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arXiv · 1806.05294

Hessian spectrum at the global minimum of high-dimensional random landscapes

Yan V Fyodorov·Pierre Le Doussal
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Abstract

Using the replica method we calculate the mean spectral density of the Hessian matrix at the global minimum of a random $N \gg 1$ dimensional isotropic, translationally invariant Gaussian random landscape confined by a parabolic potential with fixed curvature $μ>0$. Simple landscapes with generically a single minimum are typical for $μ>μ_{c}$, and we show that the Hessian at the global minimum is always {\it gapped}, with the low spectral edge being strictly positive. When approaching from above the transitional point $μ= μ_{c}$ separating simple landscapes from 'glassy' ones, with exponentially abundant minima, the spectral gap vanishes as $(μ-μ_c)^2$. For $μ<μ_c$ the Hessian spectrum is qualitatively different for 'moderately complex' and 'genuinely complex' landscapes. The former are typical for short-range correlated random potentials and correspond to 1-step replica-symmetry breaking mechanism. Their Hessian spectra turn out to be again gapped, with the gap vanishing on approaching $μ_c$ from below with a larger critical exponent, as $(μ_c-μ)^4$. At the same time in the 'most complex' landscapes with long-ranged power-law correlations the replica symmetry is completely broken. We show that in that case the Hessian remains gapless for all values of $μ<μ_c$, indicating the presence of 'marginally stable' spatial directions. Finally, the potentials with {\it logarithmic} correlations share both 1RSB nature and gapless spectrum. The spectral density of the Hessian always takes the semi-circular form, up to a shift and an amplitude that we explicitly calculate.

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Yan V Fyodorov, Pierre Le Doussal. 2018-06-16. Hessian spectrum at the global minimum of high-dimensional random landscapes. https://doi.org/10.1088/1751-8121%2Faae74f

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