Seminars & Colloquia

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Encoder-decoder networks using convolutional neural network (CNN) architecture have been extensively used in deep learning approaches for inverse problems thanks to its excellent performance. However, it is still difficult to obtain coherent geometric view why such an architecture gives the desired performance. Inspired by recent theoretical understanding on generalizability, expressivity and optimization landscape of neural networks, as well as the theory of deep convolutional framelets, here we provide a unified theoretical framework that leads to a better understanding of geometry of encoder-decoder CNNs. Our unified framework shows that encoder-decoder CNN architecture is closely related to nonlinear frame basis representation using combinatorial convolution frames, whose expressivity increases exponentially with the network depth and channels. We also demonstrate the importance of skipped connection in terms of expressivity and optimization landscape. We provide extensive experimental results from various biomedical imaging reconstruction problems to verify the performance encoder-decoder CNNs.
Host: 이창옥     Korean     2019-10-29 13:27:46
The Bayesian approach to inverse problems, in which the posterior probabil- ity distribution on an unknown eld is sampled for the purposes of computing posterior expectations of quantities of interest, is starting to become computa- tionally feasible for partial di erential equation (PDE) inverse problems. Bal- ancing the sources of error arising from nite-dimensional approximation of the unknown eld, the PDE forward solution map and the sampling of the prob- ability space under the posterior distribution are essential for the design of ecient computational Bayesian methods for PDE inverse problems. We study Bayesian inversion for a model elliptic PDE with an unknown di usion coef- cient. We consider both the case where the PDE is uniformly elliptic with respect to all the realizations, and the case where uniform ellipticity does not hold, i.e. the coecient can get arbitrarily close to 0 and arbitrarily large as in the log-normal model. We provide complexity analysis of Markov chain Monte Carlo (MCMC) methods for numerical evaluation of expectations under the Bayesian posterior distribution given data, in particular bounds on the overall work required to achieve a prescribed error level. We rst bound the computa- tional complexity of plain MCMC, based on combining MCMC sampling with linear complexity multi-level solvers for elliptic PDEs. The work versus accu- racy bounds show that the complexity of this approach can be quite prohibitive. We then present a novel multi-level Markov chain Monte Carlo strategy which utilizes sampling from a multi-level discretization of the posterior and the for- ward PDE. The strategy achieves an essentially optimal complexity level that is essentially equal to that for performing only one step on the plain MCMC. The essentially optimal accuracy and complexity of the method are mathematically rigorously proven. Numerical results con rm our analysis. This is a joint work with Jia Hao Quek (NTU, Singapore), Christoph Schwab (ETH, Switzerland) and Andrew Stuart (Warwick, England).
Host: 임미경     English     2019-11-13 09:14:09
Let $F$ be a graph. We say that a hypergraph $\mathcal H$ is an induced Berge $F$ if there exists a bijective mapping $f$ from the edges of $F$ to the hyperedges of $\mathcal H$ such that for all $xy \in E(F)$, $f(xy) \cap V(F) = \{x,y\}$. In this talk, we show asymptotics for the maximum number of edges in $r$-uniform hypergraphs with no induced Berge $F$. In particular, this function is strongly related to the generalized Turán function $ex(n,K_r, F)$, i.e., the maximum number of cliques of size $r$ in $n$-vertex, $F$-free graphs. Joint work with Zoltan Füredi.
Host: 엄상일     English     2019-11-08 15:03:23
We discuss a triangle of viewpoints for circle diffeomorphism groups: analysis, dynamics and group theory. In particular, we illustrate how the regularities (from the analytic side) of diffeomorphisms govern the dynamics and the group theoretical properties of diffeomorphisms. This line of study can be traced back to the works of Hölder, Denjoy, Tsuboi, Thurston and many more.
Host: 백형렬     Korean     2019-10-29 13:29:35
Haviv (European Journal of Combinatorics, 2019) has recently proved that some topological lower bounds on the chromatic number of graphs are also lower bounds on their orthogonality dimension over R. We show that this holds actually for all known topological lower bounds and all fields. We also improve the topological bound he obtained for the minrank parameter over R – an important graph invariant from coding theory – and show that this bound is actually valid for all fields as well. The notion of independent representation over a matroid is introduced and used in a general theorem having these results as corollaries. Related complexity results are also discussed. This is joint work with Meysam Alishahi.
Host: 엄상일     To be announced     2019-11-08 15:05:17
The debate about the correct diffusion model is related to the way to handle the randomness. In this talk, we will see an example which shows that the Stratonovitch interal is the correct way to handle it. The classical kinetic equation is related to the Ito integral. We will construct a new kinetic equation of Stratonovitch type.
Host: 권순식     Contact: 이슬기 (042-350-8111)     To be announced     2019-11-07 09:08:16
- 소스 관리 툴의 대세, 해외 IT 시장에서는 Git이 기본, 모르는 자는 취업조차 어렵다. - 프로그램 코드와 연구를 하나의 툴로 관리하자 - 나만의 Git 저장소를 만드는 법 - 목차 1) Git 배경 설명 및 필요성 2) Git 중요 개념 및 사용하기 3) Git 브랜치 관리 기법: Git-flow 4) Git 클라이언트 5) 협업하기: GitHub와 나만의 저장소 만들기
Host: 황강욱교수님     Contact: 이슬기 (042-350-8111)     To be announced     2019-11-04 09:24:20
- 소스 관리 툴의 대세, 해외 IT 시장에서는 Git이 기본, 모르는 자는 취업조차 어렵다. - 프로그램 코드와 연구를 하나의 툴로 관리하자 - 나만의 Git 저장소를 만드는 법 - 목차 1) Git 배경 설명 및 필요성 2) Git 중요 개념 및 사용하기 3) Git 브랜치 관리 기법: Git-flow 4) Git 클라이언트 5) 협업하기: GitHub와 나만의 저장소 만들기
Host: 황강욱교수님     Contact: 이슬기 (042-350-8111)     To be announced     2019-11-04 09:24:20
It is a classic result that the maximum weight stable set problem is efficiently solvable for bipartite graphs. The recent bimodular algorithm of Artmann, Weismantel and Zenklusen shows that it is also efficiently solvable for graphs without two disjoint odd cycles. The complexity of the stable set problem for graphs without disjoint odd cycles is a long-standing open problem for all other values of . We prove that under the additional assumption that the input graph is embedded in a surface of bounded genus, there is a polynomial-time algorithm for each fixed . Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes. To this end, we show that 2-sided odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed surface. This result may be of independent interest and extends a theorem of Kawarabayashi and Nakamoto asserting that odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed orientable surface. Eventually, our findings allow us to reduce the original problem to the problem of finding a minimum-cost non-negative integer circulation of a certain homology class, which we prove to be efficiently solvable in our case. This is joint work with Michele Conforti, Samuel Fiorini, Gwenaël Joret, and Stefan Weltge.
Host: 엄상일     English     2019-11-08 14:59:34
In this talk, I will discuss some recent developments on the study of singular stochastic wave equations. I also describe some similarities and differences between stochastic wave and heat equations, indicating particular difficulty of the dispersive/hyperbolic problem.
Host: 권순식     To be announced     2019-11-01 16:53:41
On page 335 in his lost notebook, Ramanujan recorded without proofs two identities involving finite trigonometric sums and doubly infinite series of Bessel functions. We proved each of these identities under three different interpretations for the double series, and showed that they are intimately connected with the classical circle and divisor problems in number theory. Furthermore, we established many analogues and generalizations of them. This is joint work with Bruce C. Berndt and Alexandru Zaharescu.
Host: 김동수     English     2019-10-28 15:53:15