Proves uniqueness of barycenters on manifolds without restrictions.
problem Finding unique barycenters on complex geometric spaces.
method Introduces new disintegrated Monge-Kantorovich metrics for barycenter problems.
result Uniqueness of barycenters on connected, complete Riemannian manifolds.
New tools for constructing disintegrations and studying their modes.
problem Difficulty in constructing disintegrations and understanding their modes.
method Developed comprehensive mathematical tools for constructing disintegrations and analyzing their modes.
result Disagreement between restricted density and disintegration density in certain cases.
The paper is accompanying "A general Duality Theorem for the Monge-Kantorovich Transport Problem". We explain the methods used in this article in an elementary setting and present two examples complementing the results obtained therein.
A new method splits diffusion operators on principal bundles, leading to disintegration theorems.
problem Diffusion operators on principal bundles with constant rank.
method Defining semi-connections and splitting diffusion operators into horizontal and vertical components.
result A disintegration theorem for the law of diffusion operators on principal bundles.
New PAC-Bayesian bounds provide practical guarantees for neural networks.
problem Loose derandomization step in PAC-Bayesian bounds for deterministic models.
method Introduce disintegrated PAC-Bayesian bounds for deterministic models.
result Significant practical improvement over state-of-the-art bounds.
New method calculates cut locus on Riemannian manifolds using optimal transport.
problem Computing the cut locus on compact Riemannian manifolds.
method Characterization via optimal transport density solution of Monge-Kantorovich equations, numerical approximation.
result Proposed novel framework for numerical approximation of cut locus.
Classifies invariant measures on specific character varieties.
problem Classifying invariant probability measures on character varieties.
method Measure disintegration along transverse Lagrangian tori fibrations.
result Ergodic measures are either counting measures on finite orbits or Liouville measures.
Unified approach solves Kyle model with dynamic information.
problem Solving a generalized Kyle model with dynamic information.
method Monge-Kantorovich duality and backward stochastic partial differential equations.
result Characterization of optimal strategies and pricing rules.
Study arbitrage theory without numéraire, generalizing NUPBR.
problem Arbitrage theory in markets without numéraire.
method Disintegration of probability space into crash times.
result Generalization of NUPBR to no unbounded profits with bounded risk.
This is a review of explicit computations of Connes distance in noncommutative geometry, covering finite dimensional spectral triples, almost-commutative geometries, and spectral triples on the algebra of compact operators. Several applications to physics are covered, like the metric interpretation of the Higgs field, …
Oral Disintegrating Tablets (ODTs) is a novel dosage form that can be dissolved on the tongue within 3min or less especially for geriatric and pediatric patients. Current ODT formulation studies usually rely on the personal experience of pharmaceutical experts and trial-and-error in the laboratory, which is inefficient…
The paper extends localisation technique to multiple constraints in Euclidean spaces.
problem Proving log-concavity of conditional measures in decomposed convex sets.
method Defining partitions of maximal closed convex sets and proving log-concavity of conditional measures.
result Existence of a partition and log-concavity of conditional measures for almost every set of the partition.
New optimal transport divergences derived from scoring functions.
problem Developing new divergences for optimal transport.
method Using scoring functions as cost functions in optimal transport.
result Comonotonic coupling is optimal for many new divergences.
Generative sampler learns velocity fields for efficient posterior inference.
problem Sampling from complex posterior distributions in high dimensions.
method Generative multivariate posterior sampler via flow matching, learning a velocity field for a deterministic transport map.
result Conditional Brenier map enables fast generation of credible sets with theoretical consistency guarantees.
In this paper we investigate model-independent bounds for exotic options written on a risky asset. Based on arguments from the theory of Monge-Kantorovich mass-transport we establish a dual version of the problem that has a natural financial interpretation in terms of semi-static hedging. In particular we prove that th…
We consider the question of learning in general topological vector spaces. By exploiting known (or parametrized) covariance structures, our Main Theorem demonstrates that any continuous linear map corresponds to a certain isomorphism of embedded Hilbert spaces. By inverting this isomorphism and extending continuously, …
New algorithm for estimating multivariate quantiles using stochastic optimal transport.
problem Estimating multivariate quantiles from data.
method Stochastic algorithm for entropic optimal transport in Banach spaces, using Fourier coefficients.
result Almost sure convergence of the stochastic algorithm in infinite-dimensional Banach spaces.
