Robust Subspace Clustering via Thresholding

The problem of clustering noisy and incompletely observed high-dimensional data points into a union of low-dimensional subspaces and a set of outliers is considered. The number of subspaces, their dimensions, and their orientations are assumed unknown. We propose a simple low-complexity subspace clustering algorithm, which applies spectral clustering to an adjacency matrix obtained by thresholding the correlations between data points. In other words, the adjacency matrix is constructed from the nearest neighbors of each data point in spherical distance. A statistical performance analysis shows that the algorithm exhibits robustness to additive noise and succeeds even when the subspaces intersect. Specifically, our results reveal an explicit tradeoff between the affinity of the subspaces and the tolerable noise level. We furthermore prove that the algorithm succeeds even when the data points are incompletely observed with the number of missing entries allowed to be (up to a log-factor) linear in the ambient dimension. We also propose a simple scheme that provably detects outliers, and we present numerical results on real and synthetic data.