Exponential Decay of Reconstruction Error From Binary Measurements of Sparse Signals

Abstract

Binary measurements arise naturally in a variety of statistical and engineering applications. They may be inherent to the problem-e.g., in determining the relationship between genetics and the presence or absence of a disease-or they may be a result of extreme quantization. A recent influx of literature has suggested that using prior signal information can greatly improve the ability to reconstruct a signal from binary measurements. This is exemplified by onebit compressed sensing, which takes the compressed sensing model but assumes that only the sign of each measurement is retained. It has recently been shown that the number of one-bit measurements required for signal estimation mirrors that of unquantized compressed sensing. Indeed, s-sparse signals in R n can be estimated (up to normalization) from Ω(s log(n/s)) one-bit measurements. Nevertheless, controlling the precise accuracy of the error estimate remains an open challenge. In this paper, we focus on optimizing the decay of the error as a function of the oversampling factor λ := m/(s log(n/s)), where m is the number of measurements. It is known that the error in reconstructing sparse signals from standard one-bit measurements is bounded below by Ω(λ −1 ). Without adjusting the measurement procedure, reducing this polynomial error decay rate is impossible. However, we show that an adaptive choice of the thresholds used for quantization may lower the error rate to e −Ω(λ) . This improves upon guarantees for other methods of adaptive thresholding as proposed in Sigma-Delta quantization. We develop * Authors are listed in alphabetical order. a general recursive strategy to achieve this exponential decay and two specific polynomialtime algorithms which fall into this framework, one based on convex programming and one on hard thresholding. This work is inspired by the one-bit compressed sensing model, in which the engineer controls the measurement procedure. Nevertheless, the principle is extendable to signal reconstruction problems in a variety of binary statistical models as well as statistical estimation problems like logistic regression.

Dense Quantum Measurement Theory

“…The application of compressed sensing covers the fields of traditional signal processing, image processing and several different fields of computational mathematics [71][72][73][74][75][76][77][78].…”

Mean estimation from adaptive one-bit measurements

“…A growing literature on one-bit measurements in high-dimensional problems [7], [18], [19] shows how to reconstruct sparse signals, where Baraniuk et al [7] show that in noiseless settings, exponential decay in MSE is possible; our results make precise the penalty for noise under one-bit sensing, showing that the error can decay (under Gaussian noise) at best as π 2 σ 2 n . In fully distributed settings (iii), the challenges are different, and there is also a substantial literature with one-bit (quantized) measurements [20], [21], [22], [23], [24].…”

“…(We shall be more formal in the sequel.) By lower bounding the quantity (1), we also provide limits on estimation of single-bit-per-measurement constrained signals in more general settings [7], [8], [9], [10], [11]. In setting (i), the estimator can evaluate any optimal estimator of location (e.g., the sample mean if the data is Gaussian), then quantize it using n bits.…”

Analysis of hard-thresholding for distributed compressed sensing with one-bit measurements

“…The field of compressed sensing emerged from the following basic question: Given some unknown and highdimensional signal x ∈ R n , what is the smallest number m of linear measurements y = Ax (1) needed to uniquely determine x, where A ∈ R m×n . From basic linear algebra we require m ≥ n in general.…”

“…In particular, real-valued measurements y i ∈ R cannot not be stored with infinite precision. The idealistic measurement model presented in (1) should be extended by a quantizer Q that maps the real-valued measurement vector Ax to a finite alphabet. The extreme case is to choose Q as the sign function acting componentwise on Ax leading to the one-bit compressed sensing model first studied in [3] y = sign(Ax) (4) i.e., y i is 1 if a i , x ≥ 0 and −1 if a i , x < 0, where a i is the i-th row of A.…”