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.
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Question: Chain graphs combine directed and undi­ rected graphs and their underlying mathe­ matics combines properties of the two. This paper gives a simplified definition of chain graphs based on a hierarchical combination of Bayesian (directed) and Markov (undirected) networks. Examples of a chain graph are multivariate feed-forward networks, cluster­ ing with conditional interaction between vari­ ables, and forms of Bayes classifiers. Chain graphs are then extended using the notation of plates so that samples and data analysis problems can be represented in a graphical model as well. Implications for learning are discussed in the conclusion.

Answer: Chain graphs for learning


Question: Reducing the amount of human supervision is a key problem in machine learning and a natural approach is that of exploiting the relations (structure) among different tasks. This is the idea at the core of multi-task learning. In this context a fundamental question is how to incorporate the tasks structure in the learning problem. We tackle this question by studying a general computational framework that allows to encode a-priori knowledge of the tasks structure in the form of a convex penalty; in this setting a variety of previously proposed methods can be recovered as special cases, including linear and non-linear approaches. Within this framework, we show that tasks and their structure can be efficiently learned considering a convex optimization problem that can be approached by means of block coordinate methods such as alternating minimization and for which we prove convergence to the global minimum.

Answer: Convex Learning of Multiple Tasks and their Structure


Question: The domain of single crossing preference profiles is a widely studied domain in social choice theory. It has been generalized to the domain of single crossing preference profiles with respect to trees which inherits many desirable properties from the single crossing domain, for example, transitivity of majority relation, existence of polynomial time algorithms for finding winners of Kemeny voting rule, etc. In this paper, we consider a further generalization of the domain of single crossing preference profiles on trees to the domain consisting of all preference profiles which can be extended to single crossing preference profiles with respect to some tree by adding more preference orderings to it. We call this domain the weakly single crossing domain on trees. It is known that most of the desirable properties of the single crossing domain on trees continue to hold for the weakly single crossing domain on trees. In this paper, we prove several interesting structural properties of the weakly single crossing domain on trees and exploit them to design efficient algorithms for recognizing and eliciting weakly single crossing preference profiles on trees. In particular, we begin with showing that, for every weakly single crossing preference profile P on trees, there exists a unique inclusion-minimal set of preference orderings Q whose inclusion in P makes the resulting profile P = P ⊎Q single crossing with respect to some tree. We call the profile P the closure of P. Using this uniqueness of closure property of weakly single crossing preference profiles on trees and the known recognition algorithm of single crossing preference profiles on trees due to Clearwater et al. [CPS15], we design a polynomial time algorithm for recognizing weakly single crossing profiles on trees. We then move on to develop a polynomial time algorithm with low query complexity for eliciting weakly single crossing preference profiles on trees even when we do not know any tree with respect to which the closure of the input preference profile is single crossing and the preferences can be queried only sequentially; moreover, the sequential order of the input preference profile is also unknown. We complement the performance of our preference elicitation algorithm by proving that our algorithm makes an optimal number of queries up to constant factors when the number of preference orderings in the input preference profile is large compared to the number of candidates, even if the

Answer:
Recognizing and Eliciting Weakly Single Crossing Profiles on Trees