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atomicarchitects / PriceofFreedom

[GRaM at ICML'24] The Price of Freedom: Exploring Tradeoffs between Expressivity and Computational Efficiency in Equivariant Tensor Products
https://openreview.net/forum?id=0HHidbjwcf
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Questions about "GRID TENSOR PRODUCT" #3

Open liangzhixin-202169 opened 1 month ago

liangzhixin-202169 commented 1 month ago

Hi! I have several questions about the grid tensor product. (1)Signal is projected to sphere directly, so how many points should be sampled in sphere? (2)How to choose quadrature method?(3) Dose it consume more memory than 2D fourier bases?

mitkotak commented 4 days ago

Hey sorry for the delay replying, regarding (3) should be similar though the 2D Fourier one currently doesn't use sparsity. Also regarding the reply on the notebook that I had on open review, yes I fixed the grid resolution there but ideally you would wanna have some kind of varying grid based on lmax. @xyuqing is planning to answer that in his reply for (1) and (2)

xyuqing commented 4 days ago

Hi, thanks for your interest in our work! (1) Based on various sampling theorems, the number of points sampled scales as O(L^2) in order to perfectly reconstruct a spherical harmonic decomposition up to degree L. The table here summarizes this for various grid types https://shtools.github.io/SHTOOLS/grid-formats.html#comparison-of-dh-and-glq-grids (2) Generally Gauss-Legendre quadrature uses fewer points to achieve perfect fidelity so it should be faster