This talk presents new methods of solving machine learning problems using probability models. For classification problems, the classifier referred to as the class probability output network (CPON) which can provide accurate posterior probabilities for the soft classification decision, is proposed. In this model, the uncertainty of decision is defined using the accuracy of estimation. The deep structure of CPON is also presented to obtain the best classification performance for the given data. Applications of CPON models are also addressed.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701

While deep learning has many remarkable success stories, finding a satisfactory mathematical explanation on why it is so effective is still considered an open challenge. One recent promising direction for this challenge is to analyse the mathematical properties of neural networks in the limit where the widths of hidden layers of the networks go to infinity. Researchers were able to prove highly-nontrivial properties of such infinitely-wide neural networks, such as the gradient-based training achieving the zero training error (so that it finds a global optimum), and the typical random initialisation of those infinitely-wide networks making them so called Gaussian processes, which are well-studied random objects in machine learning, statistics, and probability theory. These theoretical findings also led to new algorithms based on so-called kernels, which sometimes outperform existing kernel-based algorithms. The purpose of this talk is to explain these recent theoretical results on infinitely wide neural networks. If time permits, I will briefly describe my work in this domain, which aims at developing a new neural-network architecture that has multiple nice theoretical properties in the infinite-width limit. This work is jointly pursued with Fadhel Ayed, Francois Caron, Paul Jung, Hoil Lee, and Juho Lee.

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심사위원장: 임미경 / 심사위원: 김용정, 신연종, 권기운(동국대학교), 이은정(연세대학교)

In this talk, we consider the problem of minimizing multi-modal loss functions with a large number of local optima. Since the local gradient points to the direction of the steepest slope in an infinitesimal neighborhood, an optimizer guided by the local gradient is often trapped in a local minimum. To address this issue, we develop a novel nonlocal gradient to skip small local minima by capturing major structures of the loss's landscape in black-box optimization. The nonlocal gradient is defined by a directional Gaussian smoothing (DGS) approach. The key idea is to conducts 1D long-range exploration with a large smoothing radius along orthogonal directions, each of which defines a nonlocal directional derivative as a 1D integral. Such long-range exploration enables the nonlocal gradient to skip small local minima. We use the Gauss-Hermite quadrature rule to approximate the d 1D integrals to obtain an accurate estimator. We also provide theoretical analysis on the convergence of the method on nonconvex landscape. In this work, we investigate the scenario where the objective function is composed of a convex function, perturbed by a highly oscillating, deterministic noise. We provide a convergence theory under which the iterates converge to a tightened neighborhood of the solution, whose size is characterized by the noise frequency. Furthermore, if the noise level decays to zero when approaching global minimum, we prove that the DGS optimization converges to the exact global minimum with linear rates, similarly to standard gradient-based method in optimizing convex functions. We complement our theoretical analysis with numerical experiments to illustrate the performance of this approach.

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심사위원장: 임보해, 심사위원 : 김완수, 백상훈, 최도훈(고려대학교), 선해상(UNIST)

Time-series data analysis is found in various applications that deal with sequential data over the given interval of, e.g. time. In this talk, we discuss time-series data analysis based on topological data analysis (TDA). The commonly used TDA method for time-series data analysis utilizes the embedding techniques such as sliding window embedding. With sliding window embedding the given data points are translated into the point cloud in the embedding space and the method of persistent homology is applied to the obtained point cloud. In this talk, we first show some examples of time-series data analysis with TDA. The first example is from music data for which the dynamic processes in time is summarized by low dimensional representation based on persistence homology. The second is the example of the gravitational wave detection problem and we will discuss how we concatenate the real signal and topological features. Then we will introduce our recent work of exact and fast multi-parameter persistent homology (EMPH) theory. The EMPH method is based on the Fourier transform of the data and the exact persistent barcodes. The EMPH is highly advantageous for time-series data analysis in that its computational complexity is as low as O(N log N) and it provides various topological inferences almost in no time. The presented works are in collaboration with Mai Lan Tran, Chris Bresten and Keunsu Kim.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701

