Mathematical Aspects of Tensor Subspace Method
The mathematical and computational backgrounds of pattern recognition are the geometries in Hilbert space for functional analysis and applied linear algebra for numerical analysis, respectively. Organs, cells and microstructures in cells dealt with in biomedical image analysis are volumetric data. We are required to process and analyse these data as volumetric data without embedding vector space from the viewpoints of object oriented data analysis. Therefore, sampled values of volumetric data are expressed as three-way array data. These three-way array data are expressed as the third order tensor. This embedding of the data leads us to the construction of subspace method for higher-order tensors expressing multi-way array data.
- 1.Iijima, T.: Pattern Recognition. Corona-sha, Tokyo (1974). (in Japanese)Google Scholar
- 2.Otsu, N.: Mathematical studies on feature extraction in pattern recognition. Res. Electrotech. Lab. 818, 1–220 (1981). (in Japanese)Google Scholar
- 3.Grenander, U., Miller, M.: Pattern theory: from representation to inference. In: OUP (2007)Google Scholar
- 8.Mørup, M.: Applications of tensor (multiway array) factorizations and decompositions in data mining. Wiley Interdiscipl. Rev.: Data Min. Knowl. Disc. 1, 24–40 (2011)Google Scholar
- 9.Malcev, A.: Foundations of Linear Algebra, in Russian edition 1948, (English translation W.H, Freeman and Company (1963)Google Scholar
- 13.Oja, E.: Subspace Methods of Pattern Recognition. Research Studies Press, Letchworth (1983)Google Scholar