10,737 research outputs found
Robust 1-Bit Compressed Sensing via Hinge Loss Minimization
This work theoretically studies the problem of estimating a structured
high-dimensional signal from noisy -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 can be achieved with
the optimal oversampling rate in terms of the number of
measurements . Moreover, we permit a wide class of structural assumptions on
the ground truth signal, in the sense that can belong to an arbitrary
bounded convex set . 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
Improved Bounds for Universal One-Bit Compressive Sensing
Unlike compressive sensing where the measurement outputs are assumed to be
real-valued and have infinite precision, in "one-bit compressive sensing",
measurements are quantized to one bit, their signs. In this work, we show how
to recover the support of sparse high-dimensional vectors in the one-bit
compressive sensing framework with an asymptotically near-optimal number of
measurements. We also improve the bounds on the number of measurements for
approximately recovering vectors from one-bit compressive sensing measurements.
Our results are universal, namely the same measurement scheme works
simultaneously for all sparse vectors.
Our proof of optimality for support recovery is obtained by showing an
equivalence between the task of support recovery using 1-bit compressive
sensing and a well-studied combinatorial object known as Union Free Families.Comment: 14 page
Quantization and Compressive Sensing
Quantization is an essential step in digitizing signals, and, therefore, an
indispensable component of any modern acquisition system. This book chapter
explores the interaction of quantization and compressive sensing and examines
practical quantization strategies for compressive acquisition systems.
Specifically, we first provide a brief overview of quantization and examine
fundamental performance bounds applicable to any quantization approach. Next,
we consider several forms of scalar quantizers, namely uniform, non-uniform,
and 1-bit. We provide performance bounds and fundamental analysis, as well as
practical quantizer designs and reconstruction algorithms that account for
quantization. Furthermore, we provide an overview of Sigma-Delta
() quantization in the compressed sensing context, and also
discuss implementation issues, recovery algorithms and performance bounds. As
we demonstrate, proper accounting for quantization and careful quantizer design
has significant impact in the performance of a compressive acquisition system.Comment: 35 pages, 20 figures, to appear in Springer book "Compressed Sensing
and Its Applications", 201
One-bit compressive sensing with norm estimation
Consider the recovery of an unknown signal from quantized linear
measurements. In the one-bit compressive sensing setting, one typically assumes
that is sparse, and that the measurements are of the form
. Since such
measurements give no information on the norm of , recovery methods from
such measurements typically assume that . We show that if one
allows more generally for quantized affine measurements of the form
, and if the vectors
are random, an appropriate choice of the affine shifts allows
norm recovery to be easily incorporated into existing methods for one-bit
compressive sensing. Additionally, we show that for arbitrary fixed in
the annulus , one may estimate the norm up to additive error from
such binary measurements through a single evaluation of the inverse Gaussian
error function. Finally, all of our recovery guarantees can be made universal
over sparse vectors, in the sense that with high probability, one set of
measurements and thresholds can successfully estimate all sparse vectors
within a Euclidean ball of known radius.Comment: 20 pages, 2 figure
Green compressive sampling reconstruction in IoT networks
In this paper, we address the problem of green Compressed Sensing (CS) reconstruction within Internet of Things (IoT) networks, both in terms of computing architecture and reconstruction algorithms. The approach is novel since, unlike most of the literature dealing with energy efficient gathering of the CS measurements, we focus on the energy efficiency of the signal reconstruction stage given the CS measurements. As a first novel contribution, we present an analysis of the energy consumption within the IoT network under two computing architectures. In the first one, reconstruction takes place within the IoT network and the reconstructed data are encoded and transmitted out of the IoT network; in the second one, all the CS measurements are forwarded to off-network devices for reconstruction and storage, i.e., reconstruction is off-loaded. Our analysis shows that the two architectures significantly differ in terms of consumed energy, and it outlines a theoretically motivated criterion to select a green CS reconstruction computing architecture. Specifically, we present a suitable decision function to determine which architecture outperforms the other in terms of energy efficiency. The presented decision function depends on a few IoT network features, such as the network size, the sink connectivity, and other systems’ parameters. As a second novel contribution, we show how to overcome classical performance comparison of different CS reconstruction algorithms usually carried out w.r.t. the achieved accuracy. Specifically, we consider the consumed energy and analyze the energy vs. accuracy trade-off. The herein presented approach, jointly considering signal processing and IoT network issues, is a relevant contribution for designing green compressive sampling architectures in IoT networks
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