arXiv research

We derive spectral sequences for the intersection homology of stratified fibrations and approximate tubular neighborhoods in manifold stratified spaces. These neighborhoods include regular neighborhoods in PL stratified spaces.

Making an adaptive prediction based on one's input is an important ability for general artificial intelligence. In this work, we step forward in this direction and propose a semi-parametric method, Meta-Neighborhoods, where predictions are made adaptively to the neighborhood of the input. We show that Meta-Neighborhood…

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…

Study geodesics entering a fixed cusp neighborhood multiple times.

Skeleta and other pure subsets of manifold stratified spaces are shown to have neighborhoods which are teardrops of stratified approximate fibrations (under dimension and compactness assumptions). In general, the stratified approximate fibrations cannot be replaced by bundles, and the teardrops cannot be replaced by ma…

New framework distinguishes knots via neighborhood invariants.

This paper tackles selection bias in recommender systems by considering the neighborhood effect.

Maximally hyperbolic solutions contain future neighborhoods of intersecting hypersurfaces.

Many prediction problems can be phrased as inferences over local neighborhoods of graphs. The graph represents the interaction between entities, and the neighborhood of each entity contains information that allows the inferences or predictions. We present an approach for applying machine learning directly to such graph…

Understanding intrinsic patterns and predicting spatiotemporal characteristics of cities require a comprehensive representation of urban neighborhoods. Existing works relied on either inter- or intra-region connectivities to generate neighborhood representations but failed to fully utilize the informative yet heterogen…

Study on transverse knots and their neighborhoods, proving unique standard neighborhoods and destabilization results.

Revises GNN neighborhood aggregation for more accurate node classification.

The study proves a neighborhood theorem for mean curvature flow in higher dimensions.

A neighborhood homotopy is an equivalence relation on spatial graphs which is generated by crossing changes on the same component and neighborhood equivalence. We give a complete classification of all 2-component spatial graphs up to neighborhood homotopy by the elementary divisor of a linking matrix with respect to th…

Graph representation learning is of paramount importance for a variety of graph analytical tasks, ranging from node classification to community detection. Recently, graph convolutional networks (GCNs) have been successfully applied for graph representation learning. These GCNs generate node representation by aggregatin…

Neighborhood sampling affects graph neural network training outcomes.

In this paper, we consider recommender systems with side information in the form of graphs. Existing collaborative filtering algorithms mainly utilize only immediate neighborhood information and have a hard time taking advantage of deeper neighborhoods beyond 1-2 hops. The main caveat of exploiting deeper graph informa…

Dimensionality reduction is considered as an important step for ensuring competitive performance in unsupervised learning such as anomaly detection. Non-negative matrix factorization (NMF) is a popular and widely used method to accomplish this goal. But NMF do not have the provision to include the neighborhood structur…

Learning graph-structured data with graph neural networks (GNNs) has been recently emerging as an important field because of its wide applicability in bioinformatics, chemoinformatics, social network analysis and data mining. Recent GNN algorithms are based on neural message passing, which enables GNNs to integrate loc…

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…

We present a new construction of tubular neighborhoods in (possibly infinite dimensional) Riemannian manifolds M, which allows us to show that if G is an arbitrary group acting isometrically on M, then every G-invariant submanifold with locally trivial normal bundle has a G-invariant total tubular neighborhood. We appl…

This article is one of three highly influential articles on the topology of manifolds written by Robert D. Edwards in the 1970's but never published. Organizers of the Workshops in Geometric Topology (http://www.uwm.edu/~craigg/workshopgtt.htm) with the support of the National Science Foundation have facilitated the pr…

TNC learns time series representations by leveraging temporal neighborhoods.

We consider discrete graphical models Markov with respect to a graph $G$ and propose two distributed marginal methods to estimate the maximum likelihood estimate of the canonical parameter of the model. Both methods are based on a relaxation of the marginal likelihood obtained by considering the density of the variable…

This paper sets a lower limit for the size of Weinstein's Lagrangian tubular neighborhoods.

