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We are planning a Mathematical Seminar on
July 13, 2020 (Monday) 10:00-11:30, 13:00-14:30, and 15:00-16:30
via teleconferencing.
The titles and abstracts are attached at the end of this email.
These talks will be in English.
If you are interested, please register at the link below so we can send you the teleconferencing details.
https://forms.gle/RtC582Qkh2tB4Jq47
Best regards,
Mathematical Analysis Team, Mathematical Science Team, and Topological Data Analysis Team
Asuka TAKATSU (Tokyo Metropolitan University)
Title: New characterizations of log-concavity
Abstract:
A function on the real line is concave
if the line segment connecting any two points on the graph of the function
is never above the graph.
For example, a linear function is concave.
The density function of a Gaussian measure is not concave.
However, if we change the scale of the vertical axis of the graph into the
log-scale,
then the graph of the density function of a Gaussian measure is concave.
Thus the notion of concavity can be generalized by changing the scale of
the vertical axis.
A function is called log-concave if its log-scaled graph is concave.
In this talk, I characterize log-concavity as the only generalized concavity
which is closed both under positive scalar multiplication and positive
exponentiation.
This talk is based on a joint work with Kazuhiro Ishige and Paolo Salani.
Masayuki Aino, RIKEN Center for Advanced Intelligence Project (AIP)
Title: Convergence to the product of spheres and eigenvalues of the
Laplacian
Abstract: For Riemannian manifolds, the relationship between eigenvalues
of the Laplacian, the curvature and the shape of the manifold has long been
studied. In particular, Lichnerowicz gave an estimate of the first
eigenvalue of the Laplacian if the manifold is positively curved in the
sense of the Ricci curvature, and Obata showed the equality of the estimate
characterizes the sphere. Moreover, the almost equality case is well
studied, and it is known that the manifold is close to the sphere in the
Gromov-Hausdorff sense under some conditions. In this talk, we give a
Gromov-Hausdorff approximation to the product of spheres under some
conditions.
Kenji Fukumizu (The Institute of Statistical Mathematics)
Title: Smoothness and Stability in Learning Generative Adversarial Networks
Abstract:
It is known that generative adversarial networks (GANs) commonly display
unstable behavior during training. In this work, we develop a principled
theoretical framework for understanding the stability of various types of
GANs. In particular, we derive conditions that guarantee eventual
stationarity of the generator when it is trained with gradient descent,
conditions that must be satisfied by the divergence to be minimized and the
architecture of the generator. We find that existing GAN variants satisfy
some, but not all, of these conditions. Using tools from convex analysis,
optimal transport, and reproducing kernels, we construct a GAN that
fulfills these conditions simultaneously. In the derivation, we explain and
clarify the need for various existing GAN stabilization techniques,
including Lipschitz constraints, gradient penalties, and smooth activation
functions. This is a joint work with Casey Chu (Stanford) and Kentaro
Minami (Preferred Networks).
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