Optimal Transport Theory in High-Dimensional Data: Applications to Clustering and Generative Modeling

Project Abstract

In high-dimensional data analysis, optimal transport (OT) offers a principled framework to compare and align distributions, enabling robust clustering, dimensionality reduction, and generative modeling under complex geometric structures. This project investigates the theoretical and practical aspects of OT in high dimensions, emphasizing computational efficiency, statistical guarantees, and applicability to real-world datasets. We develop and analyze scalable OT formulations, including entropic regularization, sliced and projection-based methods, and unbalanced OT, to handle large sample sizes and heterogeneous mass. A central theme is the interplay between geometry and statistics we study how metric choices, such as learned Wasserstein distances and local transport plans, influence clustering stability, mode discovery, and the fidelity of synthetic data generated by OT-based models. The work advances clustering by integrating OT with hierarchical and spectral techniques to capture multi-scale structure, cluster overlap, and non-Gaussian shapes common in high-dimensional spaces. We propose novel objective functions that couple transport costs with cohesion-encouraging penalties and regularizers derived from domain knowledge. For generative modeling, we construct OT-inspired training objectives for deep generative models, including OT-GAN and variational OT flows, which align generated distributions with target empirical measures while preserving discontinuities and rare events. Our methodology also encompasses conditional and structured transport to accommodate labeled data, domain adaptation, and transfer learning scenarios, where transport maps serve as interpretable warps between source and target manifolds. From a theoretical standpoint, the project derives convergence rates for regularized OT estimators in high dimensions, analyzes the stability of transport plans under perturbations, and establishes sample complexity bounds under manifold and smoothness assumptions. We address computational trade-offs by deriving efficient algorithms with theoretical guarantees, such as stochastic Sinkhorn iterations, low-rank transport approximations, and neural network parameterizations of transport maps that leverage amortized inference. Regularization techniques are investigated to combat ill-posedness and overfitting, including sparsity-inducing transports and curvature-aware penalties that reflect intrinsic data geometry. Empirically, we validate the proposed methods on synthetic benchmarks and real-world datasets spanning image, text, and genomic modalities. We demonstrate improved clustering accuracy against baseline methods, enhanced generation quality and diversity, and increased robustness to noise and missing data. The work further assesses interpretability by visualizing transport maps, revealing latent structure shifts and transition pathways between clusters or states. Finally, we discuss practical guidelines for practitioners, including data pre-processing, choice of transport regularization, and computational considerations, to enable effective deployment of OT-based clustering and generative modeling in high-dimensional settings.

Project Overview

What This Project Is About

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This project explores how a mathematical idea called optimal transport can help us compare and move data between different shapes and distributions, especially when the data live in many dimensions. We will see how these ideas can improve clustering (grouping similar items) and generative modeling (creating new data that looks like real data).

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The Problem It Addresses

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In high-dimensional data, traditional methods often struggle to measure differences between datasets or to generate realistic new samples. The project looks for robust, scalable ways to quantify similarity and transform data that respect the underlying geometry, which can lead to better clustering results and more reliable synthetic data generation.

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Objectives of the Project

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1. Explain the basic ideas of optimal transport in simple terms.
2. Show how these ideas apply to clustering tasks in high dimensions.
3. Demonstrate how transport-based methods can improve generative models.
4. Implement a small, end-to-end workflow using accessible data sets.
5. Evaluate performance with clear, easy-to-interpret metrics.

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What You Will Do Step by Step

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1) Learn the core concepts of optimal transport and related metrics. 2) Pick simple high-dimensional data sets. 3) Implement transport-based clustering and a basic generative model. 4) Compare with standard methods using visual and numerical checks. 5) Analyze strengths, limitations, and practical considerations.

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Expected Outcome

A clear, student-friendly demonstration of how optimal transport can aid clustering and data generation in high dimensions, along with minimal code and guidelines for applying these ideas to new data sets.