The study characterizes harmonic spaces and their radial eigen-functions and vector fields.
problem Characterizing harmonic spaces and their geometric properties.
method Examining radial eigen-spaces of Laplacians and using duality.
result Results extend to spaces harmonic with respect to a single point.
Study on gradient pseudo-Ricci solitons on real hypersurfaces.
problem Characterize gradient pseudo-Ricci solitons on real hypersurfaces.
method Analyze real hypersurfaces in complex space forms with specific eigen properties of the Ricci tensor.
result Show existence of non-trivial gradient pseudo-Ricci solitons on 3D ruled real hypersurfaces.
The Complex Axis theorem states that any endomorphism of a finite-dimensional complex vector space affords an eigen-vector (or "invariant axis"). A geometric proof of this geometric result was given by A. de Medeiros, transforming the endomorphism into a topological self-map with Lefschetz number not equal to zero. We …
The paper studies Cotton solitons on specific geometric manifolds.
problem Analyzing Cotton solitons in almost Kenmotsu 3-h-manifolds. method Examined potential vector fields and their relationship with the Reeb vector field.
result Steady Cotton solitons on non-Kenmotsu manifolds are locally isometric to H2(−4)imesR. In this article we give a classification of three dimensional m-quasi Einstein manifolds with two distinct Ricci-eigen values. Our study provides explicit description of local and complete metrics and potential functions. We also describe the associated warped product Einstein manifolds in detail. For the proof we pres…
This paper is a complete study of almost α-paracosmplectic manifolds. We characterize almost α-paracosmplectic manifolds which have para Kaehler leaves. Main curvature identities which are fulfilled by any almost α-paracosmplectic manifold are found. We also proved that ξ is a harmonic vector field if and only if it is…
We prove polynomial and exponential decay at infinity of eigen-vectors of partial differential operators related to radiation problems for time-harmonic generalized Maxwell systems in an exterior domain with non-smooth inhomogeneous, anisotropic coefficients converging near infinity with a certain rate towards the iden…
EFiGP uses Fourier and eigen-decomposition for efficient ODE parameter estimation.
problem Parameter estimation and trajectory reconstruction for noisy, sparse, nonlinear ODE systems.
method EFiGP integrates Fourier transformation and eigen-decomposition into a physics-informed Gaussian Process framework.
result EFiGP efficiently estimates ODE parameters and recovers trajectories from noisy data.
The paper introduces eigen-portfolios using PCA to improve portfolio construction in finance.
problem Overfitting and poor generalization in selecting a single eigen-portfolio.
method Principal Component Analysis (PCA) to derive eigen-portfolios from asset return correlation matrices.
result An ensemble strategy combining multiple top-performing eigen-portfolios significantly improves out-of-sample performance.
The Hessian-vector product has been utilized to find a second-order stationary solution with strong complexity guarantee (e.g., almost linear time complexity in the problem's dimensionality). In this paper, we propose to further reduce the number of Hessian-vector products for faster non-convex optimization. Previous a…
Eigen-GNN enhances GNNs by preserving graph structures.
problem Existing shallow GNNs fail to effectively preserve graph structures.
method Integrates eigenspace of graph structures into GNNs as a dimensionality reduction module.
result Eigen-GNN boosts GNNs' ability to preserve graph structures without increasing depth.
Neural networks solve eigen-problems in differential equations.
problem Finding eigenpairs of self-adjoint operators.
method Using neural networks to approximate eigenfunctions and eigenvalues.
result Demonstrates potential of neural networks in solving complex eigen-problems.
IKD uses eigen-decomposition for nonlinear dimensionality reduction.
problem Lack of sophisticated and nonlinear dimensionality reduction methods.
method Inverse Kernel Decomposition (IKD) based on eigen-decomposition of sample covariance matrix.
result IKD achieves comparable performance to optimization-based methods with faster running speeds.
Eigen component analysis combines quantum mechanics with machine learning for efficient data analysis.
problem Efficiently extracting linearly separable components from complex data.
method Eigen component analysis (ECA) incorporates quantum mechanics principles into linear learning models.
result ECA outperforms classical linear models and can be integrated with deep neural networks.
In this paper, energy function is used to investigate the eigen-solutions of −Δu+Vu=λu on the Riemannian manifolds. We give a new way to prove the positivity of the initial energy of energy function, which leads to a simple way to obtain the growth of eigen-solutions.
