Extrinsic Methods for Coding and Dictionary Learning on Grassmann Manifolds

Abstract

Sparsity-based representations have recently led to notable results in various visual recognition tasks. In a separate line of research, Riemannian manifolds have been shown useful for dealing with features and models that do not lie in Euclidean spaces. With the aim of building a bridge between the two realms, we address the problem of sparse coding and dictionary learning in Grassmann manifolds, i.e., the space of linear subspaces. To this end, we propose to embed Grassmann manifolds into the space of symmetric matrices by an isometric mapping. This in turn enables us to extend two sparse coding schemes to Grassmann manifolds. Furthermore, we propose an algorithm for learning a Grassmann dictionary, atom by atom. Lastly, to handle non-linearity in data, we extend the proposed Grassmann sparse coding and dictionary learning algorithms through embedding into higher dimensional Hilbert spaces. ExperCommunicated by iments on several classification tasks (gender recognition, gesture classification, scene analysis, face recognition, action recognition and dynamic texture classification) show that the proposed approaches achieve considerable improvements in discrimination accuracy, in comparison to state-of-the-art methods such as kernelized Affine Hull Method and graphembedding Grassmann discriminant analysis.

Low-Rank Linear Dynamical Systems for Motor Imagery EEG

“…LDSs can apply the extended observability subspace 𝒪 as descriptor, but it is hard to calculate. Turaga et al [37, 38] approximate the extended observability by taking the L -order observability matrix; that is, O ( n , L ) = [ C T , ( CA ) T , …,( CA L −1 ) T ] T . In this way, an LDS model can be alternately identified as an n -dimensional subspace of R Lm .…”

Sparse Coding of Shape Trajectories for Facial Expression and Action Recognition

“…However, inferencing such a representation remains challenging due to the nonlinearity of the underlying manifolds. In the literature, two alternatives have been proposed to overcome this problem for different Riemannian manifolds -they are either Extrinsic (kernelbased) [25], [28], [34], [43] or Intrinsic [9], [10], [29], [31]. On one hand, extrinsic solutions are based on embeddings to higher dimensional Reproducing Kernel Hilbert Spaces (RKHS), which are vector spaces where Euclidean geom- etry applies.…”

Fast and accurate image and video analysis on Riemannian manifolds

“…More recently, Slama et al represented the 3D skeletal sequence as point on the Grassmann manifold and utilised tangent spaces to address the action recognition task [100]. Similarly, the geometry of Grassmann manifold also were explored in other computer vision applications such as objection recognition [60,139,158], gender classification [61,65], Hand Gesture Recognition [61,62].…”

“…The computational complexity issue in the intrinsic methods could be addressed by mapping all the data points onto a tangent space at a designated location [48,61,136,144,166]. Unfortunately, as it has been discussed multiple times, this mapping may adversely affect the performance since it will significantly distort the manifold structure in regions far from the tangent space location.…”

“…Another popular school of thought is to map manifold points onto a Reproducing Kernel Hilbert Space (RKHS) and apply kernel-based classifiers [61,70,76,148]. As Euclidean geometry applies in the RKHS, this can be thought of a decoupling between machine learning and data representation [81].…”