Robust 1-Bit Compressed Sensing via Hinge Loss Minimization

This work theoretically studies the problem of estimating a structured high-dimensional signal $x_0 \in \mathbb{R}^n$ from noisy $1$-bit Gaussian measurements. Our recovery approach is based on a simple convex program which uses the hinge loss function as data fidelity term. While such a risk minimization strategy is very natural to learn binary output models, such as in classification, its capacity to estimate a specific signal vector is largely unexplored. A major difficulty is that the hinge loss is just piecewise linear, so that its "curvature energy" is concentrated in a single point. This is substantially different from other popular loss functions considered in signal estimation, e.g., the square or logistic loss, which are at least locally strongly convex. It is therefore somewhat unexpected that we can still prove very similar types of recovery guarantees for the hinge loss estimator, even in the presence of strong noise. More specifically, our non-asymptotic error bounds show that stable and robust reconstruction of $x_0$ can be achieved with the optimal oversampling rate $O(m^{-1/2})$ in terms of the number of measurements $m$. Moreover, we permit a wide class of structural assumptions on the ground truth signal, in the sense that $x_0$ can belong to an arbitrary bounded convex set $K \subset \mathbb{R}^n$. The proofs of our main results rely on some recent advances in statistical learning theory due to Mendelson. In particular, we invoke an adapted version of Mendelson's small ball method that allows us to establish a quadratic lower bound on the error of the first order Taylor approximation of the empirical hinge loss function

A Multiple Hypothesis Testing Approach to Low-Complexity Subspace Unmixing

Subspace-based signal processing traditionally focuses on problems involving a few subspaces. Recently, a number of problems in different application areas have emerged that involve a significantly larger number of subspaces relative to the ambient dimension. It becomes imperative in such settings to first identify a smaller set of active subspaces that contribute to the observation before further processing can be carried out. This problem of identification of a small set of active subspaces among a huge collection of subspaces from a single (noisy) observation in the ambient space is termed subspace unmixing. This paper formally poses the subspace unmixing problem under the parsimonious subspace-sum (PS3) model, discusses connections of the PS3 model to problems in wireless communications, hyperspectral imaging, high-dimensional statistics and compressed sensing, and proposes a low-complexity algorithm, termed marginal subspace detection (MSD), for subspace unmixing. The MSD algorithm turns the subspace unmixing problem for the PS3 model into a multiple hypothesis testing (MHT) problem and its analysis in the paper helps control the family-wise error rate of this MHT problem at any level $\alpha \in [0,1]$ under two random signal generation models. Some other highlights of the analysis of the MSD algorithm include: (i) it is applicable to an arbitrary collection of subspaces on the Grassmann manifold; (ii) it relies on properties of the collection of subspaces that are computable in polynomial time; and ($iii$) it allows for linear scaling of the number of active subspaces as a function of the ambient dimension. Finally, numerical results are presented in the paper to better understand the performance of the MSD algorithm.Comment: Submitted for journal publication; 33 pages, 14 figure

Adaptive sensing performance lower bounds for sparse signal detection and support estimation

This paper gives a precise characterization of the fundamental limits of adaptive sensing for diverse estimation and testing problems concerning sparse signals. We consider in particular the setting introduced in (IEEE Trans. Inform. Theory 57 (2011) 6222-6235) and show necessary conditions on the minimum signal magnitude for both detection and estimation: if ${\mathbf {x}}\in \mathbb{R}^n$ is a sparse vector with $s$ non-zero components then it can be reliably detected in noise provided the magnitude of the non-zero components exceeds $\sqrt{2/s}$. Furthermore, the signal support can be exactly identified provided the minimum magnitude exceeds $\sqrt{2\log s}$. Notably there is no dependence on $n$, the extrinsic signal dimension. These results show that the adaptive sensing methodologies proposed previously in the literature are essentially optimal, and cannot be substantially improved. In addition, these results provide further insights on the limits of adaptive compressive sensing.Comment: Published in at http://dx.doi.org/10.3150/13-BEJ555 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm