Robust 1-Bit Compressive Sensing via Binary Stable Embeddings of Sparse Vectors

• Published in: IEEE transactions on information theory Vol. 59; no. 4; pp. 2082 - 2102
• Main Authors: Jacques, L., Laska, J. N., Boufounos, P. T., Baraniuk, R. G.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01-04-2013; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: 1-bit compressed sensing; Applied sciences; approximation error; compressed sensing; Computer science; control theory; systems; consistent reconstruction; Data processing. List processing. Character string processing; Decoding; Detection, estimation, filtering, equalization, prediction; dimensionality reduction; Distortion measurement; Exact sciences and technology; Information theory; Information, signal and communications theory; iterative reconstruction; Measurement errors; Measurement uncertainty; Memory organisation. Data processing; Noise measurement; Normal distribution; Quantization; reconstruction algorithms; Robustness; Sampling; Signal and communications theory; signal reconstruction; Signal, noise; Software; Sparsity; Telecommunications and information theory; Vectors; Random matrix; compressed sensing; Threshold detection; Iterative method; sparsity; iterative reconstruction; Compressive sensing; dimensionality reduction; AD converter; Sparse representation; Signal reconstruction; Robustness; Measurement error; Threshold; quantization; Lower bound; consistent reconstruction; Probabilistic approach; Subsampling; Dimension reduction; reconstruction algorithms; Gaussian process; Sampling rate; 1-bit compressed sensing; Approximation error
• ISSN: 0018-9448; 1557-9654
• DOI: 10.1109/TIT.2012.2234823

The compressive sensing (CS) framework aims to ease the burden on analog-to-digital converters (ADCs) by reducing the sampling rate required to acquire and stably recover sparse signals. Practical ADCs not only sample but also quantize each measurement to a finite number of bits; moreover, there is an inverse relationship between the achievable sampling rate and the bit depth. In this paper, we investigate an alternative CS approach that shifts the emphasis from the sampling rate to the number of bits per measurement. In particular, we explore the extreme case of 1-bit CS measurements, which capture just their sign. Our results come in two flavors. First, we consider ideal reconstruction from noiseless 1-bit measurements and provide a lower bound on the best achievable reconstruction error. We also demonstrate that i.i.d. random Gaussian matrices provide measurement mappings that, with overwhelming probability, achieve nearly optimal error decay. Next, we consider reconstruction robustness to measurement errors and noise and introduce the binary ε-stable embedding property, which characterizes the robustness of the measurement process to sign changes. We show that the same class of matrices that provide almost optimal noiseless performance also enable such a robust mapping. On the practical side, we introduce the binary iterative hard thresholding algorithm for signal reconstruction from 1-bit measurements that offers state-of-the-art performance.