Optimal transport for vector Gaussian mixtures improves efficiency and structure preservation.
problem Optimal mass transport for vector-valued Gaussian mixtures.
method Vectorizing Gaussian mixture models and studying optimal mass transport problems.
result Computational efficiency and structure preservation in optimal mass transport.
The problem for consistency between linear transports along paths and real bundle metrics in real vector bundles is stated. Necessary and/or sufficient conditions, as well as conditions for existence, for such consistency are derived. All metrics (resp. transports) consistent with a given transport (resp. metric) are e…
This research simplifies Riemannian LBFGS for SPD manifolds.
problem Optimization on Riemannian manifolds, especially SPD.
method Two mappings for tangent space, making vector transports and adjoint vector transports identity.
result RLBFGS becomes less computationally expensive and easier to analyze.
We show that finite parallel transports of vectors in Riemannian spaces, determined by the multiplication law in the deformed groups of diffeomorphisms, and sequences of infinitesimal parallel transports of vectors along geodesics are equivalent.
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.
The linear transports along paths in vector bundles introduced in Ref. [1] are applied to the special case of tensor bundles over a given differentiable manifold. Links with the transports along paths generated by derivations of tensor algebras are investigated. A possible generalization of the theory of geodesics is p…
This note addresses the construction of a notion of parallel transport along superpaths arising from the concept of a superconnection on a vector bundle over a manifold M. A superpath in M is, loosely speaking, a path in M together with an odd vector field in M along the path. We also develop a notion of parall…
A vector bundle with connection over a supermanifold leads naturally to a notion of parallel transport along superpaths. In this note we show that {\it every} such parallel transport along superpaths comes form a vector bundle with connection, at least when the base supermanifold is a manifold.
The (parallel) linear transports along paths in vector bundles are axiomatically described. Their general form and certain properties are found. It is shown that these transports are locally (i.e. along every fixed path) always Euclidean ones in a senses that there exist frames in which their matrices are unit. The inv…
Curvature and torsion of linear transports along paths in, respectively, vector bundles and the tangent bundle to a differentiable manifold are defined and certain their properties are derived.
We explain how the kind of ``parallel transport'' of a wavefunction used in discussing the Berry or Geometrical phase induces the conventional parallel transport of certain real vectors. These real vectors are associated with operators whose commutators yield diagonal operators; or in Lie algebras those operators whose…
Optimal algorithms for Riemannian optimization with reduced complexity.
problem Stochastic optimization on Riemannian manifolds with limited data.
method Zeroth-order Riemannian Averaging Stochastic Approximation algorithms using Riemannian moving-average estimators and novel geometric conditions.
result Achieves optimal sample complexities for generating approximate first-order stationary solutions.
Theory of parallel transport on non-collapsed RCD spaces established.
problem Parallel transport on non-collapsed RCD spaces.
method General theory developed for parallel transport on non-collapsed RCD spaces, including geodesics and curves via time-dependent vector fields.
result Existence and uniqueness of parallel transport results obtained.
Optimal transport is #P-hard when components are independent, even with approximate solutions.
problem Computational complexity of optimal transport with independent marginals.
method Proved #P-hardness and developed a pseudo-polynomial time approximation algorithm.
result Optimal transport is #P-hard even with independent components and approximate solutions.
The axiomatic approach to parallel transport theory is partially discussed. Bijective correspondences between the sets of connections, (axiomatically defined) parallel transports, and transports along paths satisfying some additional conditions, are constructed. In particular, the equivalence between the concepts "conn…
New method AM learns optimal vector fields for entire distribution sequences, matching OT.
problem Optimal Transport (OT) problem in generative modeling.
method Action Matching (AM) method learns optimal vector fields for a sequence of distributions.
result AM method achieves optimal transport by learning vector fields for entire distribution sequences.
Steerable neural ODEs on homogeneous spaces for equivariant feature dynamics.
problem Learning continuous-time equivariant dynamics of vector-valued features on homogeneous spaces.
method Introduces steerable neural ordinary differential equations on homogeneous spaces, interpreting features as sections of associated vector bundles over M. result Steerable NODEs are G-equivariant when the flow and connection are G-invariant, and they incorporate existing models. The general problem for consistency between arbitrary transports along paths in fibre bundles and bundle morphisms between them is formulated and investigated. The special case of one fibre bundle, its morphism and transport along paths acting in it is considered. The consistency between linear transports along paths i…
A novel method for parallel transport and geodesics on submanifolds.
problem Understanding parallel transport and geodesics on submanifolds.
method Rolling tangent space to visualize and analyze parallel transport and geodesics.
result Conditions for parallel transport and geodesics are simplified and visualized in the tangent space.
