arXiv subjects

Ben Adcock

Publications and source records attributed to Ben Adcock.

At least 73 records · Page 4Linked to original sources

arXiv2018

Uniform Recovery from Subgaussian Multi-Sensor Measurements

Parallel acquisition systems are employed successfully in a variety of different sensing applications when a single sensor cannot provide enough measurements for a high-quality reconstruction. In this paper, we consider compressed sensing (CS) for parallel acquisition systems when the individual sensors use subgaussian random sampling. Our main results are a series of uniform recovery guarantees which relate the number of measurements required to the basis in which the solution is sparse and certain characteristics of the multi-sensor system, known as sensor profile matrices. In particular, we derive sufficient conditions for optimal recovery, in the sense that the number of measurements required per sensor decreases linearly with the total number of sensors, and demonstrate explicit examples of multi-sensor systems for which this holds. We establish these results by proving the so-called Asymmetric Restricted Isometry Property (ARIP) for the sensing system and use this to derive both nonuniversal and universal recovery guarantees. Compared to existing work, our results not only lead to better stability and robustness estimates but also provide simpler and sharper constants in the measurement conditions. Finally, we show how the problem of CS with block-diagonal sensing matrices can be viewed as a particular case of our multi-sensor framework. Specializing our results to this setting leads to a recovery guarantee that is at least as good as existing results.

cs.IT↗

arXiv2017

Robustness to unknown error in sparse regularization

Quadratically-constrained basis pursuit has become a popular device in sparse regularization; in particular, in the context of compressed sensing. However, the majority of theoretical error estimates for this regularizer assume an a priori bound on the noise level, which is usually lacking in practice. In this paper, we develop stability and robustness estimates which remove this assumption. First, we introduce an abstract framework and show that robust instance optimality of any decoder in the noise-aware setting implies stability and robustness in the noise-blind setting. This is based on certain sup-inf constants referred to as quotients, strictly related to the quotient property of compressed sensing. We then apply this theory to prove the robustness of quadratically-constrained basis pursuit under unknown error in the cases of random Gaussian matrices and of random matrices with heavy-tailed rows, such as random sampling matrices from bounded orthonormal systems. We illustrate our results in several cases of practical importance, including subsampled Fourier measurements and recovery of sparse polynomial expansions.

cs.IT↗

arXiv2017

Compressed sensing approaches for polynomial approximation of high-dimensional functions

In recent years, the use of sparse recovery techniques in the approximation of high-dimensional functions has garnered increasing interest. In this work we present a survey of recent progress in this emerging topic. Our main focus is on the computation of polynomial approximations of high-dimensional functions on $d$-dimensional hypercubes. We show that smooth, multivariate functions possess expansions in orthogonal polynomial bases that are not only approximately sparse, but possess a particular type of structured sparsity defined by so-called lower sets. This structure can be exploited via the use of weighted $\ell^1$ minimization techniques, and, as we demonstrate, doing so leads to sample complexity estimates that are at most logarithmically dependent on the dimension $d$. Hence the curse of dimensionality - the bane of high-dimensional approximation - is mitigated to a significant extent. We also discuss several practical issues, including unknown noise (due to truncation or numerical error), and highlight a number of open problems and challenges.

math.NA↗

arXiv2017

Compressed sensing with local structure: uniform recovery guarantees for the sparsity in levels class

In compressed sensing, it is often desirable to consider signals possessing additional structure beyond sparsity. One such structured signal model - which forms the focus of this paper - is the local sparsity in levels class. This class has recently found applications in problems such as compressive imaging, multi-sensor acquisition systems and sparse regularization in inverse problems. In this paper we present uniform recovery guarantees for this class when the measurement matrix corresponds to a subsampled isometry. We do this by establishing a variant of the standard restricted isometry property for sparse in levels vectors, known as the restricted isometry property in levels. Interestingly, besides the usual log factors, our uniform recovery guarantees are simpler and less stringent than existing nonuniform recovery guarantees. For the particular case of discrete Fourier sampling with Haar wavelet sparsity, a corollary of our main theorem yields a new recovery guarantee which improves over the current state-of-the-art.

