tweets_main.md

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page	List of tweets	Main tweets	img/birds.jpg

• Differentiable max & argmax operators are defined through concave
• Curve approximation using Fourier series corresponds to drawing the curve
• Navier Stokes equation describes the motion of an incompressible viscous
• Lloyd algorithm can be applied on a surface using the geodesic Voronoi diagram.
• Hölder inequality generalizes Cauchy-Schwarz to arbitrary exponents.
• Approximation vs quantization: two ways to compress.
• Hopfield networks are recurrent networks minimizing an Ising-type energy
• Heat, Wave and Schrodinger equations are fundamental linear partial
• Minkowski inequality states that the l^p norm is a norm.
• Haar transform: only + and - yet surprisingly powerful. The first example
• Pullback of functions and pushforward of measures are dual one with each other.
• The english (very rough) translation of my book "The Discrete Algebra of
• The hairy ball theorem states that in odd dimension d, vector fields on
• Representation theory defines a Fourier transform on groups. Finite
• Shapiro’s inequality: for small integers, cyclic sums of quotients are
• Understanding global convergence of Newton method is hard ...
• The Wasserstein and Hellinger distances between Gaussians can be computed
• The geodesic distance extends a local metric into a global one. Geodesic
• Elastic net (Zou , Hastie, 2005) interpolates between the Lasso (l1
• Error checking code with an additional parity bit defines a simplex in the
• Pinsker inequality is one of the most fundamental inequality in
• The Levenberg-Marquardt is a standard method for non-linear least-squares,
• Non-circular gears generate non-regular motions from regular one.
• Principal component analysis and non-negative matrix factorization are two
• Minimizing sum of p-powers of the distance generalizes the mean (p=2). For
• Curse and blessing of non-smooth optimization: non-smooth parts (eg
• The average distance to origin of a Brownian motion at time t grows like
• Continued fractions are in some sense optimal ways to approximate with
• Fourier transforms comes with lots of flavors. Sampling and periodization
• Reverse mode automatic-differentiation is based on a « backward »
• The mean curvature motion is the most fundamental curve evolution.
• A Riemannian manifold is locally an Euclidean space. An embedded surface
• The integral curves of a vector field tangent to a manifold stays on the
• The primary visual cortex is organised as an oriented wavelet transform.
• Approximation using anisotropic triangulations performs optimal
• The distance field is the unique viscosity solution of the Eikonale
• The subdifferential is the set of slopes bellow the graph of a convex
• A 1 hidden layer perceptron can approximate arbitrary continuous functions
• Fourier is to convolution what Legendre is to inf-convolution. … …
• The Reeb graph of a function on a manifold is the graph of level sets
• Schatten p-norms are the l^p norms of the singular values. Define algebra
• A Brownian motion is approximation by a discrete random walk with Gaussian
• Discrete Cosine Transforms are eigenvectors of Laplacians with different
• The Fermat–Torricelli point minimizes the sum of the distance to the
• Comparison of the Wasserstein, Hellinger, Kullback-Leibler and reverse KL
• Zero crossing of the derivative of a blurred version (heat diffusion) of a
• Csiszar divergences compare the relative distributions of two histograms with 1.
• Eigenvalues of random matrices with iid entries converge to the Wigner
• Filtered back-propagation is a formula for the invert of the Radon
• A sub-group of Möbius transforms are those preserving the Poincaré upper-
• Iteratively joining the mid points of the edges of a polygon converges
• Optimization can be carried over on manifolds using gradient descent,
• Implementing a numerical function corresponds to the creation of a
• The space of metric spaces is a metric space for the Gromov-Hausdorff
• Vitali Milman's insight: convex sets in high dimension look like hedgehogs
• Taylor series expansion of sine.
• Störmer, Verlet and leapfrog are different names for the same method.
• Tutte theorem defines a valid drawing of planar graphs by solving a linear
• Varadhan formula shows that at first order, the heat kernel is equal to
• Linear approximation selects the first coefficients of a decomposition in
• The Marchenko-Pastur law is the distribution limit of eigenvalues of
• The fractional Laplacian is a differential operator generalizing the
• Summing the log of the nearest neighbors distances: a simple estimator for
• Line integral convolution is an anisotropic filtering which averages
• The heat equation can be applied on a surface and to a surface.
• Penrose tiling are aperiodic tilings obtained using typically 2 isosceles
• Code words defined using a Huffman tree generate codes with optimal
• Lorenz attractor is the set of limit trajectories for a simplified
• Radial basis functions performs interpolation and approximation by solving
• Self-organizing maps of Kohonen defines a map between a 2D space to the
• Behavior of dynamical system depends on the eigenvalues. In 2D it can be
• The convex hull is the smallest enclosing convex set. It is also the
• The binomial law is a sum of Bernoulli 0/1 random variables. After
• The heat equation can be applied to diffuse probability densities (in
• Taylor series of exp(it) up to degree 15.
• Leapfrog/Verlet are symplectic integrators which approximately conserve
• K-nearest neighbors is the baseline method for non-parametric
• Power iterations does not converge for matrices with multiple complex
• Tensor-driven anisotropic diffusion solves the heat equation on a manifold
• Stochastic gradient method: works also for infinite sums and expectation.
• The proximal operator is the implicit counterpart of the explicit gradient
• Iterative projections: converge in theory for convex sets. Works great in
• The damped harmonic oscillator (linear) vs simple gravity pendulum (non-
• Lagrangian discretization of measures (quotient space) vs. Eulerian
• Thin plate spline is one of the most popular data interpolation methods.
• Impact of the interpolation method on the level set of the resulting function.
• Dual number is a convenient way to implement forward mode of automatic
• Daubechies wavelets is a parametric family of orthogonal Wavelets.
• The mean-shift algorithm is a clustering method that progressively evolves
• The Helmholtz-Hodge decomposition split vectors fields in two orthogonal
• The logistic map can have complex patterns of adherence points depending
• The co-area formula relates the total variation to the perimeters of the
• The gradient flow of the Dirichlet energy is the heat equation, which blur
• A Sobolev ball is an ellipsoid in infinite dimension aligned along the
• John’s and Löwner’s ellipsoids are polar one from each others.
• The kernel of a domain is the (convex) set of locations from which you can
• Only harmonic spring and gravity central forces produce periodic motions.
• Poljak heavy ball method speeds up gradient descent by introducing
• Lotka–Volterra equations is a non-linear 2D ODE modeling prey/predator
• The harmonic oscillator is the canonical linear second order ODE.
• Progressive decompression of a 3D mesh.
• I have updated my course notes on automatic differentiation (last section
• I have written a text on "the mathematics of neural networks" for a
• The l1 norm achieves the best balance between convexity and sparsity-
• Kriging (aka Wiener interpolation) accounts for uncertainty in kernel
• The Game of Life is the most celebrated 2D cellular automaton.
