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.

[Q]: We start with an overview of a class of submodular functions called SCMMs (sums of concave composed with non-negative modular functions plus a final arbitrary modular). We then define a new class of submodular functions we call deep submodular functions or DSFs. We show that DSFs are a flexible parametric family of submodular functions that share many of the properties and advantages of deep neural networks (DNNs), including many-layered hierarchical topologies, representation learning, distributed representations, opportunities and strategies for training, and suitability to GPU-based matrix/vector computing. DSFs can be motivated by considering a hierarchy of descriptive concepts over ground elements and where one wishes to allow submodular interaction throughout this hierarchy. In machine learning and data science applications, where there is often either a natural or an automatically learnt hierarchy of concepts over data, DSFs therefore naturally apply. Results in this paper show that DSFs constitute a strictly larger class of submodular functions than SCMMs, thus justifying their mathematical and practical utility. Moreover, we show that, for any integer k > 0, there are k-layer DSFs that cannot be represented by a k-layer DSF for any k < k. This implies that, like DNNs, there is a utility to depth, but unlike DNNs (which can be universally approximated by shallow networks), the family of DSFs strictly increase with depth. Despite this property, however, we show that DSFs, even with arbitrarily large k, do not comprise all submodular functions. We show this using a technique that "backpropagates" certain requirements if it was the case that DSFs comprised all submodular functions. In offering the above results, we also define the notion of an antitone superdifferential of a concave function and show how this relates to submodular functions (in general), DSFs (in particular), negative second-order partial derivatives, continuous submodularity, and concave extensions. To further motivate our analysis, we provide various special case results from matroid theory, comparing DSFs with forms of matroid rank, in particular the laminar matroid. Lastly, we discuss strategies to learn DSFs, and define the classes of deep supermodular functions, deep difference of submodular functions, and deep multivariate submodular functions, and discuss where these can be useful in applications.
[A]: Deep Submodular Functions

[Q]: 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.
[A]: Least Mean Squares Estimation of Graph Signals

[Q]: The belief bias effect is a phenomenon which occurs when we think that we judge an argument based on our reasoning, but are actually influenced by our beliefs and prior knowledge. Evans, Barston and Pollard carried out a psychological syllogistic reasoning task to prove this effect. Participants were asked whether they would accept or reject a given syllogism. We discuss one specific case which is commonly assumed to be believable but which is actually not logically valid. By introducing abnormalities, abduction and background knowledge, we adequately model this case under the weak completion semantics. Our formalization reveals new questions about possible extensions in abductive reasoning. For instance, observations and their explanations might include some relevant prior abductive contextual information concerning some side-effect or leading to a contestable or refutable side-effect. A weaker notion indicates the support of some relevant consequences by a prior abductive context. Yet another definition describes jointly supported relevant consequences, which captures the idea of two observations containing mutually supportive side-effects. Though motivated with and exemplified by the running psychology application, the various new general abductive context definitions are introduced here and given a declarative semantics for the first time, and have a much wider scope of application. Inspection points, a concept introduced by Pereira and Pinto, allows us to express these definitions syntactically and intertwine them into an operational semantics.
[A]:
Contextual Abductive Reasoning with Side-Effects