All the trajectories of an extended averaged Hebbian learning equation on the quantum state space are the e-geodesics

Abstract

In this paper, two families of trajectories on the quantum state space (QSS) originating from a synaptic-neuron model and from quantum information geometry meet together. The extended averaged Hebbian learning equation (EAHLE) on the QSS developed by the author and Yuya [1] from a Hebbian synaptic-neuron model is studied from a quantum-informationgeometric point of view. It is shown that all the trajectories of the EAHLE are the egeodesics, the autoparallel curves with respect to the exponential-type parallel transport, on the QSS. As a secondary outcome, an explicit representation of solution of the averaged Hebbian learning equation, the origin of the EAHLE, is derived from that of the e-geodesics on the QSS.

Information and contact geometric description of expectation variables exactly derived from master equations

“…For constructing MCMC methods, master equations with their Markov kernels play fundamental roles. Also it should be noted that dynamical systems on statistical manifolds without contact geometry have been studied in the literature [29][30][31][32]. In addition, information geometric descriptions for Markov chains were investigated in the literature as well [33][34][35].…”