2018/11/20 by Elena Agliari, Agliari, Elena, Francesco Alemanno +5 · 1 citation
Computer Science · Neuroscience · Physics and Astronomy · #Disordered Systems and Neural Networks (cond-mat.dis-nn) #FOS: Physical sciences #Neural Networks and Applications #Neural dynamics and brain function #Quantum many-body systems #Theoretical and Computational Physics
paper · pdf · doi:10.48550/arxiv.1811.08298
openalex publication_date 2018/11/20 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this work we consider the em analog bipartite spin-glass (or em\nreal-valued restricted Boltzmann machine in a neural network jargon), whose\nvariables (those quenched as well as those dynamical) share standard Gaussian\ndistributions. First, via Guerra's interpolation technique, we express its\nquenched free energy in terms of the natural order parameters of the theory\n(namely the self- and two-replica overlaps), then, we re-obtain the same result\nby using the replica-trick: a mandatory tribute, given the special occasion.\nNext, we show that the quenched free energy of this model is the functional\ngenerator of the moments of the correlation matrix among the weights connecting\nthe two layers of the spin-glass (i.e., the Wishart matrix in random matrix\ntheory or the Hebbian coupling in neural networks): as weights are quenched\nstochastic variables, this plays as a novel tool to inspect random matrices. In\nparticular, we find that the Stieltjes transform of the spectral density of the\ncorrelation matrix is determined by the (replica-symmetric) quenched free\nenergy of the bipartite spin-glass model. In this setup, we re-obtain the\nMarchenko-Pastur law in a very simple way.\n