Displacement Convexity Research Articles

Mean Field Approximations via Log-Concavity

Abstract We propose a new approach to deriving quantitative mean field approximations for any probability measure $P$ on $\mathbb {R}^{n}$ with density proportional to $e^{f(x)}$, for $f$ strongly concave. We bound the mean field approximation for the log partition function $\log \int e^{f(x)}dx$ in terms of $\sum _{i \neq j}\mathbb {E}_{Q^{*}}|\partial _{ij}f|^{2}$, for a semi-explicit probability measure $Q^{*}$ characterized as the unique mean field optimizer, or equivalently as the minimizer of the relative entropy $H(\cdot \,|\,P)$ over product measures. This notably does not involve metric-entropy or gradient-complexity concepts which are common in prior work on nonlinear large deviations. Three implications are discussed, in the contexts of continuous Gibbs measures on large graphs, high-dimensional Bayesian linear regression, and the construction of decentralized near-optimizers in high-dimensional stochastic control problems. Our arguments are based primarily on functional inequalities and the notion of displacement convexity from optimal transport.

Characterization of the null energy condition via displacement convexity of entropy

AbstractWe characterize the null energy condition for an ‐dimensional, time‐oriented Lorentzian manifold in terms of convexity of the relative ‐Renyi entropy along displacement interpolations on null hypersurfaces. More generally, we also consider Lorentzian manifolds with a smooth weight function and introduce the Bakry–Emery ‐null energy condition that we characterize in terms of null displacement convexity of the relative ‐Renyi entropy. As application we then revisit Hawking's area monotonicity theorem for a black hole horizon and the Penrose Singularity Theorem from the viewpoint of this characterization and in the context of weighted Lorentzian manifolds.

Synthetic versus distributional lower Ricci curvature bounds

We compare two standard approaches to defining lower Ricci curvature bounds for Riemannian metrics of regularity below $C^2$. These are, on the one hand, the synthetic definition via weak displacement convexity of entropy functionals in the framework of optimal transport, and the distributional one based on non-negativity of the Ricci-tensor in the sense of Schwartz. It turns out that distributional bounds imply entropy bounds for metrics of class $C^1$ and that the converse holds for $C^{1,1}$-metrics under an additional convergence condition on regularizations of the metric.

Generalized curvature for the optimal transport problem induced by a Tonelli Lagrangian

We propose a generalized curvature that is motivated by the optimal transport problem on {mathbb {R}}^d with cost induced by a Tonelli Lagrangian L. We show that non-negativity of the generalized curvature implies displacement convexity of the generalized entropy functional on the L-Wasserstein space along C^2 displacement interpolants.

Lower N-weighted Ricci curvature bound with 𝜀-range and displacement convexity of entropies

In this paper, we provide a characterization of a lower [Formula: see text]-weighted Ricci curvature bound for [Formula: see text] with [Formula: see text]-range introduced by Lu–Minguzzi–Ohta [Comparison theorems on weighted Finsler manifolds and space-times with [Formula: see text]-range, Anal. Geom. Metr. Spaces 10(1) (2022) 1–30] in terms of a convexity of entropies over Wasserstein space. We further derive various interpolation inequalities and functional inequalities.

Equivalent Definitions of Very Strict CD(K,N) -spaces

We show the equivalence of the definitions of very strict CD(K,N) -condition defined, on one hand, using (only) the entropy functionals, and on the other, the full displacement convexity class mathcal{D}mathcal{C}_N. In particular, we show that assuming the convexity inequalities for the critical exponent implies it for all the greater exponents. We also establish the existence of optimal transport maps in very strict CD(K,N) -spaces with finite N.

Regularizing effects of the entropy functional in optimal transport and planning problems

We analyze optimal transport problems with additional entropic cost evaluated along curves in the Wasserstein space which join two probability measures m0,m1. The effect of the additional entropy functional results into an elliptic regularization for the (so-called) Kantorovich potentials of the dual problem. Assuming the initial and terminal measures to be positive and smooth, we prove that the optimal curve remains smooth for all time. We focus on the case that the transport problem is set on a convex bounded domain in the d-dimensional Euclidean space (with no-flux condition on the boundary), but we also mention the case of Gaussian-like measures in the whole space. The approach follows ideas introduced by P.-L. Lions in the theory of mean-field games [27]. The result provides with a smooth approximation of minimizers in optimization problems with penalizing congestion terms, which appear in mean-field control or mean-field planning problems. This allows us to exploit new estimates for this kind of problems by using displacement convexity properties in the Eulerian approach.

