Shuchin Aeron

Information Theoretic Bounds for Compressed Sensing

In this paper, we derive information theoretic performance bounds to sensing and reconstruction of sparse phenomena from noisy projections. We consider two settings: output noise models where the noise enters after the projection and input noise models where the noise enters before the projection. We consider two types of distortion for reconstruction: support errors and mean-squared errors. Our goal is to relate the number of measurements, m , and SNR, to signal sparsity, k, distortion level, d, and signal dimension, n . We consider support errors in a worst-case setting. We employ different variations of Fanos inequality to derive necessary conditions on the number of measurements and SNR required for exact reconstruction. To derive sufficient conditions, we develop new insights on max-likelihood analysis based on a novel superposition property. In particular, this property implies that small support errors are the dominant error events. Consequently, our ML analysis does not suffer the conservatism of the union bound and leads to a tighter analysis of max-likelihood. These results provide order-wise tight bounds. For output noise models, we show that asymptotically an SNR of ((n)) together with (k (n/k)) measurements is necessary and sufficient for exact support recovery. Furthermore, if a small fraction of support errors can be tolerated, a constant SNR turns out to be sufficient in the linear sparsity regime. In contrast for input noise models, we show that support recovery fails if the number of measurements scales as o(n(n)/SNR), implying poor compression performance for such cases. Motivated by the fact that the worst-case setup requires significantly high SNR and substantial number of measurements for input and output noise models, we consider a Bayesian setup. To derive necessary conditions, we develop novel extensions to Fanos inequality to handle continuous domains and arbitrary distortions. We then develop a new max-likelihood analysis over the set of rate distortion quantization points to characterize tradeoffs between mean-squared distortion and the number of measurements using rate-distortion theory. We show that with constant SNR the number of measurements scales linearly with the rate-distortion function of the sparse phenomena.

Wireless Ad Hoc Networks: Strategies and Scaling Laws for the Fixed SNR Regime

This paper deals with throughput scaling laws for random ad hoc wireless networks in a rich scattering environment. We develop schemes to optimize the ratio lambda(n) of achievable network sum capacity to the sum of the point-to-point capacities of source-destinations (S-D) pairs operating in isolation. Our focus in this paper is on fixed signal-to-noise ratio (SNR) networks, i.e., networks where the worst case SNR over the S-D pairs is fixed independent of n. For such fixed SNR networks, which include fixed area networks as a special case, we show that collaborative strategies yield a scaling law of lambda(n)=Omega(1/n1/3) in contrast to multihop strategies which yield a scaling law of lambda(n)=Theta(1/radicn). While networks where worst case SNR goes to zero do not preclude the possibility of collaboration, multihop strategies achieve optimal throughput. The plausible reason is that the gains due to collaboration cannot offset the effect of vanishing receive SNR. This suggests that for fixed SNR networks, a network designer should look for network protocols that exploit collaboration

Novel Methods for Multilinear Data Completion and De-noising Based on Tensor-SVD

In this paper we propose novel methods for completion (from limited samples) and de-noising of multilinear (tensor) data and as an application consider 3-D and 4- D (color) video data completion and de-noising. We exploit the recently proposed tensor-Singular Value Decomposition (t-SVD)[11]. Based on t-SVD, the notion of multilinear rank and a related tensor nuclear norm was proposed in [11] to characterize informational and structural complexity of multilinear data. We first show that videos with linear camera motion can be represented more efficiently using t-SVD compared to the approaches based on vectorizing or flattening of the tensors. Since efficiency in representation implies efficiency in recovery, we outline a tensor nuclear norm penalized algorithm for video completion from missing entries. Application of the proposed algorithm for video recovery from missing entries is shown to yield a superior performance over existing methods. We also consider the problem of tensor robust Principal Component Analysis (PCA) for de-noising 3-D video data from sparse random corruptions. We show superior performance of our method compared to the matrix robust PCA adapted to this setting as proposed in [4].

