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* [[Scaling_(geometry)|scaling]].
Incorporating the invariance to a transformation <math>T_{\theta}: \boldsymbol{x} \mapsto T_{\theta}\boldsymbol{x}</math> parametrized in <math>\theta</math> into a classifier of output <math>f(\boldsymbol{x})</math> for an input pattern <math>\boldsymbol{x}</math> corresponds to
:<math>
f(\boldsymbol{x}) = f(T_{\theta}\boldsymbol{x}), \quad \forall \boldsymbol{x}, \theta .</math>
Local invariance can also be considered for a transformation centered at <math>\theta=0</math>, so that <math>T_0\boldsymbol{x} = \boldsymbol{x}</math>, by using the constraint
:<math>
\left.\frac{\partial}{\partial \theta}\right|_{\theta=0} f(T_{\theta} \boldsymbol{x}) = 0 .
</math>
The function <math>f</math> in these equations can be either the decision function of the classifier or its real-valued output.
Another approach is to consider
:<math>
f(\boldsymbol{x}) = y_{\mathcal{P}},\ \forall \boldsymbol{x}\in \mathcal{P} ,
</math>
where <math>y_{\mathcal{P}}</math> is the membership class of the region <math>\mathcal{P}</math> of the input space.
A different type of class-invariance found in pattern recognition is
== Knowledge of the data ==
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