Tree decompositions are a powerful tool in both structural graph theory and graph algorithms. Many hard problems become tractable if the input graph is known to have a tree decomposition of bounded “width”. Exhibiting a particular kind of a tree decomposition is also a useful way to describe the structure of a graph. Tree decompositions have traditionally been used in the context of forbidden graph minors; bringing them into the realm of forbidden induced subgraphs has until recently remained out of reach. Over the last couple of years we have made significant progress in this direction, exploring both the classical notion of bounded tree-width, and concepts of more structural flavor. This talk will survey some of these ideas and results.

We present a framework of predictive modeling of unknown system from measurement data. The method is designed to discover/approximate the unknown evolution operator, i.e., flow map, behind the data. Deep neural network (DNN) is employed to construct such an approximation. Once an accurate DNN model for the evolution operator is constructed, it serves as a predictive model for the unknown system and enables us to conduct system analysis. We demonstrate that flow map learning (FML) approach is applicable for modeling a wide class of problems, including dynamical systems, systems with missing variables and hidden parameters, as well as partial differential equations (PDEs).

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KAI-X Distinguished Lecture Series

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심사위원장: 이창옥, 심사위원:김동환, 신연종, 예종철(겸임교수), 신원용(연세대학교)

Collective cell movement is critical to the emergent properties of many multicellular systems including microbial self-organization in biofilms, wound healing, and cancer metastasis. However, even the best-studied systems lack a complete picture of how diverse physical and chemical cues act upon individual cells to ensure coordinated multicellular behavior. Myxococcus xanthus is a model bacteria famous for its coordinated multicellular behavior resulting in dynamic patterns formation. For example, when starving millions of cells coordinate their movement to organize into fruiting bodies – aggregates containing tens of thousands of bacteria. Relating these complex self-organization patterns to the behavior of individual cells is a complex-reverse engineering problem that cannot be solved solely by experimental research. In collaboration with experimental colleagues, we use a combination of quantitative microscopy, image processing, agent-based modeling, and kinetic theory PDEs to uncover the mechanisms of emergent collective behaviors.

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Professor of Bioengineering & BioSciences, Associate Chair of Bioengineering, Rice U

Compressible Euler system (CE) is a well-known PDE model that was formulated in the 19th century for dynamics of compressible fluid. The most important feature of CE is the finite-time breakdown of smooth solutions, especially, the formation of shock wave as severe singularity. Therefore, a fundamental question (since Riemann 1858) is on what happens after a shock occurs. This is the problem on well-posedness (that is, existence, uniqueness, stability) of CE in a suitable class of solutions. We will discuss on the well-posedness problem, and its generalization for applications to other PDE models arising in various contexts such as magnetohydrodynamics, tumor angiogenesis, vehicular traffic flow, etc.

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첫수융합포럼 The First Wednesday Multidisciplinary Forum, May 2023 with School of Business and Technology Management ZOOM Link: https://kaist.zoom.us/j/84028206160?pwd=VzNPRGxSR2hRcnJTNk4rMHQ4Z1hiQT09

In many different areas of mathematics (such as number theory, discrete geometry, and combinatorics), one is often presented with a large "unstructured" object, and asked to find a smaller "structured" object inside it. One of the earliest and most influential examples of this phenomenon was the theorem of Ramsey, proved in 1930, which states that if n = n(k) is large enough, then in any red-blue colouring of the edges of the complete graph on n vertices, there exists a monochromatic clique on k vertices. In this talk I will discuss some of the questions, ideas, and new techniques that were inspired by this theorem, and present some recent progress on one of the central problems in the area: bounding the so-called "diagonal" Ramsey numbers. Based on joint work with Marcelo Campos, Simon Griffiths and Julian Sahasrabudhe.

