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.
A new method for asynchronous eigenspace computation on the Grassmannian.
problem Asynchronous optimization for finite-sum eigenspace computation in distributed systems.
method Grassmannian incremental aggregation method that refreshes only arriving components and reuses cached gradients.
result Two-phase linear convergence with constants controlled by component spectral spreads.
New algorithm estimates eigenspace with faulty nodes, matching performance of existing methods.
problem Estimating eigenspace in distributed systems with node failures.
method Develops an eigenspace estimation algorithm for distributed environments with arbitrary node failures.
result Matches performance of existing non-robust estimator up to an additive error.
Adaptive PCA algorithm for real-time data analysis.
problem Real-time computation of eigenspace for time-varying data.
method Online adaptive PCA algorithm with one-step update rule considering second order correlations.
result The algorithm provides an excellent approximation to the original eigenspace computed using standard PCA in batch mode.
Paper proposes an online distributed PCA algorithm for faster computation.
problem Efficiently estimating principal eigenspaces in distributed systems.
method Online distributed algorithm for principal eigenspace recovery.
result Algorithm demonstrates faster computation with comparable accuracy.
Our work connects parameter magnitudes and Hessian eigenspaces in deep neural nets.
problem Understanding the relationship between parameter magnitudes and Hessian curvature in deep learning models.
method Developed a matrix-free algorithm based on sketched SVDs to measure similarity between parameter masks and Hessian eigenspaces.
result Top Hessian eigenvectors tend to be concentrated around larger parameters, indicating a connection between parameter magnitudes and loss curvature.
A method to analyze neural network performance by measuring layer saturation.
problem Understanding which layers contribute to network performance.
method Layer saturation method: restricts layer output to eigenspace of variance matrix.
result Layer saturation indicates which layers contribute to network performance.
FedPower improves eigenspace estimation privacy in federated learning.
problem Privacy breaches and communication challenges in federated eigenspace estimation.
method FedPower uses a power method with local power iterations and global aggregation, weighted by OPT, and adds Gaussian noise for privacy.
result FedPower provides convergence bounds and demonstrates effectiveness in experiments.
Study on irreducibility of Laplacian eigenspaces in homogeneous spaces.
problem Existence of G-invariant Riemannian metrics with irreducible Laplacian eigenspaces. method Analysis of compact homogeneous spaces G/K and their metrics. result Normal metric of rank one symmetric spaces is the only one with irreducible Laplacian eigenspaces.
New algorithm reduces communication in distributed eigenspace estimation.
problem Efficiently estimating eigenspaces in distributed settings without excessive communication.
method Communication-efficient distributed algorithm using Procrustean alignment.
result Achieves similar error rate to centralized estimator for PCA.
This paper tackles distributed estimation of the top-L eigenspace in PCA for large data sets.
problem Challenges in estimating the top-L eigenspace in principal component analysis for large data sets.
method Proposes a novel multi-round algorithm using shift-and-invert preconditioning and convex optimization.
result Achieves a fast convergence rate and covers the targeted top-L eigenspace without explicit eigengap assumption.
In this note we explore a connection between finite covers of surfaces and the Teichmüller polynomial of a fibered face of a hyperbolic 3--manifold. We consider the action of a homological pseudo-Anosov homeomorphism ψ on the homology groups of a class of finite abelian covers of a surface Σg,n. Eigenspaces of t…
New measure helps identify better word embedding compression methods.
problem Challenges in evaluating compressed word embeddings for downstream tasks.
method Proposed eigenspace overlap score and developed generalization bounds.
result Eigenspace overlap score correlates with better downstream performance.
Physics-informed GP regression solves eigenvalue problems by identifying non-trivial eigenspaces.
problem Solving eigenvalue problems of linear operators with trivial solutions.
method Constructing a transfer function-type indicator using physics-informed Gaussian Process posterior.
result The posterior covariance is non-trivial only for eigenvalues of the operator, indicating non-trivial eigenspaces.
The resonant band is a useful notion for the computation of the nontrivial monodromy eigenspaces of the Milnor fiber of a real line arrangement. In this article, we develop the resonant band description for the cohomology of the Aomoto complex. As an application, we prove that real 4-nets do not exist.
A fundamental operation in many vision tasks, including motion understanding, stereopsis, visual odometry, or invariant recognition, is establishing correspondences between images or between images and data from other modalities. We present an analysis of the role that multiplicative interactions play in learning such …
SOEM clusters time series data with improved accuracy.
problem Clustering non-aligned time series data.
method Generalizes SOFM to matrix input using approximate joint diagonalisation of covariance structures.
result SOEM produces valid topological clustering of time series data.