This study compares two portfolio optimization methods on Indian stocks.
problem Designing an optimal portfolio considering stock returns and risks.
method Hierarchical Risk Parity and Eigen Portfolio approaches on NIFTY 50 sectors.
result Hierarchical Risk Parity portfolio outperforms Eigen portfolio in most sectors tested.
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.
Stratified models depend in an arbitrary way on a selected categorical feature that takes K values, and depend linearly on the other n features. Laplacian regularization with respect to a graph on the feature values can greatly improve the performance of a stratified model, especially in the low-data regime. A sign…
MOSAIC detects change points in dynamic networks with low-rank and sparse changes.
problem Detecting change points in dynamic networks with specific structural properties.
method Eigen-decomposition-based test with screened signals and residual-based adjustment.
result MOSAIC achieves minimax-optimal detection and testing rates.
This paper compares three portfolio designs for Indian stocks.
problem Designing an optimum portfolio that balances return and risk.
method Three approaches: minimum risk, optimum risk, and Eigen portfolios.
result Optimum risk portfolios and Eigen portfolios identified for each sector.
Extensions (entropies) play a central role in the theory of hyperbolic conservation laws by providing intrinsic selection criteria for weak solutions. For a given hyperbolic system u_t+f(u)_x=0, a standard approach is to analyze directly the second order PDE system for the extensions. Instead we find it advantageous to…
This paper speeds up K-FAC for deep learning by focusing on only a few eigen-modes.
problem Time-consuming computation of Kronecker factors in K-FAC for large layers.
method Theoretical analysis and randomized numerical linear algebra to approximate eigen-spectrum decay.
result Reduces time complexity from cubic to quadratic in layer width, improving efficiency.
In this paper, we introduce an algorithm for performing spectral clustering efficiently. Spectral clustering is a powerful clustering algorithm that suffers from high computational complexity, due to eigen decomposition. In this work, we first build the adjacency matrix of the corresponding graph of the dataset. To bui…
SpGAT learns graph representations using spectral attention for efficiency.
problem Efficiently capturing global graph patterns with minimal parameters.
method Introduces Spectral Graph Attention Network (SpGAT) using spectral domain attention mechanisms and a fast Chebychev approximation.
result SpGAT achieves better global pattern recognition with fewer parameters compared to GAT.
Paper addresses eigenvector perturbation in small eigen-gap scenarios.
problem Fine-grained behavior of eigenvectors in the presence of small eigen-gaps.
method Develops de-biased estimators for linear functions of an unknown eigenvector.
result Achieves minimax lower bounds for a family of scenarios, even with small eigen-gaps.
Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.
problem Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.
method Perturbations of geodesic standard double bubbles centered at critical points of the ambient scalar curvature and aligned along eigen-vectors of the ambient Ricci tensor, with general multiplicity results via Lusternik-Schnirelman theory.
result Existence of double bubbles with high constant mean curvatures in Riemannian manifolds.
In this paper, we obtain some properties of biconservative Lorentz hypersurface M1n in E1n+1 having shape operator with complex eigen values. We prove that every biconservative Lorentz hypersurface M1n in E1n+1 whose shape operator has complex eigen values with at most five distinct prin…
ASGD outperforms SGD in overparameterized linear regression, especially in subspaces of small eigenvalues.
problem Generalization of ASGD for overparameterized linear regression.
method Established instance-dependent excess risk bound for ASGD in each eigen-subspace of the data covariance matrix.
result ASGD outperforms SGD in subspaces of small eigenvalues, exhibiting faster decay of bias error.
A new kernel test reduces noise in MMD by focusing on leading eigen-directions.
problem Noise in trailing directional components degrades power of standard kernel two-sample tests.
method Truncate MMD spectral decomposition, retaining only leading eigen-directions.
result Our method achieves superior power and robustness, especially in high-dimensional and unbalanced settings.
This paper has been withdrawn by the author and it is published in AGAG
This paper aims to address two fundamental challenges arising in eigenvector estimation and inference for a low-rank matrix from noisy observations: (1) how to estimate an unknown eigenvector when the eigen-gap (i.e. the spacing between the associated eigenvalue and the rest of the spectrum) is particularly small; (2) …
New method preserves privacy while detecting communities in distributed networks.
problem Privacy-preserving community detection in locally distributed multi-layer networks.
method Privacy-preserving Distributed Spectral Clustering (ppDSC) using randomized response mechanism.
result Developed a novel algorithm that maintains community structure while protecting privacy.
