Unexplainable Explanations: Towards Interpreting tSNE and UMAP Embeddings

It has become standard to explain neural network latent spaces with attraction/repulsion dimensionality reduction (ARDR) methods like tSNE and UMAP. This relies on the premise that structure in the 2D representation is consistent with the structure in the model's latent space. However, this is an unproven assumption -- we are unaware of any convergence guarantees for ARDR algorithms. We work on closing this question by relating ARDR methods to classical dimensionality reduction techniques. Specifically, we show that one can fully recover a PCA embedding by applying attractions and repulsions onto a randomly initialized dataset. We also show that, with a small change, Locally Linear Embeddings (LLE) can reproduce ARDR embeddings. Finally, we formalize a series of conjectures that, if true, would allow one to attribute structure in the 2D embedding back to the input distribution.