Existing popular methods for semi-supervised learning with Graph Neural Networks (such as the Graph Convolutional Network) provably cannot learn a general class of neighborhood mixing relationships. To address this weakness, we propose a new model, MixHop, that can learn these relationships, including difference operat…

We develop a theory of tubular neighborhoods for the lower strata in manifold stratified spaces with two strata. In these topologically stratified spaces, manifold approximate fibrations and teardrops play the role that fibre bundles and mapping cylinders play in smoothly stratified spaces. Applications include the cla…

The $k$-NN graph has played a central role in increasingly popular data-driven techniques for various learning and vision tasks; yet, finding an efficient and effective way to construct $k$-NN graphs remains a challenge, especially for large-scale high-dimensional data. In this paper, we propose a new approach to const…

Motivated by an abstract notion of low-level edge detector filters, we propose a simple method of unsupervised feature construction based on pairwise statistics of features. In the first step, we construct neighborhoods of features by regrouping features that correlate. Then we use these subsets as filters to produce n…

Inspired by a work of Kapranov, we define the notion of Dolbeault complex of the formal neighborhood of a closed embedding of complex manifolds. This construction allows us to study coherent sheaves over the formal neighborhood via complex analytic approach, as in the case of usual complex manifolds and their Dolbeault…

Normal forms of almost complex structures in a neighborhood of pseudoholomorphic curve are considered. We define normal bundles of such curves and study the properties of linear bundle almost complex structures. We describe 1-jet of the almost complex structure along a curve in terms of its Nijenhuis tensor. For pseudo…

We characterize metric spaces $X$ whose hyperspaces $2^X$ or $Bd(X)$ of non-empty closed (bounded) subsets, endowed with the Hausdorff metric, are absolute [neighborhood] retracts.

Classifies neighborhoods around specific leaf structures.

Tubes in manifolds require wide spaces.

The local linear embedding algorithm (LLE) is a non-linear dimension-reducing technique, widely used due to its computational simplicity and intuitive approach. LLE first linearly reconstructs each input point from its nearest neighbors and then preserves these neighborhood relations in the low-dimensional embedding. W…

We show that given a point on a Finsler surface, one can always find a neighborhood of the point and isometrically embed this neighborhood into a Finsler torus without conjugate points.

Learn conditional averages in PAC framework for better predictions.

As an efficient and scalable graph neural network, GraphSAGE has enabled an inductive capability for inferring unseen nodes or graphs by aggregating subsampled local neighborhoods and by learning in a mini-batch gradient descent fashion. The neighborhood sampling used in GraphSAGE is effective in order to improve compu…

Study calculates reach and curvature of a specific geometric variety.

A multibranched surface is a 2-dimensional polyhedron without vertices. We introduce moves for multibranched surfaces embedded in a 3-manifold, which connect any two multibranched surfaces with the same regular neighborhoods in finitely many steps.

We discuss two generalizations of the collar lemma. The first is the stable neighborhood theorem which says that a (not necessarily simple) closed geodesic in a hyperbolic surface has a \lq\lq stable neighborhood\rq\rq whose width only depends on the length of the geodesic. As an application, we show that there is a lo…

The existence theorem for mapping cylinder neighborhoods is discussed as a prototypical example of controlled topology and its applications. The first of a projected series developed from lectures at the Summer School on High-Dimensional Topology, Trieste Italy 2001

New 1-parameter family of ovals identified in 4d Ricci flow classification.

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

We propose a dynamic neighborhood aggregation (DNA) procedure guided by (multi-head) attention for representation learning on graphs. In contrast to current graph neural networks which follow a simple neighborhood aggregation scheme, our DNA procedure allows for a selective and node-adaptive aggregation of neighboring …

Derives generalizations of the long neck principle and spectral width inequality.

In this purely expository note, we recall a few known properties of 3-dimensional singularity models. These properties are direct consequences of Perelman's canonical neighborhood theorem for 3-dimensional Ricci flow and compactness theorem for 3-dimensional kappa-solutions.

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.