Complexity of random smooth functions on the high-dimensional sphere

Antonio Auffinger, Gerard Ben Arous

Research output: Contribution to journalArticle

Abstract

We analyze the landscape of general smooth Gaussian functions on the sphere in dimension N, when N is large. We give an explicit formula for the asymptotic complexity of the mean number of critical points of finite and diverging index at any level of energy and for the mean Euler characteristic of level sets. We then find two possible scenarios for the bottom landscape, one that has a layered structure of critical values and a strong correlation between indexes and critical values and another where even at levels below the limiting ground state energy the mean number of local minima is exponentially large. We end the paper by discussing how these results can be interpreted in the language of spin glasses models.

Original languageEnglish (US)
Pages (from-to)4214-4247
Number of pages34
JournalAnnals of Probability
Volume41
Issue number6
DOIs
StatePublished - Nov 2013

Fingerprint

Random Function
Smooth function
High-dimensional
Critical value
Gaussian Function
Ground State Energy
Euler Characteristic
Spin Glass
Local Minima
Level Set
Explicit Formula
Critical point
Limiting
Scenarios
Energy
Model
Language

Keywords

  • Critical points
  • Parisi formula
  • Random matrices
  • Sample
  • Spin glasses

ASJC Scopus subject areas

  • Statistics and Probability
  • Statistics, Probability and Uncertainty

Cite this

Complexity of random smooth functions on the high-dimensional sphere. / Auffinger, Antonio; Arous, Gerard Ben.

In: Annals of Probability, Vol. 41, No. 6, 11.2013, p. 4214-4247.

Research output: Contribution to journalArticle

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