Gradient descent dynamics and the jamming transition in infinite dimensions

Abstract

Gradient descent dynamics in complex energy landscapes, i.e. featuring multiple minima, finds application in many different problems, from soft matter to machine learning. Here, we analyze one of the simplest examples, namely that of soft repulsive particles in the limit of infinite spatial dimension d. The gradient descent dynamics then displays a jamming transition: at low density, it reaches zero-energy states in which particles’ overlaps are fully eliminated, while at high density the energy remains finite and overlaps persist. At the transition, the dynamics becomes critical. In the d → ∞ limit, a set of self-consistent dynamical equations can be derived via mean field theory. We analyze these equations and we present some partial progress towards their solution. We also study the random Lorentz gas in a range of d = 2…22, and obtain a robust estimate for the jamming transition in d → ∞. The jamming transition is analogous to the capacity transition in supervised learning, and in the appendix we discuss this analogy in the case of a simple one-layer fully-connected perceptron.

A continuous constraint satisfaction problem for the rigidity transition in confluent tissues

“…In the following we will solve the thermodynamics of the model at zero temperature. We underline that a dynamical treatment of the model through dynamical mean field theory [29][30][31] is possible and will be developed in a forthcoming work. It will allow to include in the analysis the effect of non-conservative forces (shear drive or activity) and to discuss its differences with thermal noise (whose effect can be instead described by the formalism here below).…”

“…Anyway figure 3 shows that the two transition are very close to each other and that the shape index found in the UNSAT phase in local minima of the energy landscape is close to the thermodynamic value. A detailed analysis of out of equilibrium dynamics in the large N limit is certainly possible with dynamical mean field theory text colored [29][30][31] and will be presented in a forthcoming work. This will be also important to establish how much the thermodynamic SAT/UNSAT rigidity transition is close to the one of out-of-equilibrium greedy search algorithms.…”