Urban2Vec combines street view imagery and POIs for better urban neighborhood embeddings.
problem Lack of comprehensive representation of urban neighborhoods using heterogeneous data.
method Unsupervised multi-modal framework using CNN for visual features and bag-of-words for POI data.
result Urban2Vec achieves better performance than baseline models and comparable to fully-supervised methods.
Tubes in manifolds require wide spaces.
problem Embedding constraints in Riemannian manifolds.
method Analyzing uniformly thick tubular neighborhoods.
result Conditions for manifold embeddings with wide tubes.
New framework distinguishes knots via neighborhood invariants.
problem Distinguishing knots and knotoids.
method Study of knotoid spectra and neighborhood invariants.
result Neighborhood invariants can distinguish knots of higher Gordian distance.
Shows Euler-like vector fields come from specific embeddings.
problem Understanding Euler-like vector fields and their origins.
method Using tubular neighborhood embeddings and normal exponential maps of Riemannian metrics.
result Each Euler-like vector field originates from a specific embedding.
Survey simplifies embedding theorems for manifolds.
problem Embedding manifolds into Euclidean space.
method Combining complement and neighborhood ideas to reduce embeddability to algebraic problems.
result Clarified exposition of Browder-Levine theorem on realization of normal systems.
In this paper, we propose a Tensor Train Neighborhood Preserving Embedding (TTNPE) to embed multi-dimensional tensor data into low dimensional tensor subspace. Novel approaches to solve the optimization problem in TTNPE are proposed. For this embedding, we evaluate novel trade-off gain among classification, computation…
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…
EHNA learns node embeddings from historical network neighborhoods.
problem Capturing temporal information in evolving networks.
method Temporal random walk and deep learning model with attention mechanism.
result EHNA outperforms existing methods in network reconstruction and link prediction tasks.
In this article, we demonstrate methods for the local removal and modification of complex tangents to embeddings of S3 into C3. In particular, given any embedding of S3 and a neighborhood of the complex tangents of the embedding, we show that there exists a (C0-close) totally real embedding which a…
New graph embedding method improves link prediction and node classification.
problem Improving graph embedding methods for better node representation.
method Spectral-biased random walks with neighborhood similarity bias.
result Significantly improves link prediction and node classification.
LPL optimizes embeddings to align local neighborhoods, improving cross-lingual word alignment.
problem Aligning embeddings across different datasets and languages.
method Locality Preserving Loss (LPL) optimizes model to project embeddings while maintaining local neighborhoods and aligning them.
result LPL-based alignment leads to better and consistent accuracy, especially in small training set settings.
We continue the study the Dolbeault dga of the formal neighborhood of an arbitary closed embedding of complex manifolds previously defined by the author in \cite{DolbeaultDGA}. The special case of the diagonal embedding has been studied in \cite{Diagonal}. We describe the Dolbeault dga explicitly in terms of the formal…
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.
The study proves a tube theorem for complex hyperbolic manifolds.
problem Understanding the geometry of complex hyperbolic manifolds.
method Tubular neighborhood theorem and geometric combination theorem.
result Explicit estimates and bounds for tube widths in complex hyperbolic manifolds.
Matching datasets of multiple modalities has become an important task in data analysis. Existing methods often rely on the embedding and transformation of each single modality without utilizing any correspondence information, which often results in sub-optimal matching performance. In this paper, we propose a nonlinear…
Graph embedding is a central problem in social network analysis and many other applications, aiming to learn the vector representation for each node. While most existing approaches need to specify the neighborhood and the dependence form to the neighborhood, which may significantly degrades the flexibility of represent…
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…
Tubular neighborhoods play an important role in differential topology. We have applied these constructions to geometry of almost Hermitian manifolds. At first, we consider deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold.Further, an almost hyperHermitian struc…
In this paper, we study the global geometry of complete, constant mean curvature hypersurfaces embedded in n-manifolds. More precisely, we give conditions that imply properness of such surfaces and prove the existence of fixed size one-sided regular neighborhoods for certain constant mean curvature hypersurfaces in cer…
We present network embedding algorithms that capture information about a node from the local distribution over node attributes around it, as observed over random walks following an approach similar to Skip-gram. Observations from neighborhoods of different sizes are either pooled (AE) or encoded distinctly in a multi-s…
In this paper we prove the existence of rational homology balls smoothly embedded in regular neighborhoods of certain linear chains of smooth 2-spheres by using techniques from minimal model program for 3-dimensional complex algebraic variety.
GCNs learn by embedding similar nodes within a class and leveraging consistent neighborhood structures.
problem Understanding how GCNs perform semi-supervised node classification on both homophilous and heterophilous graphs.
method Investigated the latent node embeddings and neighborhood structures of GCNs.
result GCNs learn by embedding similar nodes within a class and leveraging consistent neighborhood structures.
PGRec improves recommendation by modeling user-item preferences as a graph and embedding it for better predictions.
problem Sparse user-item data in recommender systems.
method PGRec models user-item preferences as a PrefGraph, then uses deep learning and factorization to embed and predict user preferences.
result PGRec outperforms state-of-the-art methods by up to 3.2% in NDCG@10.
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…
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 …
Proposes a method to improve graph embedding by removing least frequent nodes.
problem Capturing global graph structure in random walk-based embeddings.
method Extends random walk-based graph embedding by removing least frequent nodes.
result Improves predictive performance slightly, if at all.
Mathematical analysis of SNE and t-SNE for dimension reduction.
problem Optimal mapping of high-dimensional data to low dimensions.
method Gradient flow of relative entropy to minimize the distance between points.
result The diameter of the evolving sets remains bounded for SNE but may blow up for t-SNE.
