Estimates tangent space variation on Riemannian submanifolds using feature size.
problem Estimating tangent space variation on Riemannian submanifolds.
method Using structural properties of local feature size and elementary Euclidean geometry.
result Asymptotically tight estimates of tangent space variation.
Geometric framework analyzes bias in variational inference for posterior functionals.
problem Analyzing the bias of posterior functionals under variational approximations.
method Developed a geometric framework to evaluate the bias of posterior functionals using the variational tangent space.
result The leading-order bias of a posterior functional is determined by its component orthogonal to the variational tangent space.
New method guarantees global convergence in variational inference.
problem Limited convergence to local optima in variational inference.
method Minimizes inclusive KL divergence using neural networks and neural tangent kernel.
result Gradient descent dynamics converge to a unique solution in function space.
Quantum neural tangent kernels help understand variational quantum circuits in machine learning.
problem Designing and predicting performance of variational quantum circuits.
method Using quantum neural tangent kernels and dynamical equations for loss functions.
result Analytical solutions for training dynamics in variational quantum circuits.
In 1931 Elie Cartan constructed a geometry which was rarely considered. Cartan proposed a way to define an infinitesimal metric ds starting from a variational problem on hypersurfaces in an n-dimensional manifold M. This distance depends not only of the point $\textsc{m}\in\mathcal{M}$ but on the orient…
In a previous paper, we have introduced a new unified description of the main equations of the gravitational and of the electromagnetic field, in terms of tidal tensors and connections on the tangent bundle TM of the space-time manifold. In the present work, we relate these equations to variational procedures on the ta…
New normalizing flows in hyperbolic space improve posterior modeling for hierarchical data.
problem Limited flexibility of existing normalizing flows in Euclidean space for hierarchical data.
method Elevated normalizing flows to hyperbolic spaces using coupling transforms and Wrapped Hyperboloid Coupling.
result Improved performance on density estimation and hierarchical graph data.
A theorem proves integrability of Fréchet tangent distributions.
problem Integrability of Fréchet tangent distributions on manifolds.
method Introduced Condition W, applied variational approach, used differential forms.
result Existence and uniqueness of maximal foliations.
The calculus of variations for lagrangians which are not functions on the tangent bundle, but sections certain affine bundles is developed. We follow a general approach to variational principles which admits boundary terms of variations.
New non-geodesic variation found in Hodge structure.
problem Finding non-geodesic variations of Hodge structures of maximum dimension.
method Using weighted projective hypersurfaces of degree 10 in a weighted projective space.
result Found a variation of Hodge structure that is nowhere tangent to geodesic variations.
TSGO optimizes gradients in tensor networks to avoid vanishing/exploding issues.
problem Gradient vanishing and exploding problems in deep learning models.
method TSGO rotates parameters towards gradient direction in tangent space of normalized state.
result TSGO naturally determines learning rate based on angle between parameters and gradient.
Minimal surfaces in harmonic conformally flat space are studied.
problem Minimal surfaces in harmonic conformally flat space.
method Variational geometry approach focusing on mean curvature and Willmore functionals.
result Critical points of mean curvature functional are homeomorphic to the sphere.
The paper defines and studies new types of submanifolds in a unit sphere.
problem Variational problems of curvature tensors for submanifolds.
method Euler-Lagrange equations for Normal-Yang-Mills and Tangent-Yang-Mills submanifolds.
result Infinitely many non-trivial examples of Normal-Yang-Mills and Tangent-Yang-Mills submanifolds are constructed.
We study natural variations of the G2 structure σ_0 \in Λ^3_+ existing on the unit tangent sphere bundle SM of any oriented Riemannian 4-manifold M. We find a circle of structures for which the induced metric is the usual one, the so-called Sasaki metric, and prove how the original structure has a preferred role in the…
For a compact riemannian manifold of negative curvature, the geodesic foliation of its unit tangent bundle is independent of the negatively curved metric, up to Holder bicontinuous homeomorphism. However, the riemannian metric defines a natural transverse measure to this foliation, the Liouville transverse measure, whi…
Dirac structures on tangent bundles provide a unified framework for Lagrange--Dirac dynamical systems.
problem Unified geometric framework for Lagrange--Dirac dynamical systems
method Introducing a Lagrange--Dirac structure on the tangent bundle
result Unified framework for nonholonomic, degenerate Lagrangian, and symmetric systems
This paper shows equivalence between SVGD and BBVI using kernel gradient flows.
problem Bayesian inference methods and their equivalence.
method Formalizes equivalence between SVGD and BBVI using kernel gradient flows.
result BBVI corresponds precisely to SVGD when using the neural tangent kernel.
