Prior knowledge for pattern recognition: Difference between revisions

Revision as of 13:56, 16 January 2012

Revision as of 14:01, 16 January 2012

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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 enforceenforcing the equality

:<math>

f(\boldsymbol{x}) = f(T_{\theta}\boldsymbol{x}), \quad \forall \boldsymbol{x}, \theta .</math>

</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 the class-invariance with respect to a "___domain of the input space" instead of a transformation. In this case, the problem becomes finding <math>f</math> so that

:<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 the '''permutation-invariance''', i.e. invariance of the class to a permutation of elements in a structured input. A typical application of this type of prior knowledge is a classifier invariant to permutations of rows inof the matrix inputs.

== Knowledge of the data ==