Given a measurement graph G = (V, E) and an unknown signal r ε ℝn, we investigate algorithms for recovering r from pairwise measurements of the form ri — ℝrj; {i, j} ε E. This problem arises in a variety of applica...
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Given a measurement graph G = (V, E) and an unknown signal r ε ℝn, we investigate algorithms for recovering r from pairwise measurements of the form ri — ℝrj; {i, j} ε E. This problem arises in a variety of applications, such as ranking teams in sports data and time synchronization of distributed networks. Framed in the context of ranking, the task is to recover the ranking of n teams (induced by r) given a small subset of noisy pairwise rank offsets. We propose a simple SVD-based algorithmic pipeline for both the problem of time synchronization and ranking. We provide a detailed theoretical analysis in terms of robustness against both sampling sparsity and noise perturbations with outliers, using results from matrix perturbation and random matrix theory. Our theoretical findings are complemented by a detailed set of numerical experiments on both synthetic and real data, showcasing the competitiveness of our proposed algorithms with other state-of-the-art methods.
The problem of partitioning a graph such that the number of edges incident to vertices in different partitions is minimized, arises in many contexts. Some examples include its recursive application for minimizing fill...
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The problem of partitioning a graph such that the number of edges incident to vertices in different partitions is minimized, arises in many contexts. Some examples include its recursive application for minimizing fill-in in matrix factorizations and load-balancing for parallel algorithms. spectral graph partitioning algorithms partition a graph using the eigenvector associated with the second smallest eigenvalue of a matrix called the graph Laplacian . The focus of this paper is the use graph theory to compute this eigenvector more quickly.
In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements ...
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In synchronization problems, the goal is to estimate elements of a group from noisy measurements of their ratios. A popular estimation method for synchronization is the spectral method. It extracts the group elements from eigenvectors of a block matrix formed from the measurements. The eigenvectors must be projected, or 'rounded', onto the group. The rounding procedures are constructed ad hoc and increasingly so when applied to synchronization problems over non-compact groups. In this paper, we develop a spectral approach to synchronization over the non-compact group $\mathrm{SE}(3)$, the group of rigid motions of $\mathbb{R}<^>{3}$. We based our method on embedding $\mathrm{SE}(3)$ into the algebra of dual quaternions, which has deep algebraic connections with the group $\mathrm{SE}(3)$. These connections suggest a natural rounding procedure considerably more straightforward than the current state of the art for spectral $\mathrm{SE}(3)$ synchronization, which uses a matrix embedding of $\mathrm{SE}(3)$. We show by numerical experiments that our approach yields comparable results with the current state of the art in $\mathrm{SE}(3)$ synchronization via the spectral method. Thus, our approach reaps the benefits of the dual quaternion embedding of $\mathrm{SE}(3)$ while yielding estimators of similar quality.
Subexponential time approximation algorithms are presented for the UNIQUE GAMES and SMALL-SET EXPANSION problems. Specifically, for some absolute constant c, the following two algorithms are presented. (1) An exp(kn(e...
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Subexponential time approximation algorithms are presented for the UNIQUE GAMES and SMALL-SET EXPANSION problems. Specifically, for some absolute constant c, the following two algorithms are presented. (1) An exp(kn(epsilon))-time algorithm that, given as input a k-alphabet unique game on n variables that has an assignment satisfying 1 - epsilon(c) fraction of its constraints, outputs an assignment satisfying 1 - epsilon fraction of the constraints. (2) An exp(n(epsilon)/delta)-time algorithm that, given as input an n-vertex regular graph that has a set S of delta n vertices with edge expansion at most epsilon(c), outputs a set S' of at most delta n vertices with edge expansion at most epsilon. Subexponential algorithms are also presented with improved approximations to MAX CUT, SPARSEST CUT, and VERTEX COVER on some interesting subclasses of instances. These instances are graphs with low threshold rank, an interesting new graph parameter highlighted by this work. Khot's Unique Games Conjecture (UGC) states that it is NP-hard to achieve approximation guarantees such as ours for UNIQUE GAMES. While the results here stop short of refuting the UGC, they do suggest that UNIQUE GAMES are significantly easier than NP-hard problems such as MAX 3-SAT, MAX 3-LIN, LABEL COVER, and more, which are believed not to have a subexponential algorithm achieving a nontrivial approximation ratio. Of special interest in these algorithms is a new notion of graph decomposition that may have other applications. Namely, it is shown for every epsilon > 0 and every regular n-vertex graph G, by changing at most delta fraction of G's edges, one can break G into disjoint parts so that the stochastic adjacency matrix of the induced graph on each part has at most n(epsilon) eigenvalues larger than 1 - eta, where eta depends polynomially on epsilon. The subexponential algorithm combines this decomposition with previous algorithms for UNIQUE GAMES on graphs with few large eigenvalues [Kolla and Tuls
Community embeddings are useful in node classification since they allow nodes to aggregate relevant information regarding the network structure. Modularity maximization -based algorithms are the most common approach t...
