Concentration of Multilinear Functions of the Ising Model with Applications to Network Data
Abstract
We prove neartight concentration of measure for polynomial functions of the Ising model under high temperature. For any degree $d$, we show that a degree$d$ polynomial of a $n$spin Ising model exhibits exponential tails that scale as $\exp(r^{2/d})$ at radius $r=\tilde{\Omega}_d(n^{d/2})$. Our concentration radius is optimal up to logarithmic factors for constant $d$, improving known results by polynomial factors in the number of spins. We demonstrate the efficacy of polynomial functions as statistics for testing the strength of interactions in social networks in both synthetic and real world data.
 Publication:

arXiv eprints
 Pub Date:
 October 2017
 DOI:
 10.48550/arXiv.1710.04170
 arXiv:
 arXiv:1710.04170
 Bibcode:
 2017arXiv171004170D
 Keywords:

 Mathematics  Probability;
 Computer Science  Machine Learning;
 Mathematical Physics;
 Mathematics  Statistics Theory;
 Statistics  Machine Learning
 EPrint:
 To appear in NIPS 2017