This paper establishes rates of universal approximation for the shallow neural tangent kernel (NTK): network weights are only allowed microscopic changes from random initialization, which entails that activations are mostly unchanged, and the network is nearly equivalent to its linearization. Concretely, the paper has …
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48 results for “Universal Transport Map”
Augmented Normalizing Flows: Bridging the Gap Between Generative Flows and Latent Variable Modelscs.LG
Enhances generative models by improving expressivity without high computational cost.
problem Improving expressivity in generative models without increasing computational complexity.
method Proposes a new family of generative flows on an augmented data space, proving they can approximate a Hamiltonian ODE as a universal transport map.
result Demonstrates state-of-the-art performance on flow-based generative modeling benchmarks.
Transportation analysis of denoising autoencoders: a novel method for analyzing deep neural networkscs.LG
The feature map obtained from the denoising autoencoder (DAE) is investigated by determining transportation dynamics of the DAE, which is a cornerstone for deep learning. Despite the rapid development in its application, deep neural networks remain analytically unexplained, because the feature maps are nested and param…
UNOT solves optimal transport problems efficiently using neural networks.
problem Computational expense in solving optimal transport problems.
method UNOT (Universal Neural Optimal Transport) uses Fourier Neural Operators to predict OT distances and plans accurately and efficiently.
result UNOT achieves up to 7.4x speedup over the Sinkhorn algorithm while maintaining accuracy.
The paper introduces surface signatures for irregular surfaces and rough surfaces.
problem Characterizing and integrating highly irregular paths and surfaces.
method Introducing surface signatures and proving extension theorems.
result Surface signatures are universal for surface holonomy and rough surfaces.
Bi-Lipschitz flows approximate a wide range of distributions.
problem Characterizing the expressivity of bi-Lipschitz normalizing flows.
method Linking score regularity to transport map bi-Lipschitzness via probability flow ODE.
result Gaussian pullbacks induced by bi-Lipschitz variance-preserving transport maps are -dense among all probability densities.
New flows model distributions on Riemannian manifolds without domain knowledge.
problem Limited modeling of distributions on Riemannian manifolds.
method Riemannian convex potential maps using optimal transport.
result These flows can model standard distributions on spheres and tori.
Paper proposes a new approach to optimal transport for vector and matrix densities.
problem Optimal transport for vector and matrix densities with positivity and action transitivity constraints.
method Gauge-theoretic approach using semi-direct product groups of diffeomorphisms and gauge transformations.
result Bures-type metrics on semi-direct product groups relate to Wasserstein-type metrics on vector and matrix densities via Riemannian submersions.
New framework formalizes estimating valid transport maps, revealing their statistical limits.
problem Estimating valid transport maps in generative modeling.
method Formalized a minimax framework for estimating valid transport maps.
result Estimating any valid transport map is as hard as estimating the optimal transport map under standard stability assumptions.
Random concave functionsmath.PR
Spaces of convex and concave functions appear naturally in theory and applications. For example, convex regression and log-concave density estimation are important topics in nonparametric statistics. In stochastic portfolio theory, concave functions on the unit simplex measure the concentration of capital, and their gr…
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.
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…
This work clarifies different transport map constructions and their causal interpretations.
problem Identifying distinct transport map constructions and their equivalence.
method Comparative analysis of three transport map constructions: cyclically monotone, quantile-preserving, and triangular monotone.
result Conditions for equivalence of different transport map constructions.
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.
Transport Score Climbing: Variational Inference Using Forward KL and Adaptive Neural Transportstat.ML
TSC uses HMC and adaptive transport maps to optimize forward KL for variational inference.
problem Variational inference underestimates uncertainty when minimizing reverse KL.
method TSC uses Hamiltonian Monte Carlo and adaptive transport maps to optimize KL(p||q).
result TSC achieves competitive performance in training variational autoencoders on large-scale data.
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.
The paper studies moduli spaces of non-smooth metric structures with non-negative Ricci curvature.
problem Understanding moduli spaces of non-smooth metric structures with non-negative Ricci curvature.
method Relating convergence of RCD(0,N)-structures to their lifts, constructing Albanese and soul maps, proving their continuity, and constructing examples.
result Construction of moduli spaces with non-trivial rational homotopy groups.
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.
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.
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.
Optimal Transport Graph Neural Networks (OT-GNN) improves graph embeddings by using optimal transport.
problem Graph Neural Networks (GNN) often lose structural or semantic information when aggregating node embeddings.
method Combines optimal transport (OT) with parametric graph models to compute graph embeddings from Wasserstein distances between node embeddings and prototype point clouds.
result OT-GNN outperforms popular methods on molecular property prediction tasks and produces smoother graph representations.
