用新型内积方法构建特征坐标，提升图像分类精度

As a key task in machine learning, data classification is essentially to find a suitable coordinate system to represent 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 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 a significant advantage of enhancing low-frequency features and suppressing high-frequency noise, compared with the Euclidean inner product. And then, a mutual-energy inner product optimization model is built to extract data features, and 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 equations including positive definite symmetric matrix 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 Gaussian classifiers are achieved on the MINST test set.