For a block upper triangular matrix, a necessary and sufficient condition has been given to let it be the sum of block upper rectangular matrices satisfying certain rank constraints; see H.Bart, A.P.M.Wagelmans (2000). The proof involves elements from integer programming and employs Farkas’ lemma. The algebra of block upper triangular matrices can be viewed as a matrix algebra determined by a pattern of zeros. The present note is concerned with the question whether the decomposition result referred to above can be extended to other zero pattern matrix algebras. It is shown that such a generalization does indeed hold for certain digraphs determining the pattern of zeros. The digraphs in question can be characterized in terms of forests, i.e., disjoint unions of rooted trees., Harm Bart, Torsten Ehrhardt, Bernd Silbermann., and Obsahuje seznam literatury
Bayesian networks are a popular model for reasoning under uncertainty. We study the problem of efficient probabilistic inference with these models when some of the conditional probability tables represent deterministic or noisy ℓ-out-of-k functions. These tables appear naturally in real-world applications when we observe a state of a variable that depends on its parents via an addition or noisy addition relation. We provide a lower bound of the rank and an upper bound for the symmetric border rank of tensors representing ℓ-out-of-k functions. We propose an approximation of tensors representing noisy ℓ-out-of-k functions by a sum of r tensors of rank one, where r is an upper bound of the symmetric border rank of the approximated tensor. We applied the suggested approximation to probabilistic inference in probabilistic graphical models. Numerical experiments reveal that we can get a gain in the order of two magnitudes but at the expense of a certain loss of precision.
In this paper a new rank test in a linear regression model is introduced. The test statistic is based on a certain minimum distance estimator, however, unlike classical rank tests in regression it is not a simple linear rank statistic. Its exact distribution under the null hypothesis is derived, and further, the asymptotic distribution both under the null hypothesis and the local alternative is investigated. It is shown that the proposed test is applicable in measurement error models. Finally, a simulation study is conducted to show a good performance of the test. It has, in some situations, a greater power than the widely used Wilcoxon rank test.
In the development of efficient predictive models, the key is to identify suitable predictors for a given linear model. For the first time, this paper provides a comparative study of ridge regression, LASSO, preliminary test and Stein-type estimators based on the theory of rank statistics. Under the orthonormal design matrix of a given linear model, we find that the rank based ridge estimator outperforms the usual rank estimator, restricted R-estimator, rank-based LASSO, preliminary test and Stein-type R-estimators uniformly. On the other hand, neither LASSO nor the usual R-estimator, preliminary test and Stein-type R-estimators outperform the other. The region of domination of LASSO over all the R-estimators (except the ridge R-estimator) is the interval around the origin of the parameter space. Finally, we observe that the L2-risk of the restricted R-estimator equals the lower bound on the L2-risk of LASSO. Our conclusions are based on L2-risk analysis and relative L2-risk efficiencies with related tables and graphs.
This work proposes an approach to tag recommendation based on a learning system. The goal of this method is to support users of current social network systems by providing a rank of new meaningful tags for a resource. This system provides a ranked tag set and it feeds on different posts depending on the resource for which the user requests the recommendation. This research studies different approaches depending on both the posts selected to form the training set and the features with which they are represented. The performance of these approaches are tested according to several evaluation measures; one of them is proposed in this paper F1+ which takes into account the positions where the system has ranked the positive tags at the same time that it considers the cases where positive tags could not be ranked. These experiments show that this learning system outperforms certain benchmark recommenders.
Raphogla rubra gen. n., sp. n., oldest representative of the (Tettigoniidea & Gryllidea) is described from the Upper Permian of the Lodève basin (France). Its phylogenetic relationships within the Ensifera are discussed. The new taxon occupies a very basal position, probably as sister group of the whole group (Tettigoniidea & Gryllidea).