lmcinnes
Interpolation in data space vs embedding space
First of all, thanks for your great work! In the Weaknesses section of your paper, you talk about the interpretability. I have a question regarding the interpolation properties. Given that UMAP is based on the distance between observations, is it safe to assume that if I interpolate linearly between samples in the data space, the resulting vectors in the UMAP embedding space are also (approximately) on a line between the embeddings of the two samples used for interpolation? In other words: Are the directions of the embedding space interpretable? Thanks in advance. Best, Anthony Mendil.
At best I think you could hope that the interpolated points are on a curve in the embedding space; that curve may be relatively straight, but it also may not. To add to the difficulties the nature of the neighborhoods and the optimization for the embedding can definitely drag things outside the curve that one would expect, so it would be, at best, a noisy curve. So, in summary, I do expect there will exist points and regions where interpolation will actually result in lines or smooth(ish) curves in the embedding space, but you certainly cannot bank on that being true in general, and should not assume it to be the case.
On Tue, Jul 4, 2023 at 8:29 AM Anthony Mendil @.***> wrote:
First of all, thanks for your great work! In the Weaknesses section of your paper, you talk about the interpretability. I have a question regarding the interpolation properties. Given that UMAP is based on the distance between observations, is it safe to assume that if I interpolate linearly between samples in the data space, the resulting vectors in the UMAP embedding space are also (approximately) on a line between the embeddings of the two samples used for interpolation? In other words: Are the directions of the embedding space interpretable? Thanks in advance. Best, Anthony Mendil.
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