Generalized displacement convexity for nonlinear mobility continuity equation and entropy power concavity on Wasserstein space over Riemannian manifolds

Generalized displacement convexity for nonlinear mobility continuity equation and entropy power concavity on Wasserstein space over Riemannian manifolds

Uniform Poincaré and logarithmic Sobolev inequalities for mean field particle systems

In this paper we consider a mean field particle systems whose confinement potentials have many local minima. We establish some explicit and sharp estimates of the spectral gap and logarithmic Sobolev constants uniform in the number of particles. The uniform Poincaré inequality is based on the work of Ledoux (In Séminaire de Probabilités, XXXV (2001) 167–194, Springer) and the uniform logarithmic Sobolev inequality is based on Zegarlinski’s theorem for Gibbs measures, both combined with an explicit estimate of the Lipschitz norm of the Poisson operator for a single particle from (J. Funct. Anal. 257 (2009) 4015–4033). The logarithmic Sobolev inequality then implies the exponential convergence in entropy of the McKean–Vlasov equation with an explicit rate, We need here weaker conditions than the results of (Rev. Mat. Iberoam. 19 (2003) 971–1018) (by means of the displacement convexity approach), (Stochastic Process. Appl. 95 (2001) 109–132; Ann. Appl. Probab. 13 (2003) 540–560) (by Bakry–Emery’s technique) or the recent work (Arch. Ration. Mech. Anal. 208 (2013) 429–445) (by dissipation of the Wasserstein distance).

Generalizations of Talagrand Inequality for Sinkhorn Distance Using Entropy Power Inequality.

The distance that compares the difference between two probability distributions plays a fundamental role in statistics and machine learning. Optimal transport (OT) theory provides a theoretical framework to study such distances. Recent advances in OT theory include a generalization of classical OT with an extra entropic constraint or regularization, called entropic OT. Despite its convenience in computation, entropic OT still lacks sufficient theoretical support. In this paper, we show that the quadratic cost in entropic OT can be upper-bounded using entropy power inequality (EPI)-type bounds. First, we prove an HWI-type inequality by making use of the infinitesimal displacement convexity of the OT map. Second, we derive two Talagrand-type inequalities using the saturation of EPI that corresponds to a numerical term in our expressions. These two new inequalities are shown to generalize two previous results obtained by Bolley et al. and Bai et al. Using the new Talagrand-type inequalities, we also show that the geometry observed by Sinkhorn distance is smoothed in the sense of measure concentration. Finally, we corroborate our results with various simulation studies.

Particle approximation of one-dimensional Mean-Field-Games with local interactions

<p style='text-indent:20px;'>We study a particle approximation for one-dimensional first-order Mean-Field-Games (MFGs) with local interactions with planning conditions. Our problem comprises a system of a Hamilton-Jacobi equation coupled with a transport equation. As we deal with the planning problem, we prescribe initial and terminal distributions for the transport equation. The particle approximation builds on a semi-discrete variational problem. First, we address the existence and uniqueness of a solution to the semi-discrete variational problem. Next, we show that our discretization preserves some previously identified conserved quantities. Finally, we prove that the approximation by particle systems preserves displacement convexity. We use this last property to establish uniform estimates for the discrete problem. We illustrate our results for the discrete problem with numerical examples.</p>

Transport proofs of some discrete variants of the Prékopa-Leindler inequality

We give a transport proof of a discrete version of the displacement convexity of entropy on integers (Z), and get, as a consequence, two discrete forms of the Prekopa-Leindler Inequality : the Four Functions Theorem of Ahlswede and Daykin on the discrete hypercube [1] and a recent result on Z due to Klartag and Lehec [16].

One dimensional weighted Ricci curvature and displacement convexity of entropies

AbstractIn the present paper, we prove that a lower bound on the 1‐weighted Ricci curvature is equivalent to a convexity of entropies on the Wasserstein space. Based on such characterization, we provide some interpolation inequalities such as the Prékopa–Leindler inequality, the Borel–Branscamp–Lieb inequality, and the Brunn–Minkowski inequality under the curvature bound.

Displacement convexity of Entropy and the distance cost Optimal Transportation

During the last decade Optimal Transport had a relevant role in the study of geometry of singular spaces that culminated with the Lott–Sturm–Villani theory. The latter is built on the characterisation of Ricci curvature lower bounds in terms of displacement convexity of certain entropy functionals along W 2 -geodesics. Substantial recent advancements in the theory (localization paradigm and local-to-global property) have been obtained considering the different point of view of L 1 -Optimal transport problems yielding a different curvature dimension CD 1 (K,N) [5] formulated in terms of one-dimensional curvature properties of integral curves of Lipschitz maps. In this note we show that the two approaches produce the same curvature-dimension condition reconciling the two definitions. In particular we show that the CD 1 (K,N) condition can be formulated in terms of displacement convexity along W 1 -geodesics.

The One‐Dimensional Log‐Gas Free Energy Has a Unique Minimizer

We prove that, at every positive temperature, the infinite‐volume free energy of the one‐dimensional log‐gas, or beta‐ensemble, has a unique minimizer, which is the Sine‐beta process arising from random matrix theory. We rely on a quantitative displacement convexity argument at the level of point processes, and on the screening procedure introduced by Sandier‐Serfaty. © 2021 The Authors. Communications on Pure and Applied Mathematics published by Wiley Periodicals LLC.

