A Model for Combinatorial Dictionary Learning and Inference

Avrim Blum; Kavya Ravichandran

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arXiv · 2407.18436

A Model for Combinatorial Dictionary Learning and Inference

Avrim Blum·Kavya Ravichandran

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Abstract

We are often interested in decomposing complex, structured data into simple components that explain the data. The linear version of this problem is well-studied as dictionary learning and factor analysis. In this work, we propose a combinatorial model in which to study this question, motivated by the way objects occlude each other in a scene to form an image. First, we identify a property we call "well-structuredness" of a set of low-dimensional components which ensures that no two components in the set are too similar. We show how well-structuredness is sufficient for learning the set of latent components comprising a set of sample instances. We then consider the problem: given a set of components and an instance generated from some unknown subset of them, identify which parts of the instance arise from which components. We consider two variants: (1) determine the minimal number of components required to explain the instance; (2) determine the correct explanation for as many locations as possible. For the latter goal, we also devise a version that is robust to adversarial corruptions, with just a slightly stronger assumption on the components. Finally, we show that the learning problem is computationally infeasible in the absence of any assumptions.

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BibTeX RIS

Avrim Blum, Kavya Ravichandran. 2024-07-26. A Model for Combinatorial Dictionary Learning and Inference. https://arxiv.org/abs/2407.18436

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