture does not fully exploit the temporally structured information in the IMU signal sequences. This is a general issue
in SMM detection, in which the segments of IMU signals
are treated as statistically independent samples. Therefore
the possible long-term dependencies stored in the longer
temporal intervals of the signal are ignored in the detection
process. Long short-term memory (LSTM) [52] as a type
of recurrent neural networks (RNN) has been effectively
used for learning long-term temporal dependencies in sequential data such as speech recognition [53], handwriting
recognition [54], and natural language modeling [55]. Recently, the LSTM has also been successfully used for human activity recognition using wearable technologies as a
classic sequence analysis problem [56, 57, 50]. Considering
these studies in human activity recognition, it is expected
that learning the temporal patterns stored in the consecutive samples of IMU data to provide higher accurate SMM
detectors. Thus, here we propose a deep architecture,
stacking an LSTM layer on top of the CNN architecture, in
2. Methods
2.1. Notation
Let S1 , S2 , . . . , Sc ∈ RL be c time-series of length L
that are recorded by a set of inertial measurement units
1 Here the architecture implies the customization of the structural
parameters of the CNN such as the number of layers, the number
and size of filters, etc.
3
(IMUs) (e.g., accelerometer, gyroscope, and magnetometer sensors) at the sampling frequency of ν Hz. Thus,
T = L/ν represents the length of the signal in seconds.
We refer to each Si as a data channel. Now consider
Xt ∈ Rc×ν for t ∈ {1, 2, . . . , T } as a sample in the raw
feature space that is constructed by concatenating timeseries of c data channels in a given time t. Let yt ∈ {0, 1}
be the label associated to Xt where yt = 1 corresponds
to an SMM sample. In this text, we use boldface capital
letters to represent matrices, boldface lowercase letters to
represent vectors, and italic lowercase letters to represent
scalars. We represent the matrix and element-wise multiplication between A and B matrices by A · B and A B,
respectively. Further, [a, b] represents vector concatenation operation between a and b vectors.
over a pooling window with size of p elements and calν
culates the reduced feature map M0t ∈ Rf × u . In fact, a
pooling layer reduces the resolution of a feature map by
factor of u1 where u is the stride (or step) size. Max-pooling
and average-pooling are two commonly used pooling functions which compute the maximum or average value among
the values in a pooling window, respectively. In averagepooling for m0i,j ∈ M0t , i ∈ {1, . . . , f }, and j ∈ {1, . . . , uν },
we have:
p
m0i,j =
1X
mi,(j−1)×u+k .
p
(3)
k=1
Alternatively, in the max-pooling each element of the
reduced feature map is the maximum value in a corresponding pooling window:
2.2. Feature Learning via Convolutional Neural Network
m0i,j = max(mi,(j−1)×u+1 , mi,(j−1)×u+2 , . . . , mi,(j−1)×u+p ).
The goal of an SMM detector is to predict the probability of being an SMM for a given sample Xt , i.e.,
P (yt = 1 | Xt ). While the raw feature space (Xt ) is sensitive to intra and inter-subject differences, feature learning
can provide the possibility to learn a new feature space
that is robust over time and across different subjects. The
aim of feature learning is to learn a linear or non-linear
mapping function F : Xt 7→ x0t , where x0t ∈ Rd is called
the learned feature space. Then a classifier can be used in
the learned feature space to estimate P (yt = 1 | x0t ).
Convolutional neural networks (CNNs) offer an effective
infrastructure for feature learning. CNN benefits from invariant local receptive fields, shared weights, and spatiotemporal sub-sampling features to provide robustness over
shifts and distortions in the input space [43]. A classic
CNN has a hierarchical architecture that alternates convolutional and pooling layers in order to summarize large
input spaces with spatio-temporal relations into a lower dimensional feature space. A 1D-convolutional layer receives
the input signal Xt ∈ Rc×ν , convolves it with a set of f
filters with the length of m, W ∈ Rf ×c×m , and produces
a feature map Mt ∈ Rf ×ν :
The reduced feature map M0t can be used as the input
to the next convolutional layer, i.e., Xt of the next layer.
In general, the reduced feature map computed by stacking
several convolution, ReLU, and pooling layers is flattened
as a vector before the classification step. The flattening
step is performed by collapsing the rows of M0t in the form
of a vector. The resulting vector is called the learned feature space x0t that represents a new representation of the
original feature space. This new representation is typically fed to a fully connected neural network followed by
a softmax layer for the classification purposes.
In this paper, and for the purpose of SMM detection on
the multi-sensor IMU data, we propose to use a three-layer
CNN to transform the time-series of multiple sensors to a
new feature space. The proposed architecture is shown in
Figure 2. Three convolutional layers are designed to have
4, 4, and 8 filters with the length of 9 samples (i.e., 0.1
seconds), respectively. The length of the pooling window
and pooling stride are fixed to 3 (p = 3) and 2 (u = 2),
respectively. The pooling stride of 2 reduces the length
of feature maps by the factor of 0.5 after each pooling
layer. The output of the third convolutional layer after
flattening provides the learned feature vector. Then, the
learned feature vector is fed to a two-layer fully-connected
network with 8 and 2 neurons that are connected to a
softmax layer. A dropout [59] rate of 0.5 is used in the fully
connected layers to avoid the overfitting problem. Since
only the information in Xt is used to compute x0t and then
predict yt , we refer to the learned feature space via this
CNN architecture as the static feature space.
