Realization of set functions as cut functions of graphs and hypergraphs

Fujishige, Satoru; Patkar, Sachin B.

doi:10.1016/s0012-365x(00)00164-3

Cited by 18 publications

(17 citation statements)

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“…We note that we need Ω(2 n ) hyperarcs in general to exactly represent the original function [13]. Corollary 1.3 implies that a slight approximation allows us to reduce the number of hyperarcs to a polynomial number in n.…”

Section: Applicationsmentioning

confidence: 99%

Spectral Sparsification of Hypergraphs

Soma¹,

Yoshida²

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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For an undirected/directed hypergraph G = (V, E), its Laplacian L G : R V → R V is defined such that its "quadratic form" x L G (x) captures the cut information of G. In particular,In this paper, we present a polynomial-time algorithm that, given an undirected/directed hypergraph G on n vertices, constructs an -spectral sparsifier of G with O(n 3 log n/ 2 ) hyperedges/hyperarcs.The proposed spectral sparsification can be used to improve the time and space complexities of algorithms for solving problems that involve the quadratic form, such as computing the eigenvalues of L G , computing the effective resistance between a pair of vertices in G, semisupervised learning based on L G , and cut problems on G. In addition, our sparsification result implies that any submodular function f : 2 V → R + with f (∅) = f (V ) = 0 can be concisely represented by a directed hypergraph. Accordingly, we show that, for any distribution, we can properly and agnostically learn submodular functions f : 2 V → [0, 1] with f (∅) = f (V ) = 0, with O(n 4 log(n/ )/ 4 ) samples.

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Section: Applicationsmentioning

confidence: 99%

Spectral Sparsification of Hypergraphs

Soma¹,

Yoshida²

2019

Proceedings of the Thirtieth Annual ACM-SIAM Symposium on Discrete Algorithms

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“…This algorithm has no known worst case complexity but in practice it usually runs in O(n 2 ). Furthermore, for certain functions which are "graph representable" [49,72], submodular minimization becomes equivalent to computing the minimum s-t cut on the corresponding graph G(V, E), which has time complexity 9Õ (|E| min{|V| 2/3 , |E| 1/2 }) [54].…”

Section: Submodular Function Minimizationmentioning

confidence: 99%

Structured Sparsity: Discrete and Convex Approaches

Kyrillidis

Baldassarre

Halabi

et al. 2015

Compressed Sensing and Its Applications

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Compressive sensing (CS) exploits sparsity to recover sparse or compressible signals from dimensionality reducing, non-adaptive sensing mechanisms. Sparsity is also used to enhance interpretability in machine learning and statistics applications: While the ambient dimension is vast in modern data analysis problems, the relevant information therein typically resides in a much lower dimensional space. However, many solutions proposed nowadays do not leverage the true underlying structure. Recent results in CS extend the simple sparsity idea to more sophisticated structured sparsity models, which describe the interdependency between the nonzero components of a signal, allowing to increase the interpretability of the results and lead to better recovery performance. In order to better understand the impact of structured sparsity, in this chapter we analyze the connections between the discrete models and their convex relaxations, highlighting their relative advantages. We start with the general group sparse model and then elaborate on two important special cases: the dispersive and the hierarchical models. For each, we present the models in their discrete nature, discuss how to solve the ensuing discrete problems and then describe convex relaxations. We also consider more general structures as defined by set functions and present their convex proxies. Further, we discuss efficient optimization solutions for structured sparsity problems and illustrate structured sparsity in action via three applications.

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“…In practice, however, the minimum-norm point algorithm is usually used, which commonly runs in O(N 2 ), but has no known complexity [12]. Furthermore, for certain functions which are "graph representable" [13,14], SFM is equivalent to the minimum s-t cut on an appropriate graph G(V, E), with time complexity…”

Section: Notation and Preliminariesmentioning

confidence: 99%

Map estimation for Bayesian mixture models with submodular priors

Halabi

Baldassarre

Cevher

2014

2014 IEEE International Workshop on Machine Learning for Signal Processing (MLSP)

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We propose a Bayesian approach where the signal structure can be represented by a mixture model with a submodular prior. We consider an observation model that leads to Lipschitz functions. Due to its combinatorial nature, computing the maximum a posteriori estimate for this model is NP-Hard, nonetheless our converging majorization-minimization scheme yields approximate estimates that, in practice, outperform state-of-the-art methods.

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Realization of set functions as cut functions of graphs and hypergraphs

Cited by 18 publications

References 0 publications

Spectral Sparsification of Hypergraphs

Spectral Sparsification of Hypergraphs

Structured Sparsity: Discrete and Convex Approaches

Map estimation for Bayesian mixture models with submodular priors

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