Given a measurement graph G = (V, E) and an unknown signal r is an element of R-n, we investigate algorithms for recovering r from pairwise measurements of the form r(i) - r(j);{i, j} is an element of E. This problem ...
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Given a measurement graph G = (V, E) and an unknown signal r is an element of R-n, we investigate algorithms for recovering r from pairwise measurements of the form r(i) - r(j);{i, j} is an element of 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.
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.
Let I be a random 3CNF formula generated by choosing a truth assignment phi for variables x(1),...,x, uniformly at random and including every clause with i literals set true by phi with probability p(i), independently...
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Let I be a random 3CNF formula generated by choosing a truth assignment phi for variables x(1),...,x, uniformly at random and including every clause with i literals set true by phi with probability p(i), independently. We show that for any constants 0 <= eta(2),eta(3) <= 1 there is a constant d(min) so that for all d >= d(min) a spectral algorithm similar to the graph coloring algorithm of Alon and Kahale will find a satisfying assignment with high probability for p(1) = d/n(2), p(2) = eta(2)d/n(2), and p(3) = eta(3)d/n(2). Appropriately setting the eta(i)'s yields natural distributions on satisfiable 3CNFs, not-all-equal-sat 3CNFs, and exactly-one-sat 3CNFs. (c) 2008 Wiley Periodicals, Inc.
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.
Imposing equilibrium restrictions provides substantial gains in the estimation of dynamic discrete games. Estimation algorithms imposing these restrictions have different merits and limitations. algorithms that guaran...
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Imposing equilibrium restrictions provides substantial gains in the estimation of dynamic discrete games. Estimation algorithms imposing these restrictions have different merits and limitations. algorithms that guarantee local convergence typically require the approximation of high-dimensional Jacobians. Alternatively, the Nested Pseudo-Likelihood (NPL) algorithm is a fixed-point iterative procedure, which avoids the computation of these matrices, but-in games-may fail to converge to the consistent NPL estimator. In order to better capture the effect of iterating the NPL algorithm in finite samples, we study the asymptotic properties of this algorithm for data generating processes that are in a neighborhood of the NPL fixed-point stability threshold. We find that there are always samples for which the algorithm fails to converge, and this introduces a selection bias. We also propose a spectral algorithm to compute the NPL estimator. This algorithm satisfies local convergence and avoids the approximation of Jacobian matrices. We present simulation evidence and an empirical application illustrating our theoretical results and the good properties of the spectral algorithm.
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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