Related papers: Adversarial flows: A gradient flow characterizatio…

We study the quantitative convergence of drift-diffusion PDEs that arise as Wasserstein gradient flows of linearly convex functions over the space of probability measures on ${\mathbb R}^d$. In this setting, the objective is in general not…

Variational inference (VI) can be cast as an optimization problem in which the variational parameters are tuned to closely align a variational distribution with the true posterior. The optimization task can be approached through vanilla…

Crafting adversarial examples for the transfer-based attack is challenging and remains a research hot spot. Currently, such attack methods are based on the hypothesis that the substitute model and the victim model learn similar decision…

Deep neural networks have demonstrated remarkable effectiveness across a wide range of tasks such as semantic segmentation. Nevertheless, these networks are vulnerable to adversarial attacks that add imperceptible perturbations to the input…

Policy optimization is a core component of reinforcement learning (RL), and most existing RL methods directly optimize parameters of a policy based on maximizing the expected total reward, or its surrogate. Though often achieving…

Neural networks trained to minimize the logistic (a.k.a. cross-entropy) loss with gradient-based methods are observed to perform well in many supervised classification tasks. Towards understanding this phenomenon, we analyze the training…

Depending on how much information an adversary can access to, adversarial attacks can be classified as white-box attack and black-box attack. For white-box attack, optimization-based attack algorithms such as projected gradient descent…

The computation of Wasserstein gradient direction is essential for posterior sampling problems and scientific computing. The approximation of the Wasserstein gradient with finite samples requires solving a variational problem. We study the…

Graph learning plays a central role in many data mining and machine learning tasks, such as manifold learning, data representation and analysis, dimensionality reduction, clustering, and visualization. In this work, we propose a highly…

We propose a new normalized Sobolev gradient flow for the Gross-Pitaevskii eigenvalue problem based on an energy inner product that depends on time through the density of the flow itself. The gradient flow is well-defined and converges to…

The training of neural networks by gradient descent methods is a cornerstone of the deep learning revolution. Yet, despite some recent progress, a complete theory explaining its success is still missing. This article presents, for…

In this work, we study the implications of the implicit bias of gradient flow on generalization and adversarial robustness in ReLU networks. We focus on a setting where the data consists of clusters and the correlations between cluster…

Adversarial training is the most successful empirical method for increasing the robustness of neural networks against adversarial attacks. However, the most effective approaches, like training with Projected Gradient Descent (PGD) are…

Diffusion-based models on continuous spaces have seen substantial recent progress through the mathematical framework of gradient flows, leveraging the Wasserstein-2 (${W}_2$) metric via the Jordan-Kinderlehrer-Otto (JKO) scheme. Despite the…

This paper contains two contributions in the study of optimal transport on metric graphs. Firstly, we prove a Benamou-Brenier formula for the Wasserstein distance, which establishes the equivalence of static and dynamical optimal transport.…

We develop a fast and scalable numerical approach to solve Wasserstein gradient flows (WGFs), particularly suitable for high-dimensional cases. Our approach is to use general reduced-order models, like deep neural networks, to parameterize…

Adversarial examples are maliciously modified inputs created to fool deep neural networks (DNN). The discovery of such inputs presents a major issue to the expansion of DNN-based solutions. Many researchers have already contributed to the…

We develop multi-step gradient methods for network-constrained optimization of strongly convex functions with Lipschitz-continuous gradients. Given the topology of the underlying network and bounds on the Hessian of the objective function,…

Many applications in machine learning involve data represented as probability distributions. The emergence of such data requires radically novel techniques to design tractable gradient flows on probability distributions over this type of…

We establish convergence results for a spatial semidiscretization of Mean Curvature Flow (MCF) for surfaces with fixed boundaries. Our analysis is based on Huisken's evolution equations for the mean curvature and the normal vector, enabling…