Connection Laplacian Research Articles

Spectral convergence under bounded Ricci curvature

For a noncollapsed Gromov–Hausdorff convergent sequence of Riemannian manifolds with a uniform bound of Ricci curvature, we establish two spectral convergence. One of them is on the Hodge Laplacian acting on differential one-forms. The other is on the connection Laplacian acting on tensor fields of every type, which include all differential forms. These are sharp generalizations of Cheeger–Colding's spectral convergence of the Laplacian acting on functions to the cases of tensor fields and differential forms. These spectral convergence have two direct corollaries. One of them is to give new bounds on such eigenvalues, in terms of bounds on volume, diameter and the Ricci curvature. The other is that we show the upper semicontinuity of the first Betti numbers with respect to the Gromov–Hausdorff topology, and give the equivalence between the continuity of them and the existence of a uniform spectral gap. On the other hand we also define measurable curvature tensors of the noncollapsed Gromov–Hausdorff limit space of a sequence of Riemannian manifolds with a uniform bound of Ricci curvature, which include Riemannian curvature tensor, the Ricci curvature, and the scalar curvature. As fundamental properties of our Ricci curvature, we show that the Ricci curvature coincides with the difference between the Hodge Laplacian and the connection Laplacian, and is compatible with Gigli's one and Lott's Ricci measure. Moreover we prove a lower bound of the Ricci curvature is compatible with a reduced Riemannian curvature dimension condition. We also give a positive answer to Lott's question on the behavior of the scalar curvature with respect to the Gromov–Hausdorff topology by using our scalar curvature.

Spectral convergence of the connection Laplacian from random samples

Spectral methods that are based on eigenvectors and eigenvalues of discrete graph Laplacians, such as Diffusion Maps and Laplacian Eigenmaps are often used for manifold learning and non-linear dimensionality reduction. It was previously shown by Belkin and Niyogi \cite{belkin_niyogi:2007} that the eigenvectors and eigenvalues of the graph Laplacian converge to the eigenfunctions and eigenvalues of the Laplace-Beltrami operator of the manifold in the limit of infinitely many data points sampled independently from the uniform distribution over the manifold. Recently, we introduced Vector Diffusion Maps and showed that the connection Laplacian of the tangent bundle of the manifold can be approximated from random samples. In this paper, we present a unified framework for approximating other connection Laplacians over the manifold by considering its principle bundle structure. We prove that the eigenvectors and eigenvalues of these Laplacians converge in the limit of infinitely many independent random samples. We generalize the spectral convergence results to the case where the data points are sampled from a non-uniform distribution, and for manifolds with and without boundary.

The Calderón problem for connections

In this paper we consider the problem of identifying a connection $\nabla$ on a vector bundle up to gauge equivalence from the Dirichlet-to-Neumann map of the connection Laplacian $\nabla^*\nabla$ over conformally transversally anisotropic (CTA) manifolds. This was proved in \cite{LCW} for line bundles in the case of the transversal manifold being simple -- we generalise this result to the case where the transversal manifold only has an injective ray transform. Moreover, the construction of suitable Gaussian beam solutions on vector bundles is given for the case of the connection Laplacian and a potential, following the works of \cite{CTA}. This in turn enables us to construct the Complex Geometrical Optics (CGO) solutions and prove our main uniqueness result. We also reduce the problem to a new non-abelian X-ray transform for the case of simple transversal manifolds and higher rank vector bundles. Finally, we prove the recovery of a flat connection in general from the DN map, up to gauge equivalence, using an argument relating the Cauchy data of the connection Laplacian and the holonomy.

Embedding Riemannian manifolds by the heat kernel of the connection Laplacian

Given a class of closed Riemannian manifolds with prescribed geometric conditions, we introduce an embedding of the manifolds into ℓ2 based on the heat kernel of the Connection Laplacian associated with the Levi-Civita connection on the tangent bundle. As a result, we can construct a distance in this class which leads to a pre-compactness theorem on the class under consideration.

Graph connection Laplacian methods can be made robust to noise

Recently, several data analytic techniques based on graph connection Laplacian (GCL) ideas have appeared in the literature. At this point, the properties of these methods are starting to be understood in the setting where the data is observed without noise. We study the impact of additive noise on these methods and show that they are remarkably robust. As a by-product of our analysis, we propose modifications of the standard algorithms that increase their robustness to noise. We illustrate our results in numerical simulations.

Alternating projection, ptychographic imaging and phase synchronization

We demonstrate necessary and sufficient conditions of the local convergence of the alternating projection algorithm to a unique solution up to a global phase factor. Additionally, for the ptychography imaging problem, we discuss phase synchronization and graph connection Laplacian, and show how to construct an accurate initial guess to accelerate convergence speed to handle the big imaging data in the coming new light source era.

