RT @marc_lelarge
𝗨-𝗡𝗲𝘁, a convolutional neural network first developed for image segmentation, is now a building block for 𝗦𝘁𝗮𝗯𝗹𝗲 𝗗𝗶𝗳𝗳𝘂𝘀𝗶𝗼𝗻.
Here is a simple implementation with max-pooling for down layers and transposed convolutions for up layers (notebook link in alt-text)
The Optimal Transport geometry of 1-D Gaussians is flat in the (mean,std) plane. In particular, the OT interpolation is a linear interpolation of mean and std.
Reproducing Kernel Hilbert spaces define norms on functions so that solutions of regularized fitting problems are linear sum of kernel functions. Defines non-parametric learning methods (complexity scales with input).
https://en.wikipedia.org/wiki/Reproducing_kernel_Hilbert_space
Reproducing kernel Hilbert space - Wikipedia
Reaction-diffusions are nonlinear PDEs which describe the formation of a surprisingly rich family of patterns. Introduced by Alan Turing in 1952.
https://en.wikipedia.org/wiki/Reaction%E2%80%93diffusion_system http://www.dna.caltech.edu/courses/cs191/paperscs191/turing.pdf
Reaction–diffusion system - Wikipedia
Avec mon collègue mais néanmoins ami @RFlamary nous faisons une conférence grand publique sur le transport optimal et l'IA, le 30 mars à Jussieu. Venez nombreux! C'est gratuit mais il faut s'inscrire. Et vive les points rouges et les cellules de Laguerre.
https://smf.emath.fr/conference-machinesMaths étonnantes - Flamary-Peyre - 2023 | Société Mathématique de France
Le transport optimal a été formulé par Gaspard Monge au 18e siècle. Il s'agit d'optimiser le coût de transport depuis un ensemble de producteurs (par exemple les boulangeries) vers des consommateurs (par exemple les cafés, le matin dans Paris). Ce problème très ancien a connu plusieurs révolutions. Léonid Kantorovitch a expliqué en 1942 comment le reformuler en un problème plus facile à résoudre et à étudier : il a obtenu le prix Nobel d’économie pour ses travaux.
The Weierstrass function is continuous if a<1 but nowhere differentiable if ab>1. The Hausdorff dimension of its graph was conjectured by Mandelbrot in 1977 and proved by Shen in 2016.
https://en.wikipedia.org/wiki/Weierstrass_function https://arxiv.org/abs/1505.03986
Weierstrass function - Wikipedia
Monotone operators (Minty, Browder) generalize monotone functions. (sub)-gradients of convex functions are (maximal) monotone. Skew-symmetric (gradients of min-max) operators are also monotone. These are the two extremal cases (Asplund).
https://en.wikipedia.org/wiki/Monotonic_function
Monotonic function - Wikipedia
The SVD decomposes the action of a matrix into rotations and scalings along the axes.
https://en.wikipedia.org/wiki/Singular_value_decomposition
Singular value decomposition - Wikipedia
The QR decomposition can be computed in a numerically stable way using planar rotation matrices.
https://en.wikipedia.org/wiki/Givens_rotationGivens rotation - Wikipedia
Non-uniform B-splines functions are smooth piecewise polynomial functions, which can be generalized to rational functions by weighting (NURBS). The backbone of computer aided geometric design and extensively used in computer graphics.
https://en.wikipedia.org/wiki/Non-uniform_rational_B-spline
Non-uniform rational B-spline - Wikipedia