Study perturbations of submodules in Drury-Arveson space, finding smooth vector bundles with Hermitian connections.
problem Geometry of perturbations in Drury-Arveson space.
method Analysis of smooth vector bundles with Hermitian connections and computation of parallel transport operators.
result Found natural Hermitian connections on perturbed submodules.
Neural optimal transport improves multivariate conformal prediction.
problem Multivariate quantile regression challenges and existing methods ignore joint distribution geometry.
method Combines neural optimal transport with amortized optimization for efficient training and faster inference.
result Constructs tighter and more informative predictive regions for multivariate conformal prediction.
POTD estimates SDR subspace using optimal transport for binary response.
problem Insufficient performance of existing SDR methods for categorical responses.
method Principal optimal transport direction (POTD) using optimal transport coupling.
result POTD exclusively estimates SDR subspace for error-free class labels.
A new copula model for multi-attribute data using optimal transport.
problem Relaxing the Gaussian assumption for multi-attribute graphical models.
method Introducing a new copula (Cyclically Monotone Copula) and using optimal transport theory.
result The model allows arbitrary continuous distributions and is more flexible than classical methods.
Solves open problem on simple surfaces with novel twistor correspondence.
problem Existence of nontrivial holomorphic vector bundles on simple surfaces.
method Novel twistor correspondence, Nash-Moser inverse function theorem, and microlocal analysis.
result Simple surface twistor space supports no nontrivial holomorphic vector bundles.
New guarantees for uniquely identifying transport maps and vector fields from finite measure-valued data.
problem Unique recovery of transport maps and vector fields from finite measure-valued data.
method Use of Whitney and Takens embedding theorems to establish conditions for unique identification.
result New metric for comparing diffeomorphisms and analogous results in infinitesimal settings.
A new method ranks uncertainty vectors from multiple measures for robust prediction.
problem Single scalar measures of model reliability are insufficient for comprehensive uncertainty quantification.
method Optimal transport ranks vectors of uncertainty measures, supporting flexible fusion of aleatoric and epistemic uncertainties.
result The method provides a robust ranking of uncertainty that supports various downstream tasks.
Develops regularity theory for Beckmann's optimal transport problem.
problem Minimizing total squared flux in continuous transport from source to target.
method Unconstrained Lagrangian formulation, variational first order optimality conditions, Schauder estimates.
result Exact Hölder regularity of potential, flux, and flow generating on bounded, regular domains.
Functional-analytic method for stochastic parallel transport in bundles.
problem Stochastic parallel transport in Hermitian bundles over Riemannian manifolds.
method Purely functional-analytic construction.
result Obtained a general Feynman-Kac formula in vector bundles.
Method predicts how probability distributions evolve over time.
problem Predicting how systems described by probability distributions evolve under different conditions.
method Wasserstein Parallel Transport
result Wasserstein Parallel Transport provides counterfactual comparisons of distributional dynamics.
We show that the metrical connection can be introduced in the two-dimensional Finsler space such that entailed parallel transports along curves joining points of the underlying manifold keep the two-vector angle as well as the length of the tangent vector, thereby realizing isometries of tangent spaces under the parall…
New bounds for optimal transport using Gaussian processes and rate-distortion functions.
problem Finding bounds for entropic optimal transport with mutual information constraints.
method Lifting technique to construct a Gaussian process and applying the majorizing measure theorem.
result Maximum expected inner product is equivalent to a truncated integral involving the rate-distortion function.
We introduce a stochastic model for noisy vector fields on manifolds.
problem Noisy vector fields violate the assumption of parallel transport in stochastic analysis.
method We define a stochastic Lie bracket that induces torsion and analyze its consequences.
result The stochastic Lie bracket induces torsion in expectation.
