Topological models confirmed for Springer fibers of even orthogonal and symplectic groups.
problem Understanding the topology of Springer fibers for even orthogonal and symplectic groups.
method Explicit construction of topological models and proving homeomorphism to Springer fibers.
result Springer fibers for even orthogonal and symplectic groups are homeomorphic to their topological models.
Paper addresses group synchronization with incomplete measurements and proves linear convergence of GPM.
problem Orthogonal group synchronization with incomplete measurements and additive noise.
method Generalized power method (GPM) with local error bound analysis.
result Linear convergence of GPM to a global maximizer under general additive noise model.
Study isotropy groups for complex orthogonal and skew-symmetric matrices.
problem Understanding isotropy subgroups of orthogonal similarity transformations.
method Analysis of group structure of nonsingular block matrices.
result Group structure of isotropy subgroups related to block Toeplitz matrices.
Explains how to combine Lie groups orthogonally.
problem Combining Lie groups orthogonally.
method Definitions and concepts in each section.
result Main idea explained.
A new method for optimizing neural networks with orthogonal constraints.
problem Optimizing neural networks with orthogonal constraints.
method Parametrization using the exponential map to transform constrained optimization into unconstrained.
result Faster, more accurate, and stable convergence in RNNs with orthogonal recurrent weights.
We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and a round 4-sphere.
Formula for Laplace-Beltrami on orthogonal group in Euclidean coords.
problem Computing Laplace-Beltrami on constrained submanifolds.
method Embedded gradient vector field method, explicit formula derivation.
result Explicit formula for Laplace-Beltrami on orthogonal group.
A Lie group is called orthogonal if it carries a bi-invariant pseudo Riemannian metric. Oscillator Lie groups constitutes a subclass of the class of orthogonal Lie groups. In this paper, we determine the Lie bialgebra structures and the solutions of the classical Yang-Baxter equation on a generic class of oscillator Li…
Study the geometric properties of skew symmetric matrices and orthogonal groups.
problem Understanding the geometric properties of skew symmetric matrices and orthogonal groups.
method Investigate the differential-geometric properties of the exponential map and Riemannian structure.
result Connections between skew symmetric matrices and orthogonal groups are revealed.
New algebraic parametrizations for harmonic maps to orthogonal group.
problem Finding solutions to harmonic maps from surfaces to O(n).
method Algebraic parametrizations and free holomorphic data.
result Reveals correspondence with null curves and minimal surfaces.
Algorithm finds isotropy subgroups of orthogonal similarity on symmetric matrices.
problem Computing isotropy subgroups of orthogonal similarity on symmetric matrices.
method Algorithmic procedure solving a Toeplitz matrix equation.
result Structure of isotropy subgroups described.
Computes isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
problem Computing isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
method Algorithm for solving a matrix equation to compute isotropy subgroups.
result Computed isotropy subgroups of orthogonal matrices acting on Hermitian matrices.
Integrable geodesics found on special orthogonal group.
problem Analyzing normal geodesics on the special orthogonal group.
method Adapted Lax pair and bi-Hamiltonian structure.
result Almost all normal geodesics are completely integrable.
Study optimizes estimation of orthogonal and rotation matrices from noisy data.
problem Estimating orthogonal and rotation matrices from noisy data.
method Iterative polar decomposition algorithm initialized by spectral methods.
result Algorithm achieves optimal error rate of $(1+o(1)) rac{σ^2 d(d-1)}{2np}$.
Modified Θ-invariant for non-vanishing cohomology groups.
problem Defining Θ-invariant for non-vanishing cohomology groups. method A slightly modified version of the Θ-invariant. result Can define Θ-invariant even when cohomology group is not vanishing. We prove that a polar orthogonal representation of a real reductive algebraic group has the same closed orbits as the isotropy representation of a pseudo-Riemannian symmetric space. We also develop a partial structural theory of polar orthogonal representations of real reductive algebraic groups which slightly generali…
New optimization algorithms on orthogonal group for machine learning.
problem Efficient optimization on the orthogonal group for machine learning tasks.
method Stochastic geometric algorithms on Lie groups.
result Strong performance on diverse machine learning tasks.
