The paper introduces a novel technique called "persistence of memory" to enhance stochastic subspace methods for large‑scale optimisation. By using a weakly correlated guidance vector that is refreshed only at wide intervals, the method provides a structured direction for random subspace descent. The authors demonstrate that this guidance can be efficiently computed in sparse or minibatch settings and present the first theoretical analysis of classical SSD methods for sparse functions, showing alignment with low‑lying Hessian eigenvectors near the optimum.
By Subhroshekhar Ghosh, Clement Z. Q. Ng, Pierre-Louis Poirion, Akiko Takeda
arXiv:2606. 00413v1 Announce Type: cross Abstract: Sufficient dimension reduction (SDR) makes high-dimensional regression tractable by projecting the covariates onto a low-dimensional subspace that preserves the conditional mean of the response.
By Thibault Pautrel, Fran\c{c}ois Portier
arXiv:2607. 25299v1 Announce Type: cross Abstract: Optimization over the Stiefel manifold plays a significant role in various machine learning tasks.
By Yuan Zhang, Jiang Hu, Zhijian Lai, Lin Lin, Zaiwen Wen
Low-rank adaptation (LoRA) is the standard way to fine-tune large models, yet when its two factors are trained independently, the update ignores the geometry of the low-rank weight change it induces. We introduce LoRA-TSD, an optimizer that treats every LoRA step as a tangent vector of the fixed-rank matrix manifold and takes the spectral-norm steepest-descent step of Muon inside that tangent space, mapping the result back to the factors through a retraction native to the LoRA parametrization.
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.
By Dmitrii Andriianov, Andrey Veprikov, Aleksandr Beznosikov
arXiv:2609.36692v1 Announce Type: cross
Abstract: Matrix optimizers have emerged as a promising direction, with Muon standing out as a prominent design. Revisiting Muon through its full-Gram represen...
By Zixuan Gong, Zeyu Gan, Jiaye Teng, Yong Liu