Proposes a transfer learning method for PCA studies.
problem Enhancing PCA estimation accuracy across multiple studies.
method Two-step algorithm integrating shared subspace information via Grassmannian barycenter.
result Knowledge transfer improves PCA estimation accuracy through enlarged eigenvalue gap.
New varifolds with capillary boundary properties studied.
problem Understanding varifolds with specific boundary conditions.
method Introducing a Radon measure on a Grassmannian bundle as a capillary boundary.
result Structural properties, monotonicity inequality, and integral compactness proved.
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.
Polynomial chaos expansions on Grassmannian submanifolds for high-dimensional stochastic systems.
problem Uncertainty quantification in high-dimensional stochastic systems.
method Principal Geodesic Analysis on the Grassmann manifold, adaptive algorithm for local submanifolds, polynomial chaos expansion.
result Efficient surrogate modeling of system behavior across different parameter spaces.
Study proves existence of precotangent bundles for Grassmannians.
problem Existence of precotangent bundles for Grassmannians.
method Proof for Grassmannians of reflexive Banach spaces and p-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.
Breathing k-means outperforms greedy k-means++ in clustering.
problem Improving k-means clustering solutions.
method Dynamic adjustment of centroids through breathing technique.
result Breathing k-means outperforms other k-means techniques, especially greedy k-means++.
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.
Constructs explicit p-harmonic functions on Grassmannians and flag manifolds.
problem Finding proper p-harmonic functions on Grassmannians and flag manifolds. method Using the method of eigenfamilies to construct explicit functions.
result Explicit complex-valued proper p-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.
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 β-functions proportional to dual Coxeter numbers. 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.
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.
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.
Improved K-Means++ and K-Means∥ with faster run-time.
problem Efficiently selecting initial seeds for K-means clustering.
method Triangle inequality pruning and dynamic priority queue.
result Up to 17x speedup for K-Means++ and 551x for K-Means$\$.
Study rectifying curves in 3D multiplicative Euclidean space.
problem Investigate rectifying curves in a non-Newtonian geometry setting.
method Apply multiplicative differential-geometric concepts to rectifying curves.
result Classify multiplicative rectifying curves using spherical curves.
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.
Investigates Darboux rectifying curves on smooth surfaces.
problem Characterizing Darboux rectifying curves on smooth surfaces.
method Analyzes the position vector under isometry and finds conformal invariance conditions.
result Identifies sufficient conditions for conformal invariance of Darboux rectifying curves.
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.
A space curve in a Euclidean 3-space E3 is called a rectifying curve if its position vector field always lies in its rectifying plane. This notion of rectifying curves was introduced by the author in [Amer. Math. Monthly {\bf 110} (2003), no. 2, 147-152]. In this present article, we introduce and study the n…
Study characterizes k-rectifiable sets in homogeneous groups.
problem Characterizing k-rectifiable sets in arbitrary homogeneous groups. method Proves characterizations using (k,G)-approximate tangent groups. result Existence of (k,G)-approximate tangent groups implies k-rectifiability. 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.
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.
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 ∣1∣--graded parabolic geometries and for almost Grassmannian structures, in particular.…
Recalls and refines the concept of algebraically rectifiable curves.
problem Classical notion of algebraically rectifiable plane curves.
method Provides new criteria, relates to quadratic differentials, and generalizes to higher order differentials.
result Generalization and new criteria for algebraic rectifiability.
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.
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.
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.
