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On the Noise Sensitivity of Monotone Functions [chapter]

Elchanan Mossel, Ryan O'Donnell
2002 Mathematics and Computer Science II  
It is natural to ask if monotone functions can be as sensitive to noise as non-monotone functions.  ...  By Parseval's identity, Sensitivity of monotone functions The parity function, f = u [n] = ⊕, is the boolean function most sensitive to noise: Z(f, η) = η n is minimal, and I(f ) = n is maximal.  ...  Since (14) happen with probability at least .85, it suffices to prove E x,x [(13)] ≥ Ω(1) conditioned on these three events holding. But in this case,  ... 
doi:10.1007/978-3-0348-8211-8_30 fatcat:6z3hiokzufaytjl3nz7t2klefq

On the noise sensitivity of monotone functions

Elchanan Mossel, Ryan O'Donnell
2003 Random structures & algorithms (Print)  
It is natural to ask if monotone functions can be as sensitive to noise as non-monotone functions.  ...  By Parseval's identity, Sensitivity of monotone functions The parity function, f = u [n] = ⊕, is the boolean function most sensitive to noise: Z(f, η) = η n is minimal, and I(f ) = n is maximal.  ...  Since (14) happen with probability at least .85, it suffices to prove E x,x [(13)] ≥ Ω(1) conditioned on these three events holding. But in this case,  ... 
doi:10.1002/rsa.10097 fatcat:x6cuxr4dsvbwbdwvqgszrf5any

Approximating the Noise Sensitivity of a Monotone Boolean Function

Ronitt Rubinfeld, Arsen Vasilyan, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization  
The noise sensitivity of a Boolean function f : {0, 1} n → {0, 1} is one of its fundamental properties. For noise parameter δ, the noise sensitivity is denoted as N S δ [f ].  ...  Therefore, we focus on estimating the noise sensitivity of monotone functions, which form an important subclass of Boolean functions, since many functions of interest are either monotone or can be simply  ...  For monotone functions, we are not aware of previous lower bounds on approximating the bias or noise sensitivity.  ... 
doi:10.4230/lipics.approx-random.2019.52 dblp:conf/approx/RubinfeldV19 fatcat:h43ul6mp55f3hej4xohirabb24

On the noise sensitivity of monotone functions

Elchanan Mossel, Ryan O'Donnell
2018
We also study the problem of achieving the best dependence on δ in the case that the noise rate ϵ is at least a small constant; the results we obtain are tight to within logarithmic factors.  ...  It is known that for all monotone functions f : {0, 1}n → {0, 1}, if x ∈ {0, 1}n is chosen uniformly at random and y is obtained from x by flipping each of the bits of x independently with probability  ...  Acknowledgments: We would like to thank Gil Kalai for providing encouragement to write this paper and for Remark 2, Yuval Peres for interesting discussions, and the anonymous referees for helpful comments  ... 
doi:10.1184/r1/6607997 fatcat:stwnb6w7ejdm3nshhpydjn4hce

Effect of background noise on neuronal coding of interaural level difference cues in rat inferior colliculus

Yasamin Mokri, Kate Worland, Mark Ford, Ramesh Rajan
2015 European Journal of Neuroscience  
To investigate the neural mechanisms underlying this, we studied the effect of background wide-band noise on neural sensitivity to variations in interaural level difference (ILD), the predominant cue for  ...  At high signal-to-noise ratios (SNR), in most ILD functions (41%), the effect of background noise appeared to be due to effects on inputs from both ears, while for a large percentage (35.8%) appeared to  ...  change in the shape, namely category, of the ILD function.  ... 
doi:10.1111/ejn.12914 pmid:25865218 pmcid:PMC4676293 fatcat:jleyelgq4rf3pett5fw6dxjn4y

Dynamical systems analysis of stack filters [article]

Matti Nykter, Juha Kesseli, Ilya Shmulevich
2009 arXiv   pre-print
of Boolean networks constructed from the corresponding monotone functions.  ...  We also examine the dynamical behavior of monotone functions corresponding to filters that are optimal in terms of their noise suppression capability, and find that the optimal filters are dynamically  ...  The class of monotone Boolean functions is one of the most widely used and studied classes of Boolean functions.  ... 
arXiv:0907.4551v1 fatcat:iputa3v2qnah7dpu2mrmg6ood4

Pivotality versus noise stability for monotone transitive functions [article]

Pál Galicza
2019 arXiv   pre-print
We show that such a sequence is volatile as well, and thus it is also an example of a volatile and noise stable sequence of transitive, monotone functions.  ...  We construct a noise stable sequence of transitive, monotone increasing Boolean functions f_n: {-1,1}^k_n⟶{-1,1} which admit many pivotals with high probability.  ...  Research supported by ERC Consolidator Grant 772466 NOISE.  ... 
arXiv:1909.05375v1 fatcat:d4btxwoam5gbzadocqq7sboedy

About the sensitivity of ordinal classifiers to non-monotone noise

Irena Milstein, Arie Ben David, Rob Potharst
2015 Artificial intelligence research  
This research investigates for the first time the effect of non-monotone noise on the accuracy related rankings of ten classifiers in a controlled manner.  ...  The findings of this experiment are reported here. They clearly show that some models are more sensitive than others to non-monotone noise.  ...  The experiment The purpose of the experiment was to check the sensitivity of various classifiers to non-monotone noise.  ... 
doi:10.5430/air.v4n2p83 fatcat:wf57nsn2i5gofh5w3c7gr2jaga

