Mark A. Iwen

Random Sampling for Analog-to-Information Conversion of Wideband Signals

We develop a framework for analog-to-information conversion that enables sub-Nyquist acquisition and processing of wideband signals that are sparse in a local Fourier representation. The first component of the framework is a random sampling system that can be implemented in practical hardware. The second is an efficient information recovery algorithm to compute the spectrogram of the signal, which we dub the sparsogram. A simulated acquisition of a frequency hopping signal operates at 33times sub-Nyquist average sampling rate with little degradation in signal quality

Group testing and sparse signal recovery

Traditionally, group testing is a design problem. The goal is to design an optimally efficient set of tests of items such that the test results contain enough information to determine a small subset of items of interest. It has its roots in the statistics community and was originally designed for the selective service during World War II to remove men with syphilis from the draft. It appears in many forms, including coin-weighing problems, experimental designs, and public health. We are interested in both the design of tests and the design of an efficient algorithm that works with the tests to determine the group of interest because many of the same techniques that are useful for designing tests are also used to solve algorithmic problems in compressive sensing, as well as to analyze and recover statistical quantities from streaming data. This article is an expository article, with the purpose of examining the relationship between group testing and compressive sensing, along with their applications and connections to sparse function learning.

Simple deterministically constructible RIP matrices with sublinear fourier sampling requirements

We present a deterministic number theoretic construction for matrices with the Restricted Isometry Property (RIP). Furthermore, we show that the number theoretic properties of our RIP matrices allow their products with Discrete Fourier Transform (DFT) matrices to be well approximated via a few highly sparse matrix multiplications. Hence, our RIP matrices may be approximately multiplied by the DFT of any input vector in sublinear-time by reading only a small fraction of its entries. As a consequence, we obtain small deterministic sample sets which are guaranteed to allow the recovery of near-optimal sparse Fourier representations for all periodic functions having an integrable second derivative over a single period. Explicit bounds are provided for the sizes of our RIP matrices, the sizes of their associated sublinear Fourier sampling sets, and the errors incurred by quickly approximating their products with DFT matrices. The Fourier sampling requirements obtained herein improve on previous deterministic Fourier sampling results in [1], [2].

Adaptive Strategies for Target Detection and Localization in Noisy Environments

This paper studies the problem of recovering a signal with a sparse representation in a given orthonormal basis using as few noisy observations as possible. Herein, observations are subject to the type of background clutter noise encountered in radar applications. Given this model, this paper proves for the first time that highly sparse signals contaminated with Gaussian background noise can be recovered by adaptive methods using fewer noisy linear measurements than required by any possible recovery method based on nonadaptive Gaussian measurement ensembles.

Group testing strategies for recovery of sparse signals in noise

We consider the recovery of sparse signals, f ∈ ℝN, containing at most k ≪ N nonzero entries using linear measurements contaminated with i.i.d. Gaussian background noise. Given this measurement model, we present and analyze a computationally efficient group testing strategy for recovering the support of f and approximating its nonzero entries. In particular, we demonstrate that group testing measurement matrix constructions may be combined with statistical binary detection and estimation methods to produce efficient adaptive sequential algorithms for sparse signal support recovery. Furthermore, when f exhibits sufficient sparsity, we show that these adaptive group testing methods allow the recovery of sparse signals using fewer noisy linear measurements than possible with non-adaptive methods based on Gaussian measurement ensembles. As a result we improve on previous sufficient conditions for sparsity pattern recovery in the noisy sublinear-sparsity regime.

Fast Phase Retrieval from Local Correlation Measurements

We develop a fast phase retrieval method which can utilize a large class of local phaseless correlation-based measurements in order to recover a given signal

A fast multiscale framework for data in high-dimensions: Measure estimation, anomaly detection, and compressive measurements

{\bf x} \in \mathbb{C}^d

On the Design of Deterministic Matrices for Fast Recovery of Fourier Compressible Functions

(up to an unknown global phase) in near-linear

Improved bounds for a deterministic sublinear-time Sparse Fourier Algorithm

\mathcal{O} \left( d \log^4 d \right)

Improved sparse fourier approximation results: faster implementations and stronger guarantees

-time. Accompanying theoretical analysis proves that the proposed algorithm is guaranteed to deterministically recover all signals