Theoretical Foundations of t-SNE for Visualizing High-Dimensional Clustered Data

TL;DR

This paper provides a comprehensive theoretical analysis of t-SNE, revealing its connection to spectral clustering, explaining its fast convergence, and offering insights for better application and parameter tuning in high-dimensional data visualization.

Contribution

It introduces a novel theoretical framework for t-SNE based on gradient descent, linking it to graph Laplacian and spectral clustering, and explains its empirical success.

Findings

t-SNE's early exaggeration stage is asymptotically equivalent to power iterations on the graph Laplacian

The analysis uncovers the phases of the low-dimensional map, including amplification and stabilization

The theory explains t-SNE's fast convergence and guides parameter selection

Abstract

This paper investigates the theoretical foundations of the t-distributed stochastic neighbor embedding (t-SNE) algorithm, a popular nonlinear dimension reduction and data visualization method. A novel theoretical framework for the analysis of t-SNE based on the gradient descent approach is presented. For the early exaggeration stage of t-SNE, we show its asymptotic equivalence to power iterations based on the underlying graph Laplacian, characterize its limiting behavior, and uncover its deep connection to Laplacian spectral clustering, and fundamental principles including early stopping as implicit regularization. The results explain the intrinsic mechanism and the empirical benefits of such a computational strategy. For the embedding stage of t-SNE, we characterize the kinematics of the low-dimensional map throughout the iterations, and identify an amplification phase, featuring the intercluster repulsion and the expansive behavior of the low-dimensional map, and a stabilization phase. The general theory explains the fast convergence rate and the exceptional empirical performance of t-SNE for visualizing clustered data, brings forth interpretations of the t-SNE visualizations, and provides theoretical guidance for applying t-SNE and selecting its tuning parameters in various applications.