Multifractal a priori probability distribution for the perceptron

Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics

Volumn 57, Issue 3, 1998, Pages 3660-3663

  Van den Broeck, C ^a   Bex, G J ^a  

^a HASSELT UNIVERSITY   (Belgium)

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EID: 4243711084     PISSN: 1063651X     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.57.3660     Document Type: Article

References (19)

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15. Note that we have used the notation c(k) rather than the more usual φ(α) because the symbols α and φ are reserved to denote other quantities in the neural network community.
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19. By blindly applying this change of variable on Eq. (7), and solving the saddle point equation for μ, one gets the result given in Eq. (9). One has, however, to remember that q is in fact an integration variable so that the change from q to μ leads to an extra constant factor exp(-2Nln2), i.e., a term -2ln2 of order N. But such a term is clearly wrong since one has to find an exponent equal to 0 for R=1. The point is that to correctly include all the finite order corrections, one has already to do this at the level of the saddle point calculation that leads to Eq. (7). It turns out that this extra correction just yields another constant factor exp(+2Nln2), which exactly cancels the one coming from the change of integration variable.