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SoS Certification for Symmetric Quadratic Functions and Its Connection to Constrained Boolean Hypercube Optimization
2021
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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
2020
The Journal of Artificial Intelligence Research
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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]
2020
arXiv
pre-print
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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]
2013
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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
2013
Foundations and Trends® in Machine Learning
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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]
2020
arXiv
pre-print
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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
2011
Automatica
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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]
2020
arXiv
pre-print
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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]
2005
arXiv
pre-print
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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]
2019
arXiv
pre-print
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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]
2016
arXiv
pre-print
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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
1994
ACM Computing Surveys
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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
2007
Foundations and Trends® in Machine Learning
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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]
2020
arXiv
pre-print
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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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