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Gabriel Peyré

@gabrielpeyre102k followers

@CNRS researcher at @ENS_ULM. One tweet a day on computational mathematics.

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1Jun 1, '23Eigenvalues of random matrices with iid entries converge to the Wigner circle law. This is universal (does not depend on the law of the entries), as proved by Tao and Vu in 2010. F16.5×330k1.5k239327
2May 24, '23Taylor series expansion of sine.15.3×306k2.9k493282
3May 27, '24Markov chain Monte Carlo methods are used to sample from a Gibbs distribution without knowing the normalizing constant. As the temperature epsilon gets small, samples cluster close14.1×281k1.4k207620
4Dec 26, '22Dijkstra’s algorithm computes the geodesic distance on a graph in O(n*log(n)) operations using a front propagation algorithm.12.8×255k1.9k265265
5Jun 12, '24Monte Carlo integration approximates integrals at a rate of 1/sqrt(n), independent of the dimension.12.5×249k1.9k340638
6Jan 3, '23Shepard interpolation is a surprisingly simple multi-dimensional interpolation method. It uses singular radial basis functions inversely proportional to the distances to the sample12.2×243k1.8k247304
7Dec 28, '23There are 3 types of non-degenerated quadrics in 3-D: Ellipsoid, Hyperboloid of one sheet, Hyperboloid of two sheets.12.1×241k2.4k459468
8Jul 10, '23Heat, Wave and Schrodinger equations are fundamental linear partial differential equations of physics.11.6×232k2.2k453393
9Jul 3, '24Mathematical modeling: part 1/2.11.6×231k3.2k4191.3k
10Apr 20, '23The mean-shift algorithm is a clustering method that progressively evolves points toward local maximum of a kernel density estimator.11.3×226k1.8k297395
11Apr 3, '24Fokker-Planck equation equivalently describes the movement of a random particle with a drift (as a stochastic ODE) and the evolution of its density (as a PDE).11.3×225k1.8k246661
12Jun 11, '23A 1 hidden layer perceptron can approximate arbitrary continuous functions using a large enough number of neutrons (Cybenko's theorem).11.3×225k1.9k332477
13Jan 12, '24The central limit theorem equivalently reads as the convergence of iterated convolutions.10.9×217k1.6k237534
14Jan 2, '23A joint distribution encodes the dependencies between its two marginal distributions (here in the simple case of a density with respect to a product of measures).10.8×215k1.4k185259
15Apr 9, '23The harmonic oscillator is the canonical linear second order ODE. Parametrized by a single damping parameter. Solutions are (possibly complex) exponentials. Interpolates between pe10.7×214k1.1k228171
16May 18, '23The fractional Laplacian is a differential operator generalizing the Laplacian. They are the only rotation/translation/scale invariant linear operator.10.7×213k1.8k294293
17Dec 29, '22My periodic reminder that my book on the discrete Fourier transforms (incl. FFT, finite fields valued FT, representation theory, and more) is available for free.10.4×207k1.8k356649
18Dec 16, '22Heat diffusion vs. wave equation on a surface.10.4×207k2.2k326210
19Jun 4, '23Comparison of the Wasserstein, Hellinger, Kullback-Leibler and reverse KL on the space of Gaussian distributions.10.4×206k1.4k252350
20Jun 10, '23Fourier is to convolution what Legendre is to inf-convolution.10.2×203k1.4k236247
21Dec 29, '22Eigenfaces are singular vectors of an image dataset. A principal components analysis is applied to reduce dimensionality and perform face recognition.9.9×197k1.1k179188
22Feb 23, '23Training a multilayer perceptron with a single hidden layer corresponds to the evolution of a sparse probability measure (sum of Dirac masses) over the neuron's parameter domain (h9.1×182k1.4k225336
23Jan 22, '25The Mathematics of Artificial Intelligence: In this introductory and highly subjective survey, aimed at a general mathematical audience, I showcase some key theoretical concepts un9.1×181k1.9k4011.9k
24Dec 22, '22Stein’s lemma is a surprisingly simple yet useful characterization of the Gaussian distribution. Enables to derive sharp bounds for convergence rates in the central limit theorem i8.9×178k1.2k150183
