Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2406.08209

Forward-Euler time-discretization for Wasserstein gradient flows can be wrong

Yewei Xu·Qin Li
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this note, we examine the forward-Euler discretization for simulating Wasserstein gradient flows. We provide two counter-examples showcasing the failure of this discretization even for a simple case where the energy functional is defined as the KL divergence against some nicely structured probability densities. A simple explanation of this failure is also discussed.

Explore related subjects

stat.MLcs.LGmath.OC

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Yewei Xu, Qin Li. 2024-06-12. Forward-Euler time-discretization for Wasserstein gradient flows can be wrong. https://arxiv.org/abs/2406.08209

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Statistical Properties of Deep Neural Networks with Dependent Data

This paper develops theory for deep neural network (DNN) estimators under dependent data. To provide theory applicable to a variety of DNN-based estimators, I first establish nonasymptotic probability bounds on the theoretical and empirical $\mathcal{L}^{2}$-errors of nonparametric sieve estimators for a general class of estimation problems under possibly nonstationary $β$-mixing data taking values in unbounded sets. I then apply the theory to fully connected and convolutional DNN estimators without bounds or sparsity restrictions on the DNN weights. For both DNN classes, I derive general results when the function to be estimated is Hölder smooth and the data are nonstationary, subgaussian, and $β$-mixing with either exponential or polynomial decay. I then specialize these to nonparametric regression, logistic regression, and quantile regression settings. Under exponential $β$-mixing, the resulting estimators attain the nonparametric minimax rate of Stone (1982) up to logarithmic factors.

stat.ML↗
arXiv2026

Regular Fourier Features for Nonstationary Gaussian Processes

Simulating a Gaussian process requires sampling from a high-dimensional Gaussian distribution, which scales cubically with the number of sample locations. Spectral methods address this challenge by exploiting the Fourier representation and treating the spectral density as a probability distribution suitable for Monte Carlo approximation. Although this probabilistic interpretation is valid for stationary processes, it is overly restrictive for the nonstationary case, where spectral densities are generally not probability measures. To avoid this limitation, we propose regular Fourier features for harmonizable processes with one-dimensional inputs. Our method discretizes the spectral representation directly, preserving the correlation structure among spectral weights without requiring probability assumptions. Assuming finite spectral support, this yields an efficient low-rank approximation that is positive semi-definite by construction and consistent under mild regularity conditions. When the spectral density is unknown, the framework also extends to kernel learning from data, which we explore as a proof of concept. We demonstrate the approximation on locally stationary and harmonizable mixture kernels, the latter with a complex-valued spectral density. As a feasibility study, we then apply the kernel-learning extension to real and synthetic data, where it matches competitive baselines.

stat.ML↗
arXiv2026

Simultaneous Latent Budget Trees for Stratified Classification

In the era of Explainable Artificial Intelligence, there is a renewed focus on single trees for their ease of interpretation. This paper introduces Simultaneous Latent Budget Trees, a probabilistic machine learning framework for classification trees in the presence of a stratification factor such as a temporal, spatial, or demographic variable, acting as a control variable or potential confounder. Standard tree growth procedures are not designed to optimize a conditional split rule. A model-based split rule is proposed in which child nodes are interpreted as latent components of a simultaneous mixture model, such as the Simultaneous Latent Budget Model and its constrained versions, fitted to the parent node. Mixing parameters drive the observations, differently for each group, to the child nodes whereas latent budgets parameters update the response classes profile of each level of the control variable. Parameters are estimated by least squares considering a neural network perspective of the model. An informative tree structure can be interactively visualized with interpretation aids on the node and the paths, including visual pruning and decision tree selection procedure. Suitable measures are proposed to handle an unbalanced response class distribution. The proposed methodology is applied to investigate gender-related differences in disease progression of Amyotrophic Lateral Sclerosis. The SLBT library with the various tree-based algorithms is available in the linked GitHub repository.

stat.ML↗