cs.IT↗

arXiv2017

Recovery guarantees for compressed sensing with unknown errors

From a numerical analysis perspective, assessing the robustness of l1-minimization is a fundamental issue in compressed sensing and sparse regularization. Yet, the recovery guarantees available in the literature usually depend on a priori estimates of the noise, which can be very hard to obtain in practice, especially when the noise term also includes unknown discrepancies between the finite model and data. In this work, we study the performance of l1-minimization when these estimates are not available, providing robust recovery guarantees for quadratically constrained basis pursuit and random sampling in bounded orthonormal systems. Several applications of this work are approximation of high-dimensional functions, infinite-dimensional sparse regularization for inverse problems, and fast algorithms for non-Cartesian Magnetic Resonance Imaging.

math.NA↗

arXiv2017

Computing reconstructions from nonuniform Fourier samples: Universality of stability barriers and stable sampling rates

We study the problem of recovering an unknown compactly-supported multivariate function from samples of its Fourier transform that are acquired nonuniformly, i.e. not necessarily on a uniform Cartesian grid. Reconstruction problems of this kind arise in various imaging applications, where Fourier samples are taken along radial lines or spirals for example. Specifically, we consider finite-dimensional reconstructions, where a limited number of samples is available, and investigate the rate of convergence of such approximate solutions and their numerical stability. We show that the proportion of Fourier samples that allow for stable approximations of a given numerical accuracy is independent of the specific sampling geometry and is therefore universal for different sampling scenarios. This allows us to relate both sufficient and necessary conditions for different sampling setups and to exploit several results that were previously available only for very specific sampling geometries. The results are obtained by developing: (i) a transference argument for different measures of the concentration of the Fourier transform and Fourier samples; (ii) frame bounds valid up to the critical sampling density, which depend explicitly on the sampling set and the spectrum. As an application, we identify sufficient and necessary conditions for stable and accurate reconstruction of algebraic polynomials or wavelet coefficients from nonuniform Fourier data.

math.NA↗

arXiv2017

Infinite-dimensional compressed sensing and function interpolation

We introduce and analyze a framework for function interpolation using compressed sensing. This framework - which is based on weighted $\ell^1$ minimization - does not require a priori bounds on the expansion tail in either its implementation or its theoretical guarantees, and in the absence of noise leads to genuinely interpolatory approximations. We also establish a new recovery guarantee for compressed sensing with weighted $\ell^1$ minimization based on this framework. This guarantee has several key benefits. First, unlike existing results, it is sharp (up to constants and log factors) for large classes of functions regardless of the choice of weights. Second, by examining the measurement condition in the recovery guarantee, we are able to suggest a good overall strategy for selecting the weights. In particular, when applied to the important case of multivariate approximation with orthogonal polynomials, this weighting strategy leads to provably optimal estimates on the number of measurements required, whenever the support set of the significant coefficients is a so-called lower set. Finally, this guarantee can also be used to theoretically confirm the benefits of alternative weighting strategies where the weights are chosen based on prior support information. This provides a theoretical basis for a number of recent numerical studies showing the effectiveness of such approaches.

math.NA↗

arXiv2016

Compressed Sensing and Parallel Acquisition

Parallel acquisition systems arise in various applications in order to moderate problems caused by insufficient measurements in single-sensor systems. These systems allow simultaneous data acquisition in multiple sensors, thus alleviating such problems by providing more overall measurements. In this work we consider the combination of compressed sensing with parallel acquisition. We establish the theoretical improvements of such systems by providing recovery guarantees for which, subject to appropriate conditions, the number of measurements required per sensor decreases linearly with the total number of sensors. Throughout, we consider two different sampling scenarios -- distinct (corresponding to independent sampling in each sensor) and identical (corresponding to dependent sampling between sensors) -- and a general mathematical framework that allows for a wide range of sensing matrices (e.g., subgaussian random matrices, subsampled isometries, random convolutions and random Toeplitz matrices). We also consider not just the standard sparse signal model, but also the so-called sparse in levels signal model. This model includes both sparse and distributed signals and clustered sparse signals. As our results show, optimal recovery guarantees for both distinct and identical sampling are possible under much broader conditions on the so-called sensor profile matrices (which characterize environmental conditions between a source and the sensors) for the sparse in levels model than for the sparse model. To verify our recovery guarantees we provide numerical results showing phase transitions for a number of different multi-sensor environments.

cs.IT↗

arXiv2016

Infinite-dimensional $\ell^1$ minimization and function approximation from pointwise data