A new method learns meaningful distances between samples using optimal transport.
problem Learning meaningful distances between samples in datasets without labeled data.
method Computes OT distances between samples and features using singular vectors of a function mapping ground metrics to OT distances.
result Wasserstein Singular Vectors provide a scalable solution for unsupervised ground metric learning.
Anisotropic connections and parallel transport defined in Finsler spacetimes.
problem Defining and characterizing anisotropic connections and parallel transport in Finsler spacetimes.
method Introducing a new covariant derivative and parallel transport, identifying vertically trivial Finsler connections with anisotropic connections, and characterizing the Levi-Civita-Chern anisotropic connection.
result Characterization of the Levi-Civita-Chern anisotropic connection as the one preserving the length of parallely propagated vectors.
Survey of diffusion and optimal transport methods in machine learning.
problem Design and analysis of time-evolving probability distributions in machine learning.
method Switch from Eulerian to Lagrangian representation through vector fields.
result Both diffusion methods and optimal transport offer computational advantages.
The displacement and deviation vectors in spaces (manifolds), the tangent bundle of which is endowed with a transport along paths, are introduced. In case these spaces are equipped with a linear connection, the deviation equations (between arbitrary, geodesic or not, paths) in such spaces are investigated.
On compact manifolds which are not simply connected, we prove the existence of "fake" solutions to the optimal transportion problem. These maps preserve volume and arise as the exponential of a closed 1 form, hence appear geometrically like optimal transport maps. The set of such solutions forms a manifold with dimensi…
The Finsleroid--induced scalar product, and hence the angle, proves to remain unchanged under the Finsleroid--type parallel transportation of involved vectors in the Landsberg case. The two--vector extension of the Finsleroid metric tensor is proposed.
A geometric interpretation of curvature and torsion of linear transports along paths is presented. A number of (Bianchi type) identities satisfied by these quantities are derived. The obtained results contain as special cases the corresponding classical ones concerning curvature and torsion of linear connections.
A JAX toolbox solves optimal transport problems for point clouds and histograms.
problem Optimal transport problems between point clouds and histograms.
method Automatic and custom reverse mode differentiation, vectorization, just-in-time compilation, and accelerators support.
result Solves a wide range of optimal transport problems including regularized OT, barycenters, Gromov-Wasserstein, and low-rank solvers.
LA-VDM accelerates VDM using landmarks to improve data analysis.
problem Efficiently analyzing complex datasets with nonuniform sampling densities.
method Landmark-constrained two-stage normalization to accelerate VDM.
result LA-VDM accurately recovers parallel transport and converges to the connection Laplacian.
New algorithm accelerates optimization on Riemannian manifolds, including Wasserstein space.
problem Accelerating optimization methods in Riemannian geometry.
method Dynamic stepsize algorithms on Riemannian manifolds with specific vector transport.
result First provable accelerated gradient method in Wasserstein space.
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.
Efficient federated algorithm for calculating transportation barycenter.
problem Efficiently calculating the free-support transportation barycenter in a federated setting.
method Single-loop dual decomposition algorithm that uses only aggregated information.
result Significantly scalable and low-complexity algorithm for federated computation.
Stein transport improves Bayesian inference with faster convergence and reduced variance.
problem Efficiently approximating posterior distributions in Bayesian inference.
method A novel Bayesian inference method using Stein transport, which pushes particles along a curve of tempered distributions.
result Stein transport reaches posterior approximations faster and more accurately than Stein variational gradient descent (SVGD).
In this paper, we introduce an inclined curves according to parallel transport frame. Also, we define a vector field called Darboux vector field of an inclined curve in and we give a new characterization such as: "α: I \subset R \rightarrow E^4 is an inclined curve \Leftrightarrow k_1 \int k_1ds + k_2 \int \k_2 +k_3ds …
We make evident a curvature tensor for every vector sub-bundle of an arbitrary manifold tangent bundle which reduces to the curvature tensor of an Ehresmann connection in the case of the horizontal sub-bundle of the tangent bundle to the total space of the nonlinear fiber bundle on which the connection is defined. Then…
Geometric framework for aligning fiber tracts across subjects.
problem Challenges in finding direct tract correspondence across multiple individuals.
method Geometric framework using intrinsic mean and deformation fields, parallel transport for registration.
result Bundle alignment results on 43 healthy adult subjects.