arXiv research

This paper tackles distributed estimation of the top-L eigenspace in PCA for large data sets.

New algorithm estimates eigenspace with faulty nodes, matching performance of existing methods.

FedPower improves eigenspace estimation privacy in federated learning.

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…

TransNet improves community detection on target networks using privacy-preserved source networks.

The paper calibrates shrinkage covariance estimators for spectral functionals in high dimensions.

New algorithm reduces communication in distributed eigenspace estimation.

We investigate numerically efficient approximations of eigenspaces associated to symmetric and general matrices. The eigenspaces are factored into a fixed number of fundamental components that can be efficiently manipulated (we consider extended orthogonal Givens or scaling and shear transformations). The number of the…

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…

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 $Σ_{g,n}$. Eigenspaces of t…

Our work connects parameter magnitudes and Hessian eigenspaces in deep neural nets.

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…

A method to analyze neural network performance by measuring layer saturation.

Physics-informed GP regression solves eigenvalue problems by identifying non-trivial eigenspaces.

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…

The $2^{nd}$ variation formula of the Seiberg-Witten functional is obtained in order to estimate the Morse index of redutible solutions $(A,0)$. It is shown that their Morse index is given by the dimension of the largest negative eigenspace of the operator $\triangle_{A} +\frac{k_{g}}{4}$, hence it is finite.

Paper identifies key function spaces for ReLU networks based on Fisher information.

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.

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 $Δ_g$. We establish algebraic criteria for the existence of left invariant metrics $g$ on $G$ such that each eigenspace of $Δ_g$, 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…

Smoothing splines provide a powerful and flexible means for nonparametric estimation and inference. With a cubic time complexity, fitting smoothing spline models to large data is computationally prohibitive. In this paper, we use the theoretical optimal eigenspace to derive a low rank approximation of the smoothing spl…

New geometric framework for positive semidefinite matrices of fixed rank.

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.

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…

Essential self-adjointness and spectrum of CR GJMS operator proved.

A theory of feature geometry using spectral analysis of weight matrices.

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…

The paper characterizes solitons and estimates scalar curvature.

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.

Study spectral properties of graph Laplacian for manifold data.

Proposes BONMI for integrating noisy matrices from multi-source data.

SU(3) instanton homology counts Tait colorings for webs and foams.

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.

Researchers create surfaces with exceptionally high Steklov eigenvalues.

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 $\mathbb{R}^n$.

New principle reveals how neural networks learn complex interactions.

New eigenvalue bounds for 3-Sasaki metrics improve previous estimates.

We show that the eigenspaces of the Laplacian $Δ_k$ 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 $Δ_k$ lies in the spectrum of $Δ_{k+1}$.

This work analyzes the role of data augmentation in self-supervised learning using RKHS approximation and regression.

Adapts EGOP to multi-class setting and proposes a simple rough estimator.

This paper uncovers the low-rank structure of neural network Hessians.

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.

Study Witten deformation on noncompact manifolds with bounded geometry.