Related papers: Metric Flows with Neural Networks

The geometry of a ball within a Riemannian manifold is coarsely controlled if it has a lower bound on its Ricci curvature and a positive lower bound on its volume. We prove that such coarse local geometric control must persist for a…

We consider the idealized setting of gradient flow on the population risk for infinitely wide two-layer ReLU neural networks (without bias), and study the effect of symmetries on the learned parameters and predictors. We first describe a…

We study the convergence of gradient flow for the training of deep neural networks. If Residual Neural Networks are a popular example of very deep architectures, their training constitutes a challenging optimization problem due notably to…

We introduce a flow of Riemannian metrics over compact manifolds with formal limit at infinite time a shrinking Ricci soliton. We call this flow the Soliton-Ricci flow. It correspond to a Perelman's modified backward Ricci type flow with…

Traditional manifold learning algorithms often bear an assumption that the local neighborhood of any point on embedded manifold is roughly equal to the tangent space at that point without considering the curvature. The curvature indifferent…

End-to-end trained convolutional neural networks have led to a breakthrough in optical flow estimation. The most recent advances focus on improving the optical flow estimation by improving the architecture and setting a new benchmark on the…

Metric and kernel learning are important in several machine learning applications. However, most existing metric learning algorithms are limited to learning metrics over low-dimensional data, while existing kernel learning algorithms are…

This paper introduces feature gradient flow, a new technique for interpreting deep learning models in terms of features that are understandable to humans. The gradient flow of a model locally defines nonlinear coordinates in the input data…

We derive identities for general flows of Riemannian metrics that may be regarded as local mean-value, monotonicity, or Lyapunov formulae. These generalize previous work of the first author for mean curvature flow and other nonlinear…

Riemannian metric learning is an emerging field in machine learning, unlocking new ways to encode complex data structures beyond traditional distance metric learning. While classical approaches rely on global distances in Euclidean space,…

We establish a general result ensuring a $C^1$ a priori bound for smooth curves of Hermitian metrics. As a main application, we obtain a new regularity result for Hermitian curvature flows, and in particular for the second Chern-Ricci flow.

The geometric evolution equations provide new ways to address a variety of non-linear problems in Riemannian geometry, and, at the same time, they enjoy numerous physical applications, most notably within the renormalization group analysis…

We consider the Ricci flow for simply connected nilmanifolds, which translates to a Ricci flow on the space of nilpotent metric Lie algebras. We consider the evolution of the inner product and the evolution of structure constants, as well…

Diffusion and flow models have become the dominant paradigm for generative modeling on Riemannian manifolds, with successful applications in protein backbone generation and DNA sequence design. However, these methods require tens to…

Motivated by a constrained minimization problem, it is studied the gradient flows with respect to Hessian Riemannian metrics induced by convex functions of Legendre type. The first result characterizes Hessian Riemannian structures on…

We study the behavior of a three-dimensional dynamical system with respect to some set $S$ given in 3-dimensional euclidian space. Geometrically such a system arises from the normalized Ricci flow on some class of generalized Wallach spaces…

This paper is devoted to the study of the evolution of positively curved metrics on the Wallach spaces $SU(3)/T_{\max}$, $Sp(3)/Sp(1)\times Sp(1)\times Sp(1)$, and $F_4/Spin(8)$. We prove that for all Wallach spaces, the normalized Ricci…

The implicit bias towards solutions with favorable properties is believed to be a key reason why neural networks trained by gradient-based optimization can generalize well. While the implicit bias of gradient flow has been widely studied…

Ricci flow deforms the Riemannian metric proportionally to the curvature, such that the curvature evolves according to a heat diffusion process and eventually becomes constant everywhere. Ricci flow has demonstrated its great potential by…

Motivated by recent progress in the problem of numerical K\"ahler metrics, we survey machine learning techniques in this area, discussing both advantages and drawbacks. We then revisit the algebraic ansatz pioneered by Donaldson. Inspired…