• The Delaunay triangulation in dimension can obtained by computing the
• J'ai rédigé un article de vulgarisation sur "les mathématiques des réseaux
• Many properties (convexity, derivative, stratifiability, partial-
• Comparing probability distributions: Csiszár f-divergences measure «
• Marden theorem states that the roots of P’’ are the focal point of the
• The marching squares/cubes is the standard algorithm to extract iso-
• The Legendre transform can be approximated in O(n*log(n)) using a Gaussian
• Lloyd algorithm can be used to progressively diffuse points, defining a
• I have released 115 animations on Wikimedia Commons, feel free to use them
• Gauss-Luca theorem shows that the roots of derivative of a polynomial
• On can map a surfaces to sphere by minimizing a Dirichlet energy.
• Interpolating linearly polynomials of a fixed degree defines an
• Rational Bezier curves generalizes Bernstein approximation using weights.
• Diffeomorphisms (warpings) are conveniently described in a Lagrangian way
• Shepard interpolation generalizes nearest neighbor interpolation.
• Power iterations converge to the leading eigenvector of a matrix.
• Taylor series approximate within its radius of convergence a function by a
• Comparing 2D linear interpolation methods of order 0, 1 and 3.
• The simple gravity pendulum is one of the simplest non-linear second order
• Jacob Bernoulli, Ars Conjectandi, 1713. Introduces many concepts central
• The Fast Fourier and Wavelet transforms correspond to two sparse
• Harmonic functions are obtained by solving Laplace equation and define
• Loyds algorithm defines a segmentation of the domain using Voronoi cells.
• The heat can be applied to diffuse probability density (in particular,
• I have put online the course notes for my master course on optimization
• « Greedy » gradient descent using optimal step choice at each iteration
• Csiszár divergences is a unifying way to define losses between arbitrary
• The QR decomposition can be computed in a numerically stable way using
• Parabolic PDEs (e.g. heat) smooth out singularities. Hyperbolic PDEs (e.g.
• Banach fixed point theorem ensures existence of a unique fixed point for
• Having partial derivatives (being differentiable along the axes) does not
• The Wasserstein-1 distance (which is a norm!) between histograms on graphs
• Bernstein polynomial can be used to approximate 1D functions but also 2D
• The medial axis (aka skeleton) generalizes the mediatrix between two
• The logistic map is the simplest dynamical system which exhibits
• The Optimal Transport geometry of 1-D Gaussians is flat in the (mean,std)
• Reproducing Kernel Hilbert spaces define norms on functions so that
• Step size selection is important for gradient descent on ill-conditioned
• Reaction-diffusions are non-linear PDEs which describe the formation of a
• The Weierstrass function is continuous if a<1 but nowhere differentiable
• Monotone operators (Minty, Browder) generalize monotone functions.
• The SVD decomposes the action of a matrix into rotations and scalings
• The QR decomposition can be computed in a numerically stable way using
• Non-uniform B-splines functions are smooth piecewise polynomial functions,
• One can encode or approximate convex shapes as intersection of the semi-
• Total variation denoising (Rudin, Osher, Fatemi 92) was studied by the
• Pinsker's inequality is a fundamental inequality of information theory.
• Training a MLP with a single hidden layer is the evolution of a sparse
• Iterative closest point is one of the basic algorithm for rigid shape
• Adding an independent Gaussian variable is the same as doing a heat
• Fixed points can be attractive or repulsive depending on the derivative of
• Parametric density fitting is a parameter estimation problem aiming at
• Harris’s methods is the most frequently used corner detector, based on the
• Schrodinger’s problem is an approximation (regularization using diffusion)
• The Travelling salesman problem is one of the most well know NP-hard
• A mixture of Gaussians model can be represented as a weighted points cloud
• « Metaballs » are levelsets of mixtures of radial basis functions, which
• Burgers’s equation is one of the simplest non-linear partial differential
• Fourier approximation of a cat.
• Cellular automata are discrete dynamical models defined by a local
• Positive 1D polynomials are sums of squares. Motzkin’s 2D polynomial is
• The Courant-Friedrichs-Lewy (CFL) condition relates time/space step sizes
• Strassen algorithm reduces the complexity of matrix multiplication in term
• Beside being useful to prove Weierstrass polynomial approximation theorem
• Lloyd’s algorithm is the continuous counterpart of k-means. Optimizes the
• 3D surface compression is achieved by projecting on Fourier-like atoms
• Can you hear the shape of a drum? The eigenvectors of the Laplacian depend
• Kalman filter defines recursively an estimator of the parameters of a
• The determinant is log-concave function. Determining the Loewner enclosing
• The cone of positive semi-definite matrices is a fundamental object of
• The Fast Fourier Transform: « the most important numerical algorithm of
• The Laplacian of a graph is a semi-definite positive operator which mimics
• The equation XAX=B is surprisingly simple to solve on the set of positive
• Solving linear systems comes with different flavors.
• Robust regression is obtained by using a l1 loss in place of least
• For saddle point problems and more general games, gradient descent can
• Exponential families hybridize convex analysis and statistics.
• A joint distribution encodes the dependencies between its two marginal
• Least square is the most fundamental data analysis method. Gauss or
• Zeros of random polynomials define point processes. For iid coefficients,
• The max and soft-max (log-sum-exp) of convex functions is convex.
• A mean field evolution is the large number limit of particles systems. The
• Birkhoff’s contraction for Hilbert’s metric is a key tool to quantify
• Gaussian scale mixtures defines heavy-tail distribution by combining
• SMACOF is the most popular stress minimization algorithm for
• The Shapley-Folkman-Starr theorem states that the Minkowski sum «
• The Optimal Transport interpolation between two Gaussians is a Gaussian.
• Runge-Kutta's formula are the workhorse for the numerical integration of
• The Kabsch-Nadas formula solves in closed form the orthogonal least square
• Sparse cholesky factorization are obtained by applying the algorithm after
• K-means++ defines a seeding strategy which is approximately optimal up to
• The continuous wavelet transform was introduced by Grossman and Morlet.
• NP-complete problems are problems in NP which are NP-hard.
• Slides for my talk "Off-the-grid Sparse Estimation". Summarizes years of
• Lee and Seung algorithm is the most popular matrix factorization
• We organise a conference (in French, sorry!) for the best PhD thesis
• When regularizing « optimal transport like » problems, you should use the
• Kolmogorov-Arnold superposition Theorem answers (in negative) Hilbert's
• The Wasserstein distance over Gaussians defines the so-called Bures
• The A^* heuristic path planing algorithm reduces the search space of
• New preprint: "Sinkhorn Divergences for Unbalanced Optimal Transport".
