arXiv subjects

Asuka Takatsu

Publications and source records attributed to Asuka Takatsu.

32 records · Page 2Linked to original sources

arXiv2020

New characterizations of log-concavity

We introduce a notion of $F$-concavity which largely generalizes the usual concavity. By the use of the notions of closedness under positive scalar multiplication and closedness under positive exponentiation we characterize power concavity and power log-concavity among nontrivial $F$-concavities, respectively. In particular, we have a characterization of log-concavity as the only $F$-concavity which is closed both under positive scalar multiplication and positive exponentiation. Furthermore, we discuss the strongest $F$-concavity preserved by the Dirichlet heat flow, characterizing log-concavity also in this connection.

math.AP↗

arXiv2020

Power concavity for elliptic and parabolic boundary value problems on rotationally symmetric domains

We study power concavity of rotationally symmetric solutions to elliptic and parabolic boundary value problems on rotationally symmetric domains in Riemannian manifolds. As applications of our results to the hyperbolic space ${\bf H}^N$ we have: $\bullet$ The first Dirichlet eigenfunction on a ball in ${\bf H}^N$ is strictly positive power concave; $\bullet$ Let $Γ$ be the heat kernel on ${\bf H}^N$. Then $Γ(\cdot,y,t)$ is strictly log-concave on ${\bf H}^N$ for $y\in {\bf H}^N$ and $t>0$.

math.AP↗

arXiv2020

Gauge freedom of entropies on $q$-Gaussian measures

A $q$-Gaussian measure is a generalization of a Gaussian measure. This generalization is obtained by replacing the exponential function with the power function of exponent $1/(1-q)$ ($q\neq 1$). The limit case $q=1$ recovers a Gaussian measure. For $1\leq q <3$, the set of all $q$-Gaussian densities over the real line satisfies a certain regularity condition to define information geometric structures such as an entropy and a relative entropy via escort expectations. The ordinary expectation of a random variable is the integral of the random variable with respect to its law. Escort expectations admit us to replace the law to any other measures. A choice of escort expectations on the set of all $q$-Gaussian densities determines an entropy and a relative entropy. One of most important escort expectations on the set of all $q$-Gaussian densities is the $q$-escort expectation since this escort expectation determines the Tsallis entropy and the Tsallis relative entropy. The phenomenon gauge freedom of entropies is that different escort expectations determine the same entropy, but different relative entropies. In this note, we first introduce a refinement of the $q$-logarithmic function. Then we demonstrate the phenomenon on an open set of all $q$-Gaussian densities over the real line by using the refined $q$-logarithmic functions. We write down the corresponding Riemannian metric.

math.QA↗

arXiv2019

Equality in the logarithmic Sobolev inequality

We investigate the rigidity problem for the logarithmic Sobolev inequality on weighted Riemannian manifolds satisfying $\mathrm{Ric}_{\infty} \ge K>0$. Assuming equality holds, we show that the $1$-dimensional Gaussian space is necessarily split off, similarly to the rigidity results of Cheng--Zhou on the spectral gap as well as Morgan on the isoperimetric inequality. The key ingredient of the proof is the needle decomposition method introduced on Riemannian manifolds by Klartag. We also present several related open problems.

math.DG↗

arXiv2019

To logconcavity and beyond

In 1976 Brascamp and Lieb proved that the heat flow preserves logconcavity. In this paper, introducing a variation of concavity, we show that it preserves in fact a stronger property than logconcavity and we identify the strongest concavity preserved by the heat flow.

math.AP↗

arXiv2018

Convergence of combinatorial Ricci flows to degenerate circle patterns

We investigate the combinatorial Ricci flow on a surface of nonpositive Euler characteristic when the necessary and sufficient condition for the convergence of the combinatorial Ricci flow is not valid. This observation addresses one of questions raised by B. Chow and F. Luo.

math.DG↗

arXiv2015

High-dimensional metric-measure limit of Stiefel and Grassmann manifolds

We study the high-dimensional limit of (projective) Stiefel and Grassmann manifolds as metric measure spaces in Gromov's topology. The limits are either the infinite-dimensional Gaussian space or its quotient by an mm-isomorphic group action, which are drastically different from the manifolds. As a corollary, we obtain some asymptotic estimates of the observable diameter of (projective) Stiefel and Grassmann manifolds.

