Complexity of random smooth functions on the high-dimensional sphere

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.Comment: Published in at http://dx.doi.org/10.1214/13-AOP862 the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Manifolds in a high-dimensional random landscape: Complexity of stationary points and depinning

“…Among other approaches, the problem of characterizing complexity of random high-dimensional landscapes by understanding statistical structure of their stationary points (minima, maxima and saddles), defined in our model by the condition δH δu(x) = 0 for all x, has attracted considerable intrinsic interest in recent years in pure and applied mathematics, see e.g. [12][13][14][15][16], as well as in theoretical physics, see [17][18][19][20][21][22][23][24] and references to earlier works in [25].For the energy functional (1)-(2) in the simplest 'toy model' limit d = 0 with no elastic interactions (when x is essentially a single point, L d = 1), the mean number of stationary points, N tot , and of stable equilibria (local minima), N st , of the landscape were investigated in the limit of large N ≫ 1 in [18,21,22]. It was found that a sharp transition occurs from a 'simple' landscape for µ 0 > µ c with typically only a single stationary point (the minimum) to a complex ('glassy') landscapes for µ < µ c with exponentially many stationary points.…”

Complexity of random smooth functions on compact manifolds

“…The distribution of critical points (or energy landscape) of isotropic random functions on R m was investigated by Fyodorov [15,16] who also relates this problem to the staistics of the eigenvalues in the ensemble GOE m+1 . Recently A. Auffinger [3,4] has investigated the distributions of critical values of certain isotropic random fields on a round sphere S m , where m → ∞, and described a connection with the distribution of eigenvalues of symmetric matrices in the ensemble GOE m+1 .…”

“…We have the following technical result whose proof is contained in Appendix A. Following the terminology on [3,4] we will refer to σ u as the variational complexity of u. Observe that supp µ u = Cr(u), supp σ u = D(u).…”

Approximate Ultrametricity for Random Measures and Applications to Spin Glasses

“…: After this paper was first submitted, progress was made by Subag [32] in the setting of spherical models. In particular, in [32], building on the work of [4,5,31,33], Subag shows approximate ultrametricity for pure p-spin models ( .t / Ď 2 t p ) at large values ofˇin a quantitative sense and in the process obtains several sharp results regarding fluctuations of free energies and temperature chaos. (In these models, consists of exactly two atoms.)…”

“…Remark 3.4. We would like to point out that there is no implication between (3) and (4). Observe that (3) does not imply (4) as it does not provide control on the intersections.…”