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A Model for Combinatorial Dictionary Learning and Inference

Avrim Blum, Kavya Ravichandran

TL;DR

The paper introduces a combinatorial, occlusion-based model for dictionary learning on 1D canvases, defining well-structured object families to enable learnability and segmentation guarantees. It develops a shotgun-sequencing inspired learning approach to recover the dictionary and presents inference algorithms (DP for arbitrary objects and a greedy signature-based method for well-structured objects) that yield correct explanations under various noise and structural assumptions. It analyzes sample complexity, proves correctness under the well-structuredness assumptions, and demonstrates robustness to adversarial corruption with epsilon-strong structures; it also shows that learning without any assumptions is computationally intractable (NP-hard). The framework provides a principled way to study non-linear, combinatorial dictionary learning and segmentation with provable guarantees in controlled settings, highlighting both theoretical guarantees and practical limitations.

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.

A Model for Combinatorial Dictionary Learning and Inference

TL;DR

The paper introduces a combinatorial, occlusion-based model for dictionary learning on 1D canvases, defining well-structured object families to enable learnability and segmentation guarantees. It develops a shotgun-sequencing inspired learning approach to recover the dictionary and presents inference algorithms (DP for arbitrary objects and a greedy signature-based method for well-structured objects) that yield correct explanations under various noise and structural assumptions. It analyzes sample complexity, proves correctness under the well-structuredness assumptions, and demonstrates robustness to adversarial corruption with epsilon-strong structures; it also shows that learning without any assumptions is computationally intractable (NP-hard). The framework provides a principled way to study non-linear, combinatorial dictionary learning and segmentation with provable guarantees in controlled settings, highlighting both theoretical guarantees and practical limitations.

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.
Paper Structure (64 sections, 26 theorems, 29 equations, 3 figures, 5 algorithms)

This paper contains 64 sections, 26 theorems, 29 equations, 3 figures, 5 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Notation, Preliminaries, and Definitions
  4. Notation
  5. Assumptions
  6. Structural Properties and Object Generation
  7. Image Generation
  8. Defining the "scene" and "view"
  9. Formal Statement of the Problem
  10. Learning:
  11. Inference:
  12. Sequencing
  13. Proof of Correctness
  14. Sample Complexity
  15. Learning
  16. ...and 49 more sections

Key Result

Lemma 2.0

[Random Objects are $w$-Well-Structured whp] A set of $m$ objects, each sampled uniformly at random from $\{ 0, 1, \hdots\,, c-1 \}^{s_i}\,,$ and $s \coloneqq \max_i s_i$ is $w$-well-structured with probability at least $1 - 3m^2s^2/c^w\,.$ In particular, $w=O(\log m s)$ is sufficient so that the $m

Figures (3)

  • Figure 1: Object recovery from pieces: Due to objects obscuring each other, we cannot hope to see all $s$ pixels of the orange object at once, but so long as we see the blue pieces shown in this figure, i.e., ones with at least $w$ pixels overlap, we can reconstruct the object.
  • Figure 2: Process used to learn objects from images in the absence of endpoint markers: This algorithm allows us to collect the pieces of length $4w$ that cover the orange object in a redundant way. However, due to the risk of problematic overlaps we must discard a length-$w$ piece (blue) on either end. Even after this discard, the pieces (green) used for sequencing must cover the object.
  • Figure 3: Consider the $L$-pixel problematic overlap string boxed in blue. There are 5 different features that uniquely define it. The first is identity of the left object (darker orange), followed by the index within it where the string of interest (boxed in blue) starts. Third, we care about the identity of the right object (lighter orange) and the index within it at which the string of interest ends. Note that these uniquely identify how much overlap there must be, since the length $L$ is fixed. Finally, we include a bit to make clear which object is on top.

Theorems & Definitions (80)

  • Definition 2.1: $w$-well-structured
  • Definition 2.2: $\epsilon$-strongly, $w$-well-structured
  • Lemma 2.0
  • Definition 2.3: Distinct Background
  • Definition 2.4: $w$-Well-Structured Background
  • Definition 2.5: Uniform Model
  • Definition 2.6: Closed Room Model
  • Definition 2.7: Open Room Model
  • Definition 2.8: Fully Random Model: Random Horizontal; Random Depth
  • Definition 2.9: Partially Random Model: Random Horizontal; Arbitrary Depth
  • ...and 70 more