Iterative Atom Refinement: A Monotonicity Principle for Dictionary Learning

Abstract:Dictionary learning seeks to recover an unknown dictionary $A$ from observations ${\bf y}_i = A{\bf x}_i$ with sparse coefficient vectors ${\bf x}_i$. We introduce the \emph{Iterative Atom Refinement} (IAR) algorithm, a simple procedure for recovering individual dictionary atoms. Starting from a random direction, IAR repeatedly selects the observations most strongly correlated with the current iterate and updates the direction by averaging the selected data. Our main contribution is a rigorous convergence theory of IAR. Using high-dimensional probabilistic estimates and a novel monotonicity principle for atom-selection probabilities, we show that a small initial advantage of one atom is amplified until that atom is isolated. Under our model assumptions, IAR identifies a generating atom after only three refinement steps. Numerical experiments support the theory and show that the resulting dynamics accurately capture the behavior observed in dictionary refinement.

Machine Learning (cs.LG); Probability (math.PR); Machine Learning (stat.ML)

arXiv:2609.23812 [cs.LG]

(or arXiv:2609.23812v1 [cs.LG] for this version)

https://doi.org/10.48550/arXiv.2609.23812 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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