# ROP inception: signal estimation with quadratic random sketching

arXiv: https://arxiv.org/abs/2205.08225 · HTML: https://mlsanity.com/paper/2205.08225/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 3 of its 17 references that are in the map.
- First posted on arXiv 17 May 2022.

## Builds on (most cited first, 3 of 3)

- [Exact and Stable Covariance Estimation from Quadratic Sampling via Convex Programming](https://mlsanity.com/paper/1310.0807/) (2013)
- [ROP: Matrix recovery via rank-one projections](https://mlsanity.com/paper/1310.5791/) (2013)
- [Random Projections through multiple optical scattering: Approximating kernels at the speed of light](https://mlsanity.com/paper/1510.06664/) (2015)

## Built on by (most cited first, 1 of 1)

- [Compressive Recovery of Sparse Precision Matrices](https://mlsanity.com/paper/2311.04673/) (2023)

## Abstract (arXiv)

Rank-one 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 sketches--equivalent to the ROPs of the "lifted signal" obtained as its outer product with itself--without 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.