One-Bit Compressive Sensing with Partial Support

The Compressive Sensing framework maintains relevance even when the available measurements are subject to extreme quantization, as is exemplified by the so-called one-bit compressed sensing framework which aims to recover a signal from measurements reduced to only their sign-bit. In applications, it is often the case that we have some knowledge of the structure of the signal beforehand, and thus would like to leverage it to attain more accurate and efficient recovery. This work explores avenues for incorporating such partial support information into the one-bit setting. Experimental results demonstrate that newly proposed methods of this work yield improved signal recovery even for varying levels of accuracy in the prior information. This work is thus the first to provide recovery mechanisms that efficiently use prior signal information in the one-bit reconstruction setting

One-Bit Compressive Sensing with Partial Support Information

This work develops novel algorithms for incorporating prior-support information into the field of One-Bit Compressed Sensing. Traditionally, Compressed Sensing is used for acquiring high-dimensional signals from few linear measurements. In applications, it is often the case that we have some knowledge of the structure of our signal(s) beforehand, and thus we would like to leverage it to attain more accurate and efficient recovery. Additionally, the Compressive Sensing framework maintains relevance even when the available measurements are subject to extreme quantization. Indeed, the field of One-Bit Compressive Sensing aims to recover a signal from measurements reduced to only their sign-bit. This work explores avenues for incorporating partial-support information into existing One-Bit Compressive Sensing algorithms. We provide both a rich background to the field of compressed sensing and in particular the one-bit framework, while also developing and testing new algorithms for this setting. Experimental results demonstrate that newly proposed methods of this work yield improved signal recovery even for varying levels of accuracy in the prior information. This work is thus the first to provide recovery mechanisms that efficiently use prior signal information in the one-bit reconstruction setting

Quantized Compressed Sensing for Partial Random Circulant Matrices

We provide the first analysis of a non-trivial quantization scheme for compressed sensing measurements arising from structured measurements. Specifically, our analysis studies compressed sensing matrices consisting of rows selected at random, without replacement, from a circulant matrix generated by a random subgaussian vector. We quantize the measurements using stable, possibly one-bit, Sigma-Delta schemes, and use a reconstruction method based on convex optimization. We show that the part of the reconstruction error due to quantization decays polynomially in the number of measurements. This is in line with analogous results on Sigma-Delta quantization associated with random Gaussian or subgaussian matrices, and significantly better than results associated with the widely assumed memoryless scalar quantization. Moreover, we prove that our approach is stable and robust; i.e., the reconstruction error degrades gracefully in the presence of non-quantization noise and when the underlying signal is not strictly sparse. The analysis relies on results concerning subgaussian chaos processes as well as a variation of McDiarmid's inequality.Comment: 15 page