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Topological signal processing (TSP) over simplicial complexes typically assumes observations associated with the simplicial complexes are real scalars. In this paper, we develop TSP theories for the case where observations belong to general abelian groups, including function spaces that are commonly used to represent time-varying signals. Our approach generalizes the Hodge decomposition and allows for signal processing tasks to be performed on these more complex observations. We propose a unified and flexible framework for TSP that expands its applicability to a wider range of signal processing applications. Numerical results demonstrate the effectiveness of this approach and provide a foundation for future research in this area.

Information-theoretic (IT) measures based on $f$-divergences have recently gained interest as a measure of privacy leakage as they allow for trading off privacy against utility using only a single-value characterization. However, their operational interpretations in the privacy context are unclear. In this paper, we relate the notion of probabilistic information privacy (IP) to several IT privacy metrics based on $f$-divergences. We interpret probabilistic IP under both the detection and estimation frameworks and link it to differential privacy, thus allowing a precise operational interpretation of these IT privacy metrics. We show that the $\chi^2$-divergence privacy metric is stronger than those based on total variation distance and Kullback-Leibler divergence. Therefore, we further develop a data-driven empirical risk framework based on the $\chi^2$-divergence privacy metric and realized using deep neural networks. This framework is agnostic to the adversarial attack model. Empirical experiments demonstrate the efficacy of our approach.

This paper explores methods for estimating or approximating the total variation distance and the chi-squared divergence of probability measures within topological sample spaces, using independent and identically distributed samples. Our focus is on the practical scenario where the sample space is homeomorphic to subsets of Euclidean space, with the specific homeomorphism remaining unknown. Our proposed methods rely on the integral probability metric with witness functions in universal reproducing kernel Hilbert spaces (RKHSs). The estimators we develop consist of learnable parametric functions mapping the sample space to Euclidean space, paired with universal kernels defined in Euclidean space. This approach effectively overcomes the challenge of constructing universal kernels directly on non-Euclidean spaces. Furthermore, the estimators we devise demonstrate asymptotic consistency, and we provide a detailed statistical analysis, shedding light on their practical implementation.

A reconfigurable intelligent surface (RIS) consists of massive meta elements, which can improve the performance of future wireless communication systems. Existing RIS-aided channel estimation methods try to estimate the cascaded channel directly, incurring high computational and training overhead especially when the number of elements of RIS is extremely large. In this paper, we propose a cost-efficient channel estimation method via rank-one matrix factorization (MF). Specifically, if the RIS is employed near base station (BS), it is found that the RIS- aided channel can be factorized into a product of low-dimensional matrices. To estimate these factorized matrices, we propose alternating minimization and gradient descent approaches to obtain the near optimal solutions. Compared to directly estimating the cascaded channel, the proposed MF method reduces training overhead substantially. Finally, the numerical simulations show the effectiveness of the proposed MF method., Comment: accepted by IEEE Wireless Communications Letters

Data is used widely by service providers as input to inference systems to perform decision making for authorized tasks. The raw data however allows a service provider to infer other sensitive information it has not been authorized for. We propose a data-driven inference privacy preserving framework to sanitize data so as to prevent leakage of sensitive information that is present in the raw data, while ensuring that the sanitized data is still compatible with the service provider's legacy inference system. We develop an inference privacy framework based on the variational method and include maximum mean discrepancy and domain adaption as techniques to regularize the domain of the sanitized data to ensure its legacy compatibility. However, the variational method leads to weak privacy in cases where the underlying data distribution is hard to approximate. It may also face difficulties when handling continuous private variables. To overcome this, we propose an alternative formulation of the privacy metric using maximal correlation and we present empirical methods to estimate it. Finally, we develop a deep learning model as an example of the proposed inference privacy framework. Numerical experiments verify the feasibility of our approach.

