arXiv subjects

Thomas Strohmer

Publications and source records attributed to Thomas Strohmer.

At least 55 records · Page 3Linked to original sources

arXiv2016

Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization

We study the question of reconstructing two signals $f$ and $g$ from their convolution $y = f\ast g$. This problem, known as {\em blind deconvolution}, pervades many areas of science and technology, including astronomy, medical imaging, optics, and wireless communications. A key challenge of this intricate non-convex optimization problem is that it might exhibit many local minima. We present an efficient numerical algorithm that is guaranteed to recover the exact solution, when the number of measurements is (up to log-factors) slightly larger than the information-theoretical minimum, and under reasonable conditions on $f$ and $g$. The proposed regularized gradient descent algorithm converges at a geometric rate and is provably robust in the presence of noise. To the best of our knowledge, our algorithm is the first blind deconvolution algorithm that is numerically efficient, robust against noise, and comes with rigorous recovery guarantees under certain subspace conditions. Moreover, numerical experiments do not only provide empirical verification of our theory, but they also demonstrate that our method yields excellent performance even in situations beyond our theoretical framework.

cs.IT↗

arXiv2015

Self-Calibration and Biconvex Compressive Sensing

The design of high-precision sensing devises becomes ever more difficult and expensive. At the same time, the need for precise calibration of these devices (ranging from tiny sensors to space telescopes) manifests itself as a major roadblock in many scientific and technological endeavors. To achieve optimal performance of advanced high-performance sensors one must carefully calibrate them, which is often difficult or even impossible to do in practice. In this work we bring together three seemingly unrelated concepts, namely Self-Calibration, Compressive Sensing, and Biconvex Optimization. The idea behind self-calibration is to equip a hardware device with a smart algorithm that can compensate automatically for the lack of calibration. We show how several self-calibration problems can be treated efficiently within the framework of biconvex compressive sensing via a new method called SparseLift. More specifically, we consider a linear system of equations y = DAx, where both x and the diagonal matrix D (which models the calibration error) are unknown. By "lifting" this biconvex inverse problem we arrive at a convex optimization problem. By exploiting sparsity in the signal model, we derive explicit theoretical guarantees under which both x and D can be recovered exactly, robustly, and numerically efficiently via linear programming. Applications in array calibration and wireless communications are discussed and numerical simulations are presented, confirming and complementing our theoretical analysis.

cs.IT↗

arXiv2015

Sparse Signal Processing Concepts for Efficient 5G System Design

As it becomes increasingly apparent that 4G will not be able to meet the emerging demands of future mobile communication systems, the question what could make up a 5G system, what are the crucial challenges and what are the key drivers is part of intensive, ongoing discussions. Partly due to the advent of compressive sensing, methods that can optimally exploit sparsity in signals have received tremendous attention in recent years. In this paper we will describe a variety of scenarios in which signal sparsity arises naturally in 5G wireless systems. Signal sparsity and the associated rich collection of tools and algorithms will thus be a viable source for innovation in 5G wireless system design. We will discribe applications of this sparse signal processing paradigm in MIMO random access, cloud radio access networks, compressive channel-source network coding, and embedded security. We will also emphasize important open problem that may arise in 5G system design, for which sparsity will potentially play a key role in their solution.

cs.IT↗

arXiv2015

Almost Eigenvalues and Eigenvectors of Almost Mathieu Operators

The almost Mathieu operator is the discrete Schrödinger operator $H_{α,β,θ}$ on $\ell^2(\mathbb{Z})$ defined via $(H_{α,β,θ}f)(k) = f(k + 1) + f(k - 1) + β\cos(2παk + θ) f(k)$. We derive explicit estimates for the eigenvalues at the edge of the spectrum of the finite-dimensional almost Mathieu operator. We furthermore show that the (properly rescaled) $m$-th Hermite function $ϕ_m$ is an approximate eigenvector of this operator, and that it satisfies the same properties that characterize the true eigenvector associated to the $m$-th largest eigenvalue. Moreover, a properly translated and modulated version of $ϕ_m$ is also an approximate eigenvector of this operator, and it satisfies the properties that characterize the true eigenvector associated to the $m$-th largest (in modulus) negative eigenvalue. The results hold at the edge of the spectrum, for any choice of $θ$ and under very mild conditions on $α$ and $β$. We also give precise estimates for the size of the "edge", and extend some of our results to the infinite dimensional case. The ingredients for our proofs comprise Taylor expansions, basic time-frequency analysis, Sturm sequences, and perturbation theory for eigenvalues and eigenvectors. Numerical simulations demonstrate the tight fit of the theoretical estimates.