Measuring Environmental Quantum Noise Exhibiting a Nonmonotonic Spectral Shape

Y. Romach, A. Lazariev, I. Avrahami, F. Kleißler, S. Arroyo-Camejo, N. Bar-Gill
2019 Physical Review Applied  
noise spectroscopy: spectral decomposition based on the CPMG sequence, the recently presented DYSCO sequence and a modified DYSCO sequence with a Gaussian envelope (gDYSCO).The performance of the sequences  ...  is quantified by analytic and numeric determination of the frequency resolution, bandwidth and sensitivity, revealing a supremacy of gDYSCO to reconstruct non-trivial features.Utilizing an ensemble of  ...  When statistical noise is added, the limited sensitivity of gDYSCO alters the results, but the peak is still clearly visible as most of the noise affects the monotonous "background".  ... 
doi:10.1103/physrevapplied.11.014064 fatcat:6ycgfxpuzzdgfpca2hitwkxv5q

Noise Sensitivity of Boolean Functions and Applications to Percolation [article]

Itai Benjamini, Gil Kalai, Oded Schramm
2000 arXiv   pre-print
On the other hand, weighted majority functions are shown to be noise-stable. Several necessary and sufficient conditions for noise sensitivity and stability are given.  ...  It is shown that a large class of events in a product probability space are highly sensitive to noise, in the sense that with high probability, the configuration with an arbitrary small percent of random  ...  Many of the results on influence and on noise sensitivity should be extendible to measures on Ω n for which the coordinate variables are positively associated, namely, measures for which every two monotone  ... 
arXiv:math/9811157v2 fatcat:akp3je2gcbamdkrpma2kq4yfma

Book Review: Noise sensitivity of Boolean functions and percolation

Eviatar B. Procaccia
2017 Bulletin of the American Mathematical Society  
The catch phrase one can learn from Benjamini, Kalai, and Schramm [3] is that functions of high frequency are noise sensitive and functions of low frequencies are noise stable.  ...  We are now in a position to state one of the main theorems in the theory of noise sensitivity: Theorem 7 ([3, Theorem 1.3]). Let {f n } be a sequence of Boolean functions.  ... 
doi:10.1090/bull/1591 fatcat:juxmi6rpqvf2lonaqdijqtugye

Optimal Noise-Adding Mechanism in Additive Differential Privacy [article]

Quan Geng, Wei Ding, Ruiqi Guo, Sanjiv Kumar
2019 arXiv   pre-print
on the origin occurs when δ > p/p+1.  ...  We explicitly derive the optimal noise distribution for general ℓ^p cost functions, including ℓ^1 (for noise magnitude) and ℓ^2 (for noise power) cost functions, and show that the probability concentration  ...  Acknowledgment We would like to thank the anonymous reviewers for their insightful comments and suggestions, which help us improve the presentation of this work.  ... 
arXiv:1809.10224v2 fatcat:gv4bygjpzzfulaqtu6jme7ev7y

The Optimal Noise-Adding Mechanism in Differential Privacy

Quan Geng, Pramod Viswanath
2016 IEEE Transactions on Information Theory  
We explicitly derive the optimal noise distribution for general p cost functions, including 1 (for noise magnitude) and 2 (for noise power) cost functions, and show that the probability concentration on  ...  Our result demonstrates an improvement over the existing Gaussian mechanisms by a factor of two and three for (0, δ)differential privacy in the high privacy regime in the context of minimizing the noise  ...  Acknowledgment We would like to thank the anonymous reviewers for their insightful comments and suggestions, which help us improve the presentation of this work.  ... 
doi:10.1109/tit.2015.2504967 fatcat:r6rbvv472jg27bmzsftjwu6wsm

Noise Sensitivity and Chaos in Social Choice Theory [chapter]

Gil Kalai
2010 Fete of Combinatorics and Computer Science  
In this paper we study the social preferences obtained from monotone neutral social welfare functions for random individual preferences.  ...  > 0, if a small fraction of individuals change their preferences (randomly) the correlation between the resulting social preferences and the original ones tends to zero as the number of individuals in  ...  [Mossel, O'Donnell and Oleszkiewicz] A sequence of strong monotone simple games (G n ) with diminishing individual Banzhaf power has property [IAOUIO].  ... 
doi:10.1007/978-3-642-13580-4_8 fatcat:2ozvqsym5jdmhd53ozwzlfz4ra

Noise Sensitivity of Boolean Functions and Applications to Percolation [chapter]

Itai Benjamini, Gn. Kalai, Oded Schramm
2011 Selected Works of Oded Schramm  
Many of the results on influence and on noise sensitivity should be extendible to measures on .on for which th e coordinate variables are positively associated, namely, measures for which every two monotone  ...  TeO and noise sensitiviry Noise sensitivity seems related to another class of boolean functions -threshold circuits of bounded depths see [36, 22) .  ... 
doi:10.1007/978-1-4419-9675-6_12 fatcat:6gdfk7sjpvbczp7jj25g37xy6u
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