In this task, you are given a part of an article. Your task is to generate headline (title) for this text. Preferred headlines are under fifteen words.

Input: Consider Input: Cooperative games model the allocation of profit from joint actions, following considerations such as stability and fairness. We propose the reliability extension of such games, where agents may fail to participate in the game. In the reliability extension, each agent only "survives" with a certain probability, and a coalition's value is the probability that its surviving members would be a winning coalition in the base game. We study prominent solution concepts in such games, showing how to approximate the Shapley value and how to compute the core in games with few agent types. We also show that applying the reliability extension may stabilize the game, making the core non-empty even when the base game has an empty core.

Output: Solving Cooperative Reliability Games

Input: Consider Input: *Correspondence: aljadda@uga.edu 1Department of Computer Science, University of Georgia, Athens,GA, USA Full list of author information is available at the end of the article Equal contributor Abstract Probabilistic Graphical Models (PGM) are very useful in the fields of machine learning and data mining. The crucial limitation of those models,however, is the scalability. The Bayesian Network, which is one of the most common PGMs used in machine learning and data mining, demonstrates this limitation when the training data consists of random variables, each of them has a large set of possible values. In the big data era, one would expect new extensions to the existing PGMs to handle the massive amount of data produced these days by computers, sensors and other electronic devices. With hierarchical data data that is arranged in a treelike structure with several levels one would expect to see hundreds of thousands or millions of values distributed over even just a small number of levels. When modeling this kind of hierarchical data across large data sets, Bayesian Networks become infeasible for representing the probability distributions. In this paper we introduce an extension to Bayesian Networks to handle massive sets of hierarchical data in a reasonable amount of time and space. The proposed model achieves perfect precision of 1.0 and high recall of 0.93 when it is used as multi-label classifier for the annotation of mass spectrometry data. On another data set of 1.5 billion search logs provided by CareerBuilder.com the model was able to predict latent semantic relationships between search keywords with accuracy up to 0.80.

Output: Mining Massive Hierarchical Data Using a Scalable Probabilistic Graphical Model

Input: Consider Input: We consider relative error low rank approximation of tensors with respect to the Frobenius norm. Namely, given an order-q tensor A R q i=1 ni , output a rank-k tensor B for which A BF (1 + ) OPT, where OPT = infrank-k A A AF . Despite much success on obtaining relative error low rank approximations for matrices, no such results were known for tensors. One structural issue is that there may be no rank-k tensor Ak achieving the above infinum. Another, computational issue, is that an efficient relative error low rank approximation algorithm for tensors would allow one to compute the rank of a tensor, which is NP-hard. We bypass these two issues via (1) bicriteria and (2) parameterized complexity solutions: 1. We give an algorithm which outputs a rank k = O((k/ )q1) tensor B for which A BF (1+ ) OPT in nnz(A)+n poly(k/ ) time in the real RAM model, whenever either Ak exists or OPT > 0. Here nnz(A) denotes the number of non-zero entries in A. If both Ak does not exist and OPT = 0, then B instead satisfies A BF < , where is any positive, arbitrarily small function of n. 2. We give an algorithm for any > 0 which outputs a rank k tensor B for which ABF (1+ ) OPT and runs in (nnz(A)+n poly(k/ )+exp(k/ )) n time in the unit cost RAM model, whenever OPT > 2O(n ) and there is a rank-k tensor B = k i=1 ui vi wi for which A BF (1 + /2) OPT and ui2, vi2, wi2 2 ). If OPT 2(n), then B instead satisfies ABF 2(n ). Our first result is polynomial time, and in fact input sparsity time, in n, k, and 1/ , for any k 1 and any 0 < < 1, while our second result is fixed parameter tractable in k and 1/ . For outputting a rank-k tensor, or even a bicriteria solution with rank-Ck for a certain constant C > 1, we show a 2 1o(1)) time lower bound under the Exponential Time Hypothesis. Our results are based on an "iterative existential argument", and give the first relative error low rank approximations for tensors for a large number of error measures for which nothing was known. In particular, we give the first relative error approximation algorithms on tensors for: column row and tube subset selection, entrywise 'p-low rank approximation for 1 p < 2, low rank approximation with respect to sum of Euclidean norms of faces or tubes, weighted low rank approximation, and low rank approximation in distributed and streaming models. We also obtain several new results for matrices, such as nnz(A)-time CUR decompositions, improving the previous nnz(A) log n-time CUR decompositions, which may be of independent interest. Work done while visiting IBM Almaden, and supported in part by UTCS TAship (CS361 Spring 17 Introduction to Computer Security). Supported in part by Simons Foundation, and NSF CCF-1617955. ar X iv :1 70 4. 08 24 6v 1 [ cs .D S] 2 6 A pr 2 01 7
Output: Relative Error Tensor Low Rank Approximation