Orthogonal iterations on Structured Pencils

TL;DR

This paper introduces fast subspace tracking algorithms for structured matrices, leveraging orthogonal iterations and LFR factorization, achieving low computational complexity for real-time applications.

Contribution

It proposes a novel class of algorithms based on orthogonal iterations and LFR factorization for efficient subspace tracking of structured matrices.

Findings

Algorithms operate with $O(nk^2)$ complexity per update.

Applicable to matrices as small rank perturbations of unitary matrices.

Achieves efficient real-time subspace tracking.

Abstract

We present a class of fast subspace tracking algorithms based on orthogonal iterations for structured matrices/pencils that can be represented as small rank perturbations of unitary matrices. The algorithms rely upon an updated data sparse factorization -- named LFR factorization -- using orthogonal Hessenberg matrices. These new subspace trackers reach a complexity of only $O(nk^2)$ operations per time update, where $n$ and $k$ are the size of the matrix and of the small rank perturbation, respectively.