论文标题
高斯完全单调猜想的重新制定：沿热流的渔民信息上的霍奇结构
A Reformulation of Gaussian Completely Monotone Conjecture: A Hodge Structure on the Fisher Information along Heat Flow
论文作者
论文摘要
在过去的十年中，J。Huh解决了在组合学中的对数凸口序列上的几个长期开放问题。这些作品中开发的开创性技术来自代数几何：“我们相信，在自然界中出现的任何对数孔序列的序列背后，都有这样的hodge结构负责对数covavity造成对数con的结构”。 如果其衍生物在符号中交替，则称为完全单调的函数。例如，$ e^{ - t} $。 A fundamental conjecture in mathematical physics and Shannon information theory is on the complete monotonicity of Gaussian distribution (GCMC), which states that $I(X+Z_t)$\footnote{The probability density function of $X+Z_t$ is called "heat flow" in mathematical physics.} is completely monotone in $t$, where $I$ is Fisher information, random variables $X$ and $ z_t $是独立的，$ z_t \ sim \ mathcal {n}（0，t）$是高斯。 受到J. Huh引入的代数几何方法的启发，GCMC以对数串联序列的形式重新制定。通常，完全单调的函数可以允许log-convex序列，并且对数符号序列可以进一步诱导log-conconcave序列。新的配方可以将GCMC引导到奇妙的代数几何神庙。此外，要使研究人员从信息理论和数学\脚注{作者不熟悉代数几何形状。该论文还旨在为人们提供有关GCMC历史和应用中GCMC历史的必要背景信息的信息理论。}以及一些新发现，详细介绍了起源，对GCMC的含义和进一步的研究。
In the past decade, J. Huh solved several long-standing open problems on log-concave sequences in combinatorics. 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 $I(X+Z_t)$\footnote{The probability density function of $X+Z_t$ is called "heat flow" in mathematical physics.} is completely monotone in $t$, where $I$ is Fisher information, random variables $X$ and $Z_t$ are independent and $Z_t\sim\mathcal{N}(0,t)$ 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\footnote{The author was not familiar with algebraic geometry. The paper is also aimed at providing people outside information theory of necessary background on the history of GCMC in theory and application.}, together with some new findings, a thorough summary of the origin, the implication and further study on GCMC is presented.
