Hessian spectrum at the global minimum of highdimensional random landscapes
Abstract
Using the replica method we calculate the mean spectral density of the Hessian matrix at the global minimum of a random dimensional isotropic, translationally invariant Gaussian random landscape confined by a parabolic potential with fixed curvature . Simple landscapes with generically a single minimum are typical for , and we show that the Hessian at the global minimum is always gapped, with the low spectral edge being strictly positive. When approaching from above the transitional point separating simple landscapes from ‘glassy’ ones, with exponentially abundant minima, the spectral gap vanishes as . For the Hessian spectrum is qualitatively different for ‘moderately complex’ and ‘genuinely complex’ landscapes. The former are typical for shortrange correlated random potentials and correspond to onestep replicasymmetry breaking mechanism. Their Hessian spectra turn out to be again gapped, with the gap vanishing on approaching from below with a larger critical exponent, as . At the same time in the ‘most complex’ landscapes with longranged powerlaw correlations the replica symmetry is completely broken. We show that in that case the Hessian remains gapless for all values of , indicating the presence of ‘marginally stable’ spatial directions. Finally, the potentials with logarithmic correlations share both 1RSB nature and gapless spectrum. The spectral density of the Hessian always takes the semicircular form, up to a shift and an amplitude that we explicitly calculate.
 Publication:

Journal of Physics A Mathematical General
 Pub Date:
 November 2018
 DOI:
 10.1088/17518121/aae74f
 arXiv:
 arXiv:1806.05294
 Bibcode:
 2018JPhA...51U4002F
 Keywords:

 Condensed Matter  Disordered Systems and Neural Networks;
 Condensed Matter  Statistical Mechanics;
 Mathematical Physics
 EPrint:
 28 pages, 1 figure