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Boolean Functions With Low Average Sensitivity Depend On Few Coordinates

Ehud Friedgut
1998 Combinatorica  
We show here that if the average sensitivity of f is k then f can be approximated by a function depending on c k coordinates where c is a constant depending only on the accuracy of the approximation but  ...  The average sensitivity of f is the average of the sensitivity of all points in f0;1g n .  ...  Acknowledgments I would like to thank Nati Linial for instructive conversations on this topic.  ... 
doi:10.1007/pl00009809 fatcat:pdidydm3uvaplazrivhp7jqswq

Average Sensitivity and Noise Sensitivity of Polynomial Threshold Functions

Ilias Diakonikolas, Prasad Raghavendra, Rocco A. Servedio, Li-Yang Tan
2014 SIAM journal on computing (Print)  
We give the first non-trivial upper bounds on the Boolean average sensitivity and noise sensitivity of degree-d polynomial threshold functions (PTFs).  ...  Our bound on Boolean noise sensitivity is achieved via a simple reduction from upper bounds on average sensitivity of Boolean PTFs to corresponding bounds on noise sensitivity. * An extended abstract merging  ...  Bounds on average sensitivity have been of use in the structural analysis of Boolean functions (see e.g.  ... 
doi:10.1137/110855223 fatcat:nts2d6cadfdbnp7dxdad52yuzq

Sensitivity as a Complexity Measure for Sequence Classification Tasks [article]

Michael Hahn, Dan Jurafsky, Richard Futrell
2021 arXiv   pre-print
To that end, we show analytically that simple lexical classifiers can only express functions of bounded sensitivity, and we show empirically that low-sensitivity functions are easier to learn for LSTMs  ...  We introduce a theoretical framework for understanding and predicting the complexity of sequence classification tasks, using a novel extension of the theory of Boolean function sensitivity.  ...  We are also grateful to Yi Liao for generous help with running u-PMLM.  ... 
arXiv:2104.10343v1 fatcat:da5pommsjnbdbhrkqhdjzchlym

Degree and Sensitivity: tails of two distributions [article]

Parikshit Gopalan, Rocco Servedio, Avishay Tal, Avi Wigderson
2016 arXiv   pre-print
The sensitivity of a Boolean function f is the maximum over all inputs x, of the number of sensitive coordinates of x.  ...  We postulate a robust analogue of the sensitivity conjecture: if most inputs to a Boolean function f have low sensitivity, then most of the Fourier mass of f is concentrated on small subsets, and present  ...  Lemma 5 . 2 . 52 Every Boolean function f : {0, 1} n → {±1} that depends on all n coordinates has a proper walk of dimension n.  ... 
arXiv:1604.07432v1 fatcat:rb7puf3tbfftphxyou4tc2karu

A tighter bound on the number of relevant variables in a bounded degree Boolean function [article]

Jake Wellens
2019 arXiv   pre-print
A classical theorem of Nisan and Szegedy says that a boolean function with degree d as a real polynomial depends on at most d2^d-1 of its variables.  ...  Here we refine their argument to show that one may take C = 4.416.  ...  for low degree boolean functions.  ... 
arXiv:1903.08214v2 fatcat:mozz6jvvrvbjjiiuykdlxv3bfq

On Graphs and the Gotsman-Linial Conjecture for d = 2 [article]

Hyo Won Kim, Chris Maldonado, Jake Wellens
2017 arXiv   pre-print
On the other hand, we establish an asymptotic form of the conjecture for quadratic threshold functions whose non-zero quadratic terms define a graph with either low fractional chromatic number or few edges  ...  Introduction An useful way to measure the complexity of a boolean function f : {−1, 1} n → {−1, 1} is through its influence or average sensitivity, defined as Identifying boolean functions with subsets  ...  This paper is concerned with upper bounding the influence of a class of boolean functions called quadratic threshold functions.  ... 
arXiv:1709.06650v1 fatcat:jyh6wyliifesvm6i2gmwpni3uy

A Low-Depth Monotone Function that is not an Approximate Junta [article]

Daniel M. Kane
2013 arXiv   pre-print
We provide an example of a monotone Boolean function on the hypercube given by a low depth decision tree that is not well approximated by any k-junta for small k.  ...  This research was done with the support of an NSF postdoctoral fellowship.  ...  Acknowledgements I would like to thank Ryan O'Donnell for making me aware of this problem, and for his help with finding appropriate references for this paper.  ... 
arXiv:1206.6541v2 fatcat:voadm5hhpfhgdngfmhn2ayntfa

