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Anna Kazeykina

Publications and source records attributed to Anna Kazeykina.

17 recordsLinked to original sources
arXiv2026

Exponential convergence of Sinkhorn algorithm for entropy martingale optimal transport

We prove the exponential convergence in relative entropy of the Sinkhorn algorithm for the entropy martingale optimal transport. We assume that the marginals have compact supports and are in strict convex order, the terminal marginal support is convex, and the initial marginal support lies in the relative interior of the terminal marginal support; the reference cost is assumed to be Lipschitz in each variable. Under these assumptions we establish uniform bounds on the dual variables modulo affine gauges. This result allows us to obtain the existence and uniqueness of the optimizer, and its exponential representation in terms of dual variables. We then establish a relative-entropy stability estimate for martingale couplings with different terminal marginals. Proving that the constant in the stability estimate stays uniform throughout the Sinkhorn iteration allows us to establish the exponential convergence of the Sinkhorn algorithm.

math.PR↗
arXiv2026

Exponential Convergence of the Sinkhorn Algorithm for the Schrödinger Bridge with Regime Switching

This paper studies the convergence of the Sinkhorn algorithm for the Schrödinger bridge problem with regime switching, as introduced in Zlotchevski and Chen (2025). We consider a class of regime-switching stochastic systems on the hybrid state space $E=\mathbb{R}^d\times\{1,\ldots,m\}$, and construct the Sinkhorn iteration through the associated equivalent entropic optimal transport formulation. The main result of this paper is the exponential convergence of the Sinkhorn algorithm in relative entropy under compactness assumptions. Our proofs are inspired by the arguments recently developed for proving exponential convergence of the classical Schrödinger problem in Chiarini, Conforti, Greco and Tamanini (2024), Eckstein (2025). We perform a similar analysis for the partially observed terminal setting, where only the marginal distribution of the continuous component is prescribed at the terminal time, while the discrete regime is unobserved.

math.PR↗
arXiv2025

Entropic Optimal Transport Problem with Convex Functional Cost

We study an entropic optimal transport problem in which the transport plan is penalized by a nonlinear convex functional acting on the coupling. We establish existence, uniqueness, and uniform a priori bounds for minimizers, and we show that each minimizer satisfies a fixed-point first-order optimality system associated with an exponentially tilted reference measure. Building on this variational structure, we introduce the Sinkhorn-Frank-Wolfe (SFW) flow, prove its global well-posedness, and derive an energy-dissipation inequality yielding exponential convergence toward the unique optimal transport plan. As an application, we implement the SFW algorithm to solve an optimal routing problem for unmanned aerial vehicles with congestion aversion.

math.OC↗
arXiv2025

Dimension-free error estimate for diffusion model and optimal scheduling

Diffusion generative models have emerged as powerful tools for producing synthetic data from an empirically observed distribution. A common approach involves simulating the time-reversal of an Ornstein-Uhlenbeck (OU) process initialized at the true data distribution. Since the score function associated with the OU process is typically unknown, it is approximated using a trained neural network. This approximation, along with finite time simulation, time discretization and statistical approximation, introduce several sources of error whose impact on the generated samples must be carefully understood. Previous analyses have quantified the error between the generated and the true data distributions in terms of Wasserstein distance or Kullback-Leibler (KL) divergence. However, both metrics present limitations: KL divergence requires absolute continuity between distributions, while Wasserstein distance, though more general, leads to error bounds that scale poorly with dimension, rendering them impractical in high-dimensional settings. In this work, we derive an explicit, dimension-free bound on the discrepancy between the generated and the true data distributions. The bound is expressed in terms of a smooth test functional with bounded first and second derivatives. The key novelty lies in the use of this weaker, functional metric to obtain dimension-independent guarantees, at the cost of higher regularity on the test functions. As an application, we formulate and solve a variational problem to minimize the time-discretization error, leading to the derivation of an optimal time-scheduling strategy for the reverse-time diffusion. Interestingly, this scheduler has appeared previously in the literature in a different context; our analysis provides a new justification for its optimality, now grounded in minimizing the discretization bias in generative sampling.

stat.ML↗
arXiv2023

Ergodicity of the underdamped mean-field Langevin dynamics

We study the long time behavior of an underdamped mean-field Langevin (MFL) equation, and provide a general convergence as well as an exponential convergence rate result under different conditions. The results on the MFL equation can be applied to study the convergence of the Hamiltonian gradient descent algorithm for the overparametrized optimization. We then provide a numerical example of the algorithm to train a generative adversarial networks (GAN).

