  This paper introduces {\em fusion subspace clustering}, a novel method to
learn low-dimensional structures that approximate large scale yet highly
incomplete data. The main idea is to assign each datum to a subspace of its
own, and minimize the distance between the subspaces of all data, so that
subspaces of the same cluster get {\em fused} together. Our method allows low,
high, and even full-rank data; it directly accounts for noise, and its sample
complexity approaches the information-theoretic limit. In addition, our
approach provides a natural model selection {\em clusterpath}, and a direct
completion method. We give convergence guarantees, analyze computational
complexity, and show through extensive experiments on real and synthetic data
that our approach performs comparably to the state-of-the-art with complete
data, and dramatically better if data is missing.
