  Deep Matrix Factorization (DMF) is an emerging approach to the problem of
matrix completion. Recent works have established that gradient descent applied
to a DMF model induces an implicit regularization on the rank of the recovered
matrix. In this work we interpret the DMF model through the lens of spectral
geometry. This allows us to incorporate explicit regularization without
breaking the DMF structure, thus enjoying the best of both worlds. In
particular, we focus on matrix completion problems with underlying geometric or
topological relations between the rows and/or columns. Such relations are
prevalent in matrix completion problems that arise in many applications, such
as recommender systems and drug-target interaction. Our contributions enable
DMF models to exploit these relations, and make them competitive on real
benchmarks, while exhibiting one of the first successful applications of deep
linear networks.
