The stochastic geometry of unconstrained one-bit data compression

TL;DR

This paper introduces a stochastic geometric model for analyzing one-bit data compression in high-dimensional spaces, focusing on the interaction between hyperplane tessellations and data separation.

Contribution

It proposes a novel stochastic geometric framework for understanding one-bit compressed sensing with stationary data and hyperplane tessellations, linking hyperplane intensity to data separation.

Findings

Hyperplane intensity must scale with dimension for effective separation.

Model applies to unconstrained stationary data sets like ^n or Poisson processes.

Results inform design of compressive sensing and source coding schemes.

Abstract

A stationary stochastic geometric model is proposed for analyzing the data compression method used in one-bit compressed sensing. The data set is an unconstrained stationary set, for instance all of $\mathbb{R}^n$ or a stationary Poisson point process in $\mathbb{R}^n$. It is compressed using a stationary and isotropic Poisson hyperplane tessellation, assumed independent of the data. That is, each data point is compressed using one bit with respect to each hyperplane, which is the side of the hyperplane it lies on. This model allows one to determine how the intensity of the hyperplanes must scale with the dimension $n$ to ensure sufficient separation of different data by the hyperplanes as well as sufficient proximity of the data compressed together. The results have direct implications in compressive sensing and in source coding.