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.
New metrics improve landing algorithms for orthogonality constraints.
problem Optimizing landing algorithms with orthogonality constraints.
method Proposed a family of metrics over full-rank matrices to enhance landing algorithms.
result Natural extension of β-metric improves landing performance.
We develop matrix models for Grassmann, flag, and Stiefel manifolds.
problem Creating efficient models for Grassmann, flag, and Stiefel manifolds.
method Orthogonally-equivariant matrix submanifold models derived for each manifold.
result Exhaustive list of orthogonally-equivariant submanifold models for the lowest dimensions.
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.
New method for identifying autoregressive systems on manifolds.
problem Identifying autoregressive systems on Stiefel and Grassmann manifolds.
method Defining parameters as orthogonal group elements, averaging over observations, conjugate gradient descent on manifolds.
result System parameters can be estimated efficiently using the proposed algorithm.
Formula derived for Laplace-Beltrami on Stiefel manifold.
problem Finding Laplace-Beltrami operator on Stiefel manifold.
method Using the general framework of Laplace operators on constraint manifolds, derived the explicit formula in terms of ambient Euclidean coordinates.
result Extended previously known formulas for sphere and special orthogonal group.
Classifies totally geodesic submanifolds and polar actions on Stiefel manifolds.
problem Classifying totally geodesic submanifolds and polar actions on Stiefel manifolds.
method Classification through polar actions and cohomogeneity-one actions.
result Classification of orbits of polar actions on Stiefel manifolds.
New algorithm finds sparse matrices on Stiefel manifold for optimisation.
problem Finding sparse matrices on Stiefel manifold for optimisation.
method Modified Orthogonal Iteration algorithm for sparse global optimality.
result Proposed method finds globally optimal sparse Stiefel matrices.
ManifoldFlow relaxes fixed-spectrum Stiefel layers to learn a positive spectrum.
problem Fixed-spectrum Stiefel layers impose rigid spectral constraints.
method Introduces ManifoldFlow, a relaxation that learns a positive spectrum while keeping the basis on the Stiefel manifold.
result Learnable SPD spectrum improves performance in various settings.
A new optimizer preserves orthogonality constraints on matrices efficiently.
problem Optimization on Stiefel manifold with orthogonality constraints.
method Interplay between continuous and discrete dynamics leading to a gradient-based optimizer with momentum.
result The method optimizes matrices on Stiefel manifold efficiently and accurately.
SOFARI improves inference on multi-task learning latent factors.
problem Challenges in precise inference on multi-task learning latent factor matrices.
method High-dimensional manifold-based Neyman near-orthogonality inference on Stiefel manifold structure.
result Easy-to-use bias-corrected estimators for latent factor vectors and singular values with asymptotic normal distributions.
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.
A new retraction on Stiefel manifold with a closed-form inverse.
problem Efficiency in Riemannian computing applications.
method Introduces a new retraction on the compact Stiefel manifold with a closed-form inverse.
result The retraction is second-order accurate and features a closed-form inverse.
Injectivity radius on Stiefel manifold is π.
problem Determining the injectivity radius of the compact Stiefel manifold.
method Utilized the property of geodesics being space curves of constant Frenet curvatures.
result The injectivity radius on the Stiefel manifold under the Euclidean metric is π.
We introduce an approach based on the Givens representation for posterior inference in statistical models with orthogonal matrix parameters, such as factor models and probabilistic principal component analysis (PPCA). We show how the Givens representation can be used to develop practical methods for transforming densit…
A new method solves optimization problems on the generalized Stiefel manifold using random estimates of B.
problem Optimization over the generalized Stiefel manifold in applications like CCA, ICA, and GEVP.
method Cheap stochastic iterative method that converges to critical points on the manifold.
result The method achieves the same convergence rates as Riemannian optimization but with lower per-iteration cost.
A new framework for PPLS combines noise estimation, optimization, and calibration.
problem Probabilistic PLS models need interpretable latent factors and calibrated uncertainty.
method End-to-end pipeline combining noise estimation, constrained optimization, and prediction calibration.
result Achieves near-nominal coverage and native calibrated uncertainty across benchmarks.
ODCGM solves non-convex optimization on manifolds with simpler projections.
problem Minimizing non-convex functions over smooth manifolds.
method Orthogonal Directions Constrained Gradient Method (ODCGM) that projects onto a vector space.
result ODCGM converges to the manifold with near-optimal oracle complexities.
