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 to characterize 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/deep linear neural network training problems, resolving an open question posed by Mulayoff & Michaeli [2020]. This expression reveals a fundamental property of the loss landscape in deep matrix factorization: *Having a constant product of the spectral norms of the left and right intermediate factors across layers is a sufficient condition for flatness*. Most notably, in both depth-$2$ matrix 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 Frobenius-norm balanced*. To complement our theory, we provide an empirical verification of an escape phenomenon during gradient-based training near a minimizer of a deep matrix factorization problem which relies on our exact expression.