Paper derives local Plücker formulas for special orthogonal groups.
problem Deriving Plücker formulas for special orthogonal groups.
method Reduction to classical A_n case.
result Local Plücker formulas for special orthogonal groups derived.
An algorithm for efficient computation of equivariant neural network layers.
problem Efficiently computing with Brauer's group equivariant neural network layers.
method Category theoretic constructions and Kronecker product matrices.
result Significant reduction in computational cost compared to naive implementation.
We study the Chern-Simons partition function of orthogonal quantum group invariants, and propose a new orthogonal Labastida-Mariño-Ooguri-Vafa conjecture as well as degree conjecture for free energy associated to the orthogonal Chern-Simons partition function. We prove the degree conjecture and some interesting cases o…
In every dimension n≥3 we introduce a class of orthogonal graph-manifolds and prove that the fundamental group of any orthogonal graph-manifold quasi-isometrically embeds into a product of n trees. As a consequence, we obtain that asymptotic and linearly-controlled asymptotic dimensions of such group are equal t…
Compact holonomy groups found in symmetric spaces.
problem Characterizing holonomy groups in symmetric spaces.
method Analyzing the holonomy group structure of locally symmetric spaces.
result Holonomy groups are compact and have finite index in the orthogonal group.
PerPCA separates unique and shared features from heterogeneous data.
problem Extracting shared and unique features from data collected from different sources with varying trends.
method Personalized PCA (PerPCA) uses orthogonal global and local principal components to encode both unique and shared features.
result PerPCA can identify and recover both unique and shared features under mild conditions.
The space Z of leftinvariant orthogonal almost complex structures, keeping the orientation, on 6-dimensional Lie groups is researched. To get explicit view of this space elements the isomorphism of Z and CP3 is used. The explicit formula for arbitrary leftinvariant orthogonal almost …
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 defines metrics from Lie groups and conjectures they are Einstein metrics.
problem Defining metrics from Lie groups.
method Analyzing metrics from specific Lie groups like unitary, orthogonal, and symplectic groups.
result Conjectures that metrics from these Lie groups are Einstein metrics.
Complete description of valuations for indefinite orthogonal groups.
problem Classifying valuations for indefinite orthogonal groups.
method Detailed analysis of continuous and generalized translation- and group-invariant valuations.
result Identification of Klain-Schneider continuous valuations within the space of translation-invariant valuations.
New convergence guarantees for learning with unknown nuisance parameters.
problem Learning problems with unknown nuisance parameters.
method Stochastic gradient optimization with Neyman orthogonality and approximately orthogonalized updates.
result Stochastic gradient algorithms can converge under conditions of nuisance parameters.
New algebraic geometry approach to classifying orthogonal separable coordinates.
problem Classifying orthogonal separable coordinates on arbitrary manifolds.
method Algebraic geometric approach using projective varieties and isometry group actions.
result New unexpected structure revealed in well-known classification.
New Einstein metrics found on orthogonal groups without natural reductivity.
problem Finding non-naturally reductive Einstein metrics on orthogonal groups.
method Using real flag manifolds and symmetry assumptions on left-invariant metrics.
result Obtained new invariant Einstein metrics on $\SO(n)$.
Characterizes group-equivariant neural networks for three groups.
problem Understanding equivariant neural networks for orthogonal, special orthogonal, and symplectic groups.
method Characterized all possible group-equivariant neural networks for three groups.
result Found spanning sets of matrices for learnable, linear equivariant layer functions.
In this paper we study contact structure on 2-step nilpotent, Heisenberg type Lie groups. We decompose this Lie groups to center and orthogonal complement, then investigate properties of both orthogonal Lie subgroups. Finally, we provide a connection between matchings in groups and field extensions and 2-step nilpotent…
New Zoll families of minimal spheres found in spheres and projective spaces.
problem Finding new Zoll families of minimal spheres in various spaces.
method Equivariant constructions using Nash-Moser-Hamilton implicit function theorem.
result First examples of metrics on real projective spaces with Zoll families of minimal projective hyperplanes.
