1-bit Measurements Research Articles

Data-Driven Method for Robust Recovery in 1-Bit Compressive Sensing with the Minimax Concave Penalty

With the advent of large-scale data, the demand for information is increasing, which makes signal sampling technology and digital processing methods particularly important. The utilization of 1-bit compressive sensing in sparse recovery has garnered significant attention due to its cost-effectiveness in hardware implementation and storage. In this paper, we first leverage the minimax concave penalty equipped with the least squares to recover a high-dimensional true signal x∈Rp with k-sparse from n-dimensional 1-bit measurements and discuss the regularization by combing the nonconvex sparsity-inducing penalties. Moreover, we give an analysis of the complexity of the method with minimax concave penalty in certain conditions and derive the general theory for the model equipped with the family of sparsity-inducing nonconvex functions. Then, our approach employs a data-driven Newton-type method with stagewise steps to solve the proposed method. Numerical experiments on the synthesized and real data verify the competitiveness of the proposed method.

Periodic Signal Recovery and Detection from 1-Bit Measurements Using Convolutional Neural Network

Periodic Signal Recovery and Detection from 1-Bit Measurements Using Convolutional Neural Network

Harmonics-Based One-Bit SNR Estimation

Signal-to-noise ratio (SNR) estimation for the one-bit analog-to-digital converter (ADC) is a challenging task due to the nonlinearity, lots of information loss, and complex probability distribution. And classical SNR estimators are unavailable in 1-bit sampling. Given this issue, we propose novel one-bit SNR estimators with the closed-form solution based on harmonic analysis to estimate the SNR from 1-bit measurements. We first derive the statistical characteristic of each-order harmonic caused by 1-bit sampling and then give the mathematical relationship between the expectation of harmonic amplitude (EoHA) and the SNR. The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> th-order harmonic-based estimator (mHE) with high precision is devised by this relationship. Further, an approximate alternative, the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${m}$ </tex-math></inline-formula> th- and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">${n}$ </tex-math></inline-formula> th-order harmonics-based ratio estimator (mnHRE), is presented to reduce the computational complexity. Moreover, two schemes for calculating the EoHA are provided to assist the proposed one-bit SNR estimators. Simulation results verify the effectiveness and superiority of our proposed estimators.

Measurement of the Parameters of Multiple Sinusoids Based on Binary Data

This article introduces a novel procedure for quick estimation of the parameters of a sum of sinusoidal signals based on 1-bit measurements. Amplitude, phases, and frequencies of the signal components are assumed unknown, as well as the threshold level of the comparator used to produce measurement results. To provide enough information at the 1-bit quantizer input, a sinewave is assumed to dither one of the two comparator's inputs. To ease the procedure's application, only the peak-to-peak amplitude of this dither signal is assumed known. Theoretical, simulation-based and experimental results validate the presented approach.

DeepFPC: A deep unfolded network for sparse signal recovery from 1-Bit measurements with application to DOA estimation

In this paper, we introduce a novel deep neural network, coined DeepFPC, and investigate its application to tackling the problem of direction-of-arrival (DOA) estimation. DeepFPC is designed by unfolding the iterations of the fixed-point continuation algorithm with one-sided ℓ1-norm (FPC-ℓ1), which has been proposed for solving the 1-bit compressed sensing problem. The network architecture resembles that of deep residual learning and incorporates prior knowledge about the signal structure (i.e., sparsity), thereby offering interpretability by design. Once DeepFPC is properly trained, a sparse signal can be recovered fast and accurately from quantized measurements. The proposed model is then applied in DOA estimation and is shown to outperform state-of-the-art solutions; namely, the iterative FPC-ℓ1 algorithm and the deep convolution network (DCN) model.

MMV-based OMP for DOA estimation with 1-bit measurement

This paper shows a novel scheme for direction-of-arrival (DOA) estimation of narrowband signals. In the scheme, the 1-bit measurement is used to reduce the system cost. A recovery compressed sensing (CS)-based algorithm with multiple measurement vector (MMV) is proposed to make up the information loss. The purposed MMV-OMP algorithm provides a new idea which make the compress sensing framework applicable directly to an MMV model. The effectiveness of the proposed approach is assessed through numerical simulations under different environment configurations. Comparisons with other methods are reported as well.