Paper identifies key function spaces for ReLU networks based on Fisher information.
problem Understanding the structure of Fisher information matrices in ReLU networks.
method Spectral decomposition of Fisher information matrices, focusing on the first three eigenspaces.
result The first three eigenspaces account for 97.7% of the trace of the Fisher information matrix, corresponding to spherical harmonic functions of order ≤2.
New methods compare Steklov eigenspaces of free boundary minimal surfaces in balls.
problem Comparing Steklov eigenspaces of free boundary minimal surfaces.
method Developed new methods to compare span of coordinate functions with Steklov eigenspace.
result Proved congruence of free boundary minimal annuli in 3D unit ball.
If G is a compact Lie group endowed with a left invariant metric g, then G acts via pullback by isometries on each eigenspace of the associated Laplace operator Δg. We establish algebraic criteria for the existence of left invariant metrics g on G such that each eigenspace of Δg, regarded as the real ve…
New method uses random signals to quickly approximate graph eigenvectors.
problem Estimating the first k eigenvectors of graph Laplacians efficiently.
method Filtering Gaussian random signals to recover eigenvectors.
result Only k random signals needed to recover k smallest eigenvectors.
In this paper, we aim at recovering an undirected weighted graph of N vertices from the knowledge of a perturbed version of the eigenspaces of its adjacency matrix W. For instance, this situation arises for stationary signals on graphs or for Markov chains observed at random times. Our approach is based on minimizi…
We study Milnor fibers of complexified real line arrangements. We give a new algorithm computing monodromy eigenspaces of the first cohomology. The algorithm is based on the description of minimal CW-complexes homotopic to the complements, and uses the real figure, that is, the adjacency relations of chambers. It enabl…
The instanton Floer homology of a knot in the three-sphere is a vector space with a canonical mod 2 grading. It carries a distinguished endomorphism of even degree,arising from the 2-dimensional homology class represented by a Seifert surface. The Floer homology decomposes as a direct sum of the generalized eigenspaces…
Analysis of DPPs and k-DPPs via spectral decomposition reveals identifiable parameters and non-identifiability gaps.
problem Identifying parameters of DPPs and k-DPPs through spectral decomposition.
method Spectral decomposition of the covariance matrix, analysis of invariances, and counting arguments.
result Identifiability of parameters changes fundamentally for k-DPPs, with specific invariances and non-identifiability gaps.
A local classification of locally conformal flat Riemannian Einstein-like four-manifolds as well as a local classification of all locally conformal flat Riemannian four-manifolds for which all Jacobi operators have parallel eigenspaces along every geodesic is given. Non-trivial explicit examples are presented. The prob…
TransNet improves community detection on target networks using privacy-preserved source networks.
problem Community detection on sensitive network data with privacy constraints.
method Spectral clustering framework leveraging locally distributed privacy-preserved auxiliary networks via randomized response and adaptive weighting.
result TransNet delivers strong gains in community detection across various privacy levels and heterogeneity patterns.
Essential self-adjointness and spectrum of CR GJMS operator proved.
problem Characterizing the spectrum of CR GJMS operator.
method Proving essential self-adjointness and closed range, analyzing spectrum.
result CR GJMS operator has discrete spectrum with finite-dimensional eigenspaces.
A theory of feature geometry using spectral analysis of weight matrices.
problem Current methods decompose neural network activations into sparse linear features, losing geometric structure.
method Develops a theory by analyzing the spectra of weight-derived matrices, introducing the frame operator.
result Features collapse onto single eigenspaces, organizing into tight frames, and admit discrete classification.
Study involutions on spaces to compute nonnegatively curved metrics dimensions.
problem Computing dimensions of spaces of nonnegatively curved metrics.
method Involution on pseudoisotopy spaces and free loop spaces.
result Explicit dimensions of manifolds with nontrivial rational homotopy groups.
This paper addresses a gap in the classifcation of Codazzi tensors with exactly two eigenfunctions on a Riemannian manifold of dimension three or higher. Derdzinski proved that if the trace of such a tensor is constant and the dimension of one of the the eigenspaces is n−1, then the metric is a warped product where t…
Proves spectral simplicity of Hodge Laplacian and curl operator along metric families.
problem Simplicity of Hodge Laplacian and curl operator eigenvalues along metric families.
method Generalized Teytel's method to compute meagre codimension of metrics with specific eigenvalue multiplicities.
result Simplicity of Hodge Laplacian and curl operator is not a meagre codimension 2 property.
