Fast (1+ε)-Approximation Algorithms for Binary Matrix Factorization
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We introduce efficient (1+ε)-approximation algorithms for the binary matrix factorization (BMF) problem, where the inputs are a matrix 𝐀∈{0,1}^n× d, a rank parameter k>0, as well as an accuracy parameter ε>0, and the goal is to approximate 𝐀 as a product of low-rank factors 𝐔∈{0,1}^n× k and 𝐕∈{0,1}^k× d. Equivalently, we want to find 𝐔 and 𝐕 that minimize the Frobenius loss 𝐔𝐕 - 𝐀_F^2. Before this work, the state-of-the-art for this problem was the approximation algorithm of Kumar et. al. [ICML 2019], which achieves a C-approximation for some constant C≥ 576. We give the first (1+ε)-approximation algorithm using running time singly exponential in k, where k is typically a small integer. Our techniques generalize to other common variants of the BMF problem, admitting bicriteria (1+ε)-approximation algorithms for L_p loss functions and the setting where matrix operations are performed in 𝔽_2. Our approach can be implemented in standard big data models, such as the streaming or distributed models.
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