Critical jammed phase of the linear perceptron above saturation

We consider the spherical perceptron problem beyond the limit of capacity and we study the landscape of local minima for the linear cost function. We show that in the replica symmetry broken region of the phase diagram, local minima are critical even far away from jamming. The gap distribution contains an isostatic delta peak of marginally satisfied gaps surrounded by two power laws for the absolute value of both small satisfied (SAT) and unsatisfied (UNSAT) gaps. We propose a scaling theory for the problem and we compute the values of the critical exponents showing that they coincide and are equal to the one that characterizes the distribution of small positive gaps at the jamming transition.