George K. Atia

Boolean Compressed Sensing and Noisy Group Testing

The fundamental task of group testing is to recover a small distinguished subset of items from a large population while efficiently reducing the total number of tests (measurements). The key contribution of this paper is in adopting a new information-theoretic perspective on group testing problems. We formulate the group testing problem as a channel coding/decoding problem and derive a single-letter characterization for the total number of tests used to identify the defective set. Although the focus of this paper is primarily on group testing, our main result is generally applicable to other compressive sensing models.

Controlled Sensing for Multihypothesis Testing

The problem of multiple hypothesis testing with observation control is considered in both fixed sample size and sequential settings. In the fixed sample size setting, for binary hypothesis testing, the optimal exponent for the maximal error probability corresponds to the maximum Chernoff information over the choice of controls, and a pure stationary open-loop control policy is asymptotically optimal within the larger class of all causal control policies. For multihypothesis testing in the fixed sample size setting, lower and upper bounds on the optimal error exponent are derived. It is also shown through an example with three hypotheses that the optimal causal control policy can be strictly better than the optimal open-loop control policy. In the sequential setting, a test based on earlier work by Chernoff for binary hypothesis testing, is shown to be first-order asymptotically optimal for multihypothesis testing in a strong sense, using the notion of decision making risk in place of the overall probability of error. Another test is also designed to meet hard risk constrains while retaining asymptotic optimality. The role of past information and randomization in designing optimal control policies is discussed.

Randomized Robust Subspace Recovery and Outlier Detection for High Dimensional Data Matrices

This paper explores and analyzes two randomized designs for robust principal component analysis employing low-dimensional data sketching. In one design, a data sketch is constructed using random column sampling followed by low-dimensional embedding, while in the other, sketching is based on random column and row sampling. Both designs are shown to bring about substantial savings in complexity and memory requirements for robust subspace learning over conventional approaches that use the full scale data. A characterization of the sample and computational complexity of both designs is derived in the context of two distinct outlier models, namely, sparse and independent outlier models. The proposed randomized approach can provably recover the correct subspace with computational and sample complexity which depend only weakly on the size of the data (only through the coherence parameters). The results of the mathematical analysis are confirmed through numerical simulations using both synthetic and real data.

Cooperative Relaying with Imperfect Channel State Information

We consider relay cooperation with imperfect channel state information (CSI) in the downlink of wireless networks. In particular, we consider a two-phase transmission where in the first phase the base station broadcasts information to the relays; the relays decode the data fully or partially depending on the transmission rate and the quality of their corresponding communication links. During the second phase, the relays cooperate by jointly beamforming information to multiple users given that channel mean and covariance are available at the transmitter side. The goal is to optimize the total network throughput (taking into account both transmission phases) by proper choice of the transmission rates, cooperation architecture and beamforming transmit vectors from the relays. The key contribution of this paper lies in the consideration of the impact of CSI imperfections in such a system. We first formulate the problem of finding the optimum throughput, which is not amenable to analytical solution. We therefore derive a suboptimum adaptive beamforming strategy that maximizes a derived upper bound on the average system throughput. Even though the relays have imperfect CSI, it is shown that relay cooperation can significantly improve the overall system throughput.

Controlled sensing for hypothesis testing

In this paper, the problem of multiple hypothesis testing with observation control is considered. The structure of the optimal controller under various asymptotic regimes is studied. First, a setup with a fixed sample size is considered. In this setup, the asymptotic quantity of interest is the optimal exponent for the maximal error probability. For the case of binary hypothesis testing, it is shown that the optimal error exponent corresponds to the maximum Chernoff information over the choice of controls. It is also shown that a pure stationary control policy, i.e., a fixed policy which does not depend on specific realizations of past measurements and past controls (open-loop), is asymptotically optimal even among the class of all causal control policies. We also derive lower and upper bounds for the optimal error exponent for the case of multiple hypothesis testing. Second, a sequential setup is considered wherein the controller can also decide when to stop taking observations. In this case, the objective is to minimize the expected stopping time subject to the constraints of vanishing error probabilities under each hypothesis. A sequential test is proposed for testing multiple hypotheses and is shown to be asymptotically optimal.

