A. Robert Calderbank

Construction of a Large Class of Deterministic Sensing Matrices That Satisfy a Statistical Isometry Property

Compressed Sensing aims to capture attributes of k-sparse signals using very few measurements. In the standard compressed sensing paradigm, the N × C measurement matrix ¿ is required to act as a near isometry on the set of all k-sparse signals (restricted isometry property or RIP). Although it is known that certain probabilistic processes generate N × C matrices that satisfy RIP with high probability, there is no practical algorithm for verifying whether a given sensing matrix ¿ has this property, crucial for the feasibility of the standard recovery algorithms. In contrast, this paper provides simple criteria that guarantee that a deterministic sensing matrix satisfying these criteria acts as a near isometry on an overwhelming majority of k-sparse signals; in particular, most such signals have a unique representation in the measurement domain. Probability still plays a critical role, but it enters the signal model rather than the construction of the sensing matrix. An essential element in our construction is that we require the columns of the sensing matrix to form a group under pointwise multiplication. The construction allows recovery methods for which the expected performance is sub-linear in C, and only quadratic in N, as compared to the super-linear complexity in C of the Basis Pursuit or Matching Pursuit algorithms; the focus on expected performance is more typical of mainstream signal processing than the worst case analysis that prevails in standard compressed sensing. Our framework encompasses many families of deterministic sensing matrices, including those formed from discrete chirps, Delsarte-Goethals codes, and extended BCH codes.

Efficient and Robust Compressed Sensing Using Optimized Expander Graphs

Expander graphs have been recently proposed to construct efficient compressed sensing algorithms. In particular, it has been shown that any <i>n</i>-dimensional vector that is <i>k</i>-sparse can be fully recovered using <i>O</i>(<i>k</i>log<i>n</i>) measurements and only <i>O</i>(<i>k</i>log<i>n</i>) simple recovery iterations. In this paper, we improve upon this result by considering expander graphs with expansion coefficient beyond ³ <i>/</i> ₄ and show that, with the same number of measurements, only <i>O</i>(<i>k</i>) recovery iterations are required, which is a significant improvement when <i>n</i> is large. In fact, full recovery can be accomplished by at most <i>2k</i> very simple iterations. The number of iterations can be reduced arbitrarily close to <i>k</i>, and the recovery algorithm can be implemented very efficiently using a simple priority queue with total recovery time <i>O</i>(<i>n</i>log(ⁿ/_k))). We also show that by tolerating a small penalty on the number of measurements, and not on the number of recovery iterations, one can use the efficient construction of a family of expander graphs to come up with explicit measurement matrices for this method. We compare our result with other recently developed expander-graph-based methods and argue that it compares favorably both in terms of the number of required measurements and in terms of the time complexity and the simplicity of recovery. Finally, we will show how our analysis extends to give a robust algorithm that finds the position and sign of the <i>k</i> significant elements of an almost <i>k</i>-sparse signal and then, using very simple optimization techniques, finds a <i>k</i>-sparse signal which is close to the best <i>k</i>-term approximation of the original signal.

Jointly optimal congestion and contention control based on network utility maximization

We study joint end-to-end congestion control and per-link medium access control (MAC) in ad-hoc networks. We use a network utility maximization formulation, in which by adjusting the types of utility functions, we can accommodate multi-class services as well as exploit the tradeoff between efficiency and fairness of resource allocation. Despite the inherent difficulties of non-convexity and non-separability of the optimization problem, we show that, with readily-verifiable sufficient conditions, we can develop, a distributed algorithm that converges to the globally and jointly optimal rate allocation and persistence probabilities.

Why Gabor frames? Two fundamental measures of coherence and their role in model selection

The problem of model selection arises in a number of contexts, such as subset selection in linear regression, estimation of structures in graphical models, and signal denoising. This paper studies non-asymptotic model selection for the general case of arbitrary (random or deterministic) design matrices and arbitrary nonzero entries of the signal. In this regard, it generalizes the notion of incoherence in the existing literature on model selection and introduces two fundamental measures of coherence — termed as the worst-case coherence and the average coherence — among the columns of a design matrix. It utilizes these two measures of coherence to provide an in-depth analysis of a simple, model-order agnostic one-step thresholding (OST) algorithm for model selection and proves that OST is feasible for exact as well as partial model selection as long as the design matrix obeys an easily verifiable property, which is termed as the coherence property. One of the key insights offered by the ensuing analysis in this regard is that OST can successfully carry out model selection even when methods based on convex optimization such as the lasso fail due to the rank deficiency of the submatrices of the design matrix. In addition, the paper establishes that if the design matrix has reasonably small worst-case and average coherence then OST performs near-optimally when either (i) the energy of any nonzero entry of the signal is close to the average signal energy per nonzero entry or (ii) the signal-to-noise ratio in the measurement system is not too high. Finally, two other key contributions of the paper are that (i) it provides bounds on the average coherence of Gaussian matrices and Gabor frames, and (ii) it extends the results on model selection using OST to low-complexity, model-order agnostic recovery of sparse signals with arbitrary nonzero entries. In particular, this part of the analysis in the paper implies that an Alltop Gabor frame together with OST can successfully carry out model selection and recovery of sparse signals irrespective of the phases of the nonzero entries even if the number of nonzero entries scales almost linearly with the number of rows of the Alltop Gabor frame.

