Structured Sparse Subspace Clustering: A Joint Affinity Learning and Subspace Clustering Framework

TL;DR

This paper introduces a joint optimization framework called Structured Sparse Subspace Clustering (S$^3$C) that simultaneously learns affinity and segmentation for subspace clustering, improving over traditional two-stage methods.

Contribution

It proposes a novel joint learning framework for affinity and segmentation in subspace clustering, incorporating structured sparse representations and optional side-information.

Findings

Outperforms traditional two-stage methods on multiple datasets.

Effectively incorporates partial side-information into the clustering process.

Demonstrates robustness across synthetic, image, motion, and cancer data sets.

Abstract

Subspace clustering refers to the problem of segmenting data drawn from a union of subspaces. State-of-the-art approaches for solving this problem follow a two-stage approach. In the first step, an affinity matrix is learned from the data using sparse or low-rank minimization techniques. In the second step, the segmentation is found by applying spectral clustering to this affinity. While this approach has led to state-of-the-art results in many applications, it is sub-optimal because it does not exploit the fact that the affinity and the segmentation depend on each other. In this paper, we propose a joint optimization framework --- Structured Sparse Subspace Clustering (S$^3$C) --- for learning both the affinity and the segmentation. The proposed S$^3$C framework is based on expressing each data point as a structured sparse linear combination of all other data points, where the structure is induced by a norm that depends on the unknown segmentation. Moreover, we extend the proposed S$^3$C framework into Constrained Structured Sparse Subspace Clustering (CS$^3$C) in which available partial side-information is incorporated into the stage of learning the affinity. We show that both the structured sparse representation and the segmentation can be found via a combination of an alternating direction method of multipliers with spectral clustering. Experiments on a synthetic data set, the Extended Yale B data set, the Hopkins 155 motion segmentation database, and three cancer data sets demonstrate the effectiveness of our approach.