This paper considers Lorden's minimax quickest change detection (QCD) problem with a privacy constraint. The goal is to sanitize a signal to satisfy inference privacy requirements while being able to detect a change quickly. We show that the Generalized Likelihood Ratio (GLR) CuSum achieves asymptotic optimality with a properly designed sanitization channel. We formulate the design of this sanitization channel as an optimization problem, which is however challenging to solve. We propose relaxations to the optimization problem and develop algorithms to obtain a solution. We also consider the privacy-aware QCD problem under a decentralized framework and propose algorithms to solve the relaxed channel design problem under this framework.

Each agent in a network makes a local observation that is linearly related to a set of public and private parameters. The agents send their observations to a fusion center to allow it to estimate the public parameters. To prevent leakage of the private parameters, each agent first sanitizes its local observation using a local privacy mechanism before transmitting it to the fusion center. We investigate the utility-privacy tradeoff in terms of the Cram\'er-Rao lower bounds for estimating the public and private parameters. We study the class of privacy mechanisms given by linear compression and noise perturbation, and derive necessary and sufficient conditions for achieving arbitrarily strong utility-privacy tradeoff in a multi-agent system for both the cases where prior information is available and unavailable, respectively. We also provide a method to find the maximum estimation privacy achievable without compromising the utility and propose an alternating algorithm to optimize the utility-privacy tradeoff in the case where arbitrarily strong utility-privacy tradeoff is not achievable.

We consider a linear dynamical system in which the state vector consists of both public and private states. One or more sensors make measurements of the state vector and sends information to a fusion center, which performs the final state estimation. To achieve an optimal tradeoff between the utility of estimating the public states and protection of the private states, the measurements at each time step are linearly compressed into a lower dimensional space. Under the centralized setting where all measurements are collected by a single sensor, we propose an optimization problem and an algorithm to find the best compression matrix. Under the decentralized setting where measurements are made separately at multiple sensors, each sensor optimizes its own local compression matrix. We propose methods to separate the overall optimization problem into multiple sub-problems that can be solved locally at each sensor. We consider the cases where there is no message exchange between the sensors; and where each sensor takes turns to transmit messages to the other sensors. Simulations and empirical experiments demonstrate the efficiency of our proposed approach in allowing the fusion center to estimate the public states with good accuracy while preventing it from estimating the private states accurately.

In a decentralized Internet of Things (IoT) network, a fusion center receives information from multiple sensors to infer a public hypothesis of interest. To prevent the fusion center from abusing the sensor information, each sensor sanitizes its local observation using a local privacy mapping, which is designed to achieve both inference privacy of a private hypothesis and data privacy of the sensor raw observations. Various inference and data privacy metrics have been proposed in the literature. We introduce the concepts of privacy implication and non-guarantee to study the relationships between these privacy metrics. We propose an optimization framework in which both local differential privacy (data privacy) and information privacy (inference privacy) metrics are incorporated. In the parametric case where sensor observations' distributions are known \emph{a priori}, we propose a two-stage local privacy mapping at each sensor, and show that such an architecture is able to achieve information privacy and local differential privacy to within the predefined budgets. For the nonparametric case where sensor distributions are unknown, we adopt an empirical optimization approach. Simulation and experiment results demonstrate that our proposed approaches allow the fusion center to accurately infer the public hypothesis while protecting both inference and data privacy.

We consider a decentralized detection network whose aim is to infer a public hypothesis of interest. However, the raw sensor observations also allow the fusion center to infer private hypotheses that we wish to protect. We consider the case where there are an uncountable number of private hypotheses belonging to an uncertainty set, and develop local privacy mappings at every sensor so that the sanitized sensor information minimizes the Bayes error of detecting the public hypothesis at the fusion center, while achieving information privacy for all private hypotheses. We introduce the concept of a most favorable hypothesis (MFH) and show how to find a MFH in the set of private hypotheses. By protecting the information privacy of the MFH, information privacy for every other private hypothesis is also achieved. We provide an iterative algorithm to find the optimal local privacy mappings, and derive some theoretical properties of these privacy mappings. Simulation results demonstrate that our proposed approach allows the fusion center to infer the public hypothesis with low error while protecting information privacy of all the private hypotheses., Comment: submitted to IEEE Transactions on Information Forensics and Security

