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Clustering with mixtures of log-concave distributions

George T. Chang, Guenther Walther
2007 Computational Statistics & Data Analysis  
We show how this algorithm can be extended to work with the flexible, nonparametric class of logconcave component distributions.  ...  The EM algorithm is a popular tool for clustering observations via a parametric mixture model.  ...  In the M-step we compute for each component (cluster) the d marginal log-concave Density contours of two distributions that belong to the multivariate log-concave model.  ... 
doi:10.1016/j.csda.2007.01.008 fatcat:5wtsgnnxozbsnet2diazr7br6a

Inference and Modeling with Log-concave Distributions

Guenther Walther
2009 Statistical Science  
Log-concave distributions are an attractive choice for modeling and inference, for several reasons: The class of log-concave distributions contains most of the commonly used parametric distributions and  ...  Due to these attractive properties, there has been considerable recent research activity concerning the theory and applications of log-concave distributions.  ...  APPLICATIONS IN MODELING AND INFERENCE One of the most fruitful applications of log-concave distributions has been in the area of clustering.  ... 
doi:10.1214/09-sts303 fatcat:jydigqfpkvejvg255firkgi6wu

Hyperprior on symmetric Dirichlet distribution [article]

Jun Lu
2017 arXiv   pre-print
In this article we introduce how to put vague hyperprior on Dirichlet distribution, and we update the parameter of it by adaptive rejection sampling (ARS).  ...  Finally we analyze this hyperprior in an over-fitted mixture model by some synthetic experiments.  ...  In view of that the product of two log-concave functions is log-concave and Theorem 2, it follows that Γ(Kα) [Γ(α)] K is log-concave. This concludes the proof.  ... 
arXiv:1708.08177v1 fatcat:hwceyletafft7eh43ukexmjdey

Inference for a two-component mixture of symmetric distributions under log-concavity

Fadoua Balabdaoui, Charles R. Doss
2018 Bernoulli  
distribution has a log-concave density.  ...  When consistent estimators for the shift locations and mixing probability are used, we show that the nonparametric log-concave Maximum Likelihood estimator (MLE) of both the mixed density and that of the  ...  . , X n are independent and identically distributed (i.i.d.) draws from a mixture distribution, with cumulative distribution function (c.d.f.)  ... 
doi:10.3150/16-bej864 fatcat:rg3fompurfg7vndups5uadtscy

Semiparametric Estimation of Symmetric Mixture Models with Monotone and Log-Concave Densities [article]

Xiao Pu, Ery Arias-Castro
2018 arXiv   pre-print
In this article, we revisit the problem of fitting a mixture model under the assumption that the mixture components are symmetric and log-concave.  ...  To this end, we first study the nonparametric maximum likelihood estimation (NPMLE) of a monotone log-concave probability density.  ...  Introduction Mixture models are a staple of statistical analysis. In this paper, we concern ourselves with semiparametric mixture models under the hypotheses of symmetry and log-concavity.  ... 
arXiv:1702.08897v3 fatcat:nv7wruc56vguffgkmpkusilmqy

The Informativeness of k-Means for Learning Mixture Models [article]

Zhaoqiang Liu, Vincent Y. F. Tan
2019 arXiv   pre-print
In this paper, we provide sufficient conditions for the closeness of any optimal clustering and the correct target clustering assuming that the data samples are generated from a mixture of log-concave  ...  Indeed, given data samples independently generated from a mixture of distributions, we often would like to find the correct target clustering of the samples according to which component distribution they  ...  Extension to Mixtures of Log-Concave Distributions In this section, we extend the results in Section 3 to mixtures of log-concave distributions.  ... 
arXiv:1703.10534v3 fatcat:etkbkdkvcrb6fiv7hgcpksa7za

Learning mixtures of structured distributions over discrete domains [chapter]

Siu-On Chan, Ilias Diakonikolas, Xiaorui Sun, Rocco A. Servedio
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms  
of all log-concave distributions.  ...  More precisely, • Log-concave distributions: We learn any mixture of k log-concave distributions over [n] using k ·Õ(1/ε 4 ) samples (independent of n) and running in timeÕ(k log(n)/ε 4 ) bit-operations  ...  These include mixtures of log-concave distributions, mixtures of monotone hazard rate (MHR) distributions, and mixtures of unimodal distributions.  ... 
doi:10.1137/1.9781611973105.100 dblp:conf/soda/ChanDSS13 fatcat:hysjwuq2xneb5k5sin35s3ocha

Learning mixtures of structured distributions over discrete domains [article]

Siu-on Chan, Ilias Diakonikolas, Rocco A. Servedio, Xiaorui Sun
2012 arXiv   pre-print
We analyze several natural types of distributions over [n], including log-concave, monotone hazard rate and unimodal distributions, and show that they have the required structural property of being well-approximated  ...  (both in terms of running time and sample complexity) algorithm that can learn any mixture of k unknown distributions from C.  ...  Are there any other natural distribution classes for which our general framework is applicable? We suspect so.  ... 
arXiv:1210.0864v1 fatcat:xy7n5midrng2llwjnj2a7oujuq

