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SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization

Adam Kurpisz, Aaron Potechin, Elias Samuel Wirth, Nikhil Bansal, Emanuela Merelli, James Worrell
2021
We connect this result to two constrained Boolean hypercube optimization problems.  ...  We study the rank of the Sum of Squares (SoS) hierarchy over the Boolean hypercube for Symmetric Quadratic Functions (SQFs) in n variables with roots placed in points k-1 and k.  ...  3s. 4.spxq ď 1 2n for all x P r3, ns.90:14 SoS Certification for SQFs and Its Connection to Boolean Hypercube OptimizationProof.  ... 
doi:10.4230/lipics.icalp.2021.90 fatcat:42e7tp6dkvaf3nc4htf7fnomzi

Representing Fitness Landscapes by Valued Constraints to Understand the Complexity of Local Search

Artem Kaznatcheev, David Cohen, Peter Jeavons
2020 The Journal of Artificial Intelligence Research  
First, we show that for fitness landscapes representable by binary Boolean valued constraints there is a minimal necessary constraint graph that can be easily computed.  ...  In the binary Boolean case, we prove that a degree 2 or treestructured constraint graph gives a quadratic bound on the number of improving moves made by any local search; hence, any landscape that can  ...  Artem Kaznatcheev was supported by the Theory Division at the Department of Translational Hematology and Oncology Research, Cleveland Clinic.  ... 
doi:10.1613/jair.1.12156 fatcat:rfzvk7jdgbekteyh5jyy4xg7dy

Representing fitness landscapes by valued constraints to understand the complexity of local search [article]

Artem Kaznatcheev, David A. Cohen, Peter G. Jeavons
2020 arXiv   pre-print
First, we show that for fitness landscapes representable by binary Boolean valued constraints there is a minimal necessary constraint graph that can be easily computed.  ...  In the binary Boolean case, we prove that a degree 2 or tree-structured constraint graph gives a quadratic bound on the number of improving moves made by any local search; hence, any landscape that can  ...  Artem Kaznatcheev was supported by the Theory Division at the Department of Translational Hematology and Oncology Research, Cleveland Clinic.  ... 
arXiv:1907.01218v4 fatcat:mwql5medxvhdpea3zs4ebldsjq

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

Francis Bach
2013 arXiv   pre-print
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  ...  functions provides a useful set of regularization functions for supervised and unsupervised learning.  ...  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

Tight Sum-of-Squares lower bounds for binary polynomial optimization problems [article]

Adam Kurpisz, Samuli Leppänen, Monaldo Mastrolilli
2016 arXiv   pre-print
For binary polynomial optimization problems of degree 2d and an odd number of variables n, we prove that n+2d-1/2 levels of the SoS/Lasserre hierarchy are necessary to provide the exact optimal value.  ...  She showed that the Sherali-Adams hierarchy requires n levels to detect the empty integer hull, and conjectured that the SoS/Lasserre rank for the same problem is n-1.  ...  The authors would like to express their gratitude to Alessio Benavoli for helpful discussions.  ... 
arXiv:1605.03019v1 fatcat:fa4fy753kndgvlk6pals7jjspq

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  ...  and convex optimization.  ...  The source is connected to all subsets G, with capacity D(G), and each subset is connected to the variables it contains, with infinite capacity.  ... 
doi:10.1561/2200000039 fatcat:kk7w6zsnsnbp3eoa6b5ol3bxbq

Positivity-preserving extensions of sum-of-squares pseudomoments over the hypercube [article]

Dmitriy Kunisky
2020 arXiv   pre-print
We trace the reason that our pseudomoments can satisfy both the hypercube and positivity constraints simultaneously to a combinatorial relationship between multiharmonic polynomials and this Möbius function  ...  This connection guides our proof that the pseudomoments satisfy the hypercube constraints.  ...  Acknowledgements I thank Afonso Bandeira for many discussions and comments on an early version of the manuscript, Alex Wein for helpful discussions about hypercontractivity and tensor networks, and Ramon  ... 
arXiv:2009.07269v1 fatcat:7vrqmubewjcwvjakd2doji5xi4

Research on probabilistic methods for control system design

Giuseppe C. Calafiore, Fabrizio Dabbene, Roberto Tempo
2011 Automatica  
A novel approach based on probability and randomization has emerged to synergize with the standard deterministic methods for control of systems with uncertainty.  ...  The focal point is on design methods, based on the interplay between uncertainty randomization and convex optimization, and on the illustration of specific control applications.  ...  The function f : R n θ × Q → {0, 1} is a (α, m)-Boolean if, for fixed q, it can be written as a Boolean expression consisting of Boolean operators involving m polynomials β 1 (θ ), . . . , β m (θ ) in  ... 
doi:10.1016/j.automatica.2011.02.029 fatcat:pyrudohsdbdglpnekzqkh4fgoi

