Rayleigh-quotient-based Data Classification Method for Machine Learning

Abstract Inspired by the ability of human ears to distinguish between different kinds of music and the ability of an ancient instrument to play countless beautiful melodies, this paper proposes the Rayleigh-quotient-based data classification method, which implements data classification by selecting an appropriate eigenfunction coordinate system to represent different categories of data and making different categories of data show significant differences in the coordinate components, meanwhile, transforming the selection problem of an eigenfunction coordinate system into a design problem of physical parameters of music instruments, such as designing a non-uniform string or membrane with distributed supports. In theory, the method is based on the eigenvalue problem for Sturm-Liouville differential equation, the eigenvalue problem for functional minimization, and the generalized eigenvalue problem for matrices, which have a time-honored history and can provide strong theoretical foundation and support. In addition, each parameter in the method has a clear physical interpretation and can be adjusted under the guidance of a set of complete theory, and their influence on classification results is definite, avoiding the large time consumption due to randomly adjusting parameters. The data classification procedure is divided into 2 steps, including mass density function design and stiffness function design, which not only simplifies the process of parameter adjustment, but also enables the optimization algorithm to be implemented by the linear programming with a low computational complexity, theoretically guaranteeing a high computational efficiency. Considering the amount of data being classified is large, two kinds of algorithms are developed to implement data classification, where one is a statistical averaging algorithm and the other is an incremental iterative algorithm. Finally, by combining the image database MNIST, the Rayleigh-quotient-based data classification method is employed to build an image classifier, which is trained on the MNIST training set including 60,000 images, and is tested on the MNIST test set including 10,000 images. The classification results show that by selecting an appropriate eigenfunction coordinate system, the Rayleigh quotient distribution of each category of image can be controlled within a very narrow range, while different categories of images show significant difference in the Rayleigh quotient distribution; the classification accuracy on the MNIST training set and on the MNIST test set are stable at about 99.50%, and the maximum accuracy can reach to 100.00% on the MNIST training set and 99.76% on the MNIST test set; and the Rayleigh-quotient threshold can be selected within a very wide range, beneficial for the robustness and stability of data classification.