[1] 인간의 질병 발생과 예방을 근본적으로 파악하려면 '인간에 대한 이해'가 필요합니다. 인간의 몸은 물질이며, 물질은 특성상 물리적 및 화학적 반복자극에 반드시 손상됩니다. 부모님께 몸을 받아 수십 년간 살다 보면 인간 내부의 태생적-구조적 요인과 외부의 환경적 요인에 의하여 부지불식중 가해지는 내부 및 외부 자극에 반복적으로 노출될 수밖에 없습니다. 이에 그와 같이 질병으로 진행될 수 밖에 없는 인간의 특성을 명화(그림)을 통하여 소개할 예정입니다. [2] 인간의 또 다른 이해로 과학적 혹은 수학적 평가로 가시화(객관화)하기 어려운 부문에 대한 내용을 역시 명화를 통하여 논의할 예정입니다.

Knowledge graphs represent human knowledge as a directed graph, representing each fact as a triplet consisting of a head entity, a relation, and a tail entity. Knowledge graph embedding is a representation learning technique that aims to convert the entities and relations into a set of low-dimensional embedding vectors while preserving the inherent structure of the given knowledge graph. Once the entities and relations in a knowledge graph are represented as a set of feature vectors, those vectors can be easily integrated into diverse downstream tasks. This talk introduces a new concept of knowledge graph called a bi-level knowledge graph, where the higher-level relationships between triplets can be represented. Learning representations on a bi-level knowledge graph, machines are allowed to solve problems requiring more advanced reasoning than simple link prediction. Also, as a practical example of knowledge graph embedding, how one can utilize the knowledge representations to operate a real robot is briefly explained. This talk discusses how knowledge graph embedding models effectively deliver human knowledge to machines, which is critical in many AI applications.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701 ACMseminar mailing list registration: https://mathsci.kaist.ac.kr/mailman/listinfo/acmseminar

We study logarithmic spiraling solutions to the 2d incompressible Euler equations which solve a nonlinear transport system on the unit circle. We show that this system is locally well-posed for L^p data as well as for atomic measures, that is logarithmic spiral vortex sheets. We prove global well-posedness for almost bounded logarithmic spirals and give a complete characterization of the long time behavior of logarithmic spirals. This is due to the observation that the local circulation of the vorticity around the origin is a strictly monotone quantity of time. We are then able to show a dichotomy in the long time behavior, solutions either blow up (in finite or infinite time) or completely homogenize. In particular, bounded logarithmic spirals converge to constant steady states. For vortex logarithmic spiral sheets the dichotomy is shown to be even more drastic where only finite time blow up or complete homogenization of the fluid can and does occur.

We first survey on nodal solutions for coupled elliptic equations, using results from nonlinear scalar field equations as motivations. Then we discuss work for constructing multiple nodal solutions using various variational methods. In particular we discuss in some details the results about solutions having componentwisely-shared nodal numbers of coupled elliptic systems. These works are done by further developing minimax type critical point theory with built-in flow invariance of the associated gradient or parabolic flows, which has been a useful tool to give locations of critical points via minimum methods, also revealing complex dynamic behavior of the flow.

다양한 소비재 중에서 유독 패션 카테고리는 그간 디지털 전환이 느린 분야였으며, 유통과 마케팅의 영역에서만 데이터를 주로 활용하는 양상을 보여왔다. 비정형적, 주관적인 의사 결정이 주를 이루는 '패션'에서 수학은 어떤 의미와 효용이 있는지 제조업을 운영하는 디자이너의 관점에서 실무 사례 위주로 이야기하려 한다.

In this talk, we provide an overview of the historical development of fast solution methods for partial differential equations, as well as their current status and potential for future advancements. We first begin with a historical survey and describe recent advances in efficient techniques, such as multigrid and domain decomposition methods. In addition, we will explore the potential of emerging methods in the realm of scientific machine learning.

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(Online participation) Zoom Link: https://kaist.zoom.us/j/87516570701

In this talk, we look at the results of various studies in which computational mathematics is used in medical imaging. Through the various scope of research from mathematical modeling to data-based methodology, we can think about the future direction by examining what we can do in data science can contribute and what contribution we can make to medical imaging.