arXiv:2607. 07204v1 Announce Type: cross Abstract: Optimization geometrodynamics views optimizer state as evolving geometry.
arXiv:2607. 07206v1 Announce Type: new Abstract: Adaptive optimizers mix several mechanisms: a metric or preconditioner maps gradients to descent directions, while estimation, memory, step-size control, constraints, stochasticity, target modification, and discretization determine which directions are available and how they are used.
arXiv:2607. 06723v1 Announce Type: cross Abstract: Most gradient-based optimization methods move parameters through a fixed background geometry, even when their internal states implicitly define changing notions of length, curvature, and preconditioning.
arXiv:2607. 07204v2 Announce Type: replace-cross Abstract: Structured preconditioners restrict optimization to a small family of positive metrics, but endpoint condition-number reachability does not measure the geometric effort required to reach a useful metric.
arXiv:2511.02821v2 Announce Type: replace-cross Abstract: We develop new accelerated first-order algorithms in the Frank-Wolfe (FW) family for minimizing smooth convex functions over compact convex s...
arXiv:2609.39301v1 Announce Type: cross Abstract: This paper considers a class of multiblock nonconvex and nonsmooth optimization problems arising in many applications. Existing methods construct pro...
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:2601. 21487v2 Announce Type: replace-cross Abstract: We study minimization of smooth functions over feasible sets that have smooth embedded-manifold structure throughout or only on selected regions, using linear minimization oracles (LMOs) to determine search directions under user-chosen norms.
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.
arXiv:2606. 00542v1 Announce Type: new Abstract: Shampoo-style optimizers approximate gradient covariance matrices using Kronecker-factored structures.
arXiv:2609. 17089v1 Announce Type: cross Abstract: The choice of Riemannian metric can strongly influence the convergence of gradient-based optimization over covariance matrices.
arXiv:2607. 22263v1 Announce Type: cross Abstract: A data-driven inverse optimization problem (DDIOP) is the problem of estimating the objective-function parameters (weights) that explain observed optimal-solution data, and it arises in many applications, including integer linear programming (ILP).
arXiv:2607. 06723v2 Announce Type: replace-cross Abstract: Adaptive optimizers carry hidden states that change how visible gradients become parameter motion.