Trading Accuracy for Numerical Stability: Orthogonalization, Biorthogonalization and Regularization
Provides new regularization tools for signal processing practitioners dealing with ill-conditioned inverse problems, but the improvement is incremental over existing methods.
The paper proposes two new regularization methods based on biorthogonal bases to trade accuracy for numerical stability in ill-conditioned inverse problems, demonstrating improved stability over standard techniques on an exponential signal basis example.
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.