Extended Lanczos bidiagonalization algorithm for low rank approximation and its applications

Low-Rank Matrix Approximation Using the Lanczos Bidiagonalization Process with Applications

Low-rank approximation of large and/or sparse matrices is important in many applications, and the singular value decomposition (SVD) gives the best low-rank approximations with respect to unitarily-invariant norms. In this paper we show that good low-rank approximations can be directly obtained from the Lanczos bidiagonalization process applied to the given matrix without computing any SVD. We ...

A Lanczos bidiagonalization algorithm for Hankel matrices

This paper presents a fast algorithm for bidiagonalizing a Hankel matrix. An m×n Hankel matrix is reduced to a real bidiagonal matrix in O((m+ n)n log(m+ n)) floating-point operations (flops) using the Lanczos method with modified partial orthogonalization and reset schemes to improve its stability. Performance improvement is achieved by exploiting the Hankel structure, as fast Hankel matrix–ve...

Structured low-rank approximation and its applications

Fitting data by a bounded complexity linear model is equivalent to low-rank approximation of a matrix constructed from the data. The data matrix being Hankel structured is equivalent to the existence of a linear timeinvariant system that fits the data and the rank constraint is related to a bound on the model complexity. In the special case of fitting by a static model, the data matrix and its ...

Extended Lanczos Bidiagonalization for Dimension Reduction in Information Retrieval

We describe an extended bidiagonalization scheme designed to compute low-rank approximations of very large data matrices. Its goal is identical to that of the truncated singular value decomposition, but it is significantly cheaper. It consists in an extension of the standard Lanczos bidiagonalization that improves its approximation capabilities, while keeping the computational cost reasonable. ...

Lanczos Bidiagonalization With Partial Reorthogonalization