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…
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48 results for “linear transport”
The parallel linear transports defined by flat linear connection are axiomatically described. On this basis a number of properties, some of which are new, of these transports and connections are derived.
pyLOT library simplifies machine learning on 3D point clouds via linearized optimal transport.
problem Performing machine learning tasks on 3D point clouds.
method Linearized optimal transport (LOT) to embed distributions into Hilbert space, enabling linear machine learning.
result Downstream tasks on embedded representations are simplified to linear operations.
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…
New algorithm solves unbalanced optimal transport on trees in quasi-linear time.
problem Efficiently solving unbalanced optimal transport problems on trees.
method Proposed an algorithm that solves a more general unbalanced optimal transport problem exactly in quasi-linear time on a tree metric.
result Solves unbalanced optimal transport on trees in quasi-linear time (less than one second for a tree with one million nodes).
The paper studies optimal transport in linear quadratic systems and derives interpolation inequalities.
problem Optimal transport problem in Linear Quadratic optimal control systems.
method Well-posedness of the Monge problem, regularity of optimal transport map, displacement interpolation of measures.
result Derivation of general interpolation inequalities for entropy functionals.
Optimal transport aligns source and target distributions for linear regression in 2D.
problem Domain adaptation for linear regression in 2D with limited target data.
method Combining K-means and optimal transport for estimating geometric transformations.
result Optimal transport recovers geometric transformations like rotations, translations, and homotheties.
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…
Extends Optimal Transport to multiple agents, aiming for equitable and optimal distribution.
problem Sharing costs or goods equitably among multiple agents with different preferences.
method Minimizes the maximum transportation cost or maximizes the minimum utility.
result Provides a new algorithm faster than standard linear programming.
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.
The Sinkhorn-Knopp derivatives converge with linear rate.
problem Optimal transport problem with entropic regularization.
method Iterative proportional fitting procedure.
result Derivatives converge with linear rate.
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.
New metrics compare rational spectra using optimal transport.
problem Comparing rational spectra efficiently and accurately.
method Optimal transport and linear-systems theory.
result Established connection to Wasserstein distance.
New scalable algorithm for non-negative linear regression with entropy-regularized OT loss.
problem Generalizing task-specific linear models to broader applications.
method Sinkhorn-like scaling iterations for convex penalty and datafit terms.
result Simple multiplicative updates for various penalty and datafit terms.
Paper relaxes optimal transport using convex functions for data science.
problem Optimal transport problem on finite spaces.
method Relaxation via strictly convex functions (Kullback-Leibler divergence, Bregman divergences). Gradient descent iterative process.
result Mathematical foundations and iterative process for the relaxed optimal transport problem.
New algorithm for linear bandits tackles Optimal Transport problems.
problem Optimal Transport problems not covered by traditional linear bandits.
method Embed actions into a Hilbertian subspace, penalize optimism, use least-squares estimation.
result Achieves same regret bounds as OFUL but interpolates between and .
Paper connects surface shape analysis and unbalanced optimal transport.
problem Computing the SRNF shape distance on piecewise linear surfaces.
method Characterizes SRNF shape distance as WFR distance pullback, proposes new algorithm for WFR distance computation.
result Direct computation of SRNF shape distance on piecewise linear surfaces.
GOAT improves graph matching speed and accuracy using optimal transport.
problem Efficiently matching large graphs in various applications.
method Replaces linear assignment with optimal transport methods.
result GOAT provides improvements in speed and accuracy.
New nonlinear smoothers improve state estimation in chaotic systems.
problem Improving state estimation in chaotic dynamical systems with non-Gaussian behavior.
method Developed nonlinear backward ensemble transport smoothers with parameterization and regularization of transport maps.
result Nonlinear smoothers yield lower estimation error than conventional methods for comparable model evaluations.
Optimal transport aligns rotated linear regression models across domains.
problem Aligning rotated linear regression models across domains with differing statistical properties.
method Combines K-means clustering, OT, and SVD to estimate rotation angle and adapt regression model.
result Optimal transport map recovers underlying rotation in .
The (parallel linear) transports in tensor spaces generated by derivations of the tensor algebra along paths are axiomatically described. Certain their properties are investigated. Transports along paths defined by derivations of the tensor algebra over a differentiable manifold are considered.
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…
Algorithm classifies point clouds using deep set linearized optimal transport.
problem Classifying point clouds efficiently and accurately.
method Deep Set Linearized Optimal Transport, ICNNs, and a discriminator network.
result Efficiently distinguishes between various classes of point clouds.
LOT framework speeds up event distance computation in collider physics.
problem Computational inefficiency in quantifying event distances.
method Linearized Optimal Transport (LOT) for efficient computation.
result LOT significantly reduces computational cost without sacrificing accuracy.
The paper extends optimal transport for linear separability of sheared distributions in supervised learning.
problem Learning on the space of probability measures using shifts and scalings.
method Embedding probability measures into spaces using optimal transport, then applying regular machine learning techniques.
result Sheared distributions can be linearly separated under certain conditions, with bounds on transformations.
Enhances RJMCMC efficiency with non-linear transport-based proposals.
problem Designing efficient RJMCMC proposals for complex models.
method Applies non-linear transport-based approach to construct efficient transdimensional jumps.
result Acceptance probability depends only on model probabilities when exact transports are used.
LOT embeds distributions for linear separability and classification.
problem Distribution discrimination in various scientific fields.
method Linear Optimal Transport (LOT) embedding into space.
result LOT embeds distributions into linearly separable spaces for certain transformations and perturbations.
