On Recovery Guarantees for One-Bit Compressed Sensing on Manifolds

Abstract

This paper studies the problem of recovering a signal from one-bit compressed sensing measurements under a manifold model; that is, assuming that the signal lies on or near a manifold of low intrinsic dimension. We provide a convex recovery method based on the Geometric Multi-Resolution Analysis and prove recovery guarantees with a near-optimal scaling in the intrinsic manifold dimension. Our method is the first tractable algorithm with such guarantees for this setting. The results are complemented by numerical experiments confirming the validity of our approach.

Robust One-bit Compressed Sensing With Manifold Data

“…To improve on this, we follow the main idea of [2] and introduce a pre-processing step, leading to Algorithm 2. As we will rigorously prove in Lemma III.6, under appropriate conditions the first step of the algorithm will identify a center c j,k satisfying (4).…”

“…Both methods are robust to noise before and during quantization. In addition, our bounds on the required number of measurements for accurate signal recovery exhibit better parameter dependencies than [2].…”

“…We wish to recover x accurately using as few measurements as possible, using a computationally efficient algorithm. The recent work [2] introduced a recovery algorithm for this problem in the setting where A is standard Gaussian, ν = 0, τ = 0 and M ⊂ S D−1 . The purpose of this note is to show that if one uses dithering in the quantizer, i.e., uses a suitably chosen random threshold vector τ , then superior results can be obtained.…”

Higher order 1-bit Sigma-Delta modulation on a circle

“…Some results for manifolds were obtained in connection with 1-bit compressed sensing. Namely, the problem of recovering an unknown data point on a given manifold from 1-bit quantized random measurements was studied in [11]. Recently, this work was further extended in [12] to incorporate Σ∆ modulation schemes.…”

“…For our numerical experiments we consider the signal f (t) = 0.1 sin(5t) cos(10t) + 0.2 with bandwidth K = 15 and the reconstruction formulas (11) and (9). (c) we show the difference between f (t), the original function, and f r (t), its reconstructed version.…”