CoSaMP: Iterative signal recovery from incomplete and inaccurate samples

This lecture note presents a new method to capture and represent compressible signals at a rate significantly below the Nyquist rate. This method, called compressive sensing, employs nonadaptive linear projections that preserve the structure of the signal; the signal is then reconstructed from these projections using an optimization process.

Compressive Sensing [Lecture Notes]

In many important statistical applications, the number of variables or parameters p is much larger than the number of observations n. Suppose then that we have observations y=Xβ+z, where β∈Rp is a parameter vector of interest, X is a data matrix with possibly far fewer rows than columns, n≪p, and the zi’s are i.i.d. N(0, σ^2). Is it possible to estimate β reliably based on the noisy data y?

https://projecteuclid.org/download/pdfview_1/euclid.aos/1201012958

The Dantzig selector: Statistical estimation when p is much larger than n

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case.

In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large.

The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.

https://faculty.washington.edu/mfazel/low-rank-v2.pdf

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

In this article, the authors present a new approach to building simpler, smaller, and cheaper digital cameras that can operate efficiently across a broader spectral range than conventional silicon-based cameras. The approach fuses a new camera architecture based on a digital micromirror device with the new mathematical theory and algorithms of compressive sampling.

/pdf/single-pixel-imaging-via-compressive-sampling-17xma709hr.pdf

Single-Pixel Imaging via Compressive Sampling

We develop a framework for analog-to-information conversion based on the theory of information recovery from random samples. The framework enables sub-Nyquist acquisition and processing of wideband signals that are sparse in a local Fourier representation. We present the random sampling theory associated with an efficient information recovery algorithm to compute the spectrogram of the signal. Additionally, we develop a hardware design for the random sampling system that demonstrates a consistent reconstruction fidelity in the presence of sampling jitter, which forms the main source of non-ideality in a practical system implementation.

Implementation models for analog-to-information conversion via random sampling

The recently developed compressive sensing (CS) framework enables the design of sub-Nyquist analog-to-digital converters. Several architectures have been proposed for the acquisition of sparse signals in large swaths of bandwidth. In this paper we consider a more flexible multi-channel signal model consisting of several discontiguous channels where the occupancy of the combined bandwidth of the channels is sparse. We introduce a new compressive acquisition architecture, the compressive multiplexer (CMUX), to sample such signals. We demonstrate that our architecture is CS-feasible and suggest a simple implementation with numerous practical advantages.

The compressive multiplexer for multi-channel compressive sensing

In this paper, we successfully demonstrate the feasibility of hardware implementation of a sub-Nyquist random- sampling based analog to information converter (RS-AIC). The RS-AIC is based on the theory of information recovery from random samples using an efficient information recovery algorithm to compute the spectrogram of the signal. Our RS-AIC enables sub-Nyquist acquisition and processing of wideband signals that are sparse in a local Fourier representation. Results from our RS-AIC hardware implementation demonstrate successful reconstruction of signals that are sampled at half the Nyquist-rate while maintaining up to a 51 dB signal-to-noise ratio (SNR), which is equivalent to an 8.5 bit resolution analog to digital converter.

On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion

A compressive sensor array (CSA) system and method uses compressive sampling techniques to acquire sensor data from an array of sensors without independently sampling each of the sensor signals. In general, the CSA system and method uses the compressive sampling techniques to combine the analog sensor signals from the array of sensors into a composite sensor signal and to sample the composite sensor signal at a sub-Nyquist sampling rate. At least one embodiment of the CSA system and method allows a single analog-to-digital converter (ADC) and single RF demodulation chain to be used for an arbitrary number of sensors, thereby providing scalability and eliminating redundant data acquisition hardware. By reducing the number of samples, the CSA system and method also facilitates the processing, storage and transmission of the sensor data.

Compressive sensor array system and method

The recently introduced theory of compressive sensing (CS) enables the reconstruction of sparse or compressible signals from a small set of nonadaptive, linear measurements. If properly chosen, the number of measurements can be significantly smaller than the ambient dimension of the signal and yet preserve the significant signal information. Interestingly, it can be shown that random measurement schemes provide a near-optimal encoding in terms of the required number of measurements. In this report, we explore another relatively unexplored, though often alluded to, advantage of using random matrices to acquire CS measurements. Specifically, we show that random matrices are democractic, meaning that each measurement carries roughly the same amount of signal information. We demonstrate that by slightly increasing the number of measurements, the system is robust to the loss of a small number of arbitrary measurements. In addition, we draw connections to oversampling and demonstrate stability from the loss of significantly more measurements.

/pdf/a-simple-proof-that-random-matrices-are-democratic-241gj408tu.pdf

Jason N. Laska

Papers

Implementation models for analog-to-information conversion via random sampling

The compressive multiplexer for multi-channel compressive sensing

On the feasibility of hardware implementation of sub-Nyquist random-sampling based analog-to-information conversion

Compressive sensor array system and method

A Simple Proof that Random Matrices are Democratic