We consider the problem of approximating a smooth function from finitely-many pointwise samples using $\ell^1$ minimization techniques. In the first part of this paper, we introduce an infinite-dimensional approach to this problem. Three advantages of this approach are as follows. First, it provides interpolatory approximations in the absence of noise. Second, it does not require \textit{a priori} bounds on the expansion tail in order to be implemented. In particular, the truncation strategy we introduce as part of this framework is independent of the function being approximated, provided the function has sufficient regularity. Third, it allows one to explain the key role weights play in the minimization; namely, that of regularizing the problem and removing aliasing phenomena. In the second part of this paper we present a worst-case error analysis for this approach. We provide a general recipe for analyzing this technique for arbitrary deterministic sets of points. Finally, we use this tool to show that weighted $\ell^1$ minimization with Jacobi polynomials leads to an optimal method for approximating smooth, one-dimensional functions from scattered data.

math.NA↗

arXiv2016

Analyzing the structure of multidimensional compressed sensing problems through coherence

Recently it has been established that asymptotic incoherence can be used to facilitate subsampling, in order to optimize reconstruction quality, in a variety of continuous compressed sensing problems, and the coherence structure of certain one-dimensional Fourier sampling problems was determined. This paper extends the analysis of asymptotic incoherence to cover multidimensional reconstruction problems. It is shown that Fourier sampling and separable wavelet sparsity in any dimension can yield the same optimal asymptotic incoherence as in one dimensional case. Moreover in two dimensions the coherence structure is compatible with many standard two dimensional sampling schemes that are currently in use. However, in higher dimensional problems with poor wavelet smoothness we demonstrate that there are considerable restrictions on how one can subsample from the Fourier basis with optimal incoherence. This can be remedied by using a sufficiently smooth generating wavelet. It is also shown that using tensor bases will always provide suboptimal decay marred by problems associated with dimensionality. The impact of asymptotic incoherence on the ability to subsample is demonstrated with some simple two dimensional numerical experiments.

cs.IT↗

arXiv2016

Density theorems for nonuniform sampling of bandlimited functions using derivatives or bunched measurements

We provide sufficient density condition for a set of nonuniform samples to give rise to a set of sampling for multivariate bandlimited functions when the measurements consist of pointwise evaluations of a function and its first $k$ derivatives. Along with explicit estimates of corresponding frame bounds, we derive the explicit density bound and show that, as $k$ increases, it grows linearly in $k+1$ with the constant of proportionality $1/\mathrm{e}$. Seeking larger gap conditions, we also prove a multivariate perturbation result for nonuniform samples that are sufficiently close to sets of sampling, e.g. to uniform samples taken at $k+1$ times the Nyquist rate. Additionally, in the univariate setting, we consider a related problem of so-called nonuniform bunched sampling, where in each sampling interval $s+1$ bunched measurements of a function are taken and the sampling intervals are permitted to be of different length. We derive an explicit density condition which grows linearly in $s+1$ for large $s$, with the constant of proportionality depending on the width of the bunches. The width of the bunches is allowed to be arbitrarily small, and moreover, for sufficiently narrow bunches and sufficiently large $s$, we obtain the same result as in the case of univariate sampling with $s$ derivatives.

math.NA↗

arXiv2016

Resolution-optimal exponential and double-exponential transform methods for functions with endpoint singularities

We introduce a numerical method for the approximation of functions which are analytic on compact intervals, except at the endpoints. This method is based on variable transforms using particular parametrized exponential and double-exponential mappings, in combination with Fourier-like approximation in a truncated domain. We show theoretically that this method is superior to variable transform techniques based on the standard exponential and double-exponential mappings. In particular, it can resolve oscillatory behaviour using near-optimal degrees of freedom, whereas the standard mappings require degrees of freedom that grow superlinearly with the frequency of oscillation. We highlight these results with several numerical experiments. Therein it is observed that near-machine epsilon accuracy is achieved using a number of degrees of freedom that is between four and ten times smaller than those of existing techniques.

math.NA↗

arXiv2016

Sparsity and Parallel Acquisition: Optimal Uniform and Nonuniform Recovery Guarantees

The problem of multiple sensors simultaneously acquiring measurements of a single object can be found in many applications. In this paper, we present the optimal recovery guarantees for the recovery of compressible signals from multi-sensor measurements using compressed sensing. In the first half of the paper, we present both uniform and nonuniform recovery guarantees for the conventional sparse signal model in a so-called distinct sensing scenario. In the second half, using the so-called sparse and distributed signal model, we present nonuniform recovery guarantees which effectively broaden the class of sensing scenarios for which optimal recovery is possible, including to the so-called identical sampling scenario. To verify our recovery guarantees we provide several numerical results including phase transition curves and numerically-computed bounds.

