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In this manuscript, we offer a gentle review of submodularity and supermodularity and their properties. We offer a plethora of submodular definitions; a full description of a number of example submodular functions and their generalizations;…
Robust optimization is becoming increasingly important in machine learning applications. In this paper, we study a unified framework of robust submodular optimization. We study this problem both from a minimization and maximization…
A new sparse SOS decomposition algorithm is proposed based on a new sparsity pattern, called cross sparsity patterns. The new sparsity pattern focuses on the sparsity of terms and thus is different from the well-known correlative sparsity…
Many large-scale machine learning problems--clustering, non-parametric learning, kernel machines, etc.--require selecting a small yet representative subset from a large dataset. Such problems can often be reduced to maximizing a submodular…
Constrained submodular function maximization has been used in subset selection problems such as selection of most informative sensor locations. While these models have been quite popular, the solutions Constrained submodular function…
We define a new class of set functions that in addition to being monotone and subadditive, also admit a very limited form of submodularity defined over a permutation of the ground set. We refer to this permutation as a submodular order.…
Maximizing a single submodular set function subject to a cardinality constraint is a well-studied and central topic in combinatorial optimization. However, finding a set that maximizes multiple functions at the same time is much less…
Symmetric submodular functions are an important family of submodular functions capturing many interesting cases including cut functions of graphs and hypergraphs. Maximization of such functions subject to various constraints receives little…
Sparse methods for supervised learning aim at finding good linear predictors from as few variables as possible, i.e., with small cardinality of their supports. This combinatorial selection problem is often turned into a convex optimization…
In a nutshell, submodular functions encode an intuitive notion of diminishing returns. As a result, submodularity appears in many important machine learning tasks such as feature selection and data summarization. Although there has been a…
With the extensive application of submodularity, its generalizations are constantly being proposed. However, most of them are tailored for special problems. In this paper, we focus on quasi-submodularity, a universal generalization, which…
Submodular set functions are undoubtedly among the most important building blocks of combinatorial optimization. Somewhat surprisingly, continuous counterparts of such functions have also appeared in an analytic line of research where they…
Malleable scheduling is a model that captures the possibility of parallelization to expedite the completion of time-critical tasks. A malleable job can be allocated and processed simultaneously on multiple machines, occupying the same time…
Submodular Functions are a special class of set functions, which generalize several information-theoretic quantities such as entropy and mutual information [1]. Submodular functions have subgradients and subdifferentials [2] and admit…
High-dimensional real-world systems can often be well characterized by a small number of simultaneous low-complexity interactions. The analysis of variance (ANOVA) decomposition and the anchored decomposition are typical techniques to find…
Efficient handling of sparse data is a key challenge in Computer Science. Binary convolutions, such as polynomial multiplication or the Walsh Transform are a useful tool in many applications and are efficiently solved. In the last decade,…
Designing and implementing systems as an interconnection of smaller subsystems is a common practice for modularity and standardization of components and design algorithms. Although not typically cast in this framework, many of these…
Deep neural networks have emerged as powerful tools for learning operators defined over infinite-dimensional function spaces. However, existing theories frequently encounter difficulties related to dimensionality and limited…
This paper considers the minimization problem of relaxed submodular functions. For a positive integer $k$, a set function is called $k$-distant submodular if the submodular inequality holds for every pair whose symmetric difference is at…
Signal decomposition and multiscale signal analysis provide many useful tools for time-frequency analysis. We proposed a random feature method for analyzing time-series data by constructing a sparse approximation to the spectrogram. The…