Open AccessGeometries of knowledge
Author(s) -
Ulf Grenander
Publication year - 1997
Publication title -
proceedings of the national academy of sciences of the united states of america
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 5.011
H-Index - 771
eISSN - 1091-6490
pISSN - 0027-8424
DOI - 10.1073/pnas.94.3.783
Subject(s) - similarity (geometry) , basis (linear algebra) , connection (principal bundle) , function (biology) , computer science , theoretical computer science , differential (mechanical device) , mathematics , variance (accounting) , algebra over a field , artificial intelligence , pure mathematics , geometry , accounting , evolutionary biology , engineering , business , image (mathematics) , biology , aerospace engineering
It is argued that knowledge representations formalized through pattern theoretic structures are geometric in nature in the following sense. The configurations and resulting patterns appearing in such representations exhibit invariances with respect to the similarity groups and are characterized topologically through their connection types. Starting with a special pattern from microbiology, it is shown how the basic pattern theoretic concepts are introduced in general and what their function is in representing knowledge. Variance/invariance of the patterns is discussed in geometric language. The measures on the configuration spaces are implemented by difference/differential equations which are used as a basis for computer algorithms.