In this paper, we aim to provide a systematic study of the relationship between Chernoff information and topological, as well as algebraic properties of the corresponding Gaussian tree graphs for the underlying graphical testing problems. We first show the relationship between Chernoff information and generalized eigenvalues of the associated covariance matrices. It is then proved that Chernoff information between two Gaussian trees sharing certain local subtree structures can be transformed into that of two smaller trees. Under our proposed grafting operations, bottleneck Gaussian trees, namely, Gaussian trees connected by one such operation, can thus be simplified into two 3-node Gaussian trees, whose topologies and edge weights are subject to the specifics of the operation. Thereafter, we provide a thorough study about how Chernoff information changes when small differences are accumulated into bigger ones via concatenated grafting operations. It is shown that the two Gaussian trees connected by more than one grafting operation may not have bigger Chernoff information than that of one grafting operation unless these grafting operations are separate and independent. At the end, we propose an optimal linear dimensional reduction method related to generalized eigenvalues.
The error exponent of an @math -ary hypothesis testing problem is the minimum pair-wise Chernoff information among them @cite_17 . So we focus on pair-wise topological differences at first. We come up with a typical topological pair, namely two trees connected by one grafting operation, which is deemed to represent the case where two Gaussian trees have a minimal topological difference. Grafting is a kind of topological operation by cutting down a subtree from another tree and pasting it to another location as shown in Fig. . In this figure, @math are the nodes in both trees, representing random variables @math . We separate subtree1 and subtree2 by cutting the edge @math with weight @math and paste subtree2 to subtree1 by adding new edge @math with the same weight @math . We call @math anchor nodes of this grafting operation. Our formulation is for the purpose of identifying the contributing factors to Chernoff information in distinguishing Gaussian trees with minimum topological difference, which can be deemed as a worst case scenario.
@cite_17
2124988974 2950981509 2943842266 1757513105
The problem of maximum-likelihood (ML) estimation of discrete tree-structured distributions is considered. Chow and Liu established that ML-estimation reduces to the construction of a maximum-weight spanning tree using the empirical mutual information quantities as the edge weights. Using the theory of large-deviations, we analyze the exponent associated with the error probability of the event that the ML-estimate of the Markov tree structure differs from the true tree structure, given a set of independently drawn samples. By exploiting the fact that the output of ML-estimation is a tree, we establish that the error exponent is equal to the exponential rate of decay of a single dominant crossover event. We prove that in this dominant crossover event, a non-neighbor node pair replaces a true edge of the distribution that is along the path of edges in the true tree graph connecting the nodes in the non-neighbor pair. Using ideas from Euclidean information theory, we then analyze the scenario of ML-estimation in the very noisy learning regime and show that the error exponent can be approximated as a ratio, which is interpreted as the signal-to-noise ratio (SNR) for learning tree distributions. We show via numerical experiments that in this regime, our SNR approximation is accurate. Let @math be a tree space (or tree network) represented by a weighted tree with @math vertices, and @math be a set of @math stochastic points in @math , each of which has a fixed location with an independent existence probability. We investigate two fundamental problems under such a stochastic setting, the closest-pair problem and the nearest-neighbor search. For the former, we study the computation of the @math -threshold probability and the expectation of the closest-pair distance of a realization of @math . We propose the first algorithm to compute the @math -threshold probability in @math time for any given threshold @math , which immediately results in an @math -time algorithm for computing the expected closest-pair distance. Based on this, we further show that one can compute a @math -approximation for the expected closest-pair distance in @math time, by arguing that the expected closest-pair distance can be approximated via @math threshold probability queries. For the latter, we study the @math most-likely nearest-neighbor search ( @math -LNN) via a notion called @math most-likely Voronoi Diagram ( @math -LVD). We show that the size of the @math -LVD @math of @math on @math is bounded by @math if the existence probabilities of the points in @math are constant-far from 0. Furthermore, we establish an @math average-case upper bound for the size of @math , by regarding the existence probabilities as i.i.d. random variables drawn from some fixed distribution. Our results imply the existence of an LVD data structure which answers @math -LNN queries in @math time using average-case @math space, and worst-case @math space if the existence probabilities are constant-far from 0. Finally, we also give an @math -time algorithm to construct the LVD data structure. Given a graph @math and a subset @math of terminals, a of @math is a tree that spans @math . In the vertex-weighted Steiner tree (VST) problem, each vertex is assigned a non-negative weight, and the goal is to compute a minimum weight Steiner tree of @math . We study a natural generalization of the VST problem motivated by multi-level graph construction, the (V-GSST), which can be stated as follows: given a graph @math and terminals @math , where each terminal @math requires a facility of a minimum grade of service @math , compute a Steiner tree @math by installing facilities on a subset of vertices, such that any two vertices requiring a certain grade of service are connected by a path in @math with the minimum grade of service or better. Facilities of higher grade are more costly than facilities of lower grade. Multi-level variants such as this one can be useful in network design problems where vertices may require facilities of varying priority. While similar problems have been studied in the edge-weighted case, they have not been studied as well in the more general vertex-weighted case. We first describe a simple heuristic for the V-GSST problem whose approximation ratio depends on @math , the number of grades of service. We then generalize the greedy algorithm of [Klein & Ravi, 1995] to show that the V-GSST problem admits a @math -approximation, where @math is the set of terminals requiring some facility. This result is surprising, as it shows that the (seemingly harder) multi-grade problem can be approximated as well as the VST problem, and that the approximation ratio does not depend on the number of grades of service. This paper is motivated by two applications, namely i) generalizations of cuckoo hashing, a computationally simple approach to assigning keys to objects, and ii) load balancing in content distribution networks, where one is interested in determining the impact of content replication on performance. These two problems admit a common abstraction: in both scenarios, performance is characterized by the maximum weight of a generalization of a matching in a bipartite graph, featuring node and edge capacities. Our main result is a law of large numbers characterizing the asymptotic maximum weight matching in the limit of large bipartite random graphs, when the graphs admit a local weak limit that is a tree. This result specializes to the two application scenarios, yielding new results in both contexts. In contrast with previous results, the key novelty is the ability to handle edge capacities with arbitrary integer values. An analysis of belief propagation algorithms (BP) with multivariate belief vectors underlies the proof. In particular, we show convergence of the corresponding BP by exploiting monotonicity of the belief vectors with respect to the so-called upshifted likelihood ratio stochastic order. This auxiliary result can be of independent interest, providing a new set of structural conditions which ensure convergence of BP.