A new method for self-attention models that improves uncertainty estimation.
problem Overconfident predictions and lack of calibrated uncertainty in Transformers.
method Kernel-Eigen Pair Sparse Variational Gaussian Processes (KEP-SVGP) with Kernel SVD (KSVD) to handle asymmetry of attention kernels.
result Reduction in time complexity and improved performance on various benchmarks.
Eigen-decomposition simplifies quadratic programming with equality constraints.
problem Optimizing solutions under linear equality constraints in quadratic programming.
method Eigenvalue decomposition of the quadratic term matrix to project optimal solutions.
result Established a linear mapping between EQP formulations with and without diagonalized Q. Repeated application of machine-learning, eigen-centric methods to an evolving dataset reveals that eigenvectors calculated by well-established computer implementations are not stable along an evolving sequence. This is because the sign of any one eigenvector may point along either the positive or negative direction of…
We investigate various structures associated with the hyperbolic Markov and homological spectra of a pseudoAnosov map φ on a surface. Each unstable eigenvalue of the action of φ on first cohomolgy yields an eigen-cocycle that is transverse and holonomy invariant to the stable foliation Fs of φ. Each …
Paper transforms torse-forming vector fields into simpler forms.
problem Generalizing vector fields and their transformations.
method Present techniques to transform torse-forming vector fields into simpler cases.
result Concrete examples of transformations are provided.
New spinorial functional connects Perelman's W- and F-functionals.
problem Unifying Perelman's functionals for spin manifolds.
method Introduced a new energy functional on spin manifolds, computed its first variation, and established a gradient flow.
result Critical points of the functional are twisted Ricci solitons and eigen-spinsors.
The paper proves that certain modified conformal vector fields are trivial on compact and non-compact manifolds.
problem Proving triviality of modified conformal vector fields on Riemannian manifolds.
method Analyzing properties of homothetic, conformal, and gradient vector fields on compact and non-compact manifolds.
result Established conditions under which m-modified conformal vector fields are trivial. The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold M. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component x^T of the position vector field is the most natural vector field tangent to the …
Conformal vector fields on LCP manifolds are orthogonal and Killing.
problem Understanding conformal vector fields on specific geometric manifolds.
method Analyzing properties of conformal vector fields on compact locally conformally product manifolds.
result Conformal vector fields are orthogonal to the flat distribution and Killing.
For a submanifold M in a Euclidean space, the tangential component x^T of the position vector field x of M is the most natural vector field tangent to the Euclidean submanifold, called the canonical vector field of M. In this article, first we prove that the canonical vector field of every Euclidean submanifold is alwa…
Study on vector fields on Lie groups reveals surprising algebraic coincidences.
problem Characterizing vector fields on Lie groups with Riemannian metrics.
method Algebraic and geometric analysis of left-invariant vector fields on nilpotent Lie groups.
result Spaces of Killing, one-harmonic, and conformal vector fields coincide with the center of the Lie algebra on nilpotent Lie groups.
Study biharmonic vector fields and unit vector fields on Riemannian manifolds.
problem Determine the equivalence of biharmonicity and harmonicity for vector fields and unit vector fields on Riemannian manifolds.
method Analyze biharmonic vector fields and unit vector fields on (M,g) with pseudo-Riemannian g-natural metrics on TM and T1M. result Contrary to Sasaki metric, biharmonicity and harmonicity are not equivalent for large classes of g-natural metrics on TM. This short report establishes some basic properties of smooth vector fields on product manifolds. The main results are: (i) On a product manifold there always exists a direct sum decomposition into horizontal and vertical vector fields. (ii) Horizontal and vertical vector fields are naturally isomorphic to smooth famil…
Defines quaternionic k-vector fields on quaternionic Kähler manifolds.
problem No specific problem stated; focuses on definition and properties.
method Introduced a modified Dirac operator to define quaternionic k-vector fields.
result Calculated the dimension of quaternionic k-vector fields on HPn. Study on generalized derivations in polynomial vector fields Lie algebras.
problem Understanding generalized derivations in specific Lie sub-algebras of polynomial vector fields.
method Analysis of Lie sub-algebras containing constant and Euler vector fields, under specified conditions.
result Characterization of generalized derivations in the studied Lie sub-algebras.
In this paper, we consider the eigen-solutions of −Δu+Vu=λu, where Δ is the Laplacian on a non-compact complete Riemannian manifold. We develop Kato's methods on manifold and establish the growth of the eigen-solutions as r goes to infinity based on the asymptotical behaviors of Δr and V(x), where r=r(x) i…