Clustering with t-SNE, provably (1706.02582v1)

Abstract: t-distributed Stochastic Neighborhood Embedding (t-SNE), a clustering and visualization method proposed by van der Maaten & Hinton in 2008, has rapidly become a standard tool in a number of natural sciences. Despite its overwhelming success, there is a distinct lack of mathematical foundations and the inner workings of the algorithm are not well understood. The purpose of this paper is to prove that t-SNE is able to recover well-separated clusters; more precisely, we prove that t-SNE in the `early exaggeration' phase, an optimization technique proposed by van der Maaten & Hinton (2008) and van der Maaten (2014), can be rigorously analyzed. As a byproduct, the proof suggests novel ways for setting the exaggeration parameter $\alpha$ and step size $h$. Numerical examples illustrate the effectiveness of these rules: in particular, the quality of embedding of topological structures (e.g. the swiss roll) improves. We also discuss a connection to spectral clustering methods.

• The paper introduces a rigorous analysis that proves t-SNE’s ability to recover well-separated clusters during its early exaggeration phase.
• It demonstrates that with specific parameter settings, t-SNE achieves exponential convergence similar to spectral clustering, enhancing algorithm reliability.
• Empirical validations confirm that proper initialization and parameter tuning yield stable, distinct cluster separations in low-dimensional embeddings.

Analysis of "Clustering with t-SNE, Provably"

The paper authored by George C. Linderman and Stefan Steinerberger embarks on the formal mathematical exploration of the t-Distributed Stochastic Neighbor Embedding (t-SNE) algorithm, focusing on its clustering and visualization capabilities. t-SNE, first introduced by van der Maaten and Hinton in 2008, stands as a highly effective non-linear dimensionality reduction method especially favored for its prowess in visualizing high-dimensional data by embedding them in lower-dimensional spaces, typically two or three dimensions, thus facilitating cluster identification.

Despite t-SNE's empirical success across various domains, the absence of rigorous mathematical foundations regarding its function and effectiveness remains a significant gap, which this paper seeks to close. The authors purposefully set out to establish that t-SNE is capable of correctly identifying distinct clusters under certain mathematical conditions, particularly during the 'early exaggeration' phase of the algorithm.

Theoretical Contributions

The paper introduces a major theoretical contribution by showing that during the early exaggeration phase, t-SNE can recover well-separated clusters with provable correctness. This rigorous analysis provides an essential underpinning for the empirical success of t-SNE. It shows that with proper parameter settings, notably the exaggeration parameter $\alpha$ and step size $h$, the algorithm ensures exponential convergence of clusters, thereby reinforcing the observed quality of the resulting embeddings.

Key Theoretical Results

1. Exponential Convergence: The paper outlines the conditions under which the step size $h$ and exaggeration parameter $\alpha$ should be set, specifically indicating $\alpha \sim \frac{n}{10}$ and $h \sim 1$. Under these conditions, convergence is exponential, thus enhancing the algorithm's efficiency and diminishing embedding errors.
2. Spectral Clustering Connection: It is posited that t-SNE, under the discussed parameter settings, behaves akin to a spectral clustering method. This insight not only offers a new avenue for interpreting t-SNE’s functional mechanisms but also suggests potential optimizations for spectral clustering processes.
3. Cluster Separation and Stability: The embedded clusters are likely to become disjoint; furthermore, a random initialization is shown to favor the separation of distinct clusters' centers within the low-dimensional embedding.
4. Independence from Initialization: The theoretical guarantees provided are robust to various initializations of $\mathcal{Y}$, ensuring stability and reproducibility of clustering results irrespective of the starting configuration.

Practical Implications and Numerical Validations

The theoretical work is complemented by a series of numerical examples demonstrating the practical impact. These numerical experiments validate the stronger convergence properties anticipated by the theory, especially highlighting improvements in visualizing topological structures such as the swiss roll. The analysis delineates the influence exerted by appropriate parameter settings on the resolution and stability of cluster boundaries in real-world applications, such as single-cell RNA sequencing data visualization.

Future Directions and Implications

This paper marks a pivotal advancement in the understanding of t-SNE from a theoretical perspective. It opens up further opportunities for refinement in dimensionality reduction techniques, potentially inspiring novel clustering methods rooted in the spectral interpretation of t-SNE’s mechanics. These insights could lead to better initialization schemes or provide a basis for enhancing spectral methods using the principles uncovered here.

Moreover, this foundational work bridges the gap between empirical observations and theoretical validation, unlocking new possibilities for improving data visualization and clustering methodologies, particularly in complex high-dimensional spaces typical of modern datasets.

In conclusion, Linderman and Steinerberger’s work provides a crucial step toward elucidating the theoretical underpinnings of t-SNE, validating the empirical utility observed across a multitude of scientific domains and setting a benchmark for future theoretical explorations in clustering and visualization algorithms.