Combining multiple distances

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spoiler alert: this is a theory question, so not directly related to the library. Sorry if it's the wrong place to ask, feel free to close/delete then

@lmcinnes I just watched your PyData Ann Arbor presentation and got curious about your point on combining distances. You say something to the tune of "as long as you define a distance for a datatype (column), you can use that to embed the object".

How does that work, exactly? Let's say that I have a reading with 3 dimensions and another two dimension specifying, say, hour and day of the week. Do I need to provide the distance between objects in this 5-dimensional type or can I just define a distance using the first 3 dimensions and a second one using the last 2? I Got the impression that it's the second case (thus the supervised embedding capabilities of UMAP), but then you're implicitly saying that you can combine multiple distances with varying underlying distributions. Is that the case? And if so, can you give any pointers regarding how that works or how/under what assumptions it makes sense?

Because that's the general case of combining multiple distances/scores without explicit weighting which is a very broad (and hard) problem that can have lots of applications by itself.

Thanks in advance and congratulations for the great research and library (and HDBSCAN, too!)

No file, test, or entry point is identified. Start by reviewing the cited PyData Ann Arbor presentation and the UMAP project's explanation of supervised embeddings and combined distances; done would be a documented, technically supported answer to the question.

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