Improved bounds on learning algorithms' performance using conditional mutual information.
problem Bounding the generalization error of learning algorithms.
method Introducing conditional mutual information and disintegrated mutual information to tighten bounds.
result New bounds are tighter than previous ones, especially for noisy, iterative algorithms.
Harmonic maps pull convex functions on metric spaces to subharmonic ones.
problem Understanding how convex functions behave under harmonic maps on metric spaces.
method Proving subharmonicity of pullbacks of convex functions by harmonic maps in metric spaces.
result The pullback of a convex function by a harmonic map is subharmonic in metric spaces.
Some optimization or equilibrium problems involving somehow the concept of optimal transport are presented in these notes, mainly devoted to applications to economic and game theory settings. A variant model of transport, taking into account traffic congestion effects is the first topic, and it shows various links with…
Monge-Kantorovich distances, otherwise known as Wasserstein distances, have received a growing attention in statistics and machine learning as a powerful discrepancy measure for probability distributions. In this paper, we focus on forecasting a Gaussian process indexed by probability distributions. For this, we provid…
In this series of lectures we introduce the Monge-Kantorovich problem of optimally transporting one distribution of mass onto another, where optimality is measured against a cost function c(x,y). Connections to geometry, inequalities, and partial differential equations will be discussed, focusing in particular on recen…
Study heat content on RCD(K,N) spaces with specific boundary conditions.
problem Analyzing heat content in RCD(K,N) spaces with irregular boundaries.
method Proved first-order asymptotics using measured interior geodesic condition.
result Established first-order heat content asymptotics on RCD(K,N) spaces.
New bounds improve neural network generalization through slicing.
problem Difficulty in evaluating mutual information in high dimensions for neural networks.
method Slicing the parameter space and using disintegrated mutual information and k-sliced mutual information.
result Slicing improves generalization and offers significant computational and statistical advantages.
Duality for robust hedging with proportional transaction costs of path dependent European options is obtained in a discrete time financial market with one risky asset. Investor's portfolio consists of a dynamically traded stock and a static position in vanilla options which can be exercised at maturity. Both the stock …
A function is exponentially concave if its exponential is concave. We consider exponentially concave functions on the unit simplex. In a previous paper we showed that gradient maps of exponentially concave functions provide solutions to a Monge-Kantorovich optimal transport problem and give a better gradient approximat…
New bounds for model generalization under deterministic gradient descent.
problem Establishing generalization bounds for models trained with gradient descent methods.
method PAC-Bayesian bounds for deterministic optimisation algorithms.
result Fully computable bounds that depend on initial distribution and Hessian.
The study proves curvature bounds for quotient spaces of isometric actions.
problem Proving curvature bounds for quotient spaces of isometric actions.
method Disintegrate absolutely continuous measures and define a functional to prove curvature bounds.
result Necessary and sufficient conditions for Ricci curvature to be bounded below.
Two probability distributions μ and ν in second stochastic order can be coupled by a supermartingale, and in fact by many. Is there a canonical choice? We construct and investigate two couplings which arise as optimizers for constrained Monge-Kantorovich optimal transport problems where only supermartingales are al…
Study SRB measures for Anosov actions on manifolds.
problem Characterize SRB measures for Anosov actions.
method Use Ruelle-Taylor resonances and properties of Sinai-Ruelle-Bowen measures.
result SRB measures have properties like smooth disintegrations, positive basins, and are unique under certain conditions.
Let (X,L) be a (semi-) polarized complex projective variety and T a real torus acting holomorphically on X with moment polytope P. Given a probability density g on P we introduce a new type of Monge-Ampere measure on X, defined for singular T-invariant metrics on the line bundle L, generalizing the ordinary Monge-Amper…
Modeling informed trading with risk-averse market makers.
problem Understanding informed trading and its impact on market liquidity and risk premia.
method Connections between optimal transport theory and Kyle's model, including new characterizations of profits and duality.
result Liquidity is lower, assets exhibit short-term reversals, and risk premia depend on market maker inventories, which are mean reverting.
The paper studies convergence rates of Tsallis entropic regularization in optimal transport.
problem Optimal transport with regularization.
method Γ-convergence and quantization/shadow arguments.
result Derives convergence rate of Tsallis entropic regularization.
