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Abstract:In this work, we study the restricted isometry property (RIP) of Kronecker-structured matrices, formed by the Kronecker product of two factor matrices. Previously, only upper and lower bounds on the restricted isometry constant (RIC) in terms of the RICs of the factor matrices were known. We derive a probabilistic measurement bound for the $s$th-order RIC. We show that the Kronecker product of two sub-Gaussian matrices satisfies RIP with high probability if the minimum number of rows among two matrices is $\mathcal{O}(s \ln \max\{N_1, N_2\})$. Here, $s$ is the sparsity level, and $N_1$ and $N_2$ are the number of columns in the matrices. We also present improved measurement bounds for the recovery of Kronecker-structured sparse vectors using Kronecker-structured measurement matrices. Finally, our analysis is further extended to the Kronecker product of more than two matrices.

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Authors:Geethu Joseph

Abstract:The convergence of expectation-maximization (EM)-based algorithms typically requires continuity of the likelihood function with respect to all the unknown parameters (optimization variables). The requirement is not met when parameters comprise both discrete and continuous variables, making the convergence analysis nontrivial. This paper introduces a set of conditions that ensure the convergence of a specific class of EM algorithms that estimate a mixture of discrete and continuous parameters. Our results offer a new analysis technique for iterative algorithms that solve mixed-integer non-linear optimization problems. As a concrete example, we prove the convergence of the EM-based sparse Bayesian learning algorithm in [1] that estimates the state of a linear dynamical system with jointly sparse inputs and bursty missing observations. Our results establish that the algorithm in [1] converges to the set of stationary points of the maximum likelihood cost with respect to the continuous optimization variables.

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Abstract:Sparsity constraints on the control inputs of a linear dynamical system naturally arise in several practical applications such as networked control, computer vision, seismic signal processing, and cyber-physical systems. In this work, we consider the problem of jointly estimating the states and sparse inputs of such systems from low-dimensional (compressive) measurements. Due to the low-dimensional measurements, conventional Kalman filtering and smoothing algorithms fail to accurately estimate the states and inputs. We present a Bayesian approach that exploits the input sparsity to significantly improve estimation accuracy. Sparsity in the input estimates is promoted by using different prior distributions on the input. We investigate two main approaches: regularizer-based MAP, and {Bayesian learning-based estimation}. We also extend the approaches to handle control inputs with common support and analyze the time and memory complexities of the presented algorithms. Finally, using numerical simulations, we show that our algorithms outperform the state-of-the-art methods in terms of accuracy and time/memory complexities, especially in the low-dimensional measurement regime.

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Abstract:In this paper, we address the problem of detecting anomalies among a given set of binary processes via learning-based controlled sensing. Each process is parameterized by a binary random variable indicating whether the process is anomalous. To identify the anomalies, the decision-making agent is allowed to observe a subset of the processes at each time instant. Also, probing each process has an associated cost. Our objective is to design a sequential selection policy that dynamically determines which processes to observe at each time with the goal to minimize the delay in making the decision and the total sensing cost. We cast this problem as a sequential hypothesis testing problem within the framework of Markov decision processes. This formulation utilizes both a Bayesian log-likelihood ratio-based reward and an entropy-based reward. The problem is then solved using two approaches: 1) a deep reinforcement learning-based approach where we design both deep Q-learning and policy gradient actor-critic algorithms; and 2) a deep active inference-based approach. Using numerical experiments, we demonstrate the efficacy of our algorithms and show that our algorithms adapt to any unknown statistical dependence pattern of the processes.

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Abstract:This paper studies the problem of Kronecker-structured sparse vector recovery from an underdetermined linear system with a Kronecker-structured dictionary. Such a problem arises in many real-world applications such as the sparse channel estimation of an intelligent reflecting surface-aided multiple-input multiple-output system. The prior art only exploits the Kronecker structure in the support of the sparse vector and solves the entire linear system together leading to high computational complexity. Instead, we break down the original sparse recovery problem into multiple independent sub-problems and solve them individually. We obtain the sparse vector as the Kronecker product of the individual solutions, retaining its structure in both support and nonzero entries. Our simulations demonstrate the superior performance of our methods in terms of accuracy and run time compared with the existing works, using synthetic data and the channel estimation application. We attribute the low run time to the reduced solution space due to the additional structure and improved accuracy to the denoising effect owing to the decomposition step.

