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Generalized linear mixed models are powerful tools for analyzing clustered data, where the unknown parameters are classically (and most commonly) estimated by the maximum likelihood and restricted maximum likelihood procedures. However, since the likelihood based procedures are known to be highly sensitive to outliers, M-estimators have become popular as a means to obtain robust estimates under possible data contamination. In this paper, we prove that, for sufficiently smooth general loss functions defining the M-estimators in generalized linear mixed models, the tail probability of the deviation between the estimated and the true regression coefficients have an exponential bound. This implies an exponential rate of consistency of these M-estimators under appropriate assumptions, generalizing the existing exponential consistency results from univariate to multivariate responses. We have illustrated this theoretical result further for the special examples of the maximum likelihood estimator and the robust minimum density power divergence estimator, a popular example of model-based M-estimators, in the settings of linear and logistic mixed models, comparing it with the empirical rate of convergence through simulation studies.

This paper studies the third-order characteristic of nonsingular discrete memoryless channels and the Gaussian channel with a maximal power constraint. The third-order term in our expansions employs a new quantity here called the \emph{channel skewness}, which affects the approximation accuracy more significantly as the error probability decreases. For the Gaussian channel, evaluating Shannon's (1959) random coding and sphere-packing bounds in the central limit theorem (CLT) regime enables exact computation of the channel skewness. For discrete memoryless channels, this work generalizes Moulin's (2017) bounds on the asymptotic expansion of the maximum achievable message set size for nonsingular channels from the CLT regime to include the moderate deviations (MD) regime, thereby refining Altu\u{g} and Wagner's (2014) MD result. For an example binary symmetric channel and most practically important $(n, \epsilon)$ pairs, including $n \in [100, 500]$ and $\epsilon \in [10^{-10}, 10^{-1}]$, an approximation up to the channel skewness is the most accurate among several expansions in the literature. A derivation of the third-order term in the type-II error exponent of binary hypothesis testing in the MD regime is also included; the resulting third-order term is similar to the channel skewness.

In this work, we study the problem of real-time tracking and reconstruction of an information source with the purpose of actuation. A device monitors an $N$-state Markov process and transmits status updates to a receiver over a wireless erasure channel. We consider a set of joint sampling and transmission policies, including a semantics-aware one, and we study their performance with respect to relevant metrics. Specifically, we investigate the real-time reconstruction error and its variance, the consecutive error, the cost of memory error, and the cost of actuation error. Furthermore, we propose a randomized stationary sampling and transmission policy and derive closed-form expressions for all aforementioned metrics. We then formulate an optimization problem for minimizing the real-time reconstruction error subject to a sampling cost constraint. Our results show that in the scenario of constrained sampling generation, the optimal randomized stationary policy outperforms all other sampling policies when the source is rapidly evolving. Otherwise, the semantics-aware policy performs the best.

Spectral density estimation is a core problem of system identification, which is an important research area of system control and signal processing. There have been numerous results on the design of spectral density estimators. However to our best knowledge, quantitative error analyses of the spectral density estimation have not been proposed yet. In real practice, there are two main factors which induce errors in the spectral density estimation, including the external additive noise and the limited number of samples. In this paper, which is a very preliminary version, we first consider a univariate spectral density estimator using covariance lags. The estimation task is performed by a convex optimization scheme, and the covariance lags of the estimated spectral density are exactly as desired, which makes it possible for quantitative error analyses such as to derive tight error upper bounds. We analyze the errors induced by the two factors and propose upper and lower bounds for the errors. Then the results of the univariate spectral estimator are generalized to the multivariate one.

We propose an alternating direction method of multipliers (ADMM) to solve an optimization problem stemming from inverse lithography. The objective functional of the optimization problem includes three terms: the misfit between the imaging on wafer and the target pattern, the penalty term which ensures the mask is binary and the total variation regularization term. By variable splitting, we introduce an augmented Lagrangian for the original objective functional. In the framework of ADMM method, the optimization problem is divided into several subproblems. Each of the subproblems can be solved efficiently. We give the convergence analysis of the proposed method. Specially, instead of solving the subproblem concerning sigmoid, we solve directly the threshold truncation imaging function which can be solved analytically. We also provide many numerical examples to illustrate the effectiveness of the method.

