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This paper deals with two related problems, namely distance-preserving binaryembeddings and quantization for compressed sensing . First, we propose fastmethods to replace points from a subset $\mathcal{X} \subset \mathbb{R}^n$,associated with the Euclidean metric, with points in the cube $\{\pm 1\}^m$ andwe associate the cube with a pseudo-metric that approximates Euclidean distanceamong points in $\mathcal{X}$. Our methods rely on quantizing fastJohnson-Lindenstrauss embeddings based on bounded orthonormal systems andpartial circulant ensembles, both of which admit fast transforms. Ourquantization methods utilize noise-shaping, and include Sigma-Delta schemes anddistributed noise-shaping schemes. The resulting approximation errors decaypolynomially and exponentially fast in $m$, depending on the embedding method.This dramatically outperforms the current decay rates associated with binaryembeddings and Hamming distances. Additionally, it is the first such binaryembedding result that applies to fast Johnson-Lindenstrauss maps whilepreserving $\ell_2$ norms.  Second, we again consider noise-shaping schemes, albeit this time to quantizecompressed sensing measurements arising from bounded orthonormal ensembles andpartial circulant matrices. We show that these methods yield a reconstructionerror that again decays with the number of measurements (and bits), when usingconvex optimization for reconstruction. Specifically, for Sigma-Delta schemes,the error decays polynomially in the number of measurements, and it decaysexponentially for distributed noise-shaping schemes based on beta encoding.These results are near optimal and the first of their kind dealing with boundedorthonormal systems.

Fast binary embeddings, and quantized compressed sensing with structured matrices