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Abstract: Techniques of matrix completion aim to impute a large portion
of missing entries in a data matrix through a small portion of observed
ones, with broad machine learning applications including collaborative
filtering, pairwise ranking, etc. In this talk, we focus on analyzing
the nonconvex matrix completion problem from the geometrical
perspective. Geometrical analysis has been conducted on various low-rank
recovery problems including matrix factorization and matrix completion
in recent few years. Taking matrix completion as an example, with
assumptions on the underlying matrix and the sampling rate, all the
local minima of the nonconvex objective function were shown to be global
minima, i.e., nonconvex optimization can recover the underlying matrix
exactly. We propose a model-free framework for nonconvex matrix
completion: We characterize how well local-minimum based low-rank
factorization can approximate the underlying matrix without any
assumption on it. As an implication, a corollary of our main theorem
improves the state-of-the-art sampling rate required for nonconvex
matrix completion to rule out spurious local minima. A unified
geometrical analysis of nonconvex matrix completion with linearly
parameterized factorization will also be discussed in this talk.

Zoom information is as following: Link: https://ucdavisdss.zoom.us/j/97296673897?pwd=Ym56emJqR0xUZ1AyN3UvQURZNG9zQT09 Meeting ID: 972 9667 3897 Password: 112358​