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Spherical Perceptron


The spherical perceptron is a random perceptron whose weight vector satisfies ||x||^2=N. For independent pattern vectors g^a in R^N with independent standard normal coordinates and M_N=|_alphaN_|, a soft-potential thermodynamic pressure is

 p_N=1/Nlnint_(||x||^2=N)exp(betasum_(a=1)^(M_N)phi(g^a·x/sqrt(N)))dsigma_N(x),

where |_x_| is the floor function and sigma_N is uniform probability on the sphere.

For hard margin constraints g^a·x/sqrt(N)>=kappa, the threshold between feasible and infeasible constraints is called jamming.


See also

Ising Perceptron, Random Perceptron, Sphere, Thermodynamic Pressure

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References

Franz, S. and Parisi, G. "The Simplest Model of Jamming." J. Phys. A: Math. Th. 49, 145001, 2016. https://doi.org/10.1088/1751-8113/49/14/145001.Talagrand, M. Mean Field Models for Spin Glasses, Vol. I: Basic Examples. Berlin: Springer, 2011. https://doi.org/10.1007/978-3-642-15202-3.

Cite this as:

Weisstein, Eric W. "Spherical Perceptron." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SphericalPerceptron.html

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