• Cauchy–Binet formula generalizes the determinant of a product of square
• Spectrahedra are matrix generalizations of polyhedra. The class of
• Lagrange and Hermite interpolations can be solved in closed form using
• Spherical interpolation (SLERP) of quaternions defines the geodesic over
• Very nice blog post from Song Mei on the replica method from statistical
• K-means algorithm computes stationary point of the quantization error by
• W. Zhou, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, Image quality
• Iterative least square minimizes a robust loss functions (eg l1 norm) by
• L2 (ridge) vs L1 (lasso) regularizations for the regularization of
• In the static regime, the electric field generated by point sources is the
• Point estimators are defined by minimizing an average risk under the
• JPEG compression performs a weighted entropic coding of discrete cosine
• Non-local means computes an adaptive filtering by comparing patches in
• Gabor and Wavelets are linear dictionaries offering different tilings of
• Smoothing splines define regularized least square whose solutions are sums
• Heat vs wave equations.
• The Johnson-Lindenstrauss lemma shows that one can project linearly m
• The dead leave model defines a stationary random distribution of sets by a
• Independent component analysis estimates a mixing matrix leveraging the
• The travelling salesman problem looks for the shortest Hamiltonian cycle
• Determinantal (resp permanental) processes are repulsive/fermionic (resp.
• The Wigner-Ville distribution mimics a probability distribution on the
• Low frequency approximation of discontinuous functions generates Gibbs
• Principal component analysis and non-negative matrix factorization are two
• The Brachistochrone problem was solved by Bernoulli and is the birth of
• Moreau’s decomposition generalizes the orthogonal decomposition from
• In the static regime, the electric / magnetic fields generated by point
• Heat diffusion vs. wave equation on a surface.
• Matrix decompositions come in many flavors!
• n-ellipses generalize ellipses with n foci. Spectrahedra (hence convex)
• Moreau's decomposition generalizes orthogonal decomposition from linear
• Reverse mode automatic differentiation computes the gradient with the same
• Non-linear approximation in a Haar wavelets basis performs an adaptive
• Gaussian functions are stable under pointwise and convolution products.
• Copula remaps the cumulative function to have uniform marginals. It is a
• Strong convexity and smoothness are the two key hypotheses to make
• A paper I enjoyed reading recently: A. P. Bartok, R. Kondor and G. Csanyi:
• Shepard interpolation is a surprisingly simple multi-dimensional
• A paper (small book) I enjoyed reading: Spectral Graph Theory, Chung.
• A paper I enjoyed reading recently: « The space of spaces (...) » KT
• Parabolic PDEs come with lots of flavors: heat, total variation,
• A paper I enjoyed reading recently: Inverse problems in spaces of
• Conditional distributions defines a parametric collection of probability
• A paper I enjoyed reading recently: Breaking the Curse of Dimensionality
• A paper I enjoyed reading recently: "From 3D models to 3D prints: an
• Photo-realistic texture synthesis methods perform pixels, patch or more
• A paper I enjoyed reading recently: Multiscale Representations for
• A paper I enjoyed reading recently: @julienmairal "Incremental
• Gradient descent on particles’ positions (Langrangian) is equivalent to an
• A paper I enjoyed reading recently: "Remarks on Toland’s duality,
• Paper I enjoyed reading recently: Learning with Fenchel-Young Losses,
• The Fourier transform diagonalizes convolution operators aka circulant
• A paper I enjoyed reading recently: The Complexity of Computing a Nash
• A paper that I enjoyed reading recently: Automatic differentiation of non-
• Calculus of variations studies infinite dimensional optimization problems
• A paper I enjoyed reading recently (poke @KyleCranmer) "Quantum optimal
• A paper I enjoyed reading recently: Rank optimality for the Burer-Monteiro
• Probability distributions can be approximated using Eulerian or Lagrangian
• A paper I enjoyed reading recently: "WHAT IS ... a Graphon?", Daniel
• A paper I enjoyed reading recently "Is computing with the finite Fourier
• Bayes formula relates the posterior distribution to the likelihood
• A paper I enjoyed reading recently: Sorting out typicality with the
• Regularity of optimal transport map is a difficult question, studied by
• A paper I enjoyed reading recently: Invariant and Equivariant Graph
• Recent paper I enjoyed reading: Hashimoto, Gifford, Jaakkola, Learning
• Durand–Kerner method: a surprisingly simple way of computing all the roots
• The dual of a triangulation is defined by joining the centroids of
• When rolling an oloid in straight line, all its surface touches the ground.
• Parametric surfaces embedded in Euclidean space define a Riemannian
• We are so glad to welcome @katecrawford for the inauguration of her
• 1D Optimal Transport distances between Gaussians is the Euclidean distance
• The push-forward operator is a linear map between measures which operates
• Perron-Frobenius is fundamental for the study of Markov chains
• Smooth functions can be decomposed as the sum of a convex and a concave
• The discrete Fourier basis is a sampling of the continuous Fourier basis.
• The gradient field defines the steepest descent direction. The gradient
• Any optimization problem is equivalent to a convex (linear) one (but
• The heat equation on polynomials (for the derivative on the complex plane)
• Brenier’s theorem solves the long standing question of existence of an
• Analyzing the global convergence of Newton is hard. Attraction areas are
• Nonlinearity matters. Linear diffusion (heat) has non-compactly supported
• Page rank is the leading eigenvector of a stochastic matrix. Can be
• The Hartley orthogonal matrix is associated to cas=cos+sin function.
• POCS, Dykstra and DR often succeed in finding points in the intersection
• Mirror-stratifiability (Drusvyatskiy/Lewis): generalizes duality of
• Support vector machines maximize the classification margin for linear
• There exists various discrete cosine transforms which correspond to
• Logistic regression defines fuzzy classification boundaries using the
• QR algorithm is a gem of numerical algebra: for a symmetric input, also
• Vizualizing zeros and poles of rational functions over the complex plane.
• The 2x2 SDP, degree 2 positive polynomials and 2-D ice-cream cones coincide.
• A new to learn about semi-discrete optimal transport, including stochastic
• The l^p functional is convex and hence a norm for p>=1. It is sparsity-
• 1D optimal transport interpolation corresponds to interpolating the
• k-nearest neighbors is the baseline non-parametric regression and
• The solution of the Eikonal equation is solved by advancing a front in the
• The Douglas-Rachford algorithm (the dual of ADMM) computes the projection
• As suggested by @sigfpe, interpolating between planar point clouds can be
• Dithering is the process of converting a real-valued image into a finite
• Eigenfaces are singular vectors of an image dataset. A principal component
• Monte Carlo integration approximates integrals at a rate 1/sqrt(n)
• Richardson–Lucy is the most celebrated deconvolution method under
• Projected gradient and Frank-Wolfe are the cornerstones of constrained
• Tutte embedding is the solution of a Poisson equation. Iterative
• Pruning the Voronoi diagram of a set of points estimates the medial axis
• Various discrepancies between 1D probability distributions are derived
• Reuleaux polygons and Meissner bodies are particular instances of bodies
• The gradient field defines the steepest descent direction. The gradient
• Tarski–Seidenberg theorem in action: the set of polynomials having a real
• Frechet, Gumbel and Weibull distributions are extreme value distributions
• Orthogonal matrices come with lots of flavours!
