Riemannian Gradient Descent for Low-Rank Architectures

In this paper, we are concerned with matrices formed by block-diagonal factors interleaved with fixed permutations -- a flexible family of structured matrices. This class has recently drawn interest in deep learning architectures for its balanced expressivity-efficiency trade-off, yet efficient computational strategies for working with it remain to be found. We approach this problem through Riemannian geometry and examine under what conditions this class admits a smooth manifold structure. For the practically important case of orthogonal two-factor matrices, we derive the essential Riemannian tools and propose efficient algorithms for their implementation. The algorithms leverage automatic differentiation, support parameter sharing within each factor, and avoid explicit dense matrix construction. We test them within the Riemannian optimization framework on the best matrix approximation problem and for parameter-efficient fine-tuning of large language models. Beyond the two-factor setting, we study the geometric and matrix-theoretic properties of factorizations with a larger number of block-diagonal factors.

The elementwise Hadamard product of two low-rank matrices provides a parameter-efficient model for data with multiplicative structure, but its modeling is challenging due to the presence of additional symmetries under coupled row/column scalings between the two factors. In order to leverage the geometry of the space, we formulate the learning of such matrices as optimization on a Riemannian quotient manifold. We propose a novel block-diagonal Riemannian metric derived from the pullback of the Frobenius inner product. The metric is shown to be invariant under these symmetries. We develop a Riemannian gradient descent algorithm that uses a tuning-free Gauss--Newton step size and scales linearly in the number of observed entries per iteration. The versatile framework of Riemannian quotient optimization enables both first-order and second-order Riemannian methods, the latter through a closed-form connection and the Riemannian Hessian. Experiments on real and synthetic datasets illustrate the efficacy of our proposed Riemannian approach.

Neural network (NN) training is inherently a large-scale matrix optimization problem, yet the matrix structure of NN parameters has long been overlooked. Recently, the optimizer Muon \citep{jordanmuon}, which explicitly exploits this structure, has gained significant attention for its strong performance in foundation model training. A key component contributing to Muon's success is matrix orthogonalization. In this paper, we propose \textit{low-rank orthogonalization}, which performs orthogonalization by leveraging the low-rank nature of gradients during NN training. Building on this, we introduce low-rank matrix-signed gradient descent (MSGD) and a low-rank variant of Muon. %Numerical experiments demonstrate the superior performance of low-rank orthogonalization, with low-rank Muon achieving promising results in GPT-2 and LLaMA pretraining---surpassing the carefully tuned vanilla Muon on tasks with large model sizes. {Numerical experiments demonstrate the advantages of low-rank orthogonalization: low-rank Muon generally matches or improves upon vanilla Muon on the GPT-2 and LLaMA pretraining tasks, with clearer improvements observed for relatively larger models.} Theoretically, we establish the iteration complexity of low-rank MSGD for finding an approximate stationary solution, and the iteration complexity of low-rank Muon for finding an approximate stochastic stationary solution under heavy-tailed noise. The code to reproduce our numerical experiments is available at https://github.com/dengzhanwang/Low-rank-Muon.