Wednesday, September 28, 2022

2022-10-05 / 15:00 ~ 16:00

by 최인혁(KAIST)

This series of talks is intended to be a gentle introduction to the random walk theory on infinite groups and hyperbolic spaces. We will touch upon keywords including hyperbolicity, stationary measure, boundaries and limit laws. Those who are interested in geometric group theory or random walks are welcomed to join.

2022-09-30 / 15:00 ~ 16:00

학과 세미나/콜로퀴엄 - 기타:

by 최인혁(KAIST)

2022-09-29 / 11:50 ~ 12:30

대학원생 세미나 - 대학원생 세미나: Kernel methods for radial transformed compositional data with many zeros

by 박준영(KAIST)

Compositional data analysis with a high proportion of zeros has gained increasing popularity, especially in chemometrics and human gut microbiomes research. Statistical analyses of this type of data are typically carried out via a log-ratio transformation after replacing zeros with small positive values. We should note, however, that this procedure is geometrically improper, as it causes anomalous distortions through the transformation. We propose a radial transformation that does not require zero substitutions and more importantly results in essential equivalence between domains before and after the transformation. We show that a rich class of kernels on hyperspheres can successfully define a kernel embedding for compositional data based on this equivalence. The applicability of the proposed approach is demonstrated with kernel principal component analysis.

2022-10-04 / 16:30 ~ 17:30

IBS-KAIST 세미나 - 이산수학: On the degenerate Turán problems

by Zixiang Xu(IBS 극단조합및확률그룹)

For a graph $F$, the Turán number is the maximum number of edges in an $n$-vertex simple graph not containing $F$. The celebrated Erdős-Stone-Simonovits Theorem gives that \[ \text{ex}(n,F)=\bigg(1-\frac{1}{\chi(F)-1}+o(1)\bigg)\binom{n}{2},\] where $\chi(F)$ is the chromatic number of $H$. This theorem asymptotically solves the problem when $\chi(F)\geqslant 3$. In case of bipartite graphs $F$, not even the order of magnitude is known in general. In this talk, I will introduce some recent progress on Turán numbers of bipartite graphs and related generalizations and discuss several methods developed in recent years. Finally, I will introduce some interesting open problems on this topic.

2022-09-29 / 16:15 ~ 17:15

학과 세미나/콜로퀴엄 - 콜로퀴엄:

by ()

We consider a deep generative model for nonparametric distribution estimation problems. The true data-generating distribution is assumed to possess a certain low-dimensional structure. Under this assumption, we study convergence rates of estimators obtained by likelihood approaches and generative adversarial networks (GAN). The convergence rate depends only on the noise level, intrinsic dimension and smoothness of the underlying structure. The true distribution may or may not possess the Lebesgue density, depending on the underlying structure. For the singular case (no Lebesgue density), the convergence rate of GAN is strictly better than that of the likelihood approaches. Our lower bound of the minimax optimal rates shows that the convergence rate of GAN is close to the optimal rate. If the true distribution allows a smooth Lebesgue density, an estimator obtained by a likelihood approach achieves the minimax optimal rate.

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