arXiv research

New method detects global factors near BBP phase transition in high-dimensional data.

We define extensions of the $L^2$-analytic invariants of closed manifolds, called delocalized $L^2$-invariants. These delocalized invariants are constructed in terms of a nontrivial conjugacy class of the fundamental group. We show that in many cases, they are topological in nature. We show that the marked length spect…

In this paper we state and prove Morse type inequalities for Morse functions as well as for closed differential 1-forms. These inequalities involve delocalized Betti numbers. As an immediate consequence, we prove the vanishing of delocalized Betti numbers of manifolds fibering over the circle.

Study delocalized eta invariants for signature operators on proper manifolds.

The paper defines higher invariants for groups of polynomial growth and proves their convergence.

Study approximates delocalized eta invariants of covering spaces.

The unadjusted Langevin algorithm converges faster for some variables in high dimensions.

New method controls bias in unadjusted Hamiltonian Monte Carlo and underdamped Langevin.

A new model explains protein interactions via electron delocalization.

The paper proves universality in optimization problems with i.i.d. random vectors.

Non-Abelian actions are resolved using equivariant K-theory and delocalized cohomology.

Improved sampling from complex distributions with reduced bias.

For any closed complex manifold $X$, we calculate the Poincaré and Hodge polynomials of the delocalized equivariant cohomology $H^*(X^n, S_n)$ with a grading specified by physicists. As a consequence, we recover a special case of a formula for the elliptic genera of symmetric products in Dijkgraaf-Moore-Verlinde-Verlin…

Let G be a finitely generated discrete group. In this paper we establish vanishing results for rho-invariants associated to (i) the spin-Dirac operator of a spin manifold with positive scalar curvature (ii) the signature operator of the disjoint union of a pair of homotopy equivalent oriented manifolds with fundamental…

A refined form of the `Folk Theorem' that a smooth action by a compact Lie group can be (canonically) resolved, by iterated blow up, to have unique isotropy type was established by the authors in the context of manifolds with corners; the canonical construction induces fibrations on the boundary faces of the resolution…

We give a description of the delocalized twisted cohomology of an orbifold and the Chern character of a twisted vector bundle in terms of supersymmetric Euclidean field theories. This includes the construction of a twist functor for $1|1$-dimensional EFTs from the data of a gerbe with connection.

Study eigenvector overlaps in large Gaussian matrices, simplifying for GOE.

Machine learning models perform better with location coordinates alone, not Moran Eigenvectors.

Paper addresses eigenvector perturbation in small eigen-gap scenarios.

We calculate eigenvector overlaps between intersecting time periods of covariance matrices.

In many applications, one has side information, e.g., labels that are provided in a semi-supervised manner, about a specific target region of a large data set, and one wants to perform machine learning and data analysis tasks "nearby" that prespecified target region. For example, one might be interested in the clusteri…

In spectral clustering, one defines a similarity matrix for a collection of data points, transforms the matrix to get the Laplacian matrix, finds the eigenvectors of the Laplacian matrix, and obtains a partition of the data using the leading eigenvectors. The last step is sometimes referred to as rounding, where one ne…

We study the problem asking if one can embed manifolds into finite dimensional Euclidean spaces by taking finite number of eigenvector fields of the connection Laplacian. This problem is essential for the dimension reduction problem in massive data analysis. Singer-Wu proposed the vector diffusion map which embeds mani…

New metric tensor field on symmetric matrices simplifies eigenvector computation.

Spectral clustering performance depends on eigenvector fluctuations, shown to be Gaussian.

New neural architectures invariant to sign flips and basis symmetries for graph representation learning.

New algorithm updates eigenvectors of evolving graphs efficiently.

Defines rho numbers for metrics with positive scalar curvature.

How many samples are sufficient to guarantee that the eigenvectors and eigenvalues of the sample covariance matrix are close to those of the actual covariance matrix? For a wide family of distributions, including distributions with finite second moment and distributions supported in a centered Euclidean ball, we prove …

New algorithm consistently orients eigenvectors for machine learning.

New theory for eigenvectors of generalized Laplacian matrices, addressing dependency issues.

Fast algorithm recovers principal eigenvector from noisy matrices.

This paper develops the exact linear relationship between the leading eigenvector of the unnormalized modularity matrix and the eigenvectors of the adjacency matrix. We propose a method for approximating the leading eigenvector of the modularity matrix, and we derive the error of the approximation. There is also a comp…

New method improves subspace iteration for eigenvectors in machine learning.

The paper explores how kernel eigenalignments affect generalization in KRR.

The problem of estimating sparse eigenvectors of a symmetric matrix attracts a lot of attention in many applications, especially those with high dimensional data set. While classical eigenvectors can be obtained as the solution of a maximization problem, existing approaches formulate this problem by adding a penalty te…

Improved spectral clustering with fewer eigenvectors performs better.

Paper tackles small eigen-gap estimation and inference for noisy symmetric matrices.

A multi-scale model predicts atomic-scale properties using both local and long-range information.

New insights into spectral clustering reveal strong connections within eigenvectors.

The original contributions of this paper are twofold: a new understanding of the influence of noise on the eigenvectors of the graph Laplacian of a set of image patches, and an algorithm to estimate a denoised set of patches from a noisy image. The algorithm relies on the following two observations: (1) the low-index e…

We characterize the contractions that are similar to the backward shift in the Hardy space $H^2$. This characterization is given in terms of the geometry of the eigenvector bundles of the operators.

Graph convolutional networks fail to use eigenvectors beyond the first, unlike spectral embedding.

We provide new examples of diffusion operators in dimension 2 and 3 which have orthogonal polynomials as eigenvectors. Their construction rely on the finite subgroups of O(3) and their invariant polynomials.

Study eigenvalues and eigenvectors in neural networks, focusing on signal propagation.

The paper tackles learning symmetries in data without expert knowledge.

Let G be a discrete group, and let M be a closed spin manifold of dimension m>3 with pi_1(M)=G. We assume that M admits a Riemannian metric of positive scalar curvature. We discuss how to use the L2-rho invariant and the delocalized eta invariant associated to the Dirac operator on M in order to get information about t…

New method improves covariance estimation for weighted samples.