Fixed-Rank Approximation of a Positive-Semidefinite Matrix from Streaming Data
This addresses the challenge of efficient fixed-rank matrix approximation for applications like streaming PCA and semidefinite programming, offering a practical solution with theoretical guarantees, though it is incremental as it builds on existing methods like Nystrom approximation.
The paper tackles the problem of approximating a large positive-semidefinite matrix from streaming data with storage constraints, developing a new algorithm that combines Nystrom approximation with rank truncation to achieve any prescribed relative error in the Schatten 1-norm, with experiments showing it outperforms alternative techniques across various examples.
Several important applications, such as streaming PCA and semidefinite programming, involve a large-scale positive-semidefinite (psd) matrix that is presented as a sequence of linear updates. Because of storage limitations, it may only be possible to retain a sketch of the psd matrix. This paper develops a new algorithm for fixed-rank psd approximation from a sketch. The approach combines the Nystrom approximation with a novel mechanism for rank truncation. Theoretical analysis establishes that the proposed method can achieve any prescribed relative error in the Schatten 1-norm and that it exploits the spectral decay of the input matrix. Computer experiments show that the proposed method dominates alternative techniques for fixed-rank psd matrix approximation across a wide range of examples.