Sparse Coding and Dictionary Learning for Symmetric Positive Definite Matrices: A Kernel Approach

By Mehrtash Harandi, Conrad Sanderson, Richard I. Hartley, Brian C. Lovell. 2012

211 citations recorded by the source corpus.

Recent advances suggest that a wide range of computer vision problems can be addressed more appropriately by considering non-Euclidean geometry. This paper tackles the problem of sparse coding and dictionary learning in the space of symmetric positive definite matrices, which form a Riemannian manifold. With the aid of the recently introduced Stein kernel (related to a symmetric version of Bregman matrix divergence), we propose to perform sparse coding by embedding Riemannian manifolds into reproducing kernel Hilbert spaces. This leads to a convex and kernel version of the Lasso problem, which can be solved efficiently. We furthermore propose an algorithm for learning a Riemannian dictionary (used for sparse coding), closely tied to the Stein kernel. Experiments on several classification tasks (face recognition, texture classification, person re-identification) show that the proposed sparse coding approach achieves notable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as tensor sparse coding, Riemannian locality preserving projection, and symmetry-driven accumulation of local features.

Subfields: Computational Mechanics, Computer Vision and Pattern Recognition

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Citations in the corpus

When Kernel Methods Meet Feature Learning: Log-Covariance Network for Action Recognition From Skeletal Data (2017, methodology): “Precisely, we define va to be the vectorization of all diagonal and lower-diagonal entries1 of logXa: as similarly done in [16, 37, 3] such intermediate representation is fully able to provide and Euclidean (vectorial) representation which keeps the powerfulness of the Riemannian encoding as SPD mat”
First-order Methods for Geodesically Convex Optimization (2016, background): “, 2013); dictionary learning (Sun et al., 2015; Harandi et al., 2012); optimization under orthogonality constraints (Edelman et al.”
Log-Euclidean Metric Learning on Symmetric Positive Definite Manifold with Application to Image Set Classification (2015, methodology): “For verification task on YTF, since RSR/LEK apply sparse representation based classifier, and CDL-PLS adopts a regression based classifier, they fail to work on this dataset and thus we do not report their results on this dataset.”
Kernel Methods on the Riemannian Manifold of Symmetric Positive Definite Matrices (2013, methodology): “For instance, covariance region descriptors are used in object detection [20], texture classification [19], object tracking, action recognition and face recognition [9].”
Bregman Divergences for Infinite Dimensional Covariance Matrices (2014, methodology): “Such descriptors take the form of, e.g., region covariance matrices for pedestrian detection [24] and texture categorization [8], human joint covariances for activity recognition [10], and covariance matrices of the local Brownian motion of water molecules in diffusion tensor imaging (DTI) [20].”
Beyond Covariance: Feature Representation with Nonlinear Kernel Matrices (2015, methodology): “Following the literature [8], each image is partitioned into 64 non-overlapping sub-images.”
From Manifold to Manifold: Geometry-Aware Dimensionality Reduction for SPD Matrices (2014, methodology): “Second, we make use of the Riemannian sparse coding algorithm of [6] (RSR).”