Near-Optimal Compression for Compressed Sensing

In this note we study the under-addressed quantization stage implicit in any compressed sensing signal acquisition paradigm. We also study the problem of compressing the bitstream resulting from the quantization. We propose using Sigma-Delta (ΣΔ) quantization followed by a compression stage comprised of a discrete Johnson-Lindenstrauss embedding, and a subsequent reconstruction scheme based on convex optimization. We show that this encoding/decoding method yields near-optimal rate-distortion guarantees for sparse and compressible signals and is robust to noise. Our results hold for sub-Gaussian (including Gaussian and Bernoulli) random compressed sensing measurements, and they hold for high bit-depth quantizers as well as for coarse quantizers including 1-bit quantization.