New algorithm computes flag mean and median on flag manifolds.
problem Computing first order flag statistics on flag manifolds.
method Transformed problem to Stiefel manifold for optimization.
result Proved convergence and effectiveness of the flag-mean computation.
This work improves disentanglement in latent space models without sacrificing generation quality.
problem Trade-off between disentanglement and generation quality in latent space models.
method Manifold optimization with a sum of autoencoder and PCA reconstruction errors, on the Stiefel manifold.
result Improves disentanglement without sacrificing generation quality.
NSA-Flow optimizes matrix representations for interpretability in complex data.
problem Balancing interpretability and model flexibility in high-dimensional data.
method Non-negative Stiefel Approximating Flow (NSA-Flow) unifies sparse matrix factorization and orthogonalization.
result NSA-Flow yields sparse, stable, and interpretable representations.
Unified analysis of multilabel Fisher discriminants with improved dimensionality and robustness.
problem Improving discriminant analysis for multilabel classification with enhanced dimensionality and robustness.
method Unified algebraic and statistical analysis of multilabel Fisher discriminants with Stiefel orthogonality constraints.
result Equivalence of four Fisher objectives under the Stiefel constraint and improved discriminant dimensionality.
New algorithms reduce orthogonality constraint enforcement time in machine learning.
problem Efficiently solving orthogonality constraints in machine learning.
method Extending the landing algorithm to Stiefel manifold, incorporating stochastic and variance reduction techniques.
result All proposed methods achieve the same convergence rate as Riemannian counterparts enforcing constraints.
New Bayesian matrix completion method using Stiefel manifolds.
problem Efficient Bayesian matrix completion with uncertainty quantification.
method Geodesic Hamiltonian Monte Carlo on Stiefel manifolds.
result Improved sampling performance and accuracy on real-world problems.
The paper studies connectivity of Schur-Horn map images in real Grassmannians.
problem Connectivity of Schur-Horn map images in real Grassmannians.
method Criterion for pre-images of vectors in \(\mathbb{R}^n\) to be connected.
result Criterion for pre-images of vectors in \(\mathbb{R}^n\) to be connected.
New model separates images into independent factors quickly and easily.
problem Separating high-dimensional data like images into independent latent factors.
method Combines bijective feature maps with linear ICA model on the Stiefel manifold.
result Models converge quickly and achieve better unsupervised latent factor discovery.
Unified analysis of multilabel Fisher discriminants with improved dimensionality and robustness.
problem Improving discriminant analysis for multilabel classification with enhanced dimensionality and robustness.
method Unified theoretical analysis of multilabel Fisher discriminants with algebraic and statistical guarantees.
result Unified characterization of multilabel Fisher objectives and their equivalence under orthogonality constraints.
Stiefel-Whitney classes of moment-angle manifolds are trivial.
problem Analyzing the topological properties of moment-angle manifolds.
method Proving triviality of Stiefel-Whitney classes for moment-angle manifolds, including partial quotients.
result Stiefel-Whitney classes of moment-angle manifolds are trivial.
A new mathematical approach detects frequency-based alterations in brain networks.
problem Understanding disease-relevant brain alterations through network analysis.
method Proposes a novel connectome harmonic analysis framework using common harmonic waves learned from Stiefel manifolds.
result Identifies more significant and reproducible network dysfunction patterns in Alzheimer's disease.
The paper calculates ranks and bounds for Stiefel manifolds over different fields.
problem Computing ranks and bounds for Stiefel manifolds over various fields.
method Computation of upper characteristic ranks and cup lengths, providing bounds and necessary conditions for maps.
result Bounds and necessary conditions for S3-maps between quaternionic Stiefel manifolds. This work proposes a novel method for interpolating ROMs without solving FEM models.
problem Interpolating ROMs for unseen parameter values without solving FEM models.
method Non-intrusive Space-Time POD interpolation on compact Stiefel manifolds.
result Robust ROMs derived for unseen parameter values with strong correlations to high-fidelity simulations.
Paper proves convergence of Kalman filter on Stiefel manifolds with measurement errors.
problem Filtering constant particle with measurement errors on Stiefel manifolds.
method Extended Kalman filter applied to Stiefel manifold-valued observations.
result Convergence of the extended Kalman filter proved for constant system process.
