List of tweets - Mathematical Tours of Data Sciences

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