What Can We Infer from t-SNE, UMAP, and Other Nonlinear 2D Maps of High-Dimensional Representations?

A common practice in machine learning and scientific data analysis is to reduce a high-dimensional representation to two dimensions using methods such as t-SNE or UMAP, then color the resulting map by semantic variables—cell type, digit identity, topic label—and read the visible organization as evidence about what the representation contains. A variable that forms clear islands in 2D may be interpreted as strongly represented, while one that appears mixed may be taken as weak or absent. We show that this backward inference can break in ways not captured by geometric fidelity alone: a variable carrying more information in the high-dimensional representation can appear less structured than a weaker one on the same fixed map — the visual ranking can reverse the information ranking. Furthermore, the mixed appearance itself is not diagnostic: nearly identical coarse 2D mixing statistics can arise from opposite underlying information states, one in which all information carried by the original noisy Y-channel is retained and another in which it is entirely absent. These results reveal a decoupling between information present before mapping and semantic expression after it. At the same time, nonlinear 2D maps can retain useful semantic structure: class-conditioned neighborhood relations — which classes tend to occur near one another — can remain similar even when most individual neighbors change. This semantic stability can coexist with substantial reorganization of between-class geometry and class-local persistence profiles. Taken together, these results highlight that semantic patterns in 2D can be informative about selected relationships in a high-dimensional representation without faithfully encoding its information content or geometry. A 2D map can therefore provide a useful view of a semantic relation without establishing the strength of the underlying information, the geometry from which it arose, or the broader scientific interpretation attached to it.