A Robust Distance Measure for Similarity-Based Classification on the SPD Manifold

DGPDR: discriminative geometric perception dimensionality reduction of SPD matrices on Riemannian manifold for EEG classification

Manifold learning with Symmetric Positive Definite (SPD) matrices has demonstrated potential for classifying Electroencephalography (EEG) in Brain-Computer Interface (BCI) applications. However, SPD matrices may lead to crucial information loss of EEG signals. This paper proposes a dimensionality reduction method based on discriminative geometric perception on the Riemannian manifold to enhance SPD matrix discriminability. Experiments on BCI Competition IV Dataset 1 and Dataset 2a show the proposed method improves accuracy by 5.0% and 19.38% respectively, demonstrating that applying discriminative geometric perception can effectively maintain robust performance associated with the dimensionality-reduced SPD matrix.

Joint Metric Learning-Based Class-Specific Representation for Image Set Classification

With the rapid advances in digital imaging and communication technologies, recently image set classification has attracted significant attention and has been widely used in many real-world scenarios. As an effective technology, the class-specific representation theory-based methods have demonstrated their superior performances. However, this type of methods either only uses one gallery set to measure the gallery-to-probe set distance or ignores the inner connection between different metrics, leading to the learned distance metric lacking robustness, and is sensitive to the size of image sets. In this article, we propose a novel joint metric learning-based class-specific representation framework (JMLC), which can jointly learn the related and unrelated metrics. By iteratively modeling probe set and related or unrelated gallery sets as affine hull, we reconstruct this hull sparsely or collaboratively over another image set. With the obtained representation coefficients, the combined metric between the query set and the gallery set can then be calculated. In addition, we also derive the kernel extension of JMLC and propose two new unrelated set constituting strategies. Specifically, kernelized JMLC (KJMLC) embeds the gallery sets and probe sets into the high-dimensional Hilbert space, and in the kernel space, the data become approximately linear separable. Extensive experiments on seven benchmark databases show the superiority of the proposed methods to the state-of-the-art image set classifiers.

U-SPDNet: An SPD manifold learning-based neural network for visual classification

With the development of neural networking techniques, several architectures for symmetric positive definite (SPD) matrix learning have recently been put forward in the computer vision and pattern recognition (CV&PR) community for mining fine-grained geometric features. However, the degradation of structural information during multi-stage feature transformation limits their capacity. To cope with this issue, this paper develops a U-shaped neural network on the SPD manifolds (U-SPDNet) for visual classification. The designed U-SPDNet contains two subsystems, one of which is a shrinking path (encoder) making up of a prevailing SPD manifold neural network (SPDNet (Huang and Van Gool, 2017)) for capturing compact representations from the input data. Another is a constructed symmetric expanding path (decoder) to upsample the encoded features, trained by a reconstruction error term. With this design, the degradation problem will be gradually alleviated during training. To enhance the representational capacity of U-SPDNet, we also append skip connections from encoder to decoder, realized by manifold-valued geometric operations, namely Riemannian barycenter and Riemannian optimization. On the MDSD, Virus, FPHA, and UAV-Human datasets, the accuracy achieved by our method is respectively 6.92%, 8.67%, 1.57%, and 1.08% higher than SPDNet, certifying its effectiveness.

Learning Non-Euclidean Representations With SPD Manifold for Myoelectric Pattern Recognition

How to learn informative representations from Electromyography (EMG) signals is of vital importance for myoelectric control systems. Traditionally, hand-crafted features are extracted from individual EMG channels and combined together for pattern recognition. The spatial topological information between different channels can also be informative, which is seldom considered. This paper presents a radically novel approach to extract spatial structural information within diverse EMG channels based on the symmetric positive definite (SPD) manifold. The object is to learn non-Euclidean representations inside EMG signals for myoelectric pattern recognition. The performance is compared with two classical feature sets using accuracy and F1-score. The algorithm is tested on eleven gestures collected from ten subjects, and the best accuracy reaches 84.85%±5.15% with an improvement of 4.04%~20.25%, which outperforms the contrast method, and reaches a significant improvement with the Wilcoxon signed-rank test. Eleven gestures from three public databases involving Ninapro DB2, DB4, and DB5 are also evaluated, and better performance is observed. Furthermore, the computational cost is less than the contrast method, making it more suitable for low-cost systems. It shows the effectiveness of the presented approach and contributes a new way for myoelectric pattern recognition.

SymNet: A Simple Symmetric Positive Definite Manifold Deep Learning Method for Image Set Classification

By representing each image set as a nonsingular covariance matrix on the symmetric positive definite (SPD) manifold, visual classification with image sets has attracted much attention. Despite the success made so far, the issue of large within-class variability of representations still remains a key challenge. Recently, several SPD matrix learning methods have been proposed to assuage this problem by directly constructing an embedding mapping from the original SPD manifold to a lower dimensional one. The advantage of this type of approach is that it cannot only implement discriminative feature selection but also preserve the Riemannian geometrical structure of the original data manifold. Inspired by this fact, we propose a simple SPD manifold deep learning network (SymNet) for image set classification in this article. Specifically, we first design SPD matrix mapping layers to map the input SPD matrices into new ones with lower dimensionality. Then, rectifying layers are devised to activate the input matrices for the purpose of forming a valid SPD manifold, chiefly to inject nonlinearity for SPD matrix learning with two nonlinear functions. Afterward, we introduce pooling layers to further compress the input SPD matrices, and the log-map layer is finally exploited to embed the resulting SPD matrices into the tangent space via log-Euclidean Riemannian computing, such that the Euclidean learning applies. For SymNet, the (2-D)²principal component analysis (PCA) technique is utilized to learn the multistage connection weights without requiring complicated computations, thus making it be built and trained easier. On the tail of SymNet, the kernel discriminant analysis (KDA) algorithm is coupled with the output vectorized feature representations to perform discriminative subspace learning. Extensive experiments and comparisons with state-of-the-art methods on six typical visual classification tasks demonstrate the feasibility and validity of the proposed SymNet.

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.