Novel approach learns optimal transport using convex neural networks.
problem Learning optimal transport between distributions from samples.
method Solving a minimax optimization to learn two convex functions, representing the optimal transport map.
result The approach finds optimal transport mappings that are independent of initialization and can handle discontinuous distributions.
Optimizes biharmonic map regularity using stratification methods.
problem Improving the known almost optimal regularity of biharmonic maps.
method Quantitative stratification method.
result Optimal regularity results for minimizing biharmonic maps.
Paper investigates optimal transport map estimation in infinite-dimensional spaces.
problem Estimating optimal transport maps in infinite-dimensional spaces is challenging.
method Characterizes γ γ γ -smoothness for optimal transport maps and develops a polynomial-rate estimator. result Shows polynomial-order minimax risk for optimal transport map estimation.
Space mapping speeds up shape optimization for PDEs.
problem Efficiently solving shape optimization problems constrained by PDEs.
method Combines fine and coarse model optimizations using Riemannian metrics.
result Space mapping methods are highly efficient for complex shape optimization problems.
Estimates optimal transport maps with known cost functions.
problem Ensuring optimal transport maps correspond to real-world usefulness.
method Differentiable neural ground costs with known Monge map forms.
result General approach for incorporating prior information.
The paper studies optimal maps between hyperbolic surfaces, focusing on their rigidity and obstructions.
problem Finding optimal Lipschitz maps between hyperbolic surfaces and understanding their rigidity and obstructions.
method Introducing deflations, optimal maps to trees that obstruct optimal maps between surfaces, and using a smooth orthogeodesic foliation.
result Deflations are the main obstructions to optimal maps between hyperbolic surfaces, and they are essentially the only ones.
Sharp inequality for p p p -harmonic maps with new optimal constant.
problem Deriving the sharp vectorial Kato inequality for p p p -harmonic mappings. method Analyzing the inequality for p p p -harmonic mappings and comparing with scalar valued cases. result Established the optimal constant for p p p -harmonic maps and enhanced the range of p p p values for regularity. A method for optimal Bayesian filtering using progressive particle flow and optimal transport maps.
problem Optimizing Bayesian filtering with deterministic particles to avoid degeneration.
method Progressive flow of particles through a sequence of sub-steps, each using an optimal transport map to replace non-equally weighted particles with equally weighted ones.
result The method avoids particle degeneration and simplifies the filtering process by not requiring inversions or monotonicity constraints.
Optimal maps exist in very strict C D ( K , ∞ ) CD(K,\infty) C D ( K , ∞ ) spaces despite plan uniqueness issues.
problem Existence of optimal transport maps in very strict C D ( K , ∞ ) CD(K,\infty) C D ( K , ∞ ) spaces. method Introduced a more restrictive C D ( K , ∞ ) CD(K,\infty) C D ( K , ∞ ) condition and showed existence of optimal maps. result Existence of optimal maps in very strict C D ( K , ∞ ) CD(K,\infty) C D ( K , ∞ ) spaces. This work broadens optimal transport map estimation theory to stochastic settings.
problem Existing theory for optimal transport map estimation is restricted to deterministic maps under specific conditions.
method Introduces a novel metric for evaluating stochastic maps, develops computationally efficient estimators with robust guarantees.
result First general-purpose theory for map estimation compatible with real-world stochastic applications.
Unique optimal map found in curved spaces.
problem Existence of optimal transport plan in curved spaces.
method Alternative proof using Alexandrov spaces.
result Existence of unique optimal map.
Paper tackles large-scale optimal transport and mapping estimation.
problem Learning optimal maps between large distributions.
method Two-step approach: first, stochastic dual regularized OT; second, Monge map estimation.
result The method scales better with large samples and converges to optimal maps.
The paper reviews advances in estimating and understanding optimal transport maps.
problem Estimating and understanding optimal transport maps from samples.
method Recent advances in statistical inference for optimal transport maps.
result Developed limit theorems for the optimal transport map using samples.
Global optimization approach for MAP clustering under Gaussian mixtures.
problem Maximum a-posteriori clustering problem under Gaussian mixture model.
method Mixed-integer nonlinear optimization (MINLP) transformed into mixed-integer quadratic program (MIQP).
result Explicit quantification of optimality gap, leading to globally optimal solutions.
Generative adversarial networks learn optimal transport maps efficiently.
problem Learning optimal transport maps between high-dimensional distributions.
method Proposed a GAN with discriminator objective as 2-Wasserstein metric.
result Generator learns optimal transport map during training.
Efficiently estimates optimal transport maps with rigorous guarantees.
problem Estimating optimal transport maps between distributions efficiently.
method Entropic version of Brenier's theorem, Sinkhorn's algorithm.
result Estimator is parallelizable and efficient for massive data sets.
