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Union of Subspaces (UoS) is a popular model to describe the underlyinglow-dimensional structure of data. The fine details of UoS structure can bedescribed in terms of canonical angles (also known as principal angles) betweensubspaces, which is a well-known characterization for relative subspacepositions. In this paper, we prove that random projection with the so-calledJohnson-Lindenstrauss (JL) property approximately preserves canonical anglesbetween subspaces with overwhelming probability. This result indicates thatrandom projection approximately preserves the UoS structure. Inspired by thisresult, we propose a framework of Compressed Subspace Learning (CSL), whichenables to extract useful information from the UoS structure of data in agreatly reduced dimension. We demonstrate the effectiveness of CSL in varioussubspace-related tasks such as subspace visualization, active subspacedetection, and subspace clustering.

Compressed Subspace Learning Based on Canonical Angle Preserving Property