k-bit Hamming compressed sensing

We consider recovering d-level quantization of a signal from k-level quantization of linear measurements. This problem has great potential in practical systems, but has not been fully addressed in compressed sensing (CS). We tackle it by proposing k-bit Hamming compressed sensing (HCS). It reduces the decoding to a series of hypothesis tests of the bin where the signal lies in. Each test equals to an independent nearest neighbor search for a histogram estimated from quantized measurements. This method is based on that the distribution of the ratio between two random projections is defined by their intersection angle. Compared to CS and 1-bit CS, k-bit HCS leads to lower cost in both hardware and computation. It admits a trade-off between recovery/measurement resolution and measurement amount and thus is more flexible than 1-bit HCS. A rigorous analysis shows its error bound. Extensive empirical study further justifies its appealing accuracy, robustness and efficiency.

I. Introduction

Recently, prosperous works in compressed sensing (CS) [1][2] show that an accurate recovery can be achieved by sampling signal at a rate proportional to its underlying “information content” rather than bandwidth. The key improvement of CS is that the sampling rate can be significantly reduced below Nyquist rate by replacing the uniform sampling with linear measurement, when the signals are sparse or compressible on certain dictionary. In particular, CS dedicates to rebuild a sparse signal from its linear measurements by solving an underdetermined system.

$$\min\limits_{x\in {\BBR}^{n}}\Vert x\Vert_{p}\quad s.t.\ y=\Phi x, \quad (0\leq p < 2)$$ where is the sensing matrix allowing and fulfilling the restricted isometry property (RIP) or incoherence condition, and norm encourages sparsity. It has been proved that a number of random matrix ensembles satisfy RIP well.