Lifting with Simple Gadgets and Applications to Circuit and Proof Complexity [article]

Susanna F. de Rezende, Or Meir, Jakob Nordström, Toniann Pitassi, Robert Robere, Marc Vinyals
2020 arXiv   pre-print
We significantly strengthen and generalize the theorem lifting Nullstellensatz degree to monotone span program size by Pitassi and Robere (2018) so that it works for any gadget with high enough rank, in  ...  length and subpolynomial line space if coefficients are restricted to be of polynomial magnitude. * We give the first explicit separation between monotone Boolean formulas and monotone real formulas.  ...  It is not hard to see that T has depth at most p and that it solves Search(Q, Z), as required. It is worth mentioning that we can prove Theorem D.1 directly, without going through Lemma D.2.  ... 
arXiv:2001.02144v1 fatcat:urgi4cd5vvczdnsqcnqloxsj6q

Limits on Efficient Computation in the Physical World [article]

Scott Aaronson
2005 arXiv   pre-print
complexity theory to help determine which are which.  ...  In this thesis I show that, while some intuitions from classical computer science must be jettisoned in the light of modern physics, many others emerge nearly unscathed; and I use powerful tools from computational  ...  So to lower-bound the deterministic query complexity, it suffices to lower-bound the size of any cut that splits the graph into two reasonably large components. 1 For the Boolean hypercube, Llewellyn  ... 
arXiv:quant-ph/0412143v2 fatcat:x6mjz4h4gzaszbfgbkshgm2v3u

The Kikuchi Hierarchy and Tensor PCA [article]

Alexander S. Wein, Ahmed El Alaoui, Cristopher Moore
2019 arXiv   pre-print
It is known that AMP, the flagship algorithm of statistical physics, has substantially worse performance than SOS for tensor PCA.  ...  The results we present here apply to tensor PCA for tensors of all orders, and to k-XOR when k is even.  ...  For helpful discussions, we thank Afonso Bandeira, Sam Hopkins, Pravesh Kothari, Florent Krzakala, Tselil Schramm, Jonathan Shi, and Lenka Zdeborová.  ... 
arXiv:1904.03858v2 fatcat:hhudilk5mzc3dd5xwjzbbtlmaq

Lecture Notes on Spectral Graph Methods [article]

Michael W. Mahoney
2016 arXiv   pre-print
So, even though L isn't positive or non-negative, we get Perron-Frobenius style results for it, in addition to the results we get for it since it is a symmetric matrix.  ...  In particular, we can apply Theorem 40 to the optimal solution for LocalSpectral(G, v {u} , 1/k) and obtain a cut T whose conductance is quadratically close to the optimal value λ(G, v {u} , 1/k).  ...  If we expand the objective function and apply the constraint z s = 1, z t = 0, then Prob. (52) becomes: Consider the optimality conditions of this quadratic problem (where s are the Lagrange multipliers  ... 
arXiv:1608.04845v1 fatcat:ppy6mlmfsvfcxedriwnndv6ztq

On randomization in sequential and distributed algorithms

Rajiv Gupta, Scott A. Smolka, Shaji Bhaskar
1994 ACM Computing Surveys  
Included with each algorithm is a discussion of its correctness and its computational complexity.  ...  to men of skdl; but tzme and chance happeneth to them all.  ...  The algorithm is optimal for w = 2, and evidence is supplied that it is optimal for larger values of w.KARGER, D. R. 1993.  ... 
doi:10.1145/174666.174667 fatcat:mwufckvt5vawlostdlhcv7rxwm

Graphical Models, Exponential Families, and Variational Inference

Martin J. Wainwright, Michael I. Jordan
2007 Foundations and Trends® in Machine Learning  
Beyond its use as a language for formulating models, graph theory also plays a fundamental role in assessing computational 3 7  ...  Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations  ...  Acknowledgments A large number of people contributed to the gestation of this survey, and it is a pleasure to acknowledge them here.  ... 
doi:10.1561/2200000001 fatcat:3f33bwasgvg5ndjfqezocaxxfa

Algorithms for Verifying Deep Neural Networks [article]

Changliu Liu, Tomer Arnon, Christopher Lazarus, Clark Barrett, Mykel J. Kochenderfer
2020 arXiv   pre-print
Deep neural networks are widely used for nonlinear function approximation with applications ranging from computer vision to control.  ...  This article surveys methods that have emerged recently for soundly verifying such properties. These methods borrow insights from reachability analysis, optimization, and search.  ...  The authors would also like to thank Christian Schilling, Marcelo Forets, and Sebastian Guadalupe, the authors of LazySets.jl, for their implementation support; Tianhao Wei for his contribution in the  ... 
arXiv:1903.06758v2 fatcat:25pqxtxpfzfz7phnnsx53q3j5y
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