Quantitative stability of optimal transport maps and linearization of the 2-Wasserstein spacestat.ML
This work studies an explicit embedding of the set of probability measures into a Hilbert space, defined using optimal transport maps from a reference probability density. This embedding linearizes to some extent the 2-Wasserstein space, and enables the direct use of generic supervised and unsupervised learning algorit…
Hodge Topology of Semiclassical Transport: A Coordinate-Free Geometric Framework for the Anomalous Hall Effect and Non-Linear Berry Dipolecond-mat.mes-hall
A new geometric framework resolves singularities in anomalous transport.
problem Mathematical singularities in quantum Berry connections.
method Hodge-de Rham decomposition of the Brillouin zone.
result A smooth geometric proxy potential for anomalous transport.
A new framework for generative modeling using value-driven transport.
problem Developing efficient methods for generative modeling.
method A discrete-time stochastic control formulation of measure transport, formulated as a linear program with dual variables corresponding to the optimal value function.
result Well-trained VDT policies lead to straight transport paths that can be simulated quickly and robustly.
Optimal transport reformulates multiple quantile hedging problem.
problem Multiple quantile hedging problem in incomplete markets.
method Reformulated as Monge optimal transport problem, introduced Kantorovitch version, proved no duality gap.
result Multiple quantile hedging problem can be seen as semi-discrete optimal transport problem.
Fast algorithm for online optimization on transport polytopes.
problem Optimizing convex objectives on transport polytopes.
method Mirror Sinkhorn algorithm combining Sinkhorn scaling and mirror descent.
result Robust and efficient online optimization for convex objectives.
Optimal transportation distances are a fundamental family of parameterized distances for histograms. Despite their appealing theoretical properties, excellent performance in retrieval tasks and intuitive formulation, their computation involves the resolution of a linear program whose cost is prohibitive whenever the hi…
Models of spatial firm competition assume that customers are distributed in space and transportation costs are associated with their purchases of products from a small number of firms that are also placed at definite locations. It has been long known that the competition equilibrium is not guaranteed to exist if the mo…
Scalable Optimal Transport in High Dimensions for Graph Distances, Embedding Alignment, and Morecs.LG
Two log-linear approximations speed up optimal transport for deep learning applications.
problem Computing optimal transport in high dimensions is computationally expensive.
method Locality-sensitive hashing (LSH) and Nyström approximation with LSH-based sparse corrections.
result Log-linear time algorithms for entropy-regularized OT perform well in high-dimensional spaces.
Computing optimal transport distances such as the earth mover's distance is a fundamental problem in machine learning, statistics, and computer vision. Despite the recent introduction of several algorithms with good empirical performance, it is unknown whether general optimal transport distances can be approximated in …
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…
New proof of Schwarzschild stability using geometric gauge.
problem Linear stability of Schwarzschild spacetime under gravitational perturbations.
method Employing a new geometric gauge and exploiting the structure of transport equations.
result Established both orbital and asymptotic stability for linearised quantities.
In a coordinate free form are found the (deviation) equations satisfied by the (infinitesimal) deviation vector, relative velocity, relative momentum, relative acceleration and relative energy of two point particles in a differentiable manifold the tangent bundle of which is endowed with a linear transport along paths,…
A new algorithm reduces the complexity of solving optimal transport problems.
problem Optimal transport problem with linear constraints.
method Primal-dual accelerated stochastic gradient descent with variance reduction (PDASGD).
result Achieves the best-known computational complexity of for OT problems.
The paper examines properties of GW optimal transport plans, showing they can be sparse and permutation-supported.
problem Properties of Gromov-Wasserstein optimal transport plans.
method Exploration of sparsity, permutation support, and cyclical monotonicity properties.
result GW optimal plans can be sparse and permutation-supported under certain conditions.
Survey of Sinkhorn algorithm for optimal transport, emphasizing its geometric origins.
problem Solving optimal transport problems efficiently and accurately.
method Discretization of a non-linear integral equation.
result Geometric interpretation and discretization of the Sinkhorn algorithm.
Noise stabilizes solutions to transport equations, preventing blow-up.
problem Proving global existence and uniqueness of solutions to stochastic transport equations.
method Characteristics-based techniques exploiting the geometric structure of transport equations.
result Noise prevents blow-up in deterministic solutions and ensures global existence and uniqueness of solutions.
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.
Paper introduces non-linearity signature to measure deep neural network performance.
problem Difficulty in explaining performance differences among similar DNN architectures.
method Affine Optimal Transport mappings to measure non-linearity.
result Signature provides better understanding of DNN inner workings.
Study non-Gaussian measures' concentration properties in metric spaces.
problem Concentration properties for non-linear Gaussian functionals with non-Gaussian tails.
method Prove generalised Transportation-Cost Inequalities (TCIs) for specific functionals.
result Extended TCIs for rough volatility and Parabolic Anderson Model.
A new method for efficient optimal partial transport in 1D.
problem Limitation of equal mass assumption in optimal transport.
method Sliced Optimal Partial Transport (Sliced-OPT) algorithm.
result Sliced-OPT demonstrates computational and accuracy benefits.
We study the asymptotic behavior of solutions to the second boundary value problem for a parabolic PDE of Monge-Ampère type arising from optimal mass transport. Our main result is an exponential rate of convergence for solutions of this evolution equation to the stationary solution of the optimal transport problem. We …