Dimensionality reduction of tensor data based on local linear embedding and mode product

Abstract

There are three contributions in this paper. (1) A tensor version of LLE (short for Local Linear Embedding algorithm) is deduced and presented. LLE is the most famous manifold learning algorithm. Since its proposal, various improvements to LLE have kept emerging without interruption. However, all these achievements are only suitable for vector data, not tensor data. The proposed tensor LLE can also be used a bridge for various improvements to LLE to transfer from vector data to tensor data. (2) A framework of tensor dimensionality reduction based on tensor mode product is proposed, in which the mode matrices can be determined according to specific criteria. (3) A novel dimensionality reduction algorithm for tensor data based on LLE and mode product (LLEMP-TDR) is proposed, in which LLE is used as a criterion to determine the mode matrices. Benefiting from local LLE and global mode product, the proposed LLEMP-TDR can preserve both local and global features of high-dimensional tenser data during dimensionality reduction. The experimental results on data clustering and classification tasks demonstrate that our method performs better than 5 other related algorithms published recently in top academic journals.

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“…LLE considers each data point to be a linearly weighted combination of its neighboring points, and therefore, the basic steps of LLE in this study were as follows. For a detailed derivation and presentation of the LLE algorithm, please refer to reference [55].…”