Sensitivity as a Complexity Measure for Sequence Classification Tasks

Michael Hahn, Dan Jurafsky, Richard Futrell
2021 Transactions of the Association for Computational Linguistics  
To that end, we show analytically that simple lexical classifiers can only express functions of bounded sensitivity, and we show empirically that low-sensitivity functions are easier to learn for LSTMs  ...  We introduce a theoretical framework for understanding and predicting the complexity of sequence classification tasks, using a novel extension of the theory of Boolean function sensitivity.  ...  On the other hand, low-sensitivity functions can be approximated with linear functions or linear combinations of functions that each only combine a few input bits (O'Donnell, 2014, Thm. 2.38) .  ... 
doi:10.1162/tacl_a_00403 fatcat:nhvhnvegufa7lo4u46j52hsmem

Phase Transitions in the Multi-cellular Regulatory Behavior of Pancreatic Islet Excitability

Thomas H. Hraha, Matthew J. Westacott, Marina Pozzoli, Aleena M. Notary, P. Mason McClatchey, Richard K. P. Benninger, Leon Glass
2014 PLoS Computational Biology  
We further explain how islets utilize this critical behavior to leverage cellular heterogeneity and coordinate a robust insulin response with high dynamic range.  ...  Despite its importance, how network dynamics generate this emergent binary on/off behavior remains to be elucidated.  ...  Data are averaged over 5000 simulations. C) Dependence of cluster activity on the cluster size for two different values of P exc , where Sp = 0.15.  ... 
doi:10.1371/journal.pcbi.1003819 pmid:25188228 pmcid:PMC4154652 fatcat:qdckol5jcfbyllbkaa5qdlsw4q

On the Sensitivity Conjecture for Disjunctive Normal Forms [article]

Karthik C. S., Sébastien Tavenas
2016 arXiv   pre-print
[ITCS '16] showed that every Boolean function f is uniquely specified by its values on a Hamming ball of radius at most 2s(f).  ...  The sensitivity conjecture of Nisan and Szegedy [CC '94] asks whether for any Boolean function f, the maximum sensitivity s(f), is polynomially related to its block sensitivity bs(f), and hence to other  ...  Acknowledgements We would like to thank Satya Lokam for discussions and encouraging us to work on the sensitivity conjecture.  ... 
arXiv:1607.05189v2 fatcat:w5lc632b6bd7jp4zmsv4yuro5q

Quantum speedups need structure [article]

Nathan Keller, Ohad Klein
2019 arXiv   pre-print
to a Boolean input and let ϵ,δ > 0.  ...  Then there exists a variable x_i whose influence on f is at least poly(Var(f)/d).  ...  For more information on the quantum-theoretic implications of Conjecture 1.5 we refer the reader to [5] .  ... 
arXiv:1911.03748v2 fatcat:2bnrj3de2jdwzhd7buqoafjeey

Page 4608 of Mathematical Reviews Vol. , Issue 99g [page]

1999 Mathematical Reviews  
boundedness of Riesz transforms.” $99:28021 28A99 0S5A16 05C80 28A60 Friedgut, Ehud (IL-HEBR-IM; Jerusalem) Boolean functions with low average sensitivity depend on few coordinates.  ...  We show here that if the average sensitivity of f is k then f can be approx- imated by a function depending on c* coordinates where ¢ is a constant depending only on the accuracy of the approximation but  ... 

Neutral Mutations and Punctuated Equilibrium in Evolving Genetic Networks

Stefan Bornholdt, Kim Sneppen
1998 Physical Review Letters  
Boolean networks may be viewed as idealizations of biological genetic networks, where each node is represented by an on-off switch which is a function of the binary output from some other nodes.  ...  We evolve connectivity in a single Boolean network, and demonstrate how the sole requirement of sequential matching of attractors may open for an evolution that exhibits punctuated equilibrium.  ...  Further the number of attractors, their length, and the length of the transients to reach these are known to depend strongly on the coordination number [13] .  ... 
doi:10.1103/physrevlett.81.236 fatcat:ebrpi3ekhbaqdoany5sewnldza

Spectral Features Of Higher-Order Side-Channel Countermeasures

Vittorio Zaccaria, Filippo Melzani, Guido Bertoni
2018 Zenodo  
This brief deals with the problem of mathematically formalizing hardware circuits' vulnerability to side-channel attacks.  ...  However, while building this new theoretical framework, we discovered that it can consistently integrate known results about linear ones as well.  ...  The question we want to 12 . each leakage depends on 9 variables, the first 3 of which are the sensitive ones.  ... 
doi:10.5281/zenodo.1134753 fatcat:wdc6nlu5dvfizescpxdoau56y4

A communication game related to the sensitivity conjecture [article]

Justin Gilmer, Michal Koucký, Michael Saks
2015 arXiv   pre-print
One of the major outstanding foundational problems about boolean functions is the sensitivity conjecture, which (in one of its many forms) asserts that the degree of a boolean function (i.e. the minimum  ...  are adjacent if they differ in exactly one coordinate and their function values are different).  ...  Note that E(Π) is exactly the set of sensitive edges for the OR function. However, the average costc(Π) is n/2 whereas the average sensitivity of the OR function is o(1).  ... 
arXiv:1511.07729v1 fatcat:3qcibujuczhhxhenplqsgrev7u
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