Dimensionality Reduction of Tensors Based on Local Homeomorphism and Global Subspace Projection Distance Minimum

Tensors are powerful tools for representing and processing multidimensional data. In this paper, an algorithm of dimension reduction for multidimensional data is proposed. We consider both the global and local structures of the data to fully extract important features. Multiplying a tensor by a matrix in mode-n can change the size of a certain dimension of the tensor. Therefore, for the global structure, we use a matrix to span a subspace to minimize the distance between the tensor data and its projection on the subspace for each mode. For local structure, we construct a dimensionality reduction algorithm for tensor data based on local homeomorphism to keep the continuous dependency relationship of the original high-dimensional tensor data unchanged after dimensionality reduced in each locality. The proposed algorithm achieves dimensionality reduction by combining the global subspace projection distance minimum and local homeomorphism, which realizes global variance maximization and maintains the local non-linear geometric structure at the same time. The proposed algorithm performs good feasibility while compared with other advanced algorithms in classification and clustering experiments.