• The distance function is smooth away from the medial axis (aka the skeleton).
• A lattice is an ordered set where any two elements have an upper and lower
• Oldie but goldie. Karl Weierstrass, Über continuirliche Functionen eines
• Error checking code with an additional parity bit defines a simplex in the
• Compactly supported orthogonal wavelets are defined using filter banks.
• Heat equation vs wave equation inside a planar domain. Boundary conditions
• Filtering by convolution complex vectors defines a sequence of polygons.
• Woodbury formula allows one to compute in two different ways the solution
• Gradient flows in metric spaces formalize the notion of descent methods on
• Partial smoothness is the correct notion of "piecewise smooth functions"
• L Page, S Brin, The PageRank citation ranking: Bringing order to the web,
• Error diffusion is the simplest dithering algorithm which quantize an
• Vizualizing the attraction bassin of Newton’s method on polynomials as the
• Stein’s lemma is a surprisingly simple yet useful characterization of the
• Interpolating coefficients of polynomials generates interpolations between
• Tarski–Seidenberg theorem: semi-algebraicity is stable by projection. The
• QR algorithm is a gem of numerical algebra: iterative QR decomposition
• Principal component analysis projects the data on the leading eigenvectors
• Auto-regressive process generates dynamical textures, and can integrate
• The wavelet transform computes an optimally sparse representation of
• There exists a single increasing map between two distributions, it is the
• Joukowsky conformal map: solving airfoil design in the pre-computer era.
• The circle, semi-circle and Marčenko–Pastur laws are three simple examples
• Gibbs distributions are maximum entropy distributions. Used in conjunction
• Difference of convex (DC) programming are non-convex problem enjoying a
• Spherical interpolation (SLERP) of quaternions defines the geodesic over
• Sampling a signal is equivalent to periodizing its Fourier transform. For
• Lloyd’s algorithm is the continuous counterpart of k-means. Optimizes the
• I wrote a short article to present numerical Optimal Transport and its
• Zonohedra are images of l^inf ball by linear map while symmetric polyhedra
• Hypotrochoids are curves obtained by rolling a circle inside another one,
• If d is a distance, then d^p for 0<p<1 is its snowflake distance. Naming
• The Delaunay triangulation is the dual of the Voronoi diagram. The most
• Dijkstra’s algorithm computes the geodesic distance on a graph in
• The bilateral filter is a non-linear edge-preserving filter where weights
• Sunflower patterns are obtained using a Fermat spiral with a golden angle
• Spatially varying blurs can be approximately computed by merging together
• Markov chain Monte Carlo methods sample from a Gibbs distribution without
• Non-circular gears are ubiquitous and generate non-constant rotation
• The space of compact sets in a metric space is a compact set for the
• Conference "Imaging and machine learning" April 1st-5th @InHenriPoincare,
• Schatten p-norms are the l^p norms of the singular values. Define algebra
• Optimal quantization is a minimum Optimal Transport projection, as
• Projected gradient is the simplest algorithm to perform constrained
• The perspective transform turns a 1D convex function into a 2D positively
• Dynamics of a system of rods (multiple pendulum) is an ODE evolution on an
• Monge and Kantorovitch Optimal Transport are equivalent when the measures
• A new Numerical Tour ( on Multilayer Perceptron with a single hidden
• My book with Marco Cuturi on computational optimal transport has just been
• Brunn-Minkowski inequality is one of the fundamental inequalities in
• Not really oldies but goldies: @keenanisalive, C. Weischedel, M. Wardetzky
• Min/max games are convex/concave saddle points. Can be solved by primal-
• The Weierstrass function is continuous if a<1 but nowhere differentiable
• Cats, bunnies and elephants are ubiquitous in graphics and applied maths.
• The Fisher-Tippett-Gnedenko theorem is the central limit theorem for the
• Poisson disk process exhibits blue noise power spectrum which is ideal for
• The Laplacian pyramid is the ancestor of the wavelet transform. Defines a
• Foveation is a spatially varying « convolution ». It is similar to human
• Randomizing the phase of the Fourier transform is the most simple texture
• The mean-shift algorithm is a clustering method that progressively evolves
• Integral curves of a vector field joins stationary points of the field.
• Holomorphic functions (differentiable over complex numbers) defines
• Quadratic splines are basic tools for vector graphics in CAD. They can be
• Generalized Apollonian gaskets are fractal domains defined by
• Maury/Roudneff-Chupin/Santambrogio model of crowd motion with congestion
• Filtered back-propagation is a formula for the inverse of the Radon
• String art generates logarithmic spiral arcs by iteratively connecting
• Reaction-diffusion with spatially varying weights.
• Subdivision curves define smooth approximating or interpolating curves
• Optimal transport flow defines particles evolutions which is a l2 gradient
• Hilbert space filling curve defines a continuous map from a segment to a
• Stationary Gaussian fields are characterized by their power spectrum.
• Mean value coordinates extend usual barycentric coordinates to arbitrary
• Poisson disk process has all points separated from each other by a minimum
• Chaikin’s corner cutting is the simplest and the most famous approximating
• (Not so) oldies but goldies: Generative Adversarial Nets, @goodfellow_ian,
• The cepstrum of a time series is the inverse Fourier transform of the log
• Mean curvature motion is the most fundamental curve evolution. Equivalent
• n-ellipses generalize ellipses with n foci. Spectrahedra (hence convex)
• Warping from an arbitrary domain and a disk can be achieved by Tutte
• Bezier curve is barycentric interpolation using Bernstein polynomial
• ⚡️ “Oldies but Goldies”
• Tensor-driven anisotropic diffusion solves the heat equation on a manifold
• Snakes aka active contours are elastic curves minimizing a potential
• Conformal maps (differentiable over the complex plane) can be defined
• The distance function defines offset of shapes and curves. Its
• The structure tensor is the local covariance matrix field of the gradient
• MAX-CUT is an NP hard problem. Goemans and Williamson SDP relaxation with
• Dynamic Time Warping aligns two time series by an increasing map. Solved
• Gestalt theory defines grouping laws explaining human vision and its
• Burgers’ equation is the prototypical non-linear advection/diffusion
• The Minkowski sum is commutative, associative and distributive wrt union.
• The geodesic in heat method of Crane, Weischedel and Wardetzky
• Wavelet transform of an image iterates low pass and high pass filtering
• Subdivision curves exist in two flavors: interpolating and approximating.
• The poisson point process has a flat power-spectrum while the poisson disk
• The medial axis (aka skeleton) is the set of points where the distance is
• A quadratic bezier spline is the envelope of regularly spaced lines.
• Unbalanced Optimal Transport (OT) generalizes OT to allow for mass
• Comparing Haar wavelet approximations on an image and on a sphere.
• Fourier approximation of a cat.
