SPD Data Dictionary Learning Based on Kernel Learning and Riemannian Metric

Abstract

The use of regional covariance descriptors to generate feature data represented by Symmetric Positive Definite (SPD) matrices from images or videos has become increasingly common in machine learning. However, SPD data itself does not constitute a vector space, and dictionary learning involves a large number of linear operations, so dictionary learning cannot be performed directly on SPD data. For this reason, a more common method is to map the SPD data to the Reproducing Kernel Hilbert Space (RKHS). The so-called kernel learning is to find the most suitable RKHS for specific tasks. RKHS can be uniquely generated by a kernel function. Therefore, RKHS learning can also be considered as kernel learning. In this article, there are two main contributions. The first contribution is to propose a framework which based on Kernel Learning and Riemannian Metric (KLRM). Usually the learnable kernel function framework is to learn some parameters in the kernel function. The second contribution is dictionary learning by applying KLRM to SPD data. The SPD data is transformed into the RKHS generated by KLRM, and RKHS after training provides the most suitable working space for dictionary learning. Under the proposed framework, we design a positive definite kernel function, which is defined by the Log-Euclidean metric. This function can be transformed into a corresponding Riemannian kernel. The experimental results provided in this paper is compared with other state-of-the-art algorithms for SPD data dictionary learning and show that the proposed algorithm achieves better results.

Robust Dictionary Learning and Sparse Coding With Riemannian Geometry Preserving Method in Symmetric Matrices Inner Product Space

“…We will compare the performance of RGP-DLSC against some state-of-the-art or baseline DLSC algorithms on SPD manifolds, namely R-KSRC [16], R-SR [26], LE-DLSC [17], R-DLSC [51] and KLRM-DL [53], in the next section.…”

“…Based on the RKHS-based DLSC [16], [17], R. Zhuang et al [53] (KLRM-DL) introduced a learnable kernel function framework to learn some parameters in kernel function. They actually combine the kernel learning based on Riemannian metric and dictionary learning to produce more discriminative kernel functions, and further generate more suitable kernel space for DLSC.…”

“…The optimization problem (35) is not a standard Lasso problem, thus we cannot use toolboxes SPAMS [56] or CVX [57] to solve it efficiently like [17], [26], and [53] did. In this paper, a sparse coding algorithm inspired by Feature-Sign Search (FSS) [58], which is a sparse coding optimization method commonly used in Euclidean space, is used to obtain the sparse codes of SPD matrices in SMIPS.…”

“…But this updating strategy has high computational complexity, and in this paper we adopt another strategy stemming from the Representer theorem [22] in RKHS. Representer theorem had been used in the RKHSbased DLSC algorithms to express the dictionary atom as a linear combination of training samples in RKHS [53], [61]. In this paper, we use a similar strategy to express the dictionary atom as a linear combination of training samples in SMIPS.…”

“…And the other key reason may be that the SPD data generated by the feature vector defined in this paper is not suitable for those methods in PIE. Because in other literatures (such as [16], [53]), researchers use higher-dimensional SPD data generated by longer feature vectors for face recognition. But in previous experiments, we used the SPD data generated by 15-dimensional feature vector described in Section V (A), so other algorithms may perform poorly.…”