Probabilistic Analysis of Least Squares, Orthogonal Projection, and QR Factorization Algorithms Subject to Gaussian Noise
This work addresses a gap in numerical linear algebra for researchers and practitioners by extending prior theoretical results to more realistic, noisy scenarios, though it is incremental as it builds directly on existing work.
The paper tackles the problem of analyzing condition number increases in QR factorization and related algorithms under imperfect orthonormality and Gaussian noise, deriving bounds without assuming perfect orthonormality or exact arithmetic.
In this paper, we extend the work of Liesen et al. (2002), which analyzes how the condition number of an orthonormal matrix Q changes when a column is added ([Q, c]), particularly focusing on the perpendicularity of c to the span of Q. Their result, presented in Theorem 2.3 of Liesen et al. (2002), assumes exact arithmetic and orthonormality of Q, which is a strong assumption when applying these results to numerical methods such as QR factorization algorithms. In our work, we address this gap by deriving bounds on the condition number increase for a matrix B without assuming perfect orthonormality, even when a column is not perfectly orthogonal to the span of B. This framework allows us to analyze QR factorization methods where orthogonalization is imperfect and subject to Gaussian noise. We also provide results on the performance of orthogonal projection and least squares under Gaussian noise, further supporting the development of this theory.