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Mathematical Seminar

Fri, 19 Feb 2021 10:00 - 16:30
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Description

We are planning a Mathematical Seminar via Zoom meeting.
The schedule is the following (See below for the details):

10:00-11:30 Koichi TOJO A method to construct q-exponential family by representation theory
13:00-14:30 Hayate SUDA Superdiffusion of energy in stochastic harmonic chains
15:00-16:30 Kei KOBAYASHI Application of metric cones to data analysis

These talks will be given in English.
Please join us if you are interested.


Talk 1 (10:00 - 11:30).

Speaker: Koichi TOJO (RIKEN AIP, Keio University)

Title: A method to construct q-exponential family by representation theory

Abstract: Exponential families play an important role in the field of information geometry
and statistics. To understand only widely used "good" exponential families such
as the family of normal distributions systematically, we introduced a method
to construct exponential families on homogeneous spaces by using representation theory in [TY18].
In this talk, we consider a generalization of this method for a q-exponential family,
which is a natural generalization of an exponential family.
We give a generalization of the method under some assumptions on homogeneous spaces,
and obtain q-exponential families (1<q<2) on the upper half plane with
explicit form including the normalizing constant.
This talk is based on a joint work with Taro Yoshino.

[TY18] K. Tojo, T. Yoshino, A method to construct exponential families by representation theory, arXiv:1811.01394


Talk 2 (13:00 - 14:30).

Speaker: Hayate SUDA (University of Tokyo)

Title : Superdiffusion of energy in stochastic harmonic chains

Abstract: Anomalous heat transport, which means the violation of
Fourier's law, has been observed in a large number of numerical
simulations for some one-dimensional systems. Stochastic harmonic chains
has been used as microscopic models for one-dimensional heat conduction.
In this talk, we review some mathematical results about derivation of
anomalous heat transport and corresponding superdiffusion of energy
from stochastic harmonic chains.


Talk 3 (15:00 - 16:30).
Speaker: Kei KOBAYASHI (Keio University)
Title: Application of metric cones to data analysis

Abstract: First I will summarize the contents of Kobayashi and Wynn (2020) with some
small facts discovered more recently.
In the paper, we proposed a method to perform data analysis (e.g.
clustering and time series analysis) after transforming the metric of the
data space using two parameters, α and β. The transformation of the
distance by parameter α is for a length space and can be implemented
approximately by exponentially transforming the length of each edge of the
empirical graph. The distance transformation by parameter β is for a
general metric space and the transformed distance corresponds to the
extrinsic distance after embedding the data space into a metric cone. We
proved both transformations monotonically change the curvature of the data
space, but in different ways.
Second, I will report on my recent work with Daisuke Takehara for
extracting hierarchical structures of network data using metric cones.

Kobayashi, K. and Wynn, H. (2020), Empirical geodesic graphs and CAT(k)
metrics for data analysis,* Statistics and Computing*, 30(1), 1-18

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