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.
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48 results for “optimal maps”
Paper investigates optimal transport map estimation in infinite-dimensional spaces.
problem Estimating optimal transport maps in infinite-dimensional spaces is challenging.
method Characterizes -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.
In this paper, we present a novel and principled approach to learn the optimal transport between two distributions, from samples. Guided by the optimal transport theory, we learn the optimal Kantorovich potential which induces the optimal transport map. This involves learning two convex functions, by solving a novel mi…
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 -harmonic maps with new optimal constant.
problem Deriving the sharp vectorial Kato inequality for -harmonic mappings.
method Analyzing the inequality for -harmonic mappings and comparing with scalar valued cases.
result Established the optimal constant for -harmonic maps and enhanced the range of 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.
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.
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.
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 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 in the semi-discrete setting.
Automatic Hyper-Parameter Optimization Based on Mapping Discovery from Data to Hyper-Parameterscs.LG
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.
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.
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.
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 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 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 -densities and boundary conditions, monotonicity formula for optimal transport maps.
result Proves -regularity for nondegenerate -densities and -regularity for boundary.
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.
Computing optimal transport maps between high-dimensional and continuous distributions is a challenging problem in optimal transport (OT). Generative adversarial networks (GANs) are powerful generative models which have been successfully applied to learn maps across high-dimensional domains. However, little is known ab…
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 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.
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.
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.
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.
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.
We give an alternative proof for the fact that in -dimensional Alexandrov spaces with curvature bounded below there exists a unique optimal transport plan from any purely -unrectifiable starting measure, and that this plan is induced by an optimal map.
Study finds weak solutions for complex map flows with optimal lifespan.
problem Existence of weak solutions for two-phase matrix-valued harmonic map flows.
method Modified minimizing movement scheme, discretizing time and interpolating solutions.
result Existence of weak solutions with optimal lifespan for the limiting system.
DPOT uses deep learning to compute optimal transport efficiently.
problem Computing optimal transport between continuous distributions from unpaired samples.
method DeepParticle methods for min-min optimization without network structure restrictions.
result Established weak convergence and error bounds between learned and optimal maps.
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.
Researchers found counterexamples to conjectures about optimal transport maps on curved spaces.
problem Extending Caffarelli's contraction theorem to curved spaces.
method Constructing counterexamples to precise conjectures.
result Found counterexamples to Milman's conjectures about optimal transport maps on curved spaces.
Optimal maps, solutions to the optimal transportation problems, are completely determined by the corresponding c-convex potential functions. In this paper, we give simple sufficient conditions for a smooth function to be c-convex when the cost is given by minimizing a Lagrangian action.
We present a global optimization approach for solving the maximum a-posteriori (MAP) clustering problem under the Gaussian mixture model.Our approach can accommodate side constraints and it preserves the combinatorial structure of the MAP clustering problem by formulating it asa mixed-integer nonlinear optimization pro…
DDPM encoder matches optimal transport for natural images.
problem Understanding theoretical properties of DDPM latent space.
method Showed DDPM encoder matches optimal transport for common distributions.
result DDPM encoder map coincides with optimal transport map for natural images.
This work builds the connection between the regularity theory of optimal transportation map, Monge-Ampère equation and GANs, which gives a theoretic understanding of the major drawbacks of GANs: convergence difficulty and mode collapse. According to the regularity theory of Monge-Ampère equation, if the support of the …
We introduce a more restrictive version of the strict -condition, the so-called very strict -condition, and show the existence of optimal maps in very strict -spaces despite the possible lack of uniqueness of optimal plans.
This paper presents a novel two-step approach for the fundamental problem of learning an optimal map from one distribution to another. First, we learn an optimal transport (OT) plan, which can be thought as a one-to-many map between the two distributions. To that end, we propose a stochastic dual approach of regularize…