Fenchel-Rockafellar duality leverages linear operators in Lagrange duality. https://en.wikipedia.org/wiki/Fenchel%27s_duality_theorem https://en.wikipedia.org/wiki/Duality_(optimization)
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Eigenvectors of the Laplacian on compact planar domains define an orthogonal basis of oscillating functions, which generalize Fourier sinusoids. https://en.wikipedia.org/wiki/Dirichlet_eigenvalue
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The Mathematics of Artificial Intelligence: In this introductory and highly subjective survey, aimed at a general mathematical audience, I showcase some key theoretical concepts underlying recent advancements in machine learning. https://arxiv.org/abs/2501.10465
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Oldies but goldies: J-J Moreau, Proximite et dualite dans un espace hilbertien, 1965. Moreau-Yosida regularization smoothes a function by inf-convolution. https://en.wikipedia.org/wiki/Convex_conjugate#Infimal_convolution
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Moreau's decomposition generalizes the orthogonal decomposition to general functions. It can also be generalized beyond Euclidean space using Bregman divergences in place of Euclidean distance. https://hal.archives-ouvertes.fr/hal-01076974/document
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Oldies but goldies: L. Greengard and V. Rokhlin, A Fast Algorithm for Particle Simulations, 1987. Evaluates in O(n) in place of O(n^2) sums involving long-range interaction kernels. https://en.wikipedia.org/wiki/Fast_multipole_method
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Analyzing the global convergence of Newton is hard. Attraction basins are fractals whose boundaries are points which do not converge. https://en.wikipedia.org/wiki/Newton_fractal
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I wrote a summary of the main ingredients of the neat proof by Hugo Lavenant that diffusion models do not generally define optimal transport. https://github.com/mathematical-tours/mathematical-tours.github.io/blob/971ddb3aab5803c7a4abef122f878292f6a6c25d/book-sources/diffusion-models/note-diffusion-ot.pdf
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K-means algorithm computes a stationary point of the quantization error by alternating between computation of Voronoi partition and computation of barycenters. https://en.wikipedia.org/wiki/K-means_clustering
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Oldies but goldies: B Cabral, L C Leedom, Imaging Vector Fields Using Line Integral Convolution, 1993. Line integral convolution is an anisotropic filtering which averages values along streamlines. Can be used for visualization of vector fields by diffusing noise. https://t.co/dRVFhkMsrJ
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A graph is planar if and only if it does not contains as minor the complete graph K5 or the bipartite graph K33. https://en.wikipedia.org/wiki/Wagner%27s_theorem
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Gradient descent on particles’ positions (Lagrangian) is equivalent to an advection linear PDE on the density of particles (Eulerian). https://en.wikipedia.org/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field
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The discrete Fourier basis is a sampling of the continuous Fourier basis. They are both orthogonal. This is (almost) magic! https://en.wikipedia.org/wiki/Discrete_Fourier_transform
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The Poisson point process has a flat power spectrum while the Poisson disk exhibits a desirable « blue noise » spectrum. Matches the arrangement of photoreceptors in the retina. https://en.wikipedia.org/wiki/Supersampling#Poisson_disc https://en.wikipedia.org/wiki/Point_process https://en.wikipedia.org/wiki/Halftone
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Compactly supported orthogonal wavelets are defined using filter banks. https://en.wikipedia.org/wiki/Daubechies_wavelet https://en.wikipedia.org/wiki/Ingrid_Daubechies
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The "eta-trick" used to represent in a variational form |x| can also be used to represent the Laplace distribution as a mixture of Gaussians. https://francisbach.com/the-%CE%B7-trick-or-the-effectiveness-of-reweighted-least-squares/ https://statisticaloddsandends.wordpress.com/2018/12/21/laplace-distribution-as-a-mixture-of-normals/
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Oldies but goldies: R. Hamming, Error detecting and error correcting codes, 1950. Introduces the first efficient binary error-correcting codes. Can correct any single-bit error, or detect all single-bit and two-bit errors. https://en.wikipedia.org/wiki/Hamming(7,4) https://en.wikipedia.org/wiki/Hamming_code
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Gaussian functions are stable under pointwise and convolution products. The Fourier transform interleaves these two algebraic structures.
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Moreau's decomposition generalizes orthogonal decomposition from linear spaces to convex cones. https://www.convexoptimization.com/wikimization/index.php/Moreau's_decomposition_theorem https://en.wikipedia.org/wiki/Convex_cone
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The l^p functional is convex and hence a norm for p>=1. It is sparsity-inducing for p<=1. The l^1 norm is the heart of the lasso, aka basis pursuit. https://en.wikipedia.org/wiki/Basis_pursuit_denoising https://en.wikipedia.org/wiki/Lasso_(statistics)
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