Sparse Recovery and Dictionary Learning From Nonlinear Compressive Measurements

Sparse coding and dictionary learning are popular techniques for linear inverse problems such as denoising or inpainting. However in many cases, the measurement process is nonlinear, for example for clipped, quantized or 1-bit measurements. These problems have often been addressed by solving constrained sparse coding problems, which can be difficult to solve, and assuming that the sparsifying dictionary is known and fixed. Here we propose a simple and unified framework to deal with nonlinear measurements. We propose a cost function that minimizes the distance to a convex feasibility set, which models our knowledge about the nonlinear measurement. This provides an unconstrained, convex, and differentiable cost function that is simple to optimize, and generalizes the linear least squares cost commonly used in sparse coding. We then propose proximal based sparse coding and dictionary learning algorithms, that are able to learn directly from nonlinearly corrupted signals. We show how the proposed framework and algorithms can be applied to clipped, quantized and 1-bit data.

Robust one-bit compressive sensing with weighted ℓ1-norm minimization

In recent years, one-bit compressive sensing (1-bit CS) has been attracting much attentions owing to its advantages of low system complexity and outstanding environmental robustness. Nevertheless, how to recover the signal from 1-bit measurements still needs deeper investigation. Inspired by the excellent performance of weighted ℓ1-norm minimization in conventional CS techniques, this paper presents robust algorithms with weighted ℓ1-norm minimization for 1-bit CS. Specifically, it is first shown that the analytical solution for Passive model can be straightforwardly extended to the weighted case, and the weights are not necessarily positive. On this basis, two representative ways are introduced to calculate the weights. Under the weighted ℓ1-norm minimization framework, the proposed weighted ℓ1-norm algorithms can be easily implemented. Numerical simulations show that the recovery fidelity can be effectively improved with the weighted ℓ1-norm minimization techniques.

Learning Tensors From Partial Binary Measurements

In this paper we generalize the 1-bit matrix completion problem to higher order tensors. We prove that when $r=O(1)$ a bounded rank-$r$, order-$d$ tensor $T$ in $\mathbb{R}^{N} \times \mathbb{R}^{N} \times \cdots \times \mathbb{R}^{N}$ can be estimated efficiently by only $m=O(Nd)$ binary measurements by regularizing its max-qnorm and M-norm as surrogates for its rank. We prove that similar to the matrix case, i.e., when $d=2$, the sample complexity of recovering a low-rank tensor from 1-bit measurements of a subset of its entries is the same as recovering it from unquantized measurements. Moreover, we show the advantage of using 1-bit tensor completion over matricization both theoretically and numerically. Specifically, we show how the 1-bit measurement model can be used for context-aware recommender systems.

Continuous-time signal recovery from 1-bit multiple measurement vectors

In this letter, the 1-bit compressive sensing (CS) technology is utilized to recover joint sparse signal. In the context of multi-bits quantization, compared to the single measurement vector (SMV) case, the successful recovery rate can be significantly improved using multiple measurement vectors (MMV) model. Therefore, we introduce the MMV model for signal recovery from 1-bit sampling. Due to the amplitude information loss using 1-bit quantizing, a new sampling framework is proposed to estimate the ℓ2-norm of each measurement vector for continuous-time signal, followed by the corresponding estimation algorithm 1-bit iterative hard thresholding (1-bit IHT). Then we develop the algorithm of binary IHT for MMV (M-BIHT) to recover joint sparse signal under uncertain noise. Compared to the 1-bit SMV model and multi-bits quantization MMV mode, the experimental results illustrate the recovery performance is improved greatly by the 1-bit MMV model.

Gridless compressive sensing method for line spectral estimation from 1-bit measurements

This paper considers the problem of recovering frequency sparse signals which consist of a few complex sinusoids and estimating the frequency components from 1-bit quantized measurements. Unlike previous grid-based 1-bit compressive sensing approaches, we present a gridless convex method to recover frequency sparse signals form 1-bit measurements via binary atomic norm minimization (BANM). And the frequencies can take any continuous values in the frequency domain, which overcomes grid mismatches caused by the off-grid problem. We further propose a dual polynomial method to achieve continuous frequency estimation. Moreover, we present an efficient algorithm to solve BANM for large scaled problem. Numerical experiments are performed to demonstrate the effectiveness of our method compared with the grid-based compressive sensing algorithm.

