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.

[EX 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.
[EX A]: Deep Submodular Functions

[EX Q]: Microarrays are made it possible to simultaneously monitor the expression profiles of thousands of genes under various experimental conditions. It is used to identify the co-expressed genes in specific cells or tissues that are actively used to make proteins. This method is used to analysis the gene expression, an important task in bioinformatics research. Cluster analysis of gene expression data has proved to be a useful tool for identifying co-expressed genes, biologically relevant groupings of genes and samples. In this paper we applied K-Means with Automatic Generations of Merge Factor for ISODATAAGMFI. Though AGMFI has been applied for clustering of Gene Expression Data, this proposed Enhanced Automatic Generations of Merge Factor for ISODATAEAGMFI Algorithms overcome the drawbacks of AGMFI in terms of specifying the optimal number of clusters and initialization of good cluster centroids. Experimental results on Gene Expression Data show that the proposed EAGMFI algorithms could identify compact clusters with perform well in terms of the Silhouette Coefficients cluster measure.
[EX A]: Performance Analysis of Enhanced Clustering Algorithm for Gene Expression Data

[EX Q]: We study sequential prediction of real-valued, arbitrary and unknown sequences under the squared error loss as well as the best parametric predictor out of a large, continuous class of predictors. Inspired by recent results from computational learning theory, we refrain from any statistical assumptions and define the performance with respect to the class of general parametric predictors. In particular, we present generic lower and upper bounds on this relative performance by transforming the prediction task into a parameter learning problem. We first introduce the lower bounds on this relative performance in the mixture of experts framework, where we show that for any sequential algorithm, there always exists a sequence for which the performance of the sequential algorithm is lower bounded by zero. We then introduce a sequential learning algorithm to predict such arbitrary and unknown sequences, and calculate upper bounds on its total squared prediction error for every bounded sequence. We further show that in some scenarios we achieve matching lower and upper bounds demonstrating that our algorithms are optimal in a strong minimax sense such that their performances cannot be improved further. As an interesting result we also prove that for the worst case scenario, the performance of randomized algorithms can be achieved by sequential algorithms so that randomized algorithms does not improve the performance.
[EX A]:
A Unified Approach to Universal Prediction: Generalized Upper and Lower Bounds