Trading Accuracy for Numerical Stability: Orthogonalization, Biorthogonalization and Regularization

TL;DR

This paper introduces two new regularization techniques based on orthogonalization principles, enabling a tradeoff between accuracy and numerical stability in ill-conditioned inverse problems, with applications demonstrated in signal processing.

Contribution

The paper proposes novel regularization methods leveraging biorthogonal bases, incorporating homotopy or tuning parameters for stability-accuracy tradeoffs, and demonstrates their effectiveness in signal processing tasks.

Findings

Effective regularization in ill-conditioned inverse problems

Improved stability-accuracy tradeoff control

Comparison shows advantages over standard techniques

Abstract

This paper presents two novel regularization methods motivated in part by the geometric significance of biorthogonal bases in signal processing applications. These methods, in particular, draw upon the structural relevance of orthogonality and biorthogonality principles and are presented from the perspectives of signal processing, convex programming, continuation methods and nonlinear projection operators. Each method is specifically endowed with either a homotopy or tuning parameter to facilitate tradeoff analysis between accuracy and numerical stability. An example involving a basis comprised of real exponential signals illustrates the utility of the proposed methods on an ill-conditioned inverse problem and the results are compared to standard regularization techniques from the signal processing literature.