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Linearly Representable Submodular Functions: An Algebraic Algorithm for Minimization

Rohit Gurjar, Rajat Rathi, Emanuela Merelli, Anuj Dawar, Artur Czumaj
2020 International Colloquium on Automata, Languages and Programming  
We give an algebraic algorithm for this problem for a special class of submodular functions that are "linearly representable".  ...  We also give reductions from two combinatorial optimization problems to linearly representable submodular minimization, and thus, get such parallel algorithms for these problems.  ...  One would expect such algebraic algorithms for submodular functions that are linear algebraic in some sense. Towards this, we define a class of linearly representable submodular functions.  ... 
doi:10.4230/lipics.icalp.2020.61 dblp:conf/icalp/GurjarR20 fatcat:o6tjumo5qfed7pp5xv7zvg7czm

Learning with Submodular Functions: A Convex Optimization Perspective [article]

Francis Bach
2013 arXiv   pre-print
This allows the derivation of new efficient algorithms for approximate and exact submodular function minimization with theoretical guarantees and good practical performance.  ...  Submodular functions are relevant to machine learning for at least two reasons: (1) some problems may be expressed directly as the optimization of submodular functions and (2) the lovasz extension of submodular  ...  The author would like to thank Rodolphe Jenatton, Armand Joulin, Simon Lacoste-Julien, Julien Mairal and Guillaume Obozinski for discussions related to submodular functions and convex optimization.  ... 
arXiv:1111.6453v2 fatcat:qsbgrxoot5f7jhss4otffr3izy

Chapter 12 Dilworth Truncation of Submodular Functions [chapter]

H. Narayanan
1997 Annals of Discrete Mathematics  
function with r,tspect to a positive weight function and the principal lattice of partitions of a submodular function.  ...  Let f o (), fi (.) be submodular functions on subsets of S.  ...  Kishi and Kajitani l° gave an algorithm for building a pair of maximally distant trees which is essentially the well-known algorithm for building a base of the union of two matroids".  ... 
doi:10.1016/s0167-5060(08)70681-2 fatcat:gge3olf3qjf4vaddfas54pmbca

Learning with Submodular Functions: A Convex Optimization Perspective

Francis Bach
2013 Foundations and Trends® in Machine Learning  
The author would like to thank Thibaut Horel, Stefanie Jegelka, Rodolphe Jenatton, Armand Joulin, Simon Lacoste-Julien, Julien Mairal and Guillaume Obozinski for discussions related to submodular functions  ...  Divide-and-conquer algorithm for proximal problems We now consider an algorithm for proximal problems, which is based on a sequence of submodular function minimizations.  ...  The first algorithm is an exact algorithm which relies on the availability of an efficient submodular function minimization algorithm, while the second set of algorithms are based on existing iterative  ... 
doi:10.1561/2200000039 fatcat:kk7w6zsnsnbp3eoa6b5ol3bxbq

Coxeter submodular functions and deformations of Coxeter permutahedra [article]

Federico Ardila, Federico Castillo, Christopher Eur, Alexander Postnikov
2020 arXiv   pre-print
Our description extends the known correspondence between generalized permutahedra, polymatroids, and submodular functions to any finite reflection group.  ...  This family of polytopes contains polyhedral models for the Coxeter-theoretic analogs of compositions, graphs, matroids, posets, and associahedra.  ...  We thank them for offering a wonderful setting to do mathematics, and the NSF and Simons Foundation for their financial support of these visits.  ... 
arXiv:1904.11029v2 fatcat:zpiebfvifbfzjfyz2d3hup3ufa

Batch greedy maximization of non-submodular functions: Guarantees and applications to experimental design [article]

Jayanth Jagalur-Mohan, Youssef Marzouk
2021 arXiv   pre-print
We propose and analyze batch greedy heuristics for cardinality constrained maximization of non-submodular non-decreasing set functions.  ...  In the context of optimal experimental design for linear Bayesian inverse problems, we bound the submodularity and supermodularity ratios when the underlying objective is based on mutual information.  ...  The authors are grateful to Daniel Ricciuto for help with the optimal sensor placement problem; to Stefanie Jegelka for an insightful discussion concerning submodular functions; to Jean-Christophe Bourin  ... 
arXiv:2006.04554v3 fatcat:6bxokkvm6jgktpv3bcznlmfpnu

Quadratic Decomposable Submodular Function Minimization [article]

Pan Li, Niao He, Olgica Milenkovic
2018 arXiv   pre-print
We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization.  ...  The problem is closely related to decomposable submodular function minimization and arises in many learning on graphs and hypergraphs settings, such as graph-based semi-supervised learning and PageRank  ...  (S), where F (S) = r∈[R] F r (S) − 2 i∈S a i [10] ; second, minimizing a submodular function decomposed into a sum of simpler components F r , r ∈ [R], is much easier than minimizing an unrestricted  ... 
arXiv:1806.09842v3 fatcat:lkgwmp4w7zay3n3jcsqd2x5r2y

Quadratic Decomposable Submodular Function Minimization: Theory and Practice (Computation and Analysis of PageRank over Hypergraphs) [article]

