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Efficient Matrix Factorization Via Householder Reflections

13 May 2024
Anirudh Dash
Aditya Siripuram
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Abstract

Motivated by orthogonal dictionary learning problems, we propose a novel method for matrix factorization, where the data matrix Y\mathbf{Y}Y is a product of a Householder matrix H\mathbf{H}H and a binary matrix X\mathbf{X}X. First, we show that the exact recovery of the factors H\mathbf{H}H and X\mathbf{X}X from Y\mathbf{Y}Y is guaranteed with Ω(1)\Omega(1)Ω(1) columns in Y\mathbf{Y}Y . Next, we show approximate recovery (in the l∞l\inftyl∞ sense) can be done in polynomial time(O(np)O(np)O(np)) with Ω(log⁡n)\Omega(\log n)Ω(logn) columns in Y\mathbf{Y}Y . We hope the techniques in this work help in developing alternate algorithms for orthogonal dictionary learning.

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