One-bit Compressed Sensing in the Presence of Noise

Many modern real-world systems generate large amounts of high-dimensional data stressing the available computing and signal processing systems. In resource-constrained settings, it is desirable to process, store and transmit as little amount of data as possible. It has been shown that one can obtain acceptable performance for tasks such as inference and reconstruction using fewer bits of data by exploiting low-dimensional structures on data such as sparsity. This dissertation investigates the signal acquisition paradigm known as one-bit compressed sensing (one-bit CS) for signal reconstruction and parameter estimation. We first consider the problem of joint sparse support estimation with one-bit measurements in a distributed setting. Each node observes sparse signals with the same but unknown support. The goal is to minimize the probability of error of support estimation. First, we study the performance of maximum likelihood (ML) estimation of the support set from one-bit compressed measurements when all these measurements are available at the fusion center. We provide a lower bound on the number of one-bit measurements required per node for vanishing probability of error. Though the ML estimator is optimal, its computational complexity increases exponentially with the signal dimension. So, we propose computationally tractable algorithms in a centralized setting. Further, we extend these algorithms to a decentralized setting where each node can communicate only with its one-hop neighbors. The proposed method shows excellent estimation performance even in the presence of noise. In the second part of the dissertation, we investigate the problem of sparse signal reconstruction from noisy one-bit compressed measurements using a signal that is statistically dependent on the compressed signal as an aid. We refer to this signal as side-information. We consider a generalized measurement model of one-bit CS where noise is assumed to be added at two stages of the measurement process- a) before quantizationand b) after quantization. We model the noise before quantization as additive white Gaussian noise and the noise after quantization as a sign-flip noise generated from a Bernoulli distribution. We assume that the SI at the receiver is noisy. The noise in the SI can be either in the support or in the amplitude, or both. This nature of the noise in SI suggests that the noise has a sparse structure. We use additive independent and identically distributed Laplacian noise to model such sparse nature of the noise. In this setup, we develop tractable algorithms that approximate the minimum mean square error (MMSE) estimator of the signal. We consider the following three different SI-based scenarios: 1. The side-information is assumed to be a noisy version of the signal. The noise is independent of the signal and follows the Laplacian distribution. We do not assume any temporal dependence in the signal.2. The signal exhibits temporal dependencies between signals at the current time instant and the previous time instant. The temporal dependence is modeled using the birth-death-drift (BDD) model. The side-information is a noisy version of the previous time instant signal, which is statistically dependent on the signal as defined by the BDD model. 3. The SI available at the receiver is heterogeneous. The signal and side-information are from different modalities and may not share joint sparse representation. We assume that the SI and the sparse signal are dependent and use the Copula function to model the dependence. In each of these scenarios, we develop generalized approximate message passing-based algorithms to approximate the minimum mean square error estimate. Numerical results show the effectiveness of the proposed algorithm. In the final part of the dissertation, we propose two one-bit compressed sensing reconstruction algorithms that use a deep neural network as a prior on the signal. In the first algorithm, we use a trained Generative model such as Generative Adversarial Networks and Variational Autoencoders as a prior. This trained network is used to reconstruct the compressed signal from one-bit measurements by searching over its range. We provide theoretical guarantees on the reconstruction accuracy and sample complexity of the presented algorithm. In the second algorithm, we investigate an untrained neural network architecture so that it acts as a good prior on natural signals such as images and audio. We formulate an optimization problem to reconstruct the signal from one-bit measurements using this untrained network. We demonstrate the superior performance of the proposed algorithms through numerical results. Further, in contrast to competing model-based algorithms, we demonstrate that the proposed algorithms estimate both direction and magnitude of the compressed signal from one-bit measurements