Similarity-Aware Spectral Sparsification by Edge Filtering

In recent years, spectral graph sparsification techniques that can compute ultra-sparse graph proxies have been extensively studied for accelerating various numerical and graph-related applications. Prior nearly-linear-time spectral sparsification methods first extract low-stretch spanning tree from the original graph to form the backbone of the sparsifier, and then recover small portions of spectrally-critical off-tree edges to the spanning tree to significantly improve the approximation quality. However, it is not clear how many off-tree edges should be recovered for achieving a desired spectral similarity level within the sparsifier. Motivated by recent graph signal processing techniques, this paper proposes a similarity-aware spectral graph sparsification framework that leverages efficient spectral off-tree edge embedding and filtering schemes to construct spectral sparsifiers with guaranteed spectral similarity (relative condition number) level. An iterative graph densification scheme is introduced to facilitate efficient and effective filtering of off-tree edges for highly ill-conditioned problems. The proposed method has been validated using various kinds of graphs obtained from public domain sparse matrix collections relevant to VLSI CAD, finite element analysis, as well as social and data networks frequently studied in many machine learning and data mining applications

Topics in Graph Construction for Semi-Supervised Learning

Graph-based Semi-Supervised Learning (SSL) methods have had empirical success in a variety of domains, ranging from natural language processing to bioinformatics. Such methods consist of two phases. In the first phase, a graph is constructed from the available data; in the second phase labels are inferred for unlabeled nodes in the constructed graph. While many algorithms have been developed for label inference, thus far little attention has been paid to the crucial graph construction phase and only recently has the importance of the graph construction for the resulting success in label inference been recognized. In this report, we shall review some of the recently proposed graph construction methods for graph-based SSL. We shall also present suggestions for future research in this area

inGRASS: Incremental Graph Spectral Sparsification via Low-Resistance-Diameter Decomposition

This work presents inGRASS, a novel algorithm designed for incremental spectral sparsification of large undirected graphs. The proposed inGRASS algorithm is highly scalable and parallel-friendly, having a nearly-linear time complexity for the setup phase and the ability to update the spectral sparsifier in $O(\log N)$ time for each incremental change made to the original graph with $N$ nodes. A key component in the setup phase of inGRASS is a multilevel resistance embedding framework introduced for efficiently identifying spectrally-critical edges and effectively detecting redundant ones, which is achieved by decomposing the initial sparsifier into many node clusters with bounded effective-resistance diameters leveraging a low-resistance-diameter decomposition (LRD) scheme. The update phase of inGRASS exploits low-dimensional node embedding vectors for efficiently estimating the importance and uniqueness of each newly added edge. As demonstrated through extensive experiments, inGRASS achieves up to over $200 \times$ speedups while retaining comparable solution quality in incremental spectral sparsification of graphs obtained from various datasets, such as circuit simulations, finite element analysis, and social networks.Comment: Accepted on DAC 202

Kernel Multivariate Analysis Framework for Supervised Subspace Learning: A Tutorial on Linear and Kernel Multivariate Methods

Feature extraction and dimensionality reduction are important tasks in many fields of science dealing with signal processing and analysis. The relevance of these techniques is increasing as current sensory devices are developed with ever higher resolution, and problems involving multimodal data sources become more common. A plethora of feature extraction methods are available in the literature collectively grouped under the field of Multivariate Analysis (MVA). This paper provides a uniform treatment of several methods: Principal Component Analysis (PCA), Partial Least Squares (PLS), Canonical Correlation Analysis (CCA) and Orthonormalized PLS (OPLS), as well as their non-linear extensions derived by means of the theory of reproducing kernel Hilbert spaces. We also review their connections to other methods for classification and statistical dependence estimation, and introduce some recent developments to deal with the extreme cases of large-scale and low-sized problems. To illustrate the wide applicability of these methods in both classification and regression problems, we analyze their performance in a benchmark of publicly available data sets, and pay special attention to specific real applications involving audio processing for music genre prediction and hyperspectral satellite images for Earth and climate monitoring

