Multi-Task Learning for Subspace Segmentation

Subspace segmentation is the process of clustering a set of data points that are assumed to lie on the union of multiple linear or affine subspaces, and is increasingly being recognized as a fundamental tool for data analysis in high dimensional settings. Arguably one of the most successful approaches is based on the observation that the sparsest representation of a given point with respect to a dictionary formed by the others involves nonzero coefficients associated with points originating in the same subspace. Such sparse representations are computed independently for each data point via ℓ1-norm minimization and then combined into an affinity matrix for use in a final spectral clustering step. The downside of this procedure is two-fold. First,unlike canonical compressive sensing scenarios with ideally randomized dictionaries, the data-dependent dictionaries here are unavoidably highly structured, disrupting many of the favorable properties of the ℓ1 norm. Secondly, by treating each data point independently, we ignore useful relationships between points that can be leveraged for jointly computing such sparse representations. Consequently, we motivate a multi-task learning-based framework for learning coupled sparse representations leading to a segmentation pipeline that is both robust against correlation structure and tailored to generate an optimal affinity matrix. Theoretical analysis and empirical tests are provided to support these claims.