# A Quotient Property for Matrices with Heavy-Tailed Entries and its Application to Noise-Blind Compressed Sensing

arXiv: https://arxiv.org/abs/1806.04261 · HTML: https://mlsanity.com/paper/1806.04261/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Jun 2018.

## Built on by (most cited first, 3 of 3)

- [A Farewell to the Bias-Variance Tradeoff? An Overview of the Theory of Overparameterized Machine Learning](https://mlsanity.com/paper/2109.02355/) (2021)
- [Tight bounds for minimum l1-norm interpolation of noisy data](https://mlsanity.com/paper/2111.05987/) (2021)
- [AdaBoost and robust one-bit compressed sensing](https://mlsanity.com/paper/2105.02083/) (2021)

## Abstract (arXiv)

For a large class of random matrices A with i.i.d. entries we show that the ℓ1-quotient property holds with probability exponentially close to 1. In contrast to previous results, our analysis does not require concentration of the entrywise distributions. We provide a unified proof that recovers corresponding previous results for (sub-)Gaussian and Weibull distributions. Our findings generalize known results on the geometry of random polytopes, providing lower bounds on the size of the largest Euclidean ball contained in the centrally symmetric polytope spanned by the columns of A. At the same time, our results establish robustness of noise-blind ℓ1-decoders for recovering sparse vectors x from underdetermined, noisy linear measurements y = Ax + w under the weakest possible assumptions on the entrywise distributions that allow for recovery with optimal sample complexity even in the noiseless case. Our analysis predicts superior robustness behavior for measurement matrices with super-Gaussian entries, which we confirm by numerical experiments.