# Polynomial classifier

Polynomial classifiers for pattern recognition were developed from statistical decision theory and have the key function in text recognition (OCR), a branch of pattern recognition.

Their main advantage lies in the possibility of solving the adaptation task directly. Building on this basic technology , complex and high-performance classifier structures (trees, networks) were developed, which were able to demonstrate their performance in very different fields of application, for example for address readers, postal automation or form readers.

## Mathematical definition

A polynomial classifier is a mapping of vectors from the -dimensional real feature space onto a set of classes: ${\ displaystyle d}$${\ displaystyle \ left \ {1, \ ldots, k \ right \}}$

${\ displaystyle f \ colon \ mathbb {R} ^ {d} \ to \ left \ {1, \ ldots, k \ right \}}$

It is defined as the most significant component of the following vector ${\ displaystyle f}$

${\ displaystyle \ mathbf {y} = \ mathbf {p} (\ mathbf {v}) = {\ begin {pmatrix} p_ {1} (\ mathbf {v}) \\\ vdots \\ p_ {k} ( \ mathbf {v}) \ end {pmatrix}}}$

The multivariate polynomials can be interpreted as probability functions that a given feature vector belongs to the class . Overall, the following applies . ${\ displaystyle p_ {i} (\ mathbf {v}) = y_ {i} \ in [0,1]}$${\ displaystyle \ mathbf {v}}$${\ displaystyle i}$${\ displaystyle \ | \ mathbf {y} \ | _ {1} = 1}$

The above notation can be simplified by calculating only one polynomial instead of many polynomials . Then it holds with a real-valued coefficient vector. Overall follows . ${\ displaystyle p_ {i} (\ mathbf {v})}$${\ displaystyle \ mathbf {x} (\ mathbf {v})}$${\ displaystyle p_ {i} (\ mathbf {v}) = \ mathbf {a} \ cdot \ mathbf {x} (\ mathbf {v})}$${\ displaystyle \ mathbf {p} (\ mathbf {v}) = A ^ {T} \ mathbf {x} (\ mathbf {v})}$

A new feature vector is thus classified by . ${\ displaystyle \ mathbf {\ hat {v}}}$${\ displaystyle \ arg \ max \ mathbf {p} (\ mathbf {\ hat {v}}) = f (\ mathbf {\ hat {v}})}$

## literature

• J. Schürmann: Pattern Classification: A Unified View of Statistical and Neural Approaches . Wiley & Sons, 1996, ISBN 0471135348 .
• H. Niemann: Classification of Patterns . Springer, Berlin 1983, ISBN 3-540-12642-2 . (2nd edition, without publisher, 2003: PDF file; 6.5 MB ).