Study harmonic maps into Hilbert-Schmidt Grassmannian.
problem Harmonic spheres conjecture.
method Twistor description of harmonic maps.
result Motivated by harmonic spheres conjecture.
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48 results for “Hilbert-Schmidt Grassmannian”
In this paper we study the action of the symplectic operators which are a perturbation of the identity by a Hilbert-Schmidt operator in the Lagrangian Grassmannian manifold.
Let be the Banach-Lie group of unitary operators in the Hilbert space which are Hilbert-Schmidt perturbations of the identity 1. In this paper we study the geometry of the unitary orbit of an infinite projection in . This orbit coincides with t…
Study of sphere bundles over Grassmann manifold with geometric properties.
problem Geometric properties of sphere bundles over Grassmann manifold.
method Analyzes the differentiable structure and metric properties of sphere bundles over a Grassmann manifold.
result Established the smooth action of unitary operators and studied geodesics.
Solves steering problem with continuous time, Hilbert-Schmidt cost, and matrix ODEs.
problem Fixed horizon linear quadratic covariance steering in continuous time with a specific terminal cost.
method Formulates necessary conditions as a coupled matrix ODE two-point boundary value problem, designs a matricial recursive algorithm, and proves convergence.
result Proposes and proves the convergence of a matricial recursive algorithm for solving the steering problem.
Classifies linear embeddings of grassmannians and ind-grassmannians.
problem Understanding linear embeddings of grassmannians and ind-grassmannians.
method Classification through isomorphism of Picard groups and direct limits.
result Most linear embeddings of grassmannians are equivariant.
Optimizes learning Hilbert-Schmidt operators between Sobolev spaces.
problem Statistical limits of learning mappings between infinite-dimensional function spaces.
method Minimax optimal regularization and multilevel training.
result Multilevel kernel operator learning achieves optimal learning rate.
Study optimizes KSD estimation from samples, revealing Hilbert-Schmidt vs trace scales.
problem Optimizing estimation of Kernel Stein Discrepancy from samples.
method Identifying and comparing minimax scales for U-statistic and V-statistic.
result Hilbert-Schmidt norm of Stein covariance operator gives optimal scale.
Optimal transport for functional data using Hilbert-Schmidt operators.
problem Optimal transport for distributions on function spaces with partially represented stochastic maps.
method Regularization technique to restrict transport maps to Hilbert-Schmidt operators, developing an efficient algorithm.
result Existence, uniqueness, and consistency of the Hilbert-Schmidt operator estimate for the transport map.
Study proves existence of precotangent bundles for Grassmannians.
problem Existence of precotangent bundles for Grassmannians.
method Proof for Grassmannians of reflexive Banach spaces and -restricted Grassmannians of polarized Hilbert space.
result Existence of bundle predual to tangent bundle (precotangent bundle).
The paper finds inequalities in Grassmannian geometry.
problem Understanding geometric properties of Grassmannians.
method Analyzes inequalities for elements in Grassmannians.
result Law of Cosines and geodesic triangle inequalities.
Curious structure of special orthogonal, unitary, and symplectic groups as products of Grassmannians discovered.
problem Understanding the structure of special orthogonal, unitary, and symplectic groups.
method Expressing these groups as products of Grassmannians realized as involution matrices.
result Special orthogonal, special unitary, and symplectic groups can be expressed as products of their corresponding Grassmannians.
The paper explores the geometry of Lagrangian Grassmannians and their connection to PDEs.
problem Understanding the geometric properties of Lagrangian subspaces.
method Thorough review of geometric properties and their relation to PDEs.
result Hypersurfaces in the Lagrangian Grassmannian correspond to second-order PDEs.
Constructs explicit -harmonic functions on Grassmannians and flag manifolds.
problem Finding proper -harmonic functions on Grassmannians and flag manifolds.
method Using the method of eigenfamilies to construct explicit functions.
result Explicit complex-valued proper -harmonic functions on compact real Grassmannians and non-descending functions on real flag manifolds.
Using the Plucker map between grassmannians, we study basic aspects of classic grassmannian geometries. For `hyperbolic' grassmannian geometries, we prove some facts (for instance, that the Plucker map is a minimal isometric embedding) that were previously known in the `elliptic' case.
Study on estimating distances between covariance operators and Gaussian processes.
problem Estimating distances between covariance operators and Gaussian processes.
method Riemannian distances, concentration results for Hilbert space-valued random variables, RKHS covariance and cross-covariance operators.
result Both distances converge in the Hilbert-Schmidt norm and can be consistently and efficiently estimated.