A method to fix radius distortion in generative models on curved spaces.
problem Distortion in geodesic radius measurements across different charts on Riemannian manifolds.
method Radial Compensation (RC) adjusts the tangent-space base distribution to match the geodesic radius law, improving model stability and interpretability.
result RC ensures that the model's geodesic radius matches the intended distribution, improving numerical stability and curvature interpretation.
In this article we introduce and investigate a new two-parameter family of knot energies TP(p,q) that contains the tangent-point energies. These energies are obtained by decoupling the exponents in the numerator and denominator of the integrand in the original definition of the tangent-point energies. We will firs…
Explicit expression for Pressure Metric on surface group representations.
problem Surface group representations into PSL(n,R) along the Fuchsian locus.
method Explicit expression in terms of holomorphic differentials and relationship with Petersson pairing.
result Generalization of classical Teichmueller theory results.
An overview of some recent results on the geometry of partial differential equations in application to integrable systems is given. Lagrangian and Hamiltonian formalism both in the free case (on the space of infinite jets) and with constraints (on a PDE) are discussed. Analogs of tangent and cotangent bundles to a diff…
The bienergy of a vector field on a Riemannian manifold (M,g) is defined to be the bienergy of the corresponding map (M,g) ---> (TM,g_S), where the tangent bundle TM is equipped with the Sasaki metric g_S. The constrained variational problem is studied, where variations are confined to vector fields, and the correspond…
Reductions of higher tangent bundles of Lie groupoids provide natural examples of geometric structures which we would like to call higher algebroids. Such objects can be also constructed abstractly starting from an arbitrary almost Lie algebroid. A higher algebroid is, in principle, a graded bundle equipped with a diff…
Paper constructs infinitely many tangent functors on diffeological spaces.
problem Tangent spaces in diffeological spaces are not uniquely defined.
method Introduced and constructed infinitely many non-isomorphic tangent functors.
result The choice of tangent functor is not unique outside smooth manifolds.
We make evident a curvature tensor for every vector sub-bundle of an arbitrary manifold tangent bundle which reduces to the curvature tensor of an Ehresmann connection in the case of the horizontal sub-bundle of the tangent bundle to the total space of the nonlinear fiber bundle on which the connection is defined. Then…
New privacy framework tailored to specific data distributions.
problem Protecting individual data points in decision-making processes.
method Introducing tangent differential privacy, a new form of differential privacy.
result Entropic regularization guarantees tangent differential privacy under general conditions.
The paper extends tangent space theory for diffeological spaces and bundles.
problem Understanding tangent spaces of diffeological bundles and spaces.
method Introducing weakly filtered and filtered diffeological spaces, extending exact sequences, and defining Hector's tangent bundle.
result Tangent bundles of filtered diffeological spaces are diffeological vector spaces.
New lower bounds improve logistic log-likelihood optimization and inference.
problem Designing computationally tractable lower bounds for logistic log-likelihoods.
method Developed a piece-wise quadratic lower bound that uniformly improves tangent quadratic minorizers.
result Improves the speed of convergence and accuracy of variational Bayes approximations.
Study of tangent spaces in diffeological spaces under Lie group actions.
problem Understanding tangent spaces in generalized spaces.
method Generalized tangent space construction and isomorphism proof.
result Internal tangent space isomorphic to stratified tangent space.
We study how the notion of tangent space can be extended from smooth manifolds to diffeological spaces, which are generalizations of smooth manifolds that include singular spaces and infinite-dimensional spaces. We focus on two definitions. The internal tangent space of a diffeological space is defined using smooth cur…
Study on Neural Tangent Kernel of Matrix Product States and their convergence.
problem Understanding the convergence of Neural Tangent Kernel of Matrix Product States.
method Analyzing the Neural Tangent Kernel of Matrix Product States and proving its convergence in the infinite bond dimensional limit.
result The Neural Tangent Kernel of Matrix Product States converges to a constant matrix during training.
Proposes an auto-encoder for Gaussian distributions using Lie group theory.
problem Generative models for Gaussian distributions.
method Lie group auto-encoder with UTDATs, incorporating geometric properties.
result Eliminates matrix exponential operator and derives intrinsic loss.
New examples of harmonic unit vector fields on hyperbolic 3-space are constructed by exploiting the reduction of symmetry arising from the foliation by horospheres. This is compared and contrasted with the analogous construction in Euclidean 3-space, using a foliation by planes, which produces some new examples of harm…
A novel method for parallel transport and geodesics on submanifolds.
problem Understanding parallel transport and geodesics on submanifolds.
method Rolling tangent space to visualize and analyze parallel transport and geodesics.
result Conditions for parallel transport and geodesics are simplified and visualized in the tangent space.
