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Clustering with mixtures of log-concave distributions
2007
Computational Statistics & Data Analysis
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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
2009
Statistical Science
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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]
2017
arXiv
pre-print
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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
2018
Bernoulli
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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]
2018
arXiv
pre-print
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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]
2019
arXiv
pre-print
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2017 pre-print arXiv:1703.10534v2 fatcat:yq7jcao6jnc6ndapwhxraskicqOther Versions
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]
2013
Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
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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]
2012
arXiv
pre-print
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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
2017
Computational Statistics & Data Analysis
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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]
2018
arXiv
pre-print
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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]
2008
arXiv
pre-print
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., 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
2010
Journal of The Royal Statistical Society Series B-statistical Methodology
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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
2016
Computational Statistics & Data Analysis
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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]
2018
arXiv
pre-print
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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]
2005
Lecture Notes in Computer Science
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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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