This paper treats the problem of minimizing a general continuously
differentiable function subject to sparsity constraints. We present and analyze
several different optimality criteria which are based on the notions of
stationarity and coordinate-wise optimality. These conditions are then used to
derive three numerical algorithms aimed at finding points satisfying the
resulting optimality criteria: the iterative hard thresholding method and the
greedy and partial sparse-simplex methods.

In this paper, we propose an efficient acquisition scheme for GPS receivers.
It is shown that GPS signals can be effectively sampled and detected using a
bank of randomized correlators with much fewer chip-matched filters than those
used in existing GPS signal acquisition algorithms. The latter use correlations
with all possible shifted replicas of the satellite-specific C/A code and an
exhaustive search for peaking signals over the delay-Doppler space.

We develop sub-Nyquist sampling systems for analog signals comprised of
several, possibly overlapping, finite duration pulses with unknown shapes and
time positions. Efficient sampling schemes when either the pulse shape or the
locations of the pulses are known have been previously developed. To the best
of our knowledge, stable and low-rate sampling strategies for a superposition
of unknown pulses without knowledge of the pulse locations have not been
derived. The goal in this two-part paper is to fill this gap.

We present a mixed analog-digital spectrum sensing method that is especially
suited to the typical wideband setting of cognitive radio (CR). The advantages
of our system with respect to current architectures are threefold. First, our
analog front-end is fixed and does not involve scanning hardware. Second, both
the analog-to-digital conversion (ADC) and the digital signal processing (DSP)
rates are substantially below Nyquist.

The Kaczmarz method is an algorithm for finding the solution to an
overdetermined system of linear equations Ax=b by iteratively projecting onto
the solution spaces. The randomized version put forth by Strohmer and Vershynin
yields provably exponential convergence in expectation, which for highly
overdetermined systems even outperforms the conjugate gradient method. In this
article we present a modified version of the randomized Kaczmarz method which
at each iteration selects the optimal projection from a randomly chosen set,
which in most cases significantly improves the convergence rate.

In this paper we revisit the sparse multiple measurement vector (MMV) problem
where the aim is to recover a set of jointly sparse multichannel vectors from
incomplete measurements. This problem has received increasing interest as an
extension of the single channel sparse recovery problem which lies at the heart
of the emerging field of compressed sensing. However the sparse approximation
problem has origins which include links to the field of array signal processing
where we find the inspiration for a new family of MMV algorithms based on the
MUSIC algorithm.

Sparse modeling is a powerful framework for data analysis and processing.
Traditionally, encoding in this framework is done by solving an l_1-regularized
linear regression problem, usually called Lasso. In this work we first combine
the sparsity-inducing property of the Lasso model, at the individual feature
level, with the block-sparsity property of the group Lasso model, where sparse
groups of features are jointly encoded, obtaining a sparsity pattern
hierarchically structured. This results in the hierarchical Lasso, which shows
important practical modeling advantages.

We consider the problem of estimating a deterministic sparse vector x from
underdetermined measurements Ax+w, where w represents white Gaussian noise and
A is a given deterministic dictionary. We analyze the performance of three
sparse estimation algorithms: basis pursuit denoising (BPDN), orthogonal
matching pursuit (OMP), and thresholding. These algorithms are shown to achieve
near-oracle performance with high probability, assuming that x is sufficiently
sparse. Our results are non-asymptotic and are based only on the coherence of
A, so that they are applicable to arbitrary dictionaries.

Time delay estimation arises in many applications in which a multipath medium
has to be identified from pulses transmitted through the channel. Various
approaches have been proposed in the literature to identify time delays
introduced by multipath environments. However, these methods either operate on
the analog received signal, or require high sampling rates in order to achieve
reasonable time resolution. In this paper, our goal is to develop a unified
approach to time delay estimation from low rate samples of the output of a
multipath channel.

The sensing matrix of a compressive system impacts the stability of the
associated sparse recovery problem. In this paper, we study the sensing matrix
of the modulated wideband converter, a recently proposed system for sub-Nyquist
sampling of analog sparse signals. Attempting to quantify the conditioning of
the converter sensing matrix with existing approaches leads to unreasonable
rate requirements, due to the relatively small size of this matrix.