Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices

By Sadeep Jayasumana, Richard I. Hartley, Mathieu Salzmann, Hongdong Li, Mehrtash Harandi. 2013 IEEE Conference on Computer Vision and Pattern Recognition, 2013

312 citations recorded by the source corpus.

Symmetric Positive Definite (SPD) matrices have become popular to encode image information. Accounting for the geometry of the Riemannian manifold of SPD matrices has proven key to the success of many algorithms. However, most existing methods only approximate the true shape of the manifold locally by its tangent plane. In this paper, inspired by kernel methods, we propose to map SPD matrices to a high dimensional Hilbert space where Euclidean geometry applies. To encode the geometry of the manifold in the mapping, we introduce a family of provably positive definite kernels on the Riemannian manifold of SPD matrices. These kernels are derived from the Gaussian kernel, but exploit different metrics on the manifold. This lets us extend kernel-based algorithms developed for Euclidean spaces, such as SVM and kernel PCA, to the Riemannian manifold of SPD matrices. We demonstrate the benefits of our approach on the problems of pedestrian detection, object categorization, texture analysis, 2D motion segmentation and Diffusion Tensor Imaging (DTI) segmentation.

Subfields: Computer Vision and Pattern Recognition, Media Technology

View Paper: https://www.semanticscholar.org/paper/c83b39d3275948512e428c7b7a6313ef3537a9db

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Citations in the corpus

Matrix Backpropagation for Deep Networks with Structured Layers (2015, background): “In this context, our work is also related to kernel learning approaches over the manifold of positive-definite matrices [23].”
Kernelized covariance for action recognition (2016, methodology): “Thus, differently from previous approaches [27], [33], [28], [29], the proposed framework is very efficient if compared to the cubic complexity of methods like [33] which require eigen-decomposition.”
Sliced Wasserstein Kernels for Probability Distributions (2015, methodology): “Based on recent works on kernel methods [17, 18, 11], we exploit the connection between the optimal transport framework and the kernel methods and introduce a family of provably positive definite kernels which we denote as Sliced Wasserstein kernels.”
From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices (2014, background): “However, as several studies have shown, undesirable phenomena may occur when Euclidean geometry is utilized to manipulate SPD matrices [16,21,8].”
Training Deep Networks with Structured Layers by Matrix Backpropagation (2015, background): “This part is also related to kernel learning approaches over the manifold of positive-deﬁnite matrices [33].”
Log-Hilbert-Schmidt metric between positive definite operators on Hilbert spaces (2014, methodology): “The third approach defines and exploits Bregman divergences on Sym++(n), such as Stein and Jeffreys divergences, see e.g. [12, 22, 8], which are not Riemannian metrics but are fast to compute and have been shown to work well on nearest-neighbor retrieval tasks.”
Domain Adaptation on the Statistical Manifold (2014, background): “This, for instance, is the case in object recognition with covariance descriptors [33, 23], action recognition on Grassmann manifolds [19], shape classification [31] and rotation averaging [20].”
Log-Euclidean Metric Learning on Symmetric Positive Definite Manifold with Application to Image Set Classification (2015, methodology): “It can be observed that our method significantly outperforms them, which demonstrates the potential superiority of LEM based metric learning framework over AIM and Stein based ones, due to the inherent advantages of the basic metric LEM, as advocated in (Wang et al., 2012; Jayasumana et al., 2013).”
Approximate Log-Hilbert-Schmidt Distances between Covariance Operators for Image Classification (2016, methodology): “For this dataset, following [15], at each pixel we extracted the 5-dimensional feature vector”
Bregman Divergences for Infinite Dimensional Covariance Matrices (2014, methodology): “To overcome the drawbacks of Euclidean geometry and better account for the Riemannian structure of CovDs, state-of-the-art methods make use of non-Euclidean metrics (e.g., [20, 11]).”
Geodesic Finite Mixture Models (2014, methodology): “Multiple category classification problems have been addressed by replacing the tangent space mapping with rolling maps [6], and by using extensions of the Kernel methods to Riemannian manifolds [16].”
Federated Transfer Learning for EEG Signal Classification (2020, background): “2) Deep Learning Methods in BCI: Symmetric Positive-Deﬁnite (SPD) matrix is one of the popular research objects encountered in a great variety of areas, such as medical imaging [12] and visual recognition [13].”
A Riemannian Network for SPD Matrix Learning (2016, methodology): “In medical imaging, they are commonly used in diffusion tensor magnetic resonance imaging (Pennec, Fillard, and Ayache 2006; Arsigny et al. 2007; Jayasumana et al. 2013).”