arXiv research

The paper improves boundary detection and density estimation on noisy data.

Heat kernel estimates on manifolds with mixed boundary conditions.

Support Vector Data Description (SVDD) is a machine-learning technique used for single class classification and outlier detection. SVDD formulation with kernel function provides a flexible boundary around data. The value of kernel function parameters affects the nature of the data boundary. For example, it is observed …

The paper compares heat kernels on manifolds with Robin boundary conditions.

The study examines Bergman kernels on complex manifolds with boundary and their asymptotic expansions.

Boundary effects inflate variance in Gaussian processes, leading to acquisition bias.

Study heat kernel on manifolds with fibred boundary metrics.

The boundary-value problem for Laplace-type operators acting on smooth sections of a vector bundle over a compact Riemannian manifold with generalized local boundary conditions including both normal and tangential derivatives is studied. The condition of strong ellipticity of this boundary-value problem is formulated. …

Kernel method learns PDEs from noisy data.

The Bergman-Szegő kernel is analyzed for weakly pseudoconvex CR manifolds of finite type.

Paper develops physics-informed, boundary-constrained Gaussian process for fluid flow field reconstruction.

Analyzes conformal anomaly in five dimensions, identifying new boundary conformal invariants.

Kernel smoothing on unknown manifolds with bounds and asymptotic normality.

In this article, we obtain a strict inequality between the conjugate Hardy $H^{2}$ kernels and the Bergman kernels on planar regular regions with $n>1$ boundary components, which is a conjecture of Saitoh.

Quantitative Sobolev extensions lead to Neumann heat kernel bounds.

Paper introduces a new Poisson kernel for strongly pseudoconvex domains.

New BdryMatérn GP model for reliable boundary integration on irregular domains.

Study on manifolds with kinks and Gaussian kernel behavior.

Derives variance kernel for reaction boundary in financial models.

Derives operational-time variance kernel for reaction boundaries in financial markets.

New method reveals corners of drum shapes.

Support Vector Data Description (SVDD) provides a useful approach to construct a description of multivariate data for single-class classification and outlier detection with various practical applications. Gaussian kernel used in SVDD formulation allows flexible data description defined by observations designated as sup…

In this talk, we review the heat kernel approach to the Atiyah-Singer index theorem for Dirac operators on closed manifolds, as well as the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary. We also discuss the odd dimensional counterparts of the above results. In particular, we describe…

On a large class of Riemannian manifolds with boundary, some dimension-free Harnack inequalities for the Neumann semigroup is proved to be equivalent to the convexity of the boundary and a curvature condition. In particular, for $p_t(x,y)$ the Neumann heat kernel w.r.t. a volume type measure $μ$ and for $K$ a constant,…

The study shows algebraic Bergman kernels imply finite type boundaries in complex domains.

We compute the leading and sub-leading terms in the asymptotic expansion of the Szegö kernel on the diagonal of a class of pseudoconvex Reinhardt domains whose boundaries are endowed with a general class of smooth measures. We do so by relating it to a Bergman kernel over projective space.

We consider the Hodge Laplacian on manifolds with incomplete edge singularities, with infinite dimensional von Neumann spaces and intricate elliptic boundary value theory. We single out a class of its algebraic self-adjoint extensions. Our microlocal heat kernel construction for algebraic boundary conditions is guided …

Novel boundary integral equations for Dirac operators in 3D Lipschitz domains.

In this paper, we prove that the L^2 Betti numbers of an amenable covering space can be approximated by the average Betti numbers of a regular exhaustion, under some hypotheses. We also prove that some L^2 spectral invariants can be approximated by the corresponding average spectral invariants of a regular exhaustion. …

Physics-informed neural networks improve by measuring effective dimensionality of constraints.

Prominently used in support vector machines and logistic regressions, kernel functions (kernels) can implicitly map data points into high dimensional spaces and make it easier to learn complex decision boundaries. In this work, by replacing the inner product function in the softmax layer, we explore the use of kernels …

Two-dimensional domains with Kähler-Einstein Bergman metrics are biholomorphic to the unit ball.

We study new heat kernel estimates for the Neumann heat kernel on a compact manifold with positive Ricci curvature and convex boundary. As a consequence, we obtain new lower bounds for the Neumann eigenvalues which are consistent with Weyl's asymptotics.

In [8] the authors introduced a pair of new de Rham complexes on a compact oriented Riemannian manifold with boundary by using a pair of new boundary conditions to discuss the refined analytic torsion on a compact manifold with boundary. In this paper we discuss the Lefschetz fixed point formula on these complexes with…

We use the Suita conjecture (now a theorem) to prove that for any domain $Ω\subset \mathbb{C}$ its Bergman kernel $K(\cdot, \cdot)$ satisfies $K(z_0, z_0) = \hbox{Volume}(Ω)^{-1}$ for some $z_0 \in Ω$ if and only if $Ω$ is either a disk minus a (possibly empty) closed polar set or $\mathbb{C}$ minus a (possibly empty) …

The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid gr…

Study reveals how to determine area and curvature from fluid flow resonances.

Characterizes kernel of linearization for minimal surfaces problem

Study Bergman kernel metrics on degenerating hyperelliptic surfaces.

We consider second-order elliptic partial differential operators acting on sections of vector bundles over a compact Riemannian manifold without boundary, working without the assumption of Laplace-like principal part $-\N^μ\N_μ$. Our objective is to obtain information on the asymptotic expansions of the corresponding r…

We implement an all-optical setup demonstrating kernel-based quantum machine learning for two-dimensional classification problems. In this hybrid approach, kernel evaluations are outsourced to projective measurements on suitably designed quantum states encoding the training data, while the model training is processed o…

A major challenge in Bayesian Optimization is the boundary issue (Swersky, 2017) where an algorithm spends too many evaluations near the boundary of its search space. In this paper, we propose BOCK, Bayesian Optimization with Cylindrical Kernels, whose basic idea is to transform the ball geometry of the search space us…

Suppose M is a compact manifold with boundary. Let N be a normal covering of M. Suppose (A,T) is an elliptic differential boundary value problem on M with lift (\tilde A,\tilde T) to N. Then the von Neumann dimension of kernel and cokernel of this lift are defined. The main result of this paper is: these numbers are fi…

Johnson kernel generated by specific Dehn twists on surfaces.

Clustering is one of the most important unsupervised problems in machine learning and statistics. Among many existing algorithms, kernel k-means has drawn much research attention due to its ability to find non-linear cluster boundaries and its inherent simplicity. There are two main approaches for kernel k-means: SVD o…

Wave equation map reveals manifold's structure.

Paper proves Torelli group's finiteness for surfaces with 2 boundaries.

Geometric theory connects machine learning classifiers to differential geometry.