Direction-Magnitude Decomposition for Low-Rank Matrix Optimization: Faster Convergence and Saddle-to-saddle Dynamics
Authors: Yudong Wei, Liang Zhang, Bingcong Li, Niao He
Organizations: Department of Mathematics, ETH Zurich, Zurich, Switzerland · Department of Computer Science, ETH Zurich, Zurich, Switzerland
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. We propose a unified framework, termed direction-magnitude decomposition (DMD), that decomposes the optimization variable to improve optimization efficiency even when the target rank is unknown. We develop two DMD-based approaches and establish their theoretical advantages on the canonical problem of matrix factorization. The first, overparameterized DMD, uses a rank r larger than necessary and enjoys faster convergence as r increases. The second, recursive DMD, is motivated by the incremental eigenpair learning, or saddle-to-saddle, behavior of overparameterized DMD. It achieves lower memory and computational costs, complementing overparameterized DMD. Both approaches are exponentially faster than gradient descent applied to the BM formulation. Numerical experiments on matrix factorization, sensing, and completion corroborate our theoretical findings and demonstrate the practical effectiveness of DMD.
We explore Riemannian optimization techniques for rank-factored matrix parameters, targeting contemporary deep learning applications. We examine ten points in the algorithm design space: two geometries for rank-r matrices, three geometries for rank-r partial isometries, and block-matrix variants of these five, where factors are shared across block-rows and block-columns. We apply our methods to the multihead attention parameters in small language models. After tuning learning rates, our methods do not conclusively outperform an AdamW baseline. Our implementations are available online.
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.
We study nonconvex methods for matrix completion, the problem of recovering a low-rank matrix from a subset of its entries. Convex methods achieve sample complexity linear in the matrix dimension and the rank, up to logarithmic factors, whereas global guarantees for commonly used nonconvex methods require a higher polynomial dependence on the rank. We close this gap by analyzing Riemannian gradient descent (RGD) and Riemannian Gauss--Newton (RGN) methods. For an n×n matrix of rank r with incoherence parameter μ and condition number κ, the two methods achieve exact recovery with high probability from O(μnrlognlog(nκ)) and O(μnrlognlog(2μrκ)) observations, respectively. The methods use a multiscale residual initialization, while the analysis simultaneously controls the spectral error and incoherence. The resulting RGD iterates converge linearly, whereas RGN eventually converges Q-quadratically.