Local selection of features and its applications to image search and annotation

In multimedia applications, direct representations of data objects typically involve hundreds or thousands of features. Given a query object, the similarity between the query object and a database object can be computed as the distance between their feature vectors. The neighborhood of the query object consists of those database objects that are close to the query object. The semantic quality of the neighborhood, which can be measured as the proportion of neighboring objects that share the same class label as the query object, is crucial for many applications, such as content-based image retrieval and automated image annotation. However, due to the existence of noisy or irrelevant features, errors introduced into similarity measurements are detrimental to the neighborhood quality of data objects. One way to alleviate the negative impact of noisy features is to use feature selection techniques in data preprocessing. From the original vector space, feature selection techniques select a subset of features, which can be used subsequently in supervised or unsupervised learning algorithms for better performance. However, their performance on improving the quality of data neighborhoods is rarely evaluated in the literature. In addition, most traditional feature selection techniques are global, in the sense that they compute a single set of features across the entire database. As a consequence, the possibility that the feature importance may vary across different data objects or classes of objects is neglected. To compute a better neighborhood structure for objects in high-dimensional feature spaces, this dissertation proposes several techniques for selecting features that are important to the local neighborhood of individual objects. These techniques are then applied to image applications such as content-based image retrieval and image label propagation. Firstly, an iterative K-NN graph construction method for image databases is proposed. A local variant of the Laplacian Score is designed for the selection of features for individual images. Noisy features are detected and sparsified iteratively from the original standardized feature vectors. This technique is incorporated into an approximate K-NN graph construction method so as to improve the semantic quality of the graph. Secondly, in a content-based image retrieval system, a generalized version of the Laplacian Score is used to compute different feature subspaces for images in the database. For online search, a query image is ranked in the feature spaces of database images. Those database images for which the query image is ranked highly are selected as the query results. Finally, a supervised method for the local selection of image features is proposed, for refining the similarity graph used in an image label propagation framework. By using only the selected features to compute the edges leading from labeled image nodes to unlabeled image nodes, better annotation accuracy can be achieved. Experimental results on several datasets are provided in this dissertation, to demonstrate the effectiveness of the proposed techniques for the local selection of features, and for the image applications under consideration

Graph Condensation for Graph Neural Networks

Given the prevalence of large-scale graphs in real-world applications, the storage and time for training neural models have raised increasing concerns. To alleviate the concerns, we propose and study the problem of graph condensation for graph neural networks (GNNs). Specifically, we aim to condense the large, original graph into a small, synthetic and highly-informative graph, such that GNNs trained on the small graph and large graph have comparable performance. We approach the condensation problem by imitating the GNN training trajectory on the original graph through the optimization of a gradient matching loss and design a strategy to condense node futures and structural information simultaneously. Extensive experiments have demonstrated the effectiveness of the proposed framework in condensing different graph datasets into informative smaller graphs. In particular, we are able to approximate the original test accuracy by 95.3% on Reddit, 99.8% on Flickr and 99.0% on Citeseer, while reducing their graph size by more than 99.9%, and the condensed graphs can be used to train various GNN architectures.Comment: 16 pages, 4 figure

Past, Present, and Future of Simultaneous Localization And Mapping: Towards the Robust-Perception Age

Simultaneous Localization and Mapping (SLAM)consists in the concurrent construction of a model of the environment (the map), and the estimation of the state of the robot moving within it. The SLAM community has made astonishing progress over the last 30 years, enabling large-scale real-world applications, and witnessing a steady transition of this technology to industry. We survey the current state of SLAM. We start by presenting what is now the de-facto standard formulation for SLAM. We then review related work, covering a broad set of topics including robustness and scalability in long-term mapping, metric and semantic representations for mapping, theoretical performance guarantees, active SLAM and exploration, and other new frontiers. This paper simultaneously serves as a position paper and tutorial to those who are users of SLAM. By looking at the published research with a critical eye, we delineate open challenges and new research issues, that still deserve careful scientific investigation. The paper also contains the authors' take on two questions that often animate discussions during robotics conferences: Do robots need SLAM? and Is SLAM solved