math.SP↗

arXiv2013

Localization of Matrix Factorizations

Matrices with off-diagonal decay appear in a variety of fields in mathematics and in numerous applications, such as signal processing, statistics, communications engineering, condensed matter physics, and quantum chemistry. Numerical algorithms dealing with such matrices often take advantage (implicitly or explicitly) of the empirical observation that this off-diagonal decay property seems to be preserved when computing various useful matrix factorizations, such as the Cholesky factorization or the QR-factorization. There is a fairly extensive theory describing when the inverse of a matrix inherits the localization properties of the original matrix. Yet, except for the special case of band matrices, surprisingly very little theory exists that would establish similar results for matrix factorizations. We will derive a comprehensive framework to rigorously answer the question when and under which conditions the matrix factors inherit the localization of the original matrix for such fundamental matrix factorizations as the LU-, QR-, Cholesky, and Polar factorization.

math.NA↗

arXiv2013

Remote sensing via $\ell_1$ minimization

We consider the problem of detecting the locations of targets in the far field by sending probing signals from an antenna array and recording the reflected echoes. Drawing on key concepts from the area of compressive sensing, we use an $\ell_1$-based regularization approach to solve this, in general ill-posed, inverse scattering problem. As common in compressed sensing, we exploit randomness, which in this context comes from choosing the antenna locations at random. With $n$ antennas we obtain $n^2$ measurements of a vector $x \in \C^{N}$ representing the target locations and reflectivities on a discretized grid. It is common to assume that the scene $x$ is sparse due to a limited number of targets. Under a natural condition on the mesh size of the grid, we show that an $s$-sparse scene can be recovered via $\ell_1$-minimization with high probability if $n^2 \geq C s \log^2(N)$. The reconstruction is stable under noise and under passing from sparse to approximately sparse vectors. Our theoretical findings are confirmed by numerical simulations.

cs.IT↗

arXiv2013

Accurate detection of moving targets via random sensor arrays and Kerdock codes

The detection and parameter estimation of moving targets is one of the most important tasks in radar. Arrays of randomly distributed antennas have been popular for this purpose for about half a century. Yet, surprisingly little rigorous mathematical theory exists for random arrays that addresses fundamental question such as how many targets can be recovered, at what resolution, at which noise level, and with which algorithm. In a different line of research in radar, mathematicians and engineers have invested significant effort into the design of radar transmission waveforms which satisfy various desirable properties. In this paper we bring these two seemingly unrelated areas together. Using tools from compressive sensing we derive a theoretical framework for the recovery of targets in the azimuth-range-Doppler domain via random antennas arrays. In one manifestation of our theory we use Kerdock codes as transmission waveforms and exploit some of their peculiar properties in our analysis. Our paper provides two main contributions: (i) We derive the first rigorous mathematical theory for the detection of moving targets using random sensor arrays. (ii) The transmitted waveforms satisfy a variety of properties that are very desirable and important from a practical viewpoint. Thus our approach does not just lead to useful theoretical insights, but is also of practical importance. Various extensions of our results are derived and numerical simulations confirming our theory are presented.

