Multidimensional KDE based on fastKDE and Laplace-Beltrami eigendecomposition (manifold Fourier) transform

[mateuszbaran]

See https://www.sciencedirect.com/science/article/pii/S0167947316300408 and https://www.math.mcgill.ca/toth/spectral%20geometry.pdf (Analysis on Manifolds via the Laplacian by Yaiza Canzani).
Just a crazy research idea.

[Affie]

FIY @dehann, maybe this is of interest in JuliaRobotics/ApproxManifoldProducts.jl#41

[kellertuer]

But be careful, I would consider this more like a long-term goal to include this. I myself have a lot of other things that are far close / easier to realise on my roadmap already and will for example not work on this any time soon.

[mateuszbaran]

This isn't really something we have the bandwidth to explore in the near future but this is interesting enough idea that I wanted to share it here, in case someone wants to work on it. Unfortunately I don't currently have an application for manifold KDE in projects I'm involved in.

KDE on manifolds looks like something that had enough work so far that the basics are theoretically worked out but so far no one has put it together in one coherent package, except for special cases like the sphere. Working out efficient implementation may be even a research-level problem still.

[dehann]

Hi Affie, thanks. I am working something related to this. The current "most coherent" version of what I got is in ApproxManifoldProducts.jl. I'm currently busy with a major overhaul of Manifolds integration. Will post here when something interesting happens on my end.

Thanks for the paper links, will take a look!