Related papers: Landscape complexity beyond invariance and the ela…
We obtain explicit expressions for the annealed complexities associated respectively with the total number of (i) stationary points and (ii) local minima of the energy landscape for an elastic manifold with internal dimension $d<4$ embedded…
This is the second of a series of three papers about the Elastic Manifold model. This classical model proposes a rich picture due to the competition between the inherent disorder and the smoothing effect of elasticity. In this paper, we…
We start with a rather detailed, general discussion of recent results of the replica approach to statistical mechanics of a single classical particle placed in a random $N (\gg 1)$-dimensional Gaussian landscape and confined by a…
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…
In a topological dynamical system the complexity of an orbit is a measure of the amount of information (algorithmic information content) that is necessary to describe the orbit. This indicator is invariant up to topological conjugation. We…
We study the landscapes of multi-species spherical spin glasses. Our results determine the phase boundary for annealed trivialization of the number of critical points, and establish its equivalence with a quenched strong topological…
We present a method to obtain the average and the typical value of the number of critical points of the empirical risk landscape for generalized linear estimation problems and variants. This represents a substantial extension of previous…
Finding the mean of the total number N_{tot} of critical points for N-dimensional random energy landscapes is reduced to averaging the absolute value of characteristic polynomial of the corresponding Hessian. For any finite N we provide the…
Systems with a complex dynamics like glasses or models of biological evolution are often pictured in terms of complex landscapes, with a large number of possible collective states. We show on the example of a stochastic spin model with…
Random neural networks are dynamical descriptions of randomly interconnected neural units. These show a phase transition to chaos as a disorder parameter is increased. The microscopic mechanisms underlying this phase transition are unknown,…
We introduce amorphic complexity as a new topological invariant that measures the complexity of dynamical systems in the regime of zero entropy. Its main purpose is to detect the very onset of disorder in the asymptotic behaviour. For…
We apply a recently developed measure of multiscale complexity to the Gaussian model consisting of continuous spins with bilinear interactions for a variety of interaction matrix structures. We find two universal behaviors of the complexity…
We use the Kac-Rice formula and results from random matrix theory to obtain the average number of critical points of a family of high-dimensional empirical loss functions, where the data are correlated $d$-dimensional Gaussian vectors,…
An anomalous mean-field solution is known to capture the non trivial phase diagram of the Ising model in annealed complex networks. Nevertheless the critical fluctuations in random complex networks remain mean-field. Here we show that a…
We consider a nonlinear autonomous random dynamical system of $N$ degrees of freedom coupled by Gaussian random interactions and characterized by a continuous spectrum $n_{\mu}(\lambda)$ of real positive relaxation rates. Using Kac-Rice…
We study the landscape complexity of the Hamiltonian $X_N(x) +\frac\mu2 \|x\|^2,$ where $X_{N}$ is a smooth Gaussian process with isotropic increments on $\mathbb R^{N}$. This model describes a single particle on a random potential in…
Topological invariants have proved useful for analyzing emergent function as they characterize a property of the entire system, and are insensitive to local details, disorder, and noise. They support boundary states, which reduce the system…
Understanding crystal growth over arbitrary curved surfaces with arbitrary boundaries is a formidable challenge, stemming from the complexity of formulating non-linear elasticity using geometric invariant quantities. Solutions are generally…
In this paper we follow up the study of 'complex complex landscapes,' rugged landscapes of many complex variables. Unlike real landscapes, the classification of saddles by index is trivial. Instead, the spectrum of fluctuations at…
We combine an analytically solvable mean-field elasto-plastic model with molecular dynamics simulations of a generic glass-former to demonstrate that, depending on their preparation protocol, amorphous materials can yield in two…