Knowledge a priori

The Pattern recognition is a field of research very active and closely related to the Apprentissage machine. Also known under the name of classification, its goal is to build a Classifieur which can determine the class (left the classifior) of a form (entered of the classifior). This procedure, so called drive (" training"), corresponds to learn a function from unknown decision starting from data of training in the shape of couples input-output $(\backslash \; boldsymbol\; \{X\}\; \_i,\; y\_i)$ called examples. Nevertheless, in the real applications, like the Recognition of the characters, a certain quantity of information on the problem is often known " a priori ". The incorporation of this knowledge a priori in the training is the key which will allow an increase in the performances of the classifior in much of applications.

Definition

The knowledge a priori, as defined by, refers to all information available on the problem in addition to the data of training. However, in this most general form, to determine a Modèle starting from a finished play of examples without knowledge a priori is a Problème badly posed, in the direction where a single model cannot exist. Many classifieurs incorporate a priori softness of the function (" Smoothness ") who implies that a form similar to a form of the base of training tends to being assigned with the same class.

In training machine, the importance of knowledge a priori can be seen by the theorem of the No Free Lunch which established that all the algorithms have the same performances on average on all the problems and who thus implies that a performance profit can be obtained only by developing a specialized algorithm and thus by using knowledge a priori.

Different the types of knowledge met a priori in pattern recognition are gathered in two principal categories: invariance of class and knowledge on the data.

Invariance of class

A very common type of knowledge a priori in pattern recognition is the invariance of the class (or the exit of classifior) compared to a Transformation of the shape of entry. This type of knowledge is known under the name of invariance by transformation (or " transformation-invariance"). The most used transformations are:

• Translation;

• Rotation;
• rectification (" Skewing ");
• Scaling.

The incorporation of an invariance to a transformation $T\_\; \{\backslash \; theta\}:\; \backslash \; boldsymbol\; \{X\}\; \backslash \; mapsto\; T\_\; \{\backslash \; theta\}\; \backslash \; boldsymbol\; \{X\}$ parameterized by $\backslash \; theta$ in a classifior of exit $f\; (\backslash \; boldsymbol\; \{X\})$ for the shape of entry $\backslash \; boldsymbol\; \{X\}$ corresponds to impose the equality

$F\; (\backslash \; boldsymbol\; \{X\})\; =\; F\; (T\_\; \{\backslash \; theta\}\; \backslash \; boldsymbol\; \{X\}),\; \backslash \; quad\; \backslash \; forall\; \backslash \; boldsymbol\; \{X\},\; \backslash \; theta$

Local invariance can also be considered for a transformation centered into $\backslash \; theta=0$, which gives $T\_0\; \backslash \; boldsymbol\; \{X\}\; =\; \backslash \; boldsymbol\; \{X\}$, by the constraint

$\backslash \; left.\; \backslash \; frac\; \{\backslash \; partial\}\; \{\backslash \; partial\; \backslash \; theta\}\; \backslash \; right|\_\; \{\backslash \; theta=0\}\; F\; (T\_\; \{\backslash \; theta\}\; \backslash \; boldsymbol\; \{X\})\; =\; 0$

Let us note that in these equations, $f$ can as well be related to decision or the exit with actual value of the classifior.

Another approach consists in considering the invariance of class compared to a " field of the space of entrée" instead of a transformation. In this case, the problem becomes: to find $f$ which makes it possible to have

$F\; (\backslash \; boldsymbol\; \{X\})\; =\; y\_\; \{\backslash \; mathcal\; \{P\}\},\; \backslash \; \backslash \; forall\; \backslash \; boldsymbol\; \{X\}\; \backslash \; in\; \backslash \; mathcal\; \{P\}$

where $y\_\; \{\backslash \; mathcal\; \{P\}\}$ is the class of membership of the area $\backslash \; mathcal\; \{P\}$ of the space of entry.

A type different of invariance of class is the invariance with the permutations , i.e the invariance of the class to the permutations of the elements in a structured entry. A typical application of this type of knowledge a priori is a classifior invariant with the permutations of lines in matric entries.

Knowledge on the data

Other forms of knowledge a priori that the invariance of class are concerned with the data more specifically and are thus of an private interest for the real applications. The three particular cases which generally occur when one gathers data of observation are:

• Of the non-labelized examples is available with supposed classes of membership;
• a imbalance of the base of training due to a strong proportion of examples of the same class;
• the quality of the data can vary from one example to another.

If it is included in the training, a knowledge a priori on the latter can améliorier the quality of the recognition. Moreover, not to take into account the bad quality of the data or a great imbalance between the classes can induce a classifior in error.

References

• , B. Scholkopf and A. Smola, " Learning with Kernels" , MIT Close 2002.

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