Low Rank Matrix Completion via Robust Alternating Minimization in Nearly Linear Time
arxiv(2023)
摘要
Given a matrix M∈ℝ^m× n, the low rank matrix completion
problem asks us to find a rank-k approximation of M as UV^⊤ for U∈ℝ^m× k and V∈ℝ^n× k by only observing a
few entries specified by a set of entries Ω⊆ [m]× [n]. In
particular, we examine an approach that is widely used in practice – the
alternating minimization framework. Jain, Netrapalli, and Sanghavi [JNS13]
showed that if M has incoherent rows and columns, then alternating
minimization provably recovers the matrix M by observing a nearly linear in
n number of entries. While the sample complexity has been subsequently
improved [GLZ17], alternating minimization steps are required to be computed
exactly. This hinders the development of more efficient algorithms and fails to
depict the practical implementation of alternating minimization, where the
updates are usually performed approximately in favor of efficiency.
In this paper, we take a major step towards a more efficient and error-robust
alternating minimization framework. To this end, we develop an analytical
framework for alternating minimization that can tolerate a moderate amount of
errors caused by approximate updates. Moreover, our algorithm runs in time
O(|Ω| k), which is nearly linear in the time to verify the
solution while preserving the sample complexity. This improves upon all prior
known alternating minimization approaches which require O(|Ω|
k^2) time.
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