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

TL;DR

This paper demonstrates that all trajectories of the extended averaged Hebbian learning equation on the quantum state space are e-geodesics, linking neural learning models with quantum information geometry.

Contribution

It establishes that the trajectories of the EAHLE are e-geodesics on the quantum state space, providing a geometric interpretation and explicit solutions.

Findings

All EAHLE trajectories are e-geodesics.

Explicit solutions of the EAHLE are derived.

Connections between neural models and quantum geometry are shown.

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 (Far East Journal of Applied Mathematics, vol.47, pp.149-167, 2010) from a Hebbian synaptic-neuron model is studied from a quantum-information-geometric point of view. It is shown that all the trajectories of the EAHLE are the e-geodesics, 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.