We introduce Block Sparse Canonical Correlation Analysis which estimates multiple pairs of canonical directions (together a "block") at once, resulting in significantly improved orthogonality of the sparse directions which, we demonstrate, translates to more interpretable solutions. Our approach builds on the sparse CC…
Researchers compute curvatures of Stiefel manifolds with new metrics.
problem Computing curvatures of Stiefel manifolds with specific metrics.
method Two approaches: global curvature formula and left-invariant metrics.
result Stiefel manifolds always carry an Einstein metric and have non-negative sectional curvature.
Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.
problem Determining the shortest geodesic loops and injectivity radius on Stiefel manifold.
method Combining bounds on sectional curvature with existing metrics.
result Exact value of the injectivity radius for a wide range of metrics.
The paper computes characteristic classes for Lie group representations.
problem Computing characteristic classes for Lie group representations.
method The paper outlines a procedure to compute characteristic classes of irreducible representations of Lie groups, expressing them as polynomial functions in the highest weight.
result The paper expresses characteristic classes of Lie group representations as polynomial functions in the highest weight.
Random matrix ensembles yield uniform distributions on manifolds.
problem Understanding distributions of vectors in random matrix ensembles.
method Analyzing eigenvalues, singular values, and Autonne-Takagi vectors of various random matrix ensembles.
result Uniform distributions on specific manifolds for different types of random matrix ensembles.
James's octonionic Stiefel spaces questions answered partially.
problem Two fundamental questions about octonionic Stiefel spaces.
method Partial answers to James's questions about octonionic Stiefel spaces.
result Partial answers to James's questions about octonionic Stiefel spaces.
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.
Upper bound on Stiefel manifold's injectivity radius found.
problem Finding the maximum distance within which the Stiefel manifold remains injective.
method Exhibited conjugate points and calculated the minimum of geodesic lengths.
result Upper bound on Stiefel manifold's injectivity radius is conjectured to be equal to the injectivity radius.
A generalized Stiefel manifold is the manifold of orthonormal frames in a vector space with a non-degenerated bilinear or hermitian form. In this article, the Isometry group of the generalized Stiefel manifolds are computed at least up to connected components in an explicit form. This is done by considering a natural n…
The problem of estimating sparse eigenvectors of a symmetric matrix attracts a lot of attention in many applications, especially those with high dimensional data set. While classical eigenvectors can be obtained as the solution of a maximization problem, existing approaches formulate this problem by adding a penalty te…
New theorem shows embedding restrictions for manifold skeletons.
problem Embedding restrictions for triangulated manifolds.
method Proves van Kampen-Flores theorem for manifolds with specific Stiefel-Whitney classes.
result Triangulated manifolds with non-trivial Stiefel-Whitney classes cannot embed into R2d. Study on determinants of unitary Brownian motion and their asymptotic laws.
problem Understanding determinants of unitary Brownian motion and their behavior over time.
method Using Stiefel fibration and skew-product decomposition of the Stiefel Brownian motion.
result Prove asymptotic laws for determinants of block entries of unitary Brownian motion.
New Morse-Bott function defined on Stiefel manifolds, revealing complex critical structures.
problem Defining Morse-Bott functions on non-linear Stiefel manifolds.
method Replacing linear height function with a quadratic one, proving it as a Morse-Bott function.
result Critical submanifolds are fibrations of products of Grassmannians, not Grassmannians themselves.
Study on rolling Stiefel manifolds with specific metrics.
problem Intrinsic and extrinsic rolling of Stiefel manifolds with α-metrics. method Investigation of intrinsic rolling of normal naturally reductive homogeneous spaces, derivation of ODEs for rolling, and explicit solutions.
result Explicit solutions for intrinsic and extrinsic rolling of Stiefel manifolds.
We discuss conditions for the integrability of an almost complex structure defined on the total space of an induced Hopf S^3-bundle over a Sasakian manifold . As an application, we obtain an uncountable family of inequivalent complex structures on the Stiefel manifolds of orthonormal 2-frames in C^{n+1}, non compatible…
Guaranteed convergence for tensor factorization using Riemannian gradient descent.
problem Recovering tensor train format from linear measurements.
method Optimization over left-orthogonal TT format using Riemannian gradient descent on Stiefel manifold.
result RGD converges linearly to the ground-truth tensor with polynomial error growth in tensor order.