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.
We denote the left unfolding operation @cite_2 of @math as the matrix @math where the last mode of the tensor becomes the column indices of the left unfolding matrix and the rest of the modes are the row indices. Similarly, for the right unfolding operation, denoting it as @math . Also, the vectorization of a tensor is denoted by @math . The F-norm of a tensor can be defined as the @math -norm of its vectorization, i.e., @math , which considers all the elements @math as an entire group and preserves the general spatial relations between elements. Besides @math -norm of a tensor is computed as @math which treats each elements separately and can probably cause the spatial information loss.
@cite_2
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Tensor train is a hierarchical tensor network structure that helps alleviate the curse of dimensionality by parameterizing large-scale multidimensional data via a set of network of low-rank tensors. Associated with such a construction is a notion of Tensor Train subspace and in this paper we propose a TT-PCA algorithm for estimating this structured subspace from the given data. By maintaining low rank tensor structure, TT-PCA is more robust to noise comparing with PCA or Tucker-PCA. This is borne out numerically by testing the proposed approach on the Extended YaleFace Dataset B.