In a discrete-time market, we study model-independent superhedging, while the semi-static superhedging portfolio consists of {\it three} parts: static positions in liquidly traded vanilla calls, static positions in other tradable, yet possibly less liquid, exotic options, and a dynamic trading strategy in risky assets …
By investigating model-independent bounds for exotic options in financial mathematics, a martingale version of the Monge-Kantorovich mass transport problem was introduced in \cite{BeiglbockHenry LaborderePenkner,GalichonHenry-LabordereTouzi}. In this paper, we extend the one-dimensional Brenier's theorem to the present…
We prove that, if Ω⊂Rn is an open bounded starshaped domain of class C2, the constancy over ∂Ω of the function φ(y)=∫0λ(y)∏j=1n−1[1−tκj(y)]dt implies that Ω is a ball. Here kj(y) and λ(y) denote respectively the principal curvatures and the cut v…
Paper develops a new generalization bound using PAC-Bayes theory and Gibbs distributions.
problem Limits of traditional generalization bounds due to complexity measures.
method Leverages PAC-Bayes bounds with Gibbs distributions to derive a flexible generalization bound.
result Derives a generalization bound that can adapt to both hypothesis class and task complexity.
New risk measures control subgroup imbalances, improving PAC-Bayesian bounds.
problem Insufficient risk bounds for subgroup imbalances in data.
method Introduce constrained f-entropic risk measures and derive PAC-Bayesian bounds.
result First disintegrated PAC-Bayesian guarantees beyond standard risks.
The curve graph and related graphs are hyperbolic and have quasi-tree fibers.
problem Understanding the structure of the curve graph and related graphs.
method Analyzing a sequence of graphs with Lipschitz maps and proving hyperbolicity and quasi-tree properties.
result The graphs in the sequence are hyperbolic and have quasi-tree fibers, leading to bounds on asymptotic dimension and acylindrical actions.
Develops a new method for sampling from Bayesian credible sets using deep generative quantile learning.
problem Sampling from posterior distributions in high-dimensional spaces with intractable likelihoods.
method Uses deep neural networks to implicitly sample from Bayesian credible sets via a push-forward mapping and Monge-Kantorovich depth.
result Demonstrates improved performance and theoretical consistency of the quantile learning framework.
For a given 1-Lipschitz map u:Rn→Rm we define a partition, up to a set of Lebesgue measure zero, of Rn into maximal closed convex sets such that restriction of u is an isometry on these sets. We consider a disintegration, with respect to this partition, of a log-concave meas…
Paper addresses the disparity between sampled and mean representations in disentangled learning.
problem Disparity between sampled and mean representations in disentangled learning.
method Proposes a method to eliminate the disparity by proving and utilizing the relationship between total correlation of sampled and mean representations for multivariate normal distributions.
result Demonstrates that a factorized mean representation can have lower total correlation than the sampled representation.
Trade finance history traced from medieval origins to modern markets.
problem Evolution and standardization of trade finance products.
method Historical analysis of market structures and regulatory changes.
result Global trade finance market evolved from local to centralized, then decentralized.
Optimal transport improves multivariate prediction uncertainty quantification.
problem Uncertainty quantification in multivariate learning tasks, especially in regression and classification.
method Introducing a novel Conformal Prediction procedure using optimal transport to handle multivariate score functions and construct flexible prediction regions.
result Ensures finite-sample, distribution-free coverage guarantees for multivariate prediction sets.
New bounds using samplewise evaluated CMI for deep neural networks.
problem Improving generalization bounds for deep neural networks.
method Introduced a new family of information-theoretic generalization bounds using samplewise evaluated conditional mutual information (CMI).
result The new bounds can be tighter than previous ones for deep neural networks.
Develops a new framework for conditional independence.
problem Generalizing previous notions of conditional independence.
method Introduces transition probability spaces and transitional random variables.
result Satisfies all desired relevance relations except symmetry.
Study approximates operators on labelled conditional distributions for non-exchangeable systems.
problem Approximating operators on constrained probability measures for non-exchangeable systems.
method Combines cylindrical approximations and DeepONet-type neural architecture for finite-dimensional representations.
result Establishes a universal approximation theorem for continuous operators on Mλ. One purpose of this article is to draw attention to the seminal work of J. Mealy in 1989 on calibrations in semi-riemannian geometry where split SLAG geometry was first introduced. The natural setting is provided by doing geometry with the complex numbers C replaced by the double numbers D, where i with i^2 = -1 is rep…