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.
Over the past few years, some embedding methods have been proposed for feature extraction and dimensionality reduction in various machine learning and pattern classification tasks. Among the methods proposed are Neighborhood Preserving Embedding (NPE), Locality Preserving Projection (LPP) and Local Discriminant Embedding (LDE) which have been used in such applications as face recognition and image video retrieval. However, although the data in these applications are more naturally represented as higher-order tensors, the embedding methods can only work with vectorized data representations which may not capture well some useful information in the original data. Moreover, high-dimensional vectorized representations also suffer from the curse of dimensionality and the high computational demand. In this paper, we propose some novel tensor embedding methods which, unlike previous methods, take data directly in the form of tensors of arbitrary order as input. These methods allow the relationships between dimensions of a tensor representation to be efficiently characterized. Moreover, they also allow the intrinsic local geometric and topological properties of the manifold embedded in a tensor space to be naturally estimated. Furthermore, they do not suffer from the curse of dimensionality and the high computational demand. We demonstrate the effectiveness of the proposed tensor embedding methods on a face recognition application and compare them with some previous methods. Extensive experiments show that our methods are not only more effective but also more efficient. Locality preserving projection (LPP) is a manifold learning method widely used in pattern recognition and computer vision. The face recognition application of LPP is known to suffer from a number of problems including the small sample size (SSS) problem, the fact that it might produce statistically identical transform results for neighboring samples, and that its classification performance seems to be heavily influenced by its parameters. In this paper, we propose three novel solution schemes for LPP. Experimental results also show that the proposed LPP solution scheme is able to classify much more accurately than conventional LPP and to obtain a classification performance that is only little influenced by the definition of neighbor samples.