Mt = Xt ∗ W =
Pc Pm
w
× xj,1+i
Pj=1 Pi=1 1,j,i
cj=1 m
w
×
xj,1+i
2,j,i
i=1
..
.
Pc Pm
j=1
i=1 wf,j,i × xj,1+i
...
...
..
.
...
Pc Pm
i=1 w1,j,i × xj,ν+i
Pcj=1 Pm
j=1
i=1 w2,j,i × xj,ν+i
..
Pc Pm .
i=1 wf,j,i × xj,ν+i
j=1
(4)
(1)
where ∗ represents the convolution operator. The feature
map is then fed to an activation function, generally the
rectified linear unit (ReLU), to add non-linearity to the
network and also to avoid the gradient vanishing problem [58], where:
(2)
2.3. Parameter Transfer Learning via Network Preinitialization
Here max(., .) represents the element-wise max operation.
Finally, in order to reduce the sensitivity of the output
to shifts and distortions, M+
t is passed through a pooling
layer which performs a local averaging or sub-sampling
The quality and characteristics of recorded IMU signals
varies not only from subject to subject but also from time
to time in a single subject. Therefore it is important that
the SMM detector system be able to adapt to new streams
f ×ν
M+
, Mt ).
t = max(0
4
(c)
X
Convolution+ReLU
c=4
W
AVG-Pooling
M
x'
M'
(e)
f=4
p=3
m=9
Softmax
f=4
c=4
f=4
1
1
1
X
CNN Layer 1
Convolution+ReLU
c
S1
W
M
CNN Layer 2
AVG-Pooling
CNN Layer 3
M'
X
c=4
f=4
f=4
W
M
AVG-Pooling
f=8
1
(b)
x'
1
1
Sc
Flattening
f=8
m=9
1
M'
Labels
f=8
c=4
p=3
m=9
c
f=4
Convolution+ReLU
Classifier
p=3
Input
Data
Learned Features
(a)
1
1
1
1
1
(d)
1
Figure 2: (a) The proposed architecture for SMM detection in the static feature space using a three-layer CNN. (b) The first CNN layer.
This layer receives the one-second long time-series of several IMU sensors at time t, i.e., Xt , and transfer it to the first level reduced feature
map M0t . (c) The second CNN layer that uses the first level reduced feature map M0t as its input, and transfer it to the second-level reduced
feature map. (d) The third CNN layer. The reduced feature map of this layer is reshaped to the learned feature vector x0t using the flattening
operation. (e) The learned feature vector is fed to a fully-connected followed by a softmax layer to classify the samples to SMM and no-SMM
classes.
of signals in longitudinal scenarios. In this paper, we explore the possibility of parameter transfer learning via network pre-initialization in order to transfer the learned patterns to the newly seen data in a different time span. In
this direction we first formalize the background theoretical
concepts.
SMM and no-SMM samples. Assume DS , DT , TS , and
TT to represent the source domain, target domain, source
task, and target task, respectively. Transfer learning aims
to benefit from the knowledge in the source domain and
task in order to improve the predictive power in the target
domain when DS 6= DT or TS 6= TT [40].
In the statistical learning theory, the goal is to learn a
task T in a certain domain D. A domain D = {X , ρX } is
defined as a possible conjunction between an input space
X and a marginal probability distribution ρX . For example in the SMM detection context, the recorded IMU
signal for different subjects can be considered as different domains as the marginal probability distribution ρX is
different from one subject to another. Similarly different
domains can be defined by time in longitudinal data collection scenarios. Given a domain D, a task T = {Y, Φ}
is defined as a predictive function Φ from D to the output space Y. For example in this study Φ is the SMM
detector, and Y represents the categorical output space of
In this study, we are interested in the application of
parameter transfer learning via pre-initializing the parameters of a deep neural network, as a well-established technique in the deep learning community, to improve the
SMM prediction performance across different subjects and
time intervals. To this end, we define the source domain
DS as the IMU signal which is collected on several subjects at the time span T1 . Similarly the target domain
DT is defined as the IMU signal which is collected on several subjects at the time span T2 . Assume ΦS be the
learned predictive function, i.e., the CNN classifier, in the
source domain. We use the learned parameters in ΦS to
pre-initialize the parameters of the predictive function in
5
the target domain ΦT . In simpler words, instead of random pre-initialization, we initialize the parameters of CNN
classifier in the target domain with the learned CNN parameters in the source domain. We hypothesize that such
a knowledge transfer via learned parameters improves the
prediction performance in the longitudinal studies where
the data are collected at different time intervals.
Classification
Long Short-Term Memory
2.4. SMM Detection in Dynamic Feature Space using
LSTM
In SMM detection using static feature space (see Section 2.2) only the data in Xt is used to predict yt . Thus
it is implicitly assumed that the sequence of samples over
time are independent and identically distributed (i.i.d).
But in reality, this assumption is not valid as the samples
in consecutive time steps are highly dependent. Therefore,
it is expected that accounting for this dependency would
improve the performance of the SMM detector. Following this hypothesis, we propose to use a long short-term
memory (LSTM) layer to model the temporal dependency
between the consecutive time steps of the recorded signal.
