This article introduces the problem of finding intrinsic torsion varieties associated to G-structures on a fixed parallelizable Riemannian manifold. As an illustration, the intrinsic torsion varieties of orthogonal almost product structures are analysed on the Iwasawa manifold.
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48 results for “intrinsic structure”
The paper classifies intrinsic torsion in various spacetime structures.
problem Classifying intrinsic torsion in different spacetime structures.
method Review and classification of intrinsic torsion in galilean, Carrollian, Aristotelian, and Bargmannian spacetime structures.
result Found 16 classes for Aristotelian structures and 27 for Bargmannian structures.
We define the intrinsic scale at which a network begins to reveal its identity as the scale at which subgraphs in the network (created by a random walk) are distinguishable from similar sized subgraphs in a perturbed copy of the network. We conduct an extensive study of intrinsic scale for several networks, ranging fro…
We discuss the construction of Sp(2)Sp(1)-structures whose fundamental form is closed. In particular, we find 10 new examples of 8-dimensional nilmanifolds that admit an invariant closed 4-form with stabiliser Sp(2)Sp(1). Our constructions entail the notion of SO(4)-structures on 7-manifolds. We present a thorough inve…
Researchers confirm integral formulas for -structures in detail.
problem Verifying integral formulas for -structures on Riemannian manifolds.
method Detailed analysis and explicit expressions of intrinsic torsion using exterior algebra.
result Agreement of integral formulas with intrinsic torsion components.
In this short note, we prove that if is a weak upper semicontinuous admissible Finsler structure on a domain in , , then the intrinsic distance and differential structures coincide.
Two new minor minimal intrinsically chiral graphs identified.
problem Identifying intrinsically chiral graphs in molecular structures.
method Analyzing graph symmetry and embedding properties.
result Found two new minor minimal intrinsically chiral graphs and .
We analyse the relationship between the components of the intrinsic torsion of an SU(3) structure on a 6-manifold and a G_2 structure on a 7-manifold. Various examples illustrate the type of SU(3) structure that can arise as a reduction of a metric with holonomy G_2.
Posterior Contraction for Sparse Neural Networks in Besov Spaces with Intrinsic Dimensionalitystat.ML
Bayesian neural networks achieve optimal posterior contraction rates in Besov spaces with intrinsic dimensionality.
problem High-dimensional structured estimation problems with unknown smoothness levels.
method Sparse Bayesian neural networks with either sparse or continuous shrinkage priors.
result Optimal posterior contraction rates are achieved, adapting to the unknown smoothness level of the true function.
Estimates intrinsic dimension of data for GANs.
problem Estimating intrinsic dimension of high-dimensional data.
method Uses Wasserstein distances for estimation.
result Provides sample complexity bounds for GANs.
A new measure of causal influence quantifies intrinsic contributions in DAGs.
problem Quantifying intrinsic causal contributions in Directed Acyclic Graphs (DAGs).
method Recursive decomposition of node contributions, structure-preserving interventions, Shapley symmetrization.
result A measure of intrinsic causal contribution that is invariant to node relabeling.
New stratification reveals intrinsic singularity types of orbit spaces.
problem Understanding the intrinsic structure of orbit spaces under Lie group actions.
method Introduced the isostabilizer decomposition and established a map to Klein strata.
result A new canonical stratification on the manifold clarifies the relationship with classical structures.
We study SU(3)-structures induced on orientable hypersurfaces of seven-dimensional manifolds with G_2-structure. Taking Gray-Hervella types for both structures into account, we relate the type of SU(3)-structure and the type of G_2-structure with the shape tensor of the hypersurface. Additionaly, we show how to compute…
Derives an integral formula for G2-structures.
problem Calculating properties of G2-structures.
method Applies an integral formula for G-structures to G2.
result Derives an integral formula relating curvatures and quadratic invariants.
Formula derived for Spin(7)-structures on 8D manifolds.
problem Deriving the intrinsic torsion for Spin(7)-structures.
method Explicit formula derivation using Levi-Civita and minimal Spin(7)-connections.
result Formula relating intrinsic torsion to Lee form and codifferential components.
New method quantifies intrinsic causal contributions in neural networks.
problem Measuring the causal influence of input features in deep neural networks.
method Proposes an identifiable generative post-hoc framework to quantify intrinsic causal contributions (ICC) as structural causal models.
result ICC generates more intuitive and reliable explanations compared to existing global explanation techniques.
