Globally local and fast explanations of t-SNE-like nonlinear embeddings

Nonlinear dimensionality reduction (NLDR) algorithms such as t-SNE are often employed to visually analyze high-dimensional (HD) data sets in the form of low-dimensional (LD) embeddings. Unfortunately, the nonlinearity of the NLDR process prohibits the interpretation of the resulting embeddings in terms of the HD features. State-of-the-art studies propose post-hoc explanation approaches to locally explain the embeddings. However, such tools are typically slow and do not automatically cover the entire LD embedding, instead providing local explanations around one selected data point at a time. This prevents users from quickly gaining insights about the general explainability landscape of the embedding. This paper presents a globally local and fast explanation framework for NLDR embeddings. This framework is fast because it only requires the computation of sparse linear regression models on subsets of the data, without ever reapplying the NLDR algorithm itself. In addition, the framework is globally local in the sense that the entire LD embedding is automatically covered by multiple local explanations. The different interpretable structures in the embedding are directly characterized, making it possible to quantify the importance of the HD features in various regions of the LD embedding. An example use-case is examined, emphasizing the value of the presented framework. Public codes and a software are available at https://github.com/PierreLambert3/glocally_explained.