• Apollonian gaskets are fractal domains defined by progressively packing
• The continued fraction approximation of pi at order 4 gives the
• The Fourier slice theorem relates the 1D Fourier transform of Radon
• Can you hear the shape of a drum? The eigenvectors of the Laplacian depend
• Hadamard–Rademacher–Walsh transform is the Fourier transform on the group
• The envelope of iterated equi-spaced connected points converges to
• Natural gradient is the gradient for the Riemannian structure on densities
• Mesh processing applies numerical methods (PDEs, optimization, etc) on 3D
• Positive 1D polynomials are sums of squares. Motzkin’s 2D polynomial is
• Fourier transform on groups turns convolution into multiplication. For
• Barycentric coordinates are uniquely defined in triangles, but there are
• Wasserstein (optimal transport) flow generalizes the evolution of a system
• Simplicial homology defines discrete derivatives (boundary operators)
• Vietoris–Rips and Čech complexes are fundamental simplicial complexes
• Regular 1:4 subdivision defines a multi-scale analysis on meshes.
• The minimum spanning tree of points for the Euclidean distance is a sub-
• Large scale N-body simulations require to evaluate sums involving long-
• Alpha-shapes is a multiscale family of simplicial complexes included in
• The basics of optical flow computation it to invert the ill-posed
• Classification and Regression Trees (CART) defines structured recursive
• Interpolating between implicit equations defines morphings between curves
• Eigenvectors of the Laplacian on compact planar domains define an
• The enveloppes of a set of curves is another curve tangent to all these
• A good triangulation to solve stably elliptic PDEs should contain
• A graph is planar if and only if it does not contain as minor the complete
• Surface parameterization computes a diffeomorphism between a 3-D surface
• Triangulated graph are maximal planar graphs, ie for which one cannot add
• The max of n independent Gaussians concentrates around sqrt(2log(n)). The
• The Marching Cubes is the the standard algorithm to extract an isosurface
• Regular 1:4 triangle subdivision allows one to define a Haar transform on
• Generative Adversarial Networks approximately solve parametric density
• The Fiedler vector of a graph is the second eigenvector of the Laplacian.
• Simplicial complexes are combinatorial objects generalizing
• The Fisher metric is the unique Riemannian structure on parametric
• Representation theory defines a Fourier transform on groups. Finite
• Convolutional neural networks are shift invariant representations obtained
• Reproducing Kernel Hilbert spaces define norms on functions so that
• Saddle points come with lots of flavors. Quadratic saddles are non-
• The cumulative function is to the max of random variable what the Fourier
• Some works of the Gauss prize winners.
• David Donoho just god the Gauss prize at ICM 2018! IMHO its four main
• The farthest point sampling is a greedy algorithm which asymptotically
• A Riemannian manifold is locally an Euclidean space. An embedded surface
• Color perception can be modeled using a Riemannian manifold. The
• Continued fractions are in some sense optimal ways to approximate with
• I will cover (at least!) these 18 algorithms for my « mathematical
• Not oldies but goldies: main papers of the 4 Fields medalists can be found
• Kantorovitch formulation of Optimal Transport is a linear program
• The heat equation can be applied to diffuse a function on a surface, but
• Particle system is a Lagrangian way to minimize energy over measure. For
• Training a multi-layer perceptron for classification. A large enough
• I put together a list of great algorithms. Thanks to everyone who replied
• Integrating flow field defines a diffeomorphism. Smooth vector fields are
• In 2000 Dongarra & Sullivan published the Top Ten Algorithms of the
• Eigenvalues of random matrices with iid entries converge to Wigner circle
• QR algorithm repeatedly applies QR decomposition to generate a sequence of
• The gradient field defines the steepest descent direction. The gradient
• Three ways to solve Fokker–Planck equations: PDE evolution, stochastic
• The stable marriage is solved by the Gale-Shapley algorithm. Shapley and
• The mean curvature motion is the most fundamental curve evolution.
• Fourier descriptors (Zahn and Roskies, 1972) are normalized Fourier
• I will start a « oldies but goldies » series starting tomorrow
• Eulerian (image, grid, finite elements, finite volumes, etc.) vs
• Histogram equalization is 1-D optimal transport. Linearly interpolating
• Curse and blessing of non-smooth optimization: non-smooth parts (eg
• Low frequency Fourier approximation of a signal and an image generates
• Edge detectors compute zero crossing of multiscales second-order
• Sinkhorn algorithm defines a smooth Optimal Transport loss function
• Parabolic PDEs come with lots of flavors: heat, total variation,
• The stochastic block model is the simplest random graph with communities
• Optimization algorithms come with many flavors depending on the structure
• Diffeomorphisms (warpings) are conveniently described in a Lagrangian way
• Edge collapse is a fundamental mesh processing primitive, at the heart of
• Total variation denoising (Rudin, Osher, Fatemi, 92) was studied in theory
• Fokker–Planck equation equivalently describes the movement of a random
• Stochastic gradient descent is the workhorse of many large scale machine
• Gradient flow of the Dirichlet energy is heat equation, which blurs edges.
• The Cucker-Smale system of ODEs is the simplest model of particles
• The l^p functional is convex and hence a norm for p>=1. It is sparsity-
• The gradient field defines the steepest descent direction. The gradient
• On Euclidean space, Maximum Mean Discrepancy norms between measures are L2
• Approximation of a function on the sphere using an increasing number of
• Optimal assignment in 1D is simply sorting the values. Allows one to
• Stable fluids of Jos Stam is a semi-Lagrangian solver for Navier-Stokes,
• Displacement Interpolation (Robert McCann) is the optimal transport
• Voronoi diagram can be defined on surface using the geodesic distance.
• f-divergences (KL, TV, Hellinger, Chi2, etc.) are all closely related and
• Color 3D quantization versus approximation are two ways to reduce the
• From Monge-Kantorovitch’s optimal transport to Schrodinger lazy gaz, there
• Laguerre (aka power) diagram is a generalization of Voronoi partitions,
• Normal mapping (aka bump mapping) computes pixel’s values using an
• The Wasserstein-1 distance between histograms on a graph can be re-written
• Lloyd algorithm can be applied on a surface using the geodesic Voronoi diagram.
• As shown by @arthurmensch and @mblondel_ml, differentiable max&argmax
• Entropic regularization (aka Sinkhorn) interpolates between a non-
• The hypercube (tesseract in dimension 4) is the d-dimensional cube. Can be
• Multilayer perceptron with 1 hidden layer breaks the curse of
• Vizualizing zeros and poles of rational functions over the complex plane.
• Solving a maze using front propagation (Fast-Marching method).
• Lambert and Phong illumination models are the most well known empirical
• Deep network comes in two flavors: discriminative (aka convolutional for
• Dual norms (aka Integral Probablity Metric) metrize weak convergence (aka
• Yann Brenier just introduced and studied the arctangential heat equation,
• Sinkhorn first proved the convergence of the diagonal scaling algorithm
• Difference of convex (DC) programming is a class of non-convex
• Voronoi diagrams can be defined over general metric spaces, for instance
• k-nearest neighbors is the baseline non-parametric regression and
• The Fast Marching algorithms performs a front propagation on the surface.
