There has been much interest in recent years in various ways of constraining the complexity of matrices based on factorizations into a product of two simpler matrices. Such measures of matrix complexity can then be used as regularizers for such tasks as matrix completion, collaborative filtering, multi-task learning and multi-class learning. In this talk I will discuss two forms of matrix regularization which constrain the norm of the factorization, namely the trace-norm (aka nuclear-norm) and the so-called max-norm (aka γ2:l1→l∞ norm). I will both argue that they are independently motivated and often better model data then rank constraints, as well as explore their relationships to the rank. In particular, I will discuss how simple low-rank matrix completion guarantees can be obtained using these measures, and without various "incoherence" assumptions. I will present both theoretical and empirical arguments for why the max-norm might actually be a better regularizer, as well as a better convex surrogate for the rank.
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