류경석 교수

Starting from where a first course in convex optimization leaves off, this text presents a unified analysis of first-order optimization methods – including parallel-distributed algorithms – through the abstraction of monotone operators. With the increased computational power and availability of big data over the past decade, applied disciplines have demanded that larger and larger optimization problems be solved. This text covers the first-order convex optimization methods that are uniquely effe

Plug-and-play (PnP) is a non-convex framework that integrates modern denoising priors, such as BM3D or deep learning-based denoisers, into ADMM or other proximal algorithms. An advantage of PnP is that one can use pre-trained denoisers when there is not sufficient data for end-to-end training. Although PnP has been recently studied extensively with great empirical success, theoretical analysis addressing even the most basic question of convergence has been insufficient. In this paper, we theoret

Given the success of Douglas--Rachford splitting (DRS), it is natural to ask\nwhether DRS can be generalized. Are there other 2 operator resolvent-splittings\nsharing the favorable properties of DRS? Can DRS be generalized to 3 operators?\nThis work presents the answers: no and no. In a certain sense, DRS is the\nunique 2 operator resolvent-splitting, and generalizing DRS to 3 operators is\nimpossible without lifting, where lifting roughly corresponds to enlarging the\nproblem size. The impossib

In this paper, we present the proximal-proximal-gradient method (PPG), a novel optimization method that is simple to implement and simple to parallelize. PPG generalizes the proximal-gradient method and ADMM and is applicable to minimization problems written as a sum of many differentiable and many non-differentiable convex functions. The non-differentiable functions can be coupled. We furthermore present a related stochastic variation, which we call stochastic PPG (S-PPG). S-PPG can be interpre

We show that the variance of the Monte Carlo estimator that is importance sampled from an exponential family is a convex function of the natural parameter of the distribution. With this insight, we propose an adaptive importance sampling algorithm that simultaneously improves the choice of sampling distribution while accumulating a Monte Carlo estimate. Exploiting convexity, we prove that the method's unbiased estimator has variance that is asymptotically optimal over the exponential family.

Despite remarkable empirical success, the training dynamics of generative adversarial networks (GAN), which involves solving a minimax game using stochastic gradients, is still poorly understood. In this work, we analyze last-iterate convergence of simultaneous gradient descent (simGD) and its variants under the assumption of convex-concavity, guided by a continuous-time analysis with differential equations. First, we show that simGD, as is, converges with stochastic sub-gradients under strict c

We propose a methodology for studying the performance of common splitting methods through semidefinite programming. We prove tightness of the methodology and demonstrate its value by presenting two applications of it. First, we use the methodology as a tool for computer-assisted proofs to prove tight analytical contraction factors for Douglas--Rachford splitting that are likely too complicated for a human to find bare-handed. Second, we use the methodology as an algorithmic tool to computational

In many applications such as color image processing, data has more than one piece of information associated with each spatial coordinate, and in such cases the classical optimal mass transport (OMT) must be generalized to handle vector-valued or matrix-valued densities. In this paper, we discuss the vector and matrix optimal mass transport and present three contributions. We first present a rigorous mathematical formulation for these setups and provide analytical results including existence of s

In this paper, we present the proximal-proximal-gradient method (PPG), a novel optimization method that is simple to implement and simple to parallelize. PPG generalizes the proximal-gradient method and ADMM and is applicable to minimization problems written as a sum of many differentiable and many non-differentiable convex functions. The non-differentiable functions can be coupled. We furthermore present a related stochastic variation, which we call stochastic PPG (S-PPG). S-PPG can be interpre

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.

Recently Stochastic Kronecker Graph (SKG), a network generation model, and vertex-centric BSP, a graph processing framework like Pregel, have attracted much attention in the network analysis community. Unfortunately the two are not very well-suited for each other and thus an implementation of SKG on vertex-centric BSP must either be done serially or in an unnatural manner. In this paper, we present a new network generation model, which we call Poisson Stochastic Kronecker Graph (PSKG), that gene