SIAM MDS20 MS13 Advances in Subspace Learning and Clustering Mini-Symposium

Abstract: Modern inference and learning often hinge on identifying low-dimensional structures that approximate large scale data. Subspace clustering achieves this through a union of linear subspaces. However, in contemporary applications data is increasingly often incomplete, rendering standard (full-data) methods inapplicable. Existing incomplete-data methods present drawbacks like lifting an already high-dimensional problem, or requiring a super polynomial number of samples. Motivated by this, we introduce a new subspace clustering algorithm inspired by fusion penalties. The main idea is to permanently 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 fused together. While our approach is mainly motivated by missing data, it is also entirely new to full-data. Our approach directly accounts for noise, it requires no liftings, it allows low, high, and even full-rank data. It performs comparably to the state-of-the-art with complete data, and significantly better if data is missing.