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 compressed sensing and in source coding.
Highlights
Introduction and motivationsOne-bit compressed sensing is a method of signal recovery from a sequence of measurements contained in {−1, 1}
As in one-bit compressed sensing, the quality can be determined by having a small error in signal recovery, which can be guaranteed if the collection of hyperplanes tessellates the signal space into cells small enough to ensure all signals within a single cell are close in Euclidean distance
While data in most directions will be separated from the typical data point, there is a set of directions of decreasing measure as dimension increases in which the compression will remain identical and where most of the volume of data compressed like the typical data point lies
Summary
One-bit compressed sensing is a method of signal recovery from a sequence of measurements contained in {−1, 1}. While data in most directions will be separated from the typical data point, there is a set of directions of decreasing measure as dimension increases in which the compression will remain identical and where most of the volume of data compressed like the typical data point lies Considering this low distortion criterion, we show that, for α = 1, there is a threshold for ρ above which the expected value of the volume in question goes to zero and below which it approaches infinity. This last representation of data sparsity is very specific and the general question of one-bit compression based on stationary Poisson hyperplanes for sparse data is far from being solved by the observations on this specific case.
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