Theoretical limits of component identification in a separable nonlinear least-squares problem

Abstract

We provide theoretical insights into component identification in a separable nonlinear least-squares problem in which the model is a linear combination of nonlinear functions (called components in this paper). Within this research, we assume that the number of components is unknown. The objective of this paper is to understand the limits of component discovery under the assumed model. We focus on two aspects. One is sensitivity analysis referring to the ability of separating regression components from noise. The second is resolution analysis referring to the ability of de-mixing components that have similar location parameters. We use a wavelet transformation that allows to zoom in at different levels of details in the observed data. We further apply these theoretical insights to provide a road map on how to detect components in more realistic settings such as a two-dimensional nuclear magnetic resonance experiment for protein structure determination.

Keywords:

• component identification
• feature detection
• Mexican Hat wavelet basis
• separable nonlinear least-squares problem
• wavelet analysis

AMS Subject Classification:

• 62G05
• 62G08
• 62H35

Acknowledgements

The authors are grateful to Professor Gordon Rule for allowing them to use the NMR data generated in his lab, for his mentorship in NMR research, and for processing the data using the software FELIX. We are also thankful to the two reviewers and the associate editor for the thorough review of the paper.