Let x0t be a set of static features that are extracted or
learned from samples in the raw feature space (i.e., from
Xt ). Here we assume the CNN architecture explained in
Section 2.2 is used to compute x0t . Then, let ct ∈ Rq and
ht ∈ Rq to represent the cell state and output of an LSTM
unit at time step t, respectively, where q is the number of
neurons in the LSTM unit. We will refer to ht as the
dynamic feature space. The LSTM unit receives x0t , ht−1 ,
and ct−1 as its inputs, and computes ct and ht as follows:
ct = ft
ht = ot
ct−1 + it
c̃t ,
(1 − e−2×ct )
Feature
Extraction
Time
Figure 3: The proposed architecture for SMM detection in the dynamic feature space using long short-term memory. Each feature extraction block contains a trained three-layer CNN architecture (see
Figure 2).
In this paper, a fully-connected layer with dropout of
0.2 is used to transfer the output of the LSTM layer at
time t, i.e., ht , to the input of the softmax layer zt =
(0) (1)
[zt , zt ]T ∈ R2 :
(1)
P (yt = 1 | xt−τ , xt−τ +1 , . . . , xt ) =
(5)
(1 + e−2×ct )−1 .
Here ft ∈ Rq is called the forget gate vector and its
elements are real numbers between 0 and 1 that decide how
much information to be passed from ct−1 to ct . During
the learning phase, the forget gate learns the forget weight
matrix Wf and the forget bias vector bf . ft is computed
by
0
0
0
(8)
where it ∈ Rq is the input gate vector with elements between 0 and 1. These values determine the level of new
information in c̃t to be transferred to the cell state ct . it
is computed based on Wi and bi as follows:
0
it = (1 + e−(Wi ·[yt−1 ,xt ]+bi ) )−1 .
(9)
Finally, ot ∈ Rq is the output gate vector that filters the
cell state ct to generate the output of the LSTM unit ht :
0
ot = (1 + e−(Wo ·[yt−1 ,xt ]+bo ) )−1 .
(1)
, (11)
Due to the random initialization and using stochastic
optimization algorithms on random mini-batches in training deep learning models, retraining the same model on the
same training set results in heterogeneous approximations
of the target classifier. This heterogeneity is the direct result of reaching different local optimums in optimizing a
complex non-convex error surface. One possible approach
to overcome this problem is ensemble learning (EL) [60].
The main idea behind EL is to combine the knowledge
learned by individual classifiers in order to achieve a more
superior and stable performance. It is shown that in general an ensemble of classifiers works better than every
single classifier due to the statistical, computational, and
representational reasons [61]. Considering the success of
deep learning ensembles in pattern recognition and signal processing applications [62, 63, 64, 65], in this study
we are interested in applying classifier selection voting approach [66] to combine an ensemble of the best base learners.
(7)
(1 + e−2×(Wc ·[yt−1 ,xt ]+bc ) )−1 ,
(0)
ezt + ezt
2.5. Ensemble of the Best Base Learners
Using a tangent hyperbolic function, c̃t ∈ Rq provides
new candidate values between −1 and 1 for ct by learning
Wc and bc :
c̃t = (1 − e−2×(Wc ·[yt−1 ,xt ]+bc ) )
ezt
where τ represents the number of previous time steps that
are used as the input to the LSTM layer. Figure 3 presents
a schematic overview of the proposed architecture.
(6)
ft = (1 + e−(Wf ·[ht−1 ,xt ]+bf ) )−1 .
Feature
Extraction
Feature
Extraction
(10)
6
Algorithm 1 The training and test procedures in the
majority voting on a set of b best models.
coding is undertaken during sessions to annotate the starting and ending time of movements. The captured data
were band-pass filtered with a cut-off frequency of 0.1Hz
to remove the DC components. Then the signal was segmented to 1 second long (i.e., 100 time-points) using a
sliding window. The sliding window was moved along the
time dimension with 10 time-steps resulting in 0.9 overlaps
between consecutive windows 2 .
1: procedure training(C,Xtr ,ytr )
2:
for all ci ∈ C do
3:
Predict ŷ using ci on Xtr .
4:
Evaluate ŷ and store the performance in αi .
5:
6:
7:
8:
9:
end
for i ← 1, l do
Store the best i classifiers in Ci∗ .
Predict ŷ1 , . . . , ŷi using classifiers in Ci∗ on Xtr .
Compute majority voting ỹ on predictions in ŷ1 , . . . , ŷi .
Evaluate ỹ and store the performance in αi .
end
Find the best value for b by b = argmaxi (αi ).
return Cb∗ .
2.6.2. Real Data
We use the data presented in [33] wherein the accelerometer data were collected for 6 subjects with autism in a
longitudinal study3 . The data were collected in the laboratory and classroom environments while the subjects wore
three 3-axis wireless accelerometers and engaged in body
rocking, hand flapping, or simultaneous body rocking and
hand flapping. The accelerometer sensors were attached
to the left and right wrists, and on the torso. Offline annotation based on a recorded video is used to annotate the
data by an expert. Two separate collections are available:
the first collection, here we call it Real Data1, was recorded
by MITes sensors at 60Hz sampling frequency [31]. The
second collection Real Data2, was recorded three years after the first recording on the same subjects using Wockets
sensors with the sampling frequency of 90Hz. The sampling rate of two recordings is equalized by re-sampling the
signal in Real Data1 to 90Hz using linear interpolation.
