← 研踪 PaperTrailarXiv:2509.25783 · v7
Sharpness of Minima in Deep Matrix Factorization
首次提交 2025-09-30 12:50
版本更新 2026-10-29 07:18
雷达观测 2026-10-30 11:25
摘要
Understanding the loss-landscape geometry near a minimum is key to explaining the implicit bias of gradient-based methods in non-convex optimization problems such as deep neural network training and deep matrix factorization. A central quantity for characterizing this geometry is the maximum eigenvalue of the Hessian of the loss. Currently, its precise role has been obfuscated because no exact expressions for this sharpness measure were known in general settings. In this paper, we present the first exact expression for the maximum eigenvalue of the Hessian of the squared-error loss at any minimizer in deep matrix factorization problems, resolving an open question posed by Mulayoff and Michaeli [2020]. This expression reveals a fundamental property of the loss landscape in deep matrix factorization: It is sufficient, but not necessary, for flatness that, across all layers, the product of the spectral norms of the left and right intermediate factors equals a constant determined by the depth and the spectral norm of the target matrix. Most notably, in both depth-2 matrix factorization and deep overparameterized scalar factorization, we show that this condition is both necessary and sufficient for flatness, which implies that flat minima are spectral-norm balanced and not necessarily balanced in depth-2 matrix factorization. To complement our theory, we provide empirical verification, relying on our exact expression, of an escape phenomenon during gradient-based training near a minimizer of a deep matrix factorization problem.
作者备注：23 pages, 5 figures
观测到的版本
相邻记录可能跳过中间版本；首次发现 v2 及以上也会标为版本更新。
v7 · 2026-10-29 07:18 · 最新观测版本
Sharpness of Minima in Deep Matrix Factorization
Understanding the loss-landscape geometry near a minimum is key to explaining the implicit bias of gradient-based methods in non-convex optimization problems such as deep neural network training and deep matrix factorization. A central quantity for characterizing this geometry is the maximum eigenvalue of the Hessian of the loss. Currently, its precise role has been obfuscated because no exact expressions for this sharpness measure were known in general settings. In this paper, we present the first exact expression for the maximum eigenvalue of the Hessian of the squared-error loss at any minimizer in deep matrix factorization problems, resolving an open question posed by Mulayoff and Michaeli [2020]. This expression reveals a fundamental property of the loss landscape in deep matrix factorization: It is sufficient, but not necessary, for flatness that, across all layers, the product of the spectral norms of the left and right intermediate factors equals a constant determined by the depth and the spectral norm of the target matrix. Most notably, in both depth-2 matrix factorization and deep overparameterized scalar factorization, we show that this condition is both necessary and sufficient for flatness, which implies that flat minima are spectral-norm balanced and not necessarily balanced in depth-2 matrix factorization. To complement our theory, we provide empirical verification, relying on our exact expression, of an escape phenomenon during gradient-based training near a minimizer of a deep matrix factorization problem.
查看此版本解读 →查看此版本原文 ↗