Abstract
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.
Highlights
The past few decades have witnessed the rapid development of machine vision, in which machine learning methods based on different mathematical platforms take various forms of image inputs
This paper proposes a dimensionality reduction (DR) algorithm of symmetric positive definite (SPD) data based on Riemannian manifold tangent spaces and isometry (RMTSISOM-SPDDR)
The proposed dimensionality reduction (DR) method based on tangent spaces and isometry first maps the original SPD data to the tangent space at the identity matrix; the bilinear transformation between manifolds is transferred to the tangent spaces
Summary
The past few decades have witnessed the rapid development of machine vision, in which machine learning methods based on different mathematical platforms take various forms of image inputs. The most common form is the vector data flattened by a twodimensional gray image or multi-channel color image [1,2,3] This method is rough and simple on data processing and, it has achieved a certain effect on some models, its problem is destroying the inherent position information and geometry structure of images. Another typical form is to keep the two-dimensional or higher-dimensional tensor data of images. This paper proposes a DR algorithm of SPD data based on Riemannian manifold tangent spaces and isometry (RMTSISOM-SPDDR).
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