EncGAN learns multi-manifold structure and abstract features using an encoder.
problem Learning multi-manifold structure and abstract features in data.
method Uses an encoder to model manifold structure and invert it for generation, with a single latent space for shared abstract features.
result Successfully learns multi-manifold structure and abstract features on MNIST, 3D-chair, and UT-Zap50k datasets.
Study graph-based algorithms for multi-manifold clustering with sufficient conditions.
problem Clustering data from a union of manifolds with different dimensions and intersections.
method Investigate sufficient conditions for similarity graphs to capture geometric information.
result High probability error bounds for spectral approximation of tensorized Laplacian.
When analyzing empirical data, we often find that global linear models overestimate the number of parameters required. In such cases, we may ask whether the data lies on or near a manifold or a set of manifolds (a so-called multi-manifold) of lower dimension than the ambient space. This question can be phrased as a (mu…
GCML preserves geometric structure in manifold clustering for diverse data types.
problem Loss functions in manifold clustering can corrupt latent space structure.
method GCML framework with isometric and ranking losses for geometric structure preservation.
result GCML outperforms other methods in latent space structure preservation and performance metrics.
This paper investigates how data augmentation improves linear separation of manifold data.
problem Understanding how data augmentation enhances linear separation of manifold data.
method Investigates the conditions under which self-supervised representations can linearly separate multi-manifold data.
result Self-supervised learning can linearly separate manifolds with a smaller distance than unsupervised learning.
Sparse coding has been popularly used as an effective data representation method in various applications, such as computer vision, medical imaging and bioinformatics, etc. However, the conventional sparse coding algorithms and its manifold regularized variants (graph sparse coding and Laplacian sparse coding), learn th…
Geometric approach clusters intersecting manifolds with high probability.
problem Clustering intersecting d-dimensional manifolds.
method Compute locality graph on d-simplices using dihedral angles, then compute LAPD to separate manifold components.
result The method separates manifold components with high probability under random sampling.
A new method for precise user targeting in advertising using hyperbolic manifold learning.
problem Improving user satisfaction in targeted advertising by overcoming skepticism and spam perception.
method Proposes a Multi-Manifold Learning framework to learn hierarchical user and ad representations in the hyperbolic space.
result Demonstrates improved performance in user targeting and prediction accuracy on both public datasets and a large-scale commercial system.
We propose a spectral clustering method based on local principal components analysis (PCA). After performing local PCA in selected neighborhoods, the algorithm builds a nearest neighbor graph weighted according to a discrepancy between the principal subspaces in the neighborhoods, and then applies spectral clustering. …
Manifold learning based methods have been widely used for non-linear dimensionality reduction (NLDR). However, in many practical settings, the need to process streaming data is a challenge for such methods, owing to the high computational complexity involved. Moreover, most methods operate under the assumption that the…
This paper advocates Riemannian multi-manifold modeling in the context of network-wide non-stationary time-series analysis. Time-series data, collected sequentially over time and across a network, yield features which are viewed as points in or close to a union of multiple submanifolds of a Riemannian manifold, and dis…
This paper advocates a novel framework for segmenting a dataset in a Riemannian manifold M into clusters lying around low-dimensional submanifolds of M. Important examples of M, for which the proposed clustering algorithm is computationally efficient, are the sphere, the set of positive definite matrices, and the…
The problem of multiple surface clustering is a challenging task, particularly when the surfaces intersect. Available methods such as Isomap fail to capture the true shape of the surface nearby the intersection and result in incorrect clustering. The Isomap algorithm uses the shortest path between points. The main draw…
The paper introduces new structures for left-symmetric algebroids.
problem Developing new mathematical structures for left-symmetric algebroids.
method Introducing Koszul-Vinberg-Nijenhuis structures and related concepts.
result Koszul-Vinberg-Nijenhuis structures provide a hierarchy of structures.
We give a notion of compatibility between a Riemannian structure and a Jacobi structure. We prove that in case of fundamental examples of Jacobi structures : Poisson structures, contact structures and locally conformally symplectic structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, …
We give a notion of compatibility between a Riemannian metric and a Jacobi structure. We prove that in case of Poisson structures, contact structures and locally conformally symplectic structures, fundamental examples of Jacobi structures, we get respectively Riemann-Poisson structures in the sense of M. Boucetta, $\fr…
Extending Jacobi and Riemannian compatibility to Lie algebroids.
problem Generalizing compatibility between Jacobi and Riemannian structures.
method Generalizing previous work on fundamental examples to Lie algebroids.
result Compatibility results for Lie algebroids.
