Global invariant for path structures and differential equations defined on torus.
problem Global invariant for path structures and differential equations.
method Computed as a secondary invariant from a Cartan connection on a canonical bundle.
result Formula for global invariant of second order differential equations on torus.
The paper classifies path structures on 3D Lie groups and reduces non-flat ones to Z/2Z-structures.
problem Classifying and reducing path structures on 3D Lie groups.
method Analyzes curvature and automorphism groups to reduce path structures to simpler forms.
result Automorphism groups of non-flat path structures are maximal dimension 3.
The abstract discusses a new causal structure on manifolds using paths and points.
problem Constructing a causal structure on manifolds using paths and points.
method Constructing a four-manifold from pairs of points and paths, and a seven-dimensional manifold from pairs of points and conics.
result The causal structure corresponds to a conformal structure only when the underlying surface is a real projective plane.
Study path geometries with constant torsion and cone structures.
problem Characterizing path geometries with nontrivial torsion.
method Introducing constant torsion, establishing correspondence with cone structures, describing in terms of integrable systems.
result Path geometries with constant torsion correspond to cone structures on homogeneous ruled surfaces.
Classifies 3D manifolds with specific structures and automorphisms.
problem Classifying compact 3D manifolds with path structures and large automorphism groups.
method Uses Cartan connections and constant curvature analysis.
result Curvature of Cartan connections is constant for these manifolds.
The paper shows that almost every path structure is not variational.
problem Determining if a path structure is variational.
method Generalized Douglas's result to higher dimensions and analyzed path geometries with infinitesimal symmetries.
result Almost every path structure is not variational.
Study of motion constraints and path-following on 3D space.
problem Path-following with non-holonomic constraints on R3. method Exploration of geometric structure and construction of guiding vector fields.
result General principles for constructing guiding vector fields for path-following.
Deep RL optimizes processing paths to desired material structures.
problem Optimizing processing paths to achieve desired material properties.
method Deep reinforcement learning guided by structure representations and reward signals.
result Algorithm learns to find optimal paths to target structures in material space.
Characterizes chains in 3D CR and para-CR structures.
problem Determining when a 3D path geometry comes from CR or para-CR chains.
method Provides necessary and sufficient conditions for a 3D path geometry to arise from chains of CR or para-CR 3-manifolds, and verifies computationally.
result Characterization of chains in 3D CR and para-CR structures.
New method identifies causal structure in count data using cumulants and path analysis.
problem Challenges in discovering causal structure from count data, especially due to non-identifiability.
method Poisson Branching Structural Causal Model (PB-SCM) with path analysis using high-order cumulants.
result Causal order is identifiable under specific conditions in PB-SCM using cumulant information.
New lattice path method for statistical inference of persistent diagrams.
problem Statistical inference on persistent diagrams.
method Lattice path representation and combinatorial enumerations.
result Topological changes observed in spike proteins of COVID-19 virus.
Deep generative networks have been widely used for learning mappings from a low-dimensional latent space to a high-dimensional data space. In many cases, data transformations are defined by linear paths in this latent space. However, the Euclidean structure of the latent space may be a poor match for the underlying lat…
New algebraic structures for topological pairs.
problem No specific problem stated; generalization of knot quandles.
method Introducing multi-quandles for topological pairs.
result New algebraic structures for topological pairs.
We show a duality which arises from distributions of Cartan type, having growth (2, 3, 5), from the view point of geometric control theory. In fact we consider the space of singular (or abnormal) paths on a given five dimensional space endowed with a Cartan distribution, which form another five dimensional space with a…
Adaptive learning, also known as adaptive teaching, relies on learning path recommendation, which sequentially recommends personalized learning items (e.g., lectures, exercises) to satisfy the unique needs of each learner. Although it is well known that modeling the cognitive structure including knowledge level of lear…
Develops Weyl structures for path geometries, simplifying their study.
problem Complexity in studying path geometries using traditional differential geometry methods.
method Defines distinguished connections and Schouten tensor, proving their dependence on line bundle sections.
result Shows a smaller subclass of Weyl structures for path geometries, with interesting connections to BGG sequences.
