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Alpha previews · Work in progress

Forthcoming posts

Gabriel Peyré

These forthcoming posts are shared as alpha previews. Their content may evolve before the finished articles appear on the blog.

Optimal transport

Averaged cost profiles, their cost transform, and the nonzero gap to the double transform for a cubic cost.Duality & geometry

c-transform as a generalized Legendre transform

Which supporting shapes survive averaging? Cost transforms, two explicit counterexamples, and the surprising case of a round sphere.

 Input translated histograms, the learned circular feature geometry, and feature/sample cost matrices after sixteen exact transport-transform cycles.Learning a ground metric

Wasserstein singular vectors

Let sample and feature distances learn one another, with a fixed-point proof and exact transport experiments on translated and rescaled bumps.

 Spectral Wasserstein coupling between two irregular point clouds and the ellipse of their displacement second moment for Schatten exponent two.Spectral geometry & robust transport

Spectral Wasserstein distances

Measure the directions of displacement, recover an adversarial quadratic cost and a dynamic action, and explore Schatten exponents down to one half.

 Gaussian Sinkhorn coupling ellipses, exact error decay, and the number of cycles required as temperature decreases.Entropic transport & convergence

Sinkhorn on Gaussians

A closed covariance recursion gives an exact scalar convergence rate and reveals how entropic temperature controls the slowdown.

 Cumulative distributions, the CDF-only Wasserstein integrand, and its quadrature convergence.One-dimensional transport

Wasserstein distance without quantiles

A triangle-area identity expresses quadratic transport directly through cumulative distribution functions.

 A noncrossing red-blue matching and the changing costs of one hundred uniformly sampled noncrossing matchings.Geometry & combinatorics

Optimal Planar Matching and Crossings

Uncrossing improves Euclidean length, but a Catalan-sized family of matchings still remains to explore.

 A quadratic optimal-transport coupling, its barycentric projection, and the conditional means of a weak-transport solution.Conditional means

Weak transport and convex order

Pay only for the movement of conditional means, and discover a projection onto mean-preserving spreads.

Flows and geometry

Entropy & convergence

Logarithmic Sobolev is Wasserstein P–L

Fisher information is an entropy gradient norm. A finite-length argument proves convergence of distributions without assuming convexity.

 Moser interpolation halfway between multimodal periodic densities: transported particles alongside speed and flux directions.Moving a prescribed density

Interpolating measures by a Lagrangian flow

Prescribe the density path, solve one Poisson equation, and follow particles through a Dacorogna–Moser transport.

 A stirred periodic Stable Fluids simulation: transported teal dye beside the velocity field and signed vorticity.Incompressibility & Fourier analysis

Stable fluids

Backtrace, diffuse, and project: stir a periodic fluid while inspecting divergence and dye-mass errors.

 Two different particle clouds approach the unit circle while their common signed-distance moment follows the same exponential decay.Moment closure

Autonomous Wasserstein statistics

When does one observable obey a closed scalar ODE? An affine squared-gradient identity gives the exact answer.

 Densities and quantile derivatives along a Wasserstein geodesic remain within the endpoint density cap.Displacement convexity

Geodesics and density caps

Transport straight lines preserve a constant upper density bound, explaining when a hard congestion constraint is inactive.

Probability and sampling

Circulating Ornstein–Uhlenbeck particles approaching the stationary Gaussian ellipse, with drift arrows and recent particle trails.Nonreversible sampling

Langevin without a potential

Confinement gives an invariant law; gradient structure identifies it. An exact Gaussian experiment reveals stationary circulation.

 Classical & quantum information

From Classical to Quantum Pinsker Inequality

A classical two-outcome argument leads to a self-contained matrix proof with the same sharp entropy constant.

 Exact Wasserstein central-limit errors and rescaled CDF errors for symmetric Bernoulli and uniform random variables.Central limit theorem

One Gaussian limit, two rates

Symmetry removes skewness but cannot remove a lattice: Bernoulli and uniform sums have different Wasserstein asymptotics.