Spectral methods achieve near-optimal performance in orthogonal and permutation group synchronization.
problem Recovering group elements from pairwise measurements in computer vision.
method Spectral methods applied with the leave-one-out technique.
result Near-optimal performance bounds for orthogonal and permutation group synchronization established.
New measures on orbit spaces for orthogonal groups identified.
problem Characterizing measures on orbit spaces of orthogonal groups.
method Constructing Hilbert measures on orbit spaces of coregular representations of orthogonal groups.
result Hilbert measures have singularities if and only if the number of copies equals the dimension.
The paper bounds the maximum group order for graph embeddings in spheres.
problem Bounding the maximum order of a group acting on a graph embedded in a sphere.
method Proved a polynomial bound on group order based on genus and dimension.
result The degree d/2 is the best possible bound in even dimensions.
Geometrically describes Higgs bundle moduli spaces for orthogonal groups.
problem Understanding moduli spaces of Higgs bundles for orthogonal groups.
method Cayley and Langlands type correspondences for real orthogonal and symplectic Higgs bundles.
result Complete abelianization of real slices for all quasi-split real forms.
NS-RGS improves orthogonal group synchronization with faster convergence.
problem Orthogonal group synchronization from pairwise measurements.
method Newton-Schulz iteration for Riemannian gradient optimization.
result NS-RGS achieves linear convergence and near-optimal accuracy.
New invariant shows 4D graph-manifolds' covers match orthogonal types.
problem Understanding covers of 4D graph-manifolds.
method Introduced a topological invariant to prove universal covers' equivalence.
result 4D graph-manifolds' universal covers are bi-Lipschitz equivalent to orthogonal graph-manifold covers.
Thin groups found in specific lattices.
problem Embedding right-angled Coxeter groups in arithmetic lattices.
method Using Agol's unpublished argument, embedding in indefinite orthogonal groups.
result Irreducible right-angled Coxeter groups embed as thin subgroups.
Spectral method for joint community detection and group synchronization.
problem Jointly detecting communities and synchronizing orthogonal groups in graphs.
method Spectral decomposition followed by CPQR factorization.
result Near-optimal guarantees for exact and stable recovery of cluster memberships and orthogonal transforms.
Local classification of 4D Ricci solitons with specific algebra properties.
problem Classifying 4D Ricci solitons with a 2D Abelian Killing algebra.
method Local classification under specific curvature and symmetry conditions.
result Classification of Ricci solitons with orthogonally intransitive 2D Abelian Killing algebra.
Improves machine learning consistency with orthogonal moment equations.
problem Improving consistency of machine learning estimates with complex nuisance parameters.
method Employing Neyman-orthogonal moment equations to improve consistency from n−1/4 to n−1/(2k+2). result Second-order orthogonality can improve consistency to n−1/(2k+2). New algorithm speeds up group equivariant neural networks computations.
problem Challenging computations in group equivariant neural networks.
method Diagrammatic framework based on category theory for matrix multiplication.
result Exponential improvement in time complexity for matrix multiplication.
We survey the existing parts of a classification of finite groups generated by orthogonal transformations in a finite-dimensional Euclidean space whose fixed point subspace has codimension one or two and extend it to a complete classification. These groups naturally arise in the study of the quotient of a Euclidean spa…
New complex orthogonal structures found on a 3D manifold.
problem Finding new geometric structures on a 3D manifold.
method Constructing uniformizable complex orthogonal and projective structures.
result A manifold has multiple uniformizable complex orthogonal structures.
A helical CR structure is a decomposition of a real Euclidean space into an even-dimensional horizontal subspace and its orthogonal vertical complement, together with an almost complex structure on the horizontal space and a marked vector in the vertical space. We prove an equivalence between such structures and step t…
When looking at Bott's original proof of his periodicity theorem for the stable homotopy groups of the orthogonal and unitary groups, one sees in the background a differential geometric periodicity phenomenon. We show that this geometric phenomenon extends to the standard inclusion of the orthogonal group into the unit…