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Online Sparse Subspace Clustering

Liam Madden, Stephen Becker, Emiliano Dall'Anese

Abstract

This paper focuses on the sparse subspace clustering problem, and develops an online algorithmic solution to cluster data points on-the-fly, without revisiting the whole dataset. The strategy involves an online solution of a sparse representation (SR) problem to build a (sparse) dictionary of similarities where points in the same subspace are considered "similar," followed by a spectral clustering based on the obtained similarity matrix. When the SR cost is strongly convex, the online solution converges to within a neighborhood of the optimal time-varying batch solution. A dynamic regret analysis is performed when the SR cost is not strongly convex.

Online Sparse Subspace Clustering

Abstract

This paper focuses on the sparse subspace clustering problem, and develops an online algorithmic solution to cluster data points on-the-fly, without revisiting the whole dataset. The strategy involves an online solution of a sparse representation (SR) problem to build a (sparse) dictionary of similarities where points in the same subspace are considered "similar," followed by a spectral clustering based on the obtained similarity matrix. When the SR cost is strongly convex, the online solution converges to within a neighborhood of the optimal time-varying batch solution. A dynamic regret analysis is performed when the SR cost is not strongly convex.

Paper Structure

This paper contains 4 sections, 4 theorems, 12 equations, 2 figures.

Table of Contents

  1. Introduction and Problem Setup
  2. Online Sparse Representation
  3. Spectral Clustering
  4. Numerical Results

Key Result

Theorem 1

Under Assumption 1, where $L_t=\max\{|1-\gamma_tm_t|,|1-\gamma_tM_t|\}$, $\Tilde{L}_t=\prod_{\tau=0}^t L_{\tau}$.

Figures (2)

  • Figure 1: Objective value of tracking sequence and actual time-varying minimum.
  • Figure 2: Clustering error of both the tracking sequence and minimizer sequence.

Theorems & Definitions (11)

  • Theorem 1
  • proof
  • Corollary 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Corollary 2
  • Remark 2
  • Remark 3
  • ...and 1 more