Critical jammed phase of the linear perceptron

Abstract

Criticality in statistical physics naturally emerges at isolated points in the phase diagram. Jamming of spheres is not an exception: varying density, it is the critical point that separates the unjammed phase where spheres do not overlap and the jammed phase where they cannot be arranged without overlaps. The same remains true in more general constraint satisfaction problems with continuous variables (CCSP) where jamming coincides with the (protocol dependent) satisfiability transition point. In this work we show that by carefully choosing the cost function to be minimized, the region of criticality extends to occupy a whole region of the jammed phase. As a working example, we consider the spherical perceptron with a linear cost function in the unsatisfiable (UNSAT) jammed phase and we perform numerical simulations which show critical power laws emerging in the configurations obtained minimizing the linear cost function. We develop a scaling theory to compute the emerging critical exponents.

arXiv comment10 pages, 4 figures

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• Critical energy landscape of linear soft spheres
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• Phase transitions in the mini-batch size for sparse and dense two-layer neural networks
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• Perspective: A Phase Diagram for Deep Learning unifying Jamming, Feature Learning and Lazy Training
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• Critical properties of the SAT/UNSAT transitions in the classification problem of structured data