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Explore the theoretical foundations and statistical aspects of optimal transport theory in this comprehensive lecture delivered at the International Centre for Theoretical Sciences. Delve into the mathematical framework that bridges probability theory, statistics, and optimization, examining how optimal transport provides powerful tools for comparing probability distributions and solving complex data science problems. Learn about the statistical properties of optimal transport, including convergence rates, computational complexity, and practical applications in machine learning and data analysis. Discover how optimal transport theory connects to various fields including generative modeling, domain adaptation, and causal inference. Understand the key challenges in statistical optimal transport, including sample complexity bounds, minimax rates, and the development of efficient algorithms for large-scale applications. Examine recent theoretical developments that have made optimal transport more accessible for practical data science applications, including entropic regularization techniques and their statistical implications. Gain insights into how optimal transport serves as a fundamental tool in modern machine learning, particularly in areas such as generative adversarial networks, Wasserstein GANs, and distribution matching problems. This lecture forms part of a comprehensive program on probabilistic and optimization methods in data science, providing essential theoretical foundations for understanding how rigorous mathematical principles inform robust and adaptable machine learning systems.
