Automatic Construction of Indoor 3D Navigation Graph from Crowdsourcing Trajectories

Lacking indoor navigation graph has become a bottleneck in indoor applications and services. This paper presents a novel automated indoor navigation graph reconstruction approach from large-scale low-frequency indoor trajectories without any other data sources. The proposed approach includes three steps: trajectory simplification, 2D floor plan extraction and 3D navigation graph construction. First, we propose a ST-Join-Clustering algorithm to identify and simplify redundant stay points embedded in the indoor trajectories. Second, an indoor trajectory bitmap construction based on a self-adaptive Gaussian filter is developed, and we then propose a new improved thinning algorithm to extract 2D indoor floor plans. Finally, we present an improved CFSFDP algorithm with time constraints to identify the 3D topological connection points between two different floors. To illustrate the applicability of the proposed approach, we conducted a real-world case study using an indoor trajectory dataset of over 4000 indoor trajectories and 5 million location points. The case study results showed that the proposed approach improves the navigation network accuracy by 1.83% and the topological accuracy by 13.7% compared to the classical kernel density estimation approach.

Graph Reconstruction from Unlabeled Edge Lengths

A d-dimensional framework is a pair (G, p), where $$G=(V,E)$$ G = ( V , E ) is a graph and p is a map from V to $$\mathbb {R}^d$$ R d . The length of an edge $$uv\in E$$ u v ∈ E in (G, p) is the distance between p(u) and p(v). The framework is said to be globally rigid in $$\mathbb {R}^d$$ R d if every other d-dimensional framework (G, q), in which the corresponding edge lengths are the same, is congruent to (G, p). In a recent paper Gortler, Theran, and Thurston proved that if every generic framework (G, p) in $$\mathbb {R}^d$$ R d is globally rigid for some graph G on $$n\ge d+2$$ n ≥ d + 2 vertices (where $$d\ge 2$$ d ≥ 2 ), then already the set of (unlabeled) edge lengths of a generic framework (G, p), together with n, determine the framework up to congruence. In this paper we investigate the corresponding unlabeled reconstruction problem in the case when the above generic global rigidity property does not hold for the graph. We provide families of graphs G for which the set of (unlabeled) edge lengths of any generic framework (G, p) in d-space, along with the number of vertices, uniquely determine the graph, up to isomorphism. We call these graphs weakly reconstructible. We also introduce the concept of strong reconstructibility; in this case the labeling of the edges is also determined by the set of edge lengths of any generic framework. For $$d=1,2$$ d = 1 , 2 we give a partial characterization of weak reconstructibility as well as a complete characterization of strong reconstructibility of graphs. In particular, in the low-dimensional cases we describe the family of weakly reconstructible graphs that are rigid but not redundantly rigid.

Multi-View Attribute Graph Convolution Networks for Clustering

Graph neural networks (GNNs) have made considerable achievements in processing graph-structured data. However, existing methods can not allocate learnable weights to different nodes in the neighborhood and lack of robustness on account of neglecting both node attributes and graph reconstruction. Moreover, most of multi-view GNNs mainly focus on the case of multiple graphs, while designing GNNs for solving graph-structured data of multi-view attributes is still under-explored. In this paper, we propose a novel Multi-View Attribute Graph Convolution Networks (MAGCN) model for the clustering task. MAGCN is designed with two-pathway encoders that map graph embedding features and learn the view-consistency information. Specifically, the first pathway develops multi-view attribute graph attention networks to reduce the noise/redundancy and learn the graph embedding features for each multi-view graph data. The second pathway develops consistent embedding encoders to capture the geometric relationship and probability distribution consistency among different views, which adaptively finds a consistent clustering embedding space for multi-view attributes. Experiments on three benchmark graph datasets show the superiority of our method compared with several state-of-the-art algorithms.

Reconstructing in the Constraint Satisfaction Problem

It is a long-standing problem in graph theory to prove or disprove the \emph{reconstruction conjecture}, also known as the Kelly-Ulam conjecture. This conjecture states that every simple graph on at least three vertices is \emph{reconstructible}, which means that the isomorphism class of such a graph is uniquely determined by the isomorphism classes of its vertex-deleted subgraphs. In this talk, the notion of reconstructing is extended from graphs to instances of the constraint satisfaction problem (CSP): an instance is \emph{reconstructible} if its isomorphism class is uniquely determined by the isomorphism classes of its constraint-deleted subinstances. Questions of interest include not only questions about reconstructible CSP instances but also about CSP instances with reconstructible properties and parameters such as the existence of solutions or the number of solutions. As shown in the talk, such questions can be answered using techniques borrowed and adapted from graph reconstruction. In particular, Lov\'{a}sz's method of counting graph homomorphisms \cite{Lov72} is adapted to characterize CSP instances for which the number of solutions is reconstructible.

GLEE: Geometric Laplacian Eigenmap Embedding

Graph embedding seeks to build a low-dimensional representation of a graph $G$. This low-dimensional representation is then used for various downstream tasks. One popular approach is Laplacian Eigenmaps (LE), which constructs a graph embedding based on the spectral properties of the Laplacian matrix of $G$. The intuition behind it, and many other embedding techniques, is that the embedding of a graph must respect node similarity: similar nodes must have embeddings that are close to one another. Here, we dispose of this distance-minimization assumption. Instead, we use the Laplacian matrix to find an embedding with geometric properties instead of spectral ones, by leveraging the so-called simplex geometry of $G$. We introduce a new approach, Geometric Laplacian Eigenmap Embedding, and demonstrate that it outperforms various other techniques (including LE) in the tasks of graph reconstruction and link prediction.