The robust EM-type algorithms for log-concave mixtures of regression models

Hao Hu, Weixin Yao, Yichao Wu
2017 Computational Statistics & Data Analysis  
Two EM-type algorithms for the mixtures of regression models with log-concave error densities are proposed.  ...  Numerical studies are made to compare the performance of our algorithms with the normal mixture EM algorithms.  ...  Wu's research is partially supported by National Institutes of Health grant R01-CA149569 and National Science Foundation grant DMS-1055210.  ... 
doi:10.1016/j.csda.2017.01.004 pmid:28947841 pmcid:PMC5609737 fatcat:glco6e3adnhyrcw4y2gizdwvne

A Mixture of Coalesced Generalized Hyperbolic Distributions [article]

Cristina Tortora, Brian C. Franczak, Ryan P. Browne, Paul D. McNicholas
2018 arXiv   pre-print
A mixture of multiple scaled generalized hyperbolic distributions (MMSGHDs) is introduced.  ...  Then, a coalesced generalized hyperbolic distribution (CGHD) is developed by joining a generalized hyperbolic distribution with a multiple scaled generalized hyperbolic distribution.  ...  ., a Collaborative Research and Development Grant from the Natural Sciences and Engineering Research Council of Canada, and an Early Researcher Award from the Ontario Ministry of Research and Innovation  ... 
arXiv:1403.2332v8 fatcat:5pn44rea7nhs5mn4jvr62co3iu

Maximum likelihood estimation of a multidimensional log-concave density [article]

Madeleine Cule, Richard Samworth, Michael Stewart
2008 arXiv   pre-print
., X_n be independent and identically distributed random vectors with a log-concave (Lebesgue) density f.  ...  We also present a real data clustering example, which shows that our methodology can be used in conjunction with the Expectation--Maximisation (EM) algorithm to fit finite mixtures of log-concave densities  ...  Panel (d) plots the fitted mixture distribution from the log-concave EM algorithm.  ... 
arXiv:0804.3989v1 fatcat:regy5akte5gr7cphljvvuvfrzy

Maximum likelihood estimation of a multi-dimensional log-concave density

Madeleine Cule, Richard Samworth, Michael Stewart
2010 Journal of The Royal Statistical Society Series B-statistical Methodology  
We also present a real data clustering example, which shows that our methodology can be used in conjunction with the expectation-maximization algorithm to fit finite mixtures of log-concave densities.  ...  We first prove that, with probability 1, there is a unique log-concave maximum likelihood estimatorf n of f.  ...  We thank Yining Chen for his help with the simulations that are reported in this rejoinder.  ... 
doi:10.1111/j.1467-9868.2010.00753.x fatcat:wtuitpjvvjhn7lkvmvneomi2ra

Maximum likelihood estimation of the mixture of log-concave densities

Hao Hu, Yichao Wu, Weixin Yao
2016 Computational Statistics & Data Analysis  
Numeric examples are also made to demonstrate that the LCMLE improves the clustering results while comparing with the traditional MLE for parametric mixture models.  ...  In this paper, a much more flexible mixture model is considered, which assumes each component density to be log-concave.  ...  Four-dimensional clustering result: normal mixture EM-algorithm vs log-concave mixture EMalgorithm by number of misclassifications. The solid lines represent the identity.  ... 
doi:10.1016/j.csda.2016.03.002 pmid:27065505 pmcid:PMC4820769 fatcat:43sskc2dozhwjpwbnhftobq6ii

An Analysis of the t-SNE Algorithm for Data Visualization [article]

Sanjeev Arora, Wei Hu, Pravesh K. Kothari
2018 arXiv   pre-print
We show that our deterministic condition is satisfied by considerably general probabilistic generative models for clusterable data such as mixtures of well-separated log-concave distributions.  ...  We then give a rigorous analysis of the performance of t-SNE under a natural, deterministic condition on the "ground-truth" clusters (similar to conditions assumed in earlier analyses of clustering) in  ...  Acknowledgments This research was done with support from NSF, ONR, Simons Foundation, Mozilla Research, Schmidt Foundation, DARPA, and SRC.  ... 
arXiv:1803.01768v2 fatcat:wgwxdvzqtzhuhksc4dx62gpmkq

On Spectral Learning of Mixtures of Distributions [chapter]

Dimitris Achlioptas, Frank McSherry
2005 Lecture Notes in Computer Science  
This second term is very small for many distributions, including Gaussians, Log-concave, and many others.  ...  We consider the problem of learning mixtures of distributions via spectral methods and derive a tight characterization of when such methods are useful.  ...  Theorem 8 Given a mixture of k Log-concave distributions with parameters {(µ i , σ i , w i )} assume that for some fixed n k(d(log d) 5 + log k)/w min the following holds: ∀i ∀j : µ i − µ j ≥ 4σ i (1/w  ... 
doi:10.1007/11503415_31 fatcat:73dbu4fv4jdqzdhx2gwhejlvrq
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