The cut-off high pass filter at 0.1Hz is applied in order to
remove the DC components of the signal. Similar to [33],
the signal is segmented to 1-second long overlapped intervals using a sliding window. The amount of overlap is set
to 10 time-points resulting in 0.87 overlap between consecutive windows. Table 1 summarizes the number of samples
in no-SMM and SMM classes for each subject. The difference in the number of samples in SMM and no-SMM
classes shows unbalanced nature of the real data, where
in Real Data1 and Real Data2 datasets 31% and 23% of
samples are in the SMM class, respectively.
10:
11:
12:
13: procedure test(Cb∗ ,Xts , yts )
14:
Predict ŷ1 , . . . , ŷb using classifiers in Cb∗ on Xts .
15:
Compute majority voting ỹ on predictions in ŷ1 , . . . , ŷb .
16:
Evaluate ỹ and store the performance in α.
17:
return α.
Let (Xtr , ytr ) and (Xts , yts ) to be the corresponding
sample/target pairs in the training and test sets, respectively. Then assume C = {c1 , c2 , . . . cl } be a set of l base
learners trained on the training set. Our goal is to first
find a set of b best classifiers C ∗ ⊆ C based on a performance measure α on the training set, and then to combine
their prediction on the test set using majority voting in the
prediction phase. Algorithm 1 summarizes this approach.
2.6. Experimental Materials
We assess the performance of the proposed methods on
both simulated and real data. In the following, we describe
the datasets and the procedures that are used for data
preparation.
2.6.1. Simulated Data
In a simulation setting, 5 healthy subjects (3 females and
2 males) are asked to emulate stereotypical movements in a
controlled environment. Each participant wore an EXLs3
sensor1 , a miniaturized electronic device with the function
of real-time IMU, fixed on the right wrist using a wristband (see Figure 4(a)). EXLs3 sensor records three-axis
accelerometer, gyroscope, and magnetometer data (it has
9 data channels in total). The sensor was set to transmit three-axis ±16g acceleration and ±2000dps angular
velocity at the 100Hz sampling rate. The participants
were instructed to perform their normal working activities
such as sitting, writing, and typing; while intermittently
performing hand flapping upon receiving a start/stop cue
from the instructor (see Figure 4(b)-(e)). The total period
of SMMs is organized somehow to keep the distribution
of two classes comparable with real datasets where 27%
of samples are in the SMM class (see Table 1 and Section 2.6.2). The total duration of each experiment was 30
minutes organized in three 10 minutes sessions. Real-time
2.7. Experimental Setups and Evaluation
To investigate the effect of static and dynamic feature
learning and parameter transfer learning on the performance of SMM detection, we conducted four experiments.
Keras library [67] is used in our implementations4 .
2.7.1. Experiment 1: Static Feature Learning
The main aim of this experiment is to compare the effectiveness of feature learning using a deep neural network
versus raw feature space and handcrafted features in an
2 The collected simulated data is made publicly available at https:
//gitlab.fbk.eu/MPBA/smm-detection.
3 The dataset and full description of data are publicly
available
at
https://bitbucket.org/mhealthresearchgroup/
stereotypypublicdataset-sourcecodes/downloads.
4 See https://gitlab.fbk.eu/MPBA/smm-detection to access the
implemented scripts and codes.
1 For the technical description see: http://www.exelmicroel.
com/elettronica_medicale-tecnologia-indossabile-exl-s3_
module.html.
7
(c)
(b)
(a)
(d)
(e)
Figure 4: (a) The configuration of the EXLs3 sensor on the right hand. (b),(c) The simulated data are collected during daily work activities
(e.g., writing, typing, etc.). (d),(e) The subjects are asked to intermittently perform hand flapping upon receiving a start/stop cue from the
instructor.
Table 1: Number of samples in SMM and no-SMM classes in three
datasets.
Data
Simulated Data
Real Data1
Real Data2
Subjects
Sub1
Sub2
Sub3
Sub4
Sub5
Total
Sub1
Sub2
Sub3
Sub4
Sub5
Sub6
Total
Sub1
Sub2
Sub3
Sub4
Sub5
Sub6
Total
No-SMM
13875
11686
13694
12428
13583
65266
21292
12763
31780
10571
17782
12207
106395
18729
22611
40557
38796
22896
2375
145964
SMM
4075
6224
4246
5532
4367
24444
5663
4372
2855
10243
6173
17725
47031
11656
4804
268
8176
6728
11178
42810
All
17950
17910
17940
17960
17950
89710
26955
17135
34635
20814
23955
29932
153426
30385
27415
40825
46972
29624
13553
188774
distribution. The stochastic gradient descent with momentum (the momentum is fixed to 0.9) is used for training the
network. All these steps are performed only on the training data to ensure unbiased error estimation. Due to the
random initialization of weights and employing stochastic
gradient descent algorithm for optimization, results can be
different from one training run to another. Therefore, we
repeated the whole procedure of learning and classification
10 times and the mean and standard variation over runs
are reported. It is important to emphasize that, similar
to [33], in all three parts of this experiment the number
of samples in minority class is used to randomly draw a
balanced training set.