Measure rigidity of synthetic lower Ricci curvature bound on Riemannian manifolds

In this paper we investigate Lott-Sturm-Villani's synthetic lower Ricci curvature bound on Riemannian manifolds with boundary. We prove several measure rigidity results related to optimal transport on Riemannian manifolds, which completely characterize CD(K,∞) condition and non-collapsed CD(K,N) condition on Riemannian manifolds with boundary. In particular, we reveal the measure rigidity of Riemannian interpolation inequality proved by Cordero-Erausquin, McCann and Schmuckenschläger. We prove that log-(semi)concave measures are the only reference measures so that displacement convexity holds on Riemannian manifolds. This is the first measure rigidity result concerning the synthetic dimension-free CD condition, which is new even on Rn. Using L1-optimal transportation theory, we also prove that CD condition yields geodesical convexity (with respect to the usual Riemannian distance).

Analysis of a two-layer neural network via displacement convexity

Fitting a function by using linear combinations of a large number $N$ of “simple” components is one of the most fruitful ideas in statistical learning. This idea lies at the core of a variety of methods, from two-layer neural networks to kernel regression, to boosting. In general, the resulting risk minimization problem is nonconvex and is solved by gradient descent or its variants. Unfortunately, little is known about global convergence properties of these approaches. Here, we consider the problem of learning a concave function $f$ on a compact convex domain $\Omega \subset {\mathbb{R}}^{d}$, using linear combinations of “bump-like” components (neurons). The parameters to be fitted are the centers of $N$ bumps, and the resulting empirical risk minimization problem is highly nonconvex. We prove that, in the limit in which the number of neurons diverges, the evolution of gradient descent converges to a Wasserstein gradient flow in the space of probability distributions over $\Omega $. Further, when the bump width $\delta $ tends to $0$, this gradient flow has a limit which is a viscous porous medium equation. Remarkably, the cost function optimized by this gradient flow exhibits a special property known as displacement convexity, which implies exponential convergence rates for $N\to \infty $, $\delta \to 0$. Surprisingly, this asymptotic theory appears to capture well the behavior for moderate values of $\delta $, $N$. Explaining this phenomenon, and understanding the dependence on $\delta $, $N$ in a quantitative manner remains an outstanding challenge.

Nonpositive curvature, the variance functional, and the Wasserstein barycenter

This paper connects nonpositive sectional curvature of a Riemannian manifold with the displacement convexity of the variance functional on the space $P(M)$ of probability measures over $M$. We show that $M$ has nonpositive sectional curvature and has trivial topology (i.e, is homeomorphic to $\mathbf{R}^n$) if and only if the variance functional on $P(M)$ is displacement convex. This is followed by a Jensen type inequality for the variance functional with respect to Wasserstein barycenters, as well as by a result comparing the variance of the Wasserstein and linear barycenters of a probability measure on $P(M)$ (that is, an element of $P(P(M))$). These results are applied to invariant measures under isometry group actions, giving a comparison for the variance functional between the Wasserstein projection and the $L^2$ projection to the set of invariant measures.

Editorial

The mathematical theory of Optimal Transport (OT) is now one of the most visible and also most active fields in nonlinear analysis, with two Fieldsmedal winners Cedric Villani and Alessio Figalli among its protagonists. The origins of OT date back to the works of GaspardMonge in the 18th century and those of Leonid Kantorovich in the ’1940s. However, what is considered as the foundations of the modern theory have mostly been established in several ground-breaking works at the end of the 20th century: among the most prominent contributors, there is Yann Brenier, who characterized optimal transport in terms of connections with PDEs and hydrodynamics in the 1980’s; there is RobertMcCann, who introduced the powerful concept of displacement convexity in the 1990’s; there is Felix Otto, who demonstrated the implications of OT to nonlinear evolution equations around the millennium.

Displacement Convexity in Spatially Coupled Scalar Recursions

We introduce a technique for the analysis of general spatially coupled systems that are governed by scalar recursions. Such systems can be expressed in variational form in terms of a potential function. We show, under mild conditions, that the potential function is displacement convex and that the minimizers are given by the fixed points (FPs) of the recursions. Furthermore, we give the conditions on the system such that the minimizing FP is unique up to translation along the spatial direction. The condition matches with that of Kudekar et al. [20] for the existence of spatial FPs. Displacement convexity applies to a wide range of spatially coupled recursions appearing in coding theory, compressive sensing, random constraint satisfaction problems, as well as statistical-mechanics models. We illustrate it with applications to low-density parity-check (LDPC) and generalized LDPC codes used for the transmission on the binary erasure channel or general binary memoryless symmetric channels within the Gaussian reciprocal channel approximation as well as compressive sensing.