Efficient Sensor Management Policies for Distributed Target Tracking in Multihop Sensor Networks

We consider the problem of distributed target tracking in a sensor network under communication constraints between the sensor nodes, a problem that has recently received significant attention. Communication constraints limit sensor data fusion in two ways. It significantly constrains sensor communication across large distances and substantially limits the number of sensors participating in data fusion at any time instant. We explore sensor management policies, i.e., sensor selection under communication constraints, for distributed target tracking. The coupled problem of track estimation and sensor management is generally intractable and significant effort has been devoted towards proposing simple strategies under various performance criteria. In this paper, we adopt a certainty equivalent approach and separate the tasks of track estimation and sensor management. Our approach is an adaptive dynamic strategy for sensor selection that seeks to optimize a tradeoff between tracking error and communications cost. We formulate the sensor management problem for the limiting case of infinite sensor density and derive sensor selection policies for different classes of target dynamics and sensor measurements. Under assumptions of a regular dense network with homogeneous sensors, the optimal strategy is a hybrid switching strategy, where the fusion center location and reporting sensors are held stationary unless the target estimates move outside of a threshold radius around the sensors. We simulate different tracking scenarios to illustrate the performance of our algorithms on sensor networks. We show that the computational as well as the communication costs are constant and do not scale with network size. We also perform different parametric studies to illustrate the validity of our approximations.

A Compressed Sensing Analog-to-Information Converter With Edge-Triggered SAR ADC Core

This paper presents the design and implementation of an analog-to-information converter (AIC) based on compressed sensing. The core of the AIC is an edge-triggered charge-sharing SAR ADC. Compressed sensing is achieved through random sampling and asynchronous successive approximation conversion using the ADC core. Implemented in 90nm CMOS, the prototype SAR ADC core achieves a maximum sample rate of 9.5MS/s, an ENOB of 9.3 bits, and consumes 550μW from a 1.2V supply. Measurement results of the compressed sensing AIC demonstrate effective sub-Nyquist random sampling and reconstruction of signals with sparse frequency support suitable for wideband spectrum sensing applications. When accounting for the increased input bandwidth compared to Nyquist, the AIC achieves an effective FOM of 10.2fJ/conversion-step.

Broadband Dispersion Extraction Using Simultaneous Sparse Penalization

In this paper, we propose a broadband method to extract the dispersion curves for multiple overlapping dispersive modes from borehole acoustic data under limited spatial sampling. The proposed approach exploits a first order Taylor series approximation of the dispersion curve in a band around a given (center) frequency in terms of the phase and group slowness at that frequency. Under this approximation, the acoustic signal in a given band can be represented as a superposition of broadband propagators each of which is parameterized by the slowness pair above. We then formulate a sparsity penalized reconstruction framework as follows. These broadband propagators are viewed as elements from an overcomplete dictionary representation and under the assumption that the number of modes is small compared to the size of the dictionary, it turns out that an appropriately reshaped support image of the coefficient vector synthesizing the signal (using the given dictionary representation) exhibits column sparsity. Our main contribution lies in identifying this feature and proposing a complexity regularized algorithm for support recovery with an l1 type simultaneous sparse penalization. Note that support recovery in this context amounts to recovery of the broadband propagators comprising the signal and hence extracting the dispersion, namely, the group and phase slownesses of the modes. In this direction we present a novel method to select the regularization parameter based on Kolmogorov-Smirnov (KS) tests on the distribution of residuals for varying values of the regularization parameter. We evaluate the performance of the proposed method on synthetic as well as real data and show its performance in dispersion extraction under presence of heavy noise and strong interference from time overlapped modes.