Defines structure constants for specific geometric structures on Lie groups.
problem No specific problem stated; focuses on defining structure constants.
method Not explicitly detailed in the abstract.
result Defines structure constants for almost complex, almost symplectic, and Riemannian structures on a local Lie group.
Study on G2∗ structures and almost para-contact structures in 7D.
problem Understanding the relation between G2∗ structures and almost para-contact structures. method Calculating projections using properties of G2∗ structures. result Determined the class of almost para-contact structures induced by G2∗ structures. Defines a new Poisson structure for generalized Sasakian spaces.
problem No specific problem stated; focuses on new structure definition.
method Defines a canonical Poisson structure on generalized contact metric spaces.
result Shows distinction between generalized Sasakian and coKähler structures.
Study on types of generalized hypercomplex structures on tori and Kodaira-Thurston surface.
problem Characterizing types of generalized hypercomplex structures.
method Analysis of S2-family of generalized complex structures and study of twistor spaces. result Existence of generalized hypercomplex structures on 4n-dimensional tori with non-maximal types. Classifies complex Dirac structures with invariants and local structure.
problem Classifying complex Dirac structures.
method Introducing invariants (order, type), proving existence and splitting theorems.
result Pointwise classification and local structure of complex Dirac structures.
Extends corner structure study to general case, constructs normal Trans-Sasakian structures.
problem Extending corner structure study to general case without conditions.
method Extends corner structure to general case, constructs Trans-Sasakian structures from non-normal corner structures.
result Constructs normal Trans-Sasakian structures from non-normal corner structures.
The paper explores geometric structures on Hom-Lie groups and algebras.
problem Exploring Kähler-Norden structures on Hom-Lie groups and algebras.
method Analyzing the relationship between holomorphic Norden structures and Kähler-Norden structures on Hom-Lie groups.
result Left-invariant holomorphic Hom-Lie groups with abelian complex structures are flat.
New metric structures generalize Sasakian and cosymplectic structures, proving rigidity and finding conditions.
problem Generalizing Sasakian and cosymplectic structures to new metric structures.
method Introducing weak structures and proving rigidity of Sasakian structures.
result Any weak Sasakian structure is homothetically equivalent to a Sasakian structure.
Study GL(2)-structures on manifolds leading to complex structures.
problem Understanding GL(2)-structures and their relation to complex structures. method Explored GL(2)-structures on differential manifolds, proving their relation to almost-complex structures and providing a canonical connection. result Established a twistor-like construction for GL(2)-geometry. 3D projective structures can be metrized with conformal structures.
problem Weyl metrizability of 3D projective structures.
method Interpreting Weyl metrizability as CR submanifolds in 7D.
result Beltrami's theorem extends to conformal structures in 3D.
Hypersymplectic structures with torsion on Lie algebroids are investigated. We show that each hypersymplectic structure with torsion on a Lie algebroid determines three Nijenhuis morphisms. From a contravariant point of view, these structures are twisted Poisson structures. We prove the existence of a one-to-one corres…
Study equivalence between Hessian and Born structures on tangent bundles.
problem Equivalence between Hessian and Born structures on tangent bundles.
method Analyzing conditions for Hessian structures and integrability of induced almost Born structures.
result Conditions for equivalence between Hessian and Born structures are established.
Introduces compatibility between Dirac structures and Nijenhuis tensors.
problem No specific problem stated; focuses on extending Poisson-Nijenhuis structures.
method Introduces compatibility between Dirac structures and (1,1)-tensor fields.
result Properties of Dirac-Nijenhuis structures studied, including connections and integrations.
We introduce generalized almost contact structures which admit the B-field transformations on odd dimensional manifolds. We provide definition of generalized Sasakain structures from the view point of the generalized almost contact structures. We obtain a generalized Sasakian structure on a non-compact manifold which…
Introduces semi-abelian generalized complex structures.
problem Deformation theory of abelian complex structures.
method Definition and examples of semi-abelian generalized complex structures.
result Illustration of new concept with examples.