The problem of quickest detection of a change in distribution is considered under the assumption that the pre-change distribution is known, and the post-change distribution is only known to belong to a family of distributions distinguishable from a discretized version of the pre-change distribution. A sequential change detection procedure is proposed that partitions the sample space into a finite number of bins, and monitors the number of samples falling into each of these bins to detect the change. A test statistic that approximates the generalized likelihood ratio test is developed. It is shown that the proposed test statistic can be efficiently computed using a recursive update scheme, and a procedure for choosing the number of bins in the scheme is provided. Various asymptotic properties of the test statistic are derived to offer insights into its performance trade-off between average detection delay and average run length to a false alarm. Testing on synthetic and real data demonstrates that our approach is comparable or better in performance to existing non-parametric change detection methods., Comment: Double-column 13-page version sent to IEEE. Transaction on Signal Processing. Supplementary material included

In an Internet of Things network, multiple sensors send information to a fusion center for it to infer a public hypothesis of interest. However, the same sensor information may be used by the fusion center to make inferences of a private nature that the sensors wish to protect. To model this, we adopt a decentralized hypothesis testing framework with binary public and private hypotheses. Each sensor makes a private observation and utilizes a local sensor decision rule or privacy mapping to summarize that observation independently of the other sensors. The local decision made by a sensor is then sent to the fusion center. Without assuming knowledge of the joint distribution of the sensor observations and hypotheses, we adopt a nonparametric learning approach to design local privacy mappings. We introduce the concept of an empirical normalized risk, which provides a theoretical guarantee for the network to achieve information privacy for the private hypothesis with high probability when the number of training samples is large. We develop iterative optimization algorithms to determine an appropriate privacy threshold and the best sensor privacy mappings, and show that they converge. Finally, we extend our approach to the case of a private multiple hypothesis. Numerical results on both synthetic and real data sets suggest that our proposed approach yields low error rates for inferring the public hypothesis, but high error rates for detecting the private hypothesis., Comment: 13 pages, 4 figures, submitted to IEEE transaction on signal processing

We study a tandem of agents who make decisions about an underlying binary hypothesis, where the distribution of the agent observations under each hypothesis comes from an uncertainty class. We investigate both decentralized detection rules, where agents collaborate to minimize the error probability of the final agent, and social learning rules, where each agent minimizes its own local minimax error probability. We then extend our results to the infinite tandem network, and derive necessary and sufficient conditions on the uncertainty classes for the minimax error probability to converge to zero when agents know their positions in the tandem. On the other hand, when agents do not know their positions in the network, we study the cases where agents collaborate to minimize the asymptotic minimax error probability, and where agents seek to minimize their worst-case minimax error probability (over all possible positions in the tandem). We show that asymptotic learning of the true hypothesis is no longer possible in these cases, and derive characterizations for the minimax error performance.

We propose a novel distributed expectation maximization (EM) method for non-cooperative RF device localization using a wireless sensor network. We consider the scenario where few or no sensors receive line-of-sight signals from the target. In the case of non-line-of-sight signals, the signal path consists of a single reflection between the transmitter and receiver. Each sensor is able to measure the time difference of arrival of the target's signal with respect to a reference sensor, as well as the angle of arrival of the target's signal. We derive a distributed EM algorithm where each node makes use of its local information to compute summary statistics, and then shares these statistics with its neighbors to improve its estimate of the target localization. Since all the measurements need not be centralized at a single location, the spectrum usage can be significantly reduced. The distributed algorithm also allows for increased robustness of the sensor network in the case of node failures. We show that our distributed algorithm converges, and simulation results suggest that our method achieves an accuracy close to the centralized EM algorithm. We apply the distributed EM algorithm to a set of experimental measurements with a network of four nodes, which confirm that the algorithm is able to localize a RF target in a realistic non-line-of-sight scenario., Comment: 30 pages, 11 figures

