A gradient system on the quantum information space realizing the averaged learning equation of Hebb type

TL;DR

This paper develops a quantum information geometric framework to realize the averaged learning equation of Hebb type, connecting quantum dynamics with classical principal component analysis.

Contribution

It introduces a gradient system on the quantum information space that extends the ALEH, linking quantum geometry with classical learning algorithms.

Findings

Gradient system on QIS extends ALEH

Flow on diagonal matrices matches ALEH

Provides geometric insight into Hebbian learning

Abstract

The averaged learning equation (ALEH) applicable to the principal component analyzer is studied from both quantum information geometry and dynamical system viewpoints. On the quantum information space (QIS), the space of regular density matrices endowed with the quantum SLD-Fisher metric, a gradient system is given as an extension of the ALEH; on the submanifold, consisting of the diagonal matrices, of the QIS, the gradient flow coincides with the ALEH up to a local diffeomorphism.