SMM/All
0.23
0.35
0.24
0.31
0.24
0.27
0.21
0.26
0.08
0.49
0.26
0.59
0.31
0.38
0.18
0.01
0.17
0.23
0.82
0.23
2.7.2. Experiment 2: Parameter Transfer Learning
As discussed before, deep neural networks provide the
capability of parameter transfer learning via network preinitialization. We applied this experiment only on two real
datasets in order to investigate the possibility of transferring learned knowledge from one dataset to another in a
longitudinal data collection setting. This experiment is
similar to Experiment 1, except for the network initialization step. Instead of random initialization, here we firstly
train the CNN on one balanced real dataset, e.g., Real
Data1, and then we use the learned parameters for preinitializing the parameters of CNN before training on another balanced real dataset, e.g., Real Data2. Similar to
previous experiment, we repeated the whole experiment
10 times to evaluate the standard deviation of the classification performance.
across-subject SMM detection setting. To evaluate the effect of both feature extraction and feature learning on the
SMM classification performance, first, without any feature
extraction the signals in raw feature space are used as the
input to a support vector machine (SVM) classifier. In this
case, all data channels of each sample Xt are collapsed into
a feature vector (with length of 900 = 9 × 100 in simulated
data and 810 = 9 × 90 in real data case). Second, to
evaluate the detection performance using handcrafted features we extracted all features mentioned in [33] including
time, frequency, and Stockwell transform features, then,
we replicated the across-subject SMM detection experiment in [33]. In this setting we used exactly the same
implementation provided by the authors 1 in the feature
extraction and classification steps. Third, a CNN architecture (see Section 2.2) is used to learn a middle representation of the multi-sensor signal. In this experiment, all effective parameters of CNN (weights and biases) are initialized by drawing small random numbers from the normal
2.7.3. Experiment 3: Training on the Unbalanced Training
Set
As explained, in Experiment 1 and 2 we balanced the
training set based on the number of samples in minority
class. Even though balancing the training set improves
the quality of the trained model but in fact it suffers from
some deficits: 1) by balancing the training set we impose
a wrong prior assumption on the original distribution of
data. As shown in Table 1 in real datasets around 0.3
of samples belong to SMM class, when by balancing the
1 The
code is available at:
https://bitbucket.org/
mhealthresearchgroup/stereotypypublicdataset-sourcecodes/
downloads.
8
Figure 5: The distribution of SMM and no-SMM samples in the 2-dimensional t-SNE space for (a) raw feature space, (b) handcrafted features,
(c) static feature space learned by CNN, (d) static feature space learned by pre-initialized CNN, and (e) dynamic feature space learned by
LSTM. Feature learning increases the separability of samples in two classes compared to raw and handcrafted features.
For all datasets, we used the same configurations for the
LSTM models by fixing τ = 25 and q = 40. We set l =
10 and used the F1-score for the performance metric α
in Algorithm 1. The experiment is repeated 10 times to
evaluate the standard deviation over the mean prediction
performance.
dataset we assume it is 0.5; 2) by balancing the training set
we cannot employ the full richness of the data as we need
to remove significant amount of samples from the training set; 3) in some practical scenarios, such as real-time
adaptation or classification on the sequence of streamed
data, balancing the training set is impractical. Considering these limitations, in this experiment in order to evaluate the effect of balancing on the performance of CNN
model we evaluate the performance of the proposed CNN
architecture in predicting SMMs when unbalanced training sets are used in the training phase.
2.7.6. Evaluation
In all experiments the leave-one-subject-out scheme is
used for model evaluation in order to measure the robustness of the trained model against inter-subject variability.
Due to the unbalanced class distributions in the test set,
we computed the F1-score to evaluate the classification
performance:
2.7.4. Experiment 4: Dynamic Feature Learning
In this experiment, we are interested in answering three
main questions: 1) what are the advantages of learning
temporal representation of IMU signals for reliable SMM
detection? 2) how long is the most informative time interval in IMU signals for detecting abnormal movements? 3)
what is the optimal configuration for the LSTM unit? To
answer these questions, we applied the proposed LSTM architecture in Section 2.4 on the three benchmark datasets
with different values for τ and q, i.e., time steps and neuron
number, respectively. We set τ = {1, 3, 5, 10, 15, 25, 50}
and q = {5, 10, 20, 30, 40, 50}. The LSTM unit is trained
on the extracted features by the CNN using the RMSProp [68] optimizer. The learned dynamic features via
LSTM (ht ) are classified to target classes using a softmax
classifier. It is worthwhile to emphasize that, in this setting, since the order of samples in the training set matters,
balancing the training set is impossible, thus we use the
original unbalanced data.
F1 = 2 ×
P recision × Recall
P recision + Recall
,
where true positive (TP), false positive (FP), and false
negative (FN) rates are used as follows to compute the
precision and recall:
TP
TP + FP
TP
Recall =
.
TP + FN
P recision =
3. Results
3.1. Feature Learning Outperforms Handcrafted Features
The classification performances summarized in Table 2
compare the quality of feature learning via CNN with raw
and handcrafted feature spaces on three datasets. In all
three datasets, the classification performance of SMM detection on the handcrafted and learned features is higher
than the classification performance on the raw feature
space. This observation demonstrates the importance of
2.7.5. Experiment 5: Ensemble of LSTMs
To explore the possible advantage of combining multiple
classifiers, we used the procedure explained in Section 2.5
in order to combine a set of b best base learners. In this
experiment, we used the LSTM models in the Experiment
4 as base learners for the classification of unbalanced data.