• Interior point methods solve approximately in polynomial time conic
• Nonlinearity matters. Linear diffusion (heat) has non-compactly supported
• The Monge-Ampère equation is a non-linear generalization of the Poisson
• Curve approximation using Fourier series corresponds to drawing the curve
• During elastic collision, particles of equal mass simply exchange their
• Network flow problem is a specific class of linear programs that finds
• Shepard interpolation generalizes nearest neighbor interpolation.
• Laguerre (aka power) diagram is an amazing tool to solve the semi-discrete
• Maximum Likelihood Estimation is a density fitting problem with a
• Csiszar divergences compare the relative densities of two measures with 1.
• Heat diffusion vs. wave equation on a surface.
• Non-circular gears generate non-regular motions from regular one. Used
• Spherical harmonics are the equivalent on the sphere of the Fourier basis.
• The Radiosity equation describes conservation of light for diffuse
• Fourier approximation of closed curves.
• Volumetric 3D heat equation smoothes all the level sets of a function.
• Multilayer perceptron with 1 hidden layer approximate functions using sums
• 3D metaballs: remind me the good old time of coding with @antoche the C++
• Tutte embedding is the solution of a Poisson equation. Iterative
• Mathematical models: part 1/2.
• Spherical harmonics is an orthogonal basis of eigenvectors of the
• Navier Stokes equation describe the motion of an incompressible viscous
• The solution of the Eikonal equation is solved by advancing a front in the
• Subdivision surfaces define a hierarchy of embedded triangulations using
• Julia sets are attraction bassins of iterated complex maps. The best known
• Heat equation vs wave equation inside a planar domain. Boundary conditions
• Interpolating coefficients of polynomials generates interpolations between
• Analyzing the global convergence of Newton is hard. Attraction bassins are
• Local averaging is consistent with the linear heat equation. Local median
• Natural neighbor interpolation (Robin Sibson) is a generalization of
• Iterative projections do not converge in general to the projection onto
• Support Vector Machine (Vapnik, Chervonenkis, Cortes) performs
• The co-area formula is the most fundamental tool of geometric measure
• A 1 hidden layer perceptron can approximate arbitrary continuous functions
• Thin plate (biharmonic) splines is an interpolation method with a closed
• Eigenvalues of random matrices with iid entries converge to Wigner circle
• « Metaballs » are levelsets of mixtures of radial basis functions, which
• The Helmholtz-Hodge decomposition split vectors fields in two orthogonal
• Gaspard Monge proved in 1781 that optimal assignment for the Euclidean
• Shepard interpolation is a surprisingly simple multi-dimensional
• Approximation vs quantization: two ways to compress.
• Lagrange and Hermite interpolations can be solved in closed form using
• Lloyd’s algorithm is the continuous counterpart of k-means. Optimizes the
• De Casteljau’s algorithm defines a subdivision’s algorithm to evaluate
• Representing complex functions using colormap and level sets produces a
• Vornoi cells on a surface using geodesic distance. Dualizes to a valid
• Parabolic PDEs (e.g. heat) smooth out singularities. Hyperbolic PDEs (e.g.
• Pruning the Voronoi diagram of a set of points is way to estimate the
• The subdifferential is the set of slopes bellow the graph of a convex
• The color palette of an image is the 3D empirical distribution of the
• The Reeb graph of a function on a manifold is the graph of level sets
• Heat equation is both L2 gradient flow of Dirichlet energy and Wasserstein
• Empirical risk minimization is a workhorse of supervised learning.
• The dual of a triangulation is obtained by joining the circumcenters of
• Metropolis-Hasting is a surprisingly simple way to sample from a density
• Finding the optimal denoising parameter is a bias-variance tradeoff.
• The geodesic distance extends a local metric into a global one. Geodesic
• The proximal point algorithm [Martinet70,Rockafellar76] is the most
• Fourier is 250 today! Fourier approximation creates Gibbs ringing
• Robert McCann's displacement interpolation is the geodesic for optimal
• The polar of a set generalizes duality between polyhedra. It is to sets
• The divergence (both continuous and discrete) is (minus) the adjoint of
• Floyd–Warshall algorithm computes all pairs shortest distances. Related to
• Fourier transforms comes with lots of flavors. Sampling and periodization
• Approximation using anisotropic triangulations performs optimal
• The gradient flow of the Dirichlet energy is the heat equation, which blur
• From a color image to its color palette. This is the empirical
• Wavelets on surfaces were introduced in 1995 by Wim Sweldens and Peter
• WGAN and VAE are respectively dual and primal approximations with deep-
• The spectrogram (aka short time Fourier transform) reveals the geometry of
• The entropy comes with lots of flavours. The relative entropy (aka
• A map is conformal (angle preserving) if and only if it is holomorphic
• 2-D Fourier atoms are tensor product of 1-D Fourier atoms. Product of a
• Reaction-diffusion are non-linear PDEs which describe the formation of a
• First official release of our book "Computational Optimal Transport"!
• Elastic net (Zou , Hastie, 2005) interpolates between the Lasso (l1
• The level set method of [Osher and Sethian, 1988] performs curves and
• Monotone operators (Minty,Browder) generalize monotone functions.
• Code words defined using a Huffman tree generate codes with optimal
• Discrete Fourier basis is simply a sampling of the continuous Fourier
• The Numerical Tours now in R! 35 @ProjectJupyter notebooks: machine
• The Brownian motion (aka Wiener process) is the scaling limit of a random
• The primary visual cortex is organised as an oriented wavelet transform
• Delaunay refinement aka Ruppert’s and Chew's algorithms. Introduced by
• Moreau-Yosida regularization smoothes a function. It is the inf-
• The Fast Marching algorithm of Sethian computes a weighted geodesic
• Lagrange and Fenchel-Rockafellar duals are (almost) the same. Crucial to
• On non-convex problems, alternating projections often stuck in local
• The medial axis (skeleton) generalizes mediatrix between 2 points. Set of
• Generalized barycentric coordinates (mean values of Floater being the
• The Delaunay triangulation is the dual of the Delaunay diagram. The most
• Histogram equalization is 1-D optimal transport. 1 line of Python code, 2
• Estimating optimal transport distance from empirical samples suffers from
• Waves on a surface!
• Be careful when training your GANs, round #2: explicit descent is
• Total variation denoising (L. Rudin, S. Osher, E. Fatemi, 92) was studied
• The l1 norm achieves the best balance between convexity and sparsity-
• Gradient descent is inefficient to find saddle point (Nash equilibrium)
• Cauchy–Binet formula generalizes the determinant of a product of square
• Optimal computation of gradients is equivalent to optimal Jacobian
• The gradient field defines the steepest descent direction. The gradient
• Only + and - operations: Walsh-Rademacher-Hadamard vs Haar. Fourier vs
• Logistic classification (regression…) defines fuzzy classification
• Heat equation on a surface and of a surface!
