TBA

Asymptoticswith consideration of ultra-high dimensional data must consider an increasing number of variables, i.e., dimensions, rather than growing the number of observations. High-dimensional asymptotic studies have revealed some unexpected characteristics of data with an exceedingly large number of variables, such as gene expressions. In the context of binary classification, i.e., supervised learning with dichotomous labels, data piling refers to the phenomenon that training data vectors from each class project to a single point for classification. This interesting phenomenon has been a key to understanding many distinctive properties of high-dimensional discrimination. In this talk, high-dimensional asymptoticsof data piling is investigated under equal covariance assumption, which reveals its close connection to the well-known ridged linear classifier. In particular, we show that a negatively ridged discriminant vector can asymptotically achieve data piling of independent test data, essentially yielding a perfect classification. Double data pilingis generalized to heterogeneous covariance models and we propose a data-splitting approach to estimate the direction for the second data piling of test data.

TBA

수학모델은 역사적으로 오래 전부터 다양한 자연현상과 사회현상을 이해하기 위해 고안되어 왔고 순수와 응용에 걸쳐 활발하게 연구되어 온 분야 중 하나이다. 데이터는 4차 산업혁명의 핵심분야인 인공지능과 머신러닝, IoT에 필수 아이템이며 실제 산업이나 일상 속에서 무수히 쏟아져 나오는 핵심 자원이다. 그렇다면 수학모델을 다루는 수학자와 공학자의 시각에서나 데이터를 다루어야 하는 실제 산업현장에서는 이 두 가지가 만났을 때의 시너지를 상상해 볼 수 있을 것이다. 본 강연에서는 데이터와 수학모델이 공존할 수 있을지, 그리고 공존 가능하다면 인공지능분야에서 수학이 풀어야 하는 숙제를, 나아가 다양한 현장의 문제를 해결할 수 있는 중요한 단초를 제공하는 데이터와 수학모델의 공존법을 소개하려고 한다.

Random simple and multigraph models based on exchangeable random measures, sometimes named graphex processes or generalised graphon models, have recently been proposed as a flexible class of sparse random graph models. This class of models can be seen as a generalisation of the popular graphon models. I will present this class of models, discuss some of their asymptotic properties, in particular the asymptotic behaviour of the degree distribution and of the clustering coefficients. I will also present some particular parametric models within this class and their use for discovering latent communities in sparse real-world networks.