Using Genetic Programming to Find Functional Mappings for UMAP Embeddings

Manifold learning is a widely used technique for reducing the dimensionality of complex data to make it more understandable and more efficient to work with. However, current state-of-the-art manifold learning techniques - such as Uniform Manifold Approximation and Projection (UMAP) - have a critical limitation. They do not provide a functional mapping from the higher dimensional space to the lower-dimensional space, instead, they produce only the lower-dimensional embedding. This means they are "black-boxes" that cannot be used in domains where explainability is paramount. Recently, there has been work on using genetic programming to perform manifold learning with functional mappings (represented by tree/s), however, these methods are limited in their performance compared to UMAP. To address this, in this work we propose utilising UMAP to create functional mappings with genetic programming-based manifold learning. We compare two different approaches: one that uses the embedding produced by UMAP as the target for the functional mapping; and the other which directly optimises the UMAP cost function by using it as the fitness function. Experimental results reinforce the value of producing a functional mapping and show promising performance compared to UMAP. Additionally, we visualise two-dimensional embeddings produced by our technique compared to UMAP to further analyse the behaviour of each of the algorithms.