• Strong convexity and smoothness are the two key hypotheses to make
• Iterative Soft Thresholding algorithm to solve the LASSO is a special case
• The Fast Marching of Sethian is a generalization of Dijkstra’s algorithm.
• Lagrangian discretization of measures: quotient space topology. Eulerian
• Douglas-Rachford (dual of ADMM) often surpasses iterative projections on
• Mean value coordinates extend usual barycentric coordinates to arbitrary
• Stochastic gradient descent: +: works also for infinite sums and
• The mean curvature motion is the most fundamental curve evolution.
• Backprop in neural-networks is reverse mode auto-diff applied to a simple
• The distance field is the unique viscosity solution of the Eikonale
• Berry–Esseen theorem provides a quantitative estimation of the convergence
• Sheep Wasserstein gradient flows: the perfect illustration of the crowd
• The soft-max is the gradient of the log-sum-exp. Central to preform
• Can you hear the shape of an elephant? Eigenvectors of the Laplace-
• What is complexity of computing gradients? The optimal answer is provided
• Tutte theorem defines a valid drawing of planar graphs by solving a linear
• The machine learning Numerical Tours now in Python! Clustering, ridge and
• The Fourier transform diagonalizes convolution operators aka circulant
• Iterative projections: converge in theory for convex sets. Works great in
• Linear approximation selects the first coefficients. Non-linear selects
• Any pair of planar triangulations of n vertices can be connected by O(n)
• The product of two Gaussians is a Gaussian. Sums of Gaussians is an
• When rolling an oloid in straight line, all its surface touches the ground.
• Varadhan formula shows that at first order, the heat kernel is equal to
• The central limit theorem equivalently reads as the convergence of
• The Fast Fourier Transform: « the most important numerical algorithm of
• Haar transform: only + and - yet surprisingly powerful. The first example
• Birkhoff's contraction for Hilbert's metric: the key tool to quantify
• Orthogonal matrices come with lots of flavours!
• The codes for most of my tweets are available here:
• Why the Central Limit Theorem cannot hold in probability. A 3 lines proof
• The Benamou-Brenier geodesic! In their landmark paper, Jean-David Benamou
• Subdivision schemes: simple and convenient way to manipulate smooth curves
• A Sobolev ball is an ellipsoid in infinite dimension aligned along the
• Non-linear approximation in a Haar wavelets basis performs an adaptive
• Density estimation: kernels vs nearest-neighbors. Parametric vs. non-parametric.
• Ingrid Daubechies Orthogonal Wavelets: a mathematical gem. One of the
• The space of metric spaces is a metric space for the Gromov-Hausdorff
• Spherical interpolation (SLERP) of quaternions defines the geodesic over
• Optimal transport to simulate the dynamics of two groups of football
• Frank-Wolfe algorithm (aka conditional gradient) works over very general
• Mirror descent generalizes gradient descent using Bregman geometries.
• Gradient descent is consistent with a first order ODE. Nesterov’s
• The Weierstrass function is continuous if a<1 but nowhere differentiable
• Toland duality for non-convex programming. Surprisingly efficient if the
• Perron-Frobenius is fundamental for the study of Markov chains … and
• Encoding points as roots of polynomials and interpolating their
• Numerical Optimal Transport: slides for a course.
• Most parametric probability distributions families can be written as
• MLE minimizes an empirical KL loss. MKE minimizes an optimal transport
• Bregman divergences are convex distance-like functionals which are locally
• Geometry of Gaussians: Optimal Transport is flat (Euclidean). Fisher-Rao
• The proximal map projects on level sets. For l^p norms, it is a
• Copula is a way to picture the dependency between two random variables.
• Displacement interpolation between continuous and discrete measures, aka
• Impact of entropic regularization (aka Sinkhorn) on optimal transport:
• Comparison of explicit stepping (unstable) vs implicit stepping (stable)
• Great video by @JustinMSolomon to give the insights of Wasserstein
• Just released course notes on deep learning (covers SGD, auto-diff and
• Computational Optimal Transport: the book. All you ever wanted to know
• Every function can be decomposed as the sum of a convex and a concave
• Reuleaux polygons and Meissner bodies are particular instances of bodies
• The proximal operator is the implicit counterpart of the explicit gradient
• Boris Polyak’s heavy ball and Yurii Nesterov’s descent use momentum to
• Convergence in law of random vectors and weak* convergence of measures are
• Spirograph (hypotrochoid) vs lissajou curves: messing around with a bunch
• Gradient flows on metric spaces: define descent methods on non-Euclidean
• Tarski–Seidenberg in action: the set of polynomials having a real root is
• Convergence of random vectors comes with lots of flavors!
• Step size selection is important for gradient descent on ill-conditioned
• Wirtinger derivatives: treat complex valued functions of complex variables
• Tarski–Seidenberg theorem: semi-algebraicity is stable by projection. The
• Checking convexity of degree 2 polynomials is trivial. Checking convexity
• Functions should be manipulated by pullback, measures by pushforward.
• Gradient descent does not always converge. Trajectories can have infinite
• Vitali Milman's insight: convex sets in high dimension look like hedgehogs
• Stochastic gradient descent dynamic on a simple 1-D example.
• Just wrote course notes on the basics of machine learning. Feedback welcome!
• Heartbroken by Wasserstein barycenters!
• Lloyd algorithm (k-means) with non-uniform sampling density.
• Stochastic gradient descent for the semi-discrete Optimal Transport,
• The circle, semi-circle and Marčenko–Pastur laws are three simple examples
• The maximum of n i.i.d. Gaussians is roughly sqrt(2log(n)). This is where
• The Marchenko-Pastur law is the distribution limit of eigenvalues of
• Signs of sparse vectors recovered by l1 minimization index a spherical
• If d(x,y) is a distance, then d(x,y)^p for 0<p<1 is its snowflake
• Optimal Transport barycenters.
• Summing the log of the nearest neighbors distances: a simple Lagrangian
• Fokker-Planck equations are Optimal Transport flows of entropies. Nice
• Eulerian vs Lagrangian discretization of a probability distribution. Two
• Rotating airfoil using Joukowsky conformal map. Code provided!
• Joukowsky conformal map: solving airfoil design in the pre-computer era.
• Durand–Kerner method: surprisingly simple way of computing all the roots
• The QR algorithm is a gem from numerical matrix analysis. Showing
• The unreasonable effectiveness of non-smooth optimization: sharp
• "Sinkhorn Autodiff" paper updated to insist on the core idea of Sinkhorn
• I knew it! Ice-creams, positive polynomials and PSD matrices all have the
• Semi-discrete Optimal Transport is a multiclass SVM. Sinkhorn version is
• I am putting here material (slides/code) related to my tweets. Will be
• n-ellipses generalize ellipses with n foci. Spectrahedra (hence convex)
• Spectrahedra are matrix generalizations of polyhedra. But the class of
• Heat eq on polynomials generates nice roots dynamics. Post by Terry Tao.
