Alignment of ordered set cartesian product

The work is devoted to the study of metric methods for analyzing objects with complex structure. It proposes to generalize the dynamic time warping method of two time series for the case of objects defined on two or more time axes. Such objects are matrices in the discrete representation. The DTW (Dynamic Time Warping) method of time series is generalized as a method of matrices dynamic alignment. The paper proposes a distance function resistant to monotonic nonlinear deformations of the Cartesian product of two time scales. The alignment path between objects is defined. An object is called a matrix in which the rows and columns correspond to the axes of time. The properties of the proposed distance function are investigated. To illustrate the method, the problems of metric classification of objects are solved on model data and data from the MNIST dataset.