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.
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.
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.
The paper solves PDEs from matrices with orthogonal columns, linking them to Hessian metrics and symmetric spaces.
problem Solving third order PDEs for strictly convex smooth functions.
method Geometric methods using Hessian metrics and symmetric spaces.
result Explicit solutions and a family of non-generic solutions with applications in Poisson geometry and Kahler structures.
A new method for efficiently computing derivatives of skew-symmetric matrix exponentials.
problem Efficient computation of derivatives for skew-symmetric matrices.
method Characterization of invertibility, construction of nearby logarithm, and efficient implementation.
result Explicit formulae for differentiation and its inverse of skew-symmetric matrix exponentials.
Study differential properties of matrix square roots in specific cases.
problem Understanding matrix square roots in semi-simple, symmetric, and orthogonal cases.
method Analysis of differential and metric structures of real square roots of matrices under specific conditions.
result Differential properties of matrix square roots in semi-simple, symmetric, and orthogonal cases.
Chevalley theorems extended to isotropic functions on matrix spaces.
problem Extending Chevalley theorems to isotropic functions on matrix spaces.
method Proving ultradifferentiable Chevalley restriction theorems for various ultradifferentiable classes.
result Isotropic functions on symmetric matrices have ultradifferentiable regularity if and only if their diagonal restrictions do.
Characterizes the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
problem Characterizing the local diffeomorphism structure of the exponential in the set of skew-symmetric matrices.
method Introduce the diffeomorphic logarithm of special orthogonal matrices and an efficient algorithm.
result The region containing the principal logarithm has a special multiplicity structure.
Researchers derive exclusive Racah matrix for complex representation with non-trivial multiplicities.
problem Constructing and understanding Racah matrices for complex representations with non-trivial multiplicities.
method Using effective field theory for arborescent knots, they deduced the exclusive Racah matrix for representation R=[3,1] with non-trivial multiplicities. result The exclusive Racah matrix $ar S$ for representation R=[3,1] is operator valued and depends on basis choices in intertwiner spaces. scoRNN improves RNN performance with simpler orthogonal weight matrices.
problem Vanishing and exploding gradients in RNNs.
method Parametrizing orthogonal recurrent weight matrices with a scaled Cayley transform.
result scoRNN achieves superior results with fewer parameters than other unitary RNNs.
The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.
problem Analyzing Riemannian Gaussian distributions on symmetric spaces.
method Analytical computation of marginals using orthogonal and skew orthogonal polynomials, and diffusion kernels.
result Riemannian Gaussian distributions are random matrix types, and their probability density functions can be computed analytically.
FedSPDnet improves federated learning for SPD matrices, outperforming existing methods.
problem Federated learning for SPD matrices with orthogonality constraints.
method Two efficient aggregation strategies: ProjAvg and RLAvg, preserving geometric structure.
result FedSPDnet outperforms federated EEGnet in F1 score and robustness to federation and partial participation.
Study reveals hidden structure behind Racah matrices for twisted knots.
problem Understanding non-associativity in representation products of twisted knots.
method Analysis of quantum R-matrices and their eigenvalues to decompose Racah matrices.
result Discovery of pentad structure (Tˉ,Sˉ,S,E,B) associated with universal R-matrix. Efficiently approximates eigenspaces for symmetric and general matrices.
problem Fast computation of eigenspaces for large matrices.
method Factor eigenspaces into fundamental components using transformations, solve minimization problems, and iteratively update.
result Improved computational efficiency for eigenspace approximation.
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.
Fast approximates orthogonal matrices for PCA.
problem Approximating orthogonal matrices for fast computation.
method Optimization over structured matrices and greedy algorithm.
result Efficient approximation balances accuracy and speed.
New method for sampling orthogonal matrices using Hamiltonian Monte-Carlo.
problem Sampling from posterior distributions of orthogonal matrices in Bayesian models.
method Proposes a new sampling scheme based on Hamiltonian Monte-Carlo and Riemannian optimization.
result New method is comparable or faster in time per iteration and more sample-efficient than conventional methods.