25Apr 19, '23The Helmholtz-Hodge decomposition split vectors fields in two orthogonal subspaces: incompressible and irrotational. In 2-D, corresponds to the sum of a gradient field and a rotate8.6×171k1.3k231214
26May 12, '23Lorenz attractor is the set of limit trajectories for a simplified mathematical model for atmospheric convection. It has a fractal geometry.8.5×169k1.5k281157
27May 28, '23Optimization can be carried over on manifolds using gradient descent, where the manifold gradient is the projection of an extrinsic gradient on the tangent plane.8.1×162k1.4k231293
28May 29, '24Convolutional neural networks are shift-invariant representations obtained by iterating convolutions and pointwise non-linearities. Introduced by Le Cun.8.1×161k1.2k132411
29Oct 28, '23Inverse of Laplacian matrix can be used to define covariance kernels with positive pairwise correlations between nodes.8.0×159k1.2k197379
30Feb 18, '23Schrodinger’s problem is an approximation (regularization using diffusion) of the Optimal Transport problem.7.9×157k1.6k273265
31Jul 30, '23The Shrödinger problem defines a dynamic interpolation between two distributions using Brownian bridges. The 0 temperature limit is Optimal Transport.7.8×155k1.3k249310
32Apr 12, '23Only harmonic spring and gravity central forces produce periodic motions.7.8×155k1.3k223163
33Jan 26, '23Sampling a signal is equivalent to periodizing its Fourier transform. For non-smooth signals (with non-compact Fourier transform), it creates aliazing.7.5×149k1.3k198252
34Mar 13, '23Parabolic PDEs (e.g. heat) smooth out singularities. Hyperbolic PDEs (e.g. wave) displace singularities.7.5×149k1.2k220188
35May 9, '23Behavior of dynamical system depends on the eigenvalues. In 2D it can be classified according to the trace and determinant.7.4×148k1.3k216229
36Jun 18, '23The mean curvature motion is the most fundamental curve evolution. It is the equivalent for curves of the heat equation. It is independent of parameterisation (intrinsic) and minim7.4×147k1.3k221253
37Feb 13, '23Burgers’ equation is one of the simplest nonlinear partial differential equations, with solutions exhibiting shocks.7.3×146k1.4k253207
38Jun 7, '23A Brownian motion is approximated by a discrete random walk with Gaussian increments.7.2×144k1.6k255252
39Mar 19, '23Harmonic functions are obtained by solving Laplace equation and define interpolating functions when using Dirichlet boundary conditions.7.1×143k1.3k198164
40Jun 26, '23The Levenberg-Marquardt is a standard method for non-linear least-squares, combining the best of gradient descent and Newton methods by replacing the Hessian by the Jacobian term o7.1×141k1.1k220318
41May 8, '23The convex hull is the smallest enclosing convex set. It is also the intersection of all half-spaces containing the points, and the set of all possible convex combinations.6.9×139k1.1k180182
42May 5, '23Taylor series of exp(it) up to degree 15.6.9×138k1.3k208125
43Mar 21, '23The simple gravity pendulum is one of the simplest nonlinear second order ODE. Exhibits elliptic-shape orbits in phase space for small enough initial speeds.6.9×137k1.4k269173
44Mar 6, '24Heat equation vs wave equation inside a planar domain. Boundary conditions matter for PDEs.6.6×131k2.5k452603
45Feb 21, '23Adding an independent Gaussian variable is the same as doing a heat diffusion on the density.6.6×131k1.1k159119
46Apr 27, '23The damped harmonic oscillator (linear) vs simple gravity pendulum (non-linear). The two most celebrated second order dynamical systems.6.5×129k1.2k234170
47Aug 5, '24"Transformers are Universal In-context Learners": in this paper, we show that deep transformers with a fixed embedding dimension are universal approximators for an arbitrarily larg6.4×128k1.5k302849
48Apr 24, '23Impact of the interpolation method on the level set of the resulting function.6.3×125k1.1k152180