Sensitivity to basis mismatch in compressed sensing

The theory of compressed sensing suggests that successful inversion of an image of the physical world (broadly defined to include speech signals, radar/sonar returns, vibration records, sensor array snapshot vectors, 2-D images, and so on) for its source modes and amplitudes can be achieved at measurement dimensions far lower than what might be expected from the classical theories of spectrum or modal analysis, provided that the image is sparse in an apriori known basis. For imaging problems in spectrum analysis, and passive and active radar/sonar, this basis is usually taken to be a DFT basis. However, in reality no physical field is sparse in the DFT basis or in any apriori known basis. No matter how finely we grid the parameter space the sources may not lie in the center of the grid cells and consequently there is mismatch between the assumed and the actual bases for sparsity. In this paper, we study the sensitivity of compressed sensing to mismatch between the assumed and the actual sparsity bases. We start by analyzing the effect of basis mismatch on the best k-term approximation error, which is central to providing exact sparse recovery guarantees. We establish achievable bounds for the l1 error of the best k -term approximation and show that these bounds grow linearly with the image (or grid) dimension and the mismatch level between the assumed and actual bases for sparsity. We then derive bounds, with similar growth behavior, for the basis pursuit l1 recovery error, indicating that the sparse recovery may suffer large errors in the presence of basis mismatch. Although, we present our results in the context of basis pursuit, our analysis applies to any sparse recovery principle that relies on the accuracy of best k-term approximations for its performance guarantees. We particularly highlight the problematic nature of basis mismatch in Fourier imaging, where spillage from off-grid DFT components turns a sparse representation into an incompressible one. We substantiate our mathematical analysis by numerical examples that demonstrate a considerable performance degradation for image inversion from compressed sensing measurements in the presence of basis mismatch, for problem sizes common to radar and sonar.

Secrecy capacity of a class of orthogonal relay eavesdropper channels

The secrecy capacity of relay channels with orthogonal components is studied in the presence of an additional passive eavesdropper node. The relay and destination receive signals from the source on two orthogonal channels such that the destination also receives transmissions from the relay on its channel. The eavesdropper can overhear either one or both of the orthogonal channels. Inner and outer bounds on the secrecy capacity are developed for both the discrete memoryless and the Gaussian channel models. For the discrete memoryless case, the secrecy capacity is shown to be achieved by a partial decode-and-forward (PDF) scheme when the eavesdropper can overhear only one of the two orthogonal channels. Two new outer bounds are presented for the Gaussian model using recent capacity results for a Gaussian multiantenna point-to-point channel with a multiantenna eavesdropper. The outer bounds are shown to be tight for two subclasses of channels. The first subclass is one in which the source and relay are clustered, and the eavesdropper receives signals only on the channel from the source and the relay to the destination, for which the PDF strategy is optimal. The second is a subclass in which the source does not transmit to the relay, for which a noise-forwarding strategy is optimal.

Modeling location uncertainty for eavesdroppers: A secrecy graph approach

In this paper, we consider end-to-end secure communication in a large wireless network, where the locations of eavesdroppers are uncertain. Our framework attempts to bridge the gap between physical layer security under uncertain channel state information of the eavesdropper and network level connectivity under security constraints, by modeling location uncertainty directly at the network level as correlated node and link failures in a secrecy graph. Bounds on the percolation threshold are obtained for square and triangular lattices, and bounds on mean degree are obtained for Poisson secrecy graphs. Both analytic and simulation results show the dramatic effect of uncertainty in location of eavesdroppers on connectivity in a secrecy graph.

Wiretap channel type II with an active eavesdropper

The wiretap channel type II with an active eavesdropper is considered in this paper. Compared with the eavesdropper model considered in much of the literature, the eavesdropper considered here can not only overhear but also modify the signal transmitted over the channel. Two modification models are considered. In the first model, the eavesdropper erases the bits it observes. In the second model, the eavesdropper modifies the bits it observes. For this channel with memory (introduced by the activity of the eavesdropper), one should conduct the worst case scenario analysis. Novel concatenated coding schemes that provide perfect security for the communications are developed for both models to give bounds on the achievable secrecy rate. The technique to modify the inner code to maintain the secrecy properties of the outer code may be of independent interest.

Novel full-diversity high-rate STBC for 2 and 4 transmit antennas

We design a new rate-5/4 full-diversity orthogonal space-time block code (STBC) for QPSK and 2 transmit antennas (TX) by enlarging the signalling set from the set of quaternions used in the Alamouti code. Selective power scaling of information symbols is used to guarantee full-diversity while maximizing the coding gain (CG) and minimizing the transmitted signal peak-to-minimum power ratio (PMPR). The optimum power scaling factor is derived analytically and shown to outperform schemes based only on constellation rotation while still enjoying a low-complexity maximum likelihood (ML) decoding algorithm. Finally, we extend our designs to the case of 4 TX by enlarging the set of quasi-orthogonal STBC with power scaling. Extensions to general M-PSK constellations are straightforward.

Reed muller sensing matrices and the LASSO

We construct two families of deterministic sensing matrices where the columns are obtained by exponentiating codewords in the quaternary Delsarte-Goethals code DG(m, r). This method of construction results in sensing matrices with low coherence and spectral norm. The first family, which we call Delsarte-Goethals frames, are 2m - dimensional tight frames with redundancy 2rm. The second family, which we call Delsarte-Goethals sieves, are obtained by subsampling the column vectors in a Delsarte-Goethals frame. Different rows of a Delsarte-Goethals sieve may not be orthogonal, and we present an effective algorithm for identifying all pairs of non-orthogonal rows. The pairs turn out to be duplicate measurements and eliminating them leads to a tight frame. Experimental results suggest that all DG(m, r) sieves with m ≤ 15 and r ≥ 2 are tight-frames; there are no duplicate rows. For both families of sensing matrices, we measure accuracy of reconstruction (statistical 0-1 loss) and complexity (average reconstruction time) as a function of the sparsity level k. Our results show that DG frames and sieves outperform random Gaussian matrices in terms of noiseless and noisy signal recovery using the LASSO.