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Geometry of Log-Concave Density Estimation
[article]

2022
*
arXiv
*
pre-print

We focus on

arXiv:1704.01910v2
fatcat:meyzmvmdorannekkqzfryabfge
*densities*on ℝ^d that are*log*-*concave*, and we study geometric properties*of*the maximum likelihood*estimator*(MLE) for weighted samples. ... Cule, Samworth, and Stewart showed that the logarithm*of*the optimal*log*-*concave**density*is piecewise linear and supported on a regular subdivision*of*the samples. ... Understand*log*-*concave**density**estimation*as a parametric optimization problem. ...##
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An Efficient Algorithm for High-Dimensional Log-Concave Maximum Likelihood
[article]

2018
*
arXiv
*
pre-print

The

arXiv:1811.03204v1
fatcat:gqyyb6vrcfd4pevm22i6e6w3zy
*log*-*concave*maximum likelihood*estimator*(MLE) problem answers: for a set*of*points $X_1,...X_n \in \mathbb R^d$, which*log*-*concave**density*maximizes their likelihood? ... We present a characterization*of*the*log*-*concave*MLE that leads to an algorithm with runtime $poly(n,d, \frac 1 \epsilon,r)$ to compute a*log*-*concave*distribution whose*log*-likelihood is at most $\epsilon ... Later work characterized the finite sample complexity*of**log*-*concave**density**estimation*. ...##
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Entropy and the hyperplane conjecture in convex geometry

2010
*
2010 IEEE International Symposium on Information Theory
*

It is also shown that the entropy per coordinate in a

doi:10.1109/isit.2010.5513619
dblp:conf/isit/BobkovM10
fatcat:k6uvkmqxwraytkastxf4yy2h4u
*log*-*concave*random vector*of*any dimension with given*density*at the mode has a range*of*just 1. ... Specifically, the hyperplane conjecture is shown to be equivalent to the assertion that all*log*-*concave*probability measures are at most a bounded distance away from Gaussianity, where distance is measured ...*Log**concavity*has been deeply studied in probability, statistics, optimization and*geometry*, and there are a number*of*results that show that*log*-*concave*random vectors resemble Gaussian random vectors ...##
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Concentration phenomena in high dimensional geometry

2014
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ESAIM: Proceedings and Surveys
*

The purpose

doi:10.1051/proc/201444002
fatcat:e6rjdgxmb5gs7nn4ltpayksgce
*of*this note is to present several aspects*of*concentration phenomena in high dimensional*geometry*. ... The topic has a broad audience going from algorithmic convex*geometry*to random matrices. We have tried to emphasize different problems relating these areas*of*research. ... Convex*geometry*and*log*-*concave*measures A function f : R n → R + is said to be*log*-*concave*if ∀x, y ∈ R n , ∀θ ∈ [0, 1], f ((1 − θ)x + θy) ≥ f (x) 1−θ f (y) θ Define a measure µ with a*log*-*concave**density*...##
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LogConcDEAD: AnRPackage for Maximum Likelihood Estimation of a Multivariate Log-Concave Density

2009
*
Journal of Statistical Software
*

Its main function is to compute the nonparametric maximum likelihood

doi:10.18637/jss.v029.i02
fatcat:3kclwxjjlfeolas2yu7kwrguqa
*estimator**of*a*log*-*concave**density*. ... In this document we introduce the R package LogConcDEAD (*Log*-*concave**density**estimation*in arbitrary dimensions). ... Here we can clearly see the structure*of*the*log*-*concave**density**estimate*. ...##
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Concentration phenomena in high dimensional geometry
[article]

2013
*
arXiv
*
pre-print

The purpose

arXiv:1310.1204v1
fatcat:ksk33kmwangqtf6rfxrnksawem
*of*this note is to present several aspects*of*concentration phenomena in high dimensional*geometry*. ... The topic has a broad audience going from algorithmic convex*geometry*to random matrices. We have tried to emphasize different problems relating these areas*of*research. ... Let P be an isotropic probability on R n with*log*-*concave**density*. Then for every t ≥ 10, P(|X| 2 ≥ t √ n) ≤ e −ct √ n . ...##
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Geometry of log-concave ensembles of random matrices and approximate reconstruction

2011
*
Comptes rendus. Mathematique
*

This parameter is

doi:10.1016/j.crma.2011.06.025
fatcat:6sl3cpb76bekjh3ay4xqz3fejm
*estimated*by means*of*new tail*estimates**of*order statistics and deviation inequalities for norms*of*projections*of*an isotropic*log*-*concave*vector. ... We study the Restricted Isometry Property*of*a random matrix Γ with independent isotropic*log*-*concave*rows. ... Recall that a random vector is isotropic*log*-*concave*if it is centered, its covariance matrix is the identity and its distribution has a*log*-*concave**density*. ...##
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Maximum likelihood estimation of a multidimensional log-concave density
[article]

