Local Algorithms for Block Models with Side Information

There has been a recent interest in understanding the power of local algorithms for optimization and inference problems on sparse graphs. Gamarnik and Sudan (2014) showed that local algorithms are weaker than global algorithms for finding large independent sets in sparse random regular graphs. Montanari (2015) showed that local algorithms are suboptimal for finding a community with high connectivity in the sparse Erd\H{o}s-R\'enyi random graphs. For the symmetric planted partition problem (also named community detection for the block models) on sparse graphs, a simple observation is that local algorithms cannot have non-trivial performance. In this work we consider the effect of side information on local algorithms for community detection under the binary symmetric stochastic block model. In the block model with side information each of the $n$ vertices is labeled $+$ or $-$ independently and uniformly at random; each pair of vertices is connected independently with probability $a/n$ if both of them have the same label or $b/n$ otherwise. The goal is to estimate the underlying vertex labeling given 1) the graph structure and 2) side information in the form of a vertex labeling positively correlated with the true one. Assuming that the ratio between in and out degree $a/b$ is $\Theta(1)$ and the average degree $(a+b) / 2 = n^{o(1)}$, we characterize three different regimes under which a local algorithm, namely, belief propagation run on the local neighborhoods, maximizes the expected fraction of vertices labeled correctly. Thus, in contrast to the case of symmetric block models without side information, we show that local algorithms can achieve optimal performance for the block model with side information.Comment: Due to the limitation "The abstract field cannot be longer than 1,920 characters", the abstract here is shorter than that in the PDF fil

Porque as algas e bactérias não são amplamente reconhecidas como seres fotossintetizantes?

Com a intenção de valorizar um conhecimento que nos permita olhar para o mundo de modo global para compreender a nossa realidade, destinamos para esse estudo indicar que as algas e bactérias não são valorizadas como organismos fotossintetizantes em alguns artigos acadêmicos. Uma das possibilidades para que esses seres não sejam contemplados, no ensino, é o fato de tentarmos fragmentar os conhecimentos tendo como finalidade simplificar a aprendizagem. Porém, isso acaba por acarretar a descontextualização e consequentemente a ausência de sentido

SE-Sync: A Certifiably Correct Algorithm for Synchronization over the Special Euclidean Group

Many important geometric estimation problems naturally take the form of synchronization over the special Euclidean group: estimate the values of a set of unknown poses given noisy measurements of a subset of their pairwise relative transforms. Examples of this class include the foundational problems of pose-graph simultaneous localization and mapping (SLAM) (in robotics), camera motion estimation (in computer vision), and sensor network localization (in distributed sensing), among others. This inference problem is typically formulated as a nonconvex maximum-likelihood estimation that is computationally hard to solve in general. Nevertheless, in this paper we present an algorithm that is able to efficiently recover certifiably globally optimal solutions of the special Euclidean synchronization problem in a non-adversarial noise regime. The crux of our approach is the development of a semidefinite relaxation of the maximum-likelihood estimation whose minimizer provides an exact MLE so long as the magnitude of the noise corrupting the available measurements falls below a certain critical threshold; furthermore, whenever exactness obtains, it is possible to verify this fact a posteriori, thereby certifying the optimality of the recovered estimate. We develop a specialized optimization scheme for solving large-scale instances of this semidefinite relaxation by exploiting its low-rank, geometric, and graph-theoretic structure to reduce it to an equivalent optimization problem defined on a low-dimensional Riemannian manifold, and then design a Riemannian truncated-Newton trust-region method to solve this reduction efficiently. Finally, we combine this fast optimization approach with a simple rounding procedure to produce our algorithm, SE-Sync. Experimental evaluation on a variety of simulated and real-world pose-graph SLAM datasets shows that SE-Sync is capable of recovering certifiably globally optimal solutions when the available measurements are corrupted by noise up to an order of magnitude greater than that typically encountered in robotics and computer vision applications, and does so more than an order of magnitude faster than the Gauss-Newton-based approach that forms the basis of current state-of-the-art techniques

Structured networks and coarse-grained descriptions: a dynamical perspective

This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We thus aim to gain a reduced description of the system that takes into account both its structure and dynamics. In the first part, we introduce the general mathematical setup for the types of dynamics we consider throughout the chapter. We provide two guiding examples, namely consensus dynamics and diffusion processes (random walks), motivating their connection to social network analysis, and provide a brief discussion on the general dynamical framework and its possible extensions. In the second part, we focus on the influence of graph structure on the dynamics taking place on the network, focusing on three concepts that allow us to gain insight into this notion. First, we describe how time scale separation can appear in the dynamics on a network as a consequence of graph structure. Second, we discuss how the presence of particular symmetries in the network give rise to invariant dynamical subspaces that can be precisely described by graph partitions. Third, we show how this dynamical viewpoint can be extended to study dynamics on networks with signed edges, which allow us to discuss connections to concepts in social network analysis, such as structural balance. In the third part, we discuss how to use dynamical processes unfolding on the network to detect meaningful network substructures. We then show how such dynamical measures can be related to seemingly different algorithm for community detection and coarse-graining proposed in the literature. We conclude with a brief summary and highlight interesting open future directions

Avaliação e seleção de progênies de cupuaçuzeiro (Theobroma grandiflorum), em Belém, Pará.

Este trabalho teve por objetivo a seleção entre e dentro de progênies de irmãos completos de cupuaçuzeiro. Foram avaliadas 21 progênies de irmãos completos, em dois experimentos, com cinco plantas por parcela. Os experimentos foram avaliados ao nível de indivíduos, em cada safra, para os caracteres: produção de frutos, polpa e semente, bem como resistência à vassoura-de-bruxa. Para efeito de seleção, também foi considerado o grau de parentesco das matrizes. As análises foram conduzidas via metodologia de modelos lineares mistos, como delineamento em blocos incompletos, desbalanceados com tratamentos comuns. Foram estimados os parâmetros genéticos e os valores genotípicos ajustados de progênies, bem como os valores genéticos aditivos e genotípicos individuais. Os resultados demonstraram que as progênies 12; 13; 18; 20; 21 e 24 tiveram os melhores desempenhos para as variáveis de produção. Entre elas, as pro- gênies 12; 13 e 18 foram as que mais se destacaram. Foram selecionadas três matrizes da progênie 18, duas matrizes da progênie 12 e uma matriz de cada uma das progênies 13; 20 e 21. É possível concluir que essas matrizes, juntamente com materiais selecionados em outras áreas experimentais, apresentam potencial para compor um pomar de sementes clonais, estabelecido em lote isolado de outros plantios, onde será produzida uma população melhorada de primeiro ciclo que se constituirá na nova cultivar de cupuaçuzeiro