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.
To preserve spatial information within tensors in the dimensionality reduction methods, @cite_17 introduces the Tucker LPP (TLPP) which is LPP based on the Tucker decomposition to analyze the high-dimensional data and has the exponential increase in storage complexity as the number of modes increases.
@cite_17
1995406764 2613951549 2154872931 2109531142
For @math -dimensional tensors with possibly large @math , an hierarchical data structure, called the Tree-Tucker format, is presented as an alternative to the canonical decomposition. It has asymptotically the same (and often even smaller) number of representation parameters and viable stability properties. The approach involves a recursive construction described by a tree with the leafs corresponding to the Tucker decompositions of three-dimensional tensors, and is based on a sequence of SVDs for the recursively obtained unfolding matrices and on the auxiliary dimensions added to the initial “spatial” dimensions. It is shown how this format can be applied to the problem of multidimensional convolution. Convincing numerical examples are given. Abstract Tensors are valuable tools to represent Electroencephalogram (EEG) data. Tucker decomposition is the most used tensor decomposition in multidimensional discriminant analysis and tensor extension of Linear Discriminant Analysis (LDA), called Higher Order Discriminant Analysis (HODA), is a popular tensor discriminant method used for analyzing Event Related Potentials (ERP). In this paper, we introduce a new tensor-based feature reduction technique, named Higher Order Spectral Regression Discriminant Analysis (HOSRDA), for use in a classification framework for ERP detection. The proposed method (HOSRDA) is a tensor extension of Spectral Regression Discriminant Analysis (SRDA) and casts the eigenproblem of HODA to a regression problem. The formulation of HOSRDA can open a new framework for adding different regularization constraints in higher order feature reduction problem. Additionally, when the dimension and number of samples is very large, the regression problem can be solved via efficient iterative algorithms. We applied HOSRDA on data of a P300 speller from BCI competition III and reached average character detection accuracy of 96.5 for the two subjects. HOSRDA outperforms almost all of other reported methods on this dataset. Additionally, the results of our method are fairly comparable with those of other methods when 5 and 10 repetitions are used in the P300 speller paradigm. Many problems in information processing involve some form of dimensionality reduction. In this paper, we introduce Locality Preserving Projections (LPP). These are linear projective maps that arise by solving a variational problem that optimally preserves the neighborhood structure of the data set. LPP should be seen as an alternative to Principal Component Analysis (PCA) – a classical linear technique that projects the data along the directions of maximal variance. When the high dimensional data lies on a low dimensional manifold embedded in the ambient space, the Locality Preserving Projections are obtained by finding the optimal linear approximations to the eigenfunctions of the Laplace Beltrami operator on the manifold. As a result, LPP shares many of the data representation properties of nonlinear techniques such as Laplacian Eigenmaps or Locally Linear Embedding. Yet LPP is linear and more crucially is defined everywhere in ambient space rather than just on the training data points. This is borne out by illustrative examples on some high dimensional data sets. Reducing the dimensionality of data without losing intrinsic information is an important preprocessing step in high-dimensional data analysis. Fisher discriminant analysis (FDA) is a traditional technique for supervised dimensionality reduction, but it tends to give undesired results if samples in a class are multimodal. An unsupervised dimensionality reduction method called locality-preserving projection (LPP) can work well with multimodal data due to its locality preserving property. However, since LPP does not take the label information into account, it is not necessarily useful in supervised learning scenarios. In this paper, we propose a new linear supervised dimensionality reduction method called local Fisher discriminant analysis (LFDA), which effectively combines the ideas of FDA and LPP. LFDA has an analytic form of the embedding transformation and the solution can be easily computed just by solving a generalized eigenvalue problem. We demonstrate the practical usefulness and high scalability of the LFDA method in data visualization and classification tasks through extensive simulation studies. We also show that LFDA can be extended to non-linear dimensionality reduction scenarios by applying the kernel trick.