aiqbench analysis
This research provides a rigorous mathematical analysis of the affinity matrix in t-SNE, demonstrating how row-wise power transforms effectively modulate local neighborhood preservation across different scales. By establishing equivalence between this smoothing technique and rescaling Gaussian bandwidths with point-dependent perplexities, the study offers practitioners a more nuanced tool for optimizing dimensionality reduction beyond standard global hyperparameters. The findings validate that specific sharpness adjustments can outperform existing multiscale methods in preserving mid-local structures without sacrificing sparsity or rank order.
How smoothing the affinity matrix affects neighborhood preservation in t-SNE
arXiv:2608.17190v1 Announce Type: new Abstract: Dimensionality reduction methods are instrumental to visualize high-dimensional data, and t-SNE stands as one of the most widely used methods due to its emphasis on local neighborhood preservation. A central component of t-SNE is the affinity matrix, which expresses pairwise similarities in the form of symmetrized probabilities, over which the optimization problem of t-SNE is defined. We study how the sharpness of this probability distribution affects neighborhood preservation at different scales. We introduce a row-wise power transform controlled by a parameter gamma that can smooth or sharpen each row of the affinity matrix while preserving sparsity and rank order. We show that this transform is equivalent to rescaling the Gaussian bandwidth and thus to changing the perplexity. However, as the sharpness of the probability distribution varies per point, a fixed gamma leads to point-dependent effective perplexities, making it distinct from changing the global perplexity. Empirically, we find that sharpening improves preservation of the very nearest neighbors, while smoothing improves preservation of broader local neighborhoods, outperforming alternative affinity constructions including multiscale methods in the mid-local range.
Abstract:Dimensionality reduction methods are instrumental to visualize high-dimensional data, and t-SNE stands as one of the most widely used methods due to its emphasis on local neighborhood preservation. A central component of t-SNE is the affinity matrix, which expresses pairwise similarities in the form of symmetrized probabilities, over which the optimization problem of t-SNE is defined. We study how the sharpness of this probability distribution affects neighborhood preservation at different scales. We introduce a row-wise power transform controlled by a parameter gamma that can smooth or sharpen each row of the affinity matrix while preserving sparsity and rank order. We show that this transform is equivalent to rescaling the Gaussian bandwidth and thus to changing the perplexity. However, as the sharpness of the probability distribution varies per point, a fixed gamma leads to point-dependent effective perplexities, making it distinct from changing the global perplexity. Empirically, we find that sharpening improves preservation of the very nearest neighbors, while smoothing improves preservation of broader local neighborhoods, outperforming alternative affinity constructions including multiscale methods in the mid-local range.
Accepted at the 29th International Conference on Discovery Science (DS 2026). 15 pages, 7 figures
Machine Learning (cs.LG); Computer Vision and Pattern Recognition (cs.CV)
arXiv:2608.17190 [cs.LG]
(or arXiv:2608.17190v1 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2608.17190 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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