Computing with Noise: Phase Transitions in Boolean Formulas

Mozeika, Alexander; Saad, David; Raymond, Jack

doi:10.1103/physrevlett.103.248701

Cited by 11 publications

(26 citation statements)

References 15 publications

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“…This result is identical to those reported in [5,6], and reduces to ϵ * (3) = 1/6 for k = 3; the non-vanishing magnetization values then become [16] …”

Section: Critical Behavior In Maj-k Based Formulaesupporting

confidence: 85%

“…While in this paper we concentrated on the majority gates, we have looked at other gate types [16], hard noise (fabrication errors) and other properties of noisy circuits [18]. We believe that much can be explored about the properties of noisy circuits using the methodology developed here, for instance, the type of functions generated depending on the gate types and the level of gate noise.…”

Section: Discussionmentioning

confidence: 99%

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On the robustness of random Boolean formulae

Mozeika

Saad

Raymond

2010

J. Phys.: Conf. Ser.

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Abstract. Random Boolean formulae, generated by a growth process of noisy logical gates are analyzed using the generating functional methodology of statistical physics. We study the type of functions generated for different input distributions, their robustness for a given level of gate error and its dependence on the formulae depth and complexity and the gates used. Bounds on their performance, derived in the information theory literature for specific gates, are straightforwardly retrieved, generalized and identified as the corresponding typical-case phase transitions. Results for error-rates, function-depth and sensitivity of the generated functions are obtained for various gate-type and noise models.

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“…This result is identical to those reported in [5,6], and reduces to ϵ * (3) = 1/6 for k = 3; the non-vanishing magnetization values then become [16] …”

Section: Critical Behavior In Maj-k Based Formulaesupporting

confidence: 85%

Section: Discussionmentioning

confidence: 99%

On the robustness of random Boolean formulae

Mozeika

Saad

Raymond

2010

J. Phys.: Conf. Ser.

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“…While no previous result provides such a formula, our work is closely related to the following. Mozeika and Saad [10,14,15] give a powerful generating function framework for analysis of Boolean networks, but do not characterize short-term stability. Seshadhri et al [11] introduced the notion of influence I(F) of transfer function distribution F, an easily computable quantity that determines the short-term behavior for a highly restricted class of balanced families F: on average, functions in F are equally likely to output +1 and −1.…”

Section: Introductionmentioning

confidence: 99%

Characterizing short-term stability for Boolean networks over any distribution of transfer functions

Seshadhri¹,

Smith²,

Vorobeychik³

et al. 2016

Phys. Rev. E

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We present a characterization of short-term stability of Kauffman's N-K (random) Boolean networks under arbitrary distributions of transfer functions. Given such a Boolean network where each transfer function is drawn from the same distribution, we present a formula that determines whether short-term chaos (damage spreading) will happen. Our main technical tool which enables the formal proof of this formula is the Fourier analysis of Boolean functions, which describes such functions as multilinear polynomials over the inputs. Numerical simulations on mixtures of threshold functions and nested canalyzing functions demonstrate the formula's correctness. To the best of our knowledge, previous formal characterizations only worked for special cases of "balanced" families.

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“…To study the entropy of functions realized by DNN [16], we adopted similar assumptions but employed the generating functional analysis [19,20], which is more general and can be applied to sparse and weight-correlated networks. The analysis of function error incurred by weight perturbations exhibits an exponential growth in error for DNN with sign activation functions, while networks with ReLU activation function are more robust to perturbations.…”

Section: Introductionmentioning

confidence: 99%

Large deviation analysis of function sensitivity in random deep neural networks

Saad

2020

J. Phys. A: Math. Theor.

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Mean field theory has been successfully used to analyze deep neural networks (DNN) in the infinite size limit. Given the finite size of realistic DNN, we utilize the large deviation theory and path integral analysis to study the deviation of functions represented by DNN from their typical mean field solutions. The parameter perturbations investigated include weight sparsification (dilution) and binarization, which are commonly used in model simplification, for both ReLU and sign activation functions. We find that random networks with ReLU activation are more robust to parameter perturbations with respect to their counterparts with sign activation, which arguably is reflected in the simplicity of the functions they generate.Keywords: large deviation theory, path integral, deep neural networks, function sensitivity ‡ The usual bias variables are omitted for simplicity, but it can be easily accommodated within the current framework.

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Computing with Noise: Phase Transitions in Boolean Formulas

Cited by 11 publications

References 15 publications

On the robustness of random Boolean formulae

On the robustness of random Boolean formulae

Characterizing short-term stability for Boolean networks over any distribution of transfer functions

Large deviation analysis of function sensitivity in random deep neural networks

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