Recent progress on geometric algorithms for approximating functions: Toward applications to data analysis

Data simplification is an extremely important issue in our current information-oriented society. Normally, a real-world database contains a massive amount of raw data, and when we consider the data as a distribution function, it has fluctuations due to sampling errors, outliers, and/or invalid inputs. Therefore, for data analysis technology such as data mining, it is important to approximate the input data by a simplified function. There are various approaches to function approximation, and functional analytical methods and learning-based techniques are quite popular. Apart from them, computational geometric approach based on optimization using discrete algorithms is widely studied. However, the conventional application of computational geometrical techniques is pattern matching, and to apply them to data analysis, their formulation and optimization criteria must be changed accordingly. Therefore, various difficulties and computational barriers arise, which must be eliminated or avoided. In this paper, we discuss data approximation in computational geometry and describe current trends centered on the author's latest research. © 2006 Wiley Periodicals, Inc. Electron Comm Jpn Pt 3, 90(3): 1–12, 2007; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/ecjc.20297