For generic torus-invariant metrics, eigenspaces are 2D and nodal sets are connected hypersurfaces.
problem Understanding spectral multiplicity and nodal sets for generic torus-invariant metrics.
method Analyzing real Δg-eigenspaces and nodal sets for generic T-invariant metrics. result For generic T-invariant metrics, real Δg-eigenspaces are irreducible and have dimension at most 2, and nodal sets are connected hypersurfaces with specific properties. The moduli space NK of infinitesimal deformations of a nearly Kähler structure on a compact 6-dimensional manifold is described by a certain eigenspace of the Laplace operator acting on co-closed primitive (1,1) forms. Using the Hermitian Laplace operator and some representation theory, we compute the space NK on all 6…
Study spectral properties of graph Laplacian for manifold data.
problem Understanding spectral properties of graph Laplacian for manifold data.
method Non-asymptotic error bounds on spectral properties of empirical graph Laplacian.
result Eigenvalues and eigenspaces of empirical graph Laplacian are close to Laplace-Beltrami operator of manifold.
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.
Consider the sum of the first N eigenspaces for the Laplacian on a Riemannian manifold. A basis for this space determines a map to Euclidean space and for N sufficiently large the map is an embedding. In analogy with a fruitful idea of Kähler geometry, we define (Riemannian) Bergman metrics of degree N to be thos…
SU(3) instanton homology counts Tait colorings for webs and foams.
problem Counting Tait colorings for webs and foams.
method SU(3) gauge theory with structure group, eigenspace decomposition, skein exact triangles.
result SU(3) homology counts Tait colorings, Euler characteristic interpretable as polynomial invariant value.
The paper explores how word embeddings affect the stability of downstream NLP models.
problem Small changes in training data can cause significant changes in model predictions.
method Empirical and theoretical analysis of embedding instability, including the introduction of eigenspace instability measure.
result Increasing embedding memory can reduce the disagreement in predictions by 5% to 37%.
Researchers create surfaces with exceptionally high Steklov eigenvalues.
problem Creating surfaces with first non-zero Steklov eigenvalue of large multiplicity.
method Constructing surfaces with specific isometry groups and gluing them based on Cayley graph structures, then analyzing the eigenspace properties.
result Surfaces with arbitrarily large multiplicity for their first non-zero Steklov eigenvalue are constructed.
The paper calculates dimensions of higher Landau levels on compact manifolds.
problem Understanding Landau levels on compact manifolds in the large magnetic field limit.
method Computing dimensions as Riemann-Roch numbers, studying Toeplitz algebras, and proving isomorphisms.
result Each Landau level is isomorphic to a quantization twisted by an auxiliary bundle.
New principle reveals how neural networks learn complex interactions.
problem Understanding neural networks' success and complexity.
method Infinite-width networks, focusing on frequency and space.
result Fine-grained eigenstructure improves network learnability.
The paper calibrates shrinkage covariance estimators for spectral functionals in high dimensions.
problem Calibrating shrinkage covariance estimators for spectral functionals in high dimensions.
method Derives first-order null laws, distribution-free Davis-Kahan bands, and calibrated tests for spectral functionals under shrinkage.
result Calibrated tests and intervals for spectral functionals are provided, addressing the issue of estimation noise and shrinkage bias.
This paper uncovers the low-rank structure of neural network Hessians.
problem Understanding the structure of Hessians in neural networks.
method Proposes a decoupling conjecture to decompose layer-wise Hessians into Kronecker products of smaller matrices.
result Proves the structure of top eigenspaces in 2-layer networks and shows high overlap in top eigenvectors across different models.
Let (E,h) be a holomorphic Hermitian vector bundle over a polarized manifold. We provide a canonical quantization of the Laplacian operator acting on sections of the bundle of Hermitian endomorphisms of E. If E is simple we obtain an approximation of the eigenvalues and eigenspaces of the Laplacian.
Let (M,g) be a compact Riemannian manifold and Pg an elliptic, formally self-adjoint, conformally covariant operator of order m acting on smooth sections of a bundle over M. We prove that if Pg has no rigid eigenspaces (see Definition 2.2), the set of functions f∈C∞(M,R) for which Pefg ha…
We use elementary methods to compute the L2-dimension of the eigenspaces of the Markov operator on the lamplighter group and of generalizations of this operator on other groups. In particular, we give a transparent explanation of the spectral measure of the Markov operator on the lamplighter group found by Grigorchuk-Z…
SGD's training dynamics align with Hessian and gradient spectra in high-dimensional classification tasks.
problem Understanding the spectra of Hessian and gradient matrices in high-dimensional classification tasks.
method Rigorous analysis of SGD dynamics and spectra of Hessian and gradient matrices.
result SGD trajectory and emergent outlier eigenspaces align with a common low-dimensional subspace in multi-class high-dimensional mixtures and neural networks.