Introduces VB-structures for geometric objects on manifolds.
problem Properties of higher tangent lifts of geometric structures.
method Introduces weighted structures for various geometric objects on a manifold with a homogeneity structure.
result Proves interesting properties of various weighted structures.
The paper studies nilpotent structures in oriented neutral vector bundles and neutral hyperKähler structures.
problem Nilpotent structures in oriented neutral vector bundles and their relation to neutral hyperKähler structures.
method Defined H-nilpotent structures for Lie subgroups of SO(2n,2n) related to neutral hyperKähler structures. result Existence of complex and paracomplex structures forming neutral hyperKähler structures if and only if there exists an H-nilpotent structure. Spin-structures on real Bott manifolds with Kähler structure are characterized.
problem Existence of spin-structures on real Bott manifolds with Kähler structure.
method Ishida characterization and techniques from \cite{PS16} using characteristic classes.
result Necessary and sufficient condition for the existence of spin-structures on M. Expanding on previous work, this note generalizes geometric structures results.
problem Generalizing geometric structures results.
method Generalization to a class of geometric structures including integrable almost-complex structures.
result Main results generalized to a broader class of geometric structures.
Unique birational structure proven on Inoue surfaces.
problem Proving uniqueness of birational structures on Inoue surfaces.
method Generalizing a result by Bruno Klingler, proving uniqueness of structures.
result The natural (Aff2(C),C2)-structure on an Inoue surface is the unique (Bir(P2),P2(C))-structure. Exotic hypercomplex structures on a torus are proven to not exist.
problem Existence of exotic hypercomplex structures on a torus.
method Classification of complete flat affine structures on real tori using the Obata connection.
result Exotic hypercomplex structures on a torus do not exist.
Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current ar…
In this paper, we show the existence of (co-oriented) contact structures on certain classes of G2-manifolds, and that these two structures are compatible in certain ways. Moreover, we prove that any seven-manifold with a spin structure (and so any manifold with G2-structure) admits an almost contact structure. We…
Two Kähler structures are PCR equivalent in the Siegel domain.
problem Equivalence of two Kähler structures in the Siegel domain.
method Construction of complex hyperbolic and Kähler structures from Sasakian structure.
result PCR Kähler equivalent structures in Siegel domain.
We characterize the Dirac structures that are parallel with respect to Gualtieri's canonical connection of a generalized Riemannian metric. On the other hand, we discuss Dirac structures that are images of generalized tangent structures. These structures turn out to be Dirac structures that, if seen as Lie algebroids, …
Introduces holed cone structures to generalize cone structures on 3-manifolds.
problem Generalizing cone structures to 3-manifolds with irreducible holonomy representations.
method Introduces holed cone structures and considers their deformation space.
result The deformation space of holed cone structures is a covering space of the character variety.
The study classifies complex symplectic structures on 4D Lie algebras and constructs hypersymplectic structures.
problem Classifying and constructing complex symplectic structures on 4D Lie algebras.
method Interpreting complex symplectic and pseudo-Kähler structures, developing a method for constructing hypersymplectic structures.
result Obtained an example of a hypersymplectic structure on a 4-step nilmanifold.
In this paper we deal with two classes of mixed metric 3-structures, namely the mixed 3-Sasakian structures and the mixed metric 3-contact structures. Firstly we study some properties of the curvature of mixed 3-Sasakian structures, proving that any manifold endowed with such a structure is Einstein. Then we prove the …
We study integrability of generalized almost contact structures, and find conditions under which the main associated maximal isotropic vector bundles form Lie bialgebroids. These conditions differentiate the concept of generalized contact structures from a counterpart of generalized complex structures on odd-dimensiona…
The paper explores conditions for tangent bundles to have Hermitian structures.
problem Conditions for tangent bundles to have Hermitian structures.
method Analyzes conditions for almost Hermitian structures on TM induced by a metric and an affine connection. result Examples of Kähler-Einstein structures on tangent bundles are provided.
Defines Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
problem No specific problem stated; focuses on new structure definition.
method Definition and properties of Jacobi-Koszul-Vinberg structures on Jacobi-left-symmetric algebroids.
result Defines a new structure on Jacobi-left-symmetric algebroids.