Generalised hyperbolic (GH) processes are a class of stochastic processes that are used to model the dynamics of a wide range of complex systems that exhibit heavy-tailed behavior, including systems in finance, economics, biology, and physics. In this paper, we present novel simulation methods based on subordination with a generalised inverse Gaussian (GIG) process and using a generalised shot-noise representation that involves random thinning of infinite series of decreasing jump sizes. Compared with our previous work on GIG processes, we provide tighter bounds for the construction of rejection sampling ratios, leading to improved acceptance probabilities in simulation. Furthermore, we derive methods for the adaptive determination of the number of points required in the associated random series using concentration inequalities. Residual small jumps are then approximated using an appropriately scaled Brownian motion term with drift. Finally the rejection sampling steps are made significantly more computationally efficient through the use of squeezing functions based on lower and upper bounds on the L\'evy density. Experimental results are presented illustrating the strong performance under various parameter settings and comparing the marginal distribution of the GH paths with exact simulations of GH random variates. The new simulation methodology is made available to researchers through the publication of a Python code repository.

Blind source separation (BSS) aims to recover an unobserved signal $S$ from its mixture $X=f(S)$ under the condition that the effecting transformation $f$ is invertible but unknown. As this is a basic problem with many practical applications, a fundamental issue is to understand how the solutions to this problem behave when their supporting statistical prior assumptions are violated. In the classical context of linear mixtures, we present a general framework for analysing such violations and quantifying their impact on the blind recovery of $S$ from $X$. Modelling $S$ as a multidimensional stochastic process, we introduce an informative topology on the space of possible causes underlying a mixture $X$, and show that the behaviour of a generic BSS-solution in response to general deviations from its defining structural assumptions can be profitably analysed in the form of explicit continuity guarantees with respect to this topology. This allows for a flexible and convenient quantification of general model uncertainty scenarios and amounts to the first comprehensive robustness framework for BSS. Our approach is entirely constructive, and we demonstrate its utility with novel theoretical guarantees for a number of statistical applications.

When is heterogeneity in the composition of an autonomous robotic team beneficial and when is it detrimental? We investigate and answer this question in the context of a minimally viable model that examines the role of heterogeneous speeds in perimeter defense problems, where defenders share a total allocated speed budget. We consider two distinct problem settings and develop strategies based on dynamic programming and on local interaction rules. We present a theoretical analysis of both approaches and our results are extensively validated using simulations. Interestingly, our results demonstrate that the viability of heterogeneous teams depends on the amount of information available to the defenders. Moreover, our results suggest a universality property: across a wide range of problem parameters the optimal ratio of the speeds of the defenders remains nearly constant.

The area of Data Analytics on graphs promises a paradigm shift as we approach information processing of classes of data, which are typically acquired on irregular but structured domains (social networks, various ad-hoc sensor networks). Yet, despite its long history, current approaches mostly focus on the optimization of graphs themselves, rather than on directly inferring learning strategies, such as detection, estimation, statistical and probabilistic inference, clustering and separation from signals and data acquired on graphs. To fill this void, we first revisit graph topologies from a Data Analytics point of view, and establish a taxonomy of graph networks through a linear algebraic formalism of graph topology (vertices, connections, directivity). This serves as a basis for spectral analysis of graphs, whereby the eigenvalues and eigenvectors of graph Laplacian and adjacency matrices are shown to convey physical meaning related to both graph topology and higher-order graph properties, such as cuts, walks, paths, and neighborhoods. Next, to illustrate estimation strategies performed on graph signals, spectral analysis of graphs is introduced through eigenanalysis of mathematical descriptors of graphs and in a generic way. Finally, a framework for vertex clustering and graph segmentation is established based on graph spectral representation (eigenanalysis) which illustrates the power of graphs in various data association tasks. The supporting examples demonstrate the promise of Graph Data Analytics in modeling structural and functional/semantic inferences. At the same time, Part I serves as a basis for Part II and Part III which deal with theory, methods and applications of processing Data on Graphs and Graph Topology Learning from data.

In this paper, we present an accurate and scalable approach to the face clustering task. We aim at grouping a set of faces by their potential identities. We formulate this task as a link prediction problem: a link exists between two faces if they are of the same identity. The key idea is that we find the local context in the feature space around an instance (face) contains rich information about the linkage relationship between this instance and its neighbors. By constructing sub-graphs around each instance as input data, which depict the local context, we utilize the graph convolution network (GCN) to perform reasoning and infer the likelihood of linkage between pairs in the sub-graphs. Experiments show that our method is more robust to the complex distribution of faces than conventional methods, yielding favorably comparable results to state-of-the-art methods on standard face clustering benchmarks, and is scalable to large datasets. Furthermore, we show that the proposed method does not need the number of clusters as prior, is aware of noises and outliers, and can be extended to a multi-view version for more accurate clustering accuracy.