Metric learning supervised UMAP explanation

Hello,

I am using your supervised UMAP variant and all works well.

In this issue i would kindly request some help about explaining (maybe with equations) or mathematically or in words what is specifically the process that takes fit and after transform:
Besides, i understand that with transform i can embed new data in a test set like in your page:
https://umap-learn.readthedocs.io/en/latest/transform.html

Training with Labels and Embedding Unlabelled Test Data (Metric Learning with UMAP)

mapper = umap.UMAP(n_neighbors=10).fit(train_data, np.array(train_labels))
test_embedding = mapper.transform(test_data)

and there is another variant to transform and it is supported in the use of labels like in your page:

https://umap-learn.readthedocs.io/en/latest/supervised.html#training-with-labels-and-embedding-unlabelled-test-data-metric-learning-with-umap

Please help me. I need to understand in detailed how the methods functions and i dont find a paper related with this supervised variant. Also i understand that the approach that i have mentioned is different to parametric umap. Remark i want to understand how the model is saved? a linear model is trained with fit? then when I use transform I would be calling what method?

Thank you in advance.

Sincerely,

Jersson Leon

Start with the supervised UMAP and transform documentation pages linked in the issue, then trace the fit and transform entry points. Document how supervised metric learning handles labels, how new test data is embedded, and how the fitted model is saved and reused, including equations or prose sufficient to distinguish it from parametric UMAP.