arXiv:2608. 07248v1 Announce Type: cross Abstract: We prove that mirror descent converges to a KKT point for the nonconvex problem without excluding boundary limits.
By Kuangyu Ding, Kim-Chuan Toh
arXiv:2609. 03762v1 Announce Type: new Abstract: The computation of the Bures-Wasserstein (BW) barycenter of an ensemble of positive definite matrices arises throughout machine learning, optimal transport, and quantum information.
By A. Afham
The paper introduces a Projected Riemannian Gradient Descent (RGD) algorithm for computing the Bures‑Wasserstein barycenter of positive definite matrices, achieving dimension‑independent linear convergence at unit step size. It resolves a previous dichotomy by showing that clipping eigenvalues to a fixed interval yields a closed‑form, non‑expansive projection in the BW metric, allowing the algorithm to match the empirical speed of unit‑step RGD while maintaining theoretical guarantees. The method also extends to the invariant matrix projection problem, providing a unified dimension‑independent analysis.
arXiv:2607. 16384v1 Announce Type: new Abstract: For stochastic gradient descent (SGD) with a constant stepsize $\alpha$, the invariant law of the iterates, centered at a minimizer, describes the behavior of the algorithm over long time horizons.
By Jingyi Zhang, Cheng Mao, Debankur Mukherjee
arXiv:2608. 05460v1 Announce Type: cross Abstract: This work introduces a proximal stochastic subgradient method for minimizing the sum of an expected cost, whose integrand is potentially nonsmooth and nonconvex, and a lower semicontinuous, prox-bounded function.
By Felipe Atenas, Alejandro Jofr\'e, Pedro P\'erez-Aros, David Torregrosa-Bel\'en
The paper investigates the geometry of full conformal prediction (FullCP) regions produced by an empirical energy‑form pairwise score. It shows that convexity of the candidate score alone does not ensure connected FullCP regions, and establishes conditions under which comparison regions share a common minimizer, making the exact conformal region star‑shaped. For power distances with exponent β≥1 the geometry is deterministic, and for β between 1 and 2 explicit Lipschitz bounds allow certified inner and outer radial envelopes with Hausdorff guarantees.
By Yiheng Feng