Visualizing stability: a sensitivity analysis framework for t-SNE embeddings

t-distributed Stochastic Neighbour Embedding (t-SNE) is a cornerstone for visualizing high-dimensional biological data, where each high-dimensional data point is represented as a point in a two-dimensional map. However, this static map provides no information about the stability of the visual layout, the features that influence it, or the impact of uncertainty in the input data. This work introduces a computational framework that allows one to extend the standard t-SNE plot by visual clues about the stability of the t-SNE embedding. First, we perform a sensitivity analysis to determine feature influence: by combining the Implicit Function Theorem with automatic differentiation, our method computes the sensitivity of the embedding w.r.t. the input data, provided in a Jacobian of first-order derivatives. Heatmap-visualizations of this Jacobian or summarizations thereof reveal which input features are most influential in shaping the embedding and identifying regions of structural instability. Second, when input data uncertainty is available, our framework uses this Jacobian to propagate error, probabilistically quantifying the positional uncertainty of each embedded point. This uncertainty is visualized by augmenting the plot with hypothetical outcomes, which display the positional confidence of each point. We apply our framework to three diverse biological datasets (bulk RNA-seq, proteomics, and single-cell transcriptomics), demonstrating its ability to directly link visual patterns to their underlying biological drivers and reveal ambiguities invisible in a standard plot. By providing this principled means to assess the robustness and interpretability of t-SNE visualizations, our work enables more rigorous and informed scientific conclusions in bioinformatics.