9
Table 2: Results for SMM detection using raw, handcrafted, static, dynamic feature spaces, and ensemble learning in three benchmarked
datasets. The results show that feature learning generally outperforms raw and handcrafted feature spaces. In addition, the parameter
transfer learning has a positive effect on the performance of the CNN classifier. Furthermore, training the CNN classifier on unbalanced
training sets causes the performance drop in feature learning and transfer learning scenarios. Using the LSTM network to extract dynamic
features from the signal alleviates this problem to some degrees. Ensemble of LSTMs shows more stable performance compared to single
LSTM classifiers.
Balanced Training Sets
Real Data2
Real Data1
Simulated
Data
Sub
Raw
Features
Handcrafted
Features
Feature
Learning
Transfer
Learning
1
2
3
4
5
Mean
1
2
3
4
5
6
Mean
1
2
3
4
5
6
Mean
0.29
0.84
0.55
0.76
0.38
0.56
0.44
0.32
0.22
0.44
0.56
0.56
0.42
0.47
0.23
0.01
0.32
0.38
0.50
0.32
0.71
0.86
0.76
0.48
0.77
0.72
0.74
0.37
0.50
0.73
0.44
0.46
0.54
0.43
0.26
0.03
0.86
0.73
0.07
0.40
0.78 ± 0.05
0.86 ± 0.03
0.80 ± 0.01
0.85 ± 0.03
0.79 ± 0.01
0.82 ± 0.03
0.74 ± 0.02
0.75 ± 0.02
0.68 ± 0.04
0.92 ± 0.01
0.51 ± 0.04
0.90 ± 0.01
0.74 ± 0.03
0.61 ± 0.11
0.20 ± 0.04
0.02 ± 0.01
0.72 ± 0.03
0.21 ± 0.09
0.36 ± 0.13
0.35 ± 0.08
0.71 ± 0.02
0.73 ± 0.01
0.70 ± 0.03
0.92 ± 0.00
0.68 ± 0.05
0.94 ± 0.01
0.78 ± 0.03
0.68 ± 0.05
0.22 ± 0.04
0.02 ± 0.01
0.77 ± 0.02
0.75 ± 0.09
0.91 ± 0.05
0.56 ± 0.05
feature extraction/learning for detecting SMMs. Furthermore, the comparison between the results achieved by
handcrafted and learned features illustrates the efficacy
of feature learning over the manual feature extraction in
SMM prediction. The learned feature space reaches on average 0.10 and 0.20 higher F1-score than the handcrafted
features in case of simulated data and real data1, respectively, while in case of real data2 its performance declines
by 0.05. These results support the overall efficacy of feature learning versus handcrafted features in extracting robust features for across-subject SMM detection. These
conclusions are even further confirmed in Figures 5(a)-(c),
where the t-distributed Stochastic Neighbor Embedding
(t-SNE) [69] technique is employed to visualize the different feature spaces in a 2-dimensional space. We used
the average of Fisher’s separability score [70] across two tSNE dimensions to quantify the separability of samples in
two classes for different feature spaces. Figure 5(a) shows
2D t-SNE distribution of SMM and no-SMM samples in
the raw feature space, where there is a high overlap between the samples of two classes. This high overlap is
also well-reflected in the low Fisher’s separability score in
raw feature space (0.02). Figure 5(b) depicts the distribution of samples of two classes in handcrafted feature space.
The samples in two classes are barely separable and the
Fisher’s separability score is 0.03. Figure 5(c) displays the
2D t-SNE space for the learned features via the CNN architecture. In this case the separability score is improved
significantly to 0.10.
Feature
Learning
(1 sec)
0.73 ± 0.13
0.78 ± 0.09
0.75 ± 0.13
0.73 ± 0.12
0.80 ± 0.04
0.76 ± 0.11
0.70 ± 0.02
0.63 ± 0.03
0.57 ± 0.08
0.88 ± 0.01
0.51 ± 0.08
0.79 ± 0.07
0.68 ± 0.06
0.33 ± 0.14
0.11 ± 0.03
0.02 ± 0.01
0.71 ± 0.14
0.14 ± 0.09
0.62 ± 0.08
0.32 ± 0.1
Unbalanced Training Sets
Feature
Dynamic
Transfer
Learning
Feature
Learning
(2.5 sec)
Learning
0.95 ± 0.01 0.95 ± 0.01
0.86 ± 0.13 0.95 ± 0.01
0.95 ± 0.03 0.97 ± 0.01
0.97 ± 0.01 0.96 ± 0.02
0.91 ± 0.01 0.90 ± 0.02
0.93 ± 0.06 0.95 ± 0.01
0.71 ± 0.03 0.73 ± 0.04 0.77 ± 0.03
0.63 ± 0.04 0.68 ± 0.04 0.71 ± 0.03
0.59 ± 0.06 0.56 ± 0.13 0.68 ± 0.05
0.88 ± 0.01 0.93 ± 0.00 0.91 ± 0.01
0.58 ± 0.07 0.51 ± 0.04 0.52 ± 0.04
0.81 ± 0.09 0.86 ± 0.12 0.90 ± 0.02
0.70 ± 0.06 0.71 ± 0.08 0.75 ± 0.03
0.36 ± 0.08 0.47 ± 0.15 0.53 ± 0.09
0.16 ± 0.04 0.13 ± 0.05 0.26 ± 0.06
0.02 ± 0.01 0.02 ± 0.01 0.02 ± 0.02
0.83 ± 0.03 0.90 ± 0.02 0.76 ± 0.09
0.09 ± 0.02 0.23 ± 0.17 0.35 ± 0.15
0.70 ± 0.16 0.68 ± 0.21 0.96 ± 0.01
0.36 ± 0.08 0.40 ± 0.13 0.48 ± 0.08
Ensemble
Learning
0.95 ± 0.00
0.96 ± 0.01
0.97 ± 0.00
0.97 ± 0.01
0.91 ± 0.00
0.95 ± 0.01
0.80 ± 0.00
0.74 ± 0.00
0.72 ± 0.01
0.93 ± 0.00
0.51 ± 0.01
0.91 ± 0.00
0.77 ± 0.01
0.59 ± 0.03
0.29 ± 0.02
0.02 ± 0.02
0.87 ± 0.02
0.43 ± 0.08
0.98 ± 0.00
0.53 ± 0.04
3.2. Parameter Transfer Learning is Beneficial in Longitudinal Studies
As mentioned in Section 2.7.2, the aim of our second
experiment was to investigate the possibility of transferring learned knowledge from one dataset to another using
parameter transfer learning. Our results in Table 2 shows
that transferring knowledge from one dataset to another
in a longitudinal study, by pre-initializing the parameters
of CNN model improves the average classification performance of the SMM detectors by 0.04 and 0.21 in Real
Data1 and Real Data2 datasets, respectively.
