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arXiv · 2501.15094

Exploring the Limitations of Structured Orthogonal Dictionary Learning

Anirudh Dash·Aditya Siripuram
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Abstract

This work is motivated by recent applications of structured dictionary learning, in particular when the dictionary is assumed to be the product of a few Householder atoms. We investigate the following two problems: 1) How do we approximate an orthogonal matrix $\mathbf{V}$ with a product of a specified number of Householder matrices, and 2) How many samples are required to learn a structured (Householder) dictionary from data? For 1) we discuss an algorithm that decomposes $\mathbf{V}$ as a product of a specified number of Householder matrices. We see that the algorithm outputs the decomposition when it exists, and give bounds on the approximation error of the algorithm when such a decomposition does not exist. For 2) given data $\mathbf{Y}=\mathbf{HX}$, we show that when assuming a binary coefficient matrix $\mathbf{X}$, the structured (Householder) dictionary learning problem can be solved with just $2$ samples (columns) in $\mathbf{Y}$.

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Anirudh Dash, Aditya Siripuram. 2025-01-25. Exploring the Limitations of Structured Orthogonal Dictionary Learning. https://arxiv.org/abs/2501.15094

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