Robust Decoding from 1-Bit Compressive Sampling with Least Squares

by   Jian Huang, et al.

In 1-bit compressive sensing (1-bit CS) where target signal is coded into a binary measurement, one goal is to recover the signal from noisy and quantized samples. Mathematically, the 1-bit CS model reads: y = ηsign (Ψ x^* + ϵ), where x^*∈R^n, y∈R^m, Ψ∈R^m× n, and ϵ is the random error before quantization and η∈R^n is a random vector modeling the sign flips. Due to the presence of nonlinearity, noise and sign flips, it is quite challenging to decode from the 1-bit CS. In this paper, we consider least squares approach under the over-determined and under-determined settings. For m>n, we show that, up to a constant c, with high probability, the least squares solution x_ls approximates x^* with precision δ as long as m ≥O(n/δ^2). For m< n, we prove that, up to a constant c, with high probability, the ℓ_1-regularized least-squares solution x_ℓ_1 lies in the ball with center x^* and radius δ provided that m ≥O( s n/δ^2) and x^*_0 := s < m. We introduce a Newton type method, the so-called primal and dual active set (PDAS) algorithm, to solve the nonsmooth optimization problem. The PDAS possesses the property of one-step convergence. It only requires to solve a small least squares problem on the active set. Therefore, the PDAS is extremely efficient for recovering sparse signals through continuation. We propose a novel regularization parameter selection rule which does not introduce any extra computational overhead. Extensive numerical experiments are presented to illustrate the robustness of our proposed model and the efficiency of our algorithm.


page 1

page 2

page 3

page 4


Robust Decoding from Binary Measurements with Cardinality Constraint Least Squares

The main goal of 1-bit compressive sampling is to decode n dimensional s...

Pinball Loss Minimization for One-bit Compressive Sensing

The one-bit quantization can be implemented by one single comparator, wh...

A Primal Dual Active Set with Continuation Algorithm for the ℓ^0-Regularized Optimization Problem

We develop a primal dual active set with continuation algorithm for solv...

Just Least Squares: Binary Compressive Sampling with Low Generative Intrinsic Dimension

In this paper, we consider recovering n dimensional signals from m binar...

One-bit compressive sensing with norm estimation

Consider the recovery of an unknown signal x from quantized linear measu...

Nonconvex penalties with analytical solutions for one-bit compressive sensing

One-bit measurements widely exist in the real world, and they can be use...

Robust Least Squares for Quantized Data

In this paper we formulate and solve a robust least squares problem for ...

Please sign up or login with your details

Forgot password? Click here to reset