5.2.3 Recall

In recall, the components of a pattern $\bf p$ are partially given, and the components of a completed pattern $\bf z$ are divided into input and output (see section 4.2). The input is the offset $\bf p$ of a hyper-plane extending into the output dimensions:

$$\bf z \bf M \eta \bf p$$ (5.10)

where $\eta$ are the free parameters of the hyper-plane. The matrix $\bf M$ defines which dimensions are input and which are output. To obtain the output for a given input, the free parameters are chosen such that the potential E($\bf z$) is minimized,

$$\eta \min_{{\boldsymbol{\eta}}}^{} \bf z \eta$$ =

$$\bf z_{{\mbox{\footnotesize opt}}}^{} \bf M \eta \bf p$$ = (5.11)

There seems to be no analytical solution for $\eta$^*. In feature space, the constraint space is not a hyper-plane. However, the minimization can be solved using standard numerical optimization algorithms.