math.NA↗

arXiv2012

Measure What Should be Measured: Progress and Challenges in Compressive Sensing

Is compressive sensing overrated? Or can it live up to our expectations? What will come after compressive sensing and sparsity? And what has Galileo Galilei got to do with it? Compressive sensing has taken the signal processing community by storm. A large corpus of research devoted to the theory and numerics of compressive sensing has been published in the last few years. Moreover, compressive sensing has inspired and initiated intriguing new research directions, such as matrix completion. Potential new applications emerge at a dazzling rate. Yet some important theoretical questions remain open, and seemingly obvious applications keep escaping the grip of compressive sensing. In this paper I discuss some of the recent progress in compressive sensing and point out key challenges and opportunities as the area of compressive sensing and sparse representations keeps evolving. I also attempt to assess the long-term impact of compressive sensing.

cs.IT↗

arXiv2012

Analysis of Sparse MIMO Radar

We consider a multiple-input-multiple-output radar system and derive a theoretical framework for the recoverability of targets in the azimuth-range domain and the azimuth-range-Doppler domain via sparse approximation algorithms. Using tools developed in the area of compressive sensing, we prove bounds on the number of detectable targets and the achievable resolution in the presence of additive noise. Our theoretical findings are validated by numerical simulations.

cs.IT↗

arXiv2011

PhaseLift: Exact and Stable Signal Recovery from Magnitude Measurements via Convex Programming

Suppose we wish to recover a signal x in C^n from m intensity measurements of the form | |^2, i = 1, 2,..., m; that is, from data in which phase information is missing. We prove that if the vectors z_i are sampled independently and uniformly at random on the unit sphere, then the signal x can be recovered exactly (up to a global phase factor) by solving a convenient semidefinite program---a trace-norm minimization problem; this holds with large probability provided that m is on the order of n log n, and without any assumption about the signal whatsoever. This novel result demonstrates that in some instances, the combinatorial phase retrieval problem can be solved by convex programming techniques. Finally, we also prove that our methodology is robust vis a vis additive noise.

cs.IT↗

arXiv2011

Phase Retrieval via Matrix Completion

This paper develops a novel framework for phase retrieval, a problem which arises in X-ray crystallography, diffraction imaging, astronomical imaging and many other applications. Our approach combines multiple structured illuminations together with ideas from convex programming to recover the phase from intensity measurements, typically from the modulus of the diffracted wave. We demonstrate empirically that any complex-valued object can be recovered from the knowledge of the magnitude of just a few diffracted patterns by solving a simple convex optimization problem inspired by the recent literature on matrix completion. More importantly, we also demonstrate that our noise-aware algorithms are stable in the sense that the reconstruction degrades gracefully as the signal-to-noise ratio decreases. Finally, we introduce some theory showing that one can design very simple structured illumination patterns such that three diffracted figures uniquely determine the phase of the object we wish to recover.

cs.IT↗

arXiv2011

Eigenvalue Estimates and Mutual Information for the Linear Time-Varying Channel

We consider linear time-varying channels with additive white Gaussian noise. For a large class of such channels we derive rigorous estimates of the eigenvalues of the correlation matrix of the effective channel in terms of the sampled time-varying transfer function and, thus, provide a theoretical justification for a relationship that has been frequently observed in the literature. We then use this eigenvalue estimate to derive an estimate of the mutual information of the channel. Our approach is constructive and is based on a careful balance of the trade-off between approximate operator diagonalization, signal dimension loss, and accuracy of eigenvalue estimates.

cs.IT↗

arXiv2011

Sparsity Enhanced Decision Feedback Equalization

For single-carrier systems with frequency domain equalization, decision feedback equalization (DFE) performs better than linear equalization and has much lower computational complexity than sequence maximum likelihood detection. The main challenge in DFE is the feedback symbol selection rule. In this paper, we give a theoretical framework for a simple, sparsity based thresholding algorithm. We feed back multiple symbols in each iteration, so the algorithm converges fast and has a low computational cost. We show how the initial solution can be obtained via convex relaxation instead of linear equalization, and illustrate the impact that the choice of the initial solution has on the bit error rate performance of our algorithm. The algorithm is applicable in several existing wireless communication systems (SC-FDMA, MC-CDMA, MIMO-OFDM). Numerical results illustrate significant performance improvement in terms of bit error rate compared to the MMSE solution.

