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Geometry of Log-Concave Density Estimation [article]

Elina Robeva, Bernd Sturmfels, Caroline Uhler
2022 arXiv   pre-print
We focus on 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.  ... 
arXiv:1704.01910v2 fatcat:meyzmvmdorannekkqzfryabfge

An Efficient Algorithm for High-Dimensional Log-Concave Maximum Likelihood [article]

Brian Axelrod, Gregory Valiant
2018 arXiv   pre-print
The 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.  ... 
arXiv:1811.03204v1 fatcat:gqyyb6vrcfd4pevm22i6e6w3zy

Entropy and the hyperplane conjecture in convex geometry

Sergey Bobkov, Mokshay Madiman
2010 2010 IEEE International Symposium on Information Theory  
It is also shown that the entropy per coordinate in a 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  ... 
doi:10.1109/isit.2010.5513619 dblp:conf/isit/BobkovM10 fatcat:k6uvkmqxwraytkastxf4yy2h4u

Concentration phenomena in high dimensional geometry

Olivier Guédon, Arnaud Guillin
2014 ESAIM: Proceedings and Surveys  
The purpose 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  ... 
doi:10.1051/proc/201444002 fatcat:e6rjdgxmb5gs7nn4ltpayksgce

LogConcDEAD: AnRPackage for Maximum Likelihood Estimation of a Multivariate Log-Concave Density

Madeleine Cule, Robert Gramacy, Richard Samworth
2009 Journal of Statistical Software  
Its main function is to compute the nonparametric maximum likelihood 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.  ... 
doi:10.18637/jss.v029.i02 fatcat:3kclwxjjlfeolas2yu7kwrguqa

Concentration phenomena in high dimensional geometry [article]

Olivier Guédon
2013 arXiv   pre-print
The purpose 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 .  ... 
arXiv:1310.1204v1 fatcat:ksk33kmwangqtf6rfxrnksawem

Geometry of log-concave ensembles of random matrices and approximate reconstruction

Radosław Adamczak, Rafał Latała, Alexander E. Litvak, Alain Pajor, Nicole Tomczak-Jaegermann
2011 Comptes rendus. Mathematique  
This parameter is 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.  ... 
doi:10.1016/j.crma.2011.06.025 fatcat:6sl3cpb76bekjh3ay4xqz3fejm

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

Madeleine Cule, Richard Samworth, Michael Stewart
2008 arXiv   pre-print
An R version 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.  ... 
arXiv:0804.3989v1 fatcat:regy5akte5gr7cphljvvuvfrzy

Dimensional behaviour of entropy and information

Sergey Bobkov, Mokshay Madiman
2011 Comptes rendus. Mathematique  
results for 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.  ... 
doi:10.1016/j.crma.2011.01.008 fatcat:qaanhubpvbafbebjdn4kw5hiyu

Learning Multivariate Log-concave Distributions [article]

Ilias Diakonikolas, Daniel M. Kane, Alistair Stewart
2017 arXiv   pre-print
We study the problem 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.  ... 
arXiv:1605.08188v2 fatcat:f6lgwwk5ezbavnf4lngkd5etva

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  
An R version 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.  ... 
doi:10.1111/j.1467-9868.2010.00753.x fatcat:wtuitpjvvjhn7lkvmvneomi2ra

Bimonotone subdivisions of point configurations in the plane

Elina Robeva, Melinda Sun
2021 Algebraic Statistics  
They correspond to statistical 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  ... 
doi:10.2140/astat.2021.12.125 fatcat:5lffatzgfvhfdjxqwbx4fz7dc4

Reverse Brunn–Minkowski and reverse entropy power inequalities for convex measures

Sergey Bobkov, Mokshay Madiman
2012 Journal of Functional Analysis  
The specialization 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).  ... 
doi:10.1016/j.jfa.2012.01.011 fatcat:nhmg3jx5hrcenoczist55dgr3i

A Polynomial Time Algorithm for Maximum Likelihood Estimation of Multivariate Log-concave Densities [article]

Ilias Diakonikolas, Anastasios Sidiropoulos, Alistair Stewart
2018 arXiv   pre-print
We study the problem 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.  ... 
arXiv:1812.05524v1 fatcat:sbkr4sgs5fdkxcloutjrzxz2ym

Revisiting maximum-a-posteriori estimation in log-concave models [article]

Marcelo Pereyra
2019 arXiv   pre-print
These results provide a new understanding 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  ... 
arXiv:1612.06149v4 fatcat:5ft576m3pbff3oyiewgzsyuq5a
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