Dimensionality Reduction of SPD Data Based on Riemannian Manifold Tangent Spaces and Isometry.

Symmetric positive definite (SPD) data have become a hot topic in machine learning. Instead of a linear Euclidean space, SPD data generally lie on a nonlinear Riemannian manifold. To get over the problems caused by the high data dimensionality, dimensionality reduction (DR) is a key subject for SPD data, where bilinear transformation plays a vital role. Because linear operations are not supported in nonlinear spaces such as Riemannian manifolds, directly performing Euclidean DR methods on SPD matrices is inadequate and difficult in complex models and optimization. An SPD data DR method based on Riemannian manifold tangent spaces and global isometry (RMTSISOM-SPDDR) is proposed in this research. The main contributions are listed: (1) Any Riemannian manifold tangent space is a Hilbert space isomorphic to a Euclidean space. Particularly for SPD manifolds, tangent spaces consist of symmetric matrices, which can greatly preserve the form and attributes of original SPD data. For this reason, RMTSISOM-SPDDR transfers the bilinear transformation from manifolds to tangent spaces. (2) By log transformation, original SPD data are mapped to the tangent space at the identity matrix under the affine invariant Riemannian metric (AIRM). In this way, the geodesic distance between original data and the identity matrix is equal to the Euclidean distance between corresponding tangent vector and the origin. (3) The bilinear transformation is further determined by the isometric criterion guaranteeing the geodesic distance on high-dimensional SPD manifold as close as possible to the Euclidean distance in the tangent space of low-dimensional SPD manifold. Then, we use it for the DR of original SPD data. Experiments on five commonly used datasets show that RMTSISOM-SPDDR is superior to five advanced SPD data DR algorithms.

Dimensionality Reduction for Tensor Data Based on Local Decision Margin Maximization.

In machine learning, the idea of maximizing the margin between two classes is widely used in classifier design. Enlighted by the idea, this paper proposes a novel supervised dimensionality reduction method for tensor data based on local decision margin maximization. The proposed method seeks to preserve and protect the local discriminant information of the original data in the low-dimensional data space. Firstly, we depart the original tensor dataset into overlapped localities with discriminant information. Then, we extract the similarity and anti-similarity coefficients of each high-dimensional locality and preserve these coefficients in the embedding data space via the multilinear projection scheme. Under the combined effect of these coefficients, each dimension-reduced locality tends to be a convex set where strongly correlated intraclass points gather. Simultaneously, the local decision margin, which is defined as the shortest distance from the boundary of each locality to the nearest point of each side, will be maximized. Therefore, the local discriminant structure of the original data could be well maintained in the low-dimensional data space. Moreover, a simple iterative scheme is proposed to solve the final optimization problem. Finally, the experiment results on 6 real-world datasets demonstrate the effectiveness of the proposed method.