Building Invariances Into Sparse Subspace Clustering
Subspace clustering is the process of assigning subspace memberships to a set of unlabeled data points assumed to have been drawn from the union of an unknown number of low-dimensional subspaces, possibly interlaced with outliers or other data corruptions. By exploiting the fact that each inlier point has a sparse representation with respect to a dictionary formed by all the other points, an <inline-formula><tex-math notation="LaTeX">$\ell _1$ </tex-math></inline-formula> regularized sparse subspace clustering method has recently shown state-of-the-art robustness and practical extensibility in a variety of applications. But there remain important lingering weaknesses. In particular, the <inline-formula><tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula>-norm solution is highly sensitive, often in a detrimental direction, to the very types of data structures that motivate interest in subspace clustering to begin with, sometimes leading to poor segmentation accuracy. However, as an alternative source of sparsity, we argue that a certain data dependent, nonconvex penalty function can compensate for dictionary structure in a way that is especially germane to subspace clustering problems. For example, we demonstrate that this proposal displays a form of invariance to feature-space transformations and affine translations that commonly disrupt existing methods, and moreover, in important settings we reveal that its performance quality is lower bounded by the <inline-formula><tex-math notation="LaTeX">$\ell _1$</tex-math></inline-formula> solution. Finally, we provide empirical comparisons on popular benchmarks that corroborate our theoretical findings and demonstrate superior performance when compared to recent state-of-the-art models.