math.MG↗

arXiv2013

Displacement convexity of generalized relative entropies.II

We introduce a class of generalized relative entropies (inspired by the Bregman divergence in information theory) on the Wasserstein space over a weighted Riemannian or Finsler manifold. We prove that the convexity of all the entropies in this class is equivalent to the combination of the nonnegative weighted Ricci curvature and the convexity of another weight function used in the definition of the generalized relative entropies. This convexity condition corresponds to Lott and Villani's version of the curvature-dimension condition. As applications, we obtain appropriate variants of the Talagrand, HWI and logarithmic Sobolev inequalities, as well as the concentration of measures. We also investigate the gradient flow of our generalized relative entropy.

math.DG↗

arXiv2012

Isoperimetric profile of radial probability measures on Euclidean spaces

We derive the isoperimetric profile of Gaussian type for an absolutely continuous probability measure on Euclidean spaces with respect to the Lebesgue measure, whose density is a radial function.The key is a generalization of the Poincaré limit which asserts that the $n$-dimensional Gaussian measure is approximated by the projections of the uniform probability measure on the Euclidean sphere of appropriate radius to the first $n$-coordinates as the dimension diverges to infinity. The generalization is done by replacing the projections with certain maps.

math.PR↗

arXiv2011

Equivalence of two orthogonalities between probability measures

Given any two probability measures on a Euclidean space with mean 0 and finite variance, we demonstrate that the two probability measures are orthogonal in the sense of Wasserstein geometry if and only if the two spaces by spanned by the supports of each probability measure are orthogonal.

math.PR↗

arXiv2011

Behaviors of $ϕ$-exponential distributions in Wasserstein geometry and an evolution equation

A $ϕ$-exponential distribution is a generalization of an exponential distribution associated to functions $ϕ$ in an appropriate class, and the space of $ϕ$-exponential distributions has a dually flat structure. We study features of the space of $ϕ$-exponential distributions, such as the convexity in Wasserstein geometry and the stability under an evolution equation. From this study, we provide the new characterizations to the space of Gaussian measures and the space of $q$-Gaussian measures.

math.MG↗

arXiv2011

Displacement convexity of generalized relative entropies

We investigate the $m$-relative entropy, which stems from the Bregman divergence, on weighted Riemannian and Finsler manifolds. We prove that the displacement $K$-convexity of the $m$-relative entropy is equivalent to the combination of the nonnegativity of the weighted Ricci curvature and the $K$-convexity of the weight function. We use this to show appropriate variants of the Talagrand, HWI and the logarithmic Sobolev inequalities, as well as the concentration of measures. We also prove that the gradient flow of the $m$-relative entropy produces a solution to the porous medium equation or the fast diffusion equation.

math.DG↗

arXiv2009

Cone structure of $L^2$-Wasserstein spaces

The purpose of this paper is to understand the geometric structure of the $L^2$-Wasserstein space $\pp$ over the Euclidean space.For this sake, we focus on its cone structure.One of our main results is that the $L^2$-Wasserstein space over a Polish space has a cone structure if and only if so does the underlying space.In particular, $\pp$ turns out to have a cone structure.It is also shown that $\pp$ splits $\R^d$ isometrically but not $\R^{d+1}$.

math.MG↗

arXiv2009

On Wasserstein geometry of the space of Gaussian measures

The space of Gaussian measures on a Euclidean space is geodesically convex in the $L^2$-Wasserstein space. This space is a finite dimensional manifold since Gaussian measures are parameterized by means and covariance matrices. By restricting to the space of Gaussian measures inside the $L^2$-Wasserstein space, we manage to provide detailed descriptions of the $L^2$-Wasserstein geometry from a Riemannian geometric viewpoint. We first construct a Riemannian metric which induces the $L^2$-Wasserstein distance. Then we obtain a formula for the sectional curvatures of the space of Gaussian measures, which is written out in terms of the eigenvalues of the covariance matrix.

math.DG↗