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.

Example Input: The theory of actual causality, defined by Halpern and Pearl, and its quantitative measure – the degree of responsibility – was shown to be extremely useful in various areas of computer science due to a good match between the results it produces and our intuition. In this paper, I describe the applications of causality to formal verification, namely, explanation of counter-examples, refinement of coverage metrics, and symbolic trajectory evaluation. I also briefly discuss recent applications of causality to legal reasoning.
Example Output: Causality and Responsibility for Formal Verification and Beyond

Example Input: In many applications spanning from sensor to social networks, transportation systems, gene regulatory networks or big data, the signals of interest are defined over the vertices of a graph. The aim of this paper is to propose a least mean square (LMS) strategy for adaptive estimation of signals defined over graphs. Assuming the graph signal to be band-limited, over a known bandwidth, the method enables reconstruction, with guaranteed performance in terms of mean-square error, and tracking from a limited number of observations over a subset of vertices. A detailed mean square analysis provides the performance of the proposed method, and leads to several insights for designing useful sampling strategies for graph signals. Numerical results validate our theoretical findings, and illustrate the performance of the proposed method. Furthermore, to cope with the case where the bandwidth is not known beforehand, we propose a method that performs a sparse online estimation of the signal support in the (graph) frequency domain, which enables online adaptation of the graph sampling strategy. Finally, we apply the proposed method to build the power spatial density cartography of a given operational region in a cognitive network environment.
Example Output: Least Mean Squares Estimation of Graph Signals

Example Input: Abstract—Spectral clustering is a powerful tool for unsupervised data analysis. In this paper, we propose a contextaware hypergraph similarity measure (CAHSM), which leads to robust spectral clustering in the case of noisy data. We construct three types of hypergraph—the pairwise hypergraph, the k-nearest-neighbor (kNN) hypergraph, and the high-order over-clustering hypergraph. The pairwise hypergraph captures the pairwise similarity of data points; the kNN hypergraph captures the neighborhood of each point; and the clustering hypergraph encodes high-order contexts within the dataset. By combining the affinity information from these three hypergraphs, the CAHSM algorithm is able to explore the intrinsic topological information of the dataset. Therefore, data clustering using CAHSM tends to be more robust. Considering the intracluster compactness and the inter-cluster separability of vertices, we further design a discriminative hypergraph partitioning criterion (DHPC). Using both CAHSM and DHPC, a robust spectral clustering algorithm is developed. Theoretical analysis and experimental evaluation demonstrate the effectiveness and robustness of the proposed algorithm.
Example Output:
Context-Aware Hypergraph Construction for Robust Spectral Clustering