Rank/Norm Regularization with Closed-Form Solutions: Application to Subspace Clustering

2012/02/14 by Yaoliang Yu, Dale Schuurmans, Yu, Yao-Liang +1 · 1 citation
Computer Science · Engineering · #FOS: Computer and information sciences #FOS: Mathematics #Face and Expression Recognition #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Numerical Analysis (math.NA) #Remote-Sensing Image Classification #Sparse and Compressive Sensing Techniques

paper · pdf · doi:10.48550/arxiv.1202.3772

openalex publication_date 2012/02/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

When data is sampled from an unknown subspace, principal component analysis (PCA) provides an effective way to estimate the subspace and hence reduce the dimension of the data. At the heart of PCA is the Eckart-Young-Mirsky theorem, which characterizes the best rank k approximation of a matrix. In this paper, we prove a generalization of the Eckart-Young-Mirsky theorem under all unitarily invariant norms. Using this result, we obtain closed-form solutions for a set of rank/norm regularized problems, and derive closed-form solutions for a general class of subspace clustering problems (where data is modelled by unions of unknown subspaces). From these results we obtain new theoretical insights and promising experimental results.

Citations

• The Approximation of One Matrix by Another of Lower Rank
• A tutorial on spectral clustering

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