Those are mostly blog posts, notes, talk slides, nice pictures and various things about mathematics, statistics, CS and machine learning.
Maths & ML Gems (Updated in 2025) - My personal curated list of old and recent outstanding papers in applied mathematics.
These series of notes focus on diffusion-based generative models, like the celebrated Denoising Diffusion Probabilistic Models; they contain the material I regularly present as lectures in some working groups for mathematicians or math graduate students, so the style is tailored for this audience. In particular, everything is fitted into the continuous-time framework (which is not exactly how it is done in practice).
In 2025 S2, I will probably completely rewrite and reorder these notes. I think that nowadays, there is no reason other than historic to present diffusion models by starting from the reversal of the SDE. The Flow Matching perspective should definitely be favored, the rest should only be anecdotic.
Flow models I: Diffusion models (March 2023) - Add noise to stuff, then remove it.
Flow models II: Score Matching Techniques (March 2025) - Learning the score of a density from its samples and Tweedie's formula.
Flow models III: Flow Matching (March 2025) - Flow straight, flow fast: velocity is everything.
Flow Models IV: What is Classifier-Free Guidance? (March 2025) - From unconditional to conditional generative models
Flow models V: consistency, flow maps and distillation (March 2025) - Solving ODEs and SDEs is long. Can we directly learn the whole map?
Flow Models V: Latent Diffusion (September 2025) - Diffusing in latent spaces is way more efficient.
The dimension of invariant and equivariant linear layers (July 2021) - We compute the dimension of equivariant linear layers in neural architectures.
The ConvMixer architecture: 🤷 (December 2021) - I am training a deep network on a GPU using the Flux.jl library. There are two takeaway messages: 1) patches are all you need, 2) in Julia, the ConvMixer *largely* fits in one Tweet.
Gradient descent I: strongly convex functions (April 2022) - For strongly convex functions, the speed of convergence is determined by the conditionning number of the Hessian.
Gradient descent II: stochastic gradient descent for convex functions (October 2023) - Stochastic Gradient Descent over strongly convex functions nearly behaves like gradient descent.
Gradient descent III: SGD for Polyak-Łojasiewicz functions (October 2023) - Convex functions are not my friends anymore. Now I am best friend with Polyak-Łojasiewicz functions.
Importance sampling ⚖️ (June 2023) - On the sample size required to get a good approximation
Importance sampling ⚖️⚖️ : The Jarzynski connection (March 2022) - Change-of-measure for out-of-equilibrium systems
🐘 The Elephant Random Walk (May 2023) - Long-time memory results in non-diffusivity
Robbins' version of the Stirling approximation (November 2022) - A handy, easy-to-remember estimate for the error in Stirling's approximation.
Super-Catalan (Janvier 2022) - Une question non-résolue, vieille de 150 ans, et probablement très inutile.
Random analytic functions: Ryll-Nardzewski's theorem (April 2021) - What happens at the boundary of the disk of convergence of random analytic series.
Théorème de Mercer et Kernel Trick (Octobre 2023) - Le théorème de représentation des noyaux positifs
🏋🏼 Heavy tails I: extremes events and randomness (November 2023) - A presentation of heavy tails, how they behave, and a short list of where they come from.
🏋🏼 Heavy-tails II: is it really heavy? (December 2023) - A presentation of Hill's estimator for the heavy-tail index.
🏋🏼 Heavy tails III: Kesten's theorem (November 2023) - The solutions of the distributional equation X = AX+B can have heavy tails: a sketch of proof, plus a presentation of the Renewal theorem.
Gaussian conditioning (September 2023) - The conditional distribution of some part of a gaussian vector given the other
The Kullback-Leibler divergence between Gaussians (June 2022) - I'll know once and for all where to find this damn formula.
Mouvement brownien I 📈 : avec une base d'ondelettes (Septembre 2023) - Une généralisation de la construction de Paul Lévy: on construit un mouvement brownien continu en s'aidant d'une base orthonormale.
Mouvement brownien II 📈📈: représentation de Karhunen-Loève (Octobre 2023) - Cette fois on construit un mouvement brownien directement dans une base orthonormale et pas implicitement comme dans la construction de Paul Lévy.
Quadratic exponentials of Gaussian random vectors (2024) - Computation of E[exp(q(X))] where q is quadratic and X is gaussian.
nothing (nothing) - nothing
An inverse visualization for the elliptic law (March 2021) - A beautiful colorplot of the characteristic polynomial of random matrices.
Random line on the plane (August 2021) - How can we draw random lines on the plane?
Waves on donuts (August 2021) - A nice plot of random Laplace eigenfuctions on the torus, also called random arithmetic waves.
Tips and tricks in the Julia language (August 2022) - A personnal collection of nice tricks in Julia.
The point of view of Professor Parapine (November 2022) - An interesting vision of Science from one century ago.