# A Reformulation of Gaussian Completely Monotone Conjecture: A Hodge Structure on the Fisher Information along Heat Flow

TL;DR: In this article , the complete monotonicity of Gaussian distribution (GCMC) was reformulated in the form of a log-convex sequence, which can admit a logconcave sequence.

Abstract: In the past decade, J. Huh solved several long-standing open problems on log-concave sequences in com-binatorics. The ground-breaking techniques developed in those work are from algebraic geometry: “We believe that behind any log-concave sequence that appears in nature there is such a Hodge structure responsible for the log-concavity”. A function is called completely monotone if its derivatives alternate in signs; e.g., e − t . A fundamental conjecture in mathematical physics and Shannon information theory is on the complete monotonicity of Gaussian distribution (GCMC), which states that I ( X + Z t ) 1 is completely monotone in t , where I is Fisher information, random variables X and Z t are independent and Z t ∼ N (0 , t ) is Gaussian. Inspired by the algebraic geometry method introduced by J. Huh, GCMC is reformulated in the form of a log-convex sequence. In general, a completely monotone function can admit a log-convex sequence and a log-convex sequence can further induce a log-concave sequence. The new formulation may guide GCMC to the marvelous temple of algebraic geometry. Moreover, to make GCMC more accessible to researchers from both information theory and mathematics 2 , together with some new ﬁndings, a thorough summary of the origin, the implication and further study on GCMC is presented.

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TL;DR: In this paper, a graded commutative algebra over the real numbers is constructed from a given combinatorial object M (a matroid), and the condition that the logarithms, log(ai), form a concave sequence is established.

Abstract: Introduction Logarithmic concavity is a property of a sequence of real numbers, occurring throughout algebraic geometry, convex geometry, and combinatorics. A sequence of positive numbers a0,... ,ad is log-concave if a2 i ≥ ai−1ai+1 for all i. This means that the logarithms, log(ai), form a concave sequence. The condition implies unimodality of the sequence (ai), a property easier to visualize: the sequence is unimodal if there is an index i such that a0 ≤ ⋯ ≤ ai−1 ≤ ai ≥ ai+1 ≥ ⋯ ≥ ad. We will discuss our work on establishing log-concavity of various combinatorial sequences, such as the coefficients of the chromatic polynomial of graphs and the face numbers of matroid complexes. Our method is motivated by complex algebraic geometry, in particular Hodge theory. From a given combinatorial object M (a matroid), we construct a graded commutative algebra over the real numbers

15 citations

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15 citations

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TL;DR: The generalized $H$ theorem for the Bobylev-Krook-Wu solution of the Boltzmann equation was verified for $nl~30$ in this article.

Abstract: The generalized $H$ theorem, ${(\ensuremath{-}\frac{d}{\mathrm{dt}})}^{n}Hg~0$, is verified for $nl~30$ for the Bobylev-Krook-Wu solution of the Boltzmann equation, and for $d$-dimensional generalizations of that solution, $1l~dl~6$.

13 citations

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TL;DR: A geometrical interpretation to the covariance-preserving transformation and study the concavity of h(√t X +√1 - t Z), revealing its connection with Costa's entropy power inequality are provided.

Abstract: Let $X$ be an arbitrary continuous random variable and $Z$ be an independent Gaussian random variable with zero mean and unit variance. For $t~>~0$, Costa proved that $e^{2h(X+\sqrt{t}Z)}$ is concave in $t$, where the proof hinged on the first and second order derivatives of $h(X+\sqrt{t}Z)$. Specifically, these two derivatives are signed, i.e., $\frac{\partial}{\partial t}h(X+\sqrt{t}Z) \geq 0$ and $\frac{\partial^2}{\partial t^2}h(X+\sqrt{t}Z) \leq 0$. In this paper, we show that the third order derivative of $h(X+\sqrt{t}Z)$ is nonnegative, which implies that the Fisher information $J(X+\sqrt{t}Z)$ is convex in $t$. We further show that the fourth order derivative of $h(X+\sqrt{t}Z)$ is nonpositive. Following the first four derivatives, we make two conjectures on $h(X+\sqrt{t}Z)$: the first is that $\frac{\partial^n}{\partial t^n} h(X+\sqrt{t}Z)$ is nonnegative in $t$ if $n$ is odd, and nonpositive otherwise; the second is that $\log J(X+\sqrt{t}Z)$ is convex in $t$. The first conjecture can be rephrased in the context of completely monotone functions: $J(X+\sqrt{t}Z)$ is completely monotone in $t$. The history of the first conjecture may date back to a problem in mathematical physics studied by McKean in 1966. Apart from these results, we provide a geometrical interpretation to the covariance-preserving transformation and study the concavity of $h(\sqrt{t}X+\sqrt{1-t}Z)$, revealing its connection with Costa's EPI.

5 citations

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07 May 2014TL;DR: It is shown that the third order derivative of h(X + √tZ) is nonnegative, which implies that the Fisher information J(X+ √z) is convex in t, and two conjectures are made on this result.

Abstract: Let X be an arbitrary continuous random variable and Z be an independent Gaussian random variable with zero mean and unit variance. In this paper, we show that the third order derivative of h(X + √tZ) is nonnegative, which implies that the Fisher information J(X+ √tZ) is convex in t. Following this result, we make two conjectures on h(X + √tZ): the first is that ∂
n
/∂t
n
h(X + √tZ) is nonnegative in t if n is odd, and negative otherwise; the second is that log J(X + √tZ) is convex in t.

4 citations