Researchers found all invariant contact structures on tangent sphere bundles of compact symmetric spaces.
problem Identifying all invariant contact metric structures on tangent sphere bundles of compact rank-one symmetric spaces.
method Explicitly obtained all structures, distinguishing K-contact, Sasakian, and 3-Sasakian structures.
result There is a unique Sasakian-Einstein metric on tangent sphere bundles of spheres and real projective spaces.
Study of holomorphic distributions on projective 3-space, focusing on stable tangent sheaves.
problem Characterizing holomorphic distributions and their tangent sheaves on projective 3-space.
method Analysis of singular schemes and tangent sheaves, classification of distributions, use of Grothendieck's Quot-scheme.
result Classification of codimension one distributions with stable tangent sheaves and description of moduli spaces.
On a Riemannian manifold Mˉm+n with an (m+1)-calibration Ω, we prove that an m-submanifold M with constant mean curvature H and calibrated extended tangent space RH⊕TM is a critical point of the area functional for variations that preserve the enclosed Ω-volume. This recovers the …
Study infinitesimal characters on semi-groups to prove interior properties and apply to Teichmüller spaces.
problem Properties of infinitesimal characters and their applications in Teichmüller spaces.
method Analyzing integrable tangent vectors on character varieties and applying to pressure forms and Teichmüller spaces.
result Non-empty interior of the cone of Jordan variations and length-normalized variations for split groups.
The paper studies singularities on parallels of tangent developable surfaces of frontal curves.
problem Understanding singularities on parallels of tangent developable surfaces.
method Generalization of tangent developable surfaces and parallel deformations for frontal curves in arbitrary dimensions.
result Classification of generic singularities on parallels of tangent developable surfaces for frontal curves in 3 or 4 dimensional Euclidean spaces.
Authors construct parallel transport for tangent cones in Wasserstein space.
problem Parallel transport in tangent cones of Wasserstein space.
method Construction of parallel transport for tangent cones in the interior of a geodesic in Wasserstein space, proving linear part and extending to full tangent cones.
result The constructed parallel transport is equivalent to previous methods.
Proposes a variational approach to shallow neural networks, bypassing optimization.
problem Theoretical understanding and optimization of shallow neural networks.
method Replaces discrete training with a continuum variational surrogate, proving global well-posedness and regularity.
result Optimal parameter density can be obtained by solving a single linear system, achieving O(1/N) generalization error. New Calabi-Yau metrics found on complex symmetric spaces.
problem Finding Calabi-Yau metrics on complex symmetric spaces.
method Complete Calabi-Yau metrics with prescribed horospherical singular tangent cone.
result First examples of Calabi-Yau smoothings of singular tangent cones.
Study of normal and tangent maps to frontals.
problem Understanding geometric and dynamical properties of frontals.
method Geometrical and dynamical analysis of normal and tangent maps.
result Parallels of the tangent map to a frontal curve are right equivalent to the tangent map of a frontal curve under certain conditions.
The paper studies metrics that match prescribed geodesics and introduces a variational problem.
problem Finding Riemannian metrics whose geodesics match given paths.
method Introduces a functional E on Riemannian metrics and computes its variational equations.
result Existence of conformally critical metrics in certain cases.
Classifies contact metric spaces using hyperquadric bundles.
problem Classifying contact metric spaces with specific invariant.
method Using tangent hyperquadric bundles of Lorentzian space forms.
result Locally classify spaces with Boeckx invariant ≤ -1.
The main new notions are the notions of tangent-like spaces and local monoids. The main result is the pasage from a local monoid to its tangent-like space which is a local Leibniz algebra.
Let L be a Lagrangian submanifold of a pseudo- or para-Kähler manifold which is H-minimal, i.e. a critical point of the volume functional restricted to Hamiltonian variations. We derive the second variation of the volume of L with respect to Hamiltonian variations. We apply this formula to several cases. In particular …
The paper provides guarantees for a tangent transform algorithm in logistic regression models.
problem Finding theoretical guarantees for statistical optimality and algorithmic convergence in non-conjugate models.
method Exploiting convex duality and minorizing the marginal likelihood, the paper derives non-asymptotic upper bounds and convergence guarantees for a tangent transform algorithm in logistic regression models.
result The tangent transform algorithm is shown to be locally asymptotically stable without assumptions on the data-generating process.