arXiv Machine Learning By Vivek S. Borkar

Stochastic gradient descent with discontinuity across a manifold

Read the original on arXiv Machine Learning →

arXiv:2608. 07618v1 Announce Type: cross Abstract: Stochastic gradient descent for a loss function discontinuous across lower dimensional manifolds is analyzed by studying its differential equation limit.

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arXiv Machine Learning
Jul 14

Riemannian Denoising Diffusion Probabilistic Models

arXiv:2505. 04338v3 Announce Type: replace Abstract: We propose Riemannian Denoising Diffusion Probabilistic Models (RDDPMs) for learning distributions on submanifolds of Euclidean space that are level sets of functions, including most of the manifolds relevant to applications.

By Zichen Liu, Wei Zhang, Christof Sch\"utte, Tiejun Li
arXiv Machine Learning
Sep 15

Stochastic Gradient Descent over P2

The paper develops a diffusion approximation for stochastic gradient descent (SGD) when the optimization target is a functional on the Wasserstein space ℝ2. By lifting the problem to a Hilbert space via Lions differentiability, the authors construct a Gaussian random-field approximation whose velocity field matches the mean and covariance of the original stochastic gradient. They prove that this Gaussian approximation achieves second‑order weak accuracy, providing a rigorous basis for replacing sample‑driven randomness with analytically tractable Gaussian fluctuations in stochastic optimization over probability measures.

By Maria Oprea, Qin Li, Yunan Yang
arXiv Machine Learning
Jul 28

Global Convergence of DGM and PINN Algorithms for Solving Nonlinear PDEs

arXiv:2607. 24726v1 Announce Type: new Abstract: The Deep Galerkin Method (DGM) and Physics Informed Neural Networks (PINNs) have become widely-used methods for solving partial differential equations (PDEs) in the rapidly growing field of scientific machine learning.

By Justin Sirignano, Konstantinos Spiliopoulos, Samuel Cohen