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.
Optimizing over the set of orthogonal matrices is a central component in problems like sparse-PCA or tensor decomposition. Unfortunately, such optimization is hard since simple operations on orthogonal matrices easily break orthogonality, and correcting orthogonality usually costs a large amount of computation. Here we…
Constructs orthogonal coordinates in curved spaces.
problem Separating variables in curved spaces.
method Explicit construction of orthogonal coordinates and transformations.
result Explicit formulas for Killing tensors and Stäckel matrices.
Proposes BONMI for integrating noisy matrices from multi-source data.
problem Integrating noisy matrices from multi-source data with block-wise missingness.
method Exploits orthogonal Procrustes problem to align eigenspaces and completes missing blocks.
result Statistical rate for eigenspace of underlying matrix comparable to independently missing assumption.
New invariants derived from random matrices for words in free groups.
problem Defining and understanding new topological invariants for words in free groups.
method Defining and analyzing invariants from w-random matrices and permutations. result Presented new topological, combinatorial, and algebraic invariants of words.
Simple matrix formulas for Grassmannian curvatures.
problem Modeling Grassmannian for curvature calculations.
method Symmetric orthogonal matrices and standard matrix operations.
result Explicit, simple formulas for various curvatures.
A new algorithm POGO optimizes thousands of orthogonal matrices efficiently.
problem Optimizing thousands of orthogonal constraints at scale is computationally expensive.
method Revisits Landing algorithm, uses modern adaptive optimizers, reduces hyperparameters.
result POGO optimizes thousands of orthogonal matrices in minutes, outperforming alternatives.
Improved machine learning performance through structured random orthogonal embeddings.
problem Improving accuracy and speed in machine learning applications.
method Structured random orthogonal matrices for dimensionality reduction and kernel approximation.
result Significant improvement in accuracy and speed compared to existing methods.
Orthogonal Random Features reduce kernel approximation error and speed up computation.
problem Gaussian kernel approximation error reduction and speed up computation.
method Replacing random Gaussian matrix with a scaled random orthogonal matrix, and using structured discrete orthogonal matrices.
result Significantly decreases kernel approximation error and reduces computation time from O(d2) to O(dlogd). New framework solves low-rank optimization problems to certifiable optimality.
problem Low-rank optimization problems with certifiable solutions.
method Mixed-Projection Conic Optimization framework using symmetric projection matrices and outer-approximation algorithms.
result Solves low-rank problems to certifiable optimality, outperforming existing methods.
This work proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
problem Proving the asymptotic freeness of layerwise Jacobians in multilayer perceptrons (MLPs).
method Replacing each layer's parameter matrix with itself multiplied by a Haar orthogonal matrix, and using the invariance of the MLP.
result Proves the asymptotic freeness of layerwise Jacobians in MLPs with Haar orthogonal matrices.
Paper introduces efficient orthonormal transformations using Householder reflectors.
problem Fast numerical procedures for orthonormal transformations are computationally expensive.
method Develops orthonormal matrices using Householder reflectors to approximate any orthonormal or symmetric transform.
result Approximations of orthonormal or symmetric transforms using a few Householder reflectors are accurate and computationally efficient.
One reflection suffices for orthogonal weights, reducing GPU usage.
problem Efficiently computing orthogonal weight matrices without high GPU utilization.
method Use an auxiliary neural network to compute one reflection instead of many.
result One reflection is sufficient for orthogonal weights, improving GPU utilization.
Minimal submanifolds in matrix spaces proven for specific ranks.
problem Minimal submanifolds in matrix spaces.
method Proving semialgebraic sets of matrices are minimal.
result Rectangular, skew-symmetric, and symmetric matrices with prescribed eigenvalues are minimal.