In this paper the notion of the intrinsic geometry of an almost contact metric manifold is introduced. Description of some classes of spaces with almost contact metric structures in terms of the intrinsic geometry is given. A new type of almost contact metric spaces, more precisely, Hermitian almost contact metric spac…
Unifying the theory of storage and the risk premium by an unobservable intrinsic electricity priceq-fin.MF
Introduces an unobservable intrinsic electricity price to link storage theory with risk premium.
problem Connecting storage theory with risk premium in electricity markets.
method Introduces an unobservable intrinsic electricity price and derives prices for various contracts.
result Finds an overall negative risk premium in empirical analysis.
Proposes eDNNs and iDNNs for deep learning on manifolds.
problem Deep learning on manifolds with geometric preservation and intrinsic geometry incorporation.
method Intrinsic and extrinsic deep neural networks (iDNNs and eDNNs) with geometric embeddings and maps.
result Empirical risk minimizers of eDNNs and iDNNs converge optimally.
We study the intrinsic structure of parametric minimal discs in metric spaces admitting a quadratic isoperimetric inequality. We associate to each minimal disc a compact, geodesic metric space whose geometric, topological, and analytic properties are controlled by the isoperimetric inequality. Its geometry can be used …
Study of potential Carroll structures and special Carrollian manifolds for null hypersurfaces.
problem Understanding intrinsic geometry of null hypersurfaces.
method Initiate study of potential Carroll structures and explore their relationship to special Carrollian manifolds.
result Initiate the study of potential Carroll structures and their relationship to special Carrollian manifolds.
We investigate quaternionic contact (qc) manifolds from the point of view of intrinsic torsion. We argue that the natural structure group for this geometry is a non-compact Lie group K containing Sp(n)H^*, and show that any qc structure gives rise to a canonical K-structure with constant intrinsic torsion, except in se…
This paper classifies chiral graphs up to size 12.
problem Understanding the chirality of simple graphs to predict molecular behavior.
method Classifying minor minimal intrinsically chiral graphs among simple graphs of size up to 12.
result Complete set of minor minimal graphs for intrinsic properties of chiral molecules.
We introduce and study a notion of invariant intrinsic torsion geometry which appears, for instance, in connection with the Bryant-Salamon metric on the spinor bundle over S^3. This space is foliated by six-dimensional hypersurfaces, each of which carries a particular type of SO(3)-structure; the intrinsic torsion is i…
Estimates scalar curvature of point clouds without embedding.
problem Estimating scalar curvature of data sets without embedding.
method Intrinsic estimator based on metric structure, consistent and stable.
result Estimator converges to scalar curvature as sample size increases.
Study of intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
problem Characterizing the intrinsic sub-Laplacian for hypersurfaces in contact sub-Riemannian manifolds.
method Construction and analysis of the intrinsic sub-Laplacian using Riemannian approximations and stochastic processes.
result The intrinsic sub-Laplacian is stochastically complete, ensuring the process does not hit characteristic points.
The present, partly expository, monograph consists of three parts. The first part treats Spin- and Pin-structures from three different perspectives and shows them to be suitably equivalent. It also introduces an intrinsic perspective on the relative Spin- and Pin-structures of Fukaya-Oh-Ohta-Ono and Solomon, establishe…
We go further on the study of harmonicity for almost contact metric structures already initiated by Vergara-Diaz and Wood. By using the intrinsic torsion, we characterise harmonic almost contact metric structures in several equivalent ways and show conditions relating harmonicity and classes of almost contact metric st…
The space of matrices of positive determinant GL^+_n inherits an extrinsic metric space structure from R^{n^2}. On the other hand, taking the infimum of the lengths of all paths connecting two points in GL^+_n gives an intrinsic metric. We prove bilipschitz equivalence for intrinsic and extrinsic metrics on GL^+_n, exp…
Low-dimensional structure in images helps deep learning models generalize better.
problem Understanding the intrinsic dimensionality of images for better model performance.
method Applied dimension estimation tools to popular image datasets and used GANs to manipulate intrinsic dimensionality.
result Natural image datasets have very low intrinsic dimensionality, which aids neural networks in learning and generalizing.
Recent theory work has found that a special type of spatial partition tree - called a random projection tree - is adaptive to the intrinsic dimension of the data from which it is built. Here we examine this same question, with a combination of theory and experiments, for a broader class of trees that includes k-d trees…
Diffusion Models Are Statistically Optimal for Learning Low-Dimensional Multi-Modal Distributionsstat.ML
Diffusion models learn multi-modal distributions with optimal efficiency.
problem Learning high-dimensional distributions with low-dimensional multi-modal structures.
method Score-based diffusion models, focusing on subgaussian distributions within subspaces.
result Diffusion models require samples for 1-Wasserstein error, improving over prior guarantees.