Contact path geometries are curved geometric structures on a contact manifold comprising smooth families of paths modeled on the family of all isotropic lines in the projectivization of a symplectic vector space. Locally such a structure is equivalent to the graphs in the space of independent and depedent variables of …
Two constructions link path geometries to almost Grassmann structures.
problem Linking path geometries to almost Grassmann structures.
method Introducing two Fefferman-type constructions.
result Characterizing conditions for almost Grassmann structures arising from these constructions.
Proves existence of longest paths in sub-Lorentzian problems.
problem Existence of longest paths in sub-Lorentzian problems.
method Generalizes classical theorem for Lorentzian manifolds to sub-Lorentzian problems.
result Longest paths exist for any left-invariant sub-Lorentzian structures on Carnot groups.
Paper presents a copula-based method to efficiently generate correlated sample paths from multi-step time series models.
problem Generating realistic correlation structures in multi-step forecast sample paths is expensive and time-consuming.
method Copula-based approach to generate correlated sample paths in one forward pass.
result Improved sample path quality and significant speedup over autoregressive sampling.
Let M be a Riemannian manifold and PM be the space of all smooth paths on M. We describe geodesics on path space PM. Normal neighbourhood structure on PM has been discussed. We identify paths on M under "back-track" equivalence. Under this identification we show that if M …
Simpler method derived for path geometries on surfaces, characterizing projective path geometries.
problem Characterizing projective path geometries on surfaces.
method Solving the equivalence problem of sub-Riemannian geometry of signature (1,1) on a contact 3-manifold.
result Characterization of projective path geometries in terms of their chains.
We introduce a novel non-parametric methodology to test for the dynamical time evolution of the lag-lead structure between two arbitrary time series. The method consists in constructing a distance matrix based on the matching of all sample data pairs between the two time series. Then, the lag-lead structure is searched…
This paper shows how path spaces on two-level manifolds can be Hilbert manifold structures.
problem Addressing the structure of path spaces on two-level manifolds.
method Introducing the notion of tameness and constructing charts on path spaces of two-level manifolds.
result Path spaces on tame two-level manifolds have the structure of a Hilbert manifold.
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…
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.
Scalable machine learning with path signatures for time series and graphs.
problem Challenges in real-world time series and graph data.
method Combines rough path theory with probabilistic, deep, and kernel methods.
result Scalable models for time series and graph data.
EntroPath learns manifold geometry from diffusion paths.
problem Learning geodesic geometry from data graphs with spurious shortcuts.
method Maximum Entropy Path Ensemble Embedding (MERW) with k-step diffusion paths.
result EntroPath converges to squared geodesic distance in the short-time limit.
Let M be any n dimensional smooth manifold and PM be the space of all smooth paths, then we showed that PM is a smooth manifold modelled over a complete normable space. We discussed many geometric structure on Path spaces and its relation to ambient space.
Study shows almost complex structures with certain tensor properties are prevalent.
problem Characterizing almost complex structures with specific tensor properties.
method Analyzes the space of almost complex structures on compact manifolds.
result The space of almost complex structures with rank at least k Nijenhuis tensor is either empty or dense in each component.
Path signatures reveal community structure in coupled oscillators' dynamics.
problem Detecting communities in multivariate dynamical processes from time series data.
method Path signatures, a mathematical framework encoding geometric and temporal properties of continuous paths.
result Achieved exact recovery of structural communities from observed time series in multiple KSBM instances.
The abstract discusses the linear and smooth structures of mapping spaces.
problem The structure of mapping spaces in differential geometry.
method Proving diffeomorphisms and fibre bundle properties.
result Path spaces and base point preserving mapping spaces are Fréchet spaces.