Harmonic maps to Euclidean buildings have rectifiable singular strata.
problem Understanding the structure of singular points for harmonic maps.
method Defining singular strata and proving rectifiability using the rectifiable Reifenberg program.
result Rectifiability of singular strata for harmonic maps into F-connected complexes. We prove (without using Federer's structure theorem) that a finite-mass flat chain over any coefficient group is rectifiable if and only if almost all of its 0-dimensional slices are rectifiable. This implies that every flat chain of finite mass and finite size is rectifiable. It also leads to a simple necessary and su…
The Grassmannian of affine subspaces is a natural generalization of both the Euclidean space, points being zero-dimensional affine subspaces, and the usual Grassmannian, linear subspaces being special cases of affine subspaces. We show that, like the Grassmannian, the affine Grassmannian has rich geometrical and topolo…
This paper contains a thorough introduction to the basic geometric properties of the manifold of Lagrangian subspaces of a linear symplectic space, known as the Lagrangian Grassmannian. It also reviews the important relationship between hypersurfaces in the Lagrangian Grassmannian and second-order PDEs.
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…
A new K-means method HT K-means uses ℓ0 penalty for sparsity.
problem Cluster center regularization for improved clustering performance.
method HT K-means with ℓ0 penalty for sparsity. result HT K-means outperforms other regularized K-means methods in simulations and real data. Modified K-means ensures local optimality with same complexity.
problem Lack of rigorous analysis on local optimality guarantees of K-means.
method Proposed modifications to K-means ensuring local optimality.
result Proposed methods provide improved locally optimal solutions.
Reformulated sigma models for complex Grassmannians using Gross-Neveu formalism.
problem Classical aspects of N=(2,2) 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.
New methods for solving hydrodynamic-type equations using quasi-rectifiable Lie algebras.
problem Solving systems of hydrodynamic-type equations.
method Introducing and analyzing quasi-rectifiable Lie algebras and vector fields.
result New methods for solving hydrodynamic-type equations.
Study cohomology rings of Grassmannians using Clifford algebras and symmetric spaces.
problem Understanding cohomology rings of Grassmannians over different fields.
method Explicit generators and relations for de Rham cohomology rings, filtered deformations related to Clifford algebras.
result Explicit generators and relations for the de Rham cohomology rings of Grassmannians.
Rectifies singular set of harmonic maps into complex.
problem Regularity of harmonic maps into complex manifolds.
method Proves (m−2)-rectifiability of singular set. result Singular set is (m−2)-rectifiable. k-means algorithm is one of the most classical clustering methods, which has been widely and successfully used in signal processing. However, due to the thin-tailed property of the Gaussian distribution, k-means algorithm suffers from relatively poor performance on the dataset containing heavy-tailed data or outlie…
The paper generalizes rectifying and normal curves in Lorentzian n-space.
problem Characterizing and classifying g−rectifying and g−normal curves in Lorentzian n-space. method Introducing a g−position vector field and defining g−rectifying and g−normal curves based on this field. result Comprehensive characterization and classification of g−rectifying and g−normal curves. Clustering is a separation of data into groups of similar objects. Every group called cluster consists of objects that are similar to one another and dissimilar to objects of other groups. In this paper, the K-Means algorithm is implemented by three distance functions and to identify the optimal distance function for c…
In this paper we investigate the performance of different types of rectified activation functions in convolutional neural network: standard rectified linear unit (ReLU), leaky rectified linear unit (Leaky ReLU), parametric rectified linear unit (PReLU) and a new randomized leaky rectified linear units (RReLU). We evalu…
Neural networks can approximate rectifiable measures with small error.
problem Approximating complex rectifiable measures using neural networks.
method Using ReLU neural networks to approximate (countably) m-rectifiable measures as push-forwards of the Lebesgue measure.
result The approximation error in terms of Wasserstein distance can be made arbitrarily small.
Paper improves clustering risk bounds for kernel k-means.
problem Improving clustering risk bounds for kernel k-means.
method Analyzes kernel k-means and Nyström approximation.
result Achieves nearly optimal excess clustering risk bound.
We show that C^1 hypersurfaces in the Heisenberg group are countably N-rectifiable. As a corollary, this shows that all C^1_H graphs over the xy-plane are countable N-rectifiable, showing the equivalence of this notion of rectifiability with that of Franchi, Serra Cassano and Serapioni for such surfaces.