Study differential properties of symmetric real matrices with a specific metric.
problem Differential geometric properties of symmetric real matrices.
method Trace metric on non-singular symmetric real matrices, isometries of positive definite matrices.
result Description of the full group of isometries for positive definite matrices.
Proposes a new RNN structure to improve expressivity without sacrificing stability.
problem Exploding and vanishing gradient problems in RNNs and reduced expressivity.
method Introduces a non-normal RNN structure using Schur decomposition and splitting.
result Enhances expressivity while maintaining stability and training speed.
Develops log-Euclidean Lie groups for SPD and correlation matrices.
problem Unifies various log-Euclidean constructions for SPD and correlation matrices.
method Theory and explicit isometries linking different log-Euclidean metrics.
result Explicit log-Euclidean metrics on SPD and correlation matrices.
Orthogonal random features approximate a Bessel kernel, offering sharper bounds than random Fourier features.
problem Approximating Gaussian kernel efficiently for large datasets.
method Use of Haar orthogonal matrices to construct orthogonal random features and analyze their bias and variance.
result Orthogonal random features approximate a Bessel kernel, not the Gaussian kernel, with sharper bounds.
A left-invariant sub-Riemannian metric d on the shortened Lorentz group SO0(2,1) under the condition that d is right-invariant relative to the orthogonal Lie subgroup 1⊗SO(2) is studied. The distance between arbitrary two elements, the cut locus (as the union of the subgroup 1⊗SO(2) with the an…
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.
A new algorithm avoids retractions to optimize orthogonal matrices efficiently.
problem Optimizing functions over the manifold of orthogonal matrices efficiently.
method Landing algorithm that avoids retractions using potential energy.
result The landing algorithm is faster and less prone to numerical errors than retraction-based methods.
Study of curvature tensor algebra with nonstandard multiplications.
problem Understanding curvature tensor algebra structures.
method Investigates orthogonally invariant commutative nonassociative multiplications on curvature tensors.
result Characterizes curvature tensor algebras in low dimensions and proves their simplicity in higher dimensions.
This work considers a computationally and statistically efficient parameter estimation method for a wide class of latent variable models---including Gaussian mixture models, hidden Markov models, and latent Dirichlet allocation---which exploits a certain tensor structure in their low-order observable moments (typically…
AuON is a linear-time optimizer that improves upon Muon's performance without approximate orthogonal matrices.
problem High memory and computational costs of orthogonal momentum updates.
method AuON uses normalized nonlinear scaling and a 'emergency brake' to handle exploding attention logits.
result AuON achieves strong performance without approximate orthogonal matrices, preserving structural alignment and reconditioning.
Efficiently optimizes orthogonal and Stiefel matrices on parallel units.
problem Optimization over orthogonal groups on parallel units.
method CWY and T-CWY transforms for parametrization and optimization.
result CWY and T-CWY methods lead to convergence on parallel units.
Improved Kalman filter for Stiefel manifold measurements.
problem Improving accuracy in measurements on Stiefel manifolds.
method Generalization of extended Kalman filter for Stiefel manifold-valued measurements.
result Significant improvement over raw measurements.
Graph alignment problem solved with convex relaxations for correlated matrices.
problem Recovering hidden vertex permutations from correlated Gaussian matrices.
method Convex relaxations of the quadratic assignment problem over doubly stochastic matrices.
result The solution of the convex relaxation concentrates around the ground-truth permutation matrix for certain correlation parameters.
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.
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.
New metric tensor field on symmetric matrices simplifies eigenvector computation.
problem Complex eigenvector computation for 2x2 symmetric matrices.
method Introducing a metric tensor field on the space of symmetric matrices, resulting in a curved manifold.
result Parallel transport simplifies eigenvector computation for one-parameter families of matrices.
Paper extends matrix inequality to Hermitian matrices.
problem Extending inequalities to Hermitian matrices.
method Using Frobenius norm of commutators for real and skew matrices, extending to Hermitian and skew-Hermitian.
result DDVV-type inequalities now apply to Hermitian matrices.