A contact projective structure is a contact path geometry the paths of which are among the geodesics of some affine connection. In the manner of T.Y. Thomas there is associated to each contact projective structure an ambient affine connection on a symplectic manifold with one-dimensional fibers over the contact manifol…
Unified approach to DP problems using Gumbel distribution and variational Bayesian inference.
problem Solving classical optimal path problems in a probabilistic framework.
method Gumbel distribution and variational Bayesian inference for latent optimal paths.
result Unified approach transforms DP problems into directed acyclic graphs with Gibbs distribution.
Defines Lewy curves in para-CR geometry and characterizes their path geometries.
problem Characterizing path geometries defined by para-CR Lewy curves.
method Definition and characterization of para-CR Lewy curves in various dimensions.
result Lewy curves determine the para-CR structure up to sign in flat cases.
Develops a new causal model for path-dependent link prediction.
problem Existing causal models assume fixed node factors, but real-world links can depend on existing ones.
method Introduces causal lifting and structural pairwise embeddings for path-dependent link prediction.
result Validated on three scenarios, demonstrating improved accuracy for causal link prediction.
New method detects and clusters market regimes in multidimensional data.
problem Detecting and clustering market regimes in complex data structures.
method Non-parametric online market regime detection and clustering using path-wise two-sample tests and maximum mean discrepancy.
result Successfully detected and clustered market regimes in various data structures.
A new method uses string method to explore diffusion models.
problem Understanding the geometry of learned distributions in diffusion models.
method String method to compute continuous paths between samples.
result The string method identifies realistic morphing sequences and transition pathways.
Graph-structured data arise ubiquitously in many application domains. A fundamental problem is to quantify their similarities. Graph kernels are often used for this purpose, which decompose graphs into substructures and compare these substructures. However, most of the existing graph kernels do not have the property of…
A Bayesian framework models dynamic probability predictions over time.
problem Dynamic probability predictions over time in various settings.
method Gaussian latent information martingale (GLIM) framework.
result GLIM outperforms baseline methods in predicting future uncertainties.
Hierarchical text classification has many real-world applications. However, labeling a large number of documents is costly. In practice, we can use semi-supervised learning or weakly supervised learning (e.g., dataless classification) to reduce the labeling cost. In this paper, we propose a path cost-sensitive learning…
New Morse theory for path homology with coefficients.
problem Defining operations on path homology with differential graded coefficients.
method Using tools from Morse theory and string topology.
result Morse-theoretic description of a product on path homology.
Although various linear log-distance path loss models have been developed, advanced models are requiring to more accurately and flexibly represent the path loss for complex environments such as the urban area. This letter proposes an artificial neural network (ANN) based multi-dimensional regression framework for path …
New topology defined from spacetime paths, reconstructing spacetime structure.
problem Reconstructing spacetime structure from path homotopy classes.
method Defining a topology on spacetime based on timelike and causal homotopy classes.
result The topology on spacetime is reconstructed from the space of homotopy classes.
We introduce a new function-preserving transformation for efficient neural architecture search. This network transformation allows reusing previously trained networks and existing successful architectures that improves sample efficiency. We aim to address the limitation of current network transformation operations that…
An almost complex structure J on a 4-manifold X may be described in terms of a rank 2 vector bundle E. A splitting of J consists of a pair of line bundles spanning E. A hypersurface M in X satisfying a nondegeneracy condition inherits a CR-structure from J and a path geometry from the splitting. Using the Cartan-Kähler…
PAN uses path integrals for graph convolution and pooling, improving GNN performance.
problem Designing efficient graph convolution and pooling for graph neural networks.
method Path integral based graph convolution and pooling using learnable weights for path lengths.
result PAN achieves state-of-the-art performance on various graph classification/regression tasks.
In this paper we use a time-evolving graph which consists of a sequence of graph snapshots over time to model many real-world networks. We study the path classification problem in a time-evolving graph, which has many applications in real-world scenarios, for example, predicting path failure in a telecommunication netw…