The paper optimizes estimating transport maps between distributions.
problem Estimating optimal transport maps between distributions.
method Plugin approach using optimal couplings and extensions.
result Minimax optimality of the proposed estimators.
New research extends optimal transport map breakdown properties to general costs.
problem Understanding robustness of optimal transport maps under contamination.
method Analyzing breakdown point of optimal transport maps for general convex costs.
result Breakdown point of optimal transport maps is independent of the cost function.
New algorithm estimates transport maps with nearly optimal error.
problem Estimating smooth transport maps efficiently and accurately.
method Solving semi-dual formulation of optimal transport with kernel sums-of-squares.
result Statistical L 2 L^2 L 2 error on maps nearly matches minimax lower-bounds. Proposes m-POT to improve m-OT's misspecified mappings issue.
problem Misspecified mappings in mini-batch optimal transport.
method Partial optimal transport (POT) between mini-batch empirical measures.
result m-POT alleviates incorrect mappings compared to current methods.
We systematically investigate the problem of representing Markov chains by families of random maps, and which regularity of these maps can be achieved depending on the properties of the probability measures. Our key idea is to use techniques from optimal transport to select optimal such maps. Optimal transport theory a…
Estimates discontinuous optimal transport maps between a discrete and continuous distribution.
problem Estimating discontinuous optimal transport maps between a discrete and continuous distribution.
method Entropic optimal transport estimator, computationally efficient.
result The estimator converges at the minimax-optimal rate n − 1 / 2 n^{-1/2} n − 1/2 in the semi-discrete setting. This paper proposes an efficient autoHPO method based on data-to-hyper-parameter mapping.
problem Manual hyper-parameter tuning is costly and dependent.
method The approach is based on mapping from data to hyper-parameters using a sophisticated network structure and effective construction algorithms.
result The proposed approach significantly outperforms state-of-the-art methods.
Estimates conditional Brenier maps using entropic optimal transport.
problem Non-parametric estimation of conditional Brenier maps.
method Entropic optimal transport for scalable non-parametric estimation.
result Entropic optimal transport maps asymptotically converge to conditional Brenier maps.
Optimal discrete harmonic maps between hyperbolic surfaces are found via minimizing energy.
problem Finding optimal discrete harmonic maps between hyperbolic surfaces.
method Minimizing Dirichlet energy over all possible hyperbolic structures and realizations within a fixed homotopy class.
result At the optimal hyperbolic structure, the discrete harmonic map and edge weights are induced from a weighted Delaunay decomposition.
Paper introduces a neural network for consistent estimation of optimal transport maps.
problem Statistically consistent estimation of optimal transport maps between probability distributions.
method Lipschitz-constrained GAN penalized by quadratic transportation cost.
result The generator converges uniformly to the optimal transport map as sample size increases.
The paper studies polar factorizations of maps on spheres, comparing algebraic and optimal mass transport methods.
problem Analyzing polar factorizations of maps on spheres using different mathematical approaches.
method Examines polar factorizations of conformal and projective maps on the sphere in the context of optimal mass transport and algebraic polar factorization.
result Agrees on the polar factorization for conformal maps but finds conditions for projective maps.
The paper connects moment maps to the stability of holomorphic fibrations.
problem Stability of holomorphic fibrations.
method Use of moment maps and K-stability criteria.
result Existence of optimal symplectic connections implies stability of fibrations.
New geometric approach gives apriori estimate for optimal transport maps.
problem Proving regularity of optimal transport maps under Ma--Trudinger--Wang condition.
method Geometric derivation using pseudo-Riemannian geometry.
result New derivation of C 1 C^1 C 1 interior estimate for optimal maps. Optimal transport for functional data using Hilbert-Schmidt operators.
problem Optimal transport for distributions on function spaces with partially represented stochastic maps.
method Regularization technique to restrict transport maps to Hilbert-Schmidt operators, developing an efficient algorithm.
result Existence, uniqueness, and consistency of the Hilbert-Schmidt operator estimate for the transport map.
Study on regularity of optimal transport maps on convex domains with quadratic cost.
problem Regularity of optimal transport maps between convex domains with quadratic cost.
method Analysis of C α C^α C α -densities and C 1 , α C^{1, α} C 1 , α boundary conditions, monotonicity formula for optimal transport maps. result Proves C 1 , 1 − ε C^{1, 1-\varepsilon} C 1 , 1 − ε -regularity for nondegenerate C α C^α C α -densities and C 2 , α C^{2, α} C 2 , α -regularity for C 1 , α C^{1, α} C 1 , α boundary. Study linearizes 2-Wasserstein space using optimal transport maps.
problem Stability and linearization of the 2-Wasserstein space.
method Explicit embedding of probability measures into a Hilbert space using optimal transport maps.
result The embedding is (bi-)Hölder continuous, with stability results for optimal transport maps.