1-Bit compressive sensing: Reformulation and RRSP-based sign recovery theory

Recently, the 1-bit compressive sensing (1-bit CS) has been studied in the field of sparse signal recovery. Since the amplitude information of sparse signals in 1-bit CS is not available, it is often the support or the sign of a signal that can be exactly recovered with a decoding method. We first show that a necessary assumption (that has been overlooked in the literature) should be made for some existing theories and discussions for 1-bit CS. Without such an assumption, the found solution by some existing decoding algorithms might be inconsistent with 1-bit measurements. This motivates us to pursue a new direction to develop uniform and nonuniform recovery theories for 1-bit CS with a new decoding method which always generates a solution consistent with 1-bit measurements. We focus on an extreme case of 1-bit CS, in which the measurements capture only the sign of the product of a sensing matrix and a signal. We show that the 1-bit CS model can be reformulated equivalently as an l 0-minimization problem with linear constraints. This reformulation naturally leads to a new linear-program-based decoding method, referred to as the 1-bit basis pursuit, which is remarkably different from existing formulations. It turns out that the uniqueness condition for the solution of the 1-bit basis pursuit yields the so-called restricted range space property (RRSP) of the transposed sensing matrix. This concept provides a basis to develop sign recovery conditions for sparse signals through 1-bit measurements. We prove that if the sign of a sparse signal can be exactly recovered from 1-bit measurements with 1-bit basis pursuit, then the sensing matrix must admit a certain RRSP, and that if the sensing matrix admits a slightly enhanced RRSP, then the sign of a k-sparse signal can be exactly recovered with 1-bit basis pursuit.

One-bit compressive sampling via ℓ 0 minimization

The problem of 1-bit compressive sampling is addressed in this paper. We introduce an optimization model for reconstruction of sparse signals from 1-bit measurements. The model targets a solution that has the least l 0-norm among all signals satisfying consistency constraints stemming from the 1-bit measurements. An algorithm for solving the model is developed. Convergence analysis of the algorithm is presented. Our approach is to obtain a sequence of optimization problems by successively approximating the l 0-norm and to solve resulting problems by exploiting the proximity operator. We examine the performance of our proposed algorithm and compare it with the renormalized fixed point iteration (RFPI) (Boufounos and Baraniuk, 1-bit compressive sensing, 2008; Movahed et al., A robust RFPI-based 1-bit compressive sensing reconstruction algorithm, 2012), the generalized approximate message passing (GAMP) (Kamilov et al., IEEE Signal Process. Lett. 19(10):607–610, 2012), the linear programming (LP) (Plan and Vershynin, Commun. Pure Appl. Math. 66:1275–1297, 2013), and the binary iterative hard thresholding (BIHT) (Jacques et al., IEEE Trans. Inf. Theory 59:2082–2102, 2013) state-of-the-art algorithms for 1-bit compressive sampling reconstruction.

HISTORY: An Efficient and Robust Algorithm for Noisy 1-Bit Compressed Sensing

We consider the problem of sparse signal recovery from 1-bit measurements. Due to the noise present in the acquisition and transmission process, some quantized bits may be flipped to their opposite states. These sign flips may result in severe performance degradation. In this study, a novel algorithm, termed HISTORY, is proposed. It consists of Hamming support detection and coefficients recovery. The HISTORY algorithm has high recovery accuracy and is robust to strong measurement noise. Numerical results are provided to demonstrate the effectiveness and superiority of the proposed algorithm.

1-Bit matrix completion

1-Bit matrix completion

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

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 e-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.

Robust 1-bit Compressive Sensing Using Adaptive Outlier Pursuit

In compressive sensing (CS), the goal is to recover signals at reduced sample rate compared to the classic Shannon-Nyquist rate. However, the classic CS theory assumes the measurements to be real-valued and have infinite bit precision. The quantization of CS measurements has been studied recently and it has been shown that accurate and stable signal acquisition is possible even when each measurement is quantized to only one single bit. There are many algorithms proposed for 1-bit compressive sensing and they work well when there is no noise in the measurements, e.g., there are no sign flips, while the performance is worsened when there are a lot of sign flips in the measurements. In this paper, we propose a robust method for recovering signals from 1-bit measurements using adaptive outlier pursuit. This method will detect the positions where sign flips happen and recover the signals using “correct” measurements. Numerical experiments show the accuracy of sign flips detection and high performance of signal recovery for our algorithms compared with other algorithms.

Relaxed maximum a posteriori fault identification

We consider the problem of estimating a pattern of faults, represented as a binary vector, from a set of measurements. The measurements can be noise corrupted real values, or quantized versions of noise corrupted signals, including even 1-bit (sign) measurements. Maximum a posteriori probability (MAP) estimation of the fault pattern leads to a difficult combinatorial optimization problem, so we propose a variation in which an approximate maximum a posteriori probability estimate is found instead, by solving a convex relaxation of the original problem, followed by rounding and simple local optimization. Our method is extremely efficient, and scales to very large problems, involving thousands (or more) of possible faults and measurements. Using synthetic examples, we show that the method performs extremely well, both in identifying the true fault pattern, and in identifying an ambiguity group, i.e., a set of alternate fault patterns that explain the observed measurements almost as well as our estimate.