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]: The shortest path between two concepts in a taxonomic ontology is commonly used to represent the semantic distance between concepts in the edge-based semantic similarity measures. In the past, the edge counting is considered to be the default method for the path computation, which is simple, intuitive and has low computational complexity. However, a large lexical taxonomy of such as WordNet has the irregular densities of links between concepts due to its broad domain but. The edge counting-based path computation is powerless for this non-uniformity problem. In this paper, we advocate that the path computation is able to be separated from the edge-based similarity measures and form various general computing models. Therefore, in order to solve the problem of non-uniformity of concept density in a large taxonomic ontology, we propose a new path computing model based on the compensation of local area density of concepts, which is equal to the number of direct hyponyms of the subsumers of concepts in their shortest path. This path model considers the local area density of concepts as an extension of the edge-based path and converts the local area density divided by their depth into the compensation for edge-based path with an adjustable parameter, which idea has been proven to be consistent with the information theory. This model is a general path computing model and can be applied in various edge-based similarity algorithms. The experiment results show that the proposed path model improves the average correlation between edge-based measures with human judgments on Miller and Charles benchmark from less than 0.8 to more than 0.85, and has a big advantage in efficiency than information content (IC) computation in a dynamic ontology, thereby successfully solving the non-uniformity problem of taxonomic ontology.
[A]: A density compensation-based path computing model for measuring semantic similarity


[Q]: Consider a weighted or unweighted k-nearest neighbor graph that has been built on n data points drawn randomly according to some density p on R. We study the convergence of the shortest path distance in such graphs as the sample size tends to infinity. We prove that for unweighted kNN graphs, this distance converges to an unpleasant distance function on the underlying space whose properties are detrimental to machine learning. We also study the behavior of the shortest path distance in weighted kNN graphs.
[A]:
Shortest path distance in random k-nearest neighbor graphs