Hessian spectrum at the global minimum of high-dimensional random landscapes
Hessian spectrum at the global minimum of high-dimensional random landscapes
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高维随机景观全局最小值处的 Hessian 谱
DOI：
10.1088/1751-8121/aae74f
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发表时间：
2018
期刊：
影响因子：
--
通讯作者：
P. Le Doussal
中科院分区：
文献类型：
--
作者：
Y. Fyodorov;P. Le Doussal
使用副本的方法，我们计算的平均谱密度的Hessian矩阵的全局最小值的随机维各向同性，几何不变的高斯随机景观所限制的抛物型潜在的固定曲率。简单的景观，一般是一个单一的最小值是典型的，我们表明，在全球最小的海森总是缺口，低光谱边缘是严格的积极。当接近从上面的过渡点分离简单的景观与“玻璃”的，指数丰富的最小值，光谱间隙消失。对于“适度复杂”和“真正复杂”的景观，海森光谱是有质的不同的。前者是典型的短程关联随机势，对应于一步复制对称破缺机制。它们的海森谱又被证明是有间隙的，当从下面以较大的临界指数接近时，差距消失。与此同时，在“最复杂”的景观与长程幂律相关的副本对称性是完全打破。我们表明，在这种情况下，海森仍然没有间隙的所有值，表明存在的“边缘稳定”的空间方向。最后，具有对数相关性的势具有1RSB性质和无隙谱。Hessian的谱密度总是采用半圆形的形式，直到我们明确计算的位移和幅度。
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 short-range correlated random potentials and correspond to one-step replica-symmetry 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 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 , 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 semi-circular form, up to a shift and an amplitude that we explicitly calculate.
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影响因子：
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作者：
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DOI：
10.3150/17-bej964
发表时间：
2018
期刊：
Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability
影响因子：
--
作者：
Cheng,Dan;Schwartzman,Armin
通讯作者：
Schwartzman,Armin