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…
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48 results for “Neural Lie Transport Operator”
This paper tackles continuous domain generalization, improving model performance across unseen domains.
problem Existing domain generalization approaches fail to capture the complex, multidimensional nature of real-world variation.
method Introduces Continuous Domain Generalization (CDG), a principled framework grounded in geometric and algebraic theories. Proposes a Neural Lie Transport Operator (NeuralLio) for structure-preserving parameter transitions and a gating mechanism for robust generalization.
result Demonstrates significant improvement in generalization accuracy and robustness across various datasets.
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 .
result Steerable NODEs are -equivariant when the flow and connection are -invariant, and they incorporate existing models.
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.
Regular Lie groups are infinite dimensional Lie groups with the property that smooth curves in the Lie algebra integrate to smooth curves in the group in a smooth way (an `evolution operator' exists). Up to now all known smooth Lie groups are regular. We show in this paper that regular Lie groups allow to push surprisi…
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.
Parallel transport defined for 2-bundles over Lie groupoids.
problem Defining parallel transport for 2-bundles over Lie groupoids.
method Using Lie 2-group torsors and pseudofunctors, extending principal 2-bundles to differentiable stacks.
result A smooth parallel transport functor defined for Haefliger paths.
Paper derives and applies a parallel transport equation on Lie groups.
problem Efficiently solving parallel transport on Lie groups with left-invariant metrics.
method Derives a parallel transport equation in Lie algebra, applies it to SE(3), and compares to existing methods.
result Stable and efficient parallel transport implementation on Lie groups.
Convolution has been playing a prominent role in various applications in science and engineering for many years. It is the most important operation in convolutional neural networks. There has been a recent growth of interests of research in generalizing convolutions on curved domains such as manifolds and graphs. Howev…
This paper develops optimal transport methods on the roto-translation group SE2.
problem Optimal transport on the roto-translation group SE2 for image analysis.
method Develops a computational framework for optimal transportation over Lie groups, focusing on SE2. Uses Sinkhorn-like algorithm with efficient distance approximations.
result Advances in image barycentric interpolation, orientation field interpolation, and Wasserstein flows on SE2.
Preparation for Gauge Theorymath-ph
Class lecture notes at a beginning graduate level on the mathematical background needed to understand classical gauge theory. Covers group actions, fiber bundles, principal bundles, connections, gauge transformations, parallel transport, curvature, covariant derivatives, pseudo-riemannian manifolds, lagrangians, cliffo…
It is shown that the heat operator in the Hall coherent state transform for a compact Lie group is related with a Hermitian connection associated to a natural one-parameter family of complex structures on . The unitary parallel transport of this connection establishes the equivalence of (geometric) quantizati…
Differential chains are a proper subspace of de Rham currents given as an inductive limit of Banach spaces endowed with a geometrically defined strong topology. Boundary is a continuous operator, as are operators that dualize to Hodge star, Lie derivative, pullback and interior product. Partitions of unity exist in thi…
Derives inequality for optimal transport on manifolds.
problem Optimal transport theory on manifolds.
method Five gradients inequality for cost functions on Lie groups and Riemannian manifolds.
result Derives inequality for optimal transport on specific manifolds.
DOODL learns shared spectral dynamics across related dynamical systems.
problem Learning independent dynamical operators for each system limits discovery of shared structure.
method DOODL learns a dictionary of characteristic spectral dynamics on a manifold of related systems.
result DOODL achieves errors one to two orders of magnitude lower than independent operator estimation methods.
Study of gauge theory and parallel transport in Lie 2-group bundles over Lie groupoids.
problem Classify and understand principal 2-bundles over Lie groupoids.
method Introduce principal Lie 2-group bundles, study connection structures, gauge transformations, and parallel transport.
result Extend classification of principal 2-bundles to differentiable stacks and establish connections between geometric and categorical parallel transport.
We present a particle flow realization of Bayes' rule, where an ODE-based neural operator is used to transport particles from a prior to its posterior after a new observation. We prove that such an ODE operator exists. Its neural parameterization can be trained in a meta-learning framework, allowing this operator to re…
Neural Local Wasserstein Regression models distribution-on-distribution regression with flexible, localized transport maps.
problem Estimating distribution-on-distribution regression with global optimal transport maps or linearization limitations.
method Proposes Neural Local Wasserstein Regression, a flexible nonparametric framework using locally defined transport maps in Wasserstein space.
result Demonstrates effective capture of nonlinear and high-dimensional distributional relationships.
Unified framework for constructing kernels for transport equations and Koopman eigenfunctions.
problem Constructing kernels for transport equations and Koopman eigenfunctions.
method Three methods: variational principle, Green's function, and resolvent operator.
result Kernels constructed via these methods are identical under mild assumptions.
A nice differential-geometric framework for (non-abelian) higher gauge theory is provided by principal 2-bundles, i.e. categorified principal bundles. Their total spaces are Lie groupoids, local trivializations are kinds of Morita equivalences, and connections are Lie-2-algebra-valued 1-forms. In this article, we const…
GeONet learns the Wasserstein geodesic without mesh discretization.
problem Computing the Wasserstein geodesic between complex data distributions.
method Mesh-invariant deep neural operator network that learns saddle point optimality conditions.
result GeONet achieves comparable accuracy to standard OT solvers with reduced computational cost.
Deep Neural Operator Driven Real Time Inference for Nuclear Systems to Enable Digital Twin Solutionsstat.ML
DeepONet accelerates nuclear DT inference with high accuracy and efficiency.
problem Real-time prediction and model evaluation in nuclear systems.
method Deep Neural Operator (DeepONet) for surrogate modeling.
result DeepONet outperforms traditional ML methods in accuracy and speed.
