*Forma,* Vol. 14 (No. 4), pp. 303-307, 1999

*Original Paper*

## Mathematical Study of Conceptualization in *N*-Dimensional Space

Masanobu Ohtsuki, Atsushi Minato and Satoru Ozawa

Division of Applied Synergetics, Graduate School of Science and Engineering, Ibaraki University, 4-12-1 Nakanarusawa, Hitachi 316-8511, Japan

(Received November 19, 1999; Accepted January 13, 2000)

**Keywords: **
Learning, *N*-Dimensional Rectangle, High Dimensional Space, Pattern Recognition

**Abstract. **
Abstract. A concept of a thing is characterized by a set of suitable parameters. It can be approximately represented as a rectangle in *N*-Dimensional space. The process of obtaining the concept from a number of observations produces a series of "sample points" in *N*-Dimensional space, where each sample point has a value of 1 or 0 depending upon whether the sample suits with the concept or not. The accuracy of guessing *N*-Dimensional rectangle from sample points is related to the way of generating samples. The HIS (Half Interval Search) method is proposed to produce the sample points. It has been shown that it is an efficient method and the expected error in the determination of the concept is
O(1/*m*), where *m* is the number of sample points. This method is successfully used for identification of a "smiling face" in a cartoon figure.

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