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Boolean Functions With Low Average Sensitivity Depend On Few Coordinates

1998
*
Combinatorica
*

We show here that if the

doi:10.1007/pl00009809
fatcat:pdidydm3uvaplazrivhp7jqswq
*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. ...##
###
Average Sensitivity and Noise Sensitivity of Polynomial Threshold Functions

2014
*
SIAM journal on computing (Print)
*

We give the first non-trivial upper bounds

doi:10.1137/110855223
fatcat:nts2d6cadfdbnp7dxdad52yuzq
*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. ...##
###
Sensitivity as a Complexity Measure for Sequence Classification Tasks
[article]

2021
*
arXiv
*
pre-print

To that end, we show analytically that simple lexical classifiers can only express

arXiv:2104.10343v1
fatcat:da5pommsjnbdbhrkqhdjzchlym
*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. ...##
###
Degree and Sensitivity: tails of two distributions
[article]

2016
*
arXiv
*
pre-print

The

arXiv:1604.07432v1
fatcat:rb7puf3tbfftphxyou4tc2karu
*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. ...##
###
A tighter bound on the number of relevant variables in a bounded degree Boolean function
[article]

2019
*
arXiv
*
pre-print

A classical theorem of Nisan and Szegedy says that a

arXiv:1903.08214v2
fatcat:mozz6jvvrvbjjiiuykdlxv3bfq
*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*. ...##
###
On Graphs and the Gotsman-Linial Conjecture for d = 2
[article]

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*. ...

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

2013
*
arXiv
*
pre-print

We provide an example of a monotone

arXiv:1206.6541v2
fatcat:voadm5hhpfhgdngfmhn2ayntfa
*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. ...##
###
Sensitivity as a Complexity Measure for Sequence Classification Tasks

2021
*
Transactions of the Association for Computational Linguistics
*

To that end, we show analytically that simple lexical classifiers can only express

doi:10.1162/tacl_a_00403
fatcat:nhvhnvegufa7lo4u46j52hsmem
*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) . ...##
###
Phase Transitions in the Multi-cellular Regulatory Behavior of Pancreatic Islet Excitability

2014
*
PLoS Computational Biology
*

We further explain how islets utilize this critical behavior to leverage cellular heterogeneity and

doi:10.1371/journal.pcbi.1003819
pmid:25188228
pmcid:PMC4154652
fatcat:qdckol5jcfbyllbkaa5qdlsw4q
*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. ...##
###
On the Sensitivity Conjecture for Disjunctive Normal Forms
[article]

2016
*
arXiv
*
pre-print

[ITCS '16] showed that every

arXiv:1607.05189v2
fatcat:w5lc632b6bd7jp4zmsv4yuro5q
*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. ...##
###
Quantum speedups need structure
[article]

2019
*
arXiv
*
pre-print

to a

arXiv:1911.03748v2
fatcat:2bnrj3de2jdwzhd7buqoafjeey
*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] . ...##
###
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

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] . ...

##
###
Spectral Features Of Higher-Order Side-Channel Countermeasures

2018
*
Zenodo
*

This brief deals

doi:10.5281/zenodo.1134753
fatcat:wdc6nlu5dvfizescpxdoau56y4
*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*. ...##
###
A communication game related to the sensitivity conjecture
[article]

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). ...

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