Online Low-Rank Subspace Clustering by Basis Dictionary Pursuit

Low-Rank Representation~(LRR) has been a significant method for segmenting data that are generated from a union of subspaces. It is also known that solving LRR is challenging in terms of time complexity and memory footprint, in that the size of the nuclear norm regularized matrix is $n$-by-$n$ (where $n$ is the number of samples). In this paper, we thereby develop a novel online implementation of LRR that reduces the memory cost from $O(n^2)$ to $O(pd)$, with $p$ being the ambient dimension and $d$ being some estimated rank~($d < p \ll n$). We also establish the theoretical guarantee that the sequence of solutions produced by our algorithm converges to a stationary point of the expected loss function asymptotically. Extensive experiments on synthetic and realistic datasets further substantiate that our algorithm is fast, robust and memory efficient.