49Nov 5, '23Generalized means interpolate between the min, the geometric mean, the arithmetic mean and the max.6.3×125k1.1k173219
50Aug 18, '23Positive semi-definite matrices come in many flavors!6.2×125k1.1k235271
51Mar 28, '23Gauss-Luca theorem shows that the roots of the derivative of a polynomial belong to the convex hull of the roots of the polynomial.6.2×123k1.2k207132
52Jun 20, '23Fourier transforms comes with lots of flavors. Sampling and periodization are dual operations which interleave these transforms.6.1×122k1.5k284258
53Nov 17, '24The discrete Fourier basis is a sampling of the continuous Fourier basis. They are both orthogonal. This is (almost) magic!6.1×121k1.2k161469
54Mar 22, '23Taylor series approximate within its radius of convergence a function by a polynomial.6.0×120k1.4k230153
55Dec 26, '23Heavy-ball dynamics (second-order ODEs) induce an advection equation (Liouville equation) of a point cloud (in the mean-field limit of an infinite number of particles: a density) o6.0×120k1.3k214319
56Nov 14, '23Oldies but goldies: F. Chung, Spectral Graph Theory. Everything you want to know about the Laplacian of a graph!6.0×119k1.1k179332
57Aug 1, '23I have written a very short mathematical summary of the main ingredients of denoising diffusion models. Most of this is directly borrowed from the amazingly clear slides of @Valent5.8×116k1.3k232745
58May 30, '23A sub-group of Möbius transforms are those preserving the Poincaré upper-half (equivalently the disk). It corresponds to PSL(2) the 2x2 matrices with determinant 1.5.7×114k1.2k221150
59Jun 17, '23A Riemannian manifold is locally an Euclidean space. An embedded surface (e.g. the Earth) is an example. It defines a Riemannian tensors field (first fundamental form) in parametri5.7×113k1.3k266196
60May 10, '23Self-organizing maps of Kohonen defines a map between a 2D space to the data, by minimizing a quantization error and a regularity penalty.5.6×111k1.2k211228
61Apr 15, '24Solving linear systems comes in different flavors.5.5×111k1.7k238778
62May 5, '23Slides of my talk of my tutorial talk "the mathematics of neural networks" at the annual meeting of Société Française d'Optique next Thursday at @ENS_ULM5.5×110k1.1k199512
63May 31, '24The gradient field defines the steepest descent direction. The gradient flow dynamic defines a segmentation of the space into attraction basins of the local minimizers.5.5×109k1.2k204352
64Dec 18, '22The Delaunay triangulation is the dual of the Voronoi diagram. The most fundamental tool of computational geometry.5.4×107k1.1k200176
65Mar 28, '24In the static regime, the electric/magnetic fields generated by point sources are the superposition of radial monopoles/dipoles.5.2×104k2.1k332599
66Mar 4, '23Reaction-diffusions are nonlinear PDEs which describe the formation of a surprisingly rich family of patterns. Introduced by Alan Turing in 1952.5.0×99k1.1k202160
67Feb 12, '23Fourier approximation of a cat.4.8×96k1.2k21293
68Feb 15, '23Comparing 2D linear interpolation methods of order 0, 1 and 3.4.8×95k1.1k139124
69Apr 21, '24The optimal computation of gradients for the composition of functions is an optimal parenthesis problem. Forward and backward (backpropagation) are two extreme cases. Backward is o4.7×94k1.4k240667
70Jan 26, '24Gradient descent does not always converge. Trajectories can have infinite length. Picture & insights from @GuillaumeG_4.6×91k1.3k202349
71Nov 30, '23Many parametric probability distribution families can be written as maximum entropy distributions.4.5×90k1.4k255365
72Jan 8, '24Spherical harmonics is an orthogonal basis of eigenvectors of the Laplacian on the sphere. Eigenspaces of fixed frequency are rotation invariant. The basic tool to perform linear a4.4×88k1.4k228374
73Jul 6, '24Lloyd's algorithm is the continuous counterpart of k-means. Optimizes the optimal quantization energy.4.4×88k1.9k302661
74Jun 16, '23The integral curves of a vector field tangent to a manifold stays on the manifold.4.4×87k1k163113
75Jan 10, '24The three ways to solve Fokker-Planck equations: PDE evolution, stochastic (independent) or deterministic (coupled) particle systems.4.3×85k1.2k236382