Sleep Control for Tracking in Sensor Networks

We study the problem of tracking an object moving through a network of wireless sensors. In order to conserve energy, the sensors may be put into a sleep mode with a timer that determines their sleep duration. It is assumed that an asleep sensor cannot be communicated with or woken up, and hence the sleep duration needs to be determined at the time the sensor goes to sleep based on all the information available to the sensor. Having sleeping sensors in the network could result in degraded tracking performance, therefore, there is a tradeoff between energy usage and tracking performance. We design sleeping policies that attempt to optimize this tradeoff and characterize their performance. As an extension to our previous work in this area, we consider generalized models for object movement, object sensing, and tracking cost. For discrete state spaces and continuous Gaussian observations, we derive a lower bound on the optimal energy-tracking tradeoff. It is shown that in the low tracking error regime, the generated policies approach the derived lower bound.

Innovation Pursuit: A New Approach to Subspace Clustering

In subspace clustering, a group of data points belonging to a union of subspaces are assigned membership to their respective subspaces. This paper presents a new approach dubbed Innovation Pursuit (iPursuit) to the problem of subspace clustering using a new geometrical idea whereby subspaces are identified based on their relative novelties. We present two frameworks in which the idea of innovation pursuit is used to distinguish the subspaces. Underlying the first framework is an iterative method that finds the subspaces consecutively by solving a series of simple linear optimization problems, each searching for a direction of innovation in the span of the data potentially orthogonal to all subspaces except for the one to be identified in one step of the algorithm. A detailed mathematical analysis is provided establishing sufficient conditions for iPursuit to correctly cluster the data. The proposed approach can provably yield exact clustering even when the subspaces have significant intersections. It is shown that the complexity of the iterative approach scales only linearly in the number of data points and subspaces, and quadratically in the dimension of the subspaces. The second framework integrates iPursuit with spectral clustering to yield a new variant of spectral-clustering-based algorithms. The numerical simulations with both real and synthetic data demonstrate that iPursuit can often outperform the state-of-the-art subspace clustering algorithms, more so for subspaces with significant intersections, and that it significantly improves the state-of-the-art result for subspace-segmentation-based face clustering.

A technical framework for light- handed regulation of cognitive radios

Light-handed regulation is discussed often in policy circles, but what it should mean technically has always been a bit vague. For cognitive radios to succeed in reducing the regulatory overhead, this has to change. For us, light-handed regulation means minimizing the mandates to be met at radio certification and relying instead on incentives to deter bad behavior. We put forth a specific technical framework in which the certification mandates are minimal - radios must modulate their transmitted waveform to embed an identity fingerprint, and radios must obey certain go-to-jail commands directed toward their identities. More specifically, the identity is represented by a temporal profile of taboo time slots in which transmission is impossible. The fraction of taboo slots represents the overhead of this approach and determines how reliably harmful interference can be attributed to the culprit(s) responsible. Meanwhile, the fraction of time that innocent radios spend in jail is the overhead for the punishment system. The analysis is carried out in the context of a real-time spectrum market, but is also applicable to opportunistic use.

Robust Energy Efficient Cooperative Spectrum Sensing in Cognitive Radios

A crucial task for a network of cognitive radios is to detect occupied frequency bands, to protect transmissions of primary users, and to identify spectrum holes (unoccupied bands) to maximize the utilization of wasted resources. This paper is motivated by the need to account for challenging constraints that naturally arise in such applications such as channel model uncertainties and demanding sensitivity constraints of the sensing devices. We propose a False Discovery Rate (FDR) based cooperative strategy to sense the occupancy of the spectrum. The proposed strategy is robust to significant uncertainties such as lack of CSI, fading and shadowing effects. Furthermore, it is shown that the cooperative sensing strategy significantly reduces sensitivity requirements. We quantify the effect of channel occupancy rate on the required cooperation degree for achieving a guaranteed level of primary user protection.

Strong Impossibility Results for Sparse Signal Processing

This letter derives strong impossibility results for several sparse signal processing problems. It is shown that regardless of the allowed error probability in identifying the salient support set (as long as this probability is below one), the required number of measurements is almost the same as that required for the error probability to be arbitrarily small. Our proof technique involves the use of the blowing-up lemma and can be applied to diverse problems from noisy group testing to graphical model selection as long as the observations are discrete.