Stephan Eckstein
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Fast Wasserstein rates for estimating probability distributions of probabilistic graphical models Open
Using i.i.d. data to estimate a high-dimensional distribution in Wasserstein distance is a fundamental instance of the curse of dimensionality. We explore how structural knowledge about the data-generating process which gives rise to the d…
Sparse Regularized Optimal Transport without Curse of Dimensionality Open
Entropic optimal transport -- the optimal transport problem regularized by KL diver\-gence -- is highly successful in statistical applications. Thanks to the smoothness of the entropic coupling, its sample complexity avoids the curse of di…
Hilbert’s projective metric for functions of bounded growth and exponential convergence of Sinkhorn’s algorithm Open
Motivated by the entropic optimal transport problem in unbounded settings, we study versions of Hilbert’s projective metric for spaces of integrable functions of bounded growth. These versions of Hilbert’s metric originate from cones which…
Exponential convergence of general iterative proportional fitting procedures Open
Motivated by the success of Sinkhorn's algorithm for entropic optimal transport, we study convergence properties of iterative proportional fitting procedures (IPFP) used to solve more general information projection problems. We establish e…
Time-Causal VAE: Robust Financial Time Series Generator Open
We build a time-causal variational autoencoder (TC-VAE) for robust generation of financial time series data. Our approach imposes a causality constraint on the encoder and decoder networks, ensuring a causal transport from the real market …
Optimal nonparametric estimation of the expected shortfall risk Open
We address the problem of estimating the expected shortfall risk of a financial loss using a finite number of i.i.d. data. It is well known that the classical plug-in estimator suffers from poor statistical performance when faced with (hea…
Hilbert's projective metric for functions of bounded growth and exponential convergence of Sinkhorn's algorithm Open
Motivated by the entropic optimal transport problem in unbounded settings, we study versions of Hilbert's projective metric for spaces of integrable functions of bounded growth. These versions of Hilbert's metric originate from cones which…
View article: Dimensionality Reduction and Wasserstein Stability for Kernel Regression
Dimensionality Reduction and Wasserstein Stability for Kernel Regression Open
In a high-dimensional regression framework, we study consequences of the naive two-step procedure where first the dimension of the input variables is reduced and second, the reduced input variables are used to predict the output variable w…
View article: Estimating the Rate-Distortion Function by Wasserstein Gradient Descent
Estimating the Rate-Distortion Function by Wasserstein Gradient Descent Open
In the theory of lossy compression, the rate-distortion (R-D) function $R(D)$ describes how much a data source can be compressed (in bit-rate) at any given level of fidelity (distortion). Obtaining $R(D)$ for a given data source establishe…
Optimal transport and Wasserstein distances for causal models Open
In this paper, we introduce a variant of optimal transport adapted to the causal structure given by an underlying directed graph $G$. Different graph structures lead to different specifications of the optimal transport problem. For instanc…
Stability and Sample Complexity of Divergence Regularized Optimal Transport Open
We study stability and sample complexity properties of divergence regularized optimal transport (DOT). First, we obtain quantitative stability results for optimizers of DOT measured in Wasserstein distance, which are applicable to a wide c…
Convergence Rates for Regularized Optimal Transport via Quantization Open
We study the convergence of divergence-regularized optimal transport as the regularization parameter vanishes. Sharp rates for general divergences including relative entropy or $L^{p}$ regularization, general transport costs and multi-marg…
Computational methods for adapted optimal transport Open
Adapted optimal transport (AOT) problems are optimal transport problems for distributions of a time series where couplings are constrained to have a temporal causal structure. In this paper, we develop computational tools for solving AOT p…
Quantitative Stability of Regularized Optimal Transport and Convergence of Sinkhorn's Algorithm Open
We study the stability of entropically regularized optimal transport with respect to the marginals. Lipschitz continuity of the value and Hölder continuity of the optimal coupling in $p$-Wasserstein distance are obtained under general cond…
Quantitative Stability of Regularized Optimal Transport Open
We study the stability of entropically regularized optimal transport with
respect to the marginals. Lipschitz continuity of the value and H\older
continuity of the optimal coupling in $p$-Wasserstein distance are obtained
under general con…
Limits of random walks with distributionally robust transition probabilities Open
We consider a nonlinear random walk which, in each time step, is free to choose its own transition probability within a neighborhood (w.r.t. Wasserstein distance) of the transition probability of a fixed Lévy process. In analogy to the cla…
Robust Pricing and Hedging of Options on Multiple Assets and Its Numerics Open
We consider robust pricing and hedging for options written on multiple assets given market option prices for the individual assets. The resulting problem is called the multi-marginal martingale optimal transport problem. We propose two num…
MinMax Methods for Optimal Transport and Beyond: Regularization, Approximation and Numerics Open
We study MinMax solution methods for a general class of optimization problems related to (and including) optimal transport. Theoretically, the focus is on fitting a large class of problems into a single MinMax framework and generalizing re…
Lipschitz neural networks are dense in the set of all Lipschitz functions Open
This note shows that, for a fixed Lipschitz constant $L > 0$, one layer neural networks that are $L$-Lipschitz are dense in the set of all $L$-Lipschitz functions with respect to the uniform norm on bounded sets.
Martingale transport with homogeneous stock movements Open
We study a variant of the martingale optimal transport problem in a multi-period setting to derive robust price bounds on a financial derivative. On top of marginal and martingale constraints, we introduce a time-homogeneity assumption, wh…
Limits of random walks with distributionally robust transition\n probabilities Open
We consider a nonlinear random walk which, in each time step, is free to\nchoose its own transition probability within a neighborhood (w.r.t. Wasserstein\ndistance) of the transition probability of a fixed L\\'evy process. In analogy\nto t…
View article: Robust risk aggregation with neural networks
Robust risk aggregation with neural networks Open
We consider settings in which the distribution of a multivariate random variable is partly ambiguous. We assume the ambiguity lies on the level of the dependence structure, and that the marginal distributions are known. Furthermore, a curr…
MinMax Methods for Optimal Transport and Beyond: Regularization, Approximation and Numerics Open
We study MinMax solution methods for a general class of optimization problems related to (and including) optimal transport. Theoretically, the focus is on fitting a large class of problems into a single MinMax framework and generalizing re…
Extended Laplace principle for empirical measures of a Markov chain Open
We consider discrete-time Markov chains with Polish state space. The large deviations principle for empirical measures of a Markov chain can equivalently be stated in Laplace principle form, which builds on the convex dual pair of relative…
Marginal and dependence uncertainty: bounds, optimal transport, and sharpness Open
Motivated by applications in model-free finance and quantitative risk management, we consider Fréchet classes of multivariate distribution functions where additional information on the joint distribution is assumed, while uncertainty in th…
On the full space--time discretization of the generalized Stokes equations: The Dirichlet case Open
In this work we treat the space-time discretization of the generalized Stokes equations in the case of Dirichlet boundary conditions. We prove error estimates in the case $p\in[\frac{2d}{d+2},\infty)$ that are independent of the degeneracy…