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The averaged learning equation (ALEH) applicable to the principal componentanalyzer is studied from both quantum information geometry and dynamical systemviewpoints. On the quantum information space (QIS), the space of regulardensity matrices endowed with the quantum SLD-Fisher metric, a gradient systemis given as an extension of the ALEH; on the submanifold, consisting of thediagonal matrices, of the QIS, the gradient flow coincides with the ALEH up toa local diffeomorphism.

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