ROP inception: signal estimation with quadratic random sketching
Abstract
Rankone projections (ROP) of matrices and quadratic random sketching of signals support several data processing and machine learning methods, as well as recent imaging applications, such as phase retrieval or optical processing units. In this paper, we demonstrate how signal estimation can be operated directly through such quadratic sketchesequivalent to the ROPs of the "lifted signal" obtained as its outer product with itselfwithout explicitly reconstructing that signal. Our analysis relies on showing that, up to a minor debiasing trick, the ROP measurement operator satisfies a generalised sign product embedding (SPE) property. In a nutshell, the SPE shows that the scalar product of a signal sketch with the "sign" of the sketch of a given pattern approximates the square of the projection of that signal on this pattern. This thus amounts to an insertion (an "inception") of a ROP model inside a ROP sketch. The effectiveness of our approach is evaluated in several synthetic experiments.
 Publication:

arXiv eprints
 Pub Date:
 May 2022
 DOI:
 10.48550/arXiv.2205.08225
 arXiv:
 arXiv:2205.08225
 Bibcode:
 2022arXiv220508225D
 Keywords:

 Electrical Engineering and Systems Science  Signal Processing;
 Computer Science  Machine Learning
 EPrint:
 9 pages, 3 figures