3.3. Training on Unbalanced Data Decreases the Performance
The results in Table 2 illustrate the negative effect of
using unbalanced training set in training CNN architecture in randomly initialized (feature learning) and preinitialized (transfer learning) scenarios. The performance
of SMM detection in feature learning scenario drops by
0.06 and 0.03 in Real Data1 and Real Data2 datasets,
respectively. This performance drop is even more pronounced in the transfer learning scenario where we observe 0.08 and 0.20 performance drop in the corresponding
datasets.
3.4. Dynamic Feature Learning Outperforms Static Feature Learning
Figure 6 compares the averaged SMM classification performance over subjects in the static feature space via the
10
Figure 6: Comparison between the classification performances of CNN and LSTM for different time-steps (τ ) and number of neurons q, on
three datasets and when an unbalanced training set is used for training the networks. The results show the superiority of the dynamic feature
space over the static feature space. While the number of neurons in the LSTM unit has little effect on the performance, using around 2.5
seconds long interval is the best choice for extracting effective dynamic features from the IMU signals.
improves the detection rates when unbalanced training
sets are used (see Table 2). This observation is consistent
across three datasets.
Figure 7 further explores the superiority of the dynamic
feature representation when the training set is unbalanced.
In the static feature space case, balancing the training
set and enforcing the wrong prior class distribution into
the classification task, despite higher recall rate, affects
negatively the precision of the classifier. In other words,
the classifiers have higher false alarm rate, which could
be problematic in real-world applications. This deficit is
recovered in the case of dynamic feature representation
where the classifier presents higher precision rate and comparable recall with respect to static features. In fact, the
LSTM-based architecture by enforcing the true prior distribution of data into the training process and, at the same
time using all the recorded samples, provides an SMM detection system with higher sensitivity and specificity.
CNN (the green dashed line for plain feature learning and
the blue dotted-dashed line for transfer learning) with the
dynamic feature space via the LSTM, the latter with different values for τ (x-axis) and q (line colors). Here in all
settings, an unbalanced training set is used in the training
phase. The results on three datasets illustrate that learning the temporal representation of signals with an LSTM
unit, consistently across datasets, improves the classification performance compared to the static feature learning
via the CNN. The classification performance improves by
increasing τ , and it reaches its highest performance around
τ = 25. Considering the consistency of the best τ value
for different subjects and different datasets, it can be concluded that using around 25 time-steps, i.e., around 2.5
seconds long interval, for extracting dynamic features is
the best choice for SMM detection purposes. On the other
hand, the results show the negligible effect of the number
of LSTM neurons (q) on the detection performance, thus,
a value around 10 can be considered a reasonable choice.
To further benchmark the advantage of dynamic feature
learning via LSTM, we used the CNN architecture for the
SMM detection on the best length for the time intervals,
i.e., on 2.5 seconds time intervals. The results on the three
datasets are summarized in Table 2 and Figure 6 (the dotted red line). The results confirm the superiority of dynamic feature learning compared to static feature learning
despite using longer time intervals for learning static features.
The effect of learning the temporal representation on
the separability of SMM and no-SMM samples is shown in
Figure 5(e). The higher Fisher’s separability score (0.17)
in the dynamic feature space compared to static feature
spaces can be considered as the basis for the higher classification performance of the proposed architecture, demonstrating the importance of learning dynamic features using
an LSTM based architecture. Furthermore, employing the
dynamic-feature representation computed by the LSTM,
3.5. Ensemble of LSTMs Stabilizes the Performance
The last column of Table 2 summarizes the results of
the ensemble approach. The results show slight boost in
the mean performance compared to single LSTM classifiers, especially on real data2 (see dashed black line in
Figure 6). Figure 7 shows that both precision and recall
contribute equally to this improvement in F1-scores. In
addition to the higher performance, the main advantage
of EL is demonstrated by the low variability of results.
