Log-Concavity and Strong Log-Concavity: a review

By Adrien Saumard, Jon A. Wellner. 2014

330 citations recorded by the source corpus.

We review and formulate results concerning log-concavity and strong-log-concavity in both discrete and continuous settings. We show how preservation of log-concavity and strongly log-concavity on ℝ under convolution follows from a fundamental monotonicity result of Efron (1969). We provide a new proof of Efron's theorem using the recent asymmetric Brascamp-Lieb inequality due to Otto and Menz (2013). Along the way we review connections between log-concavity and other areas of mathematics and statistics, including concentration of measure, log-Sobolev inequalities, convex geometry, MCMC algorithms, Laplace approximations, and machine learning.

Subfields: Artificial Intelligence, Statistics and Probability

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Citations in the corpus

A deterministic and computable Bernstein-von Mises theorem (2019, background): “Furthermore, in the large-data limit, it is known that the posterior distribution becomes simple: it tends to be almost equal to its Laplace approximation (with total variation error typically scaling as OP ( n−1/2 ) if the dimensionality p is fixed), a result known as the Bernstein-von Mises theore”
High-dimensional central limit theorems by Stein’s method (2020, background): “Recall that a probability measure μ on Rd has a log-concave density if it is supported on (the closure of) an open convex set ⊂ Rd and, on , it has a density of the form e−V with V : →R a convex (hence continuous) function; see Saumard and Wellner (2014) for more details.”
Recent Progress in Log-Concave Density Estimation (2017, background): “For alternative reviews of related topics, see Saumard and Wellner (2014), which has a greater emphasis on analytic properties, and Walther (2009), with a stronger focus on modelling and applications.”
The non-convex geometry of low-rank matrix optimization (2016, background): “Therefore, in addition to low-rank matrix recovery problems [24, 25, 36, 53, 58], it is also applicable to many other low-rank matrix optimization problems with non-quadratic objective functions, including 1-bit matrix recovery, robust PCA [24], and low-rank matrix recovery with non-Gaussian noise [”
Theoretical guarantees for approximate sampling from smooth and log‐concave densities (2014, background): “We refer the interested reader to Saumard and Wellner (2014) for a comprehensive survey on log-concave densities.”
Air traffic flow management under uncertainty using chance-constrained optimization (2017, methodology): “The required assumptions are imposed, based on the log-concavity of the distribution presented in Saumard and Wellner (2014) .”
COMBINATORIAL APPLICATIONS OF THE HODGE–RIEMANN RELATIONS (2017, background): “For an extensive discussion of log-concavity and its applications in probability and statistics, see [12, 39, 48].”
Efficient Robust Proper Learning of Log-concave Distributions (2016, background): “Log-concave distributions have been studied in a wide range of different contexts including economics [An95], statistics and probability theory (see [SW14] for a recent survey), theoretical computer science [LV07], and algebra, combinatorics and geometry [Sta89].”
Computing the quality of the Laplace approximation (2017, background): “A more realistic assumption consists in assuming that ˜ f ( θ ) is log-concave, i.e: Hφ f ( θ ) > 0 (see Saumard and Wellner [2014] for a review of the properties of log-concave densities).”
An Almost Constant Lower Bound of the Isoperimetric Coefficient in the KLS Conjecture (2020, neutral): “[20]).”