Gabriel Peyré   @gabrielpeyre

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





  Tweets by Gabriel Peyré  

Gabriel Peyré    @gabrielpeyre   ·   9/2/2021
Fokker–Planck equation equivalently describes the movement of a random particule with a drift (as a stochastic ODE) and the evolution of its density (as a PDE). https://t.co/F8up4bYD2d
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Gabriel Peyré    @gabrielpeyre   ·   9/18/2021
Calculus of variations studies infinite dimensional optimization problems over functions. The most celebrated examples are geodesics and Brachistochrone. https://t.co/FruykhRVdR https://t.co/TxykJLCr8T
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Gabriel Peyré    @gabrielpeyre   ·   9/20/2021
Optimal computation of gradients is equivalent to optimal parenthesis problem. It is NP hard. Forward and backward (backprop aka adjoint state method) are two extreme cases. Backward is optimal for scalar-valued functions. https://t.co/UVMi13PXJm https://t.co/vPZopuNMyv
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Gabriel Peyré    @gabrielpeyre   ·   10/17/2021
What is the impact on optimization of "over-parameterizing" real numbers by writing x=u*v? Turns out that gradient descent on (u,v)->f(uv) is the same as doing a mirror (aka Hessian manifold) descent for a mysterious hyperbolic entropy.
 
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Gabriel Peyré    @gabrielpeyre   ·   9/19/2021
Oldies but Goldies: E. Wigner, On the Quantum Correction for Thermodynamic Equilibrium, 1932. Introduces Wigner-Ville distributio, mimics probability distribution on space-frequency plane. Useful to define the instantaneous frequency. https://t.co/Sihl8e2blb
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Gabriel Peyré    @gabrielpeyre   ·   7/24/2021
Laguerre (aka power) diagram is a generalization of Voronoi partitions, where weights allow to move the boundaries of the cells in the orthogonal directions. https://t.co/zd3VYuq83g
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Gabriel Peyré    @gabrielpeyre   ·   18 hours
The structure tensor is the local covariance matrix field of the gradient vector field. It encodes the local anisotropy of an image. At the heart of anisotropic filtering and corner detection. https://t.co/RRVod47iMi
 
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Gabriel Peyré    @gabrielpeyre   ·   9/17/2021
Oldies but goldies: D Shepard, A two-dimensional interpolation function for irregularly-spaced data, 1968. Shepard interpolation uses singular radial basis functions inversely proportional to the distances to the samples. Nearest neighbor is limit case. https://t.co/FSKi0Tg8Zv
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Gabriel Peyré    @gabrielpeyre   ·   10/12/2021
Monge and Kantorovitch Optimal Transport are equivalent when the measures are supported on the same number of points. https://t.co/ZS3RjHc50g https://t.co/fEINE2gq3A
 
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Gabriel Peyré    @gabrielpeyre   ·   9/29/2021
Oldies but goldies: Eugene Wigner, Characteristic Vectors of Bordered Matrices with Infinite Dimensions, 1955. The empirical distribution of eigenvalues of random symmetric matrices converges to a half-circle density. https://t.co/DBYQqA1jlE
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Gabriel Peyré    @gabrielpeyre   ·   9/17/2021
Oldies but goldies: D Shepard, A two-dimensional interpolation function for irregularly-spaced data, 1968. Shepard interpolation uses singular radial basis functions inversely proportional to the distances to the samples. Nearest neighbor is limit case. https://t.co/FSKi0SZ5Xv
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Gabriel Peyré    @gabrielpeyre   ·   10/6/2021
SNE and t-SNE perform dimensionality reduction by minimizing a relative entropy (KL) between high and low dimensional spaces. t-SNE improves over SNE by replacing Gaussians by student’s t-distributions in the low dimensional space. https://t.co/xAy7GAmmJD
 
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Gabriel Peyré    @gabrielpeyre   ·   10/2/2021
The Fourier slice theorem relates the 1D Fourier transform of Radon projections to the 2D transform of the image. Useful to analyze and invert scanner medical imaging. https://t.co/w1RkoyJmtk
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Gabriel Peyré    @gabrielpeyre   ·   9/30/2021
Fast Marching algorithm is a far-reaching generalization of Dijkstra. Computes the geodesic distance in O(n*log(n)) operation. Equivalently solves the non-linear Eikonal equation in a non-iterative way by front propagation. https://t.co/6jBGNesdaH https://t.co/fwDrDiHnCK
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Gabriel Peyré    @gabrielpeyre   ·   10/7/2021
Oldies but goldies: A. S. Householder, Unitary Triangularization of a Nonsymmetric Matrix, 1958. Introduces a numerically stable way to compute the QR factorization of a matrix. https://t.co/wZzhoS8Iaa
 
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Gabriel Peyré    @gabrielpeyre   ·   9/20/2021
As pointed out by @vit_tucek, optimal parenthesis for matrix multiplication is not NP hard, it can be solved in O(n*log(n)). Deriving optimal automatic differentiation for purely feedforward graph is easy. https://t.co/vPZopuNMyv
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Gabriel Peyré    @gabrielpeyre   ·   9/25/2021
Oldies but goldies: Y Brenier, Polar factorization and monotone rearrangement of vector‐valued functions, 1991. Proved uniqueness of solution to Monge problem for the squared Euclidean cost. Showed that it is the gradient of a convex function. https://t.co/hqg0HIS6WK
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Gabriel Peyré    @gabrielpeyre   ·   10/7/2021
In "Smooth Bilevel Programming for Sparse Regularization", we show how a simple re-writing of Lasso-type problems leads to a smooth optimization problem 1/4 https://t.co/BMzAEHYE3m 1/ 1 . .
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Gabriel Peyré    @gabrielpeyre   ·   10/1/2021
Oldies but goldies: L Kantorovich, On translocation of masses, 1942. Nobel Prize in economy in 1975 for description of optimal transport as a linear program. Makes the transport problem formulated by Gaspard Monge tractable by allowing splitting of mass. https://t.co/4l9nWCNrCh
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