The Deep Linear Network – Dynamics, Riemannian Geometry and OverparametrizationdissertationsVeraszto, ZsoltcreatorMenon, GovindAdvisorCohen, NadavReaderDarbon, JeromeReaderBrown University. Department of Applied Mathematicssponsor2023ix, 76 p.born digitalThesis (Ph. D.)--Brown University, 2023thesesThe deep linear network (DLN) is a model for implicit regularization in gradient based optimization of overparametrized learning architectures. Training the DLN corresponds to a Riemannian gradient flow, where the Riemannian metric is defined by the architecture of the network and the loss function is defined by the learning task. We extend this geometric framework, obtaining explicit expressions for the volume form, including the case when the network has infinite depth. We investigate the link between the Riemannian geometry and the training asymptotics for matrix completion with rigorous analysis and numerics. We develop a stochastic model for training and report our numerical findings. We propose that under small initialization, implicit regularization is a result of bias towards high state space volume.generalizabilitygradient flowsimplicit regularizationrieamannian geometrymatrix completiondeep linear networkEnglishRPB20230602