The c-map on groupsmath.DG
We study the projective special Kaehler condition on groups, providing an intrinsic definition of homogeneous projective special Kaehler that includes the previously known examples. We give intrinsic defining equations that may be used without resorting to computations in the special cone, and emphasise certain associa…
We explore the intrinsic geometry of tangent bundles and properties of the mirror map.
problem Understanding the intrinsic geometry of tangent bundles and properties of the mirror map.
method Review and solve cohomological questions related to the mirror map and related operators.
result Solved some cohomological questions and raised other induced by the d_B operator.
In this paper, we study the intrinsic mean curvature flow on certain closed spacelike manifolds, and prove the existence of hyperbolic structures on them.
In-BO optimizes complex constrained domains using SIn-GP surrogate models.
problem Optimizing in complex constrained domains with irregular shapes.
method Sparse Intrinsic Gaussian Processes (SIn-GP) on manifolds with heat kernel estimation.
result In-BO outperforms traditional BO in complex constrained domains.
The paper refutes the manifold hypothesis for image data and proposes the union of manifolds hypothesis.
problem The manifold hypothesis fails to capture the structure of image data.
method Empirical verification of the union of manifolds hypothesis on image datasets.
result Image data lies on a disconnected set with varying intrinsic dimensions.
On a sub-Riemannian manifold we define two type of Laplacians. The \emph{macroscopic Laplacian} , as the divergence of the horizontal gradient, once a volume is fixed, and the \emph{microscopic Laplacian}, as the operator associated with a sequence of geodesic random walks. We consider a general class of rando…
Detects singularities in complex data to improve machine learning models.
problem Real-world data often contains non-manifold structures (singularities) that can mislead machine learning models.
method Develops a topological framework to quantify local intrinsic dimension and Euclidicity score for multiple scales.
result Identifies singularities and captures local geometric complexity in image data.
Efficient method estimates intrinsic dimension for big data.
problem Estimating intrinsic dimension for large datasets is costly and complex.
method Proposes a matrix-vector product-based approach for efficient intrinsic dimension estimation.
result Demonstrates superior performance compared to state-of-the-art methods.
Almost Robinson geometriesmath.DG
Study on almost Robinson geometries, focusing on their intrinsic torsion and leaf space properties.
problem Investigating the geometry of almost Robinson manifolds, a Lorentzian analog of almost Hermitian manifolds.
method Classification based on intrinsic torsion, analysis of leaf space properties, and conformal invariants.
result Comprehensive classification of almost Robinson manifolds based on their intrinsic torsion.
We establish some inequalities of Chen's type between certain intrinsic invariants (involving sectional, Ricci and scalar curvatures) and the squared mean curvature of submanifolds tangent to the structure vector fields of a generalized S-space-form and we discuss the equality cases of them. We apply the obtained resul…
Develops intrinsic Gaussian process regression for manifold-valued data.
problem Lack of intrinsic Gaussian process methods for manifold-valued response variables.
method Proposes an intrinsic covariance structure and a novel intrinsic Gaussian process regression model.
result Establishes asymptotic properties and shows posterior consistency.
New method adapts DLMs to intrinsic data dependence without prior knowledge.
problem Understanding how unmasking schedules affect DLM generation quality.
method Adapts unmasking schedule to target data distribution's dependence structure.
result Sampling convergence guarantees improve for low-complexity distributions.
A new method uncovers intrinsic data structures for unsupervised domain adaptation.
problem Learning domain-aligned features can damage intrinsic target discrimination.
method Structurally Regularized Deep Clustering (H-SRDC) integrating structural source regularization.
result H-SRDC outperforms existing methods in image classification and semantic segmentation.
BL learns interpretable optimization structures from data.
problem Learning interpretable optimization structures from data.
method BL parameterizes a compositional utility function from intrinsically interpretable modular blocks.
result BL supports architectures from single to hierarchical compositions, modeling hierarchical optimization structures.
eDCF estimates intrinsic dimension using local connectivity.
problem Challenges in estimating intrinsic dimension due to scale dependence.
method eDCF: a novel, scalable, and parallelizable method based on Connectivity Factor (CF).
result eDCF consistently matches leading estimators with comparable MAE and higher exact intrinsic dimension match rates.
Normalizes pseudo-Einstein contact forms for easier analysis.
problem Understanding pseudo-Einstein contact forms.
method Constructing intrinsic CR normal coordinates using parabolic normal coordinates.
result Normal form for pseudo-Einstein contact forms.