This reduction in the variability is well-reflected in the reduced standard deviation around the mean performance
in real datasets (0.02 and 0.04 reduction in real data1 and
real data2 datasets, respectively). In other words an ensemble of LSTMs provides more reliable SMM detector in
comparison to every single LSTM classifier.
11
Figure 7: A comparison between precision and recall rates of classifiers trained on balanced/unbalanced training sets for static/dynamic
feature representations. Using dynamic feature space provides an SMM detection system with higher sensitivity and specificity.
4. Discussion
the SMM detector in the static feature space, exploiting
the temporal patterns of multi-sensor IMU signals recovers
its performance. This advantage facilitates training highperforming models by exploiting whole data sequences in
real-time SMM detection scenarios. Our effort, for the
first time in the SMM detection context, demonstrated the
superiority of recurrent structures in extracting discriminative temporal patterns from IMU signals. As the final
contribution, considering the important role of ensemble
learning for classifying the stream data in non-stationary
environments [66, 65], we employed ensemble of the best
base learners technique to improve the reliability of the
SMM detector. In summary, our results show that feature learning, transfer learning, ensemble learning, and
learning temporal structures from multi-modal IMU signals improve the performance of SMM detection systems
especially in more realistic scenarios.
4.1. Toward Real-Time Automatic SMM Detection
In this study, we proposed an original application of
deep learning for SMM detection in ASD children using
wearable technologies. To this end, we used a three-layer
CNN architecture for learning a more discriminative and
robust feature space in an across-subject SMM detection
scenario. Our experimental results, on the simulated and
real data, support the superiority of learning middle representation of IMU signal over traditional feature extraction methods in automatic SMM detection. Further, we
showed that the parameter transfer learning via network
pre-initialization provides the infrastructure for effective
knowledge transfer from one dataset to another in a longitudinal setting. We also presented an application of LSTM
in SMM detection context for extracting dynamic features
on the sequence of IMU data. In a comparison with the
static feature space learned via CNN architecture, we illustrated the advantage of employing the temporal information in improving the separability of SMM and no-SMM
samples. We experimentally showed that using around
2.5 seconds long interval for extracting dynamic features
is the best choice for SMM detection purposes. Further,
we showed using around 10 neurons in the LSTM unit
is a reasonable choice in order to extract the dynamics
of samples over time. As a side-advantage of learning a
dynamic feature space, we experimentally demonstrated
the higher performance of our method when, in a real
world setting, the distribution of samples in SMM and
no-SMM classes is highly skewed. We showed, while the
skewness of samples negatively affects the performance of
Developing real-time mobile applications for detecting
the abnormal movements such as SMMs can be considered as a final goal in the context of automatic SMM detection using wearable sensors. At the moment there are
numerous challenges in real-time human activity recognition using wearable sensors, namely [19, 71, 72]: 1) designing effective feature extraction and inference methods; 2)
recognition on realistic data; 3) the adaptability of system
to new users. Addressing these issues demands a huge investment in research toward finding robust and effective
features that can be extracted in a reasonable time from
the stream of IMU signal. Our proposal to learn a middle
representation of the signal, that is robust to the signal
variability of a single subject data over time and also to
the across-subject variability, can be considered as an effec12
tive solution in this direction. In addition, the parameter
transfer learning capability besides the possibility of incremental training of the proposed deep architecture facilitates the online adaptation of an automatic SMM detector
in real-time scenarios. This finding overlooks the subject
specific [73], and monolithic [74, 35] activity recognition
systems opening new frontiers toward adaptable activity
recognition systems which are more appropriate for realtime usages. At the end, the high detection performance
on unbalanced training sets achieved in the dynamic feature space facilitates the application of our method on the
realistic data when the incoming data samples are highly
skewed.
Acknowledgments
The authors would like to thank the members of MPBA
lab for their kind collaboration in collecting the simulated
data.
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4.2. Limitation and Future Work
Even though the deep architecture introduced in this
study provides a significant step toward a more accurate
automatic SMM detection system in real-time scenarios,
but it suffers from a considerable limitation: the proposed
fully supervised scheme for training the SMM detection
model is problematic for its online adaptation. This problem comes from the fact that in real applications the system has no access to the labels of incoming samples during
usage by a new user. Therefore, the adaptation to new unseen data should be performed only based on the input unlabeled data. This limitation motivates future researches
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is that no labeled data in the target domain are available.
Therefore adopting a transductive transfer learning strategy in the adaptation phase can be considered as a possible
future direction to extend this work.
5. Conclusions
In this study we addressed the problem of automatic
SMM detection in the framework of deep learning. We
used a 3-layer CNN architecture to learn a robust feature space from multi-sensor/modal IMU signals. We illustrated how the proposed architecture can be employed
for parameter transfer learning in order to enhance the
adaptability of SMM detection system to new data. Further, we showed incorporating the temporal dynamics of
the signal in the feature learning process, by combining
the CNN architecture with an LSTM unit, improves the
SMM detection rate in real-world scenarios especially in
case of unbalanced data. We further illustrated the advantage of ensemble learning to provide more stable and
reliable SMM detectors. Our results demonstrate high potentials of deep learning paradigm to address the crucial
challenges toward real-time SMM detection systems using
wearable technologies.
13
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