Gabriel Peyré — BlogNotes on mathematics, optimal transport, machine learning, and scientific life, by Gabriel Peyré.https://www.gpeyre.com/blog/atom.xml2026-10-03T00:00:00+00:00Gabriel Peyréhttps://www.gpeyre.com/JekyllInverse optimisation and the convexity of the gap losshttps://www.gpeyre.com/blog/2026/10/03/inverse-optimisation-gap-loss/2026-10-03T00:00:00+00:002026-10-03T00:00:00+00:00Gabriel PeyréThe convex optimality gap as a Fenchel–Young loss, with an interactive inverse optimal transport experiment between two Gaussian mixtures and a linked bibliography.Why I love Gaussian-preserving flowshttps://www.gpeyre.com/blog/2026/09/15/gaussian-preserving-wasserstein-flows/2026-09-15T00:00:00+00:002026-09-15T00:00:00+00:00Gabriel PeyréGelbrich's Gaussian contraction and a criterion explained by Hugo Lavenant, with proofs for Gaussian relative entropy and heat flow.IA pour les maths : clarificationshttps://www.gpeyre.com/blog/2026/09/15/ia-pour-les-maths-clarifications/2026-09-15T00:00:00+00:002026-09-15T00:00:00+00:00Gabriel PeyréUne clarification de mon intervention sur France Inter, en français et en anglais.Fast gradient flows without convexityhttps://www.gpeyre.com/blog/2026/08/14/polyak-lojasiewicz-gradient-flows/2026-08-14T00:00:00+00:002026-08-14T00:00:00+00:00Gabriel PeyréWhy a gradient inequality can be enough for exponential convergence, illustrated by an interactive curved valley.The Mathematical Nexushttps://www.gpeyre.com/blog/2026/08/07/mathematical-nexus/2026-08-07T00:00:00+00:002026-08-07T00:00:00+00:00Gabriel PeyréA collection of mathematical animations and accompanying Python notebooks, now gathered in the Mathematical Nexus.From Perron–Frobenius to Sinkhornhttps://www.gpeyre.com/blog/2026/07/30/perron-frobenius-and-sinkhorn/2026-07-30T00:00:00+00:002026-07-30T00:00:00+00:00Gabriel PeyréCompare a positive Markov operator with Sinkhorn’s nonlinear scaling map in an interactive simplex.The KL barycenter of Gaussians is Gaussianhttps://www.gpeyre.com/blog/2026/07/13/kl-barycenters-of-gaussians/2026-07-13T00:00:00+00:002026-07-13T00:00:00+00:00Gabriel PeyréA short derivation of the reverse-KL Gaussian barycenter, with an animated two-dimensional example.Duhamel’s formula for matrix productshttps://www.gpeyre.com/blog/2026/07/10/discrete-duhamel-formula/2026-07-10T00:00:00+00:002026-07-10T00:00:00+00:00Gabriel PeyréDerive the discrete Duhamel identity and use it to propagate local perturbations through a matrix product.A convex functional hidden behind the Legendre transformhttps://www.gpeyre.com/blog/2026/07/07/legendre-transforms-and-moment-measures/2026-07-07T00:00:00+00:002026-07-07T00:00:00+00:00Gabriel PeyréWhy minus the log-integral of the exponential of a negative convex conjugate is convex, and how its variation produces a moment measure.Two notions of positivity, separated by a pentagonhttps://www.gpeyre.com/blog/2026/07/04/doubly-nonnegative-completely-positive/2026-07-04T00:00:00+00:002026-07-04T00:00:00+00:00Gabriel PeyréA five-by-five matrix and an elementary proof of the gap between doubly nonnegative and completely positive matrices.Optimal Transport for Machine Learnershttps://www.gpeyre.com/blog/2026/06/16/optimal-transport-for-machine-learners/2026-06-16T00:00:00+00:002026-06-16T00:00:00+00:00Gabriel PeyréThe alpha version of my new book is online, including interactive figures.Diffusion versus optimal transport: two maps from one Gaussianhttps://www.gpeyre.com/blog/2026/06/15/diffusion-versus-optimal-transport/2026-06-15T00:00:00+00:002026-06-15T00:00:00+00:00Gabriel PeyréCompare the deterministic score-based probability flow with quadratic optimal transport from a centered Gaussian to a three-Gaussian mixture, down to the three-Dirac limit.When does a covariance gauge define a matrix norm?https://www.gpeyre.com/blog/2026/05/06/covariance-gauges-and-matrix-norms/2026-05-06T00:00:00+00:002026-05-06T00:00:00+00:00Gabriel PeyréA detailed proof that a positive convex homogeneous covariance gauge lifts to a matrix norm exactly when it is monotone.Straighter flows, smaller Euler errorshttps://www.gpeyre.com/blog/2026/04/16/straight-flows-and-euler-error/2026-04-16T00:00:00+00:002026-04-16T00:00:00+00:00Gabriel PeyréA detailed Euler error bound isolating trajectory acceleration, with a proof and an interactive example.Muon as a spectral Wasserstein flowhttps://www.gpeyre.com/blog/2026/04/06/muon-spectral-wasserstein/2026-04-06T00:00:00+00:002026-04-06T00:00:00+00:00Gabriel PeyréHow Muon's polar update connects to a Wasserstein gradient flow defined through displacement covariances.Solving a positive definite congruence equationhttps://www.gpeyre.com/blog/2025/08/09/positive-definite-congruence/2025-08-09T00:00:00+00:002025-08-09T00:00:00+00:00Gabriel PeyréAn explicit solution of UYU = X, with proofs of uniqueness and smooth dependence on positive definite matrices.Which linear maps preserve matrix rank?https://www.gpeyre.com/blog/2025/07/02/linear-rank-preservers/2025-07-02T00:00:00+00:002025-07-02T00:00:00+00:00Gabriel PeyréWhy every linear rank preserver is a left-right change of basis, possibly composed with transposition in the square case.Tout comprendre (ou presque) sur l’intelligence artificiellehttps://www.gpeyre.com/blog/2025/03/28/tout-comprendre-intelligence-artificielle/2025-03-28T00:00:00+00:002025-03-28T00:00:00+00:00Gabriel PeyréA personal recommendation of Olivier Cappé and Claire Marc's illustrated book on artificial intelligence, forthcoming from CNRS Éditions.