Parallel transport map over reductive spaces is an affine submersion.
problem Understanding parallel transport in reductive homogeneous spaces with torsion.
method Generalizing previous results on affine symmetric spaces, proving compactness of shape operators, and proposing definitions for regularized mean curvatures.
result Each fiber of the parallel transport map over a reductive homogeneous space is minimal in both senses.
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.
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.
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.
Transport Quasi-Monte Carlomath.NA
Improves QMC for complex distributions using transport maps.
problem Challenges in applying QMC to general target distributions.
method Train a transport map to approximate target distributions, ensuring RQMC achieves superior error rates.
result Transport QMC achieves faster convergence rates than standard Monte Carlo under mild conditions.
New methods use transport maps to improve Langevin dynamics for sampling.
problem Sampling high-dimensional, non-Gaussian distributions efficiently.
method Apply transport maps to accelerate Langevin dynamics convergence.
result Discretized processes converge to target distribution with non-asymptotic bounds.
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.
No-collision maps improve manifold learning for image data.
problem Lack of geometric feature sensitivity in traditional distance measures.
method Developed no-collision transportation maps and distances.
result No-collision distances provide isometry for translations and dilations.
Optimal transport (OT)-based methods have a wide range of applications and have attracted a tremendous amount of attention in recent years. However, most of the computational approaches of OT do not learn the underlying transport map. Although some algorithms have been proposed to learn this map, they rely on kernel-ba…
New method uses kernel Stein discrepancy for measure transport without strict continuity constraints.
problem Minimizing Kullback-Leibler divergence for posterior approximation.
method Proposes minimizing kernel Stein discrepancy instead of Kullback-Leibler divergence.
result Demonstrates consistency and competitiveness of the new method.
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.
Neural framework for conditional OT maps learns from categorical and continuous variables.
problem Learning conditional optimal transport maps between complex distributions.
method Hypernetwork generates adaptive transport layer parameters based on conditioning variables.
result Our method outperforms simpler conditioning methods in comprehensive ablation studies.
Study develops a new method for creating fair models.
problem Ensuring equal outcomes for different protected groups.
method Introduces a new group-fair constraint based on transport maps.
result Develops a novel algorithm FTM for training group-fair models.
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.
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.
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.
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.
New method uses transport maps for efficient Bayesian inference.
problem Efficiently perform sequential Bayesian inference of static model parameters.
method Estimation of structured transport maps to extract conditional distributions.
result Gradient-based characterization of posterior density for online parameter estimation.
Develops theory for conditional optimal transport in infinite-dimensional spaces.
problem Bayesian inference with functional parameters in infinite-dimensional spaces.
method Theory of constrained optimal transport for block-triangular maps.
result Regularity estimates on conditioning maps from prior to posterior.
New algorithm improves on existing methods for solving transport problems.
problem Finding a map to transport one distribution to another.
method Iterative Markovian Fitting (IMF) and Diffusion Schrödinger Bridge Matching (DSBM).
result DSBM significantly improves over previous SB numerics and recovers various transport methods.
This paper uses normalizing flows to approximate transport maps between densities.
problem Approximating transport maps between given densities.
method Construct time-dependent controls using normalizing flows.
result Provides bounds on the number of switches for piecewise constant approximations.
Using McCann's transportation map, we establish a transport inequality on compact manifolds with positive Ricci curvature. This inequality contains the sharp spectral comparison estimates.
Generalized are the investigated in other works of the author transports along paths in fibre bundles to transports along arbitrary maps in them. Their structure and some properties are studied. Special attention is paid to the linear case and the case when the map's domain is a Cartesian product of two sets. Also cons…
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.
A new method improves Bayesian filtering in nonlinear systems.
problem Bayesian filtering in nonlinear dynamical systems with non-Gaussian posteriors.
method Transport maps with block-triangular structure and gradient flows for MMD minimization.
result Accurate approximation of non-Gaussian posteriors without particle collapse.
LazyDINO efficiently solves high-dimensional Bayesian inverse problems with fast and scalable solutions.
problem High-dimensional nonlinear Bayesian inverse problems with expensive parameter-to-observable maps.
method LazyDINO combines derivative-informed neural surrogates and lazy map variational inference for efficient posterior approximation.
result Significant cost reduction in amortized Bayesian inversion, achieving one to two orders of magnitude improvement.
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.