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Register a new userGrid Based Nonlinear Filtering Revisited: Recursive Estimation & Asymptotic Optimality
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Added by
Dionysios Kalogerias
Publication date
2016
fields
Informatics Engineering
and research's language is
English
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We revisit the development of grid based recursive approximate filtering of general Markov processes in discrete time, partially observed in conditionally Gaussian noise. The grid based filters considered rely on two types of state quantization: The textit{Markovian} type and the textit{marginal} type. We propose a set of novel, relaxed sufficient conditions, ensuring strong and fully characterized pathwise convergence of these filters to the respective MMSE state estimator. In particular, for marginal state quantizations, we introduce the notion of textit{conditional regularity of stochastic kernels}, which, to the best of our knowledge, constitutes the most relaxed condition proposed, under which asymptotic optimality of the respective grid based filters is guaranteed. Further, we extend our convergence results, including filtering of bounded and continuous functionals of the state, as well as recursive approximate state prediction. For both Markovian and marginal quantizations, the whole development of the respective grid based filters relies more on linear-algebraic techniques and less on measure theoretic arguments, making the presentation considerably shorter and technically simpler.
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This paper studies the optimal rate of estimation in a finite Gaussian location mixture model in high dimensions without separation conditions. We assume that the number of components $k$ is bounded and that the centers lie in a ball of bounded radius, while allowing the dimension $d$ to be as large as the sample size $n$. Extending the one-dimensional result of Heinrich and Kahn cite{HK2015}, we show that the minimax rate of estimating the mixing distribution in Wasserstein distance is $Theta((d/n)^{1/4} + n^{-1/(4k-2)})$, achieved by an estimator computable in time $O(nd^2+n^{5/4})$. Furthermore, we show that the mixture density can be estimated at the optimal parametric rate $Theta(sqrt{d/n})$ in Hellinger distance and provide a computationally efficient algorithm to achieve this rate in the special case of $k=2$. Both the theoretical and methodological development rely on a careful application of the method of moments. Central to our results is the observation that the information geometry of finite Gaussian mixtures is characterized by the moment tensors of the mixing distribution, whose low-rank structure can be exploited to obtain a sharp local entropy bound.
Algorithms for Massive MIMO uplink detection typically rely on a centralized approach, by which baseband data from all antennas modules are routed to a central node in order to be processed. In case of Massive MIMO, where hundreds or thousands of antennas are expected in the base-station, this architecture leads to a bottleneck, with critical limitations in terms of interconnection bandwidth requirements. This paper presents a fully decentralized architecture and algorithms for Massive MIMO uplink based on recursive methods, which do not require a central node for the detection process. Through a recursive approach and very low complexity operations, the proposed algorithms provide a sequence of estimates that converge asymptotically to the zero-forcing solution, without the need of specific hardware for matrix inversion. The proposed solution achieves significantly lower interconnection data-rate than other architectures, enabling future scalability.
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Robust estimation is an important problem in statistics which aims at providing a reasonable estimator when the data-generating distribution lies within an appropriately defined ball around an uncontaminated distribution. Although minimax rates of estimation have been established in recent years, many existing robust estimators with provably optimal convergence rates are also computationally intractable. In this paper, we study several estimation problems under a Wasserstein contamination model and present computationally tractable estimators motivated by generative adversarial networks (GANs). Specifically, we analyze properties of Wasserstein GAN-based estimators for location estimation, covariance matrix estimation, and linear regression and show that our proposed estimators are minimax optimal in many scenarios. Finally, we present numerical results which demonstrate the effectiveness of our estimators.
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