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: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: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:2607. 24518v1 Announce Type: new Abstract: Symmetric non-negative matrix factorization (SymNMF) recovers latent group structure from a dependence matrix, but its dense, quadratic-memory objective has confined prior work to moderate sizes.
The paper introduces low‑rank orthogonalization, a technique that exploits the low‑rank nature of gradients in neural network training to perform matrix orthogonalization more efficiently. Building on this, the authors present low‑rank matrix‑signed gradient descent (MSGD) and a low‑rank variant of the Muon optimizer, showing through experiments that low‑rank Muon matches or surpasses vanilla Muon on GPT‑2 and LLaMA pretraining, especially for larger models. Theoretical analysis provides iteration‑complexity bounds for both low‑rank MSGD and low‑rank Muon under heavy‑tailed noise.
arXiv:2406. 10407v3 Announce Type: replace-cross Abstract: Semidefinite programs (SDPs) and their solvers are powerful tools with many applications in machine learning and data science.
arXiv:2109. 11057v2 Announce Type: replace-cross Abstract: Weighted low-rank matrix approximation (WLRMA) generalizes classical low-rank approximation and matrix completion by allowing arbitrary elementwise weights.
arXiv:2605. 28613v2 Announce Type: replace-cross Abstract: This paper studies the stability of low-rank implicit regularization in deep matrix factorization, a tractable model for understanding how gradient-based training can favor low-complexity structure.
arXiv:2608.21607v1 Announce Type: cross Abstract: We investigate when a sparse nonnegative matrix can be recovered from a real-valued matrix of much lower rank by zeroing out its negative elements. T...
The paper introduces a deterministic Simulated Oracle Direction (SOD) framework that enables escaping spurious local minima in non‑convex low‑rank matrix sensing without explicit tensor lifting. By projecting over‑parameterized escape directions back into the original parameter space, the method guarantees a strict decrease in objective value from existing local minima. Experiments show reliable escape from local minima and convergence to global optima with minimal computational overhead compared to explicit over‑parameterization.
LoRA-TSD introduces a new optimizer for low‑rank adaptation (LoRA) that treats each update as a tangent vector on the fixed‑rank matrix manifold and applies a Muon‑style spectral‑norm steepest‑descent step within that tangent space. The method avoids costly full‑matrix operations and offers a retraction that is up to 2.8× cheaper than previous manifold approaches. The authors prove that their surrogate recovers LoRA‑Pro, identify the Riemannian gradient as the natural stationarity measure, and provide the first global convergence guarantees for both LoRA‑Pro and LoRA‑TSD, achieving superior performance across multiple benchmarks with Llama and Qwen models.
arXiv:2607. 25967v2 Announce Type: replace-cross Abstract: Singular Value Decomposition (SVD) underlies matrix factorisation tasks across many fields, with imaging applications demanding real-time processing.
arXiv:2607. 27507v1 Announce Type: new Abstract: Matrix factorisation is a fundamental tool for exploiting low-dimensional structure in high-dimensional data, with applications such as data compression, denoising, structure discovery, interpretable representation learning, and dimensionality reduction.