Optimal Transport and Its Applications to Data Analysis: Theory, Algorithms, and Applications in High-Dimensional Statistical Inference

Project Abstract

We present a comprehensive study of optimal transport (OT) frameworks and their transformative impact on data analysis in high-dimensional statistical inference, with a focus on theory, algorithms, and real-world applications. We develop a unified mathematical treatment of OT principles, including Kantorovich duality, Wasserstein metrics, and Monge–Kantorovich formulations, to quantify distributional discrepancies and inform robust statistical procedures. Our work extends classical OT to high-dimensional regimes by leveraging regularized OT (entropic and quadratic regularization), sliced and projected OT, and multi-marginal extensions, enabling scalable computation without sacrificing theoretical guarantees. We introduce novel convergence analyses and stability results for empirical OT plans under finite-sample regimes, addressing curvature, concentration, and sample complexity challenges that arise in high dimensions. On the algorithmic front, we design efficient optimization schemes based on Sinkhorn iterations, proximal point methods, and stochastic gradient techniques tailored for large-scale data, as well as adaptive regularization parameter selection to balance bias-variance trade-offs. We also explore differentiable OT pipelines for end-to-end learning, including OT-based loss functions for domain adaptation, generative modeling, and structured prediction, with particular emphasis on computational tractability in resource-constrained environments. In statistical inference, we investigate OT as a principled tool for two-sample testing, density estimation, clustering, and regression under distributional shift, providing new asymptotic results and finite-sample guarantees. Our methodology enables robust handling of heavy-tailed data, outliers, and non-Euclidean data modalities through extensions to metric measure spaces and curved geometries. We demonstrate the versatility of OT in high-dimensional data analytics through a suite of applications (i) multivariate density and distributional comparison in genomics and neuroscience; (ii) domain adaptation for cross-domain sentiment analysis and image recognition; (iii) generative modeling and unsupervised representation learning via OT-based autoencoders and GAN variants; (iv) transport-based barycenter computations for clustering and consensus analysis in heterogeneous datasets; and (v) time-evolving transport maps for dynamic networks and longitudinal data. The empirical component includes extensive simulations that validate theoretical results and benchmark OT methods against state-of-the-art alternatives in high dimensions, as well as real-world case studies in medical imaging, climate science, and finance. We discuss practical considerations such as scalability, numerical stability, and reproducibility, and provide guidelines for choosing OT variants and regularization schemes aligned with specific data characteristics and inference objectives. Finally, we outline a roadmap for integrating OT methods into mainstream data analysis pipelines, emphasizing interpretability, uncertainty quantification, and adaptability to evolving data landscapes. This work contributes to the theoretical foundations of OT in high dimensions, advances algorithmic techniques for scalable deployment, and demonstrates the broad utility of optimal transport as a unifying paradigm for modern statistical inference and data-driven decision making.

Project Overview

What This Project Is About

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The project explores a way to compare and move mass between distributions, which helps us understand data that comes from different sources or looks different. It combines simple ideas with practical algorithms to analyze high-dimensional data, like images or large datasets, in a way that respects its geometric structure and uncertainty.

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The Problem It Addresses

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Many real-world datasets are complex and high-dimensional, making direct comparisons difficult. Traditional methods may ignore geometry or fail when data shapes are mismatched. The project seeks a robust method to measure differences and transform data in a principled, interpretable way, improving tasks such as clustering, generation, and domain adaptation.

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Objectives of the Project

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1. Introduce the basic idea of transporting mass from one distribution to another (in simple terms).
2. Explain how these ideas can be turned into algorithms that work on high-dimensional data.
3. Demonstrate applications in data analysis, such as comparing datasets and improving machine learning tasks.
4. Evaluate performance on toy examples and real data to show benefits and limitations.

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What You Will Do Step by Step

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1. Learn the core concepts with gentle explanations and minimal math. 2. Implement simple transport-based algorithms on small datasets. 3. Extend to higher dimensions using scalable methods. 4. Test on real-world data and compare with standard approaches. 5. Analyze results and discuss practical usefulness and caveats.

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Expected Outcome

Clear understanding of optimal transport ideas, a runnable set of experiments showing when and why these methods help, and a practical guide for applying them to high-dimensional data.