• Discrete Cosine Transforms are eigenvectors of Laplacians with different
• Fractional derivative of a Gaussian using the Fourier transform. Code provided.
• "Twisted" Newton iterations generate nice fractal patterns! Colors display
• Understanding global convergence of Newton method is hard ... Code provided.
• Moreau's decomposition generalizes orthogonal decomposition from linear
• Reasonable properties (convexity, differential, stratifiability, partial-
• Fourier is to convolution what Legendre is to inf-convolution.
• Kurdyka-Łojasiewicz property: fundamental to show convergence of descent
• Partial smoothness [Lewis]: the correct notion of "piecewise smooth
• Mirror-stratifiability (Drusvyatskiy/Lewis): generalizes duality of
• Optimal Transport was formulated in 1930 by A.N. Tolstoi, 12 years before
• "An Introduction to Imaging Sciences": a small book for a large audience
• The "dead leaves" model of Georges Matheron applied to the Eiffel Tower.
• Introducing the "Mathematical Tours of Data Sciences". Book (work in
• Our preprint on "Multi-dimensional Sparse Super-resolution". Detailed
• The Optimal Transport geometry of 1-D Gaussians is flat in the (mean,std)
• Motzkin's polynomial is positive but is not a sum of squares. #LifeIsHarderIn2D
• The Legendre transform of the Wasserstein distance has a nice expression!
• Slides for my talk "Optimal Transport for Imaging and Learning"
• Cuturi's Sinkhorn Divergence interpolates between Optimal Transport and
• Super-resolution from M/EEG-like boundary measurements is achievable for 2
• Some insights on using Optimal Transport as registration loss in our
• Comparing distributions using kernels: heavy tail vs. local (Gaussian) kernels.
• Any optimization program is a equivalent to a convex (linear) one.
• De Boor least interpolant: the way to do multidimensional polynomial
• Evolution of interpolation as points cluster together. 1D=simple to
• Wasserstein-2 and Energy Distance (Sobolev H^-1) are equivalent for
• Numerical Tours on #MachineLearning: regression, classification,
• Last but not least Numerical Tour on Machine Learning: Stochastic Gradient
• New Numerical Tour: Linear Regression and Kernel Methods
• New Numerical Tour: PCA, Nearest-Neighbors Classification and Clustering
• New Numerical Tour: Logistic Classification. Binary, multiclasses,
• The Numerical Tours are now available in @JuliaLanguage! Wavelets, meshes,
• Oldie but Goldie, Cominetti & San Martín 94, detailed analysis of
• Python code for our @miccai2017 paper "Optimal Transport for Diffeomorphic
• Comparing probability measures: vertical vs horizontal displacent /
• Dual weak norms (aka Integral Probability Metrics): a unifying way to
• Displacement interpolation aka Optimal Transport in 1D is equivalent to
• Bures distance on PSD matrices, aka Optimal Transport between Gaussians,
• Displacement interpolation aka Optimal Transport: discrete vs continuous.
• Wasserstein-1 distance (norm!) between measures on graphs: equivalent to a
• Csiszár divergences, a unifying way to define losses between arbitrary
• From deterministic to stochastic matching: Schrödinger problem. See
• Sensitivity Analysis for Mirror-Stratifiable Convex Functions: toward
• A simple way to produce infinite zoom, run 2D auto-regressive model and
• Statistical Learning is Inverse Problems with a noisy covariance matrix /
• L2 vs L1 regularization for inverse problems / correlated designs.
• Slides for my talk "Optimal Transport and Deep Generative Models" tomorrow
• "Optimal Transport for Diffeomorphic Registration", just plug an OT loss
• Looks like the syllabus for my course next year will be quite heavy!
• GAN and VAE from an Optimal Transport Point of View
• Woodbury formula: so simple, so useful ...
• Understanding when the Continuous Basis-Pursuit is better than the Lasso
• Sinkhorn-AutoDiff: Tractable Wasserstein Learning of Generative Models
• Le poster "Le transport optimal et ses applications" (visible au salon
• Optimal Transport: from Theory to Applications!
• Reverse mode auto-diff in 1 slide. So simple. So useful.
• Le transport optimal pour des applications en informatique graphique, par
• Some more GPUs for Optimal Transport!
• Oldie but goldie: Matlab code for non-circular gear generation.
• After 2Y in review, "Model Consistency of Partly Smooth Regularizers"
• Google Scholar citations for "Optimal Transport". Soon trendier than Deep
• Going off-the-grid for super-resolution. Surprisingly effective! Slides
• Iterative soft thresholding for deconvolution: slowly but surely
• The Quantum Optimal Transport project: my largest coding effort ever...
• Unbalanced optimal transport made simple: handling mass creation &
• Quantum Optimal Transport for Tensor Field Processing: paper and code
• My new ERC² project NORIA "Numerical Optimal tRansport for ImAging"!
• View from my new office on the 4th floor @ENS_ULM
• Our preprint on "Bayesian Modeling of Motion Perception using Dynamical
• Slides et vidéos de la conférence Shannon au @_CIRM
• Sparse Support Recovery with Non-smooth Loss Functions: l^1, l^2 and l^inf
• Transparents de mon exposé grand public "compressed sensing" pour la
• Repackaging of the Wavelet Tour's book website
• 20 new Numerical Tours in Python: Wavelets, sparsity, snakes, 3D meshes
• Claude Shannon et la compression des données : mon article sur @ImagesDesMaths
• My new office @ENS_ULM !
• Matlab code and Jupyter notebook showcasing Shannon theory of entropic coding
• One Sinkhorn's algorihtm to rule them all: unbalanced optimal transport
• Mon article de vulgarisation sur Claude Shannon et la compression des données
• Sugiton's calanques processing
• "Parcimonie, problèmes inverses et échantillonnage compressé", à paraitre
• New Python "Numerical Tours" on optimization by Laurent Condat: first
• Computing barycenters over the metric space of metric spaces ... our ICML
• New paper: Optimal Transport meets Stochastic Optimization
• Our second optimal-transport-related paper at #SIGGRAPH2016: Entropic
• Optimal Transport strikes back at #SIGGRAPH2016: Wasserstein Barycentric
• Les slides de mon exposé "grand public" à l'occasion du centenaire de la
• Staircaising at its best: geometrical performance analysis of Total
• Entropy at its best: theoretical foundations of entropic optimal transport
• Optimal transport made easy using entropic regularization: a new numerical tour
• Toward a definitive answer to unbalanced optimal transport problems
• Sharp performance guarantees for sparse positive spikes deconvolution #sparsity
• Entropic optimal transport made it to SIGGRAPH! #siggraph #optimaltransport
• Web site dedicated to my grand father Enjoy the paintings !
• New paper on support (in)stability for sparse deconvolution and compressed