Graduation Semester and Year
Summer 2026
Language
English
Document Type
Dissertation
Degree Name
Doctor of Philosophy in Mathematics
Department
Mathematics
First Advisor
Li Wang
Second Advisor
Ren-Cang Li
Third Advisor
Andrzej Korzeniowski
Fourth Advisor
Shan Sun-Mitchell
Abstract
Dimensionality reduction (DR) is a fundamental tool in data science and machine learning that transforms high-dimensional data into a low-dimensional representation while preserving important structural properties of the original data. Among modern DR methods, t-distributed stochastic neighbor embedding (t-SNE) has become one of the most widely used techniques for visualization due to its strong ability to preserve local neighborhood structure and produce visually separated clusters. However, despite its popularity, t-SNE is well known to struggle with preserving global structure of data, often producing embeddings in which distances between clusters and neighborhoods do not accurately reflect relationships in the high-dimensional space. This limitation can lead to misleading visualizations and weaker interpretability.
This dissertation focuses on improving the global structure preservation capabilities of t-SNE while maintaining competitive local structure preservation. Motivated by the expected distance-preserving constraints introduced in maximum posterior manifold embedding (MPME), we propose three novel soft-constrained variants of t-SNE designed to better balance local and global information. The first proposed method, PM1, incorporates deterministic distance-preserving constraints into the t-SNE objective through hinge-loss-style penalty terms, resulting in a nonsmooth optimization problem solved using subgradient-based optimization methods. The second proposed method, PM2, introduces probabilistic distance-preserving constraints based on the conditional probabilities used in t-SNE. The third proposed method, PM2-ss, further modifies the PM2 framework by introducing an additional scaling parameter into the soft-constrained formulation, improving the balance between the KL divergence term and the distance-preserving penalty terms during optimization. For all proposed methods, constrained optimization problems are transformed into unconstrained optimization problems through penalty formulations, enabling the use of gradient-based and subgradient-based optimization algorithms.
Extensive numerical experiments are conducted on multiple benchmark datasets and compared against several widely used DR methods, including t-SNE, UMAP, TriMap, and PaCMAP. Performance is evaluated using both local and global structure preservation metrics, including nearest neighbor classification accuracy, unsupervised neighborhood preservation, random triplet accuracy, K-nearest classes preservation, Spearman correlation, and centroid distance correlation. Experimental results demonstrate that the proposed methods consistently improve global structure preservation while maintaining competitive local structure preservation performance. In many cases, the proposed methods outperform competing methods on global evaluation metrics, including methods specifically designed to preserve global structure. To improve scalability, Barnes--Hut implementations of PM2 and PM2-ss are also developed, substantially reducing computational cost while producing embeddings that are visually consistent with the exact methods. These results show that incorporating soft distance-preserving constraints into the t-SNE framework provides an effective and flexible approach for balancing local and global structure preservation.
Keywords
dimensionality reduction, t-SNE, optimization, global structure preservation
Disciplines
Mathematics
License
This work is licensed under a Creative Commons Attribution 4.0 International License.
Recommended Citation
Balderas, Joseph A., "Soft-Constrained Variants of t-Distributed Stochastic Neighbor Embedding for Global Structure Preservation" (2026). Mathematics Dissertations. 5.
https://mavmatrix.uta.edu/math_dissertations2/5