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High-dimensional data are highly correlative and redundant, making it difficult to explore and analyze. Amount of unsupervised dimensionality reduction (DR) methods has been proposed, in which constructing a neighborhood graph is the primary step of DR methods. However, there exist two problems: 1) the construction of graph is usually separate from the selection of projection direction and 2) the original data are inevitably noisy. In this article, we propose an unsupervised adaptive embedding (UAE) method for DR to solve these challenges, which is a linear graph-embedding method. First, an adaptive allocation method of neighbors is proposed to construct the affinity graph. Second, the construction of affinity graph and calculation of projection matrix are integrated together. It considers the local relationship between samples and global characteristic of high-dimensional data, in which the cleaned data matrix is originally proposed to remove noise in subspace. The relationship between our method and local preserving projections (LPPs) is also explored. Finally, an alternative iteration optimization algorithm is derived to solve our model, the convergence and computational complexity of which are also analyzed. Comprehensive experiments on synthetic and benchmark datasets illustrate the superiority of our method. [ABSTRACT FROM AUTHOR]

The linear discriminant analysis (LDA) method needs to be transformed into another form to acquire an approximate closed-form solution, which could lead to the error between the approximate solution and the true value. Furthermore, the sensitivity of dimensionality reduction (DR) methods to subspace dimensionality cannot be eliminated. In this article, a new formulation of trace ratio LDA (TRLDA) is proposed, which has an optimal solution of LDA. When solving the projection matrix, the TRLDA method given by us is transformed into a quadratic problem with regard to the Stiefel manifold. In addition, we propose a new trace difference problem named optimal dimensionality linear discriminant analysis (ODLDA) to determine the optimal subspace dimension. The nonmonotonicity of ODLDA guarantees the existence of optimal subspace dimensionality. Both the two approaches have achieved efficient DR on several data sets. [ABSTRACT FROM AUTHOR]

Dimensionality reduction (DR) technique has been frequently used to alleviate information redundancy and reduce computational complexity. Traditional DR methods generally are inability to deal with nonlinear data and have high computational complexity. To cope with the problems, we propose a fast unsupervised projection (FUP) method. The simplified graph of FUP is constructed by samples and representative points, where the number of the representative points selected through iterative optimization is less than that of samples. By generating the presented graph, it is proved that large-scale data can be projected faster in numerous scenarios. Thereafter, the orthogonality FUP (OFUP) method is proposed to ensure the orthogonality of projection matrix. Specifically, the OFUP method is proved to be equivalent to PCA upon certain parameter setting. Experimental results on benchmark data sets show the effectiveness in retaining the essential information. [ABSTRACT FROM AUTHOR]