Graph connection Laplacian and random matrices with random blocks

Graph connection Laplacian (GCL) is a modern data analysis technique that is starting to be applied for the analysis of high-dimensional and massive datasets. Motivated by this technique, we study matrices that are akin to the ones appearing in the null case of GCL, i.e. the case where there is no structure in the dataset under investigation. Developing this understanding is important in making sense of the output of the algorithms based on GCL. We hence develop a theory explaining the behavior of the spectral distribution of a large class of random matrices, in particular random matrices with random block entries of fixed size. Part of the theory covers the case where there is significant dependence between the blocks. Numerical work shows that the agreement between our theoretical predictions and numerical simulations is generally very good.

Geometric Arbitrage and Spectral Theory

Geometric Arbitrage Theory reformulates a generic asset model possibly allowing for arbitrage by packaging all assets and their forwards dynamics into a stochastic principal fibre bundle, with a connection whose parallel transport encodes discounting and portfolio rebalancing, and whose curvature measures, in this geometric language, the ''instantaneous arbitrage capability'' generated by the market itself. The cashflow bundle is the vector bundle associated to this stochastic principal fibre bundle for the natural choice of the vector space fibre. The cashflow bundle carries a stochastic covariant differentiation induced by the connection on the principal fibre bundle. The link between arbitrage theory and spectral theory of the connection Laplacian on the vector bundle is given by the zero eigenspace resulting in a parametrization of all risk neutral measures equivalent to the statistical one. This indicates that a market satisfies the no-free-lunch-with vanishing-risk condition if it is only if 0 is in the spectrum.

Inverse Boundary Problems for Systems in Two Dimensions

We prove identification of coefficients up to gauge equivalence by Cauchy data at the boundary for elliptic systems on oriented compact surfaces with boundary or domains of \({\mathbb{C}}\) . In the geometric setting, we fix a Riemann surface with boundary and consider both a Dirac-type operator plus potential acting on sections of a Clifford module and a connection Laplacian plus potential (i.e. Schrodinger Laplacian with external Yang–Mills field) acting on sections of a Hermitian bundle. In either case we show that the Cauchy data determine both the connection and the potential up to a natural gauge transformation: conjugation by an endomorphism of the bundle which is the identity at the boundary. For domains of \({\mathbb{C}}\) , we recover zeroth order terms up to gauge from Cauchy data at the boundary in first order elliptic systems.

A Cheeger Inequality for the Graph Connection Laplacian

The $O(d)$ synchronization problem consists of estimating a set of $n$ unknown orthogonal $d\times d$ matrices $O_1,\ldots,O_n$ from noisy measurements of a subset of the pairwise ratios $O_iO_j^{-1}$. We formulate and prove a Cheeger-type inequality that relates a measure of how well it is possible to solve the $O(d)$ synchronization problem with the spectra of an operator, the graph connection Laplacian. We also show how this inequality provides a worst-case performance guarantee for a spectral method to solve this problem.

Identification of a Connection from Cauchy Data on a Riemann Surface with Boundary

We consider a connection $${\nabla^X}$$ on a complex line bundle over a Riemann surface with boundary M 0, with connection 1-form X. We show that the Cauchy data space of the connection Laplacian (also called magnetic Laplacian) $${L := \nabla^X{^*\nabla^X} + q}$$ , with q a complex-valued potential, uniquely determines the connection up to gauge isomorphism, and the potential q.

Discrete connection Laplacians

SVETOSLAV ZAHARIEVAbstract. To every Hermitian vector bundle with connection over a compactRiemannian manifold M one can associate a corresponding connection Lapla-cian acting on the sections of the bundle. We define analogous combinatorial,metric dependent Laplacians associated to triangulations of M and prove thattheir spectra converge, as the mesh of the triangulations approaches zero, tothe spectrum of the connection Laplacian.

Spin geometry on quantum groups via covariant differential calculi

Let A be a cosemisimple Hopf ∗-algebra with antipode S and let Γ be a left-covariant first-order differential ∗-calculus over A such that Γ is self-dual (see Section 2) and invariant under the Hopf algebra automorphism S 2. A quantum Clifford algebra Cl( Γ, σ, g) is introduced which acts on Woronowicz’ external algebra Γ ∧. A minimal left ideal of Cl( Γ, σ, g) which is an A -bimodule is called a spinor module. Metrics on spinor modules are investigated. The usual notion of a linear left connection on Γ is extended to quantum Clifford algebras and also to spinor modules. The corresponding Dirac operator and connection Laplacian are defined. For the quantum group SL q (2) and its bicovariant 4 D ±-calculi these concepts are studied in detail. A generalization of Bochner's theorem is given. All invariant differential operators over a given spinor module are determined. The eigenvalues of the Dirac operator are computed.