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Abstract:This paper develops a channel estimation technique for millimeter wave (mmWave) communication systems. Our method exploits the sparse structure in mmWave channels for low training overhead and accounts for the phase errors in the channel measurements due to phase noise at the oscillator. Specifically, in IEEE 802.11ad/ay-based mmWave systems, the phase errors within a beam refinement protocol packet are almost the same, while the errors across different packets are substantially different. Consequently, standard sparsity-aware algorithms, which ignore phase errors, fail when channel measurements are acquired over multiple beam refinement protocol packets. We present a novel algorithm called partially coherent matching pursuit for sparse channel estimation under practical phase noise perturbations. Our method iteratively detects the support of sparse signal and employs alternating minimization to jointly estimate the signal and the phase errors. We numerically show that our algorithm can reconstruct the channel accurately at a lower complexity than the benchmarks.

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Abstract:We study the sparse recovery problem with an underdetermined linear system characterized by a Kronecker-structured dictionary and a Kronecker-supported sparse vector. We cast this problem into the sparse Bayesian learning (SBL) framework and rely on the expectation-maximization method for a solution. To this end, we model the Kronecker-structured support with a hierarchical Gaussian prior distribution parameterized by a Kronecker-structured hyperparameter, leading to a non-convex optimization problem. The optimization problem is solved using the alternating minimization (AM) method and a singular value decomposition (SVD)-based method, resulting in two algorithms. Further, we analytically guarantee that the AM-based method converges to the stationary point of the SBL cost function. The SVD-based method, though it adopts approximations, is empirically shown to be more efficient and accurate. We then apply our algorithm to estimate the uplink wireless channel in an intelligent reflecting surface-aided MIMO system and extend the AM-based algorithm to address block sparsity in the channel. We also study the SBL cost to show that the minima of the cost function are achieved at sparse solutions and that incorporating the Kronecker structure reduces the number of local minima of the SBL cost function. Our numerical results demonstrate the effectiveness of our algorithms compared to the state-of-the-art.

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Abstract:This paper presents novel cascaded channel estimation techniques for an intelligent reflecting surface-aided multiple-input multiple-output system. Motivated by the channel angular sparsity at higher frequency bands, the channel estimation problem is formulated as a sparse vector recovery problem with an inherent Kronecker structure. We solve the problem using the sparse Bayesian learning framework which leads to a non-convex optimization problem. We offer two solution techniques to the problem based on alternating minimization and singular value decomposition. Our simulation results illustrate the superior performance of our methods in terms of accuracy and run time compared with the existing works.

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Abstract:Phased arrays in near-field communication allow the transmitter to focus wireless signals in a small region around the receiver. Proper focusing is achieved by carefully tuning the phase shifts and the polarization of the signals transmitted from the phased array. In this paper, we study the impact of polarization on near-field focusing and investigate the use of dynamic polarization control (DPC) phased arrays in this context. Our studies indicate that the optimal polarization configuration for near-field focusing varies spatially across the antenna array. Such a spatial variation motivates the need for DPC phased arrays which allow independent polarization control across different antennas. We show using simulations that DPC phased arrays in the near-field achieve a higher received signal-to-noise ratio than conventional switched- or dual-polarization phased arrays.

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Abstract:We address the problem of monitoring a set of binary stochastic processes and generating an alert when the number of anomalies among them exceeds a threshold. For this, the decision-maker selects and probes a subset of the processes to obtain noisy estimates of their states (normal or anomalous). Based on the received observations, the decisionmaker first determines whether to declare that the number of anomalies has exceeded the threshold or to continue taking observations. When the decision is to continue, it then decides whether to collect observations at the next time instant or defer it to a later time. If it chooses to collect observations, it further determines the subset of processes to be probed. To devise this three-step sequential decision-making process, we use a Bayesian formulation wherein we learn the posterior probability on the states of the processes. Using the posterior probability, we construct a Markov decision process and solve it using deep actor-critic reinforcement learning. Via numerical experiments, we demonstrate the superior performance of our algorithm compared to the traditional model-based algorithms.

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