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Community embeddings are useful in node classification since they allow nodes to aggregate relevant information regarding the network structure. Modularity maximization -based algorithms are the most common approach to detect communities in networks. Moreover, spectral theory plays an important role in community detection by providing a matrix representation of the network structure. However, the literature is still scarce on spectral and modularity maximization -based community embedding methods. Besides, the node features of attributed graphs are usually not considered for producing the community embeddings. This paper introduces a community embedding algorithm based on spectral theory, called SpecRp , that has an overlapping modularity maximization -based step also herein proposed. SpecRp is a community detection method that considers node attributes and vertex proximity to obtain community embeddings. Computational experiments showed that SpecRp outperformed the literature in most of the tested benchmark datasets for the node classification task. Moreover, we observed that to detect disjoint communities, SpecRp and the reference literature algorithms presented a conflicting behavior concerning performance measures. While the reference methods achieved better results for modularity, SpecRp performed better concerning the Normalized Mutual Information to the ground -truth partitions. On detecting overlapping communities, SpecRp was considerably faster than the state-of-the-art algorithms, despite presenting worse results in most of the datasets.
For determining of the desert ecosystem based on transition zones, a quantitative approach was applied in Khorasan Razavi province in northeast of Iran. The spectral and morphological algorithms were implemented in re...
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For determining of the desert ecosystem based on transition zones, a quantitative approach was applied in Khorasan Razavi province in northeast of Iran. The spectral and morphological algorithms were implemented in remote-sensing images of MODIS surface reflectance in order to determine the ecogeomorphic thresholds between the transition zones of the semi-arid ecosystems. Furthermore, the spectral angle mapper analysis was applied for the surface conditions and the mathematical morphology algorithm of dilation was used for identifying the recovery and erodibility potential trends in the heterogeneous land surface covers. A detailed assessment of the mapping was achieved by implementation of an iterative self-organized clustering technique (ISODATA) and calculation of separability of formed typologies classes using the transformed divergence algorithm. The assessment of NDVI (normalized difference vegetation index) in each cluster indicates an increasing trend of standard deviation, which shows maximum values near desert thresholds. According to the results, several desert thresholds were detected in susceptible ecosystems to desertification in Khorasan Razavi that were affected by natural and anthropogenic factors. Further evaluation shows the location of the transition zones and the thresholds under changing climatic conditions. In synopsis, in prone areas with the high potential to desertification, the ecosystem is susceptiblt to shift to desert state.
The stochastic block model (SBM) is a random graph model with planted clusters. It is widely employed as a canonical model to study clustering and community detection, and provides generally a fertile ground to study ...
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The stochastic block model (SBM) is a random graph model with planted clusters. It is widely employed as a canonical model to study clustering and community detection, and provides generally a fertile ground to study the statistical and computational tradeoffs that arise in network and data *** note surveys the recent developments that establish the fundamental limits for community detection in the SBM, both with respect to information-theoretic and computational thresholds, and for various recovery requirements such as exact, partial and weak recovery (a.k.a., detection). The main results discussed are the phase transitions for exact recovery at the Chernoff-Hellinger threshold, the phase transition for weak recovery at the Kesten-Stigum threshold, the optimal distortion-SNR tradeoff for partial recovery, the learning of the SBM parameters and the gap between information-theoretic and computational *** note also covers some of the algorithms developed in the quest of achieving the limits, in particular two-round algorithms via graph-splitting, semi-definite programming, linearized belief propagation, classical and nonbacktracking spectral methods. A few open problems are also discussed.
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