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:2609. 17048v1 Announce Type: cross Abstract: We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries.
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:2607. 25624v1 Announce Type: new Abstract: Positive quadratic networks admit the low-rank representation f_U(x)=x^top UU^top x, where Uinmathbb{R}^{dtimes r} is identifiable only up to right orthogonal multiplication, representing a rank-r PSD matrix Q=UU^top.
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:2609.14307v1 Announce Type: new Abstract: Low-rank tensor factorization provides a flexible framework for completing multidimensional data from incomplete and corrupted observations. However, u...
The paper tackles two key gaps in streaming PCA using Oja's algorithm: it establishes sharp operator‑norm convergence for general‑rank subspaces under sub‑Gaussian data, and it provides distributional inference for the resulting subspace estimator. The authors remove non‑vanishing remainder terms from existing analyses, achieving rates that match minimax bounds in both dense‑tail and sparse‑tail regimes. They further develop a linearization of Oja’s iterates, enabling high‑dimensional Gaussian approximations and an online multiplier bootstrap for practical inference.
arXiv:1312. 0925v4 Announce Type: replace Abstract: Alternating Minimization is a widely used and empirically successful heuristic for matrix completion and related low-rank optimization problems.
arXiv:2608.19021v2 Announce Type: replace Abstract: Global Covariance Pooling (GCP) improves deep networks by capturing second-order feature statistics, and is especially effective for fine-grained r...
The paper introduces WaterKron, a method that integrates two-sided GPTQ with row- and column-dependent waterfilling scales and entropy coding for post‑training quantization. It derives a high‑rate distortion measure relative to the full Hessian, introducing a Kronecker‑Hessian mismatch factor Φ that quantifies the distortion penalty of using a Kronecker approximation. Minimizing Φ leads to a Gaussian covariance‑fitting problem solved via classical flip‑flop updates, yielding a FlipFlop Hessian that empirically improves KL divergence and perplexity compared to other Hessian choices.
arXiv:2606. 31390v1 Announce Type: cross Abstract: Low-rank matrix optimization is often carried out via the Burer-Monteiro (BM) formulation, but choosing the factorization rank $r$ is delicate and can substantially slow optimization.
arXiv:2609.07597v1 Announce Type: cross Abstract: Muon can be interpreted as optimizing a linear local objective over a spectral-norm ball. This gives a matrix-sign update that preserves the singular...