 Empirical CDFs of random and optimally placed particles, compared with their exact Wasserstein error rates.Quantization

Carefully placed versus random particles

Quantile-bin averages give the best equal-weight particles, with an exact comparison to random sampling on an interval.

 Information geometry

Two geometries of divergence

Density ratios and convex tangent gaps define different families. Their overlap singles out relative entropy.

Functions of measures

Convexity & transport

The perspective transform: convexity on measures

One mass-weighted Jensen inequality connects Csiszár divergences, the Benamou–Brenier action, and transport with creation and destruction.

 The Bernstein particle polynomial smooths the corner of the absolute-value functional of a two-point probability law.Constructive approximation

Bernstein polynomials on measures

Sample, evaluate, average: an explicit polynomial approximation with a uniform convergence proof and dimension-dependent rates.

 Transformers & transport

In-context composition

Move the query and its context together: a lifted composition rule connecting transformer layers, particle dynamics, and the continuity equation.

 Functions of measures

Universality of Wasserstein polynomials

Stone–Weierstrass on probability measures, interaction features, and the passage from point clouds to operators.

Optimization and approximation

Neural networks & invariance

Universality and invariance

Separate what a network can distinguish from what it can approximate: 1-WL for graphs, feature concatenation, and a relative Stone–Weierstrass theorem.

 Contour plots of two five-bump Gaussian mixtures and their product, with separate labelled color scales showing the product's narrower overlapping peaks.Gaussians & harmonic analysis

The hump algebra of Yves Meyer

Gaussian products, an atomic Banach-algebra norm, and a Besov description of multiscale regularity, with moving two-dimensional mixtures.

 Neural networks & symmetry

From MLP universality to ConvNet Universality

Exponential features prove shallow universality; weight sharing gives an equivariant ConvNet, and an accumulator trades width for depth.

 A Sudoku solved by product-space Douglas–Rachford reflections: teal clues and the verified completed grid, beside the violation and unrounded-residual histories.Projection algorithms

Iterative projections: Dykstra and Douglas–Rachford

Separate feasibility from nearest-point projection, then explore a genuine Douglas–Rachford Sudoku solver.

 Factor-space paths along conserved-imbalance hyperbolas and the product evolution at three different imbalances.Implicit geometry

Intrinsic dynamics of overparameterized dynamics

A conserved imbalance can close the dynamics of the represented matrix, with scalar hyperbolas and symmetric square roots.

 Exact Hessian condition number versus factor scale and local gradient-descent losses for balanced and unbalanced factorizations.Low-rank optimization

Optimal conditioning of low-rank adaptation

The same product can have very different Hessian conditioning. Balancing removes avoidable slow directions.

 Ordinary and scaled sign-gradient descent losses for two isospectral quadratics, with the soft eigendirection aligned or spread across coordinates.Norm-steepest descent

Conditioning of Non-Euclidean Gradient Descent

Two quadratics share their eigenvalues but not their coordinate geometry. Compare scaled sign descent with ordinary gradient descent.

 Tensor geometry

When the best low-rank tensor approximation does not exist

Two diverging rank-one terms approach a tensor needing three: an elementary proof of the border-rank obstruction.

 Exact covariance ellipses and transient and stationary variance curves for constant-step least-squares SGD.Stochastic optimization

The covariance of SGD iterates

An exact least-squares recursion explains an isotropic noise floor, anisotropic relaxation, and the fourth-moment correction at finite step size.

 Convex relaxation

An Axiomatic Definition of the Wasserstein Distance

Start with costs between points. Kantorovich transport is their largest convex, lower-semicontinuous extension to distributions.

 Convex duality

DC programming and Toland duality

A short duality proof, the DC algorithm, and how a convex dual can reveal global solutions of a nonconvex problem.