Kac–Rice fixed point analysis for single- and multi-layered complex systems

Ipsen, Jesper R.; Forrester, Peter J.

doi:10.1088/1751-8121/aae76d

Cited by 16 publications

(11 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Recently, a regain of interest on this subject is coming from computer science, and in particular machine learning [13] where many central questions concern the statistical properties of rough high-dimensional landscapes originating from the study of the multi-dimensional profile of loss functions. Concomitantly, advances in probability theory and mathematical physics are currently allowing to put the theoretical physics methods on a firmer basis and to obtain new results [14][15][16][17][18][19][20][21][22][23][24]. Despite this great amount of progress on enumerating and classifying local minima, the characterisation of the typical energy barriers between them is still to a large extent an open question.…”

mentioning

confidence: 99%

Complexity of energy barriers in mean-field glassy systems

Biroli¹,

Cammarota²

2019

EPL

View full text Add to dashboard Cite

We analyze the energy barriers that allow escapes from a given local minimum in a mean-field model of glasses. We perform this study by using the Kac-Rice method and computing the typical number of critical points of the energy function at a given distance from the minimum. We analyze their Hessian in terms of random matrix theory and show that for a certain regime of energies and distances critical points are index-one saddles and are associated to barriers. We find that the lowest barrier, important for activated dynamics at low temperature, is strictly lower than the "threshold" level above which saddles proliferate. We characterize how the quenched complexity of barriers, important for activated process at finite temperature, depends on the energy of the barrier, the energy of the initial minimum, and the distance between them. The overall picture gained from this study is expected to hold generically for mean-field models of the glass transition. arXiv:1809.05440v1 [cond-mat.dis-nn]

show abstract

mentioning

confidence: 99%

Complexity of energy barriers in mean-field glassy systems

Biroli¹,

Cammarota²

2019

EPL

View full text Add to dashboard Cite

show abstract

“…Different recent applications of such multiplicative processes include non-perturbative quantum gravity with space-time foliation [16], or the propagation of information in telecommunication [17,18]. Multiplying M random N × N matrices applies further to multi-layered complex networks with M layers and N degrees of freedom per layer [19], to the complexity of random maps [20], and in computer science (cf. [21]) through machine learning [22].…”

mentioning

confidence: 99%

“…For the interpolating microscopic density at the soft edge this implies lim a→∞ ρ soft (ξ; a) = 2 1/3 Ai 2 (2 1/3 ξ)−2 2/3 ξAi 2 (2 1/3 ξ) . (21) As for the Airy-kernel (20), the spectrum of (19) is not unfolded. For example, for large negative argument ξ, the limiting Airy-density (21) increases as |ξ| to the left, originating from the macroscopic semi-circular law of the GUE.…”

mentioning

confidence: 99%

From integrable to chaotic systems: Universal local statistics of Lyapunov exponents

Akemann¹,

Burda²,

Kieburg³

2019

EPL

View full text Add to dashboard Cite

Systems where time evolution follows a multiplicative process are ubiquitous in physics. We study a toy model for such systems where each time step is given by multiplication with an independent random N ×N matrix with complex Gaussian elements, the complex Ginibre ensemble. This model allows to explicitly compute the Lyapunov exponents and local correlations amongst them, when the number of factors M becomes large. While the smallest eigenvalues always remain deterministic, which is also the case for many chaotic quantum systems, we identify a critical double scaling limit N ∼ M for the rest of the spectrum. It interpolates between the known deterministic behaviour of the Lyapunov exponents for M N (or N fixed) and universal random matrix statistics for M N (or M fixed), characterising chaotic behaviour. After unfolding this agrees with Dyson's Brownian Motion starting from equidistant positions in the bulk and at the soft edge of the spectrum. This universality statement is further corroborated by numerical experiments, multiplying different kinds of random matrices. It leads us to conjecture a much wider applicability in complex systems, that display a transition from deterministic to chaotic behaviour.

show abstract

“…While these results predict large scale properties of the low energy configurations, little is known about the detailed statistical structure of the complex energy landscape of pinned manifolds. This relates to the broad effort of understanding the statistical structure of stationary points (minima, maxima and saddles) of random landscapes which is of steady interest in theoretical physics [30][31][32][33][34][35][36][37][38][39], with recent applications to statistical physics [10,[34][35][36][38][39][40][41], neural networks and complex dynamics [42][43][44][45][46], string theory [47,48] and cosmology [49,50]. It is also of active current interest in pure and applied mathematics [51][52][53][54][55][56][57][58][59][60], For the model (1)- (2) in the simplest case d = 0 (x is a single point), the mean number of stationary points and of minima of the energy function was investigated in the limit of large N 1 in [35,38,39], see also [37,…”

Section: Motivation and Goals Of The Papermentioning

confidence: 99%

Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

Fyodorov

Doussal

2020

J Stat Phys

View full text Add to dashboard Cite

We consider an elastic manifold of internal dimension d and length L pinned in a N dimensional random potential and confined by an additional parabolic potential of curvature µ. We are interested in the mean spectral density ρ(λ) of the Hessian matrix K at the absolute minimum of the total energy. We use the replica approach to derive the system of equations for ρ(λ) for a fixed L d in the N → ∞ limit extending d = 0 results of our previous work [11]. A particular attention is devoted to analyzing the limit of extended lattice systems by letting L → ∞. In all cases we show that for a confinement curvature µ exceeding a critical value µ c , the so-called "Larkin mass", the system is replicasymmetric and the Hessian spectrum is always gapped (from zero). The gap vanishes quadratically at µ → µ c . For µ < µ c the replica symmetry breaking (RSB) occurs and the Hessian spectrum is either gapped or extends down to zero, depending on whether RSB is 1-step or full. In the 1-RSB case the gap vanishes in all d as (µ c − µ) 4 near the transition. In the full RSB case the gap is identically zero. A set of specific landscapes realize the so-called "marginal cases" in d = 1, 2 which share both feature of the 1-step and the full RSB solution, and exhibit some scale invariance. We also obtain the average Green function associated to the Hessian and find that at the edge of the spectrum it decays exponentially in the distance within the internal space of the manifold with a length scale equal in all cases to the Larkin length introduced in the theory of pinning. arXiv:1903.07159v1 [cond-mat.dis-nn]

show abstract

Kac–Rice fixed point analysis for single- and multi-layered complex systems

Cited by 16 publications

References 36 publications

Complexity of energy barriers in mean-field glassy systems

Complexity of energy barriers in mean-field glassy systems

From integrable to chaotic systems: Universal local statistics of Lyapunov exponents

Manifolds Pinned by a High-Dimensional Random Landscape: Hessian at the Global Energy Minimum

Contact Info

Product

Resources

About