Mathematical Analysis Team (https://aip.riken.jp/labs/generic_tech/mathematical-analysis/) at RIKEN AIP
Speaker 1: Shin-ichi Ohta (15:00-15:40)
Title: A new transport distance on hypergraphs
I first briefly explain the activity of Mathematical Analysis Team. Then I talk about a recent work by Tomoya Akamatsu (Research Part-timer) on a new transport distance on hypergraphs. The construction of this transport distance was inspired by structured optimal transport by Alvarez-Melis et al, but is based on a different idea concentrating on capturing the structure of hyperedges. This transport distance turned out new also for usual graphs, and one can study the corresponding Ricci curvature a la Ollivier and Lin-Lu-Yau.
Speaker 2: Masayuki Aino (15:40-16:20)
Title: Self-tuning Laplacian eigenmaps and the conformal metric compatible with the measure
We introduce a theoretical analysis of the self-tuning Laplacian eigenmaps using k-NN graph and their spectral convergence to the Laplacian of the conformal metric compatible with the measure from which the sample points are taken.
Speaker 3: Taiji Marugame (16:20-17:00)
Title: The Bonnet theorem for statistical manifolds
In information geometry, spaces of probability distributions are endowed with a geometric structure called the statistical structure. A fundamental question in information geometry is when a statistical manifold can be embedded to a flat statistical manifold. An answer to this question was given by H. V. Le, who proved an analogue of the Nash embedding theorem for statistical manifolds. In this talk, as another embedding theorem, we present a Bonnet-type theorem which asserts that if a statistical manifold admits tensors satisfying the Gauss-Codazzi-Ricci equations, then it is locally embeddable to a flat statistical manifold with a fixed dimension.