arXiv research

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

Analysis of DPPs and k-DPPs via spectral decomposition reveals identifiable parameters and non-identifiability gaps.

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 $Δ_{k+1}$.

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

The decomposition of the spinor bundle of the spin Grassmann manifolds $G_{m,n}=SO(m+n)/SO(m)\times SO(n)$ into irreducible representations of $\mathfrak{so}(m)\oplus\mathfrak{so}(n)$ is presented. A universal construction is developed and the general statement is proven for $G_{2k+1,3}$, $G_{2k,4}$, and $G_{2k+1,5}$ f…

New geometric framework for positive semidefinite matrices of fixed rank.

In this note, we present a new way to associate a spectral triple to the noncommutative $C^*$-algebra $C^*(Λ)$ of a strongly connected finite higher-rank graph $Λ$. We generalize a spectral triple of Consani and Marcolli from Cuntz-Krieger algebras to higher-rank graph $C^*$-algebras $C^*(Λ)$, and we prove that these s…

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

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…

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.

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

Study finds the spectrum of a cubic Dirac operator on specific oscillator group manifolds.

FedPower improves eigenspace estimation privacy in federated learning.

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…

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…

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…

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…

Laplacian mixture models identify overlapping regions of influence in unlabeled graph and network data in a scalable and computationally efficient way, yielding useful low-dimensional representations. By combining Laplacian eigenspace and finite mixture modeling methods, they provide probabilistic or fuzzy dimensionali…

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…

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

The paper introduces polarizations in symplectic and orthogonal settings.

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.

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

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…

We show how the discovery of robust scalable numerical solvers for arbitrary bounded linear operators can be automated as a Game Theory problem by reformulating the process of computing with partial information and limited resources as that of playing underlying hierarchies of adversarial information games. When the so…

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.

New method accelerates neural network training by focusing on flat directions.

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

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 …

New algorithm reduces communication in distributed eigenspace estimation.

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.

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

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

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.

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.

Let $(M,g)$ be a compact Riemannian manifold and $P_g$ an elliptic, formally self-adjoint, conformally covariant operator of order $m$ acting on smooth sections of a bundle over $M$. We prove that if $P_g$ has no rigid eigenspaces (see Definition 2.2), the set of functions $f \in C^\infty(M, R)$ for which $P_{e^fg}$ ha…