Time 

August 19 (Mon), 2024, 11:00 ~ 16:30

Place  

Kyung-Hee University (Global Campus), Building for Colledge of Electronics and Information, Room 409

Speakers

Doheon Kim (Hanyang University) ( 11:00 ~ 11:30)

Title: On score-based diffusion models with multiplicative noise conditioning

Abstract: Score-based diffusion models learn the score associated with a diffusion process to generate new samples. Assuming that the score is well-approximated, the efficacy of these models can be theoretically explained via differential equations corresponding to sampling processes. However, empirical observation has shown that models learning the score with a neural network employing multiplicative noise conditioning can also generate satisfactory samples, even though the model capacity is evidently insufficient to learn the correct score. We provide a theoretical explanation of these models by analyzing the qualitative behavior of the differential equations associated with the diffusion processes using appropriate Lyapunov functions.

Woocheol Choi (Sungkyunkwan University) (11:30 ~ 12:00)

Title: Non-ergodic linear convergence property of the delayed gradient descent under the strongly convexity and the Polyak-Łojasiewicz condition 

Abstract: In this work, we establish the linear convergence estimate for the gradient descent involving the delay τ∈N when the cost function is μ-strongly convex and L-smooth. This result improves upon the well-known estimates in Arjevani et al. \cite{ASS} and Stich-Karmireddy \cite{SK} in the sense that it is non-ergodic and is still established in spite of weaker constraint of cost function. Also, the range of learning rate η can be extended from η≤1/(10Lτ) to η≤1/(4Lτ) for τ=1 and η≤3/(10Lτ) for τ≥2, where L>0 is the Lipschitz continuity constant of the gradient of cost function. In a further research, we show the linear convergence of cost function under the Polyak-Łojasiewicz\,(PL) condition, for which the available choice of learning rate is further improved as η≤9/(10Lτ) for the large delay τ. The framework of the proof for this result is also extended to the stochastic gradient descent with time-varying delay under the PL condition. Finally, some numerical experiments are provided in order to confirm the reliability of the analyzed results. 

Myungjoo Kang (Gachon University) (14:00 ~ 14:30)

Title: Phase-coupled models for synchronization with nonlocal temporal interactions

Abstract: In this talk, we study the emergent dynamics of the Kuramoto model with memory effect. The Kuramoto model with memory effect belongs to the system of Volterra-type integro-differential equations on unit circle. For the modeling of memory effect, we adopt nonlocal temporal interactions so that dynamic behaviors of oscillators are affected by memories of the past interactions. For the proposed model, we show the emergence of complete frequency synchronization in two different ways. One is to use a bootstrap argument and the other is to use an energy functional. In both ways, boundedness of the phase diameter is needed. In particular, we show the emergence of complete phase synchronization for the case where natural frequencies are all identical. This talk is based on the joint work with Prof. Seung-Yeal Ha (SNU) and Dr. Hangjun Cho (UW).

Seunghyeok Kim (Hanyang University) (14:30 ~ 15:00)

Title: Sharp quantitative stability estimates for critical points of fractional Sobolev inequalities

Abstract: I will introduce my recent joint work with Dr. Haixia Chen (Hanyang U.) and Prof. Juncheng Wei (UBC, CUHK) on sharp quantitative stability estimates for critical points of the fractional Sobolev inequalities induced by the embedding $\dot{H}^s(\R^n) \hookrightarrow L^{2n \over n-2s}(\R^n)$ for the whole range of $s \in (0,\frac{n}{2})$. To prove it, we develop a unified approach based on integral representations. This work fully generalizes the work I reported in the 2022 Summer Su-In Workshop.

Jinwook Jung (Hanyang University) (15:30 ~ 16:00)

Title: On the mean-field limit of Vlasov-Poisson-Fokker-Planck equations

Abstract: In this talk, we discuss the propagation of chaos results from the second-order stochastic particle system to the Vlasov-Poisson-Fokker-Planck (or to the Vlasov-Poisson) equation by using the relative entropy method developed in [Jabin-Wang, J. Func. Anal., '16]. The result is obtained in the strong $L^1$ sense with a convergence rate based on quantitative estimates provided in [Carrillo-Choi-Salem, Commun. Contemp. Math., '19] and [Huang-Liu-Pickl, J. Stat. Phys., '20]. This talk is based on the joint work with L. Chen, P. Pickl, and Z. Wang.

Junha Kim (Ajou University) (16:00 ~ 16:30)

Title: Convergence and Nonconvergence of the Euler-Maxwell Equations as the Speed of Light Tends to Infinity

Abstract: We consider the convergence of smooth solutions of the Euler-Maxwell equations to the corresponding smooth solutions of the magnetohydrodynamics as the speed of light tends to infinity. As far as we know, the strong convergence of solutions to the Euler-Maxwell equations has not been studied before. We employ a proper auxiliary linear system of the Euler-Maxwell system to obtain the convergence and nonconvergence of the Euler-Maxwell equations to the MHD equations. This is a joint work with Dong-Ha Kim(Yonsei Univ.) and Jihoon Lee(CAU).