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Register a new userPenalized Maximum Likelihood Estimator for Mixture of von Mises-Fisher Distributions
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Tin Lok James Ng
Publication date
2020
fields
Mathematical Statistics
and research's language is
English
Authors
Tin Lok James Ng
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The von Mises-Fisher distribution is one of the most widely used probability distributions to describe directional data. Finite mixtures of von Mises-Fisher distributions have found numerous applications. However, the likelihood function for the finite mixture of von Mises-Fisher distributions is unbounded and consequently the maximum likelihood estimation is not well defined. To address the problem of likelihood degeneracy, we consider a penalized maximum likelihood approach whereby a penalty function is incorporated. We prove strong consistency of the resulting estimator. An Expectation-Maximization algorithm for the penalized likelihood function is developed and simulation studies are performed to examine its performance.
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Robust estimation of location and concentration parameters for the von Mises-Fisher distribution is discussed. A key reparametrisation is achieved by expressing the two parameters as one vector on the Euclidean space. With this representation, we first show that maximum likelihood estimator for the von Mises-Fisher distribution is not robust in some situations. Then we propose two families of robust estimators which can be derived as minimisers of two density power divergences. The presented families enable us to estimate both location and concentration parameters simultaneously. Some properties of the estimators are explored. Simple iterative algorithms are suggested to find the estimates numerically. A comparison with the existing robust estimators is given as well as discussion on difference and similarity between the two proposed estimators. A simulation study is made to evaluate finite sample performance of the estimators. We consider a sea star dataset and discuss the selection of the tuning parameters and outlier detection.
A number of pattern recognition tasks, textit{e.g.}, face verification, can be boiled down to classification or clustering of unit length directional feature vectors whose distance can be simply computed by their angle. In this paper, we propose the von Mises-Fisher (vMF) mixture model as the theoretical foundation for an effective deep-learning of such directional features and derive a novel vMF Mixture Loss and its corresponding vMF deep features. The proposed vMF feature learning achieves the characteristics of discriminative learning, textit{i.e.}, compacting the instances of the same class while increasing the distance of instances from different classes. Moreover, it subsumes a number of popular loss functions as well as an effective method in deep learning, namely normalization. We conduct extensive experiments on face verification using 4 different challenging face datasets, textit{i.e.}, LFW, YouTube faces, CACD and IJB-A. Results show the effectiveness and excellent generalization ability of the proposed approach as it achieves state-of-the-art results on the LFW, YouTube faces and CACD datasets and competitive results on the IJB-A dataset.
The maximum likelihood estimator plays a fundamental role in statistics. However, for many models, the estimators do not have closed-form expressions. This limitation can be significant in situations where estimates and predictions need to be computed in real-time, such as in applications based on embedded technology, in which numerical methods can not be implemented. This paper provides a modification in the maximum likelihood estimator that allows us to obtain the estimators in closed-form expressions under some conditions. Under mild conditions, the estimator is invariant under one-to-one transformations, consistent, and has an asymptotic normal distribution. The proposed modified version of the maximum likelihood estimator is illustrated on the Gamma, Nakagami, and Beta distributions and compared with the standard maximum likelihood estimator.
Speaker Diarization (i.e. determining who spoke and when?) for multi-speaker naturalistic interactions such as Peer-Led Team Learning (PLTL) sessions is a challenging task. In this study, we propose robust speaker clustering based on mixture of multivariate von Mises-Fisher distributions. Our diarization pipeline has two stages: (i) ground-truth segmentation; (ii) proposed speaker clustering. The ground-truth speech activity information is used for extracting i-Vectors from each speechsegment. We post-process the i-Vectors with principal component analysis for dimension reduction followed by lengthnormalization. Normalized i-Vectors are high-dimensional unit vectors possessing discriminative directional characteristics. We model the normalized i-Vectors with a mixture model consisting of multivariate von Mises-Fisher distributions. K-means clustering with cosine distance is chosen as baseline approach. The evaluation data is derived from: (i) CRSS-PLTL corpus; and (ii) three-meetings subset of AMI corpus. The CRSSPLTL data contain audio recordings of PLTL sessions which is student-led STEM education paradigm. Proposed approach is consistently better than baseline leading to upto 44.48% and 53.68% relative improvements for PLTL and AMI corpus, respectively. Index Terms: Speaker clustering, von Mises-Fisher distribution, Peer-led team learning, i-Vector, Naturalistic Audio.
Gravitational-wave astronomers often wish to characterize the expected parameter-estimation accuracy of future observations. The Fisher matrix provides a lower bound on the spread of the maximum-likelihood estimator across noise realizations, as well as the leading-order width of the posterior probability, but it is limited to high signal strengths often not realized in practice. By contrast, Monte Carlo Bayesian inference provides the full posterior for any signal strength, but it is too expensive to repeat for a representative set of noises. Here I describe an efficient semianalytical technique to map the exact sampling distribution of the maximum-likelihood estimator across noise realizations, for any signal strength. This technique can be applied to any estimation problem for signals in additive Gaussian noise.
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