cs.IT↗

arXiv2016

Optimal Sparse Recovery for Multi-Sensor Measurements

Many practical sensing applications involve multiple sensors simultaneously acquiring measurements of a single object. Conversely, most existing sparse recovery guarantees in compressed sensing concern only single-sensor acquisition scenarios. In this paper, we address the optimal recovery of compressible signals from multi-sensor measurements using compressed sensing techniques, thereby confirming the benefits of multi- over single-sensor environments. Throughout the paper, we consider a broad class of sensing matrices, and two fundamentally different sampling scenarios (distinct and identical respectively), both of which are relevant to applications. For the case of diagonal sensor profile matrices (which characterize environmental conditions between a source and the sensors), this paper presents two key improvements over existing results. First, a simpler optimal recovery guarantee for distinct sampling, and second, an improved recovery guarantee for identical sampling, based on the so-called sparsity in levels signal model.

cs.IT↗

arXiv2015

Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples

In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sample points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Groechenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm---based on the existing generalized sampling framework---that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice---jittered, radial and spiral sampling schemes---and provide several examples illustrating the effectiveness of our approach when tested on these schemes.

math.NA↗

arXiv2015

On Asymptotic Incoherence and its Implications for Compressed Sensing of Inverse Problems

Recently, it has been shown that incoherence is an unrealistic assumption for compressed sensing when applied to many inverse problems. Instead, the key property that permits efficient recovery in such problems is so-called local incoherence. Similarly, the standard notion of sparsity is also inadequate for many real world problems. In particular, in many applications, the optimal sampling strategy depends on asymptotic incoherence and the signal sparsity structure. The purpose of this paper is to study asymptotic incoherence and its implications towards the design of optimal sampling strategies and efficient sparsity bases. It is determined how fast asymptotic incoherence can decay in general for isometries. Furthermore it is shown that Fourier sampling and wavelet sparsity, whilst globally coherent, yield optimal asymptotic incoherence as a power law up to a constant factor. Sharp bounds on the asymptotic incoherence for Fourier sampling with polynomial bases are also provided. A numerical experiment is also presented to demonstrate the role of asymptotic incoherence in finding good subsampling strategies.

cs.IT↗

arXiv2014

Recovering piecewise smooth functions from nonuniform Fourier measurements

In this paper, we consider the problem of reconstructing piecewise smooth functions to high accuracy from nonuniform samples of their Fourier transform. We use the framework of nonuniform generalized sampling (NUGS) to do this, and to ensure high accuracy we employ reconstruction spaces consisting of splines or (piecewise) polynomials. We analyze the relation between the dimension of the reconstruction space and the bandwidth of the nonuniform samples, and show that it is linear for splines and piecewise polynomials of fixed degree, and quadratic for piecewise polynomials of varying degree.

math.NA↗

arXiv2014

On asymptotic structure in compressed sensing

This paper demonstrates how new principles of compressed sensing, namely asymptotic incoherence, asymptotic sparsity and multilevel sampling, can be utilised to better understand underlying phenomena in practical compressed sensing and improve results in real-world applications. The contribution of the paper is fourfold: First, it explains how the sampling strategy depends not only on the signal sparsity but also on its structure, and shows how to design effective sampling strategies utilising this. Second, it demonstrates that the optimal sampling strategy and the efficiency of compressed sensing also depends on the resolution of the problem, and shows how this phenomenon markedly affects compressed sensing results and how to exploit it. Third, as the new framework also fits analog (infinite dimensional) models that govern many inverse problems in practice, the paper describes how it can be used to yield substantial improvements. Fourth, by using multilevel sampling, which exploits the structure of the signal, the paper explains how one can outperform random Gaussian/Bernoulli sampling even when the classical $l^1$ recovery algorithm is replaced by modified algorithms which aim to exploit structure such as model based or Bayesian compressed sensing or approximate message passaging. This final observation raises the question whether universality is desirable even when such matrices are applicable. Examples of practical applications investigated in this paper include Magnetic Resonance Imaging (MRI), Electron Microscopy (EM), Compressive Imaging (CI) and Fluorescence Microscopy (FM). For the latter, a new compressed sensing approach is also presented.

math.FA↗