Top: Markov chains contract the probability simplex toward the unique stationary positive eigenvector (Perron-Frobenius). Bottom: Sinkhorn contracts the simplex non-linearly to an approximate solution of optimal transport (nonlinear Perron-Frobenius).
The KL barycenter of Gaussians is Gaussian: argmin_μ ∑ᵢ λᵢ KL(μ | 𝒩(mᵢ, Σᵢ)) = 𝒩(m, Σ), with Σ⁻¹ = ∑ᵢ λᵢ Σᵢ⁻¹ m = ∑ᵢ λᵢ Σ Σᵢ⁻¹ mᵢ
Duhamel (discrete) formula for the difference of matrix products: A1 ... An − B1 ... Bn = sum_{k=1}^n A1 ... A_{k-1} (Ak − Bk) B_{k+1} ... Bn
Denoting f⁎ the Legendre transform of f, the map {f convex} ↦ −log ∫ exp(−f⁎) is convex. arxiv.org/abs/1304.0630
Moment Measures
With any convex function F on a finite-dimensional linear space X such that F goes to infinity at infinity, we associate a Borel measure on the dual space X*. This measure is obtained by pushing forwa...
arxiv.org
If θ: ℝ₊ → ℝ is concave and L: ℝᵈ → ℝ is convex, then (a,m) ↦ θ(a)L(m/θ(a)) is convex. The special case θ = id is the perspective transformation (see arxiv.org/pdf/1610.01552). A large class of exotic optimal transport extensions relies on this construction.
arxiv.org
Doubly positive matrices: D_n = { X = Xᵀ in R_+^{n×n}, eig(X) ≥ 0 }. Totally positive matrices: T_n = { X = AAᵀ : ∃p, A in R_+^{n×p} }. One has T_n ⊂ D_n. For n ≤ 4: T_n = D_n. For n ≥ 5: T_n ≠ D_n.
The Difference Between 5×5 Doubly Nonnegative and Completely Positive Matrices – Optimization Online
optimization-online.org
Slides for an introductory talk on IA for maths. speakerdeck.com/gpeyre/ia-fo...
IA for theory
Slide for an introduction talk on IA for maths.
speakerdeck.com
The alpha version of my new book "Optimal Transport for Machine Learners" is out, with in particular an online version with interactive figures www.gpeyre.com/ot4ml/
Drifting vs kernel (un-normalized) flows (yes, I know I love Gaussians). Stochastic matrices (and Sinkhorn divergences) are back in business :)
Thanks to LLMs, I think the exponent for the (refutal) of Erdos unit distance conjecture will be more famous than omega (the exponent for matrix multiplication)!
L'IA générative face au concours d'entrée à l'École normale supérieure blogs.mediapart.fr/david-monnia...
L'IA générative face au concours d'entrée à l'École normale supérieure
Dans un précédent billet, j'avais essayé un outil d'IA générative sur un commentaire historique. Passons maintenant à un énoncé de concours particulièrement difficile.
blogs.mediapart.fr
Fun fact: a norm one covariances lifts to a norm on matrices if and only if it is monotone (e.g., Shatten norms).
While it is intuitively clear that straighter trajectories should reduce discretization error when integrating an ODE (for instance, in flow matching), I could not find a precise bound. I therefore rewrote the proof of Cauchy-Lipschitz to make this explicit. github.com/gpeyre/Discr...
I wrote a short mathematical companion tutorial to my notebook on discrete diffusion models. It gives an informal derivation of the connection between maximum likelihood estimation of the backward transition kernel and denoising score matching. github.com/gpeyre/Discr...
It seems that analysts are not lagging behind ... www.scottnarmstrong.com/2026/04/form...
Formalizing De Giorgi-Nash-Moser theory in Lean - Scott Armstrong
Julia Kempe and I have just completed a Lean 4 formalization of De Giorgi--Nash--Moser theory. We formalized the full slate of interior regularity statements for weak solutions of divergence-form elli...
scottnarmstrong.com
For those interested in normalized gradient methods and optimal transport: I introduce a new class of "spectral" Wasserstein distances for which spectrally normalized gradient descent (Muon but without momentum and small step size ...) is a spectral-W gradient flow: arxiv.org/abs/2604.04891
Muon Dynamics as a Spectral Wasserstein Flow
Gradient normalization is central in deep-learning optimization because it stabilizes training and reduces sensitivity to scale. For deep architectures, parameters are naturally grouped into matrices ...
arxiv.org
Pinsker inequality (which should be your favorite inequality ever, right?!) extends verbatim to matrices!
I have updated my "Optimal Transport for Machine Learners" repository with a Pytorch illustration of Wasserstein gradient flows on pairwise interaction functionals (MMD distances) github.com/gpeyre/ot4ml
What is the set of "means" one can approximate using only arithmetic and harmonic means ? For instance the geometric mean belongs to this closure, but can one approximate any mean sandwitched between the two?
Thought of the day: It is somewhat mysterious why Gaussians remain stable under the particle-minimizing flow (i.e. the Wasserstein gradient flow) for so many widely used energies: entropy, Fisher information, quadratic interaction potentials, functionals depending only on mean and covariance,
#Communiqué 🗞️ La médaille d'or 2025 du CNRS est décernée à Stéphane Mallat, mondialement reconnu pour ses travaux autour des mathématiques appliquées au traitement du signal et à l’intelligence artificielle. 👏 👉 cnrs.fr/fr/presse/en... #TalentsCNRS 🏅
Fun (...) fact: the only linear operators on matrices that preserves the rank are X->AXB, where A and B are invertible (with X->X^T in the square case). This was apparently first proved (?) in 1959 by Marcus and Moyls.
scPRINT is now finally on the Chan Zuckerberg Institute's Model Hub! 🎉 🧬 🌈 It is one more way you can use this cell foundation model to embed, denoise, predict cell type, get gene networks from your data from scratch, or fine-tune it on your own application / usecase: virtualcellmodels.cz...
scPRINT | v1.0 | Virtual Cells Platform
scPRINT is a cell foundation model, also called a Large Cell Model (LCM), trained on single-cell RNA sequence (scRNAseq) data from more than 50M human and mouse cells available through CZ CELLxGENE. Based on the transformer architecture, the model is fully open source and reproducible, with multiple checkpoint sizes available from 2M to 100M parameters. scPRINT demonstrated high performance for genome-wide cell-specific gene network inference when benchmarked against state-of-the-art models (e.g., scGPT, Geneformer v2, GENIE3). In addition, scPRINT has various zero-shot capabilities, including cell embedding, cell label prediction (e.g., cell type, sex, disease), and gene expression imputation, highlighting its potential as a versatile tool for single-cell analysis.
virtualcellmodels.cziscience.com
Le prochain Data Science Colloquium à l'ENS, jeudi 12 juin, sera donné par David Louapre d'Ubisoft, "What modern AI and neuroscience can bring to non-playing characters in video games". David c'est bien sûr également le vulgarisateur scientifique de www.youtube.com/scienceetonn...
If one of the two distributions is an isotropic Gaussian, then flow matching is equivalent to a diffusion model. This is known as Tweedie's formula. In particular, the vector field is a gradient vector, as in optimal transport. speakerdeck.com/gpeyre/compu...