76Mar 12, '24Conformal maps (differentiable over the complex plane) can be defined infinitesimally as circle packings.4.0×80k1.5k255421
77Aug 1, '24Large-scale N-body simulations require the evaluation of sums involving long-range interaction kernels. Fast multipoles and Barnes–Hut methods approximate these sums in n*log(n) op3.7×75k1.1k195367
78Mar 20, '24Heat vs wave equations.3.3×66k1.3k208300
79Apr 5, '24The Runge-Kutta formulas are the workhorse for the numerical integration of ordinary differential equations. shttps://en.wikipedia.org/wiki/Runge%E2%80%93Kutta_methods3.1×62k1.2k189295
80Jan 28, '24The Brownian motion (aka Wiener process) is the scaling limit of a random walk. The Brownian bridge forces the path to interpolate between two points.2.9×59k1k152269
81Dec 2, '24Optimal transport computes an interpolation between two distributions using an optimal coupling. Flow matching, on the other hand, uses a simpler “independent” coupling, which is t2.9×59k1.4k197666
82Jul 14, '24The Minkowski sum is commutative, associative, and distributive with respect to union. Defines an interpolation between sets.2.9×57k1.1k157338
83Mar 24, '20I have put online the course notes for my master course on optimization for ML. Covers convex analysis, gradient descent, SGD, Lasso, autodiff, MLP.2.8×02.4k624643
84Mar 19, '24Oldies but goldies: Claude Shannon, A Mathematical Theory of Communication, 1948. Defines, among many things, the notion of entropy as a fundamental limit for compression.2.7×53k1.1k211268
85Jul 24, '24Non-circular gears are ubiquitous and generate non-constant rotation speeds. The movement of the second gear is obtained from the first one from the no-slip condition, after solvin2.4×49k1.3k226322
86Dec 5, '17Computational Optimal Transport: the book. All you ever wanted to know about OT: theoretical insights, algorithms and applications!2.1×01.9k534362
87Sep 24, '24Approximation of a function on the sphere using an increasing number of spherical harmonics.2.1×42k1.1k163295
88Jul 5, '22I wrote a Python tutorial on Conformal Prediction, which is a simple yet powerful method to compute confidence intervals for any prediction method.2.1×01.7k272664
89Jun 16The alpha version of my new book "Optimal Transport for Machine Learners" is out, with in particular an online version with interactive figures1.9×39k490104443
90Oct 30, '22Lloyd’s algorithm is the continuous counterpart of k-means. Minimizes the optimal quantization energy, which is an optimal transport distance to free Dirac masses.1.8×01.8k277399
91Nov 5, '22The uncertainty principle states that a function cannot be well concentrated both in space and in frequency.1.8×01.8k327332
92Aug 5, '21Heat equation vs wave equation inside a planar domain. Boundary conditions matter for PDEs.1.6×01.9k310153
93Aug 8, '20Taylor series expansion of sine.1.5×01.8k31886
94Sep 16, '22Linearly interpolating matrices produces intriguing interpolations of the eigenvalues.1.3×01.4k194225
95Jul 1, '22Shepard interpolation generalizes nearest neighbor interpolation.1.2×01.3k166211
96Jan 5, '20Lloyd’s algorithm is the continuous counterpart of k-means. Optimizes the optimal quantization energy, which is an optimal transport distance to free Dirac masses.1.1×01.2k235159
97Jul 23, '22Page rank is the leading eigenvector of a stochastic matrix. Can be computed efficiently for sparse graphs using power iterations.1.1×01.1k166265
98Sep 14, '21Solving linear systems comes with different flavors.1.1×01.1k175208
99Sep 8, '19Heat diffusion vs. wave equation on a surface.1.1×01.1k29653
100Aug 11, '22Spherical harmonics are the equivalent on the sphere of the Fourier basis. Useful to perform low-frequency approximation of functions.1.0×01.1k161159
101Apr 19, '20Gauss-Luca theorem shows that the roots of derivative of a polynomial belongs to the convex hull of the roots of the polynomial.1.0×01k25574

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