#Superintelligence - Add an ai prompt to any function in python sklearn or numpy like modules to get experienced params. So for ai or machine learning you add prompt describing the problem so you get experience adapted parameters and result. Gabriel Peyré

The Fourier slice theorem relates the 1D Fourier transform of Radon projections to the 2D transform of the image. Useful to analyze and invert scanner medical imaging. en.wikipedia.org/wiki/Projectio…

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Oldies but goldies: L Kantorovich, On translocation of masses, 1942. Nobel Prize in Economy in 1975 for a description of optimal transport as a linear program. en.wikipedia.org/wiki/Leonid_Ka…

The Fast Marching algorithm is a generalization of Dijkstra’s algorithm. Computes the geodesic distance in O(n*log(n)) operation. Equivalently solves the non-linear Eikonal equation in a non-iterative way by front propagation. en.wikipedia.org/wiki/Fast_marc… nbviewer.jupyter.org/github/gpeyre/…

Subdivision curves exist in two flavors: interpolating and approximating. numerical-tours.com/matlab/meshwav… ibiblio.org/e-notes/Spline… en.wikipedia.org/wiki/Spline_(m…

In case you are wondering, this paper proves that, in general, diffusion models do not define optimal transport maps. The proof is not straightforward though (diffusion maps are optimal maps in 1D, for radial measure and for Gaussians ...) cvgmt.sns.it/media/doc/pape…

Any pair of planar triangulations of n vertices can be connected by O(n) edge flips (despite the exponential number of such triangulations!). en.wikipedia.org/wiki/Klaus_Wag… eudml.org/doc/146109

The Fisher metric defines the unique Riemannian structure of parametric densities invariant by re-parameterization. For 1D Gaussians, corresponds to the Poincaré hyperbolic half plane. en.wikipedia.org/wiki/Fisher_in…

Oldies but goldies: Y Brenier, Polar factorization and monotone rearrangement of vector‐valued functions, 1991. Proved the uniqueness of the solution to Monge's problem. Showed that it is the gradient of a convex function solving the Monge-Ampère equation. en.wikipedia.org/wiki/Transport…

Gabriel Peyré Jason Lee Yeah, Charlotte's work on Gaussian SB does help a bit. but entropic regularization seems to make the flow path slightly curved. arxiv.org/pdf/2202.05722

Jason Lee Gabriel Peyré I have one paper under review (to appear soon) studying exactly this problem. The vanilla diffusion model fails to generate distributions of anisotropic shapes effectively, while a more transport-efficient diffusion model solves this problem nicely.

Oldies but goldies: Eugene Wigner, Characteristic Vectors of Bordered Matrices with Infinite Dimensions, 1955. The empirical distribution of eigenvalues of random symmetric matrices converges to a half-circle density. en.wikipedia.org/wiki/Wigner_se…

Gabriel Peyré It looks like this work of Tanana proved it: arxiv.org/abs/1709.06464

Oldies but goldies: Andrew Berry, The Accuracy of the Gaussian Approximation to the Sum of Independent Variates, 1941. Provides a quantitative estimation of the convergence speed of the central limit theorem. en.wikipedia.org/wiki/Berry%E2%…

Gabriel Peyré David Picard reminds me of arxiv.org/abs/2210.02747

Gabriel Peyré Here is an example (I made 6 months ago) of diffusion trajectories from the Gaussian to the empirical measure of a sampling from the Four corner Gaussians. It uses the analytical solution of the diffusion.

Gabriel Peyré This is a special case of the “inner-product slice theorem”. One gets the 'Fourier slice theorem' by setting H(p) to a complex exponential. The general (inner product) property makes Radon useful for applications besides medical imaging

Gabriel Peyré Are these posts archived somewhere?