2008
*
arXiv
*
pre-print

An R version

arXiv:0804.3989v1
fatcat:regy5akte5gr7cphljvvuvfrzy
*of*the algorithm is available in the package LogConcDEAD --*Log*-*Concave**Density**Estimation*in Arbitrary Dimensions. ... ., X_n be independent and identically distributed random vectors with a*log*-*concave*(Lebesgue)*density*f. ... For the case*of*highest*density*regions, the relative performance*of*the*log*-*concave**estimator*is better for the*estimation**of*smaller*density*regions. ...##
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Dimensional behaviour of entropy and information

2011
*
Comptes rendus. Mathematique
*

results for

doi:10.1016/j.crma.2011.01.008
fatcat:qaanhubpvbafbebjdn4kw5hiyu
*log*-*concave*measures, an entropic formulation*of*the hyperplane conjecture, and a new reverse entropy power inequality for*log*-*concave*measures analogous to V. ... We develop an information-theoretic perspective on some questions in convex*geometry*, providing for instance a new equipartition property for*log*-*concave*probability measures, some Gaussian comparison ... Let X and Y have*log*-*concave**densities*. Due to homogeneity*of*Theorem 1.1, assume without loss*of*generality that f ∞ 1 and g ∞ 1. ...##
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Learning Multivariate Log-concave Distributions
[article]

2017
*
arXiv
*
pre-print

We study the problem

arXiv:1605.08188v2
fatcat:f6lgwwk5ezbavnf4lngkd5etva
*of**estimating*multivariate*log*-*concave*probability*density*functions. We prove the first sample complexity upper bound for learning*log*-*concave**densities*on R^d, for all d ≥ 1. ... In more detail, we give an*estimator*that, for any d > 1 and ϵ>0, draws Õ_d ( (1/ϵ)^(d+5)/2) samples from an unknown target*log*-*concave**density*on R^d, and outputs a hypothesis that (with high probability ... During the past decade,*density**estimation**of**log*-*concave**densities*has been extensively investigated. ...##
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Maximum likelihood estimation of a multi-dimensional log-concave density

2010
*
Journal of The Royal Statistical Society Series B-statistical Methodology
*

An R version

doi:10.1111/j.1467-9868.2010.00753.x
fatcat:wtuitpjvvjhn7lkvmvneomi2ra
*of*the algorithm is available in the package LogConcDEAD-*log*-*concave**density**estimation*in arbitrary dimensions. ... We demonstrate that the*estimator*has attractive theoretical properties both when the true*density*is*log*-*concave*and when this model is misspecified. ... Finally, we record our gratitude to the Research Section for their handling*of*the paper, and the Royal Statistical Society for organizing the Ordinary Meeting. ...##
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Bimonotone subdivisions of point configurations in the plane

2021
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Algebraic Statistics
*

They correspond to statistical

doi:10.2140/astat.2021.12.125
fatcat:5lffatzgfvhfdjxqwbx4fz7dc4
*estimates**of*probability distributions*of*strongly positively dependent random variables. ... The number*of*bimonotone subdivisions compared to the total number*of*subdivisions*of*a point configuration provides insight into how often the random variables are positively dependent. ... At the time Robeva was supported by an NSF postdoctoral fellowship (DMS-170-3821) and is currently supported by a National Sciences and Engineering Research Council*of*Canada Discovery Grant (DGECR-2020 ...##
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Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures

2012
*
Journal of Functional Analysis
*

The specialization

doi:10.1016/j.jfa.2012.01.011
fatcat:nhmg3jx5hrcenoczist55dgr3i
*of*this inequality to*log*-*concave*measures may be seen as a version*of*Milman's reverse Brunn-Minkowski inequality. ... The proof relies on a demonstration*of*new relationships between the entropy*of*high dimensional random vectors and the volume*of*convex bodies, and on a study*of*effective supports*of*convex measures, ... Ball for fleshing out our understanding*of*the history*of*Corollary 4.2 (discussed in Section 4). ...##
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A Polynomial Time Algorithm for Maximum Likelihood Estimation of Multivariate Log-concave Densities
[article]

2018
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arXiv
*
pre-print

We study the problem

arXiv:1812.05524v1
fatcat:sbkr4sgs5fdkxcloutjrzxz2ym
*of*computing the maximum likelihood*estimator*(MLE)*of*multivariate*log*-*concave**densities*. Our main result is the first computationally efficient algorithm for this problem. ... To achieve this, we rely on structural results on approximation*of**log*-*concave**densities*and leverage classical algorithmic tools on volume approximation*of*convex bodies and uniform sampling from convex ... Introduction This paper is concerned with the problem*of*computing the maximum likelihood*estimator**of*multivariate*log*-*concave**densities*. ...##
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Revisiting maximum-a-posteriori estimation in log-concave models
[article]

2019
*
arXiv
*
pre-print

These results provide a new understanding

arXiv:1612.06149v4
fatcat:5ft576m3pbff3oyiewgzsyuq5a
*of*MAP and MMSE*estimation*in*log*-*concave*settings, and*of*the multiple roles that convex*geometry*plays in imaging problems. ... For*log*-*concave*models, this canonical loss is the Bregman divergence associated with the negative*log*posterior*density*. ... Acknowledgements Part*of*this work was conducted when the author held a Marie Curie Intra-European Research Fellowship for Career Development at the University*of*Bristol, and part when he was a visiting ...
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