Low-tubal-rank tensor completion using alternating minimization

The low-tubal-rank tensors have been recently proposed to model real-world multidimensional data. In this paper, we study the low-tubal-rank tensor completion problem, i.e., to recover a third-order tensor by observing a subset of elements selected uniform at random. We propose a fast iterative algorithm, called Tubal-Alt-Min, that is inspired by similar approach for low rank matrix completion. The unknown low-tubal-rank tensor is parameterized as the product of two much smaller tensors with the low-tubal-rank property being automatically incorporated, and Tubal-Alt-Min alternates between estimating those two tensors using tensor least squares minimization. We note that the tensor least squares minimization is different from its counterpart and nontrivial, and this paper gives a routine to carry out this operation. Further, on both synthetic data and real-world video data, evaluation results show that compared with the tensor nuclear norm minimization, the proposed algorithm improves the recovery error by orders of magnitude with smaller running time for higher sampling rates.

Automatic dispersion extraction using continuous wavelet transform

In this paper we present a novel framework for automatic extraction of dispersion characteristics from acoustic array data. Traditionally high resolution narrow-band array processing techniques such as Pronys polynomial method and forward backward matrix pencil method have been applied to this problem. Fundamentally these techniques extract the dispersion components frequency by frequency in the wavenumber-frequency transform domain of the array data. The dispersion curves are subsequently extracted by a supervised post processing and labelling of the extracted wavenumber estimates, making such an approach unsuitable for automated processing. Moreover, this frequency domain processing fails to exploit useful time information. In this paper we present a method that addresses both these issues. It consists in taking the continuous wavelet transform (CWT) of the array data and then applying a wide-band array processing technique based on a modified Radon transform on the resulting coefficients to extract the dispersion curve(s). The time information retained in the CWT domain is useful not only for separating the components present but also for extracting group slowness estimates. The latter help in the automated extraction of smooth dispersion curves. In this paper we will introduce this new method referred to as the exponential projected Radon transform (EPRT) in the CWT domain and limit ourselves to the analysis for the case of one dispersive mode. We will apply the method to synthetic and real data sets and compare the performance with existing methods.

Fundamental Tradeoffs between Sparsity, Sensing Diversity and Sensing Capacity

A fundamental problem in sensor networks is to determine the sensing capacity, i.e., the minimum number of sensors required to monitor a given region to a desired degree of fidelity based on noisy sensor data. This question has direct bearing on the corresponding coverage problem, wherein the task is to determine the maximum coverage region with a given set of sensors. In this paper we show that sensing capacity is a function of SNR sparsity-the inherent complexity/dimensionality of the underlying signal/information space and its frequency of occurrence-and sensing diversity, i.e., the number of independent paths from the underlying signal space to the multiple sensors. We derive fundamental tradeoffs between SNR, sparsity, diversity and capacity. We show that the capacity is a monotonic function of SNR and diversity. A surprising result is that as sparsity approaches zero so does the sensing capacity irrespective of diversity. This implies for instance that to reliably monitor a small number of targets in a given region requires an disproportionally large number of sensors.

On sensing capacity of sensor networks for a class of linear observation models

In this paper we derive fundamental information theoretic upper and lower bounds to sensing capacity of sensor networks for several classes of linear observation models under fixed SNR. We define sensing capacity as the number of signal dimensions that can be reliably identified per sensor measurement. The signal sparsity plays an important role in this context. First we derive lower bounds to probability of error by extending the Fanos inequality to handle arbitrary distortion in reconstruction and continuous signal spaces. It turns out that a necessary condition for signal reconstruction to within an average distortion level is that the rate distortion at the given level of sparsity should be less than the mutual information between the signal and the observations. Through a suitable expansion of the mutual information term we isolate the effect of structure of the sensing matrix on sensing capacity. Subsequently we analyze this effect for several interesting classes of sensing matrices that arise naturally in the context of sensor networks under different scenarios. First we show the effect of sensing diversity - which is related to the field coverage per sensor- on sensing capacity for random ensembles of sensing matrices. We show that low diversity implies low sensing capacity. However sufficiently large diversity can be traded off for SNR and signal sparsity. Then we consider deterministic sensing matrices and evaluate a general upper bound to sensing capacity. As a special case we show that a random LTI filter type structure suffers from low diversity.