A new GNN model SPIN achieves state-of-the-art performance on diverse real-world datasets.
problem Graph classification efficiency and accuracy.
method Parallel neighborhood aggregations (PA-GNNs) and SPIN model.
result SPIN model achieves state-of-the-art performance on diverse real-world datasets.
Study extends Nirenberg-Spencer's question to families of submanifolds.
problem Determine the germ of compact complex submanifolds in complex manifolds.
method Reformulate the question for families of submanifolds and their infinitesimal neighborhoods. Prove sufficient conditions for first-order neighborhoods and additional assumptions for submanifolds with nonzero vector fields.
result Affirmative answer to the reformulated question for certain submanifolds.
Proposes a novel node embedding framework for graphs using Fisher Information.
problem Lack of theoretical understanding of attention-based GNNs.
method Uses hierarchical kernels and Fisher Information to learn node embeddings.
result Proposed method outperforms existing GNNs on node classification benchmarks.
Linearizability of singular foliations is preserved under a specific equivalence relation.
problem Preserving properties of singular foliations under equivalence relations.
method Characterization of tubular neighborhood embeddings using Euler-like vector fields.
result Linearizability along a leaf is a Morita invariant.
New method constructs Stein surfaces using topological isotopy.
problem Creating Stein surfaces in complex surfaces.
method Combining Freedman's topology with Eliashberg's holomorphic theory.
result Every tame 2-complex can be isotoped to a Stein surface.
The recent proliferation of knowledge graphs (KGs) coupled with incomplete or partial information, in the form of missing relations (links) between entities, has fueled a lot of research on knowledge base completion (also known as relation prediction). Several recent works suggest that convolutional neural network (CNN…
This paper sets a lower limit for the size of Weinstein's Lagrangian tubular neighborhoods.
problem Finding the minimum size of Weinstein's Lagrangian tubular neighborhoods.
method Using the curvature tensor and second fundamental form of the submanifold.
result Explicit lower bounds for the radii of tubular neighborhoods are derived.
Survey of Locally Linear Embedding and its variants.
problem Representing high-dimensional data in a lower-dimensional space while preserving local structure.
method Explains various LLE and variant methods, including kernel LLE, inverse LLE, feature fusion, out-of-sample embedding, incremental LLE, landmark LLE, supervised LLE, robust LLE, fusion with other methods, and weighted LLE.
result Comprehensive overview of LLE and its variants.
New condition ensures submanifolds are skew in small areas.
problem Ensuring submanifolds are skew in Euclidean space.
method Introduces a third-order differential condition.
result Constructs improved totally skew embeddings for Rn. Let V be a regular neighborhood of a negative chain of 2-spheres (i.e. exceptional divisor of a cyclic quotient singularity), and let Bp,q be a rational homology ball which is smoothly embedded in V. Assume that the embedding is simple, i.e. the corresponding rational blow-up can be obtained by just a sequen…
Nodes residing in different parts of a graph can have similar structural roles within their local network topology. The identification of such roles provides key insight into the organization of networks and can be used for a variety of machine learning tasks. However, learning structural representations of nodes is a …
The first purpose of this note is to comment on a recent article of Bursztyn, Lima and Meinrenken, in which it is proved that if M is a smooth submanifold of a manifold V, then there is a bijection between germs of tubular neighborhoods of M and germs of "Euler-like" vector fields on V. We shall explain how to approach…
Let X be a complex manifold and S↪X be an embedding of complex submanifold. Assuming that the embedding is (k−1)-linearizable or (k−1)-comfortably embedded, we construct via the deformation to the normal cone a diffeomorphism F from a small neighborhood of the zero section in the normal bundle …
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…
In this paper, we show that a complete embedded minimal surface in $\Real^3$ with finite topology and one end is conformal to a once-punctured compact Riemann surface. Moreover, using the conformality and embeddedness, we examine the Weierstrass data and conclude that every such surface has Weierstrass data asymptotic …
For a given polyhedron K⊂M the notation RM(K) denotes a regular neighborhood of K in M. We study the following problem: find all pairs (m,k) such that if K is a compact k-polyhedron and M a PL m-manifold, then RM(fK)≅RM(gK), for each two homotopic PL embeddings f,g:K→M. We prove …
Paper equates torsions on wedge singularities.
problem Analyzing spaces with wedge singularities.
method Equating analytic and intersection Reidemeister torsions.
result Equation of torsions in specific wedge singularities.
Word embedding, which encodes words into vectors, is an important starting point in natural language processing and commonly used in many text-based machine learning tasks. However, in most current word embedding approaches, the similarity in embedding space is not optimized in the learning. In this paper we propose a …
A new method for efficient structural node embeddings using Von Neumann entropy.
problem Efficiently identifying structurally equivalent nodes in complex networks.
method VNEstruct: a simple approach generating low-dimensional structural node embeddings using Von Neumann entropy.
result VNEstruct achieves robustness on structural role identification and state-of-the-art performance on graph classification tasks.
Topic models are frequently used in machine learning owing to their high interpretability and modular structure. However, extending a topic model to include a supervisory signal, to incorporate pre-trained word embedding vectors and to include a nonlinear output function is not an easy task because one has to resort to…
Normal forms and isotropic embeddings via Euler-like vector fields.
problem Proving normal forms results for geometric structures.
method Construction of Euler-like vector fields compatible with geometric structures.
result Illustrated in various examples, including Morse-Bott, Weinstein, and Zung's theorems.