Mutual-Energy Inner Product Optimization Method for Constructing Feature Coordinates and Image Classification in Machine Learning

As a key task in machine learning, data classification is essential to find a suitable coordinate system to represent the data features of different classes of samples. This paper proposes the mutual-energy inner product optimization method for constructing a feature coordinate system. First, by analyzing the solution space and eigenfunctions of the partial differential equations describing a non-uniform membrane, the mutual-energy inner product is defined. Second, by expressing the mutual-energy inner product as a series of eigenfunctions, it shows the significant advantage of enhancing low-frequency features and suppressing high-frequency noise, compared to the Euclidean inner product. And then, a mutual-energy inner product optimization model is built to extract the data features, and the convexity and concavity properties of its objective function are discussed. Next, by combining the finite element method, a stable and efficient sequential linearization algorithm is constructed to solve the optimization model. This algorithm only solves positive definite symmetric matrix equations and linear programming with a few constraints, and its vectorized implementation is discussed. Finally, the mutual-energy inner product optimization method is used to construct feature coordinates, and multi-class Gaussian classifiers are trained on the MINST training set. Good prediction results of the Gaussian classifiers are achieved on the MINST test set.