cs.IT↗

arXiv2010

Performance Analysis of Spectral Clustering on Compressed, Incomplete and Inaccurate Measurements

Spectral clustering is one of the most widely used techniques for extracting the underlying global structure of a data set. Compressed sensing and matrix completion have emerged as prevailing methods for efficiently recovering sparse and partially observed signals respectively. We combine the distance preserving measurements of compressed sensing and matrix completion with the power of robust spectral clustering. Our analysis provides rigorous bounds on how small errors in the affinity matrix can affect the spectral coordinates and clusterability. This work generalizes the current perturbation results of two-class spectral clustering to incorporate multi-class clustering with k eigenvectors. We thoroughly track how small perturbation from using compressed sensing and matrix completion affect the affinity matrix and in succession the spectral coordinates. These perturbation results for multi-class clustering require an eigengap between the kth and (k+1)th eigenvalues of the affinity matrix, which naturally occurs in data with k well-defined clusters. Our theoretical guarantees are complemented with numerical results along with a number of examples of the unsupervised organization and clustering of image data.

math.NA↗

arXiv2009

General Deviants: An Analysis of Perturbations in Compressed Sensing

We analyze the Basis Pursuit recovery of signals with general perturbations. Previous studies have only considered partially perturbed observations Ax + e. Here, x is a signal which we wish to recover, A is a full-rank matrix with more columns than rows, and e is simple additive noise. Our model also incorporates perturbations E to the matrix A which result in multiplicative noise. This completely perturbed framework extends the prior work of Candes, Romberg and Tao on stable signal recovery from incomplete and inaccurate measurements. Our results show that, under suitable conditions, the stability of the recovered signal is limited by the noise level in the observation. Moreover, this accuracy is within a constant multiple of the best-case reconstruction using the technique of least squares. In the absence of additive noise numerical simulations essentially confirm that this error is a linear function of the relative perturbation.

cs.IT↗

arXiv2009

Compressed Remote Sensing of Sparse Objects

The linear inverse source and scattering problems are studied from the perspective of compressed sensing, in particular the idea that sufficient incoherence and sparsity guarantee uniqueness of the solution. By introducing the sensor as well as target ensembles, the maximum number of recoverable targets is proved to be at least proportional to the number of measurement data modulo a log-square factor with overwhelming probability. Important contributions of the analysis include the discoveries of the threshold aperture, consistent with the classical Rayleigh criterion, and the decoherence effect induced by random antenna locations. The prediction of theorems are confirmed by numerical simulations.

physics.data-an↗

arXiv2008

High-Resolution Radar via Compressed Sensing

A stylized compressed sensing radar is proposed in which the time-frequency plane is discretized into an N by N grid. Assuming the number of targets K is small (i.e., K much less than N^2), then we can transmit a sufficiently "incoherent" pulse and employ the techniques of compressed sensing to reconstruct the target scene. A theoretical upper bound on the sparsity K is presented. Numerical simulations verify that even better performance can be achieved in practice. This novel compressed sensing approach offers great potential for better resolution over classical radar.

math.NA↗

arXiv2007

Inverse-Closedness of a Banach Algebra of Integral Operators on the Heisenberg Group

Let $\mathbb{H}$ be the general, reduced Heisenberg group. Our main result establishes the inverse-closedness of a class of integral operators acting on $L^{p}(\mathbb{H})$, given by the off-diagonal decay of the kernel. As a consequence of this result, we show that if $α_{1}I+S_{f}$, where $S_{f}$ is the operator given by convolution with $f$, $f\in L^{1}_{v}(\mathbb{H})$, is invertible in $\B(L^{p}(\mathbb{H}))$, then (α_{1}I+S_{f})^{-1}=α_{2}I+S_{g}$, and $g\in L^{1}_{v}(\mathbb{H})$. We prove analogous results for twisted convolution operators and apply the latter results to a class of Weyl pseudodifferential operators. We briefly discuss relevance to mobile communications.

math.CA↗