Researchers measure distances between quantum states to speed up machine learning.
problem Calculating distances between quantum states for machine learning is complex.
method Three-step method using many-particle interference to measure Hilbert-Schmidt distance.
result The method reduces complexity in calculating Euclidean distances between quantum states.
The paper finds optimal inequalities for hypersurface curvatures in complex Grassmannians.
problem Optimizing inequalities for hypersurface curvatures in complex Grassmannians.
method Analyzing real hypersurfaces in complex Grassmannians to derive inequalities involving scalar curvature and Casorati curvatures.
result Conditions for equality in inequalities involving normalized scalar curvature and Casorati curvatures.
The paper studies geometric properties of Grassman manifolds within Euclidean spaces.
problem Understanding the geometric structure of Grassman manifolds.
method Analyzing Grassman manifold as a subset of Euclidean space and orthogonal projections.
result Explicit formulas for differential geometry of as a submanifold.
Grassmannian sigma models extend Gross-Neveu model formulations.
problem Understanding sigma models on Grassmannian targets.
method Chiral Gross-Neveu model formulations for orthogonal and symplectic Grassmannians.
result One-loop -functions proportional to dual Coxeter numbers.
Real hypersurfaces in complex Grassmannians of rank two with semi-parallel structure Jacobi operatormath.DG
No real hypersurfaces found in complex Grassmannians with specific Jacobi operators.
problem Existence of real hypersurfaces in complex Grassmannians with semi-parallel structure Jacobi operator.
method Proved nonexistence through mathematical analysis.
result No real hypersurfaces exist in complex Grassmannians of rank two with semi-parallel structure Jacobi operator.
Grassmannian packings improve CNN kernels' diversity and reduce sparsity.
problem Kernel sparsity and lack of diversity in CNNs decrease model capacity.
method Initialize CNN kernels with Grassmannian packings to maximize diversity and minimize sparsity.
result Grassmannian packings lead to diverse features and improved classification accuracy.
The affine Grassmannian generalizes Euclidean and linear subspaces with rich geometric properties.
problem Formulating machine learning and statistical problems on the affine Grassmannian.
method Showed the affine Grassmannian has multiple structures and affords an analogue of Schubert calculus.
result The affine Grassmannian serves as a concrete computational platform for various machine learning and statistical problems.
Correspondence found between exponential families and affine Grassmannians.
problem Understanding the relationship between exponential families and geometric structures.
method Established a one-to-one correspondence between exponential families and affine Grassmannians.
result Found a correspondence between minimal exponential families and affine Grassmannians.
A Fermionic Grunsky operatormath.CV
New Grunsky operator for disk maps to complex plane.
problem Characterizing domains for Hilbert-Schmidt Grunsky operators.
method Geometric treatment of Smirnov space, pull-back analysis.
result Domains with Hilbert-Schmidt Grunsky operators are Weil-Petersson quasidisks.
The paper classifies real hypersurfaces with a specific Jacobi operator in complex Grassmannians.
problem Classifying real hypersurfaces with a particular Jacobi operator.
method Introducing and classifying real hypersurfaces with a quadratic Killing structure Jacobi operator.
result A classification theorem for Hopf real hypersurfaces with quadratic Killing structure Jacobi operator.
Totally nonnegative Grassmannian and related spaces are shown to be like closed balls.
problem Understanding the topological structure of certain spaces in combinatorics.
method Proving homeomorphic to closed balls using advanced combinatorial and geometric techniques.
result Three significant spaces in combinatorics are proven to be topologically equivalent to closed balls.
Researchers describe even Clifford structures on specific Grassmannians.
problem Understanding even Clifford structures on Grassmannians.
method Explicit description of structures on real, complex, and quaternionic Grassmannians.
result Explicit description of non-flat parallel even Clifford structures of ranks 8, 6, and 5.
Differential structure on partial isometries over Grassmannian constructed.
problem No specific problem stated; abstract focuses on method and result.
method Construction of differential structure on partial isometries over restricted Grassmannian.
result Set of partial isometries over restricted Grassmannian becomes a Banach Lie groupoid.
The affine Grassmannian is a noncompact smooth manifold that parameterizes all affine subspaces of a fixed dimension. It is a natural generalization of Euclidean space, points being zero-dimensional affine subspaces. We will realize the affine Grassmannian as a matrix manifold and extend Riemannian optimization algorit…
Constructs a Morse-Bott function on symplectic Grassmannians.
problem Defines a function on symplectic Grassmannians.
method Uses a compatible linear complex structure to construct a quadratic Morse-Bott function.
result Critical loci consist of subspaces splitting into isotropic and complex parts.
Generalizes embedding complex Grassmannians into quadrics.
problem Holomorphic isometric embeddings of complex Grassmannians into quadrics.
method Generalization of do Carmo-Wallach theory for moduli spaces.
result Moduli spaces of embeddings discussed.
Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
problem Smooth structures on infinite dimensional Grassmannians and non-commutative cross-ratios.
method Analyzing and expanding the notion of non-commutative cross-ratios, proving their smoothness.
result Smoothness of non-commutative cross-ratios.
An elliptic complex is found on a Grassmannian space, with zero index.
problem Analyzing differential operators on a specific geometric space.
method Identifying an elliptic complex of length 3 starting with a 2-Dirac operator on the Grassmannian of oriented 2-planes.
result The index of the elliptic complex is zero.
Researchers compute Hochschild cohomology of Grassmannians.
problem Computing Hochschild cohomology of Grassmannians.
method Explicit description of Gerstenhaber algebra structure, vanishing of higher cohomology.
result Decomposition of Hochschild cohomology concentrated in global sections for certain Grassmannians.
New supergeometric Grassmannians created by gluing ν-domains.
problem No specific problem stated; focuses on a new construction.
method Constructing supergeometric Grassmannians by gluing ν-domains.
result Introduces a novel generalization of Grassmannians to supergeometry.
The classical concept of affine locally symmetric spaces allows a generalization for various geometric structures on a smooth manifold. We remind the notion of symmetry for parabolic geometries and we summarize the known facts for --graded parabolic geometries and for almost Grassmannian structures, in particular.…
Extends metrics for SPD matrices to infinite dimensions.
problem Lack of generalized forms for Riemannian metrics.
method Unitized Hilbert-Schmidt operators and extended Mahalanobis norm.
result Improved performance in high-dimensional comparisons.
This paper proves area-minimizing cones over Grassmannian manifolds.
problem Determine if cones over Grassmannian manifolds are area-minimizing.
method Detailed descriptions of embedding maps using Hermitian orthogonal projectors, re-proving area-minimization using Lawlor's Curvature Criterion.
result All cones over Grassmannian manifolds are area-minimizing except for oriented real Grassmannians.
Develops a correspondence between symplectic orbits and Grassmannians.
problem Understanding the homotopy types of Grassmannians of linear subspaces in symplectic vector spaces.
method Uses orbit fibrations and linear symplectic reduction to compute homotopy types.
result Recover observations from Arnold, Oh-Park, and Lee-Leung in different cases.
We prove that there does not exist any semi-parallel real hypersurface in complex two-plane Grassmannians. With this result, the nonexistence of recurrent real hypersurfaces in complex two-plane Grassmannians can also be proved.
Hyperfield Grassmanniansmath.CO
This paper generalizes Grassmannians to hyperfields.
problem Exploring Grassmannians in the context of hyperfields.
method Introducing topological hyperfields and relating R and C to matroid hyperfields.
result Relating hyperfields R and C to matroid hyperfields.
Study of geodesics on SL(n) with Hilbert-Schmidt metric, revealing complex dynamics in higher dimensions.
problem Geodesics on SL(n) with Hilbert-Schmidt metric.
method Analysis of geodesics, use of Virial-identity-based criterion, study of explicit families of solutions, classification of geodesics.
result Complex dynamics in higher dimensions, existence of bounded geodesic motions in even dimensions, instability of swirling and shear flows in even dimensions.
New supergroups created from odd involutions in supergeometry.
problem Creating new supergroups from odd involutions.
method Constructing Grassmannians by gluing domains with an odd involution.
result Introduced a supergroup associated with odd involutions.
This work improves fair tensor decomposition using a kernel criterion.
problem Learning fair low-rank tensor decompositions with statistical parity.
method Regularizes Canonical Polyadic Decomposition with KHSIC to ensure approximate statistical parity.
result The proposed algorithm achieves better fairness and fit than state-of-the-art FATR.
Complex Cayley Grassmannian is a singular variety with smooth singular locus.
problem Characterize the singularities of the Cayley Grassmannian.
method Defined a torus action and used it to prove singularity and cohomology properties.
result Singular locus is smooth and has the same cohomology as .
A nonstandard invariant fourth order operator acting on functions on a manifold equipped with an almost Grassmannian structure with an arbitrary trorsion is found by means of the curved translation principle. This operator can be viewed as a Grassmannian analogue of the Paneitz operator well known from conformal geomet…
Reformulated sigma models for complex Grassmannians using Gross-Neveu formalism.
problem Classical aspects of supersymmetric sigma models with Hermitian symmetric target spaces.
method Reformulation using Gross-Neveu formalism, proposing two types of equivalent Lagrangians.
result Proposed two types of equivalent Lagrangians for maximal isotropic Grassmannians, making either supersymmetry or geometry manifest.