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.
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…
Spectral clustering for geometric graphs achieves strong consistency in community recovery.
problem Community recovery in dense geometric graphs.
method Spectral clustering algorithm using eigenvectors of adjacency matrix.
result Strong consistency in community recovery proved.
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.
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.
Study of η invariants on lens spaces detects distinctions invisible to ordinary η.
problem Detecting distinctions in η invariants on lens spaces. method Spin-Fourier residues and equivariant η invariants. result Second derivative of the residual η germ is nonzero for some lens spaces. 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.
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…
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.
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.
Constructing an efficient parameterization of a large, noisy data set of points lying close to a smooth manifold in high dimension remains a fundamental problem. One approach consists in recovering a local parameterization using the local tangent plane. Principal component analysis (PCA) is often the tool of choice, as…
Minimal Morse functions on Poincaré dodecahedral space are selected via spectral properties.
problem Identifying minimal Morse functions on the Poincaré dodecahedral space.
method Spectral selection property P, obstruction principle, conformal variations, finite dimensional reduction.
result Restoration of minimal Morse selection on the Poincaré dodecahedral space via spectral mechanisms.
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.
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.
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.
New formulas for flag manifolds simplify eigenvector perturbation.
problem Eigenvector perturbation problem
method Explicit formulas derived from normal homogeneous spaces
result Effective formulas for flag manifolds simplify eigenvector perturbation
Compressing word embeddings is important for deploying NLP models in memory-constrained settings. However, understanding what makes compressed embeddings perform well on downstream tasks is challenging---existing measures of compression quality often fail to distinguish between embeddings that perform well and those th…
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.
For a compact homogeneous space G/K, we study the problem of existence of G-invariant Riemannian metrics such that each eigenspace of the Laplacian is a real irreducible representation of G. We prove that the normal metric of a compact irreducible symmetric space has this property only in rank one. Furthermore, w…
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…
In this paper, we present an online adaptive PCA algorithm that is able to compute the full dimensional eigenspace per new time-step of sequential data. The algorithm is based on a one-step update rule that considers all second order correlations between previous samples and the new time-step. Our algorithm has O(n) co…
We study the accuracy of estimating the covariance and the precision matrix of a D-variate sub-Gaussian distribution along a prescribed subspace or direction using the finite sample covariance. Our results show that the estimation accuracy depends almost exclusively on the components of the distribution that correspo…
Spectral methods simplify data analysis, improving accuracy and stability.
problem Extracting meaningful information from noisy, incomplete data.
method Eigenvalues and eigenvectors of matrices constructed from data.
result Spectral methods are effective and can be analyzed using modern statistical theory.
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…
This paper presents a novel time series clustering method, the self-organising eigenspace map (SOEM), based on a generalisation of the well-known self-organising feature map (SOFM). The SOEM operates on the eigenspaces of the embedded covariance structures of time series which are related directly to modes in those tim…
We focus in this work on the estimation of the first k eigenvectors of any graph Laplacian using filtering of Gaussian random signals. We prove that we only need k such signals to be able to exactly recover as many of the smallest eigenvectors, regardless of the number of nodes in the graph. In addition, we address…
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.
Principal components analysis (PCA) is a widely used dimension reduction technique with an extensive range of applications. In this paper, an online distributed algorithm is proposed for recovering the principal eigenspaces. We further establish its rate of convergence and show how it relates to the number of nodes emp…
A method to remove mean-shift noise from PCA using knockoffs.
problem High sensitivity of PCA to mean-shift contamination in high-dimensional data.
method Introducing knockoff mean-shift perturbation to separate and remove mean-shift components from PCA.
result The mean-shift spikes are spectrally separable from stable eigenvalues, allowing for robust PCA.
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…
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. A new GNN architecture called coVariance neural network (VNN) improves stability and transferability of covariance matrix analysis.
problem Stability and transferability issues in covariance matrix analysis.
method Developed coVariance neural network (VNN) that operates on sample covariance matrices.
result VNN is more stable and transferable than PCA-based approaches.
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.
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.
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 …
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%.
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.
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.
In this undergraduate thesis, we present an analytical proof of the Morse inequalities for closed smooth n-manifolds following Witten's approach. Using techniques from PDE theory, the proof is reduced to study the eigenspaces and eigenvalues of harmonic oscillators on Rn.
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.
We show that the eigenspaces of the Laplacian Δk on k-forms on a compact Kähler manifold carry Hodge and Lefschetz decompositions. Among other consequences, we show that the positive part of the spectrum of Δk lies in the spectrum of Δk+1.
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.