Intensive Lectures for

Optimal Transport and Machine Learning,

Geometric Measure Theory and Optimal Transport

Date: July - August, 2017       Place: Rm. 1503, KIAS

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Optimal Transport and Machine Learning
Lecturer: Seungjin Choi (Computer Science and Engineering, POSTECH)
Deep Generative Models for Density Estimation
Abstract: Density estimation is a core problem in machine learning, the goal of which is to construct an estimate, based on observed data, of an unobservable underlying probability density function. Generative models have played a critical role for density estimation. In this talk, I begin with linear generative models where some conditional independence structures are imposed on parameterized distributions in linear model to build a problem-specific density which is learned from a training dataset. Then, I explain rather recent advances in deep generative models where deep neural networks incorporate encoder or decoder in generative models. Two important categories are emphasized in terms of prescribed models or implicit models, including variational autoencoders and generative adversarial nets.

Lecturer: Marco Cuturi (ENSAE / CREST)
Title: Computational Optimal Transport with Applications to Machine Learning
Abstract: I will cover in these lectures fundamental aspects behind the computation of the so-called "static" optimal transport problem, as formalized by Monge first, but most importantly by Kantorovich and Hitchcok during the war, and solved numerically as a linear program by Dantzig shortly after. I will show the computational limits of this LP approach and explain how a simple regularization of the problem can result in considerably faster computations, using notably GPUs. I will then discuss recent applications where this regularized perspective has proved effective for data analysis.

Talk schedule:

[1st part] Introduction, OT as a LP, regularized OT
- Introduction to the field, brief historical review.
- Introduction to the linear programming formulation of optimal transport.
- Review of relevant algorithms to solve OT.
- Entropic regularization.
- Algorithmic properties. Connexion to matrix scaling. Convergence.
- Differentiability of regularized OT.

[2nd part] Wasserstein Variational Problems in Data Analysis
- Links with k-means
- The barycenter problem
- K-means in Wasserstein space of measures.
- Dictionary learning
- Wasserstein PCA
- Wasserstein Regression
- Minimum Kantorovich Estimation / Wasserstein GAN

Lecturer: Jinwoo Shin (Electrical Engineering, KAIST)
Title: Generative Machine Learning
Abstract: Generative models randomly generate observable data values, typically given some hidden parameters. In this talk, I will introduce most successful generative models and their underlying theory, which have developed in the machine learning community: (a) graphical models (e.g., Gaussian, Markov random fields, Boltzmann machines) and (b) neural networks (e.g., variational auto-encoders, generative adversarial networks).

Geometric Measure Theory
Lecturer: Giovanni Alberti (University of Pisa)
Title: An introduction to Geometric Measure Theory and Finite Perimeter Sets
Abstract: In this course I will introduce the basic notions of Geometric Measure Theory (Hausdorff measures, rectifiable sets) and give an overview of the theory of Finite Perimeter Sets, including the proof of De Giorgi's rectifiability theorem. 
Then I will show how this theory can be used to prove (easy) existence result for some geometric variational problems. If time allows I will also touch some additional topics, such as the theory of integral currents, the basic regularity for minimizers, and the elliptic approximation of the area functional.