math.PR↗
arXiv2020

Game on Random Environment, Mean-field Langevin System and Neural Networks

In this paper we study a type of games regularized by the relative entropy, where the players' strategies are coupled through a random environment variable. Besides the existence and the uniqueness of equilibria of such games, we prove that the marginal laws of the corresponding mean-field Langevin systems can converge towards the games' equilibria in different settings. As applications, the dynamic games can be treated as games on a random environment when one treats the time horizon as the environment. In practice, our results can be applied to analysing the stochastic gradient descent algorithm for deep neural networks in the context of supervised learning as well as for the generative adversarial networks.

cs.GT↗
arXiv2019

Anisotropic compressed sensing for non-Cartesian MRI acquisitions

In the present note we develop some theoretical results in the theory of anisotropic compressed sensing that allow to take structured sparsity and variable density structured sampling into account. We expect that the obtained results will be useful to derive explicit expressions for optimal sampling strategies in the non-Cartesian (radial, spiral, etc.) setting in MRI.

cs.IT↗
arXiv2019

Mean-field Langevin System, Optimal Control and Deep Neural Networks

In this paper, we study a regularised relaxed optimal control problem and, in particular, we are concerned with the case where the control variable is of large dimension. We introduce a system of mean-field Langevin equations, the invariant measure of which is shown to be the optimal control of the initial problem under mild conditions. Therefore, this system of processes can be viewed as a continuous-time numerical algorithm for computing the optimal control. As an application, this result endorses the solvability of the stochastic gradient descent algorithm for a wide class of deep neural networks.

math.PR↗
arXiv2016

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation. II

In this paper we continue our study on the Cauchy problem for the two-dimensional Novikov-Veselov (NV) equation, integrable via the inverse scattering transform for the two dimensional Schrödinger operator at a fixed energy parameter. This work is concerned with the case of positive energy. For the solution of the linearized equation we derive smoothing and Strichartz estimates by combining two different frequency regimes. At non-low frequencies we also derive improved smoothing estimates with gain of almost one derivative. We combine the linear estimates with the Fourier decomposition method and $ X^{s,b} $ spaces to obtain local well-posedness of NV at positive energy in $H^s$, $s>\frac12$. Our result implies, in particular, that {\it at least} for $s>\frac12$, NV does not change its behavior from semilinear to quasilinear as energy changes sign, in contrast to the closely related Kadomtsev-Petviashvili equations. As a supplement, we provide some new explicit solutions of NV at zero energy which exhibit an interesting behavior at large times.

math.AP↗
arXiv2015

Dispersive estimates for rational symbols and local well-posedness of the nonzero energy NV equation

We consider the Cauchy problem for the two-dimensional Novikov-Veselov equation integrable via the inverse scattering problem for the Schrödinger operator with fixed negative energy. The associated linear equation is characterized by a rational symbol which is not a polynomial, except when the energy parameter is zero. With the help of a complex analysis point of view of the problem, we establish uniform decay estimates for the linear solution with gain of almost one derivative, and we use this result together with Fourier decomposition methods and $X^{s,b}$ spaces to prove local well-posedness in $H^s$, $s>\frac12$.

math.AP↗
arXiv2011

A large time asymptotics for the solution of the Cauchy problem for the Novikov-Veselov equation at negative energy with non-singular scattering data

In the present paper we are concerned with the Novikov--Veselov equation at negative energy, i.e. with the $ (2 + 1) $--dimensional analog of the KdV equation integrable by the method of inverse scattering for the two--dimensional Schrödinger equation at negative energy. We show that the solution of the Cauchy problem for this equation with non--singular scattering data behaves asymptotically as $ \frac{\const}{t^{3/4}} $ in the uniform norm at large times $ t $. We also present some arguments which indicate that this asymptotics is optimal.

math.AP↗
arXiv2011

Large time asymptotics for the Grinevich-Zakharov potentials

In this article we show that the large time asymptotics for the Grinevich-Zakharov rational solutions of the Novikov-Veselov equation at positive energy (an analog of KdV in 2+1 dimensions) is given by a finite sum of localized travel waves (solitons).

math.AP↗
arXiv2010

A large time asymptotics for transparent potentials for the Novikov-Veselov equation at positive energy

In the present paper we begin studies on the large time asymptotic behavior for solutions of the Cauchy problem for the Novikov--Veselov equation (an analog of KdV in 2 + 1 dimensions) at positive energy. In addition, we are focused on a family of reflectionless (transparent) potentials parameterized by a function of two variables. In particular, we show that there are no isolated soliton type waves in the large time asymptotics for these solutions in contrast with well-known large time asymptotics for solutions of the KdV equation with reflectionless initial data.

math.AP↗