On the Visualization of Multidimensional Functions Using Canonical Decomposition

Abstract

The approximation of the multidimensional function using the high order tensor and canonical decomposition is considered for the purpose of visualization. Formally, from the standpoint of computational resources (operational memory volume and the time of computation) the canonical decomposition is beyond comparison. However, at present, the main algorithm that calculates the canonical decomposition is based on the Khatri-Rao product, which implies matrization. This circumstance restricts the domain of applicability by tensors of the relatively small order. In order to overcome this drawback, the algorithm of the computation for the canonical decomposition formmatrices is described that is composed by combination of alternating least squares and stochastic gradient descent. This algorithm has no restrictions from the standpoint of the order of the tensor under the consideration. The numerical tests are presented for the approximation of functions in six-dimensional space that demonstrate the high computational efficiency and high quality of results. The instabilities arising at the estimation of the rank of the decomposition are another trouble of the canonical decomposition. The results of the numerical tests are presented that illustrate the search for the optimal rank providing the minimum of the discrepancy of the exact solution and its approximation.