Theory And Application Researches On Geometrical Learning

Data in modern era are increasingly taking on the new characteristicsof huge data volumes, high dimension, non-structure and non-linear; whichbring up big challenges to the traditional machine learning and dataanalysis methods. Novel methodology should be developed to obtain theintrinsic structure and geometrical rule of the data information. Due to theinherent relations between the information science and thehigh-dimensional geometry, the geometrical idea can provide a new wayfor the developments of artificial intelligence and machine learning.The purpose of this paper is to investigate the intrinsic structure andrule of the high-dimensional data information via the geometrical learningviewpoint. Some innovative work has been made out:In this paper, we propose the geometrical learning ways of analyzingdata set from their geometrical properties, classify them into two groups:the manifold learning and the biomimetic pattern recognition. Then wepresent the cognitive and mathematical theories of biomimetic pattern recognition, analyze the means to its realization—the geometric figurecovering algorithm based on the multi-degree of freedom neurons;discusse the geometry properties of multi-degree of freedom neurons, anddemonstrate the boundedly convergence of a multi-degree of freedomneurons.We prove that for any given continues real function, there alwaysexists a double weighted neural network so that the output of neuralnetwork could approximate this function within the error limits.Though manifold learning has been successfully applied in wide areas,such as data visualization, dimension reduction and speech recognition;few researches have been done with the combination of the informationtheory and the geometrical learning. The paper has carried out a boldexploration in this field, raises a new approach on face recognition, theintrinsic a-Renyi entropy of the face image attained from manifoldlearning is used as the characteristic measure during recognition. The newalgorithm is tested on face recognition, and the experiments obtain thesatisfying results.