arXiv research

We study the asymptotic properties of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is in Gevrey class $G^a$ for some $a>1$, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of…

We prove a new off-diagonal asymptotic of the Bergman kernels associated to tensor powers of a positive line bundle on a compact Kähler manifold. We show that if the Kähler potential is real analytic, then the Bergman kernel accepts a complete asymptotic expansion in a neighborhood of the diagonal of shrinking size $k^…

Existing approaches to analyzing the asymptotics of graph Laplacians typically assume a well-behaved kernel function with smoothness assumptions. We remove the smoothness assumption and generalize the analysis of graph Laplacians to include previously unstudied graphs including kNN graphs. We also introduce a kernel-fr…

We present novel graph kernels for graphs with node and edge labels that have ordered neighborhoods, i.e. when neighbor nodes follow an order. Graphs with ordered neighborhoods are a natural data representation for evolving graphs where edges are created over time, which induces an order. Combining convolutional subgra…

Method estimates treatment effects in dyadic data with unknown confounders.

Develops a smooth operator framework for analyzing neural network representations.

This paper introduces a new and effective algorithm for learning kernels in a Multi-Task Learning (MTL) setting. Although, we consider a MTL scenario here, our approach can be easily applied to standard single task learning, as well. As shown by our empirical results, our algorithm consistently outperforms the traditio…

Data-driven neighborhood definitions and graph constructions are often used in machine learning and signal processing applications. k-nearest neighbor~(kNN) and $ε$-neighborhood methods are among the most common methods used for neighborhood selection, due to their computational simplicity. However, the choice of param…

We propose a new graph kernel for graph classification and comparison using Ollivier Ricci curvature. The Ricci curvature of an edge in a graph describes the connectivity in the local neighborhood. An edge in a densely connected neighborhood has positive curvature and an edge serving as a local bridge has negative curv…

Neighborhood sampling affects graph neural network training outcomes.

We give an alternate proof of the existence of the asymptotic expansion of the Bergman kernel associated to the $k$-th tensor powers of a positive line bundle $L$ in a $\frac{1}{\sqrt{k}}$-neighborhood of the diagonal using elementary methods. We use the observation that after rescaling the Kähler potential $k\varphi$ …

Unified view on random walk and Weisfeiler-Leman kernels, improving accuracy.

The study confirms a conjecture about critical points of smooth functions.

The minimal number of critical points is studied for smooth functions on closed manifolds.

Let $(M, g)$ be a real analytic Kaehler manifold. We say that a smooth map $E_p:W\to M$ from a neighborhood $W$ of the origin of $T_pM$ into $M$ is a {\em diastatic exponential} at $p$ if it satisfies $$(d \E_p)_0=\id_{T_pM},$$ $$D_p(\E_p (v))=g_p(v, v), \forall v\in W,$$ where $D_p$ is Calabi's diastasis function at $…

The estimation of probabilities of network edges from the observed adjacency matrix has important applications to predicting missing links and network denoising. It has usually been addressed by estimating the graphon, a function that determines the matrix of edge probabilities, but this is ill-defined without strong a…

Let $M$ be a compact oriented 3-dimensional smooth manifold. In this paper, we construct a moduli space consisting of pairs $(Σ, ψ)$ where $Σ$ is a $C^1$-embedding simple closed curve in $M$, $ψ$ is a $\mathbb{Z}/2$-harmonic spinor vanishing only on $Σ$, and $\|ψ\|_{L^2_1}\neq 0$. We prove that when $Σ$ is $C^2$, a nei…

Proposes a novel node embedding framework for graphs using Fisher Information.

We show that under very general assumptions the partial Bergman kernel function of sections vanishing along an analytic hypersurface has exponential decay in a neighborhood of the vanishing locus. Considering an ample line bundle, we obtain a uniform estimate of the Bergman kernel function associated to a singular metr…

Generators of the 4-sphere's smooth mapping class group via diffeomorphisms of Montesinos twins.

The paper studies asymptotic expansions of operators related to Bochner-Schrödinger on Riemannian manifolds.

We prove that a strictly stable constant-mean-curvature hypersurface in a smooth manifold of dimension less than or equal to 7 is uniquely homologically area minimizing for fixed volume in a small L^1 neighborhood.

Study on heat content for submanifolds in sub-Riemannian geometry.

The hyperbolization process affects the structure of manifolds.

MixCIT tests conditional independence for mixed data types efficiently and reliably.

Study shows harmful overfitting in Sobolev spaces even as training data grows.

This paper presents a proof of the existence of standard symplectic coordinates near a set of smooth, orthogonally intersecting symplectic submanifolds. It is a generalization of the standard symplectic neighborhood theorem. Moreover, in the presence of a compact Lie group $G$ acting symplectically, the coordinates can…

New Morse theory for shapes at distances.

We show that every negative definite configuration of symplectic surfaces in a symplectic 4--manifold has a strongly symplectically convex neighborhood. We use this to show that, if a negative definite configuration satisfies an additional negativity condition at each surface in the configuration, and if the complex si…

A new method for density estimation using nearest neighbor Dirichlet mixtures.

Constructing the adjacency graph is fundamental to graph-based clustering. Graph learning in kernel space has shown impressive performance on a number of benchmark data sets. However, its performance is largely determined by the chosen kernel matrix. To address this issue, the previous multiple kernel learning algorith…

HKConv learns hyperbolic features by aggregating kernel points.

Kernel smoothing on unknown manifolds with bounds and asymptotic normality.

In this article we construct a smooth Euler flow supported in a neighborhood of a helix. It may be considered a generalization of a similar solution found by the author for a circle.

New method for community detection in sparse directed SBMs with exact recovery guarantees.

Convolutional neural networks (CNNs) can be applied to graph similarity matching, in which case they are called graph CNNs. Graph CNNs are attracting increasing attention due to their effectiveness and efficiency. However, the existing convolution approaches focus only on regular data forms and require the transfer of …

New method improves RL in continuous spaces with kernel smoothing.

We provide a simple proof of a result of Rouby-Sjöstrand-Ngoc \cite{RSN} and Deleporte \cite{Deleporte}, which asserts that if the Kähler potential is real analytic then the Bergman kernel is an \textit{analytic kernel} meaning that its amplitude is an \textit{analytic symbol} and its phase is given by the polarization…

TNC learns time series representations by leveraging temporal neighborhoods.

This article deals with 2d almost Riemannian structures, which are generalized Riemannian structures on manifolds of dimension 2. Such sub-Riemannian structures can be locally defined by a pair of vector fields (X,Y), playing the role of orthonormal frame, that may become colinear on some subset. We denote D = span(X,Y…

Paper studies minimax optimal regression using Laplacian smoothing over graphs.

Most state-of-the-art graph kernels only take local graph properties into account, i.e., the kernel is computed with regard to properties of the neighborhood of vertices or other small substructures. On the other hand, kernels that do take global graph propertiesinto account may not scale well to large graph databases.…

Proposes NRS to find flat minima in deep neural networks.

Gradient descent in neural networks analyzed using RKBS for broader applicability.

This work incorporates the multi-modality of the data distribution into a Gaussian Process regression model. We approach the problem from a discriminative perspective by learning, jointly over the training data, the target space variance in the neighborhood of a certain sample through metric learning. We start by using…

Given a graph where every node has certain attributes associated with it and some nodes have labels associated with them, Collective Classification (CC) is the task of assigning labels to every unlabeled node using information from the node as well as its neighbors. It is often the case that a node is not only influenc…

Regular neighborhoods of singular submanifolds are isotopic to bundle morphisms.

Study the boundaries of ε-neighborhoods of planar sets, showing their structure and curvature.