A new machine learning method for Bayesian inverse problems in function spaces.
problem Bayesian inverse problems in function spaces with incompatibility of white noise sources.
method One-step generative transport with amortized neural operator and prior-aligned Gaussian random field.
result Generative operator trained on prior samples and noisy observations generates posterior samples efficiently.
Let G be a Lie group. On the trivial principal G-bundle over the Lie algebra of G there is a natural connection whose curvature is the Lie bracket. The exponential map is given by parallel transport of this connection. If G is the diffeomorphism group of a manifold, the curvature of the natural connection is the Lie br…
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.
This research proves that quadratic regularized optimal transport can approximate the Laplace-Beltrami operator on smooth manifolds.
problem Approximating the Laplace-Beltrami operator using optimal transport with quadratic regularization.
method Deriving first-order optimal potentials and analyzing the convergence of discrete Laplace operators.
result The discrete Laplace operators converge to the Laplace-Beltrami operator on smooth manifolds.
Higher Gauge Theorymath.DG
Just as gauge theory describes the parallel transport of point particles using connections on bundles, higher gauge theory describes the parallel transport of 1-dimensional objects (e.g. strings) using 2-connections on 2-bundles. A 2-bundle is a categorified version of a bundle: that is, one where the fiber is not a ma…
Develops a smooth operator framework for analyzing neural network representations.
problem Analyzing the geometry of feedforward neural network representations.
method Introduces a smooth operator-theoretic approach based on diffusion Markov operators derived from feature clouds.
result Establishes a stable operator-geometric framework for tracking training, width, and perturbation stability.
A new method uses normalizing flows to approximate optimal transport between empirical distributions.
problem Learning an optimal transport map between two empirical distributions.
method Relaxing the Monge formulation of optimal transport, using normalizing flows to approximate the solution.
result The method provides a good approximation of the true optimal transport.
Starting from the four normed division algebras - the real numbers, complex numbers, quaternions and octonions - a systematic procedure gives a 3-cocycle on the Poincare Lie superalgebra in dimensions 3, 4, 6 and 10. A related procedure gives a 4-cocycle on the Poincare Lie superalgebra in dimensions 4, 5, 7 and 11. In…
A major challenge in the training of recurrent neural networks is the so-called vanishing or exploding gradient problem. The use of a norm-preserving transition operator can address this issue, but parametrization is challenging. In this work we focus on unitary operators and describe a parametrization using the Lie al…
NOT learns optimal transport plans, kernel costs improve performance.
problem NOT algorithm learns non-optimal plans with weak quadratic costs.
method Introduced kernel weak quadratic costs to improve NOT's performance.
result Kernel costs provide improved theoretical and practical guarantees.
Egorov's theorem for transversally elliptic operators, acting on sections of a vector bundle over a compact foliated manifold, is proved. This theorem relates the quantum evolution of transverse pseudodifferential operators determined by a first order transversally elliptic operator with the (classical) evolution of it…
New framework uses cohomology to analyze probabilistic distortions and arbitrage.
problem Analyzing probabilistic distortions and arbitrage in categorical filtrations.
method Transport cohomological framework, simplicial structure, loop effects, holonomy.
result Nontrivial probabilistic distortions and obstructions generated by loops.
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.
Quantum connections replace metrics with operator inner products.
problem Quantifying geometric properties in quantum systems.
method Defining quantum connections and duals using operator fields and inner products.
result Holonomy and dual connections are equivalent in quantum geometry.
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.
Robust learning method combines kernel smoothing and robust optimization.
problem Certifying robustness against distribution shifts in machine learning models.
method Adapting integral operator using supremal convolution for robustness, leveraging optimal transport.
result The method provides theoretical guarantees for certified robustness and competitive performance.
CDOT optimizes transport between domains preserving both feature and geometric structure.
problem Optimizing transport between heterogeneous domains with preserved feature and geometric structure.
method CDOT uses operator-based regularization to align distance structures, proving pseudometric properties.
result CDOT improves robustness to local geometric variations and is provably convex.
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…
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…
Introduces a new divergence measure for optimal transport.
problem Optimal transport distances and information divergences.
method Infimal convolution formulation of proximal optimal transport divergence.
result Establishes connections to dynamic formulations and partial differential equations.
Unified formula for arbitrary liquidity operations in weighted AMMs
problem Decentralized resource allocation in intelligent transportation systems
method Weighted invariant adapted from Balancer-type AMMs
result Unified formula for four resource allocation operations
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.
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…
Paper develops Lie theory for Rota-Baxter operators on Lie algebras and groups.
problem Cohomology of relative Rota-Baxter operators on Lie algebras and groups.
method Cohomology construction, infinitesimal deformations study, differentiation and integration of Rota-Baxter operators.
result Integration of Rota-Baxter operators on Lie groups and Lie algebras.
PLOT uses optimal transport to find neural site handles for causal abstraction.
problem Finding the relevant neural site for causal analysis is computationally challenging.
method PLOT employs optimal transport to localize causal variables from neural network outputs.
result PLOT efficiently finds intervention handles for causal abstraction in neural networks.
BM learns Schrödinger bridges using neural networks.
problem Learning dynamic transport maps between two distributions.
method Coupled Bridge Matching (BM) with neural networks.
result Preliminary theoretical analysis and numerical experiments show BM's effectiveness.