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Weighed l1 on the simplex: Compressive sensing meets locality

Abiy Tasissa, Pranay Tankala, Demba Ba

TL;DR

The paper addresses the challenge of applying compressive sensing ideas to dictionary-based manifold learning where dictionary atoms are correlated and local geometry matters. It introduces weighted $\ell_0$ and weighted $\ell_1$ metrics, leveraging a unique Delaunay triangulation to enforce locality and show equivalence between weighted $\ell_0$ and $\ell_1$ representations under certain generative conditions. A novel K-Deep Simplex (KDS) framework is developed to learn both the dictionary and sparse coefficients via an autoencoder architecture, enabling efficient, GPU-accelerated optimization. Empirical results on synthetic manifolds and clustering benchmarks demonstrate that the approach yields geometrically meaningful representations and improved clustering accuracy with sparse, neighborhood-based dictionaries.

Abstract

Sparse manifold learning algorithms combine techniques in manifold learning and sparse optimization to learn features that could be utilized for downstream tasks. The standard setting of compressive sensing can not be immediately applied to this setup. Due to the intrinsic geometric structure of data, dictionary atoms might be redundant and do not satisfy the restricted isometry property or coherence condition. In addition, manifold learning emphasizes learning local geometry which is not reflected in a standard $\ell_1$ minimization problem. We propose weighted $\ell_0$ and weighted $\ell_1$ metrics that encourage representation via neighborhood atoms suited for dictionary based manifold learning. Assuming that the data is generated from Delaunay triangulation, we show the equivalence of weighted $\ell_0$ and weighted $\ell_1$. We discuss an optimization program that learns the dictionaries and sparse coefficients and demonstrate the utility of our regularization on synthetic and real datasets.

Weighed l1 on the simplex: Compressive sensing meets locality

TL;DR

The paper addresses the challenge of applying compressive sensing ideas to dictionary-based manifold learning where dictionary atoms are correlated and local geometry matters. It introduces weighted and weighted metrics, leveraging a unique Delaunay triangulation to enforce locality and show equivalence between weighted and representations under certain generative conditions. A novel K-Deep Simplex (KDS) framework is developed to learn both the dictionary and sparse coefficients via an autoencoder architecture, enabling efficient, GPU-accelerated optimization. Empirical results on synthetic manifolds and clustering benchmarks demonstrate that the approach yields geometrically meaningful representations and improved clustering accuracy with sparse, neighborhood-based dictionaries.

Abstract

Sparse manifold learning algorithms combine techniques in manifold learning and sparse optimization to learn features that could be utilized for downstream tasks. The standard setting of compressive sensing can not be immediately applied to this setup. Due to the intrinsic geometric structure of data, dictionary atoms might be redundant and do not satisfy the restricted isometry property or coherence condition. In addition, manifold learning emphasizes learning local geometry which is not reflected in a standard minimization problem. We propose weighted and weighted metrics that encourage representation via neighborhood atoms suited for dictionary based manifold learning. Assuming that the data is generated from Delaunay triangulation, we show the equivalence of weighted and weighted . We discuss an optimization program that learns the dictionaries and sparse coefficients and demonstrate the utility of our regularization on synthetic and real datasets.

Paper Structure

This paper contains 6 sections, 2 theorems, 15 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Proposed Method
  3. Dictionary Learning Algorithm
  4. Numerical Experiments
  5. Application: Clustering
  6. Conclusion

Key Result

Theorem 1

Let $\mathbf{a}_1, \ldots, \mathbf{a}_m \in \mathcal{R}^{d}$ be a set of points with a unique Delaunay triangulation $\text{DT}(\mathbf{A})$. Let $\mathbf{y} \in \mathcal{R}^{d}$ be an interior point of the $d$-simplex of $\text{DT}(\mathbf{A})$ with circumcenter $\mathbf{c}$ and radius $R$. Assume The optimal solution to the above program is $\mathbf{x}^{*}$.

Figures (2)

  • Figure 1: The red dots indicate the atoms which generate the data points. Each black dot, denoting a data point, is a convex combination of three atoms which are vertices of the triangle the point belongs to. We show that the optimal weighted $\ell_0$ and $\ell_1$ metric is based on a point representing itself using the vertices of the triangle it belongs to.
  • Figure 2: Circle and two moons. Autoencoder input (first and third) and output (second and fourth), with learned atoms marked in red.

Theorems & Definitions (7)

  • Definition 1
  • Definition 2
  • Theorem 1
  • proof
  • Definition 3
  • Theorem 2
  • proof