Abstract
We investigate the properties of local minima of the energy landscape of a continuous non-convex optimization problem, the spherical perceptron with piecewise linear cost function and show that they are critical, marginally stable and displaying a set of pseudogaps, singularities and non-linear excitations whose properties appear to be in the same universality class of jammed packings of hard spheres. The piecewise linear perceptron problem appears as an evolution of the purely linear perceptron optimization problem that has been recently investigated in [1]. Its cost function contains two non-analytic points where the derivative has a jump. Correspondingly, in the non-convex/glassy phase, these two points give rise to four pseudogaps in the force distribution and this induces four power laws in the gap distribution as well. In addition one can define an extended notion of isostaticity and show that local minima appear again to be isostatic in this phase. We believe that our results generalize naturally to more complex cases with a proliferation of non-linear excitations as the number of non-analytic points in the cost function is increased.
Highlights
Marginal stability of hard sphere packings at jamming has been the subject of an intensive line of studies in the last twenty years [2, 3]
In order to characterize local minima of the energy landscape we look at the distribution of gap variables
Our numerical results suggest that again the glassy phase is isostatic and marginally stable
Summary
Marginal stability of hard sphere packings at jamming has been the subject of an intensive line of studies in the last twenty years [2, 3]. The critical exponents controlling the excitations’ density appear to be the same (within numerical precision) to the ones of the jamming point of hard spheres It follows that jamming criticality is inherently linked to the non-analyticity of the interaction potential. We show that for each non-analytic point of the cost function, two pseudogaps emerge whose critical exponents appear to be the same as the ones controlling the jamming transition. This implies a proliferation of non-linear excitations that can trigger plastic events when the system is perturbed in some way [10]. Our results reinforce the fact that jamming criticality does not pertain only to the jamming point but it is rather related to two concomitant ingredients: the singular nature of the cost function and the non-convex nature of the problem
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