Non-Gaussian Hyperplane Tessellations and Robust One-Bit Compressed Sensing

TL;DR

This paper demonstrates that hyperplane tessellations generated by subgaussian or heavy-tailed distributions can approximate Euclidean distances in arbitrary sets and enables robust, accurate one-bit compressed sensing with non-Gaussian measurements.

Contribution

It extends hyperplane tessellation theory beyond Gaussian cases and applies it to robust one-bit compressed sensing with non-Gaussian, heavy-tailed measurement matrices.

Findings

Hyperplane tessellations approximate Euclidean distances with fewer hyperplanes.

Accurate signal reconstruction is possible from non-Gaussian one-bit measurements.

Convex programming achieves reliable recovery with subgaussian measurement matrices.

Abstract

We show that a tessellation generated by a small number of random affine hyperplanes can be used to approximate Euclidean distances between any two points in an arbitrary bounded set $T$, where the random hyperplanes are generated by subgaussian or heavy-tailed normal vectors and uniformly distributed shifts. We derive quantitative bounds on the number of hyperplanes needed for constructing such tessellations in terms of natural metric complexity measures of $T$ and the desired approximation error. Our work extends significantly prior results in this direction, which were restricted to Gaussian hyperplane tessellations of subsets of the Euclidean unit sphere. As an application, we obtain new reconstruction results in memoryless one-bit compressed sensing with non-Gaussian measurement matrices. We show that by quantizing at uniformly distributed thresholds, it is possible to accurately reconstruct low-complexity signals from a small number of one-bit quantized measurements, even if the measurement vectors are drawn from a heavy-tailed distribution. Our reconstruction results are uniform in nature and robust in the presence of pre-quantization noise on the analog measurements as well as adversarial bit corruptions in the quantization process. Moreover we show that if the measurement matrix is subgaussian then accurate recovery can be achieved via a convex program.