We propose a multi-hop diffusion strategy for a sensor network to perform distributed least mean-squares (LMS) estimation under local and network-wide energy constraints. At each iteration of the strategy, each node can combine intermediate parameter estimates from nodes other than its physical neighbors via a multi-hop relay path. We propose a rule to select combination weights for the multi-hop neighbors, which can balance between the transient and the steady-state network mean-square deviations (MSDs). We study two classes of networks: simple networks with a unique transmission path from one node to another, and arbitrary networks utilizing diffusion consultations over at most two hops. We propose a method to optimize each node's information neighborhood subject to local energy budgets and a network-wide energy budget for each diffusion iteration. This optimization requires the network topology, and the noise and data variance profiles of each node, and is performed offline before the diffusion process. In addition, we develop a fully distributed and adaptive algorithm that approximately optimizes the information neighborhood of each node with only local energy budget constraints in the case where diffusion consultations are performed over at most a predefined number of hops. Numerical results suggest that our proposed multi-hop diffusion strategy achieves the same steady-state MSD as the existing one-hop adapt-then-combine diffusion algorithm but with a lower energy budget., Comment: 14 pages, 12 figures. Submitted for publication

We consider a multihypothesis social learning problem in which an agent has access to a set of private observations and chooses an opinion from a set of experts to incorporate into its final decision. To model individual biases, we allow the agent and experts to have general loss functions and possibly different decision spaces. We characterize the loss exponents of both the agent and experts, and provide an asymptotically optimal method for the agent to choose the best expert to follow. We show that up to asymptotic equivalence, the worst loss exponent for the agent is achieved when it adopts the 0-1 loss function, which assigns a loss of 0 if the true hypothesis is declared and a loss of 1 otherwise. We introduce the concept of hypothesis-loss neutrality, and show that if the agent adopts a particular policy that is hypothesis-loss neutral, then it ignores all experts whose decision spaces are smaller than its own. On the other hand, if experts have the same decision space as the agent, then choosing an expert with the same loss function as itself is not necessarily optimal for the agent, which is somewhat counter-intuitive. We derive sufficient conditions for when it is optimal for the agent with 0-1 loss function to choose an expert with the same loss function., Comment: Submitted to IEEE Journal of Selected Topics in Signal Processing

We consider the problem of localizing two sensors using signals of opportunity from beacons with known positions. Beacons and sensors have asynchronous local clocks or oscillators with unknown clock skews and offsets. We model clock skews as random, and analyze the biases introduced by clock asynchronism in the received signals. By deriving the equivalent Fisher information matrix for the modified Bayesian Cram\'er-Rao lower bound (CRLB) of sensor position and velocity estimation, we quantify the errors caused by clock asynchronism.

We consider the decentralized binary hypothesis testing problem in networks with feedback, where some or all of the sensors have access to compressed summaries of other sensors' observations. We study certain two-message feedback architectures, in which every sensor sends two messages to a fusion center, with the second message based on full or partial knowledge of the first messages of the other sensors. We also study one-message feedback architectures, in which each sensor sends one message to a fusion center, with a group of sensors having full or partial knowledge of the messages from the sensors not in that group. Under either a Neyman-Pearson or a Bayesian formulation, we show that the asymptotically optimal (in the limit of a large number of sensors) detection performance (as quantified by error exponents) does not benefit from the feedback messages, if the fusion center remembers all sensor messages. However, feedback can improve the Bayesian detection performance in the one-message feedback architecture if the fusion center has limited memory; for that case, we determine the corresponding optimal error exponents.

We consider the problem of decentralized detection in a network consisting of a large number of nodes arranged as a tree of bounded height, under the assumption of conditionally independent, identically distributed observations. We characterize the optimal error exponent under a Neyman-Pearson formulation. We show that the Type II error probability decays exponentially fast with the number of nodes, and the optimal error exponent is often the same as that corresponding to a parallel configuration. We provide sufficient, as well as necessary, conditions for this to happen. For those networks satisfying the sufficient conditions, we propose a simple strategy that nearly achieves the optimal error exponent, and in which all non-leaf nodes need only send 1-bit messages.