Compressed Subspace Learning Based on Canonical Angle Preserving Property
This work addresses the problem of efficiently analyzing low-dimensional subspace structures in high-dimensional data for machine learning and data analysis applications, representing an incremental advancement by leveraging known properties.
The paper proves that random projection with the Johnson-Lindenstrauss property approximately preserves canonical angles between subspaces, enabling a Compressed Subspace Learning framework to extract information from Union of Subspaces structures in reduced dimensions, with effectiveness demonstrated in tasks like subspace visualization and clustering.
Union of Subspaces (UoS) is a popular model to describe the underlying low-dimensional structure of data. The fine details of UoS structure can be described in terms of canonical angles (also known as principal angles) between subspaces, which is a well-known characterization for relative subspace positions. In this paper, we prove that random projection with the so-called Johnson-Lindenstrauss (JL) property approximately preserves canonical angles between subspaces with overwhelming probability. This result indicates that random projection approximately preserves the UoS structure. Inspired by this result, we propose a framework of Compressed Subspace Learning (CSL), which enables to extract useful information from the UoS structure of data in a greatly reduced dimension. We demonstrate the effectiveness of CSL in various subspace-related tasks such as subspace visualization, active subspace detection, and subspace clustering.