Comment on "Benchmarking Compressed Sensing, Super-Resolution, and Filter Diagonalization"

TL;DR

This paper critiques a recent comparison of compressed sensing, super-resolution, and filter diagonalization, arguing that the original study misapplied FDM and contained inaccuracies, and clarifies the correct use and potential of FDM.

Contribution

The authors correct the misapplication of FDM in prior work and demonstrate its potential superiority when properly implemented for spectral estimation.

Findings

FDM was incorrectly applied in the original comparison.

Properly implemented FDM can outperform CS and SR in spectral estimation.

The original paper contained inaccuracies and misstatements.

Abstract

In a recent paper [Int. J. Quant. Chem. (2016) DOI: 10.1002/qua.25144, arXiv:1502.06579] Markovich, Blau, Sanders, and Aspuru-Guzik presented a numerical evaluation and comparison of three methods, Compressed Sensing (CS), Super-Resolution (SR), and Filter Diagonalization (FDM), on their ability of "recovering information" from time signals, concluding that CS and RS outperform FDM. We argue that this comparison is invalid for the following reasons. FDM is a well established method designed for solving the harmonic inversion problem or, similarly, for the problem of spectral estimation, and as such should be applied only to problems of this kind. The authors incorrectly assume that the problem of data fitting is equivalent to the spectral estimation problem, regardless of what parametric form is used, and, consequently, in all five numerical examples FDM is applied to the wrong problem. Moreover, the authors' implementation of FDM turned out to be incorrect, leading to extremely bad results, caused by numerical instabilities. As we demonstrate here, if implemented correctly, FDM could still be used for fitting the data, at least for the time signals composed of damped sinusoids, resulting in superior performance. In addition, we show that the published article is full of inaccuracies, mistakes and incorrect statements.