Pan Li, Niao He, Olgica Milenkovic
2020 arXiv   pre-print
The problem exhibits close ties to decomposable submodular function minimization (DSFM), yet is much more challenging to solve.  ...  We introduce a new convex optimization problem, termed quadratic decomposable submodular function minimization (QDSFM), which allows to model a number of learning tasks on graphs and hypergraphs.  ...  The authors would like to thank the reviewers for their insightful suggestions to improve the quality of the manuscript.  ... 
arXiv:1902.10132v4 fatcat:bll5yhjtwnbmzkvjuy3tegjidu

Bundle Selling by Online Estimation of Valuation Functions

Daniel Vainsencher, Ofer Dekel, Shie Mannor
2011 International Conference on Machine Learning  
We are interested in learning schemes that have a small regret compared to an agent who knows the true valuation function.  ...  We develop efficient learning algorithms that balance exploration and exploitation to achieve low regret in this setting.  ...  This research was supported in part by the Google Inter-university center for Electronic Markets and Auctions. We thank Mohit Singh for helpful discussions.  ... 
dblp:conf/icml/VainsencherDM11 fatcat:vo6tbu6kdrcv5m5s3kbqaf3gji

Rank axiom of modular supermatroids: A connection with directional DR submodular functions [article]

Takanori Maehara, So Nakashima
2020 arXiv   pre-print
We characterize supermatroids on modular lattices using the rank axiom in which the rank function is a directional DR-submodular function, which is a generalization of a submodular function introduced  ...  Barnabei, Nicoletti, and Pezzoli characterized supermatroids on distributive lattices, and Fujishige, Koshevoy, and Sano generalized the results for cg-matroids (supermatroids on lower locally distributive  ...  Acknowledgment We thank Hiroshi Hirai for fruitful discussion. This work was supported by JSPS KAKENHI Grant Number 19K20219.  ... 
arXiv:2009.00200v1 fatcat:loporohkhfh3dh5iwe6lgwiz2q

Monotone Submodular Diversity functions for Categorical Vectors with Application to Diversification of Seeds for Targeted Influence Maximization [article]

Antonio Caliò, Andrea Tagarelli
2019 arXiv   pre-print
Starting from the assumption that side-information is available at node level in the general form of categorical attribute values, we design a class of monotone submodular functions specifically conceived  ...  Surprisingly, in the classic problem of influence maximization in social networks, relatively little study has been devoted to diversity and its integration into the objective function of an influence  ...  Also, we set η = 10 −4 for DTIM, which means minimal path-pruning, and hence highest estimation accuracy for the competitor.  ... 
arXiv:1912.03727v1 fatcat:gxonyxv7hrbgzns55pbwrqjkze

Diophantine approximation on matrices and Lie groups

Menny Aka, Emmanuel Breuillard, Lior Rosenzweig, Nicolas de Saxcé
2018 Geometric and Functional Analysis  
length of an element ω ∈ Fk,G , i.e. the minimal length of a word w such that wG = ω.  ...  For now, we just give another simple example where the submodularity lemma applies, and allows to compute the diophantine exponent of an algebraic subset of matrices.  ... 
doi:10.1007/s00039-018-0436-0 fatcat:a7fm6mlxdzhf3hutomp4d6fnfq

Optimal Deterministic Polynomial-Time Data Exchange for Omniscience [article]

Nebojsa Milosavljevic and Sameer Pawar and Salim El Rouayheb and Michael Gastpar and Kannan Ramchandran
2011 arXiv   pre-print
Using established connections to the multi-terminal secrecy problem, our algorithm also implies a polynomial-time method for constructing a maximum size secret shared key in the presence of an eavesdropper  ...  Building on results from combinatorial optimization, we provide a polynomial-time algorithm (in the number of users) that, first finds the optimal rate allocations among these users, then determines an  ...  Thus, we showed that the cut-set bounds (105) for the data exchange problem with linearly coded packets can be achieved via network coding.  ... 
arXiv:1108.6046v1 fatcat:3oheuggwrfgovo675otofw2lqa

Solving Multilabel Graph Cut Problems with Multilabel Swap

Peter Carr, Richard Hartley
2009 2009 Digital Image Computing: Techniques and Applications  
Approximate solutions to labelling problems can be found using binary graph cuts and either the α-expansion or α − β swap algorithms.  ...  In some specific cases, an exact solution can be computed by constructing a multilabel graph.  ...  However, since message passing algorithms do not place conditions on the types of functions that can be minimized, Gould et al. did not consider whether the restriction of a submodular function remains  ... 
doi:10.1109/dicta.2009.90 dblp:conf/dicta/CarrH09b fatcat:kqvqkon26fcpvolqql5pkjcvtm

Discrete Energy Minimization, beyond Submodularity: Applications and Approximations [article]

Shai Bagon
2012 arXiv   pre-print
This work motivates the use of such "hard-to-optimize" non-submodular functionals, and proposes methods and algorithms to cope with the NP-hardness of their optimization.  ...  As the energies become less constrained and structured one gains more expressive power for the objective function achieving more accurate models.  ...  Experimental Results Our experiments has two main goals: first, to stress the difficulty of approximating non-submodular energies and to show the advantages of primal methods for this type of minimization  ... 
arXiv:1210.7362v2 fatcat:6txhssbravatphb7tu6tedp6rm
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