  Locality preserving projections (LPP) are a classical dimensionality
reduction method based on data graph information. However, LPP is still
responsive to extreme outliers. LPP aiming for vectorial data may undermine
data structural information when it is applied to multidimensional data.
Besides, it assumes the dimension of data to be smaller than the number of
instances, which is not suitable for high-dimensional data. For
high-dimensional data analysis, the tensor-train decomposition is proved to be
able to efficiently and effectively capture the spatial relations. Thus, we
propose a tensor-train parameterization for ultra dimensionality reduction
(TTPUDR) in which the traditional LPP mapping is tensorized in terms of
tensor-trains and the LPP objective is replaced with the Frobenius norm to
increase the robustness of the model. The manifold optimization technique is
utilized to solve the new model. The performance of TTPUDR is assessed on
classification problems and TTPUDR significantly outperforms the past methods
and the several state-of-the-art methods.