Paper introduces a novel map learning algorithm for domain translation and adaptation.
problem Learning a map between related data spaces that can be applied to out-of-sample data and satisfies application-specific constraints.
method Utilizes normalizing flows to parameterize a map that minimizes a probability distance and application-specific regularizers, solving a modified optimal transport problem.
result The proposed method (parOT) outperforms existing optimal transport approaches in domain adaptation and translation tasks.
SOS programming verifies MTW tensor non-negativity for optimal transport maps.
problem Verifying MTW tensor non-negativity for general cost functions is difficult.
method Sum-of-Squares (SOS) programming for verifying and approximating MTW non-negativity.
result SOS programming provides certificates and approximations of MTW non-negativity.
We analyze errors in filtering algorithms using optimal transport.
problem Estimation errors in optimal transport-based filtering algorithms.
method Systematic analysis of estimation errors for conditional Brenier maps.
result Demonstrates effectiveness and practical potential of the optimal transport filtering algorithm.
Mathematical analysis shows Delisle-Euler map methods are optimal.
problem Comparing ancient and modern map drawing methods.
method Analyzing similarities and differences between ancient and modern map drawing methods.
result Delisle-Euler map methods are optimal among conical maps.
This article addresses regularity of optimal transport maps for cost="squared distance" on Riemannian manifolds that are products of arbitrarily many round spheres with arbitrary sizes and dimensions. Such manifolds are known to be non-negatively cross-curved [KM2]. Under boundedness and non-vanishing assumptions on th…
The paper classifies constraint mappings in optimization problems.
problem Understanding generic behavior of constraint functions in optimization.
method Using singularity theory of smooth mappings and subgroup classification.
result Families of constraint mappings are a residual set with at most 4 parameters.
The paper compares numerical schemes for nonholonomic systems using retraction maps.
problem Optimal control of nonholonomic systems with numerical approximations.
method Retraction maps used as seed for geometric integrators of Hamilton equations.
result Performance comparison of symplectic and non-symplectic integrators.
Existence of harmonic maps from higher-dimensional manifolds to spheres proven.
problem Proving existence of nonconstant harmonic maps from arbitrary manifolds to spheres.
method Using optimal regularity and eigenvalue optimization on manifolds.
result First general existence result for harmonic maps from higher-dimensional manifolds to a large class of positively curved targets.
Optimizes energy of mappings from complex projective spaces.
problem Finding energy-minimizing mappings between complex projective spaces and Riemannian manifolds.
method Establishes optimal lower bounds for energy functionals and characterizes optimal mappings.
result Optimal lower bounds for energy functionals are characterized for mappings from real and complex projective spaces.
Contour maps show needed sample sizes for portfolio optimization under ES risk measure.
problem Determining sample size for accurate portfolio optimization under Expected Shortfall.
method Analytical calculations and statistical physics methods applied to random portfolios.
result Necessary sample sizes are often too large for practical use.
Study optimal transport on simplex boundary, proving transport map and potential regularity.
problem Regularity of transport map and potential on simplex boundary.
method Boundary regularity results for optimal transport maps, exploiting simplex symmetries.
result Regularity properties of transport map and its convex potential.
Proposes methods to compute optimal transport maps via subspace projections.
problem Computing optimal transport in high dimensions is challenging due to the curse of dimensionality.
method Develops two methods to extrapolate optimal transport plans from subspace projections to the full space.
result The best optimal transport plan is a generalization of the Knothe-Rosenblatt transport.
New method for optimizing complex composite functions with reduced variance.
problem Optimizing multi-level composite functions with nested random and smooth mappings.
method Normalized proximal approximate gradient (NPAG) method with nested stochastic variance reduction.
result Total sample complexity of O ( ε − 3 ) O(ε^{-3}) O ( ε − 3 ) in expectation and O ( N + N ε − 2 ) O(N+\sqrt{N}ε^{-2}) O ( N + N ε − 2 ) in finite-sum cases. Novel stability bounds for OT maps improve density estimation.
problem Estimating optimal transport maps between probability distributions.
method Developed novel stability bounds for OT maps, reducing the problem to density estimation.
result Stability bounds allow for sharper guarantees without smoothness assumptions.
New method optimizes kernel feature maps for better classification.
problem High computational and memory complexity of standard kernel methods.
method Discriminant Information criterion for optimizing kernel feature maps.
result Improved optimization and generalization performances over state-of-the-art methods.
The paper studies bi-slant Riemannian maps to Kenmotsu manifolds and derives inequalities.
problem Investigating bi-slant Riemannian maps and their properties.
method Introducing and studying bi-slant Riemannian maps, deriving curvature relations and inequalities.
result Construction of Chen-Ricci inequalities, DDVV inequalities, and optimal inequalities involving Casorati curvatures.